• Allen, T., , and A. R. Brown, 2002: Large-eddy simulation of turbulent separated flow over rough hills. Bound.-Layer Meteor., 102 , 177198.

    • Search Google Scholar
    • Export Citation
  • Athanassiadou, M., , and I. P. Castro, 2001: Neutral flow over a series of rough hills: A laboratory experiment. Bound.-Layer Meteor., 101 , 130.

    • Search Google Scholar
    • Export Citation
  • Belcher, S. E., , and N. Wood, 1996: Form and wave drag due to stably stratified turbulent flow over low ridges. Quart. J. Roy. Meteor. Soc., 122 , 863902.

    • Search Google Scholar
    • Export Citation
  • Bougeault, P., and Coauthors, 2001: The MAP Special Observing Period. Bull. Amer. Meteor. Soc., 82 , 433462.

  • Brown, A. R., , and N. Wood, 2001: Turbulent form drag on anisotropic three-dimensional orography. Bound.-Layer Meteor., 101 , 229241.

  • Brown, A. R., , and N. Wood, 2003: Properties and parameterization of stable boundary layer over moderate topography. J. Atmos. Sci., 60 , 27972808.

    • Search Google Scholar
    • Export Citation
  • Brown, A. R., , J. M. Hobson, , and N. Wood, 2001: Large-eddy simulation of neutral turbulent flow over rough sinusoidal ridges. Bound.-Layer Meteor., 98 , 411441.

    • Search Google Scholar
    • Export Citation
  • Ding, L., , R. J. Calhoun, , and R. L. Street, 2003: Numerical simulation of strongly stratified flow over a three-dimensional hill. Bound.-Layer Meteor., 107 , 81114.

    • Search Google Scholar
    • Export Citation
  • Doyle, J. D., , and D. R. Durran, 2002: The dynamics of mountain-wave-induced rotors. J. Atmos. Sci., 59 , 186201.

  • Doyle, J. D., , M. A. Shapiro, , Q. Jiang, , and D. Bartels, 2005: Large-amplitude mountain wave breaking over Greenland. J. Atmos. Sci., 62 , 31063126.

    • Search Google Scholar
    • Export Citation
  • Eliassen, A., , and E. Palm, 1961: On the transfer of energy in stationary mountain waves. Geofys. Publ., 22 , 123.

  • Grisogono, B., 1994: Dissipation of wave drag in the atmospheric boundary layer. J. Atmos. Sci., 51 , 12371243.

  • Hodur, R. M., 1997: The Naval Research Laboratory’s Coupled Ocean/Atmospheric Mesoscale Prediction System (COAMPS). Mon. Wea. Rev., 125 , 14141430.

    • Search Google Scholar
    • Export Citation
  • Hunt, J. C. R., , R. Richards, , and P. W. M. Brighton, 1988: Stably stratified shear flow over low hills. Quart. J. Roy. Meteor. Soc., 114 , 14351470.

    • Search Google Scholar
    • Export Citation
  • Jackson, P. S., , and J. C. R. Hunt, 1975: Turbulent wind flow over a low hill. Quart. J. Roy. Meteor. Soc., 101 , 929955.

  • Jiang, Q., , J. D. Doyle, , and R. B. Smith, 2005: Blocking, descent and gravity waves: Observations and modelling of a MAP northerly föhn event. Quart. J. Roy. Meteor. Soc., 131 , 675701.

    • Search Google Scholar
    • Export Citation
  • Jiang, Q., , J. D. Doyle, , and R. B. Smith, 2006: Interaction between trapped waves and boundary layers. J. Atmos. Sci., 63 , 617633.

  • Jiang, Q., , J. D. Doyle, , S. Wang, , and R. B. Smith, 2007: On boundary layer separation in the lee of mesoscale terrain. J. Atmos. Sci., 64 , 401420.

    • Search Google Scholar
    • Export Citation
  • Kim, Y., , S. D. Eckermann, , and H. Y. Chun, 2003: A view of past, present, and future of gravity-wave drag parameterization for numerical climate and weather prediction models. Atmos.–Ocean, 41 , 6598.

    • Search Google Scholar
    • Export Citation
  • Lettau, H., 1962: Theoretical wind spirals in the boundary layer of a barotropic atmosphere. Beitr. Phys. Atmos., 35 , 195212.

  • Louis, J. F., 1979: A parametric model of vertical eddy fluxes in the atmosphere. Bound.-Layer Meteor., 17 , 187202.

  • Louis, J. F., , M. Tiedtke, , and J. F. Geleyn, 1982: A short history of the operational PBL-parameterization at ECMWF. Proc. Workshop on Planetary Boundary Layer Parameterization, Reading, United Kingdom, ECMWF, 59–79.

    • Search Google Scholar
    • Export Citation
  • Mellor, G. L., , and T. Yamada, 1974: A hierarchy of turbulence closure models for planetary boundary layers. J. Atmos. Sci., 31 , 17911806.

    • Search Google Scholar
    • Export Citation
  • Nappo, C. J., , and G. Chimonas, 1992: Wave exchange between the ground surface and a boundary-layer critical level. J. Atmos. Sci., 49 , 10751091.

    • Search Google Scholar
    • Export Citation
  • Ólafsson, H., , and P. Bougeault, 1997: The effect of rotation and surface friction on orographic drag. J. Atmos. Sci., 54 , 193210.

  • Peng, S., , and W. T. Thompson, 2003: Some aspects of the effect of surface friction on flows over mountains. Quart. J. Roy. Meteor. Soc., 129 , 25272557.

    • Search Google Scholar
    • Export Citation
  • Raymond, D. J., , C. S. Bretherton, , and J. Molinari, 2006: Dynamics of the intertropical convergence zone of the east Pacific. J. Atmos. Sci., 63 , 582597.

    • Search Google Scholar
    • Export Citation
  • Richard, E., , P. Mascart, , and E. C. Nickerson, 1989: The role of surface friction in downslope windstorms. J. Appl. Meteor., 28 , 241251.

    • Search Google Scholar
    • Export Citation
  • Schlichting, H., , K. Gersten, , and C. Mayes, 2000: Boundary-Layer Theory. Springer-Verlag, 822 pp.

  • Scorer, R. S., 1949: Theory of waves in the lee of mountains. Quart. J. Roy. Meteor. Soc., 75 , 4156.

  • Smith, R. B., 1980: Linear theory of stratified hydrostatic flow past an isolated mountain. Tellus, 32 , 348364.

  • Smith, R. B., 2007: Interacting mountain waves and boundary layers. J. Atmos. Sci., 64 , 594607.

  • Smith, R. B., , S. Skubis, , J. D. Doyle, , A. S. Broad, , C. Kiemle, , and H. Volkert, 2002: Mountain waves over Mont Blanc: Influence of a stagnant boundary layer. J. Atmos. Sci., 59 , 20732092.

    • Search Google Scholar
    • Export Citation
  • Smith, R. B., , Q. Jiang, , and J. D. Doyle, 2006: A theory of gravity wave absorption by boundary layers. J. Atmos. Sci., 63 , 774780.

  • Stevens, B., 2006: Bulk boundary-layer concepts for simplified models of tropical dynamics. Theor. Comput. Fluid Dyn., 20 , 279304.

  • Stevens, B., , J. Duan, , J. C. McWilliams, , M. Münnich, , and J. D. Neelin, 2002: Entrainment, Rayleigh friction, and boundary layer winds over the tropical Pacific. J. Climate, 15 , 3044.

    • Search Google Scholar
    • Export Citation
  • Therry, G., , and P. Lacarrere, 1983: Improving the eddy kinetic energy model for planetary boundary layer description. Bound.-Layer Meteor., 25 , 6368.

    • Search Google Scholar
    • Export Citation
  • Thompson, W. T., , and S. D. Burk, 1991: An investigation of an Arctic front with a vertically nested mesoscale model. Mon. Wea. Rev., 119 , 233261.

    • Search Google Scholar
    • Export Citation
  • Vosper, S. B., 2004: Inversion effects on mountain lee waves. Quart. J. Roy. Meteor. Soc., 130 , 17231748.

  • Vosper, S. B., , S. D. Mobbs, , and B. A. Gardiner, 2002: Measurements of the near-surface flow over a hill. Quart. J. Roy. Meteor. Soc., 128 , 22572280.

    • Search Google Scholar
    • Export Citation
  • Vosper, S. B., , P. F. Sheridan, , and A. R. Brown, 2006: Flow separation and rotor formation beneath two-dimensional trapped lee waves. Quart. J. Roy. Meteor. Soc., 132 , 24152438.

    • Search Google Scholar
    • Export Citation
  • View in gallery

    Time–height plots of (a) u component (grayscale, interval: 1 m s−1) and isentropes (interval: 0.5 K) and (b) υ component (grayscale, interval: 1 m s−1) and TKE (dashed curves, interval: 1 m2 s−2) derived from a 1D simulation corresponding to U0 = 10 m s−1, N = 0.01 s−1, and z0 = 0.1 m. The bold curve in (b) corresponds to Ri = 0.5.

