a. The effect of a tropopause on windward ascent

We begin with a linear analysis of the vertical velocity field for a single Fourier component in the x–z plane,

for which the surface topography consists of an infinite series of sinusoidal ridges of the form

Using the Boussinesq approximation, the linear, inviscid, steady-state vertical velocity field is governed by the one-dimensional Helmholtz equation,

where l² is the so-called Scorer parameter, defined as

with U(z) representing the speed of the background flow and N(z) the Brunt–Väisälä frequency.

Here we focus on the case of a two-layer atmosphere, consisting of a troposphere with constant static stability N, capped by an infinitely deep stratosphere with constant N_s. For now we assume that U is constant throughout both layers. Applying the lower boundary condition,

along with the conventional radiation and matching conditions (e.g., Klemp and Lilly 1975), the solution to (3) within the troposphere is found to be

where

is the tropospheric vertical wavenumber, and the constants C^± are given by

In (8), H is the height of the tropopause, and

or the ratio of wavenumbers across the tropopause. Assuming k > 0, the terms exp(imz) and exp(−imz) in (6) represent upward- and downward-propagating waves with a vertical wavelength of

Because our primary interest is orographic precipitation, for now we focus on the vertical structure of w over the windward slope, where the bulk of condensation occurs. For the terrain profile described by (2), the steepest windward slope is located at x_u = 3π/2k, above which, from (1) and (6),

where A = Ukh₀ is the wave amplitude in the absence of a tropopause (i.e., ϵ = 1), and is a nondimensional tropopause height, defined as

The impact of the tropopause on windward ascent is represented by the second term inside the parentheses in (11). In the no-tropopause limit ϵ → 1, w(x_u, z) behaves like cos(mz), with ascent at the surface giving way to descent above a quarter vertical wavelength (z = λ_z/4). However, when ϵ ≠ 1, the solution includes an additional sin mz term, the sign of which is determined by the value of . At low levels [0 < z(λ_z/4)] important for orographic precipitation, sinmz and cosmz have the same sign. Therefore, ascent is enhanced in this layer if , or equivalently, if , where n is a positive integer. From (12), this condition is met when the tropopause height is a bit less than an integer multiple of a half vertical wavelength. On the other hand, low-level ascent is diminished when —that is, when the tropopause is a bit higher than an integer multiple of a half vertical wavelength.

b. Physical interpretation

To develop some intuition for the physics behind (11), it is instructive to think of the troposphere as a kind of Fabry–Pérot interferometer, with w(x_u, z) representing the interference pattern from an infinite series of waves trapped between the surface (which acts as a total reflector) and the tropopause (which acts as a partial reflector). At the tropopause, reflection is governed by the physical requirement that and match across the interface¹ (Eliassen and Palm 1961). For an upward-propagating wave e^imz, these matching conditions take the form

where r and t are coefficients representing the amplitudes of the reflected and transmitted waves. Solving these equations for r yields

where

This implies that any incident wave at the tropopause is related to its reflection by

Imposing the same conditions for surface reflections (where t = 0) gives r = −1, implying that incident and reflected waves are governed by

Applying (16) and (17) iteratively to an initial terrain-induced wave of the form A exp(imz), the resulting interference pattern is found to be

which is equivalent to (11), but in the form of an infinite geometric series of upward- and downward-propagating waves.

To understand the consequences of (18), let us consider two specific cases in detail: , which represents the midpoint of an interval over which low-level ascent is enhanced, and , which represents the midpoint of an interval over which low-level ascent is diminished. The contributions to w(x_u, z) from the first several reflected waves are shown for both cases in Fig. 1.

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Contributions to w(x_u, z) from upward- (blue) and downward- (red) propagating waves given (a) and (b) . Within the troposphere, partial reflection by the tropopause and total reflection by the surface result in an infinite series of waves, of which only those with amplitude or greater are shown. The order of waves in the series is indicated by the circled numbers, beginning with the terrain-induced wave, A cos(mz). Waves 2 and 3 reinforce each other, enhancing low-level ascent in (a) and diminishing it in (b). Waves 4 and 5 offset each other, and their net contribution to w(x_u, z) is therefore zero. Stratospheric waves result from the partial transmission of upward-propagating tropospheric waves across the tropopause.

