Diffusion in fully ionized plasma

In a fully ionized plasma, where all atoms have been stripped of their electrons, the diffusion of charged particles is primarily driven by random collisions and interactions between particles. The diffusion coefficient in such a plasma can be described using classical diffusion theory.

The first and second single fluid Magnetohydrodynamics are

In the absence of gravity equation 1 and 2 becomes for steady state, ∂v\∂t=0.

The parallel component of equation 2 or 4 is

Perpendicular component is found by taking cross product by B

The flux associated with diffusion is

Hence, the diffusion coefficient is called classical diffusion coefficient for a fully ionized plasma.

The equation of continuity

Here,

∂n/∂t represents the rate of change of the number density with respect to time.
D is the diffusion coefficient, which characterizes how fast particles diffuse in the plasma.
A is a constant that determines the strength of the diffusion process.
∇² represents the Laplacian operator applied to the number density, capturing the spatial variation of the number density.

The equation states that the rate of change of the square of the number density (∂n/∂t) is proportional to the Laplacian of the number density squared (∇²n²) with a proportionality constant given by A.

This equation implies that the diffusion of the square of the number density is taking place. The diffusion process is nonlinear because it depends on the square of the number density itself. This type of equation can arise in certain physical systems where nonlinear diffusion phenomena occur.

Solving this nonlinear diffusion equation would involve finding the spatial distribution of the number density as a function of time, considering the diffusion process governed by the given equation. The specific behaviour and solutions of this equation would depend on the boundary conditions, initial conditions, and other factors specific to the system being studied.

Time dependent solutions: By the method of separation of variables

The spatial part is different to solve n ∝ T ∝ 𝜏

This shows that density decays as 1/t as in the case of recombination.

Time independent solution:

There is one case in which diffusion can be solved simply. Consider a long plasma column with a source on the axis which maintains a steady state as plasma is lost by radial diffusion and recombination. The continuity equation is

For time independent case,

- A∇²n² = ∝n² ...........12

In phase geometry,

with solution

It's important to note that the actual solution will be specific to the boundary and initial conditions of the problem, as well as the specific diffusion coefficient and other plasma parameters. Analytical solutions are not always possible, and numerical methods may be required to solve the diffusion equation in more complex cases.

This note is a part of the Physics Repository.