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3ceb1ce0a9 Jeff* .. _tutorial_plume_on_slope:
                 
d67096e55c Jeff* Gravity Plume On a Continental Slope
                 ====================================
                 
                 (in directory: :filelink:`verification/tutorial_plume_on_slope/`)
                 
                    .. figure:: figs/billows.png
                        :width: 80%
                        :align: center
                        :alt: plume on slope image
                        :name: tut_plume_on_slop_out
                 
                        Temperature after 23 hours of cooling. The cold dense water is mixed with ambient water as it accelerates down the slope and hence is warmer than the unmixed plume.
                 
                 An important test of any ocean model is the ability to represent the
                 flow of dense fluid down a slope. One example of such a flow is a
                 non-rotating gravity plume on a continental slope, forced by a limited
                 area of surface cooling above a continental shelf. Because the flow is
                 non-rotating, a two dimensional model can be used in the across slope
                 direction. The experiment is non-hydrostatic and uses open-boundaries to
                 radiate transients at the deep water end. (Dense flow down a slope can
                 also be forced by a dense inflow prescribed on the continental shelf;
                 this configuration is being implemented by the DOME (Dynamics of
                 Overflow Mixing and Entrainment) collaboration to compare solutions in
                 different models).
                 
                 The fluid is initially unstratified. The surface buoyancy loss
                 :math:`B_0` (dimensions of L\ :math:`^2`\ T\ :math:`^{-3}`) over a
                 cross-shelf distance :math:`R` causes vertical convective mixing and
                 modifies the density of the fluid by an amount
                 
                 .. math:: \Delta \rho = \frac{B_0 \rho_0 t}{g H}
                 
                 where :math:`H` is the depth of the shelf, :math:`g` is the
                 acceleration due to gravity, :math:`t` is time since onset of cooling
                 and :math:`\rho_0` is the reference density. Dense fluid slumps under
                 gravity, with a flow speed close to the gravity wave speed:
                 
                 .. math::
                 
                    U
                    \sim \sqrt{g' H}
                    \sim \sqrt{ \frac{g \Delta \rho H}{\rho_0} }
                    \sim \sqrt{B_0 t}
                 
                 A steady state is rapidly established in which the buoyancy flux out of
                 the cooling region is balanced by the surface buoyancy loss. Then
                 
                 .. math:: U \sim (B_0 R)^{1/3} \mbox{  ;  } \Delta \rho \sim \frac{\rho_0}{g H} (B_0 R)^{2/3}
                 
                 The Froude number of the flow on the shelf is close to unity (but in
                 practice slightly less than unity, giving subcritical flow). When the
                 flow reaches the slope, it accelerates, so that it may become
                 supercritical (provided the slope angle :math:`\alpha` is steep
                 enough). In this case, a hydraulic control is established at the shelf
                 break. On the slope, where the Froude number is greater than one, and
                 gradient Richardson number (defined as :math:`Ri \sim g' h^*/U^2` where
                 :math:`h^*` is the thickness of the interface between dense and ambient
                 fluid) is reduced below 1/4, Kelvin-Helmholtz instability is possible,
                 and leads to entrainment of ambient fluid into the plume, modifying the
                 density, and hence the acceleration down the slope. Kelvin-Helmholtz
                 instability is suppressed at low Reynolds and Peclet numbers given by
                 
                 .. math:: Re \sim \frac{U h}{ \nu} \sim \frac{(B_0 R)^{1/3} h}{\nu} \mbox{  ;  } Pe = Re Pr
                 
                 where :math:`h` is the depth of the dense fluid on the slope. Hence
                 this experiment is carried out in the high Re, Pe regime. A further
                 constraint is that the convective heat flux must be much greater than
0bad585a21 Navi* the diffusive heat flux (Nusselt number :math:`\gg  1`). Then
d67096e55c Jeff* 
0bad585a21 Navi* .. math:: Nu = \frac{U h^* }{\kappa} \gg  1
d67096e55c Jeff* 
                 Finally, since we have assumed that the convective mixing on the shelf
                 occurs in a much shorter time than the horizontal equilibration, this
0bad585a21 Navi* implies :math:`H/R \ll 1`.
d67096e55c Jeff* 
                 Hence to summarize the important non-dimensional parameters, and the
                 limits we are considering:
                 
                 .. math::
                 
0bad585a21 Navi*    \frac{H}{R} \ll  1 \mbox{ ; } Re \gg  1 \mbox{  ; } Pe \gg  1 \mbox{  ; } Nu \gg  1
                    \mbox{  ;  } \mbox{  ; } Ri < 1/4 .
d67096e55c Jeff* 
                 In addition we are assuming that the slope is steep enough to provide
                 sufficient acceleration to the gravity plume, but nonetheless much less
                 that 1:1, since many Kelvin-Helmholtz billows appear on the
0bad585a21 Navi* slope, implying horizontal length scale of the slope :math:`\gg` the depth
d67096e55c Jeff* of the dense fluid.
                 
