Vlasov - Poisson sytem of equations and Landau damping

Transcription

1 7/1/212 Kinetic theory Vlasov-Poisson syste 1 Vlasov - Poisson syte of equations and Landau daping.5 The kinetic approach to finding the dispersion relation of electrostatic waves uses the Vlasov equation to find the evolution of the distribution function in one diension: E j f f f v a t x v jt where f(x, is the distribution function, x is distance, v is velocity, t is tie and a is acceleration. Poisson 's equation is: de dx nq 1 f dv The Vlasov equation can be written where the ion density is assued to have a constant value n, and q and and have their usual eaning. Df Dt where D/Dt indicates a convective derivative. If we follow a group of particles, the vaue of f(x, reains constant on the trajectory in x,v space. For exaple, if a particle oves to x fro x-dx, and siultaneously to v fro v-d the value that f(x, will have at the new tie and place is the value of f(x-dx, v-d t -d. The following prescription ipleents the change and carries us ward one tie step: f( xvt ) f( x dxv dvt d where dx = v dt, dv = a dt, E is the electric field, and a = qe/. This procedure is repeated to find the evolution of f(x,. Cheng and Knorr (see reference) developed the following ethod that is accurate to second order. The ethod takes a half step in the dx direction, calculates E, takes a full step in the dv direction, and then a half step in the dx direction: dx 1. f tep1 ( xv ) f x v dvt dt 2. E 1 f v 2 tep1 ( xv ) d 3. f tep2 ( xv ) f tep1 ( xv dv) 4. f ( xvt ) f tep2 x dx v 2 where f tep denotes a teporary function. At each step, a cubic spline function is fit to f(x,v), and the spline function is used to find the values at x - dx/2 and v - dv by interpolation. The interpolation adds significantly to the tie needed to find a solution.

2 7/1/212 Kinetic theory Vlasov-Poisson syste 2 Diensionless equations: The following substitutions will ake the Vlasov equation and Poisson's equation diensionless: v v v t x xω p v t qe E nq 2 v t ω v t ω p f v p ε t f λ D ω p 2 where v t is the theral velocity and D is defined but not used. The diensionless equations are: f f v E t x de dx f dv f v x, t Diensionless Vlasov 1 Diensionless Poisson The x and v grids In x, our grid will span 12 Debye lengths: L 12 k Define the grid using 33 points: kax 32 k kax x L Δx x x k kax 1 Δx.375 The v grid will be -3 to + 3 in diensionless units: ax kax ax v 3 6 Δv v v Δv.188 ax 1 The v and x grids ust have the sae nuber of eleents the 2-d spline to work. The electrostatic wave The diensionless wave aplitude will be δ.25 =.3 is is the largest value that does not ake f(x,v) go negative (unphysical). fstart(x k,v ) is the density-odulated distribution function that we use at the start : fstart 1 δcos k 2πx k L e v 2 There is one wavelength in the siulation region. π f is noralized to unity in 1-d. Check noralization to unity: 2 v e Δv π Check ean squared-velocity: 2 v 2 v e Δv π

3 7/1/212 Kinetic theory Vlasov-Poisson syste 3 Plot of f(x,v): x is front to back and v is left to right fstart Poisson's equation: finding E fro f(x,v) Poisson equation finds de/dx at each x k : dedx 1 k fstart Δv k E is found by integrating de/dx, using Sipson's rule, which integrates across an interval using the average of the two derivative values at the ends of the interval. This ethod has second order accuracy. Estart k 1 kax By using the cosine function the density perturbation, we have a zero electric field at the left boundary. Now all the values k >, use Sipson's rule to integrate de/dx: Estart Estart.5 dedx dedx Δx k k1 k k1 Reset range of k: k kax dedx k.2 Estart k.2 Plot of initial E and de/dx values x k

