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8679f9097b Jeff* .. include:: ../defs.hrst
                 
                 .. _sub_phys_pkg_fizhi:
                 
                 Fizhi: High-end Atmospheric Physics
                 -----------------------------------
                 
                 
                 Introduction
                 ++++++++++++
                 
                 The fizhi (high-end atmospheric physics) package includes a collection
                 of state-of-the-art physical parameterizations for atmospheric
                 radiation, cumulus convection, atmospheric boundary layer turbulence,
                 and land surface processes. The collection of atmospheric physics
                 parameterizations were originally used together as part of the GEOS-3
                 (Goddard Earth Observing System-3) GCM developed at the NASA/Goddard
                 Global Modelling and Assimilation Office (GMAO).
                 
                 Equations
                 +++++++++
                 
                 Moist Convective Processes:
                 
                 
                 .. _para_phys_pkg_fizhi_mc:
                 
                 Sub-grid and Large-scale Convection
                 ###################################
                 
                 Sub-grid scale cumulus convection is parameterized using the Relaxed
                 Arakawa Schubert (RAS) scheme of :cite:`moorsz:92`, which is a linearized Arakawa
                 Schubert type scheme. RAS predicts the mass flux from an ensemble of
                 clouds. Each subensemble is identified by its entrainment rate and level
                 of neutral bouyancy which are determined by the grid-scale properties.
                 
                 The thermodynamic variables that are used in RAS to describe the grid
                 scale vertical profile are the dry static energy, :math:`s=c_pT +gz`,
                 and the moist static energy, :math:`h=c_p T + gz + Lq`. The conceptual
                 model behind RAS depicts each subensemble as a rising plume cloud,
                 entraining mass from the environment during ascent, and detraining all
                 cloud air at the level of neutral buoyancy. RAS assumes that the
                 normalized cloud mass flux, :math:`\eta`, normalized by the cloud base
                 mass flux, is a linear function of height, expressed as:
                 
                 .. math::
                 
0bad585a21 Navi*    \pp{\eta(z)}{z} = \lambda \hspace{0.4cm} \text{or} \hspace{0.4cm} \pp{\eta(P^{\kappa})}{P^{\kappa}} = 
                    -\frac{c_p}{g} \theta \lambda
8679f9097b Jeff* 
                 where we have used the hydrostatic equation written in the form:
                 
0bad585a21 Navi* .. math:: \pp{z}{P^{\kappa}} = -\frac{c_p}{g} \theta
8679f9097b Jeff* 
                 The entrainment parameter, :math:`\lambda`, characterizes a particular
                 subensemble based on its detrainment level, and is obtained by assuming
                 that the level of detrainment is the level of neutral buoyancy, ie., the
                 level at which the moist static energy of the cloud, :math:`h_c`, is
                 equal to the saturation moist static energy of the environment,
                 :math:`h^*`. Following :cite:`moorsz:92`, :math:`\lambda` may be written as
                 
0bad585a21 Navi* .. math:: \lambda = \frac{h_B - h^*_D}{\frac{c_p}{g} \int_{P_D}^{P_B}\theta(h^*_D-h)dP^{\kappa}}
8679f9097b Jeff* 
                 where the subscript :math:`B` refers to cloud base, and the subscript
                 :math:`D` refers to the detrainment level.
                 
                 The convective instability is measured in terms of the cloud work
                 function :math:`A`, defined as the rate of change of cumulus kinetic
                 energy. The cloud work function is related to the buoyancy, or the
                 difference between the moist static energy in the cloud and in the
                 environment:
                 
                 .. math::
                 
                    A = \int_{P_D}^{P_B} \frac{\eta}{1 + \gamma}
                    \left[ \frac{h_c-h^*}{P^{\kappa}} \right] dP^{\kappa}
                 
                 where :math:`\gamma` is :math:`\frac{L}{c_p}\pp{q^*}{T}` obtained from
                 the Claussius Clapeyron equation, and the subscript :math:`c` refers to
                 the value inside the cloud.
                 
                 To determine the cloud base mass flux, the rate of change of :math:`A`
                 in time *due to dissipation by the clouds* is assumed to approximately
                 balance the rate of change of :math:`A` *due to the generation by the
                 large scale*. This is the quasi-equilibrium assumption, and results in
                 an expression for :math:`m_B`:
                 
0bad585a21 Navi* .. math:: m_B = \dfrac{- \left. \frac{dA}{dt} \right|_{\rm ls}}{K}
8679f9097b Jeff* 
                 where :math:`K` is the cloud kernel, defined as the rate of change of
                 the cloud work function per unit cloud base mass flux, and is currently
                 obtained by analytically differentiating the expression for :math:`A` in
                 time. The rate of change of :math:`A` due to the generation by the large
                 scale can be written as the difference between the current
0bad585a21 Navi* :math:`A(t+\Delta t)` and its equilibrated value after the previous
8679f9097b Jeff* convective time step :math:`A(t)`, divided by the time step.
0bad585a21 Navi* :math:`A(t)` is approximated as some critical :math:`A_{\rm crit}`, computed
                 by Lord (1982) from in situ observations.
8679f9097b Jeff* 
                 The predicted convective mass fluxes are used to solve grid-scale
                 temperature and moisture budget equations to determine the impact of
                 convection on the large scale fields of temperature (through latent
                 heating and compensating subsidence) and moisture (through precipitation
                 and detrainment):
                 
                 .. math:: \left.{\pp{\theta}{t}}\right|_{c} = \alpha \frac{ m_B}{c_p P^{\kappa}} \eta \pp{s}{p}
                 
                 and
                 
0bad585a21 Navi* .. math:: \left.{\pp{q}{t}}\right|_{c} = \alpha \frac{m_B}{L} \eta \left( \pp{h}{p}-\pp{s}{p} \right)
8679f9097b Jeff* 
                 where :math:`\theta = \frac{T}{P^{\kappa}}`, :math:`P = (p/p_0)`, and
                 :math:`\alpha` is the relaxation parameter.
                 
                 As an approximation to a full interaction between the different
                 allowable subensembles, many clouds are simulated frequently, each
                 modifying the large scale environment some fraction :math:`\alpha` of
                 the total adjustment. The parameterization thereby “relaxes” the large
0bad585a21 Navi* scale environment towards equilibrium.
8679f9097b Jeff* 
                 In addition to the RAS cumulus convection scheme, the fizhi package
                 employs a Kessler-type scheme for the re-evaporation of falling rain :cite:`sudm:88`,
                 which correspondingly adjusts the temperature assuming :math:`h` is
                 conserved. RAS in its current formulation assumes that all cloud water
                 is deposited into the detrainment level as rain. All of the rain is
                 available for re-evaporation, which begins in the level below
                 detrainment. The scheme accounts for some microphysics such as the
                 rainfall intensity, the drop size distribution, as well as the
                 temperature, pressure and relative humidity of the surrounding air. The
                 fraction of the moisture deficit in any model layer into which the rain
                 may re-evaporate is controlled by a free parameter, which allows for a
                 relatively efficient re-evaporation of liquid precipitate and larger
                 rainout for frozen precipitation.
                 
                 Due to the increased vertical resolution near the surface, the lowest
                 model layers are averaged to provide a 50 mb thick sub-cloud layer for
                 RAS. Each time RAS is invoked (every ten simulated minutes), a number of
                 randomly chosen subensembles are checked for the possibility of
                 convection, from just above cloud base to 10 mb.
                 
                 Supersaturation or large-scale precipitation is initiated in the fizhi
                 package whenever the relative humidity in any grid-box exceeds a
0bad585a21 Navi* critical value, currently 100%. The large-scale precipitation
8679f9097b Jeff* re-evaporates during descent to partially saturate lower layers in a
                 process identical to the re-evaporation of convective rain.
                 
0bad585a21 Navi* .. _fizhi_clouds:
                 
8679f9097b Jeff* Cloud Formation
                 ###############
                 
                 Convective and large-scale cloud fractons which are used for
                 cloud-radiative interactions are determined diagnostically as part of
                 the cumulus and large-scale parameterizations. Convective cloud
                 fractions produced by RAS are proportional to the detrained liquid water
                 amount given by
                 
0bad585a21 Navi* .. math:: F_{\rm RAS} = \min\left[ \frac{l_{\rm RAS}}{l_c}, 1 \right]
8679f9097b Jeff* 
                 where :math:`l_c` is an assigned critical value equal to :math:`1.25`
                 g/kg. A memory is associated with convective clouds defined by:
                 
0bad585a21 Navi* .. math:: F_{\rm RAS}^n = \min\left[ F_{\rm RAS} + \left(1-\frac{\Delta t_{\rm RAS}}{\tau}\right) F_{\rm RAS}^{n-1} \, , \, 1 \right],
8679f9097b Jeff* 
0bad585a21 Navi* where :math:`F_{\rm RAS}` is the instantaneous cloud fraction and
                 :math:`F_{\rm RAS}^{n-1}` is the cloud fraction from the previous RAS
8679f9097b Jeff* timestep. The memory coefficient is computed using a RAS cloud
                 timescale, :math:`\tau`, equal to 1 hour. RAS cloud fractions are
0bad585a21 Navi* cleared when they fall below 5%.
8679f9097b Jeff* 
                 Large-scale cloudiness is defined, following Slingo and Ritter (1985),
                 as a function of relative humidity:
                 
0bad585a21 Navi* .. math:: F_{\rm ls} = \min\left[ { \left( \frac{\textrm{RH}-\textrm{RH}_c}{1-\textrm{RH}_c} \right) }^2 \, , \, 1 \right]
8679f9097b Jeff* 
                 where
                 
0bad585a21 Navi* .. math::
                    \begin{aligned}
                    \textrm{RH}_c & = 1-s(1-s)(2-\sqrt{3}+2\sqrt{3}s)r \\
                    s & = p/p_{\rm surf} \\
                    r & = \left(\frac{1.0-\textrm{RH}_{\rm min}}{\alpha}\right) \\
                    \textrm{RH}_{\rm min} & = 0.75 \\
                    \alpha & = 0.573285 \end{aligned}
8679f9097b Jeff* 
                 These cloud fractions are suppressed, however, in regions where the
                 convective sub-cloud layer is conditionally unstable. The functional
0bad585a21 Navi* form of :math:`\textrm{RH}_c` is shown in :numref:`rhcrit`
8679f9097b Jeff* 
                 
                 
                  .. figure:: figs/rhcrit.*
                     :width: 70%
                     :align: center
                     :alt: critical relative humidity for clouds
                     :name: rhcrit
                 
                     Critical Relative Humidity for Clouds.
                 
                 
                 
                 The total cloud fraction in a grid box is determined by the larger of
                 the two cloud fractions:
                 
0bad585a21 Navi* .. math:: F_{\rm cld} = \max \left[ F_{\rm RAS} \, , \, F_{\rm ls} \right]
8679f9097b Jeff* 
                 Finally, cloud fractions are time-averaged between calls to the
                 radiation packages.
                 
                 Radiation:
                 
                 The parameterization of radiative heating in the fizhi package includes
                 effects from both shortwave and longwave processes. Radiative fluxes are
                 calculated at each model edge-level in both up and down directions. The
                 heating rates/cooling rates are then obtained from the vertical
                 divergence of the net radiative fluxes.
                 
                 The net flux is
                 
                 .. math:: F = F^\uparrow - F^\downarrow
                 
                 where :math:`F` is the net flux, :math:`F^\uparrow` is the upward flux
                 and :math:`F^\downarrow` is the downward flux.
                 
                 The heating rate due to the divergence of the radiative flux is given by
                 
                 .. math:: \pp{\rho c_p T}{t} = - \pp{F}{z}
                 
                 or
                 
                 .. math:: \pp{T}{t} = \frac{g}{c_p \pi} \pp{F}{\sigma}
                 
                 where :math:`g` is the accelation due to gravity and :math:`c_p` is the
                 heat capacity of air at constant pressure.
                 
                 The time tendency for Longwave Radiation is updated every 3 hours. The
                 time tendency for Shortwave Radiation is updated once every three hours
                 assuming a normalized incident solar radiation, and subsequently
                 modified at every model time step by the true incident radiation. The
0bad585a21 Navi* solar constant value used in the package is equal to 1365 W m\ :sup:`--2`
                 and a CO\ :sub:`2` mixing ratio of 330 ppm. For the ozone mixing ratio,
8679f9097b Jeff* monthly mean zonally averaged climatological values specified as a
                 function of latitude and height :cite:`rosen:87` are linearly interpolated to the
                 current time.
                 
                 Shortwave Radiation
                 ###################
                 
                 The shortwave radiation package used in the package computes solar
                 radiative heating due to the absoption by water vapor, ozone, carbon
                 dioxide, oxygen, clouds, and aerosols and due to the scattering by
                 clouds, aerosols, and gases. The shortwave radiative processes are
                 described by :cite:`chou:90,chou:92`. This shortwave package uses the Delta-Eddington
                 approximation to compute the bulk scattering properties of a single
                 layer following King and Harshvardhan (JAS, 1986). The transmittance and
                 reflectance of diffuse radiation follow the procedures of Sagan and
                 Pollock (JGR, 1967) and :cite:`lhans:74`.
                 
                 Highly accurate heating rate calculations are obtained through the use
                 of an optimal grouping strategy of spectral bands. By grouping the UV
                 and visible regions as indicated in :numref:`tab_phys_pkg_fizhi_solar1`, the
                 Rayleigh scattering and the ozone absorption of solar radiation can be
                 accurately computed in the ultraviolet region and the photosynthetically
                 active radiation (PAR) region. The computation of solar flux in the
                 infrared region is performed with a broadband parameterization using the
                 spectrum regions shown in :numref:`tab_phys_pkg_fizhi_solar2`. The solar radiation
                 algorithm used in the fizhi package can be applied not only for climate
                 studies but also for studies on the photolysis in the upper atmosphere
                 and the photosynthesis in the biosphere.
                 
                 
0bad585a21 Navi* .. table:: UV and visible spectral regions used in shortwave radiation package. 
8679f9097b Jeff*   :name: tab_phys_pkg_fizhi_solar1
                 
                   +----------+--------+-----------------------+
                   |     **UV and Visible Spectral Regions**   |
                   +----------+--------+-----------------------+
                   | Region   | Band   | Wavelength (micron)   |
                   +==========+========+=======================+
                   | UV-C     | 1.     | .175 - .225           |
                   +----------+--------+-----------------------+
                   |          | 2.     | .225 - .245           |
                   +----------+--------+-----------------------+
                   |          |        | .260 - .280           |
                   +----------+--------+-----------------------+
                   |          | 3.     | .245 - .260           |
                   +----------+--------+-----------------------+
                   | UV-B     | 4.     | .280 - .295           |
                   +----------+--------+-----------------------+
                   |          | 5.     | .295 - .310           |
                   +----------+--------+-----------------------+
                   |          | 6.     | .310 - .320           |
                   +----------+--------+-----------------------+
                   | UV-A     | 7.     | .320 - .400           |
                   +----------+--------+-----------------------+
                   | PAR      | 8.     | .400 - .700           |
                   +----------+--------+-----------------------+
                 
                 
                 
                 
0bad585a21 Navi* .. table:: Infrared spectral regions used in shortwave radiation package.
8679f9097b Jeff*   :name: tab_phys_pkg_fizhi_solar2
                 
                   +--------+---------------------------------+-----------------------+
                   |            **Infrared Spectral Regions**                         |
                   +--------+---------------------------------+-----------------------+
                   | Band   | Wavenumber (cm\ :sup:`--1`)     | Wavelength (micron)   |
                   +========+=================================+=======================+
                   | 1      | 1000-4400                       | 2.27-10.0             |
                   +--------+---------------------------------+-----------------------+
                   | 2      | 4400-8200                       | 1.22-2.27             |
                   +--------+---------------------------------+-----------------------+
                   | 3      | 8200-14300                      | 0.70-1.22             |
                   +--------+---------------------------------+-----------------------+
                 
                 
                 Within the shortwave radiation package, both ice and liquid cloud
                 particles are allowed to co-exist in any of the model layers. Two sets
                 of cloud parameters are used, one for ice paticles and the other for
                 liquid particles. Cloud parameters are defined as the cloud optical
                 thickness and the effective cloud particle size. In the fizhi package,
                 the effective radius for water droplets is given as 10 microns, while 65
                 microns is used for ice particles. The absorption due to aerosols is
                 currently set to zero.
                 
                 To simplify calculations in a cloudy atmosphere, clouds are grouped into
                 low (:math:`p>700` mb), middle (700 mb :math:`\ge p > 400` mb), and high
                 (:math:`p < 400` mb) cloud regions. Within each of the three regions,
                 clouds are assumed maximally overlapped, and the cloud cover of the
                 group is the maximum cloud cover of all the layers in the group. The
                 optical thickness of a given layer is then scaled for both the direct
                 (as a function of the solar zenith angle) and diffuse beam radiation so
                 that the grouped layer reflectance is the same as the original
                 reflectance. The solar flux is computed for each of eight cloud
                 realizations possible within this low/middle/high classification, and
                 appropriately averaged to produce the net solar flux.
                 
