Magnetohydrodynamics

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1 Lecture -1 磁流体力学 Magnetohydrodynamics 王赤傅绥燕 中国科学院国家空间科学中心 1

2 课程安排 课程时间内容 等离子体的流体近似 磁流体静力学 等离子体的冻结与磁重联 磁流体力学波 激波与间断面 磁流体动力学不稳定性 习题讲解和复习 期末考试 2

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4 Hierarchy of Plasma Models

5 Fluid Approach MHD Equations

6 Fluid Approach (1/2) The E, B fields are not prescribed but are determined by the positions and motions of the charges themselves. A typical plasma density might be ion-electron pairs per cm 3. If each of these paricles follows a complicated traectory and it is necessary to follow each of these, predicting the plasma s behavior would be a hopeless task. Fortunately, this is not usually necessary, the maority of plasma phenomena observed in real experiments can be explained by a rather crude fluid model.

7 Fluid Approach This model is that used in fluid mechanics, in which the indentiy of the individual particle is neglected, and only the motion of fluid elements is taken into account. Of course, the fluid contains electrical charges. It is surprising that such a model works for plasmas, which general have infrequent collisions. The ion and electron fluids will interact with each other even in the obsense of collisions, because of the E and B fields are generated.

8 Fluid Eq. of Continuity Conservation of matter requires that the number of particles (N) in a given volume (V) can only change if there is a net particle flux across the surface (S) bounding the Volume (V) N t = ndv = nu da = ( nu) dv t V s V Particle flux Divergence theorem Since this must hold for any volume, it means n + ( nu) = 0 t da da

9 Fluid Eq. of Motion A. Neglecting collisions and thermal motion m dv q( ) dt = E + v B If neglect collisions and thermal effects, all particles in a fluid element move together with average velocity u nm du nq( ) dt = E + u B d/dt is to be taken at the position of the fluid element, not very convenient. We wish to have equation for fluid elements fixed in space.

10 Transform to a fixed frame: (convective derivative) dg dt G = + ( u ) G t In a plasma with the fluid velocity u, we can have: nm[ u + ( u ) u] = nq( E + u B) t

11 B. Including Thermal Effect (Pressure) Thermal motion = the random motion of particles in and out of a fluid element == a pressure force should be added in the eq. of motion for a fluid element

12 (1) Consider only the x-component of motion though faces A and B of the fluid element centered at (X 0, 1/2dy, 1/2dz ). The number of particles per volume with velocity Vx n = v f ( v, v, v ) dv dv v x x y z y z The number of particles per second passing through the face A with velocity Vx is N = n v yz v x

13 Each particle carries a momentum. The momentum though face A, from particles with Vx>0, in the fluid centered at 1 1 ( x0 x, y, z) 2 2 P m y z v f dv dv dv n A + = = 0 fdv dv dv x y z (2) It can be written as: 2 x x y z v 2 x = v fdv dv dv 2 x x y z fdv dv dv x y z 1 P = myz n v A x x x 0 The factor ½ comes from the fact that only half the particles in the cube are going toward A.

14 Similarly, the momentum carried out though face B is 1 2 P + = myz n v B x 2 x 0 (3) The net gain in momentum for particles: 1 ( 2 ) ( 2 ) P + P + = myz n v A B x x0 x n v x x = myz( x) ( n vx ) 2 x This results is doubled by the contribution of left-moving particles, since they carry x momentum and also move in the 2 opposite direction relative to the gradient. n v x The total change of momentum of the fluid element at X 0 is therefore 2 ( nmux) xy z = m ( n vx ) xyz t x

15 (4) Let the velocity of a particle be composed by fluid velocity plus random thermal motion. For a 1D Maxwillian distribution, 2 2 x x then t v = u + kt / m v = u + v x x x xr m vxr = kt ( nmux) = m [ n( ux + kt / m)] re-group nm( ux + ux ux ) + mux ( n + nux) = ( nkt ) t x t x x Continuity n + nux = 0 t x nm( ux + ux ux ) = p t x x

