= Convective adjustment
<Convective adjustment> replaces an excessively steep vertical <temperature gradient> by an allowed profile while respecting the selected <energy> budget. It is a closure for unresolved <convection>, not a direct simulation of individual rising parcels. A prescribed critical <atmospheric lapse rate> is a modeling assumption; it need not equal the <dry adiabatic lapse rate>.
For an original dry two-slab example, take fixed equal <heat capacities> and slab centers separated by $\Delta z=2\,\mathrm{km}$. Initially $T_1=300\,\mathrm K$ below and $T_2=280\,\mathrm K$ above. Stipulate a permitted <atmospheric lapse rate> $\Gamma_c=6\,\mathrm{K\,km^{-1}}$. The initial difference $20\,\mathrm K$ exceeds the allowed $12\,\mathrm K$. Conservation of the sum of the two sensible <energies> requires $T_1'+T_2'=580\,\mathrm K$, while the adjusted gradient requires $T_1'-T_2'=12\,\mathrm K$. Hence $T_1'=296\,\mathrm K$ and $T_2'=284\,\mathrm K$. Heat removed below equals heat added above. Simply holding the lower slab fixed and raising the upper slab to $288\,\mathrm K$ would introduce external <energy>.
For unequal constant <heat capacities> $C_1,C_2$, the same two constraints instead give
$$
T_1'=\frac{C_1T_1+C_2T_2+C_2\Gamma_c\Delta z}{C_1+C_2},
\qquad T_2'=T_1'-\Gamma_c\Delta z.
$$
Apply this adjustment only when the original gradient exceeds the threshold. A moist adjustment also has to account for <latent heat> and changing composition; conservation of this dry sensible-<energy> expression alone is then insufficient.
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