                 Longwave Radiation
                 ##################
                 
                 The longwave radiation package used in the fizhi package is thoroughly
39fa6219cc Oliv* described by :cite:`chsz:94`. As described in that document, IR fluxes are
                 computed due to absorption by water vapor, carbon dioxide, and ozone. The
                 spectral bands together with their absorbers and parameterization methods,
                 configured for the fizhi package, are shown in
                 :numref:`tab_phys_pkg_fizhi_longwave`.
8679f9097b Jeff* 
0bad585a21 Navi* .. table:: IR spectral bands, absorbers, and parameterization method
8679f9097b Jeff*   :name: tab_phys_pkg_fizhi_longwave
                 
                   +----------------+------------------------------------+------------------------------+----------+
                   |                  **IR Spectral Bands**                                                        |
                   +----------------+------------------------------------+------------------------------+----------+
                   | Band           | Spectral Range (cm\ :sup:`--1`)    | Absorber                     | Method   |
                   +================+====================================+==============================+==========+
0bad585a21 Navi*   | 1              | 0-340                              | H\ :sub:`2`\ O line          | T        |
8679f9097b Jeff*   +----------------+------------------------------------+------------------------------+----------+
0bad585a21 Navi*   | 2              | 340-540                            | H\ :sub:`2`\ O line          | T        |
8679f9097b Jeff*   +----------------+------------------------------------+------------------------------+----------+
0bad585a21 Navi*   | 3a             | 540-620                            | H\ :sub:`2`\ O line          | K        |
8679f9097b Jeff*   +----------------+------------------------------------+------------------------------+----------+
0bad585a21 Navi*   | 3b             | 620-720                            | H\ :sub:`2`\ O continuum     | S        |
8679f9097b Jeff*   +----------------+------------------------------------+------------------------------+----------+
0bad585a21 Navi*   | 3b             | 720-800                            | CO\ :sub:`2`                 | T        |
8679f9097b Jeff*   +----------------+------------------------------------+------------------------------+----------+
0bad585a21 Navi*   | 4              | 800-980                            | H\ :sub:`2`\ O line          | K        |
8679f9097b Jeff*   +----------------+------------------------------------+------------------------------+----------+
0bad585a21 Navi*   |                |                                    | H\ :sub:`2`\ O continuum     | S        |
8679f9097b Jeff*   +----------------+------------------------------------+------------------------------+----------+
0bad585a21 Navi*   |                |                                    | H\ :sub:`2`\ O line          | K        |
8679f9097b Jeff*   +----------------+------------------------------------+------------------------------+----------+
0bad585a21 Navi*   | 5              | 980-1100                           | H\ :sub:`2`\ O continuum     | S        |
8679f9097b Jeff*   +----------------+------------------------------------+------------------------------+----------+
0bad585a21 Navi*   |                |                                    | O\ :sub:`3`                  | T        |
8679f9097b Jeff*   +----------------+------------------------------------+------------------------------+----------+
0bad585a21 Navi*   | 6              | 1100-1380                          | H\ :sub:`2`\ O line          | K        |
8679f9097b Jeff*   +----------------+------------------------------------+------------------------------+----------+
0bad585a21 Navi*   |                |                                    | H\ :sub:`2`\ O continuum     | S        |
8679f9097b Jeff*   +----------------+------------------------------------+------------------------------+----------+
0bad585a21 Navi*   | 7              | 1380-1900                          | H\ :sub:`2`\ O line          | T        |
8679f9097b Jeff*   +----------------+------------------------------------+------------------------------+----------+
0bad585a21 Navi*   | 8              | 1900-3000                          | H\ :sub:`2`\ O line          | K        |
8679f9097b Jeff*   +----------------+------------------------------------+------------------------------+----------+
                   | K: :math:`k`-distribution method with linear pressure scaling                                 |
                   +----------------+------------------------------------+------------------------------+----------+
                   | T: Table look-up with temperature and pressure scaling                                        |
                   +----------------+------------------------------------+------------------------------+----------+
                   | S: One-parameter temperature scaling                                                          |
                   +----------------+------------------------------------+------------------------------+----------+
                 
                 
                 The longwave radiation package accurately computes cooling rates for the
                 middle and lower atmosphere from 0.01 mb to the surface. Errors are
0bad585a21 Navi* < 0.4 C day\ :sup:`--1` in cooling rates and < 1% in
8679f9097b Jeff* fluxes. From Chou and Suarez, it is estimated that the total effect of
                 neglecting all minor absorption bands and the effects of minor infrared
0bad585a21 Navi* absorbers such as nitrous oxide (N\ :sub:`2`\ O), methane
                 (CH\ :sub:`4`), and the chlorofluorocarbons (CFCs), is an underestimate
                 of :math:`\approx 5` W m\ :sup:`--2` in the downward flux at the surface
                 and an overestimate of :math:`\approx 3` W m\ :sup:`--2` in the upward
8679f9097b Jeff* flux at the top of the atmosphere.
                 
                 Similar to the procedure used in the shortwave radiation package, clouds
                 are grouped into three regions catagorized as low/middle/high. The net
                 clear line-of-site probability :math:`(P)` between any two levels,
                 :math:`p_1` and :math:`p_2 \quad (p_2 > p_1)`, assuming randomly
                 overlapped cloud groups, is simply the product of the probabilities
                 within each group:
                 
0bad585a21 Navi* .. math:: P_{\rm net} = P_{\rm low} \times P_{\rm mid} \times P_{\rm hi}
8679f9097b Jeff* 
                 Since all clouds within a group are assumed maximally overlapped, the
                 clear line-of-site probability within a group is given by:
                 
0bad585a21 Navi* .. math:: P_{\rm group} = 1 - F_{\rm max}
8679f9097b Jeff* 
0bad585a21 Navi* where :math:`F_{\rm max}` is the maximum cloud fraction encountered between
8679f9097b Jeff* :math:`p_1` and :math:`p_2` within that group. For groups and/or levels
                 outside the range of :math:`p_1` and :math:`p_2`, a clear line-of-site
                 probability equal to 1 is assigned.
                 
                 Cloud-Radiation Interaction
                 ###########################
                 
                 The cloud fractions and diagnosed cloud liquid water produced by moist
                 processes within the fizhi package are used in the radiation packages to
                 produce cloud-radiative forcing. The cloud optical thickness associated
                 with large-scale cloudiness is made proportional to the diagnosed
                 large-scale liquid water, :math:`\ell`, detrained due to
                 super-saturation. Two values are used corresponding to cloud ice
                 particles and water droplets. The range of optical thickness for these
                 clouds is given as
                 
0bad585a21 Navi* .. math:: 0.0002 \le \tau_{\rm ice} (\text{mb}^{-1}) \le 0.002  \quad\mbox{for}\quad  0 \le \ell \le 2 \; \text{mg/kg}
8679f9097b Jeff* 
0bad585a21 Navi* .. math:: 0.02 \le \tau_{\rm H_2O} (\text{mb}^{-1}) \le 0.2  \quad\mbox{for}\quad  0 \le \ell \le 10 \; \text{mg/kg}
8679f9097b Jeff* 
                 The partitioning, :math:`\alpha`, between ice particles and water
                 droplets is achieved through a linear scaling in temperature:
                 
0bad585a21 Navi* .. math:: 0 \le \alpha \le 1 \quad\mbox{for}\quad  233.15 \le T \le 253.15
8679f9097b Jeff* 
                 The resulting optical depth associated with large-scale cloudiness is
                 given as
                 
0bad585a21 Navi* .. math:: \tau_{\rm ls} = \alpha \tau_{\rm H_2O} + (1-\alpha) \tau_{\rm ice}
8679f9097b Jeff* 
                 The optical thickness associated with sub-grid scale convective clouds
                 produced by RAS is given as
                 
0bad585a21 Navi* .. math:: \tau_{\rm RAS} = 0.16 \; \text{mb}^{-1}
8679f9097b Jeff* 
                 The total optical depth in a given model layer is computed as a weighted
                 average between the large-scale and sub-grid scale optical depths,
                 normalized by the total cloud fraction in the layer:
                 
0bad585a21 Navi* .. math:: \tau = \left( \frac{F_{\rm RAS} \,\,\, \tau_{\rm RAS} + F_{\rm ls} \,\,\, \tau_{\rm ls} }{ F_{\rm RAS}+F_{\rm ls} } \right) \Delta p
8679f9097b Jeff* 
0bad585a21 Navi* where :math:`F_{\rm RAS}` and :math:`F_{\rm ls}` are the time-averaged cloud
8679f9097b Jeff* fractions associated with RAS and large-scale processes described in
0bad585a21 Navi* :numref:`fizhi_clouds`. The optical thickness for the longwave
                 radiative feedback is assumed to be 75%  of these values.
8679f9097b Jeff* 
                 The entire Moist Convective Processes Module is called with a frequency
                 of 10 minutes. The cloud fraction values are time-averaged over the
                 period between Radiation calls (every 3 hours). Therefore, in a
                 time-averaged sense, both convective and large-scale cloudiness can
                 exist in a given grid-box.
                 
                 Turbulence
                 ##########
                 
                 Turbulence is parameterized in the fizhi package to account for its
                 contribution to the vertical exchange of heat, moisture, and momentum.
                 The turbulence scheme is invoked every 30 minutes, and employs a
                 backward-implicit iterative time scheme with an internal time step of 5
                 minutes. The tendencies of atmospheric state variables due to turbulent
                 diffusion are calculated using the diffusion equations:
                 
                 .. math::
0bad585a21 Navi*    \begin{aligned}
                    {\pp{u}{t}}_{\rm turb} &= {\pp{}{z} }{(- \overline{u^{\prime}w^{\prime}})}
                     = {\pp{}{z} }{\left(K_m \pp{u}{z}\right)} \nonumber \\
                    {\pp{v}{t}}_{\rm turb} &= {\pp{}{z} }{(- \overline{v^{\prime}w^{\prime}})}
                     = {\pp{}{z} }{\left(K_m \pp{v}{z}\right)} \nonumber \\
                    {\pp{T}{t}} = P^{\kappa}{\pp{\theta}{t}}_{\rm turb} &= 
8679f9097b Jeff*    P^{\kappa}{\pp{}{z} }{(- \overline{w^{\prime}\theta^{\prime}})}
0bad585a21 Navi*     = P^{\kappa}{\pp{}{z} }{\left(K_h \pp{\theta_v}{z}\right)} \nonumber \\
                    {\pp{q}{t}}_{\rm turb} &= {\pp{}{z} }{(- \overline{w^{\prime}q^{\prime}})}
                     = {\pp{}{z} }{\left(K_h \pp{q}{z}\right)}
                    \end{aligned}
8679f9097b Jeff* 
                 Within the atmosphere, the time evolution of second turbulent moments is
                 explicitly modeled by representing the third moments in terms of the
                 first and second moments. This approach is known as a second-order
                 closure modeling. To simplify and streamline the computation of the
                 second moments, the level 2.5 assumption of Mellor and Yamada (1974) and :cite:`yam:77`
                 is employed, in which only the turbulent kinetic energy (TKE),
                 
0bad585a21 Navi* .. math:: {\h}{q^2}={\overline{{u^{\prime}}^2}}+{\overline{{v^{\prime}}^2}}+{\overline{{w^{\prime}}^2}}
8679f9097b Jeff* 
                 is solved prognostically and the other second moments are solved
                 diagnostically. The prognostic equation for TKE allows the scheme to
                 simulate some of the transient and diffusive effects in the turbulence.
                 The TKE budget equation is solved numerically using an implicit backward
                 computation of the terms linear in :math:`q^2` and is written:
                 
                 .. math::
                 
0bad585a21 Navi*    {\dd{}{t} \left({{\h} q^2}\right)} - { \pp{}{z} \left[{ \frac{5}{3} {{\lambda}_1} q { \pp {}{z} 
                    \left({\h}q^2\right)} }\right]} =
8679f9097b Jeff*    {- \overline{{u^{\prime}}{w^{\prime}}} { \pp{U}{z} }} - {\overline{{v^{\prime}}{w^{\prime}}} 
                    { \pp{V}{z} }} + {\frac{g}{\Theta_0}{\overline{{w^{\prime}}{{{\theta}_v}^{\prime}}}}
                    - \frac{ q^3}{{\Lambda}_1} }
                 
                 where :math:`q` is the turbulent velocity, :math:`{u^{\prime}}`,
                 :math:`{v^{\prime}}`, :math:`{w^{\prime}}` and
                 :math:`{{\theta}^{\prime}}` are the fluctuating parts of the velocity
                 components and potential temperature, :math:`U` and :math:`V` are the
                 mean velocity components, :math:`{\Theta_0}^{-1}` is the coefficient of
                 thermal expansion, and :math:`{{\lambda}_1}` and :math:`{{\Lambda} _1}`
                 are constant multiples of the master length scale, :math:`\ell`, which
                 is designed to be a characteristic measure of the vertical structure of
                 the turbulent layers.
                 
                 The first term on the left-hand side represents the time rate of change
                 of TKE, and the second term is a representation of the triple
                 correlation, or turbulent transport term. The first three terms on the
                 right-hand side represent the sources of TKE due to shear and bouyancy,
                 and the last term on the right hand side is the dissipation of TKE.
                 
                 In the level 2.5 approach, the vertical fluxes of the scalars
                 :math:`\theta_v` and :math:`q` and the wind components :math:`u` and
                 :math:`v` are expressed in terms of the diffusion coefficients
                 :math:`K_h` and :math:`K_m`, respectively. In the statisically
                 realizable level 2.5 turbulence scheme of :cite:`helflab:88`, these diffusion coefficients
                 are expressed as
                 
                 .. math::
                 
                    K_h 
                     = \left\{ \begin{array}{l@{\quad\mbox{for}\quad}l} q \, \ell \, S_H(G_M,G_H) \, & \mbox{decaying turbulence}
0bad585a21 Navi*    \\ \frac{ q^2 }{ q_{\rm eq} } \, \ell \, S_{H}(G_{M_e},G_{H_e}) \, & \mbox{growing turbulence} \end{array} \right.
8679f9097b Jeff* 
                 and
                 
                 .. math::
                 
                    K_m
                     = \left\{ \begin{array}{l@{\quad\mbox{for}\quad}l} q \, \ell \, S_M(G_M,G_H) \, & \mbox{decaying turbulence}                
0bad585a21 Navi*    \\ \frac{ q^2 }{ q_{\rm eq} } \, \ell \, S_{M}(G_{M_e},G_{H_e}) \, & \mbox{growing turbulence} \end{array} \right.
8679f9097b Jeff* 
0bad585a21 Navi* where the subscript 'eq' refers to the value under conditions of
                 local equilibrium (obtained from the Level 2.0 Model), :math:`\ell` is
8679f9097b Jeff* the master length scale related to the vertical structure of the
                 atmosphere, and :math:`S_M` and :math:`S_H` are functions of :math:`G_H`
                 and :math:`G_M`, the dimensionless buoyancy and wind shear parameters,
                 respectively. Both :math:`G_H` and :math:`G_M`, and their equilibrium
                 values :math:`G_{H_e}` and :math:`G_{M_e}`, are functions of the
                 Richardson number:
                 
                 .. math::
0bad585a21 Navi*    \textrm{RI} = \frac{ \frac{g}{\theta_v} \pp{\theta_v}{z} }{ (\pp{u}{z})^2 + (\pp{v}{z})^2 }
                    =  \frac{c_p \pp{\theta_v}{z} \pp{P^ \kappa}{z} }{ (\pp{u}{z})^2 + (\pp{v}{z})^2 }
8679f9097b Jeff* 
                 Negative values indicate unstable buoyancy and shear, small positive
0bad585a21 Navi* values (<0.2) indicate dominantly unstable shear, and large
8679f9097b Jeff* positive values indicate dominantly stable stratification.
                 
                 Turbulent eddy diffusion coefficients of momentum, heat and moisture in
                 the surface layer, which corresponds to the lowest GCM level (see *—
                 missing table —*), are calculated using stability-dependant functions
                 based on Monin-Obukhov theory:
                 
0bad585a21 Navi* .. math:: {K_m} ({\rm surface}) = C_u \times u_* = C_D W_s
8679f9097b Jeff* 
                 and
                 
0bad585a21 Navi* .. math:: {K_h} ({\rm surface}) =  C_t \times u_* = C_H W_s
8679f9097b Jeff* 
                 where :math:`u_*=C_uW_s` is the surface friction velocity, :math:`C_D`
                 is termed the surface drag coefficient, :math:`C_H` the heat transfer
                 coefficient, and :math:`W_s` is the magnitude of the surface layer wind.
                 
                 :math:`C_u` is the dimensionless exchange coefficient for momentum from
                 the surface layer similarity functions:
                 
                 .. math:: {C_u} = \frac{u_* }{ W_s} = \frac{ k }{ \psi_{m} }
                 
                 where k is the Von Karman constant and :math:`\psi_m` is the surface
                 layer non-dimensional wind shear given by
                 
0bad585a21 Navi* .. math:: \psi_{m} = {\int_{\zeta_{0}}^{\zeta} \frac{\phi_{m} }{ \zeta} d \zeta}
8679f9097b Jeff* 
                 Here :math:`\zeta` is the non-dimensional stability parameter, and
                 :math:`\phi_m` is the similarity function of :math:`\zeta` which
                 expresses the stability dependance of the momentum gradient. The
                 functional form of :math:`\phi_m` is specified differently for stable
                 and unstable layers.
                 
                 :math:`C_t` is the dimensionless exchange coefficient for heat and
                 moisture from the surface layer similarity functions:
                 
                 .. math::
                 
                    {C_t} = -\frac{( \overline{w^{\prime}\theta^{\prime}}) }{ u_* \Delta \theta } =
                    -\frac{( \overline{w^{\prime}q^{\prime}}) }{ u_* \Delta q } =
                    \frac{ k }{ (\psi_{h} + \psi_{g}) }
                 
                 where :math:`\psi_h` is the surface layer non-dimensional temperature
                 gradient given by
                 
0bad585a21 Navi* .. math:: \psi_{h} = {\int_{\zeta_{0}}^{\zeta} \frac{\phi_{h} }{ \zeta} d \zeta}
8679f9097b Jeff* 
                 Here :math:`\phi_h` is the similarity function of :math:`\zeta`, which
                 expresses the stability dependance of the temperature and moisture
                 gradients, and is specified differently for stable and unstable layers
                 according to :cite:`helfschu:95`.
                 
                 :math:`\psi_g` is the non-dimensional temperature or moisture gradient
                 in the viscous sublayer, which is the mosstly laminar region between the
                 surface and the tops of the roughness elements, in which temperature and
                 moisture gradients can be quite large. Based on :cite:`yagkad:74`:
                 
                 .. math::
                 
0bad585a21 Navi*    \psi_{g} = \frac{ 0.55 ({\rm Pr}^{2/3} - 0.2) }{ \nu^{1/2} }
                    (h_{0}u_{*} - h_{0_{\rm ref}}u_{*_{\rm ref}})^{1/2}
8679f9097b Jeff* 
                 where Pr is the Prandtl number for air, :math:`\nu` is the molecular
                 viscosity, :math:`z_{0}` is the surface roughness length, and the
0bad585a21 Navi* subscript 'ref' refers to a reference value. :math:`h_{0} = 30z_{0}`
                 with a maximum value over land of 0.01.
8679f9097b Jeff* 
                 The surface roughness length over oceans is is a function of the
                 surface-stress velocity,
                 
                 .. math:: {z_0} = c_1u^3_* + c_2u^2_* + c_3u_* + c_4 + \frac{c_5 }{ u_*}
                 
                 where the constants are chosen to interpolate between the reciprocal
                 relation of :cite:`kondo:75` for weak winds, and the piecewise linear relation of :cite:`larpond:81` for
                 moderate to large winds. Roughness lengths over land are specified from
                 the climatology of :cite:`dorsell:89`.
                 
