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    One-dimensional representation of the 2D power spectrum of a section of the Coastal Range topography in western Oregon.

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    Idealized flow conditions and terrain profile for a banded precipitation event over the Coastal Range. (a) Skew T diagram of the idealized upstream sounding profile during the 12–13 Nov 2002 event and (b) smoothed terrain of northwest Oregon (h).

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    Analysis of simulations that use a strip of sinusoidal terrain perturbations centered at xb = 40 km. Rainwater mixing ratio (qr) at z = 1.5 km and t = 4 h of (a) COL20km, (b) COL10km, and (c) COL5km simulations, with contours drawn at 0.01 and 0.1 g kg−1. (d)–(f) Vertical velocities (w) at the same locations and times, with contours drawn at 0.5, 1, and 2 m s−1. Boxes in (d)–(f) indicate x-averaging areas for band counts, which are shown in the rightmost panels. For reference, solid contours of at x ≈ 40 km are shown in 25-m increments. Grayscale terrain shading is the same as in Fig. 2b.

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    Same as in Figs. 3d–f, except for (a) PA20km, (b) PA10km, (c) PA5km, (d) PA2.5km, and (e) PA1.25km simulations.

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    Same as in Fig. 4, except for (a) MPA2, (b) MPA3, (c) MPA4, (d) MPA5, and (e) RND simulations.

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    Simulated band spacings (λsim) as a function of the smallest-scale mode in (λmin) for PA20km–PA1.25km (white circles) and MPA2–MPA5 (gray circles) simulations. See text for further details.

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    Vertical cross sections of w and cloud water mixing ratio (qc) at t = 4 h for (a) PA10km (y = 9 km), (b) PA5km (y = 13 km), and (c) PA2.5km (y = 14 km) simulations. Vertical velocity (black lines) is solid for positive values and dashed for negative values, and is contoured at |w| = 0.1, 0.2, 0.5, 1, and 2 m s−1. Cloud water contours are shaded at values of qc = 0.001 and 1 g kg−1.

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    Schematic diagram of the linear model. Solid and dashed lines in w0(y, z) and w(y, z, tlin) panels denote positive and negative values of w.

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    Vertical structures of lee waves in a channel of depth d = 2 km forced by a range of topographic scales (λ).

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    Three unstable modes (p = 1, 2, and 3) of the linear-model solutions for two different cloud-base heights (H). (a) Eigenfunctions wup(z), (b) lee-wave projection coefficients Au+p(κ), and (c) unstable growth rates anp(κ), for H = 500 m. (d)–(f) Same as (a)–(c), but for H = 100 m.

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    Comparison of vertical velocity and cloud outline from the (a), (c), (e), (g) numerical and (b), (d), (f), (h) linear models at four different x locations. Vertical velocities are contoured at |w| = 0.01, 0.02, 0.05, 0.1, 0.2, 0.5, and 1 m s−1, with positive (solid lines) and negative (dashed lines) values. In the simulation, the qc value is 0.001 g kg−1.

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    Dependence of linear-model unstable vertical velocities on terrain-induced forcing scale k and cloud-base height H. Gray shade contour plots of wumax at (a) ttc = 0, (b) ttc = 750 s, and (c) ttc = 1500 s. Cuts through (a)–(c) are shown at (d) H = 500 m and (e) H = 100 m.

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    Using cloud outline at z = 1.5 km and t = 4 h to find tlin from the (a) MPA4 and (b) MPAD4 simulations. The location past which cloud-free regions are more extensive than cloudy areas is denoted as xlin, while the triggering location (determined earlier) is xc ≈ 46 km.

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    Cloud outline and negative values of moist Brunt–Väisälä frequency (N2m) at t = 1 h in smooth-terrain simulations ( = 0) for (a) RH90 and (b) RH99 soundings. Cloud outline (qc = 0.001 g kg−1) is shaded and N2m is contoured by dashed lines, with labels multiplied by 10−5 s−2.

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    Same as in Fig. 4, but for (a) PAM20km, (b) PAM10km, (c) PAM5km, (d) PAM2.5km, and (e) PAM1.25km soundings.

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    Same as in Figs. 5a–d, but for (a) MPAM2, (b) MPAM3, (c) MPAM4, and (d) MPAM5 simulations.

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    Simulated band spacings (λsim) as a function of the smallest-scale mode in (λmin) for PAM20km–PAM1.25km (white circles) and MPAM2–MPAM5 (black circles) simulations.

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    Same as in Fig. 12, but for a power spectrum characterized by Eκ−5/3, wherein (a) ttc = 0, (b) ttc = 1500 s, and (c) ttc = 3000 s.

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    Same as in Figs. 5a–d, but for (a) MPAD2, (b) MPAD3, (c) MPAD4, and (d) MPAD5 simulations.

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    Same as in Fig. 16, but for (a) MPAMD2, (b) MPAMD3, (c) MPAMD4, and (d) MPAMD5 simulations.

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    Simulated band spacings (λsim) as a function of λmin for the MPAD2–MPAD5 (gray circles) and MPAMD2–MPAMD5 (black circles) simulations.

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The Spacing of Orographic Rainbands Triggered by Small-Scale Topography

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  • 1 National Center for Atmospheric Research,* Boulder, Colorado
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Abstract

A combination of idealized numerical simulations and analytical theory is used to investigate the spacing between convective orographic rainbands over the Coastal Range of western Oregon. The simulations, which are idealized from an observed banded precipitation event over the Coastal Range, indicate that the atmospheric response to conditionally unstable flow over the mountain ridge depends strongly on the subridge-scale topographic forcing on the windward side of the ridge. When this small-scale terrain contains only a single scale (λ) of terrain variability, the band spacing is identical to λ, but when a spectrum of terrain scales are simultaneously present, the band spacing ranges between 5 and 10 km, a value that is consistent with observations. Based on the simulations, an inviscid linear model is developed to provide a physical basis for understanding the scale selection of the rainbands. This analytical model, which captures the transition from lee waves upstream of the orographic cloud to moist convection within it, reveals that the spacing of orographic rainbands depends on both the projection of lee-wave energy onto the unstable cap cloud and the growth rate of unstable perturbations within the cloud. The linear model is used in tandem with numerical simulations to determine the sensitivity of the band spacing to a number of environmental and terrain-related parameters.

* The National Center for Atmospheric Research is sponsored by the National Science Foundation

Corresponding author address: Daniel J. Kirshbaum, Department of Geology and Geophysics, Yale University, P.O. Box 208109, New Haven, CT 06511. Email: daniel.kirshbaum@yale.edu

Abstract

A combination of idealized numerical simulations and analytical theory is used to investigate the spacing between convective orographic rainbands over the Coastal Range of western Oregon. The simulations, which are idealized from an observed banded precipitation event over the Coastal Range, indicate that the atmospheric response to conditionally unstable flow over the mountain ridge depends strongly on the subridge-scale topographic forcing on the windward side of the ridge. When this small-scale terrain contains only a single scale (λ) of terrain variability, the band spacing is identical to λ, but when a spectrum of terrain scales are simultaneously present, the band spacing ranges between 5 and 10 km, a value that is consistent with observations. Based on the simulations, an inviscid linear model is developed to provide a physical basis for understanding the scale selection of the rainbands. This analytical model, which captures the transition from lee waves upstream of the orographic cloud to moist convection within it, reveals that the spacing of orographic rainbands depends on both the projection of lee-wave energy onto the unstable cap cloud and the growth rate of unstable perturbations within the cloud. The linear model is used in tandem with numerical simulations to determine the sensitivity of the band spacing to a number of environmental and terrain-related parameters.

* The National Center for Atmospheric Research is sponsored by the National Science Foundation

Corresponding author address: Daniel J. Kirshbaum, Department of Geology and Geophysics, Yale University, P.O. Box 208109, New Haven, CT 06511. Email: daniel.kirshbaum@yale.edu

1. Introduction

Quasi-stationary orographic rainbands, which form in response to moist, conditionally unstable flow over complex terrain, are meteorologically significant because they concentrate orographic precipitation over specific areas and may generate large localized precipitation amounts. Observations of such rainbands have been reported over a number of coastal mountain ranges, including the western Kyushu in Japan (Yoshizaki et al. 2000), the Cévennes region of southern France (Miniscloux et al. 2001; Cosma et al. 2002; Anquetin et al. 2003), and the Coastal Range of western Oregon (Kirshbaum and Durran 2005; Kirshbaum et al. 2007, hereafter K07). Although recent studies have started to investigate the underlying dynamics behind the rainbands, questions remain as to the precise mechanisms that give rise to the bands and control their preferred horizontal scales.

Insight into the mechanisms by which small-scale terrain obstacles trigger orographic rainbands is provided in K07, who analyzed an observed banded precipitation event over the Coastal Range in November 2002. Their numerical simulation of the observed event produced rainbands with similar locations and spatial scales as those of the observations, indicating the model’s effectiveness in capturing the underlying physics behind rainband formation. Through additional idealized simulations that preserved the mesoscale variability of the Coastal Range terrain but replaced the small-scale terrain field with simple hills and valleys, they discovered that only some small-scale obstacles were capable of triggering rainbands in the unstable cap cloud over the windward side of the mountain ridge. Those that triggered rainbands generated steady gravity waves (or lee waves) that underwent saturation at a specific phase in their stable oscillations. Specifically, the lee-wave-induced vertical velocity perturbation wc was positive and its buoyancy perturbation bc was close to zero, which allowed these perturbations to transition into growing convective motions upon entering the orographic cap cloud.

