4.5 Assignments 1474.4.3 Differences between the two-fluid and MHD descriptions
The two-fluid description shows that the slow mode (finiteEz) appears as either an inertial
or a kinetic Alfvén wave depending on the plasmaβ;the MHD description assumes that
Ez=0for this mode and does not distinguish between inertial and kinetic modes. The
two-fluid description also shows that finiteEzwill give ion acoustic modes in the parallel
direction which are decoupled. The MHD description predicts a so-called sound wave
which differs from the ion acoustic wave because the MHD sound wave doesnot have the
requirement thatTe>> Ti;the MHD sound wave is an artifact for parallel propagation
in a plasma with low collisionality (if the collisions are sufficiently large, then the plasma
would behave like a neutral gas). Then MHD description predicts a coupling between
oblique sound waves via a square root relation (see Eq.(4.85)) which does not exist in the
two-fluid model.
4.5 Assignments
- Plot frequency versus wavenumber for the electron plasma wave and the ion acoustic
wave in an unmagnetized Argon plasma which hasn=10^18 m−^3 ,Te=10eV, and
Ti=1 eV. - Let∆φbe the difference between the phase shift a Helium-Neon laser beam expe-
riences on traversing a given length of vacuum and on traversing the same length of
plasma. What is∆φwhen the laser beam passes through 10 cm of plasma having a
density ofn=10^22 m^3 ?How could this be used as a density diagnostic? - Prove that the electrostatic plasma waveω^2 =ω^2 pe+3k^2 κTe/mecan also be written
as
ω^2 =ω^2 pe(1+3k^2 λ^2 De)
and show over what range ofk^2 λ^2 De the dispersion is valid. Plot the dispersionω(k)
versuskfor both negative and positivek.Next plot on the same graph the electromag-
netic dispersionω^2 =ω^2 pe+k^2 c^2 and show the limits of validity.Plot the ion acoustic
dispersionω^2 =k^2 c^2 s/(1+k^2 λ^2 De) on this graph showing its region of validity. Fi-
nally plot the ion acoustic dispersion with a finite ion temperature. Showthe limits of
validity of the ion acoustic dispersion. - Physical picture of plasma oscillations: Suppose that a plasma is cold and initially
neutral. Consider a spherical volume of this plasma and imagine that a thin shell
of electrons at spherical radiusrhaving thicknessδrmoves radially outward by a
distance equal to its thickness. Suppose further that the ions are infinitely massive
and cannot move. What is the total ion charge acting on the electrons (consider the
charge density and volume of the ions left behind when the electron shell is moved
out)? What is the electric field due to these ions. By considering the force dueto this
electric field on an individual electron in the shell, show that the entire electron shell
will execute simple harmonic motion at the frequencyωpe.If the ions had finite mass
how would you expect the problem to be modified (hint-consider the reduced mass)? - Suppose that an MHD plasma immersed in a uniform magnetic fieldB=B 0 zˆhas