Transition From Single Fluid To Pure Electron MHD Regime Of Tearing Instability

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1 Transition From Single Fluid To Pure Electron MHD Regime Of Tearing Instability V.V.Mirnov, C.C.Hegna, S.C.Prager APS DPP Meeting, October 27-31, 2003, Albuquerque NM

2 Abstract In the most general case, the two-fluid tearing instability is driven by shear Alfven (SA), compressional Alfven (CA), and slow magneto-acoustic (MA) modes modified on short scales by two-fluid effects. In a cold plasma limit, b = 0, the only interaction occurs between the SA and CA modes. On the scales shorter than d i = c/w pi, the CA mode is converted into pure electron whistler mode which provides fast reconnection within the scope of electron MHD (EMHD). The frequency of whistlers, w / k k d I 2, tends to zero on the resonant surface k! 0, causing this branch to exhibit on this surface the properties of the CA mode with non zero frequency w = k c A determined by the guiding field and ion mass. This suppresses out-of-plane perturbations B and slows down the instability. In order to realize the conditions of EMHD d i should be much greater than the scale of equilibrium magnetic field, otherwise the instability develops in the regime similar to single fluid MHD. The transition between these two cases is investigated in the work. * Supported by U.S.D.O.E. 2

3 Results Out-of-plane perturbations of the guiding magnetic field B (1) are needed for whistler mediated tearing instability. Near resonant surface k = 0, the amplitude of B (1) is suppressed due to ion motion in compressional Alfven wave. In order to reach a whistler mediated regime of reconnection, the tearing instability should be fast enough, γ À k v A. Since γ / δ 2 d i, it requires d i À large scale L of external magnetic field. Effect of plasma compressibility is small in resistive MHD limit and in fast EMHD regime because ions are practically immovable. In transition from single fluid to electron MHD, the effect of plasma compressibility is important and may result in strong ion heating. 3

4 Tearing instabilities are important as a mechanism of relaxation to the Taylor s state Motivations for two-fluid studies in RFP High T, low guiding magnetic field B (0) two fluid effects important B (0) of major interest, absent in many Hall-MHD theories Effects of β, ρ s and d i in two-fluid MHD without kinetic closures 1 1 pe me dve E + v B = j B + η j c nec ne e dt MHD dynamo Generalized Ohm s law Hall term electron inertia resistivity Major scales in the Madison Symmetric Torus (MST) experiments External scale L (distance between resonant surfaces) 20 cm Ion skin depth d i = c / ω pi 10 cm Ion-sound gyroradius ρ s = c s / ω ci 2.5 cm Combined collisionless and resistive electron skin-depth δ 2 = c 2 / ω 2 pe + ηc 2 /4πg, 5 mm 4

5 Two-fluid tearing instability in force free equilibrium Force-free equlibrium with uniform plasma density and pressure: X radial direction, Y sheared magnetic field, Z guiding field Symmetric configuration of the current sheet Two-fluid tearing equations are derived with the treatment of plasma compressibility v and the Hall term Cold plasma case (c s << l γ ) Compressional (CA) + Shear Alfven (SA) modes Hot plasma case (c s >> L γ ) Shear Alfven (kinetic Alfven) + magnetoacoustic (MA) modes We are interested in transition from single fluid MHD to two-fluid and electron MHD in terms of dependences on guiding field B (0) and stability factor 0 5

6 MHD waves are modified on short scales by the Hall term Dispersion relation for phase velocity u = ω (1+ k 2 d e2 ) 1/2 / k v A in uniform magnetic field (β k = (1+ k 2 d e2 ) c s2 / v A 2, cos θ = B y (1) / B ) = H Hall term Plot of F(u) in hot plasma case (β > cos 2 θ) two-fluid kinetic Alfven F(u) whistlers Intersections F(u)=H(k) determine three modes of oscillations Phase velocities of the SA and CA modes increase with k good candidates for fast growth magnetoacoustic (MA) shear Alfven (SA) compressional Alfven (CA) At cos 2 θ << u 2 << β k the SA mode is similar to kinetic Alfven wave At k d i >> 1 the CA mode transfers to whistlers 6

