Observations and numerical simulations of mountain waves in the presence of directional wind shear

Transcription

1 Q. J. R. Meteorol. Soc. (2006), 132, pp doi: /qj Observations and numerical simulations of mountain waves in the presence of directional wind shear By JAMES D. DOYLE 1 and QINGFANG JIANG 2 1 Naval Research Laboratory, Monterey, USA 2 University Corporation for Atmospheric Research, Monterey, USA (Received 1 July 2005; revised 3 January 2006) SUMMARY Three research aircraft observed small-amplitude gravity waves over the south-western French Alps on 13 November 1999 during the Mesoscale Alpine Programme (MAP). Radiosonde ascents and Global Positioning System dropsondes deployed by MAP research aircraft indicate that the upstream flow was characterized by topographic blocking below 2 km and low-level directional wind shear associated with a synoptic-scale depression that extended above the Alpine crest. The in situ vertical velocity data from the three research aircraft, and backscatter from lidar onboard the low-level and upper-level aircraft, exhibit gravity-wave signatures in the lower troposphere that are characterized by relatively short horizontal wavelengths on a scale of 5 km or less. The in situ flight-level and backscatter data also suggest a rapid decrease in the wave amplitude with height, due to a directional critical layer that partially absorbs wave energy in the lower to middle troposphere in concert with a decrease in the Scorer parameter with height that traps the wave energy. Numerical simulation results, obtained from the non-hydrostatic Coupled Ocean Atmosphere Mesoscale Prediction System model using a horizontal resolution of 556 m, indicate the presence of low-level wave breaking above the highest peaks, and confirm the presence of trapped waves and a directional critical level. Results based on linear theory and nonlinear model simulations suggest that the absorption of wave energy associated with the directional critical layer has an important impact on the wave characteristics including the momentum flux. KEYWORDS: Directional shear Internal gravity waves Mesoscale Alpine Programme Numerical modelling 1. INTRODUCTION Mountain waves are generated as a consequence of stably stratified airflow over a topographic barrier. Vertically propagating internal waves may amplify, overturn, and breakdown through turbulent processes, particularly above the tropopause, due to factors such as the decrease of atmospheric density with altitude, nonlinearity, and vertical gradients of the ambient winds and stability. Mountain-wave breaking has a significant influence on the atmosphere for a number of reasons that include: clear-air turbulence (Clark et al. 2000); downslope windstorms (Peltier and Clark 1979); vertical mixing of water vapour, aerosols, and chemical constituents (Dörnbrack and Dürbeck 1998); potential-vorticity generation (Schär and Durran 1997) and associated upscale energy transfer (Aebischer and Schär 1998); and the aggregate effect of orographic drag on the large-scale circulation (Palmer et al. 1986). However, the energy of mountain waves may be partially or completely absorbed or ducted in the troposphere, reducing the transmission to higher altitudes and ultimately the likelihood of wave breaking in the upper levels of the atmosphere. Thus, obtaining a more complete characterization of the vertical propagation of mountain waves in the complex flows found in nature is a necessary step towards understanding and accurately predicting the fate of upper-level gravity waves and wave breaking. Stably stratified air flow over a relatively wide mountain, characterized by a background state with Na/U 10, where N is the Brunt Väisälä frequency, a is the half-width, and U is the incident wind velocity, typically generates hydrostatic mountain waves that propagate vertically nearly directly above the barrier and exhibit an upstream Corresponding author: Naval Research Laboratory, Marine Meteorology Division, 7 Grace Hopper Avenue, Monterey, CA , USA. doyle@nrlmry.navy.mil c Royal Meteorological Society,

2 1878 J. D. DOYLE and Q. JIANG phase tilt (e.g. Smith 1979). Non-hydrostatic waves, which can be generated by airflow over narrow terrain, Na/U 1, are dispersive in character and have group velocities that tend to propagate vertically and horizontally downstream (e.g. Zängl 2003). The depth to which mountain waves propagate is partially dependent on the environmental wind and stability profiles. For example, vertically propagating waves can be trapped beneath a wave-duct that occurs when the N/U of the incident airstream decreases rapidly with height (Scorer 1949; Sawyer 1960). Trapped lee waves are characterized by vertically oriented phase lines due to a superposition of two non-hydrostatic gravity waves, one of which propagates upward and a second that is downward propagating (Durran 1990). The horizontal wavelengths of trapped waves are typically shorter than those of hydrostatic mountain waves. As mountain waves approach a layer of zero wind velocity, referred to as a critical level, the vertical wavelength approaches zero and wave action accumulates beneath the critical layer leading to wave breaking and energy absorption (Dörnbrack et al. 1995; Grubišić and Smolarkiewicz 1997). Partial or complete reflection of gravity waves may occur at a critical level when the Richardson number, Ri = N 2 /( U/ z) 2,where z is height, is less than two (e.g. Breeding 1971; Wang and Lin 1999). A directional critical level occurs when the wind direction changes with altitude and the flow at any level is parallel to the gravity-wave phase lines, even if the wind speed does not reach a zero velocity. The wave packets stagnate below the directional critical level and wave action is subsequently advected away from the mountain (Shutts 1998; Shutts and Gadian 1999). All waves have a critical level at the zero wind velocity line in the unidirectional case. In contrast, for the situation with directional wind shear the phase line orientation of each wave number corresponds to a different critical-level height, and the wave drag is distributed correspondingly (Shutts 1995; Shutts and Gadian 1999). Sawyer (1962) presents numerical linear solutions for trapped mountain waves in the presence of multiple vertical layers of varying wind direction; however the corresponding directional critical levels were not considered in these calculations. Shutts (1998) describes an asymptotic wake, induced by a directional critical level, which extends far away from the mountain, with characteristics of decreasing kinetic energy with distance and increasing vertical wind shear variance downstream. The magnitude of the vertical flux of horizontal momentum associated with gravity waves is reduced with height, and the direction is altered, in the presence of directional critical levels (Broad 1995; Shutts and Gadian 1999; Teixeira and Miranda 2004). In a flow with mean winds that turn uniformly with height, the likelihood of gravity-wave breaking through convective overturning is enhanced as waves approach a three-dimensional (3D) critical level (e.g. Broad 1999). The investigation of directional critical levels has been limited to hydrostatic gravity waves generated by simple terrain geometries and idealized flows, which has made the theoretical interpretation more tractable (e.g. Broad 1995; Shutts 1995; Teixeira et al. 2004). The extent to which these more basic studies are applicable to mountain waves forced by complex terrain in real atmospheric flows with directional wind shear is not known. One of the objectives of the Mesoscale Alpine Programme (MAP) is to obtain a more complete understanding of the dynamics governing the turbulent breakdown of orographically generated gravity waves, particularly those forced by the complex topography of the Alps (Bougeault et al. 2001). On 13 November 1999 during the MAP Intensive Observation Period (IOP) 16, three research aircraft were deployed to measure gravity-wave breaking in the south-western Alps based on guidance from mesoscale numerical weather predictions. The background synoptic-scale flow contained considerable directional wind shear in the lower troposphere. Although the numerical models

