A Modeling Study of Nonstationary Trapped Mountain Lee Waves. Part I: Mean-Flow Variability

Louisa B. Nance Department of Atmospheric Sciences, University of Washington, Seattle, Washington

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Dale R. Durran Department of Atmospheric Sciences, University of Washington, Seattle, Washington

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Abstract

The impact of mean-flow variability on finite-amplitude trapped mountain lee waves is investigated by conducting two-dimensional mountain wave simulations for a set of idealized, time-dependent background flows. The lee-wave patterns generated by these time-dependent flows depend on two factors: 1) the degree to which the transition in the background flow changes the amplitude of the stationary trapped lee wave and 2) the difference between the group velocities of the trapped waves generated before and after the transition. When the transition in the background flow significantly reduces the amplitude of the stationary lee wave, the lee-wave pattern generated prior to the transition gradually drifts downstream away from the mountain or back over the mountain, depending on the sign of this wave packet’s group velocity after the transition. When the transition in the background flow changes the resonant wavelength while leaving the lee-wave amplitude relatively unchanged, the lee-wave train develops either 1) a smooth transition in horizontal wavelength or 2) a region of irregular variations in wavelength and amplitude, depending on the difference between the group velocities of the waves generated before and after the transition. Although linear theory is able to predict how changes in the background flow will affect the group velocities of the trapped waves, it is not able to predict whether the temporal variations in the large-scale environmental flow will amplify or dampen the resonant waves when the waves are no longer linear.

Regions of irregular variations in wavelength and amplitude may develop when stationary trapped waves generated after a transition in the background flow overtake the trapped waves generated before the transition. The fluctuations in the vertical velocities associated with such numerically simulated lee waves are compared with wind profiler observations. Estimates of the time required for the trapped waves generated after the transition to overtake those generated before the transition suggest that the temporal changes in the background flow required to qualitatively reproduce the observed vertical velocity variations are not likely to occur on a realistic timescale. In addition, the observed temporal variations in lee-wave vertical velocities appear to be the superposition of at least two distinct frequencies, whereas the temporal variations in the simulated waves are dominated by one distinct frequency.

Corresponding author address: Dr. Louisa B. Nance, NOAA/National Weather Service, 7600 Sand Point Way NE, Bin C15700, Seattle, WA 98115.

Email: nance@seawfo.noaa.gov

Abstract

The impact of mean-flow variability on finite-amplitude trapped mountain lee waves is investigated by conducting two-dimensional mountain wave simulations for a set of idealized, time-dependent background flows. The lee-wave patterns generated by these time-dependent flows depend on two factors: 1) the degree to which the transition in the background flow changes the amplitude of the stationary trapped lee wave and 2) the difference between the group velocities of the trapped waves generated before and after the transition. When the transition in the background flow significantly reduces the amplitude of the stationary lee wave, the lee-wave pattern generated prior to the transition gradually drifts downstream away from the mountain or back over the mountain, depending on the sign of this wave packet’s group velocity after the transition. When the transition in the background flow changes the resonant wavelength while leaving the lee-wave amplitude relatively unchanged, the lee-wave train develops either 1) a smooth transition in horizontal wavelength or 2) a region of irregular variations in wavelength and amplitude, depending on the difference between the group velocities of the waves generated before and after the transition. Although linear theory is able to predict how changes in the background flow will affect the group velocities of the trapped waves, it is not able to predict whether the temporal variations in the large-scale environmental flow will amplify or dampen the resonant waves when the waves are no longer linear.

Regions of irregular variations in wavelength and amplitude may develop when stationary trapped waves generated after a transition in the background flow overtake the trapped waves generated before the transition. The fluctuations in the vertical velocities associated with such numerically simulated lee waves are compared with wind profiler observations. Estimates of the time required for the trapped waves generated after the transition to overtake those generated before the transition suggest that the temporal changes in the background flow required to qualitatively reproduce the observed vertical velocity variations are not likely to occur on a realistic timescale. In addition, the observed temporal variations in lee-wave vertical velocities appear to be the superposition of at least two distinct frequencies, whereas the temporal variations in the simulated waves are dominated by one distinct frequency.

Corresponding author address: Dr. Louisa B. Nance, NOAA/National Weather Service, 7600 Sand Point Way NE, Bin C15700, Seattle, WA 98115.

Email: nance@seawfo.noaa.gov

1. Introduction

Stably stratified flow over a mountain ridge can produce internal gravity waves, commonly referred to as mountain waves. When the propagation characteristics of the airstream change rapidly with height, a portion of the energy associated with these topographically forced disturbances may be confined to a wave duct extending downstream of the mountain ridge. The resonant waves that form in these wave ducts are referred to as trapped mountain lee waves. Theoretical work pertaining to trapped lee waves has, for the most part, assumed that the background flow is independent of time in which case the linear lee waves that form near the mountain asymptotically approach a steady state on a relatively rapid timescale (Wurtele 1955; Queney et al. 1960).

Observational studies over the past three decades have documented three basic types of nonstationarity exhibited by trapped lee waves: 1) a gradual downstream drift of the entire lee-wave pattern (Colson and Lindsay 1959; Lindsay 1962), 2) a gradual change in the horizontal wavelength of the lee-wave pattern that can be as large as 30% over a time period on the order of several hours (Collis et al. 1968; Ralph et al. 1997; Sarker and Calheiros 1974; Smith 1976), and 3) temporal changes in the location and amplitude of individual ridges and troughs that generate a lee-wave pattern with irregular variations in wavelength and amplitude (Brown 1983; Starr and Browning 1972).

There are two important differences between the linear steady-state description of trapped lee waves and observed trapped lee waves. First, the background flows associated with observed trapped lee waves are never exactly steady and may undergo substantial changes during a lee-wave event (Vergeiner and Lilly 1970; Mitchell et al. 1990; Ralph et al. 1997; Smith 1976; Starr and Browning 1972). Temporal variations in the background flow transform waves forced by prior periods of steady flow into nonstationary waves (Queney et al. 1960; Rees and Mobbs 1988). Second, the amplitudes of observed trapped lee waves are usually large enough that nonlinear effects may play a key role in determining the characteristics of the lee-wave train. This paper examines the impact of mean-flow variability on finite-amplitude trapped lee waves by conducting two-dimensional mountain wave simulations for a set of idealized, time-dependent background flows. Part II of this study (Nance and Durran 1997 manuscript submitted to J. Atmos. Sci., hereafter ND97) examines the impact of nonlinearity on the stationarity of trapped lee waves by conducting two-dimensional mountain wave simulations for a family of steady background flows.

The amplitude and horizontal wavelength of linear, stationary trapped lee waves can be sensitive to small changes in the background flow (Corby and Wallington 1956; Pearce and White 1967), so temporal variations in the background flow have the potential to significantly alter the characteristics of the resonant wave generated by flow over a particular mountain ridge. Since the wavelength trend associated with observed changes in the upstream conditions is often consistent with the temporal and/or spatial wavelength trends in observed lee-wave trains, the observed changes in lee-wave structure are often attributed to changes in the upstream flow (Mitchell et al. 1990; Ralph et al. 1997; Smith 1976). It is, however, interesting to note that Vergeiner and Lilly (1970) felt many of the changes in the lee waves they observed were too large and too fast to be explained by temporal variations in the basic flow. In any case, the amplitude of linear, stationary trapped lee waves is sufficiently sensitive to the structure of the background flow that temporal changes in the wave duct can easily produce a dramatic reduction in the lee-wave amplitude so that the generation of trapped lee waves appears to cease. This amplitude sensitivity is frequently overlooked in observational studies. Thus, in addition to investigating changes in horizontal wavelength, this study will examine the changes in lee-wave amplitude that develop in response to variations in the mean flow and the timescale over which these changes occur.

2. Model description

The numerical model used in this study is based on the fully compressible numerical model developed by Durran and Klemp (1983), which is designed to calculate two-dimensional, high Rossby number airflow (f → 0) over an infinitely long uniform mountain barrier. Terrain is incorporated at the lower boundary by applying the terrain transformation
i1520-0469-54-18-2275-e1
where zt is the depth of the model domain, h(x) is the terrain elevation, z is the height in the physical grid, and ξ is the transformed vertical coordinate. The momentum equations, pressure equation, and thermodynamic equation for compressible flow in terms of this transformed vertical coordinate are
i1520-0469-54-18-2275-e2
and
i1520-0469-54-18-2275-e5
where
i1520-0469-54-18-2275-eq1
and
i1520-0469-54-18-2275-eq2
In the above, u and w represent horizontal and vertical velocities, respectively; p represents pressure; ρ represents density; θ represents potential temperature; g is the gravitational constant, cp and cυ are the specific heat of dry air at constant pressure and constant volume, respectively; Du, Dw, and Dθ represent a subgrid-scale mixing parameterization; overbars depict horizontally homogeneous, mean-state values; and carets indicate the quantity is multiplied by H (e.g., ˆρ̄ = Hρ̄). The details of the subgrid-scale mixing formulation are discussed in Durran and Klemp (1983).

