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% $Header: /u/u0/gcmpack/mitgcmdoc/part1/continuous_eqns.tex,v 1.4 2001/09/27 01:57:17 cnh Exp $ |
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% $Name: $ |
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\section{Continuous equations in `r' coordinates} |
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To render atmosphere and ocean models from one dynamical core we exploit |
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`isomorphisms' between equation sets that govern the evolution of the |
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respective fluids - see fig.4 |
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\marginpar{ |
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Fig.4. Isomorphisms}. One system of hydrodynamical equations is written down |
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and encoded. The model variables have different interpretations depending on |
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whether the atmosphere or ocean is being studied. Thus, for example, the |
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vertical coordinate `$r$' is interpreted as pressure, $p$, if we are |
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modeling the atmosphere and height, $z$, if we are modeling the ocean. A |
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complete list of the isomorphisms is given in table 1. |
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\marginpar{ |
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Table 1. Isomorphisms} |
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The state of the fluid at any time is characterized by the distribution of |
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velocity $\vec{\mathbf{v}}$, active tracers $\theta $ and $S$, a |
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`geopotential' $\phi $ and density $\rho =\rho (\theta ,S,p)$ which may |
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depend on $\theta $, $S$, and $p$. The equations that govern the evolution |
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of these fields, obtained by applying the laws of classical mechanics and |
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thermodynamics to a Boussinesq, Navier-Stokes fluid are, written in terms of |
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a generic vertical coordinate, $r$, see fig.5 |
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\marginpar{ |
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Fig.5 The vertical coordinate of model}: |
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\[ |
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\frac{D\vec{\mathbf{v}_{h}}}{Dt}+\left( 2\vec{\Omega}\times \vec{\mathbf{v}} |
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\right) _{h}+\mathbf{\nabla }_{h}\phi =\left( \mathcal{F}_{\vec{\mathbf{v}}} |
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\mathcal{+D}_{\vec{\mathbf{v}}}\right) _{h}\text{horizontal mtm} |
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\] |
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\[ |
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\frac{D\dot{r}}{Dt}+\widehat{k}\cdot \left( 2\vec{\Omega}\times \vec{\mathbf{ |
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v}}\right) +\frac{\partial \phi }{\partial r}+b=\left( \mathcal{F}_{\vec{ |
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\mathbf{v}}}\mathcal{+D}_{\vec{\mathbf{v}}}\right) _{r}\text{vertical mtm} |
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\] |
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\begin{equation} |
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\mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \dot{r}}{ |
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\partial r}=0\text{ continuity} \label{incompressible} |
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\end{equation} |
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\[ |
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b=b(\theta ,S,r)\text{ equation of state} |
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\] |
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\[ |
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\frac{D\theta }{Dt}=\mathcal{F}_{\theta }\text{ }\mathcal{+D}_{\theta }\text{ |
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potential temperature} |
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\] |
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\[ |
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\frac{DS}{Dt}=\mathcal{F}_{S}\text{ }\mathcal{+D}_{S}\text{ humidity/salinity |
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} |
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\] |
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Here: |
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\[ |
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r\text{ is the vertical coordinate} |
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\] |
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\[ |
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\frac{D}{Dt}=\frac{\partial }{\partial t}+\vec{\mathbf{v}}\cdot \nabla \text{ |
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is the total derivative} |
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\] |
