helical mhd and a -effect
DESCRIPTION
Helical MHD and a -effect. Axel Brandenburg ( Nordita, Copenhagen ) Kandaswamy Subramanian ( IUCAA, Pune ). arXiv:astro-ph/0405052 Phys. Rept. (244 pages, 62 figs). MHD equations. Magn. Vector potential. Induction Equation:. Momentum and Continuity eqns. Viscous force. - PowerPoint PPT PresentationTRANSCRIPT
Helical MHD and Helical MHD and -effect-effect
Axel Brandenburg (Nordita, Copenhagen)
Kandaswamy Subramanian (IUCAA, Pune)
arXiv:astro-ph/0405052Phys. Rept. (244 pages, 62 figs)
Brandenburg: Helical MHD 2
MHD equationsMHD equations
JBuA
t
visc2 ln
D
DFf
BJu
sc
t
utD
lnD
AB
BJ
Induction
Equation:
Magn.Vectorpotential
Momentum andContinuity eqns
ln2312
visc SuuF
Viscous force
Brandenburg: Helical MHD 3
Vector potentialVector potential
• B=curlA, advantage: divB=0• J=curlB=curl(curlA) =curl2A• Not a disadvantage: consider Alfven waves
z
uB
t
b
z
bB
t
u
00 and ,
uBt
a
z
aB
t
u02
2
0 and ,
B-formulation
A-formulation 2nd der onceis better than1st der twice!
Brandenburg: Helical MHD 4
Pencil CodePencil Code
• Started in Sept. 2001 with Wolfgang Dobler
• High order (6th order in space, 3rd order in time)
• Cache & memory efficient
• MPI, can run PacxMPI (across countries!)
• Maintained/developed by many people (CVS!)
• Automatic validation (over night or any time)
• Max resolution so far 10243
Brandenburg: Helical MHD 5
Helical versus nonhelicalHelical versus nonhelical
Inverse cascade only when Inverse cascade only when scale separationscale separation
Kida et al. (1991)Kida et al. (1991)helical forcing, but no inverse cascadehelical forcing, but no inverse cascade
Brandenburg: Helical MHD 6
Allowing for scale separationAllowing for scale separation
No inverse cascade in No inverse cascade in kinematic regimekinematic regime Decomposition in terms of Decomposition in terms of
Chandrasekhar-Kendall-Waleffe functionsChandrasekhar-Kendall-Waleffe functions
00kkkkkkk hhhA aaa
t2
peakk
Position of the peak compatible with
Brandenburg: Helical MHD 7
Kazantsev spectrum (kinematic)Kazantsev spectrum (kinematic)
Kazantsev spectrum Kazantsev spectrum confirmed (even for confirmed (even for =1) =1)
Spectrum remains highly Spectrum remains highly time-dependenttime-dependent
Opposite limit, no scale separation, forcing at kf=1-2
8
256 processor run at 1024256 processor run at 102433
EM(k) not peaked at resistive scale, as previously claimedinstead: kpeak~Rm,crit
1/2 kf ~ 6kf
Brandenburg: Helical MHD 9
Structure function exponentsStructure function exponents
agrees with She-Leveque third moment
Brandenburg: Helical MHD 10
Bottleneck effect: Bottleneck effect: 1D vs 1D vs 3D3D spectra spectra
Compensated spectra
(1D vs 3D)
Brandenburg: Helical MHD 11
Relation to ‘laboratory’ 1D spectraRelation to ‘laboratory’ 1D spectra2222
3 )(4)( kuku kdkE kD yxkyxkE zzD d d ),,(2)(
2
1 u
kkkkkkkzk
z d )(4d ),(42
0
2
uu
kk
E
zk
D d 3
0zk
222zkkk
Brandenburg: Helical MHD 12
Bottleneck in the literatureBottleneck in the literature
Porter, Pouquet, & Woodward (1998) using PPM, 10243 meshpoints
Kaneda et al. (2003) on the Earth simulator, 40963 meshpoints
Brandenburg: Helical MHD 13
Helical MHD turbulenceHelical MHD turbulence• Helically forced turbulence (cyclonic events)
• Small & large scale field grows exponentially
• Past saturation: slow evolution
Explained by magnetic helicity equation
14
AnimationsAnimations
15
Effects of magnetic helicity conservation
BJBA 2dt
d
0const BAEarly times:=0 important
Late times: steady state 0BJ
0 bjBJBJ
0 baBABA
flux term
02f
2m baBA kk
012f
2m
BABA
k
k
By the time a steady state is reached: net magnetic
helicity is generated
Brandenburg: Helical MHD 16
Slow-down explained by magnetic helicity conservation
2f
2m
21m 22 bBB kk
dt
dk
molecular value!!
