Stochastic heating in non-equilibrium plasmas J. Vranjes Von - - PowerPoint PPT Presentation

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave

Stochastic heating in non-equilibrium plasmas

• J. Vranjes

Von Karman Institute, Brussels, Belgium

Workshop on Partially Ionized Plasmas in Astrophysics, Tenerife June 19-22, 2012

Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave

Outline

1 Examples and features of stochastic heating 2 Ion acoustic wave

IA wave instability in inhomogeneous collisional plasmas IA wave instability in permeating plasmas

3 Oblique drift wave

Summary of properties of heating & consequences

4 Transverse drift wave

Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave

Outline

1 Examples and features of stochastic heating 2 Ion acoustic wave 3 Oblique drift wave 4 Transverse drift wave

Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave

Space: observed properties (solar wind; solar atmosphere)

Heating in general (chromosphere, corona, solar wind). Temperature anisotropy [T⊥ > T, T⊥ < T]. Dominant heating in the high-energy tail. Dominant bulk plasma heating. Dominant heating of ions; heavy ions better heated than light ones. Dominant heating of electrons. Transport (perpendicular to the magnetic field vector). Acceleration of particles · · ·

Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave

Some waves of interest (multi-component theory)

Plasma (Langmuir) wave. Ion acoustic wave (IA, ω > Ωi, ω < Ωi). Ion cyclotron wave (IC). Electron cyclotron. Ion-Bernstein wave (IB). Lower-hybrid wave (LH). Upper-hybrid (UH). Oblique and transverse drift (OD, TD) wave. Standing wave (various waves).

Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave

Some features of stochastic heating

Mostly electrostatic phenomena. Necessity for a large enough electric field; yet linear theory valid.

For (high frequency, ω > Ωi) IA wave: ek2

z φ|Jl(k⊥ρi)|/mi ≥ Ω2 i

16 , l = 0, ±1, ±2 · · ·

• Heating in the high-energy tail of the ion distribution.

For IC and LH wave: E B0 > 1 4 Ωi ω 1/3 ω k⊥ .

• Heating of the bulk and tail plasma.

Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave

For OD wave: k2

y ρ2 i

eφ κTi ≥ 1.

• Heating of bulk plasma.
• Better heating of heavier ions.
• Dominant perpendicular heating.

For TD wave (essentially electromagnetic mode!): k2

⊥E 2 z1

ω2B2 > 1.

• Acceleration ⇒ heating and transport.
• Acting on both ions and electrons.

Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave

Heating at ion-cyclotron harmonics.

• A. Fasoli et al., Phys.
• Rev. Lett. 70, 303

(1993).

Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave

Stochastic tail heating by IA wave. Rapid process, rate comparable to gyro-frequency Ω.

• G. R. Smith and A. N.

Kaufman, Phys. Fluids 21, 2230 (1978).

Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave

IA wave: overlapping of resonances ⇒ diffusion (loss of memory of initial conditions) . ω − kzvTi = ±lΩi. ⇒ vz = (ω ± lΩi)/kz. High frequency IA mode ω > Ωi. vz finite for kz = 0. Obliquely propagating with respect to B0! not large amplitude n1/n0 ∼ 0.1.

• G. R. Smith and A. N. Kaufman,
• Phys. Rev. Lett. 70, 303

(1993).

Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave

Drift-Alfv´ en wave heating in tokamak

Ion temperature at three different positions during the drift-Alfv´ en wave activity. Sanders et al., Phys. Plasmas 5, 716 (1998). Measured ion temperatures T⊥ and T. Wave period 230 µs. Sanders et al., Phys. Plasmas 5, 716 (1998).

Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave

Tokamak cross-section: drift-Alfv´ en wave structure (m = 2) during the heating. McChesney et al., Phys. Fluids B3, 3363 (1991). Heating during coherent (non-turbulent) wave regime.

Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave IA wave instability in inhomogeneous collisional plasmas IA wave instability in permeating plasmas

Outline

1 Examples and features of stochastic heating 2 Ion acoustic wave

IA wave instability in inhomogeneous collisional plasmas IA wave instability in permeating plasmas

3 Oblique drift wave 4 Transverse drift wave

Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave IA wave instability in inhomogeneous collisional plasmas IA wave instability in permeating plasmas

Collisional plasma; fluid description

Geometry:

• B0 = B0

ez, ∇n0 = −n′ ex. Perturbations: f (x)exp[i(kyy + kzz − ωt)], |(df /dx)/f |, |(dnj0/dx)/n0| ≪ ky The electron equations: mene ∂ ve ∂t + ( ve · ∇) ve

• = ene∇φ − ene

ve × B − κTe∇ne −meneνen( ve − vn), (1) ∂ne1 ∂t + ∇⊥(ne v⊥e) + ∇z(ne0 vez1) = 0. (2) Neutrals: ∂ vn ∂t + ( vn · ∇)

• vn = −νne(

vn − ve). (3)

Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave IA wave instability in inhomogeneous collisional plasmas IA wave instability in permeating plasmas

The momentum conservation implies that νne = meneνen/(mnnn). Ions: mini ∂ vi ∂t + ( vi · ∇) vi

• = −eni∇φ + eni

vi × B. (4) Ωe ≫ |ω| ≫ Ωi, (5) from H. J. de Blank, Plasma Physics Lecture Notes free energy in the system

Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave IA wave instability in inhomogeneous collisional plasmas IA wave instability in permeating plasmas

Dispersion equation, the oblique, density gradient driven IA mode [Vranjes and Poedts, Phys. Plasmas 16, 022101 (2009)]: k2c2

s

ω2 = ω∗e + iDp + iDz(ω2 + ν2

ne)/(ω2 − iνneω)

ω + iDp + iDz(ω2 + ν2

ne)/(ω2 − iνneω) .

(6) Dp = νenαk2

y ρ2 e,

Dz = k2

z v2

Te/νen,

ρe = vTe/Ωe. ω∗e = v∗eky,

• v∗e = −κTe

eB0

• ez × ∇n0

n0 , α = ω/(ω + iνne). α = 1 ⇐ static neutrals Dp - usually omitted without justification. ω∗e - diamagnetic frequency; introduces free energy.

Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave IA wave instability in inhomogeneous collisional plasmas IA wave instability in permeating plasmas

Vranjes and Poedts, Phys. Plasmas 16, 022101 (2009). Frequency (normalized to ω∗e), and the corresponding normalized growth-rate. Parameters: mi = 40mp, mn = mi , Te = 5 eV, nn0 = 1021 m−3, ne0 = ni0 = 5 · 1016 m−3, B0 = 0.01 T, Ln = 0.1 m, ky = 7 · 102 1/m. For these parameters σen = 8.7 · 10−20 m2. The perpendicular electron collisions drastically destabilize the mode. For small kz , the growth rate about 70 times larger. Note that for kz = 0.3 1/m we have Dp/Dz = 141, while for kz = 14 1/m this ratio is

• nly 0.06.

Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave IA wave instability in inhomogeneous collisional plasmas IA wave instability in permeating plasmas

θ [rad]

γ

• Phys. Plasmas 17, 022104 (2010).

Angle dependence - angle of preference. Frequency (full lines) and the corresponding growth rates (dashed lines), both normalized to the electron diamagnetic drift frequency, for three values of neutral number density. The lines I, II, III correspond (respectively) to nn0 = 1019, 1018, 1017 m−3. θ - arctan(kz /ky ). The line γk is the kinetic growth-rate (for the same parameters as line II). Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave IA wave instability in inhomogeneous collisional plasmas IA wave instability in permeating plasmas

Kinetic instability

Same geometry; magnetized (un-magnetized) electrons (ions); but no collisions. The perturbed number density for electrons (the same as for the drift wave): ne1 n0 = eφ1 κTe

• 1 + i

π 2 1/2 ω − ω∗e kzvTe exp

• −ω2/(2k2

z v2

Te)

• .

(7) The ion number density: ni1 ni0 = − eφ1 miv2

Ti

• 1 − J+

ωi kvTi

• .

(8) Here, J(η) = [η/(2π)1/2]

• c dζexp(−ζ2/2)/(η − ζ) is the plasma

dispersion function, and ζ = v/vTi.

Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave IA wave instability in inhomogeneous collisional plasmas IA wave instability in permeating plasmas

Frequency and the growth rate from the quasi-neutrality: ω2

k = k2c2 s

2

• 1 + (1 + 12Ti/Te)1/2

. (9) γk ≃ −(π/2)1/2ω3

k

2k2c2

s

× × ωk − ω∗e kzvTe exp

• −ω2

k/(2k2 z v2

Te)

• + Te

Ti ωk kvTi exp

• −ω2

k/(2k2v2

Ti)

• .

