Thermo-compositional Diabatic Convection in the Atmospheres of Brown Dwarfs and in Earth's Atmosphere and Oceans

For double-diffusive convection (thermal diffusion coefficient κ_T and compositional diffusion coefficient κ_μ), the source terms are given by

$$\begin{eqnarray}\begin{array}{rcl}R(X) & = & {\kappa }_{\mu }{\rm{\Delta }}X,\\ H(T) & = & {\kappa }_{T}{\rm{\Delta }}T.\end{array}\end{eqnarray} \tag{ 11 }$$

we get R_T = H_X = 0, ${H}_{T}=-{k}^{2}{\kappa }_{T}$, and ${R}_{X}=-{k}^{2}{\kappa }_{\mu }$, where k is the norm of the wavenumber of the perturbation (see the Appendix). The instability criterion thus becomes

$$\begin{eqnarray}&&({{\rm{\nabla }}}_{T}-{{\rm{\nabla }}}_{\mathrm{ad}}){\kappa }_{\mu }-{{\rm{\nabla }}}_{\mu }{\kappa }_{T}\gt 0,\end{eqnarray} \tag{ 12 }$$

which is the standard form of the instability criterion for thermohaline and fingering convection (Stern 1960). Since H_X = 0 in that case, we are in the case 1 situation, and the driving mechanism is the mean molecular weight gradient.

In Earth's oceans, the adiabatic PT profile is close to isothermal because of the incompressibility of liquid water. When oceans have hot and salty water on top of cold and fresh water, they can develop thermohaline convection through the instability criterion and form salt fingers. When a blob of hot and salty water sinks, it will thermalize with the surroundings through temperature diffusion but remain heavier because salt diffusion is slower; hence, the blob continues to sink and is unstable.

These fingers are thought to be at the origin of thermohaline fingering staircases in the oceans (Turner 1967). These structures alternate well-mixed layers at constant entropy (isothermal) and salt concentration and steps with sharp gradients of temperature and concentration prone to the fingering instability. An example of such a temperature structure is shown in Figure 1 with the alternance of isothermal layers and fingering-favorable steps. Note that thermohaline staircases can also exist with the alternance of isothermal layers and the radiative steps shown in Figure 1. This situation occurs in polar Earth oceans when cold/fresh water overlies warm/salty water (e.g., Timmermans et al. 2008). In the thermohaline fingering staircase, the well-mixed layers are prone to overturning convection, and convective plumes can also be observed transporting energy through the layer (see Figure 12 in Gregg 1988). The temperature gradients (as a function of pressure) in the fingering steps are negative (hot water on top of cold water); hence, they are reduced compared to the adiabatic (isothermal) gradient. Figure 3 of Radko (2014) shows the evolution of the average temperature profile in the formation of these steps. Starting from a given temperature gradient, the simulation evolves toward a reduced temperature gradient as a function of pressure (increased temperature gradient as a function of altitude) in the fingering layer and not toward the adiabatic isothermal profile. Hence, we can conclude that fingering convection can reduce the temperature gradient relative to pressure in Earth's oceans.

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Let us assume that the reactive terms for composition are fast and at an equilibrium given by X = X_eq (P, T). We can then relate the compositional vertical gradient to the temperature and pressure gradients using

$$\begin{eqnarray}&&\displaystyle \frac{\partial X}{\partial z}=\displaystyle \frac{\partial {X}_{\mathrm{eq}}}{\partial \mathrm{log}T}{{\rm{\nabla }}}_{T}\displaystyle \frac{\partial \mathrm{log}P}{\partial z}+\displaystyle \frac{\partial {X}_{\mathrm{eq}}}{\partial \mathrm{log}P}\displaystyle \frac{\partial \mathrm{log}P}{\partial z}.\end{eqnarray} \tag{ 13 }$$

We also assume that the reaction source term R is condensation/evaporation of water and the thermal source term is the corresponding release or pumping of latent heat L,

$$\begin{eqnarray}&&H=-\displaystyle \frac{{RL}}{{c}_{p}},\end{eqnarray} \tag{ 14 }$$

and we neglect the temperature dependence of the latent heat. We can then derive from the criterion in Equation (9) the standard form of the moist convective criterion used in Earth atmospheric physics (see Stevens 2005),

$$\begin{eqnarray}&&{{\rm{\nabla }}}_{T}-{{\rm{\nabla }}}_{\mathrm{ad}}\displaystyle \frac{1+\tfrac{{X}_{\mathrm{eq}}L}{{R}_{d}{T}_{0}}}{1+\tfrac{{X}_{\mathrm{eq}}{L}^{2}}{{c}_{p}{R}_{v}{T}_{0}^{2}}}\gt 0,\end{eqnarray} \tag{ 15 }$$

where R_v and R_d are the vapor and dry air gas constant. We add the details of this derivation in the Appendix.

Moist convection was introduced and experimentally studied during the 1950s and 1960s. A historical perspective of moist convective studies can be found in Figure 4 of Yano (2014). By taking into account the energy transported by water vapor through the potential latent heat release, it has been shown that the moist saturated PT profile has a reduced temperature gradient (the lapse rate) compared to dry adiabatic convection. In Figure 2, we show different convective adjustments for Earth's atmosphere, adapted from Manabe & Strickler (1964). Dry adiabatic convection has a lapse rate of 10 K km⁻¹, while moist convection leads to a reduced lapse rate, closer to that observed in Earth's troposphere (6.5 K km⁻¹).

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However, we insist on the fact that this PT profile at the instability limit should not be called a moist adiabat because this creates confusion on the nature of the instability. As defined in Stevens (2005), this is a pseudo-adiabat with an effective adiabatic gradient. If we were considering the gas and liquid phases for the system, there would be no energy source term, and the system would be adiabatic and obey a strict energy conservation. But since we are interested in the temperature of the gas phase for atmospheric studies, the latent heat that can be pumped or released from the liquid phase acts as an external source term. Hence, the gas phase alone is not adiabatic (also because the liquid phase can decouple from the gas phase through precipitation).

An intuitive way to describe moist convection is as follows: when a parcel is rising, it cools by expansion, thus triggering water condensation that releases latent heat and heats the atmosphere. This description fits in case 2, described in Section 2, which means that the driving mechanism is the dependence of the energy source term on composition.

