Flux-freezing breakdown observed in high-conductivity...
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1.68 × 10 -3 1.68 × 10 -1 |E Ohm | E mot E Ohm = 1 σ j E mot = - 1 c u × b 10 0 10 1 10 2 k /k f 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 E w (k )k f /E tot ∝ k -5/3 ∝ k -3/2 w = u w = b x /L u 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 Average b (x , t ) Start (x , t ) b ( x 1 , t 0 ) b ( x 2 , t 0 ) b ( x 3 , t 0 ) Transport b 1 (x , t ) b 2 (x , t ) b 3 (x , t ) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 L u t 0 /u 10 -3 10 -2 10 -1 10 0 10 1 Relative error standard N = 512 N = 1024 N = 2048 N = 4096 10 -1 10 0 10 1 t * j rms 10 -1 10 0 10 1 10 2 10 3 10 4 r 2 i (t * )j rms /λ 4t * j rms ∝ t * 8/3 t * ≡ t - t 0 i =⊥ i = || 0 1 2 3 4 5 6 ( r /( r 2 (t )) 1/2 ) 3/4 -20 -15 -10 -5 0 5 ln ( ( r 2 (t )) 1/2 p (r , t ) ) u t * /L u = 0.2716 u t * /L u = 0.6667 u t * /L u = 1.053 least squares fit Flux-freezing breakdown observed in high-conductivity magnetohydrodynamic turbulence CC Lalescu 1 , GL Eyink 1 , E Vishniac 2 , H Aluie 3 , K Kanov 1 , K B¨ urger 4 , R Burns 1 , C Meneveau 1 , A Szalay 1 1 Johns Hopkins University, 2 University of Saskatchewan, 3 Los Alamos National Laboratory, 4 Technische Universit¨ at M¨ unchen 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 ∂ t u = -(u ·∇)u + ν ∇ 2 u -∇p + j × b + F ∂ t b = ∇× (u × b)+ η ∇ 2 b ∇· u = ∇· b =0 j = c 4π ∇× b d x = u( x, τ )d τ + 2η d W(τ ) d b = b ·∇ud τ b(x, t )= b(x, t ) (Eyink, 2009) Turbulent MHD fields are “rough” u ν (x, t ) - u ν (x , t )∼ c 1 x - x h if ν < x - x < L c 2 x - x if x - x < ν E (k , t ) ∼ c 1 k -(1+2h) if 1/L < k < 1/ ν c 2 exp(-k ν ) if k > 1/ ν Is standard flux-freezing valid for infinite conductivity? Textbook derivations based on Alfv´ en’s theorem fail when velocity gradients grow with increasing conductivity. b(x, t )= b 0 (a) ·∇ a X(a, t ) det(∇ a X(a, t )) X(a,t )=x The usual derivation of the Lundquist formula assumes a smooth Lagrangian flow X(a, t ) exists. In fact, unique trajectories for initial particle locations a require finite velocity gradients: X(a, t ) - X(a , t )≤ exp(∇u ∞ (t - t 0 ))a - a Explosive separation by turbulent advection d dt (t )= δ u ()=(g 0 ε) h = ⇒ (t )= 1-h 0 +(g 0 ε) h (t - t 0 ) 1 1-h = ⇒ (t ) ∼ (g 0 ε) h t 1 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
Transcript of Flux-freezing breakdown observed in high-conductivity...
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
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