3 MOUNTAIN WAVES AND DIRECTIONAL WIND SHEAR 1879 (ii) predicted moderate mountain-wave activity, the gravity waves observed by the aircraft were small in amplitude, particularly in the middle and upper troposphere. The primary objectives of this study are to: (i) Document the characteristics of mountain waves generated by flow over narrow terrain in the presence of directional shear, through the use the available observations and high-resolution numerical simulations. Evaluate the capability of a high-resolution mesoscale model to simulate these gravity-wave characteristics. (iii) Gain a better understanding of the importance of directional critical-level absorption of mountain waves generated by complex terrain. An observational overview of the gravity-wave event is presented in section 2. The model description is contained in section 3, and the nonlinear model results are described in section 4. A linear-theory-based interpretation of the observations and model simulations is discussed in section 5, followed by a summary and conclusions in section NONLINEAR NUMERICAL MODEL AND LINEAR ANALYTIC MODEL DESCRIPTIONS (a) Nonlinear numerical model The numerical simulations of observed and idealized flows in this study are performed using the atmospheric portion of the Coupled Ocean Atmosphere Mesoscale Prediction System (COAMPS R ; Hodur 1997). This prediction system is based on a finite-difference approximation to the fully compressible, non-hydrostatic equations and uses a terrain-following vertical coordinate transformation. In this application, the finite-difference schemes are of second order accuracy in time and space, with the exception of horizontal advection which is fourth order accurate. A time splitting technique with a semi-implicit formulation for the vertical acoustic modes is used to efficiently integrate the compressible equations (Klemp and Wilhelmson 1978). A prognostic equation for the turbulence kinetic energy (TKE) budget is used to represent the planetary boundary-layer and free-atmospheric turbulent mixing and diffusion (Hodur 1997). The Louis (1979) surface-layer parametrization, which makes use of a surface energy budget based on the force-restore method, is used to represent the surface fluxes. Subgrid-scale moist convection is represented using the Kain and Fritsch (1993) parametrization. The grid-scale evolution of the moist processes is explicitly predicted from budget equations for cloud water, cloud ice, raindrops, snowflakes, and water vapour (Rutledge and Hobbs 1983). The short- and long-wave radiation processes are represented following Harshvardhan et al. (1987). The full suite of physical parametrizations is used in the real-data simulation. The idealized simulations are dry with a free-slip lower-boundary condition, and include free-atmosphere vertical diffusion derived from the explicitly predicted TKE. The initial fields for the real-data simulation are created from multivariate optimum interpolation (MVOI) analyses of upper-air sounding, surface, commercial aircraft, and satellite data that are quality controlled and blended with the 12 h COAMPS forecast fields (Barker 1992). An incremental update data assimilation procedure is used in conjunction with the MVOI, which enables mesoscale phenomena to be retained in the analysis increment fields. Lateral boundary conditions for the outermost grid mesh are based on the Navy Operational Global Analysis and Prediction System (NOGAPS) forecast fields.

4 1880 J. D. DOYLE and Q. JIANG Figure 1. The COAMPS (see text) model-simulated 500 hpa geopotential height (interval 30 m) valid at 1200 UTC 13 November 1999 (12 h forecast) from the outer grid mesh (spacing x = 45 km). The terrain contours are shown using the grey scale (250 m increment). The locations of the second ( x = 15 km), third ( x = 5km), fourth ( x = 1.7 km),andfifth( x = 0.56 km) nested grid meshes are represented by the bold rectangles. The domain configuration for the real-data simulation is shown in Fig. 1; it contains five horizontally nested grid meshes of 97 97, 97 97, , , and points, with horizontal grid increments on computational meshes of 45, 15, 5, 1.7, and 0.56 km, respectively. The model contains 55 vertical levels, on a nonuniform vertical grid consisting of an increment of 10 m at the lowest level, subsequent increments gradually increasing to 500 m at 7 km, and thereafter 500 m increments continuing to the model top at 22 km. A radiation upper-boundarycondition is applied to mitigate the reflection of vertically propagating gravity waves. The topographic data are based on the US Defense Mapping Agency (DMA) 100 m resolution dataset that enables the fine-scale topographic features of the Alps to be well represented, particularly on the innermost grid mesh shown in Fig. 2. (b) Linear analytic model The mountain-wave dynamics are examined from a linear theory perspective through the use of an analytic model. Linear theory is used to explore the characteristics of the gravity waves observed on 13 November 1999, in part because of the relatively small amplitude of the observed wave. The formulation of the 3D linear theory model, which features a generalized number of vertical layers, extends the early theoretical work of Sawyer (1962) as well as the three-layer model of Smith et al. (2002) using

