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% $Header: $ |
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% $Name: $ |
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\section{Appendix OCEAN} |
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|
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\subsection{Equations of motion for the ocean} |
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|
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We review here the method by which the standard (Boussinesq, incompressible) |
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HPE's for the ocean written in z-coordinates are obtained. The |
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non-Boussinesq equations for oceanic motion are: |
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\begin{eqnarray} |
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\frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}% |
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_{h}+\frac{1}{\rho }\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}} |
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\label{eq-zns-hmom} \\ |
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\epsilon _{nh}\frac{Dw}{Dt}+g+\frac{1}{\rho }\frac{\partial p}{\partial z} |
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&=&\epsilon _{nh}\mathcal{F}_{w} \label{eq-zns-hydro} \\ |
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\frac{1}{\rho }\frac{D\rho }{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}% |
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_{h}+\frac{\partial w}{\partial z} &=&0 \label{eq-zns-cont} \\ |
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\rho &=&\rho (\theta ,S,p) \label{eq-zns-eos} \\ |
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\frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta } \label{eq-zns-heat} \\ |
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\frac{DS}{Dt} &=&\mathcal{Q}_{s} \label{eq-zns-salt} |
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\end{eqnarray} |
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These equations permit acoustics modes, inertia-gravity waves, |
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non-hydrostatic motions, a geostrophic (Rossby) mode and a thermo-haline |
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mode. As written, they cannot be integrated forward consistently - if we |
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step $\rho $ forward in (\ref{eq-zns-cont}), the answer will not be |
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consistent with that obtained by stepping (\ref{eq-zns-heat}) and (\ref |
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{eq-zns-salt}) and then using (\ref{eq-zns-eos}) to yield $\rho $. It is |
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therefore necessary to manipulate the system as follows. Differentiating the |
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EOS (equation of state) gives: |
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|
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\begin{equation} |
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\frac{D\rho }{Dt}=\left. \frac{\partial \rho }{\partial \theta }\right| |
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_{S,p}\frac{D\theta }{Dt}+\left. \frac{\partial \rho }{\partial S}\right| |
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_{\theta ,p}\frac{DS}{Dt}+\left. \frac{\partial \rho }{\partial p}\right| |
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_{\theta ,S}\frac{Dp}{Dt} \label{EOSexpansion} |
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\end{equation} |
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|
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Note that $\frac{\partial \rho }{\partial p}=\frac{1}{c_{s}^{2}}$ is the |
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reciprocal of the sound speed ($c_{s}$) squared. Substituting into \ref |
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{eq-zns-cont} gives: |
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\begin{equation} |
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\frac{1}{\rho c_{s}^{2}}\frac{Dp}{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{% |
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v}}+\partial _{z}w\approx 0 \label{eq-zns-pressure} |
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\end{equation} |
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where we have used an approximation sign to indicate that we have assumed |
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adiabatic motion, dropping the $\frac{D\theta }{Dt}$ and $\frac{DS}{Dt}$. |
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Replacing \ref{eq-zns-cont} with \ref{eq-zns-pressure} yields a system that |
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can be explicitly integrated forward: |
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\begin{eqnarray} |
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\frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}% |
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_{h}+\frac{1}{\rho }\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}} |
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\label{eq-cns-hmom} \\ |
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\epsilon _{nh}\frac{Dw}{Dt}+g+\frac{1}{\rho }\frac{\partial p}{\partial z} |
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&=&\epsilon _{nh}\mathcal{F}_{w} \label{eq-cns-hydro} \\ |
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\frac{1}{\rho c_{s}^{2}}\frac{Dp}{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{% |
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v}}_{h}+\frac{\partial w}{\partial z} &=&0 \label{eq-cns-cont} \\ |
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\rho &=&\rho (\theta ,S,p) \label{eq-cns-eos} \\ |
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\frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta } \label{eq-cns-heat} \\ |
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\frac{DS}{Dt} &=&\mathcal{Q}_{s} \label{eq-cns-salt} |
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\end{eqnarray} |
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|
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\subsubsection{Compressible z-coordinate equations} |
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|
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Here we linearize the acoustic modes by replacing $\rho $ with $\rho _{o}(z)$ |
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wherever it appears in a product (ie. non-linear term) - this is the |
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`Boussinesq assumption'. The only term that then retains the full variation |
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in $\rho $ is the gravitational acceleration: |
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\begin{eqnarray} |
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\frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}% |
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_{h}+\frac{1}{\rho _{o}}\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}} |
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\label{eq-zcb-hmom} \\ |
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\epsilon _{nh}\frac{Dw}{Dt}+\frac{g\rho }{\rho _{o}}+\frac{1}{\rho _{o}}% |