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    Plots of (a) normalized friction velocity vs surface Rossby number (in logarithm), (b) nondimensional BL displacement thickness and depth vs normalized friction velocity, (c) BL shape factor vs normalized friction velocity, and (d) normalized relaxation time vs normalized friction velocity, derived from three groups of 1D simulations corresponding to U0 = 10 and 20 m s−1 with MY scheme and U0 = 10 m s−1 with TL scheme. The surface roughness ranges from 10−5 to 1 m. The best linear fit curves are included in Table 3 for reference.

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    (a) Plot of sea level pressure perturbation vs BL depth variation for −a < x < a, derived from a pair of control simulations with z0 = 10−2 and 1 m, respectively. (b) Plot of the amplitude and phase angle of compliance coefficient C derived from control simulations vs the normalized surface roughness. The corresponding curves computed using the bulk BL models are included for comparison.

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    Plot of terrain height and the effective terrain heights derived from a pair of COAMPS no-slip simulations with z0 = 10−4 and 10−1 m vs the alongstream distance.

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    (a) Perturbation u component at 10 m vs horizontal distance, derived from five control simulations. The corresponding free-slip simulation is included for reference. (b) The phase shift derived from the control set simulations. The two sets of symbols correspond to the phase shifts of the upstream surface wind minimum and leeside maximum relative to the free-slip run. The phase shift curves derived from the bulk BL models are included for comparison.

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    Plot of pressure drag, momentum, and effective terrain height reduction parameters, χd, χm, and χh, vs the normalized surface friction velocity u*/U0, derived from the COAMPS control simulations with a = 10 km, hm = 10 m, and z0 = 10−4, 10−3, 10−2, 10−1, and 1 m, respectively. The corresponding curves derived from the two BL models are included for comparison.

  • View in gallery

    Plot of drag and momentum flux reduction vs normalized friction velocity derived from two sets of simulations for z0 = 10−4, 10−3, 10−2, 10−1, and 1 m with MY scheme and TL scheme, respectively.

  • View in gallery

    Plot of drag and momentum flux reduction vs the normalized mountain width derived from two sets of simulations for a = 1, 2, 3, 5, 10, 20, 30, 40, and 50 km, with (a) z0 = 0.1 m and (b) z0 = 10−3 m respectively. The corresponding predictions by the two BL models are included for comparison. The parameters used in the BL models are derived from the corresponding COAMPS simulations.

  • View in gallery

    Plot of drag reduction vs the normalized ridge width from a set of COAMPS simulations with z0 = 0.1 m. The corresponding curves computed from Smith BL model (hydrostatic and nonhydrostatic), RBLM (hydrostatic and nonhydrostatic), and the inviscid shear model are included.

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    Drag reduction for a set of simulations with a = 10 km, z0 = 0.1 m, and mountain height hm = 10, 20, 50, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, 1200, and 1500 m.

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    Plot of drag and momentum flux reduction parameter vs the nondimensional mountain height for a set of simulations with a = 10 km, z0 = 0.1 m, and mountain height hm = 10, 20, 50, 100, 150, 200, 300, 400, 500, and 600 m. The estimated χd(M) using the nonlinear BL model in section 6c is indicated by the bold curve.

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    Horizontal plots of (a) the wind speed (interval: 0.03 m s−1) and wind vectors at the surface, and (b) vertical velocity (interval: 0.01 m s−1) and wind vectors at 4000 m derived from the no-slip simulation with z0 = 0.1 m and hm = 100 m. The 50-m terrain contour is included for reference. Only a center part of the domain is shown.

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    Plot of drag and momentum flux reduction parameters vs the normalized surface friction velocity derived from a set of no-slip simulations associated with flow past a 3D hill with a = 10 km and hm = 100 m.

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Impact of the Atmospheric Boundary Layer on Mountain Waves

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  • 1 University Corporation for Atmospheric Research, Monterey, California
  • 2 Yale University, New Haven, Connecticut
  • 3 Naval Research Laboratory, Monterey, California
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Abstract

Interaction between mountain waves and the atmospheric boundary layer (BL) has been investigated using a mesoscale model with BL parameterizations and analytical BL models. The impact of the BL diagnosed from the mesoscale model simulations is in qualitative agreement with the analytical BL models. In general, for stratified flow over a mesoscale ridge, the influence of the BL tends to shift wave patterns upstream, decrease the wave drag by up to 60% and the momentum flux above the BL top by up to 80%, and significantly delay the onset of wave breaking.

It has been demonstrated that the BL effect is governed by four nondimensional parameters, namely, the nondimensional BL depth, BL shape factor, nonhydrostatic parameter, and frictional adjustment parameter. The BL effect is stronger over a rougher surface, which creates a deeper BL with a smaller shape factor (i.e., slower BL flow). The scale dependence of the BL effect is governed by the nonhydrostatic and frictional adjustment parameters. For narrow terrain, the BL reduces mountain drag through rapid inertial adjustment and behaves like an inviscid shear layer, and the momentum reduction across the BL is negligible. The attenuation of the momentum flux across the BL is more substantial over a wider ridge because of the increasing importance of frictional adjustment. The BL effect decreases with increasing ridge height and for 3D terrain.

Corresponding author address: Qingfang Jiang, UCAR Visiting Scientist, Naval Research Laboratory, 7 Grace Hopper Ave., Monterey, CA 93940-5502. Email: qingfang.jiang@nrlmry.navy.mil

Abstract

Interaction between mountain waves and the atmospheric boundary layer (BL) has been investigated using a mesoscale model with BL parameterizations and analytical BL models. The impact of the BL diagnosed from the mesoscale model simulations is in qualitative agreement with the analytical BL models. In general, for stratified flow over a mesoscale ridge, the influence of the BL tends to shift wave patterns upstream, decrease the wave drag by up to 60% and the momentum flux above the BL top by up to 80%, and significantly delay the onset of wave breaking.

It has been demonstrated that the BL effect is governed by four nondimensional parameters, namely, the nondimensional BL depth, BL shape factor, nonhydrostatic parameter, and frictional adjustment parameter. The BL effect is stronger over a rougher surface, which creates a deeper BL with a smaller shape factor (i.e., slower BL flow). The scale dependence of the BL effect is governed by the nonhydrostatic and frictional adjustment parameters. For narrow terrain, the BL reduces mountain drag through rapid inertial adjustment and behaves like an inviscid shear layer, and the momentum reduction across the BL is negligible. The attenuation of the momentum flux across the BL is more substantial over a wider ridge because of the increasing importance of frictional adjustment. The BL effect decreases with increasing ridge height and for 3D terrain.

Corresponding author address: Qingfang Jiang, UCAR Visiting Scientist, Naval Research Laboratory, 7 Grace Hopper Ave., Monterey, CA 93940-5502. Email: qingfang.jiang@nrlmry.navy.mil

1. Introduction

While our understanding of mountain waves has advanced significantly over the past five decades, the effect of the atmospheric boundary layer (BL) has been largely ignored in most mountain-wave studies (e.g., Scorer 1949; Smith 1980), partially because the atmospheric BL is considered shallow relative to the depth of the troposphere, and also because it is difficult to include BL processes in analytical approaches. Over recent years, the influence of the atmospheric BL on mountain waves has been examined in a number of model-based studies. For example, the impact of surface friction on the development of Boulder, Colorado, windstorms has been studied by Richard et al. (1989), who found that bottom friction delays the onset of strong surface winds. Based on three-dimensional simulations, Ólafsson and Bougeault (1997) concluded that BL tends to reduce wave drag and delay the onset of gravity wave breaking. Peng and Thompson (2003) demonstrated that the atmospheric BL top behaves like a material surface and BL weakens mountain waves through altering the effective terrain height, the BL top. In a recent study of an observed large-amplitude breaking wave event over Greenland, Doyle et al. (2005) found that surface cooling could enhance mountain waves and prompt wave breaking.

The importance of the atmospheric BL for mountain waves has been highlighted by a few case studies of gravity wave events observed during the Mesoscale Alpine Program (Bougeault et al. 2001). The BL effect and associated flow blocking have been identified by Smith et al. (2002) and Jiang et al. (2005) as being the key mechanism that accounted for the observed relatively weak waves over the high European Alps. In addition, Smith et al. (2002) show that the BL weakens the trapped waves in the lee of Mont Blanc through partial absorption of wave energy from downward-propagating waves. BL absorption of trapped waves has been further examined by Jiang et al. (2006), who demonstrated that BL absorption could cause an exponential decay of trapped waves with distance. In a companion paper by Smith et al. (2006), a two-dimensional bulk BL model (the Smith BL model) has been proposed to interpret the BL absorption. Recently, the Smith BL model has been extended to study the interaction between a three-dimensional hydrostatic wave and the atmospheric BL (Smith 2007, hereafter S07). A number of interesting predictions have been made regarding the phase shift and the drag and momentum flux reduction of mountain waves, which will be further explored in the following sections. It is also noteworthy that similar bulk BL models have been used to study surface winds over tropical regions (e.g., Stevens et al. 2002) and ITCZ dynamics (Raymond et al. 2006), and these models have been recently reviewed by Stevens (2006).

