Editorial Type:
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Article Type:
Research Article

Impact of Elevated Kelvin–Helmholtz Billows on the Atmospheric Boundary Layer

Qingfang Jiang aNaval Research Laboratory, Monterey, California

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https://orcid.org/0000-0003-4054-577X
Online Publication:
16 Nov 2021
Print Publication:
01 Dec 2021
DOI:
https://doi.org/10.1175/JAS-D-21-0062.1
Page(s):
3983–3999
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Abstract

The impact of Kelvin–Helmholtz billows (KHBs) in an elevated shear layer (ESL) on the underlying atmospheric boundary layer (BL) is examined utilizing a group of large-eddy simulations. In these simulations, KHBs develop in the ESL and experience exponential growth, saturation, and exponential decay stages. In response, strong wavy motion occurs in the BL, inducing rotor circulations near the surface when the BL is stable. During the saturation stage, secondary instability develops in the ESL and the wavy BL almost simultaneously, followed by the breakdown of the quasi-two-dimensional KH billows and BL waves into three-dimensional turbulence. Consequently, during and after a KH event, the underlying BL becomes more turbulent with its depth increased and stratification weakened substantially, suggestive of significant lasting impact of elevated KH billows on the atmospheric BL. The eventual impact of KHBs on the BL is found to be sensitive to both the ESL and BL characteristics.

For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Qingfang Jiang, qingfang.jiang@nrlmry.navy.mil

Abstract

The impact of Kelvin–Helmholtz billows (KHBs) in an elevated shear layer (ESL) on the underlying atmospheric boundary layer (BL) is examined utilizing a group of large-eddy simulations. In these simulations, KHBs develop in the ESL and experience exponential growth, saturation, and exponential decay stages. In response, strong wavy motion occurs in the BL, inducing rotor circulations near the surface when the BL is stable. During the saturation stage, secondary instability develops in the ESL and the wavy BL almost simultaneously, followed by the breakdown of the quasi-two-dimensional KH billows and BL waves into three-dimensional turbulence. Consequently, during and after a KH event, the underlying BL becomes more turbulent with its depth increased and stratification weakened substantially, suggestive of significant lasting impact of elevated KH billows on the atmospheric BL. The eventual impact of KHBs on the BL is found to be sensitive to both the ESL and BL characteristics.

For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Qingfang Jiang, qingfang.jiang@nrlmry.navy.mil
Keywords: Instability; Turbulence; Boundary layer

1. Introduction

Kelvin–Helmholtz instability (KHI) occurs in continuously stratified flows and along interfaces between fluids associated with strong vertical shear (e.g., Helmholtz 1868; Kelvin 1871), over a wide spectrum of scales and in a variety of fluids. KHI is largely controlled by two nondimensional parameters, namely, the Reynolds number (Re) and the Richardson number (Ri). In a high Reynolds number continuously stratified fluid such as Earth’s atmosphere, KH waves or billows (KHBs) are virtually ubiquitous from the surface layer up to the mesosphere and beyond (e.g., Blumen et al. 2001; Fritts et al. 2014a; Mondal et al. 2019). In the troposphere, KHI occurs associated with a wide variety of synoptic or mesoscale processes such as cold front passage (Friedrich et al. 2008; Samelson and Skyllingstad 2016), sea-breeze circulations (e.g., Sha et al. 1991; Wood et al. 1999; Plant and Keith 2007; Lyulyukin et al. 2019; Jiang et al. 2020), airflow over mountainous areas (e.g., Jiang et al. 2010; Zhou and Chow 2013; Medina and Houze 2016; Conrick et al. 2018), low-level jets (e.g., Blumen et al. 2001; Nakanishi et al. 2014), and density currents (e.g., Sun et al. 2002). KHI is considered a major source of clear air turbulence (e.g., Sekioka 1970; Fritts et al. 2011) and a contributor to vertical mixing of momentum and scalars in the atmospheric boundary layer (e.g., Muschinski 1996). Some recent studies found that KHI may enhance precipitation as well (e.g., Medina and Houze 2016; Barnes et al. 2018).

It is believed that KHI plays a substantial role in intermittent turbulence generation and vertical mixing in a stable BL [Blumen et al. 2001; Sun et al. 2002; see Sun et al. (2015) for a review]. Possible KH billows had been frequently observed within the atmospheric BL. During the 1999 Cooperative Atmosphere–Surface Exchange Study (CASES-99) in southeast Kansas, KH billows below a low-level wind jet in a stable boundary layer (SBL) had been documented by Blumen et al. (2001) and Newsom and Banta (2003), which provided insights into the role that KHBs play in vertical momentum transfer.

Recently, wave events had been observed in the SBL by several groups using remote sensing tools (e.g., Petenko et al. 2016, 2019, 2020; Zaitseva et al. 2018; Zaitseva and Kouznetsov 2019). These wave events were attributed to either KH waves (also referred to as vorticity waves or internal shear–gravity waves) or propagating buoyancy waves (PBW; Chimonas and Fua 1984; Sun et al. 2015). While both manifest themselves as wavy motions, these two types of waves have different generation mechanisms and phase speeds. The former develop in a shear layer and propagate along the mean wind direction approximately at the mean wind speed in the shear layer. The latter are usually generated associated with vertical displacement of streamlines induced by topography, heterogeneous surfaces, or density currents. It is worth noting that buoyancy waves can be trapped in the atmospheric BL due to a sudden decrease in the stratification and increase of the wind speed aloft (e.g., Scorer 1949; Jiang et al. 2006).