                 Configuration
                 -------------
                 
                 The topography, spatial grid, forcing and initial conditions are all
                 specified in binary data files generated using matlab script
                 :filelink:`verification/tutorial_plume_on_slope/input/gendata.m` and detailed in :numref:`tut_plume_on_slope_config`. Other model
                 parameters are specified in :filelink:`input/data <verification/tutorial_plume_on_slope/input/data>` and
                 :filelink:`input/data.obcs <verification/tutorial_plume_on_slope/input/data.obcs>` and detailed in :numref:`tut_plume_on_slope_mod_parms`.
                 
                 .. _tut_plume_on_slope_config:
                 
                 Binary input data
                 -----------------
                 
                    .. figure:: figs/dx.png
                        :width: 80%
                        :align: center
                        :alt: plume on slope dx
                        :name: dx-plume-on-slope
                 
                        Horizontal grid spacing, :math:`\Delta x`, in the across-slope direction for the gravity plume experiment.
                 
                 |
                 
                    .. figure:: figs/Depth.png
                        :width: 80%
                        :align: center
                        :alt: plume on slope Depth
                        :name: depth-plume-on-slope
                 
                        Topography, :math:`h(x)`, used for the gravity plume experiment.
                 
                 |
                 
                    .. figure:: figs/Qsurf.png
                        :width: 80%
                        :align: center
                        :alt: plume on slope Qsurf
                        :name: Q-plume-on-slope
                 
                        Upward surface heat flux, :math:`Q(x)`, used as forcing in the gravity plume experiment.
                 
                 The domain is :math:`200` m deep and :math:`6.4` km across. Uniform
                 resolution of :math:`60\times3^1/_3` m is used in the vertical and
                 variable resolution of the form shown in :numref:`dx-plume-on-slope`
                 with 320 points is used in the horizontal. The formula for
                 :math:`\Delta x` is:
                 
                 .. math::
                 
                    \Delta x(i) = \Delta x_1 + ( \Delta x_2 - \Delta x_1 )
                    ( 1 + \tanh{\left(\frac{i-i_s}{w}\right)} ) /2
                 
                 where
                 
                 .. math::
                 
                    \begin{aligned}
0bad585a21 Navi*    Nx & = 320 \\
                    Lx & = 6400 \;\; \mbox{(m)} \\
                    \Delta x_1 & = \frac{2}{3} \frac{Lx}{Nx} \;\; \mbox{(m)} \\
                    \Delta x_2 & = \frac{Lx/2}{Nx-Lx/(2 \Delta x_1)} \;\; \mbox{(m)} \\
                    i_s & = Lx/( 2 \Delta x_1 ) \\
                    w & = 40\end{aligned}
d67096e55c Jeff* 
                 Here, :math:`\Delta x_1` is the resolution on the shelf,
                 :math:`\Delta x_2` is the resolution in deep water and :math:`Nx` is the
                 number of points in the horizontal.
                 
                 The topography, shown in :numref:`depth-plume-on-slope`, is given by:
                 
                 .. math:: H(x) = -H_o + (H_o - h_s) ( 1 + \tanh{\left(\frac{x-x_s}{L_s}\right)} ) / 2
                 
                 where
                 
                 .. math::
                 
                    \begin{aligned}
0bad585a21 Navi*    H_o & = 200 \;\; \mbox{(m)} \\
                    h_s & = 40 \;\; \mbox{(m)} \\
                    x_s & = 1500 + Lx/2 \;\; \mbox{(m)} \\
                    L_s & = \frac{(H_o - h_s)}{2 s} \;\; \mbox{(m)} \\
                    s & = 0.15\end{aligned}
d67096e55c Jeff* 
                 Here, :math:`s` is the maximum slope, :math:`H_o` is the maximum depth,
                 :math:`h_s` is the shelf depth, :math:`x_s` is the lateral position of
                 the shelf-break and :math:`L_s` is the length-scale of the slope.
                 
                 The forcing is through heat loss over the shelf, shown in
                 :numref:`Q-plume-on-slope` and takes the form of a fixed flux with
                 profile:
                 
                 .. math:: Q(x) = Q_o ( 1 + \tanh{\left(\frac{x - x_q}{L_q}\right)} ) / 2
                 
                 where
                 
                 .. math::
                 
                    \begin{aligned}
0bad585a21 Navi*    Q_o & = 200 \;\; \mbox{(W m$^{-2}$)} \\
                    x_q & = 2500 + Lx/2 \;\; \mbox{(m)} \\
                    L_q & = 100 \;\; \mbox{(m)}\end{aligned}
d67096e55c Jeff* 
                 Here, :math:`Q_o` is the maximum heat flux, :math:`x_q` is the
                 position of the cut-off, and :math:`L_q` is the width of the cut-off.
                 
                 The initial temperature field is unstratified but with random
                 perturbations, to induce convection early on in the run. The random
                 perturbation are calculated in computational space and because of the
                 variable resolution introduce some spatial correlations, but this does
                 not matter for this experiment. The perturbations have range
                 :math:`0-0.01` :math:`^{\circ}\mathrm{K}`.
                 
                 Code configuration
                 ------------------
                 
                 The computational domain (number of gridpoints) is specified in
                 :filelink:`code/SIZE.h <verification/tutorial_plume_on_slope/code/SIZE.h>`
                 and is configured as a single tile of dimensions
                 :math:`320\times1\times60`.
                 
