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1.3 Continuous equations in `r' coordinates

To render atmosphere and ocean models from one dynamical core we exploit `isomorphisms' between equation sets that govern the evolution of the respective fluids - see figure 1.14. One system of hydrodynamical equations is written down and encoded. The model variables have different interpretations depending on whether the atmosphere or ocean is being studied. Thus, for example, the vertical coordinate `$ r$' is interpreted as pressure, $ p$, if we are modeling the atmosphere (right hand side of figure 1.14) and height, $ z$, if we are modeling the ocean (left hand side of figure 1.14).

Figure 1.14: Isomorphic equation sets used for atmosphere (right) and ocean (left).
\resizebox{5.0in}{!}{ \includegraphics{part1/zandpcoord.eps} }

The state of the fluid at any time is characterized by the distribution of velocity $ \vec{\mathbf{v}}$, active tracers $ \theta $ and $ S$, a `geopotential' $ \phi $ and density $ \rho =\rho (\theta ,S,p)$ which may depend on $ \theta $, $ S$, and $ p$. The equations that govern the evolution of these fields, obtained by applying the laws of classical mechanics and thermodynamics to a Boussinesq, Navier-Stokes fluid are, written in terms of a generic vertical coordinate, $ r$, so that the appropriate kinematic boundary conditions can be applied isomorphically see figure 1.15.

Figure 1.15: Vertical coordinates and kinematic boundary conditions for atmosphere (top) and ocean (bottom).
\includegraphics[trim=210 80 70 140,width=.9\textwidth, clip]{part1/vertcoord.eps}

$\displaystyle \frac{D\vec{\mathbf{v}_{h}}}{Dt}+\left( 2\vec{\Omega}\times \vec{... ...f{v}} \right) _{h}+\mathbf{\nabla }_{h}\phi =\mathcal{F}_{\vec{\mathbf{v}_{h}}}$    horizontal mtm (1.1)

$\displaystyle \frac{D\dot{r}}{Dt}+\widehat{k}\cdot \left( 2\vec{\Omega}\times \vec{\mathbf{ v}}\right) +\frac{\partial \phi }{\partial r}+b=\mathcal{F}_{\dot{r}}$ vertical mtm (1.2)

$\displaystyle \mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \dot{r}}{ \partial r}=0$ continuity (1.3)

$\displaystyle b=b(\theta ,S,r)$ equation of state (1.4)

$\displaystyle \frac{D\theta }{Dt}=\mathcal{Q}_{\theta }$ potential temperature (1.5)

$\displaystyle \frac{DS}{Dt}=\mathcal{Q}_{S}$ humidity/salinity (1.6)

Here:

$\displaystyle r$ is the vertical coordinate    

$\displaystyle \frac{D}{Dt}=\frac{\partial }{\partial t}+\vec{\mathbf{v}}\cdot \nabla$    is the total derivative    

$\displaystyle \mathbf{\nabla }=\mathbf{\nabla }_{h}+\widehat{k}\frac{\partial }{\partial r}$    is the `grad' operator    

with $ \mathbf{\nabla }_{h}$ operating in the horizontal and $ \widehat{k} \frac{\partial }{\partial r}$ operating in the vertical, where $ \widehat{k}$ is a unit vector in the vertical

$\displaystyle t$ is time    

$\displaystyle \vec{\mathbf{v}}=(u,v,\dot{r})=(\vec{\mathbf{v}}_{h},\dot{r})$ is the velocity    

$\displaystyle \phi$    is the `pressure'/`geopotential'    

$\displaystyle \vec{\Omega}$ is the Earth's rotation    

$\displaystyle b$ is the `buoyancy'    

$\displaystyle \theta$    is potential temperature    

$\displaystyle S$ is specific humidity in the atmosphere; salinity in the ocean    

$\displaystyle \mathcal{F}_{\vec{\mathbf{v}}}$ are forcing and dissipation of $\displaystyle \vec{ \mathbf{v}}$    

$\displaystyle \mathcal{Q}_{\theta }\mathcal{\ }$are forcing and dissipation of $\displaystyle \theta$    

$\displaystyle \mathcal{Q}_{S}\mathcal{\ }$are forcing and dissipation of $\displaystyle S$    

The $ \mathcal{F}^{\prime }s$ and $ \mathcal{Q}^{\prime }s$ are provided by `physics' and forcing packages for atmosphere and ocean. These are described in later chapters.


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