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\[ |
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\mathbf{\nabla }=\mathbf{\nabla }_{h}+\widehat{k}\frac{\partial }{\partial r} |
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\text{ is the `grad' operator} |
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\] |
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with $\mathbf{\nabla }_{h}$ operating in the horizontal and $\widehat{k} |
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\frac{\partial }{\partial r}$ operating in the vertical, where $\widehat{k}$ |
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is a unit vector in the vertical |
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\[ |
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t\text{ is time} |
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\] |
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\[ |
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\vec{\mathbf{v}}=(u,v,\dot{r})=(\vec{\mathbf{v}}_{h},\dot{r})\text{ is the |
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velocity} |
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\] |
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\[ |
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\phi \text{ is the `pressure'/`geopotential'} |
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\] |
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\[ |
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\vec{\Omega}\text{ is the Earth's rotation} |
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\] |
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\[ |
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b\text{ is the `buoyancy'} |
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\] |
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\[ |
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\theta \text{ is potential temperature} |
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\] |
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\[ |
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S\text{ is specific humidity in the atmosphere; salinity in the ocean} |
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\] |
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\[ |
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\mathcal{F}_{\vec{\mathbf{v}}}\text{ and }\mathcal{D}_{\vec{\mathbf{v}}} |
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\text{ are forcing and dissipation of }\vec{\mathbf{v}} |
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\] |
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\[ |
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\mathcal{F}_{\theta }\mathcal{\ }\text{and }\mathcal{D}_{\theta }\text{ are |
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forcing and dissipation of }\theta |
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\] |
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\[ |
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\mathcal{F}_{S}\mathcal{\ }\text{and }\mathcal{D}_{S}\text{ are forcing and |
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dissipation of }S |
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\] |
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The $\mathcal{F}^{\prime }s$ and $\mathcal{D}^{\prime }s$ are provided by |
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extensive `physics' packages for atmosphere and ocean described in section |
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?.?. |
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\subsection{Kinematic Boundary conditions} |
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\subsubsection{vertical} |
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at fixed and moving $r$ surfaces we set (see fig.4): |
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\begin{eqnarray*} |
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\dot{r} &=&0\text{ at }r=R_{fixed}(x,y):\text{(ocean bottom, top of the |
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atmosphere)} \\ |
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\dot{r} &=&\frac{Dr}{Dt}\text{ at }r=R_{moving}\text{ (ocean surface, bottom |
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of the atmosphere)} |
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\end{eqnarray*} |
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Here |
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\[ |
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R_{moving}=R_{o}+\eta |
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\] |
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where $R_{o}(x,y)$ is the `$r-$value' (height or pressure, depending on |
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whether we are in the atmosphere or ocean) of the `moving surface' in the |
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resting fluid and $\eta $ is the departure from $R_{o}(x,y)$ in the presence |
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of motion. |
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\subsubsection{horizontal} |
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\[ |
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\vec{\mathbf{v}}\cdot \vec{\mathbf{n}}=0 |
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\] |
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where $\vec{\mathbf{n}}$ is the normal to a solid boundary. |
