1. Introduction
Topographically forced waves in the atmosphere fall essentially into two categories: vertically propagating or trapped. Vertically propagating waves where the restoring force is gravity have been studied extensively in recent years (Lin 2007; Nappo 2012) in the context of drag parameterization for global weather and climate prediction models. The momentum transported by these waves is deposited in the atmosphere at high levels (McFarlane 1987; Teixeira and Yu 2014), decelerating the mean circulation, an effect that is typically unresolved by those models (Stensrud 2009; Teixeira 2014). The importance of trapped waves propagating at temperature inversions has only been recognized more recently, in particular in connection with the occurrence of lee-wave rotors (Vosper 2004; Hertenstein 2009; Knigge et al. 2010), although early allusions to this kind of waves go back to the pioneering work of Scorer (1949, 1953, 1954). In the ocean, trapped waves are perhaps more familiar, including interfacial waves propagating at the thermocline in flow over submarine ridges and sills (Apel et al. 1985; Farmer and Armi 1999; Cummins et al. 2003), forced by drifting ice keels (Pite et al. 1995) or by ships in the context of the “dead water” phenomenon (Grue 2015a), and also, obviously, surface waves propagating at the air–water interface (Phillips 1977), including those generated by ships (Benzaquen et al. 2014; Moisy and Rabaud 2014; Rabaud and Moisy 2014).
In a recent atmospheric study, Teixeira et al. (2013) showed that waves trapped at a temperature inversion (which corresponds to a density interface) produce a drag on 2D topography. This should come as no surprise by analogy with the resistance exerted by internal or surface waves on vessels (Benzaquen et al. 2014; Rabaud and Moisy 2014; Grue 2015a) or submerged bodies (Tuck 1965). The drag produced by hydrostatic vertically propagating waves in a continuously stratified fluid is especially easy to understand and calculate, forming the basis of current orographic drag parameterizations. However, the mechanisms behind the drag due to waves trapped at low levels in the atmosphere, such as at inversions at the top of the boundary layer, are intrinsically nonhydrostatic (Vosper 2004; Yu and Teixeira 2015), since dispersion is required to produce a spatially extended wavy wake. These waves are forced by relatively narrow mountains, usually poorly represented in large-scale models. Steeneveld et al. (2008) noted that there is currently missing drag in meteorological numerical models, and this deficiency is often alleviated, without a convincing physical justification, by including a so-called long-tail formulation in the form drag parameterization of the turbulent boundary layer [see also Sandu et al. (2013)]. Those authors raise the possibility that this missing drag may be due to unresolved subgrid-scale terrain effects instead. An obvious candidate to account for these effects is the drag associated with trapped lee waves generated over narrow mountains and hills, which, like the turbulent form drag, is also exerted on the atmosphere at low levels. In the present study, this trapped lee-wave drag is investigated theoretically and compared with laboratory experiments, by considering the flow of two fluid layers of different (constant) density over a 3D topography. This setup approximates waves forced by topography at a temperature inversion (in the atmosphere), or waves at the thermocline (in the ocean), forced by, for example, drifting ice keels, ships, or flow over submarine topography (e.g., Pite et al. 1995; Grue 2015a; Farmer and Armi 1999; Bordois et al. 2016). Generically, the applicability of this model to the real atmosphere relies on the dominance of waves that may propagate at the inversion relative to those that may propagate in the layers existing above or below, if any of those are stratified. For a discussion of these conditions, and the effect of stratification aloft on such waves, the reader is referred to a recent study by Sachsperger et al. (2015).
While the drag from waves generated at a free surface in hydrostatic (sometimes called shallow water) flow over 3D topography and nonhydrostatic flow over 2D topography is well known (being given by closed analytical expressions; see Baines 1995), that is not the case with nonhydrostatic flow over 3D topography. The present study proposes deriving an expression for this drag, studying its behavior in parameter space, and comparing it with laboratory experiments carried out in a large water tank following an original idea of E. R. Johnson (University College London). In these experiments, from which results regarding interface displacements were reported by Lacaze et al. (2013), the dependence of the drag on the Froude number (Fr; a key parameter of the flow) is different from that predicted by the theories of hydrostatic flow over 3D topography (where the drag is only nonzero in so-called supercritical conditions, Fr > 1) and nonhydrostatic flow over 2D topography (where, on the contrary, it is only nonzero in subcritical conditions, Fr < 1).