5 MOUNTAIN WAVES AND DIRECTIONAL WIND SHEAR 1881 Figure 2. Alpine domain for the innermost grid mesh (spacing x = 0.56 km) with terrain height contoured every 500 m and shown by the grey scale (250 m increments). Segment AA is the mean flight path of the three research aircraft. The launch location of dropsondes D1, D2, D3, D4, and D5 are shown by the solid circles. The thin solid line corresponds country boundaries. The locations of the Massif du Pelvoux (P) and the Aigle de Chambeyron (C) are shown. the Fast Fourier transform (FFT) method of Smith (1980). Following Smith (2001) and Smith et al. (2002), the linearized mountain-wave equations using the Boussinesq approximation for the 3D (x, y, z) perturbation velocity components, u, v, w, momentum, continuity, and density, ρ, conservation equations are: U u x + V u y + w U ( ) 1 p z = ρ 0 x, (1) U v x + V v y + w V ( ) 1 p z = y, (2) U w x + V w y = ρ 0 ( ) 1 p ( ) g ρ 0 z ρ, ρ 0 (3) u x + v y + w = 0, z (4) U ρ x + V ρ y + w dρ = 0, (5) dz

6 1882 J. D. DOYLE and Q. JIANG where p,andρ are the perturbation pressure, and perturbation density, respectively. The environmental wind and density profiles are represented by U(z), V(z),andρ(z); the reference density is ρ 0 and g is the acceleration due to gravity. Equations (1) (5) are combined into a single equation, and a FFT is applied to obtain a single equation for the vertical displacement, η(k, l, z), σ 2 2 η σ η + 2σ z2 z z + (k2 + l 2 )(N 2 ασ 2 ) η = 0, (6) where σ is the intrinsic frequency defined as σ(k,l)= Uk + Vl. Non-hydrostatic effects are included when α = 1 and hydrostatic balance is imposed when α = 0. Within each layer the wind speed, direction and static stability are constant, and it follows that σ/ z = 0 within the n layers (i = 1, 2, 3,...,n) leading to the solution of (6) as: η i (k, l, z) = A i exp(im i z) + B i exp( im i z), (7) where the amplitude coefficients of the upward and downward propagating waves, with respect to wave energy, are A i and B i, respectively. In the vertically propagating case, when Ni 2 >ασi 2, the vertical wave number, m i, can be expressed as: { (N m i = (k 2 + l 2 ) 1/2 2 i ασi 2) σ 2 i } 1/2 sgn(σ i ), (8) and otherwise in the evanescent case: ( (ασ m i = i(k 2 + l 2 ) 1/2 2 i Ni 2) ) 1/2. (9) The x-component of the velocity perturbation in Fourier space can be expressed as: ( )( ) k pi û i (k, l, z) =, (10) the y-component of the velocity perturbation as: ( )( l pi v i (k, l, z) = and the vertical velocity as: The pressure is given by: p i (k, l, z) ρ 0 σ i σ i σ 2 i ρ 0 ρ 0 ), (11) ŵ i (k, l, z) = iσ i η i. (12) = i m i (N 2 i ασ 2 i ){A i exp(im i z) + B i exp( im i z)}. (13) Between each layer, a matching condition is applied across the interface based on continuity of mass and pressure such that: η = 0, (14) and it follows that integrating (6) across the layer interface and applying (14) yields: σ 2 η = 0. (15) z

7 MOUNTAIN WAVES AND DIRECTIONAL WIND SHEAR 1883 The use of a generalized number of multiple layers enables wind shear and vertically coherent structures in the static stability, such as inversions, to be represented in a piecewise manner. If the intrinsic frequency for a wave component changes sign or is zero at the interface, an absorption condition is applied. A radiation boundary condition is applied at the top (nth) layer such that B n = 0. The lower boundary condition is η(x, y, 0) = h(x, y), whereh is the terrain height, which can be transformed into Fourier space as A 1 + B 1 = ĥ(k, l). The steady-state parcel displacement field is computed from: ( ) z η(x, y, z) = exp η(k, l, z) exp(ikx + ily) dk dl, (16) 2H where H is the density-scale height. Similarly the steady-state solution for the other variables u, v, w, andp are computed using an inverse FFT. Non-Boussinesq effects are represented by the coefficient exp(z/2h). The linear theory model includes a parametrization of the dissipation of downward propagating waves within the boundary layer, which is represented by a reflection coefficient, q, in the linearized lower boundary condition, and is expressed as A 1 = ĥ(k, l) qb 1 (Smith et al. 2002). Boundary-layer dissipation of the downward propagating wave through turbulence or critical-level processes can be represented by q<1. In the application of the linear theory model to the 13 November 1999 gravity-wave event, the observations suggest that boundary-layer dissipation of downward propagating waves may not be an important issue. A reference height, Z ref, is defined as the depth of the upstream stagnant or topographically blocked layer. Thus, the effective terrain height is defined as h(x, y) Z ref,whereh(x, y) is the height of the topography. For applications using the DMA 100 m resolution dataset, the topography is reduced around the outer edges of the computational domain using a Gaussian filter with an e-folding scale of 100 km, in order to avoid spurious wave wrapping associated with the FFT method. Additionally, relatively weak dissipation at the lower boundary, specified using q = 0.9, effectively mitigates any wave wrapping that could potentially contaminate the solution. In the region of interest, the impact on the mountain waves for the case of q = 0.9 is negligible. All of the linear theory applications in this study are solved on a Cartesian grid. 3. OBSERVATIONS Several different observational sources are used to describe the synoptic-scale and mesoscale atmospheric conditions on 13 November 1999 during MAP IOP 16, as well as the mountain waves generated by the narrow topography of the western Alps. This set of observations includes the routine radiosondes, satellite imagery, and research aircraft measurements that include continuous flight-level data, lidars, and Global Positioning System (GPS) dropwindsondes. (a) Synoptic-scale overview The synoptic-scale features of interest at 500 hpa at 1200 UTC 13 November 1999 include a long-wave trough over south-western Europe, a ridge that extended from the mid-atlantic to Great Britain, and a strong north-westerly mid-tropospheric jet over northern Europe (Fig. 1). A closed low-pressure region at 500 hpa was present over the Iberian Peninsula that resulted in a broad layer of south-easterly flow impinging on the western Alps. The COAMPS 12 h simulated wind speeds at 850, 500, and 250 hpa levels

