Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 64 / 3 / d / Solution

Let $A=k^2{v}_A^2>0$ and $C=N^2+{\Omega }^2$. Solving the radially stratified magnetorotational dispersion relation as a quadratic in $s={\sigma }^2$ givesBoth roots are real. If $A+N^2-3{\Omega }^2<0$, their product is negative, so exactly one root is positive and gives exponential growth. Conversely, when $A+N^2-3{\Omega }^2\ge 0$, the coefficient $2A+N^2+{\Omega }^2\ge A+4{\Omega }^2>0$, so both roots are nonpositive. Thus the instability criterion for a specified nonzero wavenumber isIn a local continuum of allowed wavenumbers, some unstable mode exists exactly when $N^2<3{\Omega }^2$. A finite disk only permits wavelengths fitting its vertical boundaries, so this existence statement also requires an allowed mode in the interval.

The growing root is $s_{+}$, whose derivative and curvature areSetting the derivative to zero gives the fastest-growing interior mode:For a real, nonzero wavenumber, this expression requires $|N^2+{\Omega }^2|<4{\Omega }^2$, equivalentlyNonnegative right-hand side permits the endpoints, but there the stationary point is at $k=0$, outside the nonzero-wavenumber mode used above. Substituting the interior maximizing value givesThis is the maximum growth rate of radially stratified magnetorotational instability, taking $\Omega >0$. For $N^2=0$ it reduces to the usual $3\Omega /4$, at $k^2{v}_A^2=15{\Omega }^2/16$. Negative $N^2$ enhances growth and extends the unstable band; positive $N^2$ suppresses growth and eventually eliminates it at $N^2\ge 3{\Omega }^2$. Magnetic tension enables angular-momentum exchange between displaced parcels, weakening the rotational stabilization that protected the adverse hydrodynamic stratification.

If $N^2\le -5{\Omega }^2$, the interior formula is no longer the physical maximum: $s_{+}$ decreases for $A>0$, and its supremum as $k\to 0$ is $-N^2-{\Omega }^2$. The corresponding limiting growth rate is $\sqrt{-N^2-{\Omega }^2}$, dominated by the already unstable hydrodynamic branch. At $N^2=-5{\Omega }^2$ this joins continuously to $2\Omega$. For the smooth thin disk estimate $N^2\sim -(H/r{)}^2{\Omega }^2$, the ordinary interior maximum applies and its enhancement over $3\Omega /4$ is only of fractional order $(H/r{)}^2$.