## Abstract

Linear mountain gravity waves forced by a nonlinear surface boundary condition are derived for a background wind that is null at the surface and increases smoothly to reach a constant value aloft and for a constant buoyancy frequency. In this configuration, the mountain waves have a critical level just below the surface that is dynamically controlled by the surface and minimum Richardson number *J*. When the flow is very stable , and when the depth over which dissipations act is smaller than the mountain height, this critical-level dynamics easily produces large downslope winds and foehns. The downslope winds are more intense when the stability increases and much less pronounced when it decreases (when *J* goes below 1). In contrast, the trapped lee waves are very small when the flow is very stable, start to appear when , and can become pure trapped waves (e.g., not decaying downstream) when the flow is unstable (for ). For the trapped waves, these results are explained by the fact that the critical level absorbs the gravity waves downstream of the ridge when , while absorption decreases when *J* approaches 0.25. Pure trapped lee waves follow that when the absorption can become null in the nondissipative limit. In the background-flow profiles analyzed, the pure trapped lee waves also correspond to neutral modes of Kelvin–Helmholtz instability. The validity of the linear approximation used is tested a posteriori by evaluating the relative amplitude of the neglected nonlinear terms.

## 1. Introduction

The interaction between mountains and synoptic meteorology can produce intense weather situations, like rotors and downslope winds that are very uncertain and quite difficult to predict (Doyle et al. 2009; Reinecke and Durran 2009). From a dynamical point of view, the difficulty is related to the fact that the gravity waves’ response to the mountain is extremely sensitive to the upstream flow conditions near the surface, where “near” means up to around the maximum mountain height. There, small changes in the background flow can yield large changes in the response (Georgelin and Lott 2001).

Two classical examples can be used to illustrate this sensitivity. The first example is that mountain flow regimes are largely controlled by the nondimensional mountain height, , where *H* is the maximum altitude of the mountain, *N* is the low-level flow stability, and is a scale for the low-level incident wind (Smith 1979). It compares the vertical wavelength of the stationary gravity waves and the maximum mountain height. When the first is small compared to the second, the gravity waves break above the mountain, and the resulting nonlinear dynamics can yield downslope windstorms and foehns [see reviews by Smith (1985) and Durran (1990)]. A problem is that it is hard to define *N* or when there is low-level shear (Reinecke and Durran 2008): near the surface, the vertical shear of the synoptic flow makes it so that the incident wind can become quite small, and if the atmosphere is stable, can be made arbitrarily large and extremely sensitive to small changes in the choice of . From a dynamical point of view, this follows that the vertical wavelength of stationary gravity waves becomes small when the incident wind is small (e.g., it can almost always be made smaller than *H* by an appropriate choice). This near-surface behavior is strongly reminiscent of the concept of critical level introduced by Booker and Bretherton (1967), and the fact that the surface can behave as a critical level for mountain waves was first noticed by Jiang et al. (2006). Importantly also, near a critical level, the horizontal winds due to the gravity waves can become very large. We will see that this can explain the onset of downslope winds and foehns, even when there is no upper-level gravity wave breaking.

The second example is that the low-level shear controls the onset of trapped waves, which are free modes of oscillations that are resonantly excited by mountains [Scorer 1949; see also the review in Durran (1990)]. According to the conventional theory of trapped lee waves, these free modes develop along a low-level waveguide, which can result either from (i) the trapping of vertically propagating solutions between a perfectly reflecting surface and a turning point aloft or (ii) the free oscillation of a low-level density discontinuity [for recent studies, see Vosper (2004), Teixeira et al. (2013), and Sachsperger et al. (2015)]. Again, in most studies, the smallness of the incident wind near the surface and the presence of a wind shear above is not a central ingredient.