Some previous studies of related problems have considered the shallow-water equations including weak nonlinearity and weak wave dispersion (i.e., nonhydrostatic effects), resulting in a governing equation akin to the Korteweg–de Vries equation (Johnson and Vilenski 2004, 2005; Esler et al. 2007) or arbitrarily strong nonlinearity (Jiang and Smith 2000; Grue 2015b). These studies addressed flow features such as the shape of the wake behind an obstacle and the drag produced by it. They improved the prediction of the latter quantity from hydrostatic (i.e., shallow water) linear theory, among other reasons by avoiding the singular behavior of the drag at Fr = 1, and being able to produce steady waves and nonzero drag for subcritical flow (Fr < 1), as observed experimentally and simulated numerically in fully nonlinear conditions (e.g., Jiang and Smith 2000; Esler et al. 2007; Grue 2015a). However, these authors did not provide a closed-form expression for the drag which could potentially be used with minor adaptations in drag parameterizations. That is one of the aims of the present study. The following calculations will show that inclusion of arbitrarily strong nonhydrostatic and 3D effects is sufficient to produce a very substantial improvement on the predictions from hydrostatic or 2D theory, in fairly good agreement with experimental data, even if the flow is assumed to be strictly linear.
The remainder of this paper is organized as follows. Section 2 describes the linear wave model developed in this study. In section 3, the laboratory experiments are briefly described. Section 4 presents theoretical results, and their comparison with the laboratory experiments, as well as with 3D hydrostatic and 2D nonhydrostatic theories. Finally, in section 5, the main conclusions are summarized and discussed.
2. Theoretical model
Two-layer flow over an axisymmetric obstacle is considered. Each layer is assumed to have constant density: 
Profile view of the flow setup considered in the 3D nonhydrostatic linear model developed here. A lower layer, of density 
Citation: Journal of the Atmospheric Sciences 74, 6; 10.1175/JAS-D-16-0199.1
a. Solution procedure
b. Wave drag
Equation (20) is the main theoretical result of this study. In what follows, the drag is normalized by 
With this choice, the drag depends on three dimensionless parameters: Fr, 
c. Inclusion of friction
3. Laboratory experiments
The experiments were performed in the large stratified water flume at the Geophysical Fluid Mechanics Laboratory of CNRM in Toulouse, upon an original idea of E. R. Johnson (University College London). This laboratory has been used in the recent past to study a wide range of stratified flows in a geophysical context, from oceanic tides (e.g., Dossmann et al. 2011; Dossmann et al. 2014) to atmospheric stable boundary layers and orographic waves (e.g., Knigge et al. 2010). The flume was used here as a 22-m-long, 3-m-wide, and 1-m-high towing tank. The tank was filled with a stratified two-layer fluid made of saline water using a special procedure to ensure a sharp density interface between the two layers.
The flow configuration is equivalent to that shown in Fig. 1 turned upside down, with the ground replaced by the free surface of the upper fluid layer, and the frame of reference traveling with the flow. Note that, because the obstacle is the only solid boundary moving relative to layer 1, the boundary layer develops only on the obstacle and is therefore less developed than if a solid surface extended over the whole boundary. In other words, the boundary condition is no slip only on the obstacle and free slip elsewhere.
Measurements combined an optical stereoscopic method to retrieve the interface shape [reported in Lacaze et al. (2013)] and a force measurement on the obstacle being towed to estimate the drag. More specifically, drag measurements were carried out using a strain gauge able to measure a force in the range from 20 mN to 20 N with a resolution of 10 mN and a data acquisition rate of 1 Hz. Data were first filtered by removing values differing from the average by more than 1.96 times the standard deviation. The estimate of the drag was then obtained by averaging the remaining data excluding the transient periods that comprise the towing bank acceleration time necessary to attain a constant speed and the deceleration time from a constant speed to zero. Error bars are the 90% confidence interval computed from bootstrap resampling. More details regarding the experiments can be found in Lacaze et al. (2013).