The key to understanding CO/CH₄ radiative convection is to realize that moist convection is actually a particular case in our previous analysis for which the energy exchange is directly linked to the compositional reaction by Equation (14), with the energy source term being directly proportional to the compositional term. In the diabatic criterion in Equation (9), we do not need to assume such a direct link in the physics of the compositional and thermal source terms. We can thus derive an equivalent of the moist convective criterion with the two dominant processes at stake in brown dwarf atmospheres: CO/CH₄ chemistry and radiative transfer. A direct link between the two is not necessary, since composition and temperature interact with each other through either H_X $\ne$ 0 or the EOS with $\partial \mathrm{log}{\mu }_{0}/\partial X\ne 0$. When a parcel is rising, the temperature is changing because of expansion and radiative transfer; we then have two possibilities for CO/CH₄ radiative convection.

• 1.  
  It triggers chemical exchanges between CH₄+H₂O and CO+3H₂, with opacity differences in the two states that can lead to heating and cooling in the atmosphere. The driving mechanism is the dependence of the energy source term on composition.
• 2.  
  Similar to the thermohaline case, if the temperature adjustment is faster than the chemical timescale, the parcel will have more CO+3H₂ than CH₄+H₂O compared to its local environment, hence a smaller mean molecular weight, and continue to rise. In that case, the driving quantity for instability is the mean molecular weight gradient.

In the general case, for CO/CH₄ exchange and radiative transfer,

$$\begin{eqnarray}\begin{array}{rcl}R & = & -\displaystyle \frac{X-{X}_{\mathrm{eq}}}{{\tau }_{\mathrm{chem}}},\\ H & = & \displaystyle \frac{4\pi \kappa }{{c}_{p}}(J-\sigma {T}^{4}/\pi ),\end{array}\end{eqnarray} \tag{ 16 }$$

where τ_chem is the timescale of the chemical reaction, κ is the absorption of the gas, and J is the mean radiative intensity. Note that in the optically thick regime, when the radiative energy is small compared to the gas energy, the energy source terms can be written as thermal diffusion with a diffusion coefficient κ_T = c/(3κρ). For simplicity, we first assume that the opacity does not depend on composition, hence H_X = 0. In that case, the instability is only driven by the mean molecular weight gradient, and

$$\begin{eqnarray}\begin{array}{rcl}{\omega }_{X}^{{\prime} } & = & -\displaystyle \frac{1}{{\tau }_{\mathrm{chem}}},\\ {\omega }_{T}^{{\prime} } & = & -\displaystyle \frac{1}{{\tau }_{\mathrm{rad}}},\end{array}\end{eqnarray} \tag{ 17 }$$

with ${\tau }_{\mathrm{rad}}={c}_{p}/(16\pi \kappa \sigma {T}^{3})$ in the optically thin regime or ${\tau }_{\mathrm{rad}}=1/({k}^{2}{\kappa }_{T})$ in the optically thick regime.

Assuming that the chemistry is close to equilibrium in the deep atmosphere of brown dwarfs (similar to moist convection at saturation), the criterion becomes

$$\begin{eqnarray}&&{{\rm{\nabla }}}_{T}-{{\rm{\nabla }}}_{\mathrm{ad}}\displaystyle \frac{1+\tfrac{1}{{{\rm{\nabla }}}_{\mathrm{ad}}}\tfrac{\partial \mathrm{log}{\mu }_{0}}{\partial X}\tfrac{\partial {X}_{\mathrm{eq}}}{\partial \mathrm{log}P}\tfrac{{\tau }_{\mathrm{chem}}}{{\tau }_{\mathrm{rad}}}}{1-\tfrac{\partial \mathrm{log}{\mu }_{0}}{\partial X}\tfrac{\partial {X}_{\mathrm{eq}}}{\partial \mathrm{log}T}\tfrac{{\tau }_{\mathrm{chem}}}{{\tau }_{\mathrm{rad}}}}\gt 0,\end{eqnarray} \tag{ 18 }$$

which has the same form as the moist convective instability criterion; the demonstration of this expression is the same as the demonstration for moist convection in the Appendix.⁷ We show with Figure 2 that moist convection is known to reduce the temperature gradient in Earth's troposphere and that fingering convection does the same in Earth's oceans (see Figure 1). It thus becomes natural to expect CO/CH₄ radiative convection to behave in the same way, since all of these instabilities derive from the same criterion. As shown in Tremblin et al. (2015, 2016, 2017), such a temperature gradient reduction in the atmospheres of brown dwarf can explain very well the spectral reddening of field brown dwarfs and the strong reddening in the spectrum of young brown dwarfs in moving groups. However, we need to go to the nonlinear regime in order to check if the diabatic convective fluxes of CO/CH₄ radiative convection are indeed sufficiently significative to reduce the temperature gradient in the atmospheres of brown dwarfs.

Before going to the mean field approach, we will define two new conserved quantities in the linear regime. Without source terms, adiabatic convection will tend to homogenize potential temperature and composition; however, this is not the case in general with diabatic energy and compositional transports. Assuming we are in the linear regime, we can rewrite the evolution of the perturbations with

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{\partial \delta X}{\partial t}+\delta {\boldsymbol{u}}\cdot {\boldsymbol{\nabla }}{X}_{0} & = & {R}_{X}\delta X+{R}_{T}\delta T,\\ \displaystyle \frac{\partial \delta \mathrm{log}\theta }{\partial t}+\delta {\boldsymbol{u}}\cdot {\boldsymbol{\nabla }}(\mathrm{log}{\theta }_{0}) & = & \displaystyle \frac{1}{{T}_{0}}\left({H}_{T}\delta T+{H}_{X}\delta X\right).\end{array}\end{eqnarray} \tag{ 19 }$$

We make the ansatz that the density perturbations are small in the EOS, which is valid with a background that is marginally unstable, i.e., close to neutral buoyancy:

$$\begin{eqnarray}&&\displaystyle \frac{\delta \rho }{{\rho }_{0}}=\displaystyle \frac{\partial \mathrm{log}{\mu }_{0}}{\partial X}\delta X-\displaystyle \frac{\delta T}{{T}_{0}}\sim 0.\end{eqnarray} \tag{ 20 }$$

This ansatz can be verified a posteriori in the numerical simulations performed in Section 4. We can then rewrite the evolution of the perturbations in a pure transport form,