The interaction between mountain waves and the atmospheric BL has been examined by a few authors using linear theory. Nappo and Chimonas (1992) found that wave momentum flux can be reduced by the presence of a critical level within a BL. Based on linear analysis with a simple turbulence parameterization, Grisogono (1994) found that wave drag can be reduced by about 15%–20% of its surface value by a turbulent BL. In addition, turbulent form drag and parameterization have been the subject of a number of studies (e.g., Brown and Wood 2001, 2003; Kim et al. 2003). The distortion of the atmospheric BL by narrow ridges has been examined analytically by several groups using a “triple deck” approach, which separates the BL into three coupled sublayers with different dominant physical balances (e.g., Jackson and Hunt 1975; Hunt et al. 1988; Belcher and Wood 1996). Focusing on the variation of stress and wind speed near the surface associated with turbulent BL flow past narrow ridges, these studies typically assumed that gravity wave response above the BL is negligible and potential flow solutions apply. More recently, the fractional speedup of BL flow over small hills has been investigated by a number of groups using large-eddy simulations (e.g., Brown et al. 2001; Allen and Brown 2002; Ding et al. 2003), observations (Vosper et al. 2002), and laboratory experiments (Athanassiadou and Castro 2001). The BL effect has also been considered in mountain-wave-induced rotor studies, such as those of Doyle and Durran (2002), Vosper (2004), Vosper et al. (2006), and Jiang et al. (2007), focusing on wave-induced boundary layer separation.

The objective of this study is to deepen our understanding of the interaction between a propagating wave and the BL and the associated dynamics. Particularly, our goal is to identify the control parameters that govern the interaction between the BL and waves, and examine the dependence of the BL effect on BL properties and terrain geometry. The paper is organized as follows. The model configuration and domain setup are described in section 2. An analytical BL model is introduced in section 3. In section 4, the characteristics and control parameters of the simulated BL and its sensitivity to BL parameterization schemes are examined. In sections 5 and 6, the BL effect on wave drag and momentum flux is investigated and the dependence of the BL effect on surface roughness and terrain geometry are discussed. The results are summarized in section 7.

2. Numerical aspects

The atmospheric component of the Coupled Ocean– Atmospheric Mesoscale Prediction System (COAMPS1) is used for this study (Hodur 1997). COAMPS is a fully compressible, nonhydrostatic, and terrain-following mesoscale model featuring a suite of physical parameterizations. The turbulence kinetic energy (TKE) e = (u2 + υ2 + w2)/2, is a prognostic variable governed by the equation (Mellor and Yamada 1974)
i1520-0469-65-2-592-e1
where KM and KH are eddy-mixing coefficients of momentum and heat given by KM,H = SM,Hlm(2e)1/2Γ, Γ is a constant, SM,H are functions of the local Richardson number, lm is the mixing length (Mellor and Yamada 1974; Thompson and Burk 1991), and De represents the subgrid-scale TKE mixing. The subgrid-scale mixing of momentum and heat fluxes is parameterized as = −KMU/∂z, = −KMV/∂z, and = −KHθ/∂z.

Both free- and no-slip simulations have been conducted. For no-slip simulations, the surface stress is τ = ρu2*, where the friction velocity is given by u* = CdFM(zb/z0, RiB)|u|, the drag coefficient is Cd = [κ/ln(zb/z0)]2, κ = 0.4, and zb is the height of the lowest model level (Louis 1979; Louis et al. 1982). The surface roughness length z0 ranges from 10−5 m, corresponding to a smooth snow or ice surface, to 1 m, corresponding to rough surfaces over urban or mountainous areas. It is noteworthy that unlike the conventional no-slip boundary condition, zero wind speed is not explicitly imposed at the surface in our simulations. Instead, the similarity theory is applied to the unsolvable surface layer, which implies that the horizontal wind speed at the surface roughness level is zero. In addition, such no-slip boundary condition may become problematic over steep slopes. Therefore, we confine our simulations in the gentle terrain regime; throughout this study, the mean terrain slope, defined as hm/a, where hm is the maximum terrain height and a is the horizontal scale, is less than 0.01.

Three different model configurations have been used in this study. First, to characterize the structure and evolution of the simulated BL, we have carried out about 30 one-dimensional simulations. The model is initialized using an idealized sounding characterized by a uniform wind speed U0 = 10 or 20 m s−1 and a constant buoyancy frequency N = 0.01 s−1. The ambient winds are in geostrophic balance with a constant Coriolis coefficient f = 10−4 s−1 throughout the study. The results from 1D simulations are presented in section 4.

In sections 5 and 6, results from series of two-dimensional simulations are described. The computational domain is comprised of 1024 grid points in the x direction with radiation boundary conditions applied at the western and eastern boundaries. A Gaussian ridge, described by
i1520-0469-65-2-592-e2
where a is the ridge width and hm is the ridge crest height, is located at the center of the domain (i.e., x = 0), with a horizontal resolution of 0.1a. To reduce the BL spinup time, for the 2D simulations the model is initialized using the wind and potential temperature profiles derived from the corresponding 1D simulations valid at T = 6 h.
For the three-dimensional simulations presented in section 7d, the domain is comprised of 151 × 151 horizontal grid points, with a three-dimensional circular Gaussian hill,
i1520-0469-65-2-592-e3
located at the domain center, where x = 0 and y = 0. The horizontal resolution is 0.2a.

For all three configurations, there are 90 vertical levels with 50-m spacing in the lowest 1 km and coarser resolutions aloft. The model top is located approximately at 30 km AGL with Rayleigh damping applied to the top 20 levels (i.e., ∼12.5 km) to minimize downward wave reflection.

3. Analytical BL model

A number of interesting predictions have been made in S07 based on a bulk BL model regarding interactions between the BL and a hydrostatic wave. The Smith BL model assumes a momentum balance among the advection (i.e., inertial term), pressure gradient force, surface stress, and momentum mixing from either the free atmosphere above or entrainment stress. For the reason that will become clear in the next section, here we introduce a similar BL model, characterized by a momentum balance among the inertial term, the pressure gradient force, which is assumed to penetrate through the BL, Coriolis force, and surface stress. We refer to the new model as the rotational BL model (RBLM). In addition, to study the scale dependence of the BL effect, we couple the BL to a nonhydrostatic inertia–gravity wave instead of a hydrostatic wave as in S07.

For the BL, the steady-state momentum equation can be written as
i1520-0469-65-2-592-e4
where uB = (uB, υB) is the vertically averaged BL wind vector, f is the Coriolis coefficient, denotes the unit vector in the vertical direction, p is the pressure normalized by the fluid density, and CB is the BL relaxation coefficient, defined as the inverse time scale for the surface stress to remove the total momentum in the BL (see S07 and Table 1). Above the BL, the atmosphere is in geostrophic balance, that is, −f × Ug + P = 0, where the geostrophic wind is Ug = (U0, 0) and −P denotes the large-scale pressure gradient. The wind speeds in an unperturbed BL (UB, VB) can be obtained by letting UB · ∇UB = 0 and p = P in (4), which yields
i1520-0469-65-2-592-e5
The BL wind speeds are determined by the balance between the surface stress and the Coriolis force.
Now we consider that the BL and the free atmosphere are disturbed by small-amplitude wave perturbations. Using Fourier transform, that is, (u′, υ′, uB, υB, p′) = Re[(û, υ̂, ûB, υ̂B, )ei(kx+ly)], where (u′, υ′) and (uB, υB) denote the velocity perturbations in the free atmosphere and the BL, p′ is the pressure perturbation, the overcarets denote the amplitudes, and (k, l) are the horizontal wavenumbers in the wavenumber space, the perturbation momentum equations for the free atmosphere and the BL can be written as
i1520-0469-65-2-592-e6
i1520-0469-65-2-592-e7
where σ = U0k and σB = UBk + VBl are intrinsic frequencies in the free atmosphere and the BL, respectively. Using (6)(7), we obtain the velocity perturbations
i1520-0469-65-2-592-e8
i1520-0469-65-2-592-e9
It has been demonstrated in S07 that a useful parameter for describing the BL and wave interaction is the compliance coefficient C(k, l), defined by
i1520-0469-65-2-592-e10
where η̂ is the Fourier transform of the BL top displacement η. The compliance coefficient C is determined by BL properties, and once C is known, (10) provides the coupling condition between the BL and the wave above. For RBLM, C(k, l) can be obtained using (9) and the linearized continuity equation UB · η = −H · uB or σBη̂ = −H(B + lυ̂B) in the wavenumber space, where H is the unperturbed BL depth,
i1520-0469-65-2-592-e11
Over mesoscale terrain, the vertical displacement of the BL top, or the effective terrain height, is
i1520-0469-65-2-592-e12
where ĥ is the Fourier transform of the terrain height. In the free atmosphere, a linear propagating wave can be written as η̂(z) = η̂Teim(zH), where the vertical wavenumber is given by m2 = (N2σ2/σ2f2)(k2 + l2). Using (8) and the continuity equation, we obtain the pressure perturbation along the BL top,
i1520-0469-65-2-592-e13
Combining (10)(13), we obtain the effective terrain height
i1520-0469-65-2-592-e14
where the coupling parameter B is
i1520-0469-65-2-592-e15

For the purpose of comparison, we couple the Smith BL model with a nonhydrostatic inertia–gravity wave, and the resultant nondimensional compliance coefficient and coupling parameter are included in Table 2.

