1. Introduction

The impact of small- to medium-scale mountains on atmospheric dynamics is extremely sensitive to the stratification. In neutral flows, the atmospheric boundary layer stress changes the flow and hence the surface pressure on either side of the mountain. This produces a form drag that will in turn drive an exchange of momentum between the atmosphere and the Earth’s surface (Hunt et al. 1988). This pressure drop on the lee side is associated with an effect of downstream sheltering. For obstacles with small slopes, the sheltering is nonseparated, but for obstacles with larger slopes, this sheltering is separated (Reinert et al. 2007) and can cause the formation of banner clouds (Voigt and Wirth 2013). The dynamical regime in the stably stratified case is fundamentally different because internal gravity waves create a drag even in the absence of boundary layer (Durran 1990). For small mountains, the asymmetry in the fields near the surface is such that the flow decelerates upstream, and it accelerates downstream. This can cause a form of nonseparated upstream blocking with strong downslope winds (Lott et al. 2020, hereafter Part I). For large mountains, the situation is different because the associated waves approach breaking, a dynamics that produces separated upstream blocking and strong downslope winds [see recent examples in Pokharel et al. (2017)]. To summarize and from a qualitative point of view, two radically different flow regimes occur above a mountain: on the one hand, we assist to the development of strong upslope winds in neutral case, and on the other hand, we see strong downslope winds in the stratified case.

Maybe because early theories on boundary layer flow over mountains demand quite involved asymptotic analysis (Belcher and Wood 1996), subsequent theories on the interactions between boundary layer and mountain have often used simplified representation of the boundary layer to remain tractable (Smith et al. 2006; Lott 2007). To a certain extent, these simplifications mirror the simplifications made in the literature on stable boundary layer over complex terrain. In such studies, the inviscid dynamics often boils down to that above the boundary layer all the mountain waves propagate upward without being reflected back (Belcher and Wood 1996; Weng 1997; Athanassiadou 2003). There is nevertheless a growing effort in the community to analyze the interaction between boundary layers and mountain waves (Tsiringakis et al. 2017; Lapworth and Osborne 2019). These efforts are motivated by the fact that present day numerical weather prediction and climate models still make errors in the representation of subgrid-scale orography (SSO) and because these errors are at scales where neutral dynamics and stratified dynamics can no longer be treated separately [see discussion in Serafin et al. (2018) and in Part I]. Also, a remaining issue in SSO parameterizations still concerns the representation of the vertical distribution of the wave Reynolds stress (Tsiringakis et al. 2017; Lapworth and Osborne 2019), and existing theories do not tell much about this.

A first limit of Part I is that we only considered upward-propagating internal waves above the inner layer. This is a serious limitation, reflected waves potentially affecting the boundary layer when they return to the ground. A second limit is that we only studied constant shear within the hydrostatic approximation. In this situation, the properties of the inviscid solution make that we cannot study weakly stratified situations and analyze the transition from neutral to stratified flows.

The purpose of the present paper is, therefore, to work with a nonhydrostatic model in order to analyze the case where all the harmonics are reflected. As we shall see in section 2, this happens with constant infinite shear in the nonhydrostatic Boussinesq approximation. In section 3, we describe a characteristic wave field and extend the mountain-wave drag predictor proposed in Part I to the neutral case. We demonstrate that we need to substitute it by a form drag for small values of the Richardson number (J < 1). We analyze the transition from neutral to stratified situation for small slopes in section 4 and show that reflected waves can interact destructively or constructively with the surface when J ≈ 1 yielding low-drag and high-drag states. We then analyze in section 5 the action of the waves on the large-scale flow and show that this action differs between the neutral cases and the stratified cases. In section 6, we describe situations with slopes comparable to the inner-layer scale. In this case neutral flows are characterized by strong upslope winds and nonseparated sheltering on the lee side, whereas in stable case we recover the strong downslope winds and upstream blocking found in Part. I. All our results have been validated with the full nonlinear model used in Part I, the results of which are mentioned throughout the paper. We conclude and present perspectives in section 6.

