= Solution
The <magnetorotational instability> extracts energy from <differential rotation> through <magnetic tension>. It can destabilize a disk whose specific <angular momentum> increases outward, even though such a disk satisfies <Rayleigh's circulation criterion>. The distinction is that an unmagnetized displaced parcel approximately conserves its own <angular momentum>, whereas a field line transfers <angular momentum> between parcels. If <angular velocity> decreases outward, an outward-displaced parcel lags its inner partner. Magnetic tension transfers <angular momentum> to the outer parcel and removes it from the inner one, making their radial separation increase. A sufficiently stiff magnetic connection suppresses the separation instead. The instability therefore requires a finite range of field-line-bending strengths.
Make this mechanism quantitative in the local <shearing sheet> of a cylindrical rotation profile $\Omega(r)$. Put $q=-d\log\Omega/d\log r$, and take $0<q<2$ so the unmagnetized flow is stable. Its <radial epicyclic frequency> is $\kappa^2=2(2-q)\Omega^2$. Let the uniform field be vertical, $B_z\mathbf e_z$, with constant <mass density>, and use an axisymmetric horizontal disturbance proportional to $e^{st+ikz}$. Write $\mathbf h=\mathbf b/\sqrt{\mu_0\rho}$ and $\omega_A=kB_z/\sqrt{\mu_0\rho}$. The linearized <ideal magnetohydrodynamics> equations are
$$
\begin{aligned}
su_r-2\Omega u_\phi&=i\omega_Ah_r,&su_\phi+(2-q)\Omega u_r&=i\omega_Ah_\phi,\\
sh_r&=i\omega_Au_r,&sh_\phi&=-q\Omega h_r+i\omega_Au_\phi.
\end{aligned}
$$
The first pair contains <Coriolis acceleration> and the epicyclic response; the last equation is the shear-winding of a radial field perturbation into an azimuthal one. Omitting that winding term would remove the driving mechanism.
Eliminating $h_r,h_\phi$ gives $(s^2+\omega_A^2)u_r=2\Omega s u_\phi$ and $s(s^2+\omega_A^2)u_\phi+\Omega[(2-q)s^2-q\omega_A^2]u_r=0$. Their determinant is the <ideal magnetorotational dispersion relation>
$$
\boxed{s^4+(2\omega_A^2+\kappa^2)s^2+\omega_A^2(\omega_A^2-2q\Omega^2)=0.}
$$
Its <discriminant> as a quadratic in $s^2$ is $\kappa^4+16\Omega^2\omega_A^2>0$. Since $2\omega_A^2+\kappa^2>0$, a positive $s^2$ exists precisely when its constant term is negative:
$$
\boxed{0<\omega_A^2<2q\Omega^2=-\frac{d\Omega^2}{d\log r}.}
$$
Thus <angular velocity>, rather than specific <angular momentum>, must decrease outward. A Keplerian disk has $q=3/2$, $\kappa^2=\Omega^2$ and is hydrodynamically stable, yet magnetic modes with $0<k^2v_A^2<3\Omega^2$ grow.
The growing root is $s^2=-\omega_A^2-\kappa^2/2+\tfrac12\sqrt{\kappa^4+16\Omega^2\omega_A^2}$. Differentiation with respect to $\omega_A^2$ gives the <maximum growth rate of ideal magnetorotational instability>:
$$
\boxed{s_{\max}=\frac{q\Omega}2,\qquad \omega_{A,\max}^2=\frac{q(4-q)}4\Omega^2.}
$$
For Keplerian shear these are $3\Omega/4$ and $15\Omega^2/16$. The ideal maximum is set by shear; reducing field strength reduces the fastest wavelength instead of reducing this maximum growth rate. This is why the exactly zero-field problem differs from a sequence of weak-field problems with progressively larger <wave number>. At fixed <wave number>, growth does tend to zero as the field vanishes. The wavelength must fit the actual disk, and diffusion eventually matters at sufficiently short scales.
The instability has the required angular-momentum-transfer sign. In a growing mode, $u_\phi/u_r=(s^2+\omega_A^2)/(2\Omega s)>0$. At the fastest mode this ratio is one, while induction gives $h_\phi=-h_r$ after allowing for their common Fourier phase. Thus the averaged <Reynolds stress> minus <Maxwell stress tensor>, $\rho\langle u_ru_\phi-h_rh_\phi\rangle$, is positive: <angular momentum> moves outward. Its product with $q\Omega$ is the energy supplied by the background shear. These local mechanisms and coefficient conventions are also checked in https://www.damtp.cam.ac.uk/user/gio10/dad14.pdf[the primary MRI stability calculation].
