26. Non-linear effects in plasma

Transcription

1 Phys780: Plasma Physics Lecture 26. Non-linear effects. Collisionless shocks Non-linear effects in plasma Collisionless shocks ([1], p , [6], p , ; [4], p ) Collisionless shocks A typical example of collisionless shocks is the Earth s bow shock. When the supersonic solar-wind flow meets the Earth s atmosphere, a shock is launched upstream of the magnetosphere. A shock represents a discontinuous jump in the bulk parameters of the flow. At the shock the solar-wind plasma is slowed down and heated as it passes through the shock layer, transforming a supersonic flow to subsonic flow. The subsonic flow is then deflected around the magnetosphere. The bow shock represents a collisionless shock because the Coulomb

2 Phys780: Plasma Physics Lecture 26. Non-linear effects. Collisionless shocks.. 2 mean free path is about 1 AU (Earth-Sun distance) but the shock thickness is about 100 km. Figure1: SchematicpictureoftheEarth sbowshock, andtwotypesofshocks: parallel and perpendicular.

3 Phys780: Plasma Physics Lecture 26. Non-linear effects. Collisionless shocks.. 3 Figure 2: Variations of magnetic field in the parallel (top panel) and perpendicular (bottom panel) shocks.

4 Phys780: Plasma Physics Lecture 26. Non-linear effects. Collisionless shocks.. 4 We consider the structure of the perpendicular shock with B in the z-direction, and the upstream velocity v 1 in the x-direction. The structure can be described in the two-fluid model. It is adequate for describing the shock structure but unable to provide the necessary dissipation, which we will treat by adding an effective damping. Figure 3: Variables in the perpendicular shock model, v 1 is upstream velocity of electrons and ions, B 1 is upstream magnetic field.

5 Phys780: Plasma Physics Lecture 26. Non-linear effects. Collisionless shocks.. 5 The two-fluid equations in the shock wave frame are: v x,α v α x = q α m α x (n α v α ) = 0 ( E + v α B c where α is the species index i or e; q e = e,q i = e,m e = m,m i = M. B = 4π c j ) In our geometry E = 1 c B = 0 E = 0 B = (0,0,B) B t v = (v x,v y,0)

6 Phys780: Plasma Physics Lecture 26. Non-linear effects. Collisionless shocks.. 6 E = (E x,e y,0) and all parameters depend only on x (plane wave). In Maxwell equation we neglected the displacement current because we are dealing with low frequency phenomena. We consider a hydrogen plasma with n e n i. Then, from the continuity equations we get: dn e v xe dn i v xi = 0 = 0 v xi = v xe = v 1 and nv x = n 1 v 1 where n 1 and v 1 are the upstream density and velocity (the same for

7 Phys780: Plasma Physics Lecture 26. Non-linear effects. Collisionless shocks.. 7 electrons and ions). The y components of the momentum equations are: dv ey v x = e ( E y v ) xb m c dv iy v x = e ( E y v ) xb M c Adding these equations we obtain: Hence, dv ey mv x +Mv dv iy x = 0 mv ey = Mv iy v iy = m M v ey Since the ion velocity in the y direction is much smaller than the electron velocity this results in a large transverse current j y which supports the change in B across the shock front.

8 Phys780: Plasma Physics Lecture 26. Non-linear effects. Collisionless shocks.. 8 The x components of the momentum equations are dv x v x = e ( E x + v ) eyb m c dv x v x = e ( E x + v ) iyb = e ( E x m M c M M Subtracting and neglecting terms of order m/m) we get: E x = v eyb c v ey B c This means that the electrons are in the force balance in the x-direction. Substituting this in the x-component of the momentum equation for ions we obtains v x dv x = eb Mc v ey This shows that the Lorentz force on electrons ( ev y B/c) is transferred to the ions through the electric field. )

9 Phys780: Plasma Physics Lecture 26. Non-linear effects. Collisionless shocks.. 9 The y-component of the Ampere s law in Maxwell equations: gives db = 4π c j y = 4π c env ey v ey = c 4πen Thus the x-component of the momentum equation for ions becomes: db or using the continuity equation nv x dv x = 1 4πM BdB = 1 8πM db 2 n 1 v 1 dv x = 1 8πM and integrating over x from to x: v x = v 1 db 2 1 8πMn 1 v 1 (B 2 B 2 1)

10 Phys780: Plasma Physics Lecture 26. Non-linear effects. Collisionless shocks.. 10 d Finally, we apply operator: v x to the y-component of the Ampere s law: 1 db n = 4πe c v ey ( ) d 1 db v x = 4πe n c v dv ey x Using n = n 1 v 1 /v x dv ey and the equation for v x we obtain: ( ) d db v x v x = 4πe2 n 1 v 1 mc ( E y + v ) xb c For the electric field E y, from the induction equation E = 0 we have de y = 0

11 Phys780: Plasma Physics Lecture 26. Non-linear effects. Collisionless shocks.. 11 E y = const = v 1B 1 c because dv iy / = 0 at x =. For simplicity we now assume that B 1 is small and neglect it. Then, we obtain: ( ) d db v x v x = 4πe2 n 1 v 1 mc 2 v x B ω2 p c 2 v 1v x B and substituting the equation for v x we get: ( ) ( d db v x v x = ω2 p c 2 v2 1B 1 Defining a new variable τ by ω2 p c 2 v2 1B ( 1 λb 2) dτ = v x v 1 B 2 8πMn 1 v 2 1 )

12 Phys780: Plasma Physics Lecture 26. Non-linear effects. Collisionless shocks.. 12 and using we get where d v x = v d 1 dτ d 2 B dτ 2 = ω2 p c 2 B( 1 λb 2) dφ db Φ = ω2 p B 2 c 2 2 ) (1 λb2 2 We can make analogy with a particle moving along coordinate B with velocity db/dτ in potential Φ.

13 Phys780: Plasma Physics Lecture 26. Non-linear effects. Collisionless shocks.. 13 Figure 4: Potential function Φ(B).

14 Phys780: Plasma Physics Lecture 26. Non-linear effects. Collisionless shocks.. 14 Figure 5: Soliton solution. We can see that the particle coming to this potential is reflected back and never comes back. The solution of this equation represents a solitary pulse, soliton.

15 Phys780: Plasma Physics Lecture 26. Non-linear effects. Collisionless shocks.. 15 To obtain a shock-like solution we have to add a dissipative term into the momentum equation due to microturbulence. v x d ( ) db v x = 4πen 1v 1 c Then the final equation has the form: d 2 B dτ 2 = dφ db ν e db v 1 dτ ( ee y m + ev ) xb mc ν ev y The last term corresponds to a loss of energy of our hypothetical particle due to friction. Thus, this particle will be trapped and move to the bottom of the potential well. This corresponds to the collisionless shock.

16 Phys780: Plasma Physics Lecture 26. Non-linear effects. Collisionless shocks.. 16 Figure 6: The damped solution corresponding to collisionless shock.

17 Phys780: Plasma Physics Lecture 26. Non-linear effects. Collisionless shocks.. 17 Figure 7: The structure and particle motion in a perpendicular shock.

18 Phys780: Plasma Physics Lecture 26. Non-linear effects. Collisionless shocks.. 18 The structure and particle motion in collisionless shocks is quite complicated. In the absence of collisions to provide a deceleration of the upstream solar-wind ions, electric field must be responsible for their slowing down and deflection. Some ions suffer reflection from the sharp gradient and gyrate back into the upstream region before passing downstream. These reflected ions sometimes cause a broad foot in the magnetic profile about an ion gyro-radius in width.