Citation: Journal of the Atmospheric Sciences 72, 2; 10.1175/JAS-D-14-0200.1

The case is shown in Fig. 1a. The wave with the largest amplitude, A cosmz, represents the terrain-induced wave (i.e., wave 1). According to (16), the partial reflection of this wave at the tropopause results in a downward-propagating wave, r₀A sin(mz), that enhances low-level ascent (wave 2). Wave 2 is in turn reflected by the surface, producing another wave equal to r₀A sin(mz) that enhances ascent even further (wave 3). The partial reflection of wave 3 produces a downward-propagating wave equal to (wave 4), but this wave is offset by its reflection (wave 5), which is equal in amplitude but 180° out of phase from wave 4.

Continuing this line of reasoning, a pattern of alternating sines and cosines emerges, whereby each downward-propagating wave of the form cos(mz) is canceled by its reflection, while each downward-propagating wave of the form sin(mz) is reinforced by its reflection. The coefficients of the sin(mz) terms form a geometric series,

which sum to give

Combining (15) and (20) yields

which is the same result given by (11).

Repeating this exercise for the case (Fig. 1b), we find that each reflected wave of the form sin mz is 180° out of phase with its counterpart in the case. Thus, whereas the initial reflected wave is equal to r₀A sin(mz) in the previous case, here its sign is reversed, implying a reduction in low-level ascent. Following the same procedure as in the previous case, we find that

In the examples above, the impact of on windward ascent has a straightforward physical interpretation, since in both cases the initial reflected wave takes the form ±sinmz. However, the general case is not fundamentally different. For an arbitrary tropopause height, the initial reflected wave is given by (16) as , which can be decomposed into two (real) parts: and . The first of these terms is proportional to cosmz, and is therefore canceled by its reflection from the surface. However, the second term, which is proportional to sinmz, enhances low-level ascent when , and weakens it when , just as in (11). Thus, even when the initial reflected wave is not of the form sin(mz), it contains a sin(mz) component that alters low-level ascent just the same.

It is therefore clear that whether w(x_u, z) is enhanced or diminished near the surface depends only on how the initial terrain-induced wave, A cos(mz), is reflected by the tropopause. When the tropopause height is a bit less than an integer multiple of a half vertical wavelength, as in the case, the reflected wave contains a +sinmz component, which acts to enhance low-level ascent. On the other hand, when the tropopause height is a bit greater than an integer multiple of a half vertical wavelength, as in the case, the reflected wave contains a −sinmz component, which acts to diminish low-level ascent.

c. The effect of a tropopause on precipitation

Thus far we have used linear theory to investigate how the presence of a tropopause affects windward ascent for a single Fourier component. Motivating this analysis was an implicit assumption that the depth and magnitude of windward ascent are fundamentally related to the amount of orographic precipitation. Here we explore this connection more tangibly, using a few simple assumptions to approximate the total precipitation rate from saturated flow over an idealized two-dimensional ridge. The goal of this exercise is to develop a qualitative understanding of the ways in which changes in wind speed, stability, and tropopause height affect orographic precipitation, which will prove helpful for interpreting the numerical results presented in the next section.

To incorporate precipitation within a linear framework, it is necessary to account for the effects of latent heating from condensation. One simple way of doing this (e.g., Rotunno and Ferretti 2001; SB04) is to replace N with an “effective static stability,” N′, which we assume to be constant within the troposphere. Since in reality the distribution of latent heating is not uniform within the troposphere, it is perhaps best to think of N′ as a free parameter, tuned to achieve the best agreement between linear theory and observations/numerical simulations. Nevertheless, previous studies provide some indication of the factors that determine N′. If the troposphere were saturated everywhere, we might expect N′ to be close to the troposphere-mean moist Brunt–Väisälä frequency,

with N_m given by Lalas and Einaudi (1973) as

where g is the acceleration of gravity, T(z) is temperature, Γ_m(z) is the moist adiabatic lapse rate, L is the latent heat of vaporization, q(z) is the mixing ratio of water vapor, and R is the specific gas constant for dry air. In reality, descending regions in the lee are often subsaturated, resulting in an abrupt increase in stability from N_m to N as a parcel passes over the ridge (e.g., Barcilon and Fitzjarrald 1985). However, idealized simulations performed by Jiang (2003) suggest that subsaturated regions may not have much effect on N′, implying that may still hold. In the next section, we present numerical results that are broadly consistent with the notion that , except when the lower troposphere approaches convective instability (Fig. 7b).

