Applicability of Reduced-Gravity Shallow-Water Theory to Atmospheric Flow over Topography

Qingfang Jiang Naval Research Laboratory, Monterey, California

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Abstract

Applicability of the reduced-gravity shallow-water (RGSW) theory to a shallow atmospheric layer capped by an inversion underneath a deep stratified atmosphere over a two-dimensional ridge has been investigated using linear analysis and nonlinear numerical simulations. Two key nondimensional parameters are identified: namely, and , where g′ is the reduced-gravity acceleration; H0 is the RGSW layer depth; and N and U are the buoyancy frequency and wind speed, respectively, in the layer above the inversion. If J and γ are around unity or larger, the response of the RGSW flow over the ridge can be significantly modified by pressure perturbations aloft. Any jumplike perturbations in the RGSW layer rapidly decay while propagating away from the ridge as the perturbation energy radiates into the upper layer. With J and γ much less than unity, RGSW theory is more adequate for describing RGSW flows.

In addition, inversion splitting occurs downstream of a jump when , where Ni is the buoyancy frequency in the inversion and hm stands for the ridge height. A less stratified upper layer with slower winds in general has less influence on the RGSW flow below and favors the application of the RGSW theory. For a thick inversion (d), the equivalent RGSW flow depth is approximately given by H + d/2, where H is the depth of the neutral layer below the inversion.

Corresponding author address: Qingfang Jiang, Naval Research Laboratory, Marine Meteorology Division, 7 Grace Hopper Ave., Monterey, CA 93943. E-mail: jiang@nrlmry.navy.mil

Abstract

Applicability of the reduced-gravity shallow-water (RGSW) theory to a shallow atmospheric layer capped by an inversion underneath a deep stratified atmosphere over a two-dimensional ridge has been investigated using linear analysis and nonlinear numerical simulations. Two key nondimensional parameters are identified: namely, and , where g′ is the reduced-gravity acceleration; H0 is the RGSW layer depth; and N and U are the buoyancy frequency and wind speed, respectively, in the layer above the inversion. If J and γ are around unity or larger, the response of the RGSW flow over the ridge can be significantly modified by pressure perturbations aloft. Any jumplike perturbations in the RGSW layer rapidly decay while propagating away from the ridge as the perturbation energy radiates into the upper layer. With J and γ much less than unity, RGSW theory is more adequate for describing RGSW flows.

In addition, inversion splitting occurs downstream of a jump when , where Ni is the buoyancy frequency in the inversion and hm stands for the ridge height. A less stratified upper layer with slower winds in general has less influence on the RGSW flow below and favors the application of the RGSW theory. For a thick inversion (d), the equivalent RGSW flow depth is approximately given by H + d/2, where H is the depth of the neutral layer below the inversion.

Corresponding author address: Qingfang Jiang, Naval Research Laboratory, Marine Meteorology Division, 7 Grace Hopper Ave., Monterey, CA 93943. E-mail: jiang@nrlmry.navy.mil

1. Introduction

In the past six decades the reduced-gravity shallow-water (RGSW) theory has been applied to a wide variety of atmospheric phenomena associated with the interaction between topography and lower-tropospheric flows. This includes coastally trapped disturbances, California expansion fans, morning glories regularly observed offshore of northern Australia, katabatic winds, gap winds, valley winds, leeside rotors, island wakes and vortices, coastal jets, and downslope winds around the world, such as chinook in the Rocky Mountains, foehn in the European Alps, bora over the Croatia coast and Adriatic Sea, Boulder (Colorado) windstorms, and tramontane and mistral in the southeast France (see Table 1 for references). It was found that the RGSW theory can often provide concise and consistent interpretation of these complex and highly nonlinear phenomena.

Table 1.

Example phenomena and references.

Table 1.

Application of the RGSW theory is typically justified by the presence of a relatively sharp inversion or potential temperature jump (i.e., Δθ) above the less-stratified shallow flow of interest, and traditional shallow-water (TSW) theory applies after replacing the full gravity acceleration g with a fractional or reduced-gravity acceleration, . The TSW theory is attractive in part because of its simplicity. For example, a rich variety of solutions associated with a shallow-water (SW) flow past an infinite ridge can be mapped onto a F0 × M regime diagram, where is the ambient Froude number (U0 is the flow speed and H0 is the ambient SW flow depth) and is the nondimensional ridge height (hm is the height of the ridge crest; Long 1953, 1954, 1972; Baines 1995). Schaer and Smith (1993a) demonstrated that the solutions associated with a TSW flow past a three-dimensional isolated hill can be described by an F0 × M regime diagram as well. Their solution diagram was later extended to include solutions for supercritical ambient flows by Jiang and Smith (2000).

The TSW theory assumes that the SW flow is under a free surface, and there is no mass exchange across and no pressure perturbations above the free surface. The neglect of pressure perturbations above the free surface is also referred to as the passive-layer assumption. These assumptions or approximations that are adequate for an air–water interface may become questionable for flows under an inversion, above which resides a deep stratified atmosphere. For example, Marić and Durran (2009) documented strong subsidence through the mean position of a boundary layer–top inversion during a gap flow event in the Wipp Valley of the Alps, which may invalidate the mass and momentum conservation in the layer below the inversion. Based on simulations of the Chilean low-level coastal jet capped by a marine boundary layer–top inversion, Jiang et al. (2010) suggested that the RGSW theory is only applicable to coastal jets when the time scale for cross-inversion mixing is far longer than the hydraulic processes of interest. In addition to a shallow layer capped by an inversion, the RGSW has been applied to some other flow configurations as well. For example, Durran (1986) examined the analogy between large-amplitude trapped lee waves induced by a two-layer atmosphere (i.e., a stable layer underneath a deep less-stratified layer) past a ridge and hydraulic flow response based on two-dimensional simulations. Smith (1985) suggested that, as a uniformly stratified flow past a high ridge, the portion between the lee slope and the wave-breaking-induced stagnant flow aloft can be locally treated as a hydraulic flow. Multilayer solutions have been examined by Houghton and Issacson (1970), Armi (1986), and Jiang and Smith (2003). More recently, Armi and Mayr (2011) gave an observational analysis of multilayer hydraulics.

The objective of this study is to provide insight into the applicability of RGSW theory to a shallow neutral or weakly stratified layer capped by an inversion (hereafter referred to as RGSW layer), above which lies a deep uniformly stratified atmosphere. Particularly, we focus on the influence of the wave-induced pressure perturbations in the stratified upper layer on the RGSW flow, the neglect of which has been considered an a priori in previous studies. The remainder of this paper is organized as the follows. In section 2, assumptions and approximations associated with the RGSW theory and the impact of pressure perturbations aloft on linear waves in the RGSW flow are analyzed. The numerical model and setup and diagnosis methods are described in section 3. The applicability of RGSW theory to flow under an inversion past a two-dimensional ridge is investigated in section 4 using nonlinear numerical simulations. The results are summarized in section 5.