4 7/1/212 Kinetic theory Vlasov-Poisson syste 4 Define the tie grid Our diensionless tie scale is defined by the plasa period. We will follow three oscillations which will put the ending tie at 3 x (2) = 18, approxiately. EndTie 18 For accuracy, we will use a tie step of.25. Δt.25 EndTie The nuber of iterations we will need is: jax ceil jax 72 Δt Periodic boundary conditions x jaxδt 18 If the particle oves to a value of x that is outside the siulations box, we ust use the odulo function od to put the x value into the box. This function will return the reainder after dividing by the length of the box L. If the value of x is less than zero, od will return a negative nuber that is outside our box. So we ust add L to the value of x bee using the od function: This returns a value x that is always between and L. od L x.5v ΔtL k Nuber density of particles in the siulation box We can find the nuber of particles in the box by integrating f(x,v) with dx and dv. The integral is pered by a double suation. We divide by L to find the nuber per unit length. nf ( ) kax 1 L kk 1.5 f f Δv kk kk1 Δx n( fstar For accuracy, we used Sispon's rule when integratring over x. It is not necessary to do this the v integral because the values of f(x,v) are nearly zero at the end points. Total energy of particles in the siulation box For the total kinetic energy W, we integrate (su).5 v 2 with the distribution function: W( f ) kax kk 1.5 f f.5 v kk kk1 2 Δv Δx W( fstar Total energy in the electric field This is found by suing.5 E 2 with dx, using Sipson's rule: Eenergy( Estar kax kk 1 Grid point ination.5.5 Estart kk 2 Estart kk1 2 Δx Eenergy( Estar The spline fitting routine needs to know the values of the x and v grid points, so we stack these vectors into a atrix with two coluns: xv augent( xv )

5 7/1/212 Kinetic theory Vlasov-Poisson syste 5 The progra loop j is tie index, k is distance index, is velocity index M f fstart f1 kaxax f2 kaxax E ax Esave Estart j jax SplineCoeffs ax k kax f1 k SplineCoeffs cspline( xvf) od L x.5v ΔtL k interpsplinecoeffsxvf v k kax ax dedx 1 k k 1 kax cspline( xvf1) f1 Δv k E E.5 dedx dedx Δx k k1 k k1 ax k kax f2 k SplineCoeffs f k Esave x k interpsplinecoeffsxvf1 v ax k kax augent( Esavef) cspline( xvf2) E Δt k od L x.5v ΔtL k interpsplinecoeffsxvf2 v augent( EsaveE) Initialize atrices f, f1, f2 Initialize vectors E, Esave. fit spline to f find f1 fit spline to f1 find de/dx find E k find f2 fit spline to f2 find the new f

6 7/1/212 Kinetic theory Vlasov-Poisson syste 6 Augent(Esave, E) is used to stack the E values at each tie step into an array Esave. Augent (Esave,f) is used to stack the final values f(x,v) into the atrix Esave that has the E values. The first thing we do below is unstack the E values and f(x,v) into separate atrices: This stateent puts the E(x, values into the atrix Ext: Ext subatrix( M kax jax 1) This stateent recovers the values of f(x,v) into a new atrix FinalF: FinalF subatrix( M kaxjax 2cols( M) 1) This subatrix has E(x,: Ext j jax E as a function of x at 3 ties Peak value of E as a fcn of tie Ext k Ext k Ext k Ext kax j x k jδt

7 7/1/212 Kinetic theory Vlasov-Poisson syste 7 Plot of E(x,. Ext Distance is along the horizontal axis and tie is on the vertical axis. The E field "seesaws" as tie increases. This plot is of f(x,v) at the end tie ( FinalF) Check here to see if f has gone negative: If so, the wave aplitude ust be reduced. in( FinalF)

8 7/1/212 Kinetic theory Vlasov-Poisson syste 8 Did the reduced wave energy show up as increased particle energy? The final E field is: Efinal Ext k kjax The change in the E field energy is: because jax is the end tie. Eenergy( Efinal) Eenergy( Estar where: Eenergy( Estar Eenergy( Efinal) The change in the particle energy is: W( FinalF) W( fstar where: W( fstar W( FinalF) 3.6 Energy is poorly conserved because we have used a rather coarse spacing in x, v and t in order to reduce the calculation tie. Try it: Change t fro.25 to.1 and notice that energy conservation is iproved. Modification of the distribution function The daping of the waves flattens the region of f(x,v) where v is near to the phase velocity of the wave. For a nicer plot, we will average the final value of F over all x. 1 faverage kax 1 1 k FinalF k e Copare this to: Fax v 2 π.1 faverage Fax A "bulge" in the distribution function is visible at a velocity near 2.5. Try it: Exaine the E( and calculate the wave frequency. The wavenuber is 2/L. Is the phase velocity /k near to the velocity where there is a bulge in the distribution function? v

9 7/1/212 Kinetic theory Vlasov-Poisson syste 9 k dependence of Landau daping The figures below show that Landau daping is ore severe shorter wavelengths: Daping L = 12 Daping L = Ext kax j 4.5 Ext kax j jt jt Try it: How does the observed daping copare with the daping fro theory? How heavily daped is the wave with L = 6? 4? Reference: C. Z. Cheng and G. Knorr, J. Cop. Phys. 22, 33 (1976).