Precipitation over Concave Terrain

Qingfang Jiang UCAR Visiting Scientist Program, Naval Research Laboratory, Monterey, California

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Abstract

Many topographic barriers are comprised of a series of concave or convex ridges that modulate the intensity and distribution of precipitation over mountainous areas. In this model-based idealized study, stratiform precipitation associated with stratified moist airflow past idealized concave ridges is investigated with a focus on windward blocking, flow confluence, and the associated precipitation enhancement.

It is found that flow confluence and precipitation enhancement by a concave ridge are controlled by the nondimensional ridge height M (M = Nmhm/U, where Nm is the moist buoyancy frequency, hm is the maximum ridge height, and U is the wind speed), based on which three dynamical regimes can be defined. In the linear regime (M < 0.4), a flow confluence zone is present over the upwind slope of the ridge vertex, where precipitation is significantly enhanced. The precipitation enhancement is due to the additional updraft driven by the horizontal flow convergence with a considerable contribution from lateral confluence. In the blocking regime (0.4 < M < Mc), the area and intensity of the flow confluence zone decrease with increasing mountain height due to low-level blocking. The critical nondimensional ridge height (Mc) for windward flow stagnation decreases with increasing concave angle. In the two regimes, flow confluence and precipitation enhancement are more pronounced for concave ridges with a longer cross-stream dimension or a larger concave angle. In the flow reversal regime (M > Mc), no steady state can be achieved and the precipitation enhancement at the vertex is absent.

In addition, the flow confluence and precipitation enhancement upstream of a concave ridge are sensitive to the presence of a relative gap or peak at the vertex, the earth’s rotation, and the incident wind. The relevant dynamics has been examined.

Corresponding author address: Qingfang Jiang, UCAR Visiting Scientist Program, NRL, 7 Grace Hopper Ave. Monterey, CA 93943-5502. Email: jiang@nrlmry.navy.mil

Abstract

Many topographic barriers are comprised of a series of concave or convex ridges that modulate the intensity and distribution of precipitation over mountainous areas. In this model-based idealized study, stratiform precipitation associated with stratified moist airflow past idealized concave ridges is investigated with a focus on windward blocking, flow confluence, and the associated precipitation enhancement.

It is found that flow confluence and precipitation enhancement by a concave ridge are controlled by the nondimensional ridge height M (M = Nmhm/U, where Nm is the moist buoyancy frequency, hm is the maximum ridge height, and U is the wind speed), based on which three dynamical regimes can be defined. In the linear regime (M < 0.4), a flow confluence zone is present over the upwind slope of the ridge vertex, where precipitation is significantly enhanced. The precipitation enhancement is due to the additional updraft driven by the horizontal flow convergence with a considerable contribution from lateral confluence. In the blocking regime (0.4 < M < Mc), the area and intensity of the flow confluence zone decrease with increasing mountain height due to low-level blocking. The critical nondimensional ridge height (Mc) for windward flow stagnation decreases with increasing concave angle. In the two regimes, flow confluence and precipitation enhancement are more pronounced for concave ridges with a longer cross-stream dimension or a larger concave angle. In the flow reversal regime (M > Mc), no steady state can be achieved and the precipitation enhancement at the vertex is absent.

In addition, the flow confluence and precipitation enhancement upstream of a concave ridge are sensitive to the presence of a relative gap or peak at the vertex, the earth’s rotation, and the incident wind. The relevant dynamics has been examined.

Corresponding author address: Qingfang Jiang, UCAR Visiting Scientist Program, NRL, 7 Grace Hopper Ave. Monterey, CA 93943-5502. Email: jiang@nrlmry.navy.mil

1. Introduction

In spite of recent advances in computing power and numerical model skills, forecasting precipitation over complex terrain is still a challenge to the meteorological community. To some degree, the complicated interactions among orographic flow dynamics, thermodynamics, and microphysics associated with orographic precipitation are still not well understood. This is largely because complex terrain is usually comprised of multiple peaks and ridges with varying orientations, which could significantly modulate precipitation intensity and distribution.

Previous studies of orographic precipitation can be coarsely classified into two categories, namely observation-based case studies and model-based idealized studies. A great deal of our knowledge about orographic precipitation has been learned from field observations, from early rain gauge or single-radar observations (Smith 1979) to modern composite field campaigns featuring aircraft in situ measurements and remote sensing technologies, such as the Mesoscale Alpine Program (MAP; Bougeault et al. 2001) and the Improvement of Microphysical Parameterization through Observational Verification Experiment (IMPROVE; Stoelinga et al. 2003).

Our understanding of orographic precipitation has also been advanced through model-based idealized studies. Often using simple-shaped terrain and a single sounding, idealized studies allow researchers to isolate and focus on individual process through well-designed sensitivity tests in combination with conservation laws and analytical tools. For example, stratiform precipitation over a two-dimensional ridge has been examined by Colle (2004) with focus on the sensitivity of precipitation to the ridge width and height, ambient flow, moist static stability and freezing level. Precipitation associated with stratified moist airflow past a three-dimensional Gaussian hill had been examined by Jiang (2003) focusing the moist dynamics and by Jiang and Smith (2003) focusing on cloud time scales and precipitation efficiency. Flow regimes associated with moist stratified flow past isolated topography have also been investigated by Miglietta and Buzzi (2004). More recently, precipitation associated with convectively unstable airflow past a two- or three-dimensional ridge has been studied by several groups (Kirshbaum and Durran 2004, 2005; Chen and Lin 2005; Fuhrer and Schär 2005).

One natural step toward bridging the gap between complicated case studies dealing with multiple-scale complex terrain and often oversimplified idealized studies is to gradually increase the complexity of the idealized terrain and large-scale flow. In fact, some efforts have been made in this regard. For example, the flow regimes and precipitation associated with a southerly prefrontal low-level jet impinging on the southern slopes of the Alps have been investigated by Schneidereit and Schär (2000), and in their study, a simplified arc-shaped obstacle of Alpine scale has been used to represent the Alps. A similar barrier has been used in Rotunno and Ferretti (2003) as well.

Major barriers such as the Alps are typically comprised of many concave and convex ridges with different horizontal scales and orientations, which modulate precipitation distribution in mountainous areas. For example, relative to the usually moist southerly flow from the Mediterranean Sea, two concave and one convex ridges are located along the southern edge of the Alps as indicated by the white-dotted curves in Fig. 1. In fact, the main Alpine ridge is concave relative to the southerly wind and convex relative to the northerly wind. The gray shading in Fig. 1 is the monthly mean precipitation over the Alpine area for April averaged over 20 years. Intense precipitation is distributed along both the southern and northern slopes of the main Alpine ridge. In general, the southern slope is wetter than the northern slope, likely associated with moist airflow from the Mediterranean Sea. Strikingly, there are two primary precipitation maxima located along the southern edge of the Alps, adjacent to the two concave terrain shapes, between which a precipitation minimum is located upstream of the convex ridge. The maximum precipitation rate at the concave ridge vertexes is almost 3 times larger than the precipitation rate at the convex vertex. The coincidence of the precipitation maxima (minimum) and the concave (convex) terrain is found for all four seasons (Frei and Schär 1998), indicating the connection between the precipitation distribution and the concave/convex terrain shapes. The correlation between precipitation patterns along the southern Alps and terrain features has been noticed by other researchers as well (e.g., Gheusi and Davies 2004; Rakovec et al. 2004; Vrhovec et al. 2004). Although there have been some speculation regarding the funneling effect of concave terrain in precipitation enhancement, no systematic study has been done in this regard.

