Irradiated Atmosphere V: Effects of Vertical-Mixing induced Energy Transport on the Inhomogeneity

TL;DR

This paper addresses how vertical mixing and spatial inhomogeneity shape energy transport and cooling in irradiated gas-giant atmospheres. By coupling radiative transfer under a radiative-convective-mixing equilibrium (RCME) with a two-column inhomogeneous model, the authors quantify how variations in stellar flux and infrared opacity modify the internal heat flux $F_{\rm int}$ and the radiative-convective boundary. They show that Planck-function convexity and Jensen's inequality amplify the emission from inhomogeneous columns, yet vertical mixing generally reduces the planet's overall cooling even as it enhances local temperature contrasts. The findings highlight the pivotal role of vertical mixing in regulating atmospheric structure and cooling efficiency, with implications for interpreting exoplanet observations and modeling planetary evolution.

Abstract

Atmospheric variations over time and space boost planetary cooling, as outgoing internal flux responds to stellar radiation and opacity. Vertical mixing regulates this cooling. Our study examines how gravity waves or large-scale induced mixing interact with radiation transfer, affecting temperature inhomogeneity and internal flux. Through the radiative-convective-mixing equilibrium, mixing increases temperature inhomogeneity in the middle and lower atmospheres, redistributing internal flux. Stronger stellar radiation and mixing significantly reduce outgoing flux, slowing cooling. With constant infrared (IR) opacity, lower visible opacity and stronger mixing significantly reduce outgoing flux. Jensen's inequality implies that greater spatial disparities in stellar flux and opacity elevate the ratio of the average internal flux in inhomogeneous columns relative to that in homogeneous columns. This effect, particularly pronounced under high opacity contrasts, amplifies deep-layer temperature inhomogeneity and may enhance cooling. However, with mixing, overall cooling is weaker than without, as both the averaged internal flux of the inhomogeneous columns and that of the homogeneous column decline more sharply for the latter. Thus, while vertical mixing-induced inhomogeneity can enhance cooling, the overall cooling effect remains weaker than in the non-mixing case. Therefore, vertical mixing, by regulating atmospheric structure and flux, is key to understanding planetary cooling.

Figures (4)