An elusive problem that remains to be solved is the understanding of what determines the spacing between adjacent orographic rainbands. Some general ideas on convective scaling that are relevant to this topic are provided by the inviscid and Boussinesq linear theory of saturated, unsheared 2D flow in a one-layer atmosphere, which gives a normal-mode solution of the form
i1520-0469-64-12-4222-e1
where
i1520-0469-64-12-4222-e2
This solution suggests that the convective growth rates (a) inside an unstable cloud depend both on the moist stability of the cloud (N2m) and the horizontal and vertical wavenumbers (k and m) of perturbations within the layer. For fixed Nm and m, the largest values of a occur at infinitely small horizontal scales (large k), which suggests that the spacing between neighboring convective elements should be infinitely small. However, this is not the case for convective orographic rainbands, whose spacings are finite and range from around 5 to 20+ km in Yoshizaki et al. (2000), Miniscloux et al. (2001), and Kirshbaum and Durran (2005).

Additional physical processes must be considered to explain the observed spacing between orographic rainbands. Fuhrer and Schär (2005) used a similar Boussinesq analysis as that above, but extended it to 3D and added diffusional terms to shift the maximum growth rates from infinitely small to finite horizontal scales. The validity of this approach in explaining the observed spacing over the Coastal Range is evaluated in section 4. In general, however, the unstable growth rate is not the only parameter that controls the scale of the convection that develops inside the cloud. Another important parameter is ŵ0(k), which governs the initial strength of convective perturbations and may be strongly scale dependent. Cosma et al. (2002) found that the 10-km spacings of the Cévennes rainbands corresponded to a scale at which the variability of the underlying terrain, and hence ŵ0(k), was a relative maximum. This idea, however, does not explain the spacing of rainbands over the Coastal Range in K07. Although well-organized bands also developed with a mean spacing of about 10 km, the power spectrum of the terrain in Fig. 1 indicates no dominant forcing scales.

The goals of this paper are to explain the observed rainband spacings over the Coastal Range and to identify the general physical parameters that control the scales of orographic rainbands. To this end, we perform a series of idealized numerical simulations to assess the relationship between the terrain forcing scales and the spacings of the convective bands that develop. We then explain the simulation results through an analytical model that couples the gravity wave oscillations upstream of and below the orographic cloud to the moist convection within it. Finally, we use this hierarchy of models to evaluate some key physical parameters that modulate the interband spacing.

2. Numerical setup

The idealized simulations described herein are designed to systematically investigate the influence of different small-scale terrain fields (x, y) on the scales of the convective bands that form over the upslope of the mesoscale ridge h(x). These simulations are performed using the cloud-resolving numerical model described in Bryan and Fritsch (2002), and the reader is referred to that article and K07 for a detailed discussion of the governing equations, the numerical strategies used to solve them, and the parameterizations (microphysical and turbulent) that represent subgrid-scale processes. The experimental domain has dimensions of Lx = 160 km, Ly = 20 km, and Lz = 15 km, with outflow conditions applied at the x boundaries, periodic conditions at the y boundaries, and a Rayleigh wave-absorbing layer over the uppermost 5 km. In most of the simulations, the horizontal grid spacing is Δ = 250 m, but in some cases Δ is reduced to 125 m. The vertical grid in all simulations is defined as Δz = 125 m over 0 < z ≤ 5 km, which stretches to 500 m over 5 < z < 10 km and remains constant at 500 m up to Lz.

The upstream sounding (Fig. 2a) is a simplified version of the observed sounding during the 12–13 November 2002 rainband event over the Coastal Range, which, when coupled with the actual Coastal Range terrain in a numerical simulation of that event, accurately reproduced the observed rainband locations (K07). This horizontally homogeneous sounding is defined by a surface temperature Ts = 285 K, a dry Brunt–Väisälä frequency Nd = 0.01 s−1, and a wind vector u with a magnitude U = 10 m s−1 that, for simplicity, is oriented at an angle of θ = 270° measured clockwise from due north (i.e., pure westerly flow). The relative humidity (RH) is 90% from the surface to a height of z = 5 km, decreases linearly to a value of 1% at z = 10 km, and remains constant up to Lz. This sounding contains weak conditional (and potential) instability at low levels (0 ≤ z < ∼3 km), with a CAPE of 384 J kg−1.

The mesoscale component of the simulated terrain [h(x)] was created by K07, who took the Fourier transform of the full Coastal Range terrain [h(x, y)] and preserved only the x variability at scales larger than λm = 40 km (Fig. 2b). The following numerical experiments all use this h(x) profile, which lifts the potentially unstable flow to saturation and forms an unstable cap cloud over the windward, or west-facing, slope of the ridge. To trigger convection within the cloud, we add simple y-periodic small-scale terrain fields (x, y) composed of horizontal scales less than or equal to λ = 20 km to the mountain upslope. More details on the design of these small-scale terrain fields are provided in the following.

3. Response to scale-selective forcings

a. Strips of terrain perturbations at xb = 40 km

The idealized simulations of K07 demonstrated an obvious link between the small-scale topography on the ridge upslope () and the spacing of orographic rainbands. The bands that formed over the mountain scaled with the wavelength (λ = 10 km) of a y-parallel strip of small-scale, sinusoidal terrain perturbations centered at xb = 40 km. Here we investigate the scale dependence of the bands by performing a set of three simulations that differ only in their values of λ. These simulations use a slightly different expression for than K07,
i1520-0469-64-12-4222-e3
which generates a sinusoidal strip of over the mountain upslope that is always positive. In the preceding, the amplitude hcol = 100 m and x center point xcol = 40 km are identical to their values in K07, and λ = 20 km in the first simulation (COL20km), 10 km in the second simulation (COL10km), and 5 km in the third simulation (COL5km).

Horizontal cross sections of the rainwater mixing ratio (qr) at an integration time t = 4 h and a height z = 1.5 km in Figs. 3a–c indicate a single rainband in the COL20km case (Fig. 3a) and multiple rainbands in the COL10km and COL5km cases (Figs. 3b,c). Consistent with K07, the bands form downstream of locations where oscillating air parcels reach saturation with a favorable phase for band triggering, which can occur past either valleys (Fig. 3a) or hills (Figs. 3b,c) in . In each simulation the number of rainbands is equal to the number of wave crests across the width of the domain in , reinforcing the link established in Cosma et al. (2002) and K07 between the scales of the small-scale topographic forcing and that of the convective rainbands. The precipitation fields in Figs. 3a–c are closely related to the corresponding vertical velocity (w) fields (Figs. 3d–f) in that the regions of strongly positive w are collocated with, or slightly upstream of, the regions of nonzero qr. One exception is the particularly weak band in the w field of the COL20km case (Fig. 3d), which is not apparent in the qr field of that simulation (Fig. 3a).

The mean band spacing in a given simulation (λsim) is found by dividing the number of bands in the domain (Nb) by Ly. To objectively measure Nb, we adapt a formulation based on Lemone et al. (1994), who estimated the number of convective cores from the vertical velocities measured in flight segments through cumulonimbus clouds by applying a low-pass filter and counting each location that exceeded a specified threshold (1 m s−1) as an updraft. A similar procedure may be applied to the w fields in Figs. 3d–f; we average w over a 10-km-long horizontal section in x to obtain w and count each passing of the 1 m s−1 level as a rainband. This threshold value is selected because it roughly corresponds to locations where the simulated reflectivity, computed using a simple conversion algorithm in Douglas (1964), exceeds 10 dBZ, a level that usually corresponds to light precipitation. The 10-km section used for the averaging, which is determined separately for each case, is taken to be the area over which Nb is the largest.

The technique outlined above is applied to the COL20km, COL10km, and COL5km simulations in Figs. 3d–f. In each case, the boxed area over which w is computed is different, reflecting the variability in the locations of the bands as a function of λ. Through this procedure we obtain Nb = 1 in the COL20km case, Nb = 2 in the COL10km case, and Nb = 4 in the COL5km case, which are the same values that were obvious from the qr fields in Figs. 3a–c. Note that the band at y = 10 km in the w field of the COL20km case (Fig. 3d) is not counted because its w < 1 m s−1. To facilitate comparisons between the simulated data and the linear analysis of section 4, we use the w field (rather than qr) to diagnose the number of bands in all subsequent simulations. Characteristics of the COL20km–COL5km simulations, as well as most other simulations conducted in this study, are listed in Table 1.

b. Patches of terrain perturbations: Single modes

The idealized simulations in Fig. 3 suggest that the scales contained in (x, y) control the horizontal scales of the bands that develop over the mountain upslope. Because of their simplistic fields, however, they reveal very little about the scale selection of the bands over the more complicated terrain fields that characterize real mountain ranges. As a first step in our progression toward more realistic terrain, we extend the sinusoidal columns of (x, y) over a broader area so that the flow is subject to more small-scale terrain forcing as it passes over the mountain. For the same upstream sounding and h(x) used here, K07 demonstrated that (x, y) over a limited area on the mountain upslope (40 ≤ x ≤ 60 km) has a dominant effect on the convection that develops over the ridge. In keeping with this result, we extend (x, y) to span the region defined by 40 ≤ x ≤ 60 km and, as before, Ly. This is accomplished through a slight alteration of (3),
i1520-0469-64-12-4222-e4
in which hpa = 100 m, xl = 40 km and xr = 60 km are the upstream and downstream edges of the sinusoidal patch, respectively, and ϕy is an arbitrary phase shift in the y direction. Note that the signs of the cosine coefficients in (4) are negative to ensure that, as long as λ = Ly/n, where n is a positive integer, = 0 at xl and xr and the terrain field has no step discontinuities.

