Energy Conservation in Climate Models
People are often surprised to learn that climate models, specifically their atmospheric components, don’t conserve energy. Energy conservation seems like a basic constraint that we would want to build into our models, but it hasn’t been a major focus for atmospheric modelers (though recent models are putting more emphasis on it). In contrast, ocean modelers have always worried more about energy conservation, and one of their main considerations when designing model architecture is conserving energy.
Atmospheric models do obey the first law of thermodynamics, in principle, but in practice, their numerical implementations have small residual “leaks” of energy (also of angular momentum, mass).So why have atmospheric modelers tolerated these small leaks, while ocean modelers would find them unacceptable?
The first key difference between modeling the atmosphere and modeling the ocean is timescales. The slow deep-ocean circulation takes hundreds to thousands of years to equilibrate, so even a tiny energy leak integrated over a long spin-up will cause the mean temperature to drift by several degrees, completely changing the circulation. The atmosphere’s heat capacity is tiny by comparison and its radiative damping is fast (~1 month), so any small numerical energy leak is quickly radiated away and doesn’t accumulate.
The second difference is how the models are forced. To a first approximation, the ocean is forced at the surface by buoyancy fluxes and wind stress, whereas the whole atmospheric column is forced by radiative heating. The atmosphere’s strong radiative damping quickly erases any small imbalances, whereas the ocean’s internal layers can only adjust via small-scale mixing between layers or via the slow ocean circulation. A “leaky” ocean can’t radiatively correct itself. This issue is doubly important because stratification matters so much in the ocean, as it controls where mixing can occur. Small biases in the depth or location of heating can modify convection, altering the entire overturning circulation.
Finally, it’s just harder to enforce energy conservation in an atmospheric model. Processes like radiation, convection, and phase changes involve nonlinear exchanges of energy that are difficult to balance numerically, and parameterizations tend to introduce small residuals. We’ve generally decided it’s more important to get good clouds, precipitation, and circulation statistics than to close the global energy budget.
Figure: Net energy imbalances in control simulations with three CMIP5 (i.e., previous generation) climate models. Data taken from the LongRunMIP project.
Where atmospheric energy conservation does matter is when models are coupled: a bias in the atmosphere can lead to a long-term bias in the ocean. In the 80s and 90s, this was fixed by adding “flux adjustments” to climate models – small corrections to air-sea fluxes that eliminated the biases. These adjustments were controversial (should they depend on climate state?) and were eventually phased out.
Modern models are tuned more carefully and their control runs generally have top-of-atmosphere imbalances less than 1W m⁻² (compared to fluxes of 100sW m⁻²). But even today, long control integrations exhibit small drifts that must be accounted for when comparing with warming simulations (see figure for some examples). Two common strategies are to use linear regression to remove the drift in control simulations or to subtract each year of the control simulation from the corresponding year of the perturbation simulation.
Energy leaks are likely to stay with us for the foreseeable future: Irving et al. found that CMIP6 models don’t conserve energy much better than CMIP5 models. But this shouldn’t be interpreted as a major deficiency of the models, rather it’s just one of the trade-offs among the thousands of small decisions that go into building a good climate model.
