Hi there,
I just started to get interested in the Buneman instability and read the original paper of Buneman "Dissipation of Currents in Ionized Media".
However, when I recalculated the derivations of Buneman, I got stucked on page 2 at the top, where it reads:
"\beta u is real for all \omega of the form (\omega_e \omega_i cos(\theta))^1/3 exp(i \theta)"
were \beta is the wave vector and u the drift velocity of the electrons \omega_{e,i} are the plasma frequencies of the particles and \theta is the angle in the complex plane.
So, two questions:
1. Why does the product of \beta and u have to be real (I would guess this would also work for complex \beta u where the imaginary part describes growth or damping, wouldn't it?)
2. How does Buneman come up with this expicit term above?
These are probably very trivial questions but at the moment I have to admit to be clueless - maybe there is someone who also read the paper and knows the answer.
Thanks in advance
Hello Johannes,
I have read the paper some time ago and I remember bits and pieces of it.
1) β is the wave vector and in the text it is stated that β is assumed to be a real quantity. Likewise the velocity u is also a real quantity. This means that the dot product of the two vectors also has to be real.
2) Now, ω in general is a complex number but it has to satisfy equation (2). This means that the right hand side of equation two must be real because we know that the left hand side is real.
I get Buneman's result using a bit of geometry in the complex plane. I express ω in its polar representation ω=a eiθ with real number a and θ. Then I add the three terms of the RHS individually. After having added the terms, I must end up on the real axis again. The first term is the real electron plasma frequency which doesn't move away from the real axis so we can ignore it. The second and third term create two sides of a triangle with its base on the real axis. The angles of the triangle at the base are θ and 2θ. The two sides are a and b=ωpe ωpi2 / 2a2. Using the sine rule and sin2θ = 2sinθ cosθ gives the result.
I have attached an image of my quick calculation to make things clearer.
wow - thanks, that helps indeed a lot (it still doesn't explain, why \beta has to be real physically, because I think it might as well be complex, but then I will just swallow this assumption of Buneman - than it's clear that a real quantity times a real quantity has to be real again) but you nicely solved my understanding problem where this ominous term came from - sometimes I just don't see the wood because of all the trees :)
thank you very much again.
\beta is assumed real to solve the initial-value problem in a uniform plasma. The "initial value" here is trial small-amplitude plane wave.
Yuriy is correct. A complex wave number would imply that, at time 0, the wave grows exponentially in space, either as x tends to infinity or -infinity. This is not the initial condition that you want to consider when looking at small perturbations.
You may try this article which is more simple.
http://arxiv.org/pdf/1308.5952.pdf
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