Energy and Enstrophy Cascades in Numerical Models
John Thuburn, James Kent, (University of Exeter)
Nigel Wood (Met Office)
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• Energy and (potential) enstrophy are conserved by the adiabatic,
frictionless governing equations...
Meteosat water
vapour image
near the truncation limit?
Outline
• The need to remove potential enstrophy without removing too
much energy
Effect of unresolved scales on enstrophy and energy spectra
Effect of some numerical schemes on enstrophy and energy spectra
Parameterization of energy backscatter
Turbulence theory
(potential) enstrophy is transferred mainly downscale.
(Transfers weaker on scales larger than the Rossby radius.)
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Energy
Total energy (∼ 3 × 109Jm−2) is made up of available and unavailable
contributions.
Ratio: 2000 4 1
Unavailable and available energy are separately conserved.
Unavailable energy is a function of the mass per unit θ - almost
robust.
Can conserve mass in each model isentropic layer by using θ as a
vertical coordinate.
There is some evidence that about 5-10% of available energy
cascades downscale in the free atmosphere. (Mechanism?)
The rest goes upscale before being dissipated mainly by the boundary
layer.
Numerical representation of energy and potential enstrophy
transfers
dissipation such as κ∇2n
or
non-oscillatory finite volume (ILES).
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Implicit Large Eddy Simulation (ILES)
Finite resolution => need to represent effects of unresolved scales:
SG model.
At the same time, all numerical methods have truncation errors.
Can truncation errors play the role of a SG model?
Some success claimed for 3D turbulence. (Except when upscale
effects are important, e.g. near a wall.)
What about (layerwise) 2D turbulence?
Upscale energy transfers, but steeper spectrum so stronger slaving of
small scales to large.
What if we don’t remove resolved enstrophy?
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Evidence that models dissipate too much energy
Frederiksen et al. (JAS 1996): Effect on spectra of various
dissipation schemes.
comparable to boundary layer.
Bowler et al. (QJRMS 2009): Estimate of dissipation due to
semi-Lagrangian interpolation.
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Energy and Enstrophy cascades John Thuburn
If we remove enstrophy at horizontal wavenumber kdiss at a rate Z
then we necessarily remove KE at a rate E = Z/k2 diss ≥ Z/k2
max.
At current climate resolutions this is too large.
E.g. Z ∼ 10−15 s−3. Need E ∼ 10−5 m−2s−3 so kdiss ∼ 10−5 m−1.
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Energy and Enstrophy cascades John Thuburn
What does ILES or any explicit SG model need to capture?
Barotropic vorticity equation as model problem:
Dζ
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Statistically steady turbulence
scale-independent
dissipation;
−10
0 100 200 300 −0.2
0
0.2
×1 e−
0
1
×0 .0
00 1
0
0.5
×1 e−
0
0.2
×0 .0
0
0.5
×1 e−
0
0.5
×0 .0
Schematic of energy transfers
(See also Huang and Robinson 1997; Thompson and Young 2007)
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Physical space picture
Energy density ed = vi vi/2 on scales greater than some filter scale
satisfies
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Spectral interactions as represented by ∇4 and ∇8
0 50 100 150 200 −0.5
0
0.5
×1 e−
0
0.2
×0 .0
−1
0
1
×1 e−
0
0.2
×0 .0
0
0.5
×1 e−
0
0.2
×0 .0
UTOPIA advection of ζ
upwind.
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Spectral interactions as represented by UTOPIA scheme
0 50 100 150 200 −0.5
0
0.5
×1 e−
0
0.2
×0 .0
−1
0
1
×1 e−
0
0.5
×0 .0
0
1
×1 e−
0
0.5
×0 .0
Anticipated Potential Vorticity Method
Sadourny and Basdevant (1985).
Choose D = θL(v.∇ζ). Here L ≡ 1 or L ≡ −∇2
Z = −θ
∫
0
0.5
×1 e−
0
0.2
×0 .0
−0.5
0
0.5
×1 e−
0
0.2
×1 e−
0
0.5
×1 e−
0
1
×1 e−
Ringler et al. scheme
RTKS (J. Comput. Phys. 2010) developed a family of energy
conserving C-grid schemes for the SWEs on arbitrary polygonal grid
cells.
∂v
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Spectral interactions as represented by RTKS scheme
0 50 100 150 200 −0.5
0
0.5
×1 e−
0
0.2
×0 .0
−0.5
0
0.5
×1 e−
0
0.5
×0 .0
0
0.5
×1 e−
0
0.2
×0 .0
Caveat
All these diagnostics are for the instantaneous effect on truncated
high-resolution fields.
might be quite different — finite adjustent time.
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Can we represent the energy backscatter to large scales?
Scale-dependent
dissipation/anti-dissipation
Let ζ∗ = UTOPIA(ζn)
and let δE = E(ζn) − E(ζ∗)
Choose a vorticity pattern δζ and let ζn+1 = ζ∗ + αδζ.
α = − δE
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E.g.
(small scales)
δζ2 was found to work better in numerical tests, giving better energy
statistics and also a small but measurable improvement in l2 errors.
(But this is not really ‘backscatter’; more of an energy fixer!)
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Decaying turbulence E and Z time series
0 10 20 30 40 50 1.36
1.37
1.38
1.39
1.4
1.41
1.42
HR on 1282
1.38
1.4
1.42
1.44
HR on 128 2
0.04
0.045
0.05
0.055
0.06
0.065
0.07
0.075
0.08
t
Z
HR on 1282
0.04
0.05
0.06
0.07
0.08
t
Z
HR on 128 2
Possible improvements
• Use spectral dissipation characteristics of basic scheme to derive δζ
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Subgrid forcing of vorticity
Scale similarity of backscatter?
Conclusions
• For the BVE, explicit calculation of the effects of unresolved scales
shows enstrophy removal near the truncation limit and energy input
at the most energetic scales. Very robust.
• Both ILES schemes and simple explicit dissipation schemes can
remove enstrophy at small scales (but are typically not scale-selective
enough
• Neither ILES schemes nor standard SG models capture the energy
backscatter.
• A simple backscatter model can improve energy statistics and l2
errors.
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