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
• Multicolumn and global mode - pseudo-global climate via coupled 1D columns

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.

Exobase diagnostic

AGNI estimates where the exobase, the level at which the collisional mean free path equals the pressure scale height, might exist within the modelled atmosphere. It calculates this based on the $n$ is the number density, $\bar{m}$ the mean particle mass, and $\sigma = \sum_j x_j \pi \sigma_j^2$ the mole-fraction-weighted hard-sphere cross-section from the Lennard-Jones collision diameters. If none of the layers reach this criterion, the exobase variables are set equal to the TOA values.

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