(10) The electron contribution in Eq. (10) yields a kinetic instability provided that ωk < ω∗e.

Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave IA wave instability in inhomogeneous collisional plasmas IA wave instability in permeating plasmas

Plasmas with macroscopic motion

Reminder: electron current driven instability of the IA wave. Electron flow along the magnetic field, or in arbitrary z-direction without the field. Kinetic instability: γ = π 8 1/2 kcs

• me

mi 1/2ve0 cs − 1

• − τ 3/2exp
• −τ

2

• .

(11) ve0 > cs[1 + (mi/me)1/2τ 3/2exp(−τ/2)], τ = Te/Ti. (12) Threshold high; for τ = 1 it is 27cs. Two-fluid counterpart: electron-collision effect, even higher threshold! [Vranjes et al., Phys. Plasmas 13, 122103 (2006)]. γ = − νi 2(1 − χ)

• 1 − νe

νi χ

• ,

χ = me mi k2 k2

z

kzv0 ωr − 1

• .

(13)

Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave IA wave instability in inhomogeneous collisional plasmas IA wave instability in permeating plasmas

Vranjes et al., Phys. Plasmas 13, 122103 (2006). Collisional instability threshold for electron-proton plasma in a H-gas. Angle dependent (with respect to the magnetic field) due to ion collisions. Un-magnetized ions, but this time due to collisions: νi > Ωi.

Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave IA wave instability in inhomogeneous collisional plasmas IA wave instability in permeating plasmas

Permeating plasmas: IA wave instability

Two plasmas separately quasi-neutral nfi0 = nfe0 = nf 0, nsi0 = nse0 = ns0. (14) Instability threshold; below the sound speed of the static plasma. Current-less instability. The ion Doppler shift appears to play the main role. Examples:

colliding astrophysical clouds, stellar/solar winds propagation through interplanetary/interstellar space, in the solar atmosphere.

Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave IA wave instability in inhomogeneous collisional plasmas IA wave instability in permeating plasmas

Solar wind

Encyclopedia of the solar system,

• eds. L. A. McFadden et al..

1-hour average solar wind at 1 AU.

Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave IA wave instability in inhomogeneous collisional plasmas IA wave instability in permeating plasmas

The IA wave dispersion equation [Vranjes et al., Phys. Plasmas 16, 074501 (2009)]

∆(k, ω) ≡ 1 + 1 k2λ2

d

− ω2

psi

ω2 − 3k2v2

Tsi ω2 psi

ω4 +i π 2 1/2        ωω2

pse

k3v3

Tse

• c

+(ω − kvf 0)        ω2

pfe

k3v3

Tfe

• a

+ ω2

pfi

k3v3

Tfi b

       + ωω2

psi

k3v3

Tsi

exp

• −

ω2 2k2v2

Tsi

• 

      = 0. (15) 1/λ2

d = 1/λ2 dse + 1/λ2 dfe + 1/λ2 dfi ,

λdse = vTse/ωpse, etc.

The inertia of the mode provided by the static ions. a/b ≪ 1 ⇐ electron flow contribution negligible.

Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave IA wave instability in inhomogeneous collisional plasmas IA wave instability in permeating plasmas

Laboratory plasma

General case; numerical solution

Vranjes et al., Phys. Plasmas 16, 074501 (2009). The critical (threshold) values of the flowing plasma velocity for the ion-acoustic wave instability in terms of the number density of the static (target) plasma; c2

se = κTse/mi.

Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave IA wave instability in inhomogeneous collisional plasmas IA wave instability in permeating plasmas

Solar plasma

λ [m]

• 3
• 2
• 1

• Upper. Full line: IA wave
• frequency. Dashed line: the

corresponding growth rate.

• Lower. Spectrum dependence
• n the speed of the flowing

plasma.

Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave Summary of properties of heating & consequences

Outline

1 Examples and features of stochastic heating 2 Ion acoustic wave 3 Oblique drift wave

Summary of properties of heating & consequences

4 Transverse drift wave

Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave Summary of properties of heating & consequences

In cylindric geometry

A drift wave in cylindric geometry with poloidal wave number m = 2. From Vranjes and Poedts, MNRAS 398, 918 (2009).

Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave Summary of properties of heating & consequences

Cross section in cylindric geometry

The cross-section of an experimentally observed drift mode with the poloidal number m = 5 in a cylindric VINETA device (left) and the corresponding analytical solutions (right). From Grulke O., et al., Plasma

• Phys. Control. Fusion

49, B247 (2007).

Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave Summary of properties of heating & consequences

Toroidal geometry

Typical geometry of the drift wave in a torus; from Horton, Rev. Mod.

• Phys. 71, 735 (1999).

Experimentally investigated more than any other plasma mode. Naturally coupled to the ion acoustic wave and Alfv´ en wave

⇒ drift-Alfv´ en mode.

Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave Summary of properties of heating & consequences

Experiments in Japan

Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave Summary of properties of heating & consequences

Experiments in Japan

Group of Kono & Tanaka

Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave Summary of properties of heating & consequences

Experiments in Japan

Group of Kono & Tanaka

Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave Summary of properties of heating & consequences

Experiments in Japan

Vranjes et al., Phys. Rev. Lett. 89, 265002 (2002).

Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave Summary of properties of heating & consequences

Solar drift wave lab

• Fig. 3.

˚

Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave Summary of properties of heating & consequences

Oblique drift wave properties

Universally unstable: Collisional and collision-less instability within fluid theory

• J. Vranjes and S. Poedts, Astron. Astrophys. 458, 635 (2006),
• H. Saleem, J. Vranjes, and S. Poedts, Astron. Astrophys. 471, 289

(2007),

• J. Vranjes and S. Poedts, MNRAS 400, 2147 (2009).

Collision-less instability within kinetic theory

• J. Vranjes and S. Poedts, MNRAS 398, 918 (2009),
• J. Vranjes and S. Poedts, Astrophys. J. 719, 1335 (2010).

Current driven instability. Shear-flow driven instability. Trapped particle instability. Nonlinear instability.

Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave Summary of properties of heating & consequences

Minimum conditions and equations

Geometry:

• B0 = B0

ez, ∇n0 = −n′ ex. Perturbations: f (x)exp[i(kyy + kzz − ωt)]. Boltzmannian electrons (implying that ω/kz ≪ vTe, vze ≪ vTe, and on condition eφ1/(κTe) ≪ 1): ne1 n0 = eφ1 κTe . (16) Cold ions (vTi ≪ ω/kz) motion strictly perpendicular (to B0 = B0 ez) implying that ω/kz ≫ cs; low frequency perturbations ω ≪ Ωi

Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave Summary of properties of heating & consequences

mini ∂ v ∂t + v · ∇ v

• i

= nie (−∇φ + vi × B0) .

• v⊥i1 = 1

B0 ( ez × ∇φ1) − 1 ΩiB0 ∂ ∂t ∇φ1, vi = kz ω eφ1 mi . The ion continuity ∂ni ∂t + ∇ · ni vi = 0. Dispersion equation in local approximation ω2 1 + k2

y ρ2 s

• −ωω∗e−k2

z c2 s = 0,

ω∗e = kyv∗e, v∗e = −κTe eB0 1 n0 dn0 dx ey, ρs = cs/Ωi, c2

s = κTe/mi.

Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave Summary of properties of heating & consequences

Coupled drift and ion acoustic waves. Electron collisions ne1 n0 = eφ1 κTe

• 1 − i meνei

k2

z κTe

(ω∗e − ω)

• .

(17) Growth rate: γ = νeω2

r ρ2 sk2 y

k2

z v2

Te

. (18)

Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave Summary of properties of heating & consequences

Solar atmosphere: source 1 - collisional instability

Vranjes and Poedts, Astron. Astrophys. 458, 635 (2006).

Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave Summary of properties of heating & consequences

Interaction with Alfv´ en waves. growing drift wave mode → locally increased plasma-β → electromagnetic effects → excitation of (growing) AW.

exchange of identities, Vranjes and Poedts, AA 458, 635 (2006).

→ reconnection (consequence of heating)

Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave Summary of properties of heating & consequences

Source 2: shear flow instability in solar spicules

Saleem, Vranjes and Poedts, Astron.

• Astrophys. 471, 289 (2007).

