 Solution

The ratio $\lambda =H/z_s$ compares the reference constant-stratification rise length with the height at which the actual buoyancy frequency equals $N_s$. Equivalently, the actual gradient sampled around that reference height is $N^2(H)/N_s^2={\lambda }^{\beta }$. Positive $\beta$ means stratification strengthens upwards; negative $\beta$ means it weakens upwards.

For a finite-rise branch at fixed $\beta >0$, increasing $\lambda$ makes the gradient stronger at each fixed normalized height $z/H$, so $H^{\prime }/H$ decreases. A small $\lambda$ means a larger region of weaker-than-reference stratification and permits relatively greater rise. For $\beta <0$, the same comparison reverses: increasing $\lambda$ weakens the sampled stratification, increases $H^{\prime }/H$, and can permit escape when the upper stratification decays sufficiently fast. At $\beta =0$ the ratio is one. Comparison at $\lambda =1$ is not an exact equality for nonzero $\beta$, because the plume samples a whole range of heights, not only $z=H$.

Using the pure plume as a scaling estimate before appreciable buoyancy loss,when $\beta >-8/3$ and the lower integral is regular. This gives the preceding monotonic trends, but not a universal numerical height. In particular, sufficiently negative powers are singular at a literal point source at $z=0$. A finite source elevation or a regularized near-source stratification must then be specified, as in part (d); no finite positive source buoyancy can absorb an infinite near-source buoyancy loss. Once such a finite source is used, $N^{2}Q\sim z^{\beta +5/3}$ has an integrable far-field tail for $\beta <-8/3$. A sufficiently buoyant plume can then retain positive buoyancy and rise without bound. The exact decaying-buoyancy infinite-rise branch in part (d) is another, specially matched possibility.