Here we follow a similar procedure to that of section 3a by performing a series of simulations with fields designed according to (4) that are identical, except for variations in λ. In the first case (PA20km), λ = xrxl = 20 km and is composed of just one hill centered at x = 50 km, which is identical to the COL20km simulation, except that the center of , which was previously at xb = 40 km, is shifted 10 km farther downstream. In each subsequent simulation we reduce λ by a factor of 2 to 10 (PA10km), 5 (PA5km), 2.5 (PA2.5km), and 1.25 km (PA1.25km). For the PA1.25km simulation, we also reduce Δ from 250 to 125 m to better resolve the finescale features in . Except for the PA20km case where ϕy = 0, ϕy is a uniformly distributed random number between 0 and 2π.

The w field of the PA20km simulation at t = 4 h and z = 1.5 km in Fig. 4a shows that the rainband near the periodic boundary in the COL20km simulation (Fig. 3a) weakens when is shifted 10-km downstream. Compared to the COL20km case, the in-cloud residence times of unstable perturbations in the PA20km case are shorter, which limits the growth of convection before the air undergoes descent on the lee side of the ridge (Kirshbaum and Durran 2004). Strong rainbands only appear as λ is decreased in the PA10km and PA5km simulations (Figs. 4b,c), whose convective perturbations have similar residence times as the PA20km case but faster in-cloud growth rates [see (2)]. As the scale of is reduced to 2.5 and then 1.25 km in the PA2.5km and PA1.25km simulations (Figs. 4d,e), the convective updrafts, which still scale with λ, become so weak that they no longer pass the w ≥ 1 m s−1 threshold. Evidently, the band intensity is the strongest over a narrow range of horizontal scales encompassing 5 ≤ λ ≤ 10 km.

c. Patches of terrain perturbations: Multiple modes

Real terrain profiles contain not just one but an infinite number of small-scale forcing modes. As such, we may create more realistic terrain profiles by superposing the fields from two or more of the simulations in Figs. 4a–e, which are carried out in four simulations (MPA2–MPA5) that add progressively more small-scale detail to the terrain. The MPA2 case adds the fields from the PA20km and PA10km simulations, the MPA3 case adds the PA5km field to the MPA2 terrain, etc. As in the PA1.25km case, the MPA5 simulation is performed with Δ = 125 m to adequately resolve the finest horizontal forcing scale.

The w cross sections in Figs. 5a–d illustrate that the effects of adding small-scale detail to may range from dominant to minimal. Consistent with the linear theory argument in (1) for which the smallest terrain scales dominate the solution, the MPA2 case (Fig. 5a) produces nearly identical clouds as that of the λ = 10 km mode by itself (Fig. 4b). However, the MPA3, MPA4, and MPA5 simulations (Figs. 5b–d) are different from the PA5km, PA2.5km, and PA1.25km simulations (Figs. 4c–e) in that they generate bands whose spacings (λsim = 7, 7, and 5 km) are larger than λmin, the minimum λ in , and do not change substantially as extra-small-scale detail is added to the terrain.

The superposition of forcing modes in the MPA2–MPA5 simulations constitutes a single realization of (x, y) corresponding to this particular combination of random ϕy values. To ensure that the results in Figs. 5a–d are representative of all possible terrain realizations, we perform the MPA2–MPA5 simulations four extra times, with each iteration possessing a new set of random ϕy values. The statistics of these simulations are presented in Fig. 6, which plots the mean band spacing (λsim) in gray circles (with whiskers to show the maximum and minimum values of the five realizations) as a function of λmin. For reference, the results of the PA20km–PA1.25km simulations are shown by larger white circles that vanish as λmin is reduced below 5 km and the updrafts (Figs. 4d,e) are too weak to be classified as bands. A gray, dashed line of λsim = λmin is also added to indicate the scale of fastest growth predicted by (2), which is always equal to λmin. This line is consistent with the band spacings for λmin = 10 km, but diverges with the simulation results as λmin is decreased below 5 km. Past that value, the mean λsim in the multiple-mode cases is roughly constant, suggesting that it has converged to a “preferred” value of λ*sim ≈ 5 km. Clearly, the inviscid growth rates in (2) do not explain the converged band spacings in these multiple-mode simulations.

To confirm the robustness of the results in Figs. 5a–d, we show in Fig. 5e the results of an additional simulation (RND) that is identical to the MPA5 case, except that is replaced with random topographic perturbations over the entire surface of the domain. This field is created by designing a 2D field in the wavenumber domain [ˆ(kx, ky)] with a constant amplitude and uniformly distributed random phase at all values of kx and ky. We then remove the power at all scales smaller than 6Δ (or κ = k2x + k2y > 2π/6Δ), invert the field into Cartesian coordinates, and scale it so that (x, y) has a maximum value of 200 m. Despite the vast differences between this specification of and the five-mode field in the MPA5 simulation, the number of convective bands (4) and mean band spacings (λsim = 5 km) in these two cases are identical. Hence, despite the limited number of modes considered and the lack of areal coverage of in the MPA5 simulation, its results are representative of cases with fully random fields spread over the domain.

4. A linear model for rainband spacing

The rainband spacings in some, but not all, of the cases presented in Figs. 4 –5 may be easily explained by existing theories in the literature. For example, consistent with Cosma et al. (2002), the single-mode cases in Figs. 4a–c produced rainbands with a spacing of λsim = λ. Additionally, λsim = λmin in the multiple-mode MPA2 simulation (Fig. 5a), which may be easily explained by the linear Boussinesq theory of an inviscid, saturated, and unstable atmosphere in (1)(2).

Less intuitive behavior is evident in Figs. 5c–e, where the band spacing remains nearly constant as finer-scale modes are added to . One might ascribe this insensitivity to small-scale forcing to diffusional effects, which preferentially dissipate turbulent kinetic energy at small scales. Fuhrer and Schär (2005) used linear theory to show that diffusion shifts the wavelength of maximum unstable growth from infinitely small scales to a finite wavelength λ*. They applied this theory to explain the growth of embedded convection in the unstable cap cloud of one numerical simulation, which was similar to the simulations in this study, except that their mountain was smooth and convection was triggered by background thermal fluctuations instead of small-scale topographic features. By estimating a simulated mixing coefficient (α) from the in-cloud subgrid-scale turbulent coefficients and assigning that value to their linear analysis, they estimated λ* = 4 km, which is similar to the mean band spacings in Figs. 5d,e (5 km). This apparent agreement, however, is most likely a coincidence. The different convective-triggering mechanism used in Fuhrer and Schär (2005) and their focus on transient cells rather than stationary bands implies that the governing dynamics may be different from those considered here. Moreover, their value of α = 103 m2 s−1 is much larger than the mixing coefficients in the numerical simulations considered in this study. As a consequence, direct application of their procedure results in λ* = 1.6 km for the MPA4 simulation (where Δ = 250 m and α ≈ 20 m2 s−1) and 1.2 km for the MPA5 and RND simulations (where Δ = 125 m and α ≈ 10 m2 s−1), which disagree with the λsim values in Fig. 5.

Building on the arguments proposed by K07, we offer an alternate hypothesis for the scale selection of orographic rainbands that depends not on diffusional effects inside the cloud but on lee waves forced by small-scale terrain perturbations upstream of and below the cloud. This hypothesis is developed in the following through a three-stage linear model that couples the dynamics of lee-wave forcing past to the growth of moist convection inside the cap cloud. Note that the term “lee wave” in this study refers to any quasi-steady gravity wave launched by stable flow over a topographic obstacle, not specifically to trapped lee-wave “trains” that may form downwind of mesoscale terrain under certain atmospheric conditions (e.g., Durran 1990).

a. Model description

Guidance for the model setup is provided by the contours of w and cloud liquid water (qc) of the PA10km, PA5km, and PA2.5km simulations in Fig. 7. For consistency, all three of these cross sections are selected at y locations where the peaks and troughs in are maximized (9, 13, and 14 km, respectively). An obvious feature in all three panels is that the cap cloud overhangs the 40 ≤ x ≤ 60 km section where is nonzero, which suggests that both the unstable cloud region and the stable, unsaturated layer upstream of and beneath it are part of the convective triggering process. Moreover, the cloud thickens as the flow progresses from x = 40 km to x = 60 km, with the cloud top nearly constant at z ≈ 2 km, but its base lowering from about 1.5 km at x ≈ 40 km to the surface at x ≈ 55 km. Note also that most of the lee-wave energy (evident from the w contours in Fig. 7) is unable to penetrate through the cloud and is essentially trapped below the cloud top.

