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\subsection{Gridalt - Alternate Grid Package} |
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\label{sec:pkg:gridalt} |
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\begin{rawhtml} |
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<!-- CMIREDIR:package_gridalt: --> |
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\end{rawhtml} |
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\subsubsection {Introduction} |
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The gridalt package is designed to allow different components of MITgcm to |
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be run using horizontal and/or vertical grids which are different from the main |
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model grid. The gridalt routines handle the definition of the all the various |
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alternative grid(s) and the mappings between them and the MITgcm grid. |
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The implementation of the gridalt package which allows the high end atmospheric |
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physics (fizhi) to be run on a high resolution and quasi terrain-following vertical |
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grid is documented here. The package has also (with some user modifications) been used |
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for other calculations within the GCM. |
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The rationale for implementing the atmospheric physics on a high resolution vertical |
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grid involves the fact that the MITgcm $p^*$ (or any pressure-type) coordinate cannot |
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maintain the vertical resolution near the surface as the bottom topography rises above |
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sea level. The vertical length scales near the ground are small and can vary |
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on small time scales, and the vertical grid must be adequate to resolve them. |
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Many studies with both regional and global atmospheric models have demonstrated the |
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improvements in the simulations when the vertical resolution near the surface is |
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increased (\cite{bm:99,Inn:01,wo:98,breth:99}). Some of the benefit of increased resolution |
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near the surface is realized by employing the higher resolution for the computation of the |
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forcing due to turbulent and convective processes in the atmosphere. |
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The parameterizations of atmospheric subgrid scale processes are all essentially |
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one-dimensional in nature, and the computation of the terms in the equations of |
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motion due to these processes can be performed for the air column over one grid point |
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at a time. The vertical grid on which these computations take place can therefore be |
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entirely independant of the grid on which the equations of motion are integrated, and |
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the 'tendency' terms can be interpolated to the vertical grid on which the equations |
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of motion are integrated. A modified $p^*$ coordinate, which adjusts to the local |
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terrain and adds additional levels between the lower levels of the existing $p^*$ grid |
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(and perhaps between the levels near the tropopause as well), is implemented. The |
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vertical discretization is different for each grid point, although it consist of the |
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same number of levels. Additional 'sponge' levels aloft are added when needed. The levels |
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of the physics grid are constrained to fit exactly into the existing $p^*$ grid, simplifying |
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the mapping between the two vertical coordinates. This is illustrated as follows: |
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\begin{figure}[htbp] |
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\vspace*{-0.4in} |
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\begin{center} |
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\includegraphics[height=2.4in]{part6/vertical.eps} |
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\caption{Vertical discretization for MITgcm (dark grey lines) and for the |
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atmospheric physics (light grey lines). In this implementation, all MITgcm level |
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interfaces must coincide with atmospheric physics level interfaces.} |
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\end{center} |
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\end{figure} |
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The algorithm presented here retains the state variables on the high resolution 'physics' |
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grid as well as on the coarser resolution 'dynamics` grid, and ensures that the two |
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estimates of the state 'agree' on the coarse resolution grid. It would have been possible |
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to implement a technique in which the tendencies due to atmospheric physics are computed |
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on the high resolution grid and the state variables are retained at low resolution only. |
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This, however, for the case of the turbulence parameterization, would mean that the |
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turbulent kinetic energy source terms, and all the turbulence terms that are written |
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in terms of gradients of the mean flow, cannot really be computed making use of the fine |
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structure in the vertical. |
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\subsubsection{Equations on Both Grids} |
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In addition to computing the physical forcing terms of the momentum, thermodynamic and humidity |
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equations on the modified (higher resolution) grid, the higher resolution structure of the |
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atmosphere (the boundary layer) is retained between physics calculations. This neccessitates |
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a second set of evolution equations for the atmospheric state variables on the modified grid. |
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If the equation for the evolution of $U$ on $p^*$ can be expressed as: |
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\[ |
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\left . {\partial U \over {\partial t}} \right |_{p^*}^{total} = |
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\left . {\partial U \over {\partial t}} \right |_{p^*}^{dynamics} + |
