Olga Alexandrova, Universität zu Köln

Turbulence behind collisionlessMHD shocks and Alfvén vortices

Wave Turbulence workshop, IHP 10/04/2009

Examples of colissionless shocks

• Supernovae remnants

Cassiopeia A

• CME (Coronal Mass Injection)

• Planetary bow-shocks

In situ measurementswith space missions

(CLUSTER)

– forms in front of the Magnetosphere of the Earth (interactionwith supersonic solar wind)

– the closest example of collisionless MHD shock

In the solar wind: m.f.p.~1 AU ⇒ no collisions ⇒ dissipation ofkinetic energy across the bow-shock ?

Terrestrial bow-shock

solar wind

shock

!

"Bn

Planetary bow-shock and downstreamturbulence

Increase of B0 ⇒ trapped ions ⇒Anisotropic ion distributions ⇒

Source of free energy ⇒Turbulence

[Leroy et al.,1982]

Low frequency (f<fci) Q-linear instabilities

Wave Turbulence ?

transversal mode, k||B

(2) Mirror

compress. mode k ⊥ B

B0

(1) Alfven Ion Cyclotron (AIC)

!>"# 1||T

T

Plasma is in the marginal stability state,(controlled by Q-linear instabilities)

unstable

stable(2)

(1)

[Lacombe & Belmont, 1995]

upstreamdownstream

temperature anisotropy

shock

(nT)

(nT)

time (s)

Example of the Earth’s bow-shock crossing byCluster satellite

[Alexandrova et al., 2004, JGR]

Example of AIC instability observation

Solar windMagnetosheath

AIC waves??

[Alexandrova et al., 2004, JGR]

Wavelet analysis : time-scale decomposition

Turbulent spectra in the magnetosheath

!

"Bn

Kolmogorov spectrum!

[Alexandrova et al., 2008, AnGeo]

6.2!f

___ - - - ||SS!

cpf

[Czaykowska et al., 2001]

Clo

se to

the

shoc

k

Flan

ks

• Spectra with a break (like SW), butthere is a spectral knee!• Close to the shock front: spec~f-1• In the flanks: spec_perp~f-5/3

• Turbulence develops with thedistance from the shock• Spectral knee preserves with thedistance from the shock…

– Background magnetic field B0 ⇒ anisotropy– No collisions ⇒ dissipation? kmax?– Characteristic scales and frequencies ⇒ spectral breaks– Energy increase on the 1st break! What is this?

Magnetic spectrum behind bow-shock

!

fcp =1

2"

eB0

mpc

[Alexandrova et al.,2008]

kinetic scalesMHD scales

Nature of the spectral knee[Alexandrova et al., 2006, JGR]

• Intermittent events :• Alfvenic coherent fluctuations, δB⊥>δB||• Deviation of Gaussian statistics

Dimension of Alfvenic fluctuations

tube (2D)

B0

wave packet (1D)

k

B0

– Impossible to separate these 2topologies with 1 point measurements

– Cluster mission (4 identical satellites,δR ~ 700 km) ⇒ separation time/space

C1

C2

C3 C4

J

V

2D : current || B0

C1

C2

C3 C4

k

V

⇒ δT34 =0

Plane wave ⇒ infinite phase frontSpace localization⇒ finite δT betweenthe satellites

1D : Alfvenicwave packet

Dimension of Alfvenic fluctuations

Simultaneous measurementson 4 Cluster satellites

Superposed measurements;time lags between the satellitesare determined using themaximum of the cross-correlation function R:

C1

C2

C3 C4

VbV

Bo

In plasma frame:

Superposition of δB measured by4 Cluster satellites ⇒ magnetic filaments

[Alexandrova et al. 2006, JGR]

2D (k|| < k⊥) incompressible (δB || < δB⊥) Alfvenic (δV⊥ || δB⊥)

Incompressible MHD has solutions in theform of magnetic vortex (Alfvén vortex)

Properties of the observed coherent fluctuations :