For flow past a two-dimensional ridge, (11) and (15) can be written as
i1520-0469-65-2-592-e16
i1520-0469-65-2-592-e17
where = NU0C is the normalized compliance coefficients, γ = UB/U0 is referred to as the BL shape factor, = NH/U0 is the nondimensional BL depth, B = CB/σ is the friction adjustment parameter, Ñ = N/σ is the nonhydrostatic parameter, and = f/σ is the inverse Rossby number. Note that only four of the five parameters are independent, because γ, B, and are related through (5).
Following S07, for stratified flow past an isolated ridge with a dominant wavenumber k = 1/a, where a is the terrain width, the effective terrain height, the pressure drag, and the momentum flux above the BL top can be approximated as
i1520-0469-65-2-592-e18
respectively, where D is the no-slip pressure drag, F is the momentum flux evaluated at the BL top, and D0 is the corresponding free-slip drag, which is ρU0Nh2m for flow over a Gaussian ridge in the linear hydrostatic limit.
It is instructive to examine the drag and momentum flux in inviscid and weakly viscous limits. In the inviscid limit (b ≪ 1, referred to as quick inertial response in S07), the BL flow responds to pressure gradient forcing through the inertial adjustment process, that is, convergence or divergence in the BL. For a hydrostatic wave coupled with an inviscid BL, using (17)(18), we obtain
i1520-0469-65-2-592-e19
where α is the upstream shift angle of the effective terrain. Equation (19) implies that an inviscid neutral BL always tends to decrease the drag and momentum flux and that there is no momentum loss across the BL, which is consistent with Eliassen and Palm (1961). The drag reduction and upstream shift of the effective terrain are more pronounced for a deep BL with slow BL flow (i.e., small γ). When the friction is small (i.e., b ≪ 1), we obtain
i1520-0469-65-2-592-e20
which suggest that the difference between drag and momentum flux is caused by the frictional adjustment process and is proportional to b. Accordingly, (D-F)/D0, the decrease of momentum flux across the BL, is larger for a wider ridge as the frictional adjustment process becomes increasingly important. As an example, we let δ = 0.3, γ = 0.6, and b = 0.12. Using (19)(20), we obtain B, D/D0, F/D0, and α, which are 1.2i, 0.61, 0.61, 0.28π, and −0.29 + 1.2i, 0.62, 0.42, and 0.26π for the inviscid and viscous BL, respectively. Apparently, the inclusion of friction does little to the drag and the phase shift, but substantially reduces the momentum flux. Hence, we conclude that the drag reduction is primarily caused by the BL inertial adjustment and the momentum flux is further reduced across a viscous BL through the frictional adjustment process.
To evaluate the BL effect quantitatively, we define the fractional reduction parameters for the effective mountain height, drag, and momentum flux as
i1520-0469-65-2-592-e21
respectively. The BL parameters will be evaluated from 1D simulations in the next section and the BL models will be compared with COAMPS simulations in sections 5 and 6.

4. Simulated BL properties and parameters

Several groups of 1D simulations have been carried out with different geostrophic wind speeds, surface roughnesses, and BL schemes in order to (a) characterize the simulated BL properties, (b) check the consistency of the simulated BL and Rossby similarity theory, (c) estimate the time scale needed for the BL to reach quasi equilibrium, (d) test the sensitivity of the simulated BL structure to turbulence closures, (e) identify the dominant momentum balance in the simulated BL, and (f) derive the control parameters for the bulk BL models. As an example, the evolution of the BL structure derived from a 1D simulation with U0 = 10 m s−1 and z0 = 0.1 m is shown in Fig. 1. After approximately 3 h of integration, the growth of the BL depth slows down, an Ekman spiral of velocity is created due to the Coriolis effect (Fig. 1b), and the BL flow is nearly neutrally stratified. After 6 h of integration, the surface stress is approximately balanced by the vertically integrated Coriolis force across the depth of the BL. As expected, in the absence of processes such as large-scale descent or radiative cooling, no exact steady state can be achieved in the BL. In this study, we loosely refer to the slowly evolving BL after T = 6 h as being in a quasi-equilibrium state. It is noteworthy that the BL effect described in sections 5 and 6 is relatively insensitive to the integration time for T ≥ 6 h.

According to Rossby similarity theory, the only control parameter for a neutral Ekman layer is the surface Rossby number Rs = U0/( fz0), and the natural vertical length scale is given by u*/f (e.g., Lettau 1962). Shown in Fig. 2 are normalized BL properties derived from three sets of simulations corresponding to U0 = 10 and 20 m s−1, with the Mellor–Yamada scheme (MY scheme; Mellor and Yamada 1974) and U0 = 10 m s−1 with the Therry–Lacarrere scheme (TL scheme; Therry and Lacarrere 1983), which is valid at 6 h. The normalized friction velocity decreases with increasing Rs (Fig. 2a), and despite the doubling of the geostrophic wind, the difference in the normalized friction velocity for the two sets of simulations is rather small, suggesting that the simulated BL is consistent with the Rossby similarity theory. Note that the simulated BL is weakly stable, instead of completely neutral, which likely accounts for the small difference between the two curves. For the identical set of control parameters, the TL scheme produces a slightly smaller friction velocity than that of the MY scheme. The BL depth and displacement thickness normalized by u*/f are shown in Fig. 2b. The definitions of the two BL depths—an equivalent step depth (H) and a Richardson number–based depth (Hr)—and the BL displacement thickness are given in Table 1. Note that Hr depends on the initial stability of the ambient flow, and H and δ are solely determined by the BL velocity profile. It is evident that the normalized BL depths are approximately constant with the surface roughness varying over five orders of magnitude, suggesting that the simulated BL is consistent with the Ekman BL scaling. Hence, the BL depth can be approximated as c1u*/f, where c1 is a constant, taken as 0.1 through the rest of the paper. The TL scheme produces a shallower BL with weaker turbulence, which is consistent with our operational experience using COAMPS. The shape factor decreases linearly with the increase of the normalized friction velocity, with values between 0.55 and 0.85 over the range of parameters examined (Fig. 2c). Again, the difference between the two best-fit lines corresponding to the U0 = 10 and 20 m s−1 experiments is relatively small. Because the TL scheme creates a less turbulent BL, the BL shape factor simulated using the TL scheme is slightly smaller than the MY scheme. Figure 2d shows that the BL relaxation coefficients CB and f are of the same order over the range of parameters examined, and CB increases with increasing friction velocity. However, CB only increases by a factor of 2 as the surface roughness increases by five orders of magnitude. The best-fit curves of the BL depth, shape factor, and relaxation frequency as a function of u*/U0 are listed in Table 3 for reference, and these empirical formulas will be used to evaluate the BL effect in the next two sections.

5. BL effect on waves, drag, and momentum

We first examine a set of 2D simulations with a = 10 km, hm = 10 m, and z0 = 10−4, 10−3, 10−2, 10−1, and 1 m, respectively, focusing on the upstream phase shift and reduction of the effective mountain height, wave drag, and momentum flux. We refer to this set of simulations as the control set.

a. Compliance coefficient

As demonstrated in section 3, the compliance coefficient C (m Pa−1; note that the units are m−1 s2 in RBLM as well as S07, because a normalized pressure was used in these analytical models) characterizes the response of the BL depth to imposed pressure gradient and is determined by BL properties, and once C is known, it is straightforward to couple the BL and waves aloft.