                 To compile the model code for this experiment, the non-hydrostatic
                 algorithm needs to be enabled, and the open-boundaries package (:filelink:`pkg/obcs`) is required:
                 
                 -  Non-hydrostatic terms and algorithm are enabled with ``#define``
                    :varlink:`ALLOW_NONHYDROSTATIC` in :filelink:`code/CPP_OPTIONS.h <verification/tutorial_plume_on_slope/code/CPP_OPTIONS.h>`
                    and activated with
                    :varlink:`nonHydrostatic` ``=.TRUE,`` in namelist ``PARM01``
                    of :filelink:`input/data <verification/tutorial_plume_on_slope/input/data>`.
                 
                 -  Open boundaries are enabled by adding line ``obcs`` to package
                    configuration file :filelink:`code/packages.conf <verification/tutorial_plume_on_slope/code/packages.conf>`
                    and activated via
                    :varlink:`useOBCS` ``=.TRUE,`` in namelist ``PACKAGES``
                    of :filelink:`input/data.pkg <verification/tutorial_plume_on_slope/input/data.pkg>`.
                 
                 .. _tut_plume_on_slope_mod_parms:
                 
                 Model parameters
                 ----------------
                 
                 .. table:: Model parameters used in the gravity plume experiment.
                   :name: tut_plume_parm_table
                 
                   +-----------------------+---------------------------------------------------------+----------------------------------------------+
                   | **Parameter**         | **Value**                                               | **Description**                              |
                   +=======================+=========================================================+==============================================+
                   | :math:`g`             | 9.81  m s\ :sup:`-2`                                    | acceleration due to gravity                  |
                   +-----------------------+---------------------------------------------------------+----------------------------------------------+
                   | :math:`\rho_o`        | 999.8 kg m\ :sup:`-3`                                   | reference density                            |
                   +-----------------------+---------------------------------------------------------+----------------------------------------------+
                   |  :math:`\alpha`       | 2 :math:`\times` 10\ :sup:`-4` K\ :sup:`-1`             | expansion coefficient                        |
                   +-----------------------+---------------------------------------------------------+----------------------------------------------+
                   | :math:`A_h`           | 1 :math:`\times` 10\ :sup:`-2` m\ :sup:`2` s\ :sup:`-1` | horizontal viscosity                         |
                   +-----------------------+---------------------------------------------------------+----------------------------------------------+
                   | :math:`A_v`           | 1 :math:`\times` 10\ :sup:`-3` m\ :sup:`2` s\ :sup:`-1` | vertical viscosity                           |
                   +-----------------------+---------------------------------------------------------+----------------------------------------------+
                   | :math:`\kappa_h`      | 0 m\ :sup:`2` s\ :sup:`-1`                              | (explicit) horizontal diffusion              |
                   +-----------------------+---------------------------------------------------------+----------------------------------------------+
                   | :math:`\kappa_v`      | 0 m\ :sup:`2` s\ :sup:`-1`                              | (explicit) vertical diffusion                |
                   +-----------------------+---------------------------------------------------------+----------------------------------------------+
                   | :math:`\Delta t`      | 20 s                                                    | time step                                    |
                   +-----------------------+---------------------------------------------------------+----------------------------------------------+
                   | :math:`\Delta z`      | 3.33333 m                                               | vertical grid spacing                        |
                   +-----------------------+---------------------------------------------------------+----------------------------------------------+
                   | :math:`\Delta x`      |  13.3333 - 39.5 m                                       | horizontal grid spacing                      |
                   +-----------------------+---------------------------------------------------------+----------------------------------------------+
                 
                 The model parameters (:numref:`tut_plume_parm_table`) are specified in
                 :filelink:`input/data <verification/tutorial_plume_on_slope/input/data>`
                 and if not assume the default values as defined in :numref:`customize_model`.
                 A linear equation of state is used,
                 :varlink:`eosType` ``=’LINEAR’``, but only temperature is active, :varlink:`sBeta` ``=0.E-11``.
                 For the given heat flux, :math:`Q_o`, the buoyancy forcing is
                 :math:`B_o = \frac{g \alpha Q}{\rho_o c_p} \sim
                 10^{-7}` m\ :sup:`2` s\ :sup:`-3`. Using :math:`R=10^3` m, the
                 shelf width, this gives a velocity scale of
                 :math:`U\sim 5 \times 10^{-2}` m s\ :sup:`-1` for the initial front but
                 will accelerate by an order of magnitude over the slope. The temperature
                 anomaly will be of order :math:`\Delta \theta \sim 3
                 \times 10^{-2}` K. The viscosity is constant and gives a Reynolds number
                 of :math:`100`, using :math:`h=20` m for the initial front and will be
                 an order magnitude bigger over the slope. There is no explicit diffusion
                 but a non-linear advection scheme is used for temperature which adds
                 enough diffusion so as to keep the model stable. The time-step is set to
                 :math:`20` s and gives Courant number order one when the flow reaches
                 the bottom of the slope.
                 

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