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\subsection{Atmosphere} |
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In the atmosphere, see fig. we interpret: |
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\begin{eqnarray} |
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r &=&p\text{ is the pressure} \\ |
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\dot{r} &=&\frac{Dp}{Dt}=\omega \text{ is the vertical velocity in }p\text{ |
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coordinates} \\ |
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\phi &=&g\,z\text{ is the geopotential height} \\ |
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b &=&\frac{\partial \Pi }{\partial p}\theta \text{ is the buoyancy} \\ |
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\theta &=&T(\frac{p_{c}}{p})^{\kappa }\text{ is potential temperature} \\ |
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S &=&q\text{, the specific humidity} |
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\end{eqnarray} |
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where |
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\[ |
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T\text{is absolute temperature} |
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\] |
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\[ |
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p\text{ is the pressure} |
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\] |
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\begin{eqnarray*} |
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&&z\text{ is the height of the pressure surface} \\ |
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&&g\text{ is the acceleration due to gravity} |
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\end{eqnarray*} |
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In the above the ideal gas law, $p=\rho RT$, has been expressed in terms of |
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the Exner function $\Pi (p)$ given by (see Appendix Atmosphere) |
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\[ |
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\Pi (p)=c_{p}(\frac{p}{p_{c}})^{\kappa } |
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\] |
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where $p_{c}$ is a reference pressure and $\kappa =R/c_{p}$ with $R$ the gas |
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constant and $c_{p}$ the specific heat of air at constant pressure. |
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At the top of the atmosphere (which is `fixed' in our $r$ coordinate): |
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\[ |
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R_{fixed}=p_{top}=0 |
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\] |
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In a resting atmosphere the elevation of the mountains at the bottom is |
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given by |
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\[ |
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R_{moving}=R_{o}(x,y)=p_{o}(x,y) |
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\] |
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i.e. the (hydrostatic) pressure at the top of the mountains in a resting |
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atmosphere. |
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The boundary conditions at top and bottom are given by: |
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\begin{eqnarray} |
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&&\omega =0~\text{at }r=R_{fixed} \label{eq:fixed-bc-atmos} |
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\text{ (top of the atmosphere)} \\ |
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\omega &=&\frac{Dp_{s}}{Dt}\text{; at }r=R_{moving}\text{ (bottom of the |
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atmosphere)} |
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\label{eq:moving-bc-atmos} |
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\end{eqnarray} |
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|
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Then the (hydrostatic form of) eq(\ref{incompressible}) yields a consistent |
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set of atmospheric equations which, for convenience, are written out in $p$ |
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coordinates in Appendix Atmosphere - see eqs(\ref{eq-p-hmom}) to (\ref |
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{eq-p-heat}). |
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\subsection{Ocean} |
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In the ocean we interpret: |
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\begin{eqnarray} |
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r &=&z\text{ is the height} |
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\label{eq:ocean-z}\\ |
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\dot{r} &=&\frac{Dz}{Dt}=w\text{ is the vertical velocity} |
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\label{eq:ocean-w}\\ |
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\phi &=&\frac{p}{\rho _{c}}\text{ is the pressure} |
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\label{eq:ocean-p}\\ |