Shown in Fig. 3a is a plot of the surface pressure perturbation p′(x, z = 0), which has been corrected to the sea level, versus the BL depth variation derived from two control simulations with z0 = 0.01 and 1 m, respectively. The BL depth variation is evaluated by assuming mass conservation in the BL, that is, ηT = HH0h, where ∫H0 u dz = ∫H00 u dz and H0 corresponds to the BL depth at an upstream reference point. If the BL responds to the pressure gradient instantaneously, that is, there is no phase lag, the trajectories should follow straight lines. The elliptic trajectories in Fig. 3a indicate a phase lag between the pressure perturbation and the BL depth variation, suggesting a complex form of C, that is, C = ce. The amplitude c is equal to the slope of the long axis of elliptic trajectory and the phase lag ϕ is proportional to the aspect ratio of the trajectory. Apparently, Fig. 3a implies that c and ϕ are larger over a rougher surface. To estimate ϕ, we approximate the elliptic trajectory as p′ = p0 + pA sin(kx + ϕ) and η′ = η0 + ηA sin(kx), where (η0, p0) denotes the center of the ellipsis, and 2pA and 2ηA correspond to the range of pressure and BL depth variations. The phase lag is approximately given by sin(ϕ) = [p′(0) − p0]/pA. The c and ϕ values derived from the control set of simulations are shown in Fig. 3b, with the corresponding curves derived from the RBLM and the Smith BL model using the control parameters computed from the empirical formulas in Table 3 included for comparison (c has been divided by the air density ρ0 for conversion to m Pa−1). The c value derived from COAMPS is in the range between 5 and 15 m Pa−1 and increases with surface roughness. The phase lag is relatively small (∼5° or less) and increases with surface roughness as well. When ϕ = 0, the coupling parameter B becomes purely imaginary. According to (18), the BL effect still tends to shift wave patterns upstream and reduces the effective terrain height, drag, and momentum flux. However, the difference between χd and χm, which is proportional to csinϕ, becomes zero, implying that no momentum reduction and energy dissipation occur in the BL. For trapped waves, the amplitude of the complex BL reflection coefficient q = (1 + B)/(1 − B) (Smith et al. 2006) becomes unity, implying that no energy loss occurs when down-going reflected wave beams propagate across the BL. Therefore, when ϕ = 0, the BL behaves like an inviscid layer.

b. Upstream phase shift and drag reduction

Shown in Fig. 4 are the effective terrain heights derived from a pair of control simulations, with z0 = 10−1 and 10−4 m. The effective terrain height is defined as the BL top relative to that at an upstream reference point. It is evident that the BL tends to shift the wave pattern upstream and reduce the effective terrain height as predicted by the bulk models, and the BL effect is stronger over a rougher surface. The upstream phase shift is also evident in the surface wind curves (Fig. 5a). Relative to the free-slip simulation, both the wind speed minimum over the windward slope and the maximum over the lee slope shift upstream in the no-slip simulations. A more quantitative comparison is shown in Fig. 5b, where the upstream phase shift angles are estimated using 180°Δx/πa, and Δx is the upstream shift distance relative to the free-slip simulation. The phase shift of the leeside u maximum is slightly larger than that of the windward u minimum, both of which increase nearly linearly with the friction velocity. The phase shift curves computed using the two BL models and empirical formulas in Table 3 are quite close to each other and agree with COAMPS reasonably well (Fig. 5b).

To illustrate the reduction of mountain waves resulting from the BL effect, the wave drag reduction parameter and the wave momentum flux reduction parameter derived from the no-slip control simulations are shown in Fig. 6. The amplitude of the effective terrain reduction parameter χh is estimated as 1 − wmax(no slip)/wmax(free slip), where wmax is the maximum vertical velocity in the lower troposphere, based on the assumption that the linear wave amplitude is proportional to the effective terrain height. It is evident that the BL could significantly weaken waves and accordingly reduce wave drag and momentum flux aloft. As expected, the corresponding wave amplitude reduction and drag reduction parameters are very close to each other, because both wave drag and wave amplitude are proportional to the effective terrain height. The drag and momentum reduction parameters increase nearly linearly with the friction velocity. For z0 = 1 m, compared with the corresponding free-slip drag, the pressure drag is reduced by approximately 55%, and the momentum flux reduction is more than 80%. The BL effect is much less pronounced over a smoother surface. The drag is reduced by approximately 25% over a calm sea or smooth ice surface (i.e., z0 = 10−4 m), which is still quite substantial. The difference between the drag and momentum flux reduction parameters is larger over a rougher surface, indicating that the viscosity becomes increasingly important. In Fig. 6, the drag and momentum reduction parameters computed from the BL models, using the empirical formulas listed in Table 3, are shown for comparison. In general, the BL model predictions and numerical results agree with each other reasonably well.

c. Therry–Lacarrere closure

To examine the sensitivity of the BL effect to BL parameterizations, the control set of simulations have been repeated using the TL scheme. The drag and momentum reduction parameters derived from the TL scheme simulations increase nearly linearly with increasing u*/U0, which is consistent with results from the control simulations (Fig. 7). Compared to the control simulations, the reduction parameters from the TL scheme are slightly smaller over a rougher surface and larger over a smooth surface. The overall agreement between the reduction parameters from the two different BL parameterization schemes is satisfactory (Fig. 7). The agreement may seem surprising considering the difference in the shape factors and BL depths produced by the two schemes shown in section 4. Given an identical surface roughness, compared to the MY scheme, the TL scheme creates a shallower BL with weaker turbulence and a smaller shape factor. According to the BL models, a shallower BL tends to decrease the BL effect and a smaller shape factor does the opposite. The difference between the drag reduction parameters from the two schemes is small because of the two competing factors.

6. Terrain geometry

In the previous section, we focus on the BL and wave interaction in the 2D linear hydrostatic limit; issues related to the horizontal scale dependence, nonlinearity, and three-dimensionality will be examined in this section.

a. Ridge-width dependence

Shown in Fig. 8 are the drag and momentum reduction parameters derived from two sets of 2D simulations, with the ridge widths ranging from 1 to 50 km and z0 = 10−1 and 10−3 m, respectively. For narrow ridges (i.e., B < 0.05 and Ñ < 5), both the drag and momentum reduction parameters decrease rapidly with decreasing ridge width, apparently due to the nonhydrostatic effect. In addition, χd and χm tend to converge toward the narrow ridge limit, indicating that the fraction of momentum loss resulting from turbulence dissipation in the BL approaches zero. For B > 0.05, the drag reduction parameter χd decreases slightly with increasing ridge width. In contrast, the momentum reduction parameter χm shows a substantial increase, indicating that more momentum dissipation occurs in the BL over a wider ridge. In general, both analytical BL models indicate the decrease and convergence of χd and χm toward the narrow ridge limit. Over wide ridges, ranging from hydrostatic to inertia–gravity wave scales, the BL models suggest a significant increase of χmχd with increasing horizontal scale, which again is consistent with the COAMPS results. However, both BL models underpredict the increase of the momentum flux reduction parameter with B. Overall, the RBLM shows better agreement with COAMPS than the Smith BL model for wide ridges. As expected, for z0 = 10−4 m, the drag and momentum reduction parameters are substantially smaller than those of z0 = 10−1 m. The BL effect in the limit of B ≫ 1 has been discussed by S07. Apparently, because CB is of the same order as f in the neutral Ekman BL configuration, the limit of B ≫ 1 requires a ridge width to be around the order of 1000 km, which is beyond the scope of this study.

b. Narrow terrain limit

Wave drag and momentum flux created by narrow terrain are of essential importance to drag parameterization because narrow terrain is usually not well resolved by global or even regional scale models (e.g., Kim et al. 2003). Figure 9 indicates that in the narrow ridge limit, the BL effect decreases due to the nonhydrostatic effect and the difference between χd and χm diminishes. Hence, the decrease of momentum flux across the BL resulting from turbulence dissipation is virtually zero in the narrow terrain limit and the BL flow behaves like an inviscid flow. According to the bulk BL models, when B ≪ 1, the surface stress term is negligible and the primary momentum balance in the BL is between the pressure gradient term and the inertial term. For a given pressure gradient, the BL reachieves momentum balance through the inertial adjustment, and the frictional adjustment process is too slow to be significant. This inviscid assumption allows us to model a turbulent BL as a shallow inviscid shear layer. A linear solution is included in the appendix, in which the BL is approximated as an inviscid neutral shear layer. Coupling the linear solution with waves above, we obtain the coupling parameter B(k) = 1 − Ñ−2/(2γ − 1), implying that B is larger for a deeper BL with a smaller shape factor, which is consistent with the bulk BL models. A more quantitative comparison is shown in Fig. 9. All three models show a relatively constant drag reduction for 0.05 < B < 0.15 and a sharp decrease toward the narrow terrain limit for B < 0.05, clearly dominated by the nonhydrostatic effect. An underlying assumption of the BL models is that the BL maintains equilibrium while being distorted by the terrain underneath. As suggested by previous studies of BL flow over narrow ridges (∼500 m; e.g., Jackson and Hunt 1975), when TLa/UB, where TL is the turbulence Lagarangian time scale, scaled as H/u*, large eddies are advected over the ridge before they are able to adjust to equilibrium. Hence, the BL models may fail for a < HUB/u*.

