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% $Header: /u/gcmpack/manual/s_examples/global_oce_latlon/climatalogical_ogcm.tex,v 1.24 2013/05/15 22:47:12 jmc Exp $ |
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% $Name: $ |
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|
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\section[Global Ocean MITgcm Example]{Global Ocean Simulation at $4^\circ$ Resolution} |
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%\label{www:tutorials} |
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\label{sec:eg-global} |
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\begin{rawhtml} |
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<!-- CMIREDIR:eg-global: --> |
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\end{rawhtml} |
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\begin{center} |
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(in directory: {\it verification/tutorial\_global\_oce\_latlon/}) |
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\end{center} |
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|
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\bodytext{bgcolor="#FFFFFFFF"} |
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|
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\noindent {\bf WARNING: the description of this experiment is not complete. |
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In particular, many parameters are not yet described.}\\ |
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|
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%\begin{center} |
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%{\Large \bf Using MITgcm to Simulate Global Climatological Ocean Circulation |
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%At Four Degree Resolution with Asynchronous Time Stepping} |
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% |
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%\vspace*{4mm} |
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% |
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%\vspace*{3mm} |
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%{\large May 2001} |
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%\end{center} |
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|
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This example experiment demonstrates using the MITgcm to simulate the |
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planetary ocean circulation. The simulation is configured with |
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realistic geography and bathymetry on a $4^{\circ} \times 4^{\circ}$ |
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spherical polar grid. The files for this experiment are in the |
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verification directory under tutorial\_global\_oce\_latlon. Fifteen |
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levels are used in the vertical, ranging in thickness from $50\,{\rm |
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m}$ at the surface to $690\,{\rm m}$ at depth, giving a maximum |
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model depth of $5200\,{\rm m}$. |
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Different time-steps are used to accelerate the convergence to |
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equilibrium \cite[]{bryan:84} so that, at this resolution, |
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the configuration can be integrated forward for thousands of years |
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on a single processor desktop computer. |
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\\ |
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\subsection{Overview} |
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%\label{www:tutorials} |
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|
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The model is forced with climatological wind stress data from |
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\citet{trenberth90} and NCEP surface flux data from |
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\citet{kalnay96}. Climatological data \citep{Levitus94} is |
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used to initialize the model hydrography. \citeauthor{Levitus94} seasonal |
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climatology data is also used throughout the calculation to provide |
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additional air-sea fluxes. These fluxes are combined with the NCEP |
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climatological estimates of surface heat flux, resulting in a mixed |
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boundary condition of the style described in \citet{Haney}. |
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Altogether, this yields the following forcing applied in the model |
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surface layer. |
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|
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\begin{eqnarray} |
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\label{eq:eg-global-global_forcing} |
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\label{eq:eg-global-global_forcing_fu} |
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{\cal F}_{u} & = & \frac{\tau_{x}}{\rho_{0} \Delta z_{s}} |
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\\ |
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\label{eq:eg-global-global_forcing_fv} |
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{\cal F}_{v} & = & \frac{\tau_{y}}{\rho_{0} \Delta z_{s}} |
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\\ |
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\label{eq:eg-global-global_forcing_ft} |
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{\cal F}_{\theta} & = & - \lambda_{\theta} ( \theta - \theta^{\ast} ) |
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- \frac{1}{C_{p} \rho_{0} \Delta z_{s}}{\cal Q} |
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\\ |
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\label{eq:eg-global-global_forcing_fs} |
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{\cal F}_{s} & = & - \lambda_{s} ( S - S^{\ast} ) |
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+ \frac{S_{0}}{\Delta z_{s}}({\cal E} - {\cal P} - {\cal R}) |
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\end{eqnarray} |
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|
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\noindent where ${\cal F}_{u}$, ${\cal F}_{v}$, ${\cal F}_{\theta}$, |
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${\cal F}_{s}$ are the forcing terms in the zonal and meridional |