2. Theoretical considerations

The TSW theory is built upon two basic assumptions, namely, the long-wave or shallow-water (i.e., H0/L ≪ 1, where H0 is the flow depth and L is the horizontal wavelength) and free-surface assumptions (e.g., Baines 1995). Applying the RGSW theory to flow underneath an inversion, above which resides a deep stratified flow, implies three additional assumptions: 1) the inversion must be thin enough to behave like an interface, 2) any mass exchange across the inversion is negligible, and 3) the upper layer is “passive” and its impact on the lower RGSW layer is negligible. In the following analysis, we assume assumptions 1) and 2) are valid and focus on the impact of the pressure perturbation above the inversion on the RGSW flow.

a. RGSW waves under a deep continuously stratified atmosphere

Assuming that the stratified upper layer linearly responds to the vertical undulation of the thin inversion layer, which can be treated as an interface with reduced gravity g′, the linearized RGSW equations can be written as
e1
e2
where is perturbation horizontal velocity, and denotes the pressure perturbation evaluated at the inversion level normalized by the RGSW flow density ρ0. For a plane wave in the form of , using Eqs. (1) and (2), we obtain the dispersion relation for a RGSW wave under a deep stratified atmosphere,
e3
where k is the horizontal wavenumber, ω is the wave frequency, variables with hats denote the corresponding variables in the wavenumber space, and is the ratio between the pressure perturbation above the inversion and the vertical displacement of the interface for wavenumber k. It is noteworthy that could be imaginary, implying a 90° phase difference between the pressure perturbation and vertical displacement. A similar approach has been used to investigate wave–boundary layer interaction in Smith et al. (2006) and Jiang et al. (2008), boundary layer separation induced by mesoscale terrain by Jiang et al. (2007), and gravity wave effects on wind farm efficiency by Smith (2009).
The pressure perturbation above the inversion can be derived from the linearized momentum and continuity equations for the free atmosphere,
e4
e5
where the subscript f denotes perturbation variables in the free atmosphere. For a plane wave, , using Eqs. (4) and (5), we obtain , where is the intrinsic frequency and m is the vertical wavenumber given by . Here stands for the horizontal phase speed in the free atmosphere, z = 0 at the inversion level, and m and σ have opposite signs to satisfy the wave radiation condition aloft.
At the inversion level, the vertical velocity can be related to the vertical displacement of the inversion as, , or, in the spectral form,
e6
where c is the RGSW wave speed which is equal to at z = 0. Combining Eqs. (4)(6), we obtain the pressure perturbation acting on the RGSW layer in the spectral space, , or . Note that for a gravity wave aloft, m is real and accordingly α is imaginary, and for evanescent perturbations, m is imaginary and α is real. The dispersion relation for the RGSW wave becomes
e7
or in nondimensional form,
e8
where , , , and . The sign of the second term is negative for an upstream-propagating wave (i.e., k < 0) and positive for downstream-propagating waves (i.e., k > 0).
For a hydrostatic wave, Eq. (8) reduces to
e9
and the RGSW wave speed is then given by
e10
The nonzero imaginary part in the phase speed implies an amplitude decay of the RGSW wave due to leaking energy into the free atmosphere aloft. For a strong inversion and shallow lower layer, that is, the ratio of the hydrostatic wave speed in the free atmosphere and the SW wave speed, γ ≪ 1, to the first order of approximation, we have
e11
Equation (11) implies that 1) pressure perturbations associated with the hydrostatic wave response in the stratified upper layer do not alter the RGSW wave speed; 2) a propagating RGSW wave decays exponentially with the distance away from the source x as , through radiating waves into the continuously stratified atmosphere above the inversion; and 3) the vertical wind shear may enhance or weaken the wave energy leakage depending on the sign of the shear and the wave propagation direction. A forward shear (i.e., s > 0) tends to accelerate the decay of upstream-propagating waves and slow down the decay of downstream-propagating waves, and a backward shear does the opposite. To the second order of γ, the real part of Eq. (10) becomes , implying that the wave response aloft tends to slow the RGSW wave.
Next we consider a special case corresponding to U = U0 in which the upper layer is weakly stable and perturbations in the upper layer are evanescent [i.e., ]. Assuming that the RGSW flow is still hydrostatic, that is, , to the , Eq. (7) yields
e12
implying that no perturbation energy leaks into the free atmosphere, and the RGSW wave speed becomes slower than the corresponding SW wave under a free surface.

Approximate solutions for Eqs. (11) and (12) are shown in Fig. 1 along with numerical solutions to Eq. (8) for a range of γ values and kH0 = 0 and 0.1, respectively. It is evident that the modification of the RGSW wave speed by pressure perturbations aloft is small throughout the parameters examined. The amplitude of the imaginary part linearly increases with γ, and is noticeably larger (smaller) with a positive (negative) vertical wind shear.

Fig. 1.
Fig. 1.

(top) The normalized real wave speed departure from the TSW wave speed and (bottom) the corresponding normalized imaginary wave speed vs parameter γ for F0 = 0.5 and kH0 = 0 [reference, derived from Eq. (11)]. Also included are curves numerically solved from Eq. (8) for kH0 = 0.1 (red) and weak shears (s > 0, green; s < 0, blue).

Citation: Journal of the Atmospheric Sciences 71, 4; 10.1175/JAS-D-13-0101.1

b. Steady-state RGSW flow response to terrain

The steady-state response of RGSW to a ridge, h(x), can be derived by dropping the time-dependent terms in Eqs. (1), (2), (4), and (6):
e13
It is instructive to consider a RGSW flow over a sinusoidal ridge with a wavenumber k. The RGSW flow depth variation can be written as
e14
For α(k) = 0 (i.e., the upper layer is passive), Eq. (14) reduces to , according to which, the flow depth variation is in phase with the terrain profile for a supercritical flow and 180° out of phase for a subcritical flow. In the hydrostatic and irrotational limits, we have and , respectively, and Eq. (14) can be written as
e15
where and . A few aspects of Eq. (15) are worth mentioning. According to Eq. (15), for a wide ridge (i.e., ), pressure perturbations associated with a hydrostatic wave in the free atmosphere introduce a phase shift, to the RGSW flow depth variation. For a supercritical (subcritical) flow, we have χ > 0 (<0), corresponding to a downstream (upstream) shift. For a bell-shaped ridge, , Eq. (15) serves as an approximate solution over the terrain after replacing the sinusoidal wavenumber k with a dominant wavenumber . The vertical displacements (i.e., H0 + η + h) derived from Eq. (15) for F0 = 0.5 and 1.5, and J = 0, 0.2, and 1 are shown in Fig. 2. If the upper layer is completely passive (i.e., J = 0), the inversion height variation is in phase (i.e., supercritical ambient flow) or 180° out of phase (i.e., subcritical ambient flow) with the terrain underneath. For J = 0.2 or smaller, the upper layer is nearly passive and the phase shift is small. The amplitude of the vertical displacement is slightly smaller than the corresponding solution with J = 0. For J = 1 and larger, the undulation of the inversion height is characterized by a much reduced amplitude and substantial phase shift, implying that it is inappropriate to treat the atmosphere above the inversion as a passive layer. For a narrow ridge (i.e., toward the irrotational limit), pressure perturbations aloft only modify the amplitude of the vertical undulation of the inversion. Specifically, the vertical displacement over the ridge is decreased (increased) when the ambient flow is subcritical (supercritical). In addition, Eq. (15) suggests that a transition could occur over the ridge crest, where the critical Froude number is given by as opposed to predicted by the TSW theory.
Fig. 2.
Fig. 2.