8 1884 J. D. DOYLE and Q. JIANG Figure 3. The COAMPS (see text) model-simulated 12 h forecast wind vectors valid at 1200 UTC 13 November 1999 for: (a) 850, (b) 500, and (c) 250 hpa, for the second nested grid (spacing x = 15 km). The wind speed is represented by the grey scale (5 m s 1 increment). Radiosonde winds are shown by the station model wind vectors with full and half barbs representing 5 and 2.5 m s 1, respectively; and a reference vector is given in the lower right-hand corner. valid at 1200 UTC 13 November 1999 for the second grid mesh ( x = 15 km) are shown in Fig. 3. The strongest low-level winds at 850 hpa were organized in a swath from the extreme western portion of the Alps in south-eastern France extending toward the northwest (Fig. 3(a)). The winds upstream of the maritime Alps in north-western Italy were less than 10 m s 1 from an easterly direction. The middle- and upper-tropospheric winds exhibited moderate directional shear, with south-easterlies upstream of the western Alps at 500 hpa (Fig. 3(b)) veering to a southerly direction in the upper troposphere and lower stratosphere at the 250 hpa level (Fig. 3(c)). An extensive region of clouds was associated with the synoptic-scale cyclone and trough along the western Alps, as apparent in the NOAA AVHRR visible satellite image valid at 1407 UTC 13 November 1999 shown in Fig. 4. Low-level clouds were present upstream of the western Alps in Italy, as confirmed by Meteosat infrared imagery at 1400 UTC (not shown). A narrow elongated clear zone was present downstream and just to the west of the south-western Alps, consistent with lee-side descent associated with the south-easterly flow over the Alpine crest.

9 MOUNTAIN WAVES AND DIRECTIONAL WIND SHEAR 1885 Figure 4. NOAA-AVHRR visible satelliteimage valid at 1407 UTC 13 November (Courtesy of Institutfür Physik der Atmosphäre, Deutsches Zentrum für Luft- und Raumfahrt). The black and grey line segment denotes the flight path of the research aircraft. (b) Research aircraft measurements On 13 November 1999 during MAP IOP 16, a co-ordinated mission was executed involving three research aircraft: the NSF/NCAR Electra, the UK Met Office C130, and the Deutsches Zentrum für Luft- und Raumfahrt (DLR) Falcon between 1200 and 1400 UTC. The choice of flight path flown by the three research aircraft, shown in Fig. 2, was based on the forecasts from a number of mesoscale numerical weather prediction models, including COAMPS, the Canadian MC2 and the Swiss SM, that indicated the presence of moderate-amplitude mountain waves generated by the south-easterly flow over the complex terrain of the western Alps. Additionally, global prediction models such as the European Centre for Medium-Range Weather Forecasts (ECMWF) Integrated Forecasting System also forecast moderate low-level south-easterly flow, consistent with the mesoscale model predictions of mountain waves. Repeated flight segments that crossed the Massif Pelvoux (4102 m: all heights are AMSL (above mean sea level)) and Aigle de Chambeyron (3411 m) in the French Alps (see Fig. 2) were flown nearly parallel to the large-scale wind direction near the top of the Alpine crest (Figs. 3(a) and (b)). The three research aircraft deployed a total of 18 GPS dropwindsondes along the flight paths during the period UTC. The potential temperature, wind speed, and direction are shown in Fig. 5 for five representative dropsondes, deployed at locations D1 D5 (see Fig. 2). Dropsondes D1, D3, and D4 were deployed from the Electra (Figs. 5(a) and (c)), D2 from the C130 (Fig. 5(b)), and D5 from the Falcon (Fig. 5(d)). The vertical scales differ in Figs. 5(a) to (d) because the dropsondes were deployed from various altitudes. The dropsonde profile in Fig. 5(d) is only shown below 9.5 km to highlight the lower- and mid-tropospheric structure. The eastern-most deployed dropsonde, D1, located upstream of the Alps, indicates a nearly uniform static stability below 5.5 km (Fig. 5(a)) with N The D2 dropsonde, deployed from the C130, indicates that somewhat weaker static stability is present in the km layer, with N s 1. The D1 dropsonde indicates wind speeds of 5 10 m s 1 in the km layer, which are nearly constant with altitude. The upstream wind speeds are less than 5 m s 1 below 2 km, consistent with upstream flow blocking. Many of the

10 1886 J. D. DOYLE and Q. JIANG Figure 5. Measurements by dropsondes at launch locations (a) D1, (b) D2, (c) D3 and D4, and (d) D5 (see Fig. 2), deployed at 1359, 1330, 1354, 1416, and 1229 UTC 13 November 1999, respectively. Sondes at D1, D3 and D4 were deployed from the Electra, at D2 from the C130 and at D5 from the Falcon. In each panel the potentialtemperature profile (K) is shown on the left; the wind speed (m s 1, black) and direction (, grey) are on the right. Corresponding model profiles for the nearest grid point are shown by the dotted lines. The solid lines in (c) correspond to the 1354 UTC dropsonde at D3 and the dashed lines to the 1416 UTC launch at D4. The surrounding eight model grid points (selected with 2 x spacing) are shown in (a). other gravity-wave events observed during MAP were modulated by significant lowlevel blocking (e.g. Smith et al. 2002; Doyle and Smith 2003; Smith 2004; Jiang and Doyle 2004). Directional vertical shear is present below 6 km, with winds from 90 at 3 km that veer approximately linearly with altitude to a direction of 150 at 4 km (Fig. 5(a)) and 180 between 5.5 and 6 km (Fig. 5(b)). Dropsondes, D2, D3, D4, and D5 (Figs. 5(b) to (d)) indicate that several layers of increased stability were present below 4 km. Stronger low-level flow of m s 1 was measured by dropsondes D2 and D4, consistent with enhanced downslope flow that may be present in the lee of individual peaks. Further aloft, the dropsonde deployed by the Falcon (Fig. 5(d)) exhibits reversed wind shear above 7.5 km with a layer of stagnant flow at 9.4 km.

