Effects of DiscretizationP249 - Fall 2010 10/12 - 10/14

Dan Fulton

Aliasing

Force invariant under displacement.

Periodic variance depending on grid location.

Let force at x2 due to x1 be F(x1, x2).

F(x1, x2) = F(x2-x1) F(x1, x2) = F(x2-x1, .5(x2+x1))

x1 x2 x1 x2

Aliasing (cont.)

Aliasing (cont.)In actuality, force at a given point is due to forces from all points in the domain. Total force is the integral of forces acting from point to point over the whole domain. We can also take the transform of the total force.

So forces of wavenumber, k, are coupled (or aliased) to forces with wavenumbers differing by an integer multiple of kg .

If |k| << |kg| then p=0 term will be largest.

Effects of Spatial GridWe are dealing with charge, potential, field, and force in k-space. The fourier transform of quantities on the grid are periodic as:

E(k-pkg) = E(k)

Intuitively, the severity of aliasing effect will depend on the shape function, S(x) used to gather/scatter grid quantities. (Birdsall 8.6 p164 for formal argument)

xj-1 xj xj+1 xj-1 xj xj+1xj-1 xj xj+1

S(x)

Effects of Finite Timesteps

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Dispersion Relation

For discrete time steps…

Dispersion Relation

For magnetized plasma, including effects of both discrete spatial and time steps.

Zero-order orbits.

Find fields along orbits and sub back into difference eqns.

Kinetic Theory of Fluctuation, Noise, and Collisions

• Using PIC method, just have a sampling of particles.

• Understanding statistics of fluctuations in this sampling is important.

Fluctuation SpectrumStart with fourier transformed number density using a periodic delta-function

To get fluctuating charge density and energy density spectrums:

Limiting Cases• Fluctuation-dissipation theory (For Hamiltonian models)

• Spatial Spectrum (integrate prev. over )

Limiting Cases (cont.)•

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Velocity DiffusionWith leapfrog integration we get…

Using the force along the unperturbed orbit for acceleration we can calculate…

Velocity DragFind ensemble average change in velocity. Along zero-order orbit we get 0. Evaluating next order we get:

Combining above with velocity diffusion, we can express as:

Distortion of plasma by test particle also creates drag. Treat plasma as Vlasov gas and assume particle moves at constant speed.

Kinetic EquationCombine velocity drag and diffusion terms in Fokker-Planck equation to obtain total kinetic equation.