= Solution
The basic state has <velocity field>
$$
\boldsymbol U=(0,\Lambda x,0)
$$
and <buoyancy> $B=N^2z$. For disturbances independent of $y$, the <linearized equations> are
$$
u_t-fv=-p_x,\qquad
v_t+(f+\Lambda)u=0,\qquad
w_t=-p_z+b,
$$
$$
b_t+N^2w=0,\qquad u_x+w_z=0.
$$
Substituting a <plane wave> proportional to $\exp[i(kx+mz-\omega t)]$ and eliminating $p$, $v$, and $b$ gives the <dispersion relation>
$$
\omega^2=\frac{N^2k^2+f(f+\Lambda)m^2}{k^2+m^2}.
$$
Thus the requested coefficients are
$$
\widetilde f^{\,2}=f(f+\Lambda),\qquad \widetilde g=N^2.
$$
An <instability> exists precisely when some <wavenumber> pair makes $\omega^2<0$. Since the <Brunt–Väisälä frequency> satisfies $N^2>0$, this is possible exactly when
$$
f(f+\Lambda)<0.
$$
The basic <relative vorticity> is $\Lambda\boldsymbol e_z$, so its <absolute vorticity> is $(f+\Lambda)\boldsymbol e_z$. Its <Ertel potential vorticity> is therefore
$$
Q=(f+\Lambda)N^2.
$$
The instability criterion can consequently be written as $fQ<0$: the vertical absolute vorticity has the opposite sign to the planetary vorticity. This is <inertial instability>, approached most directly by disturbances with $|m/k|\gg1$.
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