This system of equations governs the dynamics of gravity and sound waves. Sound waves are of little meteorological significance, but their high propagation speeds severely limit the time step necessary to maintain stability in explicit numerical schemes. To reduce this computational burden, the model solves the governing equations in a two-time-step process. The high-speed sound waves are integrated separately with a small time step, while the bulk of the computations are carried out over a larger time step (Klemp and Wilhelmson 1978). The vertical advection, pressure gradient, and divergence terms are represented by second-order differences, while the horizontal advection terms are represented by fourth-order centered differences. The model is initialized with a hydrostatically balanced, horizontally uniform basic state. The gravity wave transients generated by this initialization process in the presence of topography are reduced by gradually increasing the horizontal wind profile and gravitational constant from zero to their specified values over the time periods tu and tg such that u(0)tu/b = 1 and u(0)tg/b = 4, where b is the half-width of the mountain profile.

The model’s computational domain represents only a small portion of the total atmosphere. The velocity normal to the lower boundary is set to zero, while open boundary conditions that allow disturbances to propagate out of the domain without generating spurious reflections are imposed at the top and lateral boundaries. The lateral boundary condition used in this model follows the procedure proposed by Orlanski (1976), but the Doppler-shifted phase speed (u ± c) is specified at each boundary (Pearson 1974; Durran et al. 1993). This lateral boundary condition is reasonably effective for most types of disturbances, but it has difficulties with standing waves. To avoid generating spurious reflections resulting from the interaction of the trapped lee waves and the downstream boundary, the simulations were terminated before the leading edge of the lee-wave train reached the downstream boundary. Upward energy propagation at the top boundary is simulated by following the procedure in Bougeault (1983) and Klemp and Durran (1983).

All of the simulations were computed on a model domain 15 km deep with a vertical grid spacing of 200 m and a horizontal grid spacing of 500 m. The small time step was 1 s and the large time step was 4 s. The width of the model domain was either 165 or 200 km. The wider domain was used for basic-state profiles that generate stationary trapped lee waves with higher group velocities or for longer simulation times in order to avoid problems with the downstream lateral boundary condition. The topography was specified using a Witch of Agnesi profile
i1520-0469-54-18-2275-eq3
with a half-width (b) of 3 km centered at x = 30 km. This smooth bell-shaped profile was chosen because its Fourier transform is relatively simple, which, in turn, makes it relatively easy to obtain linear mountain wave solutions for this profile.

3. Time-dependent background flows

The time-dependent background flows considered in this study are produced by a smooth, hydrostatically balanced transition between two horizontally uniform basic states whose linear, stationary trapped lee waves have significantly different amplitudes and/or horizontal wavelengths. Maintaining a hydrostatically balanced, horizontally uniform background flow throughout the transition ensures that all circulations will be associated with the lee waves and not some terrain-independent adjustment of the background flow. A large-scale process, such as horizontal advection parallel to the mountain ridge, acting uniformly can be viewed as the approximate source of this time-dependent behavior.

The time dependence of the background flow in each simulation is specified as follows: 1) After the start-up procedure at time tg, there is a steady background flow characterized by the initial basic-state profile for an additional time t1 (phase I); 2) the background flow undergoes a smooth, hydrostatically balanced transition from the initial basic-state profile to the final basic-state profile over a time t2 (phase II); and 3) the background flow remains steady until the end of the simulation over a time t3 (phase III). The procedure used to introduce temporal variations into the background fields is described in detail in the appendix. The periods of steady flow on either side of the basic-state transition simplify the analysis of the mean-flow variability’s impact on the trapped lee waves. To further simplify the discussion of each mountain wave simulation, t = 0 will correspond to the end of the start-up procedure.

Each basic-state profile has a constant wind speed u and a two-layer static stability profile
i1520-0469-54-18-2275-e6
where Nl and Nu are constants, Nl > Nu, and d represents the depth of the lower layer. Since the Scorer parameter for a compressible fluid can be well approximated as l2 = N2/u2 when u is constant with height (Nance 1997), these basic states have the two-layer, constant Scorer parameter configuration used by Scorer (1949) to obtain a linear solution for stationary trapped lee waves downstream of a Witch of Agnesi mountain profile. The characteristics of the five basic-state profiles used in these simulations (WS, SS, LD3, LD5, SS′) and the characteristics of the linear, stationary trapped lee waves associated with each basic-state profile are summarized in Table 1. Transitions between SS (strong static stability) and WS (weak static stability) simulate either a strengthening or a weakening in the low-level static stability, a transition from LD3 (lower layer depth of 3 km) to LD5 (lower layer depth of 5 km) simulates an increase in the depth of the resonant wave duct, and a transition from WS to SS′ simulates a strengthening of the low-level static stability combined with a weakening of the cross-mountain flow. The horizontal wavelengths in Table 1, which were obtained by solving Eq. (16) in Scorer (1949), indicate the horizontal wavelength of the stationary trapped lee wave will decrease by 50% as the background flow undergoes a transition from WS to SS, decrease by almost 20% as the background flow undergoes a transition from LD3 to LD5, and decrease by over 70% as the background flow undergoes a transition from WS to SS′. The maximum vertical velocities (W) in Table 1, which are based on a mountain height of 1 m and a mountain half-width of 3 km, indicate the amplitude of the stationary trapped lee wave will decrease by a factor of 1.7 as the background flow undergoes a transition from WS to SS, decrease by a factor of 2.7 as the background flow undergoes a transition from LD3 to LD5, and decrease by a factor of l0 as the background flow undergoes a transition from WS to SS′.

4. Amplitude tendencies due to mean-flow variability

Linear theory predicts that the temporal variations in the background flow considered in this study will have a significant impact on the strength of the stationary trapped lee wave generated by the flow over the mountain ridge. Observational evidence indicates, however, that linear theory often underestimates the strength of finite-amplitude trapped lee waves (Smith 1976). Nevertheless, it is possible that the change in amplitude predicted by linear theory may correctly indicate whether finite-amplitude trapped lee waves will strengthen or weaken in response to a change in the basic-state flow.

The dependence of lee-wave amplitude on mountain height was investigated by conducting a series of steady-flow simulations for the WS, SS, LD3, and LD5 basic-state profiles using mountain heights of 10, 200, and 400 m. The amplitudes of the stationary trapped lee waves generated by the basic states WS and SS are shown in Fig. 1a. These amplitudes have been normalized by the height of the mountain, so any departure of the plotted points from a horizontal line indicates the degree to which nonlinear effects enhance the stationary trapped lee-wave response. The impact of nonlinear effects on the lee-wave amplitude in the case WS is almost negligible, whereas nonlinear effects produce a substantial amplitude increase in the case SS. For this pair of basic states, increasing the mountain height initially reduces the amplitude difference between WS and SS, and then, as the flow becomes even more nonlinear, the amplitude difference between WS and SS begins to increase, but the case with the stronger amplitude is not the one predicted by linear theory. Hence, linear theory will not correctly identify the magnitude or the sense of the amplitude change that would occur in moderate-amplitude lee waves due to a transition between the basic states WS and SS.

The normalized lee-wave amplitudes for LD3 and LD5 are shown in Fig. 1b. When the background flow is characterized by LD3, nonlinear effects enhance the stationary trapped lee wave response, whereas nonlinear effects actually reduce the strength of the stationary trapped lee-wave response when the background flow is characterized by LD5. Thus, nonlinearity accentuates the change in amplitude that linear theory predicts would occur as the result of a transition from LD3 and LD5. In this situation, linear theory correctly identifies the sense of the amplitude change that occurs due to the transition, but it underestimates the magnitude of the change that occurs in moderate-amplitude waves.

In summary, these simulations have shown that neither the magnitude nor the sign of the change in lee-wave amplitude that occurs when the background flow changes with time can be reliably predicted by linear theory.