The fact that there is a problem with the surface flow for mountain waves is not new; it has been addressed in numerical studies, which typically show that the boundary layer reduces the generation of mountain waves, eventually suppresses gravity wave breaking aloft (Richard et al. 1989; Olafsson and Bougeault 1997; Peng and Thompson 2003), and dissipates trapped lee waves (Hills et al. 2016). Other studies emphasize the contribution of the boundary layer for more than one isolated mountain ridge (Stiperski and Grubisic 2011), in the presence of surface heating (Smith and Skyllingstad 2011), and to the formation of lee rotors (Doyle and Durran 2002). Nevertheless, in these studies, free-slip boundary conditions or limited-slip conditions are often used near the surface, so the upstream wind does not go to zero at the surface or decreases to near zero very rapidly (e.g., over few grid points). The two are nevertheless almost equivalent because few grid points cannot permit one to solve well the near-surface gravity wave dynamics. In these cases also, the wind shear is often confined to the boundary layer. In reality, however, the vertical wind shear can extend up to the midtroposphere, as shown by the upstream profiles from Fig. 1 in Doyle et al. (2011) or Fig. 2 in Sheridan et al. (2007) observed in two distinct mountain wave field experiments. There are at least two mechanisms that can cause such a gradual increase of the wind with altitude. The first is that the incident wind is coming from very far upstream, so the boundary layer is fully developed. The second is that synoptic flows are primarily forced by thermal gradients that naturally produce vertical shear spanning few kilometers in depth. The impact of very small winds near the surface has not been much analyzed in the literature, at least from a conceptual point of view, and it can be assumed that in such flows the dynamics is essentially inviscid aloft, except near the surface, where the absorption of gravity waves by a boundary layer needs to be taken into account.

The separation between an inviscid dynamics aloft interacting with a boundary layer is typically the approach followed by Smith et al. (2006) and Jiang et al. (2006) to explain why the conventional theories of trapped waves sometimes failed to explain observations (Smith et al. 2002). These papers analyze the reflection of the downward-propagating waves by the boundary layer and discuss how it affects the low-level waveguide. They confront theories with simplified turbulent boundary layers and numerical simulations to show that more unstable boundary layers eventually support larger trapped waves.

It is in this context that Lott (2007, hereinafter L07) calculates almost exactly the reflection of downward stationary gravity waves by a viscous boundary layer [e.g., the *q* factor in Jiang et al. (2006)]. In the viscous case, the wind at the surface has to be null, constant background wind shear and stratification are exact solutions, and L07 shows that the surface reflection decreases when the surface Richardson number *J* increases: perfect reflection only occurs in the inviscid limit when , suggesting again that unstable flows can give larger trapped waves (Smith et al. 2006). Note nevertheless that now the correspondence with stratified shear flow instability becomes very precise, the 0.25 value also being central in stratified shear flow stability theory (Miles 1961; Howard 1961). To support this further, L07 applies this result to evaluate the trapped solutions in a smooth tanh background wind profile (e.g., a profile with an almost linear shear above , but where the wind becomes constant in ). This is the profile used by Drazin (1958) to derive neutral modes of Kelvin–Helmholtz in the unbounded case. L07 found that, when , the longest of the two modes identified by Drazin (1958) can also be a trapped wave. With the necessary criteria for perfect surface reflection, this example establishes another strong relation between trapped waves and Kelvin–Helmholtz instabilities.

Basically, the purpose of this paper is to include a mountain forcing in the calculations of L07. Before doing so, it is worth recalling that there is a fundamental reason why this is very involved: in the inviscid linear case, both (i) the vertical velocity associated with stationary gravity waves and (ii) the mountain forcing are null at . The inclusion of dissipations can help to circumvent the first problem, but as the viscous ones are extremely complex to treat, linear ones will be used next. Note also that, from a more physical point of view, the use of dissipations is justified by the fact that dissipative effects ultimately control linear critical-level dynamics (Booker and Bretherton 1967). For the second problem, a nonlinear free-slip boundary condition can be taken at , the altitude of the mountain, rather than at only (Long 1953).

