Model Formulation, Numerics and Approximations

Vertical coordinates for climate?• What are the remaining liabilities or challenges

with z, z*, p, p*, isopycnal, and various approaches to hybrids between coordinates?

• Will sigma ever be viable for long-term global ocean climate modeling?

• Are there "coordinate free" approaches that seem viable, and if so how is the vertical resolution distributed?

• Do ice-shelves or changing coastlines introduce significant new considerations?

Strengths and Weaknesses of Terrain-following Coordinate ModelsStrengths:• Topography is represented very simply and accurately• Easy to enhance resolution near surface.• Lots of experience with atmospheric modeling to draw upon.

Traditional Weaknesses:• Pressure gradient errors are a persistent problem.

Errors are reduced with better numerics (e.g., Shchepetkin & McWilliams, 2003)• Gentle slopes (smoothed topography) must be used for consistency

Traditional requirement for stability (Beckman & Haidvogel, 1993):

ROMS requirement (Shchepetkin, pers. comm):

• Spurious diapycnal mixing due to advection may be very large. (Same issue as Z-coord.)

• Diffusion tensors may be especially difficult to rotate into the neutral direction. Strongly slopes require larger vertical stencil for the isoneutral-diffusion operator.

Myth: Near bottom resolution can be arbitrarily enhanced. Hydrostatic consistency imposes horizontal resolution-dependent constraints on

near-bottom vertical resolution, with serious implications for the ability to represent overflows

2.02

DD

DD

D

D

SSZ pp 11

83~ toz

x

dx

dz

s

Dzx /5Maximum Hydrostatically Consistent Horizontal Resolution

Horizontal Resolution (in km) Required to Permit 50m Vertical Resolution at Bottom

Dzx /5Maximum Hydrostatically Consistent Horizontal Resolution

Horizontal Resolution (in km) Required to Permit 50m Vertical Resolution at Bottom

Dzx /5Maximum Hydrostatically Consistent Horizontal Resolution

Horizontal Resolution (in km) Required to Permit 50m Vertical Resolution at Bottom

Common ApproximationsApproximation How large are errors? Consequences

Boussinesq (-0)/0~0.01 Volume conserved, not mass

Virtual salt flux (35psu-S) / S – very large? Big errors where fresh!

Rigid Lid ~1m / DOce Infinite external wave speed

Thin shell DOce / REarth ~ 0.001

Spherical Earth 10km/6370km ~ 0.0015

Constant gravitational acceleration (g)

2DOce / REarth ~ 0.0015.0025/9.8 ~ 0.00025

Hydrostatic As2*Ro, As*Ro Filter sound waves; No explicit convection

Traditional As*Ro Goes with hydrostatic

Potential temperature as Conservative temperature

(Ask Trevor McDougall)Differ by ~2 C at 0 PSU

Heat capacity varies, but agrees with old conventions

PSU as absolute salinity (35.16505/35-1) ~ 0.0047

Flow-invariant geoid 10% at amphidromic scales

Unstructured ocean grids?• Will any be IPCC-ready by AR6?

• What are the big issues?– Cost?– Conservation?– Adiabaticity?– Experience?

Pressure gradient errors?• Are there still outstanding issues with pressure

gradient force calculations with generalized (i.e. non-P, non-Z, non-in-situ-density) coordinates?

• How serious is this issue?

Numerics for Momentum Eqns• Numerical closures for the momentum

equations:– Is there anything new?– Is anything new needed?

Tracer advection• What is the state of the art?

• What is good enough at which resolutions and for which vertical coordinates?

• How serious is the problem in climate models with spurious diapycnal mixing arising from tracer advection at various resolutions?

Scaling to 1000s of PEs• Scaling to 20 pts/PE

– At 1° scales to (360/20)x(200/20) ~ 180 PEs– At 1/4° Mercator scales to ~ 3,600 PEs?– At 0.1 ° Mercator scales to ~ 25,000 Pes

• Scaling to 10 pts/PE– At 1° scales to (360/10)x(200/10) ~ 720 PEs– At 1/4° Mercator scales to ~ 14,400 PEs?– At 0.1 ° Mercator scales to ~ 100,000 Pes?

Plug & Play Software?• Is this desirable?

• What are we willing to give up for this?

• Who will pay for it?