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Queenstown, 09 Feb 2015
Transport of MHD Fluctuations in the
Solar Wind:
2-component models
Sean Oughton
Department of Mathematics
University of Waikato
Hamilton
New Zealand Bill Matthaeus
Ben Breech
Chuck Smith
Phil Isenberg
Karem Osman
Minping Wan
Sergio Servidio
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Outline
1. Solar Wind fluctuations (MHD scales)
2. Evidence for
a) turbulence
b) Alfven waves
3. 2-component models
4. Transport model
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1. Nature of SW fluctuations?
• MHD scales:
velocity v,
magnetic field b
• Turbulence and waves?
Probably
• Evidence for each (examples) • although
some quantities can be interpreted as
support for both
This talk:
incompressible MHD
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Typical Observations
• Solar wind measurements
– mean mag field: <B>
– rms fluctuation: b
• Often b / <B> ~ ½
so <B> significant, but not dominant.
• Induces: – Spectral Anisotropy [observations, models]
– waves
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Turbulence vs Waves
• No spectral transfer (linear case)
• Propagation (move energy)
• Dispersion relation w(k) – each length-scale couples to
a specific time-scale
• Inherently nonlinear => spectral transfer
• Advection [self-distortion]
• No dispersion relation – each length-scale coupled to
many time-scales (and v.v.)
• Wide range of dynamically active length/time scales
k
w(k)
Log k
Lo
g E
(k)
ASIDE:
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2a. Evidence for turbulence in SW
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2a. Evidence for turbulence in SW
• Compare
- SW observations: v, b,…
- MHD turbulence simulations
• Examples:
– v-b alignment: v.b ~ cos
– kurtosis ~ <bx4>
-5/3
-2
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Example 1. Power-law Spectra
• Power spectra of v, b
• have inertial range
power laws
~ Kolmogorov
-5/3
-2
GoldsteinEA95
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Ex 2. v-b alignment
• Prob density function
of
OsmanEA11
512^3
B0=1
sigc:
sw=0.29
sim=0.30
ACE
simulation
sig_c ~ 0.3
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Ex 3. Kurtosis (simulation)
• .
• Filtered kurtosis
kurt( kc) – same, but with
Fourier cpts k < kc
zeroed
Emagnetic(k)
kc / k_diss
kurt(Gaussian) = 3
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Kurtosis: SW data vs MHD simulation
Filtered kurtosis
Eb(k)
WanEA12, ApJ
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1st summary: lots of evidence for turbulence
• Comparing observ / MHD turb sims
– kurtosis WanEA 12
– discontinuities (PVI) GrecoEA 08…, ZhdankinEA 12
– v-b alignment OsmanEA 11
– reconnection GoslingEA 05…, ServidioEA 09
– spectra
– 3rd order moment Sorriso-ValvoEA07,MarinoEA12
MacBrideEA08, OsmanEA11,..
• Non-Gaussian statistics
• Higher-order correlations
GrecoEA09-apj
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2b. Evidence for Alfven waves in SW
(away from planetary bow shocks)
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2b. Evidence for Alfven waves in SW
• Some observ of large-ampl Alfven waves
– A-waves have v = b
– BelcherDavis71, WangEA12
• strong v-b correlation for ~15min
• arc polarization
– isolated instances?
• Other evidence? – high cross helicity intervals (?)
– transport: ~WKB agreement
(BUT lots of problems)
– …
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Cross helicity spectrum
• A-waves have v=b => 1. sc=1
2. Ev / Eb = 1
WangEA12 ApJ
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NOTE: Some observations support BOTH
waves and turbulence
eg. variance anisotropy …
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Variance Anisotropy
• <B>-aligned coords
• perp power dominates:
bx2 : by
2 : bz2 = 5 : 4 : 1
– BelcherDavis71, KleinEA91, HorburyEA95,…
• Interpretations?
A: `slab’ Alfven waves
ampl ⊥ <B>
k || <B>
– No conclusion from min variance dirn
B: quasi-2D turb
ampl AND k
⊥ <B>
<B>
Vsw
y
x
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2nd summary
• Solar wind fluctuations – lots of evidence for turbulence, and Alfven waves
• But (typically) neither
isotropic turb nor Alfven waves on their own
fit data well.
So …
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3. Two-component Models
a) Slab waves + 2D turb
b) Weak turb + Strong turb
[cf. critical balance]
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3a. Slab + 2D
• Model energy spectrum as a sum
2D turb: E(k) ~ k-5/3
plus
||-propagating Alfven waves: E(k||) ~ k||-q
• Fit to observations …
– ecliptic: 1AU 80% 2D (BieberEA96)
– high lat: 40-60% 2D (Smith 2003)
• cos ~ U.<B>
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Slab + 2D…
• Observational support: BieberEA94,96,…
Bieber et al., J. Geophys. Res., 1996
Slab-only model Fit: 5% slab
95% 2D
Magnetic power spectra
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• Fit composite freq spectrum, Pslab(f) + P2D(f)
to Ulysses data
FormanEA11 (2D: 5/3, slab: 2)
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• So,
slab + 2D “stick figure” model
has limitations
• How can we develop a better model?
