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

Outline

1. Solar Wind fluctuations (MHD scales)

2. Evidence for

a) turbulence

b) Alfven waves

3. 2-component models

4. Transport model

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

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

5

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:

2a. Evidence for turbulence in SW

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

Example 1. Power-law Spectra

• Power spectra of v, b

• have inertial range

power laws

~ Kolmogorov

-5/3

-2

GoldsteinEA95

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

Ex 3. Kurtosis (simulation)

• .

• Filtered kurtosis

kurt( kc) – same, but with

Fourier cpts k < kc

zeroed

Emagnetic(k)

kc / k_diss

kurt(Gaussian) = 3

Kurtosis: SW data vs MHD simulation

Filtered kurtosis

Eb(k)

WanEA12, ApJ

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

2b. Evidence for Alfven waves in SW

(away from planetary bow shocks)

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)

– …

Cross helicity spectrum

• A-waves have v=b => 1. sc=1

2. Ev / Eb = 1

WangEA12 ApJ

NOTE: Some observations support BOTH

waves and turbulence

eg. variance anisotropy …

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

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 …

3. Two-component Models

a) Slab waves + 2D turb

b) Weak turb + Strong turb

[cf. critical balance]

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>

Slab + 2D…

• Observational support: BieberEA94,96,…

Bieber et al., J. Geophys. Res., 1996

Slab-only model Fit: 5% slab

95% 2D

Magnetic power spectra

• Fit composite freq spectrum, Pslab(f) + P2D(f)

to Ulysses data

FormanEA11 (2D: 5/3, slab: 2)

• So,

slab + 2D “stick figure” model

has limitations

• How can we develop a better model?

• Use results from weak AND strong

turbulence theory …

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

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

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

• ~ 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

• 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]

• x-space interpretation:

wide/narrow wave

packets

• 3 classes of interactions

• Related to

- reduced MHD

- critical balance

x-space

k-space

non-resonant

wave-wave: ~Kraichnan resonant: ~ShebalinEA83

perp transfer ‘trivially’ resonant:

~hydro-like

~ unaware of <B>

B0

x-space

k-space

4. Transport of 2-cpt Models

How do flucts evolve with distance ?

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 ?

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

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

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

5. Equations: linear terms (steady)

2D energy:

wave energy:

2D corrn length:

wave corrn length:

|| scale of W:

pickup ion

driving

relax to res

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

6. Sample Solutions

Sample solutions …

Cshear = 1

= 2 = 0.25

sD = -1/3

WKB

Fixed BCs

Model vs Voyager data

• ‘mapped’ solutions: different BCs for each Voyager interval

•Voyager data: Chuck Smith, John Richardson

distance

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.

Observ/Model agreement is encouraging

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

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