12. MHD Approximation.
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1 Phys780: Plasma Physics Lecture 12. MHD approximation MHD Approximation. ([3], p ) The kinetic equation for the distribution function f( v, r, t) provides the most complete and universal description of plasma processes. In this approach we solve the kinetic equation for f and then calculate the macroscopic properties of plasma by integrating over the velocity distribution, e.g. density: n( r,t) = f( v, r,t)d 3 v and velocity: V( r,t) = 1 n vf( v, r,t)d 3 v < v >. If the macroscopic plasma properties vary much slower compared to the characteristic e-e and i-i collision times, then in small macroscopic
2 Phys780: Plasma Physics Lecture 12. MHD approximation. 2 elements of plasma the distribution functions are Maxwellian because of the frequent Coulomb collisions: ( m ) 3/2exp f M ( v, r,t) = n ( m( v ) V) 2. 2πT 2T In general, plasma properties vary in time and space: n( r,t), V( r,t),t( r,t). If at least one of these parameters of the distribution function is not a constant then such distribution is called local thermodynamic equilibrium LTE. This is reached in a characteristic time of e-e or i-i collisions. When all plasma parameters are constant in space and time and are the same for electrons and ions, then this is called the complete thermodynamic equilibrium. This requires much longer time because τ ei τ ii τ ee. The complete thermodynamic equilibrium is usually not achieved in the presence of external forces.
3 Phys780: Plasma Physics Lecture 12. MHD approximation. 3 If plasma is in LTE then the kinetic theory is not necessary. We describe the plasma processes in terms of n, V, and T. This gives a hydrodynamic theory of plasma. In the first approximation we assume that f = f M. This means that we do not consider transfer processes for these parameters, that is diffusion, viscosity and heat conduction. The deviations from the Maxwellian distribution are important for the energy and momentum transfer processes. The processes of mass, momentum and energy transfer are relatively slow. Hence we can neglect these in the first approximation. Consider a two-component plasma with particles a and b (electrons and ions). We derive equations for moments of the distribution function of particles of type a, f a ( r, v,t)... = 1 (...)fd 3 v n
4 Phys780: Plasma Physics Lecture 12. MHD approximation. 4 starting from the kinetic equation in tensor notations: f a t + (v βf a ) + ( ) Fa,β f a = St aa +St ab x β v β m a where St aa and St ab are the collision terms ( integrals ) for a-a and a-b collisions. Collision integrals describe how many particles disappear and appear at a given elementary volume in the six-dimensional phase space. Here a and b are either e or i. F a,β is the force acting on particles a: F a = e a E + e a c [ v B]. We multiply the kinetic equation by 1, m a v α, and m a v 2 α/2 and integrate over the velocity space (getting zero, first and second moments). The zero moment equation is: n a t + n av aβ x β = 0,
5 Phys780: Plasma Physics Lecture 12. MHD approximation. 5 where the integral over the third term in the kinetic equation is zero because it is reduced (due to the Gauss theorem) to a surface integral for an infinitely distant surface in the velocity space (where f is zero). The integral for the collision term is zero because of conservation of particles. We can write this equation in the vector form: n a t + (n a V a ) = 0. Calculating the first moment, we get: t (m an a V aα )+ (m a n a < v α v β >) x β e a n a (E α + 1 c [ V a B] α ) = R ab,α where R ab,α = mv α St ab d 3 v
6 Phys780: Plasma Physics Lecture 12. MHD approximation. 6 St aa does not contribute to the momentum source because of the conservation of momentum for particle collisions of the same type. Also, because of the conservation of the total momentum in collisions between a and b particles: R ba = R ab. In the third term we used integration by parts: [ v α (ee β + e v β c [ v B] ] β )f d 3 v = [ v β v α (ee β + e c [ v B] ] β )f d 3 v (ee β + e c [ v B] β ) v α fd 3 v. v β }{{} δ αβ
7 Phys780: Plasma Physics Lecture 12. MHD approximation. 7 Finally for the second moment ( (...)m a v 2 /2d 3 v) using again integration by parts we get: ( ma n ) a < v 2 > + ( ma n ) a < v 2 v β > t 2 x β 2 e a n ae V ma v 2 = St ab d 3 v 2 Separate the particle velocity v into macroscopic V and chaotic v components (for simplicity we omit the particle index a): v = v V where < v >= 0 mv 2 = T.
