## 1. Introduction

Much attention has been devoted in recent years to the problem of gravity wave drag produced by stratified flow over orography, a subgrid-scale force that must be parametrized in global weather-prediction and climate models. Following pioneering contributions, where leading-order effects of the incoming flow and orography on the drag were evaluated analytically using linear theory for an atmosphere with constant wind and static stability (Smith, 1980; Phillips, 1984), various refinements to these conditions have been considered. For example, Smith (1986) and Grubišić and Smolarkiewicz (1997) addressed the effect of a linearly varying wind profile on the drag, and Grisogono (1994) assessed boundary-layer effects, in conjunction with a more general variation of the wind with height. Teixeira *et al.* (2004) and Teixeira and Miranda (2004, 2006) applied a WKB approximation to compute corrections to the drag due to a generic, but relatively slow, variation of the wind with height. Shutts and Gadian (1999) and, more recently, Teixeira and Miranda (2009) considered effects of directional shear on the momentum flux associated with mountain waves, which is a quantity that has a direct impact on the deceleration of the large-scale atmospheric circulation. Discontinuities in the atmospheric profiles of the wind shear and static stability, and their impacts on the drag, have been examined by Wang and Lin (1999), Leutbecher (2001) and Teixeira *et al.* (2005, 2008), among others. These authors have shown that the drag can be substantially modulated by resonance associated with partial reflections of the gravity waves at sharp variations of the atmospheric parameters, or their derivatives.

Most of these investigations treated the flow as purely hydrostatic, because non-hydrostatic effects are typically relatively weak in the atmosphere at the mesoscale; it is generally believed that the main contributions to the drag arise in nearly hydrostatic conditions, and the hydrostatic approximation facilitates the calculation of closed-form analytical expressions for the drag. However, when the atmospheric parameters vary in a realistic way, in particular when the wind speed increases sufficiently, or the static stability decreases sufficiently, the flow becomes significantly non-hydrostatic at some levels. The hydrostatic approximation implies that all gravity waves in the spectrum of disturbances forced by a given orography propagate vertically. In non-hydrostatic conditions, on the other hand, there are parts of the wave spectrum (high wavenumbers) which are evanescent, and thus do not contribute to the drag, while others (low wavenumbers) propagate vertically, contributing to the drag. If an evanescent layer lies above a wave-propagating layer, wave trapping can occur, with total vertical wave reflection of some wave components at particular heights, and consequent resonant drag amplification. This phenomenon co-exists with the partial wave reflections that may occur even in hydrostatic conditions (Leutbecher, 2001; Teixeira *et al.*, 2005).

It is known that vertically trapped internal gravity waves, generally known as trapped lee waves, produce a drag force (Bretherton, 1969; Smith, 1976; Lott, 1998; Broad, 2002), but, unlike drag in a hydrostatic atmosphere, its behaviour has not been explored in detail (Wurtele *et al.*, 1996). Extending the calculations of Bretherton (1969) for flow confined vertically by a rigid lid (see also Tutiś, 1992), Smith (1976) presented a formula for trapped lee wave drag in an unbounded atmosphere, based on linear theory, where this quantity is expressed as the ratio of two terms, one involving an integral of the Fourier transform of the vertical velocity perturbation and the other the vertical derivative of this Fourier transform. However, Smith (1976) did not provide any explicit rigorous calculation based on this expression, and limited himself to giving a rough estimate of the trapped lee wave drag based on measured atmospheric parameters. More recently, Vosper (1996), Grubišić and Stiperski (2009) and Stiperski and Grubišić (2011) studied the trapped lee wave resonance that occurs in flow over two successive mountains. Using numerical simulations, they showed, among many other results, the variation of the drag with the horizontal separation of the mountains. For sufficiently large separations, and smooth atmospheric profiles, which preclude partial wave reflections, this variation can only be attributed to trapped lee waves. The amplitude that may thus be inferred for the trapped lee wave drag from the results of Stiperski and Grubišić (2011) is comparable to that of the remaining drag, which must be associated with vertically propagating waves. This is surprising, given that trapped lee waves, being highly non-hydrostatic, would be expected to produce relatively little drag. This result motivates the present study, where the behaviour of the drag associated with trapped lee waves and with vertically propagating waves will be investigated as a function of input parameters for much simpler atmospheric profiles than considered by Grubišić and Stiperski (2009) or Stiperski and Grubišić (2011).

Scorer (1949) was the first to provide a satisfactory theoretical explanation for the existence of trapped lee waves, assuming an atmosphere with two layers, where wave propagation is permitted in the lower layer, but not allowed in the upper layer for an important fraction of the wavenumbers. This author used analytical techniques to obtain the flow configuration associated with these waves relatively far from the orography that generates them, but did not calculate the drag. In the present study, a two-layer atmosphere with piecewise-constant parameters, such as adopted by Scorer (1949), will be assumed to evaluate the trapped lee wave drag and the drag associated with vertically propagating waves, and compare the magnitude of these two drag components. It will be seen that, in some circumstances, the trapped lee wave drag may be comparable, or even larger, than the drag associated with vertically propagating waves, and substantially larger than the drag for a hydrostatic atmosphere with a constant Scorer parameter equal to that existing in the lower layer. These results have implications for gravity wave drag parametrization (Lott, 1998).

This article is organized as follows. In section 2, the linear model that will be used to calculate the trapped lee wave drag and propagating wave drag is described. Section 3 presents illustrative calculations of the drag as a function of input parameters, for three different atmospheric profiles, and a brief comparison with numerical simulations. Finally, in section 4, some concluding remarks are presented.