The -model of sub-gridscale turbulence in the Parallel Ocean

Program (POP)

Matthew Hecht1, Beth Wingate1

and Mark Petersen1 with

Darryl Holm1,2 and Bernard Geurts3

1Los Alamos2Imperial College, Great Britain

3Twente University, Netherlands

LA-UR-05-0887

Ocean Modeling

• Ocean models for climate are based on the Primitive Equations– Shallow approximation– Hydrostatic

-model of sub-gridscale turbulence

-model developed within (un-approximated) Navier-Stokes Eqns– What if the velocity in the discretized NS eqns were really a

smoother, time-averaged representation of what could exist if finer scales were resolved?

• Leray had proposed something like this -- in 1934– Use of a filtered, smoother advecting velocity led to a

regularization of the NS eqs:

Kelvin’s circulation theorem

• For any closed loop embedded in and moving within a fluid, the fluid circulating around that loop only spins up or down if work is done on it:

Where (v) is some closed fluid loop moving with v(x,t).

Now, consider a smoother, filtered velocity, as Leray did:

u = g * v

and a closed fluid loop which follows this smooth velocity u:

filtered, smoother velocity

original velocity,containing finer scales

Filter, (1-2∆)-1

After manipulation, get the Kelvin-filtered Navier-Stokes Eqn

Just like Leray, but with one additional term! The difference between this and the NS eqns is what we call the -model of turbulence.

Eulerian Averaging

• Tracer concentration is averaged over some neighborhood around fixed-space cells

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Lagrangian Averaging

• Tracer concentration is averaged over some neighborhood which follows the flow

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Some Applications

Turbulent decay, direct and modeled

• Kang, Chester & Meneveau (KCM) at JHU newly performed a classic wind-tunnel experiment in turbulence decay, at 10X higher Reynolds number than was previously possible

• TWG at Los Alamos provided computational support by simulating their experimental results at 2048-cubed

• This was the largest-ever computational simulation of a turbulence experiment ever performed (It produced 11 Tbytes of data for 3 1/2 eddy turnover times)

TWG Simulation of the KCM Experiment • Pseudo spectral and spectral methods

• Resolution: 20483

• 8B grid points

• 11 TB of data (192GB per snapshot)

• 2048 CPUs

• 1 CPU century* on ASCI-Q

• R = 220 ( = 100,000)

_______________Largest computation ever, modeling a real experiment!

*800,000 CPU hours

€

R

20483 – DNS vs 2563 LANS-

Holm and Nadiga, JPO 2003

Holm & Nadiga: high res solnsecondary gyres, generated by mesoscale eddies

Holm & Nadiga: 1/4 resSecondary gyres are lost

Holm & Nadiga:1/4 res with -model

Secondary gyres recovered (but too strong)

Holm & Nadiga:1/8 res with -model

Secondary gyres are reasonable, even at 1/8 of fully-resolved res.

What to expect in 3-D ocean model?

• Baroclinic instability occurs within the curve– Onset occurs at lower wavenumber with , even

without increased forcing

k2

forc

ing

-model Eddy viscosity model

Larger time steps may be possible

• Wingate showed an easing of time step limitation in a shallow water model with increasing – “The maximum allowable time step for the

shallow water -model and its relation to time implicit differencing”, Mon. Weather Review, to appear 2004.

How does this fit in with Gent-McWilliams?

• GM was intended for “tracer eqns”– transport and mixing of temperature, salinity

and also passive tracers

• GM has a diffusive component, as well as an advective component– though it’s non-dissipative in terms of density,

adiabatic

and GM, continued

comes into momentum and tracer eqns– completely non-dissipative for constant alpha

• GM has been a major advance in ocean modeling for climate, particularly in terms of poleward heat transports– We believe the -model can be used with GM

to improve the turbulent dynamics

Test problem for -model in POP

• 4-gyre problem of Holm and Nadiga is excellent, but more “inertial” than one would see in the real ocean

• Antarctic circumpolar-like problem motivated by Karsten, Jones and Marshall, JPO, 2002:– “We argue that the eddies themselves are fundamental

in setting the stratification -- both in the horizontal and vertical.”

• Also influenced by work of Henning and Vallis (private communication).

Eddy transport across the

ACC

Karsten, Jones and Marshall, JPO, 2002

Meridional fluxes:

Ekman and Eddy vs surface

buoyancy flux

and vertical

transports

the test problem• Channel model, cyclic, with a N/S ridge

– At 60ºS, +/- 8°– 32º zonal width (re-entrant)– Meridional resolutions of 0.1º, 0.2º, 0.4º, 0.8º

• 1:1 grid aspect ratio at 60ºS

– Vertical res: 10m@surface, 250m@depth• as in CCSM ocean• 4000m max depth, N/S ridge rises to 2500m

– Buoyancy forcing through restoring of SST• 2ºC at 68ºS, 12º at 52ºS

– Zonal wind stress

just a glance at test problem…

Conclusions

• Not ready for conclusions

Discussion?-model is on a very solid footing in terms

of theory and application

• We aim to find out what it will give us in terms of the effects of unresolved turbulence on the larger scale circulation