The plan of the paper is as follows. The formalism is presented in section 2, and the results about downslope windstorms and foehns are presented in section 3. Their sensitivity to *J*, the boundary layer depth, the maximum mountain height, and their robustness against nonlinear effects are discussed in section 4. The results about trapped lee waves are given in section 5. Section 6 concludes and discusses further the relations between trapped waves and instabilities. The appendixes derive the exact vertical structure of the monochromatic solutions used to build the mountain wave field and some numerical issues.

## 2. Formalism and model

### a. Basic equations

To analyze the mountain gravity waves produced by a stably stratified shear that is null at the surface, next consider the background flow profiles

incident on a two-dimensional mountain, the height of which follows the Witch of Agnesi profile

In (1), is the background horizontal wind, *N* is the constant Brunt–Väisälä frequency, *z* is the altitude, *d* is the vertical scale of the shear, and is the incident wind maximum amplitude. In (2), *H* is the maximum mountain height, *L* its characteristic horizontal length, and *x* the horizontal coordinate. In the two-dimensional nonrotating case, the dynamics can be expressed with the nondimensional variables:

where and are the disturbance horizontal and vertical velocities, respectively, is the pressure disturbance, is the buoyancy disturbance, is the surface elevation above , and is a reference constant density. Then, assuming that inflow disturbances produced by the mountain are small enough to be described by linear equations, the steady Boussinesq set describing them is as follows:

This system will be forced by the nonlinear free-slip boundary condition:

In (4), and , where

are the surface and minimum Richardson number, the nondimensional mountain height, and the Froude number, respectively. These three parameters are central to our study; the significance of *J* and have already been discussed in the introduction, whereas we know from the literature that the Froude number measures the significance of nonhydrostatic effects: when it decreases, more harmonics are evanescent in the vertical when , and the relative contribution of the trapped harmonics to the total response increases.

The dissipative terms on the right-hand sides of (4) are introduced to regularize the near-surface critical-level dynamics. They will always be small, the parameter being a Rayleigh drag in (4a) and (4b) and a Newtonian cooling in (4c). A viscous dissipation acting in the *x* direction only is also used; it is characterized by the large “pseudo” Reynolds number . It is introduced because the Rayleigh drag alone is not efficient at large wavenumbers and to ensure that the model controls the linear critical-level dynamics of all the harmonics. From a scale analysis of (4), we can tell that these dissipative terms will compare in amplitude to the advective terms for nondimensional altitudes . To measure the significance of the dissipation, we construct from this scaling a nondimensional boundary layer depth

With this choice and above this level, the effect of the dissipations typically falls below 20% of that of the advection by the background wind.

### b. Semitheoretical model

In the following, the overlines will be dropped for conciseness. As the inflow solutions are linear, they will be expressed in terms of Fourier transforms:

The dissipative terms on the right-hand sides of (4a)–(4c) transform into , , and , respectively: they are linear in the *z* direction, and the dissipative effects are significant over altitudes that compare to the characteristic length

Sufficiently far above the surface (e.g., where ), almost satisfies the inviscid Taylor–Goldstein equation:

where is the Scorer parameter. This equation has solutions that can be expressed in terms of hypergeometric functions (see appendix A for details), and, to impose a realistic behavior in , a “canonical” solution is introduced. It corresponds to a unit amplitude exponentially decaying disturbance in :

where

and where the expression for in terms of a hypergeometric function of argument is given in the appendix A [see (A7)]. In (12), when , the trapped decaying modes in become propagating, and *m* is changed in , where ensures upward group speed. Also, when , *μ* is changed in . Near the surface, has the following asymptotic behavior:

where and are complicated functions of *k* and *J* (see appendix A).

Near , the dissipations can no longer be neglected, but at leading order the vertical structure equation

can be derived (Booker and Bretherton 1967). It has for solution

where and are as in (13) to ensure matching with the inviscid solution.