No linear model can be expected to realistically capture all of the physical processes in the nonlinear simulations of Fig. 7. However, by invoking some simplifications, we are able to design a model that is straightforward to solve and yet still captures the basic physics of the lee-wave triggering process. As shown in our schematic illustration of the model setup in Fig. 8, it is separated into the following three sections: 1) lee-wave initialization, 2) convective triggering, and 3) convective growth. This separation allows the basic dynamics to be investigated without the explicit consideration of h(x), which facilitates a much simpler analytical solution. Among the effects of h(x) are lifting the bulk flow to saturation over the ridge upslope and allowing the small-scale lee waves forced by stable flow over (x, y) to reach the leading edge of the cloud with a favorable phase for band triggering (K07). Both of these effects are represented in sections 2 and 3 of the linear model, which are described in detail below.

1) Initialization

The initialization corresponds to 40 ≤ x ≲ 50 km in Fig. 7, where lee-wave perturbations dominate the w field below the cloud top at z ≈ 2 km and no convective updrafts are apparent. The waves appear to be trapped below this level because of rapid wave decay within the cloud and/or wave reflections as the medium of wave propagation changes with height. In the linear model, we treat this region as a stable, unsaturated channel flow of depth d = 2 km, wind speed Um = U = 10 m s−1, and stability N21 = N2d = 10−4 s−2 that rides over a periodic sinusoidal lower boundary given by
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where xl and xr are the upstream and downstream edges of the sinusoidal patch, k is the wavenumber, and hm = 100 m is the peak-to-trough amplitude. Although moist effects are not explicitly considered in this section, the wave-trapping effects of the cloud are treated by imposing zero vertical velocity at the top of the channel (z = d). The steady-state solution to this problem is derived in appendix A.

A key physical feature captured by (A4) and (A5) is the dependence of the vertical lee-wave structure W(z) on λ. As shown in Fig. 9, the near-surface W increases with decreasing λ (increasing k), which is consistent with the lower boundary condition (A2). Aloft the dependence of W on λ is reversed as W weakens with decreasing λ. This is because the rate of vertical decay of the wave response [m = 2(l2k2)] is greater for smaller λ, which implies that smaller-scale terrain perturbations generate stronger lee-wave responses near the surface but weaker waves aloft. This scale-dependent lee-wave response is also obvious from the w fields over 40 ≤ x ≤ 60 km in Fig. 7.

2) Triggering

The initialization and growth sections of the model are linked at x = xr = xc, the point at which oscillating lee-wave perturbations from the initialization section saturate aloft and transition to convective updrafts and downdrafts. Across this boundary, the linear-model terrain (hlin) is set to zero, the cloud base lowers from d to H, which represents the gradual lowering of the orographic cap cloud in the numerical simulations (e.g., Fig. 7), and the independent variable x is replaced by t, such that the endpoint of the lee-wave initialization (xr = xc) is equivalent to the starting point of the growth section (t = tc).

As discussed in K07, convective updrafts are triggered when oscillating air parcels undergo saturation at x = xc with a positive lee-wave-induced w perturbation and a nearly zero b perturbation. Immediately after saturation, the positive w generates positive b in the unstable cloud (where N2m < 0) and spurs convective growth. K07 suggested that convective triggering is favored by upslope ascent over h, which gradually lifts the upstream flow closer to saturation so that only small lee-wave-induced displacements (ζ) are needed for it to saturate. As a consequence, the triggering location xc in simulations with small-scale terrain features was found to be nearly identical to the x location at which the parcels would reach saturation in the absence of small-scale terrain. This roughly corresponds to the x location at which h(x) reaches the lifting condensation level (LCL ≈ 102 m), which, for the upstream sounding considered in Fig. 2a, is xc = 46.25 km.

Although h is not explicitly considered in the linear-model equations, its effects are represented by forcing saturation to occur at a phase ϕx where the lee-wave-induced w perturbation has a maximum magnitude and the b perturbation is zero. This phase is easily found from (A8)(A9) (ϕx = π/2), and is assigned to the triggering location xr = xc in (5). For consistency with the simulations, this location is specified to match the simulated triggering location, which, as mentioned above, is xc ≈ 46.25 km. The w and b fields at xc derived from this procedure [w0(y, z) and b0(y, z)] comprise the initial conditions at t = tc in the growth section of the model. Note that because this derivation assumes a boundless domain, the upstream edge of hlin (xl) is arbitrary, but for consistency with the simulations it is selected to occur along a nodal line where hlin = 0.

3) Growth

In this section we track the time evolution of the lee-wave-induced w0(y, z) and b0(y, z) perturbations in a two-layer channel flow (again of depth d) that is unsaturated with a stability N21 > 0 below the cloud base H and saturated with a stability of N22 < 0 above, with rigid flat boundaries at z = 0 and d. This two-layer structure is analogous to the flow around x ≈ 47 km in Fig. 7b, where lee waves intersect the unstable cap cloud aloft and transition into banded updrafts farther downstream. Like our enforcement of the triggering condition in the previous subsection, our saturating the environment above H implicitly accounts for h, which lifts the simulated flow to saturation over the mountain upslope. Moreover, as mentioned above, the dependent variable x is replaced by integration time (ttc) in this section. Our working hypothesis is that one can view the passage of time in the linear model as being equivalent to the distance that the flow travels downstream through xxc = Um × (ttc).

As demonstrated by the solution in appendix B, a number of atmospheric- and terrain-related parameters govern the strength of the banded updrafts. The lee-wave vertical structure [W(z)], which is controlled by Um, N1, d, and k, influences the projection of lee-wave perturbations onto the stable and unstable modes of the solutions. The structures of these modes and their growth or oscillation rates in turn depend on H, d, N1, N2, and κ. Considering the unstable modes only, note that the scales with the largest growth rates (aup) are not necessarily those that dominate the w field for finite t, because w in (B12) is affected by both aup and the strength of the initial projection (Au+p). If the perturbations had an infinite amount time to grow, the modes with the largest aup would eventually dominate the solution because of the exponential dependence of w on aup × (ttc). However, Au+p is also crucially important because saturated air parcels undergo growth for only a short time inside the unstable cap cloud before they desaturate either in a convective downdraft or on the lee side of the ridge. Over this limited time interval (tlin), the projection of the lee-wave perturbations onto a given unstable mode p may contribute as much or more to the w field as the subsequent in-cloud convective growth of the mode.

b. A model solution

The ability of the linear model to capture the transition from lee-wave oscillations to banded convection is evaluated through direct comparison with a simulation (PA5km-xr45, not included in Table 1) that is identical to the PA5km case, except that xr = 45 km rather than 60 km. For this comparison, we assign all of the linear-model parameters to be consistent with data from the simulation, beginning with λ = 5 km and hm = 100 m in (5), to yield a hlin of the same scale and amplitude as in (4). Based on Fig. 7 we estimate the cloud-top height to be d = 2 km and use H = 500 m as an average cloud height over 40 ≤ x ≤ 60 km. The wind speed Um = 10 m s−1 and the square of the dry Brunt–Väisälä frequency N21 = N2d = 10−4 s−2 are taken from the upstream sounding, while the moist Brunt–Väisälä frequency N22 = N2m = 2 × 10−5 s−2 is characteristic of the moist instability inside the cap cloud (see Fig. 8b of K07).

Various aspects of the unstable component of the model solution (wu) are presented in Fig. 10 for H = 500 m (Figs. 10a–c) and H = 100 m (Figs. 10d–f). Beginning with H = 500 m, the structures of the unstable eigenfunctions Wup(z) determined by (B11) for p = 1–3 (Fig. 10a) are characterized by sinusoids in the upper layer and exponential decay in the lower layer, with vertical scales that decrease as p increases. The projection coefficients Au+p of the lee-wave vertical structure W(z) (Fig. 9) onto these eigenfunctions Wup(z), which are computed in (B18) and displayed in Fig. 10b, indicate that the strongest lee-wave forcing occurs at small κ and decreases rapidly as κ is increased. In contrast, the growth rates of the unstable eigenfunctions in Fig. 10c all increase with κ. Moreover, the growth rates are largest at p = 1 (the gravest mode) and decrease as p is increased, implying that shallower perturbations grow more slowly inside the cloud than do deeper perturbations.

A similar pattern of unstable eigenfunctions as that in Fig. 10a is evident for H = 100 m (Fig. 10d), except that the functions now extend over a deeper cloud layer. The unstable projection coefficients Au+p (Fig. 10e) for H = 100 m, however, differ substantially from those for H = 500 m (Fig. 10b) in that the p = 1 curve is more uniform with κ, and for p > 1 the curves do not peak at the minimum κ, but at larger κ. This indicates that the maximum unstable projection at large κ occurs for higher-order modes with shorter vertical wavelengths. Little change is evident in the convective growth rates as H is lowered to 100 m (Fig. 10f).

