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The mesosphere lies above the stratosphere and below the thermosphere. Its typical terrestrial temperature decreases upward.

The pressure axis increases downward in the sketches. Solid red segments indicate efficient convection; blue segments indicate mainly radiative transfer. The boundaries are nominal examples, since radiative-convective boundary pressure depends on opacity, gravity and intrinsic flux.

For Earth, the convective troposphere lies beneath a radiative stratosphere, with the transition near $0.1$--$0.2$ bar. The larger solar-system planets likewise have deep convective regions beneath largely radiative upper atmospheres; their tropopause/upper radiative-convective boundary is commonly of order $0.1$--$1$ bar. Their temperatures differ greatly from Earth's, and detached radiative layers can occur deeper down.

For an irradiated hot Jupiter, absorbed stellar flux maintains a hot, extended, relatively shallow-gradient radiative atmosphere. In models with a weak old-planet intrinsic flux, the deep radiative-convective boundary can lie around $10^2$--$10^3$ bar, with a useful wider model-dependent range of tens to thousands of bars. A thermal inversion can appear at low pressures if visible/UV absorption heats the upper atmosphere. Below the deep boundary the profile joins a convective adiabat.

A young, directly imaged exoplanet on a distant orbit is primarily heated from within. Its photosphere commonly joins the convective interior at order $0.1$--$10$ bar, illustrated here at one bar. The temperature generally rises monotonically with pressure over the infrared-forming layers. Strong stellar heating is absent, so a broad stellar-heated isothermal layer is unnecessary.

Two differences in thermal inversions are their absorbers and their formation conditions. Earth's ozone layer and solar-system hydrocarbon absorption can heat upper layers; the proposed hot-Jupiter absorbers include refractory titanium monoxide/vanadium monoxide or other strong visible absorbers, since ozone and methane-rich cold-planet chemistry are unsuitable at very high temperatures. Hot-Jupiter inversions depend strongly on irradiation, atmospheric cold traps and atmospheric transport, and can occur at mbar-to-sub-bar pressures; Earth and the solar-system giants have cooler, established stratospheres above their shallow tropospheres.

Two differences between the hot Jupiter and distant young-giant profiles are external versus internal heating, and the depth/shape of the radiative zone. The former can have a broad warm radiative layer, a much deeper convective boundary and sometimes an inversion; the latter typically has a steeper outward decrease toward a photosphere, a shallower boundary and no irradiation-driven inversion. These are class trends, not a unique temperature profile for every planet.

The functional form is not specific to exoplanets. With positive pressure normalization and physical temperature, a square-root exponential atmospheric profile can approximate a monotone interval of the terrestrial atmosphere. An outward-cooling interval, such as part of the troposphere or mesosphere, requires $\alpha >0$. An upward-warming interval, such as part of the stratosphere heated by ultraviolet absorption or the thermosphere heated by high-energy radiation, requires $\alpha <0$.

The local atmospheric lapse rate follows from the ideal gas relation $P=\rho k_{B}T/(\mu m_H)$ and hydrostatic equilibrium:For dry terrestrial air, approximately $\gamma =1.4$ and $\mu \simeq 29$ give the dry-adiabatic lapse rate $g/c_p\simeq 9.8\mathrm{K}\mathrm{k}{\mathrm{m}}^{-1}$. The familiar mean tropospheric value near $6.5\mathrm{K}\mathrm{k}{\mathrm{m}}^{-1}$ is less steep; a local fit must satisfy $\nabla \lesssim 2/7$ to be dry-convectively stable. Moist convection needs the moist parcel thermodynamics instead, and the terrestrial atmosphere is not uniformly dry or chemically homogeneous at all heights.

The squared-logarithm shape cannot reproduce an exactly constant nonzero lapse rate over an arbitrary thick region, all the alternating atmospheric layers, or a finite exactly isothermal region. It is a local parametrization with a fixed sign of the temperature gradient; it has no terrestrial universality.

The terrestrial atmosphere has regions with opposite signs of the atmospheric lapse rate: the troposphere usually cools upward, the stratosphere warms upward through ultraviolet heating, the mesosphere cools, and the thermosphere warms. A monotone atmospheric pressure-temperature profile can approximate one such interval, but cannot describe all their transitions with one fixed branch.