5. Nonstationary trapped waves

During phase I of each simulation, the steady background flow generates stationary trapped lee waves. The basic-state transition during phase II transforms these stationary trapped waves into nonstationary waves. For small amplitudes and negligible compressibility effects, the characteristics of these nonstationary disturbances during phase III will be governed by
i1520-0469-54-18-2275-e7
where k and c represent the horizontal wavenumber and phase speed of the disturbance, respectively, and ŵ (k, z, c) represents the amplitude of the component (k, c) of the Fourier decomposition of w(x, z, t). (This equation is the no-wind-shear version of the Taylor–Goldstein equation.)
When N is constant, a disturbance for which N2/(cu)2 < k2 will decay exponentially with height, whereas a disturbance for which N2/(cu)2 > k2 will have a sinusoidal vertical structure. Thus, the two-layer static stability profile (6) will support trapped waves with intrinsic frequencies in the range Nu < |ω̃| < Nl, where ω̃ = (cu)k. Applying the same matching and boundary conditions used by Scorer (1949) for the steady-state two-layer problem, one obtains the following resonance condition for nonstationary trapped waves in a two-layer fluid:
i1520-0469-54-18-2275-e8
where
i1520-0469-54-18-2275-eq4
and kt and ct represent the horizontal wavenumber and phase speed of the trapped mode, respectively; Sl is the vertical wavenumber in the lower layer; and 1/Su is the e-folding scale over which the trapped waves decay in the vertical. Given the characteristics of the background flow, (8) can be used to determine the phase speed of a nonstationary trapped wave of a given horizontal wavenumber. Note that (8) is also the dispersion relation for nonstationary trapped waves in a two-layer atmosphere with a uniform mean wind. Hence, the group velocity of these trapped waves,
i1520-0469-54-18-2275-e9
is obtained by differentiating (8) with respect to kt and solving for ∂ωt/∂kt, where ωt = ktct.

There are three important differences between the classical steady-state theory for trapped waves and this generalization that includes nonstationary trapped modes. First, the horizontal wavelength of a stationary trapped wave (λ = 2π/k) in a constant-wind-speed, two-layer fluid must lie between 2πu/Nl and 2πu/Nu, whereas the horizontal wavelength of a nonstationary trapped wave in the same two-layer fluid generally has an upper limit that is greater than 2πu/Nu (see Table 1). Second, a given background flow that supports only one stationary trapped wave also supports a continuous spectrum of nonstationary trapped waves over a finite range of horizontal wavenumbers that is determined by the resonance condition (8). The existence of this continuous spectrum dramatically increases the probability that an initially stationary trapped wave can remain trapped as it becomes nonstationary due to a change in the background flow. The third difference between this more general case and the classical steady-state theory involves the influence of vertical wind shear in the background flow. When c = 0, the coefficients of the Taylor–Goldstein equation for a background flow with a wind profile that is a function of z will be constant provided that the Scorer parameter associated with the flow is constant with height, whereas, when c ≠ 0, the coefficients of the Taylor–Goldstein equation for the same background flow are a function of z. Hence, when c is nonzero, (8) and (9) do not apply to an arbitrary two-layer Scorer parameter structure unless the basic-state wind speed is constant with height.

Linear theory predicts that a horizontally uniform, basic-state transition will not affect the horizontal wavelength of the waves generated during phase I, so (8) can be used to compute the phase speed of the nonstationary trapped waves that develop after the basic-state transition in the following simulations. Given the phase speed and the horizontal wavelength, (9) can be used to compute the group velocity of these waves before and after the transition. The results of these computations are summarized in Table 2 for transitions between basic states SS and WS, in Table 3 for a transition from LD3 to LD5, and in Table 4 for a transition from WS to SS′. The corresponding properties of the finite-amplitude trapped waves in the numerical simulations are listed as parenthetical values for comparison.

a. Wave patterns with uniform amplitude

The influence of a change in the background flow on the lee wavelength was investigated by considering the somewhat special case where the basic-state transition produces no significant change in the lee-wave amplitude. The transition used in this investigation is between WS and SS for the case of flow over a mountain 200 m high.

1) Downstream waves with faster group velocities

The final column in Table 2 indicates that a transition from SS to WS will cause the group velocity of waves generated during phase I to exceed the group velocity of waves generated during phase III (cgx1 > cgx3). The impact of this difference in group velocity was investigated by conducting a simulation in which there was a transition from SS to WS such that t1 = 2 h, t2 = 1 h, and t3 = 3 h. Contours of the vertical velocity field at t = 5 h are shown in Fig. 2. This lee-wave train can be divided into three basic groups: 1) the wave packet located at the leading edge of the lee-wave train that was generated during the first steady flow regime (phase I), 2) the wave packet immediately in the lee of the mountain that was generated by the second steady flow regime (phase III), and 3) the wave packet that was generated during the basic-state transition (phase II). These three groups merge to create a lee-wave pattern with a fairly uniform amplitude whose wavelength gradually decreases downstream. This smooth transition to a longer wavelength in the lee of the mountain ridge resembles the time-dependent behavior of the nonstationary trapped lee-wave events observed by Collis et al. (1968) and Ralph et al. (1997).

The transient behavior of this lee-wave pattern is illustrated in Fig. 3, which shows contours of the vertical velocity field downstream of the mountain in the xt plane at z = 3 km. The stippled region indicates the period of the basic-state transition. Just prior to the transition, the lee-wave train is composed of five fully developed stationary trapped waves with an average horizontal wavelength of approximately 7.5 km and a sixth wave developing at the leading edge. The downstream propagation of this wave train is consistent with a group velocity of approximately 6 m s−1. When the low-level static stability is modified (phase II), the flow over the mountain ridge no longer generates this particular resonant wave. As the flow generates a new resonant wave, the upstream edge of the wave pattern generated prior to the basic-state transition detaches itself from the topography and propagates downstream. This wave packet retains its original horizontal structure (i.e., the horizontal wavelength of these waves is not affected by the basic-state transition) and remains trapped as it propagates downstream. During the transition, these trapped waves gradually develop a phase speed of approximately 3 m s−1, their group velocity gradually increases to approximately 7 m s−1, and their vertical velocity maximum shifts to a slightly higher altitude (not shown). The behavior of the waves after the transition is consistent with the phase speed and group velocity obtained by solving (8) and (9) for a trapped wave with a 7.5-km horizontal wavelength propagating in the basic-state WS (see Table 2 and Fig. 3). The wave packet generated during phase I evidently responds to the modification of its environment by projecting almost all its amplitude onto the vertical structure of a nonstationary trapped wave with the same horizontal wavelength.

Shortly after the beginning of the transition, the first crest in the lee of the topography begins to drift downstream. The downstream displacement of this crest gradually increases until it is aligned with the position of the first downstream crest in the new stationary trapped wave pattern. The displacements of the next five downstream crests parallel that of the first crest as new stationary trapped waves with a horizontal wavelength of approximately 14.5 km are established at a rate consistent with a group velocity of approximately 4 m s−1. The downstream propagation of the new stationary trapped waves is 3 m s−1 slower than that of the initial wave packet, so the separation between the upstream edge of the initial wave packet and the downstream edge of the stationary trapped waves generated during phase III increases with time. The region between these diverging wave packets contains trapped waves generated by flow over the topography during phase II of the simulation. As the wave packets generated during phases I and III diverge, the number of waves in this transition zone increases and their wavelengths vary from 14.5 km at the upstream edge of the transition zone to 7.5 km at the downstream edge. The spatial variation of the wavelength within the transition zone is well predicted by the steady-state version of (8) using the basic-state conditions that were present over the mountain when each wave was generated. The local phase speed at a given time is well predicted by (8) using the current background flow and the local horizontal wavelength.

The influence of the timescale over which the transition occurs was investigated by increasing the time period over which the transition from SS to WS takes place from one hour to two hours (t1 = 2 h, t2 = 2 h, t3 = 2 h). The trapped lee waves generated by this time-dependent flow also created a wave pattern with fairly uniform amplitude whose wavelength gradually decreased downstream. Increasing the timescale of the basic-state transition reduced the curvature of the phase lines in the xt plane during the transition period, increased the initial separation of the waves generated during phases I and III, and increased the number of trapped waves generated during the transition (not shown).

2) Downstream waves with slower group velocities

A transition in the opposite sense, from WS to SS, will cause the group velocity of resonant waves generated during phase I to be less than the group velocity of resonant waves generated during phase III (cgx1 < cgx3). This case was investigated by conducting a simulation in which there is a transition from WS to SS such that t1 = 2 h, t2 = 1 h, and t3 = 4 h. As shown by the vertical velocity field at t = 5 h, contoured in Fig. 4, the trapped waves generated earlier in the simulation merge to create irregular variations in wavelength and amplitude in the region 60 km ≤ x ≤ 80 km. Qualitatively similar irregular variations in wavelength and amplitude have appeared in a number of observed lee-wave patterns (Brown 1983; Starr and Browning 1972; Shutts and Broad 1993; Reynolds et al. 1968).