Laboratory experiments are particularly appropriate to evaluate a theoretical model, as they represent a real flow, but with highly controlled conditions. To be representative of the geophysical flows they intend to simulate, laboratory experiments must be in a flow regime similar to the corresponding atmospheric or oceanic flows. This requires first of all having density stratification effects and high Reynolds numbers in the laboratory experiments. These conditions are rarely met simultaneously in the same facility, but the CNRM large stratified water flume in Toulouse was designed specifically for this purpose. In addition, the relevant dimensionless parameters controlling the physics of the problem must be similar. This means that the key parameters (such as Fr in the present case) need to be equal, whereas it is enough for other parameters to be above (or below) a given threshold (e.g., the Reynolds number in the present case).
4. Results
a. Behavior of the normalized drag
First of all, the drag behavior produced by the theoretical model described in section 2 is explored in the inviscid limit as a function of the input parameters 
Drag from (20) normalized by 
Citation: Journal of the Atmospheric Sciences 74, 6; 10.1175/JAS-D-16-0199.1
For hydrostatic flow 
Plan view of stationary interfacial waves in hydrostatic (shallow water) flow over a 3D obstacle. Resonant or free steady waves (those that produce drag) are possible if the angle 
Citation: Journal of the Atmospheric Sciences 74, 6; 10.1175/JAS-D-16-0199.1
As the flow becomes more nonhydrostatic (i.e., 
The drag becomes generally lower as 
b. Comparison with experimental data
The model described in section 2 is now compared with the experiments described in section 3. As will be seen, nonlinear processes seem to be relatively modest for obstacle A but more important for obstacle B. The balance between nonlinear and nonhydrostatic effects [as assumed in Esler et al. (2007)] might provide a better description of the flow in the latter case, as will be speculated below.
Figure 4 shows the normalized drag calculated from (20) with Rayleigh friction included (solid lines and filled circles) and from the measurements (open circles with error bars) for the lower obstacle A (Figs. 4a,c,e) and for the higher obstacle B (Figs. 4b,d,f). For comparison, the dashed–dotted lines and stars correspond to the model (without friction) where 
Comparison of the normalized drag between experimental data (open circles with error bars) and 3D nonhydrostatic two-layer theory (20) including friction (solid lines and filled circles), inviscid 3D hydrostatic theory (22) (dotted lines), inviscid 2D nonhydrostatic theory (dashed lines), inviscid 3D nonhydrostatic theory for 
Citation: Journal of the Atmospheric Sciences 74, 6; 10.1175/JAS-D-16-0199.1
For all theoretical results, the symbols correspond to drag values where the value of each model input parameter was taken from the experiments point by point, whereas the lines show the variation of the drag with Fr for averaged values of the other input parameters. No symbols were included along with the dotted, dashed, and dashed–double-dotted lines because those symbols would follow the lines fairly closely but also make the graphs too confusing.
In Figs. 4a,b the experimental data use the default definition of the Froude number based on an infinitely thin density-interface approximation (26). In Figs. 4c–f, on the other hand, Fr values that were corrected for the real finite thickness of the interface were used in the experimental data. Since this procedure allows a more accurate estimate of the phase speed of the waves trapped at the interface, which is crucial for defining a physically meaningful Froude number, it should provide a better comparison with the model developed here. In Figs. 4a,b both the model results and the data use the Fr values given by the default definition; in Figs. 4c,d the data use the corrected Fr values, whereas the model results use the default definition (which is consistent with the model’s assumptions). This should provide the fairest comparison. However, the fact that in this case the model and the data use different values of Fr precludes a comparison of the drag point by point. To avoid this, in Figs. 4e,f corrected Fr values are used both in the data and in the model, keeping all other input parameters unchanged.
