The Role of Lower Hybrid Waves in Space Plasmas

Robert BinghamRutherford Appleton Laboratory,

Space Science & Technology Department

Conference on Sun-Earth Connections: Multiscale Coupling in Sun-Earth Processes, Feb. 2004.

Collaborators: V. D. Shapiro, D. Ucer, K. B. Quest, B. J. Kellett, R. Trines, .P. K. Shukla, J. T. Mendonça, L. Silva, L. Gargate.

Summary of TalkIntroduction

• Role of Waves in the Space• Generation of waves – nonlinear propagation• Acceleration of Particles• Radiation – AKR

WavesLower Hybrid +Kinetic Alfven Nonlinear StructuresIon Acoustic e.g. CAVITONS

ParticlesElectron, Ion: BeamsElectron, Ion: Beams

⇒⇒ Instabilities Instabilities

e.g. CYCLOTRON MASERe.g. CYCLOTRON MASER

Introduction• Interactions between the solar wind and comets and non-magnetised

planets lead to the production of energetic electrons and ions which are responsible for:-– X-ray Emission from Comets – Chandra and XMM spectra– X-rays from Planets

• Venus• Earth• [Mars]• Jupiter

– Io torus and several Jovian moons• [Saturn

– Titan]

• Similar processes may also be occurring elsewhere:-– Helix Nebula – Cometary Knots– High Velocity Clouds– Supernova Shock Front/Cloud Interaction

• Theory and Modelling– Energetic electron generation by lower-hybrid plasma wave turbulence – Comet X-ray spectral fits– Io torus model

⇐ Now detected in X-rays!

⇐ Future planned targets!

MOON VENUSCOMET C/LINEAR

JUPITER SUN

CRAB NEBULA (SNR) SKY (HVCs)

radio

Sources – Optical & X-ray Images

Role of WavesIntroduction:

• Large Amplitude Waves - collisionless energy and momentum transport.

- Formation of coherent nonlinear structures. .• Kinetic Alfven waves and ion cyclotron waves

Accelerate ions parallel to the magnetic fieldPonderomotive force also creates density striation structures

Kinetic Alfven wave Solitons – SKAWs

• Lower-HybridAccelerates electrons parallel to the magnetic fieldAccelerates ions perpendicular to the magnetic fieldFormation of cavitons

• Langmuir Waves – consequences of electron beams• Ion (electron) acoustics• Whistlers + LH present at reconnection sites

• Anisotropic electron beam distributions ⇒ AKR (Auroral Kilometric Radiation)

Lower-Hybrid Waves in Space Plasmas

• Auroral regionPrecipitating electrons, perpendicular ion heating (Chang & Coppi, 1981).Ion horseshoe distributions – electron acceleration.Cavitons (Shapiro et al., 1995).

• MagnetopauseL-H turbulence – cross field e- transport, diffusion close to Bohm.Anomalous resistivity – reconnection (Sotnikov et al., 1980; Bingham et al., 2004).

• Bow ShockElectron energization (Vaisling et al., 1982).

• MagnetotailAnomalous resistivity, LHDI (Huba et al., 1993; Yoon et al., 2002).Current disruption (Lui et al., 1999).

Lower-Hybrid Waves in Space Plasmas (cont.)

• Mars, Venus (Mantle region)Accelerated electrons, heated ions (Sagdeev et al., 1990; Dobe et al., 1999).

• CometsElectron heating (Bingham et al., 1997).

• Critical velocity of ionizationDischarge ionization (Alfven, 1954; Galeev et al., 1982).

• Artificial comet release experimentsAMPTE comet simulations (Bingham et al., 1991).

Lower-Hybrid Waves in Space Plasmas

• Hence electrons are magnetized, ions are unmagnetized.• Cerenkov resonance with electrons parallel to the magnetic field:-

• Cerenkov resonance with ions perpendicular to magnetic field:-

• Simultaneous resonance with fast electrons moving along B and slow ions moving across B.