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momentum and in the potential temperature and salinity |
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equations respectively. |
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The term $\Delta z_{s}$ represents the top ocean layer thickness in |
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meters. |
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It is used in conjunction with a reference density, $\rho_{0}$ |
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(here set to $999.8\,{\rm kg\,m^{-3}}$), a |
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reference salinity, $S_{0}$ (here set to 35~ppt), |
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and a specific heat capacity, $C_{p}$ (here set to |
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$4000~{\rm J}~^{\circ}{\rm C}^{-1}~{\rm kg}^{-1}$), to convert |
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input dataset values into time tendencies of |
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potential temperature (with units of $^{\circ}{\rm C}~{\rm s}^{-1}$), |
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salinity (with units ${\rm ppt}~s^{-1}$) and |
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velocity (with units ${\rm m}~{\rm s}^{-2}$). |
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The externally supplied forcing fields used in this |
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experiment are $\tau_{x}$, $\tau_{y}$, $\theta^{\ast}$, $S^{\ast}$, |
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$\cal{Q}$ and $\cal{E}-\cal{P}-\cal{R}$. The wind stress fields ($\tau_x$, $\tau_y$) |
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have units of ${\rm N}~{\rm m}^{-2}$. The temperature forcing fields |
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($\theta^{\ast}$ and $Q$) have units of $^{\circ}{\rm C}$ and ${\rm W}~{\rm m}^{-2}$ |
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respectively. The salinity forcing fields ($S^{\ast}$ and |
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$\cal{E}-\cal{P}-\cal{R}$) have units of ${\rm ppt}$ and ${\rm m}~{\rm s}^{-1}$ |
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respectively. The source files and procedures for ingesting this data into the |
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simulation are described in the experiment configuration discussion in section |
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\ref{sec:eg-global-clim_ocn_examp_exp_config}. |
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|
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|
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\subsection{Discrete Numerical Configuration} |
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%\label{www:tutorials} |
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|
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|
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The model is configured in hydrostatic form. The domain is |
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discretised with a uniform grid spacing in latitude and longitude on |
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the sphere $\Delta \phi=\Delta \lambda=4^{\circ}$, so that there are |
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ninety grid cells in the zonal and forty in the meridional |
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direction. The internal model coordinate variables $x$ and $y$ are |
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initialized according to |
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\begin{eqnarray} |
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x=r\cos(\phi),~\Delta x & = &r\cos(\Delta \phi) \\ |
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y=r\lambda,~\Delta y &= &r\Delta \lambda |
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\end{eqnarray} |
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|
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Arctic polar regions are not |
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included in this experiment. Meridionally the model extends from |
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$80^{\circ}{\rm S}$ to $80^{\circ}{\rm N}$. |
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Vertically the model is configured with fifteen layers with the |
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following thicknesses: |
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$\Delta z_{1} = 50\,{\rm m},$\\ |
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$\Delta z_{2} = 70\,{\rm m},\, |
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\Delta z_{3} = 100\,{\rm m},\, |
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\Delta z_{4} = 140\,{\rm m},\, |
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\Delta z_{5} = 190\,{\rm m},\, |
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\Delta z_{6} = 240\,{\rm m},\, |
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\Delta z_{7} = 290\,{\rm m},\, |
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\Delta z_{8} = 340\,{\rm m},$\\ |
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$\Delta z_{9} = 390\,{\rm m},\, |
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\Delta z_{10}= 440\,{\rm m},\, |
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\Delta z_{11}= 490\,{\rm m},\, |
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\Delta z_{12}= 540\,{\rm m},\, |
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\Delta z_{13}= 590\,{\rm m},\, |
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\Delta z_{14}= 640\,{\rm m},\, |
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\Delta z_{15}= 690\,{\rm m}$\\ |
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(here the numeric subscript indicates the model level index number, ${\tt k}$) to |
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give a total depth, $H$, of $-5200{\rm m}$. |
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The implicit free surface form of the pressure equation described in |
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\citet{marshall:97a} is employed. A Laplacian operator, $\nabla^2$, provides viscous |
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dissipation. Thermal and haline diffusion is also represented by a Laplacian operator. |
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|
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Wind-stress forcing is added to the momentum equations in (\ref{eq:eg-global-model_equations}) |
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for both the zonal flow, $u$ and the meridional flow $v$, according to equations |