The minimal value of the normalized shear Γ = dvz0(x)/(Ωidx) in terms of ky and Ln. Instability for the values above the surface.

Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave Summary of properties of heating & consequences

Source 3: kinetic drift wave instability

ωr = − ω∗iΛ0(bi) 1 − Λ0(bi) + Ti/Te + k2

y λ2 di

, γ ≃ − π 2 1/2 ω2

r

|ω∗i|Λ0(bi) Ti Te ωr − ω∗e |kz|vTe exp[−ω2

r /(k2 z v2

Te)]

+ωr − ω∗i |kz|vTi exp[−ω2

r /(k2 z v2

Ti)]

• .

Λ0(bi ) = I0(bi )exp(−bi ), bi = k2

y ρ2 i , λdi = vTi /ωpi , ω∗e = −ky

v2

Te

Ωe n′

e0

ne0 ω∗i = ky v2

Ti

Ωi n′

i0

ni0 .

the presence of the energy source seen already in the real part of the frequency ωr ∝ ∇⊥n0.

Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave Summary of properties of heating & consequences Vranjes and Poedts, MNRAS 398, 918 (2009).

Kinetic growth rate normalized to the wave frequency ωr in terms of the perpendicular wavelength λy and the density scale-length Ln, for λz = s · 2 · 104 m, s ∈ (0.1, 103). Short perpendicular scale length; comparable to gyro-radius.

Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave Summary of properties of heating & consequences

Polarization drift effects within the two-fluid description

vi⊥ = 1 B0

• ez × ∇⊥φ + v2

Ti

Ωi

• ez × ∇⊥ni

ni + ez × ∇⊥ · πi miniΩi + 1 Ωi d dt ez × vi⊥, d dt ≡ ∂ ∂t + v · ∇. (19) Guiding center approximation violated:

• vpi = −

ey ωrkyφ1(t) B0Ωi

• 1 − kz

ωr dz dt

• cosϕ − γ

ωr sinϕ

• ×

×1/

• 1 − k2

y φ1(t)

B0Ωi cosϕ

• ,

ϕ = kyy(t) + kzz(t) − ωrt. (20)

Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave Summary of properties of heating & consequences

Stochastic heating: individual particle dynamics

Vranjes and Poedts, MNRAS 408, 1835 (2010).

Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave Summary of properties of heating & consequences

Numbers

Frequency limits: ω < Ωi, ω < ω∗e = v∗ek⊥; v∗e = − κTe

eB0

ez × ∇n0

n0 ∼ 1/Ln.

Ln = 100 m ⇒ ω ≃ 250 Hz; γ/ω = 0.26; growth time τg = 0.06 s; energy release rate Γ ≃ 0.7 J/m3s. Ln = 100 km ⇒ ω ≃ 0.25 Hz; γ/ω = 0.28; growth time τg = 60 s; energy release rate Γ ≃ 6 · 10−4 J/m3s. Maximum achieved particle speed: vmax =

• k2

y ρ2 i

eφ κTi + 1.9 Ωi ky ⇒ Teff .

Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave Summary of properties of heating & consequences

Properties

Crucial electrostatic nature of the wave in the given process of heating.

yet, coupling to the Alfv´ en wave included!

Highly anisotropic, takes place mainly in the direction normal to the magnetic field B0 (both the x− and y-direction velocities are stochastic). Electric fields of (tens) kV/m. Energy release rate in the range of nano-flares. Predominantly acts on ions

Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave Summary of properties of heating & consequences

Stronger heating of heavy ions The stochastically increased ion temperature Teff = miv2

max/(3κ) (in

millions K) in terms of the perpendicular wave-length λy and the ion mass. The areas with stronger background magnetic fields are subject to stronger stochastic heating.

Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave Summary of properties of heating & consequences

Particle acceleration in the parallel direction.