2. Theory

3. Transition from form drag to wave drag

In Fig. 1 we plot the flow response when the slope parameter S = 0.01, is much smaller than the inner-layer scale $\overline{\delta }\left(1\right)=0.1$ and the Richardson number J = 4. We also take a Prandtl number P = 0.5, that will stay unchanged in the remainder of the analysis. Henceforth, we will call this case the reference case. Note that these values are the same as in Part I to allow us direct comparison between Fig. 1 here and its hydrostatic counterpart (Fig. 1 of Part I).

Physical fields predicted by the viscous theory when J = 4, S = 0.01, and $\overline{\delta }=0.1$. (a) Total wind vector $\left(\overline{z}+\overline{u},\overline{w}\right)$. (b) Vertical wind $\overline{w}$. (c) Total streamfunction $\overline{\psi }$ defined by ${\partial }_{\overline{z}}\overline{\psi }=\overline{z}+\overline{u}$. (d) Vertical flux of action F^z and action flux vector (F^x, F^z). In (b) and (d), the negative values are dashed.

Citation: Journal of the Atmospheric Sciences 78, 4; 10.1175/JAS-D-20-0144.1

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Physical fields predicted by the viscous theory when J = 4, S = 0.01, and $\overline{\delta }=0.1$. (a) Total wind vector $\left(\overline{z}+\overline{u},\overline{w}\right)$. (b) Vertical wind $\overline{w}$. (c) Total streamfunction $\overline{\psi }$ defined by ${\partial }_{\overline{z}}\overline{\psi }=\overline{z}+\overline{u}$. (d) Vertical flux of action F^z and action flux vector (F^x, F^z). In (b) and (d), the negative values are dashed.

Citation: Journal of the Atmospheric Sciences 78, 4; 10.1175/JAS-D-20-0144.1

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Physical fields predicted by the viscous theory when J = 4, S = 0.01, and $\overline{\delta }=0.1$. (a) Total wind vector $\left(\overline{z}+\overline{u},\overline{w}\right)$. (b) Vertical wind $\overline{w}$. (c) Total streamfunction $\overline{\psi }$ defined by ${\partial }_{\overline{z}}\overline{\psi }=\overline{z}+\overline{u}$. (d) Vertical flux of action F^z and action flux vector (F^x, F^z). In (b) and (d), the negative values are dashed.

Citation: Journal of the Atmospheric Sciences 78, 4; 10.1175/JAS-D-20-0144.1

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The total wind at low level in Fig. 1a contours well the obstacle and is null at the surface as expected. We plot in Fig. 1b the vertical velocity field that highlights a system of gravity waves. In the upstream region x < 0, the phase lines tilt against the shear indicating upward propagation, directly above the hill the wave phase lines are more vertical, and downstream they become tilted in the direction of the shear indicating downward propagation. Such structure suggests that the mountain produces upward-propagating gravity waves, that these waves are entirely reflected in the far field (the waves phase lines tilt downstream is almost symmetric and opposite to their upstream tilt) and are almost entirely absorbed when they return to the surface (the wave amplitude rapidly decreases when horizontal distance increases). It is important to note that the amplitude of the vertical velocity is of the same order of magnitude as the amplitude predicted in Part I, which is the amplitude predicted by linear theory if we take for the incident wind at the ground the average of the incident wind over the inner-layer scale ($\overline{\delta }\left(1\right)/2$). In Part I, we interpreted that by the fact that over a distance equal to the inner-layer scale, the viscous dynamics produces a flow that streamlines have vertical displacements with amplitude near the mountain height (as we see here in Fig. 1c), as a consequence, the waves produced by the inner layer resemble the inviscid waves produced by a lower boundary located at $\overline{h}\left(\overline{x}\right)+\overline{\delta }$.

Finally, the wave action flux in Fig. 1d confirms that the waves are produced indirectly by the distortion of the inner layer rather than directly by the mountain (the wave action flux in the inner layer is oriented from one side of the mountain to the other). The orientation of the wave action flux aloft the inner layer also corroborates the fact that over the obstacle the waves propagate upward (the wave action flux points toward the surface), whereas the wave field downstream is dominated by downward-propagating wave (the wave action flux is everywhere pointing upward, F^z > 0). The fact that F^z > 0 almost everywhere on the lee side is also consistent with the fact that there is almost no surface reflection on the ground. This contrasts with Part I, where downward waves were excluded by construction, such that in the hydrostatic case, we had F^z < 0 almost everywhere above the inner layer (see Fig. 1d in Part I).

