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hwk3 - needed You can use the cold ion limit You need to...

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Physics 762 Homework #3 Spring ‘11 Dr. Drake 1. Calculate the energy of an ion acoustic wave propagating in a medium with T i = 0 using the general expression for wave energy calculated from kinetic theory W = ∂ω ∂ω | E | 2 8 π . (1) Expand the arguments of the Z function to obtain an explicit fluid-like result. Now linearize the corresponding fluid equations for the sound wave and calculate an energy-like invariant (Hint: start by multiplying the perturbed ion momen- tum equation by the velocity perturbation and integrate over space. Use the other fluid equations to find the quadratic quantity that is an invariant). Use this invariant to interpret to form of the wave energy calculated from Eq. (1). 2. Write the quasilinear equations for the ion acoustic instability with cold ions and Maxwellian electrons with thermal velocity v te drifting with a velocity U < 2 c s / 3. What is the maximum phase speed of the unstable waves? Minimum phase velocity? Hint: the ions remain linear so no equation for the ion distribution function is
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Unformatted text preview: needed. You can use the cold ion limit. You need to calculate the growth rate for the instability based on the local slope of the electron distribution function, which will vary with time. This diﬀers from the earlier assignment which simply had a drifting Maxwellian. 3. Calculate the ﬁnal electron distribution function after the instability has satu-rated. What is the minimum velocity of the plateau? The maximum velocity v ? Why was an upper limit given for U above? 4. Calculate the energy density | E k | 2 / 8 π in the waves as a function of their phase speed after saturation. Show that the wave amplitude goes to zero at either end of the plateau. Sketch the wave energy versus the phase speed of the wave. Hint: the ﬁnal answer is | E k | 2 8 π = 1 3 √ π Δ k k ω 2 ω 2 pi v 2 p ( v p-v g ) v 3 te n m e 2 ( v p-3 U/ 2) 2 , (2) where v p is the wave phase speed, v g = dω/dk is the group velocity and Δ k = 2 π/L is the mode spacing....
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