To satisfy the lower boundary condition (5), a solution is sought under the form

near the ground, it corresponds to vertical and horizontal velocities approximated by

Used in the boundary condition (5), these expressions permit us to evaluate the amplitude of each harmonics by inversion of the integral equation:

where (see appendix B for the numerical aspects).

## 3. Downslope windstorms, foehns, and low-level wave breaking

### a. Experimental setup

To expose how the model can predict downslope winds and foehns without upper-level wave breaking, consider in this section a mountain of moderate elevation, . To limit the contribution of the trapped waves, a quite-broad mountain, , and a quite-large Richardson number, , are also considered. The damping parameters are and , values that make the boundary layer depth [see (7)] much smaller than the nondimensional mountain height . Finally, the model grid has a horizontal dimension of spanned by points. For these grids and dissipations, the dissipative scales of the harmonics verify .

To interpret the results, the wave field is also compared to that predicted by the same model with uniform incident wind , and which is the solution proposed in Long (1953) [see also Laprise and Peltier (1989)]. The treatment of the nonlinear boundary condition is done as previously; the only change is now that the characteristic vertical velocity is very simple:

so the inversion (18) transforms into

### b. Results

The vertical velocity fields produced by the obstacle in the cases where *U* varies and where *U* is constant are shown in Figs. 1a and 1d, respectively. Both fields have the characteristic structure of vertically propagating mountain gravity waves, the phase lines are tilted in the direction opposite to the incident wind, and the vertical wavenumbers are comparable for large . A first important difference is that, when the incident wind varies with altitude, the wave field is essentially located on the downstream side of the mountain, whereas it stays well located above the ridge when the incident wind is uniform. When the incident wind is uniform, it is known from the literature that this almost purely vertical propagation is related to the fact that the Froude number *F*_{r} is large (Queney 1948), which is also a situation for which the hydrostatic approximation applies. Therefore, it is tempting to interpret the downstream development in Fig. 1 as a nonhydrostatic effect, but two things tell us that this cannot be the case. The first is that, when the hydrostatic approximation is made here [simply by changing *m* in (12) by ], the wave field obtained is almost identical to that in Fig. 1a (not shown). The second simply follows that near the surface the term 1/*U*^{2} largely dominates the Scorer parameter in (10): the nonhydrostatic effect becomes even less significant near the surface than in the far field. Therefore, the asymmetry in the response between the upstream and downstream sides of the ridge is clearly a consequence of combining an incident wind that goes to zero when approaching the surface and a nonlinear lower boundary condition.

This asymmetry is even more remarkable on the horizontal wind disturbance near the surface and because for each harmonic the horizontal wind amplitude varies in rather than in for [see (17a) and (17b)]. The consequences on the total wind vector near the surface are quite spectacular, with a very strong wind flowing downslope of the ridge, whereas the total upslope wind is quite small (see Fig. 1b). When the incident wind is uniform, the differences between the upslope and downslope winds are not as pronounced, at least when the nondimensional mountain height . In the uniform incident wind case, one needs to go to much higher elevations to see a pronounced downslope–upslope asymmetry [e.g., for values when breaking aloft starts (this is the configuration shown in Fig. 1e)]. Note also that when the incident wind varies, these strong downslope winds are much more confined along the mountain slope than when the incident wind is uniform (note the difference in vertical scales between Figs. 1b and 1e).

The upslope–downslope asymmetry is also very pronounced on the total buoyancy in Fig. 1c, where the Foehn effect is much more pronounced than in the uniform incident wind case (Fig. 1f). Note also that, at the tip end of the downslope wind jet, the flow becomes convectively unstable (see Fig. 1c near *x* = 30, *z* = 1), and the buoyancy fields acquire a pattern strongly reminiscent of a hydraulic jump. Also, and importantly, no wave breaking occurs aloft, and this is in strong contrast to the uniform incident wind case when in Fig. 1f. In the uniform-wind case, wave breaking only occurs above the mountain, the foehn effect downstream is not as pronounced as when the incident wind varies, and no convective overturning occurs on the lee side of the mountain.

