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 > 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). 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 > Ω i ) IA wave: z φ | J l ( k ⊥ ρ i ) | / m i ≥ Ω 2 ek 2 i 16 , l = 0 , ± 1 , ± 2 · · · -Heating in the high-energy tail of the ion distribution. For IC and LH wave: � Ω i � 1 / 3 ω E > 1 . 4 ω B 0 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: e φ k 2 y ρ 2 ≥ 1 . i κ T i -Heating of bulk plasma. -Better heating of heavier ions. -Dominant perpendicular heating. For TD wave (essentially electromagnetic mode!): k 2 ⊥ E 2 z 1 > 1 . ω 2 B 2 0 -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 − k z v T i = ± l Ω i . ⇒ v z = ( ω ± l Ω 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 0 ∼ 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 Measured ion temperatures T ⊥ drift-Alfv´ en wave activity. and T � . Wave period 230 µ s. Sanders et al. , Phys. Plasmas 5 , Sanders et al. , Phys. Plasmas 5 , 716 (1998). 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 IA wave instability in inhomogeneous collisional plasmas Oblique drift wave IA wave instability in permeating plasmas 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 4 Transverse drift wave Stochastic heating in non-equilibrium plasmas
Examples and features of stochastic heating Ion acoustic wave IA wave instability in inhomogeneous collisional plasmas Oblique drift wave IA wave instability in permeating plasmas Transverse drift wave Collisional plasma; fluid description Geometry: � ∇ n 0 = − n ′ B 0 = B 0 � e z , 0 � e x . Perturbations: f ( x )exp[ i ( k y y + k z z − ω t )], | ( df / dx ) / f | , | ( dn j 0 / dx ) / n 0 | ≪ k y The electron equations: � ∂� � v e v e × � ∂ t + ( � v e · ∇ ) � = en e ∇ φ − en e � m e n e v e B − κ T e ∇ n e − m e n e ν en ( � v e − � v n ) , (1) ∂ n e 1 + ∇ ⊥ ( n e � v ⊥ e ) + ∇ z ( n e 0 � v ez 1 ) = 0 . (2) ∂ t Neutrals: � ∂� � v n ∂ t + ( � v n · ∇ ) � v n = − ν ne ( � v n − � v e ) . (3) Stochastic heating in non-equilibrium plasmas
Examples and features of stochastic heating Ion acoustic wave IA wave instability in inhomogeneous collisional plasmas Oblique drift wave IA wave instability in permeating plasmas Transverse drift wave The momentum conservation implies that ν ne = m e n e ν en / ( m n n n ). Ions: � ∂� � v i v i × � m i n i ∂ t + ( � v i · ∇ ) � v i = − en i ∇ φ + en i � B . (4) Ω e ≫ | ω | ≫ Ω i , (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 IA wave instability in inhomogeneous collisional plasmas Oblique drift wave IA wave instability in permeating plasmas Transverse drift wave Dispersion equation, the oblique, density gradient driven IA mode [Vranjes and Poedts, Phys. Plasmas 16 , 022101 (2009)]: = ω ∗ e + iD p + iD z ( ω 2 + ν 2 ne ) / ( ω 2 − i ν ne ω ) k 2 c 2 s ne ) / ( ω 2 − i ν ne ω ) . (6) ω + iD p + iD z ( ω 2 + ν 2 ω 2 D p = ν en α k 2 y ρ 2 D z = k 2 z v 2 e , T e /ν en , ρ e = v T e / Ω e . v ∗ e = − κ T e e z × ∇ n 0 ω ∗ e = v ∗ e k y , � � , α = ω/ ( ω + i ν ne ) . eB 0 n 0 α = 1 ⇐ static neutrals D p - 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 IA wave instability in inhomogeneous collisional plasmas Oblique drift wave IA wave instability in permeating plasmas Transverse drift wave Vranjes and Poedts, Phys. Plasmas 16 , 022101 (2009). 0.6 IA frequency Frequency (normalized to ω ∗ e ), and the corresponding 0.5 D =0 normalized growth-rate. p w ith D p Parameters: m i = 40 m p , m n = m i , T e = 5 eV, 0.4 n n 0 = 10 21 m − 3 , n e 0 = n i 0 = 5 · 10 16 m − 3 , 0 5 10 15 k z B 0 = 0 . 01 T, L n = 0 . 1 m, k y = 7 · 10 2 1/m. For these 0.12 parameters σ en = 8 . 7 · 10 − 20 m 2 . D =0 IA growth rate p w ith D p The perpendicular electron collisions drastically 0.06 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 0.00 0 5 10 15 k z only 0.06. Stochastic heating in non-equilibrium plasmas
Examples and features of stochastic heating Ion acoustic wave IA wave instability in inhomogeneous collisional plasmas Oblique drift wave IA wave instability in permeating plasmas Transverse drift wave 0.5 Angle dependence - angle of preference. Frequency (full lines) III II 0.4 I and the corresponding growth rates (dashed lines), both *e frequency/ ω 0.3 normalized to the electron diamagnetic drift frequency, for three 0.2 γ values of neutral number density. The lines I, II, III correspond k 0.1 (respectively) to n n 0 = 10 19 , 10 18 , 10 17 m − 3 . I II III 0.0 0.000 0.005 0.010 θ - arctan( k z / k y ). θ [rad] The line γ k is the kinetic growth-rate (for the same parameters as line II). Phys. Plasmas 17 , 022104 (2010). Stochastic heating in non-equilibrium plasmas
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