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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 jmc 1.12 The gridalt package \citep{mol:09}
10     is designed to allow different components of MITgcm to
11 molod 1.5 be run using horizontal and/or vertical grids which are different from the main
12     model grid. The gridalt routines handle the definition of the all the various
13     alternative grid(s) and the mappings between them and the MITgcm grid.
14     The implementation of the gridalt package which allows the high end atmospheric
15     physics (fizhi) to be run on a high resolution and quasi terrain-following vertical
16     grid is documented here. The package has also (with some user modifications) been used
17     for other calculations within the GCM.
18    
19     The rationale for implementing the atmospheric physics on a high resolution vertical
20     grid involves the fact that the MITgcm $p^*$ (or any pressure-type) coordinate cannot
21     maintain the vertical resolution near the surface as the bottom topography rises above
22     sea level. The vertical length scales near the ground are small and can vary
23     on small time scales, and the vertical grid must be adequate to resolve them.
24     Many studies with both regional and global atmospheric models have demonstrated the
25     improvements in the simulations when the vertical resolution near the surface is
26     increased (\cite{bm:99,Inn:01,wo:98,breth:99}). Some of the benefit of increased resolution
27     near the surface is realized by employing the higher resolution for the computation of the
28     forcing due to turbulent and convective processes in the atmosphere.
29    
30     The parameterizations of atmospheric subgrid scale processes are all essentially
31     one-dimensional in nature, and the computation of the terms in the equations of
32     motion due to these processes can be performed for the air column over one grid point
33     at a time. The vertical grid on which these computations take place can therefore be
34     entirely independant of the grid on which the equations of motion are integrated, and
35     the 'tendency' terms can be interpolated to the vertical grid on which the equations
36     of motion are integrated. A modified $p^*$ coordinate, which adjusts to the local
37     terrain and adds additional levels between the lower levels of the existing $p^*$ grid
38     (and perhaps between the levels near the tropopause as well), is implemented. The
39     vertical discretization is different for each grid point, although it consist of the
40     same number of levels. Additional 'sponge' levels aloft are added when needed. The levels
41     of the physics grid are constrained to fit exactly into the existing $p^*$ grid, simplifying
42     the mapping between the two vertical coordinates. This is illustrated as follows:
43    
44 molod 1.1 \begin{figure}[htbp]
45     \vspace*{-0.4in}
46     \begin{center}
47 jmc 1.13 \includegraphics[height=2.4in]{s_phys_pkgs/figs/vertical.eps}
48 edhill 1.8 \caption{Vertical discretization for MITgcm (dark grey lines) and for the
49 molod 1.5 atmospheric physics (light grey lines). In this implementation, all MITgcm level
50     interfaces must coincide with atmospheric physics level interfaces.}
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 \[
72 jmc 1.14 \left . \frac{\partial U}{\partial t} \right |_{p^*}^{total} =
73     \left . \frac{\partial U}{\partial t} \right |_{p^*}^{dynamics} +
74     \left . \frac{\partial U}{\partial t} \right |_{p^*}^{physics}
75 molod 1.1 \]
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 \[
79 jmc 1.14 \left . \frac{\partial U}{\partial t} \right |_{p^{*m}}^{total} =
80     \left . \frac{\partial U}{\partial t} \right |_{p^{*m}}^{dynamics} +
81     \left . \frac{\partial U}{\partial t} \right |_{p^{*m}}^{physics} +
82 molod 1.1 \gamma ({\left . U \right |_{p^*}} - {\left . U \right |_{p^{*m}}})
83     \]
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 molod 1.6 \item{Advance the physics state: ${{T^{n+1}}^{**}}_{phys}(l) = {T^n}_{phys}(l) + \delta T_{phys}$.}
101 molod 1.5
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 molod 1.6 ${{T^{n+1}}^*}_{phys}(l)$ = ${{T^{n+1}}^{**}}_{phys}(l) + {\delta T_{dyn}(L)}(l)$.}
109 molod 1.5
110     \item{Apply correction term to physics state to account for divergence from dynamics state:
111 molod 1.6 ${T^{n+1}}_{phys}(l)$ = ${{T^{n+1}}^*}_{phys}(l) + \gamma \{ T_{dyn}(L) - [T_{phys}(l)](L) \}(l)$.} \\
112 molod 1.5 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 molod 1.7 The simple scheme employed to achieve this is:\\
121 molod 1.5
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 molod 1.7 For all physics layers l in dynamics layer L, $T_{dyn}(L) = [T_{phys}(l)] = \int{T_{phys} dp } $.\\
127 molod 1.5
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 1.7 \noindent
134     One of the central elements of the gridalt package is the routine which
135     is called from subroutine gridalt\_initialise to define the grid to be
136     used for the high end physics calculations. Routine make\_phys\_grid
137     passes back the parameters which define the grid, ultimately stored
138     in the common block gridalt\_mapping.
139    
140     \begin{verbatim}
141     subroutine make_phys_grid(drF,hfacC,im1,im2,jm1,jm2,Nr,
142     . Nsx,Nsy,i1,i2,j1,j2,bi,bj,Nrphys,Lbot,dpphys,numlevphys,nlperdyn)
143     c***********************************************************************
144     c Purpose: Define the grid that the will be used to run the high-end
145     c atmospheric physics.
