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

Diff of /manual/s_overview/continuous_eqns.tex

Parent Directory | Revision Log | View Revision Graph Revision Graph | View Patch Patch

-revision 1.1.1.1 by adcroft,
Wed Aug  8 16:16:18 2001 UTC
+revision 1.5 by cnh,
Wed Oct 10 16:48:40 2001 UTC
 Line 5
  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 fig.4%
+ respective fluids - see fig.4
  \marginpar{
  Fig.4. Isomorphisms}. 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 and height, $z$, if we are modeling the ocean. A
- complete list of the isomorphisms is given in table 1.%
+ complete list of the isomorphisms is given in table 1.
  \marginpar{
  Table 1. Isomorphisms}
 Line 22 
 velocity $\vec{\mathbf{v}}$, active trac
  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, $r$, see fig.5%
+ a generic vertical coordinate, $r$, see fig.5
  \marginpar{
  Fig.5 The vertical coordinate of model}:
  \[
- \frac{D\vec{\mathbf{v}_{h}}}{Dt}+\left( 2\vec{\Omega}\times \vec{\mathbf{v}}%
+ \frac{D\vec{\mathbf{v}_{h}}}{Dt}+\left( 2\vec{\Omega}\times \vec{\mathbf{v}}
- \right) _{h}+\mathbf{\nabla }_{h}\phi =\left( \mathcal{F}_{\vec{\mathbf{v}}}%
+ \right) _{h}+\mathbf{\nabla }_{h}\phi =\left( \mathcal{F}_{\vec{\mathbf{v}}}
  \mathcal{+D}_{\vec{\mathbf{v}}}\right) _{h}\text{horizontal mtm}
  \]
  \[
- \frac{D\dot{r}}{Dt}+\widehat{k}\cdot \left( 2\vec{\Omega}\times \vec{\mathbf{%
+ \frac{D\dot{r}}{Dt}+\widehat{k}\cdot \left( 2\vec{\Omega}\times \vec{\mathbf{
- v}}\right) +\frac{\partial \phi }{\partial r}+b=\left( \mathcal{F}_{\vec{%
+ v}}\right) +\frac{\partial \phi }{\partial r}+b=\left( \mathcal{F}_{\vec{
  \mathbf{v}}}\mathcal{+D}_{\vec{\mathbf{v}}}\right) _{r}\text{vertical mtm}
  \]
  \begin{equation}
- \mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \dot{r}}{%
+ \mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \dot{r}}{
  \partial r}=0\text{ continuity}  \label{incompressible}
  \end{equation}
 Line 53 
 potential temperature}
  \]
  \[
- \frac{DS}{Dt}=\mathcal{F}_{S}\text{ }\mathcal{+D}_{S}\text{ humidity/salinity%
+ \frac{DS}{Dt}=\mathcal{F}_{S}\text{ }\mathcal{+D}_{S}\text{ humidity/salinity
  }
  \]
 Line 69 
 is the total derivative}
  \]
  \[
- \mathbf{\nabla }=\mathbf{\nabla }_{h}+\widehat{k}\frac{\partial }{\partial r}%
+ \mathbf{\nabla }=\mathbf{\nabla }_{h}+\widehat{k}\frac{\partial }{\partial r}
  \text{ is the `grad' operator}
  \]
- with $\mathbf{\nabla }_{h}$ operating in the horizontal and $\widehat{k}%
+ 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
 Line 106 
 S\text{ is specific humidity in the atmo
  \]
  \[
- \mathcal{F}_{\vec{\mathbf{v}}}\text{ and }\mathcal{D}_{\vec{\mathbf{v}}}%
+ \mathcal{F}_{\vec{\mathbf{v}}}\text{ and }\mathcal{D}_{\vec{\mathbf{v}}}
  \text{ are forcing and dissipation of }\vec{\mathbf{v}}
  \]
 Line 201 
 atmosphere.
  The boundary conditions at top and bottom are given by:
- \begin{eqnarray*}
+ \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)}
- \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 $p$
-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 231 
 At the bottom of the ocean: $R_{fixed}(x
+Line 237 
 At the bottom of the ocean: $R_{fixed}(x
  The surface of the ocean is given by: $R_{moving}=\eta $
- The position of the resting free surface of the ocean is given by $%
+ The position of the resting free surface of the ocean is given by $
  R_{o}=Z_{o}=0$.
  Boundary conditions are:
- \begin{eqnarray*}
+ \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}\\