In interpreting the following comparison between the linear model and numerical simulation, note that the linear-model terrain hlin(x, y) differs from the simulated terrain [h(x, y) = h(x) + (x, y)] because it lacks the mean component in (see 4) as well as the mesoscale component h. For simplicity, we neglect the mean term in our design of hlin in (5), which has no effect on ∂hlin/∂x and hence the forcing at the surface (A2), but causes hlin to be lower than everywhere. Thus, although the phases of hlin and are identical, the nodal x locations where (x, y) = 0 and hlin(x, y) = 0 are different. By design, the simulated terrain in (4) is continuous in x because (x, y) = 0 at xl = 40 km and xr = 45 km. In the linear model, however, we force xr = xc = 46.25 km in (5) to ensure that the terrain ends at the same location where convection is triggered. As will be seen, this slight downstream extension of the linear-model terrain results in stronger subcloud stable oscillations than in the simulation, but these oscillations have little impact on the bands because they have virtually no effect on the unstable convective growth above H. In addition, h(x) adds mesoscale variability to the simulated w field that is absent from the linear model. Thus, to directly compare simulated and linear-model data, we remove the contribution from h on the simulated w by subtracting the “mean” or y-averaged component of w [w(x, z)] from the full w, which isolates the small-scale response w′(x, y, z).

In Fig. 11 we compare w′ and the cloud outline (qc = 0.001 g kg−1) in the numerical simulation to that of the linear model in y–z cross sections at four different x locations. General agreement is apparent between the left (simulation) and right (linear model) panels at these locations, suggesting that the model is able to capture the evolution of perturbations as they oscillate over and transition to moist convection inside the cloud. Focusing on x = 43.75 km in Figs. 11a,b, the PA5km-xr45 simulation (Fig. 11a) exhibits a lee-wave pattern across the domain characterized by descent in the center and edges and ascent in between, along with an overhanging cap cloud above z ≈ 1.5 km. Because x = 43.75 km lies within the initialization section of the linear model, no cloud is present in Fig. 11b, but a similar pattern of lee waves as that in the simulation is evident across the width of the domain.

Farther downstream at the triggering location (xc = 46.25 km in Figs. 11c,d), the phase of the lee waves has reversed so that upward motion is present at the center and edges with subsidence in between. Clearly, the basic phase and amplitude of the lee waves in the linear model still agree with the simulation, but some details differ in the responses below the cloud base. Whereas w′ reaches a maximum at z ≈ 1 km in the simulation (Fig. 7c), its maximum in the linear model is at the surface (Fig. 7d). This discrepancy is a direct result of the aforementioned differences between the simulated and linear-model terrain at xxc. In the simulation, (x, y) = 0 past x = 45 km, which forces w′ to be zero at the surface and reach its maximum magnitude aloft. In the linear model, however, hlin(x, y) continues to oscillate until xr = 46.25 km, which maintains the surface-based forcing of lee waves until convective triggering occurs.

Comparing the growth section of the linear model with the simulation at x = 50 km (Figs. 11e,f) and x = 60 km (Figs. 11g,h), strong agreement is apparent between the locations of the banded updrafts and their vertical velocities. Some differences also exist, however, the first being that the subcloud oscillations in the linear model at x = 50 km (Fig. 11f) are much stronger than in the simulation (Fig. 11e). As in Figs. 11c,d, this discrepancy is likely tied to the slight differences between and hlin near x = xc, which cause stronger lee-wave forcing in the linear model and project this extra energy onto the stable, oscillating modes in the solution. Another contributing factor to this difference is our approximation of a horizontal cloud base in the linear model, which is more conducive to high-amplitude resonant oscillations below it than the uneven cloud base in the numerical simulation. We also note that the linear model (Figs. 11f,h) does not capture the apparent widening of the downdrafts and narrowing of the updrafts as the simulated flow progresses downstream (Figs. 11e,g). This difference is likely caused by nonlinearities introduced by desaturation in the simulated downdrafts that cannot be captured by the linear model.

5. Sensitivity tests

Having demonstrated that the linear model provides a reasonable estimation of the dynamics of banded convective updrafts in one test case, we now use the model in conjunction with numerical model simulations to evaluate the sensitivity of the bands to various physical parameters. Although, for the sake of brevity, we restrict our analysis to only a few atmospheric- and terrain-related parameters, this examination provides unique insight into the physics of the scale-selection process.

a. Forcing scale κ and cloud-base height H

1) Linear model

To investigate the sensitivity of the rainbands to the terrain forcing scale and the cloud-base height, we compute linear-model solutions for a range of λ (20 ≤ λ ≤ 0.6 km) and H (50 ≤ H ≤ 1000 m), with hm held constant at 100 m for all experiments. We focus on the convective response of the unstable cloud (which ultimately dominates the stable subcloud response) by limiting our examination to the unstable vertical velocity (wu) in (B9), which is given by
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The contour plots in Figs. 12a–c illustrate the dependence of the maximum wu over the y–z domain (wumax) on κ and H at three different output times: 1) t = tc, 2) ttc = 750 s, and 3) ttc = 1500 s. At t = tc (Fig. 12a), wumax is simply the projection of the lee waves at x = xc onto the unstable cloud. The strongest projections occur for small values of κ because their lee waves penetrate the deepest into the atmosphere and most effectively perturb the unstable cloud aloft (Fig. 9). For large κ, on the other hand, the evanescent waves forced by the surface decay rapidly with height and only faintly perturb the cloud at H. The strength of the projections increases as the cloud base is lowered because more of the lee-wave signal is fed into the cloud. In particular, the lee-wave projections at large κ increase because the strong near-surface amplitudes of the waves project more strongly onto the lowered cloud layer. This explains the increases in Au+p at large κ for H = 100 m (Fig. 10e) over that for H = 500 m (Fig. 10b).
As the convection evolves, a scale emerges that maximizes the combined contributions of the initial lee-wave projection and the subsequent unstable growth in (6). We interpret this scale (λ*lin) as the “preferred” rainband spacing in the linear model, which decreases over time because smaller-scale perturbations grow faster and eventually become stronger than the larger-scale perturbations. Because of the sensitivity of λ*lin to model integration time, we carefully select tlin using guidance from the numerical simulations. The qc outline of the MPA4 simulation (Fig. 13a) indicates that more than half of the cloud area is desaturated by x = 61 km, after which nonlinear effects may be expected to dominate the solution. This location (xlin) is an estimate of how long the simulated flow remains quasi linear, and corresponds to an integration time of
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Returning to Fig. 12a, the strongest initial forcing occurs at the largest scale allowed by the model λ*lin = 20 km for virtually all values of H. Once convection begins, the faster growth of smaller-scale perturbations causes λ*lin to decrease over time in Figs. 12b,c. This time-evolving shift to smaller scales is clearly displayed in cuts through Figs. 12a–c at H = 500 m and H = 100 m in Figs. 12d,e. At H = 500 m (Fig. 12d), λ*lin decreases from 20 km at t = tc to ∼5 km at ttc = 1500 s. Similarly, at H = 100 m (Fig. 12e), λ*lin decreases from 20 km at t = tc to ∼1.6 km at ttc = 1500 s.

The differences between λ*lin at H = 500 m and H = 100 m in Fig. 12 are explained by the strong dependence of W(z) on k in Fig. 9. The initial (ttc = 0) spectra of lee-wave projections in Figs. 12d,e peak at the largest forcing scales where the lee waves extend the deepest into the flow above. At small scales, the strong vertical attenuation of the lee-wave signal projects little to no energy onto the higher-based cloud at H = 500 m (Fig. 12d), and the ensuing convection is weak. However, the same small-scale lee waves are able to project far greater energy onto the cloud when it is lowered to H = 100 m (Fig. 12e). With comparable initial power at all scales, the faster in-cloud growth of smaller-scale perturbations at large κ (see Fig. 10f) allows them to quickly dominate the larger-scale perturbations.

2) Simulations

To evaluate the linear-model result that lower cloud bases favor smaller rainband spacings, we perform two sets of simulations (PAM20km–PAM1.25km and MPAM2–MPAM5) that are identical to the PA20km–PA1.25km and MPA2–MPA5 simulations of section 3, except for a different upstream sounding that contains more low-level moisture and consequently generates a cap cloud with a lower base. Only two changes are made to the sounding profile in Fig. 2a—the low-level RH is increased from 90% to 99.5% and, to maintain a similar degree of moist instability in the cloud, Ts is reduced from 285 to 283 K.

The modifications in the background environment caused by the above changes to the upstream sounding are shown in Fig. 14, which compares the cloud outlines and N2m fields at t = 1 h of simulations using these two soundings (RH90 and RH99) over a smooth mountain ( = 0). The cloud in the RH99 case (Fig. 14b) is clearly more expansive than that in the RH90 case (Fig. 14a), extending higher, lower, and farther upstream. Despite this difference, the unstable moist layers (dashed lines) have similar depths and stabilities, and differ mainly in the position of the cloud base over 40 ≤ x ≤ 60 km, which is as high as z = 2 km in the RH90 case but rests near the surface in the RH99 case. Thus, terrain-induced waves over 40 ≤ x ≤ 60 km perturb a completely saturated and unstable layer in the RH99 case rather than only a partially saturated and unstable layer in the RH90 case.