We begin our precipitation analysis with linear theory as before, assuming a two-layer atmosphere with constant U and a saturated troposphere, flowing over an isolated “witch-of-Agnesi” ridge of the form

where h₀ is the ridge height and a is the half-width. In this case, w(x, z) can be found within the troposphere by replacing h₀ in (6) with the Fourier transform of (25),

and taking the inverse transform, given by

In general, a closed-form expression for w(x, z) does not exist owing to the dependence of m on the horizontal wavenumber k [see (7)]. However, if the ridge is broad enough that N′a/U ≫ 1, the dominant wavenumbers in (27) satisfy the hydrostatic condition, k² ≪ N′²/U², and the solution within the troposphere can be approximated as

where

is the hydrostatic vertical wavenumber,

is the hydrostatic nondimensional tropopause height, and²

As proof of the relevance of our earlier analysis to the present isolated ridge, note that at the point x = −a, which lies one-half width upwind of the crest, (28) simplifies to

which matches the expression for w(x_u, z) for the single Fourier component (11) to within a constant factor.

To estimate the precipitation rate under saturated conditions, let us assume, following SB04, that the condensation rate S can be approximated as

where H_w is the moisture-scale height and S₀ is the condensation per unit vertical displacement at the surface. Let us further assume—very crudely—that total precipitation at the surface is equal to the vertical integral of condensation upstream of the crest:

In nature, moisture tends to be concentrated in the lower troposphere, implying that

As a result, the stratospheric component of w can be neglected in (33). Combining (28), (33), and (34), we arrive at the following estimate for precipitation:

where is a nondimensional moisture-scale height, which determines the extent to which condensation depends on mountain-wave dynamics.³

To understand the full implications of (36), it is helpful first to consider the limit ϵ → 1 in which the tropopause vanishes. In this case, the quantity within parentheses approaches unity, and (36) simplifies to

Since we are specifically interested in the relationship between mountain-wave dynamics and precipitation, we assume that the thermodynamic quantities S₀ and H_w are constant and focus instead on two ways in which U and N′ influence (37).

First, U and N′ together determine the vertical wavelength λ_z, which is inversely proportional to in the denominator of (37). An increase in U, or a decrease in N′, will cause to decrease, thereby enhancing precipitation. Physically, this change in P_est results from an increase in the depth of windward ascent, which in the absence of a tropopause is proportional to λ_z.

Second, in addition to influencing the depth of ascent via λ_z, U also controls the magnitude of ascent via the lower boundary condition in (5), as indicated in the numerator of (37). In a one-layer atmosphere, these two effects reinforce each other: an increase in U enhances the magnitude as well as the depth of windward ascent, while a decrease in U does the opposite. On the other hand, a change in N′ only alters the depth of ascent, and will therefore have a smaller impact on P_est than a change in U of the same magnitude.

Returning to the more general expression for precipitation in (36), we find that, like low-level windward ascent, P_est is diminished by wave reflection at the tropopause when and is enhanced when . Furthermore, because depends on U and N′, the actual relationship between U, N′, and P_est may be quite different than expected from the one-layer approximation in (37). The influence of these factors is illustrated in Fig. 2, where each curve plots P_est as a function of , but with different variables (H, N′, or U) serving as the independent variable in (36). To facilitate comparison, each curve has been normalized to one at . In dimensional terms, it may be helpful to think of as representing H = 12 km, U = 15 m s⁻¹, and N′ = 0.008 s⁻¹. The black lines show the dependence of P_est on H assuming fixed U and N′. Since H is proportional to , the x axis spans values of H ranging from 6 km at to 18 km at . For the red lines, U and H are fixed while N′ ranges from 0.004 to 0.012 s⁻¹. For the blue lines, N′ and H are fixed while U ranges from 30 m s⁻¹ at to 10 m s⁻¹ at , reflecting the inverse relationship between U and . Dashed lines represent the one-layer solution [see (37)] and serve both to highlight the effect of the tropopause and to illustrate the general slope of each curve.

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The quantity P_est as a function of , but with three different variables (H, N′, or U) acting as the independent variable in (36). To facilitate comparison, each curve has been normalized to 1 at = 1. Black curves: variable H, fixed U and N′. Red curves: variable N′, fixed U and H. Blue curves: variable U, fixed N′ and H. Solid lines represent a two-layer atmosphere with ϵ = 0.5, while dashed lines represent the limit ϵ → 1 [see (37)]. All curves use = 1 for the nondimensional moisture-scale height.

