A straight cylindrical plasma column is in static equilibrium with and pressure…

PHYSICS · Subjective · Fermi University Physics · Plasma Physics · Fluid and magnetohydrodynamic description · Fluid and magnetohydrodynamic description

Plasma Physics, Ch Holding a plasma still: pinches, flux surfaces and the Grad-Shafranov equation

Problem

A straight cylindrical plasma column is in static equilibrium with B=Bz(r)z^\mathbf B = B_z(r)\hat{\mathbf z} and pressure p(r)p(r) decreasing monotonically outward from the axis, satisfying ∇p=j×B\nabla p = \mathbf j\times\mathbf B with the azimuthal current this requires. Someone argues: "I want to add a small radial current jr(r)j_r(r) on top of the existing azimuthal current, to model a slow radial leakage. Charge continuity for a steady state only requires ∇⋅j=0\nabla\cdot\mathbf j = 0, and a suitable choice of jr(r)j_r(r) together with the existing currents keeps ∇⋅j=0\nabla\cdot\mathbf j = 0 satisfied everywhere. Since force balance ∇p=j×B\nabla p = \mathbf j\times\mathbf B only fixes j\mathbf j through this one vector equation, and continuity is the only other constraint on j\mathbf j, this modified current distribution is still consistent with the same static equilibrium." Find the error in this reasoning.

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