Horizontal cross sections of w for the PAM20km–PAM1.25km simulations in Fig. 15 reveal a different dependence on λ than those presented in Fig. 4. Whereas the responses for λ = 20 and 10 km are weaker than their RH = 90% equivalents, the responses for λ = 2.5 and 1.25 km are stronger. The relatively weak bands at large λ in the RH = 99% simulations are probably attributable to the stronger nonlinear effects of desaturation in the RH = 90% cases, which, by narrowing the updrafts, may increase the convective growth rates relative to the more symmetric updraft–downdraft pairs in the RH = 99% cases. The strengthened responses of the PAM2.5km–PAM1.25km simulations compared to the PA2.5km–PA1.25km simulations is attributable to the stronger lee-wave projections onto the lowered cap cloud, which allow these fast-growing modes to intensify into convective plumes more rapidly. In contrast with the weak bands in the PA2.5km–PA1.25km simulations (Figs. 4d,e), the PAM2.5km–PAM1.25km simulations (Figs. 15d,e) generate strong rainbands before desaturating in the lee of the ridge.

Turning to the MPAM2–MPAM5 simulations in Fig. 16, the addition of low-level moisture causes λ*sim to converge to half the value (2.5 km) of the MPA2–MPA5 cases (5 km). This convergence is seen in Figs. 16c,d because the addition of the λ = 1.25 km mode is not accompanied by any reduction in λ*sim. As in section 3, we evaluate the robustness of the MPAM2–MPAM5 simulations by performing four extra sets of simulations with different combinations of ϕy. The statistics of these simulations in Fig. 17 demonstrate that the single-mode cases lie on the λ*sim = λ*min line, which is consistent with the faster inviscid growth rates at smaller scales in (2). However, when the modes are superposed in Fig. 16, the dominant scale approaches λ = 2.5 km, which is not the smallest λ in (Fig. 16d). As in the RH = 90% simulations, this is caused by the decrease of the initial wumax with κ (Fig. 12e), which, though not as scale dependent as that in the H = 500 m case (Fig. 12d), still prevents the smallest-scale perturbations from dominating the solution over the short time period of model integration.

b. Terrain forcing spectrum

Another key parameter governing the spacing of orographic rainbands is the scale dependence of the small-scale terrain amplitudes, which thus far have been constant at 100 m for all horizontal wavenumbers (κ) and correspond to a power spectrum that, when represented in 1D, follows E(κ) ∼ κ1. As shown in Fig. 1, however, the real Coastal Range terrain is characterized not by E(κ) ∼ κ1 but by E(κ) ∼ κ−5/3. Accordingly, the linear-model forcing spectrum is made more realistic by maintaining hpa = 100 m at λ = 20 km but diminishing hpa at smaller scales to generate a power spectrum that exactly follows E(κ) ∼ κ−5/3. Note that, because the terrain forcing amplitude is reduced at nearly all horizontal scales, its overall power is much less than that of the κ1 spectrum. As before, we compute the linear-model solution to this new terrain spectrum and evaluate it through comparison with a set of numerical simulations.

1) Linear model

The rapid loss of the terrain-induced forcing amplitude as κ is increased causes the lee-wave projection coefficients (Au+p) to decrease much faster with κ than in the previous case. As a result, larger scales are provided an even greater “head start” over smaller scales than for a κ1 terrain spectrum, which, for a given value of tlin, leads to somewhat larger band spacings. This is seen by comparing the contour plot of wumax at ttc = 1500 s (Fig. 18b) to that of the Eκ1 case at the same time (Fig. 12c). For a cloud-base height of H = 500 m, convection past the κ1 terrain is the strongest at λ = 5 km, while the convection past the κ−5/3 terrain is maximized at the largest perturbation scale allowed (λ = 20 km).

Another consequence of the reduced lee-wave forcing in the κ−5/3 case is that the convective perturbations inside the cloud require more time to develop into strong downdrafts that are capable of desaturating the flow. Although the larger-scale perturbations have similar amplitudes as before, their in-cloud growth is slow (Figs. 10c,f). Smaller-scale perturbations grow faster, but their initial lee-wave projections are now so weak that they need extra time to grow into perturbations of the same strength as those in the κ1 case. As a result, the cloud remains largely saturated for a longer period than in the κ1 case. This is seen by comparing the qc fields of two simulations in Figs. 13a,b, the first of which is the MPA4 case described earlier, and the second (MPAD4, discussed more below) is identical except that its terrain spectrum follows E(κ) ∼ κ−5/3. Because of its weakened lee-wave forcing, the largely saturated flow in the κ−5/3 case (Fig. 13b) extends about 15 km farther downstream than in the MPA4 case (Fig. 13), which doubles tlintc to around 3000 s. Thus, although smaller-scale perturbations are initially weaker in the κ−5/3 case, they have more time to “catch up” with the initially dominant larger-scale perturbations.

When the Eκ−5/3 solution is integrated an additional 1500 s (Fig. 18c), the wumax contours still peak at larger scales than those of the corresponding κ1 simulations at t = tlin. This holds for H = 500 m (Fig. 18d), where λ*lin = 6 km is slightly larger than 5 km in Fig. 12d, and for H = 100 m (Fig. 18e), where λ*lin = 3.5 km is larger than 1.6 km in Fig. 12e. We also note that, although wumax peaks at λ*lin ≈ 6 km for H = 500 m (Fig. 18d), the curve is relatively flat for λ > λ*lin which implies that the rainbands are nearly as likely to form at scales larger than λ*lin than they are to form at λ*lin. This point is useful in interpreting the simulation results below.

2) Simulations

The linear-model results above suggest that the band spacing increases slightly when the terrain forcing spectrum is changed from E(κ) ∼ κ1 to κ−5/3. We evaluate this result through two sets of numerical simulations (MPAD2–MPAD5 and MPAMD2–MPAMD5) that are identical to the MPA2–MPA5 and MPAM2–MPAM5 simulations except that the power spectrum of follows E(κ) ∼ κ−5/3 rather than κ1.

Consistent with the linear-model solution for H = 500 m, the cross sections of the MPAD2–MPAD5 simulations in Fig. 19 indicate that the bands are spaced slightly farther apart than in the κ1 case (Fig. 5). Whereas the band spacings ranged from 5 to 7 km in Figs. 5b–d, they range from 5 to 10 km in Figs. 19b–d. This increase in both the mean and variability of the spacing may be traced to Fig. 18d, where wumax at t = tlin is nearly constant over 20 ≤ λ ≤ 5 km, suggesting that not just one but multiple scales in this range may influence the band spacing, and consequently the preferred scale of the bands is less clear. Also, as mentioned above, the bands in the κ−5/3 simulations form farther downstream (Fig. 19) than their κ1 counterparts (Fig. 5) as a result of their weakened lee-wave forcing.

Consistencies with the linear-model results for H = 100 m are also apparent as the low-level relative humidity is increased to 99.5% in the MPAMD2–MPAMD5 simulations. As expected, the spacing of the bands forced by the κ−5/3 terrain power spectra (λsim ≈ 3 km in Figs. 20c,d) is larger than that for the κ1 terrain in the MPAM2–MPAM5 simulations (λsim = 2.5 km in Figs. 16c,d). Another obvious difference between the convective responses to the two terrain spectra is seen in Figs. 16a and 20a, where the two bands in the MPAM2 simulation are replaced by disorganized convective cells in the MPAMD2 case. The weakened lee-wave forcing imposed by the κ−5/3 spectrum, combined with the relatively slow convective growth at larger scales, causes the terrain-induced convection to be so weak that it is overwhelmed by disorganized cells arising from random numerical round-off errors.

As with previous comparisons, we perform four additional sets of MPAD2–MPAD5 and MPAMD2–MPAMD5 simulations with different values of ϕy. The statistics of these simulations in Fig. 21 reinforce the linear-model trend for the bands to be spaced farther apart in the E(κ) ∼ κ−5/3 case than in the κ1 case. When RH = 90% at low levels, λsim fluctuates between 5 and 10 km as λmin is decreased rather than converging to a single value as before. We thus set λ*sim = 5–10 km to cover this range of scales, which is consistent with the linear-model band spacing λ*lin ≈ 6 km for H = 500 m (Fig. 18d). In addition, λ*sim converges to around 3 km for RH = 99.5%, which is consistent with λ*lin ≈ 3.5 km for H = 100 m (Fig. 18e). Finally, another consequence of the less clear scale selection in the simulations with E(κ) ∼ κ−5/3 is that, as seen by the longer whiskers in Fig. 21 compared to those in Fig. 5, there tends to be higher variability in the spacing of the bands for a given test case.

6. Conclusions

A combination of simulations performed with a fully nonlinear mesoscale model and solutions from a simple linear model are used to investigate and understand the spacings of quasi-stationary convective orographic rainbands triggered by small-scale topography. The simulations use a smoothed, 1D version of the Coastal Range terrain in western Oregon [h(x)] with simple 2D small-scale terrain features [(x, y)] superposed over the upslope, and an upstream sounding representative of that during a real banded convective precipitation event over the Coastal Range. As the flow passes over the upslope of the mountain ridge, it forms a statically unstable cap cloud that gives rise to convective bands when perturbed by small-scale lee waves forced by . In simulations with a single mode of terrain variability (λ), the band spacings scale with λ but are the strongest over a limited range of scales (5 ≤ λ ≤ 10 km). Additional simulations that randomly superpose the forcing modes of different scales indicate that, as more and more small-scale detail is added to the terrain, the band spacings converge to a “preferred” value λ*sim ≈ 5 km that is independent of the numerical resolution.

