Flux tubes, magnetic reconnection and turbulence in the solar wind Z.Vörös 1 1.Space Research Institute, Austrian Acad. Sciences, Graz, Austria 2.INAF-IAPS,

Flux tubes, magnetic reconnection and turbulence in the solar wind

Z.Vörös1

1. Space Research Institute, Austrian Acad. Sciences, Graz, Austria

2. INAF-IAPS, Rome, Italy

3. Faculty of Physics, Sofia University, Bulgaria

T. Zaqarashvili1, J. Sasunov1, R. Bruno2, G. Consolini2

Y. Narita1, I. Zheliazkov3, STORM team

Project support: EU-FP7-STORM-313038 & FWF P26181-N27

Solar system plasma turbulence, intermittency and multifractals, 06-13 September, 2015, Mamaia, Romania

•Turbulence statistics: PDFs

•Turbulence, intermittency and structures

•Reconnection outflow generated turbulence

•Flux tubes within reconnection outflows • Unstable twisted flux tubes

OutlineOutlineImage Credit: NASA/Goddard Space Flight Center/SDO


Mamaia beach, 9.9.2015

Intermittency in neutral fluidsIntermittency in neutral fluids


“Intermittency is the non-uniform distribution of eddy formationsin a stream. The modulus or the square of the vortex field, the energy dissipation velocity or related quantities quadratic in the gradients of velocity and temperature (of the concentration of passive admixture) may serve as indicators.” (Novikov, 1971)

Leo

nar

do

PDFs: strong gradients (long tails) develop when the Re number increases.Spectra: High Re number turbulence has a wide range of scales.

e.g. Jimenez et al., 1993

~

Frisch, 1995

N-S equation nonlinear interactions turbulent cascade

Intermittency in the solar wind: PDFs as in Intermittency in the solar wind: PDFs as in neutral fluids neutral fluids


1 h

16 h

1.3 d

85 d

Correlation length: 0.015 AU- turbulence0.25 AU – large-scale struct.

Is there any coupling between these scales?

1. Kappa2. Log-normal3. Gaussian 4. else?

1.

2.

2.

3.

Burlaga & Vinas, 2004Leubner & Vörös, 2005Consolini et al., 2009

Kappa distribution: correlations, memoryKappa distribution: correlations, memory


Log-normal: multiplicationsLog-normal: multiplications

Multi-scale PDFs in the solar windMulti-scale PDFs in the solar wind


Harward-SmithonianCtr for Astrophysics

Wan et al, 2009.

Additive, multiplicative, correlativeAdditive, multiplicative, correlative

Log-kappa PDF – solar cycle minimumLog-kappa PDF – solar cycle minimum

Conditional statisticsConditional statistics

Cross-scale coupling: multi-scale vs. small-scaleCross-scale coupling: multi-scale vs. small-scale

Vörös et al., 2015

Conclusions I.Conclusions I.

• Conditional statistics is necessary for understanding ofmulti-scale multi-component processes

• Large-scale structures modify turbulent statistics overthe small-scales

• Solar wind fluctations: mixture of additive, multiplicativeprocesses exhibiting correlations

In the limit of large kappa ´• kappa = normal• log-kappa = log-normal

Intermittency and structure formation

Intermittency in the solar wind: challenges Intermittency in the solar wind: challenges


The nonlinear term is responsible for the local transfer of energy between adjacent scales. However, additional correlations can compete or supressthe nonlinear term (Matthaeus et al, 2015).

Elsässer variables

reduces inductionequation

the Lorentz force

Simulation results: turbulence and structuresSimulation results: turbulence and structures

2D ideal compressibleMHD

Magn. fieldWu & Chang, 2000

Ideal Resistive MHD

Current density: sheet-likestructures; Wan et al., 2009;Coherent structures ofnon-dissipative origin

Simulation results: turbulence and structuresSimulation results: turbulence and structures

2D MHD, moderate Reelectric current density + B

Matthaeus et al.,2015

The partial varianceof increments (PVI)

allows to detect thecurrent sheets ordiscontinuities, orflux tube walls(Greco et al., 2008),at which heatingoccurs (Osman et al.2011).

Conclusions II.Conclusions II.


