Boundary Conditions

Mathematically, a well-posed problem requires both governing equations and a complete set of boundary conditions (the Cauchy data for the problem). The most common boundary conditions we use are perfectly conducting walls (flux surfaces) or a vacuum region.

Perfectly Conducting Wall

For the case where the plasma extends out to a perfectly conducting (impermeable) wall. Perfectly conducting walls do not support tangential electric field:

$\left. \vec E_t \right|_{wall} = 0 \quad \rightarrow \quad \left. \vu n \cross \vec E \right| _{wall} = 0$

Applying Faraday's law at the wall,

$\left. \vu n \cdot \pdv{\vec B}{t} \right|_{wall} = \left. - \vu n \cdot \curl \vec E \right|_{wall} = \left. \div (\vu n \cross \vec E) \right| _{wall} = 0$

$\pdv{}{t} \vu n \cdot \vec B |_{wall} = 0$

If initially there is no normal magnetic field, then

$\vu n \cdot \vec B|_{wall} = 0 \quad \text{if initially true}$

And of course, for an impermeable wall,

$\vu n \cdot \vec v |_{wall} = 0$

Is this a sufficient set of boundary conditions? Think back to the governing equations in conservation form

$\pdv{}{t} \vec Q + \div \vec F = 0$

The boundary conditions come into play when defining $\vec F$ at the boundary. In particular, we need to know what $\dd \vec S \cdot \vec F |_{wall}$ is. In our governing equations, this will involve $\vec E$, $\vec B$, and $\vec v$.

Insulating Boundary

As a slight modification, an insulating boundary can have a tangential electric field. Consider a simple geometry of parallel electrodes with an insulating wall between them.

From Ohm's law

$\vec E + \vec v \cross \vec B = 0$

so the only way an electric field tangential to the wall can exist is if $\vu n \cdot \vec v \neq 0$.

For either a perfectly conducting or an insulating boundary, the other variables are arbitrary: $\rho$ , $p$, $\vec v_t$, $\vec B_t$.

Vacuum Region

The plasma (radius $R_p$) is supported by a region of vacuum out to a perfectly conducting wall at some radius $R_w$. We assume that there is no plasma in the vacuum region. The governing equations in vacuum are just Maxwell's equations

$\curl \vec B_{vac} = 0 \qquad \text{and} \qquad \div \vec B_{vac} = 0$

At the wall,

$\vu n \cross \vec E |_{wall} = 0$

$\left. \vu n \cdot \pdv{\vec B}{t} \right|_{wall} = 0$

What happens at the plasma-vacuum interface? We need to specify jump conditions and continuity conditions. Let's use square brackets to signify a jump:

$\left[ X \right] = \left. X \right|_{R_p + dr} - \left. X \right|_{R_p - dr}$

The normal magnetic field has to be continuous.

$[\vu n \cdot \vec B]_{R_p} = 0$

The tangential magnetic field jump is given by the surface current density at the jump

$\left[ \vu n \cross \vec B \right] _{R_p} = \mu_0 \vec K$

Integrating $\grad_\perp (p + \frac{B^2}{2 \mu_0}) = \frac{B^2}{\mu_0} \vec \kappa$ over a differential volume across the surface gives

$\left[ p + \frac{B^2}{2 \mu_0} \right] _{R_p} = 0$

The plasma shape is determined self-consistently by the wall shape and surface current. This is a free-boundary problem. Another option is to specify the plasma shape, and then determine the required wall shape. This is a fixed-boundary problem.

The most realistic case includes externally applied magnetic fields coming from source coils, perhaps computed by Biot-Savart law. The vacuum magnetic field is then $\vec B_{vac} = \vec B_{ext} + \vec B_{plasma}$. The crazy coil shapes in the stellarator design come from the 3D geometry computations solving this problem.

Conservation of Magnetic Flux ("Frozen-In" Flux)

Locally, $\vec E + \vec v \cross \vec B = 0$ with Faraday's law

$\pdv{B}{t} = - \curl \vec E = - B \div \vec v + \vec B \cdot \grad \vec v - \vec v \cdot \grad B$

From the continuity equation,

$\pdv{\rho}{t} + \vec v \cdot \grad \rho = - \rho \div \vec v$

Combining we find that

$\dv{\vec B}{t} = \frac{\vec B}{\rho} \dv{\rho}{t} + \vec B \cdot \grad \vec v$

$\rightarrow \dv{}{t} \left( \frac{\vec B}{\rho} \right) = \frac{\vec B}{\rho} \cdot \grad \vec v$

This says that the field and plasma density move together. Locally, if the magnetic field increases then mass density increases, such that the ratio $\vec B / \rho$ remains constant. In the direction parallel to the magnetic field we have a term that involves field line twisting, which is a bit more complicated, but in the perpendicular direction

$\dv{}{t} \left( \frac{\vec B}{\rho} \right) _\perp = 0$

If we consider globally the magnetic flux through a moving surface S at velocity $\vec u$. The magnetic flux penetrating the surface is

$\Psi = \int \vec B \cdot \dd \vec S$

or

$\dv{\Psi}{t} = \int \dv{\vec B}{t} \cdot \vu n \dd S$

$= \int \pdv{\vec B}{t}\cdot \vu n \dd S + \oint \vec B \cross \vec u \dd \vec l$

Using Faraday's law

$\dv{\Psi}{t} = \int - \curl \vec E \cdot \vu n \dd S + \oint \vec B \cross \vec u \cdot \dd \vec l$

$= \oint (- \vec E + \vec B \cross \vec u) \cdot \dd \vec l$

Using the electric field from Ohm's law

$\dv{\Psi}{t} = \oint(\vec v - \vec u) \cross \vec B \cdot \dd \vec l$

This tells us that if the surface moves with the plasma $\vec u = \vec v$ then

$\dv{\Psi}{t} = 0$

the flux through the surface is constant, and the flux is a constant of the topology. This is a direct consequence of ideal MHD. If we add even a small amount of resistivity, we dramatically alter the results in a process called "tearing" where the magnetic field "tears" and reconnects with itself.