In all model results with friction displayed in Fig. 4, the Rayleigh damping parameter was adjusted to produce the best possible agreement of the drag between model and experimental data for obstacle A, taking the value 
Given the measurement error bars, and the numerous assumptions in the model that are not strictly or even approximately satisfied, the agreement is surprisingly good in Figs. 4a,c,e (obstacle A). In Fig. 4a, the two-layer model with friction (solid line and filled circles) predicts the magnitude of the drag maximum accurately (which is not surprising, since the friction coefficient was adjusted to achieve this), while the model for 
Figure 4c shows that when the Fr used in the experimental data is the corrected one, agreement with the two-layer model including friction improves very substantially, and becomes unquestionably the best one. This agreement is preserved in Fig. 4e, where the model also uses the corrected Fr values. Since the values of 
The contrast of all the model results described above with those from 3D hydrostatic theory (dotted lines), which severely overestimate the drag for all Fr > 1 and obviously underestimates it for Fr < 1, is striking. Two-dimensional nonhydrostatic theory (dashed lines), on the other hand, severely overestimates the drag for Fr < 1 and underestimates it for Fr ≥ 1, as would be expected.
Figures 4b,d,f shows that the agreement between theory and measurements is not as satisfactory for the higher obstacle B. As mentioned previously, the choice made to normalize the drag may not give the best scaling when nonlinear effects are important, as seems to happen here. Note also that the error bars in the experimental data are considerably smaller than in Figs. 4a,c,e owing to a larger signal-to-noise ratio in the measurements. The measured drag has a somewhat flatter distribution as a function of Fr and, hence, is more substantially overestimated at the maximum by all models and underestimated at low Fr (only one data point). The agreement between the model for 
Near the main drag maximum, the experimental data in Figs. 4b,d,f are substantially less overestimated by the discrete model data than by the continuous line obtained with averaged flow parameters, except for a few data points that are overestimated even more. In Fig. 4f, the average ratio between the theoretical and experimental values is 2.24, with a large contribution from the points with higher Fr. Among 28 data points available, the two-layer model with friction is within a factor of 2 of the data for 9 (32%) of these points and within a factor of 3 for 24 (86%) of them.
Again, 3D hydrostatic theory and 2D nonhydrostatic theory produce much worse agreement, with much more severe overestimation of the data for Fr > 1 or Fr < 1, respectively.
Differences between Figs. 4a,c,e and 4b,d,f can be largely attributed to nonlinear effects. Both Jiang and Smith (2000) and Johnson and Vilenski (2004) showed that these effects lead to an overestimation of the drag maximum as a function of Fr by linear theory (without friction) and an underestimation of the drag away from the maximum. However, Fig. 7 of Jiang and Smith (2000) shows a migration of the drag maximum to higher values of Fr as nonlinearity (quantified by M) increases. This does not explain the drag behavior seen in Fig. 4, described above. The numerical model used by Jiang and Smith (2000) was based on the shallow-water equations, so nonhydrostatic effects were not taken into account. This suggests that nonhydrostatic effects may be dominant over nonlinear effects to explain the value of Fr at which the maximum drag is attained in Fig. 4. Besides, Fig. 7 of Johnson and Vilenski (2004) shows a very substantial flattening of the drag distribution as a function of Fr [or, equivalently, with the related “detuning parameter,” defined in their paper as 
Some speculative comments may be made on the role of nonlinear effects in these cases. Esler et al. (2007) developed a scaling for the drag as a function of the detuning parameter, showing from nonlinear theory that the drag maximum scales as 
The main conclusion to take from this comparison is that 3D and nonhydrostatic effects appear to explain the drag behavior observed in the experiments to a certain extent, accounting for the nonzero drag for Fr < 1 and the migration of the drag maximum to Fr < 1. Frictional effects act in the same direction. The effect of nonlinearity, which appears to be much stronger for obstacle B (Figs. 4b,d,f), consistent with the corresponding value of M, may explain the worse performance of the model in that case.
