= Solution
The <magnetohydrodynamic total pressure> is
$$
\boxed{\Pi=p+\frac{B^2}{2\mu_0}.}
$$
The <Lorentz force> has been split into the <gradient> of <magnetic pressure> and the remaining <magnetic tension> term. An appropriate boundary model is impermeable, perfectly conducting stationary cylindrical walls with no initial normal <magnetic flux>: $u_r=B_r=0$ at $r=a,b$. In an inviscid fluid this is a slip condition, not a no-slip condition on $u_\phi$ or $u_z$. It also makes the tangential ideal <electric field> vanish at the walls. Take <perturbations> periodic in $\phi$ and periodic or Fourier-resolved in $z$; the wall reaction supplies the normal <pressure> force without an extra prescribed <pressure> value.
Write the steady <velocity> and <magnetic field> as $U(r)\mathbf e_\phi$ and $B(r)\mathbf e_\phi$. They satisfy both <divergence> constraints. Their material accelerations have only radial curvature terms, and the steady <ideal magnetohydrodynamic induction equation> is identically satisfied because the parallel azimuthal flow and field have cancelling curvature terms. The radial momentum equation is
$$
-\rho\frac{U^2}r=-\frac{d\Pi}{dr}-\frac{B^2}{\mu_0r}.
$$
Hence the most general such equilibrium is
$$
\boxed{U(r),B(r)\text{ arbitrary smooth functions},\qquad
\Pi(r)=\Pi_0+\int_{r_0}^r\left(\frac{\rho U(s)^2}s-\frac{B(s)^2}{\mu_0s}\right)ds.}
$$
Its <gas pressure> is $p=\Pi-B^2/(2\mu_0)$; on a bounded annulus the additive constant can be chosen to keep it positive. Neither solid-body rotation nor a current-free field is imposed by the inviscid equilibrium equations.
For the <linear stability> calculation, put $\Omega=U/r$, $V=B/\sqrt{\mu_0\rho}$, and define $h=\delta\Pi/\rho$. Use axisymmetric <normal modes> proportional to $e^{ikz-i\omega t}$. Denote the <velocity> amplitudes by $(u,v,w)$ and the magnetic amplitudes in <velocity> units by $(b_r,b_\phi,b_z)=\delta\mathbf B/\sqrt{\mu_0\rho}$. <Linearization> of the cylindrical momentum equations gives
$$
\begin{aligned}
-i\omega u-2\Omega v&=-h'-\frac{2V}r b_\phi,\\
-i\omega v+(2\Omega+r\Omega')u&=(V'+V/r)b_r,\\
-i\omega w&=-ikh.
\end{aligned}
$$
The linearized <ideal magnetohydrodynamic induction equation> and <divergence> equations are
$$
\begin{aligned}
-i\omega b_r&=0,\\
-i\omega b_\phi&=r\Omega'b_r+(V/r-V')u,\\
-i\omega b_z&=0,\\
\frac{(ru)'}r+ikw&=0,\qquad
\frac{(rb_r)'}r+ikb_z=0.
\end{aligned}
$$
The curvature terms are essential: although the <perturbations> have no azimuthal dependence, the cylindrical unit vectors do.
For $\omega\ne0$, $b_r=b_z=0$. Introduce the radial <Lagrangian fluid displacement> by $u=-i\omega\xi$. The azimuthal amplitudes reduce to
$$
v=-(2\Omega+r\Omega')\xi,\qquad
b_\phi=(V/r-V')\xi.
$$
Consequently the radial and vertical equations, combined with the <incompressible flow> constraint, give
$$
h'=(\omega^2-\mathcal D)\xi,\qquad
h=\frac{\omega^2}{k^2}\frac{(r\xi)'}r,
$$
where the <Michael criterion for axisymmetric toroidal-field interchange> coefficient is
$$
\boxed{\mathcal D(r)=\kappa^2-r\frac{d}{dr}\left(\frac{V^2}{r^2}\right),\qquad
\kappa^2=4\Omega^2+2r\Omega\Omega'.}
$$
For $k\ne0$, the required single displacement equation is therefore
$$
\boxed{\omega^2\left[-\frac{d}{dr}\left(\frac{(r\xi)'}r\right)+k^2\xi\right]
=k^2\mathcal D\xi,\qquad \xi(a)=\xi(b)=0.}
$$
The same differential equation holds for $u$, which is a constant multiple of $\xi$.