c. Nonlinear effect

A set of simulations with z0 = 0.1 m, a = 10 km, and a ridge height ranging from 10 to 1500 m (i.e., the nondimensional mountain height, M = Nhm/U0, ranging from 0.01 to 1.5) have been conducted to examine the sensitivity of the BL-induced mountain-wave drag and momentum reduction with nonlinearity. The pressure drag derived from the no-slip and the corresponding free-slip simulations, normalized by the corresponding linear hydrostatic drag, is shown in Fig. 10. For the free-slip simulations, the normalized pressure drag increases slowly with increasing ridge height from unity for M = 0.01 to 1.08 for M = 0.5. We refer to M < 0.5 as the linear wave regime because the normalized drag increase is less than 10%. A sharp increase of the normalized drag occurs between M = 0.5 and 0.6, corresponding to gravity wave breaking at z = 2.25λz, where λz = 2πU0/N is the vertical hydrostatic wavelength, primarily resulting from non-Boussinesq effects. Wave breaking occurs at the z = 1.25λz level as well for a ridge height greater than 0.85, and accordingly the normalized drag increases almost linearly with M between M = 0.6 and 1.0. For M > 1, regardless of the increase of M, the normalized drag is almost constant with a value around 2.1. The normalized drag for the no-slip simulations is approximately 0.55 in the low-terrain limit and increases steadily with increasing M. Wave breaking occurs at z = 2.25λz around M = 0.65 and at the z = 1.25λz level around M = 1.2, indicating that the BL tends to delay the onset of gravity wave breaking.

The drag and momentum flux reduction parameters derived from the pair of no- and free-slip simulations are shown in Fig. 11. Both parameters show a substantial decrease with increasing ridge height. From M = 0.01 to M = 0.5, the momentum flux and drag reduction parameters decreases by 25% and 30%, respectively. For M > 0.6, the two parameters are not well defined because of the sharp increase of free-slip drag associated with wave breaking. The linear regime, defined as the range of M within which the decrease of the reduction parameters is less than 10%, is approximately given by M < 0.1, which is much smaller than the linear wave regime for the free-slip solutions. The smaller linear regime associated with the BL effect can be understood using scaling arguments. The presence of the BL introduces a new vertical scale, the BL depth, which is much smaller than a typical vertical wavelength of a hydrostatic wave (i.e., 2πU0/N). Accordingly, the linear regime shrinks significantly. Note that the velocity perturbation in the BL is typically comparable to or larger than that in the free stream (e.g., Fig. 5a), and the mean velocity in the BL is significantly smaller. Hence, if we use the ratio of the velocity perturbation and mean velocity (i.e., u′/U) as a nonlinearity index, u′/U is substantially larger in the BL than in the free atmosphere, and consequently the nonlinearity in the BL is stronger than in the free atmosphere. In the narrow terrain limit, we can estimate the variation of c by retaining the nonlinear inertial term and ignoring the surface stress and Coriolis terms in the bulk BL momentum equation, that is,
i1520-0469-65-2-592-e22
Integrating (22), and using mass conservation uBH = UBH and c = p′/(HH), where UB and H are the BL flow velocity and depth at certain reference point, we obtain either
i1520-0469-65-2-592-e23
where c is the compliance coefficient corresponding to UB and H. According to (23), c is quite sensitive to the BL depth or velocity; c becomes smaller when the BL is shallower and faster. For the no-slip set of simulations with z0 = 0.1, the average enhancement of BL speed over the lee slope between x = 0.5a and x = 1.5a can be approximated as uB/UB = 1 + 1.2M. Using (18) and (23) with c = 0.5 and B = iNU0C, the estimated χd(M) curve is included in Fig. 11, which shows satisfactory agreement with the COAMPS results.

It is noteworthy that there is apparent inconsistency between the above results and Jiang et al. (2006), who concluded that the BL effect for decaying trapped waves is relatively insensitive to the mountain height. This is likely because the variation of the BL depth over the lee slope associated with a propagating wave is much larger than that induced by a trapped wave over a flat surface. Additionally, the thickening and thinning of the BL induced by trapped waves is almost symmetric, and the increase of c over the deceleration segment of the trapped wave is partially cancelled out by the decrease of c over the acceleration segment. Over the ridge, the BL thinning and the acceleration over the lee slope is much stronger than the windward deceleration and BL thickening, and therefore leads to a rapid decrease of c with increasing mountain height.

d. Three-dimensionality

Finally, we examine the BL response to three-dimensional waves by diagnosing a set of COAMPS simulations of flow past a three-dimensional Gaussian hill with a = 10 km and hm = 100 m. Shown in Fig. 12 are horizontal plots of surface winds and the vertical velocity above the BL (i.e., 4000 m AGL), respectively. Due to the impact of the Coriolis force, the mean winds are oriented from the southwest to northeast. Accordingly, the surface wind maximum is located over the northern flank of the lee slope. Compared to the corresponding free-slip simulation (not shown), the surface wind maximum is approximately 3 km closer to the peak. In addition, a weak wake extends farther downstream, likely due to the increased surface stress over the lee slope associated with the enhanced surface winds. Above the BL, the axis through the vertical motion minimum and maximum is oriented along the geostrophic winds. However, the wave beams extend farther away to the south side of the axis. The left–right asymmetric wave patterns are in qualitative agreement with RBLM.

Similar to the 2D solutions, the pressure drag and momentum flux reduction parameters increase nearly linearly with the friction velocity, as does the difference between the two (Fig. 13). However, both the drag and momentum flux reduction parameters are only approximately 60% of the corresponding 2D cases, indicating that the BL effect is more pronounced over a 2D ridge than over a 3D hill in terms of drag and momentum reduction.

7. Summary

Interaction between propagating mountain waves and the near-neutrally stratified atmospheric BL has been investigated using COAMPS and analytical BL models. The simulated BL is consistent with the Rossby similarity theory and characterized by a momentum balance between the surface stress and the vertically integrated Coriolis force. The COAMPS simulations of flow past a 2D ridge or a 3D hill indicate that, in general, the atmospheric BL tends to move mountain-wave patterns upstream, weaken waves aloft, and consequently reduce the drag and momentum flux, which are consistent with the predictions of the rotational BL model (RBLM) as well as the predictions by S07.

According to the RBLM, the interaction between the BL and gravity waves can be described by the coupling parameter B, which is a function of the nondimensional BL height , BL shape factor γ, frictional adjustment parameter B, the nonhydrostatic parameter Ñ, and the inverse terrain Rossby number . The BL responds to mesoscale perturbations primarily through two dynamical processes—inertial adjustment and frictional adjustment. The upstream shift of wave patterns and the reduction of wave drag are predominantly induced by the inertial adjustment process, which is governed by two nondimensional parameters, and γ. More phase shift and drag reduction occur over a rougher surface related to a larger and smaller γ. The frictional adjustment process tends to decrease momentum flux and accounts for the difference between the pressure drag and momentum flux evaluated at the BL top from the COAMPS simulations. The control parameter for the frictional adjustment process B is sensitive to the horizontal scale. Over narrow terrain (B < 0.05), the BL is nearly inviscid and the pressure drag is approximately equal to the momentum flux at the BL top. The frictional adjustment becomes increasingly important for wider terrain, and a substantial decrease of momentum flux across the BL occurs over terrain with B ≥ 0.1. In addition, the BL effect is sensitive to Ñ and in the narrow and wide terrain limits, respectively.

The BL effect tends to decrease with increasing nonlinearity, implying the nonlinear nature of the BL response. The derived drag and momentum reduction parameters are relatively constant for M < 0.1 and decrease substantially with further increasing ridge height, indicating that the inclusion of the BL significantly shrinks the linear wave regime. For flow over a three-dimensional hill, the Ekman BL breaks the left–right symmetry of the wave patterns in the BL with a surface wind maximum located over the northern flank of the lee slope. Similar to flow past a 2 D ridge, both the drag and momentum flux decrease substantially because of the presence of the BL. However, the derived drag and momentum reduction parameters are, in general, smaller than the corresponding ones for flow over a 2D ridge, indicating that the BL effect is more pronounced in the 2D limit.

Finally, the importance of including gravity wave drag in modeling the general circulation has been well established (e.g., Kim et al. 2003). This study shows that the atmospheric BL could significantly reduce mountain-wave drag, implying that the BL effect should be properly taken into account in gravity wave parameterizations. This study suggests three methods to estimate the BL-induced drag and momentum flux reduction: (i) one can estimate the BL compliance coefficient from mesoscale model simulations, as described in section 5a, and compute the drag and momentum flux using Eqs. (16)(18); (ii) one can directly compute the drag and momentum flux using the empirical formulas in Table 4, which only requires u*/U0 or B, both of which are functions of the surface Rossby number (Table 3); or (iii) one can estimate χm and χd using the bulk BL models, which requires three steps—(a) deriving the four governing parameters using the empirical formulas in Table 3, (b) computing the coupling parameter B(k) using Table 2, and (c) estimating the drag or momentum flux reduction using Eq. (18).