16 Generalizing to three dimensions, we have nm u t + u u = p Add the pressure-gradient force with the electromagnetic forces: nm[ u + ( u ) u] = nq( E + u B) P t

17 What we have derived is only a special case: the transfer of x momentum by motion in the x direction; and we have assumed that the fluid is isotropic, so that the same result holds in the y and z directions. But it is possible to transfer y momentum by motion in the x direction. Suppose that the y-velocities of particles at x 0 - Δx and x 0 + Δx were a maximum, and that vy = 0 at x 0. Then particles passing through Faces A and B would bring more y-momentum into the fluid element at x0 than they take out. This would give rise to a shear stress on the fluid element at x 0, which must be described in general by a stress tensor, P, The off-diagonal elements of P are usually associated with viscosity.

18 C. Including Collisions If a neutral gas is present, the charged fluid can exchange momentum with it through collisions. The momentum lost per collision will be proportional to the relative velocity between the changed fluid and the neutral gas (u-u0), where u0 is the velocity of the neutral fluid. u nm[ + ( u ) u] = nq( E + u B) P mn( u u0) t

19 (A)When we took the velocity distribution to be Maxwellian, and therefore the averages of velocity, we assumed that there were collisions. However, the fluid theory is not sensitive to the distribution function, as long as we can use the same average velocity. (B) The other reason for the fluid theory to work for the plasma is that magnetic field and wave particle interaction may play the role as collisions in a certain sense.

20 Fluid Eq. of State We use the thermodynamic equation of state to close the equations. p = C = C / C p v === p n = = p n p n = const

21 Fluid Eq. of State (1) For isothermal compression: (2) Adiabatic compression (T also changes) (3) More general (adiabatic), γ = (2+N)/N where N is the number of degrees of freedom, it is valid for negligible heat flow.

22 Isotropic plasma and Anisotropic plasma A plasma is called isotropic, if its pressure tensor is diagonal with all diagonal elements having the same value P= p I Particle distribution in a plasma often anisotropic.

23

24 MultiFluid Equation: = n q + n q = e( n + Zn ) e e i i e i = n q v + n q v = e( n v + Zn v ) = en ( v v ) e e e i i i e e i i e e i n t + ( n v ) = 0 v n m [ + ( v ) v ] = n q ( E + v B) p kn m ( v vk ) t p n = const Poisson s equation E = 0 B E = t Faraday s law 0 0 E B= + 0 t B = 0 Ampere s law Maxwell s Equations

25

26 Magnetohydrodynamics (MHD) Description of plasma as a multi-fluid system is often too complicated. Simplified to single fluid description (MHD) The MHD model is applicable only when charge separation (e.g. plasma oscillations or electromagnetic waves in plasmas) is negligible. The condition for it is that the length scales should be larger than the Debye length and the time scales larger than the inverse of plasma frequency. Assuming that plasma consists of electrons (m e, q e =-e) and one component of ions (m i, q i =e)

27 Single-Fluid approximation Mass density: = n m + n m m e e i i Velocity: V = ( n m v + n m v ) / ( n m + n m ) e e e i i i e e i i Charge density: = n q + n q q e e i i Current density: Total pressure: p = p + p i e

28 m e (Ce) + m i (Ci) ne ni me + me ( nev e) = 0 m + i m i ( ni vi ) = 0 t t ( mini + mene ) + ( mini vi + mene ve) t ( mini + mene ) mini vi + mene ve = + [( mini + mene ) ] t m n + m n i i e e Mass Conservation m t + ( V) = 0 m

29 q e (Ce) + q i (Ci) n n e i qe + qe ( nev e) = 0 q + i q i ( ni vi ) = 0 t t Charge Conservation t q + = 0