Concave and convex ridges are commonly seen along coastal mountain ranges such as the Cascades in North America and major barriers such as the Andes and Alps. These concave and convex ridges have different horizontal scales and ridge heights, and usually comprise relative peaks or valleys. In this idealized study, precipitation modulated by concave ridges over a range of geometries and horizontal scales is examined and the relevant dynamics is discussed. The rest of the paper is organized as follows. In section 2, the numerical setup and diagnosis indices are introduced. Windward flow blocking and lateral flow confluence are examined in section 3. The precipitation enhancement by concave ridges and the relevant dynamics are investigated in section 4. The sensitivity of the precipitation enhancement to terrain geometries is demonstrated in section 5. In section 6, the rotation effect on precipitation is discussed, and the results are summarized in section 7.

2. Numerical setup and control parameters

a. Numerical aspects

The atmospheric component of the Navy’s Coupled Ocean/Atmospheric Mesoscale Prediction System (COAMPS;1 Hodur 1997) is used in this idealized study. COAMPS is a fully compressible, nonhydrostatic model with a suite of physical parameterizations. For simplicity, Kessler’s warm rain scheme (Kessler 1969) is used for this study, which includes condensation of cloud water when the air is saturated, autoconversion of cloud water to rainwater, accretion of cloud water by rain, evaporation/condensation of rainwater, and fallout of rainwater (Rutledge and Hobbs 1983).

The computational domain is configured with two nested grids and the corresponding horizontal spatial resolutions are 15 and 5 km, respectively. The domain sizes are 1800 km × 1800 km and 750 km × 750 km respectively with a concave ridge located at the domain center. There are 55 levels in the vertical and the terrain-following coordinate is stretched with ΔZmin = 20 m near the surface. The flat model top is located at 25 km where a radiation boundary condition is applied. For the sake of simplicity, a free-slip boundary condition is applied at the bottom. Radiation boundary conditions are applied along the lateral boundaries of the coarse domain as well. Additional simulations have been carried out to test the sensitivity of solutions to domain sizes and horizontal resolution, which indicate that with the current domain sizes the lateral boundary effect on the dynamics and precipitation features in the vicinity of the terrain is negligible. Simulations with finer horizontal resolutions (i.e., 2 and 4 km) only produce slightly more precipitation. The model is initialized using a single sounding that will be described later.

b. Idealized concave ridge

The idealized concave ridge is shown in Fig. 2 and described by the following equations.
i1520-0469-63-9-2269-e1
where
i1520-0469-63-9-2269-e2
The concave ridge described by Eqs. (1)(2) comprises a pair of tilting straight ridges with a half-width a, which are mirror symmetric with respect to the centerline (i.e., y = 0), and connected by a parabolic ridge near the vertex (i.e., where the two ridges intercept each other, see Fig. 2) and between y = ∓d. In Eqs. (1)(2), x′ is the distance from the ridge crest. The parameters used to specify the terrain include the maximum ridge height hm (m), concave angle α (degree, the angle between the tilting ridge lines and the y axis), ridge half-width a (km), ridge half-length b (km), gap parameter β, gap half-width c (km), and parabolic ridge half-length d (km). When the concave angle α = 0, Eqs. (1)(2) correspond to a straight ridge with a half-length b, and when α < 0, they represent a convex ridge relative to the incoming flow oriented along the x axis (i.e., westerly). For β = 0, the height of the ridge crest between y = ∓b is a constant (i.e., hm) and the vertical interception area with respect to westerly flow is 2bhm, which is independent of the concave angle α. The maximum terrain slope in x direction varies with the cross-stream distance (Fig. 2). For a straight ridge (i.e., α = 0), the slope is constant between y = ∓b, and decreases sharply away from the ridge. For a concave ridge, the slope reaches maxima at the two ridge ends (i.e., y = ∓b), and reduces by a factor of cos(α) between the two maxima due to the tilting. The gap factor −1 < β < 0 (1 > β > 0) corresponds to a relative gap (peak) located between y = ∓c. The terrain parameters used in this study are listed in Table 1 for reference.

There are three characteristic horizontal length scales for a concave ridge with a constant crest height, namely, the local ridge width a, the ridge concavity b tan(α) and the cross-stream length scale Ly = 2b. For b = 80 km, a = 15 km, and α = 30°, the three scales are approximately, 15, 50, and 160 km. Therefore, the fine and coarse domain sizes are approximately 15Lx × 5Ly and 36Lx × 12Ly, respectively, where Lx = max[a, b tan(α)].

c. Idealized sounding profiles

The profiles of potential temperature and relative humidity of the idealized sounding are shown in Fig. 3. The idealized sounding is modified from the Milano (Italy) 1200 UTC sounding of 21 October 1999. Intense and persistent stratiform precipitation over the southern Alps, especially over the Po Valley, was documented by MAP scientists on 21 October 1999, and has been studied by several groups (e.g., Bousquet and Smull 2003; Rotunno and Ferretti 2003). For this study, the stability in the lower troposphere is increased and some details are smoothed out in both the potential temperature and relative humidity profiles. The air is saturated in the lowest 3 km, above which the relative humidity decreases gradually with the altitude (Fig. 3b). The dry Brunt–Väisälä frequency is approximately 0.014 s−1 in the lowest 7 km and less stable in the upper troposphere. It has been demonstrated in previous studies that in a saturated environment the moist Brunt–Väisälä frequency, Nm, defined by (Lalas and Einaudi 1974)
i1520-0469-63-9-2269-e3
should be used to replace its dry counterpart. Here qw = qυ + qc is the total water mixing ratio, qυ and qc are mixing ratios of water vapor and cloud water, respectively; L is the latent heat of evaporation and R is the ideal gas constant for dry air; Γm is the moist-adiabatic lapse rate given by
i1520-0469-63-9-2269-e4
where cp, c, and cw are heat capacities of dry air, water vapor, and liquid water under a constant pressure, and ε = 0.622 is the ratio of the gas constants for dry air and water vapor.

Using the above definition and the sounding profiles shown in Fig. 3, we find that the average Nm in the lowest 3 km is approximately 0.01 s−1. For simplicity, the wind we use is westerly with a constant wind speed Uo = 15 m s−1. For our control runs (i.e., a = 15 km, b = 80 km, and α = 30°), the corresponding time scale for an air parcel to drift across a concave ridge is τa = Lx/Uo ≃ 3000 s. In most simulations, the dynamical and precipitation fields reach a quasi steady state in the vicinity of the terrain after 10 h integration, which is approximately 12τa. The integration time is 16 h for most simulations and 24 h for some simulations with high ridges to check the steadiness of solutions. With the surface pressure Po = 1013 hPa and surface temperature To = 282.5 K, the freezing level is located approximately at 2.5 km above the surface. As will be shown, most of the condensation associated with upslope ascent, which directly contributes to the windward precipitation, occurs below the freezing level. A set of simulations with ice-phase microphysics (Rutledge and Hobbs 1983) have been carried out for b = 80 km, α = 0° and 30°, and moderate ridge heights, which produce approximately 5%–10% more precipitation than the corresponding warm rain simulations. The corresponding precipitation patterns are almost identical with or without ice-phase microphysics.

d. Indices and variables

Before reporting numerical results, we define some variables and indices as listed in Table 2, which will be used in the model diagnosis in the following sections.

The nondimensional mountain height M is defined as Nmhm/Uo, where Nm is the moist Brunt–Väisälä frequency. It has been demonstrated by Smith (1989) that in the absence of rotation and in hydrostatic limit, the dynamics of airflow past an isolated hill is solely controlled by M. The normalized zonal wind speed minimum over the upwind slope, ûm, is defined as umin/Uo, where umin is the zonal wind minimum at the surface of the upwind slope associated with terrain-blocking effect. For stratified airflow past a two- or three-dimensional mountain, ûm decreases with increasing nondimensional mountain height M, and ûm = 0 indicates the onset of windward flow stagnation and flow reversal (Smith 1989). Another terrain related parameter is the nondimensional ridge width, Nma/Uo, which is of the order of 10. Therefore, the dominant wave response to terrain forcing is hydrostatic in this study.