• Figure 1: Panels (A)--(D) show how vertical mixing affects outgoing internal heat flux ($F_{\rm int}$) in giant planets under varying stellar flux ($F_{\odot}$), IR opacity ($\kappa_{\rm IR}$), and opacity ratio ($\alpha$). Parameters: mixing strength $K_{\rm zz}=10^{6}-10^{6.6}\ \mathrm{cm^{2}\,s^{-1}}$, diffusivity $D=2$, adiabatic slope $\beta=0.76$, and baseline-model opacity $\kappa_{\rm IR,baseline}=0.1\ \mathrm{cm^{2}\,g^{-1}}$ and temperature parameter $K_{\rm baseline}=1$. Black lines show no mixing; red lines show mixing. $K \propto \kappa_{\rm IR}^{-\beta}$ adjusts $K$ with opacity. (A)$F_{\rm int}$ vs. $F_{\odot}$ for $\alpha = 0.5$ (solid) and $\alpha = 5$ (dashed). The response is convex. (B)$F_{\rm int}$ vs. $\kappa_{\rm IR}$ at $F_{\odot} = 1$. The convex response varies with opacity. (C)$F_{\rm int}$ vs. $\alpha$ at $\kappa_{\rm IR} = 0.01 \mathrm{cm}^2/\mathrm{g}$, $K = 0.2$. The response is concave. (D)$F_{\rm int}$ vs. $\kappa_{\rm IR}$ with fixed visible opacity ($\alpha$ constant). The convex response mirrors Panel (B).
• Figure 2: Panels (A)--(D) show vertical mixing effects on a giant planet's atmosphere with $\alpha = 0.5$, comparing two inhomogeneous RCME models and one homogeneous model. Parameters: $K = 0.2$, $D = 2$, $\kappa_{\rm IR} = 0.1 \mathrm{cm}^2/\mathrm{g}$. Dashed/solid pairs (blue--cyan, red--orange, green--lime) denote $F_{\odot}=1.0,2.0$ and $3.0$ for mixing (no-mixing). Dashed or red lines show mixing ($K_{\rm zz} = 10^{6.8} \mathrm{cm}^2/\mathrm{s}$ in (A)--(B), $10^{7.5} \mathrm{cm}^2/\mathrm{s}$ in (D)). (A) Temperature vs. pressure ratio ($P/P_{\rm s}$) for varying $F_{\odot}$. An inset highlights mixing onset. (B) Temperature ratio ($T_{\rm homo}^4 / T_{\rm mean}^4 - 1$) vs. pressure ratio ($P/P_{\rm s}$). (C) Normalized internal flux ($F_{\rm int}$) vs. mixing strength ($K_{\rm zz}$). Dotted lines show no mixing. (D) Internal flux ratio ($\bar{F}_{\rm int} / F_{\rm int,homo}$) vs. $F_{\odot}$ with fixed opacity. Dots show no mixing; crosses show mixing. The colorbar indicates normalized $F_{\rm int,homo}$.
• Figure 3: Panels (A)–(C) show temperature and mixing effects for two inhomogeneous and one homogeneous column ($\alpha = 0.5$, $K = 0.2$, $D = 2$, $\kappa_{\rm IR} = 0.1 \, \text{cm}^2/\text{g}$, $F_\odot = 1.0$). Blue, green, and orange lines represent $F_{\rm int} = 0.001$, 0.005, and 0.003, respectively. Solid lines indicate no vertical mixing; dashed or red lines show mixing ($K_{\rm zz} = 10^{6.3} \, \text{cm}^2/\text{s}$ in (A)). (A) Temperature vs. pressure profiles, with an inset zooming in on details. (B) Temperature ratio ($T_{\rm homo}^4 / T_{\rm mean}^4 - 1$) vs. pressure ratio ($P/P_{\rm s}$). Black lines: no mixing; red lines: mixing. (C) Internal temperature parameter ($K$) vs. vertical mixing strength ($K_{\rm zz}$).
• Figure 4: Panels (A)–(D) show vertical mixing effects in two inhomogeneous and one homogeneous RCME column of a giant planet’s atmosphere ($\alpha = 0.5$, $\kappa_{\rm IR} = 0.1$, 1.0 for inhomogeneous columns, 0.55 for homogeneous column, in $\text{cm}^2/\text{g}$). (A) Temperature vs. pressure ratio ($P/P_{\rm s}$) profiles. Blue, green, and orange lines show $\kappa_{\rm IR} = 0.1$, 1.0, and 0.55, affecting temperature parameter $K$. Solid lines: no mixing; dashed lines: mixing ($K_{\rm zz} = 10^{6.8} \, \text{cm}^2/\text{s}$). (B) Temperature ratio ($T_{\rm homo}^4 / T_{\rm mean}^4 - 1$) vs. pressure ratio ($P/P_{\rm s}$). Black lines: no mixing; red lines: mixing ($K_{\rm zz} = 10^{6.8} \, \text{cm}^2/\text{s}$). (C) Internal temperature parameter ($K$) vs. mixing strength ($K_{\rm zz}$). Blue, green, and orange lines show $\kappa_{\rm IR} = 0.1$, 1.0, and 0.55. Solid lines: no mixing; dashed lines: mixing. (D) Mean internal flux ratio ($\bar{F}_{\rm int}/F_{\rm int,homo}$) at $F_\odot = 3.0$, baseline-model opacity $\kappa_{\rm IR,baseline} = 0.1 \, \text{cm}^2/\text{g}$. Dots: no mixing; crosses: mixing ($K_{\rm zz} = 10^{7} \, \text{cm}^2/\text{s}$). Colorbar shows normalized $F_{\rm int,homo}$.