s_phys_pkgs/text/gridalt.tex

\subsection{Gridalt - Alternate Grid Package}
\label{sec:pkg:gridalt}
\begin{rawhtml}
<!-- CMIREDIR:package_gridalt: -->
\end{rawhtml}

\subsubsection {Introduction} 

The gridalt package is designed to allow different components of the MITgcm to
be run using horizontal and/or vertical grids which are different from the main 
model grid. The gridalt routines handle the definition of the all the various
alternative grid(s) and the mappings between them and the MITgcm grid.
The implementation of the gridalt package which allows the high end atmospheric 
physics (fizhi) to be run on a high resolution and quasi terrain-following vertical 
grid is documented here.  The package has also (with some user modifications) been used 
for other calculations within the GCM. 

The rationale for implementing the atmospheric physics on a high resolution vertical
grid involves the fact that the MITgcm $p^*$ (or any pressure-type) coordinate cannot 
maintain the vertical resolution near the surface as the bottom topography rises above
sea level. The vertical length scales near the ground are small and can vary 
on small time scales, and the vertical grid must be adequate to resolve them.
Many studies with both regional and global atmospheric models have demonstrated the 
improvements in the simulations when the vertical resolution near the surface is 
increased (\cite{bm:99,Inn:01,wo:98,breth:99}). Some of the benefit of increased resolution 
near the surface is realized by employing the higher resolution for the computation of the 
forcing due to turbulent and convective processes in the atmosphere.  

The parameterizations of atmospheric subgrid scale processes are all essentially
one-dimensional in nature, and the computation of the terms in the equations of
motion due to these processes can be performed for the air column over one grid point 
at a time.  The vertical grid on which these computations take place can therefore be 
entirely independant of the grid on which the equations of motion are integrated, and 
the 'tendency' terms can be interpolated to the vertical grid on which the equations
of motion are integrated. A modified $p^*$ coordinate, which adjusts to the local 
terrain and adds additional levels between the lower levels of the existing $p^*$ grid 
(and perhaps between the levels near the tropopause as well), is implemented. The 
vertical discretization is different for each grid point, although it consist of the 
same number of levels. Additional 'sponge' levels aloft are added when needed. The levels 
of the physics grid are constrained to fit exactly into the existing $p^*$ grid, simplifying 
the mapping between the two vertical coordinates.  This is illustrated as follows:

\begin{figure}[htbp]
\vspace*{-0.4in}
\begin{center}
\includegraphics[height=2.4in]{part6/vertical.eps}
\caption{Vertical discretization for the MITgcm (dark grey lines) and for the
atmospheric physics (light grey lines). In this implementation, all MITgcm level
interfaces must coincide with atmospheric physics level interfaces.}
must be entirely
\end{center}
\end{figure}

The algorithm presented here retains the state variables on the high resolution 'physics'
grid as well as on the coarser resolution 'dynamics` grid, and ensures that the two 
estimates of the state 'agree' on the coarse resolution grid.  It would have been possible 
to implement a technique in which the tendencies due to atmospheric physics are computed 
on the high resolution grid and the state variables are retained at low resolution only. 
This, however, for the case of the turbulence parameterization,  would mean that the 
turbulent kinetic energy source terms, and all the turbulence terms that are written 
in terms of gradients of the mean flow, cannot really be computed making use of the fine 
structure in the vertical. 

