Plasma waves and instabilities : from drift waves to ... waves and instabilities : from drift waves to kinetic MHD ... Plasma response: kinetic or ... • Quasi-linear theory works

Plasma waves and instabilities : from drift waves to kinetic MHD

modes

Xavier GarbetCEA/IRFMCadarache

X. Garbet, Les Houches March, 27 2015 | PAGE 1

Outline

• Introduction to waves and instabilities – reactiveand kinetic instabilities

• MHD instabilities – kinetic MHD

• Drift waves

• Non linear saturation processes (sketchy)

| PAGE 2X. Garbet, Les Houches March, 27 2015

• Small sinusoidal perturbation of the electromagnetic field

• Linear response of current of charge and current densities

• Self-consistent problem

Plasma waves: basics


Plasma response: kinetic or fluid equations

Maxwell equations

e.m. fieldE,B

charge andcurrent

densities ρ, jφ

t

Leads to a dispersion relation

• Maxwell + plasma response leads to a dispersion relation

• Solution usually complex

ω(k)= ωr(k) + iγ(k)

• Real solution ωr(k) → wave

• Complex solution γ(k) >0 → instability


φ

t

φ

t

• Convenient to calculate the Lagrangian of the electromagnetic field

• Electric field

• Magnetic field

A bit more on the dispersion relation

| PAGE 5

Current density Charge density

Electromagnetic field Wave-particle interaction

X. Garbet, Les Houches March, 27 2015

A bit more on the dispersion relation

• Maxwell equations

• Small perturbations: ρkω and Jkω are linear functions of φkω and Akω .

• Dispersion relation


• Two types of instabilities

→ “reactive”

→ “kinetic”

• Marginal stability

Energetics of an instability

• Energy exchanged between e.m. field and particles (real ωωωω)


Reactive mode R=0Damping R>0“Kinetic” instability R<0

Kinetic instability

• Situation close marginal stability γ(k)<<ωr(k)

• Taylor development of ε(k,ω)=0

- Lowest order → pulsation ωr(k)

- Next order → growth rate γ(k)

• Energy density

• Instability γ(k)>0 if Pk<0 : energy transferred from particles to wave


Reactive instability

• Since P(k,ω)=0, if ωk is solution, ωk* is solution too

• At threshold

and

→ energy Wk=0

• Reactive instability sometimes called “negative energy” wave


<Stable

Unstable

Neutral

Analogy with particle motion in a potential

• Vlasov equation, 1D and electrostatic

• Response function

A simple illustration of the reactive vs kinetic character of an instability


Plasma frequency

Unperturbed dist. function

Hydrodynamic limit: Buneman instability


• Hydrodynamic limit ω>>kv for ions and electron beam f0=neδ(v-v0)


• Negative energy wave is

unstable if

Baumjohann & Treumann 2012

Kinetic limit : bump on tail

| PAGE 12

• Hydrodynamic limit for all thermal species ω>>kv+ hot beam


• Dielectric

• Kinetic instability

F(v)

v

v=vph=ω/k

Single fluid ideal MHD

| PAGE 13

• MHD equations

• Lagrangian derivative

• MHD displacement

Velocity

Pressure

Adiabatic index x

y

z

B0

kk//

k⊥


Magnetic field

Alfvén waves in incompressible medium ρρρρ=cte

| PAGE 14

• 3 solutions: shear Alfvénwave, fast and slow magneto-acoustic waves

• Shear Alfvén wave

• Alfvén velocity

ξ


MHD instabilities

| PAGE 15

• In single fluid MHD, two main destabilizing terms:

1) Pressure gradient → interchange instability

2) Current density gradient → kink mode


MHD instabilities : interchange

| PAGE 16

B

Field linelength dl

Flux tube 1P1, V1

Flux tube 2P2, V2

Flux tube 2P’2, V1

Flux tube 1P’1, V2

InterchangeB dσ


PVΓ=cte

MHD instabilities : interchange (cont.)

