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f67abf1ee3 Jeff* Overview
                 ********
                 
                 This document provides the reader with the information necessary to
                 carry out numerical experiments using MITgcm. It gives a comprehensive
                 description of the continuous equations on which the model is based, the
                 numerical algorithms the model employs and a description of the associated
                 program code. Along with the hydrodynamical kernel, physical and
                 biogeochemical parameterizations of key atmospheric and oceanic processes
                 are available. A number of examples illustrating the use of the model in
                 both process and general circulation studies of the atmosphere and ocean are
                 also presented.
                 
                 Introduction
                 ============
                 
                 MITgcm has a number of novel aspects:
                 
                  - it can be used to study both atmospheric and oceanic phenomena; one hydrodynamical kernel is used to drive forward both atmospheric and oceanic models - see :numref:`onemodel`
                 
                   .. figure:: figs/onemodel.*
                     :width: 80%
                     :align: center
                     :alt: One model for atmospheric and oceanic simulations
                     :name: onemodel
                 
                     MITgcm has a single dynamical kernel that can drive forward either oceanic or atmospheric simulations.
                 
                 
                  - it has a non-hydrostatic capability and so can be used to study both small-scale and large scale processes - see :numref:`all-scales`
                 
                   .. figure:: figs/scales.png
                     :width: 90%
                     :align: center
                     :alt: MITgcm can simulate a wide range of scales
                     :name: all-scales
                 
                     MITgcm has non-hydrostatic capabilities, allowing the model to address a wide range of phenomenon - from convection on the left, all the way through to global circulation patterns on the right.
                 
                  - finite volume techniques are employed yielding an intuitive discretization and support for the treatment of irregular geometries using orthogonal curvilinear grids and shaved cells - see :numref:`fvol`
                 
                   .. figure:: figs/fvol.*
                     :width: 80%
                     :align: center
                     :alt: Finit volume techniques
                     :name: fvol
                 
                     Finite volume techniques (bottom panel) are used, permitting a treatment of topography that rivals :math:`\sigma` (terrain following) coordinates.
                 
                  - tangent linear and adjoint counterparts are automatically maintained along with the forward model, permitting sensitivity and optimization studies.
                 
                  - the model is developed to perform efficiently on a wide variety of computational platforms.
                 
                 
                 Key publications reporting on and charting the development of the model are Hill and Marshall (1995), Marshall et al. (1997a), 
                 Marshall et al. (1997b), Adcroft and Marshall (1997), Marshall et al. (1998), Adcroft and Marshall (1999), Hill et al. (1999),
                 Marotzke et al. (1999), Adcroft and Campin (2004), Adcroft et al. (2004b), Marshall et al. (2004) (an overview on the model formulation can also be found in Adcroft et al. (2004c)):
                 
                 Hill, C. and J. Marshall, (1995)
                 Application of a Parallel Navier-Stokes Model to Ocean Circulation in 
                 Parallel Computational Fluid Dynamics,
                 In Proceedings of Parallel Computational Fluid Dynamics: Implementations 
                 and Results Using Parallel Computers, 545-552.
                 Elsevier Science B.V.: New York :cite:`hill:95`
                 
                 Marshall, J., C. Hill, L. Perelman, and A. Adcroft, (1997a)
                 Hydrostatic, quasi-hydrostatic, and nonhydrostatic ocean modeling,
                 J. Geophysical Res., **102(C3)**, 5733-5752 :cite:`marshall:97a`
                 
                 Marshall, J., A. Adcroft, C. Hill, L. Perelman, and C. Heisey, (1997b)
                 A finite-volume, incompressible Navier Stokes model for studies of the ocean
                 on parallel computers, J. Geophysical Res., **102(C3)**, 5753-5766 :cite:`marshall:97b`
                 
                 Adcroft, A.J., Hill, C.N. and J. Marshall, (1997)
                 Representation of topography by shaved cells in a height coordinate ocean
                 model, Mon Wea Rev, **125**, 2293-2315 :cite:`adcroft:97`
                 
                 Marshall, J., Jones, H. and C. Hill, (1998)
                 Efficient ocean modeling using non-hydrostatic algorithms,
                 Journal of Marine Systems, **18**, 115-134 :cite:`mars-eta:98`
                 
                 Adcroft, A., Hill C. and J. Marshall: (1999)
                 A new treatment of the Coriolis terms in C-grid models at both high and low
                 resolutions,
                 Mon. Wea. Rev., **127**, 1928-1936 :cite:`adcroft:99`
                 
                 Hill, C, Adcroft,A., Jamous,D., and J. Marshall, (1999)
                 A Strategy for Terascale Climate Modeling,
                 In Proceedings of the Eighth ECMWF Workshop on the Use of Parallel Processors
                 in Meteorology, 406-425
                 World Scientific Publishing Co: UK :cite:`hill:99`
                 