• Simulations indicate that turbulence can generatestructures, current sheets, discontinuities, whichcan be flux rope walls,

• The structures are associated with the long tailsof the PDFs,

• The structures in both simulations and experimentsare places of reconnection and heating,

Although turbulence can generate coherent structures in the solar wind, there is a large variety of structures which are not generated by turbulence.

Reconnection generated turbulenceReconnection generated turbulence


The newly reconnected field linesassociated with plasma inflow throughthe separatrix are kinked.

The kinks representing a pair of Alfvenic discontinuitie or RDs in the inflow regions are accelerating plasmasto the exhaust.

Usually the exhausts contain plasmaswith decreased B and enhanced Tp and Np, the transition to the exhaust isa slow-mode shock. (Gosling, 2013).

Enhanced fluctuations near the exhaustboundaries (Huttunen et al., 2007).


Reconnection generated turbulenceReconnection generated turbulence


An unstable current sheet, via reconnection, decays into a systemof moving large amplitude fast, slow,Alfven waves.

The waves propagate along the currentsheet together with the reconnectedplasma and magnetic flux,collectively forming the outflow regionwith embedded discontinuities and apropagating flux tube. (Sasunov et al.2012).

The flux tube can be K-H unstableTD generating turbulence and heating.


Example: reconnection outflowExample: reconnection outflow

Vörös et al. 2014


Reconnection outflow: spectral featuresReconnection outflow: spectral features


Vörös et al. 2014field-aligned fluctuations

Conclusions III.Conclusions III.


• In both reconnection scenarios turbulence can be generated locally,

• The turbulent layer contains a slowly cooling-mixingplasma exhibiting a fluctuating temperature-densityprofile with embedded directional changes of B,

• In the reconnection database of Phan et al., (2009)we found that 27 outflow events (out of 51) were accompanied by a turbulent boundary layer.

Magnetic flux tubesMagnetic flux tubes

Laboratory, space and astrophysical plasmasself-organize themselvesto non-space-fillingcoherent structures (Alfven 1951).

Flux tubes are volumesenclosed by magneticfield lines which intersect closed curves(Priest, 1991, 2014).

L1 L2

Markidis et al., 2014

TokamakYun et al., 2012

Laboratory(electron cyclotronemission imaging)

Space(remote observation ofa coherent structure withina white-light CME)

Cheng et al., 2014

Space(in-situ observations:trajectories of the tipsof B in the solar wind)

Bruno et al., 2001





L1 L2


Space(in-situ observations:Flux transfer events atthe magnetopause)

Russel & Elphic, 1978,1979

Space(in-situ observations:Transport of flux ropesfrom the dayside to the flank tail)

Eastwood et al., 2014

Themis

Artemis





L1 L2


Space(in-situ observations:Flux tubes/plasmoids in the Earth‘s magnetotail)

Hietala et al., 2014


Reconnection

Astrophysics(remote observation:Double helix nebula)

Morris et al., 2006Spitzer Space Telescope

80 li

ght

year

s


Twisted magnetic flux tubesTwisted magnetic flux tubes

Multi-wavelength observations:magnetic flux can be transported from the solar interior into the atmosphere and corona in the form of TWISTEDflux tubes (Okamoto et al. 2008; Srivastava et al., 2010).

Simulations:The bodily emergence of coherent flux tubes requires a nonzero TWIST (Hood et al. 2009).

Observations:A CME is the eruption of a coherent magnetic, TWISTcarrying coronal structure... (Vourlidas et al., 2013)

Loop with helical twist

ACTIVE REGION

SOHO/MDI contours onTRACE 171 A imageon June4, 2007.White/black contours showpositive/negative polarity.Srivastava et al. 2010.

Vourlidas et al., 2013


Twisted tubes in the solar Twisted tubes in the solar windwind

Moldwin et al., 2000

Force free: plasma << 0.5

0

Force free solution (Lundquist, 1950)in cylindrical coordinates:

Observations:Lynch et al.,2005; DURATION:Feng et al. 2007,2008; 0.5 - 40 hrsLepping and Wu, 2010; ORIGIN:Cartwright and Moldwin, 2010; Sun, HCS, MR,?Janvier et al., 2014

Model:


( )

The force-free model has limitations regarding the real magnetic field structure mainly for small-scale twisted tubes for which > 0.5.

We propose a model for the twisted magnetic tubes based on the observations of the total (thermal+magnetic) pressure. This removes the limitations of the force-free approach.