- \end{eqnarray*}
+ 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
-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}}{%
+ \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.
  The $\left( \vec{\mathbf{G}}_{\vec{v}},G_{\dot{r}}\right) $ in eq(\ref
  {hor-mtm}) and (\ref{vertmtm}) represent advective, metric and Coriolis
- terms in the momentum equations. In spherical coordinates they take the form%
+ terms in the momentum equations. In spherical coordinates they take the form
- \footnote{%
+ \footnote{
  In the hydrostatic primitive equations (\textbf{HPE}) all underlined terms
  in (\ref{Gu}), (\ref{Gv}) and (\ref{Gw}) are omitted; the singly-underlined
  terms are included in the quasi-hydrostatic model (\textbf{QH}). The fully
-Line 285 
 non-hydrostatic model (\textbf{NH}) incl
+Line 293 
 non-hydrostatic model (\textbf{NH}) incl
  \left.
  \begin{tabular}{l}
  $G_{u}=-\vec{\mathbf{v}}.\nabla u$ \\
- $-\left\{ \underline{\frac{u\dot{r}}{{r}}}-\frac{uv\tan lat}{{r}}%
+ $-\left\{ \underline{\frac{u\dot{r}}{{r}}}-\frac{uv\tan lat}{{r}}
  \right\} $ \\
- $-\left\{ -2\Omega v\sin lat+\underline{\underline{2\Omega \dot{r}\cos lat}}%
+ $-\left\{ -2\Omega v\sin lat+\underline{\underline{2\Omega \dot{r}\cos lat}}
  \right\} $ \\
- $+\mathcal{F}_{u}\mathcal{+D}_{u}$%
+ $+\mathcal{F}_{u}\mathcal{+D}_{u}$
  \end{tabular}
  \right\} \left\{
  \begin{tabular}{l}
-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}
  \left.
  \begin{tabular}{l}
  $G_{v}=-\vec{\mathbf{v}}.\nabla v$ \\
- $-\left\{ \underline{\frac{v\dot{r}}{{r}}}-\frac{u^{2}\tan lat}{{r}%
+ $-\left\{ \underline{\frac{v\dot{r}}{{r}}}-\frac{u^{2}\tan lat}{{r}
  }\right\} $ \\
  $-\left\{ -2\Omega u\sin lat\right\} $ \\
- $+\mathcal{F}_{v}\mathcal{+D}_{v}$%
+ $+\mathcal{F}_{v}\mathcal{+D}_{v}$
  \end{tabular}
  \right\} \left\{
  \begin{tabular}{l}
-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 325 
 $+\mathcal{F}_{v}\mathcal{+D}_{v}$%
+Line 333 
 $+\mathcal{F}_{v}\mathcal{+D}_{v}$%
  \left.
  \begin{tabular}{l}
  $G_{\dot{r}}=-\vec{\mathbf{v}}.\nabla \dot{r}$ \\
- $+\left\{ \frac{u^{_{^{2}}}+v^{2}}{{{r}}}%
+ $+\left\{ \frac{u^{_{^{2}}}+v^{2}}{{{r}}}
  \right\} $ \\
  ${+2\Omega u\cos lat}$ \\
- $\mathcal{F}_{\dot{r}}\mathcal{+D}_{\dot{r}}$%
+ $\mathcal{F}_{\dot{r}}\mathcal{+D}_{\dot{r}}$
  \end{tabular}
  \right\} \left\{
  \begin{tabular}{l}
-Line 337 
 $\mathcal{F}_{\dot{r}}\mathcal{+D}_{\dot
+Line 345 
 $\mathcal{F}_{\dot{r}}\mathcal{+D}_{\dot
  \textit{Coriolis} \\
  \textit{\ Forcing/Dissipation}
  \end{tabular}
- \right.  \label{Gw}
+ \right.  \label{eq:gw-spherical}
  \end{equation}
  \qquad \qquad \qquad \qquad \qquad
- In the above `${r}$' is the distance from the center of the earth and `$%
+ In the above `${r}$' is the distance from the center of the earth and `$
  lat$' is latitude.
  Grad and div operators in spherical coordinates are defined in appendix
- OPERATORS.%
+ OPERATORS.
  \marginpar{
  Fig.6 Spherical polar coordinate system.}
-Line 358 
 Fig.6 Spherical polar coordinate system.
+Line 366 
 Fig.6 Spherical polar coordinate system.
  These are discussed at length in Marshall et al (1997a).
  In the `hydrostatic primitive equations' (\textbf{HPE)} all the underlined
- terms in Eqs. (\ref{Gu} $\rightarrow $\ \ref{Gw}) are neglected and `${r%
+ terms in Eqs. (\ref{Gu} $\rightarrow $\ \ref{Gw}) are neglected and `${r
  }$' is replaced by `$a$', the mean radius of the earth. Once the pressure is
  found at one level - e.g. by inverting a 2-d Elliptic equation for $\phi