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(\ref{eq:eg-global-global_forcing_fu}) and (\ref{eq:eg-global-global_forcing_fv}). |
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Thermodynamic forcing inputs are added to the equations |
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in (\ref{eq:eg-global-model_equations}) for |
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potential temperature, $\theta$, and salinity, $S$, according to equations |
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(\ref{eq:eg-global-global_forcing_ft}) and (\ref{eq:eg-global-global_forcing_fs}). |
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This produces a set of equations solved in this configuration as follows: |
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|
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\begin{eqnarray} |
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\label{eq:eg-global-model_equations} |
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\frac{Du}{Dt} - fv + |
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\frac{1}{\rho}\frac{\partial p^{'}}{\partial x} - |
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\nabla_{h}\cdot A_{h}\nabla_{h}u - |
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\frac{\partial}{\partial z}A_{z}\frac{\partial u}{\partial z} |
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& = & |
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\begin{cases} |
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{\cal F}_u & \text{(surface)} \\ |
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0 & \text{(interior)} |
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\end{cases} |
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\\ |
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\frac{Dv}{Dt} + fu + |
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\frac{1}{\rho}\frac{\partial p^{'}}{\partial y} - |
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\nabla_{h}\cdot A_{h}\nabla_{h}v - |
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\frac{\partial}{\partial z}A_{z}\frac{\partial v}{\partial z} |
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& = & |
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\begin{cases} |
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{\cal F}_v & \text{(surface)} \\ |
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0 & \text{(interior)} |
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\end{cases} |
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\\ |
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\frac{\partial \eta}{\partial t} + \nabla_{h}\cdot \vec{u} |
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&=& |
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0 |
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\\ |
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\frac{D\theta}{Dt} - |
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\nabla_{h}\cdot K_{h}\nabla_{h}\theta |
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- \frac{\partial}{\partial z}\Gamma(K_{z})\frac{\partial\theta}{\partial z} |
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& = & |
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\begin{cases} |
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{\cal F}_\theta & \text{(surface)} \\ |
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0 & \text{(interior)} |
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\end{cases} |
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\\ |
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\frac{D s}{Dt} - |
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\nabla_{h}\cdot K_{h}\nabla_{h}s |
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- \frac{\partial}{\partial z}\Gamma(K_{z})\frac{\partial s}{\partial z} |
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& = & |
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\begin{cases} |
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{\cal F}_s & \text{(surface)} \\ |
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0 & \text{(interior)} |
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\end{cases} |
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\\ |
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g\rho_{0} \eta + \int^{0}_{-z}\rho^{'} dz & = & p^{'} |
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\end{eqnarray} |
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|
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\noindent where $u=\frac{Dx}{Dt}=r \cos(\phi)\frac{D \lambda}{Dt}$ and |
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$v=\frac{Dy}{Dt}=r \frac{D \phi}{Dt}$ |
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are the zonal and meridional components of the |
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flow vector, $\vec{u}$, on the sphere. As described in |
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MITgcm Numerical Solution Procedure \ref{chap:discretization}, the time |
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evolution of potential temperature, $\theta$, equation is solved prognostically. |
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The total pressure, $p$, is diagnosed by summing pressure due to surface |
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elevation $\eta$ and the hydrostatic pressure. |
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\\ |
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|
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\subsubsection{Numerical Stability Criteria} |
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%\label{www:tutorials} |
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|
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The Laplacian dissipation coefficient, $A_{h}$, is set to $5 \times 10^5 m s^{-1}$. |
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This value is chosen to yield a Munk layer width \citep{adcroft:95}, |
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\begin{eqnarray} |
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\label{eq:eg-global-munk_layer} |
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&& M_{w} = \pi ( \frac { A_{h} }{ \beta } )^{\frac{1}{3}} |
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\end{eqnarray} |
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|
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\noindent of $\approx 600$km. This is greater than the model |
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resolution in low-latitudes, $\Delta x \approx 400{\rm km}$, ensuring that the frictional |
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boundary layer is adequately resolved. |
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\\ |
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|