\subsubsection{Equations on Both Grids}

In addition to computing the physical forcing terms of the momentum, thermodynamic and humidity 
equations on the modified (higher resolution) grid, the higher resolution structure of the 
atmosphere (the boundary layer) is retained between physics calculations. This neccessitates
a second set of evolution equations for the atmospheric state variables on the modified grid. 
If the equation for the evolution of $U$ on $p^*$ can be expressed as:
\[
\left . {\partial U \over {\partial t}} \right |_{p^*}^{total} = 
\left . {\partial U \over {\partial t}} \right |_{p^*}^{dynamics} + 
\left . {\partial U \over {\partial t}} \right |_{p^*}^{physics}
\]
where the physics forcing terms on $p^*$ have been mapped from the modified grid, then an additional 
equation to govern the evolution of $U$ (for example) on the modified grid is written:
\[
\left . {\partial U \over {\partial t}} \right |_{p^{*m}}^{total} = 
\left . {\partial U \over {\partial t}} \right |_{p^{*m}}^{dynamics} + 
\left . {\partial U \over {\partial t}} \right |_{p^{*m}}^{physics} +
\gamma ({\left . U \right |_{p^*}} - {\left . U \right |_{p^{*m}}})
\]
where $p^{*m}$ refers to the modified higher resolution grid, and the dynamics forcing terms have 
been mapped from $p^*$ space.  The last term on the RHS is a relaxation term, meant to constrain
the state variables on the modified vertical grid to `track' the state variables on the $p^*$ grid 
on some time scale, governed by $\gamma$. In the present implementation, $\gamma = 1$, requiring
an immediate agreement between the two 'states'.

\subsubsection{Time stepping Sequence}
If we write $T_{phys}$ as the temperature (or any other state variable) on the high
resolution physics grid, and $T_{dyn}$ as the temperature on the coarse vertical resolution
dynamics grid, then:

\begin{enumerate}
%\itemsep{-0.05in}

\item{Compute the tendency due to physics processes.}

\item{Advance the physics state: ${T^{n+1}^{**}}_{phys}(l) = {T^n}_{phys}(l) + \delta T_{phys}$.}

\item{Interpolate the physics tendency to the dynamics grid, and advance the dynamics
state by physics and dynamics tendencies:
${T^{n+1}}_{dyn}(L) = {T^n}_{dyn}(L) + \delta T_{dyn}(L) + [\delta T _{phys}(l)](L)$.}

\item{Interpolate the dynamics tendency to the physics grid, and update the physics
grid due to dynamics tendencies: 
${T^{n+1}^*}_{phys}(l)$ = ${T^{n+1}^{**}}_{phys}(l) + {\delta T_{dyn}(L)}(l)$.}

\item{Apply correction term to physics state to account for divergence from dynamics state:
${T^{n+1}}_{phys}(l)$ = ${T^{n+1}^*}_{phys}(l) + \gamma \{  T_{dyn}(L) - [T_{phys}(l)](L) \}(l)$.} \\
Where $\gamma=1$ here. 

\end{enumerate}

\subsubsection{Interpolation}
In order to minimize the correction terms for the state variables on the alternative,
higher resolution grid, the vertical interpolation scheme must be constructed so that
a dynamics-to-physics interpolation can be exactly reversed with a physics-to-dynamics mapping.
The simple scheme employed to achieve this is:

Coarse to fine:\
For all physics layers l in dynamics layer L, $ T_{phys}(l) = \{T_{dyn}(L)\} = T_{dyn}(L) $.

Fine to coarse:\
For all physics layers l in dynamics layer L, $T_{dyn}(L) = [T_{phys}(l)] = \int{T_{phys} dp } $.

Where $\{\}$ is defined as the dynamics-to-physics operator and $[ ]$ is the physics-to-dynamics operator, $T$ stands for any state variable, and the subscripts $phys$ and $dyn$ stand for variables on
the physics and dynamics grids, respectively.