Over the past two decades, large-eddy simulations (LES) and direct numerical simulations play an instrumental role in advancing our understanding of nonlinear KHI physics and dynamics in low-to-moderate Reynolds number fluids (e.g., Smyth et al. 2005; Rahmani et al. 2014). KHI in the stratosphere had been subject to a number of LES studies (e.g., Fritts et al. 2014a). There were only a handful of LES studies of KHI related to atmospheric BL processes although KHI is considered a major source of internal waves and intermittent turbulence in the SBL (e.g., Sun et al. 2015). For example, Zhou and Chow (2013) reproduced some KH waves over real topography documented during CASES-99 using nested large-eddy simulations. Using wind and potential temperature profiles modified from field observations, Na et al. (2014) simulated three-dimensional shear instability in a SBL associated with an inflection point in a low-level jet. More recently, intermittent bursting in the SBL associated with KHI near the surface was studied by van der Linden et al. (2020) using high-resolution large-eddy simulations.

KHI may occur near or above the BL top as well. For example, plausible KH waves were observed in sea breezes over Black Sea associated with the wind shear between the offshore flow and return flow aloft (e.g., Lyulyukin et al. 2019; Petenko et al. 2020), and in an elevated inversion (e.g., Atlas et al. 1970). Often KH billows manifest themselves as beautiful billow clouds (e.g., Smyth and Moum 2012) associated with vertical undulation of the BL top. While it is clear that KHI may frequently occur in an elevated shear layer, the impact of elevated KH billows on the underlying BL is largely unknown. The objective of this study is to explore the impact of KH billows developed in an ESL on the atmospheric BL utilizing idealized large-eddy simulations. The remainder of this paper is organized as follows. The LES code and modeling strategy are illustrated in section 2. The general characteristics associated with interaction between KH billows and the SBL are presented in section 3. The sensitivity in response of a balanced BL to the presence of an imposed ESL of particular magnitude and layer depth is examined in section 4. Section 5 includes discussion and concluding remarks.

2. LES code and model configuration

The LES code used here is the NCAR pseudospectral LES, which solves a set of spatially filtered three-dimensional Boussinesq equations in a rectangular domain with periodic boundary conditions applied along the four sidewalls (e.g., Moeng 1984; Sullivan et al. 2016). The subgrid-scale (SGS) stress and fluxes are parameterized following the two-part SGS scheme illustrated in Sullivan et al. (1994). For the simulations presented in this study, the domain length and width are Lx = 4480 m and Ly = 560 m, respectively, with horizontal grid spacings of Δx = Δy = 4.375 m. There are 220 vertical grid levels with the model top located at z = 1500 m above ground level (AGL), where a radiation boundary condition (Klemp and Durran 1983) is applied. The lowest grid level is at z = 3 m AGL and the vertical grid interval increases with height at a constant stretching ratio, Δzi+1/Δzi = 1.007, where Δzi and Δzi+1 denote the vertical spacings at two adjacent levels. The initial horizontal winds are unidirectional and vertically uniform, V0⁡(z)=(U0,0), where U0 = 6 m s−1 is the wind component in the x direction (referred to as u wind hereafter). The atmosphere is initially in geostrophic balance with a constant Coriolis parameter f = 1.2 × 10−4 s−1. The potential temperature is given by θ(z) = θ0 + Γz, where θ0 = 265 K is the potential temperature at the surface and Γ = 6 K km−1 is the constant lapse rate.

The surface stress (τ) and kinematic heat flux (Fθ) are evaluated at each grid point using the bulk formula, namely,
τ=ρaCD⁡(u12+υ12) and
Fθ=−Cθu12+υ12⁡(θ1−θs),
where the subscripts s and 1 denote values at the surface and the first grid level, respectively. The drag and heat flux exchange coefficients (i.e., CD and Cθ) are computed following the Monin–Obukhov similarity theory with the surface roughness length z0 = 0.1 m for both stress and the kinematic heat flux. The ground temperature linearly decreases (i.e., stable boundary layer) or increases (i.e., convective BL) with the integration time, i.e., θs(t) = θ0 − αt, where θ0 = 265 K is the initial ground temperature, t is the integration time, and α = 0.5 K h−1 is the surface cooling rate in the SBL simulations. The integration time interval (Δt) is evaluated at every time step from the maximum wind speed to ensure that the maximum Courant–Friedrichs–Lewy number is less than 0.3 across the domain for the sake of computational stability.
The model is initially integrated for 55 000 time step (i.e., t = T0 ~ 3.2 h with an average time interval of Δt ~ 0.21 s) to allow for the BL to develop. Afterward, the model is restarted with a shear layer, characterized by a hyperbolic-tangent variation of the u-wind component with height z, i.e.,
Us⁡(z)=ΔU{1+tanh[⁡(z−zc)/D]},
superimposed onto the u wind, i.e., urs(x, y, z) = u(x, y, z, t = T0) + Us(z), where urs denotes the initial u wind at restart. Equation (2) represents an elevated shear layer (ESL) or vortex sheet centered at level zc with a half depth D (Fig. 1). An additional term of −fUs is added to the υ-wind momentum equation to maintain the geostrophic balance. According to (2), Us varies from ~0 far below the ESL (i.e., zc − z ≫ D) to 2ΔU above the ESL (i.e., z − zc ≫ D). For the control simulation (CONTROL), we have ΔU = −U0, zc = 400 m, and D = 150 m. Near the BL top (Zi ~ 100 m), |Us/ΔU| < 4% and at the surface |Us/ΔU| ~ 1%. Therefore, the insertion of the shear layer has minimal impact on the winds and momentum balance in the BL. For CONTROL, the shear or vorticity in the ESL is negative (i.e., dUs/dz < 0, referred to as backward shear hereafter) and the vortex sheet is centered at z = zc, where zc is a critical level (i.e., u¯rs=0 at restart) and an inflection point level (i.e., d2Us/dz2 = 0). The magnitude of the spanwise vorticity, |dUs/dz|, is maximum at the critical level and the gradient Richardson number is minimum [Rim=g/θ0⁡(ΓD2/U02)]. For CONTROL, we have Rim = 0.12. Previous studies indicated that the necessary conditions for KHI to occur in a high Reynolds number continuously stratified fluid such as Earth’s atmosphere include the existence of an inflection point and a minimum Richardson number less than the critical Richardson number (i.e., Ric = 0.25) in the shear layer (e.g., Miles 1961; Howard 1961). Both conditions are satisfied in the ESL.
Fig. 1.
Fig. 1.