The normalized lateral confluence is defined ψ(x, y, z) = ∂υ/∂y/ψo, where υ is the meridional wind component and ψo = 10−5 s−1. For westerly flows, ψ(x, y, z) describes lateral flow confluence, and ψ > 0 (<0) corresponds to flow diffluence (confluence). In addition, we use Vmax, the maximum centerwise motion upstream of a concave ridge, as a measure of flow confluence as well; a larger Vmax indicates stronger confluence and Vmax is zero in the absence of confluence.

Three indices are used to illustrate the upslope ascent, namely the windward surface vertical velocity (w) maximum ws, the surface w maximum along the centerline wc, and the w maximum at the hm + 500 m level wa. The upslope ascent includes contributions from both the direct upslope ascent (DUA) and the vertical circulation (VC) driven by the horizontal flow convergence. The contribution from DUA, measured by ws and wc, is proportional to the product of the surface horizontal wind vector and the mountain slope as forced by the impermeable bottom boundary condition and decays with altitude to zero approximately at one-quarter vertical wavelength of the hydrostatic wave. The contribution from VC associated with the horizontal flow convergence is approximately zero at the surface and reaches a maximum approximately at the mountain-top level. The ratio of wa and wc is a measure of the relative contribution to the updraft from VC.

The cross-stream precipitation distribution function P(y) is defined as P(y) = ∫t1+Tt1x2x1 p(x, y, t) dx dt/(aT), where p(x, y, t) is local precipitation rate. With t1 = 10 h, T = 6 h, the operator ∫t1+Tt1 dt/T represents an average over the 6-h period from t = 10 to t = 16 h. The operator ∫x2(y)x1(y) dx/a represents the upslope precipitation rate averaged along the wind direction. For a given y, the two points x1(y) and x2(y) are obtained by first identifying the maximum precipitation rate over the windward slope and then searching upstream for x1 and downstream for x2, where the precipitation rate decreases to 5% of the maximum. The reason to use x1 and x2 instead of −∞ and ∞ is to separate the upslope precipitation from leeside wave induced precipitation and precipitation associated with upstream propagating patterns detached from the ridge slope. To evaluate the precipitation enhancement effect by a concave ridge, we define two precipitation enhancement factors, γ1, as the ratio of the maximum P(y) values derived from a concave and the corresponding straight ridge simulations, γ2 as the ratio of the maximum ∫t1+Tt1 p(x, y, t) dt/T values over the windward slope derived from a concave and the corresponding straight ridge simulations. For a concave ridge with a relative gap or peak at the vertex, γ1 is defined as the ratio of the maximum streamwise-averaged precipitation rates within the gap/peak width and over its neighboring ridge arms.

3. Blocking and confluence

a. Flow blocking and flow confluence

Shown in Fig. 4 are some surface fields derived from three simulations with α = 30° and M = 0.4, 0.8, and 1.2, respectively. For low terrain, the surface minima of the zonal wind are located over the upwind slope of the two ridge ends with a relative jet located upstream of the vertex (Fig. 4a). The flow deceleration zone extends upstream over a distance comparable to the ridge length (i.e., ∼2b), which is consistent with previous studies of irrotational stratified flow past topography (e.g., Pierrehumbert and Wyman 1985, hereafter PW85). Assuming that the diabatic heating is negligible over the flat surface upstream and the perturbation velocity u′, υ′ ≪ Uo, following a streamline near the surface, to the leading order, we obtain the perturbation pressure p′ ≃ −ρoUou′, implying that a negative zonal wind perturbation corresponds to a positive pressure perturbation. Here ρo is the air density. Therefore, according to Fig. 4a, there are two surface pressure maxima located upstream of the two ridge ends and a relative low pressure zone located upstream of the vertex. Consistent with the lateral pressure gradient force pointing toward the centerline, a confluence zone is located upstream of the vertex. With the strongest flow confluence occurring over the upwind slope of the vertex, the flow confluence zone extends approximately 100 km upstream with a maximum width of approximately 80 km, half of the ridge length. Flow diffluence occurs near the concave terrain ends (Fig. 4b) associated with the pressure maxima. Weak and widespread confluence also occurs far upstream of the terrain. Although the ridge slope is larger near the two ends (Fig. 2), w is quite uniform along the upwind slope of the concave ridge, likely due to the stronger cross-ridge wind speed at the vertex.

For M = 0.8, the zonal wind minimum shifts to the upwind slope of the vertex, apparently associated with stronger flow blocking, and accordingly the surface updraft shows two maxima near the two ridge ends, where the ridge slope is larger and the zonal wind is stronger. Compared to the M = 0.4 run, the flow diffluence zones upstream of the two ridge ends expand significantly toward the centerline, and accordingly, upstream of the vertex, flow confluence weakens and the area of the confluence zone shrinks. In the lee side, wave breaking occurs with a pair of eddies trailing downstream, as shown in previous studies of dry stratified flow past three-dimensional topography (e.g., Schär and Durran 1997).

For M = 1.2, flow reversal occurs over the windward slope of the vertex, and a flow confluence zone is located at the vertex associated with the reversed downslope flow (Fig. 5). Compared to the runs with M = 0.4 and 0.8, the surface vertical velocity is significantly weaker due to low-level blocking. In addition, a bow-shaped precipitation band is located far upstream and detached from the concave ridge, the formation of which will be further discussed in the next section.

Based on Figs. 4 and 6, the upstream flow response to a concave ridge can be classified into three regimes, namely, linear, blocking, and flow reversal regimes. In the linear regime (M < 0.4), the windward blocking index ûm is close to unity and decreases slowly with increasing mountain height M (Fig. 6a), a large flow confluence zone is present upstream of the vertex (Fig. 4b) with the intensity increasing with M (Fig. 6b), and the zonal wind shows minima near the ridge ends with the surface updraft relatively uniform along the upwind ridge slope. In the blocking regime (0.4 < M < Mc, where Mc is the critical mountain height to have windward flow stagnation), ûm decreases sharply with M (Fig. 6a), indicating that flow blocking is significant, the size and intensity of the confluence zone decrease with increasing M (Figs. 4d and 6b), and both the surface zonal wind and updraft are characterized by a minimum at the vertex (Figs. 4c,d and 6c). The ratio between wc and wa in Figs. 6c,d indicates how fast the updraft decays with the altitude. Apparently, the vertical motion decays much slower over the vertex of a concave ridge than over the corresponding straight ridge, indicating that the VC driven by the horizontal convergence is significantly enhanced by the lateral flow confluence induced by concave ridges. In the flow reversal regime (M > Mc), a flow confluence zone is present corresponding to the downslope reversed flow, and the surface updraft is relatively weak associated with weak cross-ridge flow. For α = 30°, the critical ridge height for windward flow stagnation Mc is around 1.1 (Fig. 6a).

b. Bow-shaped precipitation band

As shown in Figs. 4e and 5, for M = 1.2, precipitation over the windward slope of the concave ridge is relatively weak and a bow-shaped precipitation zone is located further upstream and detached from the ridge slope, coincided with a zonal wind minimum. A similar precipitation band upstream of a high three-dimensional Gaussian hill has been examined by Jiang (2003). To further illustrate the dynamics associated with the formation of the detached precipitation zone and its dependence on the mountain height, a pair of distance–time diagrams is shown in Fig. 7, in which signatures parallel to the time axis are stationary and signatures with negative slopes correspond to upstream-propagat-ing modes. For M = 0.8, at the beginning of the integration, a narrow precipitation zone coincided with a zonal wind minimum propagates upstream approximately at a speed of 2.7 m s−1. The precipitation intensity as well as the zonal wind minimum in the bow-shaped band weakens while propagating away from the ridge, and the propagation speed decreases gradually. After 10 h, both the precipitation and the zonal wind minimum associated with the propagating band become nearly indistinguishable and a quasi steady state is reached over the windward slope of the ridge.

