[1] Two formulations for the radiative dissipation of temperature fluctuations, in the spatial and the spectral domains, are reconciled. Both can be written in terms of a first-order decay of temperature variance. The decay constant was obtained originally by Brutsaert (1972) for stable conditions from considerations on the temperature autocovariance function (in the spatial domain). Here it is obtained directly from a spectral dissipation function, using the appropriate Monin-Obukhov dimensionless functions for a stable atmosphere. The general behavior of the constant with stability is the same for the two formulations, with Brutsaert's original formulation producing stronger dissipation than its spectral counterpart. [2] Although radiative cooling of the mean temperature field is widely recognized as the main physical cause of the nighttime stable layer close to the surface, longwave radiation also plays a role in the dampening of the temperature fluctuations. This has importance in the maintenance of turbulence itself as shown in the pioneering study of Brutsaert [1972], and on the stable temperature spectrum [Coantic and Simonin, 1984;Dias and Brutsaert, 1998].[3] Dias and Brutsaert's [1998] conclusion was that, close to the surface, these radiative effects were relatively unimportant. Most of the time, more recent works on stable conditions have not given much attention to radiation effects on the dissipation of temperature fluctuations [e.g., Vickers and Mahrt, 2004;Edwards, 2009].[4] However, more and better data under stable conditions continue to become available, along with important developments in understanding the collapse of turbulence in stable conditions [Acevedo et al., 2012;van de Wiel et al., 2012avan de Wiel et al., , 2012b, and a better understanding of radiative dissipation of temperature fluctuations remains in order. In this work, we reconcile two approaches to model this process.[5] The maintenance of temperature fluctuations in a stable surface layer depends in various ways on their production, transfer, and dissipation. In addition to molecular dissipation of temperature variance 0 0 , an additional dissipation term exists which is due to the emission of longwave radiation. Radiative dissipation can be modeled both using the spectral budget of temperature fluctuations,and the budget of temperature variance[6] We use the standard symbols x, y, z for the coordinate axes and u, v, w for the corresponding velocity components. In (1), is the mean air temperature ; E w and E are spherical shell averages (with wave number radius k) of the corresponding 3-D turbulence spectra and cospectra ; T is the inertial transfer term, v is the molecular diffusivity of heat, and N(k) is the spectral dissipation function. A detailed description of N(k) and its calculation can be found in Coantic and Simonin [1984] and Dias and Brutsaert [1998]. In (2), stationarity and horizontal homogeneity are assumed ; in (1), 3-D homogeneity is assumed.[7] Brutsaert [1972] modeled the radiative dissipation term E R i...