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b(\theta ,S,r) &=&\frac{g}{\rho _{c}}\left( \rho (\theta ,S,r)-\rho |
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_{c}\right) \text{ is the buoyancy} |
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\label{eq:ocean-b} |
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\end{eqnarray} |
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where $\rho _{c}$ is a fixed reference density of water and $g$ is the |
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acceleration due to gravity.\noindent |
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In the above |
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At the bottom of the ocean: $R_{fixed}(x,y)=-H(x,y)$. |
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The surface of the ocean is given by: $R_{moving}=\eta $ |
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The position of the resting free surface of the ocean is given by $ |
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R_{o}=Z_{o}=0$. |
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Boundary conditions are: |
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\begin{eqnarray} |
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w &=&0~\text{at }r=R_{fixed}\text{ (ocean bottom)} |
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\label{eq:fixed-bc-ocean}\\ |
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w &=&\frac{D\eta }{Dt}\text{ at }r=R_{moving}=\eta \text{ (ocean surface) |
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\label{eq:moving-bc-ocean}} |
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\end{eqnarray} |
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where $\eta $ is the elevation of the free surface. |
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Then eq(\ref{incompressible}) yields a consistent set of oceanic equations |
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which, for convenience, are written out in $z$ coordinates in Appendix Ocean. |
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\subsection{Hydrostatic, Quasi-hydrostatic, Quasi-nonhydrostatic and |
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Non-hydrostatic forms} |
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Let us separate $\phi $ in to surface, hydrostatic and non-hydrostatic terms: |
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\begin{equation} |
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\phi (x,y,r)=\phi _{s}(x,y)+\phi _{hyd}(x,y,r)+\phi _{nh}(x,y,r) |
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\label{eq:phi-split} |
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\end{equation} |
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and write eq(\ref{incompressible}a) in the form: |
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\begin{equation} |
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\frac{\partial \vec{\mathbf{v}_{h}}}{\partial t}+\mathbf{\nabla }_{h}\phi |
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_{s}+\mathbf{\nabla }_{h}\phi _{hyd}+\epsilon _{nh}\mathbf{\nabla }_{h}\phi |
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_{nh}=\vec{\mathbf{G}}_{\vec{v}_{h}} \label{eq:mom-h} |
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\end{equation} |
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\begin{equation} |
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\frac{\partial \phi _{hyd}}{\partial r}=-b \label{eq:hydrostatic} |
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\end{equation} |
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\begin{equation} |
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\epsilon _{nh}\frac{\partial \dot{r}}{\partial t}+\frac{\partial \phi _{nh}}{ |
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\partial r}=G_{\dot{r}} \label{eq:mom-w} |
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\end{equation} |
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Here $\epsilon _{nh}$ is a non-hydrostatic parameter. |
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The $\left( \vec{\mathbf{G}}_{\vec{v}},G_{\dot{r}}\right) $ in eq(\ref |
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{hor-mtm}) and (\ref{vertmtm}) represent advective, metric and Coriolis |
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terms in the momentum equations. In spherical coordinates they take the form |
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\footnote{ |
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In the hydrostatic primitive equations (\textbf{HPE}) all underlined terms |
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in (\ref{Gu}), (\ref{Gv}) and (\ref{Gw}) are omitted; the singly-underlined |
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terms are included in the quasi-hydrostatic model (\textbf{QH}). The fully |
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non-hydrostatic model (\textbf{NH}) includes all terms.}: |
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292 |
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\begin{equation} |
293 |
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\left. |
294 |
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\begin{tabular}{l} |
295 |
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$G_{u}=-\vec{\mathbf{v}}.\nabla u$ \\ |
296 |
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1.4 |