Schematics of the mean wind profiles before restart (i.e., U), and at restart (i.e., ⟨Urs⟩) along with the imposed shear profiles (i.e., Us). The dashed and dotted profiles of Us and ⟨Urs⟩ correspond to backward shear (i.e., ΔU = −U0) and forward shear (i.e., ΔU = U0) in Eq. (2), respectively. The ESL is located between Zc − D and Zc + D, where Zc denotes the inflection point level (also a critical level for backward shear) and D is the half depth of the ESL.

Citation: Journal of the Atmospheric Sciences 78, 12; 10.1175/JAS-D-21-0062.1

The KH modes in a hyperbolic tangent shear layer along with a constant potential temperature lapse rate above a solid surface had been examined by Tanaka (1975). Their linear analysis suggested that (i) the wave phase speed is zero (i.e., stationary) as the shear is symmetric with respect to the critical level where the mean wind speed is zero; (ii) the fastest KH wave modes and their growth rates are relatively insensitive to the vertical distance of the shear layer from the rigid ground surface (note: they used zc ~3D in their analysis), and (iii) for Rim < 0.25, the fastest growth mode is λ ~ 4πD and the e-folding growth rate decreases with increasing Rim.

In addition to CONTROL, results from four other simulations (Table 1) are presented in the next two sections. The governing parameters for LOWER and HIGHER are identical to those for CONTROL except that zc is 300 and 600 m, respectively. For CBL, a prescribed surface warming of 0.25 K h−1 is applied throughout the simulation and accordingly the BL is weakly convective. In FWSHR, we have ΔU = U0 (i.e., dUs/dz > 0, hereafter referred to as forward shear) and accordingly the spanwise vorticity is positive in both the ESL and the SBL. While zc remains an inflection point, there is no critical level in FWSHR. The mean wind speed at z = zc is 12 m s−1, which is the phase speed of the linear KH waves as well.

Table 1.

List of parameters for the five simulations. They are, from left to right, shear strength (ΔU), the inflection point level (zc), surface cooling rate (α), BL responsive ratio (R = IBmax/IEmax), IB ratio before–after the KH event [RBA = IB(TA)/IB(TB), where TB and TA denote the times before and after the KH event], the maximum vertical integrated PKE above the BL (IEmax), nondimensional horizontal wind fluctuation (sz = U0Δu/Δp) at z =50 m (s50), and dimensionless ESL level [h^=k⁡(zc−D)].

Table 1.

Several aspects are worth noting about the LES configuration. First, the model configuration has been chosen based on a suite of experimental simulations with different grid spacings (e.g., Δx = Δy = 2.188 or 8.75 m), domain sizes, and ESL characteristics. Over the parameters examined in this study, the simulated KH wave modes are relatively insensitive to the grid spacings. The use of Δx = Δy = 4.375 m is consistent with previous LES studies of the SBL. For example, in a multiple LES intercomparison study, Beare et al. (2006) found that the LES simulated SBL profiles tend to converge when the grid spacing is 6.25 m or less. The number of KH waves and the dominant wavelength vary some with the domain length due to the constraint of periodic lateral boundary conditions. Nevertheless, the dynamics around the interaction between the elevated KH billows and the BL examined in this study remains the same regardless of the domain size. Second, no initial perturbations are introduced to the ESL at restart and KHI is initiated by random perturbations from the BL turbulence. Finally, in test runs with the ESL included in the initial wind profile, KHI starts developing in less than 20 min, well before the BL fully develops. The restart modeling strategy allows the BL to develop before KHI kicks in, which is crucial for investigating the impact of elevated KHBs on the BL.

