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