The vertical displacement of the RGSW flow interface over a bell-shaped ridge with a hydrostatic wave above [i.e., Eq. (15)] vs horizontal distance. Two sets of solutions are included corresponding to a subcritical (F0 = 0.5, solid) and supercritical (F0 = 1.5, dashed) examples, and J = 0 (black), 0.2 (red), and 1 (green), respectively.

Citation: Journal of the Atmospheric Sciences 71, 4; 10.1175/JAS-D-13-0101.1

c. Governing nondimensional parameters

Besides F0 and M, the linear analysis above identifies two additional nondimensional parameters that measure the impact of the stratified atmosphere above the inversion on the RGSW flow, namely, and . As noted in Jiang et al. (2008), for a given vertical displacement of the inversion layer (η) associated with a linear stationary hydrostatic wave, the amplitudes of the corresponding pressure perturbations due to the density jump across the inversion layer and the hydrostatic wave aloft are (appendix A) and , respectively. Therefore, J also denotes the ratio between the perturbation pressure associated with hydrostatic waves above the inversion and the density jump across the inversion. A smaller J implies that the wave-induced pressure perturbation above the inversion is less important to the RGSW layer. Accordingly, local stationary response of a RGSW flow to the topography underneath is in better agreement with that of a shallow-water flow. It is noteworthy that the derivation in appendix A demonstrates that the definition of is valid for an inversion of a finite depth. The linear dependence of the pressure perturbation on serves as a foundation for the RGSW theory.

The second parameter corresponds to the ratio between the horizontal hydrostatic wave speed in the stratified atmosphere and the RGSW wave speed. Under a stronger inversion, γ is smaller and the leak of perturbation energy from the RGSW layer into the stratified atmosphere is slower. Consequently, waves or jumps in the RGSW flow decay slower as they propagate away from the wave source. It is worth noting that the two parameters can be thought as an inverse measure of the inversion strength as well. This can be seen by rewriting the two parameters into the following form:
e16
where represents a wave-scale height, which is proportional to the vertical hydrostatic wavelength. It is evident that J and represent the ratios between the potential temperature increase in the upper layer over a vertical wavelength or a vertical distance of the RGSW flow depth and the potential temperature jump across the inversion, respectively. The two are related as , where is often referred to as Froude number in the stratified upper layer.

In the presence of vertical wind shear across the inversion, the nondimensional shear defined by becomes relevant, and its role in the applicability of the RGSW theory is discussed in section 4c. Another relevant nondimensional parameter is (Ni is the buoyancy frequency in the inversion), which provides a measure of nonlinearity of the inversion response to the terrain underneath.

3. Numerical model configuration

The atmospheric component of the Coupled Ocean–Atmosphere Mesoscale Prediction System (COAMPS)1 (Hodur 1997) is used for this study. COAMPS is a fully compressible, nonhydrostatic and terrain-following mesoscale model with a suite of physical parameterizations such as a 1.5-order turbulence closure following Mellor and Yamada (1974).

The computation domain comprises 1001 grid points along flow direction with radiation boundary conditions applied at the eastern and western boundaries. There are 90 vertical levels with a 50-m spacing in the lowest 1 km and coarser resolutions aloft. The model top is located approximately at 30 km AGL with Rayleigh damping applied to the top 20 levels (i.e., ~12.5 km) to minimize downward wave reflection. There is no heat or momentum flux from the ground surface (i.e., free-slip condition). A Gaussian ridge, described by h(x) = hm exp(−x2/a2), where, a = 10 km, is the ridge width and hm is the ridge crest height, is located at the center of the domain, where x = 0. The model horizontal spacing is 0.05a. The model is initialized using an idealized sounding characterized by a three-layer atmosphere, namely, a shallow nearly neutral lower layer with a depth H, a uniform flow speed U0 and a constant buoyancy frequency N0, a deep layer aloft with a uniform flow speed U and a constant buoyancy frequency N, and between the two, a thin inversion layer with a depth d, potential temperature jump Δθ, and a linear shear of (U − U0)/d, unless specified otherwise. All the simulations are carried out for 6 h.

Before proceeding to numerical results, we shall discuss the criteria for testing the applicability of RGSW. According to the TSW theory, the response of a hydraulic flow over a two-dimensional ridge is governed by the ambient Froude number and the nondimensional mountain height. Specifically, the solutions fall into one of the following six regimes on an F0 × M regime diagram (Baines 1995), namely, subcritical (I), stationary lee jump (II), propagating jumps (III), supercritical (IV), hysteresis (V), and blocking (VI) regimes, respectively (Fig. 3). The flow characteristics in each regime and the derivation of the regime boundaries can be found in Baines (1995). Simulations over a wide range of control parameters are presented in the next section. For each simulation, we check the qualitative agreement between the simulated RGSW flow response and corresponding TSW solution in terms of the following features: supercritical (subcritical) response over the terrain, a state transition over the ridge crest, stationary jumplike structure over the lee slope, or propagating bore or lee jump. In addition, we compare the horizontal velocity perturbations in the RGSW layer and in the lower-tropospheric portion of the free atmosphere. It is concluded that that the RGSW theory is applicable if (i) the simulated RGSW flow assumes the general form of the corresponding TSW solution, and (ii) the horizontal velocity perturbation in the RGSW layer is substantially larger than that above the inversion.

Fig. 3.
Fig. 3.

The F0 × M regime diagram for a TSW flow past an infinity ridge and the regime boundaries as derived from TSW theory. The definition of the six solution regimes (i.e., I–VI) and derivation of the regime boundary curves (i.e., AB, BC, AD, AE, and AG) can be found in Baines (1995). The symbols represent the ambient flow conditions for some numerical simulations. Specifically, crosses, filled triangles, and open diamonds correspond to the three categories in the control simulations. The open circles and open triangles correspond to the sensitivity simulations with increasing inversion thickness and RGSW flow depth in section 4d.

Citation: Journal of the Atmospheric Sciences 71, 4; 10.1175/JAS-D-13-0101.1

4. Results

Several groups of simulations have been carried out to examine the response of a shallow neutral layer capped by an inversion to an underlying infinite ridge, including the control simulations presented in section 4a with emphasis on the influence of the inversion strength and ambient wind speed, two groups of simulations focusing on the sensitivity of the RGSW layer to the wind speed and stratification in the free atmosphere (i.e., sections 4b,c), and additional simulations with varying inversion height and depth and mountain height, respectively (section 4d).

a. Control simulations

We start with a group of 30 simulations (hereafter referred to as the control simulations) with a uniform wind speed U0 and a three-layered atmosphere, namely, a nearly neutral layer (i.e., N0 ~ 0.001 s−1) below H = 500 m, an inversion between 500 and 700 m (i.e., d = 200 m) and a uniformly stratified atmosphere aloft with N = 0.01 s−1. The ridge width is a = 10 km and crest height is hm = 300 m. The inversion strength is Δθ = 3, 5, 10, 15, and 20 K, and the ambient wind speed is U0 = U = 5, 10, 15, 20, and 30 m s−1, respectively. Using H0 = H + d/2 = 600 m as the RGSW layer depth, we obtain M = 0.5 and F0 between 0.2–3 (Fig. 3). Based on characteristics of the solutions, the control simulations can be broadly divided into three categories, namely, the nonpassive (J > 0.5, γ > 0.45), inversion-splitting (J < 0.5, γ < 0.45, Mi >1.5), and passive (J < 0.5, γ < 0.45, Mi < 1.5) categories (Fig. 4). Examples of the numerical solutions in these categories are shown in Figs. 57, respectively.