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\left . {\partial U \over {\partial t}} \right |_{p^*}^{physics} |
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\] |
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where the physics forcing terms on $p^*$ have been mapped from the modified grid, then an additional |
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equation to govern the evolution of $U$ (for example) on the modified grid is written: |
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\[ |
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\left . {\partial U \over {\partial t}} \right |_{p^{*m}}^{total} = |
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\left . {\partial U \over {\partial t}} \right |_{p^{*m}}^{dynamics} + |
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\left . {\partial U \over {\partial t}} \right |_{p^{*m}}^{physics} + |
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\gamma ({\left . U \right |_{p^*}} - {\left . U \right |_{p^{*m}}}) |
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\] |
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where $p^{*m}$ refers to the modified higher resolution grid, and the dynamics forcing terms have |
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been mapped from $p^*$ space. The last term on the RHS is a relaxation term, meant to constrain |
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the state variables on the modified vertical grid to `track' the state variables on the $p^*$ grid |
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on some time scale, governed by $\gamma$. In the present implementation, $\gamma = 1$, requiring |
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an immediate agreement between the two 'states'. |
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\subsubsection{Time stepping Sequence} |
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If we write $T_{phys}$ as the temperature (or any other state variable) on the high |
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resolution physics grid, and $T_{dyn}$ as the temperature on the coarse vertical resolution |
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dynamics grid, then: |
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\begin{enumerate} |
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%\itemsep{-0.05in} |
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\item{Compute the tendency due to physics processes.} |
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\item{Advance the physics state: ${{T^{n+1}}^{**}}_{phys}(l) = {T^n}_{phys}(l) + \delta T_{phys}$.} |
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\item{Interpolate the physics tendency to the dynamics grid, and advance the dynamics |
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state by physics and dynamics tendencies: |
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${T^{n+1}}_{dyn}(L) = {T^n}_{dyn}(L) + \delta T_{dyn}(L) + [\delta T _{phys}(l)](L)$.} |
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\item{Interpolate the dynamics tendency to the physics grid, and update the physics |
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grid due to dynamics tendencies: |
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${{T^{n+1}}^*}_{phys}(l)$ = ${{T^{n+1}}^{**}}_{phys}(l) + {\delta T_{dyn}(L)}(l)$.} |
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\item{Apply correction term to physics state to account for divergence from dynamics state: |
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${T^{n+1}}_{phys}(l)$ = ${{T^{n+1}}^*}_{phys}(l) + \gamma \{ T_{dyn}(L) - [T_{phys}(l)](L) \}(l)$.} \\ |
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Where $\gamma=1$ here. |
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\end{enumerate} |
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\subsubsection{Interpolation} |
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In order to minimize the correction terms for the state variables on the alternative, |
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higher resolution grid, the vertical interpolation scheme must be constructed so that |
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a dynamics-to-physics interpolation can be exactly reversed with a physics-to-dynamics mapping. |
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The simple scheme employed to achieve this is:\\ |
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Coarse to fine:\ |
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For all physics layers l in dynamics layer L, $ T_{phys}(l) = \{T_{dyn}(L)\} = T_{dyn}(L) $. |
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Fine to coarse:\ |
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For all physics layers l in dynamics layer L, $T_{dyn}(L) = [T_{phys}(l)] = \int{T_{phys} dp } $.\\ |
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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 |
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the physics and dynamics grids, respectively. |
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\subsubsection {Key subroutines, parameters and files } |
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\noindent |
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One of the central elements of the gridalt package is the routine which |
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is called from subroutine gridalt\_initialise to define the grid to be |
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used for the high end physics calculations. Routine make\_phys\_grid |
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passes back the parameters which define the grid, ultimately stored |
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in the common block gridalt\_mapping. |
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\begin{verbatim} |
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subroutine make_phys_grid(drF,hfacC,im1,im2,jm1,jm2,Nr, |
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. Nsx,Nsy,i1,i2,j1,j2,bi,bj,Nrphys,Lbot,dpphys,numlevphys,nlperdyn) |
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c*********************************************************************** |
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c Purpose: Define the grid that the will be used to run the high-end |
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c atmospheric physics. |
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c |
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c Algorithm: Fit additional levels of some (~) known thickness in |
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c between existing levels of the grid used for the dynamics |
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c |
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c Need: Information about the dynamics grid vertical spacing |
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c |
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c Input: drF - delta r (p*) edge-to-edge |
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c hfacC - fraction of grid box above topography |
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c im1, im2 - beginning and ending i - dimensions |
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c jm1, jm2 - beginning and ending j - dimensions |
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c Nr - number of levels in dynamics grid |
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c Nsx,Nsy - number of processes in x and y direction |
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c i1, i2 - beginning and ending i - index to fill |
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c j1, j2 - beginning and ending j - index to fill |
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c bi, bj - x-dir and y-dir index of process |
