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TRANSCRIPT
Small scales turbulence and LES modelling with GENE
P. Morel, A. Banon Navarro, M. Albrecht-‐Marc and D. Cara8 Université Libre de Bruxelles
F. Merz, T. Gorler and F. Jenko Max Planck Ins8tut fur PlasmaPhysik -‐ Garching
PIERRE MOREL Gyokine?cs for ITER – WPI Wien 22/03/2010
outline
Sub grid and modelling
Free energy balance
Hyperdiffusion as a first model
Dynamic procedure
PIERRE MOREL Gyokine?cs for ITER – WPI Wien 22/03/2010
PIERRE MOREL Gyokine?cs for ITER – WPI Wien 22/03/2010
On sub grid modelling
Applying a filter to gyrokine?cs equa?ons solved in GENE leads to:
With the sub grid term:
To gain in resolu?on and numerical effort, this term has to be modelled and analysed.
PIERRE MOREL Gyokine?cs for ITER – WPI Wien 22/03/2010
On sub grid modelling
First analysis: under resolved study
Principle: decrease resolu?on without model, see what is lost:
0 0.1 0.2 0.3 0.4 0.5100
101
102
103
104
kx s
||
cyclonen(x,y) x 3/4n(x,y)/2n(x,y)/4
0 0.5 1 1.50
2
4
6
8
10
12
14
16
18
kx s|n
|
cyclonen(x,y) x 3/4n(x,y)/2n(x,y)/4
CBC case with decreasing perpendicular resolu8on electrosta8c poten8al (leN) and density (right) along kx (radial): crucial need of a model!
PIERRE MOREL Gyokine?cs for ITER – WPI Wien 22/03/2010
On sub grid modelling
Basic idea: sub grid must insert some dissipa?ons
A good candidate for quan?fying these dissipa?on processes is the free energy
outline
Sub grid and modelling
Free energy balance
Hyperdiffusion as a first model
Dynamic procedure
PIERRE MOREL Gyokine?cs for ITER – WPI Wien 22/03/2010
PIERRE MOREL Gyokine?cs for ITER – WPI Wien 22/03/2010
Free energy balance
In local version of GENE, free energy balance can be expressed:
By applying the free energy operator:
to Vlasov equa?on.
NB: the drive is a L.C of eq. gradients and fluxes NBB: dissipa?on in collisionless regime ensured by hyper-‐diffusion operators
PIERRE MOREL Gyrokine?cs for ITER – WPI Wien 22/03/2010
Free energy balance – terms (adiaba6c electrons)
TdS: Entropy
Electrosta?c
Gradients: Free energy
sources
Hyperdiffusions: Free energy sinks
PIERRE MOREL Gyokine?cs for ITER – WPI Wien 22/03/2010
Free energy balance – terms: lhs
−2−1
01
2
0
0.2
0.4
0.6
0.8−1
0
1
2
3
4
5
6
7
8
−2−1
01
2
0
0.2
0.4
0.6
0.8−6
−4
−2
0
2
4
6
Entropy electrosta?c energy
Perp spectra of ?me averaged quan??es:
PIERRE MOREL Gyokine?cs for ITER – WPI Wien 22/03/2010
Free energy balance – terms: rhs
−2−1
01
2
0
0.2
0.4
0.6
0.8−0.5
0
0.5
1
1.5
2
2.5
−2−1
01
2
0
0.2
0.4
0.6
0.85.5
6
6.5
7
7.5
8
8.5
Fluxes drive (eq. gradients) velocity hyperdiffusion (log scale)
PIERRE MOREL Gyokine?cs for ITER – WPI Wien 22/03/2010
Free energy balance – conserva6on in 6me
Formally, the parallel deriva?ves terms should cancel under v-‐z integra?on – Not always verified numerically:
D. Hatch – first free energy analysis with GENE
PIERRE MOREL Gyrokine?cs for ITER – WPI Wien 22/03/2010
Free energy balance – conserva6on in 6me
modified velocity boundaries (based on H0 trajectories)
Source: G
Parallel deriva?ves: Very closed to 0
Sinks: hyp terms
PIERRE MOREL Gyrokine?cs for ITER – WPI Wien 22/03/2010
Free energy balance -‐ nonlinearity
Nonlinear term has no ac?on in the total entropy balance (thanks to PB)
Is responsible of the mode to mode transfer:
PIERRE MOREL Gyrokine?cs for ITER – WPI Wien 22/03/2010
Free energy balance -‐ nonlinearity
The free energy balance depends on resolu?on
The filtering introduces the sub grid term T must act as a dissipa?on of the resolved scales:
Verified numerically:
outline
Sub grid and modelling
Free energy balance
Hyperdiffusion as a first model
Dynamic procedure
PIERRE MOREL Gyokine?cs for ITER – WPI Wien 22/03/2010
PIERRE MOREL Gyrokine?cs for ITER – WPI Wien 22/03/2010
Hyper-‐diffusion as model
In collisionless regime, dissipa?on is ensured by hyper-‐diffusion terms.
Such terms depends on the numerics, in GENE:
Prac?cally, prefactor is a fixed parameter in most of cases
NB: a detailed study of these terms can be found in M. Pueschel Thesis
PIERRE MOREL Gyokine?cs for ITER – WPI Wien 22/03/2010
Hyper-‐diffusion as model
Analyze correla?on between hyp’s and sub grids:
With hyp’s as model:
PIERRE MOREL Gyokine?cs for ITER – WPI Wien 22/03/2010
Hyper-‐diffusion as model
Correla?on betw T and hyp’s during ?me
Reach stable but low level
PIERRE MOREL Gyokine?cs for ITER – WPI Wien 22/03/2010
Hyper-‐diffusion as model
Velocity plane correla?ons
Good correla?ons at mid domains in velocity plane
PIERRE MOREL Gyokine?cs for ITER – WPI Wien 22/03/2010
Hyper-‐diffusion as model
Free energy: velocity hyperdiffusion act on f/F0
Ac?on: damps free energy ?ll perturbed df reaches a constant ra?o of equilibrium
outline
Sub grid and modelling
Free energy balance
Hyperdiffusion as a first model
Dynamic procedure
PIERRE MOREL Gyokine?cs for ITER – WPI Wien 22/03/2010
PIERRE MOREL Gyokine?cs for ITER – WPI Wien 22/03/2010
consider a simula?on with model:
(1)
introduce an intermediate scale, keep same resolu?on, complete with zeroes:
filter eq(1) at intermediate scale:
minimiza?on leads to the value of c:
Dynamic procedure
PIERRE MOREL Gyokine?cs for ITER – WPI Wien 22/03/2010
Dynamic procedure
sim. domain “ar?f.” filter