--- manual/s_overview/continuous_eqns.tex 2001/08/08 16:16:18 1.1.1.1 +++ manual/s_overview/continuous_eqns.tex 2001/09/26 14:53:10 1.3 @@ -1,4 +1,4 @@ -% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/Attic/continuous_eqns.tex,v 1.1.1.1 2001/08/08 16:16:18 adcroft Exp $ +% $Header: /home/ubuntu/mnt/e9_copy/manual/s_overview/Attic/continuous_eqns.tex,v 1.3 2001/09/26 14:53:10 cnh Exp $ % $Name: $ \section{Continuous equations in `r' coordinates} @@ -201,11 +201,13 @@ The boundary conditions at top and bottom are given by: -\begin{eqnarray*} -&&\omega =0~\text{at }r=R_{fixed}\text{ (top of the atmosphere)} \\ +\begin{eqnarray} +&&\omega =0~\text{at }r=R_{fixed} \label{eq:fixed-bc-atmos} +\text{ (top of the atmosphere)} \\ \omega &=&\frac{Dp_{s}}{Dt}\text{; at }r=R_{moving}\text{ (bottom of the atmosphere)} -\end{eqnarray*} +\label{eq:moving-bc-atmos} +\end{eqnarray} Then the (hydrostatic form of) eq(\ref{incompressible}) yields a consistent set of atmospheric equations which, for convenience, are written out in $p$ @@ -216,11 +218,15 @@ In the ocean we interpret: \begin{eqnarray} -r &=&z\text{ is the height} \\ -\dot{r} &=&\frac{Dz}{Dt}=w\text{ is the vertical velocity} \\ -\phi &=&\frac{p}{\rho _{c}}\text{ is the pressure} \\ +r &=&z\text{ is the height} +\label{eq:ocean-z}\\ +\dot{r} &=&\frac{Dz}{Dt}=w\text{ is the vertical velocity} +\label{eq:ocean-w}\\ +\phi &=&\frac{p}{\rho _{c}}\text{ is the pressure} +\label{eq:ocean-p}\\ b(\theta ,S,r) &=&\frac{g}{\rho _{c}}\left( \rho (\theta ,S,r)-\rho _{c}\right) \text{ is the buoyancy} +\label{eq:ocean-b} \end{eqnarray} where $\rho _{c}$ is a fixed reference density of water and $g$ is the acceleration due to gravity.\noindent @@ -236,10 +242,12 @@ Boundary conditions are: -\begin{eqnarray*} -w &=&0~\text{at }r=R_{fixed}\text{ (ocean bottom)} \\ -w &=&\frac{D\eta }{Dt}\text{ at }r=R_{moving}=\eta \text{ (ocean surface)} -\end{eqnarray*} +\begin{eqnarray} +w &=&0~\text{at }r=R_{fixed}\text{ (ocean bottom)} +\label{eq:fixed-bc-ocean}\\ +w &=&\frac{D\eta }{Dt}\text{ at }r=R_{moving}=\eta \text{ (ocean surface) +\label{eq:moving-bc-ocean}} +\end{eqnarray} where $\eta $ is the elevation of the free surface. Then eq(\ref{incompressible}) yields a consistent set of oceanic equations @@ -252,23 +260,23 @@ \begin{equation} \phi (x,y,r)=\phi _{s}(x,y)+\phi _{hyd}(x,y,r)+\phi _{nh}(x,y,r) -\label{pressuresplit} +\label{eq:phi-split} \end{equation} and write eq(\ref{incompressible}a) in the form: \begin{equation} \frac{\partial \vec{\mathbf{v}_{h}}}{\partial t}+\mathbf{\nabla }_{h}\phi _{s}+\mathbf{\nabla }_{h}\phi _{hyd}+\epsilon _{nh}\mathbf{\nabla }_{h}\phi -_{nh}=\vec{\mathbf{G}}_{\vec{v}_{h}} \label{hor-mtm} +_{nh}=\vec{\mathbf{G}}_{\vec{v}_{h}} \label{eq:mom-h} \end{equation} \begin{equation} -\frac{\partial \phi _{hyd}}{\partial r}=-b \label{hydro} +\frac{\partial \phi _{hyd}}{\partial r}=-b \label{eq:hydrostatic} \end{equation} \begin{equation} \epsilon _{nh}\frac{\partial \dot{r}}{\partial t}+\frac{\partial \phi _{nh}}{% -\partial r}=G_{\dot{r}} \label{vertmtm} +\partial r}=G_{\dot{r}} \label{eq:mom-w} \end{equation} Here $\epsilon _{nh}$ is a non-hydrostatic parameter. @@ -298,7 +306,7 @@ \textit{Coriolis} \\ \textit{\ Forcing/Dissipation} \end{tabular} -\right. \qquad \label{Gu} +\right. \qquad \label{eq:gu-speherical} \end{equation} \begin{equation} @@ -317,7 +325,7 @@ \textit{Coriolis} \\ \textit{\ Forcing/Dissipation} \end{tabular} -\right. \qquad \label{Gv} +\right. \qquad \label{eq:gv-spherical} \end{equation} \qquad \qquad \qquad \qquad \qquad @@ -337,7 +345,7 @@ \textit{Coriolis} \\ \textit{\ Forcing/Dissipation} \end{tabular} -\right. \label{Gw} +\right. \label{eq:gw-spherical} \end{equation} \qquad \qquad \qquad \qquad \qquad @@ -409,7 +417,7 @@ here) by: \begin{equation} -\dot{r}=\frac{Dp}{Dt}=\frac{Dp_{hyd}}{Dt} \label{quasinonhydro} +\dot{r}=\frac{Dp}{Dt}=\frac{Dp_{hyd}}{Dt} \label{eq:quasi-nh-w} \end{equation} where $p_{hy}$ is the hydrostatic pressure. @@ -469,13 +477,8 @@ dividing the total (pressure/geo) potential in to three parts, a surface part, $\phi _{s}(x,y)$, a hydrostatic part $\phi _{hyd}(x,y,r)$ and a non-hydrostatic part $\phi _{nh}(x,y,r)$, as in (\ref{pressuresplit}), and -writing the momentum equation in the form -\begin{equation} -\frac{\partial }{\partial t}\vec{\mathbf{v}_{h}}+\mathbf{\nabla }_{h}\phi -_{s}+\mathbf{\nabla }_{h}\phi _{hyd}+\epsilon _{nh}\mathbf{\nabla }\phi -_{nh}=\vec{\mathbf{G}}_{\vec{v}} \label{mtm-split} -\end{equation} -as in (\ref{hor-mtm}). +writing the momentum equation +as in (\ref{eq:mom-h}). \subsubsection{Hydrostatic pressure} @@ -488,9 +491,10 @@ \] and so -\[ +\begin{equation} \phi _{hyd}(x,y,r)=\int_{r}^{R_{o}}bdr -\] +\label{eq:hydro-phi} +\end{equation} \subsubsection{Surface pressure} @@ -515,14 +519,15 @@ \begin{equation} \frac{\partial \eta }{\partial t}+\mathbf{\nabla }_{h}\cdot \int_{R_{fixed}}^{R_{moving}}\vec{\mathbf{v}}_{h}dr=0 -\label{integralcontinuity} +\label{eq:free-surface} \end{equation} Whether $\phi $ is pressure (ocean model, $p/\rho _{c}$) or geopotential (atmospheric model), in (\ref{mtm-split}), the horizontal gradient term can be written \begin{equation} -\mathbf{\nabla }_{h}\phi _{s}=\mathbf{\nabla }_{h}b\eta \label{link} +\mathbf{\nabla }_{h}\phi _{s}=\mathbf{\nabla }_{h}b\eta +\label{eq:phi-surf} \end{equation} where $b$ is the buoyancy. @@ -540,7 +545,7 @@ \begin{equation} \nabla _{3}^{2}\phi _{nh}=\nabla .\vec{\mathbf{G}}_{\vec{v}}-\left( \mathbf{% \nabla }_{h}^{2}\phi _{s}+\mathbf{\nabla }^{2}\phi _{hyd}\right) =\nabla .% -\vec{\mathbf{F}} \label{3dinvert} +\vec{\mathbf{F}} \label{eq:3d-invert} \end{equation} For a given rhs this 3-d elliptic equation must be inverted for $\phi _{nh}$ @@ -569,7 +574,7 @@ \begin{equation} \widehat{n}.\nabla \phi _{nh}=\widehat{n}.\vec{\mathbf{F}} -\label{inhomneumann} +\label{eq:inhom-neumann-nh} \end{equation} where @@ -598,7 +603,7 @@ {inhomneumann}) the modified boundary condition becomes: \begin{equation} -\widehat{n}.\nabla \phi _{nh}=0 \label{homneuman} +\widehat{n}.\nabla \phi _{nh}=0 \label{eq:hom-neumann-nh} \end{equation} If the flow is `close' to hydrostatic balance then the 3-d inversion @@ -622,10 +627,11 @@ Many forms of momentum dissipation are available in the model. Laplacian and biharmonic frictions are commonly used: -\[ +\begin{equation} D_{V}=A_{h}\nabla _{h}^{2}v+A_{v}\frac{\partial ^{2}v}{\partial z^{2}}% +A_{4}\nabla _{h}^{4}v -\] +\label{eq:dissipation} +\end{equation} where $A_{h}$ and $A_{v}\ $are (constant) horizontal and vertical viscosity coefficients and $A_{4}\ $is the horizontal coefficient for biharmonic friction. These coefficients are the same for all velocity components. @@ -635,17 +641,18 @@ The mixing terms for the temperature and salinity equations have a similar form to that of momentum except that the diffusion tensor can be non-diagonal and have varying coefficients. $\qquad $% -\[ +\begin{equation} D_{T,S}=\nabla .[\underline{\underline{K}}\nabla (T,S)]+K_{4}\nabla _{h}^{4}(T,S) -\] +\label{eq:diffusion} +\end{equation} where $\underline{\underline{K}}\ $is the diffusion tensor and the $K_{4}\ $% horizontal coefficient for biharmonic diffusion. In the simplest case where the subgrid-scale fluxes of heat and salt are parameterized with constant horizontal and vertical diffusion coefficients, $\underline{\underline{K}}$, reduces to a diagonal matrix with constant coefficients: -\[ +\begin{equation} \qquad \qquad \qquad \qquad K=\left( \begin{array}{ccc} K_{h} & 0 & 0 \\ @@ -653,7 +660,8 @@ 0 & 0 & K_{v} \end{array} \right) \qquad \qquad \qquad -\] +\label{eq:diagonal-diffusion-tensor} +\end{equation} where $K_{h}\ $and $K_{v}\ $are the horizontal and vertical diffusion coefficients. These coefficients are the same for all tracers (temperature, salinity ... ). @@ -667,7 +675,7 @@ \frac{D\vec{\mathbf{v}}}{Dt}=\frac{\partial \vec{\mathbf{v}}}{\partial t}% +\left( \nabla \times \vec{\mathbf{v}}\right) \times \vec{\mathbf{v}}+\nabla \left[ \frac{1}{2}(\vec{\mathbf{v}}\cdot \vec{\mathbf{v}})\right] -\label{vecinvariant} +\label{eq:vi-identity} \end{equation} This permits alternative numerical treatments of the non-linear terms based on their representation as a vorticity flux. Because gradients of coordinate