                 For an unstable surface layer, the stability functions, chosen to
                 interpolate between the condition of small values of :math:`\beta` and
                 the convective limit, are the KEYPS function :cite:`pano:73` for momentum, and its
                 generalization for heat and moisture:
                 
                 .. math::
                 
                    {\phi_m}^4 - 18 \zeta {\phi_m}^3 = 1 \hspace{1cm} ; \hspace{1cm} 
0bad585a21 Navi*    {\phi_h}^2 - 18 \zeta {\phi_h}^3 = 1 \hspace{1cm}
8679f9097b Jeff* 
                 The function for heat and moisture assures non-vanishing heat and
                 moisture fluxes as the wind speed approaches zero.
                 
                 For a stable surface layer, the stability functions are the
                 observationally based functions of :cite:`clarke:70`, slightly modified for the momemtum
                 flux:
                 
                 .. math::
                 
                    {\phi_m} = \frac{ 1 + 5 {{\zeta}_1} }{ 1 + 0.00794 {\zeta}_1
                    (1+ 5 {\zeta}_1) } \hspace{1cm} ; \hspace{1cm}
                    {\phi_h} = \frac{ 1 + 5 {{\zeta}_1} }{ 1 + 0.00794 {\zeta}
0bad585a21 Navi*    (1+ 5 {{\zeta}_1}) }
8679f9097b Jeff* 
                 The moisture flux also depends on a specified evapotranspiration
                 coefficient, set to unity over oceans and dependant on the
                 climatological ground wetness over land.
                 
                 Once all the diffusion coefficients are calculated, the diffusion
                 equations are solved numerically using an implicit backward operator.
                 
                 Atmospheric Boundary Layer
                 ##########################
                 
                 The depth of the atmospheric boundary layer (ABL) is diagnosed by the
                 parameterization as the level at which the turbulent kinetic energy is
                 reduced to a tenth of its maximum near surface value. The vertical
                 structure of the ABL is explicitly resolved by the lowest few (3-8)
                 model layers.
                 
                 Surface Energy Budget
                 #####################
                 
                 The ground temperature equation is solved as part of the turbulence
                 package using a backward implicit time differencing scheme:
                 
0bad585a21 Navi* .. math:: C_g\pp{T_g}{t} = R_{\rm sw} - R_{\rm lw} + Q_{\rm ice} - H - LE
8679f9097b Jeff* 
0bad585a21 Navi* where :math:`R_{\rm sw}` is the net surface downward shortwave radiative
                 flux and :math:`R_{\rm lw}` is the net surface upward longwave radiative
8679f9097b Jeff* flux.
                 
                 :math:`H` is the upward sensible heat flux, given by:
                 
                 .. math::
0bad585a21 Navi*    {H} = P^{\kappa}\rho c_{p} C_{H} W_s (\theta_{\rm surface} - \theta_{\rm NLAY})
                    \hspace{1cm}\text{where}: \hspace{.2cm}C_H = C_u C_t
8679f9097b Jeff* 
                 where :math:`\rho` = the atmospheric density at the surface,
                 :math:`c_{p}` is the specific heat of air at constant pressure, and
                 :math:`\theta` represents the potential temperature of the surface and
                 of the lowest :math:`\sigma`-level, respectively.
                 
0bad585a21 Navi* The upward latent heat flux, :math:`\textrm{LE}`, is given by
8679f9097b Jeff* 
                 .. math::
                 
0bad585a21 Navi*    \textrm{LE} =  \rho \beta L C_{H} W_s (q_{\rm surface} - q_{\rm NLAY})
                    \hspace{1cm}\text{where}: \hspace{.2cm}C_H = C_u C_t
8679f9097b Jeff* 
                 where :math:`\beta` is the fraction of the potential evapotranspiration
                 actually evaporated, L is the latent heat of evaporation, and
0bad585a21 Navi* :math:`q_{\rm surface}` and :math:`q_{\rm NLAY}` are the specific humidity of
8679f9097b Jeff* the surface and of the lowest :math:`\sigma`-level, respectively.
                 
0bad585a21 Navi* The heat conduction through sea ice, :math:`Q_{\rm ice}`, is given by
8679f9097b Jeff* 
0bad585a21 Navi* .. math:: {Q_{\rm ice}} = \frac{C_{\rm ti} }{ H_i} (T_i-T_g)
8679f9097b Jeff* 
0bad585a21 Navi* where :math:`C_{\rm ti}` is the thermal conductivity of ice, :math:`H_i` is
                 the ice thickness, assumed to be 3 m where sea ice
8679f9097b Jeff* is present, :math:`T_i` is 273 degrees Kelvin, and :math:`T_g` is the
                 surface temperature of the ice.
                 
                 :math:`C_g` is the total heat capacity of the ground, obtained by
                 solving a heat diffusion equation for the penetration of the diurnal
0bad585a21 Navi* cycle into the ground (Blackadar 1977), and is given by:
8679f9097b Jeff* 
                 .. math::
                 
                    C_g = \sqrt{ \frac{\lambda C_s }{ 2\omega} } = \sqrt{(0.386 + 0.536W + 0.15W^2)2\times10^{-3}
0bad585a21 Navi*    \frac{86400}{2\pi} }
8679f9097b Jeff* 
                 Here, the thermal conductivity, :math:`\lambda`, is equal to
0bad585a21 Navi* :math:`2\times10^{-3}` :math:`\frac{\text{ly}}{\text{sec}}\frac{\text{cm}}{\text{K}}`,
                 the angular velocity of the earth, :math:`\omega`, is
                 written as 86400 sec day\ :sup:`--1` divided by :math:`2 \pi`
                 radians day\ :sup:`--1`, and the expression for :math:`C_s`, the heat capacity per unit
8679f9097b Jeff* volume at the surface, is a function of the ground wetness, :math:`W`.
                 
                 Land Surface Processes:
                 
                 Surface Type
                 ############
                 
                 The fizhi package surface Types are designated using the Koster-Suarez
                 :cite:`ks:91,ks:92` Land Surface Model (LSM) mosaic philosophy which allows multiple
                 “tiles”, or multiple surface types, in any one grid cell. The
                 Koster-Suarez LSM surface type classifications are shown in :numref:`tab_phys_pkg_fizhi_surface_type_designation`. The surface types and the percent of the grid cell
                 occupied by any surface type were derived from the surface
                 classification of :cite:`deftow:94`, and information about the location of permanent ice
                 was obtained from the classifications of :cite:`dorsell:89`. The surface type map for a
                 :math:`1^\circ` grid is shown in :numref:`fig_phys_pkg_fizhi_surftype`. The
                 determination of the land or sea category of surface type was made from
                 NCAR’s 10 minute by 10 minute Navy topography dataset, which includes
                 information about the percentage of water-cover at any point. The data
                 were averaged to the model’s grid resolutions, and any grid-box whose
                 averaged water percentage was :math:`\geq 60 \%` was defined as a water
                 point. The Land-Water designation was further modified subjectively to
                 ensure sufficient representation from small but isolated land and water
                 regions.
                 
                 .. table:: Surface Type Designation
                     :name: tab_phys_pkg_fizhi_surface_type_designation
                 
                     +--------+-----------------------------+
                     | Type   | Vegetation Designation      |
                     +========+=============================+
                     | 1      | Broadleaf Evergreen Trees   |
                     +--------+-----------------------------+
                     | 2      | Broadleaf Deciduous Trees   |
                     +--------+-----------------------------+
                     | 3      | Needleleaf Trees            |
                     +--------+-----------------------------+
                     | 4      | Ground Cover                |
                     +--------+-----------------------------+
                     | 5      | Broadleaf Shrubs            |
                     +--------+-----------------------------+
                     | 6      | Dwarf Trees (Tundra)        |
                     +--------+-----------------------------+
                     | 7      | Bare Soil                   |
                     +--------+-----------------------------+
                     | 8      | Desert (Bright)             |
                     +--------+-----------------------------+
                     | 9      | Glacier                     |
                     +--------+-----------------------------+
                     | 10     | Desert (Dark)               |
                     +--------+-----------------------------+
                     | 100    | Ocean                       |
                     +--------+-----------------------------+
                 
                 
                 
                 .. figure:: figs/surftype.*
                     :width: 70%
                     :align: center
                     :alt: surface type combinations
                     :name: fig_phys_pkg_fizhi_surftype
                 
                     Surface type combinations 
                 
                 
                 
                 Surface Roughness
                 #################
                 
                 The surface roughness length over oceans is computed iteratively with
                 the wind stress by the surface layer parameterization :cite:`helfschu:95`. It employs an
                 interpolation between the functions of :cite:`larpond:81` for high winds and of :cite:`kondo:75` for weak
                 winds.
                 
                 
                 Albedo
                 ######
                 
                 The surface albedo computation, described in , employs the “two stream”
                 approximation used in Sellers’ (1987) Simple Biosphere (SiB) Model which
                 distinguishes between the direct and diffuse albedos in the visible and
                 in the near infra-red spectral ranges. The albedos are functions of the
                 observed leaf area index (a description of the relative orientation of
                 the leaves to the sun), the greenness fraction, the vegetation type, and
                 the solar zenith angle. Modifications are made to account for the
                 presence of snow, and its depth relative to the height of the vegetation
                 elements.
                 
                 Gravity Wave Drag
                 #################
                 
                 The fizhi package employs the gravity wave drag scheme of :cite:`zhouetal:95`. This scheme
                 is a modified version of Vernekar et al. (1992), which was based on
                 Alpert et al. (1988) and Helfand et al. (1987). In this version, the
                 gravity wave stress at the surface is based on that derived by
                 Pierrehumbert (1986) and is given by:
                 
                 .. math:: 
0bad585a21 Navi*   |\vec{\tau}_{\rm sfc}| = \frac{\rho U^3}{N \ell^*} \left( \frac{F_r^2}{1+F_r^2}\right)
8679f9097b Jeff* 
                 
                 where :math:`F_r = N h /U` is the Froude number, :math:`N` is the *Brunt
                 - V�is�l�* frequency, :math:`U` is the surface wind speed, :math:`h` is
                 the standard deviation of the sub-grid scale orography, and
                 :math:`\ell^*` is the wavelength of the monochromatic gravity wave in
                 the direction of the low-level wind. A modification introduced by Zhou
                 et al. allows for the momentum flux to escape through the top of the
                 model, although this effect is small for the current 70-level model. The
                 subgrid scale standard deviation is defined by :math:`h`, and is not
                 allowed to exceed 400 m.
                 
                 The effects of using this scheme within a GCM are shown in :cite:`taksz:96`. Experiments
                 using the gravity wave drag parameterization yielded significant and
                 beneficial impacts on both the time-mean flow and the transient
                 statistics of the a GCM climatology, and have eliminated most of the
                 worst dynamically driven biases in the a GCM simulation. An examination
                 of the angular momentum budget during climate runs indicates that the
                 resulting gravity wave torque is similar to the data-driven torque
                 produced by a data assimilation which was performed without gravity wave
                 drag. It was shown that the inclusion of gravity wave drag results in
                 large changes in both the mean flow and in eddy fluxes. The result is a
                 more accurate simulation of surface stress (through a reduction in the
                 surface wind strength), of mountain torque (through a redistribution of
                 mean sea-level pressure), and of momentum convergence (through a
                 reduction in the flux of westerly momentum by transient flow eddies).
                 
0bad585a21 Navi* Boundary Conditions and other Input Data
                 ########################################
8679f9097b Jeff* 
                 Required fields which are not explicitly predicted or diagnosed during
                 model execution must either be prescribed internally or obtained from
                 external data sets. In the fizhi package these fields include: sea
                 surface temperature, sea ice estent, surface geopotential variance,
                 vegetation index, and the radiation-related background levels of: ozone,
                 carbon dioxide, and stratospheric moisture.
                 
                 Boundary condition data sets are available at the model’s resolutions
                 for either climatological or yearly varying conditions. Any frequency of
                 boundary condition data can be used in the fizhi package; however, the
                 current selection of data is summarized in :numref:`tab_phys_pkg_fizhi_inputs`. The
                 time mean values are interpolated during each model timestep to the
                 current time.
                 
                 .. table:: Boundary conditions and other input data used in the fizhi package. Also noted are the current years and frequencies available.
                     :name: tab_phys_pkg_fizhi_inputs
                 
                     +-----------------------------------------+-----------+-----------------------------+
                     | **Fizhi Input Datasets**                                                          |
                     +-----------------------------------------+-----------+-----------------------------+
                     | Sea Ice Extent                          | monthly   | 1979-current, climatology   |
                     +-----------------------------------------+-----------+-----------------------------+
                     | Sea Ice Extent                          | weekly    | 1982-current, climatology   |
                     +-----------------------------------------+-----------+-----------------------------+
                     | Sea Surface Temperature                 | monthly   | 1979-current, climatology   |
                     +-----------------------------------------+-----------+-----------------------------+
                     | Sea Surface Temperature                 | weekly    | 1982-current, climatology   |
                     +-----------------------------------------+-----------+-----------------------------+
                     | Zonally Averaged Upper-Level Moisture   | monthly   | climatology                 |
                     +-----------------------------------------+-----------+-----------------------------+
                     | Zonally Averaged Ozone Concentration    | monthly   | climatology                 |
                     +-----------------------------------------+-----------+-----------------------------+
                 
                 
                 Topography and Topography Variance
                 ##################################
                 
                 Surface geopotential heights are provided from an averaging of the Navy
                 10 minute by 10 minute dataset supplied by the National Center for
                 Atmospheric Research (NCAR) to the model’s grid resolution. The original
                 topography is first rotated to the proper grid-orientation which is
                 being run, and then averages the data to the model resolution.
                 
                 The standard deviation of the subgrid-scale topography is computed by
                 interpolating the 10 minute data to the model’s resolution and
                 re-interpolating back to the 10 minute by 10 minute resolution. The
                 sub-grid scale variance is constructed based on this smoothed dataset.
                 
                 
                 Upper Level Moisture
                 ####################
                 
                 The fizhi package uses climatological water vapor data above 100 mb from
                 the Stratospheric Aerosol and Gas Experiment (SAGE) as input into the
                 model’s radiation packages. The SAGE data is archived as monthly zonal
                 means at :math:`5^\circ` latitudinal resolution. The data is
                 interpolated to the model’s grid location and current time, and blended
                 with the GCM’s moisture data. Below 300 mb, the model’s moisture data is
                 used. Above 100 mb, the SAGE data is used. Between 100 and 300 mb, a
                 linear interpolation (in pressure) is performed using the data from SAGE
                 and the GCM.
                 
9ce7d74115 Jeff* 
                 .. _fizhi_diagnostics:
                 
8679f9097b Jeff* Fizhi Diagnostics
                 +++++++++++++++++
                 
0bad585a21 Navi* Fizhi Diagnostic Menu:
8679f9097b Jeff*     
                 +--------+----------------------------------+---------+--------------------------------------------------+
                 | NAME   |  UNITS                           |  LEVELS | DESCRIPTION                                      |
                 +--------+----------------------------------+---------+--------------------------------------------------+
                 | UFLUX  |  N m\ :sup:`--2`                 |  1      | Surface U-Wind Stress on the atmosphere          |
                 +--------+----------------------------------+---------+--------------------------------------------------+
                 | VFLUX  |  N m\ :sup:`--2`                 |  1      | Surface V-Wind Stress on the atmosphere          |
                 +--------+----------------------------------+---------+--------------------------------------------------+
                 | HFLUX  |  W m\ :sup:`--2`                 |  1      | Surface Flux of Sensible Heat                    |
                 +--------+----------------------------------+---------+--------------------------------------------------+
                 | EFLUX  |  W m\ :sup:`--2`                 |  1      | Surface Flux of Latent Heat                      |
                 +--------+----------------------------------+---------+--------------------------------------------------+
                 | QICE   |  W m\ :sup:`--2`                 |  1      | Heat Conduction through Sea-Ice                  |
                 +--------+----------------------------------+---------+--------------------------------------------------+
                 | RADLWG |  W m\ :sup:`--2`                 |  1      | Net upward LW flux at the ground                 |
                 +--------+----------------------------------+---------+--------------------------------------------------+
                 | RADSWG |  W m\ :sup:`--2`                 |  1      | Net downward SW flux at the ground               |
                 +--------+----------------------------------+---------+--------------------------------------------------+
                 | RI     |  dimensionless                   |  Nrphys | Richardson Number                                |
                 +--------+----------------------------------+---------+--------------------------------------------------+
                 | CT     |  dimensionless                   |  1      | Surface Drag coefficient for T and Q             |
                 +--------+----------------------------------+---------+--------------------------------------------------+
                 | CU     |  dimensionless                   |  1      | Surface Drag coefficient for U and V             |
                 +--------+----------------------------------+---------+--------------------------------------------------+
                 | ET     |  m\ :sup:`2` s\ :sup:`--1`       |  Nrphys | Diffusivity coefficient for T and Q              |
                 +--------+----------------------------------+---------+--------------------------------------------------+
                 | EU     |  m\ :sup:`2` s\ :sup:`--1`       |  Nrphys | Diffusivity coefficient for U and V              |
                 +--------+----------------------------------+---------+--------------------------------------------------+
                 | TURBU  |  m s\ :sup:`--1` day\ :sup:`--1` |  Nrphys | U-Momentum Changes due to Turbulence             |
                 +--------+----------------------------------+---------+--------------------------------------------------+
                 | TURBV  |  m s\ :sup:`--1` day\ :sup:`--1` |  Nrphys | V-Momentum Changes due to Turbulence             |
                 +--------+----------------------------------+---------+--------------------------------------------------+
                 | TURBT  |  deg day\ :sup:`--1`             |  Nrphys | Temperature Changes due to Turbulence            |
                 +--------+----------------------------------+---------+--------------------------------------------------+
                 | TURBQ  |  g/kg/day                        |  Nrphys | Specific Humidity Changes due to Turbulence      |
                 +--------+----------------------------------+---------+--------------------------------------------------+
                 | MOISTT |  deg day\ :sup:`--1`             |  Nrphys | Temperature Changes due to Moist Processes       |
                 +--------+----------------------------------+---------+--------------------------------------------------+
                 | MOISTQ |  g/kg/day                        |  Nrphys | Specific Humidity Changes due to Moist Processes |  
                 +--------+----------------------------------+---------+--------------------------------------------------+
                 | RADLW  |  deg day\ :sup:`--1`             |  Nrphys | Net Longwave heating rate for each level         |
                 +--------+----------------------------------+---------+--------------------------------------------------+
                 | RADSW  |  deg day\ :sup:`--1`             |  Nrphys | Net Shortwave heating rate for each level        |
                 +--------+----------------------------------+---------+--------------------------------------------------+
                 | PREACC |  mm/day                          |  1      | Total Precipitation                              |
                 +--------+----------------------------------+---------+--------------------------------------------------+
                 | PRECON |  mm/day                          |  1      | Convective Precipitation                         |
                 +--------+----------------------------------+---------+--------------------------------------------------+
                 | TUFLUX |  N m\ :sup:`--2`                 |  Nrphys | Turbulent Flux of U-Momentum                     |
                 +--------+----------------------------------+---------+--------------------------------------------------+
                 | TVFLUX |  N m\ :sup:`--2`                 |  Nrphys | Turbulent Flux of V-Momentum                     |
                 +--------+----------------------------------+---------+--------------------------------------------------+
                 | TTFLUX |  W m\ :sup:`--2`                 |  Nrphys | Turbulent Flux of Sensible Heat                  |
                 +--------+----------------------------------+---------+--------------------------------------------------+
                             
