= Solution
The <Batchelor entrainment hypothesis> takes the inflow speed through the plume's exposed outer edge to be $\alpha W$, where $\alpha$ is the <entrainment coefficient>. A wall plume has only one such edge, so $dV/dz=\alpha W$.
The <Boussinesq approximation> replaces density by a constant reference value $\rho_0$ in inertia and mass flux while retaining the small density deficit in <buoyancy>. It requires $|\widehat\rho-\rho|/\rho_0\ll1$. A sufficiently hot radiator can violate this near the source, where thermal expansion is large and the developed-plume description may also fail.
Put $B_0=F_0/\rho_0$ for the kinematic <buoyancy flux> per unit length. Dimensional analysis for a <line plume> gives
$$
W\sim B_0^{1/3},
\qquad
b\sim z,
\qquad
V\sim B_0^{1/3}z.
$$
The plume rise time is therefore $t_{\rm rise}\sim H/B_0^{1/3}$. Changing the room stratification requires a plume volume comparable with $XH$, so $t_{\rm room}\sim XH/V(H)\sim X/B_0^{1/3}$. Hence
$$
\frac{t_{\rm rise}}{t_{\rm room}}\sim\frac HX\ll1.
$$
The plume consequently follows the slowly changing ambient through a <quasi-steady approximation> when $X\gg H$.
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