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f67abf1ee3 Jeff* .. _atmos_appendix:
                 
                 Hydrostatic Primitive Equations for the Atmosphere in Pressure Coordinates
                 --------------------------------------------------------------------------
                 
                 The hydrostatic primitive equations (**HPE**’s) in :math:`p-`\coordinates are:
                 
                 .. math::
0bad585a21 Navi*    \frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\boldsymbol{k}}\times \vec{\mathbf{v}}_{h}+ \nabla _{p}\phi = \vec{\boldsymbol{\mathcal{F}}}
f67abf1ee3 Jeff*    :label: atmos-mom
                  
                 .. math::
                    \frac{\partial \phi }{\partial p}+\alpha = 0
                    :label: eq-p-hydro-start
                 
                 .. math::
0bad585a21 Navi*     \nabla _{p}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \omega }{\partial p} = 0
f67abf1ee3 Jeff*    :label: atmos-cont
                 
                 .. math::
                    p\alpha = RT  
                    :label: atmos-eos
                 
                 .. math::
                    c_{v}\frac{DT}{Dt}+p\frac{D\alpha }{Dt} = \mathcal{Q}
                    :label: atmos-heat
                 
                 where :math:`\vec{\mathbf{v}}_{h}=(u,v,0)` is the ‘horizontal’ (on pressure surfaces) component of velocity,
0bad585a21 Navi* :math:`\frac{D}{Dt}=\frac{\partial}{\partial t}+\vec{\mathbf{v}}_{h}\cdot  \nabla _{p}+\omega \frac{\partial }{\partial p}`
f67abf1ee3 Jeff* is the total derivative, :math:`f=2\Omega \sin \varphi` is the Coriolis
                 parameter, :math:`\phi =gz` is the geopotential, :math:`\alpha =1/\rho`
                 is the specific volume, :math:`\omega =\frac{Dp }{Dt}` is the vertical
                 velocity in the :math:`p-`\ coordinate. Equation :eq:`atmos-heat` is the
                 first law of thermodynamics where internal energy :math:`e=c_{v}T`,
                 :math:`T` is temperature, :math:`Q` is the rate of heating per unit mass
                 and :math:`p\frac{D\alpha }{Dt}` is the work done by the fluid in
                 compressing.
                 
                 It is convenient to cast the heat equation in terms of potential
                 temperature :math:`\theta` so that it looks more like a generic
                 conservation law. Differentiating :eq:`atmos-eos` we get:
                 
                 .. math:: p\frac{D\alpha }{Dt}+\alpha \frac{Dp}{Dt}=R\frac{DT}{Dt}
                 
                 which, when added to the heat equation :eq:`atmos-heat` and using
                 :math:`c_{p}=c_{v}+R`, gives:
                 
                 .. math::
                    c_{p}\frac{DT}{Dt}-\alpha \frac{Dp}{Dt}=\mathcal{Q}
                    :label: eq-p-heat-interim
                 
                 Potential temperature is defined:
                 
                 .. math:: \theta =T(\frac{p_{c}}{p})^{\kappa }
                    :label: potential-temp
                 
                 where :math:`p_{c}` is a reference pressure and
                 :math:`\kappa =R/c_{p}`. For convenience we will make use of the Exner
                 function :math:`\Pi (p)` which is defined by:
                 
                 .. math:: \Pi (p)=c_{p}(\frac{p}{p_{c}})^{\kappa }
                    :label: Exner
                 
                 The following relations will be useful and are easily expressed in
                 terms of the Exner function:
                 
                 .. math::
                    c_{p}T=\Pi \theta \;\;;\;\;\frac{\partial \Pi }{\partial p}=\frac{\kappa \Pi 
                    }{p}\;\;;\;\;\alpha =\frac{\kappa \Pi \theta }{p}=\frac{\partial \ \Pi }{
                    \partial p}\theta \;\;;\;\;\frac{D\Pi }{Dt}=\frac{\partial \Pi }{\partial p}
                    \frac{Dp}{Dt}
                 
                 where :math:`b=\frac{\partial \ \Pi }{\partial p}\theta` is the buoyancy.
                 