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{\partial \delta {X}^{{\prime} }}{\partial t}+\delta {\boldsymbol{u}}\cdot ({\boldsymbol{\nabla }}{X}_{0}^{{\prime} }) & = & 0,\\ \displaystyle \frac{\partial \delta \mathrm{log}{\theta }^{{\prime} }}{\partial t}+\delta {\boldsymbol{u}}\cdot {\boldsymbol{\nabla }}(\mathrm{log}{\theta }_{0}^{{\prime} }) & = & 0,\end{array}\end{eqnarray} \tag{ 21 }$$

with

$$\begin{eqnarray}\begin{array}{rcl}{X}^{{\prime} } & = & X-\mathrm{log}\theta {\left(\displaystyle \frac{\partial \mathrm{log}{\mu }_{0}}{\partial X}\right)}^{-1}\displaystyle \frac{{\omega }_{X}^{{\prime} }}{{\omega }_{T}^{{\prime} }},\\ \mathrm{log}{\theta }^{{\prime} } & = & \mathrm{log}\theta -X\displaystyle \frac{\partial \mathrm{log}{\mu }_{0}}{\partial X}\displaystyle \frac{{\omega }_{T}^{{\prime} }}{{\omega }_{X}^{{\prime} }}.\end{array}\end{eqnarray} \tag{ 22 }$$

This is the generalization of the moist potential temperature; in the case of moist convection, it reduces to

$$\begin{eqnarray}&&\mathrm{log}{\theta }^{{\prime} }=\mathrm{log}\theta -X\displaystyle \frac{L}{{c}_{p}T}.\end{eqnarray} \tag{ 23 }$$

This new potential temperature includes a term that depends on composition that takes into account the potential release of energy from a compositional change through the energy and chemical source terms in the small-scale fluctuations. When the energy source term is proportional to the compositional source term (e.g., for moist convection), we do not need to linearize the system, and we can define these conserved quantities directly in the nonlinear regime. For arbitrary source terms, these quantities are only strictly conserved during the linear regime.

In Section 2.3, we have assumed that the chemistry is at equilibrium in order to derive a criterion for CO/CH₄ radiative convection. This is valid for the deep atmospheres of brown dwarfs. It is well known, however, that CO/CH₄ chemistry is not at equilibrium in the upper part of the atmosphere. We therefore need to take into account out-of-equilibrium chemistry to compute the convective fluxes, and we can use mixing-length theory derived from a mean field approach to do that.

We start from the equations of potential temperature and composition transport,

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{\partial X}{\partial t}+{\boldsymbol{u}}\cdot {\boldsymbol{\nabla }}X & = & R(X,T),\\ \displaystyle \frac{\partial \mathrm{log}\theta }{\partial t}+{\boldsymbol{u}}\cdot {\boldsymbol{\nabla }}(\mathrm{log}\theta ) & = & \displaystyle \frac{H(X,T)}{T},\end{array}\end{eqnarray} \tag{ 24 }$$

and we decompose the fields into a background component ($\bar{X}$, $\bar{T}$, $\bar{{\boldsymbol{u}}}=0$) and a small-scale fluctuation part (δX, δT, $\delta {\boldsymbol{u}}$). As in the Boussinesq approximation, we neglect the pressure perturbation δP = 0 and choose the reference pressure in the potential temperature ${P}_{\mathrm{ref}}=\bar{P}$. We do not assume the amplitude of the fluctuations to be small compared to the background.

We can then rewrite the evolution of the large-scale system with

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{\partial \bar{X}}{\partial t}+\displaystyle \frac{1}{\rho }\displaystyle \frac{\partial {X}_{d}}{\partial z} & = & R(\bar{X},\bar{T}),\\ \displaystyle \frac{\partial \bar{T}}{\partial t}+\displaystyle \frac{1}{\rho {c}_{p}}\displaystyle \frac{\partial {F}_{d}}{\partial z} & = & H(\bar{X},\bar{T}),\end{array}\end{eqnarray} \tag{ 25 }$$

with

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{1}{\rho {c}_{p}}\displaystyle \frac{\partial {F}_{d}}{\partial z} & = & \displaystyle \frac{\partial \delta T}{\partial t}+\delta {\boldsymbol{u}}\cdot {\boldsymbol{\nabla }}(\mathrm{log}\theta )T-H(X,T)+H(\bar{X},\bar{T}),\\ \displaystyle \frac{1}{\rho }\displaystyle \frac{\partial {X}_{d}}{\partial z} & = & \displaystyle \frac{\partial \delta X}{\partial t}+\delta {\boldsymbol{u}}\cdot {\boldsymbol{\nabla }}X-R(X,T)+R(\bar{X},\bar{T}),\end{array}\end{eqnarray} \tag{ 26 }$$

where F_d and X_d are the turbulent convective fluxes that will contain the nonlinear interactions between the background and the fluctuations and between the fluctuations themselves. The term $H(X,T)-H(\bar{X},\bar{T})$ can be Taylor-expanded and contains all of the nonlinear interactions through the source terms represented by ${H}_{{X}^{k},{T}^{l}}{(\delta X)}^{k}/k!{(\delta T)}^{l}/l!$, with ${H}_{{X}^{k},{T}^{l}}$ being the kth and lth derivative with respect to composition and temperature. We then need to find a closure relation expressing X_d and F_d as a function of the background variables. It can be obtained through numerical simulations by taking a space and time average of the fluxes defined with Equation (26). We point out that in general, it will be needed to evaluate all of the nonlinear terms. To avoid this complicated procedure, we propose using an argument similar to mixing-length theory.

When the source terms are zero, i.e., in the adiabatic case, the closure relation that leads to the standard mixing-length theory is

$$\begin{eqnarray}\begin{array}{rcl}{X}_{\mathrm{ad}} & = & -{h}_{p}\rho w\displaystyle \frac{\partial \bar{X}}{\partial z},\\ {F}_{\mathrm{ad}} & = & -{h}_{p}\rho {c}_{p}\bar{T}w\displaystyle \frac{\partial \mathrm{log}\bar{\theta }}{\partial z},\end{array}\end{eqnarray} \tag{ 27 }$$

with $w=\omega {l}_{\mathrm{conv}}$, l_conv being the mixing length and ω the growth rate of the convective instability. These relations also correspond to the flux gradient laws used for thermohaline convection (e.g., Radko 2014).