7 In cold plasma b! 0, the SA and CA modes determine dynamics of two-fluid tearing instability Plot of F(u) in cold plasma (cos 2 θ À β! 0) F(u) shear Alfven (SA) whistlers compressional Alfven (CA) u Phase velocity increases with k for the CA mode and decreases for the SA mode The CA branch is a good candidate for fast reconnection nonresonant at k = 0 term k 2 v A2 0 reduces effect of whistlers whistlers 7

8 Two-fluid effects are described by the perturbation of out-of-plane magnetic field B (1) B z Linearization for pure growing tearing mode with k = (0, k, 0) B z is determined by the z-component of the induction equation Plasma compressibility r v is driven by the magnetic pressure B z (0) B z / 4 π Fast compressional Alfven wave results in reduction of B z and slowing of two-fluid tearing instability down to the time scale of resistive MHD 8

9 Tearing instability driven by two-fluid SA and CA modes at b = 0 B z / t div v magnetic pressure Hall term Hall term parallel Ohm s law vorticity equation diffusion of B z effect of compressional Alfven mode δ eff2 = δ / γ 2 τ a2 ε 2 finite ion mass and guiding field / B 02 / m i 9

10 In pure electron MHD (m i! 1), the tearing instability is driven by whistlers In the limit m i! 1, the growth rate γ is determined by the electrons and does not depend on m i, τ a / m i 1/2, d i / m i 1/2 The EMHD tearing equations In the EMHD, the SA and CA modes are decoupled and the instability is driven by whistlers ( S.V.Bulanov, F.Pegoraro, A.S.Sakharov, 1992) 10

11 Solution for the tearing instability driven by the SA and CA modes ( constant- y approximation ) Layer width l ' 0 δ 2 << δ, δ eff! δ eff 2 d 2 B z /dx 2 >> B z! B z (x) = v(x) d i / k δ eff 2 ( A.Fruchtman, H.R.Strauss, 1993) Γ = γ τ a / k d i, u = dv / dx. Since Γ ' ( 0 δ ) 2 δ / δ eff! x >> δ eff Γ! one can ignore 1 / x 2 in the r.h.s. ( constant - ψ approximation). Integrating over x yields Dispersion relation Characteristic scale 11

12 Sharp transition from single fluid to pure electron MHD takes place at critical value of D Growth rate γ τ a /k δ γ rmhd Stability factor δ γ emhd Joint effect of decoupled electron and ions Transition requires J / ne & v Due to suppression of B z by the nonresonant CA mode it takes place at d i2 & δ eff2 = δ 2 + (1 / γ 2 τ a2 ε 2 ) If guiding field B 0 is small this condition is satisfied, d i >> δ In magnetically confined plasmas with strong guiding field, the mechanism of whistlers is suppressed 0 threshold for whistler mediated tearing instability 12

13 Realization of fast whistler mediated reconnection with guiding magnetic field requires c / w pi À L Critical value of 0 for fast whistled mediated tearing instability Constant - ψ approximation 0 δ 1 Using δ ' d e / L = d i ( m e / m i ) 1/2 / L yields condition of constant - ψ whistler mediated instability ( 0 c δ << 0 δ << 1) The EMHD regime of tearing instability can exist in cold plasma with sufficiently large c / ω pi À L 13

14 Summary Out-of-plane perturbations of the guiding magnetic field B (1) are needed for whistler mediated tearing instability. Near resonant surface k = 0, the amplitude of B (1) is suppressed due to ion motion in compressional Alfven wave. In order to reach a whistler mediated regime of reconnection, the tearing instability should be fast enough, γ À k v A. Since γ / δ 2 d i, it requires d i À large scale L of external magnetic field. Effect of plasma compressibility is small in resistive MHD limit and in fast EMHD regime because ions are practically immovable. In transition from single fluid to electron MHD, the effect of plasma compressibility is important and may result in strong ion heating. 14