$-\left\{ \underline{\frac{u\dot{r}}{{r}}}-\frac{uv\tan lat}{{r}} |
297 |
adcroft |
1.1 |
\right\} $ \\ |
298 |
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1.4 |
$-\left\{ -2\Omega v\sin lat+\underline{\underline{2\Omega \dot{r}\cos lat}} |
299 |
adcroft |
1.1 |
\right\} $ \\ |
300 |
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$+\mathcal{F}_{u}\mathcal{+D}_{u}$ |
301 |
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1.1 |
\end{tabular} |
302 |
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\right\} \left\{ |
303 |
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\begin{tabular}{l} |
304 |
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\textit{advection} \\ |
305 |
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\textit{metric} \\ |
306 |
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\textit{Coriolis} \\ |
307 |
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\textit{\ Forcing/Dissipation} |
308 |
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\end{tabular} |
309 |
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1.2 |
\right. \qquad \label{eq:gu-speherical} |
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1.1 |
\end{equation} |
311 |
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312 |
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\begin{equation} |
313 |
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\left. |
314 |
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\begin{tabular}{l} |
315 |
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$G_{v}=-\vec{\mathbf{v}}.\nabla v$ \\ |
316 |
cnh |
1.4 |
$-\left\{ \underline{\frac{v\dot{r}}{{r}}}-\frac{u^{2}\tan lat}{{r} |
317 |
adcroft |
1.1 |
}\right\} $ \\ |
318 |
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$-\left\{ -2\Omega u\sin lat\right\} $ \\ |
319 |
cnh |
1.4 |
$+\mathcal{F}_{v}\mathcal{+D}_{v}$ |
320 |
adcroft |
1.1 |
\end{tabular} |
321 |
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\right\} \left\{ |
322 |
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\begin{tabular}{l} |
323 |
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\textit{advection} \\ |
324 |
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\textit{metric} \\ |
325 |
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\textit{Coriolis} \\ |
326 |
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\textit{\ Forcing/Dissipation} |
327 |
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\end{tabular} |
328 |
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1.2 |
\right. \qquad \label{eq:gv-spherical} |
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adcroft |
1.1 |
\end{equation} |
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\qquad \qquad \qquad \qquad \qquad |
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|
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\begin{equation} |
333 |
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\left. |
334 |
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\begin{tabular}{l} |
335 |
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$G_{\dot{r}}=-\vec{\mathbf{v}}.\nabla \dot{r}$ \\ |
336 |
cnh |
1.4 |
$+\left\{ \frac{u^{_{^{2}}}+v^{2}}{{{r}}} |
337 |
adcroft |
1.1 |
\right\} $ \\ |
338 |
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${+2\Omega u\cos lat}$ \\ |
339 |
cnh |
1.4 |
$\mathcal{F}_{\dot{r}}\mathcal{+D}_{\dot{r}}$ |
340 |
adcroft |
1.1 |
\end{tabular} |
341 |
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\right\} \left\{ |
342 |
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\begin{tabular}{l} |
343 |
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\textit{advection} \\ |
344 |
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\textit{metric} \\ |
345 |
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\textit{Coriolis} \\ |
346 |
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\textit{\ Forcing/Dissipation} |
347 |
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\end{tabular} |
348 |
cnh |
1.2 |
\right. \label{eq:gw-spherical} |
349 |
adcroft |
1.1 |
\end{equation} |
350 |
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\qquad \qquad \qquad \qquad \qquad |
351 |
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|
352 |
cnh |
1.4 |
In the above `${r}$' is the distance from the center of the earth and `$ |
353 |
adcroft |
1.1 |
lat$' is latitude. |
354 |
|
|
|
355 |
|
|
Grad and div operators in spherical coordinates are defined in appendix |
356 |
cnh |
1.4 |
OPERATORS. |
357 |
adcroft |
1.1 |
\marginpar{ |
358 |
|
|
Fig.6 Spherical polar coordinate system.} |
359 |
|
|
|
360 |
|
|
\subsubsection{Shallow atmosphere approximation} |
361 |
|
|
|
362 |
|
|
............................ |
363 |
|
|
|
364 |
|
|
\subsubsection{Hydrostatic and quasi-hydrostatic forms} |
365 |
|
|
|
366 |
|
|
These are discussed at length in Marshall et al (1997a). |
367 |
|
|
|
368 |
|
|
In the `hydrostatic primitive equations' (\textbf{HPE)} all the underlined |
369 |
cnh |
1.4 |
terms in Eqs. (\ref{Gu} $\rightarrow $\ \ref{Gw}) are neglected and `${r |
370 |
adcroft |
1.1 |
}$' is replaced by `$a$', the mean radius of the earth. Once the pressure is |
371 |
|
|
found at one level - e.g. by inverting a 2-d Elliptic equation for $\phi |
372 |
|
|
_{s} $ at $r=R_{moving}$ - the pressure can be computed at all other levels |
373 |
|
|
by integration of the hydrostatic relation, eq(\ref{hydro}). |
374 |
|
|
|
375 |
|
|
In the `quasi-hydrostatic' equations (\textbf{QH)} strict balance between |
