The model of subgridscale turbulence in the Parallel Ocean Program POP

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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 Alamos
• 2Imperial College, Great Britain
• 3Twente University, Netherlands
• LA-UR-05-0887

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Ocean Modeling

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

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?-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

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Kelvins 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).
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• Now, consider a smoother, filtered velocity, as
  Leray did
• u g v
• and a closed fluid loop which follows this smooth
  velocity u

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.
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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
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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)

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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)

_______________
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20483 DNS vs 2563 LANS-a
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Holm and Nadiga, JPO 2003
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Holm Nadiga high res soln
secondary gyres, generated by mesoscale eddies
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Holm Nadiga 1/4 res
Secondary gyres are lost
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Holm Nadiga1/4 res with ?-model
Secondary gyres recovered (but too strong)
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Holm Nadiga1/8 res with ?-model
Secondary gyres are reasonable, even at 1/8 of
fully-resolved res.
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What to expect in 3-D ocean model?
Eddy viscosity model
?-model
forcing
k2

• Baroclinic instability occurs within the curve
• Onset occurs at lower wavenumber with ?, even
  without increased forcing

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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.

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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 its non-dissipative in terms of density,
  adiabatic

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? 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

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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).

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Eddy transport across the ACC
Karsten, Jones and Marshall, JPO, 2002
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Meridional fluxes Ekman and Eddy vs surface
buoyancy flux
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and vertical transports
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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
• 11 grid aspect ratio at 60ºS
• Vertical res 10m_at_surface, 250m_at_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

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just a glance at test problem
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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