Stochastic heating in non-equilibrium plasmas

Transcription

1 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

2 Outline Examples and features of stochastic heating 1 Examples and features of stochastic heating 2 IA wave instability in inhomogeneous collisional plasmas IA wave instability in permeating plasmas 3 Summary of properties of heating & consequences 4

3 Outline Examples and features of stochastic heating 1 Examples and features of stochastic heating 2 3 4

4 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

5 Some waves of interest (multi-component theory) Plasma (Langmuir) wave. (IA, ω > Ω i, ω < Ω i ). 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).

6 Some features of stochastic heating Mostly electrostatic phenomena. Necessity for a large enough electric field; yet linear theory valid. For (high frequency, ω > Ω i ) IA wave: ekz 2 φ J l (k ρ i ) /m i Ω2 i, l = 0, ±1, ±2 16 -Heating in the high-energy tail of the ion distribution. For IC and LH wave: E B 0 > 1 4 ( ) 1/3 Ωi ω. ω k -Heating of the bulk and tail plasma.

7 For OD wave: k 2 y ρ 2 i -Heating of bulk plasma. -Better heating of heavier ions. -Dominant perpendicular heating. eφ κt i 1. For TD wave (essentially electromagnetic mode!): k 2 E 2 z1 ω 2 B 2 0 > 1. -Acceleration heating and transport. -Acting on both ions and electrons.

8 Heating at ion-cyclotron harmonics. A. Fasoli et al., Phys. Rev. Lett. 70, 303 (1993).

9 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).

10 IA wave: overlapping of resonances diffusion (loss of memory of initial conditions). ω k z v Ti = ±lω i. v z = (ω ± lω i )/k z. High frequency IA mode ω > Ω i. v z finite for k z 0. Obliquely propagating with respect to B 0! not large amplitude n 1 /n G. R. Smith and A. N. Kaufman, Phys. Rev. Lett. 70, 303 (1993).

11 Drift-Alfvén wave heating in tokamak Ion temperature at three different positions during the drift-alfvén 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).

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

13 Outline Examples and features of stochastic heating IA wave instability in inhomogeneous collisional plasmas IA wave instability in permeating plasmas 1 Examples and features of stochastic heating 2 IA wave instability in inhomogeneous collisional plasmas IA wave instability in permeating plasmas 3 4

14 Collisional plasma; fluid description Geometry: IA wave instability in inhomogeneous collisional plasmas IA wave instability in permeating plasmas B 0 = B 0 e z, n 0 = n 0 e x. Perturbations: f (x)exp[i(k y y + k z z ωt)], (df /dx)/f, (dn j0 /dx)/n 0 k y The electron equations: Neutrals: m e n e [ ve t + ( v e ) v e n e1 ] = en e φ en e v e B κt e n e m e n e ν en ( v e v n ), (1) + (n e v e ) + z (n e0 v ez1 ) = 0. (2) t [ ] vn t + ( v n ) v n = ν ne ( v n v e ). (3)

15 IA wave instability in inhomogeneous collisional plasmas IA wave instability in permeating plasmas The momentum conservation implies that ν ne = m e n e ν en /(m n n n ). Ions: [ ] vi m i n i t + ( v i ) v i = en i φ + en i v i B. (4) Ω e ω Ω i, (5) from H. J. de Blank, Plasma Physics Lecture Notes free energy in the system

16 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, (2009)]: k 2 c 2 s ω 2 = ω e + id p + id z (ω 2 + ν 2 ne)/(ω 2 iν ne ω) ω + id p + id z (ω 2 + ν 2 ne)/(ω 2 iν ne ω). (6) D p = ν en αk 2 y ρ 2 e, D z = k 2 z v 2 Te/ν en, ρ e = v Te /Ω e. ω e = v e k y, v e = κt e eb 0 e z n 0 n 0, α = ω/(ω + iν ne ). α = 1 static neutrals D p - usually omitted without justification. ω e - diamagnetic frequency; introduces free energy.

17 IA wave instability in inhomogeneous collisional plasmas IA wave instability in permeating plasmas IA growth rate 0.5 IA frequency D p =0 with D p 0 5 k z D p =0 with D p k z Vranjes and Poedts, Phys. Plasmas 16, (2009). Frequency (normalized to ω e ), and the corresponding normalized growth-rate. Parameters: m i = 40m p, m n = m i, T e = 5 ev, n n0 = m 3, n e0 = n i0 = m 3, B 0 = 0.01 T, L n = 0.1 m, k y = /m. For these parameters σ en = m 2. The perpendicular electron collisions drastically destabilize the mode. For small k z, the growth rate about 70 times larger. Note that for k z = 0.3 1/m we have D p/d z = 141, while for k z = 14 1/m this ratio is only 0.06.