376 |
|
|
gravity and vertical pressure gradients is not imposed. The $2\Omega u\cos |
377 |
|
|
\phi $ Coriolis term are not neglected and are balanced by a non-hydrostatic |
378 |
cnh |
1.4 |
contribution to the pressure field: only the terms underlined twice in Eqs. ( |
379 |
adcroft |
1.1 |
\ref{Gu} $\rightarrow $\ \ref{Gw}) are set to zero and, simultaneously, the |
380 |
|
|
shallow atmosphere approximation is relaxed. In \textbf{QH}\ \textit{all} |
381 |
|
|
the metric terms are retained and the full variation of the radial position |
382 |
|
|
of a particle monitored. The \textbf{QH}\ vertical momentum equation (\ref |
383 |
|
|
{vertmtm}) becomes: |
384 |
|
|
|
385 |
|
|
\[ |
386 |
|
|
\frac{\partial \phi _{nh}}{\partial r}=2\Omega u\cos lat |
387 |
|
|
\] |
388 |
|
|
making a small correction to the hydrostatic pressure. |
389 |
|
|
|
390 |
|
|
\textbf{QH} has good energetic credentials - they are the same as for |
391 |
|
|
\textbf{HPE}. Importantly, however, it has the same angular momentum |
392 |
|
|
principle as the full non-hydrostatic model (\textbf{NH)} - see Marshall |
393 |
|
|
et.al., 1997a. As in \textbf{HPE }only a 2-d elliptic problem need be solved. |
394 |
|
|
|
395 |
|
|
\subsubsection{Non-hydrostatic and quasi-nonhydrostatic forms} |
396 |
|
|
|
397 |
|
|
The MIT model presently supports a full non-hydrostatic ocean isomorph, but |
398 |
|
|
only a quasi-non-hydrostatic atmospheric isomorph. |
399 |
|
|
|
400 |
|
|
\paragraph{Non-hydrostatic Ocean} |
401 |
|
|
|
402 |
cnh |
1.4 |
In the non-hydrostatic ocean model all terms in equations (\ref{Gu} $ |
403 |
adcroft |
1.1 |
\rightarrow $\ \ref{Gw}) are retained. A three dimensional elliptic equation |
404 |
|
|
must be solved subject to Neumann boundary conditions (see below). It is |
405 |
|
|
important to note that use of the full \textbf{NH} does not admit any new |
406 |
|
|
`fast' waves in to the system - the incompressible condition (\ref |
407 |
|
|
{incompressible}) has already filtered out acoustic modes. It does, however, |
408 |
|
|
ensure that the gravity waves are treated accurately with an exact |
409 |
|
|
dispersion relation. The \textbf{NH} set has a complete angular momentum |
410 |
|
|
principle and consistent energetics - see White and Bromley, 1995; Marshall |
411 |
|
|
et.al.\ 1997a. |
412 |
|
|
|
413 |
|
|
\paragraph{Quasi-nonhydrostatic Atmosphere} |
414 |
|
|
|
415 |
cnh |
1.4 |
In the non-hydrostatic version of our atmospheric model we approximate $\dot{ |
416 |
adcroft |
1.1 |
r}$ in the vertical momentum eqs(\ref{vertmtm}) and (\ref{Gw}) (but only |
417 |
|
|
here) by: |
418 |
|
|
|
419 |
|
|
\begin{equation} |
420 |
cnh |
1.2 |
\dot{r}=\frac{Dp}{Dt}=\frac{Dp_{hyd}}{Dt} \label{eq:quasi-nh-w} |
421 |
adcroft |
1.1 |
\end{equation} |
422 |
|
|
where $p_{hy}$ is the hydrostatic pressure. |
423 |
|
|
|
424 |
|
|
........................................ |
425 |
|
|
|
426 |
|
|
\subsubsection{Summary of equation sets supported by model} |
427 |
|
|
|
428 |
|
|
The key equation sets and isomorphisms are summarised in fig.4. |
429 |
|
|
|
430 |
|
|
\paragraph{Atmosphere} |
431 |
|
|
|
432 |
|
|
\subparagraph{Hydrostatic and quasi-hydrostatic} |
433 |
|
|
|
434 |
|
|
Hydrostatic, and quasi-hydrostatic forms of the compressible non-Boussinesq |
435 |
|
|
equations in $p-$coordinates are supported\ref{eq-p} - see appendix |
436 |
|
|
Atmosphere, where they are written out in $p-$coordinates. |
437 |
|
|
|
438 |
|
|
\subparagraph{Quasi-nonhydrostatic} |
439 |
|
|
|
440 |
|
|
A quasi-nonhydrostatic form is also supported - see appendix Ocean. |
441 |
|
|
|
442 |
|
|
\paragraph{Ocean} |
443 |
|
|
|
444 |
|
|
\subparagraph{Hydrostatic and quasi-hydrostatic} |
445 |
|
|
|
446 |
|
|
Hydrostatic, and quasi-hydrostatic forms of the incompressible Boussinesq |
447 |
|
|
equations in $z-$coordinates are supported |
448 |
|
|
|
449 |
|
|
\subparagraph{Non-hydrostatic } |
450 |
|
|
|
451 |
cnh |
1.4 |
Non-hydrostatic forms of the incompressible Boussinesq equations in $z-$ |
452 |
adcroft |
1.1 |
coordinates are supported. |
453 |
|
|
|
454 |
|
|
\subsection{Solution strategy} |
455 |
|
|
|
456 |
cnh |
1.4 |
The method of solution employed in the \textbf{HPE}, \textbf{QH} and \textbf{ |
457 |
|
|
NH} models are summarized in Fig.7. |
458 |
adcroft |
1.1 |
\marginpar{ |
459 |
|
|
Fig.7 Solution strategy} |
460 |
|
|
|
461 |
|
|
Overview paragraph...... |
462 |
|
|
|
463 |
|
|
There is no penalty in implementing \textbf{QH} over \textbf{HPE} except, of |
464 |
cnh |
1.4 |
course, some complication that goes with the inclusion of $\cos \phi \ $ |
465 |
adcroft |
1.1 |
Coriolis terms and the relaxation of the shallow atmosphere approximation. |
466 |
|
|
But this leads to negligible increase in computation. In \textbf{NH}, in |
467 |
|
|
contrast, one additional elliptic equation - a three-dimensional one - must |
468 |
|
|
be inverted for $p_{nh}$. However the `overhead' of the \textbf{NH} model is |
469 |
|
|
essentially negligible in the hydrostatic limit (see detailed discussion in |
470 |
|
|
Marshall et al, 1997) resulting in a non-hydrostatic algorithm that, in the |
471 |
|
|
hydrostatic limit, is as computationally economic as the \textbf{HPEs}. |
472 |
|
|
|
473 |
|
|
\subsection{Finding the pressure field} |
474 |
|
|
|
475 |
|
|
Unlike the prognostic variables $u$, $v$, $w$, $\theta $ and $S$, the |
476 |
|
|
pressure field must be obtained diagnostically. We proceed, as before, by |