More general fields require separating field-line bending from field gradients and curvature. At leading local order, for a stationary compatible field in an <incompressible flow> with axisymmetric wavevector $(k_r,0,k_z)$, the bending frequency is
$$
\omega_A=\frac{\mathbf k\cdot\mathbf B}{\sqrt{\mu_0\rho}},\qquad k^2=k_r^2+k_z^2.
$$
<Pressure> elimination projects the radial force by $f=k_z^2/k^2$. The radial momentum equation becomes $su_r-2f\Omega u_\phi=i\omega_Ah_r$; the azimuthal and induction equations retain the form above. Eliminating the amplitudes gives the <oblique axisymmetric magnetorotational dispersion relation>
$$
\boxed{\frac{k^2}{k_z^2}(s^2+\omega_A^2)^2+\kappa^2(s^2+\omega_A^2)-4\Omega^2\omega_A^2=0.}
$$
Here $k_z\ne0$. The unstable range is $0<\omega_A^2<-f\,d\Omega^2/d\log r$. The <poloidal magnetic field> therefore determines whether an axisymmetric perturbation bends a field line. A locally uniform <toroidal magnetic field> does not enter $\mathbf k\cdot\mathbf B$ for axisymmetry, and can contribute <magnetic pressure> that is projected out of this incompressible leading-order problem. Thus a toroidal component does not suppress the weak-field MRI merely by being present. Conversely a purely toroidal field has no ordinary axisymmetric local bending MRI in this approximation. This does not imply that every globally curved toroidal equilibrium is stable.
A steady general <poloidal magnetic field> must also be compatible with the equilibrium rotation. The azimuthal <ideal magnetohydrodynamic induction equation> contains $\partial_tB_\phi=r\mathbf B_p\cdot\nabla\Omega$. A stationary axisymmetric equilibrium with no meridional flow consequently requires $\mathbf B_p\cdot\nabla\Omega=0$, <Ferraro's law of isorotation>. A radial field crossing cylindrical differential-rotation surfaces would otherwise wind secularly; it cannot simply be substituted into a stationary normal-mode calculation. Boundary conditions and the along-field eigenvalue problem replace a freely chosen local $k$ in nonuniform or bounded equilibria. Strong enough field-line tension can then stabilize all admissible wavelengths.
An explicit axisymmetric effect of nonuniform field strength and curvature is obtained with a purely toroidal field $B_\phi(r)\mathbf e_\phi$, constant <mass density> and cylindrical rotation. A radial <fluid displacement> changes the field by
$$
b_\phi=\left(\frac{B_\phi}{r}-B_\phi'\right)\xi_r.
$$
This follows directly from $\mathbf b=\nabla\times(\boldsymbol\xi\times\mathbf B)$ and incompressibility. Its radial curvature force, after absorbing the magnetic-pressure perturbation, is $-2B_\phi b_\phi/(\mu_0\rho r)$. The axisymmetric azimuthal momentum equation is $su_\phi+(2\Omega+r\Omega')u_r=0$. Combining it with the radial force and then projecting out <pressure> gives
$$
\boxed{\omega^2=\frac{k_z^2}{k^2}\mathcal D,\qquad \mathcal D=\kappa^2-\frac r{\mu_0\rho}\frac{d}{dr}\left(\frac{B_\phi}{r}\right)^2.}
$$
This is the local <Michael criterion for axisymmetric toroidal-field interchange>. The new term can either stabilize or destabilize: for $B_\phi\propto r^p$, $\mathcal D=\kappa^2+2(1-p)v_{A\phi}^2/r^2$. A current-free $B_\phi\propto r^{-1}$ is stabilizing; $B_\phi\propto r$ leaves this axisymmetric interchange coefficient unchanged; a sufficiently steep increase with $p>1$ can destabilize even a centrifugally stable rotation law. That last instability can persist without differential rotation, so its energy source is the magnetic-current equilibrium rather than the ordinary MRI. Nonaxisymmetric current-driven modes are outside the requested essay scope.
For mixed poloidal and toroidal fields, the full induction perturbation contains $(\mathbf B\cdot\nabla)\boldsymbol\xi-(\boldsymbol\xi\cdot\nabla)\mathbf B-\mathbf B\nabla\cdot\boldsymbol\xi$, and the linear <Lorentz force> contains both $(\nabla\times\mathbf b)\times\mathbf B$ and $(\nabla\times\mathbf B)\times\mathbf b$. These terms explain why gradients and curvature cannot be replaced everywhere by a scalar $k^2v_A^2$: they can draw on equilibrium-current energy and couple the radial and vertical motions. Compressibility introduces additional <magnetic pressure> and magnetosonic responses; a strong toroidal field can alter the dispersion even though it disappeared from the leading incompressible weak-field result.
Finally, a vertically or radially stratified general field also supports part of the equilibrium weight through <magnetic pressure>. A displaced flux tube carries its frozen-in mass-to-flux relation, so its <mass density> at the surrounding total <pressure> can differ from its new environment. This introduces <magnetic buoyancy instability> and competition with ordinary <buoyancy>. Axisymmetric interchange displacements can access this energy without an azimuthal <wave number>, while poloidal tension resists variation along the field. Growth that survives when $d\Omega/dr=0$ must not be attributed to shear-driven MRI. The general-field stability problem therefore combines shear, field-line tension, current/curvature forces, compressibility, <buoyancy> and boundary constraints; the uniform-field result isolates the basic angular-momentum feedback mechanism.
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