The transient behavior of the trapped waves generated by this time-dependent background flow is illustrated in Fig. 5. Just prior to the transition in the background flow, the lee-wave train is composed of two fully developed stationary trapped waves with a horizontal wavelength of approximately 14.5 km and a third trapped wave in the development stage at the leading edge. The downstream propagation of this wave packet is consistent with a group velocity of approximately 4 m s−1. During the transition, these waves gradually develop a phase speed of −5 m s−1, their group velocity slows to approximately 2 m s−1, and their vertical velocity maximum shifts to a slightly lower altitude, but their horizontal wavelength remains relatively constant. The stationary trapped lee waves that develop after the transition propagate downstream at a group velocity of 6 m s−1, which is 4 m s−1 faster than that of the initial wave packet, so the leading edge of the new stationary trapped waves overtakes the initial wave packet and eventually appears at the leading edge of the lee-wave train (see Fig. 5). The transient behavior of the vertical velocity perturbations in this simulation can be reproduced by the simple superposition of two wave groups, as illustrated in Fig. 6. The two modes plotted in Fig. 6 have the form
i1520-0469-54-18-2275-e10
where
i1520-0469-54-18-2275-eq5
and
t
In the above, the subscripts w and s represent the stationary trapped waves generated by WS and SS, respectively; ŵi is an amplitude function that determines the width and location of the wave packet; ϕi is a constant that is determined by the initial phase of each wave packet; li and ri represent the location of each wave packet’s left and right boundaries, respectively; and λi, ci, and cgi represent the theoretical wavelength, phase speed, and group velocity characteristics of each wave packet during phase III of the simulation. The left and right boundaries for WS at = 0 recreate the configuration of the initial wave packet at the beginning of phase III. Note the similarity between the wave pattern in Fig. 5 between three and seven hours and the wave pattern in Fig. 6.

A trapped wave with a peak to peak wavelength of approximately 10.5 km forms in the lee of the mountain during the one-hour transition from WS to SS. Since the stationary trapped waves generated during phase III of this simulation overtake the wave packet generated during phase I, it is difficult to isolate the behavior of the wave packet generated during the transition, but this wave packet must contain a superposition of trapped modes with horizontal wavelengths ranging from 7.5 to 14.5 km. Equation (9) indicates a wave packet composed of nonstationary trapped waves with these horizontal wavelengths would overtake the wave packet generated during phase I, while simultaneously being overtaken by the stationary trapped waves generated during phase III. Hence, the region of irregular variations in wavelength and amplitude is actually produced by the superposition of three basic groups of waves: 1) trapped waves generated during the first steady flow regime (phase I), 2) trapped waves generated during the transition from WS to SS (phase II), and 3) trapped waves generated during the second steady flow regime (phase III).

The influence of the timescale over which the transition occurs on the evolution of this lee-wave pattern was investigated by increasing the length of the transition, t2 from 1 to 3 hours (t1 = 2 h, t2 = 3 h, t3 = 2 h). A comparison of the lee-wave trains generated by the 1-h and 3-h transitions at equivalent intervals after the end of the basic-state transition (not shown) revealed that the development of irregular variations in wavelength and amplitude during phase III of the simulation is delayed by a little over an hour. Hence, increasing the transition by two hours, delayed the development of irregular variations in wavelength and amplitude by over three hours. This delay in the development of irregular variations in wavelength and amplitude is a consequence of a more gradual downstream decrease in the group velocity of the trapped waves, which is related to the more gradual increase in the horizontal wavelength of the trapped waves.

The timescale on which mean-flow variability can generate irregular variations in wavelength and amplitude depends on how rapidly the group velocity of the trapped waves decreases downstream of the mountain, which in turn depends upon the magnitude of the change in the background flow and the timescale on which these changes occur. The preceding simulations demonstrate that trapped lee waves can develop irregular variations in wavelength and amplitude within three hours of the time that the flow becomes nonsteady. On the other hand, the time-dependence of the background flow in these simulations does not necessarily represent a realistic evolution of the large-scale flow. To put this time-dependence into a more physical context, consider the potential temperature distribution that would be necessary to achieve a 1-hour transition from WS to SS via advection by a uniform mean flow parallel to the mountain crest of 10 m s−1 (Fig. 7). The maximum slopes of the isentropic surfaces in this cross section lie between 1/15 and 1/10. In comparison, the slope of a frontal surface varies within wide limits around an average slope of 1/150, where a slope of 1/50 is usually regarded as steep and a slope of 1/300 is considered to be slight (Petterssen 1956; Byers 1974).1 To bring the slope of the isentropic surfaces associated with a transition from WS to SS within this range of frontal slopes, the timescale of the transition must be increased from 1 to at least 5 h, which, in turn, suggests that the timescale on which mean-flow variability would generate irregular variations in wavelength and amplitude would be on the order of 10 h. Since the longevity of many lee-wave events is also on the order of 10 h (Vergeiner and Lilly 1970; Pearce and White 1967), the timescale on which irregular variations in wavelength and amplitude develop in response to mean-flow variability may often be too long to account for this type of variability in actual lee waves. It is, however, important to note that this analysis has not considered temporal variations in the mean wind profile. Temporal variations in the cross-mountain flow combined with temporal variations in the static stability profile might be able to generate wave patterns with irregular variations in wavelength and amplitude on more realistic timescales.

b. Wave patterns with large amplitude variations

In general, one would expect changes in the background flow to produce changes in both amplitude and wavelength of the resonant waves generated by flow over a particular mountain ridge. This more typical situation was investigated by conducting a simulation for flow over a mountain 200 m high in which there is a transition from LD3 to LD5 such that t1 = 2.5 h, t2 = 1 h, and t3 = 2.5 h. Contours of the vertical velocity fields at t = 3.5 h (the beginning of phase III) and t = 5.5 h are shown in Fig. 8. The generation of trapped lee waves appears to cease during phase III due to the large decrease in the amplitude of the stationary trapped wave, and the lee waves forced during phase I gradually drift downstream, while maintaining their original horizontal structure. This downstream drift of the lee-wave pattern resembles the time-dependent behavior of the observed lee wave pattern discussed in Colson and Lindsay (1959) and Lindsay (1962).

The propagation characteristics of the trapped waves generated by this time-dependent background flow are once again in good agreement with the theoretical predictions obtained by solving (8) and (9) (see Table 3). In this case, the group velocity of the stationary trapped wave generated during phase III exceeds that of the nonstationary trapped waves in the initial wave packet by 1 m s−1, so the leading edge of this stationary trapped wave will eventually overtake phase I’s trapped waves. The range of group velocities within the entire wave train is rather small, so a long time is required before the waves generated during phase III overtake the waves generated during phase I. Hence, irregular variations in wavelength and amplitude similar to those observed in Fig. 4 do not develop before the end of this simulation. Even when the waves generated during phase III overtake those generated during phase I, there will be only a modest distortion of the phase I waves because the phase III waves are much lower amplitude.

c. Waves upstream of the mountain ridge

A number of observational studies over the past three decades have noted the presence of waves upstream of the topography whose origin is unknown (Brown 1983; Clark and Gall 1982; Hoinka 1984; Reynolds et al. 1968). Temporal changes in the mean flow may be responsible for at least some of the trapped waves observed upstream of the topography since temporal changes in the background flow can produce nonstationary trapped waves with negative group velocities. The impact of nonstationary trapped waves with upstream group velocities on trapped wave patterns generated by a time-dependent background flow was investigated by conducting a simulation for flow over a mountain 1 m high in which there is a transition from WS to SS′ such that t1 = 2 h, t2 = 1 h, and t3 = 4 h. The mountain profile in this simulation was centered at 80 km, instead of 30 km, to avoid spurious reflections resulting from interaction between the trapped waves and the upstream lateral boundary. The vertical velocity fields at t = 3 h and t = 7 h are shown in Fig. 9. Once again, the generation of trapped lee waves appears to cease after the basic-state transition due to the large decrease in the amplitude of the stationary trapped wave, but this time the lee waves forced during phase I gradually drift upstream. The wave pattern is distorted slightly as it passes over the mountain ridge, but the wave pattern returns to its original horizontal structure upstream of the mountain (see Fig. 9b). Thus, the distortion appears to be due to a superposition of the hydrostatic mountain wave and the resonant waves. The propagation characteristics of the trapped waves generated by this time-dependent background flow are in agreement with the predictions obtained by solving (8) and (9) (see Table 4), so the stationary trapped wave has once again responded to the basic-state transition by projecting almost all of its amplitude onto a nonstationary trapped wave with the same horizontal wavelength.

A simulation was also conducted for this time-dependent background flow over a mountain 200 m high. The finite-amplitude trapped lee waves generated during phase I of this simulation also responded to the basic-state transition by projecting most of their amplitude onto upstream-propagating trapped waves with the same wavelength, but the interaction between these upstream-propagating waves and the topography was slightly more complicated. A portion of the energy associated with the initial wave packet propagates across the ridge, but the horizontal wavelength of the waves that appear upstream of the topography (16–18 km) is slightly longer than that of the original trapped waves generated during phase I. In addition, as the phase I trapped waves propagate back across the ridge, they appear to scatter some energy into nonstationary trapped waves with a downstream group velocity and a horizontal wavelength of approximately 6 km. The exact nature of this interaction between the topography and the finite-amplitude trapped waves with a negative group velocity requires further investigation.