Citation: Journal of the Atmospheric Sciences 72, 2; 10.1175/JAS-D-14-0200.1

When H is the independent variable (black curve), P_est is periodic with zero overall slope, indicating that changes in the height of the tropopause only affect the phase of the reflected mountain waves, not their vertical wavelength. In contrast, when N′ is the independent variable (red curve), P_est exhibits a negative slope on top of periodic fluctuations as N′ increases from left to right. This behavior is due to the fact that N′ influences both the wavelength and the phase of the reflected waves. The negative slope is even greater in the case where U is the independent variable (blue curve), as a result of weaker ascent at the lower boundary as U decreases from left to right. Yet even with the blue curve, the general tendency for P_est to vary inversely with is sometimes overcome as the effects of wave reflection at the tropopause generate local regions with positive slope. Such instances of positive slope suggest that decreases in U (blue curve) or increases in N′ (red curve) can sometimes increase orographic precipitation. This surprising result will be explored further in the next section.

The direct impact of the tropopause on orographic precipitation is evident in the difference between the dashed and solid lines in Fig. 2. Over the range of typical atmospheric conditions the one- and two-layer approximations can differ by more than a factor of 2, suggesting that wave reflection by the tropopause should be considered among the most important factors controlling orographic precipitation. Still, it is important to keep the preceding analysis in perspective. Though concise, (36) is without question a very crude approximation of orographic precipitation in nature. In addition to linear dynamics, (36) is based on extremely simple thermodynamics [see (33)] and ignores cloud microphysics altogether. To better estimate the impact of the tropopause on orographic precipitation, we must resort to more sophisticated numerical simulations.

a. 100-m ridge, dry troposphere

We begin with simulations of dry flow over a low two-dimensional witch-of-Agnesi ridge with dimensions h₀ = 100 m and a = 25 km. To isolate the impact of wave reflection at the tropopause, we hold N and U constant while varying H. For the upstream sounding, we set N = 0.01 s⁻¹, N_s = 0.02 s⁻¹, and U = 15.92 m s⁻¹, which implies a hydrostatic vertical wavelength in the troposphere of λ_z = 10 km. Simulations are performed with tropopause heights of 8.5, 9.5, 10.5, and 11.5 km, corresponding to nondimensional tropopause heights of 0.85, 0.95, 1.05, and 1.15.

Figure 3 shows the vertical velocity fields for each simulation (right column), along with equivalent fields predicted by linear theory [see (28)] (left column). To compare the linear and numerical solutions directly, it is necessary to account for our use of the Boussinesq approximation in deriving the linear solution. To do so, we have scaled the fields from the numerical model by a factor of (ρ/ρ₀)^1/2, where ρ₀ is the air density at the surface and ρ(z) is the density of the background flow as a function of height. Although this scaling is only exact as a transformation between Boussinesq and isothermal atmospheres, it is sufficiently accurate for our purposes, as demonstrated by the similarity in the magnitude of vertical velocities within the linear and numerical solutions in Fig. 3.

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Vertical velocities (m s⁻¹) over a 100-m-high witch-of-Agnesi ridge in a two-layer atmosphere, (left) derived from linear theory [see (28)] and (right) a numerical model. Red (blue) colors indicate regions of ascent (descent), with thick black lines representing the zero contour. Green lines represent the tropopause. Solutions are shown for tropopause heights: (top to bottom) H = 8.5, 9.5, 10.5, and 11.5 km and = 0.85, 0.95, 1.05, and 1.15.

Citation: Journal of the Atmospheric Sciences 72, 2; 10.1175/JAS-D-14-0200.1

Significant changes to the flow field occur as varies between 0.85 and 1.15. Upstream of the ridge crest (i.e., x < 0), low-level ascent is deeper and more vigorous when (i.e., when H = 8.5 and 9.5 km) than when (i.e., H = 10.5 and 11.5 km), which is consistent with our analysis of windward ascent discussed in the previous section (Fig. 1). Other differences are also apparent, most notably in the magnitude and location of maximum leeside descent. Such differences will turn out to be important in later simulations, when we compare the distributions of precipitation across the ridge.

Overall, the numerical model does a good job of capturing variability in ascent associated with changes in tropopause height. The strong agreement between the linear and numerical fields in Fig. 3 demonstrates both the ability of the model to simulate wave reflection at the tropopause and the diversity of flow-field patterns that can result from it.

b. 1-km ridge, saturated troposphere

Our next series of simulations involves a two-dimensional ridge of the same 25-km half-width as before, but with a height of 1 km capable of generating significant precipitation. We consider separately three scenarios, in which one of the variables H, N, or U is varied while the other two variables are held fixed. All simulations were performed using a surface temperature of 5°C and relative humidities of 100% in the troposphere and 20% in the stratosphere.