18 IA wave instability in inhomogeneous collisional plasmas IA wave instability in permeating plasmas frequency/ω *e III II I γ k I II III θ [rad] 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 n n0 = 10 19, 10 18, m 3. θ - arctan(k z /k y ). The line γ k is the kinetic growth-rate (for the same parameters Phys. Plasmas 17, (2010). as line II).

19 Kinetic instability IA wave instability in inhomogeneous collisional plasmas IA wave instability in permeating plasmas Same geometry; magnetized (un-magnetized) electrons (ions); but no collisions. The perturbed number density for electrons (the same as for the drift wave): n e1 n 0 = eφ 1 κt e The ion number density: { ( π ) 1/2 ω ω e 1 + i exp [ ω 2 /(2kz 2 v 2 k z v Te) 2 ] }. (7) Te n i1 n i0 = eφ 1 m i v 2 Ti [ ( )] ωi 1 J +. (8) kv Ti Here, J(η) = [η/(2π) 1/2 ] c dζexp( ζ2 /2)/(η ζ) is the plasma dispersion function, and ζ = v/v Ti.

20 IA wave instability in inhomogeneous collisional plasmas IA wave instability in permeating plasmas Frequency and the growth rate from the quasi-neutrality: ω 2 k = k2 c 2 s 2 [ 1 + (1 + 12T i /T e ) 1/2]. (9) γ k (π/2)1/2 ωk 3 2k 2 cs 2 { ωk ω e exp [ ωk 2 k z v /(2k2 z vte) 2 ] + T e Te T i ω k exp [ ωk 2 kv /(2k2 vti) 2 ] Ti The electron contribution in Eq. (10) yields a kinetic instability provided that ω k < ω e. }. (10)

21 Plasmas with macroscopic motion IA wave instability in inhomogeneous collisional plasmas IA wave instability in permeating plasmas 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: ( π γ = 8 ) 1/2kcs [ (me m i ) 1/2 ( ) ve0 ( 1 τ 3/2 exp τ ) ]. (11) c s 2 v e0 > c s [1 + (m i /m e ) 1/2 τ 3/2 exp( τ/2)], τ = T e /T i. (12) Threshold high; for τ = 1 it is 27c s. Two-fluid counterpart: electron-collision effect, even higher threshold! [Vranjes et al., Phys. Plasmas 13, (2006)]. ν i γ = 2(1 χ) ( 1 ν ) e χ, χ = m e k 2 ( ) kz v 0 ν i m i kz 2 1. (13) ω r

22 IA wave instability in inhomogeneous collisional plasmas IA wave instability in permeating plasmas v 0 /v s k z /k Vranjes et al., Phys. Plasmas 13, (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: ν i > Ω i.

23 Permeating plasmas: IA wave instability IA wave instability in inhomogeneous collisional plasmas IA wave instability in permeating plasmas Two plasmas separately quasi-neutral n fi0 = n fe0 = n f 0, n si0 = n se0 = n s0. (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.

24 Solar wind Examples and features of stochastic heating IA wave instability in inhomogeneous collisional plasmas IA wave instability in permeating plasmas Encyclopedia of the solar system, eds. L. A. McFadden et al.. 1-hour average solar wind at 1 AU.

25 IA wave instability in inhomogeneous collisional plasmas IA wave instability in permeating plasmas The IA wave dispersion equation [Vranjes et al., Phys. Plasmas 16, (2009)] (k, ω) k 2 λ 2 d ω2 psi ω 2 3k 2 v 2 T si ωpsi 2 ω 4 ( ) π 1/2 ωω 2 pse ω 2 ( ) pfe +i 2 k 3 v T 3 +(ω kv f 0 ) se k 3 v T 3 + ω2 pfi k fe 3 v 3 + ωω2 psi T fi k 3 v 3 exp ω2 2k T si 2 v 2 = 0. (15) T si }{{}}{{}}{{} c a b 1/λ 2 d = 1/λ2 dse + 1/λ2 dfe + 1/λ2 dfi, λ dse = v T se /ω pse, etc. The inertia of the mode provided by the static ions. a/b 1 electron flow contribution negligible.

26 Laboratory plasma IA wave instability in inhomogeneous collisional plasmas IA wave instability in permeating plasmas General case; numerical solution v f0 /c se n s0 /N 0 T f i = 10 5 K T f i = K T f i = K Vranjes et al., Phys. Plasmas 16, (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; c 2 se = κt se /m i.