6. Wind profiler observations

Wind-profiler-observed vertical motions from recent observational studies of trapped mountain lee waves have exhibited temporal oscillations with periods ranging from slightly less than 1 h to almost 4 h (Ralph et al. 1992; Bougeault et al. 1993). Figure 10 shows an example of wind-profiler-observed vertical motions obtained downstream of the Pyrenees mountains during the PYREX experiment (Bougeault et al. 1993). The strong low-level vertical motions recorded between 0500 and 0900 UTC were associated with a lee-wave event whose horizontal wavelength, determined from aircraft data, was approximately 10 km. The temporal variations in these low-level vertical motions are dominated by oscillations with a period of roughly 2 h, which, as noted by Bougeault et al. (1993), could have been produced by a nonstationary wave packet with a horizontal wavelength of 10 km and a phase speed of 1.3 m s−1.

The evolution of the vertical velocity field above a fixed spatial location in the preceding simulations is presented in the same format as the VHF profiler observations in Figs. 11 and 12. Figure 11a shows the vertical velocity signature that would have been observed by a profiler located 30 km downstream of the mountain crest (at x = 60 km) during the simulation in which the flow underwent a 1-h transition from SS to WS (see Fig. 3). Two full oscillations occur above this location as the original lee-wave train becomes nonstationary and is replaced by a new set of stationary waves. Figure 11b shows the vertical velocity signature that would have been observed by a profiler located 40 km downstream of the mountain crest (at x = 70 km) during the simulation in which the flow underwent a 1-h transition from LD3 to LD5. Although there are approximately five wave crests upstream of this location at the end of the transition (see Fig. 8a), less than two full oscillations in the vertical velocity are observed as these waves pass over this location. The difference between the number of wave crests present in a spatial cross section and the number of crests that appear in a time series at a fixed spatial location depends on the ratio of the phase speed of the waves to their group velocity. Suppose there are n wave crests in a spatial cross section and the horizontal wavelength is λ. The time it takes this wave packet to traverse a fixed location is nλ/|cg|. The number of oscillations observed above a fixed location is obtained by simply dividing this time by the period of the waves, 2π/ω. Thus, the number of oscillations observed above a fixed spatial location is given by |nλω/2πcg| or |nc/cg|. As illustrated in both panels of Fig. 11, |c| may be substantially less than |cg|, and, as a consequence, a long train of horizontally propagating trapped waves can pass over a fixed point without generating many temporal oscillations.

The opposite case is illustrated in Fig. 12, which shows the vertical velocities that would have been observed by a profiler 33 and 35 km downstream of the mountain crest (at x = 63 km and x = 65 km) during the simulation in which the flow underwent a 1-h transition from WS to SS (see Fig. 5). In this case, the phase speed of the waves that are rendered nonstationary by the transition is significantly larger than their group velocity, so a large number of oscillations are observed above a fixed spatial location. The sampling location for Fig. 12b is at a nodal point of the vertical velocity field associated with the stationary wave train forced after the transition, so the vertical velocity above this point is completely determined by the packet of nonstationary trapped waves. On the other hand, the sampling location for Fig. 12a is at a maximum of the vertical velocity field associated with the stationary wave train forced after the transition, so the total vertical velocity above this point is biased upward after the arrival of the new stationary trapped wave. The case shown in Fig. 12, in which the waves generated after the transition overtake the waves generated before the transition, comes the closest to qualitatively replicating the actual profiler observations shown in Fig. 10. Note, however, that the profiler observations suggest the presence of at least two distinct wave periods (of 30 min and 2 h), whereas the simple basic-state transition that generates the time series shown in Fig. 12 only produces oscillations at a single frequency. As will be discussed in ND97, it seems likely that the vertical velocity fluctuations observed by the profiler downstream of the Pyrenees were generated by nonlinear wave dynamics rather than changes in the background flow.

7. Conclusions

The lee-wave trains generated by the time-dependent background flows considered in this study are composed of three basic wave groups: 1) waves generated by the steady flow before the transition, 2) waves generated while the background flow is undergoing the transition, and 3) waves generated by the steady flow after the transition. The wave pattern created by these wave groups depends on two factors: the relative amplitudes of the waves in each group and the difference between the group velocities of the waves in each group.

Since variations in the background wind speed and static stability can significantly modify both wavelength and amplitude of linear resonant waves, one might expect that changes in the background flow will weaken or amplify the resonant waves. Indeed, if the resonant waves are already well tuned for a given topographic forcing, changes in the background flow are unlikely to preserve this optimal tuning, and therefore the new background flow is likely to generate weaker lee waves. Two examples were considered in this paper in which only a very weak resonant wave was generated after the transition in the background flow. In the first case, the group velocity of the waves generated prior to the transition remained positive after the transition, and the packet of resonant waves propagated downstream away from the mountain. In the second case, the group velocity of the waves generated prior to the transition became negative after the transition, and the packet of resonant waves propagated back over the mountain. The trapped waves that are occasionally observed upstream of an isolated ridge crest may be generated by this mechanism.

Although it is known that linear theory does not accurately predict the amplitude of observed finite-amplitude trapped lee waves, one might, nevertheless, hope that linear theory can be used to predict the sign of the change in the lee-wave amplitude that accompanies a change in the background flow. Unfortunately, this is not the case. As demonstrated in this paper, the impact of nonlinear effects on the lee-wave amplitude is highly dependent on the characteristics of the background flow, so linear theory is not capable of reliably predicting whether a change in the background flow will amplify or dampen the resonant waves generated by flow over a particular mountain ridge.

In those cases where a change in the background flow generates a change in the resonant wavelength while leaving the wave amplitude relatively unchanged, either 1) a smooth transition between or 2) a superposition of, the wave groups generated before and after the transition develops depending on the difference between the group velocities of the waves. As is the case with all stationary lee waves, the group velocity of the lee waves generated after the transition is directed downstream. If this group velocity is less than that of the resonant wave packet generated before the transition, the two wave trains are connected by an ever-widening zone filled with trapped waves whose wavelengths vary smoothly between the original resonant wavelength and the resonant wavelength after the transition. On the other hand, if the group velocity of the waves generated after the transition is greater than that of the resonant wave packet generated before the transition, the newly generated waves eventually overtake the waves generated before the transition, creating a complex pattern of irregular variations in wavelength and amplitude where the two wave trains are superimposed.

The time required for waves generated after the transition in the background flow to overtake the waves generated before the transition depends on the difference between the group velocities of these two wave groups and the duration of the transition itself. It is important to note that one rarely observes more than six to eight well-defined trapped waves downstream from an isolated mountain ridge because real atmospheric waves continually leak energy into the stratosphere. The simple two-layer profiles considered in this study fail to account for the influence of the stability increase associated with the stratosphere on the characteristics of the trapped lee-waves. If the timescale required for the waves generated after the transition to overtake the waves generated before the transition is too long, then the region in which the waves superimpose will occur so far downstream from the mountain ridge that the resulting disturbance will be too small to be observed. Estimates of the timescale associated with large-scale changes in lower-tropospheric stable layers and the accompanying changes in wave propagation characteristics suggest that changes in the static stability alone are not likely to occur with sufficient rapidity to generate readily observable regions of irregular variations in wavelength and amplitude like those shown in Fig. 4. Changes in the background wind speed might be able to reduce the timescale on which the waves can superimpose. Additional sources of temporal variability, such as the advection of cross-mountain gradients or spatially nonuniform forcing in an Eulerian reference frame, might also decrease the timescale required to generate significant changes in the basic-state flow. There is, nevertheless, an additional constraint that makes it difficult to generate irregular lee-wave patterns no matter what mechanism is responsible for the temporal variations in the basic-state flow. Irregularities in the lee-wave structure will not develop unless the basic states before and after the transition generate lee waves of comparable amplitude.

Observational documentation of the temporal variations in lee-wave-induced vertical velocities above a fixed spatial location have been obtained by several investigators using VHF profilers (Ralph et al. 1992; Bougeault et al. 1993). Several features appearing in the wind profiler trace shown in Fig. 10, which is representative of similar measurements made elsewhere, are potentially explicable as a consequence of changes in the background flow. Changes in the background flow can force a preexisting lee-wave train to translate upstream or downstream, and, as noted by Bougeault et al. (1993), a series of sinusoidal oscillations in the vertical velocity field will be recorded by the profiler as these nonstationary lee waves pass over the instrument. It is, however, important to appreciate that the number of oscillations recorded as the waves pass over the profiler is equal to the ratio |c/cg| times the number of horizontal wavelengths in the wave train, which can be very different from the number of wave crests visible in a satellite photograph of the wave train. In the preceding experiments, the most complex vertical velocity time series are produced when the waves generated after the transition overtake the waves generated before the transition, in which case the oscillations associated with the propagating trapped wave are superimposed on the steady vertical velocity associated with the new stationary trapped wave. Provided that the sampling point is not located at a node of the stationary wave, the superposition of these two waves creates a modestly irregular oscillation in the vertical velocity as the new stationary wave propagates into the area, but the additional frequencies that appear in the actual profiler observations (Fig. 10) are not present in these modestly irregular fluctuations (Fig. 12a).