1) The response of precipitation to changing tropopause height

Let us first consider how changes in H affect precipitation while U and N are held constant. Thirteen simulations were performed with tropopause heights between 7.5 and 13.5 km, each with upstream soundings of U = 15 m s⁻¹, N = 0.012 s⁻¹, and N_s = 0.02 s⁻¹. From (24), we find that these conditions imply a moist stability, N_m, ranging from about 0.005 s⁻¹ at the surface to 0.012 s⁻¹ above 10 km.

The blue dots in Fig. 4 show the cross-mountain-integrated precipitation rate for each simulation. The shape of the dots is approximately periodic, as predicted by linear theory [see (36)]. The effective stability, N′, can be estimated from the phase of the dots, which suggests that near H = 10.5 km, with more (less) precipitation for tropopause heights below (above) that level. Since U = 15 m s⁻¹, this implies an effective static stability of N′ ≈ 0.009 s⁻¹, which is very close to the average of across all simulations (0.0091 s⁻¹). Significantly, this result appears to validate our earlier prediction—based on Jiang (2003)—that when there are widespread regions of saturation and N_m is significantly greater than zero. Overall, precipitation varies by more than a factor of two across the simulations (53–110 m² h⁻¹), with the greatest precipitation occurring when H = 9.5 km and the least when H = 11.5 km.

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Cross-ridge-integrated precipitation rates produced by two-dimensional saturated flow over a 1-km-high ridge as a function of tropopause height H when ϵ = 0.6. Results are from nonlinear numerical simulations (blue dots) and the linear estimate [see (36)] with H_w = 1.5 (red line) or 2.5 km (green line). For reference, the precipitation in the one-layer limit, ϵ = 1, (37) is also shown (dotted black line).

Citation: Journal of the Atmospheric Sciences 72, 2; 10.1175/JAS-D-14-0200.1

How do these results compare to the linear approximation [see (36)] derived in the previous section? Given S₀ = 1.9 × 10⁻³ g m⁻⁴ (see appendix), an effective stability of N′ = 0.009 s⁻¹, and a moisture-scale height of H_w = 1.5 km, the linear approximation agrees well with the numerical simulations (red line). However, for the conditions simulated (T_s = 5°C, N = 0.012 s⁻¹), the actual e-folding height of water vapor is around 2.5 km (see appendix). Using this more realistic value for H_w, (36) significantly overpredicts the sensitivity of precipitation to changes in tropopause height (green line).

There are at least two possible reasons why using a lower value of H_w in (36) produces better agreement with the numerical simulations, both of which are evident in Fig. 5, which shows the linear and numerical vertical velocity fields for the cases of maximum and minimum precipitation (H = 9.5 and 11.5 km). First, (36) is based on the assumption that precipitation is equal to upstream condensation, which neglects the microphysical processes and time scales involved in converting condensation into precipitation. In reality, some of the condensation that forms aloft evaporates in the lee before reaching the surface, especially when windward ascent is deepest (e.g., when H = 9.5 km; top-left panel). Second, (33) [from which (36) is derived] is indifferent to the sign of w, treating regions of descent as negative sources of condensation (i.e., evaporation). Therefore, when windward ascent is shallow and topped by a layer of vigorous descent (as in the 11.5-km case; bottom-left panel), the linear column-integrated condensation is negative over portions of the windward slope near the crest. In such cases, (36) gives too much weight to evaporation aloft at the expense of condensation near the surface. Both of these problems are mitigated by using a lower moisture-scale height, likely explaining the improved agreement between the linear and numerical results.

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As in Fig. 3, except for a 1-km-high ridge; and solutions are shown for the two numerical cases: (top) H = 9.5 and (bottom) 11.5 km, exhibiting the most and least precipitation, respectively. Units are m s⁻¹.

Citation: Journal of the Atmospheric Sciences 72, 2; 10.1175/JAS-D-14-0200.1

To better understand the connection between vertical velocity and precipitation within the numerical simulations, it is useful to compare the H = 9.5- and 11.5-km cases in greater detail. Figures 6a and 6b show the concentration of cloud water (blue lines) and rainwater (shading), in addition to streamlines of parcels originating 100 km upstream of the ridge crest (red lines). Focusing first on the streamlines, we find that for a parcel that begins at 1 km altitude, the differences between the two cases are modest, with only about 150 m more ascent in the 9.5-km case. However, for a parcel originating at 3 or 4 km, where the flow is less influenced by the free-slip lower boundary condition, the difference in maximum ascent between the two cases is much larger (~400 m). In the 11.5-km case, in fact, the 4-km streamline actually descends over the windward slope, falling more than 150 m by the time it crosses the ridge crest. This explains why the cloud layer is both deeper and thicker in the 9.5-km case, resulting in more than twice as much precipitation.

