--- manual/s_phys_pkgs/text/gridalt.tex 2005/07/18 20:45:27 1.4 +++ manual/s_phys_pkgs/text/gridalt.tex 2005/08/02 21:15:00 1.5 @@ -6,59 +6,130 @@ \subsubsection {Introduction} -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: +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} -\vspace*{-0.5in} -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 calculations. This neccessitates a second -set of evolution equations for the atmospheric state variables on the -modified grid. If the equations for the evolution of the state -on $p^*$ can be expressed as: +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 computed from a -mapping from the modified grid, then an additional set of equations -to govern the evolution of $U$ on the modified grid are written: +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 the $p^*$ 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, $\gamma$. +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} -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 - \subsubsection {Gridalt Reference}