As argued previously, it is far from obvious that the irregular variations observed in trapped mountain lee waves are due to temporal changes in the background flow. Part II of this study (ND97) examines a second possible mechanism for the generation of nonstationary trapped mountain lee waves: nonlinear wave dynamics. The results from Part II suggest that nonlinear wave dynamics are likely to be responsible for at least some of the irregular variations observed in real-world trapped mountain lee waves. On the other hand, systematic changes in the wavelength of a lee-wave train do appear to be the product of changes in the background flow. Changes in the background flow also appear to be responsible for the behavior of those trapped waves that are observed to gradually drift upstream or downstream. It should be noted that the two-dimensional framework of this study does not allow the consideration of temporal variability in lee-wave-induced vertical velocities that are the result of changes in the orientation of a three-dimensional lee-wave pattern due to small changes in the upstream wind direction (Ralph et al. 1992).

Acknowledgments

We would like to thank Dr. James Holton and Dr. Christopher Bretherton for their suggestions and comments throughout this project. We would also like to thank Jean Luc Caccia for his help with Fig. 10. This work was supported by the National Science Foundation Grants ATM-8813971, ATM-9218376, and ATM-9322480.

REFERENCES

  • Bougeault, P., 1983: A non-reflective upper boundary condition for limited-height hydrostatic models. Mon. Wea. Rev.,111, 420–429.

  • ——, A. Jansa, J. L. Attie, I. Beau, B. Benech, R. Beniot, P. Bessemoulin, J. L. Caccia, J. Campins, B. Carissimo, J. L. Champeaux, M. Crochet, A. Druilhet, P. Durand, A. Elkhalfi, P. Flamant, A. Genoves, M. Georgelin, K. P. Hoinka, V. Klaus, E. Koffi, V. Kotroni, C. Mazaudier, J. Peon, M. Petitdidier, Y. Pointin, D. Puech, E. Richard, T. Satomura, J. Stein, and D. Tannhauser, 1993: The atmospheric momentum budget over a major mountain range: First results of the PYREX field program. Ann. Geophys.,11, 395–418.

  • Brown, P. R. A., 1983: Aircraft measurements of mountain waves and their associated momentum flux over the British Isles. Quart. J. Roy. Meteor. Soc.,109, 849–865.

  • Byers, H. R., 1974: General Meteorology. McGraw Hill, 461 pp.

  • Clark, T. L., and R. Gall, 1982: Three-dimensional numerical model simulations of airflow over mountainous terrain: A comparison with observations. Mon. Wea. Rev.,110, 766–791.

  • Collis, R. T. H., F. G. Fernald, and J. E. Alder, 1968: Lidar observations of Sierra wave conditions. J. Appl. Meteor.,7, 227–233.

  • Colson, D., and C. V. Lindsay, 1959: Unusual wave cloud over Washington DC, Nov. 30, 1959. Mon. Wea. Rev.,87, 451–452.

  • Corby, G. A., and C. E. Wallington, 1956: Airflow over the mountains: The lee wave amplitude. Quart. J. Roy. Meteor. Soc.,82, 266–274.

  • Durran, D. R., and J. B. Klemp, 1983: Compressible model for the simulation of moist mountain waves. Mon. Wea. Rev.,111, 2341–2361.

  • ——, M. Yang, D. N. Slinn, and R. G. Brown, 1993: Toward more accurate wave permeable boundary conditions. Mon. Wea. Rev.,121, 604–620.

  • Hoinka, K. A., 1984: Observations of a mountain-wave event over the Pyrenees. Tellus,36A, 369–383.

  • Klemp, J. B., and R. Wilhelmson, 1978: The simulation of three-dimensional convective storm dynamics. J. Atmos. Sci.,35, 1070–1096.

  • ——, and D. R. Durran, 1983: An upper boundary condition permitting internal gravity wave radiation in numerical mesoscale models. Mon. Wea. Rev.,111, 430–444.

  • Lindsay, C. V., 1962: Mountain waves in the Appalachians. Mon. Wea. Rev.,90, 271–276.

  • Mitchell, R. M., R. P. Cechet, P. J. Turner, and C. C. Elsum, 1990: Observation and interpretation of wave clouds over Macquarie Island. Quart. J. Roy. Meteor. Soc.,116, 741–752.

  • Nance, L. B., 1997: On the inclusion of compressibility effects in the Scorer parameter. J. Atmos. Sci.,54, 362–367.

  • Orlanski, I., 1976: A simple boundary condition for unbounded hyperbolic flows. J. Comput. Phys.,21, 251–269.

  • Pearce, R. P., and P. W. White, 1967: Lee wave characteristics derived from a three layer model. Quart. J. Roy. Meteor. Soc.,93, 155–165.

  • Pearson, R. A., 1974: Consistent boundary conditions for numerical models of systems that admit dispersive waves. J. Atmos. Sci.,31, 1481–1489.

  • Petterssen, S., 1956: Weather Analysis and Forecasting. Vol. I: Motion and Motion Systems, McGraw Hill, 428 pp.

  • Queney, P., G. Corby, N. Gerbier, H. Koschmieder, and J. Zierep, 1960: The airflow over mountains. World Meteorological Organization Tech. Note 34, 135 pp. [Available from WMO, Case Postale 2300, CH-1211 Geneva 2, Switzerland.].

  • Ralph, F. M., M. Crochet, and S. V. Venkateswaran, 1992: A study of mountain lee waves using clear-air radar. Quart. J. Roy. Meteor. Soc.,118, 597–627.

  • ——, P. J. Neiman, T. L. Keller, D. Levinson, and L. Fedor, 1997: Observations, simulations, and analysis of nonstationary trapped lee waves. J. Atmos. Sci.,54, 1308–1333.

  • Rees, J. M., and S. D. Mobbs, 1988: Studies of internal gravity waves at Halley Base Antarctica, using wind observation. Quart. J. Roy. Meteor. Soc.,114, 939–966.

  • Reynolds, R. D., R. L. Lamberth, and M. G. Wurtele, 1968: Investigation of a complex mountain wave situation. J. Appl. Meteor.,7, 353–358.

  • Sarker, R. P., and R. V. Calheiros, 1974: Theoretical analysis of lee waves over the Andes as seen by satellite pictures. Pure Appl. Geophys.,112, 301–319.

  • Scorer, R., 1949: Theory of waves in the lee of mountains. Quart. J. Roy. Meteor. Soc.,75, 41–56.

  • Shutts, G. J., and A. Broad, 1993: A case study of lee waves over the Lake District in northern England. Quart. J. Roy. Meteor. Soc.,119, 377–408.

  • Smith, R. B., 1976: The generation of lee waves by the Blue Ridge. J. Atmos. Sci.,33, 507–519.

  • Starr, J. R., and K. A. Browning, 1972: Observations of lee waves by high-power radar. Quart. J. Roy. Meteor. Soc.,98, 73–85.

  • Vergeiner, I., and D. K. Lilly, 1970: The dynamic structure of lee wave flow as obtained from balloon and airplane observations. Mon. Wea. Rev.,98, 220–232.

  • Wurtele, M. G., 1955: The transient development of a lee wave. J. Mar. Res.,14, 1–13.

APPENDIX

Achieving a Hydrostatically Balanced, Time-Dependent Background Flow

The time-dependent background flows considered in this study are based on a scheme that smoothly transitions between two horizontally uniform basic states, while maintaining hydrostatic balance. Let (ρi, pi, θ̄i) and (ρf, pf, θ̄f) represent the thermodynamic characteristics of the background flow before and after the basic-state transition, respectively. A smooth transition from ρi to ρf can be obtained by expressing the change in ρ over one time step, Δt, as
ρ(t + Δt) = ρ(t) + ΔρδF,
where
i1520-0469-54-18-2275-eqa1
and
δFFttFt
In the expression for F(t), τs represents the time at which the transition starts and τ represents the time period over which the transition occurs. To maintain hydrostatic balance during the basic-state transition, p(t + Δt) and p(t) must satisfy the relationships
i1520-0469-54-18-2275-eqa3
respectively. Combining these two expressions and using the relationship (A1) produces the expression
i1520-0469-54-18-2275-eqa5
Since the initial and final basic states also satisfy the hydrostatic relationship, this expression can be rewritten as
i1520-0469-54-18-2275-ea2
where
ppfpi
Integrating (A2) with respect to z produces the relationship
pttptpδF,
which describes the change in p over one time step. Taking the logarithm of the equation of state, differentiating this equation with respect to time, and approximating the time derivatives with finite differences produces the relationship
i1520-0469-54-18-2275-ea4
By applying (A1) and (A3), the change in θ̄ over one time step can be rewritten in terms of the pressure and density of the initial and final basic states,
i1520-0469-54-18-2275-ea5
The basic-state transitions in the numerical simulations were implemented by using (A3) and (A5) to update p and θ̄, respectively, and then computing a new ρ from the equation of state. The transition in the speed of the cross-mountain flow associated with the transition from WS to SS′ was achieved by updating the wind speed of the background flow in a manner similar to that used for the mean-state pressure field.

