| 326 |
\begin{equation*} |
\begin{equation*} |
| 327 |
\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}} |
| 328 |
\right) _{h}+\mathbf{\nabla }_{h}\phi =\mathcal{F}_{\vec{\mathbf{v}_{h}}} |
\right) _{h}+\mathbf{\nabla }_{h}\phi =\mathcal{F}_{\vec{\mathbf{v}_{h}}} |
| 329 |
\text{ horizontal mtm} |
\text{ horizontal mtm} \label{eq:horizontal_mtm} |
| 330 |
\end{equation*} |
\end{equation*} |
| 331 |
|
|
| 332 |
\begin{equation*} |
\begin{equation} |
| 333 |
\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{ |
| 334 |
v}}\right) +\frac{\partial \phi }{\partial r}+b=\mathcal{F}_{\dot{r}}\text{ |
v}}\right) +\frac{\partial \phi }{\partial r}+b=\mathcal{F}_{\dot{r}}\text{ |
| 335 |
vertical mtm} |
vertical mtm} \label{eq:vertical_mtm} |
| 336 |
\end{equation*} |
\end{equation} |
| 337 |
|
|
| 338 |
\begin{equation} |
\begin{equation} |
| 339 |
\mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \dot{r}}{ |
\mathbf{\nabla }_{h}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \dot{r}}{ |
| 340 |
\partial r}=0\text{ continuity} \label{eq:continuous} |
\partial r}=0\text{ continuity} \label{eq:continuity} |
| 341 |
\end{equation} |
\end{equation} |
| 342 |
|
|
| 343 |
\begin{equation*} |
\begin{equation} |
| 344 |
b=b(\theta ,S,r)\text{ equation of state} |
b=b(\theta ,S,r)\text{ equation of state} \label{eq:equation_of_state} |
| 345 |
\end{equation*} |
\end{equation} |
| 346 |
|
|
| 347 |
\begin{equation*} |
\begin{equation} |
| 348 |
\frac{D\theta }{Dt}=\mathcal{Q}_{\theta }\text{ potential temperature} |
\frac{D\theta }{Dt}=\mathcal{Q}_{\theta }\text{ potential temperature} |
| 349 |
\end{equation*} |
\label{eq:potential_temperature} |
| 350 |
|
\end{equation} |
| 351 |
|
|
| 352 |
\begin{equation*} |
\begin{equation} |
| 353 |
\frac{DS}{Dt}=\mathcal{Q}_{S}\text{ humidity/salinity} |
\frac{DS}{Dt}=\mathcal{Q}_{S}\text{ humidity/salinity} |
| 354 |
\end{equation*} |
\label{eq:humidtity_salt} |
| 355 |
|
\end{equation} |
| 356 |
|
|
| 357 |
Here: |
Here: |
| 358 |
|
|
| 525 |
atmosphere)} \label{eq:moving-bc-atmos} |
atmosphere)} \label{eq:moving-bc-atmos} |
| 526 |
\end{eqnarray} |
\end{eqnarray} |
| 527 |
|
|
| 528 |
Then the (hydrostatic form of) eq(\ref{eq:continuous}) yields a consistent |
Then the (hydrostatic form of) equations (\ref{eq:horizontal_mtm}-\ref{eq:humidity_slainty}) |
| 529 |
set of atmospheric equations which, for convenience, are written out in $p$ |
yields a consistent set of atmospheric equations which, for convenience, are written out in $p$ |
| 530 |
coordinates in Appendix Atmosphere - see eqs(\ref{eq:atmos-prime}). |
coordinates in Appendix Atmosphere - see eqs(\ref{eq:atmos-prime}). |
| 531 |
|
|
| 532 |
\subsection{Ocean} |
\subsection{Ocean} |
| 562 |
\end{eqnarray} |
\end{eqnarray} |
| 563 |
where $\eta $ is the elevation of the free surface. |
where $\eta $ is the elevation of the free surface. |
| 564 |
|
|
| 565 |
Then eq(\ref{eq:continuous}) yields a consistent set of oceanic equations |
Then equations (\ref{eq:horizontal_mtm}-\ref{eq:humidity_slainty}) yield a consistent set |
| 566 |
|
of oceanic equations |
| 567 |
which, for convenience, are written out in $z$ coordinates in Appendix Ocean |
which, for convenience, are written out in $z$ coordinates in Appendix Ocean |
| 568 |
- see eqs(\ref{eq:ocean-mom}) to (\ref{eq:ocean-salt}). |
- see eqs(\ref{eq:ocean-mom}) to (\ref{eq:ocean-salt}). |
| 569 |
|
|
| 576 |
\phi (x,y,r)=\phi _{s}(x,y)+\phi _{hyd}(x,y,r)+\phi _{nh}(x,y,r) |
\phi (x,y,r)=\phi _{s}(x,y)+\phi _{hyd}(x,y,r)+\phi _{nh}(x,y,r) |
| 577 |
\label{eq:phi-split} |
\label{eq:phi-split} |
| 578 |
\end{equation} |
\end{equation} |
| 579 |
and write eq(\ref{incompressible}a,b) in the form: |
and write eq(\ref{eq:incompressible}) in the form: |