| PAGE 17

• Exchange of two flux tubes

• Released energy

• Interchange instability ∼∇P aligned with ∇B


Energy principle


• Force balance equation

• MHD energy δW combines wave character and instability sources

• MHD instabilities are reactive

real for real ω

The MHD description usually fails at low frequency

• Two reasons for breakdown of the MHD description:

1) Landau resonances ω = k....v

2) Effects of finite orbit width δ

• Ideal MHD approach valid when

ω >> k....v and k⊥δ<<1

• Difficulties occur at lowfrequencies


Particle

gyro-center

Fieldline B

Low frequency limit

• Momentum equation for each species

• Strong guide field B


Diamagnetic velocityEXB velocity

Parallel flow

Polarization drift

Field line

Potential δφ+++ +++

- - -

Charge -δnee

E E

ne≈n0 exp(eφ/Te)

Drift wave

• Simple slab geometry, electrostatic, fluid ions

• fast motion along field lines → adiabatic electrons

• electro-neutrality (kλD<<1)


Drift wave (cont.)

• Density gradient in the x direction, uniform B, y,z periodic

• Phase velocity = electron diamagnetic velocity


x

neq

x

y

z

BV*ne

Drift wave (cont.) – schematic picture

• Start with a density ni=ne corrugation

• Fast electron response along field lines → potential adjusts → electric field E

• ExB drifts shifts the perturbation along V*e


Grulke&Klinger 02

V*e

neq

x

y


The ion inertia plays a crucial for non linear saturation

• Drawbacks of the previous model: no instability, infinity of non linear solutions

• Add the polarization drift (ion inertia)

Lagrangianderivative

Divergence of polarization current

• Same assumptions+ polarization drift

• Charney-Hasegawa-Mima (CHM) equation



A paradigm : the CHM equation

Ion gyroradius

B

x

y

z

φ∇×=2B

BvE

vE

Vortex

Example of a tokamak:electrostatic modes


15

10

5

06420

-Rd rL

og(T

)

-RdrLog(ne)

Ion Mode (ITG) Electron

Mode (TEM)

Ion + Electron Mode (ITG+TEM)

Stable

• Dominant instabilities at low frequency : drift waves mainly driven by interchange

• Kinetic type: driven via resonances by electrons or ions.

•Threshold in temperature and density gradients

k⊥ρs<1

Growth rates


• Several branches may co-exist.

• Electron branch at k⊥ρs<1

Jenko 13

Example of a tokamak: electromagnetic modes


• Ballooning mode: shear Alfvén wave coupled to interchange instability

• Low frequency limit: diamagnetic drifts matter

Huysmans 2009

Connection with MHD

• Drift waves dominate at low beta

• At high beta, kinetic ballooning modesbecome unstable


Pueschel 2010

ITG

Kinetic ballooning mode

Two options:

1) Solve the full kinetic problem (high k)

2) Hybrid formulation (low k)

Hybrid formulation for kinetic MHD modes

Non thermal particle stress tensor


→ imaginary part of δW

→ kinetic instabilities

Alfvén waves driven by fast particles

• Driving mechanism similar to bump on tail

• Quasi-coherent modes

• In some cases, frequency chirping


Turnbull 1993

Non linear saturation

| PAGE 32

• Variety of non linear dynamics:

1) Few modes : steady saturated state, relaxation oscillations, explosive behavior, …

2) Many coupled modes : usually evolve towardsturbulence. Turbulent state is different if waves are involved.

• Bump on tail instability is the testbed here


One mode only: particle trapping

| PAGE 33

• Particle energy in the wave frame of reference

• Similar to a pendulum – trapping time

v

x


Plateau effect

| PAGE 34

• F(E) is solution of the Vlasov equation

• Flattening of the distribution in the unstable region → stabilisation Berk & Breizman 97



Analogy with vortex mixing :mixing-length estimate

Mixing of the pressureprofile by vortex of size ℓ→ “mixing length

estimate” of the fluctuation level

pLpp l≈δ

r

p Vortex of size ℓ

Lp

Plateau can generate a secondary instability


• Edges of plateau can be unstable Lilley 15

• Plateau splits in holes and clumps

• Motion of holes/clumps in phase space → chirping Berk&Breizman 97

Lilley 15

Lilley 15

Frequency chirping observed both in experiments and simulations

| PAGE 37

Lesur 12

Time (ms)

