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revision 1.4 by molod, Mon Jul 18 20:45:27 2005 UTC revision 1.5 by molod, Tue Aug 2 21:15:00 2005 UTC
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6    
7  \subsubsection {Introduction}  \subsubsection {Introduction}
8    
9  The gridalt package was developed to allow the high end atmospheric physics  The gridalt package is designed to allow different components of the MITgcm to
10  (fizhi) physics to be run on a separate grid from the hydrodynamics. The package  be run using horizontal and/or vertical grids which are different from the main
11  could (with some user modification) be used in conjunction with other packages  model grid. The gridalt routines handle the definition of the all the various
12  or for other calculations within the GCM. For the case of the atmospheric  alternative grid(s) and the mappings between them and the MITgcm grid.
13  physics, a modified $p^*$ coordinate, which adds additional levels between  The implementation of the gridalt package which allows the high end atmospheric
14  the lower levels of the existing $p^*$ grid (and perhaps between the levels near  physics (fizhi) to be run on a high resolution and quasi terrain-following vertical
15  the tropopause as well), is implemented. The vertical discretization is  grid is documented here.  The package has also (with some user modifications) been used
16  different for each grid point, although it consist of the same number of  for other calculations within the GCM.
17  levels. This is illustrated as follows:  
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  \begin{figure}[htbp]  \begin{figure}[htbp]
44  \vspace*{-0.4in}  \vspace*{-0.4in}
45  \begin{center}  \begin{center}
46  \includegraphics[height=2.4in]{part6/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    must be entirely
51  \end{center}  \end{center}
52  \end{figure}  \end{figure}
53    
54  \vspace*{-0.5in}  The algorithm presented here retains the state variables on the high resolution 'physics'
55  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
56  thermodynamic and humidity equations on the modified (higher resolution)  estimates of the state 'agree' on the coarse resolution grid.  It would have been possible
57  grid, the higher resolution structure of the atmosphere (the boundary  to implement a technique in which the tendencies due to atmospheric physics are computed
58  layer) is retained between calculations. This neccessitates a second  on the high resolution grid and the state variables are retained at low resolution only.
59  set of evolution equations for the atmospheric state variables on the  This, however, for the case of the turbulence parameterization,  would mean that the
60  modified grid. If the equations for the evolution of the state  turbulent kinetic energy source terms, and all the turbulence terms that are written
61  on $p^*$ can be expressed as:  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  \[  \[
72  \left . {\partial U \over {\partial t}} \right |_{p^*}^{total} =  \left . {\partial U \over {\partial t}} \right |_{p^*}^{total} =
73  \left . {\partial U \over {\partial t}} \right |_{p^*}^{dynamics} +  \left . {\partial U \over {\partial t}} \right |_{p^*}^{dynamics} +
74  \left . {\partial U \over {\partial t}} \right |_{p^*}^{physics}  \left . {\partial U \over {\partial t}} \right |_{p^*}^{physics}
75  \]  \]
76  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
77  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:  
78  \[  \[
79  \left . {\partial U \over {\partial t}} \right |_{p^{*m}}^{total} =  \left . {\partial U \over {\partial t}} \right |_{p^{*m}}^{total} =
80  \left . {\partial U \over {\partial t}} \right |_{p^{*m}}^{dynamics} +  \left . {\partial U \over {\partial t}} \right |_{p^{*m}}^{dynamics} +
81  \left . {\partial U \over {\partial t}} \right |_{p^{*m}}^{physics} +  \left . {\partial U \over {\partial t}} \right |_{p^{*m}}^{physics} +
82  \gamma ({\left . U \right |_{p^*}} - {\left . U \right |_{p^{*m}}})  \gamma ({\left . U \right |_{p^*}} - {\left . U \right |_{p^{*m}}})
83  \]  \]
84  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
85  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
86  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
87  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
88  state variables on the $p^*$ grid on some time scale, $\gamma$.  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    
131  \subsubsection {Key subroutines, parameters and files }  \subsubsection {Key subroutines, parameters and files }
132    
133  \subsubsection {Dos and donts}  \subsubsection {Dos and donts}
134    
 In the context of a Held-Suarez type of model experiment (located  
 in the fizhi-gridalt-hs 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  
   
135  \subsubsection {Gridalt Reference}  \subsubsection {Gridalt Reference}
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