| 580 |
|
|
| 581 |
\begin{equation} |
\begin{equation} |
| 582 |
\frac{\partial \vec{\mathbf{v}_{h}}}{\partial t}+\mathbf{\nabla }_{h}\phi |
\frac{\partial \vec{\mathbf{v}_{h}}}{\partial t}+\mathbf{\nabla }_{h}\phi |
| 669 |
|
|
| 670 |
Grad and div operators in spherical coordinates are defined in appendix |
Grad and div operators in spherical coordinates are defined in appendix |
| 671 |
OPERATORS. |
OPERATORS. |
|
\marginpar{ |
|
|
Fig.6 Spherical polar coordinate system.} |
|
| 672 |
|
|
| 673 |
%%CNHbegin |
%%CNHbegin |
| 674 |
\input{part1/sphere_coord_figure.tex} |
\input{part1/sphere_coord_figure.tex} |
| 731 |
three dimensional elliptic equation must be solved subject to Neumann |
three dimensional elliptic equation must be solved subject to Neumann |
| 732 |
boundary conditions (see below). It is important to note that use of the |
boundary conditions (see below). It is important to note that use of the |
| 733 |
full \textbf{NH} does not admit any new `fast' waves in to the system - the |
full \textbf{NH} does not admit any new `fast' waves in to the system - the |
| 734 |
incompressible condition eq(\ref{eq:continuous})c has already filtered out |
incompressible condition eq(\ref{eq:continuity}) has already filtered out |
| 735 |
acoustic modes. It does, however, ensure that the gravity waves are treated |
acoustic modes. It does, however, ensure that the gravity waves are treated |
| 736 |
accurately with an exact dispersion relation. The \textbf{NH} set has a |
accurately with an exact dispersion relation. The \textbf{NH} set has a |
| 737 |
complete angular momentum principle and consistent energetics - see White |
complete angular momentum principle and consistent energetics - see White |
| 780 |
\subsection{Solution strategy} |
\subsection{Solution strategy} |
| 781 |
|
|
| 782 |
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{ |
| 783 |
NH} models is summarized in Fig.7. |
NH} models is summarized in Figure \ref{fig:solution-strategy}. |
| 784 |
\marginpar{ |
Under all dynamics, a 2-d elliptic equation is |
|
Fig.7 Solution strategy} Under all dynamics, a 2-d elliptic equation is |
|
| 785 |
first solved to find the surface pressure and the hydrostatic pressure at |
first solved to find the surface pressure and the hydrostatic pressure at |
| 786 |
any level computed from the weight of fluid above. Under \textbf{HPE} and |
any level computed from the weight of fluid above. Under \textbf{HPE} and |
| 787 |
\textbf{QH} dynamics, the horizontal momentum equations are then stepped |
\textbf{QH} dynamics, the horizontal momentum equations are then stepped |
| 838 |
|
|
| 839 |
\subsubsection{Surface pressure} |
\subsubsection{Surface pressure} |
| 840 |
|
|
| 841 |
The surface pressure equation can be obtained by integrating continuity, ( |
The surface pressure equation can be obtained by integrating continuity, |
| 842 |
\ref{eq:continuous})c, vertically from $r=R_{fixed}$ to $r=R_{moving}$ |
(\ref{eq:continuity}), vertically from $r=R_{fixed}$ to $r=R_{moving}$ |
| 843 |
|
|
| 844 |
\begin{equation*} |
\begin{equation*} |
| 845 |
\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} |
| 864 |
where we have incorporated a source term. |
where we have incorporated a source term. |
| 865 |
|
|
| 866 |
Whether $\phi $ is pressure (ocean model, $p/\rho _{c}$) or geopotential |
Whether $\phi $ is pressure (ocean model, $p/\rho _{c}$) or geopotential |
| 867 |
(atmospheric model), in (\ref{mtm-split}), the horizontal gradient term can |
(atmospheric model), in (\ref{eq:mom-h}), the horizontal gradient term can |
| 868 |
be written |
be written |
| 869 |
\begin{equation} |
\begin{equation} |
| 870 |
\mathbf{\nabla }_{h}\phi _{s}=\mathbf{\nabla }_{h}\left( b_{s}\eta \right) |