By using this closure relation, the convective fluxes will tend to homogenize the composition and potential temperature and give the standard form of the convective flux for adiabatic convection:

$$\begin{eqnarray}&&{F}_{\mathrm{ad}}=\rho {c}_{p}w\bar{T}({{\rm{\nabla }}}_{T}-{{\rm{\nabla }}}_{\mathrm{ad}}).\end{eqnarray} \tag{ 28 }$$

We have to make a clear distinction now between the convective instability that constrains the velocity in the flux (Schwarzschild, Ledoux, or diabatic) and the energy and composition that are transported, which can be impacted by the source terms (diabatic transport) or not (adiabatic transport). This will depend on the averaging timescale in the mean field approach: if the source terms' typical timescale is shorter than the averaging timescale, they can impact the transport of energy and composition even in the Schwarzschild or Ledoux convective regimes.

In the linear regime, diabatic energy and compositional transports do not tend to homogenize potential temperature and composition, as shown in Section 3.1. This is also likely the case in the nonlinear regime. For that reason, we extend the analysis presented there into a mean field model, in which we suggest using the quantities we identified as conserved during the linear regime. Using this ansatz, the new closure relation for the small-scale diabatic processes is then given by

$$\begin{eqnarray}\begin{array}{rcl}{X}_{{\rm{d}}} & = & -{h}_{p}\rho w\displaystyle \frac{\partial \bar{{X}^{{\prime} }}}{\partial z},\\ {F}_{{\rm{d}}} & = & -{h}_{p}\rho {c}_{p}\bar{T}w\displaystyle \frac{\partial \mathrm{log}\bar{{\theta }^{{\prime} }}}{\partial z},\end{array}\end{eqnarray} \tag{ 29 }$$

with the prime quantities defined in Equation (22). The diabatic convective transport is similar to the parameterization used for moist convection (Arakawa & Jung 2011). In the thermohaline context, using the standard potential temperature and composition in the flux gradient laws can lead to inconsistencies, such as an ultraviolet catastrophe, when trying to predict the growth rates of the staircase modes at small scales (Radko 2014). The new parameterization proposed here could provide a way to solve these issues.

The convective diabatic fluxes can be written in a more usual form:

$$\begin{eqnarray}\begin{array}{rcl}{X}_{{\rm{d}}} & = & \rho w{\left(\displaystyle \frac{\partial \mathrm{log}{\mu }_{0}}{\partial X}\right)}^{-1}({{\rm{\nabla }}}_{\mu }-({{\rm{\nabla }}}_{T}-{{\rm{\nabla }}}_{\mathrm{ad}}){\omega }_{X}^{{\prime} }{/\omega }_{T}^{{\prime} }),\\ {F}_{{\rm{d}}} & = & \rho {c}_{p}w\bar{T}({{\rm{\nabla }}}_{T}-{{\rm{\nabla }}}_{\mathrm{ad}}-{{\rm{\nabla }}}_{\mu }{\omega }_{T}^{{\prime} }/{\omega }_{X}^{{\prime} }).\end{array}\end{eqnarray} \tag{ 30 }$$

For the convective velocities in the Schwarzschild and Ledoux regimes, we get (for ${k}_{z}\ll {k}_{x,y}$ )

$$\begin{eqnarray}&&w={l}_{\mathrm{conv}}\sqrt{\displaystyle \frac{g}{{h}_{p}}({{\rm{\nabla }}}_{T}-{{\rm{\nabla }}}_{\mathrm{ad}}-{{\rm{\nabla }}}_{\mu })},\end{eqnarray} \tag{ 31 }$$

while for the diabatic convective instability, we can approximate the growth rate and convective velocity with (see the Appendix for details)

$$\begin{eqnarray}&&w={l}_{\mathrm{conv}}\displaystyle \frac{{{\rm{\nabla }}}_{\mu }{\omega }_{T}^{{\prime} }-({{\rm{\nabla }}}_{T}-{{\rm{\nabla }}}_{\mathrm{ad}}){\omega }_{X}^{{\prime} }}{({H}_{T}{R}_{X}-{H}_{X}{R}_{T}){h}_{p}/g-({{\rm{\nabla }}}_{T}-{{\rm{\nabla }}}_{\mathrm{ad}}-{{\rm{\nabla }}}_{\mu })}.\end{eqnarray} \tag{ 32 }$$

Ideally, the mixing length l_conv can be estimated from the typical scale at which the most unstable linear mode saturates nonlinearly (e.g., the typical finger length in the fingering convective case). This estimation does not seem easy to obtain in the general case, and we take l_conv as an adjustable parameter, as often done in the mixing-length theory. The growth rate can also be evaluated numerically directly from the dispersion relation. Of course, this convective velocity is always positive and set to zero as soon as the instability criterion is not met. The existence of the adiabatic and diabatic convective fluxes, in addition to the nonconvective radiative flux, can give rise to spatial or temporal bifurcations between the three energy transport regimes:

• 1.  
  spatial bifurcation with, e.g., the formation of the thermohaline staircases (Turner 1967); and
• 2.  
  temporal bifurcation, which could explain the L/T transition during the cooling sequence of brown dwarfs (Tremblin et al. 2016).

Using the small-scale parameterization of the convective transport, we can now solve for the energy and compositional conservation equations in the whole atmosphere of a brown dwarf in steady state:

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{\partial {F}_{{\rm{d}}}}{\partial z} & = & \rho {c}_{p}H,\\ \displaystyle \frac{\partial {X}_{{\rm{d}}}}{\partial z} & = & \rho R.\end{array}\end{eqnarray} \tag{ 33 }$$

Note that this system automatically includes out-of-equilibrium effects through the reactive equation, since the compositional convective flux can quench chemical source terms. We apply this mixing-length model to the CO/CH₄ radiative convection with the source terms described by Equation (16). Unfortunately, this system is quite stiff numerically because the factor ${\omega }_{T}^{{\prime} }/{\omega }_{X}^{{\prime} }\,={\tau }_{\mathrm{chem}}/{\tau }_{\mathrm{rad}}$ can become very large when the chemistry starts to be slow. We have included the molecular diffusion timescale given by ${l}_{\mathrm{conv}}^{2}/{\kappa }_{\mu }$ to limit ${\tau }_{\mathrm{chem}}$ (with κ_μ the molecular diffusion coefficient). We have also implemented this mixing-length parameterization in a toy model with gray radiative transfer and used a limiter approach to incrementally increase ${\tau }_{\mathrm{chem}}/{\tau }_{\mathrm{rad}}$ in the atmosphere. We show the result of this model in Figure 3 on a gray atmosphere with an effective temperature of 825 K, a gravity of 10⁴ cm s⁻², and a gray opacity profile of the form