To emphasize that the model can predict downslope windstorms and foehns much more systematically than when the incident flow is uniform, the two experiments in Fig. 1 have been repeated dividing the mountain height by a factor of 2. The dissipations are divided by 2 as well, which leaves the ratio between the mountain height and the boundary layer depth unchanged. The results (Fig. 2) show that when the wind varies with altitude almost the same results occur as in Fig. 1: a strong downslope wind in Fig. 2a, accompanied by a pronounced foehn and convective instability at the low level in the lee side (Fig. 2b). These phenomena are almost absent or quite weak when the wind is uniform (Figs. 2c,d).

## 4. Sensitivity tests and validity of the linear approximation

### a. Sensitivity test

As suggested by the previous results, the intensity of the downslope winds predicted by the theory is somehow related to the relative amplitude between the mountain height and the boundary layer depth. It also depends on the Richardson number *J*, an issue that has not yet been shown. These sensitivities are next analyzed more systematically by varying *J*, , and . To measure the intensity of the response, a measure of the amplitude of the downslope wind effect is defined as the ratio between the maximum of the horizontal wind disturbance along the foothills (define as the region with and ) and the background wind at the top of the hill []:

Typically, when this factor approaches and exceeds 1, the dynamics induces along the foothill wind amplitudes that compare and exceed the winds at the summit of the ridge. This measure of the amplitude—that the dynamics produce intense downslope winds—is then calculated as a function of the penetration of the mountain into the shear , for four values of *J*, six values of , and six values of (e.g., an ensemble of 144 realizations). The results are in Fig. 3a and first show that the downslope winds become much more substantial when the flow becomes more and more stable. When , the downslope wind factor is always below 1, whatever the boundary layer depth or the mountain height is. When , the downslope wind factors stay limited around 1, again whatever or is. When the flow becomes more stable, downslope winds start to become quite spectacular, with the downslope wind factor reaching values up to 4 when and when . This maximum value reaches more than 8 when . Of course in these extreme cases we cannot expect our model to stay valid (see next subsection), but it nevertheless tells us that strong downslope winds are strongly favored in stable situations when .

Figure 3a also shows an interesting sensitivity to the boundary layer depth. In general, and as soon as the flow stability permits (e.g., for typically), there are strong downslope winds for boundary layer depth up to at least half the mountain height. Nevertheless, the intensity of the downslope wind is clearly increasing when the boundary layer depth decreases. Finally, the results show a rather weak sensitivity to the penetration of the mountain into the shear , telling us that, downslope, the disturbance winds are, in general, proportional to the background wind at the summit. It is the proportionality factor that is strongly sensitive, first to *J* and second to .

### b. Amplitude of nonlinearities

The predictions of downslope winds that have been done so far are obviously limited by the fact that the equations used are linear. There is no theory that justifies this approximation in the presence of a nonlinear boundary, except when the incident flow is uniform (Long 1953). To see if some aspects of the model can nevertheless be trusted, it is possible to proceed a posteriori and to evaluate the error made when the nonlinear terms are neglected. For this purpose, the model can be used to evaluate the amplitude of the nonlinear advection term (e.g., the term if the analysis is on the horizontal momentum equation) and to compare it to the amplitude of the horizontal pressure gradient. To avoid divisions by zero, it is better to take for the latter the amplitude of its associated complex form:

The ratio between the two is shown in Figs. 3b–d when and for *J* = 0.5, 1, and 2, respectively. In the case , which corresponds to weak downslope winds according to Fig. 3a, Fig. 3b shows that the nonlinear term is everywhere quite small. It reaches slightly more than 25% of the amplitude of the linear pressure gradient in a small region located near the surface along the foothill of the ridge. As expected, the nonlinear terms become more substantial when the downslope winds and foehns intensify (e.g., for larger values of *J* in Figs. 3c,d). Still, and at least in the moderate downslope wind case in Fig. 3b, strong nonlinearities only occur in a small region, always located at low level, well downstream on the foothill. When the mountain height is changed, these results stay essentially valid; the foothill and alongslope domain where the nonlinear terms are significant increase with , but not in a dramatic way, compared to Figs. 3b–d (not shown, but this was tested up to ). Finally, it is also likely that the changes produced by these nonlinear terms will affect the fluid essentially downstream of where the errors are located, the absolute group velocity of the gravity waves involved in the dynamics being essentially positive.

## 5. Trapped lee waves and low-level shear flow stability

### a. Theoretical prediction with linear dampings

According to L07, when , the stationary gravity waves propagating toward the ground are always absorbed, with the absorption increasing when *J* increases. According to Jiang et al. (2006), this absorption results in the decay of the trapped waves’ amplitudes downstream. In the cases analyzed here, this decay should therefore increase with *J*. When the flow is unstable, , L07 also shows that the absorption vanishes in the inviscid limit so “pure” (e.g., nondecaying downstream) trapped waves can exist. L07 also suggests that these pure trapped waves should correspond to the longer of the two neutral modes derived by Drazin (1958). As the results in L07 are based on a viscous treatment of the boundary layer, they are derived here for the case of a linear dissipation.

Fundamentally, the derivation consists of finding if there exist harmonics that can satisfy the lower boundary condition and that can therefore be resonantly excited by the mountain. With linear dissipations, the condition means that the ratio *q* between the two solutions in (15) has amplitude

For and in the inviscid limit , this implies

When , (24) suggests that the reflection increases as *J* increases. Nevertheless, as the identification between and reflection is only rigorously justified in the limit, a more precise measure of the absorption is given by the momentum flux. When , it is given by

and is always positive, which means that the downward-propagating waves are absorbed. When , its amplitude

As in the inviscid limit, (24) shows that when , its imaginary part as well: the ground reflection becomes total in the inviscid limit when the flow is unstable at the surface.

When the wind varies with altitude, partial or total reflections also occur aloft, and it is necessary to analyze if, in the presence of the subsequent multiple reflections, there are solutions that satisfy (24). To establish this, Fig. 4a shows the values of and as a function of , corresponding to the canonical solution near the surface [see (13)]. The results are shown for different values of *J*; when , , so the condition (24) cannot be satisfied, because . Note in particular that, for propagating modes [e.g., modes with ; see (12)] , whereas for trapped ones exactly. For , the situation for the propagating modes is now reversed, since but , so the condition (24) can still not be satisfied. On the other hand, there are two points in the trapped sector that deserve attention: they are the points where and . These two points correspond to the wavenumbers satisfying [e.g., the wavenumbers of the neutral solutions of the unbounded problem found by Drazin (1958)]. For trapped waves, it is the first one that matters, since it satisfies in (24). It will next be shown that this neutral solution can become a pure trapped wave.

### b. Model results

Next, to treat more specifically the case of the trapped waves, simulations are made for a narrower mountain than in section 3 (e.g., for *F*_{r} = 2) and where *J* varies between 0 and 1. The contribution of the dissipations is also reduced by taking . For completeness, the horizontal size of the domain now equals , and the number of horizontal grid points is still .

Figure 4b shows the amplitude , characterizing the contribution of each harmonic to the wave field in (16). When , there is for each *J* a pronounced narrow peak exactly where . These peaks broaden when *J* becomes larger than 0.25 and almost disappear when *J* approaches 1. The responses on the vertical velocity fields are in Fig. 4c and are displayed as a function of horizontal distance. As long as there are well-defined pure trapped solutions (e.g., when ), oscillations develop far downstream of the ridge with little attenuation. These oscillations decay substantially with downstream distance when , but the transition is not brutal. For and up to in Fig. 4c, substantial trapped waves are present as far downstream as (e.g., an order of magnitude farther than *F*_{r}). It is only when *J* approaches 1 that the disturbance stays entirely confined along the downstream flank of the mountain and does not propagate downstream.