Fig. 4.
Fig. 4.

The control simulations mapped onto a g′ × U0 diagram. Simulations in the nonpassive, inversion-splitting, and RGSW categories are shown as squares, dots, and triangles, respectively. The solid, dashed, gray squares, and dot–dashed curves correspond to F0 = 1, J = 0.5 and 1, γ = 0.3, and Mi = 1.5 and 3, respectively.

Citation: Journal of the Atmospheric Sciences 71, 4; 10.1175/JAS-D-13-0101.1

Fig. 5.
Fig. 5.

(a),(b) Vertical cross sections of u (grayscale in increments of 1 m s−1) and isentropes (contours in increments of 2 K) for two examples in the nonpassive category valid at t = 1 h, corresponding to: Δθ = 5 K and U0 = 15 m s−1, and Δθ = 10 K and U0 = 30 m s−1, respectively. (c),(d) The corresponding distance–time diagrams of u (grayscale) at 200 m. The slopes of the dashed lines indicate the speeds of the upstream and downstream jumps from the TSW solutions. The corresponding time for the vertical cross sections is indicated by the arrows in (c),(d).

Citation: Journal of the Atmospheric Sciences 71, 4; 10.1175/JAS-D-13-0101.1

Fig. 6.
Fig. 6.

As in Fig. 5, but for two examples in the inversion-splitting category valid at 3 h with (a),(c) Δθ = 10 K and U0 = 5 m s−1 and (b),(d) Δθ = 20 K and U0 = 10 m s−1. The u increment in (b),(d) is 2 m s−1.

Citation: Journal of the Atmospheric Sciences 71, 4; 10.1175/JAS-D-13-0101.1

Fig. 7.
Fig. 7.

As in Fig. 6, but for the passive category with (a),(c) Δθ = 20 K and U0 = 15 m s−1 and (b),(d) Δθ = 20 K and U0 = 30 m s−1. The potential temperature contours (increments of 3 K) at the 600-m level are included in (c),(d) as well.

Citation: Journal of the Atmospheric Sciences 71, 4; 10.1175/JAS-D-13-0101.1

The simulations in the nonpassive category are characterized by strong ambient wind speed and a weak inversion. Consequently, the RGSW flow is typically supercritical and these solutions fall into regimes III–V on the classical F0 × M diagram (Fig. 3). In this category, J and γ are relatively large, the pressure perturbation above the inversion is comparable to that due to the inversion. Accordingly, the atmosphere above the inversion has significant impact on the RGSW flow and the passive layer assumption becomes problematic. A pair of the nonpassive category solutions is shown in Fig. 5, corresponding to F0 = 1.5 and 2.1, J = 0.9, and γ = 0.6 and 0.42, respectively. Consistent with the TSW theory, a downstream-propagating jump is present in the lee, above which gravity waves are evident, indicative of energy leak from the RGSW layer (Fig. 5a). Consequently, the jump weakens rapidly while propagating away from the ridge, as evidenced in the Hovmöller diagram (Fig. 5c). However, the propagation speeds are in agreement with the TSW theory prediction, implying that the modification of the RGSW wave speeds by the pressure perturbations above the inversion is negligible. Over the ridge, the response of the RGSW layer resembles that of a supercritical flow except that the vertical displacement maximum of the inversion is located upstream of the ridge crest, a clear upstream phase shift as predicted by the linear analysis (e.g., supercritical examples in Fig. 2). Fairly strong waves are evident above the inversion with the horizontal wind speed perturbations comparable below and above the inversion. Although the potential temperature jump is doubled in the second example, the amplitude of the velocity perturbation in the free atmosphere directly above the ridge remains comparable to that in the RGSW layer, due to the doubled ambient wind speed which keeps J near unity. However, the upstream shift of the inversion dome is noticeably smaller in the second example, due to the larger F0. The downwind-propagating speed of the lee jump is consistent with the TSW theory and the jump amplitude decays slower with the distance than in the first example in accordance to a smaller γ.

In summary, one should be cautious when applying the RGSW theory to an atmosphere in this category for the following reasons: 1) a significant fraction of the perturbation energy leaks through the inversion, 2) any jumplike structure decays rapidly with distance while propagating away from the terrain, 3) the response of the RGSW layer over the terrain departs substantially from the corresponding TSW solution, and 4) velocity perturbations in the free atmosphere and in the RGSW layer are comparable.

The inversion-splitting category corresponds to a weak ambient wind and a moderate-to-strong inversion. Accordingly, F0 is small (i.e., subcritical, see Fig. 3). While both J and γ are small, the nonlinear parameter Mi is above 1.5, suggesting that the flow is highly nonlinear (Fig. 4). The COAMPS simulations in this category are characterized by an upstream-propagating jump, a subcritical–supercritical transition near the ridge crest, and a leeside jump. The lee jump could be stationary residing over the lee slope of the ridge or downstream propagating depending on whether F0 is below or above the BC curve (Fig. 3). Two examples are shown in Fig. 6, corresponding to K and U0 = 5 m s−1 (i.e., F0 = 0.35, J = 0.15, and γ = 0.42) and K and U0 = 10 m s−1 (i.e., F0 = 0.5, J = 0.15, and γ = 0.3), respectively. The first example falls clearly under the AD curve and the COAMPS simulation shows plunging of the inversion and acceleration of the RGSW flow over the lee slope followed by an abrupt ascent of the upper portion of the inversion (Fig. 6a). The leading edge of this hydraulic jumplike structure is nearly stationary throughout the simulation, which is consistent with the TSW theory. The second example is slightly below the AD curve (Fig. 3) and accordingly, the lee jump is located immediately downstream of the ridge at 3 h and is nearly stationary through the rest of the simulation (Figs. 6b,d). For both examples, in the vicinity of the ridge, the wind speed perturbation in the RGSW layer is substantially larger than that right above the inversion. The amplitudes of the upstream jumps show little decay over the 6-h period, consistent with the relatively small γ values, and their speeds are in good agreement with the TSW theory prediction as well.