Fig. 1.
Fig. 1.

Amplitude, normalized by the mountain height, of the trapped lee waves generated by the steady basic states (a) SS (⋄) and WS (□), and (b) LD3 (⋄) and LD5 (□) flowing over a mountain of height 10 m, 200 m, and 400 m.

Citation: Journal of the Atmospheric Sciences 54, 18; 10.1175/1520-0469(1997)054<2275:AMSONT>2.0.CO;2

Fig. 2.
Fig. 2.

Contour plot of the vertical velocity field in the xz plane at t = 5 h for flow over a 200-m-high mountain in which there is a 1-h transition from SS to WS. The contour interval is 0.25 m s−1. The heavy lines demarcate the boundaries of the three basic wave groups generated by this time-dependent background flow.

Citation: Journal of the Atmospheric Sciences 54, 18; 10.1175/1520-0469(1997)054<2275:AMSONT>2.0.CO;2

Fig. 3.
Fig. 3.

Contour plot of the vertical velocity field in the xt plane at z = 3 km for flow over a mountain 200 m high in which there is a 1-h transition from SS to WS. The contour interval is 0.25 m s−1. The transition period is stippled. The heavy solid line indicates the phase speed obtained by solving (8) for the horizontal wavelength of the phase I linear stationary trapped wave and the characteristics of the background flow during phase III. The heavy dashed lines in the unstippled regions indicate the group velocities obtained by solving (9) for the horizontal wavelength of the phase I linear stationary trapped wave, the basic-state properties during the respective time periods, and the appropriate phase speed. The heavy dashed line in the stippled region represents the average of the group velocities associated with phase I’s trapped waves during phases I and III.

Citation: Journal of the Atmospheric Sciences 54, 18; 10.1175/1520-0469(1997)054<2275:AMSONT>2.0.CO;2

Fig. 4.
Fig. 4.

Contour plot of the vertical velocity field in the xz plane at t = 5 h for flow over a mountain 200 m high in which there is a 1-h transition from WS to SS. The contour interval is 0.25 m s−1.

Citation: Journal of the Atmospheric Sciences 54, 18; 10.1175/1520-0469(1997)054<2275:AMSONT>2.0.CO;2

Fig. 5.
Fig. 5.

Same as Fig. 3 except the basic-state transition is from WS to SS.

Citation: Journal of the Atmospheric Sciences 54, 18; 10.1175/1520-0469(1997)054<2275:AMSONT>2.0.CO;2

Fig. 6.
Fig. 6.

Contour plot of w(x, t) = ww(x, t) + ws(x, t) where ww and ws are defined by (10).

Citation: Journal of the Atmospheric Sciences 54, 18; 10.1175/1520-0469(1997)054<2275:AMSONT>2.0.CO;2

Fig. 7.
Fig. 7.

Cross section of the mean state potential temperature in the plane parallel to the mountain ridge (see inset) that corresponds to a mean wind of 10 m s−1 parallel to the mountain ridge and a 1-h transition from WS to SS. The corresponding time axis has been included for reference. The heavy solid line denotes a slope of 1/10, while the heavy dashed line denotes a slope of 1/15. The inset illustrates the relationship between the mountain ridge, represented by stippling, and the orientation of the potential temperature cross section.

Citation: Journal of the Atmospheric Sciences 54, 18; 10.1175/1520-0469(1997)054<2275:AMSONT>2.0.CO;2

Fig. 8.
Fig. 8.

Contour plots of the vertical velocity field in the xz plane at (a) t = 3.5 h, and (b) t = 5.5 h for flow over a mountain 200 m high in which there is a 1-h transition from LD3 to LD5. The contour interval is 0.2 m s−1.

Citation: Journal of the Atmospheric Sciences 54, 18; 10.1175/1520-0469(1997)054<2275:AMSONT>2.0.CO;2

Fig. 9.
Fig. 9.

Contour plots of the vertical velocity field in the xz plane at (a) t = 3 h, and (b) t = 7 h for flow over a mountain 1 m high centered at x = 80 km in which there is a 1-h transition from WS to SS′. The contour interval is 0.00075 m s−1. The location of the mountain is indicated by a solid triangle.

Citation: Journal of the Atmospheric Sciences 54, 18; 10.1175/1520-0469(1997)054<2275:AMSONT>2.0.CO;2

Fig. 10.
Fig. 10.

Time evolution of the vertical velocity observed by the VHF profiler in Lannemezan on 15 October 1981. The scale is indicated to the right of the picture. The vertical velocities were obtained every 4 min 50 s. Each individual measurement is representative of a 750-m vertical window, and the horizontal window varies from 145 m at 2250 m to 1325 m at 15750 m. The mean error is estimated at 0.12 m s−1. In order to remove the slow variations, and to reduce the noise, a bandpass FFT filtering technique has been applied to the raw data, keeping only the timescale from 30 min to 3 h 30 min (from Bougeault et al. 1993).

Citation: Journal of the Atmospheric Sciences 54, 18; 10.1175/1520-0469(1997)054<2275:AMSONT>2.0.CO;2

Fig. 11.
Fig. 11.

Time series of the vertical velocity at eight different elevations above a site located at (a) x = 60 km in Fig. 3 and (b) x = 70 km in Fig. 8.

Citation: Journal of the Atmospheric Sciences 54, 18; 10.1175/1520-0469(1997)054<2275:AMSONT>2.0.CO;2

Fig. 12.
Fig. 12.

Time series of the vertical velocity at eight different elevations above a site located at (a) x = 63 km and (b) x = 65 km in Fig. 5.

Citation: Journal of the Atmospheric Sciences 54, 18; 10.1175/1520-0469(1997)054<2275:AMSONT>2.0.CO;2

Table 1. 

Characteristics of the two-layer, basic-state profiles used to generate time-dependent background flows, the characteristics of the linear, stationary trapped lee waves associated with each basic-state profile, and the maximum horizontal wavelength, λmax, for which each profile can support a nonstationary trapped wave. The horizontal wavelength of each stationary trapped wave, λs, was determined by solving Eq. (16) in Scorer (1949). Equation (12) from Corby and Wallington (1956) was used to determine the maximum vertical velocity W, which is based on a mountain height of 1 m and a mountain half-width of 3 km. Equation (8) was used to determine λmax.

Table 1. 
Table 2. 

Characteristics of the linear trapped waves associated with the pair of two-layer Scorer parameter profiles used to simulate a time-dependent, low-level static stability. The parenthetical values represent the properties of the finite-amplitude trapped waves in the numerical simulations.

Table 2. 
Table 3. 

Characteristics of the linear trapped waves associated with the pair of two-layer Scorer parameter profiles used to simulate a time-dependent, lower-layer depth. The parenthetical values represent the properties of the finite-amplitude trapped waves in the numerical simulation.

Table 3. 
Table 4. 

Characteristics of the linear trapped waves associated with the pair of two-layer Scorer parameter profiles used to simulate a time-dependent, lower-level static stability and a reduction in the cross-mountain flow. The parenthetical values represent the properties of the trapped waves in the numerical simulation.

Table 4. 

1

Surface fronts can have slopes in the lowest 1–2 km that are much steeper than 1/50, but the passage of such fronts would be accompanied by dramatic weather changes that are not observed in the more moderate episodes of lee-wave variability being considered in this paper.

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  • Bougeault, P., 1983: A non-reflective upper boundary condition for limited-height hydrostatic models. Mon. Wea. Rev.,111, 420–429.

  • ——, A. Jansa, J. L. Attie, I. Beau, B. Benech, R. Beniot, P. Bessemoulin, J. L. Caccia, J. Campins, B. Carissimo, J. L. Champeaux, M. Crochet, A. Druilhet, P. Durand, A. Elkhalfi, P. Flamant, A. Genoves, M. Georgelin, K. P. Hoinka, V. Klaus, E. Koffi, V. Kotroni, C. Mazaudier, J. Peon, M. Petitdidier, Y. Pointin, D. Puech, E. Richard, T. Satomura, J. Stein, and D. Tannhauser, 1993: The atmospheric momentum budget over a major mountain range: First results of the PYREX field program. Ann. Geophys.,11, 395–418.

