Trapped lee wave interference in presence of surface friction
Ivana Stiperski1, Vanda Grubišić2
1 Meteorological and Hydrological Service, Zagreb, Croatia
E-mail:
2 University of Vienna, Vienna, Austria
Abstract: Lee-wave interference over twin peaks in the presence of a frictional boundary layer is investigated by means of idealized high-resolution two-dimensional simulations with the NRL COAMPS model. The interference is examined under linear to highly nonlinear flow regimes. The results show that the pattern of positive and negative interference, evident in the wave amplitude and rotor strength, is consistent withthe predictions of linear interference theory. Interference is found to most significantly affect the flow in the lee of the downstream ridge, preventing boundary-layer separation there under conditions of destructive interference and amplifying rotor strength under constructive interference in different flow regimes. Only understrongly nonlinear conditions is the rotor strength amplified by constructive interference beyond that obtained in the lee of a single mountain.
Keywords: boundary layer separation, critical mountain height, nonlinearity, rotors
1 INTRODUCTION
Trapped lee waves form downwind of a mountain ridge under conditions conducive to wave energy confinement within the lower troposphere. The study of nonlinear lee-wave resonances over twin ridges under free slip conditions(Grubišić and Stiperski 2009)shows that the resonance is governed by the ratio of the intrinsic trapped lee wave wavelength s and the ridge-separation distance V. The positive and negative interference patterns are evident in the wave amplitudes and total wave drag. In the presence of a frictional boundary layer, the adverse pressure gradients associated with nonlinear, trapped lee waves are expected to lead to boundary layer separation and formation of recirculating regions (or rotors) underneathlee wave crests (Doyle and Durran 2002). Here we investigate the effects of a frictional boundary layer on the lee wave resonance and rotor strength over twin peaks.
2 MODEL SETUP
Two-dimensional numerical simulations were carried out using the nonhydrostatic COAMPS model initialized with an idealized sounding. The surface windspeed is 10 m s-1, vertical wind shearequal to 6 m s-1 km-1, and the stability N=0.012 s-1is constant within the troposphere (Fig. 1). Friction at the lower boundary is imposed via a partial-slip condition with surface roughnessz0=0.01m. A set of idealized simulationswas conducted for single and double bell-shaped obstacles for mountain heights H=300-1500 mand valley widths V=30-65 km.The mountain half-width is 5 km. The simulations were run for 10 hours during which time a quasi-steady state was reached in the simulations. Figure 1. Initial upstream profileofpotential temperature (K) and horizontal windspeed (m s-1)
3 RESULTS
3.1. Linear regime H< Hc=600m
For twin peaks lower than Hc the wave interference pattern follows the linear interference theory predictions with constructive and destructive interference occurring, respectively, for V=n*s, and V=(2n-1)/2*s, where s is the intrinsic horizontal wavelength downwind of a single obstaclefor the given upstream profile (s=21.8 km) and n>2 is an integer (Fig. 2). The positive and negative interferences are barely evident in the wave amplitudes and horizontal velocity perturbations in the lee of the first obstacle (A1 and U1 over the valley), but leave a clear imprint in the lee of the second obstacle, where the wave amplitude and horizontal velocity perturbation (A2and U2)are enhanced or reduced relative to the value in the lee of a single mountain (As). The non-dimensional wavelengths cluster around integer values. The flow structure over the downstream obstacle is different under constructive and destructive interference (not shown). Whereas under constructive interference the flow resembles that over a single ridge, for destructive interference the potential temperature and horizontal windspeed are symmetric around the downstream peak. For H>Hs=370m, the wave-induced adverse pressure gradientsare strong enough to cause boundary-layer separation and rotor formation in the lee of a single obstacle (Us<0). Within the valley, the rotor strength (U1)remains equal to Us for all V, whereas downstream of the secondary obstacle the decrease in amplitude due to destructive interference is strong enough to prevent reversed flow formation,leading to large differences between the flows under constructive and destructive interference.
3.2. Nonlinear regime Hc<H<Hn=1000m
For H>Hc=600m the regime becomes nonlinear as the mountains are high enough for flow reversals to occur underneath the first lee wave crests, downstream of both obstaclesirrespective of the interference pattern. In the weakly nonlinear regime, Hc<H<Hn=1000 m, the rotors over the valley remain unaffected by the second ridge. The rotors downstream of the second ridge do not attain velocities larger than those downwind of a single ridge, even under constructive interference (not shown).
3.3. Highly nonlinear regime HHn
For H≥Hn=1000m, the nonlinearities start to affect the solutionssignificantly.Even though the interference still retains its linear pattern of occurrence,it is no longer evident in the variations of total drag (Fig.2), and the V/ attainsalso non-integer values The flow within the valley, unaffected by the secondary obstacle with the linear and weakly non-linear regimes, is weaker than in the lee of a single obstacle and exhibits some variation with V. Also,, A2>A1 for all V. For H=1000 m constructive interference significantly enhances the rotors developing in the lee of the second ridge, while the rotors within the valley are weaker. For H=1500m, the enhancement of the rotor circulation is somewhat weaker, possibly due to the nonlinear interaction between the wave field and the PBL flow.
Figure 2. Amplitudes: A2 (solid), A1 (dashed); rotor strength U2 (solid), U1 (dashed); normalized wave drag (D); non- dimensional wavelength (V/), as function of valley width V for H=400 m (left) and H=1000 m (right). Solid red horizontal lines denote values downstream of a single ridge. Vertical linesdenote V for which constructive (orange) and destructive (blue) interference is predicted by the linear interference theory.
Figure 3. Amplitudes and rotor strength downstream of the first (A1,U1) and second (A2,U2) obstacle as function of mountain height H. Different V for which constructive (orange) and destructive (blue) interference occurs are indicated, with the values for single ridge in red. Vertical lines denote threshold mountain heights Hs=370 m, Hc=600 m, and Hn=1000m.
4 CONCLUSIONS
Figure 3 summarizes the influence of mountain height and interference pattern on the wave amplitudes and rotor strength. The significance of the three threshold mountain heights (Hs, Hc and Hn) is best evident in the rotor strength diagrams. Rotors start to form (Ui<0, i=1,2) oncethe critical mountain height Hsfor boundary layer separation is reached. Up to Hc = 600m, the wave amplitudesand rotor strength increaselinearlywith H. Within the valley, the U and A values coincide with those for a single ridge, whereas downwind of the second obstacle a clear distinction between constructive and destructive interference is evident in both of these values. Weaker wave amplitudes under destructive interference require a higher critical mountain height for the rotor formation there. In the strongly nonlinear regime, for H>Hc,, while the distinction between the interference patterns is becoming more apparent over the valley, where an indication of a limit to the rotor strength is also obtained, downstream of the second ridge the distinction is getting smaller and the rotor strength increases with H reduced.
REFERENCES
Doyle, J.D., and D.R. Durran, 2002:The dynamics of mountain-wave-induced rotors. J. Atmos. Sci.,59,186–201.
Grubišić, V., and I. Stiperski, 2009: Lee wave resonances over double bell-shaped obstacles. J. Atmos. Sci., In press.