Acknowledgments

This research was supported by the Office of Naval Research (ONR) Program Element 0601153 N. At Yale, support came from the National Science Foundation (ATM-0112354). The first author has greatly benefited from discussions with Dr. Shouping Wang at the Naval Research Laboratory. The simulations were made using the Coupled Ocean–Atmospheric Mesoscale Prediction System (COAMPS), developed by the Naval Research Laboratory.

REFERENCES

  • Allen, T., , and A. R. Brown, 2002: Large-eddy simulation of turbulent separated flow over rough hills. Bound.-Layer Meteor., 102 , 177198.

    • Search Google Scholar
    • Export Citation
  • Athanassiadou, M., , and I. P. Castro, 2001: Neutral flow over a series of rough hills: A laboratory experiment. Bound.-Layer Meteor., 101 , 130.

    • Search Google Scholar
    • Export Citation
  • Belcher, S. E., , and N. Wood, 1996: Form and wave drag due to stably stratified turbulent flow over low ridges. Quart. J. Roy. Meteor. Soc., 122 , 863902.

    • Search Google Scholar
    • Export Citation
  • Bougeault, P., and Coauthors, 2001: The MAP Special Observing Period. Bull. Amer. Meteor. Soc., 82 , 433462.

  • Brown, A. R., , and N. Wood, 2001: Turbulent form drag on anisotropic three-dimensional orography. Bound.-Layer Meteor., 101 , 229241.

  • Brown, A. R., , and N. Wood, 2003: Properties and parameterization of stable boundary layer over moderate topography. J. Atmos. Sci., 60 , 27972808.

    • Search Google Scholar
    • Export Citation
  • Brown, A. R., , J. M. Hobson, , and N. Wood, 2001: Large-eddy simulation of neutral turbulent flow over rough sinusoidal ridges. Bound.-Layer Meteor., 98 , 411441.

    • Search Google Scholar
    • Export Citation
  • Ding, L., , R. J. Calhoun, , and R. L. Street, 2003: Numerical simulation of strongly stratified flow over a three-dimensional hill. Bound.-Layer Meteor., 107 , 81114.

    • Search Google Scholar
    • Export Citation
  • Doyle, J. D., , and D. R. Durran, 2002: The dynamics of mountain-wave-induced rotors. J. Atmos. Sci., 59 , 186201.

  • Doyle, J. D., , M. A. Shapiro, , Q. Jiang, , and D. Bartels, 2005: Large-amplitude mountain wave breaking over Greenland. J. Atmos. Sci., 62 , 31063126.

    • Search Google Scholar
    • Export Citation
  • Eliassen, A., , and E. Palm, 1961: On the transfer of energy in stationary mountain waves. Geofys. Publ., 22 , 123.

  • Grisogono, B., 1994: Dissipation of wave drag in the atmospheric boundary layer. J. Atmos. Sci., 51 , 12371243.

  • Hodur, R. M., 1997: The Naval Research Laboratory’s Coupled Ocean/Atmospheric Mesoscale Prediction System (COAMPS). Mon. Wea. Rev., 125 , 14141430.

    • Search Google Scholar
    • Export Citation
  • Hunt, J. C. R., , R. Richards, , and P. W. M. Brighton, 1988: Stably stratified shear flow over low hills. Quart. J. Roy. Meteor. Soc., 114 , 14351470.

    • Search Google Scholar
    • Export Citation
  • Jackson, P. S., , and J. C. R. Hunt, 1975: Turbulent wind flow over a low hill. Quart. J. Roy. Meteor. Soc., 101 , 929955.

  • Jiang, Q., , J. D. Doyle, , and R. B. Smith, 2005: Blocking, descent and gravity waves: Observations and modelling of a MAP northerly föhn event. Quart. J. Roy. Meteor. Soc., 131 , 675701.

    • Search Google Scholar
    • Export Citation
  • Jiang, Q., , J. D. Doyle, , and R. B. Smith, 2006: Interaction between trapped waves and boundary layers. J. Atmos. Sci., 63 , 617633.

  • Jiang, Q., , J. D. Doyle, , S. Wang, , and R. B. Smith, 2007: On boundary layer separation in the lee of mesoscale terrain. J. Atmos. Sci., 64 , 401420.

    • Search Google Scholar
    • Export Citation
  • Kim, Y., , S. D. Eckermann, , and H. Y. Chun, 2003: A view of past, present, and future of gravity-wave drag parameterization for numerical climate and weather prediction models. Atmos.–Ocean, 41 , 6598.

    • Search Google Scholar
    • Export Citation
  • Lettau, H., 1962: Theoretical wind spirals in the boundary layer of a barotropic atmosphere. Beitr. Phys. Atmos., 35 , 195212.

  • Louis, J. F., 1979: A parametric model of vertical eddy fluxes in the atmosphere. Bound.-Layer Meteor., 17 , 187202.

  • Louis, J. F., , M. Tiedtke, , and J. F. Geleyn, 1982: A short history of the operational PBL-parameterization at ECMWF. Proc. Workshop on Planetary Boundary Layer Parameterization, Reading, United Kingdom, ECMWF, 59–79.

    • Search Google Scholar
    • Export Citation
  • Mellor, G. L., , and T. Yamada, 1974: A hierarchy of turbulence closure models for planetary boundary layers. J. Atmos. Sci., 31 , 17911806.

    • Search Google Scholar
    • Export Citation
  • Nappo, C. J., , and G. Chimonas, 1992: Wave exchange between the ground surface and a boundary-layer critical level. J. Atmos. Sci., 49 , 10751091.

    • Search Google Scholar
    • Export Citation
  • Ólafsson, H., , and P. Bougeault, 1997: The effect of rotation and surface friction on orographic drag. J. Atmos. Sci., 54 , 193210.

  • Peng, S., , and W. T. Thompson, 2003: Some aspects of the effect of surface friction on flows over mountains. Quart. J. Roy. Meteor. Soc., 129 , 25272557.

    • Search Google Scholar
    • Export Citation
  • Raymond, D. J., , C. S. Bretherton, , and J. Molinari, 2006: Dynamics of the intertropical convergence zone of the east Pacific. J. Atmos. Sci., 63 , 582597.

    • Search Google Scholar
    • Export Citation
  • Richard, E., , P. Mascart, , and E. C. Nickerson, 1989: The role of surface friction in downslope windstorms. J. Appl. Meteor., 28 , 241251.

    • Search Google Scholar
    • Export Citation
  • Schlichting, H., , K. Gersten, , and C. Mayes, 2000: Boundary-Layer Theory. Springer-Verlag, 822 pp.

  • Scorer, R. S., 1949: Theory of waves in the lee of mountains. Quart. J. Roy. Meteor. Soc., 75 , 4156.

  • Smith, R. B., 1980: Linear theory of stratified hydrostatic flow past an isolated mountain. Tellus, 32 , 348364.

  • Smith, R. B., 2007: Interacting mountain waves and boundary layers. J. Atmos. Sci., 64 , 594607.

  • Smith, R. B., , S. Skubis, , J. D. Doyle, , A. S. Broad, , C. Kiemle, , and H. Volkert, 2002: Mountain waves over Mont Blanc: Influence of a stagnant boundary layer. J. Atmos. Sci., 59 , 20732092.

    • Search Google Scholar
    • Export Citation
  • Smith, R. B., , Q. Jiang, , and J. D. Doyle, 2006: A theory of gravity wave absorption by boundary layers. J. Atmos. Sci., 63 , 774780.

  • Stevens, B., 2006: Bulk boundary-layer concepts for simplified models of tropical dynamics. Theor. Comput. Fluid Dyn., 20 , 279304.

  • Stevens, B., , J. Duan, , J. C. McWilliams, , M. Münnich, , and J. D. Neelin, 2002: Entrainment, Rayleigh friction, and boundary layer winds over the tropical Pacific. J. Climate, 15 , 3044.

    • Search Google Scholar
    • Export Citation
  • Therry, G., , and P. Lacarrere, 1983: Improving the eddy kinetic energy model for planetary boundary layer description. Bound.-Layer Meteor., 25 , 6368.

    • Search Google Scholar
    • Export Citation
  • Thompson, W. T., , and S. D. Burk, 1991: An investigation of an Arctic front with a vertically nested mesoscale model. Mon. Wea. Rev., 119 , 233261.

    • Search Google Scholar
    • Export Citation
  • Vosper, S. B., 2004: Inversion effects on mountain lee waves. Quart. J. Roy. Meteor. Soc., 130 , 17231748.