  • Brown, P. R. A., 1983: Aircraft measurements of mountain waves and their associated momentum flux over the British Isles. Quart. J. Roy. Meteor. Soc.,109, 849–865.

  • Byers, H. R., 1974: General Meteorology. McGraw Hill, 461 pp.

  • Clark, T. L., and R. Gall, 1982: Three-dimensional numerical model simulations of airflow over mountainous terrain: A comparison with observations. Mon. Wea. Rev.,110, 766–791.

  • Collis, R. T. H., F. G. Fernald, and J. E. Alder, 1968: Lidar observations of Sierra wave conditions. J. Appl. Meteor.,7, 227–233.

  • Colson, D., and C. V. Lindsay, 1959: Unusual wave cloud over Washington DC, Nov. 30, 1959. Mon. Wea. Rev.,87, 451–452.

  • Corby, G. A., and C. E. Wallington, 1956: Airflow over the mountains: The lee wave amplitude. Quart. J. Roy. Meteor. Soc.,82, 266–274.

  • Durran, D. R., and J. B. Klemp, 1983: Compressible model for the simulation of moist mountain waves. Mon. Wea. Rev.,111, 2341–2361.

  • ——, M. Yang, D. N. Slinn, and R. G. Brown, 1993: Toward more accurate wave permeable boundary conditions. Mon. Wea. Rev.,121, 604–620.

  • Hoinka, K. A., 1984: Observations of a mountain-wave event over the Pyrenees. Tellus,36A, 369–383.

  • Klemp, J. B., and R. Wilhelmson, 1978: The simulation of three-dimensional convective storm dynamics. J. Atmos. Sci.,35, 1070–1096.

  • ——, and D. R. Durran, 1983: An upper boundary condition permitting internal gravity wave radiation in numerical mesoscale models. Mon. Wea. Rev.,111, 430–444.

  • Lindsay, C. V., 1962: Mountain waves in the Appalachians. Mon. Wea. Rev.,90, 271–276.

  • Mitchell, R. M., R. P. Cechet, P. J. Turner, and C. C. Elsum, 1990: Observation and interpretation of wave clouds over Macquarie Island. Quart. J. Roy. Meteor. Soc.,116, 741–752.

  • Nance, L. B., 1997: On the inclusion of compressibility effects in the Scorer parameter. J. Atmos. Sci.,54, 362–367.

  • Orlanski, I., 1976: A simple boundary condition for unbounded hyperbolic flows. J. Comput. Phys.,21, 251–269.

  • Pearce, R. P., and P. W. White, 1967: Lee wave characteristics derived from a three layer model. Quart. J. Roy. Meteor. Soc.,93, 155–165.

  • Pearson, R. A., 1974: Consistent boundary conditions for numerical models of systems that admit dispersive waves. J. Atmos. Sci.,31, 1481–1489.

  • Petterssen, S., 1956: Weather Analysis and Forecasting. Vol. I: Motion and Motion Systems, McGraw Hill, 428 pp.

  • Queney, P., G. Corby, N. Gerbier, H. Koschmieder, and J. Zierep, 1960: The airflow over mountains. World Meteorological Organization Tech. Note 34, 135 pp. [Available from WMO, Case Postale 2300, CH-1211 Geneva 2, Switzerland.].

  • Ralph, F. M., M. Crochet, and S. V. Venkateswaran, 1992: A study of mountain lee waves using clear-air radar. Quart. J. Roy. Meteor. Soc.,118, 597–627.

  • ——, P. J. Neiman, T. L. Keller, D. Levinson, and L. Fedor, 1997: Observations, simulations, and analysis of nonstationary trapped lee waves. J. Atmos. Sci.,54, 1308–1333.

  • Rees, J. M., and S. D. Mobbs, 1988: Studies of internal gravity waves at Halley Base Antarctica, using wind observation. Quart. J. Roy. Meteor. Soc.,114, 939–966.

  • Reynolds, R. D., R. L. Lamberth, and M. G. Wurtele, 1968: Investigation of a complex mountain wave situation. J. Appl. Meteor.,7, 353–358.

  • Sarker, R. P., and R. V. Calheiros, 1974: Theoretical analysis of lee waves over the Andes as seen by satellite pictures. Pure Appl. Geophys.,112, 301–319.

  • Scorer, R., 1949: Theory of waves in the lee of mountains. Quart. J. Roy. Meteor. Soc.,75, 41–56.

  • Shutts, G. J., and A. Broad, 1993: A case study of lee waves over the Lake District in northern England. Quart. J. Roy. Meteor. Soc.,119, 377–408.

  • Smith, R. B., 1976: The generation of lee waves by the Blue Ridge. J. Atmos. Sci.,33, 507–519.

  • Starr, J. R., and K. A. Browning, 1972: Observations of lee waves by high-power radar. Quart. J. Roy. Meteor. Soc.,98, 73–85.

  • Vergeiner, I., and D. K. Lilly, 1970: The dynamic structure of lee wave flow as obtained from balloon and airplane observations. Mon. Wea. Rev.,98, 220–232.

  • Wurtele, M. G., 1955: The transient development of a lee wave. J. Mar. Res.,14, 1–13.

  • Fig. 1.

    Amplitude, normalized by the mountain height, of the trapped lee waves generated by the steady basic states (a) SS (⋄) and WS (□), and (b) LD3 (⋄) and LD5 (□) flowing over a mountain of height 10 m, 200 m, and 400 m.

  • Fig. 2.

    Contour plot of the vertical velocity field in the xz plane at t = 5 h for flow over a 200-m-high mountain in which there is a 1-h transition from SS to WS. The contour interval is 0.25 m s−1. The heavy lines demarcate the boundaries of the three basic wave groups generated by this time-dependent background flow.

  • Fig. 3.

    Contour plot of the vertical velocity field in the xt plane at z = 3 km for flow over a mountain 200 m high in which there is a 1-h transition from SS to WS. The contour interval is 0.25 m s−1. The transition period is stippled. The heavy solid line indicates the phase speed obtained by solving (8) for the horizontal wavelength of the phase I linear stationary trapped wave and the characteristics of the background flow during phase III. The heavy dashed lines in the unstippled regions indicate the group velocities obtained by solving (9) for the horizontal wavelength of the phase I linear stationary trapped wave, the basic-state properties during the respective time periods, and the appropriate phase speed. The heavy dashed line in the stippled region represents the average of the group velocities associated with phase I’s trapped waves during phases I and III.

  • Fig. 4.

    Contour plot of the vertical velocity field in the xz plane at t = 5 h for flow over a mountain 200 m high in which there is a 1-h transition from WS to SS. The contour interval is 0.25 m s−1.

  • Fig. 5.

    Same as Fig. 3 except the basic-state transition is from WS to SS.

  • Fig. 6.

    Contour plot of w(x, t) = ww(x, t) + ws(x, t) where ww and ws are defined by (10).

  • Fig. 7.

    Cross section of the mean state potential temperature in the plane parallel to the mountain ridge (see inset) that corresponds to a mean wind of 10 m s−1 parallel to the mountain ridge and a 1-h transition from WS to SS. The corresponding time axis has been included for reference. The heavy solid line denotes a slope of 1/10, while the heavy dashed line denotes a slope of 1/15. The inset illustrates the relationship between the mountain ridge, represented by stippling, and the orientation of the potential temperature cross section.

  • Fig. 8.

    Contour plots of the vertical velocity field in the xz plane at (a) t = 3.5 h, and (b) t = 5.5 h for flow over a mountain 200 m high in which there is a 1-h transition from LD3 to LD5. The contour interval is 0.2 m s−1.

  • Fig. 9.

    Contour plots of the vertical velocity field in the xz plane at (a) t = 3 h, and (b) t = 7 h for flow over a mountain 1 m high centered at x = 80 km in which there is a 1-h transition from WS to SS′. The contour interval is 0.00075 m s−1. The location of the mountain is indicated by a solid triangle.

  • Fig. 10.

    Time evolution of the vertical velocity observed by the VHF profiler in Lannemezan on 15 October 1981. The scale is indicated to the right of the picture. The vertical velocities were obtained every 4 min 50 s. Each individual measurement is representative of a 750-m vertical window, and the horizontal window varies from 145 m at 2250 m to 1325 m at 15750 m. The mean error is estimated at 0.12 m s−1. In order to remove the slow variations, and to reduce the noise, a bandpass FFT filtering technique has been applied to the raw data, keeping only the timescale from 30 min to 3 h 30 min (from Bougeault et al. 1993).

  • Fig. 11.

    Time series of the vertical velocity at eight different elevations above a site located at (a) x = 60 km in Fig. 3 and (b) x = 70 km in Fig. 8.

  • Fig. 12.

    Time series of the vertical velocity at eight different elevations above a site located at (a) x = 63 km and (b) x = 65 km in Fig. 5.

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