A few aspects of the lee jumps in the inversion-splitting category are worth mentioning. First, in Figs. 6a,b, the inversion splits after the jump, and only the upper portion of the inversion ascends and experiences a sudden deceleration. The structure of weak reversed flow near the leading edge of the jump and accelerated flow below resembles a roller or rollers in a traditional hydraulic jump for flows with an upstream Froude number greater than 1.7 (e.g., Hager 1992). The Froude numbers upstream of the lee jumps in these two examples are around 1.8–1.9, in agreement with the TSW theory. It is also worth noting that similar hydraulic jumps have been documented by several numerical (e.g., Vosper 2004; Jiang and Doyle 2005) and observational (Jiang and Doyle 2004) studies of tropospheric flows over topography. Second, the stationary jump in Fig. 6a bears resemblance to a breaking internal wave near the inversion top as well. While the two correspond to different dynamics (i.e., breaking of a horizontally propagating surface wave versus breaking of a vertically propagating internal wave), it is difficult to distinguish them here. Third, in Fig. 6b, the surface jet extends far downstream from the leading edge of the jump, and slowly weakens and thins with distance. This is likely associated with vertical turbulence mixing of momentum owing to the strong vertical wind shear, a process not well resolved by mesoscale models such as COAMPS. It is noteworthy that whether there is a surface jet or wake downstream of coastal mountains or islands has dramatically different implication on the downwind air–sea interaction (e.g., Jiang and Doyle 2005; Pullen et al. 2006). In addition, surface friction, which is ignored in this study and TSW theory, may become important for the extended surface jet.

The solutions in the passive category correspond to a moderate-to-strong inversion and a relatively strong ambient wind (Fig. 4) and fall into regime III on the TSW regime diagram (Fig. 3). In this category, J and γ are relatively small and Mi < 1.5. Solutions in this category are characterized by well-defined slowly decaying downstream and upstream-propagating jumps, and a subcritical–supercritical state transition near the ridge crest. Velocity perturbations in the RGSW layer are substantially larger than above the inversion. Two examples from this category are shown in Fig. 7, corresponding to subcritical (i.e., K, U0 = 15 m s−1, F0 = 0.75, J = 0.23, and γ = 0.3) and supercritical (i.e., K, U0 = 30 m s−1, F0 = 1.5, J = 0.45, and γ = 0.3) ambient flows, respectively. For the subcritical example, shooting flow develops over the lee slope and extends downstream with little deceleration until a hydraulic jump occurs, which approximately doubles the flow depth and halves the speed. Similarly, for the supercritical example, the upstream and downstream jumps decay slowly as they propagate away from the ridge with speeds comparable to those predicted by the TSW theory (Figs. 7b,d). The upstream jump assumes the form of undular bores and the shooting flow developed over the lee slope extends more than 150 km.

In summary, for the passive category, with J and γ much less than unity, the passive layer assumption appears to be a reasonable approximation. The solutions are characterized by a subcritical–supercritical transition over the ridge crest and slowly decaying propagating jumps, and are in qualitative agreement with the TSW theory. The applicability of the RGSW theory becomes problematic when J or γ is near unity or larger, and the passive-upper-layer assumption becomes inadequate. However, the transition from the nonpassive to passive category is rather gradual.

b. Upper-layer stability

To examine the impact of the stratification in upper layer on the RGSW flow, we compare a group of four simulations, including the control simulation corresponding to Δθ = 20 K, U0 = U = 15 m s−1, and three additional simulations with the identical governing parameters except that the buoyancy frequency in the upper layer is N = 0.0, 0.005, and 0.015 s−1, respectively. In accordance to the increase of N from 0 to 0.015 s−1, we have J = 0, 0.11, 0.23, and 0.34, and γ = 0, 0.15, 0.3, and 0.45.

All four simulations reproduce the gross features predicted by the TSW theory (e.g., Baines 1995), including upstream and leeside propagating jumps, a subcritical–supercritical transition near the ridge crest, and the shooting flow between the ridge crest and the lee jump (Figs. 8 and 9). Careful inspection reveals substantial difference between these simulations in terms of both the characteristics of the jumps and the wave patterns aloft. With a less stratified atmosphere above (i.e., N = 0 or 0.005 s−1) there are no discernible wave patterns in the upper layer. Perturbations associated with vertical undulation of the inversion layer monotonically decrease with height (Fig. 8a). In the lee side, a hydraulic jump propagates downstream, across which wind speed below the inversion is substantially decreased and the RGSW flow depth is approximately doubled. Localized perturbations appear right over the jump in response to the abrupt ascent of the inversion and decay aloft. A series of undular bores propagate upstream, behind which the RGSW layer becomes thicker and slower. Acceleration and thinning of the RGSW layer occurs over the lee slope of the hill and the resulting jet or shooting flow extends downstream until a lee jump takes place. Both the upstream- and downstream-propagating jumps approximately maintain their amplitudes over the integration time (Fig. 8c), implying little energy leak from the RGSW layer. According to linear analysis, the pressure perturbation associated with an evanescent wave in the upper layer tends to increase the RGSW wave speed by [see Eq. (12)]. The characteristic wavenumber k corresponding to the narrow leading edge of the lee jump is substantially larger than that corresponding to the upstream bore and therefore, the lee jump moves noticeably faster than predicted by the TSW theory. With a neutrally stratified layer above (i.e., N = 0), perturbations become evanescent in the upper layer and the perturbation energy is trapped in the RGSW layer (not shown). A stratified upper layer can support stationary gravity waves with a wavelength longer than the cutoff wavelength given by . For U = 20 m s−1 and N = 0.005 s−1, the cutoff horizontal wavelength is ~25 km, and accordingly a large fraction of the perturbations becomes evanescent in the upper layer.

Fig. 8.
Fig. 8.

As in Fig. 7, but for Δθ = 20 K and U0 = 15 m s−1, and the buoyancy frequency in the upper layer is (a),(c) 0.005 and (b),(d) 0.015 s−1.

Citation: Journal of the Atmospheric Sciences 71, 4; 10.1175/JAS-D-13-0101.1

Fig. 9.
Fig. 9.

(top to bottom) The depth, vertically averaged horizontal wind speed, and local Froude number of the RGSW layer valid at 3 h derived from four simulations with same wind speed in lower and upper layers (U0 = U = 15 m s−1) and Δθ = 20 K; N = 0.0, 0.005, 0.01, and 0.015 s−1 plotted as a function of the horizontal distance.

Citation: Journal of the Atmospheric Sciences 71, 4; 10.1175/JAS-D-13-0101.1

For N = 0.01 s−1 and larger, waves are evident in the upper layer over the ridge and the lee jump (Fig. 8b). In response to the wave-induced pressure gradient above the inversion, the shooting flow between the lee slope and the lee jump thickens and slows down downstream. Consistent with linear theory, both the upstream and downstream jumps decay with the distance away from the ridge crest (Fig. 8d). For N = 0.015 s−1, the cutoff wavelength (i.e., ) is much shorter than that with N = 0.005 s−1, implying that the upper layer become less passive. From 3 to 6 h, the lee jump becomes slower in accordance to the decrease in the jump amplitude attributed to the perturbation energy leak.