477 |
|
|
dividing the total (pressure/geo) potential in to three parts, a surface |
478 |
|
|
part, $\phi _{s}(x,y)$, a hydrostatic part $\phi _{hyd}(x,y,r)$ and a |
479 |
|
|
non-hydrostatic part $\phi _{nh}(x,y,r)$, as in (\ref{pressuresplit}), and |
480 |
cnh |
1.2 |
writing the momentum equation |
481 |
|
|
as in (\ref{eq:mom-h}). |
482 |
adcroft |
1.1 |
|
483 |
|
|
\subsubsection{Hydrostatic pressure} |
484 |
|
|
|
485 |
|
|
Hydrostatic pressure is obtained by integrating (\ref{hydro}) vertically |
486 |
|
|
from $r=R_{o}$ where $\phi _{hyd}(r=R_{o})=0$, to yield: |
487 |
|
|
|
488 |
|
|
\[ |
489 |
|
|
\int_{r}^{R_{o}}\frac{\partial \phi _{hyd}}{\partial r}dr=\left[ \phi |
490 |
|
|
_{hyd}\right] _{r}^{R_{o}}=\int_{r}^{R_{o}}-bdr |
491 |
|
|
\] |
492 |
|
|
and so |
493 |
|
|
|
494 |
cnh |
1.2 |
\begin{equation} |
495 |
adcroft |
1.1 |
\phi _{hyd}(x,y,r)=\int_{r}^{R_{o}}bdr |
496 |
cnh |
1.2 |
\label{eq:hydro-phi} |
497 |
|
|
\end{equation} |
498 |
adcroft |
1.1 |
|
499 |
|
|
\subsubsection{Surface pressure} |
500 |
|
|
|
501 |
cnh |
1.4 |
The surface pressure equation can be obtained by integrating continuity, ( |
502 |
adcroft |
1.1 |
\ref{incompressible})c, vertically from $r=R_{fixed}$ to $r=R_{moving}$ |
503 |
|
|
|
504 |
|
|
\[ |
505 |
cnh |
1.4 |
\int_{R_{fixed}}^{R_{moving}}\left( \mathbf{\nabla }_{h}\cdot \vec{\mathbf{v} |
506 |
adcroft |
1.1 |
}_{h}+\partial _{r}\dot{r}\right) dr=0 |
507 |
|
|
\] |
508 |
|
|
|
509 |
|
|
Thus: |
510 |
|
|
|
511 |
|
|
\[ |
512 |
|
|
\frac{\partial \eta }{\partial t}+\vec{\mathbf{v}}.\nabla \eta |
513 |
cnh |
1.4 |
+\int_{R_{fixed}}^{R_{moving}}\mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}} |
514 |
adcroft |
1.1 |
_{h}dr=0 |
515 |
|
|
\] |
516 |
cnh |
1.4 |
where $\eta =R_{moving}-R_{o}$ is the free-surface $r$-anomaly in units of $ |
517 |
adcroft |
1.1 |
r $. The above can be rearranged to yield, using Leibnitz's theorem: |
518 |
|
|
|
519 |
|
|
\begin{equation} |
520 |
|
|
\frac{\partial \eta }{\partial t}+\mathbf{\nabla }_{h}\cdot |
521 |
|
|
\int_{R_{fixed}}^{R_{moving}}\vec{\mathbf{v}}_{h}dr=0 |
522 |
cnh |
1.2 |
\label{eq:free-surface} |
523 |
adcroft |
1.1 |
\end{equation} |
524 |
|
|
|
525 |
|
|
Whether $\phi $ is pressure (ocean model, $p/\rho _{c}$) or geopotential |
526 |
|
|
(atmospheric model), in (\ref{mtm-split}), the horizontal gradient term can |
527 |
|
|
be written |
528 |
|
|
\begin{equation} |
529 |
cnh |
1.2 |
\mathbf{\nabla }_{h}\phi _{s}=\mathbf{\nabla }_{h}b\eta |
530 |
|
|
\label{eq:phi-surf} |
531 |
adcroft |
1.1 |
\end{equation} |
532 |
|
|
where $b$ is the buoyancy. |
533 |
|
|
|
534 |
|
|
In the hydrostatic limit ($\epsilon _{nh}=0$), Eqs(\ref{mtm-split}), (\ref |
535 |
|
|
{integralcontinuity}) and (\ref{link}) can be solved by inverting a 2-d |
536 |
|
|
elliptic equation for $\phi _{s}$ as described in section ?.?. Both `free |
537 |
|
|
surface' and `rigid lid' approaches are available. |
538 |
|
|
|
539 |
|
|
\subsubsection{Non-hydrostatic pressure} |
540 |
|
|
|
541 |
cnh |
1.4 |
Taking the horizontal divergence of (\ref{hor-mtm}) and adding $\frac{ |
542 |
adcroft |
1.1 |
\partial }{\partial r}$ of (\ref{vertmtm}), invoking the continuity equation |
543 |
|
|
(\ref{incompressible}), we deduce that: |
544 |
|
|
|
545 |
|
|
\begin{equation} |
546 |
cnh |
1.4 |
\nabla _{3}^{2}\phi _{nh}=\nabla .\vec{\mathbf{G}}_{\vec{v}}-\left( \mathbf{ |
547 |
|
|
\nabla }_{h}^{2}\phi _{s}+\mathbf{\nabla }^{2}\phi _{hyd}\right) =\nabla . |
548 |
cnh |
1.2 |
\vec{\mathbf{F}} \label{eq:3d-invert} |
549 |
adcroft |
1.1 |
\end{equation} |
550 |
|
|
|
551 |
|
|
For a given rhs this 3-d elliptic equation must be inverted for $\phi _{nh}$ |
552 |
|
|
subject to appropriate choice of boundary conditions. This method is usually |
553 |
|
|
called \textit{The Pressure Method} [Harlow and Welch, 1965; Williams, 1969; |
554 |
|
|
Potter, 1976]. In the hydrostatic primitive equations case (\textbf{HPE}), |
555 |
|
|
the 3-d problem does not need to be solved. |
556 |
|
|
|
557 |
|
|
\paragraph{Boundary Conditions} |
558 |
|
|
|
559 |
|
|
We apply the condition of no normal flow through all solid boundaries - the |
560 |
|
|
coasts (in the ocean) and the bottom: |
561 |
|
|
|
562 |
|
|
\begin{equation} |
563 |
|
|
\vec{\mathbf{v}}.\widehat{n}=0 \label{nonormalflow} |
564 |
|
|
\end{equation} |
565 |
|
|
where $\widehat{n}$ is a vector of unit length normal to the boundary. The |
566 |
|
|
kinematic condition (\ref{nonormalflow}) is also applied to the vertical |
567 |
cnh |
1.4 |
velocity at $r=R_{moving}$. No-slip $\left( v_{T}=0\right) \ $or slip $ |
568 |
adcroft |
1.1 |
\left( \partial v_{T}/\partial n=0\right) \ $conditions are employed on the |
569 |
|
|
tangential component of velocity, $v_{T}$, at all solid boundaries, |
570 |
|
|
depending on the form chosen for the dissipative terms in the momentum |
571 |
|
|
equations - see below. |
572 |
|
|
|
573 |
|
|
Eq.(\ref{nonormalflow}) implies, making use of (\ref{mtm-split}), that: |
574 |
|
|
|
575 |
|
|
\begin{equation} |
576 |
|
|
\widehat{n}.\nabla \phi _{nh}=\widehat{n}.\vec{\mathbf{F}} |
577 |
cnh |
1.2 |
\label{eq:inhom-neumann-nh} |
578 |
adcroft |
1.1 |
\end{equation} |
579 |
|
|
where |
580 |
|
|
|
581 |
|
|
\[ |
582 |
|
|
\vec{\mathbf{F}}=\vec{\mathbf{G}}_{\vec{v}}-\left( \mathbf{\nabla }_{h}\phi |
583 |
|
|
_{s}+\mathbf{\nabla }\phi _{hyd}\right) |
584 |
|
|
\] |
585 |
|
|
presenting inhomogeneous Neumann boundary conditions to the Elliptic problem |
586 |
|
|
