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