For M = 1.2, the upstream velocity perturbation is stronger and the associated precipitation band propagates faster. Again, the precipitation and the zonal wind minima weaken and slow down gradually while propagating upstream. However, different from the M = 0.8 run, approximately at t = 7 h, a second band forms over the windward slope and propagates away, implying that in the presence of upstream flow reversal, a steady state may not be achievable even in the vicinity of the terrain. It is noteworthy that upstream-propagating signatures are not observed in a similar distance–time diagram created for M = 0.4 (not shown).

Apparently the ridge height is a key parameter here that controls the upstream propagation of perturbations, the dynamics of which can be illustrated using linear wave theory. The dispersion relation for a hydrostatic wave in continuously stratified flow characterized by a uniform wind speed Uo and a constant Brunt–Väisälä frequency Nm can be written as (e.g., Smith 1979)
i1520-0469-63-9-2269-e5
where ω is the intrinsic frequency, and k and l are the horizontal wavenumbers along the x and y directions. For waves to propagate upstream, the horizontal group velocity relative to the terrain, Cgx, given by
i1520-0469-63-9-2269-e6
should be negative; that is,
i1520-0469-63-9-2269-e7
To excite such a wave mode, the vertical extension of the perturbed layer D should be comparable to one-quarter vertical hydrostatic wavelength. Using kl, condition (7) can be written as D > (πU0/2Nm), where D is proportional to hm; that is, D = ĉhm, and ĉ is a constant. Therefore, the formation of a upstream propagating precipitation band is only possible for high terrain, and the propagation speed is larger for a higher ridge. Based on the propagation speed estimated from Fig. 7, the constant ĉ is approximately 2.3, which yields a critical mountain height for upstream propagation, Mcp ≃ 0.68, or a critical mountain Froude number Fr = U/(Nmhm) ≃ 1.46.

Over the range of parameters examined, the upstream propagation of the precipitation band shows moderate sensitivity to the variation of the concave angle α. Due to strong blocking, the air upstream of the vertex is virtually stagnant and can be treated as an extension of the terrain (e.g., Marwitz 1980). As a result, the concave terrain acts more like a straight ridge to the incoming flow. It is noteworthy that the three-dimensional upstream-propagating waves discussed here are different from the upstream surge examined by PW85. The two-dimensional columnar mode discussed by PW85 is nondispersive in the horizontal and trapped in the lowest layer. The three-dimensional waves examined in this study are dispersive and propagate in the vertical, and as a result, the precipitation band weakens as it propagates upstream. The critical nondimensional mountain height Mcp is considerably smaller than that required to excite a two-dimensional columnar mode (∼1.33 by PW85).

4. Precipitation enhancement

In both linear and blocking regimes, a precipitation maximum is located over the upwind slope of the vertex (Figs. 4a,c), slightly downstream of the flow confluence zone, implying the connection between precipitation enhancement at the vertex and the horizontal flow confluence. To further quantify the enhancement of precipitation by concave ridges, the cross-stream distributions of the streamwise-averaged precipitation rates derived from simulations over a range of mountain heights and concave angles are shown in Fig. 8.

For a low concave ridge (i.e., M = 0.4), a precipitation maximum is located at the ridge vertex with a characteristic width of approximately 50 km (Fig. 8a), which is wider for a higher concave ridge (Fig. 8b). Relative to the corresponding straight ridge simulation, the precipitation at the vertex is significantly enhanced by the concave ridge with an identical vertical interception area. Compared to M = 0.4, while the precipitation is much stronger for M = 0.8, the precipitation enhancement seems to be less pronounced. It is evident in Fig. 9 that the precipitation enhancement factors tend to decrease with increasing mountain height, implying that the strongest precipitation enhancement by a concave ridge occurs in the linear regime with a small nondimensional mountain height. Figure 9 also indicates that, in general, γ1 > γ2, implying that the precipitation at the vertex is more widespread along the wind direction. For high ridges with M > Mc, no steady state can be achieved in the vicinity of the ridge. If we still compute P(y), γ1 and γ2 using definitions given in section 2d for M = 1.2, the P(y) is relatively uniform along the ridge with γ1 close to and γ2 less than unity.

The dynamical mechanisms associated with precipitation enhancement by concave ridges can be illustrated by comparing the vertical sections from three simulations with M = 0.8, and α = 0°, 30°, and 45° (Fig. 10). Figure 10 shows that as moist airflow ascends over a ridge, the zonal wind speed decreases because of blocking, rainwater forms over the upwind slope associated with upslope ascent. For a straight ridge (Fig. 10a), the w maximum is located at the surface and decreases with the altitude rapidly to zero at one-quarter vertical wavelength of the hydrostatic wave; that is, λz/4 = πU/(2N) ∼2.5 km. The rapid decrease of the vertical velocity implies that the DUA dominates and the VC driven by the horizontal convergence is relatively small. For α = 30°, although the vertical velocity maximum is still located at the surface, the decrease of the vertical velocity is much slower, likely due to the contribution from the horizontal flow convergence. In addition, Fig. 10b shows a stronger zonal wind speed gradient and a broader precipitation zone over the vertex slope. Figure 10c shows that for α = 45°, two vertical velocity maxima are located at approximately the 1.8-km level and detached from the terrain surface, suggesting that the contribution from the convergence-induced VC dominates. The primary w maximum is located right above the surface w maximum, likely produced by the superposition of DUA and VC. The secondary maximum is located approximately 15 km farther upstream, where the surface vertical velocity is small and the u gradient shows a maximum, primarily driven by the horizontal flow convergence. In a vertical section away from the vertex, compared to the centerline cross section, although the surface updraft is much stronger, the decrease of the vertical velocity with the altitude is much faster and accordingly, the rainwater mixing ratio is much smaller (Fig. 10d).

The contribution to the vertical velocity from the horizontal convergence can be estimated from Figs. 4, 10 and 11 using −(ux + υy)hm. For α = 0°, the maximum zonal wind gradient (i.e., ux) is approximately −5 × 10−5 s−1 (Fig. 10a) and the lateral flow diffluence is υy ∼5 × 10−5 s−1 (Fig. 11b); the two terms with opposite signs and comparable amplitude largely cancel each other and the contribution from the horizontal convergence is negligible. For α = 30°, the maximum gradient of the zonal wind is approximately −10−4 s−1 (Fig. 10b) and υy is approximately −2 × 10−5 s−1 (Fig. 4b), and the contribution from the horizontal convergence driven updraft is approximately 0.14 m s−1, approximately 30% of the contribution from DUA. For α = 45°, with ux ≃ −1.5 × 10−4 s−1 (Fig. 10c) and υy ≃ −0.5 × 10−4 s−1 (Fig. 11d), the contribution from the horizontal convergence is approximately 0.24 m s−1, accounting for more than 50% of the primary maximum and 75% of the secondary maximum. Clearly, |υy| is much smaller than |ux| for α = 30°. Still the concave ridge enhances precipitation at the vertex by significantly decreasing the flow diffluence, which is usually comparable to |ux| for a straight ridge of a finite length. For α = 45°, flow confluence |υy| and deceleration |ux| are comparable. In the linear regime, for both α = 30° and 45°, |υy| is larger than |ux| at the vertex, indicating that flow confluence is more pronounced for low terrain.

In the steady-state solutions shown in Fig. 10, the equivalent potential temperature contours are equivalent to parcel trajectories. Clearly, associated with a broader updraft zone and slower decay of w aloft, the low-level air parcels travel in the centerline cross section of a concave ridge experience significantly more ascent (Figs. 10b,c) compared to parcels ascending over a straight ridge (Fig. 10a) or ascending over a concave ridge in a vertical section away from the vertex (Fig. 10d), and more condensation occurs accordingly. In addition, with a much broader updraft zone, the time scale for a parcel to move across the upslope clouds is larger, and the precipitation efficiency should be higher accordingly (Jiang and Smith 2003).

