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4. Fundamental equations
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Introduction to MOM and its use
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Basic formulation
IV.
Basic formulation
4. Fundamental equations
4.1 Assumptions
4.2 The primitive equations
4.2.1 Basic constants and parameters
4.2.2 Hydrostatic pressure and the equation of state
4.2.3 Horizontal momentum equations
4.2.3.1 Coriolis force
4.2.3.2 Horizontal pressure gradient
4.2.3.3 Advection
4.2.3.4 Nonlinear advective ``metric'' term
4.2.3.5 Vertical friction
4.2.3.6 Horizontal friction
4.2.4 Tracer equations
4.3 Boundary and initial conditions
4.3.1 Bottom kinematic boundary condition
4.3.2 Surface kinematic boundary condition
4.3.3 Dynamic boundary conditions
4.3.4 Tracer fluxes through the model boundaries
4.3.5 Open boundaries and sponges
4.3.6 Initial conditions
4.4 Comments on volume versus mass conservation
4.4.1 Volume conservation
4.4.2 Mass conservation
4.4.3 Surface kinematic boundary conditions revisited
4.5 Flux form and finite volumes
4.6 Some basic formulae and notation
4.6.1 Differential operators
4.6.2 Leibnitz's Rule
4.6.3 Cross-products and the Levi-Civita symbol
4.6.4 Area element and volume element on a sphere
4.6.5 Vertical grid levels
5. Momentum equation methods
5.1 Separation into vertical modes
5.1.1 Vertical modes in MOM and their relation to eigenmodes
5.1.2 Motivation for separating the modes
5.2 Methods for solving the separated equations
5.2.1 The fixed surface / rigid lid method in brief
5.2.1.1 Fixed surface height
5.2.1.2 Vanishing velocity at the ocean surface
5.2.1.3 Fresh water forcing in the rigid lid
5.2.1.4 Two rigid lid methods in MOM
5.2.2 The free surface / non-rigid lid method in brief
5.2.2.1 The barotropic equation and its two solution methods
5.2.2.2 The non-rigid lid approximation
6. Rigid lid streamfunction method
6.1 The barotropic streamfunction
6.2 Streamfunction and volume transport
6.3 Hydrostatic pressure with the rigid lid
6.4 The barotropic vorticity equation
6.4.1 Tendencies for the vertically averaged velocities
6.4.2 The barotropic vorticity equation
6.4.3 Caveat: inversions with steep topography
6.5 Boundary conditions and island integrals
6.5.1 Dirichlet boundary condition on the streamfunction
6.5.2 Separating the streamfunction's boundary value problem
6.5.3 Island integrals for the volume transport
6.6 The baroclinic mode
6.7 Summary of the rigid lid streamfunction method
6.8 Rigid lid surface pressure method
7. Free surface method
7.1 Hydrostatic pressure with the free surface
7.2 The barotropic system
7.2.1 Vertically integrated transport
7.2.2 Bottom and surface kinematic boundary conditions
7.2.3 Free surface height equation
7.2.4 Vertically integrated momentum equations
7.2.5 Global water budget
7.3 A linearized barotropic system
7.3.1 The barotropic system
7.3.2 The shallow water limit
7.3.3 The linearized free surface height equation
7.3.4 Summary of the linear barotropic system
7.4 Stresses at the ocean surface and bottom
7.4.1 Bottom stress
7.4.2 Surface stress
7.4.3 Revisiting the surface stress
7.5 A comment about atmospheric pressure
7.6 Vertically integrated transport
7.6.1 General considerations
7.6.2 An approximate streamfunction
8. The tracer budget
8.1 The continuum tracer concentration budget
8.2 Finite volume budget for the total tracer
8.3 Surface tracer flux
8.4 Comments on the surface tracer fluxes
8.4.1 Fresh water flux into the free surface model
8.4.2 Heat flux into the free surface model
8.5 River runoff
9. Momentum friction
9.1 History of friction in MOM
9.2 Basic properties of the stress tensor
9.2.1 The deformation or rate of strain tensor
9.2.2 Relating strain to stress
9.2.3 Angular momentum and symmetry of the stress tensor
9.3 The stress tensor in Cartesian coordinates
9.3.1 Generalized Hooke's law form
9.3.2 Angular momentum
9.3.3 Dissipation of total kinetic energy
9.3.4 Transverse isotropy
9.3.5 Trace-free frictional stress
9.3.6 Summary of the frictional stress tensor
9.3.7 Quasi-hydrostatic assumption
9.3.8 Cartesian form of the friction vector
9.3.9 The case of nonconstant viscosity
9.4 Orthogonal curvilinear coordinates
9.4.1 Some rules of tensor analysis on manifolds
9.4.2 Orthogonal coordinates
9.4.3 Physical components of tensors
9.4.4 General form of the frictional stress tensor
9.4.5 Horizontal tension and shearing rate of strain
9.4.6 The friction vector
9.4.7 Effects on kinetic energy
9.4.8 Summary of second order friction
9.5 Biharmonic friction
9.5.1 General formulation
9.5.2 Effects on kinetic energy
9.6 Comments on frictional and advective metric terms
9.6.1 Motion on an infinite plane
9.6.2 Conservation of angular momentum about the north pole
9.6.3 The advective and frictional metric terms
9.7 Functional formalism
9.7.1 Continuum formulation
9.7.2 Discrete formulation
9.8 Old friction implementation
9.8.1 Spherical form of second order friction
9.8.2 Zonal friction
9.8.3 Meridional friction
9.8.4 Old biharmonic algorithm
RC Pacanowski and SM Griffies, GFDL, Jan 2000