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\noindent The model is stepped forward with a time step $\Delta |
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t_{\theta}=24~{\rm hours}$ for thermodynamic variables and $\Delta |
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t_{v}=30~{\rm minutes}$ for momentum terms. With this time step, |
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the stability parameter to the horizontal Laplacian friction |
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\citep{adcroft:95} |
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\begin{eqnarray} |
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\label{eq:eg-global-laplacian_stability} |
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&& S_{l} = 4 \frac{A_{h} \Delta t_{v}}{{\Delta x}^2} |
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\end{eqnarray} |
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|
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\noindent evaluates to 0.6 at a latitude of $\phi=80^{\circ}$, which |
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is above the 0.3 upper limit for stability, but the zonal grid spacing |
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$\Delta x$ is smallest at $\phi=80^{\circ}$ where $\Delta |
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x=r\cos(\phi)\Delta \phi\approx 77{\rm km}$ and the stability |
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criterion is already met 1 grid cell equatorwards (at $\phi=76^{\circ}$). |
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|
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|
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\noindent The vertical dissipation coefficient, $A_{z}$, is set to |
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$1\times10^{-3} {\rm m}^2{\rm s}^{-1}$. The associated stability limit |
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\begin{eqnarray} |
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\label{eq:eg-global-laplacian_stability_z} |
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&& S_{l} = 4 \frac{A_{z} \Delta t_{v}}{{\Delta z}^2} |
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\end{eqnarray} |
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|
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\noindent evaluates to $0.0029$ for the smallest model |
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level spacing ($\Delta z_{1}=50{\rm m}$) which is well below |
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the upper stability limit. |
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\\ |
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|
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% The values of the horizontal ($K_{h}$) and vertical ($K_{z}$) diffusion coefficients |
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% for both temperature and salinity are set to $1 \times 10^{3}~{\rm m}^{2}{\rm s}^{-1}$ |
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% and $3 \times 10^{-5}~{\rm m}^{2}{\rm s}^{-1}$ respectively. The stability limit |
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% related to $K_{h}$ will be at $\phi=80^{\circ}$ where $\Delta x \approx 77 {\rm km}$. |
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% Here the stability parameter |
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% \begin{eqnarray} |
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% \label{eq:eg-global-laplacian_stability_xtheta} |
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% S_{l} = \frac{4 K_{h} \Delta t_{\theta}}{{\Delta x}^2} |
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% \end{eqnarray} |
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% evaluates to $0.07$, well below the stability limit of $S_{l} \approx 0.5$. The |
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% stability parameter related to $K_{z}$ |
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% \begin{eqnarray} |
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% \label{eq:eg-global-laplacian_stability_ztheta} |
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% S_{l} = \frac{4 K_{z} \Delta t_{\theta}}{{\Delta z}^2} |
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% \end{eqnarray} |
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% evaluates to $0.005$ for $\min(\Delta z)=50{\rm m}$, well below the stability limit |
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% of $S_{l} \approx 0.5$. |
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% \\ |
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|
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\noindent The numerical stability for inertial oscillations |
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\citep{adcroft:95} |
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|
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\begin{eqnarray} |
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\label{eq:eg-global-inertial_stability} |
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&& S_{i} = f^{2} {\Delta t_v}^2 |
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\end{eqnarray} |
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|
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\noindent evaluates to $0.07$ for |
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$f=2\omega\sin(80^{\circ})=1.43\times10^{-4}~{\rm s}^{-1}$, which is |
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below the $S_{i} < 1$ upper limit for stability. |
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\\ |
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|
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\noindent The advective CFL \citep{adcroft:95} for a extreme maximum |
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horizontal flow |
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speed of $ | \vec{u} | = 2 ms^{-1}$ |
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|
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\begin{eqnarray} |
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\label{eq:eg-global-cfl_stability} |
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&& S_{a} = \frac{| \vec{u} | \Delta t_{v}}{ \Delta x} |
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\end{eqnarray} |
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|
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\noindent evaluates to $5 \times 10^{-2}$. This is well below the stability |
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limit of 0.5. |
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\\ |
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|
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\noindent The stability parameter for internal gravity waves propagating |