Ω

Normalized perturbed velocity in the direction parallel to the magnetic field. Because the mean free path ∼ v4 ⇒ kappa distribution. Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave

Outline

1 Examples and features of stochastic heating 2 Ion acoustic wave 3 Oblique drift wave 4 Transverse drift wave

Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave

Properties of the transverse drift wave

• B0 = B0

ez, ∇n0 = (dn0/dx) ex,

• k = k

ey,

• k ·

B0 = 0,

• E1

B0,

• B1

B0,

• B1 = (kE1/ω)

ex. In the limit kyρi < 1 [Krall & Rosenbluth, Phys. Fluids 6, 254 (1963)] ωr = −kyκTe en0B0 dn0 dx 1 1 + k2

y c2/ω2 pe

, ǫn,b = 1/Ln,b, γ ωr = πme mi ǫn ǫb

• 1 + k2

y c2

ω2

pe

−2 1 + k2

y c2

ω2

pe

+ Te Ti

• ×exp
• −ǫn

ǫb Te Ti 1 1 + k2

y c2/ω2 pe

• ,

⇐ no threshold!

Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave

Motivation: transport

Particle transport to high latitudes.

• C. G. Maclennan et al.,
• Geophys. Res. Lett. 30,

198033 (2003). ACE = Advanced Composition Explorer

Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave

• D. Lario et al., Adv. Space Res. 32, 579 (2003).

Delays longer than for a spiral magnetic connection. Rise-time at ACE much faster ⇒ more diffusive transport at high lat.

Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave

Motivation: solar wind heating

• J. C. Kasper et al., Phys. Rev.
• Lett. 101, 261103 (2008).

Ion anisotropy T⊥α/Tα, T⊥p/Tp in terms of △Vαp ≡ Vα − Vp.

Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave

Favorable properties

• J. Lima, K. Tsinganos, Geophys.
• Res. Lett. 23, 117 (1996).

Observations; the latitude dependent density in the solar wind.

Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave

• D. J. McComas et al., J.
• Geophys. Res. 105(A5), 10419

(2000). Ulysses observations - the latitude dependent density in the solar wind.

Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave

Transverse drift wave in the corona

γ/ω

β

• J. Vranjes, Month. Not. Roy.
• Astron. Soc. 415, 1543 (2011).

Growth rate for coronal plasma parameters. n0 = 5 · 1014 m−3; T0 = 106 K. For λy = 100 m, Ln = 105 m we have ωr = 0.27 Hz. For Ln = 106 m ⇒ ωr = 0.027 Hz.

Acceleration: v = (e E1/mi) τ/2 sin(ωrt) = 2e E1/(miωr); τ = 2π/ωr. For E1 = 0.001 V/m ⇒ v = 700 km/s; fast solar wind. For E1 = 0.02 V/m ⇒ MeV proton energy!

Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave

Used solar wind parameters

At 1 AU: Ti ≃ Te = 1.5·105 K, ni0 = ne0 = n0 = 5·106 m−3, B0 = 5·10−9 T, Ln = 108 m, λ = 106 m, plasma-beta = 1.04, ρi = 74·103 m, Ωi ≃ 0.5 Hz, ωr = 0.00016 Hz, Tw ≃ 3.8 · 104 s, [≃ 10.7 hours] The ions are singly charged protons. Plasma magnetized; the local approximation well satisfied. The drift approximation ωr/Ωi ≪ 1 well satisfied. Because Ln/LB = β/2 ⇒ Lb = 1.9 · 108 m. νii = 1.4 · 10−7 Hz, λf = 2.6 · 1011 m[≃ 1.7 AU!] The classic perpendicular ion diffusion coefficient D⊥ ≈ κTiνi/(miΩ2

i ) = 736 m2/s.

Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave

• 10
• 5

• 6

• J. Vranjes, Astron. Astrophys. 532, A137 (2011).
• Upper. Full line: proton velocity along the magnetic

field, in the wave field Ez1sin(−ωt + ky) for

• Ez1 = 2 · 10−8 V/m, ωr = 0.00016 Hz. Dashed line:

the corresponding displacement along the magnetic field. The particle is subject to continuous directed drift along the magnetic field vector although the wave parallel electric field is oscillatory; around half million kilometers within one wave-period.

• Lower. The displacement in the x, y-plane,

corresponding to the upper figure. Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave

• 2
• 1

• 6
• 3

• J. Vranjes, Astron. Astrophys. 532, A137

(2011).

Order of magnitude stronger electric field Ez1 = 2 · 10−7 V/m. Stochastic motion develops after 8.9 hours. Displacement along the density gradient of the same order as the displacement in the z-direction. The x-displacement remains constant after the development of the stochastic motion. The particle already moved a few million kilometers. The average drift (diffusion) velocity is around vD = 1.2 · 105 m/s ⇒ effective diffusion coefficient D⊥ = vDLn = 1.2 · 1013 m2/s. [Note that this estimate is for the least favorable case, i.e., for the particles with zero starting velocity in the z-direction.] Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave

• 2
• 1

• J. Vranjes, Astron. Astrophys. 532, A137 (2011).

