Post on 29-Sep-2020
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Dynamics and structure of rotating MHD turbulenceat high magnetic Reynolds number
with B. Favier & C. Cambon
Fabien S. Godeferd
Laboratoire de M ecanique des Fluides et d’AcoustiqueUniversit e de Lyon — Ecole Centrale de Lyon, France
Ecole de physique des Houches - 2/2011
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Contents
A numerical approach of the idealized rotating turbulent flow of conducting fluid placed in
a uniform magnetic field. From an initial isotropic state, the flow evolves to a structure
characterized by anisotropic energy transfers mediated by both the kinematic cascade
and the kinematic/magnetic exchange.
– Introduction
– Linear regime of magneto-inertial waves
– Structuration and dynamics of energy, transfers
– Spectral characterization of anisotropy
– Conclusions
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Motivation
Homogeneous anisotropic turbulence as a part
of a complex dynamical system, due to :– Boundary conditions : interfaces, solid walls
– Geometry : enclosure, shape, topography
– Added phenomena due to body forces :
gravity & buoyancy, Coriolis, Lorentz
– Forcing : instabilities, mechanical, large
scalesA model for studying phenomena in liquid metal
flows within an external magnetic field, e.g. the
Earth’s core dynamics
ΩB0
-
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Homogeneous Isotropic Turbulence
@@@R
⇒ Homogeneous Anisotropic Turbulence
Ω
6B0
[3D views created using VAPOR by NCAR]
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Equations
Navier-Stokes equation with Coriolis and Lorentz-Laplace forces
∂u
∂t− ν∇2
u = u× (ω + 2Ω) + j × (B0 + b)−∇P
∇ · u = 0
– vorticity ω = ∇× u
– normalized electrical current j = ∇× b
– pressure P modified by magnetic pressure and centrifugal terms
Induction equation∂b
∂t− η∇2
b = ∇× (u× (B0 + b))
∇ · b = 0
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Non dimensional parameters
Charact. times : eddy turno’r τ = l0u0
magn. damping τη = η
B2
0
Coriolis param. τΩ = 12Ω
Reynolds number Re = u0l0ν
≫ 1
Magnetic Reynolds number RM = u0l0η
> 1 ≃ 100 100
Magnetic interaction parameter N =B2
0l0
u0η> 1 ≃ 10−5 20
Rossby number Ro = u0
2Ωl0< 1 ≃ 10−5 0.05
Elsasser number Λ =B2
0
2Ωη= NRo ≃ 1 ≃ 1 1
Lundquist number S = B0l0η
=√NRM
Lehnert number L = B0
2Ωl0
Magnetic Prandtl number Pr = RM
Re ≃ 10−7 1
Earth’s our
core DNS
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Linear response without dissipation
[Lehnert, 1954 ; Moffatt, 1971]
Toroidal/poloidal decomposition (also Craya-Herring) of fluctuating
fields : u(1,2), b(1,2)
Dispersion relations : inertial waves : ωi = 2Ωcos θ ; Alfven waves ωa = B0k cos θ
∂t
u(1)
u(2)
b(1)
b(2)
+
0 −ωi −iωa 0
ωi 0 0 −iωa
−iωa 0 0 0
0 −iωa 0 0
u(1)
u(2)
b(1)
b(2)
= 0
Magneto-inertial waves dispersion relation
ω =1
2
(
±ωi ±√
ω2i + 4Ω2
a
)
with simplification of the solution if ωi ≫ ωa.
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Illustration of the wave propagation from an impulse
Alven
waves
B0 = 1
Ω = 0
B0 = 1
Ω = 3
B0 = 1
Ω = 10
Inertial
waves
B0 = 0
Ω = 10
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Kinetic and magnetic energies Velocity/magnetic field alignment
Magnetic quantities rescaled
in Alfven speed units viz.
B0 = B/√ρ0µ0
ρ(x) = 2u(x)·b(x)
u2(x)+b2(x)
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Direct Numerical Simulations
– Pseudo spectral method with full de-aliasing
– Implicit viscous/magnetic dissipation terms
– Third-order Adams-Bashforth time
advancement scheme
– 2563 Fourier modes
– Step 1 : forced isotropic hydrodynamic
simulation
– Step 2 : forcing is turned off, Ω and/or B0
are turned on.
– b(x, t = 0) = 0– Decaying turbulence
– Ω //B0 (axisymmetric configuration)
– L < 1→ inertial waves rapid wrt Alfven
waves
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Nonlinear structuration at a glance
Non rotating Rotating
B0 Ω6B06
0
j2m
Current density is shown for cases at high RM and Λ ≃ 0.5
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Statistics of orientation
Velocity/magnetic field alignment Magnetic field components
ρ(x) = 2u(x)·b(x)
u2(x)+b2(x)
reverse trend wrt linear
b‖, b⊥ with symbols
horizontal intermittency w/
rotation
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Energies
Velocity/magnetic energies Ratio
Rotation impedes equipartition
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Dissipations
KE dissipation Magnetic energy dissipation
ǫK = ν⟨ω2
⟩ǫM = η
⟨j2
⟩
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Intermodal and interscale energy transfers (1/2) - Global balance
Kinetic energy equation K = u · u/2∂K
∂t−νu·∇2
u = u·(u×ω)+2u·(u×Ω)+u·(j ×B0)︸ ︷︷ ︸
cancellation
+u·((∇× b)× b)︸ ︷︷ ︸
∂i(ujbjbi)−∂juibibj−∂i(b
2ui)/2
−u·∇P
Magnetic energy equation M = b · b/2∂M
∂t− ηb·∇2
b = b·(∇× (u×B0))︸ ︷︷ ︸
cancellation
+ b·(∇× (u× b))︸ ︷︷ ︸
∂juibibj−∂i(b
2ui)/2
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Intermodal and interscale energy transfers (2/2) - Spectral tran sfers
KE transfer
[u·(u× ω)]Total energy transfer
[u·((∇×b)×b)+b·(∇×(u× b))]
Magnetic fluctuations advec-
tion
[b·(∇× (u× b))]
Reduced “classical” cascade Reduced total transfer (and
dissipation)
Intermodal transfer (non zero-
integral) reduced by rotation
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Spectral equilibrium cut-off
ωi = 2Ω · kk
ωa = B0 · k
L =B0k
2Ω≈ 1
⇒ kc =2Ω
B0
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Spectral anisotropy (1/3) - Shebalin angles
Shebalin angles isotropy : θQ = 55
tan2 θQ =
∑
kk2h|Q(k, t)|2
∑
kk2z |Q(k, t)|2
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Spectral anisotropy (2/3) - Directional spectra
No return to isotropy at small scales :
“Osmidov” scale kΩ =√
Ω3
ǫK≃ 280
Rotation dominant at k < kΩ
Scalings in Galtier [Nonlin. Proc.
Geoph. 16, 2009] E(k⊥, k‖) ∝k−5/3⊥ with k‖ ∼ k
2/3⊥ ?
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Spectral anisotropy (3/3) - Horizontal/vertical partition of energ y
Λ → ∞Non-rotating results
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Conclusions
– Role of inertial waves, dominant at small Elsasser number Λ, as expected, from u, b
alignment
– Structure of rotating MHD turbulence at high RM : current sheets, as in non rotating
case, but with additional folds and small-scale structures re-aligned with rotation axis
– Turbulent scales affected differently depending on the equilibrium between inertial
waves and Alfven waves, parameter kc– Equipartition between kinetic and magnetic energies at scales < kc
– Energy cascade and intermodal transfers damped by rotation, as in hydrodynamic
turbulence
– Higher resolution DNS required for spectral scalings
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