Kinetic plasma description Distribution function Boltzmann and Vlaso equations Soling the Vlaso equation Examples of distribution functions plasma element t 1 r t 2 r Different leels of plasma description Exact microphysical description follow all particles and calculate the resulting fields practically impossible in plasma useful in strong external fields, e.g., acceleration of indiidual particles in predescribed fields Kinetic theory (this lecture) consider particle distribution functions Boltzmann and Vlaso equations Macroscopic theory (next seeral lectures) calculate macroscopic ariables (density, flux, pressure, temperature, ) from the distribution function seeral different approaches, e.g., magnetohydrodynamics (MHD) 1
d 2D Distribution function (x,) phase space d 3 6D (r,) dx A plasma particle (i) is at time t in location x d 3 r and has elocity r The distribution function gies the particle number density in the (r,) phase space element dxdydzd x d y d z at time t The units of f : olume 1 x (olume of elocity space) 1 = s 3 m 6 Normalization: total number of particles Sometimes (in particular in mathematical physics) the distribution function is normalized to 1. Aerage density: ; density at location r: Example: Maxwellian distribution Exercise: Integrate this oer the 3D elocity space to get n Hint: thermal elocity (elocity spread) 2
Velocity moments of f Density is the zeroth moment; [n] = m 3 The first moment: Particle flux; [] = m 2 s 1 Aerage elocity = flux/density, [V] = m s 1 DO NOT EVER MIX V(r,t) and (t)!! Electric current density, [J] = C m 2 s 1 = A m 2 Pressure tensor Pressure and temperature from the second elocity moments dyadic product tensor If where is the unit tensor, we find the scalar pressure introducing the temperature Assume V = 0: T K.E. Thus we can calculate a temperature also in non-maxwellian plasma! Magnetic pressure (i.e. magnetic energy density) Plasma beta B dominates oer plasma plasma dominates oer B thermal pressure / magnetic pressure 3 rd elocity moment heat flux (temperature x elocity), etc. to higher orders 3
Vlaso and Boltzmann equations equation(s) of motion for f plasma element Each point in the element moes according to t 1 r t 2 r Let V be some phase space olume (6D) containing particles Conseration of particles in a olume moing with the particles gies Diergence theorem in 6D space ds 5D surface element in 6D space The conseration law is independent of the phase space olume selected If F F() Coulomb force and graitation OK, but the magnetic force is Fortunately cf. the analogous situation in the deriation of Lagrangian for the EM potential! Vlaso equation (VE) Compare with the Boltzmann equation in statistical physics (BE) Boltzmann deried for strong short-range collisions In plasmas most collisions are long-range small-angle collisions. They are taken care by the aerage Lorentz force term Ludwig Boltzmann VE is often called collisionless Boltzmann equation (M. Rosenbluth: It is actually a Bolzmann-less collision equation!) large-angle collisions only e.g., charge s. neutral 4
On the solutions of Vlaso equation (products of E & B with f ) E and B must satisfy Maxwell s equations, where and J depend on f the Vlaso equation is non-linear Consider the most simple case: - free electrons - B = 0 : 1-dimensional; 2D phase space (x,) - small perturbation in E and no background E: E 0 = 0 (in order to linearize) Assume a harmonic perturbation (plane wae) Now VE is and As f 1 is small, we can linearize it, i.e., take 1 st order terms only Recall the linearization of the electron continuity equation 0 0 ( u 0 = 0 electrons are assumed cold ) 0 2 nd order The same for the Vlaso equation Now the only Maxwell eq. is in 1D: 5
Around 1940 Vlaso tried to sole the equation with a plane wae assumption for f 1 as well (cf. Fourier-transform method to sole partial differential equations) Linearized VE Diiding by we get the dispersion equation dispersion function or dielectric function; note the formal similarity with dielectrics: Vlaso could not figure out how to handle the pole in the integral In 1945 Vlaso presented a solution at the long waelength limit using the expansion: Assume a 1D Maxwellian background The two leading terms of the dispersion equation are now where thermal speed plasma frequency with the solution Langmuir wae Thus in finite temperature the plasma oscillation propagates as a wae 6
In 1946 Le Landau found the way to handle the pole at He used the Fourier method in space but treated the problem as an initial alue problem and used Laplace transform in time. (see some of the adanced plasma physics books or attend the course Adanced Space Physics in the spring 2012) The solution for E is: For Maxwellian f 0 < 0 and the wae is damped: Landau damping Le Landau Fast electrons speed up the wae (energy to the wae wae growth) Slow electrons are pushed by the wae (energy to the electrons damping) In Maxwellian plasma there are more slow than fast particles net damping Examples of distribution functions Maxwellian Maxwellian in a frame of reference that moes with elocity V 0 Anisotropic (pancake) distribution ( B) Can also be cigar-shaped (elongated in the direction of B) Drifting Maxwellian 7
Magnetic field-aligned beam (e.g., particles causing the aurora): Loss-cone distribution in a magnetic bottle: In reality plasma distributions are not really Maxwellian. The next approximation: Maxwellian with a high-energy tail The kappa distribution The tail follows a power law flux Maxwellian distribution Kappa distribution energy -function energy at the peak of the distribution Obsered particle distributions often resemble kappa distributions; a signature that non-thermal acceleration has taken place somewhere 8
Distribution as a function of energy Consider a Maxwellian distribution where How to replace with a function of W? (U is here just a placeholder for a elocity independent potential) Howeer, distribution functions are not measured directly. The measured quantity is the particle flux to the detector is the differential particle flux per unit area for gien energy, pitch-angle and location. The particle flux thorugh the surface is elocity normal to the surface Consider particles in the elocity interal d and arrie in the solid angle d Then the number density of particles moing with elocity in the unit phase space olume is Multiplyingthis with, we get the for the differential flux the relation As we hae the relationship between the differential flux and distribution function Recall the representation of Maxwellian and the kappa distribution as flux s. energy flux Maxwellian distribution Kappa distribution energy 9