  _{s} $ at $r=R_{moving}$ - the pressure can be computed at all other levels
-Line 367 
 by integration of the hydrostatic relati
+Line 375 
 by integration of the hydrostatic relati
  In the `quasi-hydrostatic' equations (\textbf{QH)} strict balance between
  gravity and vertical pressure gradients is not imposed. The $2\Omega u\cos
  \phi $ Coriolis term are not neglected and are balanced by a non-hydrostatic
- contribution to the pressure field: only the terms underlined twice in Eqs. (%
+ contribution to the pressure field: only the terms underlined twice in Eqs. (
  \ref{Gu} $\rightarrow $\ \ref{Gw}) are set to zero and, simultaneously, the
  shallow atmosphere approximation is relaxed. In \textbf{QH}\ \textit{all}
  the metric terms are retained and the full variation of the radial position
-Line 391 
 only a quasi-non-hydrostatic atmospheric
+Line 399 
 only a quasi-non-hydrostatic atmospheric
  \paragraph{Non-hydrostatic Ocean}
- In the non-hydrostatic ocean model all terms in equations (\ref{Gu} $%
+ In the non-hydrostatic ocean model all terms in equations (\ref{Gu} $
  \rightarrow $\ \ref{Gw}) are retained. A three dimensional elliptic equation
  must be solved subject to Neumann boundary conditions (see below). It is
  important to note that use of the full \textbf{NH} does not admit any new
-Line 404 
 et.al.\ 1997a.
+Line 412 
 et.al.\ 1997a.
  \paragraph{Quasi-nonhydrostatic Atmosphere}
- In the non-hydrostatic version of our atmospheric model we approximate $\dot{%
+ In the non-hydrostatic version of our atmospheric model we approximate $\dot{
  r}$ in the vertical momentum eqs(\ref{vertmtm}) and (\ref{Gw}) (but only
  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 440 
 equations in $z-$coordinates are support
+Line 448 
 equations in $z-$coordinates are support
  \subparagraph{Non-hydrostatic }
- Non-hydrostatic forms of the incompressible Boussinesq equations in $z-$%
+ Non-hydrostatic forms of the incompressible Boussinesq equations in $z-$
  coordinates are supported.
  \subsection{Solution strategy}
- The method of solution employed in the \textbf{HPE}, \textbf{QH} and \textbf{%
+ The method of solution employed in the \textbf{HPE}, \textbf{QH} and \textbf{
- NH} models are summarized in Fig.7.%
+ NH} models are summarized in Fig.7.
  \marginpar{
  Fig.7 Solution strategy}
  Overview paragraph......
  There is no penalty in implementing \textbf{QH} over \textbf{HPE} except, of
- course, some complication that goes with the inclusion of $\cos \phi \ $%
+ course, some complication that goes with the inclusion of $\cos \phi \ $
  Coriolis terms and the relaxation of the shallow atmosphere approximation.
  But this leads to negligible increase in computation. In \textbf{NH}, in
  contrast, one additional elliptic equation - a three-dimensional one - must
-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}
- The surface pressure equation can be obtained by integrating continuity, (%
+ The surface pressure equation can be obtained by integrating continuity, (
  \ref{incompressible})c, vertically from $r=R_{fixed}$ to $r=R_{moving}$
  \[
- \int_{R_{fixed}}^{R_{moving}}\left( \mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}%
+ \int_{R_{fixed}}^{R_{moving}}\left( \mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}
  }_{h}+\partial _{r}\dot{r}\right) dr=0
  \]
-Line 506 
 Thus:
+Line 510 
 Thus:
  \[
  \frac{\partial \eta }{\partial t}+\vec{\mathbf{v}}.\nabla \eta
- +\int_{R_{fixed}}^{R_{moving}}\mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}}%
