/[MITgcm]/manual/s_overview/continuous_eqns.tex

Diff of /manual/s_overview/continuous_eqns.tex

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-revision 1.1.1.1 by adcroft,
Wed Aug  8 16:16:18 2001 UTC
+revision 1.2 by cnh,
Tue Sep 11 14:03:33 2001 UTC
 Line 202 
 atmosphere.
  The boundary conditions at top and bottom are given by:
  \begin{eqnarray*}
- &&\omega =0~\text{at }r=R_{fixed}\text{ (top of the atmosphere)} \\
+ &&\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)}
+ \label{eq:moving-bc-atmos}
  \end{eqnarray*}
  Then the (hydrostatic form of) eq(\ref{incompressible}) yields a consistent
-Line 216 
 coordinates in Appendix Atmosphere - see
+Line 218 
 coordinates in Appendix Atmosphere - see
  In the ocean we interpret:
  \begin{eqnarray}
- r &=&z\text{ is the height} \\
+ r &=&z\text{ is the height}
- \dot{r} &=&\frac{Dz}{Dt}=w\text{ is the vertical velocity} \\
+ \label{eq:ocean-z}\\
- \phi &=&\frac{p}{\rho _{c}}\text{ is the pressure} \\
+ \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
-Line 237 
 R_{o}=Z_{o}=0$.
+Line 243 
 R_{o}=Z_{o}=0$.
  Boundary conditions are:
  \begin{eqnarray*}
- w &=&0~\text{at }r=R_{fixed}\text{ (ocean bottom)} \\
+ w &=&0~\text{at }r=R_{fixed}\text{ (ocean bottom)}
- w &=&\frac{D\eta }{Dt}\text{ at }r=R_{moving}=\eta \text{ (ocean surface)}
+ \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.
-Line 252 
 Let us separate $\phi $ in to surface, h
+Line 260 
 Let us separate $\phi $ in to surface, h
  \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.
-Line 298 
 $+\mathcal{F}_{u}\mathcal{+D}_{u}$%
+Line 306 
 $+\mathcal{F}_{u}\mathcal{+D}_{u}$%
  \textit{Coriolis} \\
  \textit{\ Forcing/Dissipation}
  \end{tabular}
- \right. \qquad  \label{Gu}
+ \right. \qquad  \label{eq:gu-speherical}
  \end{equation}
  \begin{equation}
-Line 317 
 $+\mathcal{F}_{v}\mathcal{+D}_{v}$%
+Line 325 
 $+\mathcal{F}_{v}\mathcal{+D}_{v}$%
  \textit{Coriolis} \\
  \textit{\ Forcing/Dissipation}
  \end{tabular}
- \right. \qquad  \label{Gv}
+ \right. \qquad  \label{eq:gv-spherical}
  \end{equation}
  \qquad \qquad \qquad \qquad \qquad
-Line 337 
 $\mathcal{F}_{\dot{r}}\mathcal{+D}_{\dot
+Line 345 
 $\mathcal{F}_{\dot{r}}\mathcal{+D}_{\dot
  \textit{Coriolis} \\
  \textit{\ Forcing/Dissipation}
  \end{tabular}
- \right.  \label{Gw}
+ \right.  \label{eq:gw-spherical}
  \end{equation}
  \qquad \qquad \qquad \qquad \qquad
-Line 409 
 r}$ in the vertical momentum eqs(\ref{ve
+Line 417 
 r}$ in the vertical momentum eqs(\ref{ve
  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.
-Line 469 
 pressure field must be obtained diagnost
+Line 477 
 pressure field must be obtained diagnost
  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
+ writing the momentum equation
- \begin{equation}
+ as in (\ref{eq:mom-h}).
- \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}).
  \subsubsection{Hydrostatic pressure}
-Line 488 
 _{hyd}\right] _{r}^{R_{o}}=\int_{r}^{R_{
+Line 491 
 _{hyd}\right] _{r}^{R_{o}}=\int_{r}^{R_{
  \]
  and so
- \[
+ \begin{equation}
  \phi _{hyd}(x,y,r)=\int_{r}^{R_{o}}bdr
- \]
+ \label{eq:hydro-phi}
+ \end{equation}
  \subsubsection{Surface pressure}
-Line 515 
 r $. The above can be rearranged to yiel
+Line 519 
 r $. The above can be rearranged to yiel
  \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.
-Line 540 
 Taking the horizontal divergence of (\re
+Line 545 
 Taking the horizontal divergence of (\re
  \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 $\phi _{nh}$
-Line 569 
 Eq.(\ref{nonormalflow}) implies, making
+Line 574 
 Eq.(\ref{nonormalflow}) implies, making
  \begin{equation}
  \widehat{n}.\nabla \phi _{nh}=\widehat{n}.\vec{\mathbf{F}}
- \label{inhomneumann}
+ \label{eq:inhom-neumann-nh}
  \end{equation}
  where
-Line 598 
 that $\widetilde{\vec{\mathbf{F}}}.\wide
+Line 603 
 that $\widetilde{\vec{\mathbf{F}}}.\wide
  {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
-Line 622 
 The forcing terms $\mathcal{F}$ on the r
+Line 627 
 The forcing terms $\mathcal{F}$ on the r
  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.
-Line 635 
 friction. These coefficients are the sam
+Line 641 
 friction. These coefficients are the sam
  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:
- \[
+ \begin{equation}
  \qquad \qquad \qquad \qquad K=\left(
  \begin{array}{ccc}
  K_{h} & 0 & 0 \\
-Line 653 
 K_{h} & 0 & 0 \\
+Line 660 
 K_{h} & 0 & 0 \\
 & 0 & K_{v}
  \end{array}
  \right) \qquad \qquad \qquad
- \]
+ \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 ... ).
-Line 667 
 For some purposes it is advantageous to
+Line 675 
 For some purposes it is advantageous to
  \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

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