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Cloud water mixing ratio (blue contours, intervals of 0.2 g kg⁻¹), rainwater mixing ratio (blue shading, g kg⁻¹), and parcel streamlines (red lines) from numerical simulations in which H is (a) 9.5 and (b) 11.5 km. Dashed red lines show the streamlines from the other simulation. (c) Precipitation rates as a function of the cross-ridge coordinate for both cases (mm h⁻¹).

Citation: Journal of the Atmospheric Sciences 72, 2; 10.1175/JAS-D-14-0200.1

A further contrast between the two cases is evident downwind of the crest, where each streamline in the 9.5-km case descends lower than its counterpart in the 11.5-km case. This behavior is caused by much stronger leeside descent in the 9.5-km case (Fig. 5), and it has important consequences for leeside evaporation: in the 9.5-km case, no cloud water is present beyond 15 km downwind of the crest, while in the 11.5-km case, cloud water persists more than 30 km downwind of the crest owing to weaker descent and lower evaporation rates.

These differences in leeside descent–evaporation also affect precipitation, as shown in Fig. 6c. While the 9.5-km case exhibits greater precipitation overall, the ratio of leeward-to-windward precipitation is significantly lower in the 9.5-km case than in the 11.5-km case (0.35 versus 0.51), indicative of a stronger orographic rain shadow. In section 4, we discuss the possible implications of this result for rain-shadow variability in realistic terrain.

2) The response of precipitation to changing wind speed and static stability

In addition to varying H, further simulations were performed varying U (from 10 to 20 m s⁻¹, with N = 0.012 s⁻¹) and N (from 0.009 to 0.015 s⁻¹, with U = 15 m s⁻¹). All other parameters where held constant, including N_s = 0.02 s⁻¹, H = 10.5 km, and the surface temperature at 5°C.

Figure 7a shows the cross-mountain-integrated precipitation rate from the variable-U simulations (blue dots), alongside the linear approximation [see (36)] (red line), calculated using S₀ = 1.9 × 10⁻³ g m⁻⁴, H_w = 1.5 km, and s⁻¹. Overall, the linear approximation does a good job of capturing both the amplitude and phase of precipitation variability—further evidence that is accurate under these conditions. While faster wind speeds generally result in greater precipitation, this tendency is reversed over certain intervals of U (between 11 and 13 and 18 and 20 m s⁻¹), just as predicted in the linear approximation. That such behavior would be impossible in a one-layer atmosphere (dotted line) demonstrates the significant impact the tropopause can have on orographic precipitation.

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Cross-ridge-integrated precipitation rates produced by saturated flow over a 1-km-high ridge as a function of (a) wind speed and (b) static stability, from numerical simulations (blue dots) and (36) (red lines), calculated assuming .

Citation: Journal of the Atmospheric Sciences 72, 2; 10.1175/JAS-D-14-0200.1

Calculating P_est is trickier when N is the independent variable (Fig. 7b), because N′ can no longer be identified from the phase of the numerical results (blue dots). If we assume, based on previous results, that , the linear approximation (red line) performs reasonably well when N ≥ 0.012 s⁻¹ but rather poorly when N < 0.012 s⁻¹, suggesting that linear theory breaks down as N decreases and the lower troposphere approaches convective instability. Overall, the influence of the tropopause is clear in both the linear and numerical results, which show precipitation increasing as N is raised from 0.013 and 0.015 s⁻¹—a result that would not be expected in the absence of a tropopause (dotted line).

