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Choose the length convention ${\ell }_m=-U_{\ast }^3/B_0$; a dimensionless multiplier can be absorbed into the similarity functions. Dimensional analysis under the stated local dependence givesTo obtain a log-linear expansion one also needs a regular neutral limit: take $f_M=a_M+d_M\zeta +O({\zeta }^2)$ and $f_B=a_B+d_B\zeta +O({\zeta }^2)$, with positive $a_M,a_B$. Dimensions alone do not specify these constants or guarantee differentiability. Let $s_u=-J_M/U_{\ast }^2$ denote the shear sign. In the high-number regime of part (a), integrate the reciprocal expansions from a reference height $z_r>0$ to obtain the near-neutral log-linear surface-layer profilesTheir leading terms are logarithmic; the first buoyancy-dependent corrections are linear in height. The additive constants require boundary or reference data and cannot be obtained from the prescribed fluxes alone.

The gradient Richardson number isThus $|z/{\ell }_m|\ll 1$ is nearly neutral, shear-dominated turbulence: the buoyancy contribution is small relative to shear production. Stable flux has $B_0<0$, ${\ell }_m>0$ and positive Richardson number; heating has $B_0>0$, ${\ell }_m<0$ and negative Richardson number. The smallness condition is on the magnitude, not an automatically satisfied signed inequality for every negative $z/{\ell }_m$.