3. Interaction between KHBs and BL

We start by examining the characteristics and evolution of the KH billows and their impact on the underlying BL in CONTROL.

a. Visualization of KH billows in ESL and BL response

The life cycle of the simulated KH event is illustrated in Fig. 2 by a sequence of potential temperature vertical cross-section snapshots for T between 32 and 95 min. Here T denotes the integration time from the restart. There are three KH billows in the 4.48 km long domain, implying a predominant wavelength of λ = 1.49 km, which is about 10 times of the shear-layer half depth. The KH billows are centered along the critical level and grow rapidly from smooth undulations at 32 min (Fig. 2a) to mature KH billows at 53 min characterized by well-developed KHB cores and overturning of isentropic surfaces (Fig. 2d). The KHB cores reach their maximum depth of ~300 m (i.e., between 250 and 550 m) at T = 53 min (Fig. 2d). The depth and wavelength ratio is around 0.2, which is comparable to those from previous studies (e.g., Fritts et al. 2014a). Fine-scale spikes with horizontal scales of ~150 m first appear along the perimeter of the KHB cores in Fig. 2c, implying development of secondary instability (SI; e.g., Smyth 2003). SI is more evident in Fig. 2d with coiling of isentropic surfaces followed by rapid breakdown of the KH billows in the next several minutes. After the breakdown, the KHB cores becomes less stratified, while vigorous mixing occurs along the lower and upper edges of the cores as revealed by fine-scale spikes (Fig. 2f). As a result, the turbulent zone centered at the critical level expands outward, which is consistent with the lidar and radar observations of KH billows in the midtroposphere by Luce et al. (2018) and the direct numerical simulations by Fritts et al. (2014b). The KH waves continue weakening while the originally stratified shear layer becomes turbulent (Figs. 2g,h). The whole life cycle lasts for about 90 min.

Fig. 2.
Fig. 2.

Snapshots of potential temperature (i.e., θ − θ0; K) vertical cross sections at y = Ly/2 from CONTROL valid at T = (a) 32, (b) 39, (c) 46, (d) 53, (e) 60, (f) 67, (g) 81, and (h) 95 min, respectively. Only the lowest 800 m is shown.

Citation: Journal of the Atmospheric Sciences 78, 12; 10.1175/JAS-D-21-0062.1

Several interesting aspects of this KH event are worth noting. First, the nondimensional wavenumber of the primary KH waves is k^=kD=2πD/λ~0.63, which is slightly larger than that the fastest growth mode predicted by linear theory (e.g., Fig. 2 of Tanaka 1975), presumably due to the periodic boundary conditions. Second, the KH waves are nearly stationary before the breakdown (Figs. 2a–c), which is consistent with the linear theory prediction. Third, the influence of the KH billows extends all the way to the surface, implying their potentially substantial impact on the BL. Corresponding to each KH wave crest in the ESL, there is a wave crest in the BL with a noticeable phase shift across the critical level, as predicted by the linear theory (e.g., Newsom and Banta 2003). Furthermore, the undulation in the BL grows simultaneously with the elevated KHBs and by T = 53 min (Fig. 2d) the vertical undulation of the BL top exceeds 150 m with some isentropes in the downwind side of the BL waves vertically oriented or overturned. The simultaneous growth of the wavy motion in the BL (referred to as BL waves hereafter) implies tight coupling or interaction between the ESL and the SBL. SI and breakdown of the BL waves occur near the surface between 53 and 60 min, and the BL flow underneath wave crests becomes more chaotic and less stratified (Figs. 2e and 2f). Animation of the similar second-by-second vertical cross sections reveals that turbulent breakdown of the BL waves first occurs on the downwind side of the wave crests and lags behind the breakdown of the KH billows aloft by a few minutes (see online supporting materials). Finally, the KH waves dissipate rapidly after the breakdown of KHBs resulting in a turbulent and less stratified BL even in the presence of surface cooling.

The characteristics of the KH billows and BL response are further revealed by the vertical cross sections of θ − θ0, u, υ, w, the spanwise vorticity (i.e., η = uz − wx), and the buoyancy frequency squared [i.e., N2=(g/θ0)⁡(dθ/dz)] at T = 50 min (Fig. 3). The KHBs are fully developed with overturning of isentropes in the ESL and underneath the BL wave crests (Fig. 3a). The u wind in the SBL responds strongly to the elevated KHB forcing with rotors forming in the wave crests associated with wind reversal (i.e., u < 0, Fig. 3b). According to the linear theory, KH waves propagate along the shear layer with associated undulations decaying with the vertical distance from the ESL [e.g., Gossard et al. (1973) and appendix]. The large u-wind fluctuation in the BL implies active interaction between the KHBs and the BL waves. The υ wind is generally positive in the BL and exhibits similar oscillation as the u wind. Above the BL, υ-wind perturbations are small (Fig. 3c), implying that the elevated KHBs remain virtually two-dimensional before their breakdown. The w field shows sharp phase shift across the wavy critical level (Fig. 3d), which is in qualitative agreement with linear solutions and is due to the sign change in the mean u wind across the critical level (e.g., Newsom and Banta 2003). The active interaction between the ESL and the SBL is more evident in the spanwise vorticity section (i.e., Fig. 3e). In the ESL, the negative vorticity filaments coil around with some fine-scale features riding along the braids, which are clear indication of SI. As suggested by Corcos and Sherman (1976), SI tends to occur along KH braids where the strain field results in stronger shear across the braids.

Fig. 3.
Fig. 3.

Vertical cross-section snapshots of (a) θ − θ0 (K), (b) u (m s−1), (c) υ (m s−1), (d) w (m s−1), (e) spanwise vorticity, η = ux − wz normalized by η0=dUs/dz|z=zc=0.04 s−1, and (f) 104 N2 (s−2) from CONTROL valid at T = 50 min (i.e., between c and d in Fig. 1).