• Transfers energy from ions to electrons (or vice versa)

, with small pp LH ce kω ω ω ⊥ 0k B

evkω

=

iv k kkω

⊥ ⊥⊥

=

1 12 22 2

2 2 2 22 2

12 2

21

pp pe pp pe

epe

ce

k kk k

ω ω ω ωω

ε ωω

⎛ ⎞ ⎛ ⎞+ +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠= =⎛ ⎞+⎜ ⎟⎜ ⎟

⎝ ⎠

204where ppp

n emπω =

Generation of Lower-Hybrid Waves at Shocks

As the plasma flows through the interacting medium, pick-up ions form an anisotropic distribution

y

x

z B

Shock Wave

These ions move in the crossed E and B fields - following cycloidalpaths:-

where is the shock wave velocity ⊥ to the magnetic field B.

( )( )0cos1v ttu cix −−= ⊥ ω ( )( )0sinv ttu ciy −= ⊥ ω

/u E B⊥ =

The beam is unstable ⇒ modified two stream instability (MTSI) (McBride et al. 1972).

Ion Pick-Up in Comets (also Mars & Venus)

• If B ⊥ vSW the heavy ions are carried away by the streaming plasma (pick-up)

• Energy

keV for N, v⊥ ~ 400 km/seci.e. a beam is formed

2v2

iim=⊥ε

20=

B

vb

The Modified Two Stream Instability

• A cross-field ion beam generates lower-hybrid waves

Lower-hybrid dispersion relation

vi

B

01 2

2

2

2

2

2

2

2

2 =−++ ⊥

ωω

ωω

ωω pipez

ce

pe

kk

kk

||||2

||222 , , , kkkkkkkkkk zyx >>=+=+= ⊥⊥⊥

||||v ,v

kkωω

==⇒⊥

⊥

||v v <<∴ ⊥

• Dispersion relation changes when an ion beam is present -

ui is ⊥ ion beam, kz = k⎪⎪(Fluid limit )

For

Then (1) becomes

Let ω' is complex

⇒ growth rate

( ) (1) 01 2

2

2

2

2

2

2

2

2

2

=⋅−

−−+i

pipez

ce

pex

ukkk

kk

ωω

ωω

ωω

ωωρ ≤⋅−<<< eiie kukkk v ,v ,1 ||

1 ,~ 2

221

≅⎟⎟⎠

⎞⎜⎜⎝

⎛kk

mm

kk x

i

ez

( ) 01 2

2

2

2

2

2

=⋅−

−−+i

pipe

ce

pe

ukωω

ωω

ωω

iuk ⋅−=′21ωω iyx +=′ω

( ) LH

cepe

pi ωωω

ωγ 2

1

2122

21

1≅

+=

LHωγ 21≅

Electron Acceleration by Lower-Hybrid Waves

• Parallel component resonates with electrons

• Perpendicular component resonates with ions

– Note

• Model LH waves transfer energy between ions andelectrons

k⊥k||

B (magnetic field)

||

vkeω

≅

⊥

≅kiωv

21

~||

⎟⎟⎠

⎞⎜⎜⎝

⎛⊥

e

i

mm

kk

ION BEAM

vi= vi⊥

Energetic Electrons

ve= ve⊥

Lower-Hybrid Waves

⇒ ⇒

Energy Balance at Shocks

• The free energy is the relative motion between the shock wave and the newly created interaction ions.

• Characteristic energy and density of suprathermal electrons obtained using conservation equations between energy flux of ions into wave turbulence and absorption of wave energy by electrons.

• In steady state energy flux of pick-up ions equals energy flux carried away by suprathermal electrons.Energy flux balance equation:-

• Balancing growth rate of lower-hybrid waves γi due to interaction pick-up ions with Landau damping rate γe on electrons.

