By Prof. R. P. Sharma

Centre for Energy Studies

Nonlinear evolution of dispersive Alfvén wave and turbulent spectra

Alfvén waves

• Pure em wave• Mixed mode• Magnetosonic• Hall MHD

Alfvén waves

Low frequency waves

Electromagnetic waves

Propagating along

the magnetic field

Japan's Hinode solar telescope observes Alfvén waves in the Sun's

corona. Credit: Hinode/JAXA/NASA

Dispersion relation

where

Alfvén speed

Alfvén waves modes

Inertial Alfvén waves Kinetic Alfvén waves

Where β is the ratio of plasma pressure to the background magnetic pressure

Alfvén waves applications

Coronal heating

Solar wind turbulence

Plasma heating in Earth’s magnetosphere

Plasma heating in fusion devices

Solar Corona

Tenuous coronaParameters(at 0.01AU)

Coronal Holes

Earth’s Magnetosphere

Cusp RegionParameters(5-6 Earth’s radii)

Solar wind

Heliocentric distances(0.3AU≤ r ≤1AU)parameters

Solar wind/magnetosphere

Alfven wavesAlfvenic

turbulence

Particle acceleration

Atmospheric O and N

aurora

Fusion Research (Tokamak)

Fusion Research (Tokamak)

Representative values of the e- density ne, temperature, magnetic field B, Alfven speed Va sound speed Cs, and plasma β, in different regimes

10-41061081031041012Solar atmosphere

10-210510610-51021Interstellar gas

110510510-5102103Gaeous nebula

10-21071081106107Solar corona

<10-4105109-10710-0.5103102-106Ionosphere

10-4-102107109-1061041061012-1018Laboratory plasma

βcs(cm/s)va(cm/s)B(G)T(K)ne(/cc)

Forc

ing R

ange

Inertial Range

Dissipation RangeLog[

E(k

)]

k

How the energy is distributed over the multiplicity of scales (Energy spectrum)

Kolomogorov scaling

A typical turbulent spectrum

Observational spectra

(a) (b)

(a) Horbury et al., Plasma Phys. Control. Fusion 47, B703 (2005).

(b) Nykyri et al., Ann. Geophys. 24, 1057 (2006)

Solar wind (1 AU) Polar cusp (5-Earth radii)

Observational spectra (Solar wind 1 AU)

Sahraoui et al., Phys. Rev. Lett. 102, 231102 (2009)

Cluster spacecraft

Observational spectra (Earth’s Auroral region)

Chaston et al., Phys. Rev. Lett. 100, 175003 (2008)

Observational spectrum (Earth’s magnetotail region 3-4 )

Eastwood et al., Phys. Rev. Lett. 102, 035001 (2009)

Observational spectrum (Solar wind, 19 )

Bale et al., Phys. Rev. Lett. 94, 215002 (2005)

Solar Wind Turbulence• k-5/3 (incompressible, nonmagnetised,

Kolomogorov scaling)• k-3/2 (magnetic fluid, isotropic, IK theory)• k⊥

-5/3(incompressible MHD,Goldrich and Sridhar model k

∝ k⊥2/3)

• ?? (compressible MHD modes)

• MHD Model• Kinetic Model/fluid Model

MHD

• Incompressible + Infinite conductivity• Incompressible + generalized Ohm's

law• Compressible + Infinite conductivity

+ generalized Ohm’s law• Hall MHD

• Non-linear effects associated with Alfvén wave

• Filamentation (hot spot formation)• Effect on turbulence - spectrum • Heating Fokker Planck equation Diffusion coefficient (velocity space)

Introduction

Model Equations

The perpendicular component of the electron and ion fluid velocities are given by

Kinetic Alfvén wave

On the other hand parallel component of the electron fluid velocities is given by

the y-component of the KAWs magnetic field and the KAW electric fields are related by Faraday’s law

On the other hand, by inserting drift velocities into the conservation of the current density equation

and eliminating the parallel component of the plasma current density from the Ampere’s law, we have

where

Now using the parallel component of Ampere’s law, the parallel electron drift velocity and equation of continuity, the time derivative of parallel electric field is given by

Using the above Eqns. of parallel and perpendicular electric field in the time derivative of Ampere’s law, we get the dynamical equation for nonlinear KAWs propagating in plane in intermediate-beta plasmas, is governed by

If the density pertubation then the above dynamical equation satisfies the well known Dispersion relation of KAWs

Ion acoustic wave

Ponderomotive force

and

Taking the time derivative of continuity equation and substituting the values of perpendicular and Parallel ion velocities

Model Equations

Kinetic Alfvén wave Ion acoustic wave

If R. H. S. is zero

Coupling

Modified Zakharov SystemOf Equations (non-paraxialregime)

Solar wind parameters (0.3AU≤ r ≤1AU)

Normalizations

Numerical Simulation

Initial condition

Filamentation

Contd…..

Contd…..

Particle heating

Density dipoles

Turbulent spectra

Evolution of power spectra of fluctuations of the magnetic field at β=0.5 and t= 30

Turbulent spectra

Evolution of power spectra of fluctuations of the magnetic field at β=0.5 and t= 40

Observational spectra

(a)

(a) Horbury et al., Plasma Phys. Control. Fusion 47, B703 (2005). (b) Sahraoui et al., Phys. Rev. Lett. 102, 231102 (2009)

(a) (b)

Conclusions

Chaoticity of the filaments increases as the time increases

Magnetic field intensity increases as plasma beta increases

Electron heating rate is increased Density fluctuations (dips and humps) Turbulent spectra (having multiple scaling laws) and

supportive to the observational claims Plasma heating in the solar wind

Model Equation (Adiabatic Case)

Low β (upper sign)

intermediate β (lower sign)

Numerical Simulation for steady

Filamentation

Turbulent Spectra

Landau damped Kinetic Alfvén Wave

On considering the plane wave solution

where

[1] A. Hasegawa and L. Chen, Phys. Fluids 19, 1924 (1976).

We get modified nonlinear Schrödinger equation

Solar corona parameters (at 0.01AU)

Normalizations

Numerical Simulation

Filamentation (IC-A)

Turbulent spectra

Observational spectrum

Sahraoui et al., Phys. Rev. Lett. 102, 231102 (2009)

Cluster spacecraft

Filamentation and turbulent spectra (IC-B)

Conclusions

Damped filaments at different times The spectral index is deviated from Kolmogorov law Spectral index found in dissipation range Random perturbation can cause the multiple filaments Landau damping plays significant role in plasma heating in the solar

corona

Overall Conclusion

These turbulent structures (filaments, spectra) of kinetic Alfvénwaves can be responsible for plasma heating in many astrophysical

plasmas.

Scope for future

Coupling of kinetic Alfvén wave (KAW) and ion acoustic wave with Landau damping may be important in plasma heating in space plasmas.

KAW turbulence in edge region TOKAMAK plasmas may be important

THANK YOU...

Turbulence

Turbulence or turbulent flow is a fluid regime characterized by chaotic, stochastic property changes.

Kolmogorov introduces the hypothesis:

For very high Re, the small scale turbulence are universally determined by the viscosity and energy dissipation

Dissipation rateof the total energy

Kinetic energy per gramper unit wave vector

Kolmogorov scaling law