= Solution
Choose the length convention $\ell_m=-U_*^3/B_0$; a dimensionless multiplier can be absorbed into the similarity functions. <Dimensional analysis> under the stated local dependence gives
$$
K_M=U_*z f_M(\zeta),\qquad K_B=U_*z f_B(\zeta),\qquad \zeta=z/\ell_m.
$$
To 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 $s_u=-J_M/U_*^2$ 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 profiles>
$$
\boxed{\bar u(z)-\bar u(z_r)=\frac{s_uU_*}{a_M}
\left[\ln\frac z{z_r}-\frac{d_M}{a_M}\frac{z-z_r}{\ell_m}+O(\zeta^2+\zeta_r^2)\right],}
$$
$$
\boxed{\bar b(z)-\bar b(z_r)=-\frac{B_0}{a_BU_*}
\left[\ln\frac z{z_r}-\frac{d_B}{a_B}\frac{z-z_r}{\ell_m}+O(\zeta^2+\zeta_r^2)\right].}
$$
Their 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> is
$$
\boxed{\mathrm{Ri}=\frac{\bar b_z}{\bar u_z^2}=-\frac{B_0z}{U_*^3}\frac{f_M^2}{f_B}
=\frac z{\ell_m}\frac{f_M^2}{f_B}.}
$$
Thus $|z/\ell_m|\ll1$ 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 $z/\ell_m$.
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