BJBA 2dt
d
)(2
m
f22 s2m1 ttke
k
k bB
Brandenburg: Helical MHD 17
Connection with Connection with effect: effect: writhe with writhe with internalinternal twist as by-product twist as by-product
clockwise tilt(right handed)
left handedinternal twist
Yousef & BrandenburgA&A 407, 7 (2003)
Brandenburg: Helical MHD 18
Internal twist as feedback on Internal twist as feedback on (Pouquet, Frisch, Leorat 1976)(Pouquet, Frisch, Leorat 1976)
031 / bjuω
How can this be used in practice?
Need a closure for <j.b>
Brandenburg: Helical MHD 19
Rm dependence of PFL formulaRm dependence of PFL formula
031 / bjuω
St = urms kf not
suppressed in Rm dependent fashion
is suppressed in Rm dependent fashion
Brandenburg: Helical MHD 20
Revised nonlinear dynamo theoryRevised nonlinear dynamo theory(originally due to Kleeorin & Ruzmaikin 1982)(originally due to Kleeorin & Ruzmaikin 1982)
BJBA 2d
d
t
BJBBA 22d
dE
t
bjBba 22d
dE
t
Two-scale assumption JB t E
Production of large scale helicity comes at the priceof producing also small scale magnetic helicity
Brandenburg: Helical MHD 21
Express in terms of Express in terms of
bjBba 22d
dE
t
M
eqmfM B
Rkt
2
22d
d BE
Dynamical -quenching (Kleeorin & Ruzmaikin 1982)
22
20
/1
/
eqm
eqmt
BR
BR
B
BJ
Steady limit: consistent with VC92
no additional free parameters
(algebraicquenching)
Brandenburg: Helical MHD 22
Is Is tt quenched? quenched?can be in models with shearcan be in models with shear
Larger mean field
Slow growthbut short cycles:
Depends onassumption aboutt-quenching!
Brandenburg: Helical MHD 23
Additional effect of shear
Negative shear
Positive shear
Consistent with g=3 andeq
t0t /1 Bg B
Kitchatinov et al (1996), Kleeorin & Rogachevskii (1999)
Brandenburg: Helical MHD 24
Effect of surface losses of Effect of surface losses of current helicitycurrent helicity
• Large scale (LS) field:– Drainage on LS dynamo
• Rm-dependent cutoff
– Shortens saturation time
• Small scale (SS) field– Enhancement of LS dynamo
SJE d2
Sje d2
25
The need for small scale lossesThe need for small scale losses
Numerical experiment:
remove field for k>4every 1-3 turnover
times
2) higher saturation level3) still slow time scale
1) large scale losses:lower saturation level
2ff
2mm
21m 22 bBB kk
dt
dk
)(2
mm
ff22 s2mm1 ttke
k
k
bB
2smff
2 )(2 bB ttkk initial slope
Brandenburg: Helical MHD 26
How do magnetic helicity losses How do magnetic helicity losses look like?look like?
N-shaped (north)S-shaped (south)(the whole loop corresponds to CME)
Brandenburg: Helical MHD 27
Sigmoidal filamentsSigmoidal filaments
(from S. Gibson)
Brandenburg: Helical MHD 28
Examples ofExamples ofhelical structures helical structures
Brandenburg: Helical MHD 29
Simulating solar-like Simulating solar-like differential rotationdifferential rotation
Brandenburg: Helical MHD 30
Results for current helicity fluxResults for current helicity flux
kjikji BBuF ,C
First order smoothing, and tau approximation
Vishniac & Cho (2001
Expected to be finite on when there is shear
Brandenburg: Helical MHD 31
ConclusionsConclusions• Homogeneous dynamos saturate resistively
– Entirely magnetic helicity controlled
• Inhomogeneous dynamo– Open surface, equator– Still many issues to be addressed– Current helicity flux important
• Finite if there is shear
– Avoid magnetic helicity, use current helicity