In summary, a neutral or weakly stratified upper layer is in general more passive, as most perturbation energy is trapped in the RGSW layer. However, it is noteworthy that even with N = 0, the upper layer is still not completely passive, and the hydraulic jump speed can be modified by pressure perturbations associated with evanescent perturbations in the upper layer. With increasing stability in the upper layer, the impact of waves aloft on the RGSW flow becomes more pronounced for the following reasons. First, according to linear theory, the amplitude of pressure perturbations associated with a hydrostatic wave in the upper layer is proportional to the buoyancy frequency aloft. Second, the cutoff wavelength, , decreases with an increasing N, allowing for shorter-wave components of perturbations to leak into the upper layer. With a more stable upper layer, both the upstream and downstream jumps become weaker and slower (Fig. 9). Especially the leading edge of the upstream jump becomes gentler as N increases and the shooting flow over the lee slope and beyond slows down. Further inspection indicates that the transition from subcritical to supercritical flows occur approximately over the ridge crest for N = 0 and 0.005 s−1, and about 3 and 5 km upstream of the ridge crest for N = 0.01 and 0.015 s−1 associated with gravity waves aloft. In addition, a more stable upper layer increases the possibility of wave breaking above the inversion.

c. Vertical wind shear effect

To examine the influence of the wind speed in the upper layer or the vertical wind shear on the RGSW layer, we carry out four additional simulations with Δθ = 20 K, U0 = 15 m s−1, N = 0.01 s−1, and U = 5, 10, 20, and 30 m s−1 (i.e., J = 0.075, 0.15, 0.23, and 0.45), respectively. The parameter γ = 0.3 remains the same for all these simulations.

We start by comparing two examples corresponding to U = 5 (i.e., backward shear) and 30 (i.e., forward shear) m s−1, respectively (Fig. 10). In both simulations, the response of the RGSW layer qualitatively resembles a TSW flow, including the upstream- and downstream-propagating jumps, and the plunging of the inversion and acceleration of RGSW flow over the lee slope. However, gravity waves are evident above the inversion in both simulations. With a backward shear, there is a series of well-defined upstream bores in the RGSW layer, propagating at a speed comparable to the TSW prediction and shows little decay in amplitude. In the lee side, the backward shear prompts the occurrence of inversion-splitting (note, no inversion-splitting occurs in the corresponding control simulation). With a stronger wind in the upper layer, the upstream jump becomes weaker, smoother, and slower (Figs. 10c,d). The lee jump propagates at a speed comparable to the TSW solution with multiple finescale trapped waves trailing behind. The jump and trapped waves show little decay in amplitudes (Fig. 10d), and above the trapped waves, the finescale perturbations decay monotonically with the altitude (Fig. 10c).

Fig. 10.
Fig. 10.

As in Fig. 8, but for (a),(c) U = 5 and (b),(d) U = 30 m s−1.

Citation: Journal of the Atmospheric Sciences 71, 4; 10.1175/JAS-D-13-0101.1

The sensitivity of the jump characteristics to the upper-layer wind speed is more evident in Fig. 11. As U increases, the upstream bore becomes progressively slower and weaker, and over the lee slope, the shooting flow weakens noticeably as well. The lee jump is most sensitive to the vertical wind shear. For U = 5 m s−1, the lee jump manifests as a gradual decrease (increase) in the wind speed (RGSW depth) due to the inversion splitting. For U = 10 and 15 m s−1, the lee jump shows an abrupt increase in depth, across which the flow transitions into a subcritical flow. For U = 20 and 30 m s−1, the lee jump becomes substantially weaker with trailing finescale trapped waves. It is noteworthy that the transition from subcritical to supercritical RGSW flow occurs approximately over the ridge crest regardless of the vertical wind shear.

Fig. 11.
Fig. 11.

(top to bottom) The RGSW flow depth, mean velocity, and local Froude number valid at 3 h plotted as a function of the horizontal distance for four simulations corresponding to U0 = 15 m s−1, Δθ = 20 K, N = 0.01 s−1, and U = 5, 10, 20, and 30 m s−1.

Citation: Journal of the Atmospheric Sciences 71, 4; 10.1175/JAS-D-13-0101.1

The complexity of the upper-layer wind speed effect can be partially seen from a theoretical perspective. The pressure perturbation associated with a hydrostatic wave is proportional to the wind speed U. In the meantime, as U increases from 5 to 30 m s−1, the cutoff wavelength, , increases from ~3 to ~19 km, implying a substantially larger fraction of the perturbation energy is trapped in the RGSW layer. These two countereffects partially cancel each other out. The increase in U results in the formation of trapped waves trailing behind the lee jump in accordance to the decrease of Scorer parameter aloft (i.e., l ~ N/U; Scorer 1949). The vertical wind shear also contributes to the fast decay of a propagating jump as demonstrated in section 2. The backward shear tends to promote low-level wave breaking and inversion splitting.

Finally, it is well known that the existence of a critical level help decouple low-level flow from perturbations above. One such example (i.e., Δθ = 10 K between 500 and 700 m; U0 = 10 m s−1 below 700 m, which decreases linearly to 0 at 3 km and −10 m s−1 at 5.3 km, and U = −10 m s−1 above) is shown in Fig. 12. With a critical level at 3 km MSL in the ambient flow, gravity waves are virtually absorbed by the critical level, and pressure perturbations above the inversion are inhibited. Accordingly, the RGSW layer assumes the form of a TSW flow, characterized by an upstream jump, subcritical-to-supercritical transition over the crest, shooting flow developing over the lee slope, and a lee jump. Both the upstream and lee jumps show little sign of decaying while propagating away from the ridge.

Fig. 12.
Fig. 12.

(a) Vertical cross section of u (grayscale, increments of 2 m s−1) and θ (contour, increments of 1 K) for a critical-level simulation (see text for specifics) valid at 3 h. (b) The corresponding Hovmöller diagram of u at 200 m (increments of 2 m s−1) and θ at 600 m (contour, increments of 3 K).

Citation: Journal of the Atmospheric Sciences 71, 4; 10.1175/JAS-D-13-0101.1

d. Inversion depth and RGSW layer depth

For the sake of comparison with linear theory, the inversion is kept thin (i.e., d = 200 m) in the preceding simulations. In reality, the depth of a low-level inversion varies from less than 100 m for stratocumulus-topped marine boundary layers to a kilometer or thicker for cloud-free boundary layers over land. The control simulation of Δθ = 10 K and U = 10 m s−1 has been repeated with an increasing inversion depth (i.e., d = 400, 600, 800, and 1000 m), in order to answer the following questions: 1) is RGSW still applicable when the inversion is thick, 2) does a thicker inversion tend to promote inversion splitting across a jump, and 3) what is the proper equivalent RGSW depth for calculating Froude number and nondimensional ridge height in the presence of a thick inversion?

As shown in Fig. 13, in general, the simulated RGSW flow exhibits primary characteristics of a TSW flow. The control simulation (i.e., d = 200 m) shows a typical passive solution with a slowly decaying jump propagating downstream (Figs. 13a,c), and for d = 1000 m, the lee jump becomes stationary and is located over the lee slope (Figs. 13b,d). In fact, as d increases from 200 to 600 m, the propagation speed of the lee jump gradually decreases, and for the two simulations with d = 800 and 1000 m the lee jump becomes stationary. For the two examples, the acceleration of the RGSW flow over the lee slope is substantially more pronounced than that above the inversion, implying that the RGSW theory is qualitatively applicable over the range of inversion depths examined. Using , the nondimensional inversion depth can be written as , which corresponds to the phase shift of a hydrostatic wave across the inversion. In theory, to be approximated as a single layer, the phase difference between the ground and the inversion top should be at least less than π (i.e., if the layer below the inversion is neutral).