  • Vosper, S. B., , S. D. Mobbs, , and B. A. Gardiner, 2002: Measurements of the near-surface flow over a hill. Quart. J. Roy. Meteor. Soc., 128 , 22572280.

    • Search Google Scholar
    • Export Citation
  • Vosper, S. B., , P. F. Sheridan, , and A. R. Brown, 2006: Flow separation and rotor formation beneath two-dimensional trapped lee waves. Quart. J. Roy. Meteor. Soc., 132 , 24152438.

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APPENDIX

Inviscid Formulation

In a narrow terrain limit, the BL can be approximated as an inviscid neutral shear layer. To test this hypothesis, here we consider a shallow inviscid neutral layer with a constant shear αs = (U0Us)/D, where Us is the wind speed at the surface and D is the neutral layer depth, which satisfies kD ≪ 1. The flow above the neutral layer is characterized by a constant buoyancy frequency N and a uniform wind speed U0. For a hydrostatic wave with a wavelength k, the linear solutions in the neutral layer and the free atmosphere above can be written as
i1520-0469-65-2-592-ea1
i1520-0469-65-2-592-ea2
where ŵ1 and ŵ2 are vertical motion in the shear layer and the free atmosphere, respectively, and m is the vertical wavenumber in the stratified free atmosphere. The coefficients Â, , and Âf can be determined using the following boundary conditions:
i1520-0469-65-2-592-ea3
i1520-0469-65-2-592-ea4
i1520-0469-65-2-592-ea5
Using (A1)(A5), we obtain the three unknown coefficients,
i1520-0469-65-2-592-ea6
i1520-0469-65-2-592-ea7
Using (A1)(A5), η̂ = ŵ(0)/(ikU0), and ekD ≈ 1 + kD, we obtain
i1520-0469-65-2-592-ea8
which yields the coupling parameter B = imDU0/Us for the shallow inviscid neutral layer. Using Us/U0 = 2γ − 1, we obtain the coupling parameter B = iD̃ 1 − Ñ−2/(2γ − 1), where = ND/U0. It is evident that the BL effect increases with increasing BL depth and decreasing BL wind speed or BL shape factor. Although the neutral shear layer decreases the effective terrain height and wave drag, the drag and momentum flux reduction parameters are equal, that is, χd = χm, implying zero momentum attenuation through the BL, which is consistent with the Eliassen and Palm (1961) theorem.
Fig. 1.
Fig. 1.

Time–height plots of (a) u component (grayscale, interval: 1 m s−1) and isentropes (interval: 0.5 K) and (b) υ component (grayscale, interval: 1 m s−1) and TKE (dashed curves, interval: 1 m2 s−2) derived from a 1D simulation corresponding to U0 = 10 m s−1, N = 0.01 s−1, and z0 = 0.1 m. The bold curve in (b) corresponds to Ri = 0.5.

Citation: Journal of the Atmospheric Sciences 65, 2; 10.1175/2007JAS2376.1

Fig. 2.
Fig. 2.

Plots of (a) normalized friction velocity vs surface Rossby number (in logarithm), (b) nondimensional BL displacement thickness and depth vs normalized friction velocity, (c) BL shape factor vs normalized friction velocity, and (d) normalized relaxation time vs normalized friction velocity, derived from three groups of 1D simulations corresponding to U0 = 10 and 20 m s−1 with MY scheme and U0 = 10 m s−1 with TL scheme. The surface roughness ranges from 10−5 to 1 m. The best linear fit curves are included in Table 3 for reference.

Citation: Journal of the Atmospheric Sciences 65, 2; 10.1175/2007JAS2376.1

Fig. 3.
Fig. 3.

(a) Plot of sea level pressure perturbation vs BL depth variation for −a < x < a, derived from a pair of control simulations with z0 = 10−2 and 1 m, respectively. (b) Plot of the amplitude and phase angle of compliance coefficient C derived from control simulations vs the normalized surface roughness. The corresponding curves computed using the bulk BL models are included for comparison.

Citation: Journal of the Atmospheric Sciences 65, 2; 10.1175/2007JAS2376.1

Fig. 4.
Fig. 4.

Plot of terrain height and the effective terrain heights derived from a pair of COAMPS no-slip simulations with z0 = 10−4 and 10−1 m vs the alongstream distance.

Citation: Journal of the Atmospheric Sciences 65, 2; 10.1175/2007JAS2376.1

Fig. 5.
Fig. 5.

(a) Perturbation u component at 10 m vs horizontal distance, derived from five control simulations. The corresponding free-slip simulation is included for reference. (b) The phase shift derived from the control set simulations. The two sets of symbols correspond to the phase shifts of the upstream surface wind minimum and leeside maximum relative to the free-slip run. The phase shift curves derived from the bulk BL models are included for comparison.

Citation: Journal of the Atmospheric Sciences 65, 2; 10.1175/2007JAS2376.1

Fig. 6.
Fig. 6.

Plot of pressure drag, momentum, and effective terrain height reduction parameters, χd, χm, and χh, vs the normalized surface friction velocity u*/U0, derived from the COAMPS control simulations with a = 10 km, hm = 10 m, and z0 = 10−4, 10−3, 10−2, 10−1, and 1 m, respectively. The corresponding curves derived from the two BL models are included for comparison.

Citation: Journal of the Atmospheric Sciences 65, 2; 10.1175/2007JAS2376.1

Fig. 7.
Fig. 7.

Plot of drag and momentum flux reduction vs normalized friction velocity derived from two sets of simulations for z0 = 10−4, 10−3, 10−2, 10−1, and 1 m with MY scheme and TL scheme, respectively.

Citation: Journal of the Atmospheric Sciences 65, 2; 10.1175/2007JAS2376.1

Fig. 8.
Fig. 8.

Plot of drag and momentum flux reduction vs the normalized mountain width derived from two sets of simulations for a = 1, 2, 3, 5, 10, 20, 30, 40, and 50 km, with (a) z0 = 0.1 m and (b) z0 = 10−3 m respectively. The corresponding predictions by the two BL models are included for comparison. The parameters used in the BL models are derived from the corresponding COAMPS simulations.

Citation: Journal of the Atmospheric Sciences 65, 2; 10.1175/2007JAS2376.1

Fig. 9.
Fig. 9.

Plot of drag reduction vs the normalized ridge width from a set of COAMPS simulations with z0 = 0.1 m. The corresponding curves computed from Smith BL model (hydrostatic and nonhydrostatic), RBLM (hydrostatic and nonhydrostatic), and the inviscid shear model are included.

Citation: Journal of the Atmospheric Sciences 65, 2; 10.1175/2007JAS2376.1

Fig. 10.
Fig. 10.

Drag reduction for a set of simulations with a = 10 km, z0 = 0.1 m, and mountain height hm = 10, 20, 50, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, 1200, and 1500 m.

Citation: Journal of the Atmospheric Sciences 65, 2; 10.1175/2007JAS2376.1

Fig. 11.
Fig. 11.

Plot of drag and momentum flux reduction parameter vs the nondimensional mountain height for a set of simulations with a = 10 km, z0 = 0.1 m, and mountain height hm = 10, 20, 50, 100, 150, 200, 300, 400, 500, and 600 m. The estimated χd(M) using the nonlinear BL model in section 6c is indicated by the bold curve.

Citation: Journal of the Atmospheric Sciences 65, 2; 10.1175/2007JAS2376.1

Fig. 12.
Fig. 12.

Horizontal plots of (a) the wind speed (interval: 0.03 m s−1) and wind vectors at the surface, and (b) vertical velocity (interval: 0.01 m s−1) and wind vectors at 4000 m derived from the no-slip simulation with z0 = 0.1 m and hm = 100 m. The 50-m terrain contour is included for reference. Only a center part of the domain is shown.

Citation: Journal of the Atmospheric Sciences 65, 2; 10.1175/2007JAS2376.1

Fig. 13.
Fig. 13.

Plot of drag and momentum flux reduction parameters vs the normalized surface friction velocity derived from a set of no-slip simulations associated with flow past a 3D hill with a = 10 km and hm = 100 m.

Citation: Journal of the Atmospheric Sciences 65, 2; 10.1175/2007JAS2376.1

Table 1.

Definitions of BL parameters.

Table 1.
Table 2.

BL models. Control parameters for Smith BL model: = NH/U0, B = aCB/U0, T = aCT/U0, Ñ = aN/U0, = l/k, and γ = σB/σ for 3D; and γ = UB/U0 for 2D. Control parameters for RBLM are , B, = af/U0, Ñ, , and γ; in 2D limit, = 0. The normalized compliance coefficient is = NU0C.

Table 2.
Table 3.

Least squares best-fit curves for BL structure parameters.

Table 3.
Table 4.

Least squares best-fit curves for drag and momentum reduction.

Table 4.
1

COAMPS is a registered trademark of the Naval Research Laboratory.

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