= Solution
The three basic modes are \b[radiation, <convection> and <thermal conduction>]. <Radiative transfer> dominates transparent atmospheric layers; <convection> carries heat through sufficiently unstable fluid regions; <thermal conduction> matters especially in solids or highly conducting dense material.
Consider a chemically homogeneous <ideal gas> atmosphere and a small, dry parcel displacement, with rapid pressure equilibration and negligible heat exchange during the displacement. The parcel's <first law of thermodynamics> per unit mass is
$$
T\,ds=C_p\,dT-\frac{dP}{\rho}.
$$
For an <adiabatic process>, $ds=0$. Using <hydrostatic equilibrium>, $dP/dz=-\rho g$, the parcel follows
$$
\left(\frac{dT}{dz}\right)_{\rm ad}=-\frac{g}{C_p}.
$$
After an upward displacement $\delta z>0$, its temperature relative to its new surroundings is
$$
T_{\rm parcel}-T_{\rm env}=\left(-\frac{g}{C_p}-\frac{dT}{dz}\right)\delta z.
$$
Here $C_p$ is the <specific heat capacity at constant pressure>, and $N$ is the <buoyancy frequency>. At the same pressure, a warmer parcel has lower density and accelerates upward. Thus \b[strict instability requires $dT/dz<-g/C_p$]. Equivalently, the parcel equation is $\ddot{\delta z}=-N^2\delta z$, with
$$
\boxed{N^2=\frac{g}{T}\left(\frac{dT}{dz}+\frac{g}{C_p}\right).}
$$
Negative $N^2$ gives growing displacements. The printed non-strict inequality includes the onset boundary: \b[equality is neutrally stable, not a strictly growing instability]. This <dry parcel buoyancy in a homogeneous atmosphere> is the dry <Schwarzschild criterion>; condensation and composition gradients require modified criteria.
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