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with a maximum speed of $c_{g}=10~{\rm ms}^{-1}$ |
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\citep{adcroft:95} |
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|
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\begin{eqnarray} |
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\label{eq:eg-global-gfl_stability} |
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&& S_{c} = \frac{c_{g} \Delta t_{v}}{ \Delta x} |
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\end{eqnarray} |
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|
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\noindent evaluates to $2.3 \times 10^{-1}$. This is close to the linear |
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stability limit of 0.5. |
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|
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\subsection{Experiment Configuration} |
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%\label{www:tutorials} |
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\label{sec:eg-global-clim_ocn_examp_exp_config} |
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|
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The model configuration for this experiment resides under the |
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directory {\it tutorial\_global\_oce\_latlon/}. The experiment files |
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|
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\begin{itemize} |
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\item {\it input/data} |
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\item {\it input/data.pkg} |
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\item {\it input/eedata}, |
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\item {\it input/trenberth\_taux.bin}, |
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\item {\it input/trenberth\_tauy.bin}, |
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\item {\it input/lev\_s.bin}, |
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\item {\it input/lev\_t.bin}, |
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\item {\it input/lev\_sss.bin}, |
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\item {\it input/lev\_sst.bin}, |
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\item {\it input/bathymetry.bin}, |
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%\item {\it code/CPP\_EEOPTIONS.h} |
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%\item {\it code/CPP\_OPTIONS.h}, |
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\item {\it code/SIZE.h}. |
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\end{itemize} |
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contain the code customizations and parameter settings for these |
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experiments. Below we describe the customizations |
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to these files associated with this experiment. |
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|
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\subsubsection{Driving Datasets} |
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%\label{www:tutorials} |
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|
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%% New figures are included before |
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%% Relaxation temperature |
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%\begin{figure} |
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%\centering |
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%\includegraphics[]{relax_temperature.eps} |
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%\caption{Relaxation temperature for January} |
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%\label{fig:relax_temperature} |
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%\end{figure} |
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|
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%% Relaxation salinity |
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%\begin{figure} |
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%\centering |
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%\includegraphics[]{relax_salinity.eps} |
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%\caption{Relaxation salinity for January} |
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%\label{fig:relax_salinity} |
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%\end{figure} |
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|
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%% tau_x |
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%\begin{figure} |
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%\centering |
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%\includegraphics[]{tau_x.eps} |
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%\caption{zonal wind stress for January} |
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%\label{fig:tau_x} |
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%\end{figure} |
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|
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%% tau_y |
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%\begin{figure} |
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%\centering |
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%\includegraphics[]{tau_y.eps} |
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%\caption{meridional wind stress for January} |
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%\label{fig:tau_y} |
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%\end{figure} |
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|
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%% Qnet |
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%\begin{figure} |
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%\centering |
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%\includegraphics[]{qnet.eps} |
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%\caption{Heat flux for January} |
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%\label{fig:qnet} |
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%\end{figure} |
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|
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%% EmPmR |
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%\begin{figure} |
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%\centering |
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%\includegraphics[]{empmr.eps} |
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%\caption{Fresh water flux for January} |