                 
                 +--------+---------------------+---------+-------------------------------------------------------------------------------------+
                 | NAME   |  UNITS              |  LEVELS | DESCRIPTION                                                                         |
                 +--------+---------------------+---------+-------------------------------------------------------------------------------------+
                 | TQFLUX | W m\ :sup:`--2`     | Nrphys  | Turbulent Flux of Latent Heat                                                       |
                 +--------+---------------------+---------+-------------------------------------------------------------------------------------+
                 | CN     | dimensionless       | 1       | Neutral Drag Coefficient                                                            |
                 +--------+---------------------+---------+-------------------------------------------------------------------------------------+
                 | WINDS  | m s\ :sup:`--1`     | 1       | Surface Wind Speed                                                                  |
                 +--------+---------------------+---------+-------------------------------------------------------------------------------------+
                 | DTSRF  | deg                 | 1       | Air/Surface virtual temperature difference                                          |
                 +--------+---------------------+---------+-------------------------------------------------------------------------------------+
                 | TG     | deg                 | 1       | Ground temperature                                                                  |
                 +--------+---------------------+---------+-------------------------------------------------------------------------------------+
                 | TS     | deg                 | 1       | Surface air temperature (Adiabatic from lowest model layer)                         |
                 +--------+---------------------+---------+-------------------------------------------------------------------------------------+
                 | DTG    | deg                 | 1       | Ground temperature adjustment                                                       |
                 +--------+---------------------+---------+-------------------------------------------------------------------------------------+
                 | QG     | g kg\ :sup:`--1`    | 1       | Ground specific humidity                                                            |
                 +--------+---------------------+---------+-------------------------------------------------------------------------------------+
                 | QS     | g kg\ :sup:`--1`    | 1       | Saturation surface specific humidity                                                |
                 +--------+---------------------+---------+-------------------------------------------------------------------------------------+
                 | TGRLW  | deg                 | 1       | Instantaneous ground temperature used as input to the Longwave radiation subroutine |
                 +--------+---------------------+---------+-------------------------------------------------------------------------------------+
                 | ST4    | W m\ :sup:`--2`     | 1       | Upward Longwave flux at the ground (:math:`\sigma T^4`)                             |
                 +--------+---------------------+---------+-------------------------------------------------------------------------------------+
                 | OLR    | W m\ :sup:`--2`     | 1       | Net upward Longwave flux at the top of the model                                    |
                 +--------+---------------------+---------+-------------------------------------------------------------------------------------+
                 | OLRCLR | W m\ :sup:`--2`     | 1       | Net upward clearsky Longwave flux at the top of the model                           |
                 +--------+---------------------+---------+-------------------------------------------------------------------------------------+
                 | LWGCLR | W m\ :sup:`--2`     | 1       | Net upward clearsky Longwave flux at the ground                                     |
                 +--------+---------------------+---------+-------------------------------------------------------------------------------------+
                 | LWCLR  | deg day\ :sup:`--1` | Nrphys  | Net clearsky Longwave heating rate for each level                                   |
                 +--------+---------------------+---------+-------------------------------------------------------------------------------------+
                 | TLW    | deg                 | Nrphys  | Instantaneous temperature used as input to the Longwave radiation subroutine        |
                 +--------+---------------------+---------+-------------------------------------------------------------------------------------+
                 | SHLW   | g g\ :sup:`--1`     | Nrphys  | Instantaneous specific humidity used as input to the Longwave radiation subroutine  |
                 +--------+---------------------+---------+-------------------------------------------------------------------------------------+
                 | OZLW   | g g\ :sup:`--1`     | Nrphys  | Instantaneous ozone used as input to the Longwave radiation subroutine              |
                 +--------+---------------------+---------+-------------------------------------------------------------------------------------+
                 | CLMOLW | :math:`0-1`         | Nrphys  | Maximum overlap cloud fraction used in the Longwave radiation subroutine            |
                 +--------+---------------------+---------+-------------------------------------------------------------------------------------+
                 | CLDTOT | :math:`0-1`         | Nrphys  | Total cloud fraction used in the Longwave and Shortwave radiation subroutines       |
                 +--------+---------------------+---------+-------------------------------------------------------------------------------------+
                 | LWGDOWN| W m\ :sup:`--2`     | 1       | Downwelling Longwave radiation at the ground                                        |
                 +--------+---------------------+---------+-------------------------------------------------------------------------------------+
                 | GWDT   | deg day\ :sup:`--1` | Nrphys  | Temperature tendency due to Gravity Wave Drag                                       |
                 +--------+---------------------+---------+-------------------------------------------------------------------------------------+
                 | RADSWT | W m\ :sup:`--2`     | 1       | Incident Shortwave radiation at the top of the atmosphere                           |
                 +--------+---------------------+---------+-------------------------------------------------------------------------------------+
                 | TAUCLD | per 100 mb          | Nrphys  | Counted Cloud Optical Depth (non-dimensional) per 100 mb                            |
                 +--------+---------------------+---------+-------------------------------------------------------------------------------------+
                 | TAUCLDC| Number              | Nrphys  | Cloud Optical Depth Counter                                                         |
                 +--------+---------------------+---------+-------------------------------------------------------------------------------------+
                 
                 +--------+-----------------+----------+---------------------------------------------------------------+
                 | NAME   | UNITS           | LEVELS   | Description                                                   |
                 +--------+-----------------+----------+---------------------------------------------------------------+
                 | CLDLOW | 0-1             | Nrphys   | Low-Level ( 1000-700 hPa) Cloud Fraction (0-1)                |
                 +--------+-----------------+----------+---------------------------------------------------------------+
                 | EVAP   | mm/day          | 1        | Surface evaporation                                           |
                 +--------+-----------------+----------+---------------------------------------------------------------+
                 | DPDT   | hPa/day         | 1        | Surface Pressure time-tendency                                |
                 +--------+-----------------+----------+---------------------------------------------------------------+
                 | UAVE   | m/sec           | Nrphys   | Average U-Wind                                                |
                 +--------+-----------------+----------+---------------------------------------------------------------+
                 | VAVE   | m/sec           | Nrphys   | Average V-Wind                                                |
                 +--------+-----------------+----------+---------------------------------------------------------------+
                 | TAVE   | deg             | Nrphys   | Average Temperature                                           |
                 +--------+-----------------+----------+---------------------------------------------------------------+
                 | QAVE   | g/kg            | Nrphys   | Average Specific Humidity                                     |
                 +--------+-----------------+----------+---------------------------------------------------------------+
                 | OMEGA  | hPa/day         | Nrphys   | Vertical Velocity                                             |
                 +--------+-----------------+----------+---------------------------------------------------------------+
                 | DUDT   | m/sec/day       | Nrphys   | Total U-Wind tendency                                         |
                 +--------+-----------------+----------+---------------------------------------------------------------+
                 | DVDT   | m/sec/day       | Nrphys   | Total V-Wind tendency                                         |
                 +--------+-----------------+----------+---------------------------------------------------------------+
                 | DTDT   | deg/day         | Nrphys   | Total Temperature tendency                                    |
                 +--------+-----------------+----------+---------------------------------------------------------------+
                 | DQDT   | g/kg/day        | Nrphys   | Total Specific Humidity tendency                              |
                 +--------+-----------------+----------+---------------------------------------------------------------+
                 | VORT   | 10^{-4}/sec     | Nrphys   | Relative Vorticity                                            |
                 +--------+-----------------+----------+---------------------------------------------------------------+
                 | DTLS   | deg/day         | Nrphys   | Temperature tendency due to Stratiform Cloud Formation        |
                 +--------+-----------------+----------+---------------------------------------------------------------+
                 | DQLS   | g/kg/day        | Nrphys   | Specific Humidity tendency due to Stratiform Cloud Formation  |
                 +--------+-----------------+----------+---------------------------------------------------------------+
                 | USTAR  | m/sec           | 1        | Surface USTAR wind                                            |
                 +--------+-----------------+----------+---------------------------------------------------------------+
                 | Z0     | m               | 1        | Surface roughness                                             |
                 +--------+-----------------+----------+---------------------------------------------------------------+
                 | FRQTRB | 0-1             | Nrphys-1 | Frequency of Turbulence                                       |
                 +--------+-----------------+----------+---------------------------------------------------------------+
                 | PBL    | mb              | 1        | Planetary Boundary Layer depth                                |
                 +--------+-----------------+----------+---------------------------------------------------------------+
                 | SWCLR  | deg/day         | Nrphys   | Net clearsky Shortwave heating rate for each level            |
                 +--------+-----------------+----------+---------------------------------------------------------------+
                 | OSR    | W m\ :sup:`--2` | 1        | Net downward Shortwave flux at the top of the model           |
                 +--------+-----------------+----------+---------------------------------------------------------------+
                 | OSRCLR | W m\ :sup:`--2` | 1        | Net downward clearsky Shortwave flux at the top of the model  |
                 +--------+-----------------+----------+---------------------------------------------------------------+
                 | CLDMAS | kg / m^2        | Nrphys   | Convective cloud mass flux                                    |
                 +--------+-----------------+----------+---------------------------------------------------------------+
                 | UAVE   | m/sec           | Nrphys   | Time-averaged :math:`u`-Wind                                  |
                 +--------+-----------------+----------+---------------------------------------------------------------+
                 
                 
                 
                 +--------+-------------------+--------+---------------------------------------------------------------+
                 | NAME   | UNITS             | LEVELS | DESCRIPTION                                                   |
                 +--------+-------------------+--------+---------------------------------------------------------------+
                 | VAVE   | m/sec             | Nrphys | Time-averaged :math:`v`-Wind                                  |
                 +--------+-------------------+--------+---------------------------------------------------------------+
                 | TAVE   | deg               | Nrphys | Time-averaged Temperature`                                    |
                 +--------+-------------------+--------+---------------------------------------------------------------+
                 | QAVE   | g/g               | Nrphys | Time-averaged Specific Humidity                               |
                 +--------+-------------------+--------+---------------------------------------------------------------+
                 | RFT    | deg/day           | Nrphys | Temperature tendency due Rayleigh Friction                    |
                 +--------+-------------------+--------+---------------------------------------------------------------+
                 | PS     | mb                | 1      | Surface Pressure                                              |
                 +--------+-------------------+--------+---------------------------------------------------------------+
                 | QQAVE  | (m/sec)\ :sup:`2` | Nrphys | Time-averaged Turbulent Kinetic Energy                        |
                 +--------+-------------------+--------+---------------------------------------------------------------+
                 | SWGCLR | W m\ :sup:`--2`   | 1      | Net downward clearsky Shortwave flux at the ground            |
                 +--------+-------------------+--------+---------------------------------------------------------------+
                 | PAVE   | mb                | 1      | Time-averaged Surface Pressure                                |
                 +--------+-------------------+--------+---------------------------------------------------------------+
                 | DIABU  | m/sec/day         | Nrphys | Total Diabatic forcing on :math:`u`-Wind                      |
                 +--------+-------------------+--------+---------------------------------------------------------------+
                 | DIABV  | m/sec/day         | Nrphys | Total Diabatic forcing on :math:`v`-Wind                      |
                 +--------+-------------------+--------+---------------------------------------------------------------+
                 | DIABT  | deg/day           | Nrphys | Total Diabatic forcing on Temperature                         |
                 +--------+-------------------+--------+---------------------------------------------------------------+
                 | DIABQ  | g/kg/day          | Nrphys | Total Diabatic forcing on Specific Humidity                   |
                 +--------+-------------------+--------+---------------------------------------------------------------+
                 | RFU    | m/sec/day         | Nrphys | U-Wind tendency due to Rayleigh Friction                      |
                 +--------+-------------------+--------+---------------------------------------------------------------+
                 | RFV    | m/sec/day         | Nrphys | V-Wind tendency due to Rayleigh Friction                      |
                 +--------+-------------------+--------+---------------------------------------------------------------+
                 | GWDU   | m/sec/day         | Nrphys | U-Wind tendency due to Gravity Wave Drag                      |
                 +--------+-------------------+--------+---------------------------------------------------------------+
                 | GWDU   | m/sec/day         | Nrphys | V-Wind tendency due to Gravity Wave Drag                      |
                 +--------+-------------------+--------+---------------------------------------------------------------+
                 | GWDUS  | N m\ :sup:`--2`   | 1      | U-Wind Gravity Wave Drag Stress at Surface                    |
                 +--------+-------------------+--------+---------------------------------------------------------------+
                 | GWDVS  | N m\ :sup:`--2`   | 1      | V-Wind Gravity Wave Drag Stress at Surface                    |
                 +--------+-------------------+--------+---------------------------------------------------------------+
                 | GWDUT  | N m\ :sup:`--2`   | 1      | U-Wind Gravity Wave Drag Stress at Top                        |
                 +--------+-------------------+--------+---------------------------------------------------------------+
                 | GWDVT  | N m\ :sup:`--2`   | 1      | V-Wind Gravity Wave Drag Stress at Top                        |
                 +--------+-------------------+--------+---------------------------------------------------------------+
                 | LZRAD  | mg/kg             | Nrphys | Estimated Cloud Liquid Water used in Radiation                |
                 +--------+-------------------+--------+---------------------------------------------------------------+
                 
                 +--------+-------------------+--------+-----------------------------------------------------+
                 | NAME   | UNITS             | LEVELS | DESCRIPTION                                         |
                 +--------+-------------------+--------+-----------------------------------------------------+
                 | SLP    | mb                | 1      | Time-averaged Sea-level Pressure                    |
                 +--------+-------------------+--------+-----------------------------------------------------+
                 | CLDFRC | 0-1               | 1      | Total Cloud Fraction                                |
                 +--------+-------------------+--------+-----------------------------------------------------+
                 | TPW    | gm cm\ :sup:`--2` | 1      | Precipitable water                                  |
                 +--------+-------------------+--------+-----------------------------------------------------+
                 | U2M    | m/sec             | 1      | U-Wind at 2 meters                                  |
                 +--------+-------------------+--------+-----------------------------------------------------+
                 | V2M    | m/sec             | 1      | V-Wind at 2 meters                                  |
                 +--------+-------------------+--------+-----------------------------------------------------+
                 | T2M    | deg               | 1      | Temperature at 2 meters                             |
                 +--------+-------------------+--------+-----------------------------------------------------+
                 | Q2M    | g/kg              | 1      | Specific Humidity at 2 meters                       |
                 +--------+-------------------+--------+-----------------------------------------------------+
                 | U10M   | m/sec             | 1      | U-Wind at 10 meters                                 |
                 +--------+-------------------+--------+-----------------------------------------------------+
                 | V10M   | m/sec             | 1      | V-Wind at 10 meters                                 |
                 +--------+-------------------+--------+-----------------------------------------------------+
                 | T10M   | deg               | 1      | Temperature at 10 meters                            |
                 +--------+-------------------+--------+-----------------------------------------------------+
                 | Q10M   | g/kg              | 1      | Specific Humidity at 10 meters                      |
                 +--------+-------------------+--------+-----------------------------------------------------+
                 | DTRAIN | kg m\ :sup:`--2`  | Nrphys | Detrainment Cloud Mass Flux                         |
                 +--------+-------------------+--------+-----------------------------------------------------+
                 | QFILL  | g/kg/day          | Nrphys | Filling of negative specific humidity               |
                 +--------+-------------------+--------+-----------------------------------------------------+
                 | DTCONV | deg/sec           | Nr     | Temp Change due to Convection                       |
                 +--------+-------------------+--------+-----------------------------------------------------+
                 | DQCONV | g/kg/sec          | Nr     | Specific Humidity Change due to Convection          |
                 +--------+-------------------+--------+-----------------------------------------------------+
                 | RELHUM | percent           | Nr     | Relative Humidity                                   |
                 +--------+-------------------+--------+-----------------------------------------------------+
                 | PRECLS | g/m^2/sec         | 1      | Large Scale Precipitation                           |
                 +--------+-------------------+--------+-----------------------------------------------------+
                 | ENPREC | J/g               | 1      | Energy of Precipitation (snow, rain Temp)           |
                 +--------+-------------------+--------+-----------------------------------------------------+
                 
                 
                 Fizhi Diagnostic Description
                 ++++++++++++++++++++++++++++
                 
                 In this section we list and describe the diagnostic quantities available
                 within the GCM. The diagnostics are listed in the order that they appear
                 in the Diagnostic Menu, Section [sec:pkg:fizhi:diagnostics]. In all
                 cases, each diagnostic as currently archived on the output datasets is
                 time-averaged over its diagnostic output frequency:
                 
                 .. math:: {\bf DIAGNOSTIC} = \frac{1}{TTOT} \sum_{t=1}^{t=TTOT} diag(t)
                 
                 where :math:`TTOT = \frac{ {\bf NQDIAG} }{\Delta t}`, **NQDIAG** is the
                 output frequency of the diagnostic, and :math:`\Delta t` is the timestep
                 over which the diagnostic is updated.
                 