5. Terrain geometry impact

a. Concave angle and ridge length impact

To examine the impact of the concave angle on windward flow confluence and precipitation more systematically, we compare four simulations with an identical vertical interception area (i.e., b = 80 km and hm = 1.2 km) and a concave angle, α = −30°, 0°, 30°, and 45° respectively (Figs. 4c,d and 11).

For α = 0°, with a u minimum centered at the ridge vertex (Fig. 11a), flow diffluence occurs upstream of the ridge (Fig. 11b). In general, the zonal wind, diffluence strength, surface updraft, and rainwater mixing ratio are relatively uniform along the upwind slope of the straight ridge. For α = 45°, a u minimum is located over the upwind slope of the vertex upstream of which a u maximum is present, indicating a lateral pressure gradient force pointing toward the centerline. Correspondingly, compared to the simulation with α = 30° (Figs. 4b,c), the confluence is much stronger, the confluence zone is considerably larger (Figs. 11c,d), and the centerwise lateral motion is almost twice as strong (Fig. 6b). As an example, a simulation with moist airflow past a convex ridge is shown in Figs. 11e,f. Different from a concave ridge, the slope maximum (along the x direction) of a convex ridge is located at the vertex, where the cross-mountain wind shows a minimum. The surface vertical velocity is relatively uniform along the upwind ridge slope, likely due to the balance between the slower zonal flow and larger ridge slope at the vertex. Associated with blocking-induced deceleration, strong flow diffluence occurs upstream of the convex ridge with a maximum located at the vertex, where the rainwater mixing ratio shows a minimum accordingly.

The windward blocking seems to be stronger for a concave ridge with a larger α (Fig. 6a). According to Fig. 6a, the critical nondimensional mountain height for windward stagnation decreases from 1.1 for α = 30° to ∼0.9 for α = 45°. The updraft may increase with the altitude as the contribution from VC dominates (Fig. 6d), and a deeper and broader updraft zone forms with significantly stronger precipitation at the vertex. Associated with stronger flow confluence, the precipitation enhancement factors are much larger for α = 45° than for α = 30° (Figs. 8 and 9) over the examined parameter ranges. Figure 8 also shows that as moist flow passes an isolated convex ridge, the strongest precipitation occurs near the ends of the terrain and a precipitation minimum is located upstream of the vertex associated with strong flow diffluence. In terms of the total precipitation [i.e., ∫−∞ P(y) dy], Fig. 8 indicates that, relative to the corresponding straight ridge, a concave ridge could produce significantly more precipitation, and a convex ridge may produce more or approximately same amount of precipitation depending on the ridge height. Therefore, a major barrier with alternating concave and convex ridges such as the southern edge of the Alps is likely more efficient in removing water from cross-barrier moist flows.

Simulations have been carried out for M = 0.4 and 0.8 with the half-ridge length b ranging from 40 to 160 km. In the linear regime, a confluence zone is present in all the simulations with a typical width of approximately half of the ridge length, and the confluence intensity increases with the ridge length. Accordingly, precipitation enhancement occurs at ridge vertex with both precipitation maximum and the enhancement factors increase with the ridge length. In the blocking regime (i.e., M = 0.8), the blocking is stronger (i.e., ûm is smaller) for a longer ridge, and the size of the confluence zone decreases with the decreasing ridge length. In fact, the flow confluence zone disappears for b = 40 km. Still, precipitation enhancement at the vertex is observed for all the ridge length and the enhancement factors increases slowly with increasing ridge length.

b. Gap and peak impact

In the previous sections, the heights of the concave or convex ridges examined are constant between y = ∓b. However, the crest height of real complex terrain usually varies, resulting in relative gaps or peaks. The role of a relative gap or peak located at the vertex of a concave ridge in modulating the dynamics and precipitation is examined based on a set of simulations with the gap factor −0.6 < β < 0.6 and a range of mountain heights, and the results are summarized in Figs. 12 and 13.

Relative to the corresponding simulations with a flat ridge (i.e., β = 0), it is evident that the presence of a gap (peak) tends to weaken (enhance) windward flow blocking and promote (inhibit) confluence (Figs. 12a,d). For a low concave ridge with a relative gap, three precipitation maxima are evident: two are located over the windward slopes of the tilting ridges and the third is located right in the middle of the gap (Fig. 12b), apparently associated with flow confluence upstream (Fig. 12a). For a higher ridge, the third precipitation maximum tends to shift upstream.

Again, it is instructive to decompose the vertical motion into DUA and VC. The gentler (steeper) terrain slope associated with a gap (peak) tends to decrease (increase) DUA, and on the other hand, the stronger (weaker) zonal wind associated with less (more) blocking in the presence of a gap (peak) tends to increase (decrease) DUA. Similarly, the change of VC in the presence of a gap (peak) depends on two competing factors, increase (decrease) of lateral confluence and decrease (increase) of zonal wind gradient. Clearly, for low and moderate ridges with β = −0.4, w at the surface shows a minimum at the vertex (Figs. 12a,c), indicating that the decrease of the local slope overpowers the increase of surface wind speed at the vertex. For a high ridge (M ≥ 1), a relative gap at the vertex tends to increase DUA by significantly increasing the zonal wind upstream of the gap, and therefore, enhance precipitation upstream or in the gap. For a low concave ridge (M < 0.4) with a relative peak at the vertex, the maximum w is located upstream of the peak due to increase of the slope (not shown), and for a higher concave ridge with a peak, at the surface, w is characterized by a minimum upstream of the peak due to the decrease of the cross-ridge wind speed associated with more severe blocking (Fig. 12d).

In terms of cross-stream precipitation distribution, for a low concave ridge (M = 0.4), the presence of a gap appears to significantly weaken the confluence-induced precipitation enhancement (Fig. 13a). The enhancement factor, γ1, is close to unity for the simulation with M = 0.4 and β = −0.4, indicating that the increase of vertical motion associated with VC is largely canceled out by the decrease of DUA caused by the decrease of the mountain slope at the gap. On the contrary, the presence of a peak over a low concave ridge tends to increase the precipitation over windward slope of the peak, indicating that the DUA dominates the VC for low ridges. As expected, the width of the precipitation maxima and the gap width are comparable. For high concave ridges, precipitation in the vicinity of the vertex could be significantly enhanced in the presence of a gap and weakened by the presence of a relative peak (Fig. 13b). Additional simulations have been carried out with α ranging from 0° to 45° and β ranging from −0.6 to 0.6 and the derived γ1 tends to increase with increasing α. Even for α = 0, the derived γ1 is larger than unity for M > 1, indicating that the presence of a gap in a high straight ridge tends to cause stronger precipitation in or upstream of the gap. The enhancement factors only increase slightly corresponding to the decrease of β from −0.4 to −0.6 (i.e., a deeper gap).