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\frac{\partial p}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w} |
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\label{eq-zcb-hydro} \\ |
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\frac{1}{\rho _{o}c_{s}^{2}}\frac{Dp}{Dt}+\mathbf{\nabla }_{z}\cdot \vec{% |
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\mathbf{v}}_{h}+\frac{\partial w}{\partial z} &=&0 \label{eq-zcb-cont} \\ |
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\rho &=&\rho (\theta ,S,p) \label{eq-zcb-eos} \\ |
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\frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta } \label{eq-zcb-heat} \\ |
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\frac{DS}{Dt} &=&\mathcal{Q}_{s} \label{eq-zcb-salt} |
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\end{eqnarray} |
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These equations still retain acoustic modes. But, because the |
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``compressible'' terms are linearized, the pressure equation \ref |
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{eq-zcb-cont} can be integrated implicitly with ease (the time-dependent |
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term appears as a Helmholtz term in the non-hydrostatic pressure equation). |
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These are the \emph{truly} compressible Boussinesq equations. Note that the |
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EOS must have the same pressure dependency as the linearized pressure term, |
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ie. $\left. \frac{\partial \rho }{\partial p}\right| _{\theta ,S}=\frac{1}{% |
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c_{s}^{2}}$, for consistency. |
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|
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\subsubsection{`Anelastic' z-coordinate equations} |
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|
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The anelastic approximation filters the acoustic mode by removing the |
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time-dependency in the continuity (now pressure-) equation (\ref{eq-zcb-cont}% |
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). This could be done simply by noting that $\frac{Dp}{Dt}\approx -g\rho _{o}% |
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\frac{Dz}{Dt}=-g\rho _{o}w$, but this leads to an inconsistency between |
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continuity and EOS. A better solution is to change the dependency on |
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pressure in the EOS by splitting the pressure into a reference function of |
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height and a perturbation: |
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\[ |
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\rho =\rho (\theta ,S,p_{o}(z)+\epsilon _{s}p^{\prime }) |
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\] |
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Remembering that the term $\frac{Dp}{Dt}$ in continuity comes from |
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differentiating the EOS, the continuity equation then becomes: |
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\[ |
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\frac{1}{\rho _{o}c_{s}^{2}}\left( \frac{Dp_{o}}{Dt}+\epsilon _{s}\frac{% |
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Dp^{\prime }}{Dt}\right) +\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+% |
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\frac{\partial w}{\partial z}=0 |
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\] |
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If the time- and space-scales of the motions of interest are longer than |
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those of acoustic modes, then $\frac{Dp^{\prime }}{Dt}<<(\frac{Dp_{o}}{Dt},% |
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\mathbf{\nabla }\cdot \vec{\mathbf{v}}_{h})$ in the continuity equations and |
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$\left. \frac{\partial \rho }{\partial p}\right| _{\theta ,S}\frac{% |
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Dp^{\prime }}{Dt}<<\left. \frac{\partial \rho }{\partial p}\right| _{\theta |
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,S}\frac{Dp_{o}}{Dt}$ in the EOS (\ref{EOSexpansion}). Thus we set $\epsilon |
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_{s}=0$, removing the dependency on $p^{\prime }$ in the continuity equation |
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and EOS. Expanding $\frac{Dp_{o}(z)}{Dt}=-g\rho _{o}w$ then leads to the |
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anelastic continuity equation: |
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\begin{equation} |
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\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial w}{\partial z}-% |
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\frac{g}{c_{s}^{2}}w=0 \label{eq-za-cont1} |
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\end{equation} |
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A slightly different route leads to the quasi-Boussinesq continuity equation |
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where we use the scaling $\frac{\partial \rho ^{\prime }}{\partial t}+% |
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\mathbf{\nabla }_{3}\cdot \rho ^{\prime }\vec{\mathbf{v}}<<\mathbf{\nabla }% |
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_{3}\cdot \rho _{o}\vec{\mathbf{v}}$ yielding: |
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\begin{equation} |
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\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{1}{\rho _{o}}\frac{% |
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\partial \left( \rho _{o}w\right) }{\partial z}=0 \label{eq-za-cont2} |
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\end{equation} |
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Equations \ref{eq-za-cont1} and \ref{eq-za-cont2} are in fact the same |
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equation if: |
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\begin{equation} |
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\frac{1}{\rho _{o}}\frac{\partial \rho _{o}}{\partial z}=\frac{-g}{c_{s}^{2}} |
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\end{equation} |
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Again, note that if $\rho _{o}$ is evaluated from prescribed $\theta _{o}$ |
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and $S_{o}$ profiles, then the EOS dependency on $p_{o}$ and the term $\frac{% |
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g}{c_{s}^{2}}$ in continuity should be referred to those same profiles. The |
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full set of `quasi-Boussinesq' or `anelastic' equations for the ocean are |
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then: |
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\begin{eqnarray} |
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\frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}% |