                 The heat equation is obtained by noting that
                 
                 .. math::
                    c_{p}\frac{DT}{Dt}=\frac{D(\Pi \theta )}{Dt}=\Pi \frac{D\theta }{Dt}+\theta 
                    \frac{D\Pi }{Dt}=\Pi \frac{D\theta }{Dt}+\alpha \frac{Dp}{Dt}
                 
                 and on substituting into :eq:`eq-p-heat-interim` gives:
                 
                 .. math::
                    \Pi \frac{D\theta }{Dt}=\mathcal{Q}
                    :label: potential-temperature-equation
                 
                 which is in conservative form.
                 
                 For convenience in the model we prefer to step forward
                 :eq:`potential-temperature-equation` rather than :eq:`atmos-heat`.
                 
                 Boundary conditions
                 ~~~~~~~~~~~~~~~~~~~
                 
                 The upper and lower boundary conditions are:
                 
                 .. math::
                    \begin{aligned}\mbox{at the top:}\;\;p=0 &\text{,  }\omega =\frac{Dp}{Dt}=0\end{aligned}
                    :label: boundary-condition-atmosphere-top
                 
                 .. math::
0bad585a21 Navi*    \begin{aligned}\mbox{at the surface:}\;\;p=p_{s} &\text{,  }\phi =\phi _{\rm topo}=g~Z_{\rm topo}\end{aligned}
f67abf1ee3 Jeff*    :label: boundary-condition-atmosphere-bot
                 
                 In :math:`p-`\coordinates, the upper boundary acts like a solid boundary
                 (:math:`\omega=0` ); in :math:`z-`\coordinates the lower boundary is analogous to a
                 free surface (:math:`\phi` is imposed and :math:`\omega \neq 0`).
                 
                 .. _hpe-p-geo-potential-split:
                 
                 Splitting the geopotential
                 ~~~~~~~~~~~~~~~~~~~~~~~~~~
                 
                 For the purposes of initialization and reducing round-off errors, the
                 model deals with perturbations from reference (or ‘standard’) profiles.
                 For example, the hydrostatic geopotential associated with the resting
                 atmosphere is not dynamically relevant and can therefore be subtracted
                 from the equations. The equations written in terms of perturbations are
                 obtained by substituting the following definitions into the previous
                 model equations:
                 
                 .. math::
                    \theta = \theta _{o}+\theta ^{\prime }
                    :label: atmos-ref-prof-theta 
                 
                 .. math::
                    \alpha = \alpha _{o}+\alpha ^{\prime }
                    :label: atmos-ref-prof-alpha
                 
                 .. math::
                    \phi  = \phi _{o}+\phi ^{\prime }
                    :label: atmos-ref-prof-phi
                 
                 The reference state (indicated by subscript ‘*o*’) corresponds to
                 horizontally homogeneous atmosphere at rest
                 (:math:`\theta _{o},\alpha _{o},\phi_{o}`) with surface pressure :math:`p_{o}(x,y)` that satisfies
0bad585a21 Navi* :math:`\phi_{o}(p_{o})=g~Z_{\rm topo}`, defined:
f67abf1ee3 Jeff* 
                 .. math:: \theta _{o}(p) = f^{n}(p) \\
                 .. math:: \alpha _{o}(p)  = \Pi _{p}\theta _{o} \\
0bad585a21 Navi* .. math:: \phi _{o}(p)  = \phi _{\rm topo}-\int_{p_{0}}^{p}\alpha _{o}dp
f67abf1ee3 Jeff* 
                 The final form of the **HPE**’s in :math:`p-`\coordinates is then:
                 
                 .. math::
0bad585a21 Navi*    \frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\boldsymbol{k}}\times \vec{\mathbf{v}}
                    _{h}+ \nabla _{p}\phi ^{\prime } = \vec{\boldsymbol{\mathcal{F}}} 
f67abf1ee3 Jeff*    :label: atmos-prime
                 
                 .. math::
                    \frac{\partial \phi ^{\prime }}{\partial p}+\alpha ^{\prime }  = 0
                    :label: atmos-prime2
                  
                 .. math::
0bad585a21 Navi*     \nabla _{p}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \omega }{
f67abf1ee3 Jeff*    \partial p} = 0
                    :label: atmos-prime3
                  
                 .. math::
                    \frac{\partial \Pi }{\partial p}\theta ^{\prime } = \alpha ^{\prime }
                    :label: atmos-prime4
                 
                 .. math::
                    \frac{D\theta }{Dt} = \frac{\mathcal{Q}}{\Pi }
                    :label: atmos-prime5
                 

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