11 MOUNTAIN WAVES AND DIRECTIONAL WIND SHEAR 1887 It is noteworthy that dropsondes D3 and D4 were deployed at 1354 and 1416 UTC, respectively, and are in reasonably close agreement (Fig. 5(c)), although some variations over the 22 minute period are apparent, presumably associated with non-stationarity of the flow. The dropsondes do not provide conclusive evidence that the low-level flow in the lee was sufficiently weak or stagnant to induce a boundary-layer critical level (e.g. Smith et al. 2002). For example, data from dropsondes D2, D3, and D4 contain wind speeds of 3 5 m s 1 or greater within the boundary layer, which is evident from the near-neutral lapse rate below 2500 m in D2, or 2750 m in the case of D3 and D4 (Fig. 5(c)). Resonant lee waves will develop if the Scorer Parameter, l S, given by: l 2 S (z) = N 2 U 2 1 U 2 U z 2, (17) decreases rapidly with height (e.g. Scorer 1949), where U(z)is the cross-mountain wind speed and z is the vertical coordinate. The Scorer parameter profile is computed based on smoothed mean profiles of cross-mountain wind speed and potential temperature from dropsondes D1 and D2. The Scorer parameter calculations suggests that mountain waves with horizontal wavelengths of approximately 5 km or less are trapped, while longer wavelengths than 5 km can permeate the wave duct. Backscatter from the downward looking Scanning Aerosol Backscatter Lidar (SABL) corresponding to the first Electra flight segment is shown in Fig. 6. Enhanced backscatter from three layers of clouds between 2 and 4 km AMSL are present upstream of the large peaks. A single layer of clouds that undulates with altitude is apparent in the lee of several peaks, consistent with the presence of low-level mountain waves. The largest vertical excursion of the cloud layer is approximately 800 m in altitude and located in the lee of the first major peak, Aigle de Chambeyron; this is followed by several smaller-amplitude waves trailing downstream. Other peaks downstream force additional smaller-amplitude mountain waves. The horizontal wavelength of the dominant gravity-wave signature superimposed on the low-level clouds is approximately 3 5 km. The vertical velocity (w) obtained from the in situ Electra and C130 flight-level measurements is shown in Fig. 6(a). The Electra executed repeated segments at altitudes of 5.5 and 6.4 km, while the C130 performed measurements at 7.6 and 8.3 km. The vertical velocities for the repeated segments, flown at altitudes of 5.5 and 7.6 km, differ substantially between passes in contrast to several other MAP events that exhibited less transience (e.g. Smith et al. 2002; Doyle and Smith 2003; Jiang et al. 2005). At the 5.5 km altitude, the vertical velocity data contain several short-wavelength features with maxima as large as 3 m s 1, particularly prominent above and downstream of the easternmost peak above 3000 m, the Aigle de Chambeyron Massif. At the 6.4 km altitude, the amplitude of the vertical velocity signature from the Electra is damped considerably, with maxima of approximately 1 m s 1 that are not well correlated with the terrain. The vertical velocities at segments with altitudes of 7.6 and 8.3 km exhibit high-frequency waves characterized by relatively small amplitudes that have vertical velocities generally less than ±1 ms 1. A few waves of larger horizontal wavelength and amplitude are present at the 7.6 km altitude in the km region of Fig. 6(a), apparently linked with flow over the Aigle de Chambeyron Massif and the complex terrain located immediately to the west. The horizontal wind speed and direction corresponding to the Electra and C130 flight segments are shown in Fig. 6(b). The eastern portions of the flight segments

12 1888 J. D. DOYLE and Q. JIANG Figure 6. Cross-section of the backscatter coefficient (db) from the Scanning Aerosol Backscatter Lidar for the first Electra flight segment valid UTC 13 November 1999, represented by grey shading (grey scale on right-hand side) with terrain height shown by the black contour. Also shown are: (a) vertical velocities (w,ms 1 ); and (b) wind speeds (black, m s 1 ) and wind directions (grey, ) based on in situ measurements. The flight transects shown from the top down correspond to a C-130 segment at 8.3 km (AMSL), three C-130 segments at 7.4 km, an Electra segment at 6.4 km, and three Electra flight segments at 5.5 km. The mean low-level wind direction is from right to left. indicate that the upstream velocity was approximately 5 10 m s 1.Thein situ flightlevel data indicate veering of the wind direction with altitude from 140 at 5.5 km to 200 at altitudes 7.6 and 8.3 km. Downstream of the Aigle de Chambeyron Massif, the wind speeds for the 5.4, 7.6, and 8.3 km segments are reduced to 3 m s 1 or less, with localized regions of cross-mountain wind component that are reversed. The flight segment at the 6.4 km altitude exhibits much less variability in the wind direction and speed.

13 MOUNTAIN WAVES AND DIRECTIONAL WIND SHEAR 1889 Figure 6. Continued. Backscatter from the downward-looking Differential Absorption Lidar (DIAL) onboard the DLR Falcon is shown in Fig. 7 for the third flight segment. The DIAL has proved to be a valuable instrument capable of identifying gravity-wave signatures in cloud and aerosol layers (e.g. Smith et al. 2002; Volkert et al. 2003). The DIAL backscatter indicates the presence of a deck of cirrus clouds in the layer 9 11 km downstream of the eastern portion of the section. Three layers of low-level clouds are present upstream of the topography with bases at approximately 2.2, 3.0 and 3.9 km. The altitude of the middle cloud layer undulates in the lee of the larger peaks, in particular the Aigle de Chambeyron Massif, which is consistent with the presence of mountain waves. A series of low-level waves are present in the lee of Aigle de Chambeyron, as indicated by the DIAL and SABL (Fig. 6) backscatter, and in situ flight-level data (Fig. 6(a)), which is suggestive of trapped lee waves. Several layers of