Field aligned vortex: monopole Inclined vortex: dipole

B0y

Alfvén vortex

monopole

dipole

ω - vorticity & Ψ - stream function

HD vortices: localized solutions of2D Navier-Stokes equation

Particular case: slow variations & vorticity is localized in a circle of the radius a

- Helmholtz’s equation- Laplace’s equation

Vector potential, A, ~ to stream function ⇒field lines || stream lines & current || vorticity

[Petviashvilli & Pokhotelov, 1992]

Alfvén vortices ~ 2D HD vortices

Monopole ~ force freecurrent, standing structure

Dipole ~ two inversedcurrents, propagates

Magnetic fluctuations of Alfvén vortices

Satellite

[Alexandrova et al., JGR, 2006]

In the Earth’s magnetosheath weobserve both vortex types !

monopole

dipole

Observ.

Model

MHD turbulence behind the bow-shock

Why the vortices appear on the spectral break?Stability in compressible plasma?Distribution: random or network?Universality in space plasmas?…

Superposition of waves & coherent structures, inthe form of Alfven vortices (k||=0)

Universality of Alfven vortices?Saturn’s Magnetosheath (Cassini)

Plasma and field are very different with the Earth’s conditions!

We have no 4 s/c, but still we can try to find vortex signatures

BIMF = 0.3 nT Bmsh = 1.2 nT nmsh ~ 0.5 cm-3 (Voyager-2) Vb,msh ~ 130 km/s (Voyager-2) c/ωpi ~ 300 km Mach ~ 15

Localized Alfvénic events at Saturn:Cassini observations

Scalograms of compressive δB||and transverse δB⊥fluctuations:

δB2⊥(t) δB2|| (t)

δB2||(t,τ)

δB2⊥(t,τ)

[Alexandrova & Saur, 2008, GRL]

We observe signatures of the Alfven vortices! Same scales a~10c/ωpi Indication of universality of the phenomena

LH: Vϕ ↓ with k

RH

Alfvén waves dispersion in Hall MHD

– low frequencies : ε = vl /vϕ << 1 ⇒ weak turbulence is possible– at f~fci Vϕ ~ 0 and τNL< τA ⇒ generation of coherent structures?

– Direct cascade from large scale and inverse one from small scales?– Change of turbulence nature ⇒ Bottleneck effect?– From weak (ε = τA/τNL<< 1) to strong turbulence?

ε=τA/τNL<1 ε>1 τnew

Why vortices appear at the spectral break?

Vortex distribution ?

- random or network ?

Distance between the vortices, λ,varies between 1200 and 1800 km;a ~ 500 km

B

λ

2a

7 sec 10 sec 9 sec7 sec

Distance between two closest vortices

λ/a ~ 3

Cross-correlation function:

C2

C4

C1

C3

Spectral properties of Alfvén vortices

a-1 a-1

• Spectral knee at k=a-1 ; power law spectra above it• Monopole ⇒ δB2~k-4 (due to discontinuity of the current)• Dipole ⇒ δB2~k-6 (due to discont. of the current derivative)

Vortex radius a=1

Spectral properties of a vortex network(model)

Plateau appears at the scales between λ-1 and a-1

B

λ

2a

[Alexandrova, NPG, 2008]

Vortex radius a=1Network parameter λ=10

Vortex network signatures in themagnetosheath spectra (2 cases)

• a ~700 km = 13 c/ωpi• λ ~ 2300 km = 42 c/ωpi

16/12/2001

• a ~ 300 km ~ 10 c/ωpi• λ ~ 1000 km ~ 30 c/ωpi

31/03/2001

λ-1 a-1

λ/a ~ 3

MHD turbulence behind the bow-shock

Why the vortices appear on the spectral break?Stability in compressible plasma?Distribution: random or network?Universality in space plasmas?…

Superposition of q-linear waves & coherentstructures, in the form of Alfven vortices (k||=0)

Solar wind M/sheath

Shock