(2) 21

3⎟⎟⎠

⎞⎜⎜⎝

⎛≅

e

eeTeSSWcici m

numn εεα

0=+ ei γγ ⇒ (3) 2umnn

ci

ci

e

Te ≅ε

Example

25

1

52

SSWcici

ee umm

m⎟⎠⎞⎜

⎝⎛≅ αε

(4) vvvv

v||||||

2

||2

2

||

e||

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

∂∂⋅

⋅∆∂∂

=∂∂ e

e

fk

E

mef

; v 2|||| Ek LHω=∆

(5) m 2

e 3

2

21

2

e

2

emax ⎥⎦

⎤⎢⎣

⎡≅ fEl

πε

From 2 and 3 the average energy of the suprathermal electrons is given by

• For nitrogen ions mciu2 ≈ 1 keVAverage energy εe ~ 100 eVDensity nTe ≈ 1 cm-3

⇒ Powerful source of suprathermal electrons⇒ 1019 erg/sec through X-ray region.

• Electron energy spectrum obtained by solving the quasi-linear velocity diffusion equation.

Note L-H wave field.

Max. electron energy

Quasi-Linear Model• The electron energy spectrum can be obtained by solving the standard quasilinear

diffusion equation. Since electrons are magnetised in the lower-hybrid oscillation only field aligned motion is allowed leading to the 1-D stationary case (e.g. Davidson, 1977):

Here, fe(z,v⎜⎜) is the one dimensional electron distribution function, |E⎜⎜k|2 is the one dimensional electric field spectral density, z is the magnetic field direction and v⎜⎜denotes the electron velocity parallel to B. Rewriting the electric field spectral density in the electrostatic limit

and then integrating over k⎜⎜ with the help of the δ function, leads to the Fokker-Planck equation:

( )2 2

22e e

k ke

f fev dk E k vz m v v

δ ω⎡ ⎤∂ ∂∂

= −⎢ ⎥∂ ∂ ∂⎢ ⎥⎣ ⎦

∫

22 2

2k k

kE E

k=

2 2 22

2

k

ke e

e k v

E k kf fevz m v vv k

ωω

=

⎡ ⎤∂ ∂∂ ⎢ ⎥=⎢ ⎥∂ ∂ ∂− ∂ ∂⎣ ⎦

• Using

• And assuming

• We can then write the resonant electron-wave interaction in the form

• And the denominator as

• Resonance between the electron parallel velocity and the group velocity of the waves then strongly enhances the particle diffusion.

• Using a gaussian wave spectral distribution with ε = mev2⎜⎜/2Tp has the field aligned

energy of the resonant electrons normalised to the proton temperature

k ce

kk

ω ω=

and 1e

p

k mk m

β>

ce kvω =

2

232

kv vk kω∂− ∂

( )22 2

2

1 expp ek

e LH

TE E

mπ ε εε

ω

⎡ ⎤−= −⎢ ⎥

∆ ∆⎢ ⎥⎣ ⎦

• With some typical scale length R0, we can normalise the acceleration distance as ξ = z / R0.

• The energy diffusion equation can then be written

• The diffusion coefficient G(ε) is of the form

• Assuming a Lorentz shape for the spectral distribution

• Leads to

( )f fG εξ ε ε∂ ∂ ∂

=∂ ∂ ∂

( ) ( )2220

2 20

1 exp4

cp p pe e

e ce pp p

ER mG

m n TT mω ω ε εεε

ωπ

⎡ ⎤−= −⎢ ⎥

∆ ∆⎢ ⎥⎣ ⎦

( )

2 32 2

22 2

12 pk

e ce LHe

TE E

m uε

ω ω ε ε

∆=

⎡ ⎤− + ∆⎣ ⎦

( )( )

22 2 30

22 2 20

12

p pe

cp e ce pe

EmR uGm n T

ω εεπ ω ω ε ε

∆=

⎡ ⎤− + ∆⎣ ⎦

Solutions of the Fokker-Planck Eq.