27 Solar plasma IA wave instability in inhomogeneous collisional plasmas IA wave instability in permeating plasmas 20 spectrum spectrum ω [x 10 5 Hz] γ [x10 4 Hz] λ [m] ω [x10 5 Hz] γ [x10 4 Hz] v f0 /c s Upper. Full line: IA wave frequency. Dashed line: the corresponding growth rate. Lower. Spectrum dependence on the speed of the flowing plasma.

28 Outline Examples and features of stochastic heating Summary of properties of heating & consequences 1 Examples and features of stochastic heating 2 3 Summary of properties of heating & consequences 4

29 In cylindric geometry Summary of properties of heating & consequences A drift wave in cylindric geometry with poloidal wave number m = 2. From Vranjes and Poedts, MNRAS 398, 918 (2009).

30 Cross section in cylindric geometry Summary of properties of heating & consequences 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).

31 Toroidal geometry Summary of properties of heating & consequences 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 Alfvén wave drift-alfvén mode.

32 Experiments in Japan Summary of properties of heating & consequences

33 Experiments in Japan Summary of properties of heating & consequences Group of Kono & Tanaka

34 Examples and features of stochastic heating Summary of properties of heating & consequences Experiments in Japan Group of Kono & Tanaka

35 Experiments in Japan Summary of properties of heating & consequences Vranjes et al., Phys. Rev. Lett. 89, (2002).

36 Summary of properties of heating & consequences Solar drift wave lab The heating of the solar corona 5 Fig. 3. SOHO/EIT 171Å global image of the solar disc with a close-up snapshot of active region loops from the higher resolution TRACE instrument in the same spectral line but not confined within these brighter loops (of course the entire corona is filled with magnetic field; Fig. 5b)? Or what form of coronal heating is operating in plumes that are observed in coronal emission lines with their open magnetic-field structure within coronal holes (Fig. 5c)? Maybe one should concentrate upon the formation of many

37 properties Summary of properties of heating & consequences 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.

38 Minimum conditions and equations Summary of properties of heating & consequences Geometry: B 0 = B 0 e z, n 0 = n 0 e x. Perturbations: f (x)exp[i(k y y + k z z ωt)]. Boltzmannian electrons (implying that ω/k z v Te, v ze v Te, and on condition eφ 1 /(κt e ) 1): n e1 n 0 = eφ 1 κt e. (16) Cold ions (v Ti ω/k z ) motion strictly perpendicular (to B 0 = B 0 e z ) implying that ω/k z c s ; low frequency perturbations ω Ω i

39 Summary of properties of heating & consequences m i n i ( v t ) + v v = n i e ( φ + v i B 0 ). i v i1 = 1 B 0 ( e z φ 1 ) 1 Ω i B 0 t φ 1, v i = k z eφ 1. ω m i The ion continuity n i t + n i v i = 0. Dispersion equation in local approximation ω 2 ( 1 + ky 2 ρ 2 ) s ωω e kz 2 cs 2 = 0, ω e = k y v e, v e = κt e 1 dn 0 eb 0 n 0 dx e y, ρ s = c s /Ω i, c 2 s = κt e /m i.

40 Summary of properties of heating & consequences Coupled drift and ion acoustic waves. Electron collisions n e1 n 0 = eφ 1 κt e [ 1 i m ] eν ei kz 2 (ω e ω). (17) κt e Growth rate: γ = ν eω 2 r ρ 2 s k 2 y k 2 z v 2 Te. (18)

41 Summary of properties of heating & consequences Solar atmosphere: source 1 - collisional instability Vranjes and Poedts, Astron. Astrophys. 458, 635 (2006).

42 Summary of properties of heating & consequences Interaction with Alfvén 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)

43 Summary of properties of heating & consequences Source 2: shear flow instability in solar spicules The minimal value of the normalized shear Γ = dv z0 (x)/(ω i dx) in terms of k y and L n. Instability for the values above the surface. Saleem, Vranjes and Poedts, Astron. Astrophys. 471, 289 (2007).

