Model description

AGNI is a 1D radiative-convective atmosphere model designed to simulate the climates of rocky exoplanets under extreme conditions [1, 2].

It models a planetary atmosphere as a single column split into $N$ pressure levels (cell-centres) bounded by $N+1$ cell-edge interfaces. Levels are distributed logarithmically in pressure between the surface and the top of the atmosphere. The atmosphere is assumed to be plane-parallel.

Thermodynamic quantities (temperature, pressure, heat capacity, density, composition) are calculated at level-centres; energy fluxes are calculated at level-edges.

Physics overview

AGNI accounts for the following energy transport processes:

The total net upward-directed energy flux at each cell edge $l$ is the sum of all contributions:

\[F_l = F^\text{sns}_l + F^\text{cvt}_l + F^\text{lat}_l + F^\text{rad}_l + F^\text{cdt}_l\]

For a detailed description of each process, see the corresponding pages in this section:

• Radiative transfer - correlated-k RT with SOCRATES, opacity sources, greenhouse physics
• Atmospheric convection - mixing-length theory, Schwarzschild criterion, convective shutdown
• Thermodynamics and EOS - equations of state, heat capacities, saturation pressures, latent heats
• Composition and chemistry - condensation scheme, ocean model, FastChem equilibrium chemistry
• Sensible heat flux - turbulent surface exchange
• Height structure - hydrostatic integration, self-gravity
• Advective heating - deep heat deposition profile

Obtaining a solution

AGNI is designed for modelling planetary atmospheres with high surface pressures and temperatures. Radiative timescales can differ by many orders of magnitude across the column, which makes time-stepping methods slow to converge. AGNI instead solves for the temperature structure as an optimisation problem - finding the $T(p)$ profile that exactly conserves energy at all levels simultaneously.

Solution types

The solution_type configuration variable specifies the boundary condition for the surface temperature $T_s$:

• (1) Fixed surface temperature. $T_s$ is held constant. Useful for computing the OLR for a prescribed surface temperature.
• (2) Conductive boundary layer. $T_s$ is determined by energy balance through a thin conductive boundary layer (CBL) of thickness $d_c$ and conductivity $\kappa_c$: $F^\text{atm} = \kappa_c (T_m - T_s) / d_c$, where $T_m$ is the prescribed mantle temperature. This is used when coupling AGNI to a magma ocean interior model within the PROTEUS framework [2, 3].
• (3) Free surface temperature. $T_s$ is solved for alongside the rest of the atmosphere, subject to a prescribed net flux $F^\text{atm} = \sigma T_\text{int}^4$, where $T_\text{int}$ is an internal (intrinsic) temperature representing interior heat production.
• (4) Target OLR. $T_s$ and the temperature profile are solved such that the outgoing longwave radiation at the top of the atmosphere matches a user-specified target value.

Construction

The energy flux schemes in AGNI all take cell-centre temperatures $T_l$, pressures $p_l$, geometric heights, and mixing ratios as input variables at each level. In return, they provides energy fluxes $F_l$ at all $N+1$ interfaces.

The total upward-directed energy flux $F_l$ describes the total upward-directed energy transport (units of $\text{W m}^{-2}$) from cell $l$ into cell $l-1$ above (or into space for $l=1$). For energy to be conserved throughout the column, it must be true that $F_l = F_t \text{ } \forall \text{ } l$ where $F_t$ is the total amount of energy being transported out of the planet. In global radiative equilibrium, $F_t = 0$.

Definition of residuals

We solve for the temperature profile of the atmosphere as an $N+1$-dimensional optimisation problem. This directly solves for $T(p)$ at radiative-convective equilibrium without invoking heating-rate calculations, thereby avoiding slow convergence in regions with long radiative timescales. The residuals vector (length $N+1$)

\[\bm{r} = \begin{pmatrix} r_l \\ r_{l+1} \\ \vdots \\ r_N \\ r_{N+1} \end{pmatrix} = \begin{pmatrix} F_{l+1} - F_l \\ F_{l+2} - F_{l+1} \\ \vdots \\ F_{N+1} - F_N \\ F_{N+1} - F^\text{atm} \end{pmatrix}\]

is what we minimise as our objective function, subject to the solution vector of cell-centre temperatures

\[\bm{x} = \begin{pmatrix} T_l \\ T_{l+1} \\ \vdots \\ T_N \\ T_s \end{pmatrix}\]

where $T_s$ is the surface temperature. Cell-edge temperatures in the bulk atmosphere are interpolated from cell-centres. The bottom- and top-most cell-edge temperatures are extrapolated by estimation of $dT/d\log p$. Cell properties (heat capacity, gravity, density, mean molecular weight) are consistently updated at each evaluation of $\bm{r}$. Condensation and rainout are also handled at each evaluation to avoid supersaturation.

The model converges when the cost function

\[\mathcal{C}(\bm{x}) = \left(\sum_l |r_l|^3\right)^{1/3}\]

satisfies

\[\mathcal{C}(\bm{x}) < \mathcal{C}_a + \mathcal{C}_r \cdot \underset{l}{\max}\ |F_l|\]

which represents a state where fluxes are sufficiently conserved. The quantities $\mathcal{C}_a$ and $\mathcal{C}_r$ are the absolute and relative tolerances provided by the user (parameters conv_atol and conv_rtol in solver.solve_energy!()).

Iterative steps

The model solves for $\bm{x}$ iteratively starting from some initial guess. The flowchart below broadly outlines the solution process.

The Jacobian matrix $\bm{J}$ represents the directional gradient of the residuals with respect to the solution vector:

\[J_{uv} = \frac{\partial r_u}{\partial x_v}\]

AGNI estimates $\bm{J}$ using a finite-difference scheme, requiring $2(N+1)$ evaluations of $\bm{r}$. To reduce the total number of calculations, AGNI retains Jacobian columns between iterations when the local residuals are small - this assumes the second derivative of the residuals is small in converged regions.

With a Jacobian constructed, the Newton–Raphson update step is

\[\bm{d} = -\bm{J}^{-1} \bm{r}\]

Gauss-Newton and Levenberg–Marquardt methods are also supported as alternatives.

Three stabilisation strategies are applied to $\bm{d}$:

1. Nudging: when the model is stuck at a local minimum ($|\bm{d}|$ is on a plateau), $\bm{d}$ is scaled up.
2. Clipping: when $|\bm{d}| > d_\text{max}$, the step is scaled back to $d_\text{max} \hat{\bm{d}}$ to prevent instabilities from the non-convex solution space.
3. Line search: if the full step would increase the cost, $\bm{d}$ is scaled by a factor $\alpha$ found by golden-section or backtracking line search. This avoids oscillation around the solution.

Other features

Emission spectra

AGNI can calculate emission spectra from a given $T(p)$ profile and gas volume mixing ratios. The longwave contribution function can also be computed.

Transparent atmospheres

For bare-rock planets, or to determine a planet's surface temperature in the absence of an atmosphere, AGNI supports a transparent atmosphere mode (composition.transparent = true). This sets the atmospheric pressure to a small value and disables all gas opacity and absorption in SOCRATES. Use the dedicated transparent solver in this case.

Julia and Fortran

AGNI is primarily written in Julia [4], while SOCRATES is written in Fortran. Julia's precompilation allows the SOCRATES Fortran binaries to be packaged inside the precompiled Julia code, which significantly improves startup performance.

Bibliography for this page