nonlinear evolution of dispersive alfvén wave and...
TRANSCRIPT
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