\mathbf{\nabla }_{h}\phi _{s}=\mathbf{\nabla }_{h}\left( b_{s}\eta \right) |
| 872 |
\end{equation} |
\end{equation} |
| 873 |
where $b_{s}$ is the buoyancy at the surface. |
where $b_{s}$ is the buoyancy at the surface. |
| 874 |
|
|
| 875 |
In the hydrostatic limit ($\epsilon _{nh}=0$), Eqs(\ref{eq:mom-h}), (\ref |
In the hydrostatic limit ($\epsilon _{nh}=0$), equations (\ref{eq:mom-h}), (\ref |
| 876 |
{eq:free-surface}) and (\ref{eq:phi-surf}) can be solved by inverting a 2-d |
{eq:free-surface}) and (\ref{eq:phi-surf}) can be solved by inverting a 2-d |
| 877 |
elliptic equation for $\phi _{s}$ as described in Chapter 2. Both `free |
elliptic equation for $\phi _{s}$ as described in Chapter 2. Both `free |
| 878 |
surface' and `rigid lid' approaches are available. |
surface' and `rigid lid' approaches are available. |
| 879 |
|
|
| 880 |
\subsubsection{Non-hydrostatic pressure} |
\subsubsection{Non-hydrostatic pressure} |
| 881 |
|
|
| 882 |
Taking the horizontal divergence of (\ref{hor-mtm}) and adding $\frac{ |
Taking the horizontal divergence of (\ref{eq:mom-h}) and adding |
| 883 |
\partial }{\partial r}$ of (\ref{vertmtm}), invoking the continuity equation |
$\frac{\partial }{\partial r}$ of (\ref{eq:mom-w}), invoking the continuity equation |
| 884 |
(\ref{incompressible}), we deduce that: |
(\ref{eq:continuity}), we deduce that: |
| 885 |
|
|
| 886 |
\begin{equation} |
\begin{equation} |
| 887 |
\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{ |
| 911 |
depending on the form chosen for the dissipative terms in the momentum |
depending on the form chosen for the dissipative terms in the momentum |
| 912 |
equations - see below. |
equations - see below. |
| 913 |
|
|
| 914 |
Eq.(\ref{nonormalflow}) implies, making use of (\ref{mtm-split}), that: |
Eq.(\ref{nonormalflow}) implies, making use of (\ref{eq:mom-h}), that: |
| 915 |
|
|
| 916 |
\begin{equation} |
\begin{equation} |
| 917 |
\widehat{n}.\nabla \phi _{nh}=\widehat{n}.\vec{\mathbf{F}} |
\widehat{n}.\nabla \phi _{nh}=\widehat{n}.\vec{\mathbf{F}} |
| 951 |
converges rapidly because $\phi _{nh}\ $is then only a small correction to |
converges rapidly because $\phi _{nh}\ $is then only a small correction to |
| 952 |
the hydrostatic pressure field (see the discussion in Marshall et al, a,b). |
the hydrostatic pressure field (see the discussion in Marshall et al, a,b). |
| 953 |
|
|
| 954 |
The solution $\phi _{nh}\ $to (\ref{eq:3d-invert}) and (\ref{homneuman}) |
The solution $\phi _{nh}\ $to (\ref{eq:3d-invert}) and (\ref{eq:inhom-neumann-nh}) |
| 955 |
does not vanish at $r=R_{moving}$, and so refines the pressure there. |
does not vanish at $r=R_{moving}$, and so refines the pressure there. |
| 956 |
|
|
| 957 |
\subsection{Forcing/dissipation} |
\subsection{Forcing/dissipation} |
| 959 |
\subsubsection{Forcing} |
\subsubsection{Forcing} |
| 960 |
|
|
| 961 |
The forcing terms $\mathcal{F}$ on the rhs of the equations are provided by |
The forcing terms $\mathcal{F}$ on the rhs of the equations are provided by |
| 962 |
`physics packages' described in detail in chapter ??. |
`physics packages' and forcing packages. These are described later on. |
| 963 |
|
|
| 964 |
\subsubsection{Dissipation} |
\subsubsection{Dissipation} |
| 965 |
|
|
| 1007 |
\subsection{Vector invariant form} |
\subsection{Vector invariant form} |
| 1008 |
|
|
| 1009 |
For some purposes it is advantageous to write momentum advection in eq(\ref |
For some purposes it is advantageous to write momentum advection in eq(\ref |
| 1010 |
{hor-mtm}) and (\ref{vertmtm}) in the (so-called) `vector invariant' form: |
{eq:horizontal_mtm}) and (\ref{eq:vertical_mtm}) in the (so-called) `vector invariant' form: |
| 1011 |
|
|
| 1012 |
\begin{equation} |
\begin{equation} |