c. Waves at the density interface
It is useful to understand what kinds of waves are associated with each drag regime. Figure 5 presents the normalized streamline (or density interface) vertical displacement field at 
Normalized vertical displacement of the streamlines (or density interface) at 
Citation: Journal of the Atmospheric Sciences 74, 6; 10.1175/JAS-D-16-0199.1
In each row except Figs. 5m–p, as Fr increases, the resonant wake downstream of the obstacle shifts from being dominated by transverse waves, with phase lines almost perpendicular to the incoming flow, to being dominated by divergent waves, with crests and troughs making a smaller angle with the flow. For intermediate values of Fr where both types of waves are important, the well-known “Kelvin ship-wave” pattern can be seen (e.g., Figs. 5b,f). As would be expected, the magnitude of the wave disturbance peaks near the drag maximum: for Fr between 0.6 and 0.8 in Figs. 5a–d and for Fr ≈ 0.8 in Figs. 5e–h and 5i–l (in the hydrostatic situation of Figs. 5m–p the drag maximum is infinite and occurs at Fr = 1, as discussed previously). More specifically, the minimum in streamline elevation (corresponding to an interface depression) immediately downstream of 
Variation of the normalized drag with the normalized value of the minimum of the interface elevation occurring immediately downstream of the obstacle in the flows illustrated in Fig. 5. Each data point corresponds to one of the panels in that figure. Squares: 
Citation: Journal of the Atmospheric Sciences 74, 6; 10.1175/JAS-D-16-0199.1
Some additional physical interpretation of the results can be obtained from (19), which gives the orientation angle of the resonant wavenumber vector 
It can be noticed that there are many qualitative similarities between the wave patterns displayed in Figs. 5a–l and in Fig. 8 of Lacaze et al. (2013); for example, there is a transition from the Kelvin ship-wave pattern to a wake dominated by divergent waves as Fr increases. A detailed comparison is beyond the scope of this article, but quantitative agreement seems less obvious. A reason for this could be that the shape to be taken into account in the theory to force the waves (using Fourier transforms) may not be exactly the topography shape itself. Instead the shape “seen” by the flow may include the effect of the boundary layer developing over the obstacle (Peng and Thompson 2003), which can separate from the topography under certain conditions or generate an upstream/downstream asymmetry, among other possibilities. However, this should in theory also affect the drag, which does not seem to occur, at least for obstacle A.
5. Discussion and conclusions
Theoretical calculations were developed for the drag force produced by an obstacle at the boundary of a stratified flow comprising two layers with different densities, owing to the generation of waves at the interface between them. The theoretical predictions were then tested using data from laboratory experiments. The conditions considered here are representative of either an atmosphere with a sharp temperature inversion at the top of the boundary layer flowing over an isolated hill, the motion of drifting ice keels at the surface of an ocean with a sharp thermocline, or an ocean flow with a sharp thermocline over a sill. The problem also has much in common with that of waves generated either at the thermocline or at the sea surface by ships (Grue 2015a; Moisy and Rabaud 2014), and their corresponding resistance force, which have been studied extensively. Results both from ship-wave theory (Rabaud and Moisy 2013) and from geophysical studies (e.g., Steeneveld et al. 2008) suggest that this gravity wave drag may be a substantial fraction of the total drag exerted on these obstacles. However its effect is usually neglected in a meteorological or oceanographic context (Lott and Miller 1997; Pite et al. 1995) and possibly misrepresented as turbulent form drag.
Two essential differences between the present study and the ship-wave problem are that the density discontinuity is much smaller than at an air–water interface (and can be in practice neglected except in the definition of buoyancy), hydrostatic (or shallow water) effects may be relevant, and the waves are generated remotely rather than at the same interface where the obstacle sits. The first difference allows us to use the Boussinesq approximation without substantial loss of accuracy. The second difference means that a theory for waves in a fluid of arbitrary depth must be used, which allows us to take both the “deep water” and the “shallow water” (hydrostatic) limits, the latter of which yields the hydrostatic drag expression from linear theory that was used in most previous studies. All of these differences are absent in the generation of waves by ships or ice keels at the thermocline, which is essentially similar to the problem being addressed here turned upside-down.