In Part I, we noticed that predicting the wave amplitude with linear inviscid theory was also useful to scale the mountain waves stress and drag,

$\overline{\overline{u}\hspace{0.17em}\overline{w}}\left(\overline{z}\right)={\displaystyle {\int }_{-\infty }^{+\infty }\overline{u}\left(\overline{x},\overline{z}\right)\overline{w}\left(\overline{x},\overline{z}\right)d\overline{x},\text{Dr}}=-{\displaystyle {\int }_{-\infty }^{+\infty }\overline{p}\left(\overline{x},\overline{h}\right){\displaystyle \frac{\partial \overline{h}}{\partial \overline{x}}}d\overline{x}},$(21)

More precisely, we found that the predictor

${\text{Dr}}_{\text{GWP}}=\sqrt{J}\hspace{0.17em}\overline{\delta }\left(1\right)S^2/2$(22)

provides a good description of the drag for a large range of slopes S and for Richardson numbers J > 0.25. This scaling was, however, based on hydrostatic theory, such that we cannot use it for neutral cases (J ≪ 1). In neutral cases, the mountain drag becomes a form drag due to dissipative loss of pressure when the air passes over the obstacle. To estimate this drag we next make the conventional hypothesis that in the inner layer the pressure varies little in the vertical direction and that the horizontal pressure gradient balances the divergence of the viscous stress,

${\partial }_{\overline{x}}\overline{p}\approx \overline{\nu }\hspace{0.17em}{\partial }_{\overline{z}}^2\overline{u}.$(23)

If we then remark that in the inner layer the wind increases from 0 (at the surface) to $\overline{h}$ (at the top of the inner layer), then the surface wind shear should be on the order of $\overline{h}/\overline{\delta }\left(1\right)$. We can then estimate the form drag as a vertical integral of (23) over the inner layer. We get $\overline{\delta }\left(1\right){\partial }_{\overline{x}}\overline{p}\approx -\overline{\nu }\overline{h}/\overline{\delta }\left(1\right)=-\overline{\delta }{\left(1\right)}^2\overline{h}$. We can thus estimate the form drag as

${\displaystyle {\int }_{-\infty }^{+\infty }\overline{h}{\partial }_{\overline{x}}\overline{p}d\overline{x}}\approx -{\displaystyle {\int }_{-\infty }^{+\infty }\overline{\delta }\left(1\right){\overline{h}}^2}=-\sqrt{\pi }\overline{\delta }\left(1\right)S^2.$(24)

Because this evaluation is qualitative and because the transition between stratified cases and near-neutral cases is more likely occurring near J = 1 we simplify the form drag predictor in

${\text{Dr}}_{\text{FDP}}=\overline{\delta }\left(1\right){\displaystyle \frac{S^2}{2}}.$(25)

Then, following Belcher and Wood (1996) we take as predictor of the mountain drag and stress the maximum between (22) and (25):

$\text{D}{\text{r}}_P=\text{Max}\left(1,\sqrt{J}\right)\hspace{0.17em}\overline{\delta }\left(1\right){\displaystyle \frac{S^2}{2}}.$(26)

We plot in Fig. 2 the mountain drag normalized by this predictor for several values of J and S. We see that the predictor is quite accurate (the ratio is around 1) at least when the flow is stable (J > 3) or neutral (J < 0.1). But there is a transition zone when J ≈ 1, which seems quite rich dynamically. This transition is characterized by a relative maximum of the drag near J = 1.6, and a relative minimum near J = 0.7 that were completely absent in the hydrostatic case [see the thin gray lines in Fig. 2 and remember again that, in Part I, (i) the cases with J < 0.25 were not treated, and (ii) that the reflected waves were absent by construction]. To understand the physics behind the minimum and maximum values of the drag for intermediate values of J, it is important to include the reflected waves in the discussion. We recall that the altitude of dominant turning point of the wave field, which is the turning point above which the dominant wavenumber $\overline{k}=1$ becomes evanescent is ${\overline{z}}_T\left(1\right)=\sqrt{J}$ and so increases with J. As J diminishes, waves are reflected closer to the surface. The local minimum and maximum of the drag in Fig. 2 correspond to a situation where the reflections occur at altitudes close to the mountain horizontal scale [in dimensional units $z_T\left(1/L\right)=\sqrt{J}L$)]. In these situations, the reflected waves interact destructively and constructively with the emitted waves to produce low-drag and high-drag states, respectively. When the reflections occur higher, the reflected waves return to the surface farther on the lee side, so their effect on the surface pressure becomes small over the hill compared to that of the upward-propagating waves.