6. Rotation effect

In the previous examples, with the cross-ridge wind Uo = 15 m s−1, the ridge length L = 2b = 160 km, and Coriolis coefficient f = 10−4 s−1, we obtain the mountain Rossby number Ro = U/( fL) ≃ 0.93 < 1, implying that the earth’s rotation effect should be taken into account. To examine the rotation effect on flow dynamics, which may further modulate precipitation, a series of simulations have been carried out with a constant Coriolis coefficient f = 10−4 s−1 and the ambient flow in geostrophic balance and the results are summarized in Figs. 14 and 15. As expected, the inclusion of rotation breaks the left–right symmetry of the flow field. Because of blocking, the decelerated upstream flow is subgeostrophic and turns predominantly to the north under the influence of the pressure gradient force (Fig. 14b), which is consistent with previous studies (Smith 1982; PW85). For α = 0°, associated with the establishment of an along-ridge jet (i.e., υ > 0), the zonal wind component shows an along-ridge gradient: stronger to the north and weaker to the south of the ridge (Fig. 14b). The rotation effect can be illustrated by the following simple scaling. Assuming that the mass adjustment is slow relative to the velocity adjustment, we can write the linearized momentum equations as
i1520-0469-63-9-2269-e8
In response to a mean deceleration in the zonal wind Uou′ over a distance L, the along-ridge wind component due to the Coriolis acceleration is υ = fL(1 − u′/Uo), which further causes an increase in the zonal wind speed u″ = fυL/Uo, or u″/Uo = [1 − g(M)]/R2o, where g(M) = u′/Uo represents blocking effect. Clearly, parcels that flow around the northern end of the ridge travel the longest distance in the meridional direction and their corresponding Rossby number Ro is smallest, and subsequently acquire more acceleration in the zonal direction. For a concave ridge, in addition to the increase of the zonal component by the Coriolis force, the cyclonic turning of the upstream flow increases the incident angle with respect to the northern arm of the ridge, which significantly increases the direct upslope ascent. Accordingly, precipitation is significantly enhanced over the slope of the northern ridge arm. The surface zonal wind minimum is now located slightly to the north of the vertex (Fig. 14d).

The cross-stream precipitation distributions derived from four pairs simulations, corresponding to M = 0.8 and 1 and α = 0 and 30°, are shown in Fig. 15. Several aspects of Fig. 15 deserve mention. First, relative to the corresponding nonrotational simulations, the total precipitation in the rotational runs is significantly stronger. For the straight ridge runs, rotation increases the average precipitation rate between y = ∓b approximately by 20% and 110% for M = 0.8 and 1.0, respectively, implying that for the same Rossby number, the enhancement of precipitation by the earth’s rotation increases with increasing mountain height. In addition, consistent with Hunt et al. (2001), precipitation is substantially stronger over the northern portion of the ridge (Figs. 14b and 15), where the cross-mountain wind component (Fig. 14b) and therefore the upslope ascent is stronger. With rotation, increasing M from 0.8 to 1 results in a significant increase of precipitation, likely due to the sharp decrease of g(M) with increasing M in the blocking regime as shown in Fig. 4a, which increases u″ considerably. For a concave ridge with rotation, although, the precipitation over the northern ridge arm is significantly enhanced, the primary precipitation maximum is still located at the vertex, indicating that the precipitation enhancement by a concave ridge discussed in the previous sections is relevant.

7. Summary

Flow confluence and precipitation enhancement associated with moistly stratified flow past concave ridges have been examined in this model-based theoretical study. As stratified moist airflow past an isolated concave ridge, flow diffluence occurs far upstream of the ridge and in the vicinity of the two ridge ends, and a relative low pressure zone is located upstream of the ridge vertex where flow confluence occurs in response to the center-pointing horizontal pressure gradient force. The size and intensity of the flow confluence zone is controlled largely by the nondimensional mountain height, based on which three dynamical flow regimes can be defined, namely, linear (M < 0.4), blocking (0.4 < M < Mc), and flow reversal (M > Mc) regimes, where the critical nondimensional mountain height for windward flow stagnation, Mc ∼ 1, varies with the concave angle and ridge length. In the linear regime, the flow confluence zone extends upstream over a distance comparable to the ridge length with a width of approximately one-half of the ridge length. With a relative jet located upstream of the vertex, w at the surface is relatively uniform along the upwind slope. Over the vertex, the decrease of w with the altitude is much slower than over the ridge arms due to the vertical circulation driven by the horizontal flow convergence with a significant contribution from lateral flow confluence. Precipitation is considerably enhanced at the vertex accordingly. The flow confluence intensity increases with increasing mountain height. In the blocking regime, the size of the confluence zone shrinks and the intensity decreases with increasing terrain height associated with windward flow blocking. The cross-ridge wind minimum is located at the vertex, and as a result, w shows a strong gradient along the ridge; stronger near the two ridge ends and much weaker at the vertex. Again, the updraft over the vertex decreases with the altitude much slower compared to over the ridge arms, and even increases with the altitude when the VC driven by horizontal convergence is comparable to the DUA, which usually occurs associated with a relatively large concave angle and a moderate ridge height. Associated with the presence of a deeper and broader updraft zone at the vertex, precipitation over the upwind slope reaches a maximum at the vertex. For higher ridges (i.e., M > Mc), flow reversal occurs over the windward slope of the vertex. A bow-shaped precipitation band forms upstream of and propagates away from the ridge, and no steady state can be achieved. Consequently, no pronounced precipitation enhancement is observed over the windward slope of the concave ridge. In general, the precipitation enhancement factors decrease with increasing M through the parameters examined.

In the linear and blocking regimes, the flow confluence at the vertex generally increases with the concave angle and the cross-stream ridge length over the range of parameters examined, and the precipitation enhancement factors increase accordingly. Especially in the linear regime, the variation of precipitation along a concave ridge is comparable to the variation of the precipitation climatology along the southern edge of the Alps shown in Fig. 1. The simulations also indicate that concave ridges not only modulate the along-ridge precipitation distribution, but also increase the total precipitation significantly, implying that, in general, a concave ridge is more effective in removing water vapor from moist cross-mountain airflow than the corresponding straight ridge with an identical cross-stream ridge length and height.

The impact of a relative gap or peak located at the vertex of a concave ridge on flow confluence and precipitation enhancement is examined. Depending on the ridge height, the presence of a gap may enhance or weaken precipitation at the vertex. For low terrain, the decrease of the terrain slope due to the presence of a gap significantly weakens the DUA and therefore, weakens the precipitation enhancement effect. In this limit, the presence of a peak causes an increase of DUA, which overpowers the decrease of flow confluence and leads to stronger precipitation. For high terrain, the presence of a gap tends to increase the cross-ridge wind speed in the vicinity of the gap and enhances the flow confluence upstream. The decrease of DUA due to the decrease of the mountain slope is partially compensated by the increase of the cross-mountain wind speed. As a result, precipitation is enhanced at the gap. The precipitation enhancement by a relative gap increases with increasing mountain height over the range of parameters examined. In the high ridge limit, the presence of a peak weakens the precipitation near the vertex through weakening the cross-mountain wind and causing flow diffluence.

As geostrophically balanced moist airflow impinges on a ridge with a length comparable to or larger than the Rossby radius of deformation Uo/f (i.e., Rossby number is close or smaller than unity), the upslope precipitation can be significantly enhanced by the earth’s rotation effect. Dynamically, rotation tends to increase the cross-mountain wind component and help parcels to ascend over the ridge, and therefore increases the DUA. The ratio of the total precipitation from a rotational and the corresponding nonrotational simulations increases with increasing ridge height and a sudden increase occurs around M = Mc, where ûm decreases sharply with increasing M. The strongest precipitation enhancement by the rotation effect occurs near the northern end of the ridge, where the cross-ridge wind reaches a maximum. However, for the concave ridges examined here, the precipitation maximum is still located at the ridge vertex, implying that the enhancement of precipitation by concave ridges we learn from the nonrotational simulations presented in sections 35 is relevant in the presence of the Coriolis force.

Additional simulations have been performed with incident wind direction varying between 270° ± 45°. While the streamwise-averaged precipitation maximum is still located at the ridge vertex, precipitation over the ridge arm that is more normal to the incident wind is stronger and precipitation over the ridge arm that is oriented more along the wind direction is significantly weaker. The precipitation enhancement factors computed by averaging over all the wind directions between 270° ± 45° are fairly close to those derived from the corresponding control runs, indicating that on average, the precipitation enhancement by a concave ridge is relatively insensitive to the incident wind direction (Fig. 16).