$$\begin{eqnarray}&&\kappa ={\kappa }_{\max }{e}^{-{az}/{z}_{\max }},\end{eqnarray} \tag{ 34 }$$

where z_max is the thickness of the atmosphere in the model (1% of Jupiter radius), z is the local elevation from the bottom, a is a dimensionless parameter fixed at 5, and κ_max is 0.1 cm² g⁻¹ in order to get a temperature profile representative of a nongray brown dwarf model. The pressure at the bottom of the atmosphere is fixed at 1 kbar. The temperature gradient of the purely radiative model corresponds to γ_eff = 1.4 with $\partial \mathrm{log}T/\partial \mathrm{log}P=(1-{\gamma }_{\mathrm{eff}}^{-1})$. For the mixing-length model, we have used a mixing length at 0.016 times the local pressure scale height, and the limiter is set at ${({\tau }_{\mathrm{chem}}/{\tau }_{\mathrm{rad}})}_{\max }$ = 10⁶. The model with this value of the limiter is converged for this choice of mixing length, but we point out that with a higher mixing length, it might be needed to increase the value of the limiter to get a converged model.

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Figure 3 shows that we can indeed reduce the temperature gradient with CO/CH₄ radiative convection in a large part of the atmosphere, from 50 bars to approximately 1 bar, compared with the radiative or adiabatic profiles. The corresponding γ_eff value is going down to 1.1 in that model (to be compared with the adiabatic exponent γ ≈ 1.4), which is qualitatively what is needed to explain the reddening of brown dwarf spectra as shown in Figures 4 (γ_eff = 1.03 for PSO J318.5338–22.8603) and 5 (γ_eff = 1.15 for HN Peg b).

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The nature of the temperature gradient reduction is relatively clear in the case of moist convection. The direct release or pumping of latent heat clearly leads to such an effect. Even though we can demonstrate with the generalized mixing-length theory that the same phenomenon should arise for thermohaline and CO/CH₄ radiative convection, the explanation of this effect seems more subtle. In order to illustrate this mechanism, we present a series of idealized 2D stratified hydrodynamic simulations in the Ledoux unstable regime using the code ARK⁸ developed in Padioleau et al. (2019). The code is fully compressible and explicit, which means that we have a strong constraint on the size of the timestep that is limited by the speed of sound. It is therefore difficult to reach the long timescales needed to explore the diabatic or fingering instability directly. Nevertheless, we can already perform simulations in the Ledoux unstable regime and show that compositional source terms can impact and reduce the temperature gradient even in that regime.

The system is composed of the Euler equations (Equation (1)) with a gravitational and radiative source term and a reactive scalar field. The source terms are given by Equation (16). Here X_eq is a linear profile between 1 and 0 from the bottom to the top of the atmosphere. The mean molecular weight is given by 1/μ = X/μ₁ + (1 − X)/μ₂, and we will force a mean molecular weight gradient with μ₁ < μ₂. When the chemical source term is included, we have used τ_chem = 20 s. The radiative transfer is solved with a gray two-stream scheme adapted from the 1D/2D code ATMO (Drummond et al. 2016; Tremblin et al. 2016) with the method from Bueno & Bendicho (1995). The incoming radiative flux at the bottom is set to a radiative temperature of T_rad,zmin = 1100 K, and the downward radiative intensity at the top is set to zero. The opacity κ is set to 5 × 10⁻⁴ cm² g⁻¹, i.e., H_X = 0, and we explore the situation when the mean molecular weight gradient is the driving mechanism. The hydrodynamic is solved using a finite-volume "all-regime" scheme well-suited for low- and high-Mach flows (Chalons et al. 2016a) and well-balanced for gravity (Chalons et al. 2016b; Padioleau et al. 2019). The well-balanced property of the solver allows the scheme to capture the hydrostatic balance $\partial {P}_{0}/\partial z=-{\rho }_{0}g$ down to machine precision. With radiative transfer, we precompute the discretized PT profile, satisfying both the hydrostatic balance and H(X, P, T) = 0 for the discretized numerical scheme. By initializing the simulation on this profile, we are able to preserve the equilibrium state with vertical velocities lower than 10⁻⁵ cm s⁻¹, even with an equilibrium state unstable to convection. As a consequence, the quality of this numerical scheme allows us to precisely study the unstable convective mechanism, and we need to explicitly introduce a velocity perturbation to trigger the instability,

$$\begin{eqnarray}&&{u}_{0,z}(x,z)={{Ac}}_{s}\sin (m\pi x/{x}_{\max }){e}^{-{\left(z-0.5{z}_{\max }\right)}^{2}/{\left({{Wz}}_{\max }\right)}^{2}},\end{eqnarray} \tag{ 35 }$$

where c_s is the local sound speed, and x_max and z_max are the horizontal and vertical extent of the box, respectively. For all of the simulations in this paper, we have used the same perturbation with A = 10⁻⁴, W = 0.25, and m = 2 in a box with x_max = 2.5 × 10⁴ cm and ${z}_{\max }=5\times {10}^{4}\,\mathrm{cm}$, and the pressure at the bottom of the atmosphere is fixed at P_zmin = 10 bars. The total time of the simulations is 1600 s, i.e., 80 times the chemical timescale. All of the simulations have reached a quasi-steady state at 800 s, and we averaged the final profiles between 800 and 1600 s.⁹ The resolution is fixed at 100 × 50 cells, and the scheme has a spatial and temporal discretization at first order.

We have performed the first two simulations without a mean molecular weight gradient (μ₁ = μ₂ = 2.4) and without a reactive source term (R(X, P, T) = 0). The initial PT profile of this setup corresponds to an equivalent gamma index of γ_eff = 1.33 (red curve in Figure 7). We present the averaged temperature profile of the simulations in Figure 6. The first simulation is done with an adiabatic index of the gas γ = 1.4, i.e., stable to Schwarzschild convection. The velocity perturbation is damped, and the atmosphere stays at rest. The second simulation is done with an adiabatic index of the gas γ = 1.2, unstable to Schwarzschild convection. Convective motions mix the atmosphere, and the final PT profile has an equivalent gamma index of γ_eff = 1.19. These two simulations show that the code properly captures Schwarzschild convection, similar to Padioleau et al. (2019) with Rayleigh–Benard convection.

