## 1. What Does it Mean for a Parameterization to be Accurate?

[2] Unlike many closure problems faced in models of the atmosphere, the environmental factors that control the distribution of radiation in the atmosphere are very well understood, so the solution to fully specified problems is known to great accuracy. Radiation parameterizations therefore seek primarily to find an acceptable compromise between accuracy and computational cost. The accuracy of radiative transfer calculations may be measured via comparison to benchmark models [*Oreopoulos et al*., 2012] which are themselves known to be in excellent agreement with observations [*Mlawer et al*., 2000; *Turner et al*., 2004]. Comparisons are normally made for clear-sky conditions, consistent with the way the parameterizations of absorption by gases are developed.

[3] State-of-the-art radiation parameterizations can reproduce benchmark calculations to within 1% for shortwave fluxes and fractions of a percent for longwave fluxes [*Oreopoulos et al*., 2012] but this accuracy is so computationally expensive that radiation parameterizations cannot be applied at every time step of the model. Instead, radiative heating and cooling rates are normally updated less frequently than are model dynamics and, in most cases, other physical parameterizations. The choice to update radiative heating rates less frequently than other fields is an approximation made, not in the radiation parameterization, but in the coupling to the rest of the model. The simulation errors caused by this approximation may range from modest changes in temperature fields [*Xu and Randall*, 1995; *Morcrette*, 2000] to the introduction of more dramatic instabilities [*Pauluis and Emanuel*, 2004] but are generally difficult to quantify. To minimize simulation errors prudence dictates that the radiation time step be as close to the dynamical time step as can be afforded, although precisely how close is a subjective choice.

[4] Several approaches have been developed to accelerate the calculation of radiative fluxes to allow for more frequent calculation. One is to use physically based radiative transfer models to train fast statistical models (normally artificial neural networks) to emulate fluxes based on the state of the atmosphere [e.g., *Chevallier et al*., 1998; *Krasnopolsky et al*., 2008]. An intermediate tactic is to apply physical models sparsely in space and/or time, use simple statistical models (e.g., regression) to predict changes since the last radiation time step, and selectively update calculations based on the expected error [*Venema et al*., 2007]. A third alternative exploits two facts, that cloud properties vary much more quickly in the atmosphere than does the concentration of gases, and that variations in clouds and gases affect fluxes in different, roughly disjoint spectral regions, to motivate updating only the cloud-affected portions of the spectrum at high frequency [*Manners et al*., 2009]. (*Räisänen and Barker* [2004] and *Hill et al*. [2011] take a similar approach to the related problem of representing cloud variability.) The calculation of cloud-affected fluxes can be further accelerated by reducing the spectral detail used to treat absorption by gases [*Manners et al*., 2009].

[5] Each of these methods, including infrequent radiation calculations, represent approximations which introduce errors in radiative heating rates. These errors depend on many factors including how quickly the optical properties of the atmosphere are changing. But the error characteristics of an approximation can be crucially important in determining whether the approximation affects model evolution. Since radiative fluxes at the top of the atmosphere are essentially in balance (after accounting for ocean heat storage), for example, even small (1<W/m^{2}) biases in radiative fluxes affect multidecadal simulations and must be “tuned” away [*Mauritsen et al*., 2012] and/or balanced by compensating errors. Random, uncorrelated noise, on the other hand, does not affect the statistical evolution of most models, whether that noise comes from parameterizations of gravity wave drag [*Eckermann*, 2012; *Lott et al*., 2012] or radiation [*Pincus et al*., 2003] or is externally applied in an effort to diversify ensembles of medium-range forecasts [*Buizza et al*., 1999]. For the purposes of parameterization development this implies that unbiased algorithms, even if they introduce even quite substantial noise in heating rates, can be more accurate, in the sense of introducing smaller changes in model evolution, than other approximations including detailed algorithms used infrequently.

[6] Here we describe an approach to radiative transfer parameterization that emphasizes the accuracy relevant for hydrodynamic models, including both the radiation calculations and the ways those calculations are coupled to the rest of the model. The approach takes advantage of the local homogenization of heating rates arising from small-scale fluid dynamical processes. We have implemented these ideas in a new radiation package, PSrad (named because it is a postscript to the RRTMG package from which it descends), and initially implemented in the ECHAM climate model. PSrad is unique in that it allows only a small sample of the full broadband spectral integration to be performed, with the idea that these calculations should be performed at each time step. This spectral sampling introduces grid-scale noise in radiative fluxes, as does the more common use of stochastic samples to represent the subgrid-scale distribution of cloud properties [*Pincus et al*., 2003]. Experiments show that ECHAM is insensitive to even large grid-scale perturbations to radiative heating rates within the atmosphere, but that significant perturbations in surface fluxes can introduce systematic biases in the model trajectory. Simulation bias can be limited by bounding errors in surface fluxes using carefully selected subsets of the broadband calculation. The approach is applicable to dynamical models at all scales even as significant noise is introduced into individual calculations.

[7] The next section details several strategies for coupling radiation calculations to model integrations, including infrequent radiation calculations and spectral subsets computed at higher frequency. Section 'PSrad/RRTMG: A New Radiation Code for Climate Models' describes the code we have developed to implement these strategies and section 'Assessing Approximation Impacts in a Global Model' the impact of two classes of approximations on forecasting and quasi-climatological time scales in climate model integrations. Section 'Conclusions: Parameterization Error, Simulation Error, and the Coupling of Radiative Transfer to Atmospheric Models' discusses implications for weather forecasting and climate projection applications.