The simulation results appear to defy existing notions regarding the scale selection of orographic convection in that in-cloud diffusional effects are too weak to explain the simulated band spacings and that no dominant mode of terrain variability is required for evenly spaced bands to form over the mountain upslope. Our hypothesis for the band spacing is fundamentally different from previous work in that we attribute the scale selection of the bands to lee waves generated by stable flow over small-scale topographic perturbations lying upstream of and/or beneath the unstable cap cloud. This hypothesis is developed through a linear model that is specifically designed to capture the transition from stable lee-wave oscillations to moist in-cloud convection. While this model is far simpler than the fully nonlinear numerical simulations, it provides a reasonably accurate depiction of the dynamics that gave rise to the rainbands.

The inviscid linear model offers unique physical insight into the physical processes that control the scales of topographically triggered convective bands. Lee waves forced by stable flow over oscillate until undergoing saturation at the leading edge of the cap cloud. The projection of lee-wave energy onto this unstable cloud governs the initial strength of the convection and is maximized at large λ, where the lee waves penetrate the deepest into the atmosphere aloft. Although smaller values of λ have stronger surface-based forcing, their associated lee waves decay rapidly with height and project only weakly onto the cloud. As the perturbations pass through the unstable cap cloud, the faster growth at small scales shifts the scale of the strongest response (λ*lin) to smaller values over time.

A number of linear-model solutions are computed and evaluated through comparisons with numerical simulations to investigate the sensitivity of the band spacings to various atmospheric- and terrain-related parameters. In the first experiment, the cloud-base height in the linear model (H = 500 m) is chosen to represent the mean cloud-base height of the cap cloud in the numerical simulations, and the small-scale topographic forcing amplitude is held constant for all λ. The preferred band spacing computed by the linear model (λ*lin = 5 km) agrees with the simulated band spacing from the nonlinear numerical model. This spacing is not equal to the largest forcing scale (λ = 20 km), which projects the most lee-wave energy onto the cloud, or the smallest forcing scale (λ = 1.25 km), where perturbations grow the fastest. Rather, the bands are the strongest at an intermediate scale that maximizes the combined contributions of these two effects.

To investigate the influence of the cloud-base height H on the band spacings, we lowered H to 100 m in the linear model and added more low-level moisture to the upstream flows in the simulations to produce a similarly lowered cloud base. In both the linear model (λ*lin = 1.6 km) and the simulations (λ*sim = 2.5 km), the preferred band spacing decreases when the cloud base is lowered. This closer spacing is explained by the increased ability of lee waves forced by smaller-scale terrain features to perturb the orographic cloud when it was positioned closer to the ground. For H = 500 m, the projections of the small-scale lee waves onto the cloud are extremely weak, but when H is lowered to 100 m, these same waves are able to project substantial energy onto the cloud, which shifts the band spacings to smaller scales.

When the terrain forcing spectrum is changed from the simplest case of equal forcing at all scales [E(κ) ∼ κ1] to a κ−5/3 spectrum that is designed to mimic the terrain spectrum of the Coastal Range, the weakened small-scale forcing affords the larger scales an even greater head start in controlling the scales of the rainbands. However, the reduced terrain forcing power also weakens the initial convective perturbations inside the cloud, and more growth time is required for the convection to reach a similar strength as in the κ1 case. This causes the bands to form farther downstream and allows the faster-growing smaller-scale perturbations more time catch up with the larger-scale perturbations. The net result of these two offsetting effects is to increase the band spacing slightly from that in the κ1 case.

The strong agreement between the rainband spacings in the linear model and the numerical simulations suggest that the model accurately captures the fundamental processes controlling the scales of the bands. Moreover, the values produced by these experiments are consistent with those from observations over the Coastal Range. Kirshbaum and Durran (2005) found that the spacings between adjacent bands in precipitation events over the Coastal Range varied from 5 to 15 km, the lower half of which matches the linear model and simulated results in this paper.

In this study we have analyzed the sensitivity of the bands to just a few parameters of interest. Ongoing work has demonstrated that the interband spacings are also sensitive to the atmospheric stability in the moist and dry regions of the flow, as well as to the speed of the wind and strength of the terrain-induced forcing. Future work in this area will help to quantify these effects as well as to evaluate the performance of our analytical model in explaining the spacings of rainbands that form over other mountain ranges around the world.

Acknowledgments

The work performed by Daniel Kirshbaum is supported by the Advanced Study Program (ASP) at the National Center of Atmospheric Research in Boulder, Colorado. We also thank the two anonymous reviewers for their insightful comments and suggestions.

REFERENCES

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APPENDIX A

Perturbation Initialization

The steady lee-wave response to the linear-model terrain field hlin is computed by combining the linear, inviscid, steady-state Boussinesq equations of motion into a single partial differential equation for the vertical velocity w, which is given by (8.8.13) of Gill (1982),
i1520-0469-64-12-4222-ea1
where l2 is the square of the Scorer parameter, which for a uniform background wind speed Um is Nd/Um, where Nd is the basic-state Brunt–Väisälä frequency. Lateral boundaries are treated as periodic, while lower and upper boundary conditions are given by
i1520-0469-64-12-4222-ea2
i1520-0469-64-12-4222-ea3
where z = 0 and z = d are the bottom and top of the channel flow. The steady lee-wave response to the hlin specified in (5), which possesses a single mode of variability at wavenumber k, is found using separation of variables
i1520-0469-64-12-4222-ea4
where W is found by substituting (A4) into (A1) and invoking the boundary conditions in (A2),
i1520-0469-64-12-4222-ea5
where m2 = 2 (l2k2). Note that for l2 > k2, m is real and the solution is part of a sine wave, while for l2 < k2, m is imaginary and the solution is a decaying exponential. Waves produced by smaller scales (i.e., larger values of k) are thus associated with faster vertical decay, which is seen in the series of W plots for different k in Fig. 9.
Initial conditions for the time-dependent growth section of the model are determined by specifying a triggering location xc and matching xr to be the same value. The location of this point satisfies the condition of K07 that lee-wave triggering occurs when the wave undergoes saturation at a phase of its oscillations where w is strongly positive and b is zero. This combination, which allows the ascent at cloud entry to immediately generate positive b and convective motions within the unstable cloud, is expressed formally either as
i1520-0469-64-12-4222-ea6
i1520-0469-64-12-4222-ea7
or equivalently
i1520-0469-64-12-4222-ea8
i1520-0469-64-12-4222-ea9
Note that w0 as defined above is nonzero at the surface, and as such is inconsistent with the growth section boundary condition (see appendix B) that w0(y, z = 0, t) = 0. To enforce consistency between these two sections without altering the basic structure of W(z), we replace W(0 ≤ z < δ) with z/δ × W(δ). For δH the linear-model solutions are found to be highly insensitive to the choice of δ, so we arbitrarily select δ = 50 m for all of the linear-model experiments conducted in this study. Another ad hoc adjustment intended to smooth the transition between the initialization and growth sections is our replacement of the y wavenumber (k) in (A4) with the full horizontal wavenumber (κ = k2x + k2y = 2k) in (A8). This collapses the 3D perturbations in (A4) into a 2D field that varies only in y as demanded by the growth section in appendix B.

APPENDIX B

Perturbation Growth

We analyze the unstable evolution of the initial lee-wave perturbations in a moist, 2D, two-layer atmosphere with an unsaturated and stable lower layer of stability N21 > 0 and a saturated and unstable upper layer of stability N22 < 0. The governing equations (incompressible, Boussinesq, linearized) in this section are
i1520-0469-64-12-4222-eb1
i1520-0469-64-12-4222-eb2
i1520-0469-64-12-4222-eb3
i1520-0469-64-12-4222-eb4
where P is the ratio of the pressure p to a reference density ρ0. Boundary conditions are w = 0 at z = 0 and z = d, and initial conditions are (A8)(A9). Using standard techniques, we may combine (B1)(B4) to form a single partial differential equation for w [(8.4.11) of Gill 1982],
i1520-0469-64-12-4222-eb5
We look for solutions of the form
i1520-0469-64-12-4222-eb6
where the “m” subscript denotes one of the infinite number of modal solutions. Substituting (B6) into (B5), the governing partial differential equations reduce to ordinary differential equations of the form
i1520-0469-64-12-4222-eb7
which are subject to the boundary conditions
i1520-0469-64-12-4222-eb8
The equations formulated above are a regular Sturm–Liouville eigenvalue problem with eigenvectors Wm(z), eigenvalues ωm, and Dirichlet boundary conditions (e.g., Haberman 1998). Eigenvalues are found by invoking (B8) and applying two matching conditions at z = H to solve for ω2m. Values of ω2m range from N21 to N22, implying that both stable oscillating modes and unstable growing modes coexist in the solution. This is reflected by the general solution
i1520-0469-64-12-4222-eb9
where f2n = (ωsm)2(ω2m > 0), a2p = −(ωup)2(ω2m < 0), and the eigenfunctions are given by
i1520-0469-64-12-4222-eb10
i1520-0469-64-12-4222-eb11
In the preceding,
i1520-0469-64-12-4222-eqb1
and
i1520-0469-64-12-4222-eqb2
The stable (As±n) and unstable (Au±p) coefficients in (B9), which determine the initial weights of the eigenfunctions, are found from the initial conditions (A8)(A9) with xc replaced by tc. Substitution of (A9) into (B9) yields As+n = Asn and Au+p = Aup, which simplifies (B9) to
i1520-0469-64-12-4222-eb12
Invoking (A8) then yields
i1520-0469-64-12-4222-eb13
To solve (B13) for As+n and Au+p, we must also exploit the orthogonality of the eigenfunctions. For the Sturm–Loiuville system in (B7) and (B8), the orthogonality condition between two different eigenfunctions may be written (Haberman 1998)
i1520-0469-64-12-4222-eb14
where m and q are two arbitrary modal indices and mq. For mathematical convenience, an equivalent orthogonality expression is found by substituting (B7) into (B14) to give
i1520-0469-64-12-4222-eb15
This relation is applied to (B13) by multiplying both sides of (B13) by [−κ2Wsq + (Wsq)″] and integrating over the depth of the channel d
i1520-0469-64-12-4222-eb16
Substituting (B15) leads to
i1520-0469-64-12-4222-eb17
An identical procedure as that followed above, except that the multiplication of (B13) by [−κ2Wsq + (Wsq)″] leads to an analogous expression for Au+q,
i1520-0469-64-12-4222-eb18

Fig. 1.
Fig. 1.