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In the three other simulations in Figure 7, the initial PT profile also corresponds to an equivalent gamma index of γ_eff = 1.33, and we have kept γ = 1.4 so that the atmosphere is stable to Schwarzschild convection. We introduce in these simulations a mean molecular weight gradient with μ₁ = 2.4 and μ₂ = 2.41 or 2.42 so that the atmosphere is unstable to the Ledoux criterion, and we have verified that we get back the Ledoux growth rate in the linear regime. The first simulation is done with no reactive source term, R(X, P, T) = 0. In this situation, the final averaged PT profile has a temperature gradient similar to the initial one with γ_eff ≈ 1.33. The second simulation is done with the chemical source term and μ₂ = 2.41. In that case, an exchange of energy can be performed through the compositional change induced by the chemistry, which then can lead to heating and cooling by the work of pressure adjustments due to expansion/contraction induced by the EOS (increase/decrease of temperature at constant entropy). In this simulation, the final averaged PT profile has a significantly reduced temperature gradient with an equivalent gamma index of γ_eff = 0.99 (magenta curve in Figure 7). In the third simulation, we have increased the mean molecular weight gradient with μ₂ = 2.42. In that case, the temperature gradient reduction is stronger, with γ_eff = 0.77 (green curve in Figure 7) showing that the magnitude of the effect is a function of the mean molecular weight gradient.

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A blob description helps explain why the temperature gradient can be decreased. Figure 8 illustrates the behavior of a perturbed hot parcel with a high concentration of heavy elements. Because the blob is Ledoux unstable, its entropy is lower than the surroundings. On a compositional timescale for the source term, this excess of composition will be dissipated in the nonlinear regime. If we ignore the first energy source terms (no source term on the entropy equation), the compositional change will happen at constant entropy; hence, it will lead to a temperature of the blob that is smaller than the temperature of the surroundings because of the expansion induced by the change in concentration (similar to haline expansion in the oceanic case) and the work of the pressure forces. If we reintroduce the thermal source terms in the analysis, the blob will pump energy from the environment, since it has a lower temperature; hence, it will cool the deep cold atmosphere. That is the reason why the temperature gradient relative to pressure can be decreased even more in the presence of diabatic processes (even in the Ledoux regime). Of course, the intermediate state (${T}_{{b}^{{\prime} }},{S}_{{b}^{{\prime} }}$ in Figure 8) in which we ignored the energy source terms is a fictive state: the energy pumping by energy source terms will happen continuously during the compositional change phase if the energy source terms are faster than the compositional source terms. We suggest that this picture might also apply in the case of the formation of fingering steps in the thermohaline context.

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Leconte (2018) tried to argue that any type of convection should always lead to an adiabatic PT profile. In general, an adiabatic PT profile requires at least an adiabatic energy equation; i.e., the energy source term should be zero or relatively small. This is not necessarily the case in the radiative part of an atmosphere or in the presence of latent heat release for Earth moist convection. We point out that this is also not the case for thermohaline convection (see Figure 3 in Radko 2014). The 2D simulations presented in this paper also clearly show that as soon as a compositional source term is included in the problem, which is absent in the analysis presented in Leconte (2018), the temperature gradient can be reduced compared with an adiabatic one.

The 2D simulations we have performed are very idealized and in the Ledoux unstable regime. This is why we only use them to illustrate that a compositional source term can significantly impact the thermal profile of an atmosphere. The extension of the simulation box is 1% of a scale height, the mean molecular weight gradient is 10 times more than the gradient induced by the CO/CH₄ transition, the opacity is relatively small, and we so far neglect the dependence of the energy source term on composition. Going to more realistic conditions will be the challenge for future works because it will require large computational resources to really explore the diabatic instability at stake in brown dwarf atmospheres.

To derive the Schwarzschild criterion, we can eliminate δX from the system and assume H = 0 in an adiabatic environment. The matrix reduces to

$$\begin{eqnarray}\left(\begin{array}{cccc}{k}_{z}g & 0 & 0 & {{ik}}^{2}\\ g & \omega {\rho }_{0} & 0 & {{ik}}_{z}\\ 0 & -\tfrac{{T}_{0}}{{h}_{p}}\left({{\rm{\nabla }}}_{T}-{{\rm{\nabla }}}_{\mathrm{ad}}\right) & \omega & 0\\ -1/{\rho }_{0} & 0 & -1/{T}_{0} & 0\end{array}\right),\end{eqnarray} \tag{ 40 }$$

whose determinant is given by

$$\begin{eqnarray}&&\begin{array}{l}{k}_{z}{{gik}}_{z}\displaystyle \frac{g}{{h}_{p}}({{\rm{\nabla }}}_{T}-{{\rm{\nabla }}}_{\mathrm{ad}})-{{ik}}^{2}\displaystyle \frac{g}{{h}_{p}}({{\rm{\nabla }}}_{T}-{{\rm{\nabla }}}_{\mathrm{ad}})+{{ik}}^{2}{\omega }^{2}=0\\ {\omega }^{2}-({{\rm{\nabla }}}_{T}-{{\rm{\nabla }}}_{\mathrm{ad}})\displaystyle \frac{{k}_{x}^{2}+{k}_{y}^{2}}{{k}^{2}}\displaystyle \frac{g}{{h}_{p}}=0.\end{array}\end{eqnarray} \tag{ 41 }$$

This equation admits a positive real solution (which corresponds to the onset of adiabatic convective instability) if and only if ${{\rm{\nabla }}}_{T}-{{\rm{\nabla }}}_{\mathrm{ad}}\gt 0$.

We derive the Ledoux criterion for convection assuming an adiabatic environment H = 0 and a nonreactive/diffusive fluid R = 0.

The determinant of the matrix is given by

$$\begin{eqnarray}&&\begin{array}{l}-\,i({k}_{x}^{2}+{k}_{y}^{2})\omega \displaystyle \frac{g}{{h}_{p}}({{\rm{\nabla }}}_{T}-{{\rm{\nabla }}}_{\mathrm{ad}})+{{ik}}^{2}{\omega }^{3}\\ \quad -\,i({k}_{x}^{2}+{k}_{y}^{2})\displaystyle \frac{\partial \mathrm{log}{\mu }_{0}}{\partial z}g\omega =0\\ {\omega }^{2}-\displaystyle \frac{g}{{h}_{p}}({{\rm{\nabla }}}_{T}-{{\rm{\nabla }}}_{\mathrm{ad}}-{{\rm{\nabla }}}_{\mu })\displaystyle \frac{{k}_{x}^{2}+{k}_{y}^{2}}{{k}^{2}}=0\end{array}\end{eqnarray} \tag{ 42 }$$

with ${{\rm{\nabla }}}_{\mu }=-{h}_{p}\partial \mathrm{log}{\mu }_{0}/\partial z$, and we get the Ledoux criterion for convective instabilities ${{\rm{\nabla }}}_{T}-{{\rm{\nabla }}}_{\mathrm{ad}}-{{\rm{\nabla }}}_{\mu }\gt 0$. So, a region unstable to the Schwarzschild criterion can be stabilized by a μ gradient if ${{\rm{\nabla }}}_{\mu }\gt {{\rm{\nabla }}}_{T}-{{\rm{\nabla }}}_{\mathrm{ad}}\gt 0$.