Surface pressure drag normalized by the predictor Dr_p in (26). The hydrostatic pressure drag normalized by Dr_GWP from Part I is also shown for comparison (thin gray lines). The gray dots are from the MITgcm with S = 0.15.

Citation: Journal of the Atmospheric Sciences 78, 4; 10.1175/JAS-D-20-0144.1

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Surface pressure drag normalized by the predictor Dr_p in (26). The hydrostatic pressure drag normalized by Dr_GWP from Part I is also shown for comparison (thin gray lines). The gray dots are from the MITgcm with S = 0.15.

Citation: Journal of the Atmospheric Sciences 78, 4; 10.1175/JAS-D-20-0144.1

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Surface pressure drag normalized by the predictor Dr_p in (26). The hydrostatic pressure drag normalized by Dr_GWP from Part I is also shown for comparison (thin gray lines). The gray dots are from the MITgcm with S = 0.15.

Citation: Journal of the Atmospheric Sciences 78, 4; 10.1175/JAS-D-20-0144.1

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4. Low-drag and high-drag states

To better appreciate what occurs when the flow is weakly or moderately stratified, we plot in Fig. 3 the vertical velocity and action flux in a weakly stratified case (J = 0.1), and in the two moderately stratified cases (J = 0.7 and J = 1.7) where the drag is respectively lower and larger than the predictor. To ease comparison, we keep all the other parameters similar to those of the reference case (Fig. 1). In the weekly stratified case, the vertical velocity is positive on the upstream side of the ridge and negative on the downstream side. This pattern is similar to the neutral solutions in the inviscid case with no vertical tilt. We also see in Fig. 3b that the wave action flux stays confined inside the inner layer: there is almost no flux of action through the height $z=5\overline{\delta }\left(1\right)$, which measures the inner-layer depth (see Part I). We conclude that in the neutral case, the drag cannot have an inviscid wave origin.

(a),(c),(e) Vertical velocity and (b),(d),(f) action flux [vertical component F^z and vector (F^x, F^z) for S = 0.01. Contour interval for w in (a), (c), and (e) is as in Fig. 1b]. Contour interval for vertical component of the wave action flux is as in Fig. 1d.

Citation: Journal of the Atmospheric Sciences 78, 4; 10.1175/JAS-D-20-0144.1

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(a),(c),(e) Vertical velocity and (b),(d),(f) action flux [vertical component F^z and vector (F^x, F^z) for S = 0.01. Contour interval for w in (a), (c), and (e) is as in Fig. 1b]. Contour interval for vertical component of the wave action flux is as in Fig. 1d.

Citation: Journal of the Atmospheric Sciences 78, 4; 10.1175/JAS-D-20-0144.1

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(a),(c),(e) Vertical velocity and (b),(d),(f) action flux [vertical component F^z and vector (F^x, F^z) for S = 0.01. Contour interval for w in (a), (c), and (e) is as in Fig. 1b]. Contour interval for vertical component of the wave action flux is as in Fig. 1d.

Citation: Journal of the Atmospheric Sciences 78, 4; 10.1175/JAS-D-20-0144.1

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For J = 0.7 in Fig. 3c one sees that the vertical velocity field has still quite vertical phase lines but it extends significantly higher above the inner layer than in the case with J = 0.1. Above the inner layer, one sees in Fig. 3d that there is substantial pseudomomentum fluxes, pointing upward on the windward side and downward on the leeward side. Although the local directions of pseudomomentum fluxes do not quantify directions of propagation without ambiguity (in theory an action flux is proportional to action times group velocity after averaging over a wave phase), it is quite systematic that for mountain waves a negative vertical component of the wave action flux (F^z < 0) indicate upward propagation (although there are variations from one wave crest to the other, as seen in Fig. 1d of Part I). Accordingly, we state that regions above the inner layer where F^z > 0 correspond to downward-propagating waves, as seen in Fig. 3d on the downwind side of the hill. Still in Fig. 3d, we notice that regions with F^z > 0 occupy about the same area as regions with F^z < 0, as if the downward-propagating waves were balancing almost exactly the upward-propagating waves in terms of vertical flux of momentum. This balance probably explains the minimum in pressure drag seen when J ≈ 0.7 in Fig. 2.