Fre

quen

cy (

kZ)


Several pathways towards relaxation


Electric field damping

Dis

trib

utio

n fu

nctio

nre

laxa

tion

rate

Vann 03, Lesur 09

• Bump on tail: variety of dynamics depending on dissipation and drive

• Limit of strong drive still to be explored Zonca 15

Stochasticity


• Multiple modes: islands localized around v=ωp/k

• Trajectory becomes stochastic → ergodization → flattening Chirikov 59, see Lichtenberg &Liberman,1983

Quasi-Linear Theory

| PAGE 40

• Linear solution Vlasov

• Evolution equation of f0 in velocity space


Average distribution function Flux in phase space

Quasi-Linear Theory (cont.)

| PAGE 41

Besse 2011


• Linear solution of Vlasovequation → flux

• Quasi-linear diffusion coefficient

• Often effective beyond validity conditions cf lecture Gürcan

Mode coupling is needed to compute the turbulence intensity


• Generic form of a non linear equation in Fourier space

• Triad

• Coupling leads to energy transfer between waves – if coupling is ‘’local’’ → cascade

k

k’

k’’

Case of drift waves

| PAGE 43

• Coupling term for CHM equation

• Conserves energy and enstrophy


• Decay of a « pump » mode (k0,ω0) into two« daughter » waves (k1,ω1), (k2,ω2)

• Constraint k0= k1+k2 ; ω0= ω1+ ω2

• Drift waves: decay possible if

3 modes only : decay instability

| PAGE 44

MergingDecay



Decay instability (cont.)

| PAGE 45

• If energy is strictly conserved : pump recovers

→ dissipative processes make it irreversible

Pump

Daughter waves



Example of a parametric decay of an acoustic wave into two drift waves

| PAGE 46

• Geodesic acoustic mode driven by energetic ions

• Parametric decay into drift (ITG) waves


drift waves

Geodesic acoustic mode

Zarzoso 13

• If many modes are unstable : the system evolves towards a turbulent state Waltz 83,Horton 86

• Can be seen as a strange attractor in the phase space {φk}


Many unstable modes: turbulence

Horton 86

Frequency spectra

| PAGE 48

• Frequency spectra are broad

• Predictive model of the frequency spectrum shape still an open question

Horton 86


A few remarks on spectra and transport

| PAGE 49

• A dual cascade is expected CHM

• Does not fit observation in tokamaks (see lecture

Vermare) – many reasons:- no inertial range- coupling to large scale

flows- kinetic effects kinj


• Quasi-linear theory works well for drift waves

A few words on wave turbulence

| PAGE 50

• Waves change the nature of turbulence see lecture Nazarenko

• Drift or Rossby waves:

- not isotropic

- generation of large scale shear flows → self-regulation

Dif-Pradalier 15


Conclusions


• Linear stability is well documented – not that simple though …

• Non linear dynamics much more complex – No simple recipe !

• For turbulent states and wave/particle interaction via Landau resonances: quasi-linear theory often works well

• Predicting a level of fluctuations is trickier

• For quasi-coherent modes: variety of non linear dynamics – parameter dependent.

Some useful textbooks

• D. B. Melrose “Instabilities in Space and Laboratory Plasmas” Cambridge UP 1986

• W. Baumjohann and R.A. Treumann “Basic Space Plasma Physics” Imperial UP 2012 vol I and II

• P. H. Diamond , S.-I. Itoh, K. Itoh, “Physical Kinetics of Turbulent Plasmas” Cambridge UP 2010

• D. Biskamp “Nonlinear Magnetohydrodynamics », 1997 Cambridge UP

• A.J. Lichtenberg and M.A. Liberman, “Regular and stochastic motion” Springer 1983

• Y. Elskens and D. Escande “Microscopic dynamics of plasmas and chaos”, IOP 2003


Documents

Plasma waves and instabilities : from drift waves to ... waves and instabilities : from drift waves to kinetic MHD ... Plasma response: kinetic or ... • Quasi-linear theory works