• Use results from weak AND strong
turbulence theory …
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Recall: Weak Turb explanation for spectral anisotropy
• MHD with ~strong mean field
develops anisotropy
– sharper gradients across B0
– strong k┴ transfer
• Why?
– B0 causes suppression of || transfer
– perturbation theory for Alfven waves
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Improving on slab + 2D
• Fourier space: slab + 2D is – k|| axis and k|| = 0 plane
– neglects most of Fourier space
• Generalise:
2D -> quasi-2D
slab -> wave-like
~ strong turb
~ weak turb
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3b. Weak and Strong turb (Fourier space)
• Two timescales (incomp MHD):
– Alfven tA (k)
– nonlinear tNL (k)
• Which t is faster at a given k ?
– controls k-space dynamics
B0
k
weak
turb
strong
turb
k||
k
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• ~ 2 coupled components:
– wave-like weak turb. flucts: W
– quasi-2D (low-freq) turbulence: Z
Distinguished by which timescale
is shorter
at each k: tNL(k) < tA(k)
• W-region: strong perp transfer
and weak ||-transfer
• Z-region: local “isotropic” transfer
(OughtonEA04,06,11)
Z W
B0
CONCEPTS:
Reduced MHD
KadomtsevPogutse76,
Strauss78,
MontgomeryTurner81,
Montgomery82
Critical Balance
Higdon86
GoldreichSridhar95
k||
k
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• Point: for any
1. Have a k-space region
where nonlinear couplings
dominate over ~wave ones:
* strong turb
* spectrum ~ Kolmogorov
2. For other k, get
~weak turb
with ~perp transfer
• Compressible case?
– simulations (low Mach #) show similar || suppression to incomp case
[Oughton Matthaeus ‘05]
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• x-space interpretation:
wide/narrow wave
packets
• 3 classes of interactions
• Related to
- reduced MHD
- critical balance
x-space
k-space
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non-resonant
wave-wave: ~Kraichnan resonant: ~ShebalinEA83
perp transfer ‘trivially’ resonant:
~hydro-like
~ unaware of <B>
B0
x-space
k-space
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4. Transport of 2-cpt Models
How do flucts evolve with distance ?
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1. Observational B/gnd
• Spacecraft data: fluctuations v, b, r
Voyager, ACE, Ulysses, …
Voyager data
WKB R-4/3
QSN: How do flucts evolve with distance ?
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2. Objective
• Model radial evoln of SW flucts
• Treat flucts as 2 coupled components:
- wave-like (high-freq) flucts: W
- quasi-2D (low-freq) turbulence: Z
Z W
k||
E(k_par, k_perp)
B0
k
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3. Why 2 components?
• Earlier transport models assumed 1 type of fluctuation
• But, physics is • shear drives at low-freq (non-WKB)
• pickup ions drive high-freq flucts (Alfven waves)
So 2 types of flucts improvement
• Equations for
– energy, cross helicity, corrn length
of each component
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4. Processes: What causes the evolution?
• Z,W:
– expansion, advection
– stream-shear [shocks, large-scale inhomog]
• W driven by pickup ions (outer heliosphere)
• Energy exchange between cpts: Z W
• Nonlinear cascades of W, Z heating
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5. Equations: linear terms (steady)
2D energy:
wave energy:
2D corrn length:
wave corrn length:
|| scale of W:
pickup ion
driving
relax to res
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Nonlinear terms: Modeling
• ~von Karman-Howarth phenomenology
2D:
waves:
• Cascades proton heating
• Also eqns for lengths:
• Include cross helicity effects
cascades
~Kraichnan
Oughton et al 06, Phys. Plasma
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6. Sample Solutions
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Sample solutions …
Cshear = 1
= 2 = 0.25
sD = -1/3
WKB
Fixed BCs
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Model vs Voyager data
• ‘mapped’ solutions: different BCs for each Voyager interval
•Voyager data: Chuck Smith, John Richardson
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Parameters
• Model has various parameters, controlling – strength of stream-shear
[forces energies & lengthscales]
– pickup ion driving [reasonably constrained/understood. IsenbergEA03, 05]
– local conserv laws for Z or W nonlinear dynamics
• Solns are typically stable to small changes in these params.
• Similarly for small changes in
boundary conditions for Z2, W2, etc.
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Observ/Model agreement is encouraging
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SUMMARY (OughtonEA 2011, JGR)
Extension of models for radial evoln of SW fluctuations---
2 types of flucts: waves + turbulence
• Allows driving physics to be included more consistently.
• Like 1-cpt models, turbulence cascade terms give
heating/energy levels in ~agreement with observations
• Corrn length:
2-cpt model ~better fit than 1-cpt. Z W
k_par
B0
k_
perp
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Main Points
• MHD turbulence simulations produce many features
similar to SW observations
• Many observations can’t be explained using just
linear waves
• MHD with a B0 = combination of strong and weak
turb
• 2-cpt transport model fits observ data reasonably well