+ +\int_{R_{fixed}}^{R_{moving}}\mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}}
  _{h}dr=0
  \]
- where $\eta =R_{moving}-R_{o}$ is the free-surface $r$-anomaly in units of $%
+ where $\eta =R_{moving}-R_{o}$ is the free-surface $r$-anomaly in units of $
  r $. The above can be rearranged to yield, using Leibnitz's theorem:
  \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 533 
 surface' and `rigid lid' approaches are
+Line 538 
 surface' and `rigid lid' approaches are
  \subsubsection{Non-hydrostatic pressure}
- Taking the horizontal divergence of (\ref{hor-mtm}) and adding $\frac{%
+ Taking the horizontal divergence of (\ref{hor-mtm}) and adding $\frac{
  \partial }{\partial r}$ of (\ref{vertmtm}), invoking the continuity equation
  (\ref{incompressible}), we deduce that:
  \begin{equation}
- \nabla _{3}^{2}\phi _{nh}=\nabla .\vec{\mathbf{G}}_{\vec{v}}-\left( \mathbf{%
+ \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 .%
+ \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 559 
 coasts (in the ocean) and the bottom:
+Line 564 
 coasts (in the ocean) and the bottom:
  \end{equation}
  where $\widehat{n}$ is a vector of unit length normal to the boundary. The
  kinematic condition (\ref{nonormalflow}) is also applied to the vertical
- velocity at $r=R_{moving}$. No-slip $\left( v_{T}=0\right) \ $or slip $%
+ velocity at $r=R_{moving}$. No-slip $\left( v_{T}=0\right) \ $or slip $
  \left( \partial v_{T}/\partial n=0\right) \ $conditions are employed on the
  tangential component of velocity, $v_{T}$, at all solid boundaries,
  depending on the form chosen for the dissipative terms in the momentum
-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 579 
 _{s}+\mathbf{\nabla }\phi _{hyd}\right)
+Line 584 
 _{s}+\mathbf{\nabla }\phi _{hyd}\right)
  \]
  presenting inhomogeneous Neumann boundary conditions to the Elliptic problem
  (\ref{3dinvert}). As shown, for example, by Williams (1969), one can exploit
- classical 3D potential theory and, by introducing an appropriately chosen $%
+ classical 3D potential theory and, by introducing an appropriately chosen $
  \delta $-function sheet of `source-charge', replace the inhomogenous
  boundary condition on pressure by a homogeneous one. The source term $rhs$
  in (\ref{3dinvert}) is the divergence of the vector $\vec{\mathbf{F}}.$ By
-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:
- \[
+ \begin{equation}
- D_{V}=A_{h}\nabla _{h}^{2}v+A_{v}\frac{\partial ^{2}v}{\partial z^{2}}%
+ D_{V}=A_{h}\nabla _{h}^{2}v+A_{v}\frac{\partial ^{2}v}{\partial z^{2}}
  +A_{4}\nabla _{h}^{4}v
- \]
+ \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 634 
 friction. These coefficients are the sam
+Line 640 
 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 $%
+ 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}
- where $\underline{\underline{K}}\ $is the diffusion tensor and the $K_{4}\ $%
+ \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 664 
 For some purposes it is advantageous to
+Line 672 
 For some purposes it is advantageous to
  {hor-mtm}) and (\ref{vertmtm}) in the (so-called) `vector invariant' form:
  \begin{equation}
- \frac{D\vec{\mathbf{v}}}{Dt}=\frac{\partial \vec{\mathbf{v}}}{\partial t}%
+ \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

 Legend:



Removed from v.1.1.1.1
 


changed lines


 
Added in v.1.5
 Legend:



Removed from v.1.1.1.1
 


changed lines


 
Added in v.1.5
-Removed from v.1.1.1.1
+Added in v.1.5

	ViewVC Help
Powered by ViewVC 1.1.22