Finally, we should remind readers that relative to the almost infinite parameter space, only a small parameter space is examined in this study. For example, we use a single sounding, which yields Nm = 0.01 s−1. Using hm = 3 km as the mean Alpine ridge height, and Uo =15 m s−1, we obtain that Nm should be 0.005 s−1 or smaller in order to satisfy M < 1, the condition for concave ridge enhancement of precipitation to be significant. The ridge width we use is a = 15 km, comparable to the width of the concave ridges along the southern Alps, but much smaller than the width of the main Alpine ridge (∼100 km). The variation of a may modify the flow regime boundaries through changing the ridge aspect ratio (Smith 1989), influence the near-terrain dynamics in the presence of rotation through changing the Burger number (PW85), and change the precipitation efficiency though changing the advection time (Jiang and Smith 2003). Convective precipitation associated with convectively unstable moist air past concave ridges will be investigated in a future study.

Acknowledgments

This research was supported by the Office of Naval Research (ONR) program element 0601153N. The simulations were made using the Coupled Ocean/Atmosphere Mesoscale Prediction System (COAMPS) developed by U.S. Naval Research Laboratory. The author wishes to thank Drs. Ronald Smith and James Doyle for helpful discussions and suggestions. Comments from two anonymous reviewers substantially improved an earlier version of this manuscript.

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Fig. 1.
Fig. 1.

Monthly mean precipitation (grayscale, increment = 0.6 mm day−1) for April over the Alpine area for the two years from 1971 to 1990. The 1000 m MSL contour is shown as bold curves. Some concave or convex ridges are highlighted by dotted white curves. Courtesy of Institute for Atmospheric and Climate Science at the Swiss Federal Institute of Technology Zurich (ETH).

Citation: Journal of the Atmospheric Sciences 63, 9; 10.1175/JAS3761.1

Fig. 2.
Fig. 2.

(left) Contours of an idealized concave ridge described by Eqs. (1) and (2) with α = 30° and β = 0. The contour interval is hm/8. (middle) Same as (left) except for β = −0.4. (right) The maximum along-stream slopes of three ridges normalized by hm/a as a function of the cross-stream distance. The three curves correspond to a straight ridge, a concave ridge with a constant ridge height, and a concave ridge with a relative gap located at the vertex.

Citation: Journal of the Atmospheric Sciences 63, 9; 10.1175/JAS3761.1

Fig. 3.
Fig. 3.

Profiles of potential temperature (K) and relative humidity (RH, %) derived from the idealized sounding. Only the lowest 15 km is shown.

Citation: Journal of the Atmospheric Sciences 63, 9; 10.1175/JAS3761.1

Fig. 4.
Fig. 4.

Plan views of surface fields derived from three simulations with α = 30°. The mountain height is (a), (b) 0.6, (c), (d) 1.2, and (e), (f) 1.8 km. (a) Rainwater mixing ratio (grayscale, increment = 0.04 g kg−1, u (dashed contours, only for u < Uo with increment = 0.2 m s−1) and streamlines; (b) updraft (grayscale, increment = 0.1 m s−1), flow confluence (contours, from −4 to 4 with increment = 1, areas with negative values hatched), and horizontal wind vectors; (c) same as (a) except the contour increment for u is 0.5 m s−1; (d) same as (b); (e) same as (c); and (f) same as (b) except the updraft increment is 0.05 m s−1. The bold contour corresponds to terrain height h = hm/2. Only a 200 km × 200 km portion of the inner domain is plotted and the horizontal tick-mark interval is 50 km.

Citation: Journal of the Atmospheric Sciences 63, 9; 10.1175/JAS3761.1

Fig. 5.
Fig. 5.

Vertical cross section of streamlines, updraft (grayscale, interval = 0.02 m s−1), and rainwater mixing ratio (bold contours, interval = 0.02 g kg−1) along the centerline derived from a control simulation with α = 30°, and hm = 1.8 km.

Citation: Journal of the Atmospheric Sciences 63, 9; 10.1175/JAS3761.1

Fig. 6.
Fig. 6.

Plots of indices as a function of nondimensional mountain height M, derived from three groups of simulations with respectively. (a) Plot of ûm vs M; (b) plot of Vmax vs M; (c) plot of ws, wc, and wa for α = 0° and 30°; (d) Same as (c) but for α = 0° and 45°. See section 2c for the definitions of the indices.

Citation: Journal of the Atmospheric Sciences 63, 9; 10.1175/JAS3761.1

Fig. 7.
Fig. 7.

Distance–time diagrams of the rainwater mixing ratio (in grayscale, increment = 0.05 g kg−1) and u (dashed contours with increment = 2 m s−1) along the centerline and at the surface, derived from a pair of simulations with α = 30° and M = 0.8 and 1.2, respectively. The horizontal tick mark interval is 50 km.

Citation: Journal of the Atmospheric Sciences 63, 9; 10.1175/JAS3761.1

Fig. 8.
Fig. 8.

Cross-stream precipitation distribution, P(y), derived from two sets of simulations corresponding to (a) M = 0.4 and (b) M = 0.8, respectively. In addition, curves from a pair of simulations with b = 40 km and α = 30° are included for comparison.

Citation: Journal of the Atmospheric Sciences 63, 9; 10.1175/JAS3761.1

Fig. 9.
Fig. 9.

Plot of the precipitation enhancement factors vs the nondimensional mountain height M for α = 30° and 45°.

Citation: Journal of the Atmospheric Sciences 63, 9; 10.1175/JAS3761.1

Fig. 10.
Fig. 10.

Vertical cross sections of u (dashed contours, increment = 0.5 m s−1), updraft (grayscale, increment = 0.1 m−1), and rainwater mixing ratio (solid contours, increment = 0.2 g kg−1) oriented along the centerline for hm = 1.2 km and (a) α = 0°, (b) α = 30°, and (c) α = 45°, respectively. The two bold-dashed contours correspond to equivalent potential temperature θe = 305 and 309 K, respectively. (d) Same as (c) except that the cross section is located 75 km to the north of the centerline.

Citation: Journal of the Atmospheric Sciences 63, 9; 10.1175/JAS3761.1

Fig. 11.
Fig. 11.

Same set of surface fields as shown in Fig. 4 but for three simulations corresponding to M = 0.8 and α = (a), (b) 0°, (c), (d) 45°, and (e), (f) −30°.

Citation: Journal of the Atmospheric Sciences 63, 9; 10.1175/JAS3761.1

Fig. 12.
Fig. 12.

Plan views of surface fields derived from three simulations with a relative gap or peak at the vertex respectively: (a) updraft (grayscale, increment = 0.1 m s−1), confluence (contours from −4 to 4 with increment = 1 and regions with negative values are hatched), and horizontal wind vectors derived from the simulation with hm = 600 m and β = −0.4; (b) rainwater mixing ratio (grayscale, increment = 0.04 g kg−1), u (dashed contours for u < Uo with increment = 0.25 m s−1), and streamlines for the same simulation as (a); (c) same as (a) except for hm = 1200 m; and (d) same as (c) except for β = 0.4. Terrain is indicated by bold contours with increment = hm/3.

Citation: Journal of the Atmospheric Sciences 63, 9; 10.1175/JAS3761.1

Fig. 13.
Fig. 13.

Cross-stream precipitation distribution, P(y), derived from (a) three pairs of simulations with β = −0.4, 0, and 0.4, and hm = 0.6 and 1.2 km, and (b) five simulations corresponding to β = −0.4, 0, 0.4, and hm = 1.5 and 1.8 km.

Citation: Journal of the Atmospheric Sciences 63, 9; 10.1175/JAS3761.1

Fig. 14.
Fig. 14.

