= Solution
Vigorous <convection> keeps the interior nearly uniform, so the upper <thermal boundary layer> carries a <temperature> drop of order $\theta$. A <boundary layer> becomes unstable when its local <Rayleigh number> reaches an order-fixed critical value $R_c$:
$$
\operatorname{Ra}_\delta=\frac{\rho_0g\alpha\theta\delta^3}{\eta\kappa}\sim R_c.
$$
This is <marginal-stability thermal boundary-layer scaling> using the <Local Rayleigh-number closure>: boundary-layer <buoyancy> overcomes viscous and thermal-diffusive stabilization. Equivalently, a buoyancy-driven velocity scale $u\sim\rho_0g\alpha\theta\delta^2/\eta$ makes $u\delta/\kappa$ order $R_c$. Thus
$$
\boxed{\delta\sim\left(\frac{R_c\eta\kappa}{\rho_0g\alpha\theta}\right)^{1/3},\qquad
E(t)\sim\frac{k\theta}{\delta}=C\theta^{4/3},\quad
C=k\left(\frac{\rho_0g\alpha}{R_c\eta\kappa}\right)^{1/3}.}
$$
There is no $d$ in this closure, as required for a thin <boundary layer> controlled by its own stability. The exponent is fixed by scaling; its absolute coefficient is not. Here $R_c$ represents the geometry and <boundary conditions>, together with any order-one boundary-layer coefficient. The question supplies no numerical value for it. The assumption requires $\delta\ll d$, positive $\theta$, and properties sufficiently constant over the <temperature> contrast.
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