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_{h}+\frac{1}{\rho _{o}}\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}} |
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\label{eq-zab-hmom} \\ |
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\epsilon _{nh}\frac{Dw}{Dt}+\frac{g\rho }{\rho _{o}}+\frac{1}{\rho _{o}}% |
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\frac{\partial p}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w} |
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\label{eq-zab-hydro} \\ |
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\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{1}{\rho _{o}}\frac{% |
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\partial \left( \rho _{o}w\right) }{\partial z} &=&0 \label{eq-zab-cont} \\ |
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\rho &=&\rho (\theta ,S,p_{o}(z)) \label{eq-zab-eos} \\ |
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\frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta } \label{eq-zab-heat} \\ |
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\frac{DS}{Dt} &=&\mathcal{Q}_{s} \label{eq-zab-salt} |
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\end{eqnarray} |
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|
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\subsubsection{Incompressible z-coordinate equations} |
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|
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Here, the objective is to drop the depth dependence of $\rho _{o}$ and so, |
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technically, to also remove the dependence of $\rho $ on $p_{o}$. This would |
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yield the ``truly'' incompressible Boussinesq equations: |
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\begin{eqnarray} |
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\frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}% |
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_{h}+\frac{1}{\rho _{c}}\mathbf{\nabla }_{z}p &=&\vec{\mathbf{\mathcal{F}}} |
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\label{eq-ztb-hmom} \\ |
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\epsilon _{nh}\frac{Dw}{Dt}+\frac{g\rho }{\rho _{c}}+\frac{1}{\rho _{c}}% |
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\frac{\partial p}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w} |
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\label{eq-ztb-hydro} \\ |
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\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial w}{\partial z} |
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&=&0 \label{eq-ztb-cont} \\ |
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\rho &=&\rho (\theta ,S) \label{eq-ztb-eos} \\ |
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\frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta } \label{eq-ztb-heat} \\ |
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\frac{DS}{Dt} &=&\mathcal{Q}_{s} \label{eq-ztb-salt} |
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\end{eqnarray} |
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where $\rho _{c}$ is a constant reference density of water. |
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|
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\subsubsection{Compressible non-divergent equations} |
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|
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The above ``incompressible'' equations are incompressible in both the flow |
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and the density. In many oceanic applications, however, it is important to |
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retain compressibility effects in the density. To do this we must split the |
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density thus: |
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\[ |
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\rho =\rho _{o}+\rho ^{\prime } |
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\] |
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We then assert that variations with depth of $\rho _{o}$ are unimportant |
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while the compressible effects in $\rho ^{\prime }$ are: |
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\[ |
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\rho _{o}=\rho _{c} |
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\] |
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\[ |
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\rho ^{\prime }=\rho (\theta ,S,p_{o}(z))-\rho _{o} |
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\] |
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This then yields what we can call the semi-compressible Boussinesq |
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equations: |
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\begin{eqnarray} |
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\frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}}% |
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_{h}+\frac{1}{\rho _{c}}\mathbf{\nabla }_{z}p^{\prime } &=&\vec{\mathbf{% |
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\mathcal{F}}} \label{eq-zpe-hmom} \\ |
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\epsilon _{nh}\frac{Dw}{Dt}+\frac{g\rho ^{\prime }}{\rho _{c}}+\frac{1}{\rho |
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_{c}}\frac{\partial p^{\prime }}{\partial z} &=&\epsilon _{nh}\mathcal{F}_{w} |
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\label{eq-zpe-hydro} \\ |
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\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial w}{\partial z} |
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&=&0 \label{eq-zpe-cont} \\ |
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\rho ^{\prime } &=&\rho (\theta ,S,p_{o}(z))-\rho _{c} \label{eq-zpe-eos} \\ |
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\frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta } \label{eq-zpe-heat} \\ |
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\frac{DS}{Dt} &=&\mathcal{Q}_{s} \label{eq-zpe-salt} |
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\end{eqnarray} |
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Note that the hydrostatic pressure of the resting fluid, including that |
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associated with $\rho _{c}$, is subtracted out since it has no effect on the |
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dynamics. |
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|
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Though necessary, the assumptions that go into these equations are messy |
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since we essentially assume a different EOS for the reference density and |
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the perturbation density. Nevertheless, it is the hydrostatic ($\epsilon |
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_{nh}=0$ form of these equations that are used throughout the ocean modeling |
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community and referred to as the primitive equations (HPE). |
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