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%\label{fig:empmr} |
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%\end{figure} |
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|
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%% Bathymetry |
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%\begin{figure} |
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%\centering |
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%\includegraphics[]{bathymetry.eps} |
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%\caption{Bathymetry} |
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%\label{fig:bathymetry} |
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%\end{figure} |
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|
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|
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Figures (\ref{fig:sim_config_tclim_pcoord}-\ref{fig:sim_config_empmr_pcoord}) |
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%(\ref{fig:sim_config_tclim}-\ref{fig:sim_config_empmr}) |
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show the relaxation temperature ($\theta^{\ast}$) and salinity ($S^{\ast}$) |
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fields, the wind stress components ($\tau_x$ and $\tau_y$), the heat flux ($Q$) |
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and the net fresh water flux (${\cal E} - {\cal P} - {\cal R}$) used |
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in equations |
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(\ref{eq:eg-global-global_forcing_fu}-\ref{eq:eg-global-global_forcing_fs}). |
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The figures also indicate the lateral extent and coastline used in the |
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experiment. Figure ({\it --- missing figure --- }) %ref{fig:model_bathymetry}) |
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shows the depth contours of the model domain. |
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|
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\subsubsection{File {\it input/data}} |
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%\label{www:tutorials} |
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|
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\input{s_examples/global_oce_latlon/inp_data} |
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|
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\subsubsection{File {\it input/data.pkg}} |
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%\label{www:tutorials} |
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|
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This file uses standard default values and does not contain |
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customisations for this experiment. |
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|
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\subsubsection{File {\it input/eedata}} |
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%\label{www:tutorials} |
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|
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This file uses standard default values and does not contain |
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customisations for this experiment. |
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|
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\subsubsection{Files{\it input/trenberth\_taux.bin} and {\it |
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input/trenberth\_tauy.bin}} |
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%\label{www:tutorials} |
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|
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The {\it input/trenberth\_taux.bin} and {\it |
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input/trenberth\_tauy.bin} files specify a three-dimensional |
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($x,y,time$) map of wind stress, $(\tau_{x},\tau_{y})$, values |
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\citep{trenberth90}. The units used are $Nm^{-2}$. |
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|
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\subsubsection{File {\it input/bathymetry.bin}} |
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%\label{www:tutorials} |
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|
| 435 |
The {\it input/bathymetry.bin} file specifies a two-dimensional |
| 436 |
($x,y$) map of depth values. For this experiment values range |
| 437 |
between~$0$ and $-5200\,{\rm m}$, and have been derived from |
| 438 |
ETOPO5. The file contains a raw binary stream of data that is |
| 439 |
enumerated in the same way as standard MITgcm two-dimensional, |
| 440 |
horizontal arrays. |
| 441 |
|
| 442 |
\subsubsection{File {\it code/SIZE.h}} |
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%\label{www:tutorials} |
| 444 |
|
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\input{s_examples/global_oce_latlon/cod_SIZE.h} |
| 446 |
|
| 447 |
%\subsubsection{File {\it code/CPP\_OPTIONS.h}} |
| 448 |
%\label{www:tutorials} |
| 449 |
|
| 450 |
%This file uses standard default values and does not contain |
| 451 |
%customisations for this experiment. |
| 452 |
|
| 453 |
|
| 454 |
%\subsubsection{File {\it code/CPP\_EEOPTIONS.h}} |
| 455 |
%\label{www:tutorials} |
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|
| 457 |
%This file uses standard default values and does not contain |
| 458 |
%customisations for this experiment. |
| 459 |
|
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\subsubsection{Other Files } |
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%\label{www:tutorials} |
| 462 |
|
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% Other files relevant to this experiment are |
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% \begin{itemize} |
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% \item {\it model/src/ini\_cori.F}. This file initializes the model |
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% coriolis variables {\bf fCorU}. |
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% \item {\it model/src/ini\_spherical\_polar\_grid.F} |
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% \item {\it model/src/ini\_parms.F}, |
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% \item {\it input/windx.sin\_y}, |
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% \end{itemize} |
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% contain the code customisations and parameter settings for this |
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% experiments. Below we describe the customisations |
| 473 |
% to these files associated with this experiment. |
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