| 1013 |
\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} |
| 1025 |
|
|
| 1026 |
\subsection{Adjoint} |
\subsection{Adjoint} |
| 1027 |
|
|
| 1028 |
Tangent linear and adjoint counterparts of the forward model and described |
Tangent linear and adjoint counterparts of the forward model are described |
| 1029 |
in Chapter 5. |
in Chapter 5. |
| 1030 |
|
|
| 1031 |
% $Header$ |
% $Header$ |
| 1147 |
The final form of the HPE's in p coordinates is then: |
The final form of the HPE's in p coordinates is then: |
| 1148 |
\begin{eqnarray} |
\begin{eqnarray} |
| 1149 |
\frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}} |
\frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}} |
| 1150 |
_{h}+\mathbf{\nabla }_{p}\phi ^{\prime } &=&\vec{\mathbf{\mathcal{F}}} \\ |
_{h}+\mathbf{\nabla }_{p}\phi ^{\prime } &=&\vec{\mathbf{\mathcal{F}}} \label{eq:atmos-prime} \\ |
| 1151 |
\frac{\partial \phi ^{\prime }}{\partial p}+\alpha ^{\prime } &=&0 \\ |
\frac{\partial \phi ^{\prime }}{\partial p}+\alpha ^{\prime } &=&0 \\ |
| 1152 |
\mathbf{\nabla }_{p}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \omega }{ |
\mathbf{\nabla }_{p}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \omega }{ |
| 1153 |
\partial p} &=&0 \\ |
\partial p} &=&0 \\ |
| 1154 |
\frac{\partial \Pi }{\partial p}\theta ^{\prime } &=&\alpha ^{\prime } \\ |
\frac{\partial \Pi }{\partial p}\theta ^{\prime } &=&\alpha ^{\prime } \\ |
| 1155 |
\frac{D\theta }{Dt} &=&\frac{\mathcal{Q}}{\Pi } \label{eq:atmos-prime} |
\frac{D\theta }{Dt} &=&\frac{\mathcal{Q}}{\Pi } |
| 1156 |
\end{eqnarray} |
\end{eqnarray} |
| 1157 |
|
|
| 1158 |
% $Header$ |
% $Header$ |
| 1171 |
\epsilon _{nh}\frac{Dw}{Dt}+g+\frac{1}{\rho }\frac{\partial p}{\partial z} |
\epsilon _{nh}\frac{Dw}{Dt}+g+\frac{1}{\rho }\frac{\partial p}{\partial z} |
| 1172 |
&=&\epsilon _{nh}\mathcal{F}_{w} \\ |
&=&\epsilon _{nh}\mathcal{F}_{w} \\ |
| 1173 |
\frac{1}{\rho }\frac{D\rho }{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}} |
\frac{1}{\rho }\frac{D\rho }{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}} |
| 1174 |
_{h}+\frac{\partial w}{\partial z} &=&0 \\ |
_{h}+\frac{\partial w}{\partial z} &=&0 \label{eq-zns-cont}\\ |
| 1175 |
\rho &=&\rho (\theta ,S,p) \\ |
\rho &=&\rho (\theta ,S,p) \label{eq-zns-eos}\\ |
| 1176 |
\frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta } \\ |
\frac{D\theta }{Dt} &=&\mathcal{Q}_{\theta } \label{eq-zns-heat}\\ |
| 1177 |
\frac{DS}{Dt} &=&\mathcal{Q}_{s} \label{eq:non-boussinesq} |
\frac{DS}{Dt} &=&\mathcal{Q}_{s} \label{eq-zns-salt} |
| 1178 |
|
\label{eq:non-boussinesq} |
| 1179 |
\end{eqnarray} |
\end{eqnarray} |
| 1180 |
These equations permit acoustics modes, inertia-gravity waves, |
These equations permit acoustics modes, inertia-gravity waves, |
| 1181 |
non-hydrostatic motions, a geostrophic (Rossby) mode and a thermo-haline |
non-hydrostatic motions, a geostrophic (Rossby) mode and a thermo-haline |
| 1194 |
\end{equation} |
\end{equation} |
| 1195 |
|
|
| 1196 |
Note that $\frac{\partial \rho }{\partial p}=\frac{1}{c_{s}^{2}}$ is the |
Note that $\frac{\partial \rho }{\partial p}=\frac{1}{c_{s}^{2}}$ is the |
| 1197 |
reciprocal of the sound speed ($c_{s}$) squared. Substituting into \ref |
reciprocal of the sound speed ($c_{s}$) squared. Substituting into \ref{eq-zns-cont} gives: |
|
{eq-zns-cont} gives: |
|
| 1198 |
\begin{equation} |
\begin{equation} |
| 1199 |
\frac{1}{\rho c_{s}^{2}}\frac{Dp}{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{ |
\frac{1}{\rho c_{s}^{2}}\frac{Dp}{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{ |
| 1200 |
v}}+\partial _{z}w\approx 0 \label{eq-zns-pressure} |
v}}+\partial _{z}w\approx 0 \label{eq-zns-pressure} |
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