In order to model objects that exsist between the Ion Gyroradius and the Electron Gyroradius, we must make a set of assumptions similar to that used in Electron Magnetohydrodynamic (EMHD) models. We assume that perturbations to the equilibrium are small (order epsilon) and that gradients parallel to the magnetic field are much smaller than those perpendicular to the magnetic field. We also assume that in this picture, the ions are so large that they do not have an opportunity to move on the time scale of the changing perturbations. Thus, any velocities in our model are those associated with the movement of the electrons.

Our model does modify standard EMHD theory by including the particle effect of ions in EM fields on the density. We then use quasineutrality to assume that the ion and electron densities are the same.

To validate my code in 2D, I chose to look at the problem of 2D Electron MHD (EMHD) turbulence and compare my resuts to those obtained by Biskamp, Schwarz, Zeiler, Celani and Drake in March 1999, reported in Physics of Plasmas, Vol.6, p. 751-758. In order to do so, I needed to compare normalizations of codes, etc. The work I did on this can be found here .

---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

1. This is the eta = nu graph, looking at the dissipation rate, Gamma:   [eta/nu vs gamma]
2. This is the eta fixed at 1.0e-8 and nu varied graph:   [nu vs gamma]
3. This is the nu fixed and eta varied graph:   [eta vs gamma]

1. This is beta vs. omega:   beta.ps
2. This is Lz vs. omega:   LZ.ps
3. This is tratio vs omega:   tratio.ps
4. Here is Lx vs omega   Lx.ps
5. Here is Ly vs omega (which should be the same as Lx):   Ly.ps
6. Here is omega*dt vs omega:  
   1. Nothing fancy:   dtold.ps
   2. Here's one with the stability line included:   dt.ps
   3. And here's one with the stability line that also includes zero:   dtstab.ps
7. Here is the kperp vs omega plot using the Gamma0 function (now corrected):   kperp.ps
8. There are a few different views with no log in the y direction:   kperpnology.ps
9. And with no logs anywhere (truncated):   kperpnologs.ps
10. Here is the Gamma0 function(now corrected as well):   gamma0.ps

Descriptions of plasmas exist at both the long and short wavelength extremes. However, the region in-between where large scale perturbations break up into smaller and smaller eddies is not well studied. Using a pseudospectral explicit code, we plan to obtain relations between energy and wave number in this regime. Validation of the linear equations can be done analytically. For the non-linear work, we will need to compare results obtained by others, notably, Jim Drake and colleagues.

Here are a few documents that include my derivations of some of the relevant equations for this project. This is essentially my "scratch paper" work.

LinearWaves:   LinearWaves.pdf
Ideal MHD:   ideal.pdf
Linear EMHD:   EMHD.pdf
EMHD Take 2:   EMHD2.pdf
This is a document I created as I was working to understand the code I inherited from Nuno Loureiro. It includes details such as the derivation of the equations and an explanation of some the normalizations:   NunosMHD.pdf
This includes the same equations Nuno used originally, only it includes terms that makes the equations 3D:   From_2D_to_3D.pdf
Here is a new file that derives my equations, but includes the perturbation in the parallel direction:   VelocityN.pdf
Here is a copy of the source code for my crude boundary finder program:   [bf_2.f90]
This is the accompanying input file:   constants_s1.f90
And the makefile I use to compile it:   makefile.bf