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

Use perturbations $\mathbf{v},\mathbf{b},p,\vartheta$ for velocity, magnetic field, magnetohydrodynamic total pressure and buoyancy displacement variable. SetThis is advection by the background Keplerian shearing sheet. Subtracting the equilibrium before retaining first-order terms gives the nine linearized equations:The azimuthal coefficient $\Omega /2$ combines the Coriolis acceleration with perturbation advection of the background shear. The $-3\Omega b_x/2$ term is the winding of radial magnetic field by that shear. The tidal force cancels when the equations at a fixed position are subtracted; it does not add a separate Eulerian perturbation force. Magnetic pressure is already included in $p$, leaving only linear magnetic tension. In this normalization $\vartheta$ has dimensions of length because $D_0\vartheta =v_x$.

For an axisymmetric vertical mode of a shearing sheet with real $k\ne 0$, the two divergence constraints give $ik{v}_z=ik{b}_z=0$. The vertical momentum equation then yields $ikp=0$. Thus all three amplitudes vanish:Define the magnetic amplitudes $a_i=b_i/\sqrt{4\pi {\rho }_0}$ and the signed vertical Alfvén velocity $v_A=B_0/\sqrt{4\pi {\rho }_0}$. Write $A=k^2{v}_A^2$. The remaining equations for the normal mode areFor a growing or oscillatory mode with $\sigma \ne 0$, eliminating $a_x,a_y,\vartheta$ givesA nonzero velocity requires the determinant to vanish. After removing its factor $\sigma$, the dynamical dispersion relation isThis is the radially stratified magnetorotational dispersion relation, with $A=k^2{B}_0^2/(4\pi {\rho }_0)$. Keeping the original five-amplitude system instead gives characteristic polynomial $\sigma$ times this quartic. There is also a stationary balanced normal mode; division by $\sigma$ excludes it but loses no exponentially growing mode. At $k=0$ the divergence argument for vanishing vertical components does not apply, so that spatially uniform case must be treated separately.

When the imposed magnetic field vanishes, the dynamical dispersion relation becomesThe nonstationary hydrodynamic branch therefore has ${\sigma }^2=-(N^2+{\Omega }^2)$. The radial Solberg–Høiland instability criterion in this local model isThe radial buoyancy frequency supplies either restoration or a driving force, while the radial epicyclic frequency $\kappa =\Omega$ supplies rotational restoration. Zero sum is marginal, and positive sum gives stable oscillations. The neutral roots do not themselves signify exponential growth.

Because both equilibrium pressure and dimensionless specific entropy decrease outward, their radial gradients have the same sign. The leading minus sign in the expression for $N^2$ therefore makes $N^2<0$: the radial stratification is adverse. For smooth profiles varying over radius $r$, $|{\partial }_{r}P|\sim P/r$ and $|{\partial }_{r}S|\sim 1/r$, soFor a thin disk, the magnitude of this negative radial buoyancy frequency squared is much less than ${\Omega }^2$. Rotation stabilizes the hydrodynamic mode despite the adverse entropy gradient. This estimate assumes gradients on the global radial scale; a sharp thermal feature may be different. It addresses the ideal nondiffusive equations here. Processes such as convective overstability require additional thermal relaxation and are not ruled out by this particular criterion.

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 $(H/r{)}^2$.