\subsubsection {Key subroutines, parameters and files } 

\subsubsection {Dos and donts}

\subsubsection {Gridalt Reference} 
1	molod	1.4	\subsection{Gridalt - Alternate Grid Package}
2	edhill	1.2	\label{sec:pkg:gridalt}
3			\begin{rawhtml}
4			<!-- CMIREDIR:package_gridalt: -->
5			\end{rawhtml}
6	molod	1.1
7	molod	1.4	\subsubsection {Introduction}
8	molod	1.1
9	molod	1.5	The gridalt package is designed to allow different components of the MITgcm to
10			be run using horizontal and/or vertical grids which are different from the main
11			model grid. The gridalt routines handle the definition of the all the various
12			alternative grid(s) and the mappings between them and the MITgcm grid.
13			The implementation of the gridalt package which allows the high end atmospheric
14			physics (fizhi) to be run on a high resolution and quasi terrain-following vertical
15			grid is documented here. The package has also (with some user modifications) been used
16			for other calculations within the GCM.
17
18			The rationale for implementing the atmospheric physics on a high resolution vertical
19			grid involves the fact that the MITgcm $p^*$ (or any pressure-type) coordinate cannot
20			maintain the vertical resolution near the surface as the bottom topography rises above
21			sea level. The vertical length scales near the ground are small and can vary
22			on small time scales, and the vertical grid must be adequate to resolve them.
23			Many studies with both regional and global atmospheric models have demonstrated the
24			improvements in the simulations when the vertical resolution near the surface is
25			increased (\cite{bm:99,Inn:01,wo:98,breth:99}). Some of the benefit of increased resolution
26			near the surface is realized by employing the higher resolution for the computation of the
27			forcing due to turbulent and convective processes in the atmosphere.
28
29			The parameterizations of atmospheric subgrid scale processes are all essentially
30			one-dimensional in nature, and the computation of the terms in the equations of
31			motion due to these processes can be performed for the air column over one grid point
32			at a time. The vertical grid on which these computations take place can therefore be
33			entirely independant of the grid on which the equations of motion are integrated, and
34			the 'tendency' terms can be interpolated to the vertical grid on which the equations
35			of motion are integrated. A modified $p^*$ coordinate, which adjusts to the local
36			terrain and adds additional levels between the lower levels of the existing $p^*$ grid
37			(and perhaps between the levels near the tropopause as well), is implemented. The
38			vertical discretization is different for each grid point, although it consist of the
39			same number of levels. Additional 'sponge' levels aloft are added when needed. The levels
40			of the physics grid are constrained to fit exactly into the existing $p^*$ grid, simplifying
41			the mapping between the two vertical coordinates. This is illustrated as follows:
42
43	molod	1.1	\begin{figure}[htbp]
44			\vspace*{-0.4in}
45			\begin{center}
46	molod	1.3	\includegraphics[height=2.4in]{part6/vertical.eps}
47	molod	1.5	\caption{Vertical discretization for the MITgcm (dark grey lines) and for the
48			atmospheric physics (light grey lines). In this implementation, all MITgcm level
49			interfaces must coincide with atmospheric physics level interfaces.}
50			must be entirely
51	molod	1.1	\end{center}
52			\end{figure}
53
54	molod	1.5	The algorithm presented here retains the state variables on the high resolution 'physics'
55			grid as well as on the coarser resolution 'dynamics` grid, and ensures that the two
56			estimates of the state 'agree' on the coarse resolution grid. It would have been possible
57			to implement a technique in which the tendencies due to atmospheric physics are computed
58			on the high resolution grid and the state variables are retained at low resolution only.
59			This, however, for the case of the turbulence parameterization, would mean that the
60			turbulent kinetic energy source terms, and all the turbulence terms that are written