Upper.The velocity along the magnetic field vector for vy [0] = 104 m/s.

• Lower. Drift perpendicular to the magnetic field, for

three different starting velocities vz (0) in the direction of the background magnetic field. A similar test done by taking vz (0) = 106 m/s; within 19 hours the particle drifts to x ≃ 3 · 1010 m, i.e., to 0.2 AU. Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave

New stochastic heating mechanism

Follow two particles in the wave field; initially positions r1, r2;

• r2 =

r1 + δr; in general different velocities v1, v2: d v1 dt = q m

• E(

r1, t) + v1 × B( r1, t)

• ,

(21) d v2 dt = q m

• E(

r2, t) + v2 × B( r2, t)

• .

(22) Calculate the distance between the two. Subtracting them gives the displacement (forced harmonic oscillations due to the term on the right-hand side): d2δy dt2 + Ω2

• 1 − v2,z

Ω ∂B1( r1, t)/B0 ∂y

• δy = Ωdδz

dt B1( r1, t) B0 . (23)

Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave

Introduced notation:

• B(

r1, t) = B0 + B1( r1, t),

• B(

r2, t) = B0 + B1( r1, t)+(δ r ·∇) B1( r1, t),

• E(

r2, t) = E1( r1, t) + (δ r · ∇) E1( r1, t). The perpendicular distance δy between the two particles is determined by the parallel velocity of one of the particles. The distance can grow in time and this is equivalent to stochastic heating. Condition: δ ≡ v2,z ΩB0 ∂B1( r1, t) ∂y > 1. (24) Some critical value for the parallel velocity v2,z is required!

Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave

In the first approximation, from the parallel equation of motion v2,z = eEz1/(mωr) δ ≃ k2 E 2

z1

ω2

r B2

> 1. (25) The condition very easily satisfied, and, as a result, the particle motion in the wave field becomes stochastic for very small amplitude of the wave. In the given examples this happens for Ez1 ≃ 2 · 10−7 V/m. Completely new stochastic heating mechanism.

Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave

Drift wave in solar plasma community

A letter from Editor Dear Dr. Vranjes,

• Re. XX-09-0376-XX - ”The universally growing mode in the solar

atmosphere: coronal heating by drift waves” by Vranjes, Poedts. We have contacted twelve potential reviewers of your submission so far without success. If you wish to maintain your submission I would appreciate a list of at least six reviewers that you would consider competent to assess your submission. My apologies for the delay, which neither I nor the editorial staff could have foreseen.

Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave

THANK YOU!

More details in:

• J. Vranjes and S. Poedts, A new pradigm for solar coronal heating, Europhys. Lett. 86, 39001 (2009).
• J. Vranjes and S. Poedts, Solar nanoflares and smaller energy release events as growing drift waves, Phys. Plasmas 16,

092902 (2009).

• J. Vranjes and S. Poedts, The universally growing mode in the solar atmosphere: coronal heating by drift wave, MNRAS

398, 918 (2009).

• J. Vranjes and S. Poedts, Electric field in solar magnetic structures due to gradient driven instabilities: heating and

acceleration of particles, MNRAS 400, 2147 (2009).

• J. Vranjes and S. Poedts, Drift waves in the corona: heating and acceleration of ions at frequencies far below the gyro

frequency, MNRAS 408, 1835 (2010).

• J. Vranjes and S. Poedts, Kinetic instability of drift-Alfv´

en waves in solar corona and stochastic heating, ApJ 719, 1335 (2010).

• J. Vranjes, Growing electric field parallel to magnetic filed due to transverse kinetic drift wave in inhomogeneous corona,

MNRAS 415, 1543 (2011).

• J. Vranjes, Transport and diffusion of particles in solar wind due to transverse drift wave, Astron. Astrophys. 532, A137

(2011). Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave Stochastic heating in non-equilibrium plasmas

Examples and features of stochastic heating Ion acoustic wave Oblique drift wave Transverse drift wave

“I am not young enough to know everything” (Oscar Wilde).

Stochastic heating in non-equilibrium plasmas