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 eq ns, 3 diagnostic eq ns 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 eq ns, 2 diagnostic eq ns –8 th equation: prescribe

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

5. Using the continuity equation 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 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 eq ns for 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)? Lagrangian phase (Optional) Eulerian phase (remapping) To make this N-H, we have to already know the flow by this point. 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 (p s ) –Hydrostatic (p h ) –Non-hydrostatic (p nh )

11. Non-hydrostatic mode 2D + 3D elliptic problem Hydrostatic Can use EVD or LVD up until this point 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 Ocean Atmosphere OceanAtmosphere

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