One important result in Fig. 4b is that the resonance for when occurs for trapped modes with *k* very near 1, which are modes that decay in the vertical very slowly according to (11) and (12). The consequences for the vertical velocity field are shown on Fig. 5a. It shows over the mountain a well-defined vertically propagating wave, and in the lee well-developed trapped waves, the lee waves amplitude being substantial up to (e.g., well above the low-level shear zone). For larger values of *J* (Fig. 5b), the trapped modes become more confined at low levels, because the now-less-pronounced maxima in in Fig. 4b occur for *k* values substantially larger than 1. Figure 5c also shows that the downstream development falls off rapidly when *J* becomes large.

As for the downslope windstorms in section 3, the degree of nonlinearities of the response has been evaluated. As the foehn effect is not very strong for the values of the Richardson number considered , it is found that the relative importance of the nonlinearities can be controlled by the dissipative scale. For the trapped waves developing downstream, the nonlinear term becomes significant near the surface when the nondimensional mountain height is large compared to the boundary layer depth, as in the examples shown in Figs. 4 and 5, as predicted by nonlinear dissipative critical-level theory (Lott and Teitelbaum 1992).

## 6. Discussion

In the midlatitudes, the synoptic-scale flows produced by horizontal thermal gradients or associated with low-level fronts systematically present vertical wind shears. These shears extend from the surface, where the wind is small, to the mid- or high troposphere, where the westerly jet cores are located. Such increase of the wind with altitude can also correspond to well-developed boundary layers. Starting from well-understood theoretical tools, like the inviscid theory of mountain waves and the theory of stratified-shear-flow instabilities, this paper has analyzed the impact of these low-level shears on mountain waves. The fact that a null background wind at the surface yields a degenerated problem in the inviscid linear case is solved by adding a small dissipation and by considering a nonlinear boundary condition.

The central result of the paper is that the critical-level dynamics near the surface plays a crucial role in the response. It produces large horizontal wind and buoyancy disturbances at low levels that result in intense downslope winds and foehns. These phenomena occur almost systematically when the flow is stable and when the boundary layer depth is smaller than the maximum mountain height. The downslope wind intensity also rapidly increases when the flow stability increases. This behavior is consistent with the fact that in nature downslope winds more likely occur during night [e.g., during more statically stable situations (Jiang and Doyle 2008)]. Interestingly, the mechanism proposed here does not call for upper-level wave breaking or trapped wave resonance, as often suggested by other theories (Smith 1985; Durran 1990). Note, nevertheless, that the purpose here is not to invalidate these theories, but rather to shown that an important ingredient is missing in them: the low-level shear of the large-scale winds. In fact, and for completeness, it should be kept in mind that strong downslope winds can occur without upper-level wave breaking in the full 3D case (Miranda and James 1992) and that hydraulic theories of downslope winds have also revealed the significance of having incoming flows that are slow at low levels (Winters and Armi 2014). Finally, it is also important to recall that, in real flow, the wind shear can be very strong near the surface in a fairly shallow layer and less intense aloft; the model presented here does not capture this.

The critical-level dynamics also impacts the onset of trapped lee waves. Their downstream extension also becomes conditional to the value of the Richardson number at the surface, with trapped lee waves occurring more easily when and decreases. The connection with shear-flow stability is further supported by the fact that, in our profiles, and when , some well-known neutral modes of Kelvin–Helmholtz can become pure trapped mountain waves. Another more historical analogy may be worth mentioning. In the early literature about Kelvin–Helmholtz instabilities, some authors have used discontinuous wind profiles and found that these profiles are unconditionally unstable (Lindzen 1974). This, in fact, does not contradict the Miles (1961) and Howard (1961) theorem, since a jump in the wind corresponds to a null Richardson number. In mountain wave theory, when the upper-level wind just above the surface is finite, it should be considered that the Richardson number at the surface is null, because the wind exactly at the surface is null, which yields an infinite surface shear. As found in many papers, the onset of trapped lee waves in this case is no longer conditional to the low-level shear-flow stability. It was shown here that it becomes so when there is a finite shear at the surface. Such finite surface shear can be more realistic than the infinite ones: they naturally occur when a fully developed boundary layer with a no-slip boundary condition is taken into account or when the incident flow is produced by thermal horizontal gradients.

