= Solution
Choose the <poloidal magnetic field> convention $\mathbf B_p=\nabla\times(A\hat{\mathbf y})=(-A_z,0,A_x)$. The term $\alpha B$ regenerates this poloidal flux function from the <toroidal magnetic field> through the <alpha effect>. It represents a component of the <mean-field electromotive force> produced by correlated <velocity> and magnetic fluctuations. The shear $G=\pi V/d$ stretches the poloidal vertical component $B_{p,z}=A_x$ into toroidal field, giving $GA_x$: this is the <Omega effect>. Both $\eta\nabla^2$ terms diffuse and damp the field. In a turbulent mean-field model, $\eta$ can denote an effective diffusivity including turbulent transport. The <alpha-Omega dynamo> approximation retains the shear-dominated toroidal source, and the <kinematic dynamo> assumption prescribes the <velocity> and coefficients without their <Lorentz force> feedback.
For a horizontal <wavenumber> $k$ and vertical <wavenumber> $\pi/d$, set $K^2=k^2+\pi^2/d^2$ and seek amplitudes proportional to $e^{st}$. The two modal equations are
$$
(s+\eta K^2)\widehat A=\alpha\widehat B,
\qquad(s+\eta K^2)\widehat B=ikG\widehat A.
$$
A nonzero mode requires the <dispersion relation>
$$
(s+\eta K^2)^2=i\alpha Gk.
$$
For real nonzero $\alpha Gk$, the root with positive real square-root part is
$$
s_+=-\eta K^2+\sqrt{\frac{|\alpha Gk|}{2}}
\left(1+i\operatorname{sgn}(\alpha Gk)\right).
$$
The other root has strictly negative real part. For $k=0$ there is no oscillatory feedback and both roots have real part $-\eta\pi^2/d^2$; the possible triangular-system prefactor cannot yield exponential growth.
Let $h=|k|d/\pi>0$ and $D=\alpha Vd^2/(\pi^2\eta^2)$. The growing-branch exponent satisfies
$$
\frac{\Re s_+}{\eta\pi^2/d^2}
=-(1+h^2)+\sqrt{\frac{|D|h}{2}}.
$$
Consequently the <plane-layer alpha-Omega dynamo threshold> at fixed $h$ is
$$
|D|>D_c(h):=\frac{2(1+h^2)^2}{h}.
$$
To find its least value, differentiate $D_c(h)=2(h^{-1}+2h+h^3)$:
$$
D_c'(h)=2(-h^{-2}+2+3h^2)=0
\quad\Longleftrightarrow\quad3h^4+2h^2-1=0.
$$
The unique positive solution is $h^2=1/3$. The threshold diverges at both ends $h\downarrow0$ and $h\to\infty$, so this is its global minimum. Hence
$$
\boxed{|D|_c=\frac{32}{3\sqrt3},\qquad
|k_c|=\frac{\pi}{\sqrt3\,d}.}
$$
With the usual positive-$D$ convention, the onset value is $D_c=32/(3\sqrt3)$ and exponentially growing waves exist for $D>D_c$. Exactly at the threshold the real part is zero, so it is a neutral dynamo wave: the quoted minimum is the onset threshold, or infimum for strict growth, not a growing solution at equality. If signed $D$ is allowed, the growth condition concerns $|D|$; there is no smallest signed negative value among all growing cases.
The real field is obtained by taking the real part of a <Fourier mode>. Its phase is $kx+(\Im s_+)t+\pi z/d$, so the horizontal phase speed is $-(\Im s_+)/k$. Reversing the sign of $\alpha V$ reverses propagation, while leaving the growth threshold unchanged. Reversing $k$ gives the conjugate real-field representation.
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