44 Source 3: kinetic drift wave instability Summary of properties of heating & consequences ω i Λ 0 (b i ) ω r = 1 Λ 0 (b i ) + T i /T e + ky 2 λ 2, di ( π ) [ 1/2 ω 2 γ r Ti ω r ω e exp[ ωr 2 /(kz 2 v 2 ω i Λ 0 (b i ) T e k z v Te)] 2 Te + ω r ω i k z v Ti exp[ ω 2 r /(k 2 z v 2 Ti)] ]. Λ 0 (b i ) = I 0 (b i )exp( b i ), b i = k 2 y ρ2 i, λ di = v T i /ω pi, ω e = k y v 2 T e Ω e ne0 n e0 2 vt i ni0 ω i = k y. Ω i n i0 the presence of the energy source seen already in the real part of the frequency ω r n 0.

45 Summary of properties of heating & consequences Vranjes and Poedts, MNRAS 398, 918 (2009). Kinetic growth rate normalized to the wave frequency ω r in terms of the perpendicular wavelength λ y and the density scale-length L n, for λ z = s m, s (0.1, 10 3 ). Short perpendicular scale length; comparable to gyro-radius.

46 Summary of properties of heating & consequences Polarization drift effects within the two-fluid description v i = 1 B 0 e z φ + v 2 Ti Ω i e z n i n i + 1 Ω i d dt e z v i, + e z π i m i n i Ω i d dt + v. (19) t Guiding center approximation violated: [( ω r k y φ 1 (t) v pi = e y 1 k z dz B 0 Ω i ω r dt ( ) 1/ 1 k2 y φ 1 (t) cosϕ B 0 Ω i ) cosϕ γ ] sinϕ ω r, ϕ = k y y(t) + k z z(t) ω r t. (20)

47 Summary of properties of heating & consequences Stochastic heating: individual particle dynamics Vranjes and Poedts, MNRAS 408, 1835 (2010).

48 Numbers Examples and features of stochastic heating Summary of properties of heating & consequences Frequency limits: ω < Ω i, ω < ω e = v e k ; v e = κte eb 0 e z n 0 n 0 1/L n. L n = 100 m ω 250 Hz; γ/ω = 0.26; growth time τ g = 0.06 s; energy release rate Γ 0.7 J/m 3 s. L n = 100 km ω 0.25 Hz; γ/ω = 0.28; growth time τ g = 60 s; energy release rate Γ J/m 3 s. Maximum achieved particle speed: ( v max = ky 2 ρ 2 eφ i κt i ) Ωi k y T eff.

49 Properties Examples and features of stochastic heating Summary of properties of heating & consequences Crucial electrostatic nature of the wave in the given process of heating. yet, coupling to the Alfvén wave included! Highly anisotropic, takes place mainly in the direction normal to the magnetic field B 0 (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

50 Summary of properties of heating & consequences Stronger heating of heavy ions The stochastically increased ion temperature T eff = m i v 2 max/(3κ) (in millions K) in terms of the perpendicular wave-length λ y and the ion mass. The areas with stronger background magnetic fields are subject to stronger stochastic heating.

51 Summary of properties of heating & consequences Particle acceleration in the parallel direction. 0.8 vz (t) 0.4 Normalized perturbed velocity in the direction parallel to the magnetic field x x10 4 Ω i t Fk(v) Maxwellian k = 2 k = 4 Because the mean free path v 4 kappa distribution. 0 1x10 6 2x10 6 3x10 6 v [m/s]

52 Outline Examples and features of stochastic heating 1 Examples and features of stochastic heating 2 3 4

53 Properties of the transverse drift wave B 0 = B 0 e z, n 0 = (dn 0 /dx) e x, k = k ey, k B0 = 0, E 1 B 0, B1 B 0, B1 = (ke 1 /ω) e x. In the limit k y ρ i < 1 [Krall & Rosenbluth, Phys. Fluids 6, 254 (1963)] ω r = k y κt e dn 0 en 0 B 0 dx γ ω r = π m e m i ɛ n exp ɛ b ( ky 2 c 2 /ωpe 2, ɛ n,b = 1/L n,b, 1 + k2 y c 2 ω 2 pe { ɛ n ɛ b T e T i k 2 y c 2 /ω 2 pe ) 2 ( 1 + k2 y c 2 ω 2 pe + T e T i ) }, no threshold!

54 Motivation: transport Particle transport to high latitudes. C. G. Maclennan et al., Geophys. Res. Lett. 30, (2003). ACE = Advanced Composition Explorer

55 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.

56 Motivation: solar wind heating J. C. Kasper et al., Phys. Rev. Lett. 101, (2008). Ion anisotropy T α /T α, T p /T p in terms of V αp V α V p.

57 Favorable properties J. Lima, K. Tsinganos, Geophys. Res. Lett. 23, 117 (1996). Observations; the latitude dependent density in the solar wind.

