= Solution
Keep the background <magnetic field>, thermal gradient and <wavevector> fixed as $|\Omega|\to\infty$. In particular, $k_z/K$ is nonzero and of order one. Put $H=g\alpha\beta$ and $X=\omega^2$. Multiplying the nonzero-frequency <dispersion relation> by $\omega^2$ gives
$$
K^2(X-a^2)^2-4\Omega^2k_z^2X+Hh^2(X-a^2)=0,
$$
that is,
$$
K^2X^2-\left(4\Omega^2k_z^2+2K^2a^2-Hh^2\right)X
+a^2(K^2a^2-Hh^2)=0.
$$
The sum and product of its two roots are
$$
X_f+X_s=\frac{4\Omega^2k_z^2}{K^2}+2a^2-\frac{Hh^2}{K^2},\qquad
X_fX_s=\frac{a^2(K^2a^2-Hh^2)}{K^2}.
$$
For large <rotation>, the sum is of order $\Omega^2$ while the product is independent of $\Omega$. The quadratic therefore has one large root and one small root. More explicitly,
$$
X_f=\frac{4\Omega^2k_z^2}{K^2}+O(1),\qquad
X_s=\frac{a^2(K^2a^2-Hh^2)}{4\Omega^2k_z^2}+O(|\Omega|^{-4}).
$$
The fast frequencies are consequently
$$
\boxed{\omega_f\sim\pm\frac{2|\Omega k_z|}{K}},
$$
which are <inertial waves> at leading order. For nonzero $a$ away from the zero-frequency boundary, the <rapid-rotation slow magneto-Coriolis branch> has
$$
\boxed{\omega_s\sim\pm\frac{|a|}{2|\Omega k_z|}\sqrt{K^2a^2-Hh^2}}.
$$
Its frequency is of order $|\Omega|^{-1}$, much smaller than the fast <inertial wave> frequency. <Coriolis force> balances most of the magnetic and buoyant forces in this slow motion, with inertia supplying only a small correction.
For $a^2>0$, \b[the slow wave has real nonzero frequency when]
$$
\boxed{\frac{(\mathbf B\cdot\mathbf k)^2}{\mu_0\rho}
>g\alpha\beta\frac{k_x^2+k_y^2}{|\mathbf k|^2}}.
$$
Stable thermal stratification has $H<0$ and automatically satisfies this condition. With adverse stratification, $H>0$, sufficiently strong <magnetic tension> along the <wavevector> is needed. If the inequality is reversed, $X_s<0$ and the slow branch has imaginary frequencies, including an exponentially growing mode. Equality gives a stationary limit, rather than a nonzero oscillation. If $\mathbf B\cdot\mathbf k=0$, the proposed slow wave degenerates into zero-frequency disturbances: there is no magnetic restoring coupling. The remaining nonzero branch has $\omega^2=(4\Omega^2k_z^2-Hh^2)/K^2$. These qualifications also explain why zero roots introduced by clearing denominators must not be interpreted as ordinary propagating waves.
Back to article page