The case with J = 1.7 in Fig. 3e presents substantial phase line tilt, and a system of internal waves with two crests and throughs. Upstream and above the ridge, the pseudomomentum flux is quite strong and points downward, as expected for upward-propagating waves. There is also large pseudomomentum flux above the inner layer that points upward but this flux is located well on the downwind side, that is, as if the reflected wave were returning to the surface further downstream than in the case with J = 0.7. This is again consistent with the fact that the characteristic altitude of the turning points where the waves are reflected (${\overline{Z}}_T\left(1\right)=\sqrt{J}$) increases with J. Interestingly, it seems that the downward waves in this case return to the surface near enough downstream the mountain to interfere with the surface boundary condition and to produces large pseudomomentum fluxes and drag.

5. Waves Reynolds stress

The predictors of the surface pressure drag may not be very useful if we take them as a measure of the effect of the mountain on the large-scale flow, as generally done in mountain meteorology (see discussion in Part I). The reason is that, in a steady state, the wave pseudomomentum flux vector within the inner layer is oriented from the upstream side of the ridge toward the downstream side. This situation differs from the inviscid case where this flux goes through the surface and produces an exchange of momentum between the fluid and the solid ground in the form of a pressure drag. In the hydrostatic case, we concluded that the acceleration that balances the gravity wave drag is not communicated to the Earth surface but rather to the flow below around the inner-layer scale. As we shall see, this is even more problematic in the nonhydrostatic case because mountain drag does not necessarily lead to flow deceleration above the inner-layer scale.

To understand how mountains interact with the large-scale flow, we plot in Fig. 4 the vertical profile of the wave Reynolds stress (in black), the pressure stress (gray) and the viscous stress (dashed) acting along displaced streamlines. These are the three terms of the balance equation derived in Part I:

$\overline{\overline{u}\hspace{0.17em}\overline{w}}=-\overline{\overline{p}{\partial }_{\overline{x}}\overline{\eta }}-\overline{\nu }\hspace{0.17em}\overline{\left(\overline{\eta }\hspace{0.17em}{\partial }_{\overline{z}}^2\overline{u}\right)},\text{where}\overline{z}{\partial }_{\overline{x}}\overline{\eta }=\overline{w},$(27)

and that can only be estimated above the mountain top S. We see in Fig. 4 that at low level, the Reynolds stress is small and there is a balance between pressure and viscous stress. In the inner layer, the magnitude of the Reynolds stress increases with height, reaches an extreme and vanishes when $\overline{z}\to \infty$ (as expected because all harmonics are evanescent in $\overline{z}\to \infty$). What is remarkable is that in the near-neutral case J = 0.1 as well as in the low-drag case J = 0.7, the Reynolds stress $\overline{\overline{u}\hspace{0.17em}\overline{w}}>0$ is positive in the inner layer such that it should produce a deceleration of the large-scale flow in the lower part of the inner layer [e.g., around $\overline{z}\approx \overline{\delta }\left(1\right)$] and an acceleration of the large-scale flow in the upper part [e.g., around $\overline{z}\approx 3\overline{\delta }\left(1\right)$]. In the stratified case (J > 1), we recover the standard result that waves accelerate the large-scale flow in the lower part of the inner layer and decelerate the large-scale flow above, as expected for mountain gravity wave drag (Figs. 4c,d).

Vertical profiles of the Reynolds stress (thick line), pressure drag through streamlines (thick gray), and viscous drag through streamlines (thick dashed); see the balance equation [Eq. (27)] and for S = 0.01 and $\overline{\delta }=0.1$.

Citation: Journal of the Atmospheric Sciences 78, 4; 10.1175/JAS-D-20-0144.1

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Vertical profiles of the Reynolds stress (thick line), pressure drag through streamlines (thick gray), and viscous drag through streamlines (thick dashed); see the balance equation [Eq. (27)] and for S = 0.01 and $\overline{\delta }=0.1$.

