## 1. Introduction

[2] This paper proposes a radiative scaling for the nocturnal boundary layer (NBL) over land, based on the 24-h mean surface net longwave radiation flux (LW_{net24}). The same radiative scaling is also applied to the diurnal temperature range (DTR), an important climate parameter. Worldwide, the DTR has been decreasing in recent decades [*Karl et al.*, 1993; *Horton*, 1995; *Easterling et al.*, 1997]. Coincident increases of cloud cover, which would reduce outgoing LW_{net}, have been noted by many authors [e.g., *Henderson-Sellers*, 1992; *Dessens and Bucher*, 1995; *Jones*, 1995]. Indeed, the analysis of *Dai et al.* [1999] provides evidence of the close coupling of cloud cover with DTR.

[3] There has been extensive development of similarity theory for the stable boundary layer (BL), but the role of radiative cooling is generally not considered in this scaling [e.g., *Nieuwstadt*, 1984; *Derbyshire*, 1994; *Stull*, 1988]. A nighttime surface energy balance model was developed by *Holtslag and De Bruin* [1988], in which the so-called isothermal net radiation [*Monteith*, 1981] plays an important role. Over land, however, the diurnal cycle is driven by daytime radiative heating and the nocturnal longwave cooling of the surface [e.g., *Betts*, 2003]. The development of mixed layer models were a major breakthrough in understanding the unstable daytime dry and cumulus BLs [*Deardorff et al.*, 1969; *Betts*, 1973; *Carson*, 1973; *Tennekes*, 1973], as well as mixed stratocumulus layers [*Lilly*, 1968]. These recognized the role of the radiative forcing at the surface and at cloud-top in generating the turbulence, which produces nearly well-mixed layers. Surprisingly, comparatively little attention has been paid to the corresponding role of radiative forcing in determining the strength and depth of the NBL; so this is the focus of this paper. Conceptually, it is a different starting point from the traditional wind and stability framework, although we will show that wind stress remains a significant independent factor, as it affects the coupling between surface and atmosphere. This analysis will show that the strength of the NBL at sunrise (which will be defined later in section 2.3, equation (5a) as a temperature difference, ΔT_{N}) and the DTR are directly related to a radiative temperature scale, associated with the slope of the Stefan-Boltzmann equation. We start using the framework of model data from the European Centre for Medium-range Forecasts (ECMWF) reanalysis (known as ERA-40 [*Uppala et al.*, 2005]) to illustrate the scaling, and quantify relationships using multiple linear regression. In the model data, after scaling, ΔT_{N} and DTR increase with the growth-time of the NBL and decrease weakly with increasing wind stress. A conceptual model for the depth, **h**, of the NBL is then developed in terms of a radiative velocity scale, and the scaled surface heat flux which increases steeply with wind stress. This simple model is then applied to estimate the nocturnal rise in concentration of gases such as CO_{2} and radon that are emitted at the surface and trapped within the NBL.

[4] Modeling the stable boundary layer (BL) has presented considerable difficulties in global models. In the ECMWF model, the stable BL was reformulated [*Viterbo et al.*, 1999] to reduce cold temperature biases in winter over continental areas. Changing the stability functions introduced more diffusion at high stable Richardson numbers, which in turn increased the coupling between surface and atmosphere, and reduced the fall of surface temperature at night, driven by radiative cooling. However, more diffusion gives deeper stable BLs, which may now be too deep in the model (Beljaars, 2005, personal communication). *Viterbo et al.* [1999] also addressed the effect of the thermal inertia of soil freezing in damping the response of the surface temperature in winter. The ERA-40 reanalysis model uses this same formulation of the stable BL (documentation is available at http://www.ecmwf.int/research/ifsdocs/CY23r4/index.html), although the model data presented here will be for the warm season. Since this analysis is based entirely on model output data, our conclusions do depend on the model physics parameterizations, particularly the stable BL and ground heat flux parameterizations. The model data we use is averaged over river basins (see next section). This smooths and simplifies the analysis, but this means that the evaluation of our results against observations is not an easy task, and it is left to future work.