Twisted tubes in the solar Twisted tubes in the solar windwind

Bz

0In ,4

20

r

B

dr

dPT

8

22

00

BBPP z

T

.

4

10

0

2

00 dss

sBPrP

r

TT

Totalpressure

Pressure balancecondition insidethe tube

The total pressure is not constant across the tube andits structure depends on twist!

= 0


Wind observations of twisted tubeWind observations of twisted tube

WIND observation of a twisted magnetic tube. The same event was analyzed by Feng et al. (2007) and Telloni et al. (2012) .

Zaqarashvili, Vörös, Narita and Bruno, ApJL 2014


Total pressure profile

ArrB )(


Red solid line: theoretical profile of total pressure in the tubes with Black line: Feng et al. (2007)Green, magenta, blue and purple lines: Moldwin et al. (2000)


Kink instability of twisted tubes

The twisted tubes are subjects to the kink instability when the twist exceeds a critical threshold value.

The instability may trigger the magnetic reconnection near boundary owing to kinking motion, which may lead to remove the additional twist from the tube.

Therefore, it is interesting to find the critical twist angle for solar wind tubes.

Török and Kliem 2004

An external magn.fieldcan increase the thresholdand stabilize the instability(Benett et al. 1999).However, a flow along thetube may decrease thecritical threshold(Zaqarashvili et al., 2010)


The bounded solution at the tube axis is

Solutions inside and outside the tubeSolutions inside and outside the tube

,01 2

02

2

2

2

t

tt pmr

m

dr

dp

rdr

pd

,)(4

41

220

2222

0

A

AAkm

.4 0

2

z

A

kBmA

Inside the tube:constant density , ).,,0( zBAr

where is the longitudinal wavenumber, is the frequency

),0,0( zeBOutside the tube: constant density ,

).( 00 rmIap mt ).(krKap met

The bounded solution at infinity is

The magnetic tube is moving along Z by speed –U. To obtain the dispersion relationgoverning the dynamics of the tube, the linearized MHD equations are perturbed by and Bessel equations for total pressure are obtained insideand outside of the tube (Dungey and Loughhead, 1954)


Marginal stability analysis in the thin tube approximation gives the criterion for the kink (m=1) instability as

Dispersion equationDispersion equation

,)(/4

)(4

44

2)(4

022

0

0

222220

022

0

kaPkU

kaP

A

mamF

meAe

m

AA

AmA

Continuity of Langrangian displacement and Lagrangian pressure at the tube surface leads to the transcendental dispersion equation

where

.;'

00

0'

00 kaK

kakaKamP

amI

amaImamF

m

mm

m

mm

,112)( 22

0

A

ezz M

A

kBBaB

where z

e

B

B is the Alfvén Mach number.

0AA V

UM and


Critical twist angleCritical twist angle

The critical twist angle for the kink instability can be approximated as

.12arctan)(

arctan 22

0

Ae

zc M

B

aB

The critical twist angle decreases when μ decreases and MA increases.

Maximum critical twist angle (~700) occurs for μ=1 and MA=0.

The tubes twisted with >700 angle are always unstable for the kink instability.


Red surface:

8.00

e

Blue surface -

5.00

e

z

e

B

B

0AA V

UM


Conclusions IV: twisted tubes & kink in the SW

Three from five tubes have the twist angle of 500 – 550, which is less than the critical angle for the kink instability of the tubes with M

A<0.5. But it may become

larger than the critical angle for tubes with MA>0.5,

therefore the magnetic reconnection can occur near boundary owing to kinking motion.

Suppose a magnetic flux tube is twisted with sub-critical angle near the Sun. The Alfvén Mach number can increase during the transport of the flux tube towards the Earth due to the decrease of the Alfvén speed.

Consequently, initially stable flux tube may become unstable to the kink instability at some distance from the Sun.


First we consider the situation when magnetic tubes move along the Parker spiral.

Therefore, the external magnetic field is axial.

Then we have the dispersion equation which is already presented above.

Kelvin-Helmholtz instability: external untwisted Kelvin-Helmholtz instability: external untwisted fieldfield


Then we get

Kelvin-Helmholtz instability: external untwisted Kelvin-Helmholtz instability: external untwisted fieldfield

Thin flux tube approximation yields the polynomial dispersion relation

Let us consider the modes propagating perpendicular to magnetic field i.e.

