= Isothermal interchange criterion for magnetic buoyancy
{title2=$B\,d(B/\rho)/dz<0$}
A horizontal <flux tube> displaced without bending preserves $B/\rho$ by <magnetic flux freezing>. If thermal exchange keeps its temperature equal to its new surroundings, lateral <magnetic pressure> balance gives $(c_T^2+v_A^2)(\rho_i'-\rho')=\rho B(B/\rho)'/\mu_0$. Here $c_T^2=p/\rho$, $v_A^2=B^2/(\mu_0\rho)$, and primes denote the response to a small vertical displacement from an equilibrium. Its linear <buoyancy> acceleration is $-gB(B/\rho)'\xi/[\mu_0(c_T^2+v_A^2)]$. Thus local interchange is unstable when $B(B/\rho)'<0$, equivalently $(B^2/\rho^2)'<0$. This fast-thermal-exchange criterion differs from the adiabatic <localized interchange criterion for a magnetized atmosphere>.
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