146     c
147     c Algorithm: Fit additional levels of some (~) known thickness in
148     c between existing levels of the grid used for the dynamics
149     c
150     c Need: Information about the dynamics grid vertical spacing
151     c
152     c Input: drF - delta r (p*) edge-to-edge
153     c hfacC - fraction of grid box above topography
154     c im1, im2 - beginning and ending i - dimensions
155     c jm1, jm2 - beginning and ending j - dimensions
156     c Nr - number of levels in dynamics grid
157     c Nsx,Nsy - number of processes in x and y direction
158     c i1, i2 - beginning and ending i - index to fill
159     c j1, j2 - beginning and ending j - index to fill
160     c bi, bj - x-dir and y-dir index of process
161     c Nrphys - number of levels in physics grid
162     c
163     c Output: dpphys - delta r (p*) edge-to-edge of physics grid
164     c numlevphys - number of levels used in the physics
165     c nlperdyn - physics level number atop each dynamics layer
166     c
167     c NOTES: 1) Pressure levs are built up from bottom, using p0, ps and dp:
168     c p(i,j,k)=p(i,j,k-1) + dp(k)*ps(i,j)/p0(i,j)
169     c 2) Output dp's are aligned to fit EXACTLY between existing
170     c levels of the dynamics vertical grid
171     c 3) IMPORTANT! This routine assumes the levels are numbered
172     c from the bottom up, ie, level 1 is the surface.
173     c IT WILL NOT WORK OTHERWISE!!!
174     c 4) This routine does NOT work for surface pressures less
175     c (ie, above in the atmosphere) than about 350 mb
176     c***********************************************************************
177     \end{verbatim}
178    
179     \noindent In the case of the grid used to compute the atmospheric physical
180     forcing (fizhi package), the locations of the grid points move in time with
181     the MITgcm $p^*$ coordinate, and subroutine gridalt\_update is called during
182     the run to update the locations of the grid points:
183    
184     \begin{verbatim}
185     subroutine gridalt_update(myThid)
186     c***********************************************************************
187     c Purpose: Update the pressure thicknesses of the layers of the
188     c alternative vertical grid (used now for atmospheric physics).
189     c
190     c Calculate: dpphys - new delta r (p*) edge-to-edge of physics grid
191     c using dpphys0 (initial value) and rstarfacC
192     c***********************************************************************
193     \end{verbatim}
194    
195     \noindent The gridalt package also supplies utility routines which perform
196     the mappings from one grid to the other. These routines are called from the
197     code which computes the fields on the alternative (fizhi) grid.
198    
199     \begin{verbatim}
200     subroutine dyn2phys(qdyn,pedyn,im1,im2,jm1,jm2,lmdyn,Nsx,Nsy,
201     . idim1,idim2,jdim1,jdim2,bi,bj,windphy,pephy,Lbot,lmphy,nlperdyn,
202     . flg,qphy)
203     C***********************************************************************
204     C Purpose:
205     C To interpolate an arbitrary quantity from the 'dynamics' eta (pstar)
206     C grid to the higher resolution physics grid
207     C Algorithm:
208     C Routine works one layer (edge to edge pressure) at a time.
209     C Dynamics -> Physics retains the dynamics layer mean value,
210     C weights the field either with the profile of the physics grid
211     C wind speed (for U and V fields), or uniformly (T and Q)
212     C
213     C Input:
214     C qdyn..... [im,jm,lmdyn] Arbitrary Quantity on Input Grid
215     C pedyn.... [im,jm,lmdyn+1] Pressures at bottom edges of input levels
216     C im1,2 ... Limits for Longitude Dimension of Input
217     C jm1,2 ... Limits for Latitude Dimension of Input
218     C lmdyn.... Vertical Dimension of Input
219     C Nsx...... Number of processes in x-direction
220     C Nsy...... Number of processes in y-direction
221     C idim1,2.. Beginning and ending i-values to calculate
222     C jdim1,2.. Beginning and ending j-values to calculate
223     C bi....... Index of process number in x-direction
224     C bj....... Index of process number in x-direction
225     C windphy.. [im,jm,lmphy] Magnitude of the wind on the output levels
226     C pephy.... [im,jm,lmphy+1] Pressures at bottom edges of output levels
227     C lmphy.... Vertical Dimension of Output
228     C nlperdyn. [im,jm,lmdyn] Highest Physics level in each dynamics level
229     C flg...... Flag to indicate field type (0 for T or Q, 1 for U or V)
230     C
231     C Output:
232     C qphy..... [im,jm,lmphy] Quantity at output grid (physics grid)
233     C
234     C Notes:
235     C 1) This algorithm assumes that the output (physics) grid levels
236     C fit exactly into the input (dynamics) grid levels
237     C***********************************************************************
238     \end{verbatim}
239    
240     \noindent And similarly, gridalt contains subroutine phys2dyn.
241    
242 molod 1.9 \subsubsection {Gridalt Diagnostics}
243     \label{sec:pkg:gridalt:diagnostics}
244    
245 edhill 1.10 {\footnotesize
246 molod 1.9 \begin{verbatim}
247    
248     ------------------------------------------------------------------------
249     <-Name->|Levs|<-parsing code->|<-- Units -->|<- Tile (max=80c)
250     ------------------------------------------------------------------------
251     DPPHYS | 20 |SM ML |Pascal |Pressure Thickness of Layers on Fizhi Grid
252     \end{verbatim}
253 edhill 1.10 }
254 molod 1.9
255 molod 1.4 \subsubsection {Dos and donts}
256 molod 1.1
257 molod 1.4 \subsubsection {Gridalt Reference}
258 molod 1.11
259     \subsubsection{Experiments and tutorials that use gridalt}
260     \label{sec:pkg:gridalt:experiments}
261    
262     \begin{itemize}
263     \item{Fizhi experiment, in fizhi-cs-32x32x10 verification directory }
264     \end{itemize}

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