8 Phys780: Plasma Physics Lecture 12. MHD approximation. 8 Then, in the first-moment equation: v α v β = V α V β + v αv β where t (mnv α)+ (mnv α V β )+ mn v x β x αv β FMα = β Consider the first two terms: mn V α t = R α ( F Mα = en E + 1 c [ V B ] ) n + mv α t +mv nv β V α α +mnv β x β x β }{{}}{{} =0,because of continuty eq. α. mn( V ) V The third term is the pressure tensor: P αβ = mv αv βfd 3 v = mn v αv β = pδαβ +π αβ,.
9 Phys780: Plasma Physics Lecture 12. MHD approximation. 9 where p is the scalar pressure: p = mn < v x2 >= mn < v y 2 >= mn < v z 2 >= = 1 3 mn < v 2 >= nt. For real systems the distribution function can be quite anisotropic. In the isotropic case, the momentum equation is: where mn dv α dt = p x α +en[ E + 1 c V B] α +R α, d dt = t +V β x β. This is so-called full (or material) derivative. Now, we consider the second term of the second-moment (energy)
10 Phys780: Plasma Physics Lecture 12. MHD approximation. 10 equation: v 2 2 v β = V 2 2 V β +V α v α v β ( V 2 = ) p V β + V απ αβ 2mn mn + v 2 V β + v v 2 v β Hence, for the energy equation we get: ( nmv 2 + 3nT ) + t [( nmv 2 + 5nT ] )V β +π αβ V α +q β x β v β. = = where q β = = en E V + R V +Q, mv 2 2 v βfd 3 v = mn v 2 2 v β
11 Phys780: Plasma Physics Lecture 12. MHD approximation. 11 is called the heat flux. Q = mv 2 2 St abd 3 v is the heat transfer rate from particles b to particles a. The term en E V is the work of the electric field on charged particles (in unit volume and in unit time); R V is the work of the friction force R. Using mathematical identities and the momentum equation the energy equation can be reduced to: 3 2 ndt dt +p V β x β = q β x β +Q+ R V.
12 Phys780: Plasma Physics Lecture 12. MHD approximation. 12 MHD equations Consider two-fluid continuity and momentum equations for electrons and ions: n e t + (n e V e ) = 0 n i t + (n i V i ) = 0 m e n e d V e dt = p e en e [ E + 1 c V e B]+ R ei m i n i d V i dt = p i +Zen i [ E + 1 c V e B]+ R ie where the drag force R ei = R ie. It can be approximated as R ei m e n e ( V i V e )ν ei. The plasma is described in terms of two mutually penetrating fluids. Quite often plasma can be considered as a single fluid. Consider a
13 Phys780: Plasma Physics Lecture 12. MHD approximation. 13 plasma with singly ionized ions: Z = 1, n e = n i = n. Introduce mass density ρ and hydrodynamic velocity V: ρ = n i m i +m e n e = n(m i +m e ) V = 1 ρ (n im ivi +n e m eve ) = = m i V i +m e Ve m i +m e Then multiplying the continuity equations by m e and m i and taking their sum we get: ρ t + (ρ V) = 0 - the conservation of mass.
14 Phys780: Plasma Physics Lecture 12. MHD approximation. 14 Then, we take sum of the momentum equations: where ρ d V dt = p+ 1 c j B, j = e(n i Vi n e Ve ) = en e ( V i V e ) is the electric current density, p = p i +p e is the total pressure, j B/c is so-called ponderomotive force.
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