Fig. 13.
Fig. 13.

As in Fig. 8, but from a pair of simulations with Δθ = 10 K, U0 = 10 m s−1, and (a),(c) d = 200 and (b),(d) d = 1000 m.

Citation: Journal of the Atmospheric Sciences 71, 4; 10.1175/JAS-D-13-0101.1

For a thick inversion, a question arises regarding what is the proper equivalent RGSW flow depth for calculating parameters such as F0 and M. As demonstrated in appendix B, an adequate choice is H0 = H + d/2, which is an approximation consistent with the vertically averaged linear RGSW equations. Using H0 = H + d/2 as the RGSW layer depth, as d increases from 200 to 1000 m, the ambient flow changes from regimes III to II, associated with the decrease of both F0 and M (Fig. 3). The corresponding COAMPS simulations are in qualitative agreement with the TSW theory prediction.

Associated with the increase of d from 200 to 1000 m, increases from 0.82 to 1.83 and Mi decreases from 1.2 to 0.55. The lack of inversion splitting in these simulations suggests that Mi rather than the inversion depth govern the occurrence of inversion splitting. This is supported by two additional groups of simulations with Δθ = 10 K, U0 = 10 m s−1, and d = 200 m, but varying H and (or) hm. In the first group, the ridge height hm = 300 m is fixed while the RGSW flow depth increases from 600 to 1600 m. Accordingly, F0 and M decrease. These simulations span regimes I–III on the F0 × M regime diagram (Fig. 3). As shown in the two examples (Figs. 14a,b), the simulated RGSW flow response resembles the corresponding TSW flow without inversion splitting, as the nonlinearity parameter Mi remains less than 1.5. In the second group, the nondimensional ridge height, M = 0.5, is fixed while the RGSW flow depth increases from 400 to 1600 m. Two examples are shown in Figs. 14c,d, corresponding to , respectively. While the solutions transition from propagating jump to stationary lee jump regimes, inversion splitting occurs in all simulations with Mi > 1.5.

Fig. 14.
Fig. 14.

As in Fig. 8a, but from four simulations with Δθ = 10 K, U0 = 10 m s−1, and (a) H0 = 1 km, hm = 300 m; (b) H0 = 1.4 km, hm = 300 m; (c) H0 = 1 km, hm = 500 m; and (d) H0 = 1.6 km, hm = 800 m.

Citation: Journal of the Atmospheric Sciences 71, 4; 10.1175/JAS-D-13-0101.1

5. Discussion and conclusions

The applicability of the reduced-gravity shallow-water (RGSW) theory to a shallow nearly neutral layer underneath an inversion and a deep stratified atmosphere past a ridge has been examined using linear analysis and nonlinear numerical simulations. A broad range of inversion strength and depth, the stratification in the atmosphere above the inversion, and vertical wind shear has been examined. The following two nondimensional parameters are found to be critical in evaluating the applicability of the RGSW theory: and . A numerical solution falls into one of the three categories, namely, the nonpassive category characterized by strong winds but a weak inversion (J or γ near unity or larger), the inversion-splitting category characterized by relatively weak winds (J or γ much less than unity but ), and the passive category characterized by moderate-to-strong winds and a moderate-to-strong inversion (J or γ much less than unity and ).

One should be cautious when applying the RGSW theory to flows in the nonpassive category. With J or γ near unity or larger, perturbations in the RGSW layer decay rapidly while propagating away from the ridge, and over the ridge the RGSW response is substantially modified by wave-induced pressure perturbations aloft. For flows in the other two categories with J or γ much less than unity, the passive layer assumption is adequate and the RGSW theory provides solutions in qualitative agreement with traditional shallow-water (TSW) theory. For a flow in the inversion-splitting category with , the response of the RGSW flow to the terrain is highly nonlinear, and the inversion splits near the leading edge of the jump with weakly reversed flow in upper portion of the inversion and a surface jet underneath. It is worth noting that, in some simulations, a surface jet may extent far downstream. In reality the characteristic length scale of the surface jet is likely controlled by the vertical turbulence mixing and bottom friction, which is either not resolved or not included in these simulations.

Both the wind speed and stratification in the upper layer have a substantial impact on the RGSW flow response. In general, a neutrally stratified upper layer serves as the best approximation of a passive layer. Perturbations excited by the vertical undulation of the inversion decay rapidly with height in a neutral layer, and little perturbation energy leaks through the inversion into the upper layer. A more stratified upper layer tends to increase J and γ. In addition, only waves with a wavelength longer than the cutoff wavelength given by are supported by the upper layer, and therefore, a more stable upper layer also allows for shorter-wave perturbations to leak through the inversion. Consequently, propagating perturbations in the RGSW layer decay faster under a more stable upper layer and in general the passive layer assumption becomes more problematic. The impact of the wind speed in the upper layer is more complicated: wave-induced pressure perturbations are proportional to the wind speed, and in the meantime, a stronger wind also increases the cutoff wavelength and prohibits the energy leak associated with shorter waves. The presence of a critical level above the inversion helps decouple the RGSW layer from perturbations above and facilitate the application of RGSW theory. Nonlinear simulations also show that RGSW theory is applicable over a fairly wide range of inversion depth and inversion height. For a neutral layer (H) capped by a thick inversion (d), the equivalent RGSW flow depth is approximately given by H0 = H + d/2.

Although this study focuses on flows over a two-dimensional ridge, we believe that the conclusions should hold for interaction between a RGSW flow under an inversion and a wide range of mesoscale terrain such as three-dimensional hills, coastal ridge, valley, and islands. Finally, for the sake of simplicity, some potentially important processes such as bottom friction and wave–tropopause interaction are not considered in this study, and will be examined in future studies.

Acknowledgments

This research is supported by NRL Base Program PE 0601153N. The author has greatly benefited from discussions with Drs. James Doyle at Naval Research Laboratory and Ronald Smith at Yale University. We want to thank three anonymous reviewers for their helpful comments. The simulations were made using the Coupled Ocean–Atmosphere Mesoscale Prediction System (COAMPS) developed by U.S. Naval Research Laboratory.

APPENDIX A

Estimate the Reduced-Gravity Acceleration

Complexity associated with how to estimate the reduced-gravity acceleration in a two-layer atmosphere had been illustrated in Durran (1986). Here we try to derive an equivalent reduced-gravity acceleration for the three-layer atmosphere examined in this study. Combining the hydrostatic equation, , equation of state for dry air, , and potential temperature definition, , we obtain
ea1
where R = 287 J (kg K)−1 is the gas constant, is the Poisson constant, and p0 = 1000 kPa is the reference surface pressure. Consider a three-layer atmosphere with a potential profile as the following:
ea2
where D is the top of the atmosphere and is the potential temperature at the surface. Substituting Eqs. (A2) into (A1) and integrating vertically, we obtain
ea3
where ps is the surface pressure. Assuming , we have
ea4
where is the potential temperature increase across the inversion layer.
Assuming the inversion depth is constant and differentiating Eq. (A4) with respect to the inversion height H, we obtain the surface pressure change, , due to a small increase in the inversion height η:
eq1
Letting , the above equation reduces to
ea5
where the reduced-gravity acceleration .