30 (Me) + (Mi) RHS:==== LHS== : mn [ + v ] v t (1) ignore the contribution of electron momentum (m e << m i ) (2) V is approximated as v i Momentum Eq. { n q ( E + v B) p + Fk} = qe + B ( pe + pi ) mn [ + ] m( + ) t v v t V V v n m [ + ( v ) v ] = n q ( E + v B) p kn m ( v vk ) t m [ + V ] V = + qe B p t ( F ei = -F ie, so no net friction)

31 q e m e (Me) + q i m i (Mi) v n m [ + ( v ) v ] = n q ( E + v B) p kn m ( v vk ) t n q q q [ ] [ ( ) ] 2 n q + t v v = E p k m + v B m + F m

32 = n m + n m m e e i i = n q v + n q v = n q ( v v ) e e e i i i e e e i vi ve n q [ t + v ] v n q [ tv ] = niqi + neqe t t For simplicity, we treat v as small and neglect the term v v neqe ve + niqi vi ( ) = ( n m + n m ) ( ) t t n m + n m m e e i i m e e i i n m n q v + n q v n q v + n q v = ( ni + ) ni mi t nm e e n i i + t n mi LHS== : nq [ t + v ] v m ( ) t e e e e e i i i e e e i i i m

33 RHS:==== [I] nq E ( ) E = = nq e e + m neme nimi = nq 2 2 e e qq nimi + neme ( ) E n m n m e i m mm e i e e i i E

34 [II] Left for excise nq 2 2 ve 2 vi v = ne qe + niqi m me mi qeqi neqemi niqime = { ve + vi} m m q q e i i e qq e i = { nimi ve + nemi vi} mm e i qeqi mi me = { neme ve + nimi vi ( + )( qene ve + qini vi )} m m q q e i i e qeqi mi me = { mv ( + ) } m m q q e i i e

35 [III] F = nm( v v ) = F ei ei e e e i ie q m e Fk = ei ( neqe neqi )( ve vi ) m m i qm e e = ei (1 ) qm i i

36 LHS=RHS qeqi mi me m ( ) = [ me + { mv ( + ) } B t m m q q m e i i e qe qi qeme ( p + p ) (1 ) m m q m e i ei e i i i memi 1 mi me E + V B = ( ) + ( + ) B q q t q q e i m m i e qe qi memi qeme memi ( p + p ) (1 ) m m q q q m q q e i ei e i e i m i i e i m

37 General Ohm s Law memi 1 mi me E + V B = ( ) + ( + ) B q q t q q qe qi memi ( pe + pi) m m q q e i e i m q m (1 ) q m m m q q e e e i i i e i m e i m m i e ei Hall term electron inertia, negligible for low enough frequency ~ the resistivity 1 1 me E + V B = + B P + e 2 ne ne ne t

38 General Ohm s Law Last term in J has a coefficient, ignoring m e /m i c.f. 1 The resistivity (σ ei = ) Hence dropping electron inertia, Hall term, pressure term, the Ohm s law becomes:

39 Equation for the State = = e i m nmi m p = const

40 Summary of MHD equatons (I) m Mass Conservation + ( mv) = 0 t Momentum ( + V ) V = E + B t Eq. State p = const m q p m Charge Conservation t q + = 0 Ohm s Law

41 Since the fluid is assumed to be electrically neutral, the charge density ρ q is taken to be zero. We shall not need Poisson s equation because that is taken care of by quasi-neutrality Normally, one uses MHD only for low frequency phenomena, so the Maxwell displacement current can be ignored in comparison with the conduction current.

42 Summary of MHD Equations (II) In order to obtain a compete and self-consistent description of a resistive MHD plasma, the fluid equations and Ohm s law must be combined with Maxwell s equations: Ampere's Law Faraday s law B = μo J B = 0 E B = t Gauss s law E = 0

43 Ideal MHD equations As the collision frequency goes to zero, the condctivity goes to infinity: ideal MHD plasma E + V B = 0 The fluid velocity component perpendicular to B is then give by E x B/ B 2, which is identical to the E x B drift velocity encountered in single particle orbit theory.

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