Citation: Journal of the Atmospheric Sciences 78, 12; 10.1175/JAS-D-21-0062.1

In the BL, the positive vortex sheet rolls up with smaller-scale negative vortices generated in the wave crests, resembling subrotors observed underneath trapped mountain waves (e.g., Doyle et al. 2009). The buoyancy frequency squared indicates that the KH cores centered at the critical level are less stratified (i.e., smaller N2), encompassed by a pair of negative and positive N2 rings (Fig. 3f), which is consistent with the coiling of isentropic surfaces in Fig. 3a. The existence of negative N2 rings implies that the KH billows are locally convectively unstable. The fact that fine-scale features appear away from the negative N2 ring indicates that shear-induced SI initially grows faster than the convective instability. Similarly, in the BL, the airflow is less stratified capped by a thin wavy inversion along the BL top, suggesting enhanced vertical mixing in the BL associated with wind reversal and the more turbulent air in rotors.

b. Evolution of KHBs

Shown in Fig. 4 are the time–height cross sections of horizontally averaged fields, fluxes, and variances from the control simulation. The KH event is evidenced by the maxima in the perturbation kinetic energy (PKE), e=⁡(u′2+υ′2+w′2)/2¯, kinematic temperature flux (i.e., w′θ′¯), u-wind momentum flux (i.e., w′u′¯), and potential temperature variances (θ′2¯, Figs. 4a–d). Here PKE and fluxes include contributions from both the KHB-induced coherent undulations and resolved turbulence. In the ESL, a PKE maximum first appears at the critical level and rapidly expands through the whole depth of the ESL while growing in magnitude. A more pronounced maximum develops simultaneously in the BL and reaches its peak around T = 60 min (Fig. 4a), about 5 min after the turbulent breakdown occurs. Later, the PKE maximum in the ESL branches into two separate maxima while weakening with time. The two branches, corresponding to the lower and upper edges of the KHB cores (Fig. 2), respectively, diverge away from the critical level with a relative minimum between them. The momentum flux is positive in the ESL, implying predominantly upward momentum transfer by the KHBs. This is consistent with the negative wind shear in the ESL, and in qualitative agreement with the analysis of tower observations by Newsom and Banta (2003). Physically, KHBs grow by extracting energy from the sheared flow while consuming the wind shear through the upward momentum transfer.

Fig. 4.
Fig. 4.

Time–height cross sections of domain-averaged (a) PKE (e, m2 s−2), (b) u-wind momentum flux (u′w′¯, m2 s−2), (c) kinematic temperature flux (w′θ′¯, K m s−1), (d) potential temperature variance (θ′2¯, K2), (e) local Richardson number (Ri), (f) third-order vertical velocity moment (w′3¯, m3 s−3), (g) mean potential temperature (θ¯−θ0, K), and (h) mean u-wind (u¯, m s−1) . Here overbars denote horizontally averaged variables or fluxes. The time period between 35 and 105 min is shown.

Citation: Journal of the Atmospheric Sciences 78, 12; 10.1175/JAS-D-21-0062.1

The positive momentum flux extends into the upper portion of the BL where the local vertical shear is nearly zero or positive. The momentum flux remains negative near the surface (i.e., z < 20 m) in accordance with the generally positive vertical wind shear, suggesting that local downgradient turbulent mixing dominates the KHB-induced upward transfer. The momentum flux decreases sharply for T > 60 min, presumably due to the breakdown of KH billows. The KHB-induced heat flux is negative with its magnitude much larger than its counterpart in the BL due to downward turbulent mixing, implying that KHBs are effective in vertical transfer of the momentum and scalars (Fig. 4c). This is in contrast to steady neutral (i.e., zero growth rate) linear buoyancy waves, which, with the vertical velocity and scalar perturbation in quadrature [see Eq. (A5)], carry no scalar flux. After the breakdown of the KH billows, episodic positive and negative heat flux maxima appear along the lower and upper PKE maximum branches with a characteristic time scale of a few minutes, presumably associated with SI and local overturning of isentropic surfaces. The potential temperature variance (θ′2¯) exhibits two maxima, which diverge away from the critical level with time, resembling the PKE maxima (Fig. 4d). The double θ′2¯ maxima are coincided with the thin layers characterized by enhanced stratification (Fig. 4g) below and above the KH cores. This phenomenon, referred to as “double-layer structure” in the literature, had been observed in the atmosphere (e.g., Browning and Watkins 1970; Luce et al. 2018) and oceans (e.g., Bourgault et al. 2001) associated with KH events.

Before and during the early stage of the KH event, the Ri minima in both the ESL and BL are less than unity (Fig. 4e). Near the critical level, Ri increases progressively with time in accordance to the weakening of the vertical wind shear and exceeds unity after the breakdown of the KH billows. The shear maxima propagate away from zc and consequently the low Richardson number zone is divided into two branches. Secondary instability occurs in the two low Ri branches and accounts for the diverging maxima in PKE and θ′2¯ along with episodic maxima and minima in the fluxes. It is worth noting that, although the horizontally averaged Ri in the two low Ri zones is larger than Ric = 0.25, the local Ri, regulated by the KH waves, could fall below Ric in wave crests as demonstrated in Jiang et al. (2010). During the KH event, the w skewness is negative above the critical level and positive below (Fig. 4f), implying that the KH billows tend to transfer w′2¯ toward the critical level in the ESL. Approximately below 200 m, the w skewness becomes negative, in accordance with a w′2¯ maximum located near z = 200 m, suggesting that KHBs tend to transfer PKE into the BL. The mean wind and scalar fields have been regulated by the KH event as well (Figs. 4g,h). The ESL expands substantially during the KH event with the stratification reduced and vertical shear weakened, owing to the large positive momentum and negative sensible heat fluxes. Similarly, the BL expands as well with weaker u wind after the KH event (Fig. 4h), highlighting the substantial impact of the KH event on the BL underneath. The thickening of the BL after the KH event is also evident in the potential temperature cross sections. The BL becomes less stratified capped by an inversion (Fig. 4g). It is worth noting that while the KH event only lasts for about 90 min, it likely takes hours for the BL to recover its pre-KH state.