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c Nrphys - number of levels in physics grid |
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c |
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c Output: dpphys - delta r (p*) edge-to-edge of physics grid |
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c numlevphys - number of levels used in the physics |
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c nlperdyn - physics level number atop each dynamics layer |
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c |
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c NOTES: 1) Pressure levs are built up from bottom, using p0, ps and dp: |
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c p(i,j,k)=p(i,j,k-1) + dp(k)*ps(i,j)/p0(i,j) |
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c 2) Output dp's are aligned to fit EXACTLY between existing |
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c levels of the dynamics vertical grid |
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c 3) IMPORTANT! This routine assumes the levels are numbered |
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c from the bottom up, ie, level 1 is the surface. |
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c IT WILL NOT WORK OTHERWISE!!! |
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c 4) This routine does NOT work for surface pressures less |
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c (ie, above in the atmosphere) than about 350 mb |
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c*********************************************************************** |
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\end{verbatim} |
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\noindent In the case of the grid used to compute the atmospheric physical |
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forcing (fizhi package), the locations of the grid points move in time with |
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the MITgcm $p^*$ coordinate, and subroutine gridalt\_update is called during |
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the run to update the locations of the grid points: |
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\begin{verbatim} |
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subroutine gridalt_update(myThid) |
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c*********************************************************************** |
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c Purpose: Update the pressure thicknesses of the layers of the |
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c alternative vertical grid (used now for atmospheric physics). |
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c |
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c Calculate: dpphys - new delta r (p*) edge-to-edge of physics grid |
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c using dpphys0 (initial value) and rstarfacC |
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c*********************************************************************** |
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\end{verbatim} |
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\noindent The gridalt package also supplies utility routines which perform |
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the mappings from one grid to the other. These routines are called from the |
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code which computes the fields on the alternative (fizhi) grid. |
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\begin{verbatim} |
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subroutine dyn2phys(qdyn,pedyn,im1,im2,jm1,jm2,lmdyn,Nsx,Nsy, |
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. idim1,idim2,jdim1,jdim2,bi,bj,windphy,pephy,Lbot,lmphy,nlperdyn, |
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. flg,qphy) |
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C*********************************************************************** |
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C Purpose: |
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C To interpolate an arbitrary quantity from the 'dynamics' eta (pstar) |
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C grid to the higher resolution physics grid |
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C Algorithm: |
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C Routine works one layer (edge to edge pressure) at a time. |
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C Dynamics -> Physics retains the dynamics layer mean value, |
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C weights the field either with the profile of the physics grid |
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C wind speed (for U and V fields), or uniformly (T and Q) |
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C |
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C Input: |
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C qdyn..... [im,jm,lmdyn] Arbitrary Quantity on Input Grid |
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C pedyn.... [im,jm,lmdyn+1] Pressures at bottom edges of input levels |
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C im1,2 ... Limits for Longitude Dimension of Input |
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C jm1,2 ... Limits for Latitude Dimension of Input |
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C lmdyn.... Vertical Dimension of Input |
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C Nsx...... Number of processes in x-direction |
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C Nsy...... Number of processes in y-direction |
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C idim1,2.. Beginning and ending i-values to calculate |
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C jdim1,2.. Beginning and ending j-values to calculate |
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C bi....... Index of process number in x-direction |
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C bj....... Index of process number in x-direction |
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C windphy.. [im,jm,lmphy] Magnitude of the wind on the output levels |
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C pephy.... [im,jm,lmphy+1] Pressures at bottom edges of output levels |
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C lmphy.... Vertical Dimension of Output |
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C nlperdyn. [im,jm,lmdyn] Highest Physics level in each dynamics level |
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C flg...... Flag to indicate field type (0 for T or Q, 1 for U or V) |
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C |
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C Output: |
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C qphy..... [im,jm,lmphy] Quantity at output grid (physics grid) |
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C |
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C Notes: |
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C 1) This algorithm assumes that the output (physics) grid levels |
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C fit exactly into the input (dynamics) grid levels |
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C*********************************************************************** |
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\end{verbatim} |
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\noindent And similarly, gridalt contains subroutine phys2dyn. |
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\subsubsection {Dos and donts} |
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\subsubsection {Gridalt Reference} |
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