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1  \section{Gridalt - Alternate Grid Package}  \subsection{Gridalt - Alternate Grid Package}
2    \label{sec:pkg:gridalt}
3    \begin{rawhtml}
4    <!-- CMIREDIR:package_gridalt: -->
5    \end{rawhtml}
6    
7    \subsubsection {Introduction}
8    
9    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    
 \subsection {Introduction}  
 To take advantage of a `high end' turbulence parameterization  
 (and convection parameterization), the vertical resolution near the surface  
 must be increased substantially as compared to the vertical resolution needed  
 aloft. This cannot be accomplished if the high end physics is computed using  
 the $p^*$ coordinate currently in use in the MIT gcm.  
   
 The gridalt package was developed to allow the high end atmospheric physics  
 (fizhi) physics to be run on a separate grid from the hydrodynamics. The package  
 could (with some user modification) be used in conjunction with other packages  
 or for other calculations within the GCM. For the case of the atmospheric  
 physics, a modified $p^*$ coordinate, which 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. This is illustrated as follows:  
43  \begin{figure}[htbp]  \begin{figure}[htbp]
44  \vspace*{-0.4in}  \vspace*{-0.4in}
45  \begin{center}  \begin{center}
46  \includegraphics[height=2.4in]{vertical.eps}  \includegraphics[height=2.4in]{part6/vertical.eps}
47    \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  \end{center}  \end{center}
51  \end{figure}  \end{figure}
52    
53  \vspace*{-0.5in}  The algorithm presented here retains the state variables on the high resolution 'physics'
54  In addition to computing the physical forcing terms of the momentum,  grid as well as on the coarser resolution 'dynamics` grid, and ensures that the two
55  thermodynamic and humidity equations on the modified (higher resolution)  estimates of the state 'agree' on the coarse resolution grid.  It would have been possible
56  grid, the higher resolution structure of the atmosphere (the boundary  to implement a technique in which the tendencies due to atmospheric physics are computed
57  layer) is retained between calculations. This neccessitates a second  on the high resolution grid and the state variables are retained at low resolution only.
58  set of evolution equations for the atmospheric state variables on the  This, however, for the case of the turbulence parameterization,  would mean that the
59  modified grid. If the equations for the evolution of the state  turbulent kinetic energy source terms, and all the turbulence terms that are written
60  on $p^*$ can be expressed as:  in terms of gradients of the mean flow, cannot really be computed making use of the fine
61    structure in the vertical.
62    
63    \subsubsection{Equations on Both Grids}
64    
65    In addition to computing the physical forcing terms of the momentum, thermodynamic and humidity
66    equations on the modified (higher resolution) grid, the higher resolution structure of the
67    atmosphere (the boundary layer) is retained between physics calculations. This neccessitates
68    a second set of evolution equations for the atmospheric state variables on the modified grid.
69    If the equation for the evolution of $U$ on $p^*$ can be expressed as:
70  \[  \[
71  \left . {\partial U \over {\partial t}} \right |_{p^*}^{total} =  \left . {\partial U \over {\partial t}} \right |_{p^*}^{total} =
72  \left . {\partial U \over {\partial t}} \right |_{p^*}^{dynamics} +  \left . {\partial U \over {\partial t}} \right |_{p^*}^{dynamics} +
73  \left . {\partial U \over {\partial t}} \right |_{p^*}^{physics}  \left . {\partial U \over {\partial t}} \right |_{p^*}^{physics}
74  \]  \]
75  where the physics forcing terms on $p^*$ have been computed from a  where the physics forcing terms on $p^*$ have been mapped from the modified grid, then an additional
76  mapping from the modified grid, then an additional set of equations  equation to govern the evolution of $U$ (for example) on the modified grid is written:
 to govern the evolution of $U$ on the modified grid are written:  
77  \[  \[
78  \left . {\partial U \over {\partial t}} \right |_{p^{*m}}^{total} =  \left . {\partial U \over {\partial t}} \right |_{p^{*m}}^{total} =
79  \left . {\partial U \over {\partial t}} \right |_{p^{*m}}^{dynamics} +  \left . {\partial U \over {\partial t}} \right |_{p^{*m}}^{dynamics} +
80  \left . {\partial U \over {\partial t}} \right |_{p^{*m}}^{physics} +  \left . {\partial U \over {\partial t}} \right |_{p^{*m}}^{physics} +
81  \gamma ({\left . U \right |_{p^*}} - {\left . U \right |_{p^{*m}}})  \gamma ({\left . U \right |_{p^*}} - {\left . U \right |_{p^{*m}}})
82  \]  \]
83  where $p^{*m}$ refers to the modified higher resolution grid, and  where $p^{*m}$ refers to the modified higher resolution grid, and the dynamics forcing terms have
84  the dynamics forcing terms have been mapped from the $p^*$ space.  been mapped from $p^*$ space.  The last term on the RHS is a relaxation term, meant to constrain
85  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
86  the state variables on the modified vertical grid to `track' the  on some time scale, governed by $\gamma$. In the present implementation, $\gamma = 1$, requiring
87  state variables on the $p^*$ grid on some time scale, $\gamma$.  an immediate agreement between the two 'states'.
   
