MHD RELATED TO 2-FLUID THEORY, KINETIC THEORY AND MAGANETIC RECONNECTION

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1 MHD RELATED TO 2-FLUID THEORY, KINETIC THEORY AND MAGANETIC RECONNECTION Marty Goldman University of Colorado Spring 2017 Physics 5150

2 Issues 2 How is MHD related to 2-fluid theory Level of MHD depends on terms kept in Generalized Ohm's Law: Ideal MHD (magnetic flux frozen in to fluid) Resistive MHD (e.g, electron-ion collisions retained) Hall MHD (J x B retained) "Collisionless" MHD (pressure and/or inertia retained) Under what conditions can magnetic reconnection occur in Earth's magnetosphere? MMS: collisionless reconnection in electron diffusion regions Reconnection is not properly described by resistive MHD based on the magnetosphere's true resistivity

3 Where does magnetic reconnection occur? What happens in magnetic reconnection

4 4 Two-fluid eqns with pressure and electron-ion collisions one-fluid MHD Cont. eqn: t n s + ( n s u s ) = 0, Mom. eqn: m s n s ( t + u s )u s = s q s n s E+ u s c B P+ K s P s = pressure tensor from kinetic theory. Often ad hoc scalar pressure P sij = p s δ ij, p s n γ K e = m e n e υ e ( u e u i ) = m e e υ ej = collisional momentum transfer (electrons to ions) K i = K e (momentum lost by electrons = momentum gained by ions) Eight fluid variables, n s, u s (and fields) Eight MHD fluid variables (plus fields) ρ M mass density = m e n e + m i n i ( ) ρ charge density = e n i n e V flow velocity = n im i u i + n e m e u e ρ M ( ) J current density = e n i u i n e u e Assuming quasineutrality, n e = n i = n (low ω) ρ M nm i ρ 0 V u i + m e m i u e ( ) J en u i u e

5 Two-fluid eqns with pressure and collisions gives one-fluid MHD eqns 5 t n s + ( n s u s ) = 0, m s n s ( t + u s )u s = q s n s E+ u s c B P + K s s ρ M nm i ρ 0 V u i + m e m i u e ( ) J en u i u e From 2-fluid continuity eqns, Mass conservation: t ρ M + ( ρ M V) = 0 Charge density conservation: t ρ + J = 0 Add electron and ion momentum eqns to get Force eqn. Force eqn: ρ M V t = P+ J c B, P P e + P i (K-terms cancel; ρ = 0)

6 MHD eqns so far plus approximate Maxwell's eqns for fields 6 t ρ M + ( ρ M V) = 0, J = 0 ρ M V t = P + J c B, t ( γ P / ρ ) M = 0 Faraday's Law: c E = t B, B fields are closed loops: B = 0 Poisson's eqn: E = 0 (E fields are closed loops) Ampere's Law with no displacement current ( v A << c): B = 4π c J Need add'l eqn: Generalized Ohm's law for MHD

7 There is a generalized Ohm's Law for the electric field, E, in any magnetized fluid! 7 Electron momentum eqn gives Ohm's law in electron fluid Electron fluid term R e from kinetic simulation or measured Ion fluid Ohm's law can be derived from electron fluid Ohm's law E+ u e c B = R e 1 ne P e pressure + m e e Du e Dt inertial Substitute u e = J en + u i to yield ion fluid Ohm's Law: E+ u i c B = R i J B nec + R e

8 MHD Generalized Ohm's law follows from electron and ion fluid Ohm's laws 8 Include collisional momentum transfer E+ u e B / c = R e = 1 ne P e m η = ν e e ne 2 = resistivity E+ u i B / c = R i = J B nec + R e pressure + m e e Du e Dt inertial + [ ηj] resistive, E+ V B / c = R MHD J B nec + R e V m e M i u e + u i

9 Different levels of MHD depending on what terms are kept in Gen. Ohm's Law 9 Ideal MHD Magnetic flux frozen-into fluid (flow velocity moves with magnetic field lines) Magnetic diffusion and magnetic reconnection cannot occur Resistive MHD Collisions kept in 2-fluid eqns so resistivity appears in MHD in (generalized Ohm's law); magnetic diffusion and reconnection enabled Often no justification for ad hoc resistivity in plasma which is collisionless (physical collisional terms << other terms Ohm's Law) Hall MHD Enables separate diffusion regions for electron fluid and ion fluid n Equivalent to two-fluid theory Magnetic diffusion and reconnection may occur

10 Generalized Ohm's law for electron, ion or MHD fluids and velocity slippage 10 { E+ u B / c} = R, u = u e, u i, V; R = R e, R i, R MHD velocity slippage occurs if R 0: { } B B 2 = u s c E B = R B B 2 B, E 2 = R s Ideal fluid: R = 0 E = u B / c convective field u = ce B / B 2 field lines frozen-in to fluid, E = 0

11 Generalized Ohm's law for electron, ion or MHD fluids and magnetic flux conservation 11 { E+ u B / c} = R, u = u e, u i, V; R = R e, R i, R MHD magnetic flux: { E+ u B / c} = R = t B+ u s B / c Ideal fluid R = 0 E = u B / c convective field t B = u B / c ( ) (using Faraday's Law) ( ) flux conservation (when integrated - next slide) Resistive MHD: R MHD =ηj t B = V B ( ) / c η 2 B Magnetic flux not conserved diffusion of B lines Flow, V, no longer frozen in to B-lines However even with no resistivity, collisionless terms in R MHD can break frozen-in condition (collisionless electron and ion diffusion regions sometimes called dissipative regions even though plasma is collisionless).

12 Frozen-in condition follows from ideal MHD condition 12

13 Hall MHD separate electron and ion diffusion regions (not one fluid but two) 13 Ion diffusion region (can verify with MMS measurements): E+ u e B / c = R e = 0 Electrons still frozen-in (ideal electron fluid) J E+ u i B / c = R i = n e c B J E = n e c B Hall Hall + R e = Hall electric field. Current carried mainly by electrons. In electron diffusion region R e 0 and R i 0. Can verify with MMS measurements.

14 Electron and ion diffusion regions in reconnection geometry 14 ion diffusion region electron diffusion region

15 Central goal of MMS mission is to study 15 diffusion regions in magnetic reconnection. Electron diffusion region is where magnetic reconnection occurs Many physically different measures. None sufficient for reconnection Collisionless kinetic plasma can be treated as electron fluid, ion fluid, MHD Generalized Ohm's laws studied in kinetic simulations of magnetotail reconnection (Goldman, et al, Sp. Sc. Rev, 2015) n "Slippage" of particle motion from field line motions n Conservation of magnetic flux PIC simulations agree with measured (electron) diffusion regions during magnetic reconnection Tail 2D, 3D; Dayside

16 First measurements of electron diffusion region in magnetopause 16

17 17 Extras

18 18 E x not bipolar in asymmetric MP crossing be- cause ion Hall term is smaller on sheath side sheath sphere Gen. Ohm's laws: E+ u e B = R e P e ne D Dt + E+ u i B = R i = J, - nec B. / 0 E x + u iy B z J y B z ne % ' & Hall m e u e e + R e ( * ) Gen Ohm's law verified sheath B z /n smaller in sheath