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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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
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).
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