Together, Figs. 4 and 7 demonstrate two things. First, despite its simplicity, (36) is nevertheless a useful tool for understanding the connection between windward ascent and orographic precipitation. Second, within a nonlinear, nonhydrostatic numerical model, the amount and distribution of orographic precipitation is significantly affected by wave reflection at the tropopause, just as one would expect based on linear theory.

c. 3D ridge, linear shear

Our third set of simulations incorporates two changes intended to better approximate nature: three-dimensional terrain and linear wind shear. We focus on a simple forward-shear scenario, in which U increases linearly with height in the troposphere and is constant in the stratosphere. The linear flow field under these conditions has been discussed elsewhere (e.g., Klemp and Lilly 1975) and will not be repeated here. For our purposes, it is sufficient to note that the presence of wind shear does not change the essential result from section 2—that low-level windward ascent will either be enhanced or diminished depending on the phase of the mountain waves reflected by the tropopause. Using the same notation as before, and following the same procedure described by Klemp and Lilly (1975), it can be shown that for a tropospheric wind profile given by

the hydrostatic nondimensional tropopause height takes the form

While this equation is considerably more complicated than its no-shear equivalent [see (30)], plays essentially the same role in both cases, indicating diminished low-level windward ascent when and enhanced ascent when . Therefore, whether U is constant or sheared, we expect precipitation to be similarly sensitive to variations in tropopause height.

To test this hypothesis, we have performed a series of 3D numerical simulations with tropopause heights between 7.5 and 13.5 km, but with constant upstream soundings of U₀ = 15 m s⁻¹, α = 1.67 × 10⁻⁴ m⁻¹, N = 0.012 s⁻¹, 100% tropospheric relative humidity, and a surface temperature of 5°C. From (38), these parameters imply that U increases 2.5 m s⁻¹ per kilometer of altitude within the troposphere. In the stratosphere, N_s = 0.02 s⁻¹ and U = U₀(1 + αH). The terrain consists of a finite ridge given by

where h₀ = 1 km and a = 25 km are the height and half-width of the ridge as before, while 2l = 300 km is the length of the ridge in the y dimension. At the upstream boundary, the direction of the background flow is assumed to be perpendicular to the ridge axis at all levels. Simulations were performed on the f plane using Coriolis parameters of 0 and 10⁻⁴ s⁻¹. Since the Coriolis term did not change the results substantially, here we discuss only the f = 0 case.

As expected, precipitation within these simulations is found to vary significantly with H, just as it did in previous 2D simulations with constant U. Overall, the greatest precipitation occurs when H = 8.5 km and the least when H = 13 km. Figure 8 shows the precipitation rate in each of these simulations on top of topographic contours (black lines). Along the bisect shown in green, the integrated cross-ridge precipitation rate differs by a factor of 1.75 between the two simulations (80.2 vs 45.9 m² h⁻¹).

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Precipitation rate (shaded contours, mm h⁻¹) and terrain height (black contours, 200-m intervals) for the 3D simulations with (a) H = 8.5 and (b) H = 13 km. The flow is across the ridge from left to right. Along the green line bisecting the ridge, the cross-ridge-integrated precipitation rate is (a) 80.2 and (b) 45.9 m² h⁻¹.

Citation: Journal of the Atmospheric Sciences 72, 2; 10.1175/JAS-D-14-0200.1

The reason for this difference in precipitation is evident in Figs. 9a,b, which show the concentrations of cloud water (blue lines), rainwater (blue shading), and parcel streamlines (red lines) in the vertical plane intersecting the green line in Fig. 8. When H = 8.5 km, parcels ascend higher over the windward slope (red lines), leading to greater concentrations of liquid water and ultimately precipitation (Fig. 9c), just as we saw in the 2D simulations (Fig. 6). Leeside descent also differs between the two simulations, though rain-shadow strength is less affected because the contrast in descent is greatest beyond 25 km downwind of the crest, where there is little liquid water to evaporate.

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Cloud water mixing ratio (blue contours, intervals of 0.2 g kg⁻¹), rainwater mixing ratio (blue shading, g kg⁻¹), and parcel streamlines (red lines) along the bisecting plane of the ridge (green line, Fig. 8) in the 3D numerical simulations exhibiting the least and most precipitation: (a) H = 8.5 and (b) 13 km. Dashed red lines indicate streamlines from the other simulation. (c) Precipitation rates for the two cases (mm h⁻¹).