The evolution of some variables is further revealed in the distance–time (i.e., x–T) sections (i.e., Hovmöller diagrams) for z = 30, 300, and 500 m, respectively (Fig. 5). At the early stage of the event (approximately T < 50 min) the KH waves are nearly stationary at all levels as predicted by the linear theory. Later they propagate leftwards (i.e., upwind with respect to the BL winds) with a phase speed of ~0.5 m s−1, presumably due to the weakening of the mean u wind below the critical level. It is worth noting that while θ′ and w undulations associated with the KHBs have larger amplitudes in the ESL than in the BL, the opposite is true for u′. This is consistent with the pronounced PKE maximum (Fig. 5a) and large u-wind fluctuations (Fig. 3b) in the BL. At the two selected levels in the ESL, large eddies first appear around T = 45 min (approximately 10 min before the breakdown occurs) riding on the quasi-two-dimensional KH billows, and become progressively stronger, presumably owing to the SI. For T > 50 min, the BL turbulence is substantially enhanced, especially over the descent portion of the BL waves, implying that BL becomes more turbulent even before breakdown occurs. For T > 70 min, the quasi-two-dimensional KH undulations in the ESL become weaker while chaotic perturbations become more significant.

Fig. 5.
Fig. 5.

Distance (x, km)–time (T, min) sections of (a)–(c) θ′, (d)–(f) u′, and (g)–(i) w′ at z = (left) 30, (center) 300, and (right) 500 m, respectively. In (a), θ′ is multiplied by 2. A dashed curve approximately corresponding to the location of a wave trough in θ′ at z = 30 m is included in all the panels to highlight phase differences between levels and variables.

Citation: Journal of the Atmospheric Sciences 78, 12; 10.1175/JAS-D-21-0062.1

The evolution of some domain-averaged variables are shown in Fig. 6. These variables are derived by first horizontally averaging across the domain and then vertically integrating over the depth of the BL and the rest separately (i.e., ϕB=∫0ZBϕ dz and ϕE=∫ZBZTϕ dz, where ZB = 150 m is the approximate depth of the BL and ZT = 1500 m is the domain top). Figure 6 suggests that the KH event can be divided into three stages, namely, the growth (i.e., approximately between 20 and 50 min), saturation (between 50 and 70 min), and decay (between 70 and 110 min) stages. The total PKE in the ESL (i.e., IE=∫ZBZTe dz) starts growing exponentially approximately at T ~ 20 min and exceeds its counterpart in the BL (i.e., IB=∫0ZBe dz) at T ~ 30 min. Between 30 and 50 min, both IE and IB grow exponentially with an e-folding growth time of ~3.3 min, which is comparable to the linear theory prediction (i.e., ~3.6 min; e.g., Tanaka 1975). The saturation stage is manifested by a plateau in the IE and IB curves. As shown in Fig. 5, during the saturation stage, the KH waves are either highly nonlinear with emerging large eddies associated with SI or in the process of breakdown (i.e., for T > 55 min). Afterward, the KH event enters its decay or dissipation stage characterized by a rapid decrease of IE and IB with time. Between 70 and 110 min, IE and IB exhibit an exponential decay with an estimated e-folding decay time of ~11 min. For T > 110 min, the decrease of IE becomes much slower and the KH billows virtually transition into neutral waves. Shown in Fig. 6b are the vertically integrated shear production, total production, and PKE dissipation rates. The shear production rate peaks at T = 55 min, approximately the time when the KHB breakdown takes place. The total production is sizably smaller than the shear production due to the negative contribution from buoyancy. Afterward, the shear production rate decreases sharply in accordance to the reduction in the wind shear. The dissipation rates in the BL and the ESL reach their maxima at 62 and 70 min, respectively, lagging behind the production maximum by ~7–15 min.

Fig. 6.
Fig. 6.

Evolution of domain averaged variables over the first 150 min, including (a) the vertically integrated PKE (IB and IE), (b) the PKE production rates (i.e., shear production, SPR, and total production, TPR = SPR + BPR, where BPR denotes the buoyancy production rate), and dissipation rates in the BL (εB) and the ESL (εE), (c) vertical aspect ratios (ARwB and ARwE), (d) meridional aspect ratios (ARυB and ARυE), and (e) normalized surface friction velocity (u*/u*0; where the subscript “0” denotes the pre-KH mean) and normalized sensible heat flux (Hs/Hs0). The saturation stage is brown shaded.