With ${R}_{X,T}\ne 0$ and ${H}_{T,X}\ne 0$, the determinant is now given by a third-degree polynomial ${\omega }^{3}+{a}_{2}{\omega }^{2}+{a}_{1}\omega +{a}_{0}=0$ with

$$\begin{eqnarray}\begin{array}{rcl}{a}_{2} & = & -{R}_{X}-{H}_{T}\\ {a}_{1} & = & {H}_{T}{R}_{X}-{H}_{X}{R}_{T}-\displaystyle \frac{{k}_{x}^{2}+{k}_{y}^{2}}{{k}^{2}}\displaystyle \frac{g}{{h}_{p}}({{\rm{\nabla }}}_{T}-{{\rm{\nabla }}}_{\mathrm{ad}}-{{\rm{\nabla }}}_{\mu })\\ {a}_{0} & = & \displaystyle \frac{{k}_{x}^{2}+{k}_{y}^{2}}{{k}^{2}}\displaystyle \frac{g}{{h}_{p}}(({{\rm{\nabla }}}_{T}-{{\rm{\nabla }}}_{\mathrm{ad}}){R}_{X}-{{\rm{\nabla }}}_{\mu }{H}_{T})\\ & & +\displaystyle \frac{{k}_{x}^{2}+{k}_{y}^{2}}{{k}^{2}}{T}_{0}\displaystyle \frac{g}{{h}_{p}}({{\rm{\nabla }}}_{T}-{{\rm{\nabla }}}_{\mathrm{ad}})\displaystyle \frac{\partial \mathrm{log}{\mu }_{0}}{\partial X}{R}_{T}\\ & & +\displaystyle \frac{{k}_{x}^{2}+{k}_{y}^{2}}{{k}^{2}}g\displaystyle \frac{1}{{T}_{0}}{H}_{X}\displaystyle \frac{\partial {X}_{0}}{\partial z}.\end{array}\end{eqnarray} \tag{ 43 }$$

According to the Hurwitz criterion (e.g., Kato 1966), one of the roots of the polynomial has a positive real part if at least one of the coefficients is negative (when they are all nonzero), a₀ < 0, a₁ < 0, or a₂ < 0. The conditions a₂ < 0, a₁ < 0 lead to the following inequalities:

$$\begin{eqnarray}&&\begin{array}{l}-{R}_{X}-{H}_{T}\lt 0\\ {{\rm{\nabla }}}_{T}-{{\rm{\nabla }}}_{\mathrm{ad}}-{{\rm{\nabla }}}_{\mu }\gt ({H}_{T}{R}_{X}-{H}_{X}{R}_{T})\displaystyle \frac{{k}^{2}}{{k}_{x}^{2}+{k}_{y}^{2}}\displaystyle \frac{{h}_{p}}{g}.\end{array}\end{eqnarray} \tag{ 44 }$$

For realistic physical conditions, R_X < 0 and H_T < 0; hence, the first inequality is never met. The second inequality reduces to the Ledoux criterion because when ${{\rm{\nabla }}}_{T}-{{\rm{\nabla }}}_{\mathrm{ad}}-{{\rm{\nabla }}}_{\mu }\gt 0$, we can always find wavenumbers $({k}_{x},{k}_{y},{k}_{z})$ that will satisfy the inequality. Note that if all of the coefficients are positive, we can still have an instability if ${a}_{2}{a}_{1}\lt {a}_{0}$, according to the Hurwitz criterion; the interpretation of this instability condition remains an open question.

The instability condition a₀ < 0 leads to the following inequality:

$$\begin{eqnarray}&&({{\rm{\nabla }}}_{T}-{{\rm{\nabla }}}_{\mathrm{ad}}){\omega }_{X}^{{\prime} }-{{\rm{\nabla }}}_{\mu }{\omega }_{T}^{{\prime} }\lt 0,\end{eqnarray} \tag{ 45 }$$

with

$$\begin{eqnarray}\begin{array}{rcl}{\omega }_{X}^{{\prime} } & = & {R}_{X}+{T}_{0}{R}_{T}\displaystyle \frac{\partial \mathrm{log}{\mu }_{0}}{\partial X}\\ {\omega }_{T}^{{\prime} } & = & {H}_{T}+\displaystyle \frac{1}{{T}_{0}}{H}_{X}{\left(\displaystyle \frac{\partial \mathrm{log}{\mu }_{0}}{\partial X}\right)}^{-1}.\end{array}\end{eqnarray} \tag{ 46 }$$

This criterion describes the diabatic instability linked to the presence of source terms and is the one that encompasses thermohaline/fingering and moist convection. Note that the instability criterion remains well defined in the limit $\partial \mathrm{log}{\mu }_{0}/\partial X=0$. In that case, the inequality tends to

$$\begin{eqnarray}&&({{\rm{\nabla }}}_{T}-{{\rm{\nabla }}}_{\mathrm{ad}}){R}_{X}-{{\rm{\nabla }}}_{X}{X}_{0}{H}_{X}/{T}_{0}\lt 0.\end{eqnarray} \tag{ 47 }$$

We can also approximate the growth rate of the instability by assuming that the polynomial reduces to ${a}_{1}\omega +{a}_{0}\sim 0$ for the diabatic instability in the limit of small growth rates when a₁ > 0 and a₀ < 0. We then get an approximate formula for the growth rate for k_z ≪ k_{x, y}:

$$\begin{eqnarray}&&\omega \sim \displaystyle \frac{{{\rm{\nabla }}}_{\mu }{\omega }_{T}^{{\prime} }-({{\rm{\nabla }}}_{T}-{{\rm{\nabla }}}_{\mathrm{ad}}){\omega }_{X}^{{\prime} }}{({H}_{T}{R}_{X}-{H}_{X}{R}_{T}){h}_{p}/g-({{\rm{\nabla }}}_{T}-{{\rm{\nabla }}}_{\mathrm{ad}}-{{\rm{\nabla }}}_{\mu })}.\end{eqnarray} \tag{ 48 }$$

When this approximation is not valid, the growth rate can be evaluated numerically directly from the dispersion relation.

