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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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