Citation: Journal of the Atmospheric Sciences 78, 4; 10.1175/JAS-D-20-0144.1

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Vertical profiles of the Reynolds stress (thick line), pressure drag through streamlines (thick gray), and viscous drag through streamlines (thick dashed); see the balance equation [Eq. (27)] and for S = 0.01 and $\overline{\delta }=0.1$.

Citation: Journal of the Atmospheric Sciences 78, 4; 10.1175/JAS-D-20-0144.1

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It is clear from Fig. 4 that the interesting quantity is the extreme value of the wave Reynolds stress rather than the pressure drag itself. In fact, these extremes are always smaller in amplitude, and even of opposite sign to the pressure drag. We further explore the parameter space, and we plot in Fig. 5 these extremes normalized by the predictor of the pressure drag (26) for different values of the slope and stability. We conclude that our predictors overestimate by a factor of 3 the extreme value of the Reynolds stress and more importantly that the sign of the Reynolds stress extreme changes around J = 1: there is flow acceleration above the inner-layer scale $\overline{\delta }$ when J < 1 and deceleration due to gravity wave drag when J > 1. These acceleration/deceleration are balanced by opposing deceleration/acceleration below $\overline{\delta }\left(1\right)$, at least when $S\ll \overline{\delta }\left(1\right)$, but these start to be partly transferred to the ground when $S\approx \overline{\delta }\left(1\right)$, as in Part I (not shown).

Extrema in Reynolds stress normalized by the predictor Dr_p. Hydrostatic values normalized by D_GWP from Part I are also shown for comparison (thin gray lines). Gray dots are from the MITgcm with S = 0.15.

Citation: Journal of the Atmospheric Sciences 78, 4; 10.1175/JAS-D-20-0144.1

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Extrema in Reynolds stress normalized by the predictor Dr_p. Hydrostatic values normalized by D_GWP from Part I are also shown for comparison (thin gray lines). Gray dots are from the MITgcm with S = 0.15.

Citation: Journal of the Atmospheric Sciences 78, 4; 10.1175/JAS-D-20-0144.1

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Extrema in Reynolds stress normalized by the predictor Dr_p. Hydrostatic values normalized by D_GWP from Part I are also shown for comparison (thin gray lines). Gray dots are from the MITgcm with S = 0.15.

Citation: Journal of the Atmospheric Sciences 78, 4; 10.1175/JAS-D-20-0144.1

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6. Transition from downstream sheltering to upstream blocking when $S\approx \overline{\delta }$

To analyze further what occurs in the more nonlinear situations we next consider cases where the slope parameter becomes comparable to the inner-layer scale $\overline{\delta }\left(1\right)$. We first consider the upper limit S = 0.185 beyond which our theoretical model often diverges when $\overline{\delta }\left(1\right)=0.1$. We choose J = 0.01 to illustrate the neutral case and J = 9 to illustrate the stratified case. We plot in Fig. 6 the streamfunction and the wind field for these two cases as well as in the intermediate case where the reflected waves impact strongly the surface conditions near the mountain (J = 1.7). In the near-neutral case (Figs. 6a,b), the wind is intensified on the windward side and small downwind, which correspond to a form of nonseparated sheltering. When stratification increases, this upslope/downslope asymmetry reduces, up to around J = 1: the low-drag case with J ≈ 0.7, for instance, is almost symmetric between the upstream and the downstream side (not shown).

(a),(c),(e) Streamfunction defined by $\partial \overline{\psi }/\partial \overline{z}=\overline{u}+\overline{z}$. (b),(d),(f) Total wind. In all panels S = 0.185 and $\overline{\delta }=0.1$. In (a) and (b) J = 0.01, in (c) and (d) J = 1.70, and in (e) and (f) J = 9.00.

Citation: Journal of the Atmospheric Sciences 78, 4; 10.1175/JAS-D-20-0144.1

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(a),(c),(e) Streamfunction defined by $\partial \overline{\psi }/\partial \overline{z}=\overline{u}+\overline{z}$. (b),(d),(f) Total wind. In all panels S = 0.185 and $\overline{\delta }=0.1$. In (a) and (b) J = 0.01, in (c) and (d) J = 1.70, and in (e) and (f) J = 9.00.

