Direct heating rates associated with gravity wave saturation

Abstract

Analysis of filtering out subscale motions is applied for internal gravity waves. This leads to a new perspective of the planetary-scale sensible heat budget of the upper mesosphere/lower thermosphere. In line with previous results of Becker and Schmitz, the present paper recapitulates that the dissipation of gravity wave kinetic energy and the local adiabatic conversion of mean enthalpy into gravity wave kinetic energy cannot be neglected, and that the net effect of both cools the upper mesosphere/lower thermosphere. In addition, the importance of the wave entropy flux—an effect which is ignored in customary gravity wave parameterizations for global circulation models—is stressed. We show that, when evaluated on the basis of Lindzen's saturation assumption, the wave entropy flux convergence behaves like a vertical diffusion of the mean stratification, where the wave-induced diffusion coefficient is involved with a Prandtl number of 2. This result imposes an upper bound of 2 for the effective Prandtl number which scales the combined entropy flux owing to turbulence and gravity waves. The direct heating rates generated by gravity wave saturation are assessed quantitatively, using an idealized general circulation model completed by a Lindzen-type gravity wave parameterization.

Introduction

Internal gravity waves (hereafter IGWs) which are generated in the troposphere and lower stratosphere control the general circulation of the upper mesosphere/lower thermosphere (hereafter MLT). The adiabatic cooling and heating rates associated with the IGW-driven mean meridional circulation are strong enough to account for a reversal of the summer-to-winter-pole temperature gradient (Lindzen, 1981). This is the usual explanation for the peculiar thermal structure of the MLT. Indeed, the high-latitude summer mesopause is the coldest place of the terrestrial atmosphere, with observed temperatures as low as $\text{∼130}\text{K}$ (e.g., Lübken et al., 1999).

However, the dynamics of the MLT is complicated by several other processes. First, the breakdown of atmospheric tides accounts for wave-mean flow interactions at low latitudes. Second, planetary wave activity in the underlying atmospheric layers can efficiently modulate the propagation and breakdown of IGWs on both the intraseasonal scale, for example during sudden warming events (Holton, 1983), and on the climatological scale as well (Becker and Schmitz, 2003, hereafter BS03). Third, the direct heating rates associated with gravity wave-mean flow interaction and turbulent diffusion are known to give important contributions to the planetary-scale sensible heat budget of the MLT. While corresponding model estimates appear less certain, it is important to mention state-of-the-art rocket-borne in-situ measurements of Lübken 1992, Lübken 1997. These experiments indicate that in the summer MLT the turbulent dissipation (frictional heating) amounts on average to 10–$\text{20}\text{Kd}{}^{\mathrm{-1}}$. Such values are comparable with the radiative heating.

At present, IGWs cannot be resolved reasonably in general circulation models (hereafter GCMs) of the atmosphere, even not in high-resolution simulations (Koshyk and Hamilton, 2001). Therefore, since the nonconservative propagation of IGWs is ultimately linked to small-scale turbulence (Lindzen, 1981; Matsuno, 1982; Liu et al., 1999; Fritts and Werne, 2000), both subscale processes must be parameterized simultaneously in simulations of the general circulation of the MLT. This represents a conceptional difficulty with regard to the direct sensible heating rates owing to wave-mean flow interaction, as well as with regard to the dissipation and diffusion generated by small-scale turbulent motions.

Several concepts to parameterize these heating rates can be found in the literature. As concerns the turbulent dissipation of gravity wave kinetic energy, some authors assume that the heating of the mean flow owing to wave-mean flow interaction, i.e. the energy deposition, is available for dissipation (e.g. Fritts and VanZandt, 1993; Fritts and Luo, 1995; Hines 1997, Hines 1999; Fritts and Werne, 2000; Akmaev, 2001), whereas, others propose that the dissipation should be identified with the frictional heating associated with turbulent momentum diffusion (Walterscheid, 1981; Liu, 2000; Becker and Schmitz, 2002, hereafter BS02). With regard to wave-mean flow interaction, Hines and Reddy (1967), Lindzen (1973), Lindzen (1990, Chapter 8.5), Hines (1997), or BS02 noted that the wave pressure flux gives the main contribution to the wave energy flux, i.e. the vertical flux of enthalpy (sensible heat) plus kinetic energy. On the other hand, some accounts of the enthalpy budget of the mean flow employ the log-pressure coordinate system, in which wave pressure perturbations vanish by definition (e.g. Walterscheid, 1981; Weinstock, 1983; Schoeberl et al., 1983), or they neglect the wave pressure flux (e.g. Fritts and Luo, 1995; Liu, 2000). Other existing views of the planetary-scale heat budget of the MLT ignore both the local adiabatic conversion of mean enthalpy into IGW kinetic energy and the wave entropy flux convergence (Chandra, 1980; Lübken et al., 1993), while the latter is usually neglected in gravity wave parameterizations (e.g. Holton and Zhu, 1984; Hines 1997, Hines 1999; BS02; Akmaev, 2001). Summarizing, a generally accepted theoretical concept of the direct heating rates associated with gravity wave-mean flow interaction and gravity wave-induced turbulent diffusion is not yet available.

The goal of the present study is to derive an energetically consistent formulation of the planetary-scale enthalpy budget of the MLT region, and to provide corresponding numerical estimates. In this respect, previous results of BS02 are recapitulated. In addition, we derive a precise relationship between the energy deposition and the turbulent dissipation, and we recover estimates of Fritts and Dunkerton (1985) of the wave entropy flux in the case of uniform turbulent diffusion. As in BS02, we shall quantitatively assess these terms on the basis of a simplified general circulation model (SGCM) completed by a Lindzen-type gravity wave parameterization.

In Section 2, we apply some general conceptions of the action of subscale motions on the resolved flow in order to derive general forms of the direct sensible heating rates associated with gravity wave breakdown, and to identify the specific approximations that apply to small-scale turbulence on the one hand and to IGWs on the other. In Section 3, the direct IGW heating rates are specified for linear monochromatic waves subject to turbulent diffusion according to Lindzen's saturation assumption (Lindzen, 1981; Holton, 1982; Fritts, 1984). Corresponding SGCM results are presented in Section 4, which is followed by the conclusions.