Citation: Journal of the Atmospheric Sciences 72, 2; 10.1175/JAS-D-14-0200.1

In summary, whether in two or three dimensions, uniform or sheared flow, the simulations presented in this section have demonstrated that the tropopause has a first-order impact on the amount and distribution of orographic precipitation. In the next section, we consider the possible implications of this result for precipitation variability in realistic terrain.

a. Linear model description

The SB04 model utilizes the same approximation for column-integrated condensation [see (33)] that we used to approximate precipitation over a 2D ridge [see (36)]. However, rather than integrating in space to calculate the total precipitation [see (34)], the SB04 model estimates the spatial distribution of precipitation using the following differential equations governing the column-integrated densities of cloud water q_c and rainwater q_r:

Here, U is the two-dimensional wind vector and S(x, y) is the two-dimensional, column-integrated form of the condensation rate defined in (33). The constants τ_c and τ_f represent characteristic time scales for cloud water conversion and rainwater fallout and are, therefore, a simple way to incorporate cloud microphysics within a linear framework. In the second equation, the sink term q_r/τ_f is equivalent to the precipitation rate at the surface. By Fourier transforming these equations in the horizontal and rearranging the terms, the precipitation rate is found to be

where represents the two-dimensional Fourier transform of the column-integrated condensation,

with S₀ and H_w defined in (33).⁴

In the original SB04 model, in (44) is calculated assuming a one-layer atmosphere with constant N′ and U. An extension of the model, introduced by Barstad and Schüller (2011), accommodates multiple layers with different N′ and U and has been used to demonstrate how changes in wind speed and microphysical time scales affect precipitation. However, if we assume that (35) holds, and that U, τ_c, and τ_f are constant everywhere, it is sufficient for our purposes to use the original model, but with in (44) replaced by its two-layer analog within the troposphere. With this single modification, our model retains the simplicity of the original model, while still incorporating the effects of a tropopause.

Following the same procedure described in section 2 for a single Fourier component, the three-dimensional transformed tropospheric vertical velocity is found to be

where the constants C^± are defined in (8), is the transformed height of the terrain, and σ = Uk + Vl, with V and l representing the wind speed and wavenumber in the y dimension, analogous to U and k in the x dimension. Given constant N′ and U, the (nonhydrostatic) 2D vertical wavenumber is given by

Combining (43)–(45), we arrive at the full expression for the Fourier transformed precipitation rate,

which is equivalent to (49) from SB04, but with the effects of a tropopause included. As in the original model, if we allow for a background precipitation rate, P_∞, and exclude the possibility of negative precipitation, the precipitation rate in real space becomes

where

b. Application of the linear model to the Washington Cascades

To illustrate the potential for tropopause height to influence precipitation patterns in realistic terrain, here we apply the modified linear model to the Cascades of Washington State, where significant variability in rain-shadow strength has been observed (Siler et al. 2013). We present results for two different tropopause heights, with τ_c = τ_f = 1000 s, and with other parameters unchanged from the 2D simulations presented in section 3 (S₀ = 1.9 × 10⁻³ g m⁻⁴, N′ = 0.009 s⁻¹). With microphysics now implicitly accounted for, the factors that necessitated a reduction in H_w in section 3 no longer apply, and we therefore use the empirically derived value of H_w = 2.5 km. Based on an analysis of Cascade storms by Siler et al. (2013), we set P_∞ = 1 mm h⁻¹ and assume a wind speed of 15 m s⁻¹ with a west-southwesterly orientation of 250°. Calculations were made using a 1024 × 1024 grid with 1-km resolution centered on the Washington Cascades (not shown).

Figure 10 shows the precipitation patterns in the Washington Cascades predicted by the linear model for tropopause heights of 9.5 and 11.5 km. These patterns exhibit similar differences as the 2D numerical simulations (Fig. 6c). First, precipitation is significantly greater in the 9.5-km case as a result of enhanced ascent over the windward slope. Second, the distribution of precipitation is more evenly distributed between eastern and western slopes in the 11.5-km case than in the 9.5-km case, with ratios of eastern-to-western precipitation equal to 0.42 and 0.30, respectively. As in the 2D simulations, the contrast in precipitation patterns is a result of both weaker windward ascent and weaker leeside descent in the 11.5-km case, which allows cloud water to penetrate further downstream. These results suggest that wave reflection by the tropopause may account for some of the variability in rain-shadow strength observed among storms in the Cascades (Siler et al. 2013), though more observations are needed to evaluate this theory with any confidence.

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The precipitation rate (mm h⁻¹) in the Washington Cascades given tropopause heights of (a) 9.5 and (b) 11.5 km, calculated using a modified version of the linear model of SB04. Input parameters are N′ = 0.009 s⁻¹, S₀ = 1.9 × 10⁻³ g m⁻⁴, H_w = 2.5 km, |U| = 15 m s⁻¹ (from 250°/west-southwest), τ_c = τ_f = 1000 s, and P_∞ = 1 mm h⁻¹. The black line represents the crest of the range.

Citation: Journal of the Atmospheric Sciences 72, 2; 10.1175/JAS-D-14-0200.1