Particle-in-Cell Simulations of Cross Field Ion Flow

Ion energy increased by x30-100

Final electron distribution

v||

v⊥

v⊥/c

t=380

f (v||)

tv||

fe (v||)

Time evolution of the parallel component of the electron distribution

ωpe t

x

Test Particle Trajectories & Caviton Formation

B0

y

Strong Turbulence/Caviton Formation

Simulations

Lower-Hybrid Dissipative Cavitons• Lower-hybrid cavitons in the auroral zone.

– Observations reveal E ≥ 300mV/m – this is greater than the threshold (~50 mV/m) for Modulational Instability (Kintner et al., 1992; Wahlund et al., 1994).

• The modulational interaction is created by the ponderomotive force exerted by the waves on the plasma particles.

• The ponderomotive force is due to the Reynolds stress and creates density modulations.

• Density modulations lead to the formation of unipolar and dipolar cavitons.

• Density modulation δn/n0 produces two corrections to the lower-hybrid frequency

( ).B v v∇

δn

LH waveLH wave

Cavitons• The equation for the slow variation of the lower-hybrid electric potential

is

• The equation for the slow density variation is

• Driven by the ponderomotive force of the lower-hybrid wave.– 1st term on RHS is the scalar non-linearity– 2nd term on RHS is the vector non-linearity

• Similar process can occur for lower-hybrid waves coupled to kinetic Alfven waves. In this case the density perturbation of the Alfven wave is expressed through the vector potential.

• Observed lower-hybrid waves with fields of order 300mV/m can excite Alfven waves with field strength B ~ 1nT.

222 2 4

2 2

2 pei

lh e

mi Rt m z c

ωϕϕ ϕ ϕω

∂ ∂− ∇ − ∇ + + =

∂ ∂

( )2 22

22 *2 2

116 16

pe pe

i ce i ce

n T int m m m

ω ωδ δ ϕ ϕ ϕπ ω π ωω

∂− ∇ = − ∆ ∇ + ∇ ×∇ ⋅

∂e

( ) ( ) ( )2 22

0 02

1 2

1

lh i

pece e lh

ce

m nn din m n iω δϕ δ ϕ ϕ

ωω ωω

′ ′∇ ×∇ ⋅ − ∇ + Γ − ∇+

∫e r r r r

Lower Hybrid Drift Instability

• We assume the lower hybrid drift instability is operating in the kinetic regime (VDi << VTi ). Its maximum growth rate is:

• with a perpendicular wavenumber of

• Where

• and

22 8

D ilh

T i

VV

πγ ω⎛ ⎞

= ⎜ ⎟⎝ ⎠

1 2

2 2 lh e cem

T i i T i

mkV m Vω ⎛ ⎞ Ω

= = ⎜ ⎟⎝ ⎠

21 2

D i L i

T i

V rV x⊥

⎛ ⎞= ⎜ ⎟∆⎝ ⎠

1 ln ln Bxx xρ−

⊥⊥ ⊥

∂ ∂= ≈

∂ ∂

Lower-Hybrid Drift Instability at Current Sheets

Basic Theory of Magnetic Reconnection

• The Lower Hybrid Drift Instability (LHDI) is an excellent mechanism for producing LHWs and is relatively easy to excite by slow moving MHD flows found at reconnection sites

Dissipation RegionVout Vout

Vin

Vin

• Demanding the waves have time to grow places a constraint on thedamping rate, we require

• The characteristic scale length required to maintain the drift current at marginal stability is given by

10 gD

Vx

γ ⊥

⊥

≈∆

1 25 32

iLi

e

mx rm

π⊥∆ ≈

• If we now demand a balance between the Poynting flux of electromagnetic energy flowing into the tail and the rate of Ohmicdissipation, we find the resistivity required

• Now which implies

• but

• or

• where we have used an earlier result and λpe = c/ωpe is the plasma skin depth.

