= Solution
In the <oceanic plate cooling model>, $T$ is <temperature>, $z$ height above the plate base, $a$ plate thickness, and $t$ age since formation at the spreading axis. With <half-spreading rate> $v$, a parcel at distance $x$ has $t=x/v$. The surface and basal <temperatures> are $T_0$ and $T_1$ respectively. The positive integer $n$ labels a sine <diffusion> mode. For constant <thermal diffusivity> $\kappa$,
$$
\boxed{\tau_0=\frac{a^2}{\pi^2\kappa}.}
$$
Substituting one sine mode into $T_t=\kappa T_{zz}$ fixes this normalization; $a^2/\kappa$ itself is the <diffusion> timescale before the factor $\pi^2$.
\b[Source qualification.] The printed transient coefficients have the wrong sign for the intended initially hot plate. Indeed, in the interior,
$$
\sum_{n\geq1}\frac{2(-1)^n}{n\pi}\sin\frac{n\pi z}{a}=-\frac za.
$$
Thus the literal zero-age profile is $T_0+\Delta T(1-2z/a)$, where $\Delta T=T_1-T_0$, rather than $T_1$. It would even approach $T_0-\Delta T$ from inside the upper boundary. The intended hot-ridge cooling solution reverses the transient sign, using $2(-1)^{n+1}/(n\pi)$. Both versions have the same boundary <temperatures> and old-age limit. Part (b) gives both surface-flux expressions; the physical ridge-elevation and ridge-push calculations explicitly use the hot initial plate. This sign is present in the original PDF as well as the held transcription.
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