What does “Frozen-In” Mean?
For highly conductive plasma (conductivity (σ) → ∞), we often say that the plasma is ‘frozen-in’ the magnetic field lines, but what does this mean physically?
Let’s begin by examining the mathematical (forgive me) behavior of the flow.
From Ohm’s Law and Ampere’s Law1
J = σ(E + u × B) = \nabla × \frac{B}{µ_{0}}
Solve for E
E = -u × B + \frac{1}{µ_{0}σ}[\nabla × B]
Take the curl of both sides
\nabla × E = \nabla × [-u × B] + \frac{1}{µ_{0}σ} [ \nabla × [ \nabla × B]]
By Faraday’s Law and the vector identify for \nabla × [ \nabla × F]
- \frac{\partial B}{\partial t} = - \nabla × (u × B) + \frac{1}{µ_{0}σ} [ \nabla [ \nabla \cdot B] - \nabla^2 B]
Recall the Maxwell equation that says that there can be no magnetic monopoles [\nabla \cdot B] = 0 and distribute the negative sign
\frac{\partial B}{\partial t} = \nabla × (u × B) + \frac{1}{µ_{0}σ} [\nabla^2 B]
\frac{\partial B}{\partial t} = \nabla × (u × B) + η\nabla^2 B]
where we have defined η as the magnetic diffusivity (η = = \frac{1}{µ_{0}σ})
So we have now described how the plasma’s magnetic field will evolve over time.
Now we can apply the restriction that σ → ∞ so η → 0, which leaves us with
\frac{\partial B}{\partial t} = \nabla × (u × B)
From the Frozen-In Flux theorem, the magnetic flux must be zero [\frac{\partial [B \cdot A]}{\partial t} = 0] , so let’s take our final time evolution at a high conductivity and apply it over an arbitrary area A.
\frac{\partial B}{\partial t}A = A[\nabla × (u × B)] = 0
\nabla × (u × B) = 0
This is only true is u is parallel to B everywhere (by definition, u × B = 0 if u || B). So from this we have a mathematical description of what it means for the plasma to be ‘frozen-in’ the magnetic field! Specifically, it says that any flow of the plasma perpendicular (a.k.a away) from the magnetic field lines is prohibited!
In short: Frozen-in plasma will follow the magnetic field.
This process can be viewed in reverse (i.e. it is equivalent to say that the plasma is frozen-in to the field and the field is frozen-in to the plasma).
1. Here we will disregard displacement current because the processes in MHD (magneto-hydrodynamics) take a long time with respect to the gyrofrequency and occur on length scales larger than the gyroradious. It is safe to assume the plasma velocity (u) is much smaller than the speed of light, so the displacement current is very small and can be neglected.