2

4 inV xc

πη ⊥∆=

24 / pevη π ω⊥=2 2

44 pein inpe

pe

V x Vv xc c c

ωππ ωω

⊥⊥ ⊥

∆= = ∆

2

8pe

Ev

nkTδ

ωπ⊥ =

21 2

8 5 32i Li in

e pe

E m r VnkT m c

δ ππ λ

=

• Using the same set of parameters as used by Huba, Gladd, and Papadopoulos (1977), that is, B ≈ 2.0 x 10-4 gauss, n ≈ 10, and kTi ≈10-3mic2 we find ∆x⊥ ≈ 0.67rLi, rLi ≈ 1.16 x 107 cm, VTi ≈ 2.2 x107 cm/s, λpe ≈ 1.68 x 106 cm, and VA ≈ 1.38 x 107 cm/s.

• Taking Vin ≈ 0.1VA we find

• We can compute the energies and fluxes of non-thermal electrons.

242.13 10

8EnkT

δ

π−≈ ×

Electron Temperature

• A rough estimate of the change in temperature of the electrons due to the Ohmic heating can be obtained by noting that as the plasma flows through and out of the region of instability electrons will experience Ohmic heating for a time ∆t ≈ L/VA , hence the change in electron temperature during this period is

• We find

2

BA

J Lk Tn V

η∆ ≈

25 3242

eB

i A Li

m u B Lk Tm V rn ππ

⋅∆ ≈

• Using the previous parameters we find

• In the magneto-tail rLi ≈ 1.16 x 107 cm, VA ≈ 1.38 x 107 cm/s and ∆x|| ≈VA∆t ≈ 1000 – 6000 km and using

• results in electron temperatures of order a few hundred or more eV

1( ) 2.9 10Li

LT evr

∆ ≈ ×

1( ) 2.9 10Li

LT evr

∆ ≈ ×

Ion Temperature• The LHDI leads to strong ion heating perpendicular to the

magnetic field. Equipartition of energy indicates that 2/3 of the available free energy ends up in ion heating and 1/3 ends up in heating the electrons parallel to the magnetic field. The simplecalculation below confirms this argument. To estimate the ion temperature we utilize the fact that the ratio of ion and electron drifts is given by

• together with Ampere's equation yields

Di i

De e

V TV T

=

1 2 Ai e

V xT Tu x

⊥⎛ ⎞∆

= +⎜ ⎟⎜ ⎟∆⎝ ⎠

The LHD instability can generate relatively large amplitude lower hybrid waves in current sheets. These LH waves are shown to be capable ofaccelerating electrons and heating ions in the current sheet during reconnection. The process describes the micro-physics of particle heating and generation during reconnection. The mechanism is important in the magnetosphere and other space environments where reconnection is invoked to explain particle heating and acceleration.

Wave kinetics: a novel approach to the interaction of waves with plasmas, leading to a new description of turbulence in plasmas and atmospheres [1-5].

A monochromatic wave is described as a compact distribution of quasi-particles.

Broadband turbulence is described as a gas of quasi-particles (photons, plasmons, driftons, etc.)

[1] R. Trines et al., in preparation (2004).[2] J. T. Mendonça, R. Bingham, P. K. Shukla, Phys. Rev. E 68, 0164406 (2003).[3] L.O. Silva et al., IEEE Trans. Plas. Sci. 28, 1202 (2000).[4] R. Bingham et al., Phys. Rev. Lett. 78, 247 (1997).[5] R. Bingham et al., Physics Letters A 220, 107 (1996).