The drag estimated from laboratory experiments of a two-density-layer flow across an axisymmetric Gaussian hill attains a maximum at a Froude number (Fr) slightly lower than 1 and is severely overestimated by 3D hydrostatic linear theory, which predicts the maximum to be infinite and exactly centered at Fr = 1. The experimental data are much better predicted by the 3D nonhydrostatic linear calculation developed here, in particular its two-layer version with a rigid lid including the effect of friction. However, even the model that is more directly applicable to the atmosphere, in which the upper layer is infinite and friction is neglected, produces much better results than hydrostatic theory. The inviscid two-layer version of the model provides an explicit expression for the drag in terms of a 1D integral. This expression shows that, for the conditions in the experiments, the drag depends essentially on three dimensionless parameters: Fr, the ratio of the obstacle width to the depth of the layer in contact with it; 
Agreement with our 3D nonhydrostatic linear model, at least for obstacle A, may have been facilitated by the fact that the depths of the two fluid layers in the experiments are not equal. As noted by Johnson and Vilenski (2005) and Esler et al. (2007), this excludes cubic nonlinearities in the weakly nonlinear wave equation adopted by them, which can amplify the drag by a factor larger than 10 [see also Grue (2015b)]. Departures of the drag behavior from 3D nonhydrostatic linear theory, which are especially salient for the higher obstacle B considered in the experiments, are consistent with what is known from previous studies about the impact of nonlinearity on the flow—namely, causing a flattening of the drag variation with Fr.
The drag behavior in 3D nonhydrostatic linear theory is associated with characteristic wave signatures at the density interface. For Fr < 1, and more precisely for an Fr lower than that where the drag maximum occurs, the flow is dominated by transverse waves with crests almost perpendicular to the flow, which are the only ones that exist in 2D nonhydrostatic conditions (where the drag is only nonzero for Fr < 1). In contrast, when Fr > 1 the flow is dominated by divergent waves, with crests at smaller angles to the flow, which are the only ones that exist in 3D hydrostatic flow (where the drag is only nonzero for Fr > 1). When Fr ≲ 1 and 
Feasible improvements to the 3D nonhydrostatic linear calculations developed here include a more accurate representation of friction—for example, using a bulk boundary layer approach akin to that developed by Smith et al. (2006) and Smith (2007). While in the data from laboratory experiments used here the boundary layer developed only over the obstacle, boundary layer influence on the drag may be more pervasive in the atmosphere—for example, when it is associated with the formation of rotors downstream of the obstacle (Teixeira 2017), which can substantially complicate the flow topology and lead to additional drag.
Part of the effect of the boundary layer is to modify the way the obstacle is “seen” by the flow, so that the actual shape to take into account in the theory is not the topography shape (Peng and Thompson 2003) but what the inviscid part of the atmosphere above “sees.” This is an important lead to improve both drag and wave-pattern predictions. Another feasible improvement would be considering the effects of stratification of the upper fluid layer, as in Teixeira et al. (2013) or Sachsperger et al. (2015).
Future work could include a combination of theory, numerical modeling, and laboratory experiments to explore further the combined influence of nonlinearity and nonhydrostatic effects. This could be based on the theoretical framework developed by Johnson and Vilenski (2005) and Esler et al. (2007), which includes weak nonlinearity and weak nonhydrostatic effects, or the more recent fully nonlinear framework developed by Grue (2015a, 2015b).
Acknowledgments
The authors acknowledge the financial support of the European Commission, under Marie Curie Career Integration Grant GLIMFLO, Contract PCIG13-GA-2013-618016. The drag data used in this study came from experiments supported by the European Union’s Sixth Framework Programme, through the budget of the Integrating Activity HYDRALAB III, Contract 022441(RII3), based upon an original idea of E. R. Johnson. METEO-FRANCE and CNRS also contributed significantly to the support of these experiments. We thank B. Beaudoin, B. Bourdelles, J.-C. Boulay, J.-C. Canonici, F. Murguet, M. Morera, S. Lassus Pigat, and H. Schaffner from the CNRM Geophysical Fluid Mechanics Laboratory in Toulouse for their kind support. We also thank F. Marie and F. Stoop for their help with data processing and S. Cazin, E. Cid, O. Eiff, J. G. Esler, O. Kryeziu, L. Lacaze, and J. D. Pearce for their participation in these experiments. We thank L. Lacaze and M. Mercier for a helpful discussion on wave pattern prediction. Finally we thank E. R. Johnson, who led the HYDRALAB III Transnational Access Project that supported the laboratory experiments used in this article.
APPENDIX
Complex Integration for Evaluating the Drag
A similar type of procedure, based on the original approach of Scorer (1949) but following more directly Sawyer (1962), may be used to obtain (23) from (8) and (14).
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