                 Surface Zonal Wind Stress on the Atmosphere (:math:`Newton/m^2`)
                 ################################################################
                 
                 The zonal wind stress is the turbulent flux of zonal momentum from the
                 surface.
                 
                 .. math:: {\bf UFLUX} =  - \rho C_D W_s u \hspace{1cm}where: \hspace{.2cm}C_D = C^2_u
                 
                 where :math:`\rho` = the atmospheric density at the surface,
                 :math:`C_{D}` is the surface drag coefficient, :math:`C_u` is the
                 dimensionless surface exchange coefficient for momentum (see diagnostic
                 number 10), :math:`W_s` is the magnitude of the surface layer wind, and
                 :math:`u` is the zonal wind in the lowest model layer.
                 
                 Surface Meridional Wind Stress on the Atmosphere (:math:`Newton/m^2`)
                 ######################################################################
                 
                 The meridional wind stress is the turbulent flux of meridional
                 momentum from the surface.
                 
                 .. math:: {\bf VFLUX} =  - \rho C_D W_s v \hspace{1cm}where: \hspace{.2cm}C_D = C^2_u
                 
                 where :math:`\rho` = the atmospheric density at the surface,
                 :math:`C_{D}` is the surface drag coefficient, :math:`C_u` is the
                 dimensionless surface exchange coefficient for momentum (see diagnostic
                 number 10), :math:`W_s` is the magnitude of the surface layer wind, and
                 :math:`v` is the meridional wind in the lowest model layer.
                 
                 Surface Flux of Sensible Heat (W m\ :sup:`--2`)
                 ################################################
                 
                 The turbulent flux of sensible heat from the surface to the atmosphere
                 is a function of the gradient of virtual potential temperature and the
                 eddy exchange coefficient:
                 
                 .. math::
                 
0bad585a21 Navi*    {\bf HFLUX} =  P^{\kappa}\rho c_{p} C_{H} W_s (\theta_{\rm surface} - \theta_{Nrphys})
8679f9097b Jeff*    \hspace{1cm}where: \hspace{.2cm}C_H = C_u C_t
                 
                 where :math:`\rho` = the atmospheric density at the surface,
                 :math:`c_{p}` is the specific heat of air, :math:`C_{H}` is the
                 dimensionless surface heat transfer coefficient, :math:`W_s` is the
                 magnitude of the surface layer wind, :math:`C_u` is the dimensionless
                 surface exchange coefficient for momentum (see diagnostic number 10),
                 :math:`C_t` is the dimensionless surface exchange coefficient for heat
                 and moisture (see diagnostic number 9), and :math:`\theta` is the
                 potential temperature at the surface and at the bottom model level.
                 
                 Surface Flux of Latent Heat (:math:`Watts/m^2`)
                 ###############################################
                 
                 The turbulent flux of latent heat from the surface to the atmosphere
                 is a function of the gradient of moisture, the potential
                 evapotranspiration fraction and the eddy exchange coefficient:
                 
                 .. math::
                 
0bad585a21 Navi*    {\bf EFLUX} =  \rho \beta L C_{H} W_s (q_{\rm surface} - q_{Nrphys})
8679f9097b Jeff*    \hspace{1cm}where: \hspace{.2cm}C_H = C_u C_t
                 
                 where :math:`\rho` = the atmospheric density at the surface,
                 :math:`\beta` is the fraction of the potential evapotranspiration
                 actually evaporated, L is the latent heat of evaporation, :math:`C_{H}`
                 is the dimensionless surface heat transfer coefficient, :math:`W_s` is
                 the magnitude of the surface layer wind, :math:`C_u` is the
                 dimensionless surface exchange coefficient for momentum (see diagnostic
                 number 10), :math:`C_t` is the dimensionless surface exchange
                 coefficient for heat and moisture (see diagnostic number 9), and
0bad585a21 Navi* :math:`q_{\rm surface}` and :math:`q_{Nrphys}` are the specific humidity at
8679f9097b Jeff* the surface and at the bottom model level, respectively.
                 
                 Heat Conduction Through Sea Ice (:math:`Watts/m^2`)
                 ###################################################
                 
                 Over sea ice there is an additional source of energy at the surface due
                 to the heat conduction from the relatively warm ocean through the sea
                 ice. The heat conduction through sea ice represents an additional energy
                 source term for the ground temperature equation.
                 
                 .. math:: {\bf QICE} = \frac{C_{ti}}{H_i} (T_i-T_g)
                 
                 where :math:`C_{ti}` is the thermal conductivity of ice, :math:`H_i` is
                 the ice thickness, assumed to be :math:`3 \hspace{.1cm} m` where sea ice
                 is present, :math:`T_i` is 273 degrees Kelvin, and :math:`T_g` is the
                 temperature of the sea ice.
                 
                 NOTE: QICE is not available through model version 5.3, but is
                 available in subsequent versions.
                 
                 
                 Net upward Longwave Flux at the surface (:math:`Watts/m^2`)
                 ###########################################################
                 
                 .. math::
                 
                    \begin{aligned}
                    {\bf RADLWG} & =  & F_{LW,Nrphys+1}^{Net} \\
                                 & =  & F_{LW,Nrphys+1}^\uparrow - F_{LW,Nrphys+1}^\downarrow\end{aligned}
                 
                 where Nrphys+1 indicates the lowest model edge-level, or
                 :math:`p = p_{surf}`. :math:`F_{LW}^\uparrow` is the upward Longwave
                 flux and :math:`F_{LW}^\downarrow` is the downward Longwave flux.
                 
                 
                 Net downard shortwave Flux at the surface (:math:`Watts/m^2`)
                 #############################################################
                 
                 .. math::
                 
                    \begin{aligned}
                    {\bf RADSWG} & =  & F_{SW,Nrphys+1}^{Net} \\
                                 & =  & F_{SW,Nrphys+1}^\downarrow - F_{SW,Nrphys+1}^\uparrow\end{aligned}
                 
                 where Nrphys+1 indicates the lowest model edge-level, or
                 :math:`p = p_{surf}`. :math:`F_{SW}^\downarrow` is the downward
                 Shortwave flux and :math:`F_{SW}^\uparrow` is the upward Shortwave flux.
                 
                 Richardson number (:math:`dimensionless`)
                 #########################################
                 
                 The non-dimensional stability indicator is the ratio of the buoyancy
                 to the shear:
                 
                 .. math::
                 
                    {\bf RI} = \frac{ \frac{g}{\theta_v} \pp {\theta_v}{z} }{ (\pp{u}{z})^2 + (\pp{v}{z})^2 }
                     =  \frac{c_p \pp{\theta_v}{z} \pp{P^ \kappa}{z} }{ (\pp{u}{z})^2 + (\pp{v}{z})^2 }
                 
                 where we used the hydrostatic equation:
                 
                 .. math:: {\pp{\Phi}{P^ \kappa}} = c_p \theta_v
                 
                 Negative values indicate unstable buoyancy **AND** shear, small positive
                 values (:math:`<0.4`) indicate dominantly unstable shear, and large
                 positive values indicate dominantly stable stratification.
                 
                 CT - Surface Exchange Coefficient for Temperature and Moisture (dimensionless)
                 ###############################################################################
                 
                 The surface exchange coefficient is obtained from the similarity
                 functions for the stability dependant flux profile relationships:
                 
                 .. math::
                 
                    {\bf CT} = -\frac{( \overline{w^{\prime}\theta^{\prime}} ) }{ u_* \Delta \theta } = 
                    -\frac{( \overline{w^{\prime}q^{\prime}} ) }{ u_* \Delta q } = 
                    \frac{ k }{ (\psi_{h} + \psi_{g}) }
                 
                 where :math:`\psi_h` is the surface layer non-dimensional temperature
                 change and :math:`\psi_g` is the viscous sublayer non-dimensional
                 temperature or moisture change:
                 
                 .. math::
                 
                    \psi_{h} = \int_{\zeta_{0}}^{\zeta} \frac{\phi_{h} }{ \zeta} d \zeta \hspace{1cm} and 
                    \hspace{1cm} \psi_{g} = \frac{ 0.55 (Pr^{2/3} - 0.2) }{ \nu^{1/2} } 
                    (h_{0}u_{*} - h_{0_{ref}}u_{*_{ref}})^{1/2}
                 
                 and: :math:`h_{0} = 30z_{0}` with a maximum value over land of 0.01
                 
                 :math:`\phi_h` is the similarity function of :math:`\zeta`, which
                 expresses the stability dependance of the temperature and moisture
                 gradients, specified differently for stable and unstable layers
                 according to . k is the Von Karman constant, :math:`\zeta` is the
                 non-dimensional stability parameter, Pr is the Prandtl number for air,
                 :math:`\nu` is the molecular viscosity, :math:`z_{0}` is the surface
                 roughness length, :math:`u_*` is the surface stress velocity (see
                 diagnostic number 67), and the subscript ref refers to a reference
                 value.
                 
                 CU - Surface Exchange Coefficient for Momentum (dimensionless)
                 ##############################################################
                 
                 The surface exchange coefficient is obtained from the similarity
                 functions for the stability dependant flux profile relationships:
                 
                 .. math:: {\bf CU} = \frac{u_* }{ W_s} = \frac{ k }{ \psi_{m} }
                 
                 where :math:`\psi_m` is the surface layer non-dimensional wind shear:
                 
                 .. math:: \psi_{m} = {\int_{\zeta_{0}}^{\zeta} \frac{\phi_{m} }{ \zeta} d \zeta}
                 
                 :math:`\phi_m` is the similarity function of :math:`\zeta`, which
                 expresses the stability dependance of the temperature and moisture
                 gradients, specified differently for stable and unstable layers
                 according to . k is the Von Karman constant, :math:`\zeta` is the
                 non-dimensional stability parameter, :math:`u_*` is the surface stress
                 velocity (see diagnostic number 67), and :math:`W_s` is the magnitude of
                 the surface layer wind.
                 
                 ET - Diffusivity Coefficient for Temperature and Moisture (m^2/sec) 
                 ###################################################################
                 
                 In the level 2.5 version of the Mellor-Yamada (1974) hierarchy, the
                 turbulent heat or moisture flux for the atmosphere above the surface
                 layer can be expressed as a turbulent diffusion coefficient :math:`K_h`
                 times the negative of the gradient of potential temperature or moisture.
                 In the :cite:`helflab:88` adaptation of this closure, :math:`K_h` takes the form:
                 
                 .. math::
                 
                    {\bf ET} = K_h = -\frac{( \overline{w^{\prime}\theta_v^{\prime}}) }{ \pp{\theta_v}{z} } 
                     = \left\{ \begin{array}{l@{\quad\mbox{for}\quad}l} q \, \ell \, S_H(G_M,G_H) & \mbox{decaying turbulence}
                    \\ \frac{ q^2 }{ q_e } \, \ell \, S_{H}(G_{M_e},G_{H_e}) & \mbox{growing turbulence} \end{array} \right.
                 
                 where :math:`q` is the turbulent velocity, or
                 :math:`\sqrt{2*turbulent \hspace{.2cm} kinetic \hspace{.2cm} 
                 energy}`, :math:`q_e` is the turbulence velocity derived from the more
                 simple level 2.0 model, which describes equilibrium turbulence,
                 :math:`\ell` is the master length scale related to the layer depth,
                 :math:`S_H` is a function of :math:`G_H` and :math:`G_M`, the
                 dimensionless buoyancy and wind shear parameters, respectively, or a
                 function of :math:`G_{H_e}` and :math:`G_{M_e}`, the equilibrium
                 dimensionless buoyancy and wind shear parameters. Both :math:`G_H` and
                 :math:`G_M`, and their equilibrium values :math:`G_{H_e}` and
                 :math:`G_{M_e}`, are functions of the Richardson number.
                 
                 For the detailed equations and derivations of the modified level 2.5
                 closure scheme, see :cite:`helflab:88`.
                 
                 In the surface layer, :math:`{\bf {ET}}` is the exchange coefficient
                 for heat and moisture, in units of :math:`m/sec`, given by:
                 
                 .. math:: {\bf ET_{Nrphys}} =  C_t * u_* = C_H W_s
                 
                 where :math:`C_t` is the dimensionless exchange coefficient for heat and
                 moisture from the surface layer similarity functions (see diagnostic
                 number 9), :math:`u_*` is the surface friction velocity (see diagnostic
                 number 67), :math:`C_H` is the heat transfer coefficient, and
                 :math:`W_s` is the magnitude of the surface layer wind.
                 
                 
                 EU - Diffusivity Coefficient for Momentum (m^2/sec)
                 ###################################################
                 
                 In the level 2.5 version of the Mellor-Yamada (1974) hierarchy, the
                 turbulent heat momentum flux for the atmosphere above the surface layer
                 can be expressed as a turbulent diffusion coefficient :math:`K_m` times
                 the negative of the gradient of the u-wind. In the :cite:`helflab:88` adaptation of this
                 closure, :math:`K_m` takes the form:
                 
                 .. math::
                 
                    {\bf EU} = K_m = -\frac{( \overline{u^{\prime}w^{\prime}} ) }{ \pp{U}{z} }
                     = \left\{ \begin{array}{l@{\quad\mbox{for}\quad}l} q \, \ell \, S_M(G_M,G_H) & \mbox{decaying turbulence}
                    \\ \frac{ q^2 }{ q_e } \, \ell \, S_{M}(G_{M_e},G_{H_e}) & \mbox{growing turbulence} \end{array} \right.
                 
                 where :math:`q` is the turbulent velocity, or
                 :math:`\sqrt{2*turbulent \hspace{.2cm} kinetic \hspace{.2cm}
                 energy}`, :math:`q_e` is the turbulence velocity derived from the more
                 simple level 2.0 model, which describes equilibrium turbulence,
                 :math:`\ell` is the master length scale related to the layer depth,
                 :math:`S_M` is a function of :math:`G_H` and :math:`G_M`, the
                 dimensionless buoyancy and wind shear parameters, respectively, or a
                 function of :math:`G_{H_e}` and :math:`G_{M_e}`, the equilibrium
                 dimensionless buoyancy and wind shear parameters. Both :math:`G_H` and
                 :math:`G_M`, and their equilibrium values :math:`G_{H_e}` and
                 :math:`G_{M_e}`, are functions of the Richardson number.
                 
                 For the detailed equations and derivations of the modified level 2.5
                 closure scheme, see :cite:`helflab:88`.
                 
                 In the surface layer, :math:`{\bf {EU}}` is the exchange coefficient
                 for momentum, in units of :math:`m/sec`, given by:
                 
                 .. math:: {\bf EU_{Nrphys}} = C_u * u_* = C_D W_s
                 
                 where :math:`C_u` is the dimensionless exchange coefficient for momentum
                 from the surface layer similarity functions (see diagnostic number 10),
                 :math:`u_*` is the surface friction velocity (see diagnostic number 67),
                 :math:`C_D` is the surface drag coefficient, and :math:`W_s` is the
                 magnitude of the surface layer wind.
                 
                 
                 
                 TURBU - Zonal U-Momentum changes due to Turbulence (m/sec/day) 
                 ##############################################################
                 
                 The tendency of U-Momentum due to turbulence is written:
                 
                 .. math::
                 
0bad585a21 Navi*    {\bf TURBU} = {\pp{u}{t}}_{\rm turb} = {\pp{}{z} }{(- \overline{u^{\prime}w^{\prime}})}
8679f9097b Jeff*     = {\pp{}{z} }{(K_m \pp{u}{z})}
                 
                 The Helfand and Labraga level 2.5 scheme models the turbulent flux of
                 u-momentum in terms of :math:`K_m`, and the equation has the form of a
                 diffusion equation.
                 
                 TURBV - Meridional V-Momentum changes due to Turbulence (m/sec/day) 
                 ###################################################################
                 
                 The tendency of V-Momentum due to turbulence is written:
                 
                 .. math::
                 
0bad585a21 Navi*    {\bf TURBV} = {\pp{v}{t}}_{\rm turb} = {\pp{}{z} }{(- \overline{v^{\prime}w^{\prime}})}
8679f9097b Jeff*     = {\pp{}{z} }{(K_m \pp{v}{z})}
                 
                 The Helfand and Labraga level 2.5 scheme models the turbulent flux of
                 v-momentum in terms of :math:`K_m`, and the equation has the form of a
                 diffusion equation.
                 
                 
                 TURBT - Temperature changes due to Turbulence (deg/day) 
                 #######################################################
                 
                 The tendency of temperature due to turbulence is written:
                 
                 .. math::
                 
0bad585a21 Navi*    {\bf TURBT} = {\pp{T}{t}} = P^{\kappa}{\pp{\theta}{t}}_{\rm turb} = 
8679f9097b Jeff*    P^{\kappa}{\pp{}{z} }{(- \overline{w^{\prime}\theta^{\prime}})}
                     = P^{\kappa}{\pp{}{z} }{(K_h \pp{\theta_v}{z})}
                 
                 The Helfand and Labraga level 2.5 scheme models the turbulent flux of
                 temperature in terms of :math:`K_h`, and the equation has the form of a
                 diffusion equation.
                 
                 
                 TURBQ - Specific Humidity changes due to Turbulence (g/kg/day) 
                 ###############################################################
                 
                 The tendency of specific humidity due to turbulence is written:
                 
                 .. math::
                 
0bad585a21 Navi*    {\bf TURBQ} = {\pp{q}{t}}_{\rm turb} = {\pp{}{z} }{(- \overline{w^{\prime}q^{\prime}})}
8679f9097b Jeff*     = {\pp{}{z} }{(K_h \pp{q}{z})}
                 
                 The Helfand and Labraga level 2.5 scheme models the turbulent flux of
                 temperature in terms of :math:`K_h`, and the equation has the form of a
                 diffusion equation.
                 