Wave Kinetic Approach to Drift Mode Turbulence

Lower-hybrid drift modes at the magnetopause

Strong plasma turbulence – Langmuir wave collapse

Zonal flows in a “drifton gas” – anomalous transport driven by density/temperature gradients

Turbulence in planetary, stellar atmospheres – Rossby waves

Intense laser-plasma interactions, e.g. plasma accelerators

Plasma waves driven by neutrino bursts

The list goes on

Applications

We define the quasiparticle density as the waveenergy density divided by the quasiparticle energy:

The number of quasiparticles is conserved:

Then we can apply Liouville’s theorem:

where andare obtained from the q.p. dispersion relation

)(/),,(),,( ktrkWtrkN ω=

∫ = 0),,( kdrdtrkNdtd

0),,(),,( =⎟⎠⎞

⎜⎝⎛

∂∂

+∂∂

+∂∂

= trkNk

Fr

vt

trkNdtd

kdtxdv ∂∂== // ω xddtkdF // ω−∂==

Liouville Theory

Lower hybrid drift modes…

control plasma transport at the magnetopause,

provide anomalous resistivity necessary for

reconnection at tangential discontinuities,

control the transport determining the thickness of the

magnetopause boundary layer.

We aim to study them through the quasi-particle method.

Application: Drift Waves

We use the model for 2-D drift waves by Smolyakov, Diamond, and Shevchenko [6]

Fluid model for the plasma (el. static potential Φ(r)):

Particle model for the “driftons”:

Drifton number conservation;

Hamiltonian:

Equations of motion: from the Hamiltonian

( ) kdNkk

kkt k

r

r 22221∫ ++

=∂Φ∂

ϑ

ϑ

( ) ,1 22*

ϑ

ϑϑω

kkVk

rk

ri ++

+∂Φ∂

=rn

nV

∂∂

−= 0

0*

1

[6] A.I. Smolyakov, P.H. Diamond, and V.I. Shevchenko, Phys. Plasmas 7, 1349 (2000).

Condensed Theory of Drift Waves

Two spatial dimensions, cylindrical geometry,

Homogeneous, broadband drifton distribution,

A 2-D Gaussian plasma density distribution around the origin.

We obtained the following results:

Modulational instability of drift modes,

Excitation of a zonal flow,

Solitary wave structures drifting outwards.

Simulations

Radial e.s. field and plasma density fluctuations versus radius r :

Excitation of a zonal flow for small r, i.e. small background density gradients,

Propagation of “zonal” solitons towards larger r, i.e. regions with higher gradients.

Simulation Results – 1

-0.005

0

0.005

0.01

20 40 60 80

t=200t=300

-0.005

0

0.005

0.01

20 40 60 80

t=400t=500

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

20 40 60 80

t=400t=500

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

20 40 60 80

t=200t=300

r

E

r

E

δn

δn

-0.2

0

0.2

0.4

0 20 40 60 80 100 120

t=300

Radial and azimuthal wave numbers versus r:

Bunching and drift of driftons under influence of zonal flow,

Some drifton turbulence visible at later times.

r

kr

r

kθ

-4

-2

0

2

4

0 20 40 60 80

t=500

-4

-2

0

2

4

0 20 40 60 80

t=300

-0.2

0

0.2

0.4

0 20 40 60 80 100 120

t=500

kr

kθ

Simulation Results – 2

Wave kineticsWave kinetics……

Allows one to study the interaction of a vast spectrum of ‘fast’ waves with underdense plasmas,

Provides an easy description of broadband, incoherent waves and beams, e.g. L-H drift, magnetosonic modes, solitons

Helps us to understand drift wave turbulence, e.g. in the magnetosphere,

Future work:Future work: progress towards 2-D/3-D simulations and even more applications of the wave kinetic scheme

Wave kinetics: a valuable new perspective on wave-plasma interactions!

Lower-Hybrid Waves in Space Plasmas – Wave Kinetics

Conclusions • Lower-Hybrid waves play a unique role in all areas of space

plasmas.• Ideal for energizing ions/electrons and transferring free

energy from ions (electrons) to electrons (ions).• Non-Linear physics very rich

– e.g. cavitons– e.g. strong turbulence theories