Citation: Journal of the Atmospheric Sciences 78, 12; 10.1175/JAS-D-21-0062.1

The three-dimensionality of KH billows and BL turbulence can be measured by the vertical and meridional aspect ratios of PKE defined as ARw=w′2¯/⁡(2e) andARυ=υ′2¯/⁡(2e) (the primed variables are deviations from their means at a given level). In the BL, ARw ~ 0.1 and ARυ ~ 0.3 before the KH event (Figs. 6c,d) due to the stable stratification. During the growth stage, while ARw increases moderately with time, ARυ in the BL decreases substantially, implying that the PKE maximum in the BL comes mostly from the u-wind undulations. Later, the fractional contribution from υ wind increases substantially, in accordance with the turbulent breakdown of the quasi-two-dimensional BL waves. The evolution of ARw and ARυ in the ESL is similar to those in the BL except that ARw is much larger during the KH event than its counterpart in the BL. The ARυ is small in the ESL during the growth stage as expected of the virtually 2D KH billows, and increases substantially after their breakdown. It is interesting that the occurrence of KH instability aloft also has substantial impact on the air–land interaction. During this KH event, the surface friction velocity nearly doubles associated with the increase in the surface wind speed. The surface sensible heat flux increases by a factor of 5–9 during the saturation stage, owing to downward heat transfer by KHBs.

The influence of the KH event on the mean profiles is evident in Fig. 7. The wind shear and stratification in the ESL have been markedly regulated by the KH event. Most noticeably, the wind shear near the critical level is gradually reduced during and after the KH event. Similarly, the potential temperature profile is modified by the KH event with a less stable layer near the critical level in accordance with the KHB-induced vertical heat transfer and associated turbulent mixing after breakdown. A pair of positive and negative maxima in υ wind forms below and above the critical level, in response to the vertical redistribution of the u-wind component. The BL capped by an inversion becomes substantially deeper and less stratified after the KH event. During the KH event, the PKE exhibits two distinct maxima in the ESL and the BL, respectively. Afterward, the elevated PKE maximum is split into two, while the BL maximum becomes smaller than its peak value in the saturation stage but is still substantially larger than the pre-KH maximum.

Fig. 7.
Fig. 7.

Domain-averaged profiles of (left to right) U, V, θ − θ0, and PKE from CONTROL valid at T = 18 (i.e., before KH), 60 (i.e., during KH), and 105 min (i.e., after the KH event). The mean profiles at t = T0 are included for comparison. The critical level in the mean winds is indicated by a dashed line.

Citation: Journal of the Atmospheric Sciences 78, 12; 10.1175/JAS-D-21-0062.1

c. Power spectra and cospectra in the BL

The power spectra of the u wind in the BL are shown in Fig. 8 for five different time periods. Each spectrum is averaged over 1000 time steps (i.e., T ± 1.75 min). Before the KH event, the BL is much less energetic with a well-defined −5/3 slope for the dimensionless wavenumber Kh approximately between 30 and 300. At later times, as expected, the spectra exhibit pronounced maxima centered at Kh = 3, corresponding to the primary KH waves. For smaller scales (i.e., Kh > 30 or wavelength <150 m), the spectra follow the −5/3 slope reasonably well. From 45.5 to 52.5 min, the primary maximum increases by a factor 3, while the turbulence (i.e., Kh > 30) becomes approximately 8 times more energetic, implying the significant impact of KH billows on the turbulence generation in the BL even before vigorous KHB breakdown occurs. The primary maxima are nearly identical for T = 52.5 and 59.5 min as the KHBs are in their saturation stage. However, there is noticeable increase in the turbulent kinetic energy. It is interesting to compare the spectra for T = 45.5 and 87.5 min. While the primary maximum at T = 87.5 min is only ~1/5 of its counterpart at T = 45.5 min, the power for KH > 30 is about 5 times of its counterpart, implying that significant percentage of the energy in KH billows has been transferred to turbulent scales.

Fig. 8.
Fig. 8.

Power spectra density for the u wind (Euu, m3 s−2) in the BL at z = 30.9 m valid at T = 45.5, 52.2, 59.5, and 87.5 min. The same spectrum at the time of restart multiplied by a factor 10 is included for comparison. The horizontal axis is the total wavenumber normalized by 2π/Lx.

Citation: Journal of the Atmospheric Sciences 78, 12; 10.1175/JAS-D-21-0062.1

The corresponding cospectra of w–u and w–θ are shown in Fig. 9. A primary positive maximum is centered at Kh = 3 in the w–u spectrum, suggestive of a large upward momentum transfer by the KH waves even within the BL. The positive primary maximum grows between 46 and 53 min, decays between 53 and 60 min, and becomes barely noticeable by 88 min (not shown), in accordance with the breakdown of KH billows and BL waves. A secondary positive maximum appears for T = 53 and 60 min, presumably due to SI. Over the turbulent range, the momentum flux is negative, being consistent with generally positive shear in the mean u wind inside the BL.

Fig. 9.
Fig. 9.

Cospectra of (a) u and w and (b) w and θ at z = 30.9 m AGL valid at T = 45.5 (black), 52.5 (red), and 59.5 (green) min. The horizontal axis is as in Fig. 8.