Set $y=r\xi$. Multiplication by $y^*$ and <integration by parts> with the wall conditions gives the <global variational form of the Michael criterion>:
$$
\boxed{\omega^2=k^2\frac{\displaystyle\int_a^b\mathcal D\,|y|^2/r\,dr}
{\displaystyle\int_a^b(|y'|^2+k^2|y|^2)/r\,dr}.}
$$
The denominator is strictly positive for a nonzero displacement, so every mode's $\omega^2$ is real. If $\mathcal D\geq0$ everywhere, no exponentially growing mode exists. Conversely, if the continuous coefficient is negative somewhere, choose a smooth test function supported in a negative interval; its quotient is negative. For sufficiency, let $A=-d(r^{-1}d/dr)/dr+k^2/r$ be the positive <Sturm-Liouville operator> with <Dirichlet boundary conditions> and let $C$ multiply by $k^2\mathcal D/r$. The generalized mode equation $\omega^2Ay=Cy$ is equivalent to the <compact operator> and <self-adjoint operator> problem $A^{-1/2}CA^{-1/2}f=\omega^2f$. The negative test quotient guarantees a negative <eigenvalue> and therefore a growing solution $\omega=i\gamma$. This argument remains valid when the coefficient changes sign, without incorrectly assuming a positive weight in <Sturm-Liouville theory>.
Thus, for any fixed nonzero vertical <wavenumber>,
$$
\boxed{\text{axisymmetric exponential instability}\quad\Longleftrightarrow\quad
\kappa^2-r\frac{d}{dr}\left(\frac{B_\phi^2}{\mu_0\rho r^2}\right)<0\ \text{somewhere}.}
$$
Equality is marginal. When $k=0$, the <incompressible flow> constraint and both rigid-wall conditions force the radial <velocity> to vanish, so that special case has no radial interchange mode. When the field vanishes, the coefficient reduces to $\kappa^2=r^{-3}(r^4\Omega^2)'$, recovering the <Rayleigh centrifugal stability criterion>.
For a <Keplerian disk>, $\kappa^2=\Omega^2$. If $B_\phi\propto r^p$, the <toroidal interchange field threshold in a thin Keplerian disk> is
$$
\mathcal D=\Omega^2-2(p-1)V^2/r^2<0,
\qquad\boxed{p>1,\quad V^2>\frac{r^2\Omega^2}{2(p-1)}.}
$$
For ordinary radial gradients on the scale $r$, the <Alfvén speed> must therefore be comparable to the orbital speed. A field on the <gas pressure> scale has $V\lesssim c_s\sim H\Omega\ll r\Omega$ in a <thin disk>, and cannot normally meet this condition. Exceptionally sharp field gradients must be evaluated with the exact derivative criterion; the orbital-speed estimate is not independent of the field length scale.
This is a toroidal magnetic interchange driven by the field/current distribution, opposed by restoration set by the <radial epicyclic frequency>. It is \b[not the usual weak vertical-field <magnetorotational instability>]. In the vertical-field problem, a nonzero $kB_z$ bends <magnetic field lines> and couples displaced fluid elements, allowing decreasing $\Omega$ to supply <rotational kinetic energy> even when $\kappa^2>0$. Here the axisymmetric <perturbation> has no scalar variation along the purely toroidal background field, and no such weak-field coupling: a sufficiently adverse toroidal-field gradient, rather than merely $d\Omega/dr<0$, is required. Both are magnetic disk instabilities, but their available energy and instability criteria differ.
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