                 
                 MOISTT - Temperature Changes Due to Moist Processes (deg/day) 
                 #############################################################
                 
                 .. math:: {\bf MOISTT} = \left. {\pp{T}{t}}\right|_{c} + \left. {\pp{T}{t}} \right|_{ls}
                 
                 where:
                 
                 .. math::
                 
                    \left.{\pp{T}{t}}\right|_{c} = R \sum_i \left( \alpha \frac{m_B}{c_p} \Gamma_s \right)_i 
                    \hspace{.4cm} and 
                    \hspace{.4cm} \left.{\pp{T}{t}}\right|_{ls} = \frac{L}{c_p} (q^*-q)
                 
                  and
                 
                 .. math:: \Gamma_s = g \eta \pp{s}{p}
                 
                 The subscript :math:`c` refers to convective processes, while the
                 subscript :math:`ls` refers to large scale precipitation processes, or
                 supersaturation rain. The summation refers to contributions from each
                 cloud type called by RAS. The dry static energy is given as :math:`s`,
                 the convective cloud base mass flux is given as :math:`m_B`, and the
                 cloud entrainment is given as :math:`\eta`, which are explicitly defined
                 in :numref:`para_phys_pkg_fizhi_mc`, the description of the convective
                 parameterization. The fractional adjustment, or relaxation parameter,
                 for each cloud type is given as :math:`\alpha`, while :math:`R` is the
                 rain re-evaporation adjustment.
                 
                 MOISTQ - Specific Humidity Changes Due to Moist Processes (g/kg/day)
                 ####################################################################
                 
                 .. math:: {\bf MOISTQ} = \left. {\pp{q}{t}}\right|_{c} + \left. {\pp{q}{t}} \right|_{ls}
                 
                 where:
                 
                 .. math::
                 
                    \left.{\pp{q}{t}}\right|_{c} = R \sum_i \left( \alpha \frac{m_B}{L}(\Gamma_h-\Gamma_s) \right)_i 
                    \hspace{.4cm} and 
                    \hspace{.4cm} \left.{\pp{q}{t}}\right|_{ls} = (q^*-q)
                 
                 and
                 
                 .. math:: \Gamma_s = g \eta \pp{s}{p}\hspace{.4cm} and \hspace{.4cm}\Gamma_h = g \eta \pp{h}{p}
                 
                 The subscript :math:`c` refers to convective processes, while the
                 subscript :math:`ls` refers to large scale precipitation processes, or
                 supersaturation rain. The summation refers to contributions from each
                 cloud type called by RAS. The dry static energy is given as :math:`s`,
                 the moist static energy is given as :math:`h`, the convective cloud base
                 mass flux is given as :math:`m_B`, and the cloud entrainment is given as
                 :math:`\eta`, which are explicitly defined in :numref:`para_phys_pkg_fizhi_mc`,
                 the description of the convective parameterization. The fractional
                 adjustment, or relaxation parameter, for each cloud type is given as
                 :math:`\alpha`, while :math:`R` is the rain re-evaporation adjustment.
                 
                 
                 RADLW - Heating Rate due to Longwave Radiation (deg/day)
                 ########################################################
                 
                 The net longwave heating rate is calculated as the vertical divergence
                 of the net terrestrial radiative fluxes. Both the clear-sky and
                 cloudy-sky longwave fluxes are computed within the longwave routine. The
                 subroutine calculates the clear-sky flux, :math:`F^{clearsky}_{LW}`,
                 first. For a given cloud fraction, the clear line-of-sight probability
                 :math:`C(p,p^{\prime})` is computed from the current level pressure
                 :math:`p` to the model top pressure, :math:`p^{\prime} = p_{top}`, and
                 the model surface pressure, :math:`p^{\prime} = p_{surf}`, for the
                 upward and downward radiative fluxes. (see Section
                 [sec:fizhi:radcloud]). The cloudy-sky flux is then obtained as:
                 
0bad585a21 Navi* .. math:: F_{LW} = C(p,p') \cdot F^{clearsky}_{LW}
8679f9097b Jeff* 
                 Finally, the net longwave heating rate is calculated as the vertical
                 divergence of the net terrestrial radiative fluxes:
                 
0bad585a21 Navi* .. math:: \pp{\rho c_p T}{t} = - \p{z} F_{LW}^{NET}
8679f9097b Jeff* 
                 or
                 
0bad585a21 Navi* .. math:: {\bf RADLW} = \frac{g}{c_p \pi} \p{\sigma} F_{LW}^{NET}
8679f9097b Jeff* 
                 where :math:`g` is the accelation due to gravity, :math:`c_p` is the
                 heat capacity of air at constant pressure, and
                 
                 .. math:: F_{LW}^{NET} = F_{LW}^\uparrow - F_{LW}^\downarrow
                 
                 
                 RADSW - Heating Rate due to Shortwave Radiation (deg/day) 
                 #########################################################
                 
                 The net Shortwave heating rate is calculated as the vertical divergence
                 of the net solar radiative fluxes. The clear-sky and cloudy-sky
                 shortwave fluxes are calculated separately. For the clear-sky case, the
                 shortwave fluxes and heating rates are computed with both CLMO (maximum
                 overlap cloud fraction) and CLRO (random overlap cloud fraction) set to
                 zero (see Section [sec:fizhi:radcloud]). The shortwave routine is then
                 called a second time, for the cloudy-sky case, with the true
                 time-averaged cloud fractions CLMO and CLRO being used. In all cases, a
                 normalized incident shortwave flux is used as input at the top of the
                 atmosphere.
                 
                 The heating rate due to Shortwave Radiation under cloudy skies is
                 defined as:
                 
0bad585a21 Navi* .. math:: \pp{\rho c_p T}{t} = - \p{z} F(cloudy)_{SW}^{NET} \cdot {\rm RADSWT}
8679f9097b Jeff* 
                 or
                 
0bad585a21 Navi* .. math:: {\bf RADSW} = \frac{g}{c_p \pi} \p{\sigma} F(cloudy)_{SW}^{NET}\cdot {\rm RADSWT}
8679f9097b Jeff* 
                 where :math:`g` is the accelation due to gravity, :math:`c_p` is the
                 heat capacity of air at constant pressure, RADSWT is the true incident
                 shortwave radiation at the top of the atmosphere (See Diagnostic #48),
                 and
                 
                 .. math:: F(cloudy)_{SW}^{Net} = F(cloudy)_{SW}^\uparrow - F(cloudy)_{SW}^\downarrow
                 
                 
                 PREACC - Total (Large-scale + Convective) Accumulated Precipition (mm/day) 
                 ###########################################################################
                 
                 For a change in specific humidity due to moist processes,
                 :math:`\Delta q_{moist}`, the vertical integral or total precipitable
                 amount is given by:
                 
                 .. math::
                 
                    {\bf PREACC} = \int_{surf}^{top} \rho \Delta q_{moist} dz = - \int_{surf}^{top} \Delta  q_{moist}
                    \frac{dp}{g} = \frac{1}{g} \int_0^1 \Delta q_{moist} dp
                 
                 A precipitation rate is defined as the vertically integrated moisture
                 adjustment per Moist Processes time step, scaled to :math:`mm/day`.
                 
                 
                 PRECON - Convective Precipition (mm/day)
                 ########################################
                 
                 For a change in specific humidity due to sub-grid scale cumulus
                 convective processes, :math:`\Delta q_{cum}`, the vertical integral or
                 total precipitable amount is given by:
                 
                 .. math::
                 
                    {\bf PRECON} = \int_{surf}^{top} \rho \Delta q_{cum} dz = - \int_{surf}^{top} \Delta  q_{cum}
                    \frac{dp}{g} = \frac{1}{g} \int_0^1 \Delta q_{cum} dp
                 
                 A precipitation rate is defined as the vertically integrated moisture
                 adjustment per Moist Processes time step, scaled to :math:`mm/day`.
                 
                 TUFLUX - Turbulent Flux of U-Momentum (Newton/m^2)
                 ##################################################
                 
                 The turbulent flux of u-momentum is calculated for
                 :math:`diagnostic \hspace{.2cm} purposes 
                 \hspace{.2cm} only` from the eddy coefficient for momentum:
                 
                 .. math::
                 
                    {\bf TUFLUX} =  {\rho } {(\overline{u^{\prime}w^{\prime}})} =  
                    {\rho } {(- K_m \pp{U}{z})}
                 
                 where :math:`\rho` is the air density, and :math:`K_m` is the eddy
                 coefficient.
                 
                 TVFLUX - Turbulent Flux of V-Momentum (Newton/m^2)
                 ###################################################
                 
                 The turbulent flux of v-momentum is calculated for
                 :math:`diagnostic \hspace{.2cm} purposes 
                 \hspace{.2cm} only` from the eddy coefficient for momentum:
                 
                 .. math::
                 
                    {\bf TVFLUX} =  {\rho } {(\overline{v^{\prime}w^{\prime}})} = 
                     {\rho } {(- K_m \pp{V}{z})}
                 
                 where :math:`\rho` is the air density, and :math:`K_m` is the eddy
                 coefficient.
                 
                 
                 TTFLUX - Turbulent Flux of Sensible Heat (Watts/m^2) 
                 ####################################################
                 
                 The turbulent flux of sensible heat is calculated for
                 :math:`diagnostic \hspace{.2cm} purposes 
                 \hspace{.2cm} only` from the eddy coefficient for heat and moisture:
                 
                 .. math::
                 
                    {\bf TTFLUX} = c_p {\rho }  
                    P^{\kappa}{(\overline{w^{\prime}\theta^{\prime}})}
                     = c_p  {\rho } P^{\kappa}{(- K_h \pp{\theta_v}{z})}
                 
                 where :math:`\rho` is the air density, and :math:`K_h` is the eddy
                 coefficient.
                 
                 
                 TQFLUX - Turbulent Flux of Latent Heat (Watts/m^2)
                 ###################################################
                 
                 The turbulent flux of latent heat is calculated for
                 :math:`diagnostic \hspace{.2cm} purposes 
                 \hspace{.2cm} only` from the eddy coefficient for heat and moisture:
                 
                 .. math::
                 
                    {\bf TQFLUX} = {L {\rho } (\overline{w^{\prime}q^{\prime}})} = 
                    {L {\rho }(- K_h \pp{q}{z})}
                 
                 where :math:`\rho` is the air density, and :math:`K_h` is the eddy
                 coefficient.
                 
                 
                 CN - Neutral Drag Coefficient (dimensionless)
                 #############################################
                 
                 The drag coefficient for momentum obtained by assuming a neutrally
                 stable surface layer:
                 
                 .. math:: {\bf CN} = \frac{ k }{ \ln(\frac{h }{z_0}) }
                 
                 where :math:`k` is the Von Karman constant, :math:`h` is the height of
                 the surface layer, and :math:`z_0` is the surface roughness.
                 
                 WINDS - Surface Wind Speed (meter/sec)
                 ######################################
                 
                 The surface wind speed is calculated for the last internal turbulence
                 time step:
                 
                 .. math:: {\bf WINDS} = \sqrt{u_{Nrphys}^2 + v_{Nrphys}^2}
                 
                 where the subscript :math:`Nrphys` refers to the lowest model level.
                 
                 The air/surface virtual temperature difference measures the stability of
                 the surface layer:
                 
                 .. math:: {\bf DTSRF} = (\theta_{v{Nrphys+1}} - \theta{v_{Nrphys}}) P^{\kappa}_{surf}
                 
                 where
                 
                 .. math::
                 
                    \theta_{v{Nrphys+1}} = \frac{ T_g }{ P^{\kappa}_{surf} } (1 + .609 q_{Nrphys+1}) \hspace{1cm}
                    and \hspace{1cm} q_{Nrphys+1} = q_{Nrphys} + \beta(q^*(T_g,P_s) - q_{Nrphys})
                 
                 :math:`\beta` is the surface potential evapotranspiration coefficient
                 (:math:`\beta=1` over oceans), :math:`q^*(T_g,P_s)` is the saturation
                 specific humidity at the ground temperature and surface pressure, level
                 :math:`Nrphys` refers to the lowest model level and level
                 :math:`Nrphys+1` refers to the surface.
                 
                 
                 TG - Ground Temperature (deg K)
                 ################################
                 
                 The ground temperature equation is solved as part of the turbulence
                 package using a backward implicit time differencing scheme:
                 
                 .. math::
                 
                    {\bf TG} \hspace{.1cm} is \hspace{.1cm} obtained \hspace{.1cm} from: \hspace{.1cm}
0bad585a21 Navi*    C_g\pp{T_g}{t} = R_{sw} - R_{lw} + Q_{\rm ice} - H - LE
8679f9097b Jeff* 
                 where :math:`R_{sw}` is the net surface downward shortwave radiative
                 flux, :math:`R_{lw}` is the net surface upward longwave radiative flux,
0bad585a21 Navi* :math:`Q_{\rm ice}` is the heat conduction through sea ice, :math:`H` is the
8679f9097b Jeff* upward sensible heat flux, :math:`LE` is the upward latent heat flux,
                 and :math:`C_g` is the total heat capacity of the ground. :math:`C_g` is
                 obtained by solving a heat diffusion equation for the penetration of the
                 diurnal cycle into the ground (), and is given by:
                 
                 .. math::
                 
                    C_g = \sqrt{ \frac{\lambda C_s }{ 2 \omega } } = \sqrt{(0.386 + 0.536W + 0.15W^2)2x10^{-3}
0bad585a21 Navi*    \frac{86400.}{2\pi} }
8679f9097b Jeff* 
                 Here, the thermal conductivity, :math:`\lambda`, is equal to
                 :math:`2x10^{-3}` :math:`\frac{ly}{sec} 
                 \frac{cm}{K}`, the angular velocity of the earth, :math:`\omega`, is
                 written as :math:`86400` :math:`sec/day` divided by :math:`2 \pi`
                 :math:`radians/
                 day`, and the expression for :math:`C_s`, the heat capacity per unit
                 volume at the surface, is a function of the ground wetness, :math:`W`.
                 
                 
                 TS - Surface Temperature (deg K)
                 #################################
                 
                 The surface temperature estimate is made by assuming that the model’s
                 lowest layer is well-mixed, and therefore that :math:`\theta` is
                 constant in that layer. The surface temperature is therefore:
                 
                 .. math:: {\bf TS} = \theta_{Nrphys} P^{\kappa}_{surf}
                 
                 
                 DTG - Surface Temperature Adjustment (deg K)
                 ############################################
                 
                 The change in surface temperature from one turbulence time step to the
                 next, solved using the Ground Temperature Equation (see diagnostic
                 number 30) is calculated:
                 
                 .. math:: {\bf DTG} = {T_g}^{n} - {T_g}^{n-1}
                 
                 where superscript :math:`n` refers to the new, updated time level, and
                 the superscript :math:`n-1` refers to the value at the previous
                 turbulence time level.
                 
                 
                 QG - Ground Specific Humidity (g/kg)
                 #####################################
                 
                 The ground specific humidity is obtained by interpolating between the
                 specific humidity at the lowest model level and the specific humidity of
                 a saturated ground. The interpolation is performed using the potential
                 evapotranspiration function:
                 
                 .. math:: {\bf QG} = q_{Nrphys+1} = q_{Nrphys} + \beta(q^*(T_g,P_s) - q_{Nrphys})
                 
                 where :math:`\beta` is the surface potential evapotranspiration
                 coefficient (:math:`\beta=1` over oceans), and :math:`q^*(T_g,P_s)` is
                 the saturation specific humidity at the ground temperature and surface
                 pressure.
                 
                 
                 QS - Saturation Surface Specific Humidity (g/kg)
                 ################################################
                 
                 The surface saturation specific humidity is the saturation specific
                 humidity at the ground temprature and surface pressure:
                 
                 .. math:: {\bf QS} = q^*(T_g,P_s)
                 
                 TGRLW - Instantaneous ground temperature used as input to the Longwave radiation subroutine (deg)
                 #################################################################################################
                 
                 .. math:: {\bf TGRLW}  = T_g(\lambda , \phi ,n)
                 
                 where :math:`T_g` is the model ground temperature at the current time
                 step :math:`n`.
                 
                 ST4 - Upward Longwave flux at the surface (Watts/m^2)
                 #####################################################
                 
                 .. math:: {\bf ST4} = \sigma T^4
                 
                 where :math:`\sigma` is the Stefan-Boltzmann constant and T is the
                 temperature.
                 
                 
                 OLR - Net upward Longwave flux at :math:`p=p_{top}` (Watts/m^2)
                 ################################################################
                 
                 .. math:: {\bf OLR}  =  F_{LW,top}^{NET}
                 
                 where top indicates the top of the first model layer. In the GCM,
                 :math:`p_{top}` = 0.0 mb.
                 
                 
                 OLRCLR - Net upward clearsky Longwave flux at :math:`p=p_{top}` (Watts/m^2)
                 ###########################################################################
                 
                 .. math:: {\bf OLRCLR}  =  F(clearsky)_{LW,top}^{NET}
                 
                 where top indicates the top of the first model layer. In the GCM,
                 :math:`p_{top}` = 0.0 mb.
                 
                 
                 LWGCLR - Net upward clearsky Longwave flux at the surface (Watts/m^2)
                 ######################################################################
                 
                 .. math::
                 
                    \begin{aligned}
                    {\bf LWGCLR} & =  & F(clearsky)_{LW,Nrphys+1}^{Net} \\
                                 & =  & F(clearsky)_{LW,Nrphys+1}^\uparrow - F(clearsky)_{LW,Nrphys+1}^\downarrow\end{aligned}
                 
                 where Nrphys+1 indicates the lowest model edge-level, or
                 :math:`p = p_{surf}`. :math:`F(clearsky)_{LW}^\uparrow` is the upward
                 clearsky Longwave flux and the :math:`F(clearsky)_{LW}^\downarrow` is
                 the downward clearsky Longwave flux.
                 