.0)(4

42

)(4

||2

0

00

0

22222

022

0

0

0

02

e

A

eAeA

ee

AmAUkUk

kU

.1||1||2

202

e

e

eA B

BmMm

The criterion means that only super-Alfvénic motions are unstable as magnetic field stabilizes the Kelvin-Helmholtz instability.

.0Bk

.0)(4

||2

0

222222

0

0

0

02

e

Aeee

mAUkUk

kU

The Kelvin-Helmholtz instability yields the complex frequency, therefore the instability criterion is


Numerical solution of dispersion equationNumerical solution of dispersion equation

Zaqarashvili, Vörös and Zheliazkov, A&A, 2014

The figure shows m = −3 unstable harmonics for different values of MA

after the solution of general dispersion equation.

It is seen that the critical Alfvén Mach number equals to 1.491 for m = −3 harmonics.

The analytically estimated critical Mach number is MA ≈ 1.4907 for m = −3

harmonics.


If the tubes move at an angle to the Parker spiral, then the external magnetic field will have a transverse component.

Therefore, we now consider that the external magnetic field is also twisted and has a form

2

,,0r

aB

r

aB eze

Kelvin-Helmholtz instability: external twisted Kelvin-Helmholtz instability: external twisted fieldfield


where

,)(4

41

222

2222

a

Bkm

Aee

ee

.4 2

consta

kaBmB

e

ezeAe

The external density can be taken as

so that the Alfvén frequency is constant

),(2

2

rmKr

aap eet

4

r

ae

.4

8

4

44

22222

222

Aee

Aee

Aee

e

a

mB

a

Bmm

Then the solution of total pressure perturbation outside the tube is



.)(

)(

4/4

4/2)(222

222

222220

0022

GamQaHL

GamQa

AkU

mAamFkU

eAe

eAe

AA

AmA

The we get the following transcendental dispersion equation

Thin flux tube approximation yields the polynomial dispersion relation as

.04

2||2

||1

2222

0

e

e mAUkkU

m

m

The Kelvin-Helmholtz instability yields the complex frequency, therefore the instability criterion is

.||

||21|| 02

eA m

mMm

Harmonics with sufficiently high m are always unstable for any value of MA .




Zaqarashvili, Vörös and Zheliazkov, A&A, 2014

The figure shows unstable harmonics for different values of MA

after the solution of dispersion equation.

It is seen that there are unstable harmonics with a sufficiently high azimuthal mode number m for any value of Alfvén Mach number.

Kelvin-Helmholtz vortices may contribute into the solar wind turbulence.


•Even slightly twisted magnetic tubes moving with an angle to the Parker spiral are unstable to the Kelvin-Helmholtz instability for sufficiently large azimuthal wave number - m.

•The Kelvin-Helmholtz vortices near the tube boundary may significantly contribute to solar wind turbulence.

Conclusions V: Kelvin-Helmholtz instabilityConclusions V: Kelvin-Helmholtz instability

.1||1||2

202

e

e

eA B

BmMm

.||

||21|| 02

eA m

mMm

Instability criteria:

Only super-Alfvénic motions are unstable as magnetic field stabilizes the Kelvin-Helmholtz instability.

Harmonics with sufficiently high m are unstable.


Flux tubes within reconnection outflowsFlux tubes within reconnection outflows


The 3D evolution of a guide field reconnection is dominated by the formation and interactions oftwisted flux tubes (Daughton et al., 2011).

Moldwin et al. (1995,2000) suggested

reconnection flux tubes

Gosling (2013)

reconnection flux tubes

Kinetic simulations

Flux tubes within reconnection outflowsFlux tubes within reconnection outflows


Non-force-free flux tube embedded in high-beta reconnection outflow.

In the Phan et al. (2009) reconnection database (51 events)we found 2 embedded flux tubes.

CONCLUSIONSCONCLUSIONS


RECONNECTION

STRUCTURESTURBULENCE

Radial evolution? Scales?Occurrence frequency?Heating?

Documents

Flux tubes, magnetic reconnection and turbulence in the solar wind Z.Vörös 1 1.Space Research Institute, Austrian Acad. Sciences, Graz, Austria 2.INAF-IAPS,