= Solution
Let $\tau_t$ be the tropopause <optical depth> and linearize $B$ with positive <derivative> $B_T$. In the troposphere $T=T_t+\Gamma H_\tau\log(\tau/\tau_t)$, so the requested schematic is
$$
B(\tau)=\begin{cases}B_t,&\tau\leq\tau_t,\\B_t+B_T\Gamma H_\tau\log(\tau/\tau_t),&\tau>\tau_t.\end{cases}
$$
It is constant in the isothermal stratosphere and rises linearly with $\log\tau$ below it. At unchanged <Temperature>[temperatures] and a multiplicative abundance change $a$, <optical depth> at each height changes to $a\tau$, hence $\boxed{B_a(\tau)=B_{\mathrm{old}}(\tau/a)}$. Doubling moves the tropopause kink to $2\tau_t$ and gives colder emission <Temperature>[temperatures] at a fixed <optical depth>; halving does the opposite.
\Image[/past-exam-of-the-short-options-paper-shared-by-prelims-and-parts-a-and-b-of-the-physics-course-of-the-university-of-oxford/2017-atmospheric-emission-and-feedback.png]
{title=Optical-depth remapping of atmospheric emission and stable versus unstable climate-feedback equilibria}
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