Conservation Properties

ClimaAtmos conserves total energy, air mass, and water to floating-point precision, without ad hoc fixers. It does so in moist atmospheres, in the presence of subgrid-scale parameterizations, over complex topography, and in a deep atmosphere [1].

Conservation is a design goal, and it constrains choices throughout the model. This page collects those choices in one place and describes what is conserved, what is not, and how to check.

Why it matters

An energy leak of a fraction of a watt per square meter is small against the roughly 240 W m⁻² of top-of-atmosphere radiation. Integrated over a century-long simulation, it produces a temperature drift that cannot be distinguished from a real climate signal [15].

Models that do not conserve by construction often add energy fixers [12, 35]: corrections that redistribute the imbalance to close the budget. A fixer hides the error instead of removing it, and it obscures how energy moves through the model. Here the discrete equations conserve to floating-point precision, so no fixer is needed.

Where conservation comes from

Conservation rests on four ingredients, and it fails if any one of them is missing.

Flux-conservative equations. The governing equations for mass, energy, and total water are written in flux form: their right-hand sides contain only flux-divergence terms, with no non-conservative sources or sinks. The domain-integrated quantities can then change only through fluxes across the boundaries. This structure holds even for the subgrid-scale fluxes that parameterizations contribute. See Governing Equations.

Specific total energy as a prognostic variable. The model prognoses $e_{tot} = I + \Phi + \kappa + \kappa_{SGS}$, the specific total energy of moist air, instead of a temperature-like variable. Total energy is an extensive quantity satisfying a volume-integrated conservation law, so a flux-form equation for it conserves it.

A consistent thermodynamic formulation. The internal energies, enthalpies, and latent heats must be mutually consistent for the energy budget to close in the presence of phase changes. Thermodynamics and the Working Fluid develops this, including the reference-temperature invariance that any new energy flux must respect.

A mimetic discretization. A discretization can break conservation that the continuous equations have. The spectral element method's weak-form flux divergences satisfy discrete analogues of the divergence and Stokes theorems; direct stiffness summation preserves the discrete inner product; the vertical averaging and difference operators satisfy discrete integration-by-parts and averaging-by-parts identities; and the reconstructions between cell centers and faces are chosen so that combinations of these operators satisfy discrete conservation laws. See Discretization and Operators.

Consequences

Because momentum advection in vector-invariant form conserves kinetic energy and vorticity globally, and because total energy is separately conserved, any numerical conversion between kinetic and non-kinetic energy is performed solely by the discretized pressure-gradient term and by the physical sources and sinks. The advection scheme contributes none. Energy therefore moves between the resolved scales and the thermodynamic reservoirs along one path, in moist conditions as well as dry.

The subgrid-scale and numerical terms follow the same rule. Hyperdiffusion is written as flux divergences along coordinate surfaces, so it redistributes without creating or destroying anything; its enthalpy flux is decomposed to account for the energy carried by the hyperdiffusive water flux, so total energy and water are conserved together. The same decomposition governs the turbulent enthalpy flux and the sponge layer.

What is not conserved

There is a trade-off. Potential temperature, which some models conserve discretely along material trajectories, has no such guarantee here.

Potential temperature is conserved only for dry, adiabatic dynamics, and it is not an extensive quantity satisfying a volume-integrated conservation law. Under the moist, diabatic conditions of the real atmosphere, it is not conserved physically either. Total energy is, so that is what the model conserves.

The effective moisture source in the mass continuity equation, which comes from subgrid-scale water fluxes, is retained for generality. In Earth's atmosphere, it is two or more orders of magnitude smaller than the other terms. It matters where the condensable species is a major atmospheric constituent, such as carbon dioxide on Mars.

Checking conservation in a run

The model's test suite checks conservation, and you can measure it in any simulation. Enable the conservation callback with the check_conservation configuration key, then compare the first and last saved states:

import ClimaAtmos as CA

errors = CA.check_conservation(simulation)

This returns dimensionless relative errors:

• energy_conservation: the change in atmospheric plus surface energy, less the net radiative input at the top, divided by the initial total energy.
• mass_conservation: the change in total dry-plus-moist mass, divided by the initial mass.
• water_conservation: the change in atmospheric plus surface water, divided by the final total atmospheric water. Zero for dry runs.

The surface terms matter: energy and water leave the atmosphere through surface fluxes and precipitation, so a closed budget has to account for what the surface receives. With a slab ocean the surface reservoir is explicit; otherwise the accumulated surface fluxes stand in for it.

Where this is implemented