14 1890 J. D. DOYLE and Q. JIANG Figure 7. Backscatter coefficient (db, colour scale) from the Differential Absorption Lidar (DIAL) for the third Falcon flight segment valid UTC 13 November The terrain height is represented by the black contour. The two red lines are used to illustrate representative isopleths of backscatter ratio. The dashed black lines highlight several schematic equiphase contours. The mean low-level wind direction is from right to left. enhanced backscatter, associated with aerosols or moisture, are present above the lowlevel cloud layers and are highlighted in red in Fig. 7. The schematic equiphase contours shown in Fig. 7 suggest that the waves had little tilt with altitude, which is consistent with horizontal energy propagation and trapping. An exception is the upstream tilted primary wave in the lee of Aigle de Chambeyron. The layers with enhanced backscatter indicate that the amplitude of the waves decays rapidly with height, this is likely to be associated with absorption due to the directional critical layer acting in concert with the evanescent wave characteristics. Co-existence of these two wave processes can only occur for wave vectors of very different azimuth angles, which is generally the case for waves forced by 3D topography. At 3.5 km altitude, the peak-to-trough amplitude of the wave immediately in the lee of Aigle de Chambeyron is 800 m, as delineated by the backscatter from the cloud layer. However, the amplitude is reduced by 50% or more to m at the 6 km level. The amplitude decay suggested by the lidar backscatter is also consistent with the in situ flight-level data (Fig. 6). It is noteworthy that for a finite time period, the concept of critical-level absorption may be more appropriately referred to as critical-level dispersion, as there is no certainty that absorption will take place because the vertical group velocity approaches zero near the critical level and the packet never reaches the critical level. The vertical velocity power spectrum for the first and second Electra flight segments at 5.5 km AMSL is shown in Fig. 8(a) based on the 25 Hz dataset. The power spectrum indicates the existence of multiple maxima with the most prominent corresponding to wavelengths of approximately 28 and 9 km. Additionally, a few short wavelength maxima are present at scales of 1 km or less. The two flight segments are in relatively good agreement in terms of the spectral characteristics. The power spectral density of the terrain derived from a 100 m resolution digital elevation model and interpolated to the research aircraft flight path is shown in Fig. 8(b). Dominant spectral maxima exist in the terrain field along the flight path at wavelengths of 27, 8, and 2 km, very similar to the dominant wavelengths noted in the vertical velocity.

15 MOUNTAIN WAVES AND DIRECTIONAL WIND SHEAR 1891 Figure 8. Power spectral density as a function of frequency for: (a) vertical velocity (m 2 s 2 Hz 1 )forthefirst two Electra flight segments, and (b) terrain height (m 2 Hz 1 ) along the flight segment. Wavelengths corresponding to selected peaks in the spectrum are given in (a) and (b). 4. NUMERICAL SIMULATION A numerical simulation using COAMPS was performed using five nested grid meshes with a minimum horizontal resolution of 0.56 km and 55 vertical levels. The model-simulated winds for the second nested grid ( x = 15 km) valid at 1200 UTC 13 November 1999 (12 h) are shown in Fig. 3 with the available observations for 850, 500, and 250 hpa levels. The simulated winds are in reasonable agreement with the radiosonde observations, and accurately depict the directional wind shear over the south-western portion of the Alps, which varies from an east-south-easterly in the hpa layer to a southerly direction in the hpa layer. The simulated 500 hpa geopotential height valid at 1200 UTC, shown in Fig. 1, is in very close agreement with the ECMWF analysis and corresponding 12 h forecast. The 12 h COAMPS simulation indicates a closed low over the Iberian Peninsula with an enhanced geopotential-height gradient positioned over the western Alps, which results in a moderate south-easterly low-level flow impinging on the Alpine barrier (Fig. 3). Additionally, the simulated temperature and winds valid at 1200 UTC are in close agreement with the ECMWF analyses and radiosonde observations, which implies that the COAMPS simulation accurately captures the synoptic-scale flow. The model-simulated potential temperature, wind speed and wind direction interpolated to the dropsonde locations D1 D5 are shown in Fig. 5 together with the dropsonde observations valid between 1229 and 1416 UTC. The model profiles are valid to the nearest 15 minutes of the dropsonde deployment time. In order to illustrate the spatial variability of the model simulation in the vicinity of the dropsonde location, the eight model grid points surrounding D1 using a 2 x grid spacing are shown in Fig. 5(a). The model-simulated potential temperature is in reasonable agreement with the upstream dropsonde D1, however, the wind speed is 2 4 m s 1 too strong in the 3 5 km layer. The model profiles surrounding dropsonde D1 exhibit relatively little variance,

16 1892 J. D. DOYLE and Q. JIANG Figure 9. Backscatter coefficient (db, grey scale) from the Scanning Aerosol Backscatter Lidar for the fourth Electra flight segment valid at UTC 13 November The model-simulated isentropes (2 K interval) are shown from the innermost grid mesh (spacing x = 0.56 km) valid at 1415 UTC and interpolated to the flight path. Air parcel displacements for the four Electra flight segments, derived from flight-level data and computed from Eq. (18), are represented by the bold black contours. The mean low-level wind direction is from right to left. with the exception of the wind direction below 2.25 km where wind speeds are less than 5 m s 1. Overall the model simulation captures the gross features of the profiles including the presence of low-level and elevated inversions, and directional and speed shear. However, there are several noteworthy discrepancies between the model and observed profiles. For example, the simulated low-level inversion and wind speed maxima are too strong, and the wind direction has too strong a southerly component above 6 km at location D2. Dropsondes D3 and D4, shown in Fig. 5(c), were deployed nearly in the same location (see Fig. 2) and approximately 20 minutes apart. These two profiles illustrate some notable variations in the wind speed profiles and within the elevated inversions; these are presumably due to the non-stationary character of the flow and horizontal small-scale variability, and underscore the challenges for a mesoscale model to capture the vertical structure in complex terrain. A vertical cross-section of the model-simulated isentropes from the 0.56 km resolution grid mesh projected along the flight path for the Electra valid at 1415 UTC 13 November is shown in Fig. 9, together with the backscatter coefficient from the SABL for the fourth Electra flight segment corresponding to the time period UTC. The simulated dry static stability is largest below approximately 3 km, decreases slightly