61			in terms of gradients of the mean flow, cannot really be computed making use of the fine
62			structure in the vertical.
63
64			\subsubsection{Equations on Both Grids}
65
66			In addition to computing the physical forcing terms of the momentum, thermodynamic and humidity
67			equations on the modified (higher resolution) grid, the higher resolution structure of the
68			atmosphere (the boundary layer) is retained between physics calculations. This neccessitates
69			a second set of evolution equations for the atmospheric state variables on the modified grid.
70			If the equation for the evolution of $U$ on $p^*$ can be expressed as:
71	molod	1.1	\[
72			\left . {\partial U \over {\partial t}} \right \|_{p^*}^{total} =
73			\left . {\partial U \over {\partial t}} \right \|_{p^*}^{dynamics} +
74			\left . {\partial U \over {\partial t}} \right \|_{p^*}^{physics}
75			\]
76	molod	1.5	where the physics forcing terms on $p^*$ have been mapped from the modified grid, then an additional
77			equation to govern the evolution of $U$ (for example) on the modified grid is written:
78	molod	1.1	\[
79			\left . {\partial U \over {\partial t}} \right \|_{p^{*m}}^{total} =
80			\left . {\partial U \over {\partial t}} \right \|_{p^{*m}}^{dynamics} +
81			\left . {\partial U \over {\partial t}} \right \|_{p^{*m}}^{physics} +
82			\gamma ({\left . U \right \|_{p^}} - {\left . U \right \|_{p^{m}}})
83			\]
84	molod	1.5	where $p^{*m}$ refers to the modified higher resolution grid, and the dynamics forcing terms have
85			been mapped from $p^*$ space. The last term on the RHS is a relaxation term, meant to constrain
86			the state variables on the modified vertical grid to `track' the state variables on the $p^*$ grid
87			on some time scale, governed by $\gamma$. In the present implementation, $\gamma = 1$, requiring
88			an immediate agreement between the two 'states'.
89
90			\subsubsection{Time stepping Sequence}
91			If we write $T_{phys}$ as the temperature (or any other state variable) on the high
92			resolution physics grid, and $T_{dyn}$ as the temperature on the coarse vertical resolution
93			dynamics grid, then:
94
95			\begin{enumerate}
96			%\itemsep{-0.05in}
97
98			\item{Compute the tendency due to physics processes.}
99
100			\item{Advance the physics state: ${T^{n+1}^{**}}_{phys}(l) = {T^n}_{phys}(l) + \delta T_{phys}$.}
101
102			\item{Interpolate the physics tendency to the dynamics grid, and advance the dynamics
103			state by physics and dynamics tendencies:
104			${T^{n+1}}_{dyn}(L) = {T^n}_{dyn}(L) + \delta T_{dyn}(L) + [\delta T _{phys}(l)](L)$.}
105
106			\item{Interpolate the dynamics tendency to the physics grid, and update the physics
107			grid due to dynamics tendencies:
108			${T^{n+1}^}_{phys}(l)$ = ${T^{n+1}^{*}}_{phys}(l) + {\delta T_{dyn}(L)}(l)$.}
109
110			\item{Apply correction term to physics state to account for divergence from dynamics state:
111			${T^{n+1}}_{phys}(l)$ = ${T^{n+1}^*}_{phys}(l) + \gamma \{ T_{dyn}(L) - [T_{phys}(l)](L) \}(l)$.} \\
112			Where $\gamma=1$ here.
113
114			\end{enumerate}
115
116			\subsubsection{Interpolation}
117			In order to minimize the correction terms for the state variables on the alternative,
118			higher resolution grid, the vertical interpolation scheme must be constructed so that
119			a dynamics-to-physics interpolation can be exactly reversed with a physics-to-dynamics mapping.
120			The simple scheme employed to achieve this is:
121
122			Coarse to fine:\
123			For all physics layers l in dynamics layer L, $ T_{phys}(l) = \{T_{dyn}(L)\} = T_{dyn}(L) $.
124
125			Fine to coarse:\
126			For all physics layers l in dynamics layer L, $T_{dyn}(L) = [T_{phys}(l)] = \int{T_{phys} dp } $.
127
128			Where $\{\}$ is defined as the dynamics-to-physics operator and $[ ]$ is the physics-to-dynamics operator, $T$ stands for any state variable, and the subscripts $phys$ and $dyn$ stand for variables on
129			the physics and dynamics grids, respectively.
130	molod	1.1
131	molod	1.4	\subsubsection {Key subroutines, parameters and files }
132	molod	1.1
133	molod	1.4	\subsubsection {Dos and donts}
134	molod	1.1
135	molod	1.4	\subsubsection {Gridalt Reference}