Atmospheric convection

Physical background

Convection is a turbulent process that transfers heat by bulk fluid motion. In a planetary atmosphere, a parcel of gas that is warmer (less dense) than its surroundings will rise, expand, and cool adiabatically. If it remains warmer than the environment throughout its ascent, buoyancy drives continued upward motion (the atmosphere is convectively unstable) and heat is transported upward. The atmosphere is convectively stable in regions where energy transported primarily by radiation.

Convection can be driven by internal heat production or the absorption of shortwave stellar radiation in the deep atmosphere. Atmospheres are not always unstable to convection, leaving a stable radiative layers [2, 5]. The general pattern of a convective troposphere below a radiative stratosphere is found across Solar System bodies with thick atmospheres and may hold broadly for planetary atmospheres [6]. AGNI does not enforce this structure a priori; the solver determines which layers are convective and which are radiative self-consistently.

Dry adiabat and Schwarzschild criterion

For an ideal gas parcel undergoing dry (non-condensing) adiabatic ascent, the dry adiabatic lapse rate is

\[\nabla_\text{ad} := \left.\frac{d \ln T}{d \ln p}\right|_\text{ad} = \frac{R}{\mu \, c_{pm}}\]

where $R = 8.314\ \mathrm{J\ mol^{-1}\ K^{-1}}$ is the universal gas constant, $\mu$ is the local mean molecular weight of the gas, and $c_{pm}$ is the molar heat capacity at constant pressure.

The Schwarzschild criterion for convective instability is then

\[\left|\frac{d \ln T}{d \ln p}\right|_\text{atm} > \left|\frac{d \ln T}{d \ln p}\right|_\text{ad}\]

In this case, the atmosphere becomes unstable when its lapse rate exceeds that of an adiabatically rising parcel [7, 8]. This criterion applies in the absence of composition gradients.

The more general Ledoux criterion reduces the tendency to convect due to MMW gradients [7]. In the ideal-gas limit it can be written

\[\nabla_\text{ld} = \nabla_\text{ad} + \frac{d \ln \mu}{d \ln p}\]

AGNI supports checking against either criterion, selected via the physics.convection_crit configuration option ('s' for Schwarzschild, 'l' for Ledoux). The Ledoux criterion is not self-consistently supported when a real-gas equation of state is in use (see below) since the $d\ln\mu/d\ln p$ correction term neglects the non-ideal contributions to the mixture's thermodynamics. The dry adiabatic lapse rate above assumes an ideal gas compressibility. For a general equation of state, the thermodynamic identity $dT/dp|_\text{ad} = 1/(\rho \, c_{pm})$ still adopts

\[\nabla_\text{ad} = \frac{p}{T \, \rho \, c_{pm}}\]

where $\rho$ is evaluated from the real-gas equation of state (see Gas densities) rather than the ideal-gas law.

Below a minimum pressure (CONVECT_MIN_PRESSURE, $10^{-9}$ bar by default), the convective stability check is skipped entirely and the column is assumed radiative; this avoids evaluating the criterion in the numerically ill-conditioned uppermost layers of the model, where pressures and densities become vanishingly small.

Mixing-length theory

Atmospheric convection is fundamentally multidimensional, so it must be parameterised in a 1D column model. AGNI applies mixing-length theory (MLT) to directly calculate the convective heat flux [9–11]. MLT represents convection by a single characteristic eddy: a rising parcel of gas that diffuses its excess enthalpy over a mixing length $\lambda$ before dissolving back into the environment. This is analogous to treating only the dominant wavenumber of turbulence [12]. MLT remains a common and well-validated approach in the literature [13].

The convective heat flux is [2]:

\[F^\text{cvt} = \frac{1}{2} \rho \, c_{pm} \, w \, T \, \frac{\lambda}{H} \, (\nabla_T - \nabla_\text{ad})\]

where $T$ is the local temperature, $c_{pm}$ is the heat capacity per unit mass, $H = RT/(\mu g)$ is the pressure scale height, $\nabla_T = d\ln T / d\ln p$ is the local atmospheric lapse rate, and

\[w = \lambda \sqrt{\frac{g}{H} \, (\nabla_T - \nabla_\text{ad})}\]

is the characteristic convective velocity. The convective flux is positive (upward) wherever $\nabla_T > \nabla_\text{ad}$.

Two mixing-length parametrisations are available, selected via the mlt_asymptotic option:

• Asymptotic (default): a near-surface formulation

\[\lambda = \frac{k_v z}{1 + k_v z / (\alpha_\text{MLT} H)}\]

where $k_v \approx 1/\sqrt{2\pi}$ is the von Kármán constant and $z$ is the height above the surface [14, 15]. This ensures $\lambda \to 0$ as $z \to 0$, respecting the Law of the Wall for the near-surface turbulent boundary layer, and $\lambda \to \alpha_\text{MLT} H$ aloft.

• Fixed: $\lambda = \alpha_\text{MLT} H$ everywhere, independent of height above the surface.

\[\alpha_\text{MLT}\]

is a dimensionless mixing-length efficiency parameter, analogous to that used in stellar structure models [11]; it is currently fixed at $\alpha_\text{MLT}=1$. In practice the resulting temperature structure and convective flux are not sensitive to the specific choice of $\lambda$ parametrisation.

Eddy diffusion coefficient $K_{zz}$

AGNI calculates the vertical eddy diffusion coefficient $K_{zz}$ (units of m² s⁻¹) to parameterise vertical mixing processes in the atmosphere. This coefficient is used by chemical kinetics codes (e.g. VULCAN). Three parametrisations are available, selectable via the Kzz_type configuration parameter:

Type 1: Constant value

\[K_{zz} = K_{\text{break}}\]

where $K_{\text{break}}$ is a user-specified constant diffusion coefficient.

Type 2: Simple MLT scaling (default)

\[K_{zz} = \lambda_{\text{conv}} \, w_{\text{conv}}\]

where $\lambda_{\text{conv}}$ is the mixing length and $w_{\text{conv}}$ is the convective velocity from MLT, calculated only in convective regions.

Type 3: MLT convective flux scaling

Following Charnay et al. (2015), Equation 16:

\[K_{zz} = \frac{H_p}{3} \left(\frac{\lambda_{\text{conv}}}{H_p}\right)^{4/3} \left(\frac{R_{\text{gas}} F_c}{\mu \rho c_p}\right)^{1/3}\]

where $H_p$ is the pressure scale height, $R_{\text{gas}}$ is the gas constant, $F_c$ is the convective energy flux, $\mu$ is the mean molecular weight, $\rho$ is the density, and $c_p$ is the specific heat capacity at constant pressure.

In non-convective (radiative) regions, $K_{zz}$ is extended using a power-law scaling with pressure:

\[K_{zz}(p) = K_{\text{ref}} \left(\frac{p}{p_{\text{ref}}}\right)^{\alpha}\]

where $K_{\text{ref}}$ and $p_{\text{ref}}$ are the reference diffusion coefficient and pressure at the convective region boundary, and $\alpha$ is the power-law index (default: $\alpha = -0.4$), following [10]. Below convective regions, $K_{zz}$ is held constant at its deepest convective value.

If the atmosphere contains multiple detached convective layers (e.g. separated by an intervening radiative zone), the power-law extension is anchored by the shallowest region with constant extension to the deepest one. Any radiative gaps between separate convective regions are filled with the minimum $K_{zz}$ value found within the convective regions, floored at a configurable minimum Kzz_floor. If the column is entirely radiative, $K_{zz}$ is instead set to a single user-specified reference value Kzz_kbreak at the reference pressure Kzz_pbreak.

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