Perspectives on general coordinate models

1. Perspectives on general coordinate models • Motivation: • Single model (or framework/environment) for both global scale • Adiabatic interior (hybrid coordinates) and process studies • Non-hydrostatic • Regional impact of global change, super-parameterization … e.g. 5-20km global resolution, 100m nested regional resolution (“Mosaics”) This will be a reality within the decade (or a few years). • Can a Lagrangian (layered) class ocean model include non-hydrostatic effects? • Pertinent issues originally noted by Bleck, Schopf and others • Recently discussed in note: “On methods for solving the oceanic equations of motion in general coordinates”, Adcroft and Hallberg (2005), Ocean Modell. 8 (?) with which we hope to re-invigorate the discussion.

2. Hydrostatic (Boussinesq) equations in height coordinates: “z” • 7 unknowns, 4 prognostic eqns, 3 diagnostic eqns • 2 x Gravity mode • 1 x Planetary mode • 1 x thermo-haline mode • Free-surface equation obtained from continuity + B.C.s

3. Hydrostatic (Boussinesq) equations in isopycnal coordinates: “ρ” • 8 unknowns, 7 equations • 5 prognostic eqns, 2 diagnostic eqns • 8th equation: prescribe

4. Hydrostatic (Boussinesq) equations in general coordinates: “r” • Coordinate transformation: • is “thickness” • 8 unknowns, 7 equations 8th equation? or

5. Eulerian Vertical Dynamics Method (EVD) Specifies Uses continuity diagnostically to find Lagrangian Vertical Dynamics Method (LVD) Specifies (Inconsistent with a N-H vertical momentum equation?) Uses continuity to predict Using the continuity equation Unlikely to recover “adiabatic” properties of isopycnal models

6. Non-hydrostatic (Boussinesq) equations in height coordinates (z) Momentum (3d) Continuity (Volume) Temperature, salt and E.O.S. • Seven degrees of freedom • u,v,w,ρ,θ,s,p • Seven equations • 5 prognostic + 2 relations • No eqnsfor p

7. Solving the non-hydrostatic equations in height coordinates: “projection method” Momentum (3d) Continuity (Volume) Essential Algorithm Constraint on flow = Equation for pressure!

8. Projection method in LVD mode? Momentum (3d)(as before) Continuity Using the Eulerian approach: If this is prescribed we can not insert the vertical momentum equation here

9. Arbitrary Lagrangian-Eulerian method (ALE)? To make this N-H, we have to already know the flow by this point. • Lagrangian phase • (Optional) Eulerian phase (remapping) The EVD approach tries to constrain the N-H pressure with the final form of continuity

10. Hydrostatic/non-hydrostatic decomposition • Decompose pressure into parts • Surface (ps) • Hydrostatic (ph) • Non-hydrostatic (pnh)

11. Non-hydrostatic mode • 2D + 3D elliptic problem Can use EVD or LVD up until this point Hydrostatic N-H update

12. Non-hydrostatic modeling in general coordinates • Explicit solution of Navier-Stokes equations • Continuity leads to a prognostic equation for pressure • Can be integrated in any coordinate system • Separation of time scales in ocean is prohibitive Atmosphere Ocean Ocean Atmosphere

13. Points to take home • Some hybrid coordinate models use Eulerian paradigm (not HyCOM, HyPOP, Poseidon) • need to assess adiabaticity • Lagrangian paradigm • Easy to make adiabatic • Harder to make non-hydrostatic (not impossible) • Breaks symmetry between horizontal and vertical directions