Discussion
Which balance combination gives the thermal wind relation?
Answer: B
Differentiating the geostrophic relation with respect to pressure and substituting the hydrostatic relation gives the thermal wind directly. Both ingredients are therefore required, and the relation inherits the validity conditions of each. Because gradient wind balance involves curvature, a corresponding thermal relation for it is much more complicated and is rarely used, which is why the geostrophic version is the one universally quoted. Working one numerical case through both cases side by side fixes the distinction better than any rule of thumb.
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