Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 314 / 4 / c

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- Solution c

Solution

Although the Lorentz force can exchange energy between poloidal and toroidal motion, it does no net work in the frame rotating with a magnetic surface. Dotting steady momentum with the poloidal field and using the mass-loading, isorotation, and angular-momentum invariants combines the magnetic work with the toroidal kinetic term to give

B_{p} \cdot \nabla ϵ = 0.

Therefore

ϵ (ψ) = \frac{1}{2} u_{p}^{2} + \frac{1}{2} (u_{ϕ} - r ω)^{2} + Φ_{c g} + h

is constant on each magnetic surface. Here

Φ_{c g} = Φ - \frac{1}{2} ω^{2} r^{2}

is the gravitational potential plus the centrifugal potential in the frame corotating at the field-line angular velocity. This is the modified Bernoulli function of the wind.

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