17 MOUNTAIN WAVES AND DIRECTIONAL WIND SHEAR 1893 in the 3 5 km layer, and then decreases substantially in the 5 7 km layer. The descent and ascent of the simulated isentropes associated with the first of several prominent waves immediately in the lee of Aigle de Chambeyron and Massif du Pevoux are in reasonable agreement in terms of vertical displacements and horizontal positions relative to the cloud-layer backscatter undulations. However, the simulated smaller-scale waves that are apparent downstream of the major peaks are not as well correlated with the wave signatures in the backscatter data. Air-parcel vertical displacements (η) for the four Electra flight segments are also shown in Fig. 9; these are computed from the aircraft-derived vertical velocity (w) and horizontalvelocity (U) through the application of the steady-state vertical displacement equation following Smith et al. (2002a): η(x) = x 0 (w/u) dx. (18) The mean vertical motion along the flight segment is removed and the integration of (18) is initiated at the upstream point of the flight segment. Two well-defined prominent mountain-wave crests and troughs are apparent in the displacement heights just to the lee of Aigle de Chambeyron in all three Electra legs at 5.5 km altitude. Farther downstream, the displacement heights are much smaller in amplitude and less well correlated between the three flight legs. The in situ flight-level data and displacement height calculations at 6.4 km indicate much less wave activity than at 5.5 km (see Fig. 6). A direct comparison of the in situ vertical velocity for flight legs 2 and 4 with the COAMPS simulation interpolated to the aircraft location is shown in Fig. 10. The model simulation and observations both indicate large-amplitude waves in the lee of the two major peaks. In general, the model-simulated vertical velocity is too large, particularly in the lee of the prominent terrain peaks in leg 2. The low-level winds in the km layer upstream of the Alpine crest are in general 1 5 m s 1 stronger than those observed (Figs. 5(a) and (b)), which is consistent with simulated vertical velocity extrema that are larger than observed in the lee of the highest terrain. The model appears to capture the character of the wave response including the horizontal wavelength, however, the simulated individual wave crests and troughs are not particularly well correlated with the observations. It is noteworthy that the model produces the strongest upward vertical velocity, approximately 3 m s 1, immediately to the lee of Aigle de Chambeyron during flight segment 2, which is the location of the strongest observed upward vertical velocity of 2.5 m s 1. The model-simulated horizontal wind vectors and vertical velocity at 5 km AMSL valid at 1200 UTC 13 November is shown in Fig. 11 for the 0.56 km grid mesh. The flow at this level is generally south-easterly with the mountain-wave crests oriented southwest to north-east. The vertical velocity pattern is quite complex, in part forced by the 3D terrain below. The simulated vertical velocity field indicates that the strongest wave activity is associated with flow over the two most prominent massifs, in general agreement with the flight-level data. Two vertical cross-sections valid at 1200 UTC oriented along the aircraft flight path and midtropospheric flow direction are shown in Figs. 12(a) and (b). The simulated along-cross-section wind speed and potential temperature (Fig. 12(a)) indicate that the rather weak upstream flow forces smallamplitude mountain waves below 5 km. The amplitudes of the waves generated by the highest peaks decrease with increasing altitude below the weak-stability layer (e.g. Fig. 12(b)). Regions where Ri is less than unity are generally located in the boundary layer and within the largest-amplitude waves below 5 km that are generated by the two prominent massifs, suggestive of shear-induced wave breaking within the directional critical layer. The simulated wind direction (Fig. 12(b)) indicates that the

18 1894 J. D. DOYLE and Q. JIANG Figure 10. Vertical velocity (m s 1 ) derived from the Electra in situ observations (dashed) and COAMPS simulation interpolated to the aircraft location (solid) for: (a) leg 2 and (b) leg 4; (c) shows the digital elevation model terrain interpolated to the aircraft location for legs 2 and 4. model captures the directionally sheared flow in the lower troposphere, which ranges from easterly flow below 2 km to southerly flow near 6 km. Within the weak-stability region aloft in the 6 8 km layer, the vertical velocity exhibits little correlation with the underlying terrain, similar in character to the in situ C130 research aircraft observations (Fig. 6(a)). In order to further illustrate some of the characteristics of the gravity waves, the vertical flux of horizontal momentum was computed from the Electra and C-130 research aircraft flight-level data, and from the model simulation projected along the aircraft flight paths. The horizontal momentum flux components are defined as: d M x = ρ u w dx, (19) d 0 and M y = ρ d v w dx, (20) d 0 where d is the length of the aircraft flight track, and primes represent deviations from the along-flight-track mean. A band-pass filter on the aircraft data is used to

19 MOUNTAIN WAVES AND DIRECTIONAL WIND SHEAR 1895 Figure 11. Model-simulated vertical velocity and horizontal wind vectors at 5 km AMSL valid at 1200 UTC 13 November 1999 for the innermost grid mesh ( x = 0.56 km). The contour interval is 1 m s 1 ; dashed contours indicate down-wind motion, with the zero contour suppressed. The model terrain height is shown by the grey scale (500 m increment). The bold AA segment is the mean flight path of the three research aircraft and corresponds to the location of model vertical cross-sections. The thin solid line corresponds to country boundaries. select wavelengths between 2 and 30 km. The momentum flux profiles for model simulation times of 1200, 1230, 1300, and 1330 UTC, which approximately span the time period of the aircraft measurements, are shown in Fig. 13. Overall, the momentum flux is relatively small with considerable transience apparent in both components of the simulated fluxes. The model-simulated zonal momentum flux, M x, is positive below approximately 6.5 km, which is consistent with our expectations, having large-scale easterly flow in the low levels (Fig. 13(a)). The momentum-flux profiles oscillate about zero aloft. The meridional momentum flux, M y, is generally negative and smaller in the low-levels, and approaches zero at 5.5 km (Fig. 13(b)). The profile at 1330 UTC exhibits a negative flux throughout the troposphere, which may be a reflection of strengthening southerlies aloft during this time period in the simulation. The aircraft momentum fluxes for the eight repeated flight segments are also shown in Figs. 13(a) and (b). The momentum flux magnitude is larger for the flight legs at 5500 m relative to the fluxes aloft which are near zero. Overall, the simulated momentum fluxes are larger in magnitude than the aircraft momentum fluxes, particularly at higher altitudes. The model simulation exhibits a greater spread in the momentum flux for the various times during the aircraft measurement period. It should be noted that, since measurements were made at only a few different altitudes in the 2D transect, the observed momentum flux profile

20 1896 J. D. DOYLE and Q. JIANG Figure 12. Vertical cross-section along the research aircraft flight path (AA in Fig. 11) valid at 1200 UTC 13 November 1999, of: (a) wind speed (grey scale, m s 1 ), and isentropes (contours, 2 K interval), with regions where the Richardson number, Ri, is unity or less shown by hatching; (b) wind direction (grey scale, ) and vertical velocity (contours, 1 m s 1 interval; dashed contours indicate downward motion with zero contour suppressed). The mean low-level wind direction is from right to left.