                 
                 LWCLR - Heating Rate due to Clearsky Longwave Radiation (deg/day)
                 #################################################################
                 
                 The net longwave heating rate is calculated as the vertical divergence
                 of the net terrestrial radiative fluxes. Both the clear-sky and
                 cloudy-sky longwave fluxes are computed within the longwave routine. The
                 subroutine calculates the clear-sky flux, :math:`F^{clearsky}_{LW}`,
                 first. For a given cloud fraction, the clear line-of-sight probability
                 :math:`C(p,p^{\prime})` is computed from the current level pressure
                 :math:`p` to the model top pressure, :math:`p^{\prime} = p_{top}`, and
                 the model surface pressure, :math:`p^{\prime} = p_{surf}`, for the
                 upward and downward radiative fluxes. (see Section
                 [sec:fizhi:radcloud]). The cloudy-sky flux is then obtained as:
                 
0bad585a21 Navi* .. math:: F_{LW} = C(p,p') \cdot F^{clearsky}_{LW}
8679f9097b Jeff* 
                 Thus, **LWCLR** is defined as the net longwave heating rate due to the
                 vertical divergence of the clear-sky longwave radiative flux:
                 
0bad585a21 Navi* .. math:: \pp{\rho c_p T}{t}_{clearsky} = - \p{z} F(clearsky)_{LW}^{NET}
8679f9097b Jeff* 
                 or
                 
0bad585a21 Navi* .. math:: {\bf LWCLR} = \frac{g}{c_p \pi} \p{\sigma} F(clearsky)_{LW}^{NET}
8679f9097b Jeff* 
                 where :math:`g` is the accelation due to gravity, :math:`c_p` is the
                 heat capacity of air at constant pressure, and
                 
                 .. math:: F(clearsky)_{LW}^{Net} = F(clearsky)_{LW}^\uparrow - F(clearsky)_{LW}^\downarrow
                 
                 
                 TLW - Instantaneous temperature used as input to the Longwave radiation subroutine (deg)
                 ########################################################################################
                 
                 .. math:: {\bf TLW}  = T(\lambda , \phi ,level, n)
                 
                 where :math:`T` is the model temperature at the current time step
                 :math:`n`.
                 
                 
                 SHLW - Instantaneous specific humidity used as input to the Longwave radiation subroutine (kg/kg)
                 #################################################################################################
                 
                 .. math:: {\bf SHLW}  = q(\lambda , \phi , level , n)
                 
                 where :math:`q` is the model specific humidity at the current time step
                 :math:`n`.
                 
                 
                 OZLW - Instantaneous ozone used as input to the Longwave radiation subroutine (kg/kg)
                 #####################################################################################
                 
                 .. math:: {\bf OZLW}  = {\rm OZ}(\lambda , \phi , level , n)
                 
                 where :math:`\rm OZ` is the interpolated ozone data set from the
                 climatological monthly mean zonally averaged ozone data set.
                 
                 
                 CLMOLW - Maximum Overlap cloud fraction used in LW Radiation (0-1)
                 ##################################################################
                 
                 **CLMOLW** is the time-averaged maximum overlap cloud fraction that has been
                 filled by the Relaxed Arakawa/Schubert Convection scheme and will be
                 used in the Longwave Radiation algorithm. These are convective clouds
                 whose radiative characteristics are assumed to be correlated in the
                 vertical. For a complete description of cloud/radiative interactions,
                 see Section [sec:fizhi:radcloud].
                 
                 .. math:: {\bf CLMOLW} = CLMO_{RAS,LW}(\lambda, \phi,  level )
                 
                 
                 CLDTOT - Total cloud fraction used in LW and SW Radiation (0-1)
                 ###############################################################
                 
                 **CLDTOT** is the time-averaged total cloud fraction that has been
                 filled by the Relaxed Arakawa/Schubert and Large-scale Convection
                 schemes and will be used in the Longwave and Shortwave Radiation
                 packages. For a complete description of cloud/radiative interactions,
                 see Section [sec:fizhi:radcloud].
                 
                 .. math:: {\bf CLDTOT} = F_{RAS} + F_{LS}
                 
                 where :math:`F_{RAS}` is the time-averaged cloud fraction due to
                 sub-grid scale convection, and :math:`F_{LS}` is the time-averaged cloud
                 fraction due to precipitating and non-precipitating large-scale moist
                 processes.
                 
                 
                 CLMOSW - Maximum Overlap cloud fraction used in SW Radiation (0-1)
                 ##################################################################
                 
                 **CLMOSW** is the time-averaged maximum overlap cloud fraction that has been
                 filled by the Relaxed Arakawa/Schubert Convection scheme and will be
                 used in the Shortwave Radiation algorithm. These are convective clouds
                 whose radiative characteristics are assumed to be correlated in the
                 vertical. For a complete description of cloud/radiative interactions,
                 see Section [sec:fizhi:radcloud].
                 
                 .. math:: {\bf CLMOSW} = CLMO_{RAS,SW}(\lambda, \phi,  level )
                 
                 
                 CLROSW - Random Overlap cloud fraction used in SW Radiation (0-1)
                 #################################################################
                 
                 **CLROSW** is the time-averaged random overlap cloud fraction that has been
                 filled by the Relaxed Arakawa/Schubert and Large-scale Convection
                 schemes and will be used in the Shortwave Radiation algorithm. These are
                 convective and large-scale clouds whose radiative characteristics are
                 not assumed to be correlated in the vertical. For a complete description
                 of cloud/radiative interactions, see Section [sec:fizhi:radcloud].
                 
                 .. math:: {\bf CLROSW} = CLRO_{RAS,Large Scale,SW}(\lambda, \phi,  level )
                 
                 
                 RADSWT - Incident Shortwave radiation at the top of the atmosphere (Watts/m^2)
                 ##############################################################################
                 
                 .. math:: {\bf RADSWT} = {\frac{S_0}{R_a^2}} \cdot cos \phi_z
                 
                 where :math:`S_0`, is the extra-terrestial solar contant, :math:`R_a` is
                 the earth-sun distance in Astronomical Units, and :math:`cos \phi_z` is
                 the cosine of the zenith angle. It should be noted that **RADSWT**, as
                 well as **OSR** and **OSRCLR**, are calculated at the top of the
                 atmosphere (p=0 mb). However, the **OLR** and **OLRCLR** diagnostics are
                 currently calculated at :math:`p= p_{top}` (0.0 mb for the GCM).
                 
                 
                 EVAP - Surface Evaporation (mm/day)
                 ###################################
                 
                 The surface evaporation is a function of the gradient of moisture, the
                 potential evapotranspiration fraction and the eddy exchange coefficient:
                 
0bad585a21 Navi* .. math:: {\bf EVAP} =  \rho \beta K_{h} (q_{\rm surface} - q_{Nrphys})
8679f9097b Jeff* 
                 where :math:`\rho` = the atmospheric density at the surface,
                 :math:`\beta` is the fraction of the potential evapotranspiration
                 actually evaporated (:math:`\beta=1` over oceans), :math:`K_{h}` is the
                 turbulent eddy exchange coefficient for heat and moisture at the surface
                 in :math:`m/sec` and :math:`q{surface}` and :math:`q_{Nrphys}` are the
                 specific humidity at the surface (see diagnostic number 34) and at the
                 bottom model level, respectively.
                 
                 
                 DUDT - Total Zonal U-Wind Tendency  (m/sec/day)
                 ###############################################
                 
                 **DUDT** is the total time-tendency of the Zonal U-Wind due to Hydrodynamic,
                 Diabatic, and Analysis forcing.
                 
                 .. math:: {\bf DUDT} = \pp{u}{t}_{Dynamics} + \pp{u}{t}_{Moist} + \pp{u}{t}_{Turbulence} + \pp{u}{t}_{Analysis}
                 
                 
                 DVDT - Total Zonal V-Wind Tendency  (m/sec/day)
                 ###############################################
                 
                 **DVDT** is the total time-tendency of the Meridional V-Wind due to
                 Hydrodynamic, Diabatic, and Analysis forcing.
                 
                 .. math:: {\bf DVDT} = \pp{v}{t}_{Dynamics} + \pp{v}{t}_{Moist} + \pp{v}{t}_{Turbulence} + \pp{v}{t}_{Analysis}
                 
                 
                 DTDT - Total Temperature Tendency  (deg/day)
                 ############################################
                 
                 **DTDT** is the total time-tendency of Temperature due to Hydrodynamic, Diabatic,
                 and Analysis forcing.
                 
                 .. math::
                 
                    \begin{aligned}
0bad585a21 Navi*    {\bf DTDT} & = \pp{T}{t}_{Dynamics} + \pp{T}{t}_{Moist Processes} + \pp{T}{t}_{Shortwave Radiation} \\
                               & + \pp{T}{t}_{Longwave Radiation} + \pp{T}{t}_{Turbulence} + \pp{T}{t}_{Analysis} \end{aligned}
8679f9097b Jeff* 
                 
                 DQDT - Total Specific Humidity Tendency  (g/kg/day)
                 ###################################################
                 
                 **DQDT** is the total time-tendency of Specific Humidity due to Hydrodynamic,
                 Diabatic, and Analysis forcing.
                 
                 .. math::
                 
                    {\bf DQDT} = \pp{q}{t}_{Dynamics} + \pp{q}{t}_{Moist Processes} 
                    + \pp{q}{t}_{Turbulence} + \pp{q}{t}_{Analysis}
                 
                 
                 USTAR -  Surface-Stress Velocity (m/sec)
                 ########################################
                 
                 The surface stress velocity, or the friction velocity, is the wind speed
                 at the surface layer top impeded by the surface drag:
                 
                 .. math::
                 
                    {\bf USTAR} = C_uW_s \hspace{1cm}where: \hspace{.2cm} 
                    C_u = \frac{k}{\psi_m}
                 
                 :math:`C_u` is the non-dimensional surface drag coefficient (see
                 diagnostic number 10), and :math:`W_s` is the surface wind speed (see
                 diagnostic number 28).
                 
                 
                 Z0 - Surface Roughness Length (m)
                 #################################
                 
                 Over the land surface, the surface roughness length is interpolated to
                 the local time from the monthly mean data of . Over the ocean, the
                 roughness length is a function of the surface-stress velocity,
                 :math:`u_*`.
                 
                 .. math:: {\bf Z0} = c_1u^3_* + c_2u^2_* + c_3u_* + c_4 + {c_5}{u_*}
                 
                 where the constants are chosen to interpolate between the reciprocal
                 relation of for weak winds, and the piecewise linear relation of for
                 moderate to large winds.
                 
                 
                 FRQTRB - Frequency of Turbulence (0-1)
                 ######################################
                 
                 The fraction of time when turbulence is present is defined as the
                 fraction of time when the turbulent kinetic energy exceeds some minimum
                 value, defined here to be :math:`0.005 \hspace{.1cm}m^2/sec^2`. When
                 this criterion is met, a counter is incremented. The fraction over the
                 averaging interval is reported.
                 
                 
                 PBL - Planetary Boundary Layer Depth (mb)
                 #########################################
                 
                 The depth of the PBL is defined by the turbulence parameterization to be
                 the depth at which the turbulent kinetic energy reduces to ten percent
                 of its surface value.
                 
0bad585a21 Navi* .. math:: {\bf PBL} = P_{PBL} - P_{\rm surface}
8679f9097b Jeff* 
                 where :math:`P_{PBL}` is the pressure in :math:`mb` at which the
                 turbulent kinetic energy reaches one tenth of its surface value, and
                 :math:`P_s` is the surface pressure.
                 
                 
                 SWCLR - Clear sky Heating Rate due to Shortwave Radiation (deg/day)
                 ###################################################################
                 
                 The net Shortwave heating rate is calculated as the vertical divergence
                 of the net solar radiative fluxes. The clear-sky and cloudy-sky
                 shortwave fluxes are calculated separately. For the clear-sky case, the
                 shortwave fluxes and heating rates are computed with both CLMO (maximum
                 overlap cloud fraction) and CLRO (random overlap cloud fraction) set to
                 zero (see Section [sec:fizhi:radcloud]). The shortwave routine is then
                 called a second time, for the cloudy-sky case, with the true
                 time-averaged cloud fractions CLMO and CLRO being used. In all cases, a
                 normalized incident shortwave flux is used as input at the top of the
                 atmosphere.
                 
                 The heating rate due to Shortwave Radiation under clear skies is defined
                 as:
                 
0bad585a21 Navi* .. math:: \pp{\rho c_p T}{t} = - \p{z} F(clear)_{SW}^{NET} \cdot {\rm RADSWT}
8679f9097b Jeff* 
                 or
                 
0bad585a21 Navi* .. math:: {\bf SWCLR} = \frac{g}{c_p } \p{p} F(clear)_{SW}^{NET}\cdot {\rm RADSWT}
8679f9097b Jeff* 
                 where :math:`g` is the accelation due to gravity, :math:`c_p` is the
                 heat capacity of air at constant pressure, RADSWT is the true incident
                 shortwave radiation at the top of the atmosphere (See Diagnostic #48),
                 and
                 
                 .. math:: F(clear)_{SW}^{Net} = F(clear)_{SW}^\uparrow - F(clear)_{SW}^\downarrow
                 
                 
                 OSR - Net upward Shortwave flux at the top of the model (Watts/m^2)
                 ###################################################################
                 
                 .. math:: {\bf OSR}  =  F_{SW,top}^{NET}
                 
                 where top indicates the top of the first model layer used in the
                 shortwave radiation routine. In the GCM, :math:`p_{SW_{top}}` = 0 mb.
                 
                 
                 OSRCLR - Net upward clearsky Shortwave flux at the top of the model (Watts/m^2)
                 ###############################################################################
                 
                 .. math:: {\bf OSRCLR}  =  F(clearsky)_{SW,top}^{NET}
                 
                 where top indicates the top of the first model layer used in the
                 shortwave radiation routine. In the GCM, :math:`p_{SW_{top}}` = 0 mb.
                 
                 
                 CLDMAS - Convective Cloud Mass Flux (kg/m^2) 
                 ############################################
                 
                 The amount of cloud mass moved per RAS timestep from all convective
                 clouds is written:
                 
                 .. math:: {\bf CLDMAS} = \eta m_B
                 
                 where :math:`\eta` is the entrainment, normalized by the cloud base mass
                 flux, and :math:`m_B` is the cloud base mass flux. :math:`m_B` and
                 :math:`\eta` are defined explicitly in :numref:`para_phys_pkg_fizhi_mc`, the
                 description of the convective parameterization.
                 
                 
                 UAVE - Time-Averaged Zonal U-Wind (m/sec)
                 #########################################
                 
                 The diagnostic **UAVE** is simply the time-averaged Zonal U-Wind over
                 the **NUAVE** output frequency. This is contrasted to the instantaneous
                 Zonal U-Wind which is archived on the Prognostic Output data stream.
                 
                 .. math:: {\bf UAVE} = u(\lambda, \phi, level , t)
                 
                 Note, **UAVE** is computed and stored on the staggered C-grid.
                 
                 
                 VAVE - Time-Averaged Meridional V-Wind (m/sec)
                 ##############################################
                 
                 The diagnostic **VAVE** is simply the time-averaged Meridional V-Wind
                 over the **NVAVE** output frequency. This is contrasted to the
                 instantaneous Meridional V-Wind which is archived on the Prognostic
                 Output data stream.
                 
                 .. math:: {\bf VAVE} = v(\lambda, \phi, level , t)
                 
                 Note, **VAVE** is computed and stored on the staggered C-grid.
                 
                 
                 TAVE - Time-Averaged Temperature (Kelvin)
                 #########################################
                 
                 The diagnostic **TAVE** is simply the time-averaged Temperature over
                 the **NTAVE** output frequency. This is contrasted to the instantaneous
                 Temperature which is archived on the Prognostic Output data stream.
                 
                 .. math:: {\bf TAVE} = T(\lambda, \phi, level , t)
                 
                 
                 QAVE - Time-Averaged Specific Humidity (g/kg)
                 #############################################
                 
                 The diagnostic **QAVE** is simply the time-averaged Specific Humidity
                 over the **NQAVE** output frequency. This is contrasted to the
                 instantaneous Specific Humidity which is archived on the Prognostic
                 Output data stream.
                 
                 .. math:: {\bf QAVE} = q(\lambda, \phi, level , t)
                 
                 
                 PAVE - Time-Averaged Surface Pressure - PTOP (mb)
                 #################################################
                 
                 The diagnostic **PAVE** is simply the time-averaged Surface Pressure -
                 PTOP over the **NPAVE** output frequency. This is contrasted to the
                 instantaneous Surface Pressure - PTOP which is archived on the
                 Prognostic Output data stream.
                 
                 .. math::
                 
                    \begin{aligned}
                    {\bf PAVE} & =  & \pi(\lambda, \phi, level , t) \\
                               & =  & p_s(\lambda, \phi, level , t) - p_T\end{aligned}
                 
                 QQAVE - Time-Averaged Turbulent Kinetic Energy (m/sec)^2
                 ########################################################
                 
                 The diagnostic **QQAVE** is simply the time-averaged prognostic
                 Turbulent Kinetic Energy produced by the GCM Turbulence parameterization
                 over the **NQQAVE** output frequency. This is contrasted to the
                 instantaneous Turbulent Kinetic Energy which is archived on the
                 Prognostic Output data stream.
                 
                 .. math:: {\bf QQAVE} = qq(\lambda, \phi, level , t)
                 
                 Note, **QQAVE** is computed and stored at the “mass-point” locations
                 on the staggered C-grid.
                 
                 
                 SWGCLR - Net downward clearsky Shortwave flux at the surface (Watts/m^2)
                 ########################################################################
                 
                 .. math::
                 
                    \begin{aligned}
                    {\bf SWGCLR} & =  & F(clearsky)_{SW,Nrphys+1}^{Net} \\
                                 & =  & F(clearsky)_{SW,Nrphys+1}^\downarrow - F(clearsky)_{SW,Nrphys+1}^\uparrow\end{aligned}
                 
                  
                 where Nrphys+1 indicates the lowest model edge-level, or
                 :math:`p = p_{surf}`. :math:`F(clearsky){SW}^\downarrow` is the downward
                 clearsky Shortwave flux and :math:`F(clearsky)_{SW}^\uparrow` is the
                 upward clearsky Shortwave flux.
                 
                 
                 DIABU - Total Diabatic Zonal U-Wind Tendency  (m/sec/day)
                 #########################################################
                 
                 **DIABU** is the total time-tendency of the Zonal U-Wind due to Diabatic
                 processes and the Analysis forcing.
                 