One-dimensional representation of the 2D power spectrum of a section of the Coastal Range topography in western Oregon.

Citation: Journal of the Atmospheric Sciences 64, 12; 10.1175/2007JAS2335.1

Fig. 2.
Fig. 2.

Idealized flow conditions and terrain profile for a banded precipitation event over the Coastal Range. (a) Skew T diagram of the idealized upstream sounding profile during the 12–13 Nov 2002 event and (b) smoothed terrain of northwest Oregon (h).

Citation: Journal of the Atmospheric Sciences 64, 12; 10.1175/2007JAS2335.1

Fig. 3.
Fig. 3.

Analysis of simulations that use a strip of sinusoidal terrain perturbations centered at xb = 40 km. Rainwater mixing ratio (qr) at z = 1.5 km and t = 4 h of (a) COL20km, (b) COL10km, and (c) COL5km simulations, with contours drawn at 0.01 and 0.1 g kg−1. (d)–(f) Vertical velocities (w) at the same locations and times, with contours drawn at 0.5, 1, and 2 m s−1. Boxes in (d)–(f) indicate x-averaging areas for band counts, which are shown in the rightmost panels. For reference, solid contours of at x ≈ 40 km are shown in 25-m increments. Grayscale terrain shading is the same as in Fig. 2b.

Citation: Journal of the Atmospheric Sciences 64, 12; 10.1175/2007JAS2335.1

Fig. 4.
Fig. 4.

Same as in Figs. 3d–f, except for (a) PA20km, (b) PA10km, (c) PA5km, (d) PA2.5km, and (e) PA1.25km simulations.

Citation: Journal of the Atmospheric Sciences 64, 12; 10.1175/2007JAS2335.1

Fig. 5.
Fig. 5.

Same as in Fig. 4, except for (a) MPA2, (b) MPA3, (c) MPA4, (d) MPA5, and (e) RND simulations.

Citation: Journal of the Atmospheric Sciences 64, 12; 10.1175/2007JAS2335.1

Fig. 6.
Fig. 6.

Simulated band spacings (λsim) as a function of the smallest-scale mode in (λmin) for PA20km–PA1.25km (white circles) and MPA2–MPA5 (gray circles) simulations. See text for further details.

Citation: Journal of the Atmospheric Sciences 64, 12; 10.1175/2007JAS2335.1

Fig. 7.
Fig. 7.

Vertical cross sections of w and cloud water mixing ratio (qc) at t = 4 h for (a) PA10km (y = 9 km), (b) PA5km (y = 13 km), and (c) PA2.5km (y = 14 km) simulations. Vertical velocity (black lines) is solid for positive values and dashed for negative values, and is contoured at |w| = 0.1, 0.2, 0.5, 1, and 2 m s−1. Cloud water contours are shaded at values of qc = 0.001 and 1 g kg−1.

Citation: Journal of the Atmospheric Sciences 64, 12; 10.1175/2007JAS2335.1

Fig. 8.
Fig. 8.

Schematic diagram of the linear model. Solid and dashed lines in w0(y, z) and w(y, z, tlin) panels denote positive and negative values of w.

Citation: Journal of the Atmospheric Sciences 64, 12; 10.1175/2007JAS2335.1

Fig. 9.
Fig. 9.

Vertical structures of lee waves in a channel of depth d = 2 km forced by a range of topographic scales (λ).

Citation: Journal of the Atmospheric Sciences 64, 12; 10.1175/2007JAS2335.1

Fig. 10.
Fig. 10.

Three unstable modes (p = 1, 2, and 3) of the linear-model solutions for two different cloud-base heights (H). (a) Eigenfunctions wup(z), (b) lee-wave projection coefficients Au+p(κ), and (c) unstable growth rates anp(κ), for H = 500 m. (d)–(f) Same as (a)–(c), but for H = 100 m.

Citation: Journal of the Atmospheric Sciences 64, 12; 10.1175/2007JAS2335.1

Fig. 11.
Fig. 11.

Comparison of vertical velocity and cloud outline from the (a), (c), (e), (g) numerical and (b), (d), (f), (h) linear models at four different x locations. Vertical velocities are contoured at |w| = 0.01, 0.02, 0.05, 0.1, 0.2, 0.5, and 1 m s−1, with positive (solid lines) and negative (dashed lines) values. In the simulation, the qc value is 0.001 g kg−1.

Citation: Journal of the Atmospheric Sciences 64, 12; 10.1175/2007JAS2335.1

Fig. 12.
Fig. 12.

Dependence of linear-model unstable vertical velocities on terrain-induced forcing scale k and cloud-base height H. Gray shade contour plots of wumax at (a) ttc = 0, (b) ttc = 750 s, and (c) ttc = 1500 s. Cuts through (a)–(c) are shown at (d) H = 500 m and (e) H = 100 m.

Citation: Journal of the Atmospheric Sciences 64, 12; 10.1175/2007JAS2335.1

Fig. 13.
Fig. 13.

Using cloud outline at z = 1.5 km and t = 4 h to find tlin from the (a) MPA4 and (b) MPAD4 simulations. The location past which cloud-free regions are more extensive than cloudy areas is denoted as xlin, while the triggering location (determined earlier) is xc ≈ 46 km.

Citation: Journal of the Atmospheric Sciences 64, 12; 10.1175/2007JAS2335.1

Fig. 14.
Fig. 14.

Cloud outline and negative values of moist Brunt–Väisälä frequency (N2m) at t = 1 h in smooth-terrain simulations ( = 0) for (a) RH90 and (b) RH99 soundings. Cloud outline (qc = 0.001 g kg−1) is shaded and N2m is contoured by dashed lines, with labels multiplied by 10−5 s−2.

Citation: Journal of the Atmospheric Sciences 64, 12; 10.1175/2007JAS2335.1

Fig. 15.
Fig. 15.

Same as in Fig. 4, but for (a) PAM20km, (b) PAM10km, (c) PAM5km, (d) PAM2.5km, and (e) PAM1.25km soundings.

Citation: Journal of the Atmospheric Sciences 64, 12; 10.1175/2007JAS2335.1

Fig. 16.
Fig. 16.

Same as in Figs. 5a–d, but for (a) MPAM2, (b) MPAM3, (c) MPAM4, and (d) MPAM5 simulations.

Citation: Journal of the Atmospheric Sciences 64, 12; 10.1175/2007JAS2335.1

Fig. 17.
Fig. 17.

Simulated band spacings (λsim) as a function of the smallest-scale mode in (λmin) for PAM20km–PAM1.25km (white circles) and MPAM2–MPAM5 (black circles) simulations.

Citation: Journal of the Atmospheric Sciences 64, 12; 10.1175/2007JAS2335.1

Fig. 18.
Fig. 18.

Same as in Fig. 12, but for a power spectrum characterized by Eκ−5/3, wherein (a) ttc = 0, (b) ttc = 1500 s, and (c) ttc = 3000 s.

Citation: Journal of the Atmospheric Sciences 64, 12; 10.1175/2007JAS2335.1

Fig. 19.
Fig. 19.

Same as in Figs. 5a–d, but for (a) MPAD2, (b) MPAD3, (c) MPAD4, and (d) MPAD5 simulations.

Citation: Journal of the Atmospheric Sciences 64, 12; 10.1175/2007JAS2335.1

Fig. 20.
Fig. 20.

Same as in Fig. 16, but for (a) MPAMD2, (b) MPAMD3, (c) MPAMD4, and (d) MPAMD5 simulations.

Citation: Journal of the Atmospheric Sciences 64, 12; 10.1175/2007JAS2335.1

Fig. 21.
Fig. 21.

Simulated band spacings (λsim) as a function of λmin for the MPAD2–MPAD5 (gray circles) and MPAMD2–MPAMD5 (black circles) simulations.

Citation: Journal of the Atmospheric Sciences 64, 12; 10.1175/2007JAS2335.1

Table 1.

List of numerical simulations and their properties, categorized by the simulation name, perturbation type (strip or patch), scales of forcing contained in (km), slope of the small-scale terrain power spectrum E as a function of the horizontal wavenumber κ, basic-state RH, number of otherwise identical runs with randomized ϕy, and mean and std dev σ(λsim) (where applicable) of the band spacings in those runs (km).

Table 1.
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