(\ref{3dinvert}). As shown, for example, by Williams (1969), one can exploit |
587 |
cnh |
1.4 |
classical 3D potential theory and, by introducing an appropriately chosen $ |
588 |
adcroft |
1.1 |
\delta $-function sheet of `source-charge', replace the inhomogenous |
589 |
|
|
boundary condition on pressure by a homogeneous one. The source term $rhs$ |
590 |
|
|
in (\ref{3dinvert}) is the divergence of the vector $\vec{\mathbf{F}}.$ By |
591 |
|
|
simultaneously setting $ |
592 |
|
|
\begin{array}{l} |
593 |
|
|
\widehat{n}.\vec{\mathbf{F}} |
594 |
|
|
\end{array} |
595 |
|
|
=0$\ and $\widehat{n}.\nabla \phi _{nh}=0\ $on the boundary the following |
596 |
|
|
self-consistent but simpler homogenised Elliptic problem is obtained: |
597 |
|
|
|
598 |
|
|
\[ |
599 |
|
|
\nabla ^{2}\phi _{nh}=\nabla .\widetilde{\vec{\mathbf{F}}}\qquad |
600 |
|
|
\] |
601 |
|
|
where $\widetilde{\vec{\mathbf{F}}}$ is a modified $\vec{\mathbf{F}}$ such |
602 |
|
|
that $\widetilde{\vec{\mathbf{F}}}.\widehat{n}=0$. As is implied by (\ref |
603 |
|
|
{inhomneumann}) the modified boundary condition becomes: |
604 |
|
|
|
605 |
|
|
\begin{equation} |
606 |
cnh |
1.2 |
\widehat{n}.\nabla \phi _{nh}=0 \label{eq:hom-neumann-nh} |
607 |
adcroft |
1.1 |
\end{equation} |
608 |
|
|
|
609 |
|
|
If the flow is `close' to hydrostatic balance then the 3-d inversion |
610 |
|
|
converges rapidly because $\phi _{nh}\ $is then only a small correction to |
611 |
|
|
the hydrostatic pressure field (see the discussion in Marshall et al, a,b). |
612 |
|
|
|
613 |
|
|
The solution $\phi _{nh}\ $to (\ref{3dinvert}) and (\ref{homneuman}) does |
614 |
|
|
not vanish at $r=R_{moving}$, and so refines the pressure there. |
615 |
|
|
|
616 |
|
|
\subsection{Forcing/dissipation} |
617 |
|
|
|
618 |
|
|
\subsubsection{Forcing} |
619 |
|
|
|
620 |
|
|
The forcing terms $\mathcal{F}$ on the rhs of the equations are provided by |
621 |
|
|
`physics packages' described in detail in section ?.?. |
622 |
|
|
|
623 |
|
|
\subsubsection{Dissipation} |
624 |
|
|
|
625 |
|
|
\paragraph{Momentum} |
626 |
|
|
|
627 |
|
|
Many forms of momentum dissipation are available in the model. Laplacian and |
628 |
|
|
biharmonic frictions are commonly used: |
629 |
|
|
|
630 |
cnh |
1.2 |
\begin{equation} |
631 |
cnh |
1.4 |
D_{V}=A_{h}\nabla _{h}^{2}v+A_{v}\frac{\partial ^{2}v}{\partial z^{2}} |
632 |
adcroft |
1.1 |
+A_{4}\nabla _{h}^{4}v |
633 |
cnh |
1.2 |
\label{eq:dissipation} |
634 |
|
|
\end{equation} |
635 |
adcroft |
1.1 |
where $A_{h}$ and $A_{v}\ $are (constant) horizontal and vertical viscosity |
636 |
|
|
coefficients and $A_{4}\ $is the horizontal coefficient for biharmonic |
637 |
|
|
friction. These coefficients are the same for all velocity components. |
638 |
|
|
|
639 |
|
|
\paragraph{Tracers} |
640 |
|
|
|
641 |
|
|
The mixing terms for the temperature and salinity equations have a similar |
642 |
|
|
form to that of momentum except that the diffusion tensor can be |
643 |
cnh |
1.4 |
non-diagonal and have varying coefficients. $\qquad $ |
644 |
cnh |
1.2 |
\begin{equation} |
645 |
adcroft |
1.1 |
D_{T,S}=\nabla .[\underline{\underline{K}}\nabla (T,S)]+K_{4}\nabla |
646 |
|
|
_{h}^{4}(T,S) |
647 |
cnh |
1.2 |
\label{eq:diffusion} |
648 |
|
|
\end{equation} |
649 |
cnh |
1.4 |
where $\underline{\underline{K}}\ $is the diffusion tensor and the $K_{4}\ $ |
650 |
adcroft |
1.1 |
horizontal coefficient for biharmonic diffusion. In the simplest case where |
651 |
|
|
the subgrid-scale fluxes of heat and salt are parameterized with constant |
652 |
|
|
horizontal and vertical diffusion coefficients, $\underline{\underline{K}}$, |
653 |
|
|
reduces to a diagonal matrix with constant coefficients: |
654 |
|
|
|
655 |
cnh |
1.2 |
\begin{equation} |
656 |
adcroft |
1.1 |
\qquad \qquad \qquad \qquad K=\left( |
657 |
|
|
\begin{array}{ccc} |
658 |
|
|
K_{h} & 0 & 0 \\ |
659 |
|
|
0 & K_{h} & 0 \\ |
660 |
|
|
0 & 0 & K_{v} |
661 |
|
|
\end{array} |
662 |
|
|
\right) \qquad \qquad \qquad |
663 |
cnh |
1.2 |
\label{eq:diagonal-diffusion-tensor} |
664 |
|
|
\end{equation} |
665 |
adcroft |
1.1 |
where $K_{h}\ $and $K_{v}\ $are the horizontal and vertical diffusion |
666 |
|
|
coefficients. These coefficients are the same for all tracers (temperature, |
667 |
|
|
salinity ... ). |
668 |
|
|
|
669 |
|
|
\subsection{Vector invariant form} |
670 |
|
|
|
671 |
|
|
For some purposes it is advantageous to write momentum advection in eq(\ref |
672 |
|
|
{hor-mtm}) and (\ref{vertmtm}) in the (so-called) `vector invariant' form: |
673 |
|
|
|
674 |
|
|
\begin{equation} |
675 |
cnh |
1.4 |
\frac{D\vec{\mathbf{v}}}{Dt}=\frac{\partial \vec{\mathbf{v}}}{\partial t} |
676 |
adcroft |
1.1 |
+\left( \nabla \times \vec{\mathbf{v}}\right) \times \vec{\mathbf{v}}+\nabla |
677 |
|
|
\left[ \frac{1}{2}(\vec{\mathbf{v}}\cdot \vec{\mathbf{v}})\right] |
678 |
cnh |
1.2 |
\label{eq:vi-identity} |
679 |
adcroft |
1.1 |
\end{equation} |
680 |
|
|
This permits alternative numerical treatments of the non-linear terms based |
681 |
|
|
on their representation as a vorticity flux. Because gradients of coordinate |
682 |
|
|
vectors no longer appear on the rhs of (\ref{vecinvariant}) (???), explicit |
683 |
|
|
representation of the metric terms in (\ref{Gu}), (\ref{Gv}) and (\ref{Gw}), |
684 |
|
|
can be avoided: information about the geometry is contained in the areas and |
685 |
|
|
lengths of the volumes used to discretize the model. |
686 |
|
|
|
687 |
|
|
\subsection{Adjoint} |
688 |
|
|
|
689 |
|
|
...... |