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-revision 1.3 by cnh,
Wed Sep 26 14:53:10 2001 UTC
+revision 1.4 by cnh,
Thu Sep 27 01:57:17 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 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:
 Line 275 
 _{nh}=\vec{\mathbf{G}}_{\vec{v}_{h}}  \l
  \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{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 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 313 
 $+\mathcal{F}_{u}\mathcal{+D}_{u}$%
  \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 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 349 
 $\mathcal{F}_{\dot{r}}\mathcal{+D}_{\dot
  \end{equation}
  \qquad \qquad \qquad \qquad \qquad
- In the above `${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 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 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 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 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:
 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 498 
 and so
  \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 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}
 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{eq:3d-invert}
  \end{equation}
 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 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 628 
 Many forms of momentum dissipation are a
  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}
 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}
  \end{equation}
- where $\underline{\underline{K}}\ $is the diffusion tensor and the $K_{4}\ $%
+ 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, $\underline{\underline{K}}$,
 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{eq:vi-identity}

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