1.68× 10−3
1.68× 10−1
|EOhm|E ′mot
EOhm = 1σj
Emot = − 1c u× b
100 101 102
k/kf
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
Ew(k)k
f/Etot
∝ k−5/3
∝ k−3/2
w = uw = b
x/Lu
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
y/L u
3.0
3.5
4.0
z/L u
8.9
9.0
9.1
9.2
9.3
9.4
9.5
9.6
Averageb(x , t)
Start (x , t)
b(x1, t0)
b(x2, t0)
b(x3, t0)
Transport
b1(x , t)
b2(x , t)
b3(x , t)
0.0 0.2 0.4 0.6 0.8 1.0 1.2
Lut0/u′
10−3
10−2
10−1
100
101
Rela
tive
erro
r
standardN = 512N = 1024N = 2048N = 4096
10−1 100 101t∗jrms
10−1
100
101
102
103
104
〈r2 i〉(t ∗)j r
ms/λ
4t∗jrms
∝ t∗8/3
t∗ ≡ t − t0i =⊥i = ||
0 1 2 3 4 5 6(r/(〈 r2〉(t))1/2
)3/4−20
−15
−10
−5
0
5
ln( (〈
r2〉(t))1/2 p(r,t))
u′t∗/Lu = 0.2716u′t∗/Lu = 0.6667u′t∗/Lu = 1.053least squares fit
Flux-freezing breakdown observed in high-conductivity magnetohydrodynamic turbulenceCC Lalescu1, GL Eyink1, E Vishniac2, H Aluie3, K Kanov1, K Burger4, R Burns1, C Meneveau1, A Szalay1
1Johns Hopkins University, 2University of Saskatchewan, 3Los Alamos National Laboratory, 4Technische Universitat Munchen
MHD simulation data is stored in a public, web-accessible database, please visit http://turbulence.pha.jhu.edu for more information.Work supported by NSF grant CDI-II: CMMI 094153, OCI-108849, JHU’s IDIES, and the NSERCC.
Magnetic field lines in a resistive plasma “move” stochastically
∂tu = −(u · ∇)u+ ν∇2u−∇p + j× b+ F
∂tb = ∇× (u× b) + η∇2b∇ · u = ∇ · b = 0j = c
4π∇× b
d x = u(x, τ)dτ +√2ηdW(τ)
d b = b · ∇udτb(x, t) = 〈b(x, t)〉
(Eyink, 2009)
Turbulent MHD fields are “rough”
‖uν(x, t)− uν(x′, t)‖ ∼{c1 ‖x− x′‖h if `ν < ‖x− x′‖ < Lc2 ‖x− x′‖ if ‖x− x′‖ < `ν
E (k, t) ∼{c ′1k−(1+2h) if 1/L < k < 1/`νc ′2 exp(−k`ν) if k > 1/`ν
Is standard flux-freezing valid for infinite conductivity?
Textbook derivations based on Alfven’s theorem failwhen velocity gradients grow with increasing conductivity.
b(x, t) = b0(a) · ∇aX(a, t)det(∇aX(a, t))
∣∣∣∣∣X(a,t)=x
The usual derivation of the Lundquist formula assumes a smoothLagrangian flow X(a, t) exists. In fact, unique trajectories for initialparticle locations a require finite velocity gradients:
‖X(a, t)− X(a′, t)‖ ≤ exp(‖∇u‖∞(t − t0))‖a− a′‖
Explosive separation by turbulent advection
ddt `(t) = δu(`) = (g0ε`)h
=⇒ `(t) =[`1−h0 + (g ′0ε)h(t − t0)
] 11−h
=⇒ `(t) ∼ (g ′0ε)ht1
1−h
initial separation is forgotten
`ν → 0
while ν
η= const
=⇒ above result still holds,
even for `(0) = 0!
(Bernard, Gawedzki & Kupiainen, 1998)
Standard flux-freezing is wrong
by many orders of magnitude
in high conductivity MHD turbulence
Top Related