Parker instability

Parker instability is an undular magnetic buoyancy instability of a gravitationally stratified atmosphere supported partly by magnetic pressure. A bent horizontal magnetic field can permit gas to drain along its magnetic field lines, leaving buoyant rising regions. Magnetic tension resists bending, so wavelengths and boundary conditions affect the instability threshold. For a magnetized isothermal atmosphere, the unrestricted-wavevector threshold is given by the magnetic buoyancy energy criterion.

At nonzero along-field wavenumber, put $q=i{k}_y{\xi }_y$, $P=\gamma p$ and $C=B^2/{\mu }_0$. After pressure minimization the energy density completes to $CP/(P+C)|q-\rho g{\xi }_z/P{|}^2+[C{k}_y^2-g{\rho }^{\prime }-{\rho }^2{g}^2/P]|{\xi }_z{|}^2$. Choosing the square to vanish and then taking a sufficiently small nonzero $k_y$ gives negative energy exactly when the displayed strict criterion holds somewhere. A fixed prescribed wavenumber retains the stabilizing $C{k}_y^2$ tension term; the criterion asserts existence of an allowed nonzero wavenumber. The associated along-field displacement lets material drain while magnetic flux bends.

For the magnetized isothermal atmosphere, set $a=1/(2H)$, $s=v_s^2>0$, $b=v_a^2>0$ and $\Delta =i{k}_x{\xi }_x+i{k}_y{\xi }_y+(i{k}_z+a){\xi }_z$. After weighting the displacement by $e^{z/(2H)}$, the restoring Hermitian matrix $\mathsf{K}$ satisfiesThus $gH\le s$ makes the form nonnegative for every wavevector. If $gH>s$, choose $k_z=0$, $k_x\ne 0$, ${\xi }_z=1$, ${\xi }_x=-a/(i{k}_x)$ and ${\xi }_y=g/(si{k}_y)$. The squares vanish, and sufficiently small nonzero $k_y$ gives a negative form. The Rayleigh quotient proves instability when these wavelengths are admissible. Since $s=\gamma c_s^2$ and $\beta =2c_s^2/b$, the condition is precisely $\beta (\gamma -1)<1$.

For a horizontal field, set $C=B^2/{\mu }_0$. Static force balance gives $\delta \Pi =\rho g{\xi }_z-(\gamma p+C)\nabla \cdot \xi +Ci{k}_y{\xi }_y$. Given smooth compactly supported vertical and along-field displacements, the displayed choice makes total-pressure perturbation zero while ${\xi }_x=O(k_x^{-1})$. The nonnegative pressure and transverse-tension terms vanish as $|k_x|\to \infty$. This constructs the infimum needed for the magnetohydrodynamic energy principle; a strictly negative limit is already negative at some finite wavenumber.