Let us assume that the reactive terms for composition are fast and at an equilibrium given by $X={X}_{\mathrm{eq}}(P,T)$.

The criterion can be written in the following form:

$$\begin{eqnarray}&&\begin{array}{l}{{\rm{\nabla }}}_{T}-{{\rm{\nabla }}}_{\mathrm{ad}}-{{\rm{\nabla }}}_{\mu }\displaystyle \frac{{\omega }_{T}^{{\prime} }}{{\omega }_{X}^{{\prime} }}\gt 0\\ {{\rm{\nabla }}}_{T}-{{\rm{\nabla }}}_{\mathrm{ad}}-{\left(\displaystyle \frac{\partial \mathrm{log}{P}_{0}}{\partial z}\right)}^{-1}\displaystyle \frac{\partial \mathrm{log}{\mu }_{0}}{\partial X}\displaystyle \frac{\partial {X}_{\mathrm{eq}}}{\partial z}\displaystyle \frac{{\omega }_{T}^{{\prime} }}{{\omega }_{X}^{{\prime} }}\gt 0\\ {{\rm{\nabla }}}_{T}-{{\rm{\nabla }}}_{\mathrm{ad}}-\displaystyle \frac{\partial \mathrm{log}{\mu }_{0}}{\partial X}\left(\displaystyle \frac{\partial {X}_{\mathrm{eq}}}{\partial \mathrm{log}T}{{\rm{\nabla }}}_{T}+\displaystyle \frac{\partial {X}_{\mathrm{eq}}}{\partial \mathrm{log}P}\right)\displaystyle \frac{{\omega }_{T}^{{\prime} }}{{\omega }_{X}^{{\prime} }}\gt 0\\ {{\rm{\nabla }}}_{T}-{{\rm{\nabla }}}_{\mathrm{ad}}\displaystyle \frac{1+\tfrac{1}{{{\rm{\nabla }}}_{\mathrm{ad}}}\tfrac{\partial \mathrm{log}{\mu }_{0}}{\partial X}\tfrac{\partial {X}_{\mathrm{eq}}}{\partial \mathrm{log}P}\tfrac{{\omega }_{T}^{{\prime} }}{{\omega }_{X}^{{\prime} }}}{1-\tfrac{\partial \mathrm{log}{\mu }_{0}}{\partial X}\tfrac{\partial {X}_{\mathrm{eq}}}{\partial \mathrm{log}T}\tfrac{{\omega }_{T}^{{\prime} }}{{\omega }_{X}^{{\prime} }}}\gt 0,\end{array}\end{eqnarray} \tag{ 49 }$$

assuming that ${\omega }_{X}^{{\prime} }\lt 0$ and $1-\tfrac{\partial \mathrm{log}{\mu }_{0}}{\partial X}\tfrac{\partial {X}_{\mathrm{eq}}}{\partial \mathrm{log}T}\tfrac{{\omega }_{T}^{{\prime} }}{{\omega }_{X}^{{\prime} }}\gt 0$ (always verified for moist convection).

We assume that the reaction source term R is condensation/evaporation of water and the thermal source term is the corresponding release or pumping of latent heat L, and we neglect the temperature dependence of the latent heat. In that case, the criterion becomes

$$\begin{eqnarray}\begin{array}{rcl}{\omega }_{X}^{{\prime} } & = & {R}_{X}+{T}_{0}{R}_{T}\displaystyle \frac{\partial \mathrm{log}{\mu }_{0}}{\partial X}\\ {\omega }_{T}^{{\prime} } & = & -\displaystyle \frac{1}{{T}_{0}}{\left(\displaystyle \frac{\partial \mathrm{log}{\mu }_{0}}{\partial X}\right)}^{-1}\displaystyle \frac{{\omega }_{X}^{{\prime} }L}{{c}_{p}}\\ {{\rm{\nabla }}}_{T} & - & {{\rm{\nabla }}}_{\mathrm{ad}}\displaystyle \frac{1-{\rho }_{0}\tfrac{\partial {X}_{\mathrm{eq}}}{\partial P}L}{1+\tfrac{\partial {X}_{\mathrm{eq}}}{\partial T}\tfrac{L}{{c}_{p}}}\gt 0.\end{array}\end{eqnarray} \tag{ 50 }$$

For Earth's atmosphere, we can then express ${X}_{\mathrm{eq}}(P,T)$ as a function of vapor pressure at saturation f(T), and R_v and R_d are the vapor and dry air gas constant:

$$\begin{eqnarray}\begin{array}{rcl}{X}_{\mathrm{eq}}(P,T) & \approx & \displaystyle \frac{{R}_{d}}{{R}_{v}}\displaystyle \frac{f(T)}{P}\\ f(T) & = & 6.11\exp \left(\displaystyle \frac{L}{{R}_{v}}\left(\displaystyle \frac{1}{273}-\displaystyle \frac{1}{T}\right)\right)\quad \mathrm{hPa}\\ \displaystyle \frac{\partial {X}_{\mathrm{eq}}}{\partial T} & \approx & \displaystyle \frac{{{LX}}_{\mathrm{eq}}}{{R}_{v}{T}^{2}}\\ \displaystyle \frac{\partial {X}_{\mathrm{eq}}}{\partial P} & \approx & -\displaystyle \frac{{X}_{\mathrm{eq}}}{\rho {R}_{d}T},\end{array}\end{eqnarray} \tag{ 51 }$$

which gives the standard criterion for Earth moist convection (see Stevens 2005),

$$\begin{eqnarray}&&{{\rm{\nabla }}}_{T}-{{\rm{\nabla }}}_{\mathrm{ad}}\displaystyle \frac{1+\tfrac{{X}_{\mathrm{eq}}L}{{R}_{d}{T}_{0}}}{1+\tfrac{{X}_{\mathrm{eq}}{L}^{2}}{{c}_{p}{R}_{v}{T}_{0}^{2}}}\gt 0.\end{eqnarray} \tag{ 52 }$$