Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 75 / 2

For the theta pinch, the electric current is azimuthal:Its Lorentz force is ${\mathbf{j}}_0\times {\mathbf{B}}_0/c=-B_0{B}_0^{\prime }\hat{\mathbf{r}}/(4\pi )$. Static radial balance with the pressure gradient therefore givesThe constant is the equilibrium magnetohydrodynamic total pressure.

Suppress the common Fourier mode factor and define the transverse and full divergences of the fluid Lagrangian displacement byThe perturbed magnetic field is $\mathbf{Q}=({\mathbf{B}}_0\cdot \nabla )\mathbf{\xi }-(\mathbf{\xi }\cdot \nabla ){\mathbf{B}}_0-{\mathbf{B}}_{0}D$. Its components areThus the magnetic part of the energy integrand isUse $p_0^{\prime }=-B_0{B}_0^{\prime }/(4\pi )$ from force balance. The pressure-gradient cross term isThe current term has the azimuthal cross-product component ${\xi }_z^{\ast }{Q}_r-{\xi }_r^{\ast }{Q}_z$, givingEvery term involving $B_0^{\prime }$ cancels in the sum, including the possibly complex ${\xi }_z^{\ast }{\xi }_r$ terms. The magnetohydrodynamic energy principle therefore reduces exactly toFor physical $p_0\ge 0$ and positive adiabatic index $\gamma$, every surviving term is nonnegative. On the admissible displacement domain for the supplied functional, squared ideal-mode frequencies cannot be negative. This is the positive ideal energy of a theta pinch, proving absence of exponentially growing ideal modes. The qualification is stable or marginal, not strictly positive energy for every motion: at $k=0$, suitable incompressible displacements can have zero energy. As usual, the energy principle presumes regularity and boundary conditions eliminating unwanted surface terms; additional external systems are not part of the given functional.