Citation: Journal of the Atmospheric Sciences 78, 4; 10.1175/JAS-D-20-0144.1

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(a),(c),(e) Streamfunction defined by $\partial \overline{\psi }/\partial \overline{z}=\overline{u}+\overline{z}$. (b),(d),(f) Total wind. In all panels S = 0.185 and $\overline{\delta }=0.1$. In (a) and (b) J = 0.01, in (c) and (d) J = 1.70, and in (e) and (f) J = 9.00.

Citation: Journal of the Atmospheric Sciences 78, 4; 10.1175/JAS-D-20-0144.1

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In situations with high drag (J = 1.7), the upslope/downslope asymmetry is not much pronounced, at least on the streamlines in Fig. 6c near the surface. The most remarkable behavior is the pronounced ridge occurring downstream around $\overline{x}=4$, which corresponds to the strong positive vertical wind anomaly already present in the case with small slope and around the same place (Fig. 3e). This pronounced oscillation cannot be attributed to trapped lee waves because these waves are not present in our configuration: trapped waves are always related to neutral modes of Kelvin–Helmholtz (KH) instability when the wind vanishes at the surface (Lott 2016), and these modes do not exist when the Richardson number is constant according the Miles–Howard theorem (Miles 1961; Howard 1961). The absence of trapped modes differs from the study of Keller (1994), who first solved the Bessel’s equation to analyze inviscid trapped waves in constant shear cases. In Keller (1994) nevertheless, the wind at the surface is nonzero. Lott (2016) proposes that when the surface wind does not vanish, the surface wind shear is infinite and the surface Richardson number is null, so downward-propagating stationary waves can be entirely reflected and neutral modes can exist.

In situations with strong stratification (J = 9, Figs. 6e,f), we recover the upstream blocking and downslope winds present in the hydrostatic case in Part I, although in this case all the waves are reflected toward the ground. We do not discuss the results from the MITgcm, but we have used this model in all the configurations with S = 0.15 and S = 0.185 presented in this paper and the solutions from the nonlinear model are almost identical to those shown in Fig. 6 (see also the thorough comparison in Part I, where the validation of the theory by the model was excellent).

We propose one last index to characterize the downstream sheltering versus upstream blocking as a function of S and J. We define this index as the ratio between the wind amplitude along the downwind slope and the upwind slope of the ridge defined as

${\displaystyle \frac{\underset{\overline{z}<2\overline{h}/3,0<\overline{x}<2}{\underbrace{\max }}\sqrt{{\left(\overline{z}+\overline{u}\right)}^2+{\overline{w}}^2}}{\underset{\overline{z}<2\overline{h}/3,-2<\overline{x}<0}{\underbrace{\max }}\sqrt{{\left(\overline{z}+\overline{u}\right)}^2+{\overline{w}}^2}}},$(28)

and we plot this index for several values of J and S in Fig. 7. As in the hydrostatic case and for large values of J, this index can easily reach values around 4 or 5 for slopes near the inner-layer depth and larger. This ratio is always around 1 when J ≈ 1, as in the hydrostatic case, except near the critical value J = 1.7, which corresponds to the high-drag scenario. For J < 1, the ratio becomes smaller than 1, which corresponds to nonseparated sheltering. The smallest values we obtain are around 0.5 for J = 0.01 and slopes S ≈ 0.15.

Downslope sheltering vs upstream blocking index defined as the ratio between the max downslope wind amplitude and the max upslope wind amplitude [see Eq. (28)]. Gray dots are from the MITgcm with S = 0.15.

Citation: Journal of the Atmospheric Sciences 78, 4; 10.1175/JAS-D-20-0144.1

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Downslope sheltering vs upstream blocking index defined as the ratio between the max downslope wind amplitude and the max upslope wind amplitude [see Eq. (28)]. Gray dots are from the MITgcm with S = 0.15.

Citation: Journal of the Atmospheric Sciences 78, 4; 10.1175/JAS-D-20-0144.1

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Downslope sheltering vs upstream blocking index defined as the ratio between the max downslope wind amplitude and the max upslope wind amplitude [see Eq. (28)]. Gray dots are from the MITgcm with S = 0.15.

Citation: Journal of the Atmospheric Sciences 78, 4; 10.1175/JAS-D-20-0144.1

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7. Conclusions