                 Marotzke, J, Giering,R., Zhang, K.Q., Stammer,D., Hill,C., and T.Lee, (1999)
                 Construction of the adjoint MIT ocean general circulation model and 
                 application to Atlantic heat transport variability,
                 J. Geophysical Res., **104(C12)**, 29,529-29,547 :cite:`maro-eta:99`
                 
                 A. Adcroft and J.-M. Campin, (2004a)
                 Re-scaled height coordinates for accurate representation of free-surface flows in ocean circulation models, 
                 Ocean Modelling, **7**, 269–284 :cite:`adcroft:04a`
                 
                 A. Adcroft, J.-M. Campin, C. Hill, and J. Marshall, (2004b)
                 Implementation of an atmosphere-ocean general circulation model on the expanded 
                 spherical cube, 
                 Mon Wea Rev , **132**, 2845–2863 :cite:`adcroft:04b`
                 
                 J. Marshall, A. Adcroft, J.-M. Campin, C. Hill, and A. White, (2004)
                 Atmosphere-ocean modeling exploiting fluid isomorphisms, Mon. Wea. Rev., **132**, 2882–2894 :cite:`marshall:04`
                 
                 A. Adcroft, C. Hill, J.-M. Campin, J. Marshall, and P. Heimbach, (2004c)
                 Overview of the formulation and numerics of the MITgcm, In Proceedings of the ECMWF seminar series on Numerical Methods, Recent developments in numerical methods for atmosphere and ocean modelling, 139–149. URL: http://mitgcm.org/pdfs/ECMWF2004-Adcroft.pdf :cite:`adcroft:04c`
                 
                 We begin by briefly showing some of the results of the model in action to
                 give a feel for the wide range of problems that can be addressed using it.
                 
                 
                 Illustrations of the model in action
                 ====================================
                 MITgcm has been designed and used to model a wide range of phenomena,
                 from convection on the scale of meters in the ocean to the global pattern of
                 atmospheric winds - see :numref:`all-scales`. To give a flavor of the
                 kinds of problems the model has been used to study, we briefly describe some
                 of them here. A more detailed description of the underlying formulation,
                 numerical algorithm and implementation that lie behind these calculations is
                 given later. Indeed many of the illustrative examples shown below can be
                 easily reproduced: simply download the model (the minimum you need is a PC
                 running Linux, together with a FORTRAN\ 77 compiler) and follow the examples
                 described in detail in the documentation.
                 
                 
                 .. toctree::
                    :maxdepth: 3
                 
                    global_atmos_hs.rst
                    ocean_gyres.rst
                    global_ocean_circ.rst
                    cvct_mixing_topo.rst
                    bound_forc_inter_waves.rst
                    parm_sens.rst
                    global_state_est.rst
                    ocean_biogeo_cyc.rst
                    sim_lab_exp.rst
                 
                  
                 Continuous equations in ‘r’ coordinates
                 =======================================
                 To render atmosphere and ocean models from one dynamical core we exploit
                 ‘isomorphisms’ between equation sets that govern the evolution of the
                 respective fluids - see :numref:`isomorphic-equations`. One system of
                 hydrodynamical equations is written down and encoded. The model
                 variables have different interpretations depending on whether the
                 atmosphere or ocean is being studied. Thus, for example, the vertical
                 coordinate ‘:math:`r`’ is interpreted as pressure, :math:`p`, if we are
                 modeling the atmosphere (right hand side of :numref:`isomorphic-equations`) and height, :math:`z`, if we are modeling
                 the ocean (left hand side of :numref:`isomorphic-equations`).
                 
                 
                   .. figure:: figs/zandpcoord.png
                     :width: 80%
                     :align: center
                     :alt: isomorphic-equations
                     :name: isomorphic-equations
                 
                     Isomorphic equation sets used for atmosphere (right) and ocean (left).
                 
                 
                 The state of the fluid at any time is characterized by the distribution
                 of velocity :math:`\vec{\mathbf{v}}`, active tracers :math:`\theta` and
                 :math:`S`, a ‘geopotential’ :math:`\phi` and density
                 :math:`\rho =\rho (\theta ,S,p)` which may depend on :math:`\theta`,
                 :math:`S`, and :math:`p`. The equations that govern the evolution of
                 these fields, obtained by applying the laws of classical mechanics and
                 thermodynamics to a Boussinesq, Navier-Stokes fluid are, written in
                 terms of a generic vertical coordinate, :math:`r`, so that the
                 appropriate kinematic boundary conditions can be applied isomorphically
                 see :numref:`zandp-vert-coord`.
                 
                 
                   .. figure:: figs/vertcoord.*
                     :width: 60%
                     :align: center
                     :alt: zandp-vert-coord
                     :name: zandp-vert-coord
                 
                     Vertical coordinates and kinematic boundary conditions for atmosphere (top) and ocean (bottom).
                 