                 .. math:: {\bf DIABU} = \pp{u}{t}_{Moist} + \pp{u}{t}_{Turbulence} + \pp{u}{t}_{Analysis}
                 
                 
                 
                 DIABV - Total Diabatic Meridional V-Wind Tendency  (m/sec/day)
                 ##############################################################
                 
                 **DIABV** is the total time-tendency of the Meridional V-Wind due to Diabatic
                 processes and the Analysis forcing.
                 
                 .. math:: {\bf DIABV} = \pp{v}{t}_{Moist} + \pp{v}{t}_{Turbulence} + \pp{v}{t}_{Analysis}
                 
                 
                 DIABT Total Diabatic Temperature Tendency (deg/day)
                 ###################################################
                 
                 **DIABT** is the total time-tendency of Temperature due to Diabatic processes and
                 the Analysis forcing.
                 
                 .. math::
                 
                    \begin{aligned}
0bad585a21 Navi*    {\bf DIABT} & = \pp{T}{t}_{Moist Processes} + \pp{T}{t}_{Shortwave Radiation} \\
                               & + \pp{T}{t}_{Longwave Radiation} + \pp{T}{t}_{Turbulence} + \pp{T}{t}_{Analysis} \end{aligned}
8679f9097b Jeff* 
                 If we define the time-tendency of Temperature due to Diabatic
                 processes as
                 
                 .. math::
                 
                    \begin{aligned}
0bad585a21 Navi*    \pp{T}{t}_{Diabatic} & = \pp{T}{t}_{Moist Processes} + \pp{T}{t}_{Shortwave Radiation} \\
                                         & + \pp{T}{t}_{Longwave Radiation} + \pp{T}{t}_{Turbulence}\end{aligned}
8679f9097b Jeff* 
                 then, since there are no surface pressure changes due to Diabatic
                 processes, we may write
                 
                 .. math:: \pp{T}{t}_{Diabatic} = \frac{p^\kappa}{\pi}\pp{\pi \theta}{t}_{Diabatic}
                 
                 where :math:`\theta = T/p^\kappa`. Thus, **DIABT** may be written as
                 
                 .. math:: {\bf DIABT} = \frac{p^\kappa}{\pi} \left( \pp{\pi \theta}{t}_{Diabatic} + \pp{\pi \theta}{t}_{Analysis} \right)
                 
                 
                 DIABQ - Total Diabatic Specific Humidity Tendency (g/kg/day)
                 ############################################################
                 
                 **DIABQ** is the total time-tendency of Specific Humidity due to Diabatic
                 processes and the Analysis forcing.
                 
                 .. math:: {\bf DIABQ} = \pp{q}{t}_{Moist Processes} + \pp{q}{t}_{Turbulence} + \pp{q}{t}_{Analysis}
                 
                 If we define the time-tendency of Specific Humidity due to Diabatic
                 processes as
                 
                 .. math:: \pp{q}{t}_{Diabatic} = \pp{q}{t}_{Moist Processes} + \pp{q}{t}_{Turbulence}
                 
                 then, since there are no surface pressure changes due to Diabatic
                 processes, we may write
                 
                 .. math:: \pp{q}{t}_{Diabatic} = \frac{1}{\pi}\pp{\pi q}{t}_{Diabatic}
                 
                  Thus, **DIABQ** may be written as
                 
                 .. math:: {\bf DIABQ} = \frac{1}{\pi} \left( \pp{\pi q}{t}_{Diabatic} + \pp{\pi q}{t}_{Analysis} \right)
                 
                 
                 VINTUQ - Vertically Integrated Moisture Flux (m/sec  g/kg)
                 ##########################################################
                 
                 The vertically integrated moisture flux due to the zonal u-wind is
                 obtained by integrating :math:`u q` over the depth of the atmosphere at
                 each model timestep, and dividing by the total mass of the column.
                 
                 .. math:: {\bf VINTUQ} = \frac{ \int_{surf}^{top} u q \rho dz  } { \int_{surf}^{top} \rho dz  }
                 
                 Using
                 :math:`\rho \delta z = -\frac{\delta p}{g} = - \frac{1}{g} \delta p`, we
                 have
                 
                 .. math:: {\bf VINTUQ} = { \int_0^1 u q dp  }
                 
                 
                 VINTVQ - Vertically Integrated Moisture Flux (m/sec g/kg)
                 #########################################################
                 
                 The vertically integrated moisture flux due to the meridional v-wind
                 is obtained by integrating :math:`v q` over the depth of the atmosphere
                 at each model timestep, and dividing by the total mass of the column.
                 
                 .. math:: {\bf VINTVQ} = \frac{ \int_{surf}^{top} v q \rho dz  } { \int_{surf}^{top} \rho dz  }
                 
                 Using
                 :math:`\rho \delta z = -\frac{\delta p}{g} = - \frac{1}{g} \delta p`, we
                 have
                 
                 .. math:: {\bf VINTVQ} = { \int_0^1 v q dp  }
                 
                 
                 VINTUT - Vertically Integrated Heat Flux (m/sec deg)
                 ####################################################
                 
                 The vertically integrated heat flux due to the zonal u-wind is
                 obtained by integrating :math:`u T` over the depth of the atmosphere at
                 each model timestep, and dividing by the total mass of the column.
                 
                 .. math:: {\bf VINTUT} = \frac{ \int_{surf}^{top} u T \rho dz  } { \int_{surf}^{top} \rho dz  }
                 
                 Or,
                 
                 .. math:: {\bf VINTUT} = { \int_0^1 u T dp  }
                 
                 
                 VINTVT - Vertically Integrated Heat Flux (m/sec deg)
                 ####################################################
                 
                 The vertically integrated heat flux due to the meridional v-wind is
                 obtained by integrating :math:`v T` over the depth of the atmosphere at
                 each model timestep, and dividing by the total mass of the column.
                 
                 .. math:: {\bf VINTVT} = \frac{ \int_{surf}^{top} v T \rho dz  } { \int_{surf}^{top} \rho dz  }
                 
                 Using :math:`\rho \delta z = -\frac{\delta p}{g}`, we have
                 
                 .. math:: {\bf VINTVT} = { \int_0^1 v T dp  }
                 
                 
                 CLDFRC - Total 2-Dimensional Cloud Fracton (0-1)
                 ################################################
                 
                 If we define the time-averaged random and maximum overlapped cloudiness
                 as CLRO and CLMO respectively, then the probability of clear sky
                 associated with random overlapped clouds at any level is (1-CLRO) while
                 the probability of clear sky associated with maximum overlapped clouds
                 at any level is (1-CLMO). The total clear sky probability is given by
                 (1-CLRO)\*(1-CLMO), thus the total cloud fraction at each level may be
                 obtained by 1-(1-CLRO)\*(1-CLMO).
                 
                 At any given level, we may define the clear line-of-site probability by
                 appropriately accounting for the maximum and random overlap cloudiness.
                 The clear line-of-site probability is defined to be equal to the product
                 of the clear line-of-site probabilities associated with random and
                 maximum overlap cloudiness. The clear line-of-site probability
                 :math:`C(p,p^{\prime})` associated with maximum overlap clouds, from the
                 current pressure :math:`p` to the model top pressure,
                 :math:`p^{\prime} = p_{top}`, or the model surface pressure,
                 :math:`p^{\prime} = p_{surf}`, is simply 1.0 minus the largest maximum
                 overlap cloud value along the line-of-site, ie.
                 
                 .. math:: 1-MAX_p^{p^{\prime}} \left( CLMO_p \right)
                 
                 Thus, even in the time-averaged sense it is assumed that the maximum
                 overlap clouds are correlated in the vertical. The clear line-of-site
                 probability associated with random overlap clouds is defined to be the
                 product of the clear sky probabilities at each level along the
                 line-of-site, ie.
                 
                 .. math:: \prod_{p}^{p^{\prime}} \left( 1-CLRO_p \right)
                 
                 The total cloud fraction at a given level associated with a line-
                 of-site calculation is given by
                 
                 .. math::
                 
                    1-\left( 1-MAX_p^{p^{\prime}} \left[ CLMO_p \right] \right)
                        \prod_p^{p^{\prime}} \left( 1-CLRO_p \right)
                 
                 The 2-dimensional net cloud fraction as seen from the top of the
                 atmosphere is given by
                 
                 .. math::
                 
                    {\bf CLDFRC} = 1-\left( 1-MAX_{l=l_1}^{Nrphys} \left[ CLMO_l \right] \right)
                        \prod_{l=l_1}^{Nrphys} \left( 1-CLRO_l \right)
                 
                 For a complete description of cloud/radiative interactions, see
                 Section [sec:fizhi:radcloud].
                 
                 
                 QINT - Total Precipitable Water (gm/cm^2)
                 #########################################
                 
                 The Total Precipitable Water is defined as the vertical integral of the
                 specific humidity, given by:
                 
                 .. math::
                 
                    \begin{aligned}
0bad585a21 Navi*    {\bf QINT} & = \int_{surf}^{top} \rho q dz \\
                               & = \frac{\pi}{g} \int_0^1 q dp
8679f9097b Jeff*    \end{aligned}
                 
                 where we have used the hydrostatic relation
                 :math:`\rho \delta z = -\frac{\delta p}{g}`.
                 
                 
                 U2M  Zonal U-Wind at 2 Meter Depth (m/sec)
                 ##########################################
                 
                 The u-wind at the 2-meter depth is determined from the similarity
                 theory:
                 
                 .. math::
                 
                    {\bf U2M} = \frac{u_*}{k} \psi_{m_{2m}} \frac{u_{sl}}{W_s} =
                    \frac{ \psi_{m_{2m}} }{ \psi_{m_{sl}} }u_{sl}
                 
                 where :math:`\psi_m(2m)` is the non-dimensional wind shear at two
                 meters, and the subscript :math:`sl` refers to the height of the top of
                 the surface layer. If the roughness height is above two meters,
                 :math:`{\bf U2M}` is undefined.
                 
                 
                 V2M - Meridional V-Wind at 2 Meter Depth (m/sec)
                 ################################################
                 
                 The v-wind at the 2-meter depth is a determined from the similarity
                 theory:
                 
                 .. math::
                 
                    {\bf V2M} = \frac{u_*}{k} \psi_{m_{2m}} \frac{v_{sl}}{W_s} =
                    \frac{ \psi_{m_{2m}} }{ \psi_{m_{sl}} }v_{sl}
                 
                 where :math:`\psi_m(2m)` is the non-dimensional wind shear at two
                 meters, and the subscript :math:`sl` refers to the height of the top of
                 the surface layer. If the roughness height is above two meters,
                 :math:`{\bf V2M}` is undefined.
                 
                 
                 T2M - Temperature at 2 Meter Depth (deg K)
                 ##########################################
                 
                 The temperature at the 2-meter depth is a determined from the similarity
                 theory:
                 
                 .. math::
                 
                    {\bf T2M} = P^{\kappa} (\frac{\theta*}{k} ({\psi_{h_{2m}}+\psi_g}) + \theta_{surf} ) = 
                    P^{\kappa}(\theta_{surf} + \frac{ \psi_{h_{2m}}+\psi_g }{ \psi_{h_{sl}}+\psi_g }
                    (\theta_{sl} - \theta_{surf}) )
                 
                 where:
                 
                 .. math:: \theta_* = - \frac{ (\overline{w^{\prime}\theta^{\prime}}) }{ u_* }
                 
                 where :math:`\psi_h(2m)` is the non-dimensional temperature gradient
                 at two meters, :math:`\psi_g` is the non-dimensional temperature
                 gradient in the viscous sublayer, and the subscript :math:`sl` refers to
                 the height of the top of the surface layer. If the roughness height is
                 above two meters, :math:`{\bf T2M}` is undefined.
                 
                 
                 Q2M - Specific Humidity at 2 Meter Depth (g/kg)
                 ###############################################
                 
                 The specific humidity at the 2-meter depth is determined from the
                 similarity theory:
                 
                 .. math::
                 
                    {\bf Q2M} = P^{\kappa} \frac({q_*}{k} ({\psi_{h_{2m}}+\psi_g}) + q_{surf} ) = 
                    P^{\kappa}(q_{surf} + \frac{ \psi_{h_{2m}}+\psi_g }{ \psi_{h_{sl}}+\psi_g }
                    (q_{sl} - q_{surf}))
                 
                 where:
                 
                 .. math:: q_* = - \frac{ (\overline{w^{\prime}q^{\prime}}) }{ u_* }
                 
                 where :math:`\psi_h(2m)` is the non-dimensional temperature gradient
                 at two meters, :math:`\psi_g` is the non-dimensional temperature
                 gradient in the viscous sublayer, and the subscript :math:`sl` refers to
                 the height of the top of the surface layer. If the roughness height is
                 above two meters, :math:`{\bf Q2M}` is undefined.
                 
                 
                 U10M - Zonal U-Wind at 10 Meter Depth (m/sec)
                 #############################################
                 
                 The u-wind at the 10-meter depth is an interpolation between the surface
                 wind and the model lowest level wind using the ratio of the
                 non-dimensional wind shear at the two levels:
                 
                 .. math::
                 
                    {\bf U10M} = \frac{u_*}{k} \psi_{m_{10m}} \frac{u_{sl}}{W_s} =
                    \frac{ \psi_{m_{10m}} }{ \psi_{m_{sl}} }u_{sl}
                 
                 where :math:`\psi_m(10m)` is the non-dimensional wind shear at ten
                 meters, and the subscript :math:`sl` refers to the height of the top of
                 the surface layer.
                 
                 
                 V10M - Meridional V-Wind at 10 Meter Depth (m/sec)
                 ##################################################
                 
                 The v-wind at the 10-meter depth is an interpolation between the surface
                 wind and the model lowest level wind using the ratio of the
                 non-dimensional wind shear at the two levels:
                 
                 .. math::
                 
                    {\bf V10M} = \frac{u_*}{k} \psi_{m_{10m}} \frac{v_{sl}}{W_s} =
                    \frac{ \psi_{m_{10m}} }{ \psi_{m_{sl}} }v_{sl}
                 
                 where :math:`\psi_m(10m)` is the non-dimensional wind shear at ten
                 meters, and the subscript :math:`sl` refers to the height of the top of
                 the surface layer.
                 
                 
                 T10M - Temperature at 10 Meter Depth (deg K)
                 ############################################
                 
                 The temperature at the 10-meter depth is an interpolation between the
                 surface potential temperature and the model lowest level potential
                 temperature using the ratio of the non-dimensional temperature gradient
                 at the two levels:
                 
                 .. math::
                 
                    {\bf T10M} = P^{\kappa} (\frac{\theta*}{k} ({\psi_{h_{10m}}+\psi_g}) + \theta_{surf} ) = 
                    P^{\kappa}(\theta_{surf} + \frac{\psi_{h_{10m}}+\psi_g}{\psi_{h_{sl}}+\psi_g}
                    (\theta_{sl} - \theta_{surf}))
                 
                 where:
                 
                 .. math:: \theta_* = - \frac{ (\overline{w^{\prime}\theta^{\prime}}) }{ u_* }
                 
                 where :math:`\psi_h(10m)` is the non-dimensional temperature gradient
                 at two meters, :math:`\psi_g` is the non-dimensional temperature
                 gradient in the viscous sublayer, and the subscript :math:`sl` refers to
                 the height of the top of the surface layer.
                 
                 
                 Q10M - Specific Humidity at 10 Meter Depth (g/kg)
                 #################################################
                 
                 The specific humidity at the 10-meter depth is an interpolation between
                 the surface specific humidity and the model lowest level specific
                 humidity using the ratio of the non-dimensional temperature gradient at
                 the two levels:
                 
                 .. math::
                 
                    {\bf Q10M} = P^{\kappa} (\frac{q_*}{k} ({\psi_{h_{10m}}+\psi_g}) + q_{surf} ) = 
                    P^{\kappa}(q_{surf} + \frac{\psi_{h_{10m}}+\psi_g}{\psi_{h_{sl}}+\psi_g}
                    (q_{sl} - q_{surf}))
                 
                 where:
                 
                 .. math:: q_* =  - \frac{ (\overline{w^{\prime}q^{\prime}}) }{ u_* }
                 
                 where :math:`\psi_h(10m)` is the non-dimensional temperature gradient
                 at two meters, :math:`\psi_g` is the non-dimensional temperature
                 gradient in the viscous sublayer, and the subscript :math:`sl` refers to
                 the height of the top of the surface layer.
                 
                 
                 DTRAIN - Cloud Detrainment Mass Flux (kg/m^2)
                 #############################################
                 
                 The amount of cloud mass moved per RAS timestep at the cloud
                 detrainment level is written:
                 
                 .. math:: {\bf DTRAIN} = \eta_{r_D}m_B
                 
                 where :math:`r_D` is the detrainment level, :math:`m_B` is the cloud
                 base mass flux, and :math:`\eta` is the entrainment, defined in :numref:`para_phys_pkg_fizhi_mc`.
                 
                 
                 QFILL - Filling of negative Specific Humidity (g/kg/day)
                 ########################################################
                 
                 Due to computational errors associated with the numerical scheme used
                 for the advection of moisture, negative values of specific humidity may
                 be generated. The specific humidity is checked for negative values after
                 every dynamics timestep. If negative values have been produced, a
                 filling algorithm is invoked which redistributes moisture from below.
                 Diagnostic **QFILL** is equal to the net filling needed to eliminate
                 negative specific humidity, scaled to a per-day rate:
                 
                 .. math:: {\bf QFILL} = q^{n+1}_{final} - q^{n+1}_{initial}
                 
                 where
                 
                 .. math:: q^{n+1} = (\pi q)^{n+1} / \pi^{n+1}
                 
                 Key subroutines, parameters and files
                 +++++++++++++++++++++++++++++++++++++
                 
                 
                 Dos and don'ts
                 ++++++++++++++
                 
                 
                 Fizhi Reference
                 +++++++++++++++
                 
                 
                 Experiments and tutorials that use fizhi
                 ++++++++++++++++++++++++++++++++++++++++
                 
                 -  Global atmosphere experiment with realistic SST and topography in
                    fizhi-cs-32x32x10 verification directory.
                 
                 -  Global atmosphere aqua planet experiment in fizhi-cs-aqualev20
                    verification directory.
                 
                 

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