58 D. J. McComas et al., J. Geophys. Res. 105(A5), (2000). Ulysses observations - the latitude dependent density in the solar wind.

59 in the corona γ/ω r β J. Vranjes, Month. Not. Roy. Astron. Soc. 415, 1543 (2011). Growth rate for coronal plasma parameters. n 0 = m 3 ; T 0 = 10 6 K. For λ y = 100 m, L n = 10 5 m we have ω r = 0.27 Hz. For L n = 10 6 m ω r = Hz. Acceleration: v = (eê1/m i ) τ/2 0 sin(ω r t) = 2eÊ1/(m i ω r ); τ = 2π/ω r. For Ê1 = V/m v = 700 km/s; fast solar wind. For Ê1 = 0.02 V/m MeV proton energy!

60 Used solar wind parameters At 1 AU: T i T e = K, n i0 = n e0 = n 0 = m 3, B 0 = T, L n = 10 8 m, λ = 10 6 m, plasma-beta = 1.04, ρ i = m, Ω i 0.5 Hz, ω r = Hz, T w s, [ 10.7 hours] The ions are singly charged protons. Plasma magnetized; the local approximation well satisfied. The drift approximation ω r /Ω i 1 well satisfied. Because L n /L B = β/2 L b = m. ν ii = Hz, λ f = m[ 1.7 AU!] The classic perpendicular ion diffusion coefficient D κt i ν i /(m i Ω 2 i ) = 736 m 2 /s.

61 z, vz x(t), y(t) [m] 0-5 velocity v z (x 10 4 m/s) displacement z (x 10 8 m) x [x 10 7 ] y [x 10 3 ] t [s] t [s] J. Vranjes, Astron. Astrophys. 532, A137 (2011). Upper. Full line: proton velocity along the magnetic field, in the wave field Êz1sin( ωt + ky) for Ê z1 = V/m, ω r = 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.

62 vz [x 10 5 m/s] y [x 10 5 m] t [s] t [s] J. Vranjes, Astron. Astrophys. 532, A137 (2011). x [x 10 9 m] t [s] Order of magnitude stronger electric field Êz1 = 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 v D = m/s effective diffusion coefficient D = v D L n = m 2 /s. [Note that this estimate is for the least favorable case, i.e., for the particles with zero starting velocity in the z-direction.]

63 vz [x 10 5 m/s] x [x 10 9 m] v z [0]= 10 4 [m/s] v z [0]= [m/s] v z [0]= 10 5 [m/s] t [s] t [s] J. Vranjes, Astron. Astrophys. 532, A137 (2011). Upper.The velocity along the magnetic field vector for v y [0] = 10 4 m/s. Lower. Drift perpendicular to the magnetic field, for three different starting velocities v z (0) in the direction of the background magnetic field. A similar test done by taking v z (0) = 10 6 m/s; within 19 hours the particle drifts to x m, i.e., to 0.2 AU.

64 New stochastic heating mechanism Follow two particles in the wave field; initially positions r 1, r 2 ; r 2 = r 1 + δr; in general different velocities v 1, v 2 : d v 1 dt [ E( r1, t) + v 1 B( r 1, t)], (21) = q m d v 2 = q [ E( r2, t) + v 2 B( r dt m 2, 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): d 2 δy dt 2 + Ω2 [ 1 v 2,z Ω ] B 1 ( r 1, t)/b 0 δy = Ω dδz B 1 ( r 1, t). (23) y dt B 0

65 Introduced notation: B( r 1, t) = B 0 + B 1 ( r 1, t), B( r2, t) = B 0 + B 1 ( r 1, t)+(δ r ) B 1 ( r 1, t), E( r 2, t) = E 1 ( r 1, t) + (δ r ) E 1 ( r 1, 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 v 2,z B 1 ( r 1, t) > 1. (24) ΩB 0 y Some critical value for the parallel velocity v 2,z is required!

66 In the first approximation, from the parallel equation of motion v 2,z = ee z1 /(mω r ) δ k2 Ê 2 z1 ω 2 r B 2 0 > 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 Ê z V/m. Completely new stochastic heating mechanism.

67 Drift wave in solar plasma community A letter from Editor Dear Dr. Vranjes, Re. XX 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.

68 THANK YOU! More details in: J. Vranjes and S. Poedts, A new pradigm for solar coronal heating, Europhys. Lett. 86, (2009). J. Vranjes and S. Poedts, Solar nanoflares and smaller energy release events as growing drift waves, Phys. Plasmas 16, (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-alfvén 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).

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