                 .. math::
0bad585a21 Navi*    \frac{D\vec{\mathbf{v}}_{h}}{Dt}+\left( 2\vec{\boldsymbol{\Omega}}\times \vec{\mathbf{v}}
                    \right) _{h}+ \nabla _{h}\phi = \vec{\boldsymbol{\mathcal{F}}}_h\text{  horizontal momentum}
f67abf1ee3 Jeff*    :label: horiz-mtm
                 
                 .. math::
0bad585a21 Navi*    \frac{D\dot{r}}{Dt}+\hat{\boldsymbol{k}}\cdot \left( 2\vec{\boldsymbol{\Omega}}\times \vec{\mathbf{
f67abf1ee3 Jeff*    v}}\right) +\frac{\partial \phi }{\partial r}+b=\mathcal{F}_{\dot{r}}\text{  vertical momentum}
                    :label: vert-mtm
                 
                 .. math::
0bad585a21 Navi*     \nabla _{h}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial \dot{r}}{
f67abf1ee3 Jeff*    \partial r}=0\text{  continuity}
                    :label: continuity
                 
                 .. math:: 
                    b=b(\theta ,S,r)\text{  equation of state} 
                    :label: eos
                  
                 .. math::
                    \frac{D\theta }{Dt}=\mathcal{Q}_{\theta }\text{  potential temperature}
                    :label: pot-temp
                 
                 .. math::
                    \frac{DS}{Dt}=\mathcal{Q}_{S}\text{  humidity/salinity}
                    :label: humidity-salt
                 
                 Here:
                 
                 .. math:: r\text{ is the vertical coordinate}
                 
                 .. math::
                 
0bad585a21 Navi*    \frac{D}{Dt}=\frac{\partial }{\partial t}+\vec{\mathbf{v}}\cdot  \nabla  \text{ is the total derivative}
f67abf1ee3 Jeff* 
                 .. math::
                 
0bad585a21 Navi*     \nabla = \nabla _{h}+\hat{\boldsymbol{k}}\frac{\partial }{\partial r}
f67abf1ee3 Jeff*    \text{  is the ‘grad’ operator}
                 
0bad585a21 Navi* with :math:`\nabla _{h}` operating in the horizontal and
                 :math:`\hat{\boldsymbol{k}}
f67abf1ee3 Jeff* \frac{\partial }{\partial r}` operating in the vertical, where
0bad585a21 Navi* :math:`\hat{\boldsymbol{k}}` is a unit vector in the vertical
f67abf1ee3 Jeff* 
                 .. math:: t\text{ is time}
                 
                 .. math::
                 
                    \vec{\mathbf{v}}=(u,v,\dot{r})=(\vec{\mathbf{v}}_{h},\dot{r})\text{ is the velocity}
                 
                 .. math:: \phi \text{ is the ‘pressure’/‘geopotential’}
                 
0bad585a21 Navi* .. math:: \vec{\boldsymbol{\Omega}}\text{ is the Earth's rotation}
f67abf1ee3 Jeff* 
                 .. math:: b\text{ is the ‘buoyancy’}
                 
                 .. math:: \theta \text{ is potential temperature}
                 
                 .. math:: S\text{ is specific humidity in the atmosphere; salinity in the ocean}
                 
                 .. math::
                 
0bad585a21 Navi*    \vec{\boldsymbol{\mathcal{F}}}\text{ are forcing and dissipation of }\vec{
f67abf1ee3 Jeff*    \mathbf{v}}
                 
                 .. math:: \mathcal{Q}_{\theta }\mathcal{\ }\text{ are forcing and dissipation of }\theta
                 
                 .. math:: \mathcal{Q}_{S}\mathcal{\ }\text{are forcing and dissipation of }S
                 
0bad585a21 Navi* The terms :math:`\vec{\boldsymbol{\mathcal{F}}}` and :math:`\mathcal{Q}` 
f67abf1ee3 Jeff* are provided by ‘physics’ and forcing packages for atmosphere and ocean.
                 These are described in later chapters.
                 
                 
                 .. toctree::
                    :maxdepth: 3
                 
                    kinematic_bound.rst
                    atmosphere.rst
                    ocean.rst
                    hydrostatic.rst
                    soln_strategy.rst
                    finding_pressure.rst
                    forcing_dissip.rst
                    vector_invar.rst
                    adjoint.rst
                 
                 
                 Appendix ATMOSPHERE
                 ===================
                 
                 .. toctree::
                    :maxdepth: 3
                 
                    hydro_prim_eqn.rst
                 
                 
                 Appendix OCEAN
                 ==============
                 
                 .. toctree::
                    :maxdepth: 3
                 
                    eqn_motion_ocn.rst
                 
                 
                 Appendix OPERATORS
                 ==================
                 
                 .. toctree::
                    :maxdepth: 3
                 
                    coordinate_sys.rst

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