Same as Fig. 4c but for four simulations corresponding to hm = 1.5 km, and (a) α = 0° and f = 0; (b) α = 0° and f = 10−4 s−1; (c) α = 30° and f = 0; and (d) α = 30° and f = 10−4 s−1.

Citation: Journal of the Atmospheric Sciences 63, 9; 10.1175/JAS3761.1

Fig. 15.
Fig. 15.

Cross-stream precipitation distribution, P(y), derived from simulations corresponding to mountain height hm = 1.2 and 1.5 km, concave (C, α = 30°) and straight (S) ridges, and with (R) and without (N) Coriolis force.

Citation: Journal of the Atmospheric Sciences 63, 9; 10.1175/JAS3761.1

Fig. 16.
Fig. 16.

Cross-stream precipitation distribution, P(y), derived from simulations corresponding to hm = 1.2 km and the incident wind direction = 225°, 240°, 255°, and 270°, respectively. The dotted curve corresponds to an average over seven wind directions.

Citation: Journal of the Atmospheric Sciences 63, 9; 10.1175/JAS3761.1

Table 1.

List of simulations and the corresponding control parameters

Table 1.
Table 2.

Variables and indices used for model diagnosis.

Table 2.

1

COAMPS is a registered trademark of the Naval Research Laboratory.

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  • Bougeault, P., and Coauthors, 2001: The MAP special observation period. Bull. Amer. Meteor. Soc., 82 , 433462.

  • Bousquet, O., and B. F. Smull, 2003: Observations and impacts of upstream blocking during a widespread orographic precipitation event. Quart. J. Roy. Meteor. Soc., 129 , 391410.

    • Search Google Scholar
    • Export Citation
  • Chen, S-H., and Y-L. Lin, 2005: Effects of moist Froude number and CAPE on a conditionally unstable flow over a mesoscale mountain ridge. J. Atmos. Sci., 62 , 331350.

    • Search Google Scholar
    • Export Citation
  • Colle, B. A., 2004: Sensitivity of orographic precipitation to changing ambient conditions and terrain geometries: An idealized modeling perspective. J. Atmos. Sci., 61 , 588606.

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    • Export Citation
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  • Fig. 1.

    Monthly mean precipitation (grayscale, increment = 0.6 mm day−1) for April over the Alpine area for the two years from 1971 to 1990. The 1000 m MSL contour is shown as bold curves. Some concave or convex ridges are highlighted by dotted white curves. Courtesy of Institute for Atmospheric and Climate Science at the Swiss Federal Institute of Technology Zurich (ETH).

  • Fig. 2.

    (left) Contours of an idealized concave ridge described by Eqs. (1) and (2) with α = 30° and β = 0. The contour interval is hm/8. (middle) Same as (left) except for β = −0.4. (right) The maximum along-stream slopes of three ridges normalized by hm/a as a function of the cross-stream distance. The three curves correspond to a straight ridge, a concave ridge with a constant ridge height, and a concave ridge with a relative gap located at the vertex.

  • Fig. 3.

    Profiles of potential temperature (K) and relative humidity (RH, %) derived from the idealized sounding. Only the lowest 15 km is shown.

  • Fig. 4.

    Plan views of surface fields derived from three simulations with α = 30°. The mountain height is (a), (b) 0.6, (c), (d) 1.2, and (e), (f) 1.8 km. (a) Rainwater mixing ratio (grayscale, increment = 0.04 g kg−1, u (dashed contours, only for u < Uo with increment = 0.2 m s−1) and streamlines; (b) updraft (grayscale, increment = 0.1 m s−1), flow confluence (contours, from −4 to 4 with increment = 1, areas with negative values hatched), and horizontal wind vectors; (c) same as (a) except the contour increment for u is 0.5 m s−1; (d) same as (b); (e) same as (c); and (f) same as (b) except the updraft increment is 0.05 m s−1. The bold contour corresponds to terrain height h = hm/2. Only a 200 km × 200 km portion of the inner domain is plotted and the horizontal tick-mark interval is 50 km.

  • Fig. 5.

    Vertical cross section of streamlines, updraft (grayscale, interval = 0.02 m s−1), and rainwater mixing ratio (bold contours, interval = 0.02 g kg−1) along the centerline derived from a control simulation with α = 30°, and hm = 1.8 km.

  • Fig. 6.

    Plots of indices as a function of nondimensional mountain height M, derived from three groups of simulations with respectively. (a) Plot of ûm vs M; (b) plot of Vmax vs M; (c) plot of ws, wc, and wa for α = 0° and 30°; (d) Same as (c) but for α = 0° and 45°. See section 2c for the definitions of the indices.

  • Fig. 7.

    Distance–time diagrams of the rainwater mixing ratio (in grayscale, increment = 0.05 g kg−1) and u (dashed contours with increment = 2 m s−1) along the centerline and at the surface, derived from a pair of simulations with α = 30° and M = 0.8 and 1.2, respectively. The horizontal tick mark interval is 50 km.

  • Fig. 8.

    Cross-stream precipitation distribution, P(y), derived from two sets of simulations corresponding to (a) M = 0.4 and (b) M = 0.8, respectively. In addition, curves from a pair of simulations with b = 40 km and α = 30° are included for comparison.

  • Fig. 9.

    Plot of the precipitation enhancement factors vs the nondimensional mountain height M for α = 30° and 45°.

  • Fig. 10.

    Vertical cross sections of u (dashed contours, increment = 0.5 m s−1), updraft (grayscale, increment = 0.1 m−1), and rainwater mixing ratio (solid contours, increment = 0.2 g kg−1) oriented along the centerline for hm = 1.2 km and (a) α = 0°, (b) α = 30°, and (c) α = 45°, respectively. The two bold-dashed contours correspond to equivalent potential temperature θe = 305 and 309 K, respectively. (d) Same as (c) except that the cross section is located 75 km to the north of the centerline.

  • Fig. 11.

    Same set of surface fields as shown in Fig. 4 but for three simulations corresponding to M = 0.8 and α = (a), (b) 0°, (c), (d) 45°, and (e), (f) −30°.

  • Fig. 12.

    Plan views of surface fields derived from three simulations with a relative gap or peak at the vertex respectively: (a) updraft (grayscale, increment = 0.1 m s−1), confluence (contours from −4 to 4 with increment = 1 and regions with negative values are hatched), and horizontal wind vectors derived from the simulation with hm = 600 m and β = −0.4; (b) rainwater mixing ratio (grayscale, increment = 0.04 g kg−1), u (dashed contours for u < Uo with increment = 0.25 m s−1), and streamlines for the same simulation as (a); (c) same as (a) except for hm = 1200 m; and (d) same as (c) except for β = 0.4. Terrain is indicated by bold contours with increment = hm/3.

  • Fig. 13.

    Cross-stream precipitation distribution, P(y), derived from (a) three pairs of simulations with β = −0.4, 0, and 0.4, and hm = 0.6 and 1.2 km, and (b) five simulations corresponding to β = −0.4, 0, 0.4, and hm = 1.5 and 1.8 km.

  • Fig. 14.

    Same as Fig. 4c but for four simulations corresponding to hm = 1.5 km, and (a) α = 0° and f = 0; (b) α = 0° and f = 10−4 s−1; (c) α = 30° and f = 0; and (d) α = 30° and f = 10−4 s−1.

  • Fig. 15.

    Cross-stream precipitation distribution, P(y), derived from simulations corresponding to mountain height hm = 1.2 and 1.5 km, concave (C, α = 30°) and straight (S) ridges, and with (R) and without (N) Coriolis force.

  • Fig. 16.

    Cross-stream precipitation distribution, P(y), derived from simulations corresponding to hm = 1.2 km and the incident wind direction = 225°, 240°, 255°, and 270°, respectively. The dotted curve corresponds to an average over seven wind directions.

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