21 MOUNTAIN WAVES AND DIRECTIONAL WIND SHEAR 1897 Figure 13. Vertical flux of horizontal momentum based on the model simulation projected along the aircraft transect (Fig. 2) for: (a) the zonal component, and (b) the meridional component. The simulated momentum flux components are shown for the simulation times valid at 1200 UTC (12 h forecast, full lines), 1230 UTC (12.5 h, dashed), 1300 UTC (13 h, dotted), and 1330 UTC (13.5 h, dash-dotted). The momentum flux components computed from the in situ aircraft data are shown by X for the Electra and + for the C130. is under-sampled, which makes the quantitative comparison between the model and observed momentum fluxes difficult. 5. LINEAR THEORY PERSPECTIVE In order to aid the interpretation of the 13 November gravity-wave observations and nonlinear numerical simulation, and the more general problem of gravity waves in directionally shear flows, we apply steady-state linear theory in 3D for several simplified flows that are relevant to this case. In the linear model, waves may be absorbed at interfaces associated with vertical directional shear of the horizontal wind. If the horizontal wind vectors in the two adjacent layers are V i = (U i,v i ) and V i+1 = (U i+1,v i+1 ), respectively, waves with horizontal wave vectors satisfying σ i σ i+1 < 0are absorbed at the interface between layers i and i + 1, where σ is the intrinsic frequency given by σ Uk + Vl. For stationary waves, σ i = V i K i is the intrinsic frequency in the ith layer, where K = (k, l) is the horizontal wave vector. According to linear theory, a vertically propagating wave is absorbed at a critical level, which is defined as V K = 0. The consistency between our interfacial absorption condition and the linear criticallayer theory can be illustrated conceptually by considering an interface between two adjacent layers, with horizontal wind vectors V i and V i+1, as a layer of a finite depth characterized by continuous wind turning from V i to V i+1.asv i K and V i+1 K are of opposite sign in the shear layer, there must be a level where the wind vector V satisfies V K = 0, which is a critical level to wave K. In the first test, we compare our linear calculations with those of Shutts and Gadian (1999) for a hydrostatic wave generated by stratified flow over a 200 m bell-shaped hill

22 1898 J. D. DOYLE and Q. JIANG Figure 14. Vertical velocity at 6.3 km obtained from linear theory with a mean state consisting of turning winds and constant stability based on the Shutts and Gadian (1999) hydrostatic-wave case. The dashed contours are negative. The contour interval is 0.01 m s 1. The 60 m terrain contour is shown by the solid circle. The arrow in the lower left indicates the wind direction at this level. Tick marks along the axes are shown every 10 km. The domain shown is km. of half-width 20 km in the presence of turning flow. In this case, the mean-state v-wind component is 0 at the surface, and has a constant shear of s 1. The mean state also includes a u wind component of 10 m s 1 that is independent of altitude, and N = 0.01 s 1. The linear solution is obtained using the FFT method with a grid increment of 2 km and 30 vertical layers. The vertical velocity at 6.3 km, shown in Fig. 14, exhibits a pattern similar to the constant-wind solution with a clockwise rotation, as discussed in Shutts and Gadian (1999). With the piecewise representation of the shear in the approach used in this study, the linear solution is very similar to the model simulation and the linear analytic solution of Shutts and Gadian (1999; see their Figs. 8 and 9). This agreement provides some confidence that the linear solutions using the FFT approach with vertically discrete layers are a viable tool to further explore the dynamics of non-hydrostatic waves in the presence of directional shear. A second case considered, assumes a two-layer stratified flow over a bell-shaped mountain with half-width 5 km, and a terrain height of 100 m. The background state consists of a layer between the surface and 3 km with N = s 1,andN = 0.01 s 1 aloft above 3 km, with a constant U = 25 m s 1 in both layers. A two-layer linear theory solution, shown in Fig. 15(a), yields a trapped wave pattern in the vertical velocity commonly referred to as ship waves (e.g. Sharman and Wurtele 1983). A linear theory solution is obtained using the FFT model with a horizontal resolution of 1 km. The linear resonant wavelength for this case is approximately 9.3 km. The linear solution compares well with the COAMPS simulated vertical velocity at 3 km after 4 h using the identical background state and terrain shape parameters shown in Fig. 15(c). The shear layer in the reference state in COAMPS is vertically smoothed over 2 km to avoid shear instability issues. The amplitudes of the waves are damped somewhat in the COAMPS simulation, particularly in the far field; this is likely to be due to numerical reasons. A second linear solution is obtained using a wind direction that is rotated counterclockwise by 90 (Fig. 15(b)). The ship waves comprised of the transverse and divergent waves that are apparent in the constant wind direction flows (Figs. 15(a) and (c))

23 MOUNTAIN WAVES AND DIRECTIONAL WIND SHEAR 1899 Figure 15. Vertical velocity at 3 km obtained from linear theory, with a mean state consisting of a two layer stratification and: (a) constant wind direction, and (b) wind direction rotated by 90 counter-clockwise for flow over a bell-shaped 100 m high hill. A COAMPS (see text) simulation after 4 h is also shown for identical terrain with: (c) constant wind direction, and (d) wind direction rotated by 90 counter-clockwise. The red contours are positive and the blue dashed contours are negative. The contour interval is 0.01 m s 1 following the colour scale on the right. The 25 m terrain height is shown by the solid circle. Tick marks along the axes are shown every 10 km. The domain shown is km. Figure 16. Vertical velocity over the western Alps at 5.5 km obtained from linear theory using a three-layer reference state derived from dropsonde observations for 13 November 1999 using the western Alpine topography. The cases shown are: (a) constant wind direction, and (b) wind direction rotated by 90 clockwise. The reference state wind direction at this level is easterly in (a) and southerly in (b). The 2500 m terrain level is shown by the black contours. The contour interval is 0.2 m s 1 following the colour scale to the right (zero contour suppressed). Tick marks along the axes are shown every 10 km. The domain shown is km.