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For $-5{\Omega }^2<N_r^2<3{\Omega }^2$ and a continuum of vertical wavenumbers, the growing branch of the radially stratified magnetorotational dispersion relation is maximized at $k^2{v}_A^2={\Omega }^2-(N_r^2+{\Omega }^2{)}^2/(16{\Omega }^2)$, with ${\sigma }_{\max }=(3{\Omega }^2-N_r^2)/(4\Omega )$. For $N_r^2\le -5{\Omega }^2$ the optimum moves to the $k\to 0$ boundary and the limiting rate is $\sqrt{-N_r^2-{\Omega }^2}$. For $N_r^2\ge 3{\Omega }^2$ there is no exponential instability. Boundaries that discretize the allowed wavenumbers can alter the actual maximum.

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.

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