The theory that inflow linear equations are not adapted when a nonlinear boundary condition is used has also been tested. In fact, from a theoretical point of view, Long (1953) only shows that such a combination is valid when the incident wind is uniform. Theory also shows that, near critical levels, nonlinear effects can easily become significant but also that their significance can be mitigated by dissipations (Lott and Teitelbaum 1992). This issue has been discussed by comparing the nonlinear contribution to the horizontal momentum equation to the linear contribution of the pressure gradient . It was found that the first only dominates the second at the downstream foothills and when the downslope winds are strong (e.g., places where it is certain that the nonlinear effects will influence the flow evolution anyway). Upstream and above the ridge, such a comparison shows that the linear hypothesis is essentially valid. Although this partially validates the model a posteriori, the fact that, in places, nonlinearities become important now calls for further tests with full numerical models.

In this context, some preliminary numerical results have shown that some of the results are reproducible, at least when viscous dissipations are included. The onset of downslope winds without upper-level breaking when *J* is large seems robust; the onset of trapped lee waves when *J* becomes small also seems robust. These simulations also show that the transitional value , above which downslope winds easily occur and below which trapped waves start to be substantial, can shift to larger values. For trapped waves, this is consistent with the viscous treatment in L07, the surface critical level absorptions being less intense with true viscosities than with the linear damping used here.

## Acknowledgments

This work was supported by the European Commission’s project ARISE2 (Grant Agreement 653980). The author acknowledges Riwal Plougonven for helpful discussions and the three anonymous reviewers for their useful comments.

### APPENDIX A

#### Canonical Solution

Taking , the Taylor–Goldstein equation [see (10)] becomes

Equation (A1) has three regular singular points; when , its exponent pairs are

with *μ* and *m* being given by (12). Following Olver (1974), the change , (A1) transforms into the hypergeometric equation:

where

To impose a realistic behavior in , the solution (15.5.6) in Abramowitz and Stegun (1964, hereinafter AS) is used:

where *F* is the hypergeometric function, and the “canonical” solution in (11) that corresponds to a unit-amplitude exponentially decaying disturbance in (or to an upward-propagating wave) is given by

To evaluate near the surface, the transformation (15.3.6) in AS is used to express (A7) in terms of the solutions (15.5.3) and (15.5.4) in AS: that is,

where

This yields

When approaching the surface, this inviscid solution behaves as (13), providing that

### APPENDIX B

#### Numerical aspects

To build a solution valid over the full domain, a uniform approximation of the characteristic function is built by using the inviscid solution form in (11) when and with the damped solution [see (15)] near . More precisely, a smooth transition is made between the two by using tanh vertical tapers centered on and of characteristic extension . The other disturbance fields (e.g., and ) are obtained from the uniform approximation of and using the Fourier transforms of (4). Finally, and for the buoyancy *b*, the profiles are adjusted vertically in the physical space so that the predicted total buoyancy field is statically stable.

To evaluate the wave amplitude in (16), the inverse of (18) is made numerically. More specifically, considering a domain of length *D* sampled by an odd number *M* of points, the same horizontal wavenumbers and physical location as when fast Fourier transforms can be used [e.g., and ] for *j* and . In this discrete formalism, (18) transforms into

where the matrix component

Equation (B1) is then solved by inverting the matrix with a conventional inversion software.

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