 \subsection {Key subroutines, parameters and files }  
   
 \subsection {Dos and donts}  
   
 In the context of a Held-Suarez type of model experiment (located  
 in the fizhi-hs.cs-32x32x10 verification experiment) with  
 topography, the forcing terms which represent the physics are computed on  
 the modified grid. The forcing terms are computed as functions of the  
 state variables on the modified grid. The tendencies are then interpolated  
 to the standard grid  
88    
89  \subsection {Gridalt Reference}  \subsubsection{Time stepping Sequence}
90    If we write $T_{phys}$ as the temperature (or any other state variable) on the high
91    resolution physics grid, and $T_{dyn}$ as the temperature on the coarse vertical resolution
92    dynamics grid, then:
93    
94    \begin{enumerate}
95    %\itemsep{-0.05in}
96    
97    \item{Compute the tendency due to physics processes.}
98    
99    \item{Advance the physics state: ${{T^{n+1}}^{**}}_{phys}(l) = {T^n}_{phys}(l) + \delta T_{phys}$.}
100    
101    \item{Interpolate the physics tendency to the dynamics grid, and advance the dynamics
102    state by physics and dynamics tendencies:
103    ${T^{n+1}}_{dyn}(L) = {T^n}_{dyn}(L) + \delta T_{dyn}(L) + [\delta T _{phys}(l)](L)$.}
104    
105    \item{Interpolate the dynamics tendency to the physics grid, and update the physics
106    grid due to dynamics tendencies:
107    ${{T^{n+1}}^*}_{phys}(l)$ = ${{T^{n+1}}^{**}}_{phys}(l) + {\delta T_{dyn}(L)}(l)$.}
108    
109    \item{Apply correction term to physics state to account for divergence from dynamics state:
110    ${T^{n+1}}_{phys}(l)$ = ${{T^{n+1}}^*}_{phys}(l) + \gamma \{  T_{dyn}(L) - [T_{phys}(l)](L) \}(l)$.} \\
111    Where $\gamma=1$ here.
112    
113    \end{enumerate}
114    
115    \subsubsection{Interpolation}
116    In order to minimize the correction terms for the state variables on the alternative,
117    higher resolution grid, the vertical interpolation scheme must be constructed so that
118    a dynamics-to-physics interpolation can be exactly reversed with a physics-to-dynamics mapping.
119    The simple scheme employed to achieve this is:
120    
121    Coarse to fine:\
122    For all physics layers l in dynamics layer L, $ T_{phys}(l) = \{T_{dyn}(L)\} = T_{dyn}(L) $.
123    
124    Fine to coarse:\
125    For all physics layers l in dynamics layer L, $T_{dyn}(L) = [T_{phys}(l)] = \int{T_{phys} dp } $.
126    
127    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
128    the physics and dynamics grids, respectively.
129    
130    \subsubsection {Key subroutines, parameters and files }
131    
132    \subsubsection {Dos and donts}
133    
134    \subsubsection {Gridalt Reference}
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