It is noteworthy that Eq. (A5) is valid for a two-layer atmosphere with a shallow and more stable layer underneath similar to the one considered in Durran (1986) as well. For the two-layer atmosphere, η is the change in the stable-layer depth. Equation (A5) can be obtained by letting H = 0 and differentiating Eq. (A4) with respect to the inversion depth d in Eq. (A4).

APPENDIX B

Equivalent Depth of the RGSW Layer

A key question arises from applying the RGSW theory to a neutral layer underneath an inversion of a finite depth is how to determine the equivalent depth of the RGSW layer. Consider a RGSW layer consisting of a neutral layer H and an inversion of a finite depth d. The linear one-dimensional shallow-water equations can be written as
eb1
eb2
where H0 is the equivalent RGSW flow depth, U is the mean wind, and vertically integrated perturbation variables are denoted by overbars. The pressure perturbation associated with vertical undulation of the inversion height η is below the inversion, and is at a level z within the inversion, which becomes zero at the top of the inversion. Then the layer-averaged pressure p′ in the last term in Eq. (B1) is given by
eb3
Further assuming the velocity perturbation takes the form of in the neutral layer and in the inversion, which decreases to zero above, we obtain
eb4
Substituting Eqs. (B3) and (B4) into Eqs. (B1) and (B2) and letting H0 = H + d/2 recovers traditional linear SW equations. This derivation suggests that H0 = H + d/2 serves as a good approximation of the equivalent RGSW layer depth, at least in the linear limit.

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  • Armi, L., 1986: The hydraulics of two flowing layers with different densities. J. Fluid Mech., 163, 2758.

  • Armi, L., and G. M. Mayr, 2011: The descending stratified flow and internal hydraulic jump in the lee of the Sierras. J. Appl. Meteor. Climatol., 50, 19952011.

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  • Fig. 1.

    (top) The normalized real wave speed departure from the TSW wave speed and (bottom) the corresponding normalized imaginary wave speed vs parameter γ for F0 = 0.5 and kH0 = 0 [reference, derived from Eq. (11)]. Also included are curves numerically solved from Eq. (8) for kH0 = 0.1 (red) and weak shears (s > 0, green; s < 0, blue).

  • Fig. 2.

    The vertical displacement of the RGSW flow interface over a bell-shaped ridge with a hydrostatic wave above [i.e., Eq. (15)] vs horizontal distance. Two sets of solutions are included corresponding to a subcritical (F0 = 0.5, solid) and supercritical (F0 = 1.5, dashed) examples, and J = 0 (black), 0.2 (red), and 1 (green), respectively.

  • Fig. 3.

    The F0 × M regime diagram for a TSW flow past an infinity ridge and the regime boundaries as derived from TSW theory. The definition of the six solution regimes (i.e., I–VI) and derivation of the regime boundary curves (i.e., AB, BC, AD, AE, and AG) can be found in Baines (1995). The symbols represent the ambient flow conditions for some numerical simulations. Specifically, crosses, filled triangles, and open diamonds correspond to the three categories in the control simulations. The open circles and open triangles correspond to the sensitivity simulations with increasing inversion thickness and RGSW flow depth in section 4d.

  • Fig. 4.

    The control simulations mapped onto a g′ × U0 diagram. Simulations in the nonpassive, inversion-splitting, and RGSW categories are shown as squares, dots, and triangles, respectively. The solid, dashed, gray squares, and dot–dashed curves correspond to F0 = 1, J = 0.5 and 1, γ = 0.3, and Mi = 1.5 and 3, respectively.

  • Fig. 5.

    (a),(b) Vertical cross sections of u (grayscale in increments of 1 m s−1) and isentropes (contours in increments of 2 K) for two examples in the nonpassive category valid at t = 1 h, corresponding to: Δθ = 5 K and U0 = 15 m s−1, and Δθ = 10 K and U0 = 30 m s−1, respectively. (c),(d) The corresponding distance–time diagrams of u (grayscale) at 200 m. The slopes of the dashed lines indicate the speeds of the upstream and downstream jumps from the TSW solutions. The corresponding time for the vertical cross sections is indicated by the arrows in (c),(d).

  • Fig. 6.

    As in Fig. 5, but for two examples in the inversion-splitting category valid at 3 h with (a),(c) Δθ = 10 K and U0 = 5 m s−1 and (b),(d) Δθ = 20 K and U0 = 10 m s−1. The u increment in (b),(d) is 2 m s−1.

  • Fig. 7.

    As in Fig. 6, but for the passive category with (a),(c) Δθ = 20 K and U0 = 15 m s−1 and (b),(d) Δθ = 20 K and U0 = 30 m s−1. The potential temperature contours (increments of 3 K) at the 600-m level are included in (c),(d) as well.

  • Fig. 8.

    As in Fig. 7, but for Δθ = 20 K and U0 = 15 m s−1, and the buoyancy frequency in the upper layer is (a),(c) 0.005 and (b),(d) 0.015 s−1.

  • Fig. 9.

    (top to bottom) The depth, vertically averaged horizontal wind speed, and local Froude number of the RGSW layer valid at 3 h derived from four simulations with same wind speed in lower and upper layers (U0 = U = 15 m s−1) and Δθ = 20 K; N = 0.0, 0.005, 0.01, and 0.015 s−1 plotted as a function of the horizontal distance.

  • Fig. 10.

    As in Fig. 8, but for (a),(c) U = 5 and (b),(d) U = 30 m s−1.

  • Fig. 11.

    (top to bottom) The RGSW flow depth, mean velocity, and local Froude number valid at 3 h plotted as a function of the horizontal distance for four simulations corresponding to U0 = 15 m s−1, Δθ = 20 K, N = 0.01 s−1, and U = 5, 10, 20, and 30 m s−1.

  • Fig. 12.

    (a) Vertical cross section of u (grayscale, increments of 2 m s−1) and θ (contour, increments of 1 K) for a critical-level simulation (see text for specifics) valid at 3 h. (b) The corresponding Hovmöller diagram of u at 200 m (increments of 2 m s−1) and θ at 600 m (contour, increments of 3 K).

  • Fig. 13.

    As in Fig. 8, but from a pair of simulations with Δθ = 10 K, U0 = 10 m s−1, and (a),(c) d = 200 and (b),(d) d = 1000 m.

  • Fig. 14.

    As in Fig. 8a, but from four simulations with Δθ = 10 K, U0 = 10 m s−1, and (a) H0 = 1 km, hm = 300 m; (b) H0 = 1.4 km, hm = 300 m; (c) H0 = 1 km, hm = 500 m; and (d) H0 = 1.6 km, hm = 800 m.

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