Citation: Journal of the Atmospheric Sciences 78, 12; 10.1175/JAS-D-21-0062.1

4. Sensitivity study

Four additional simulations (see Table 1 for the control parameters) have been conducted to explore the sensitivity of the KHB–BL interaction to the ESL height, the BL type, and the sign of the elevated shear. In these simulations, KHBs develop in the ESL similar to those in CONTROL. On the other hand, the BL response is substantially different from simulation to simulation. In terms of the u-wind fluctuations near the surface, the wind reversal and rotor formation only occur in LOWER with the negative u wind about half the speed as in CONTROL (Fig. 10). There is no wind reversal near the surface in HIGHER, implying weaker forcing on the underlying SBL if KHBs are higher. The BL top (i.e., the level where the potential temperature and vorticity exhibit sharp gradients) undulation in HIGHER is markedly weaker than its counterparts in LOWER and CONTROL (Figs. 10e,f). While the BL top in CBL undulates with an amplitude comparable to that in CONTROL, the u-wind fluctuations near the surface are substantially smaller than in the latter. There is no flow reversal (i.e., no rotor) under wave crests in CBL and the spanwise vorticity is confined in a much shallower layer immediately above the surface (Figs. 10c,h). The absence of rotor circulations in the convective BL is likely due to the following reasons: (i) the BL is much deeper and KH induced fluctuations suffer from stronger turbulent dissipation before they can reach the surface layer, (ii) the turbulent mixing of momentum is much stronger than in the SBL, which tends to reduce the u-wind fluctuation, and (iii) the vertical shear is confined in a very thin layer above the surface. The weakest BL response occurs in FWSHR and the amplitude of the wavy isentropes appears to decay faster toward the surface than in the others.

Fig. 10.
Fig. 10.

Vertical cross-section snapshots of (a)–(d) u (m s−1), (e)–(h) θ − θ0 (K), and (i)–(l) spanwise vorticity (i.e., η = ux − wz, normalized by η0) for (first column) LOWER, (second column) HIGHER, (third column) CBL, and (fourth column) FWSHR valid at T = 45 min, approximately 10 min before KHB breakdown, are shown.

Citation: Journal of the Atmospheric Sciences 78, 12; 10.1175/JAS-D-21-0062.1

The differences in statistics between simulations are further illustrated in the time–height sections (Fig. 11). The vertical distribution of PKE in LOWER and HIGHER (Figs. 11a,b) is qualitatively similar to their counterpart in CONTROL (Fig. 4a), characterized by a pronounced maximum in the BL. Quantitatively, the PKE maximum above the BL is the largest in CONTROL and smallest in FWSHR, which is consistent with Fig. 10. The vertical variation of the PKE maxima in CBL resembles those in CONTROL except that the PKE maximum in the BL is less pronounced but distributed throughout a deeper BL than in CONTROL (Fig. 11c). For FWSHR, there is only one well-defined maximum located above the BL (Fig. 11d) suggesting that the impact of the elevated KH billows on the BL is much less pronounced than the others.

Fig. 11.
Fig. 11.

Time–height sections of (a)–(d) PKE (e, m2 s−2), (e)–(h) u′w′¯ (m2 s−2), and (i)–(l) w′θ′¯ (K m s−1) for T between 20 and 85 min derived from (first column) LOWER, (second column) HIGHER, (third column) CBL, and (fourth column) FWSHR. The momentum flux for FWSHR is multiplied by −1 (i.e., −u′w′¯) for convenience of comparison.

Citation: Journal of the Atmospheric Sciences 78, 12; 10.1175/JAS-D-21-0062.1

The vertical distribution of the momentum and sensible heat fluxes is qualitatively similar in these simulations characterized by a broad primary positive or negative maximum centered at the critical level (Figs. 11e–l). In FWSHR, while the heat flux is predominantly negative as in others, the momentum flux is generally negative, implying downward momentum transfer in accordance to the positive mean shear in its ESL. Again, the fluxes in FWSHR have the smallest magnitudes and accordingly the KHBs exert least influence on the underlying BL.

The vertically integrated PKE from these simulations exhibits similar three-stage evolution (Fig. 12). While the IE and IB growth and decay rates are generally comparable, there are some interesting discrepancies that are worth discussing. First, the KHBs in CBL develop about 15 min earlier than in the others, presumably due to stronger perturbations from the convective BL. This difference in timing is likely minimal if the ESL is seeded with small sinusoidal perturbations at restart. Second, IE in LOWER grows slightly slower than in the others, likely due to the negative impact of the ground surface and the BL on the elevated KHBs. Furthermore, the IE maxima vary substantially from simulation to simulation (Table 1), due to the interaction between the ESL and the BL. The IE maximum is the largest in CONTROL and CBL and smallest in FWSHR. For FWSHR, IE maximum is around half that in CONTROL, suggesting the importance of the sign of the elevated shear.

Fig. 12.
Fig. 12.

The vertically integrated PKE in the BL (IB, solid) and ESL (IE, dashed) are plotted as a function of time for CONTROL (gray), LOWER (black), CBL (red), FWSHR (green), and HIGHER (blue).

Citation: Journal of the Atmospheric Sciences 78, 12; 10.1175/JAS-D-21-0062.1

For each simulation, while IB exhibits similar exponential growth and decay as IE, it is substantially smaller than the latter. The ratio between the IE and IB maxima (i.e., R = IBmax/IEmax referred to as responsive ratio) provides a crude measure of the BL response to a given KHB forcing above. The R values (Table 1) tend to decrease with the ESL level, implying that the impact of the KHBs on the BL is generally less significant when the ESL is higher. The ratio R is dramatically lower for CBL and FWSHR. This is consistent with the vertical cross sections in Fig. 10 and the underlying dynamics will be further explored below. Another useful parameter is the IB ratio before and after the KH event, defined as RBA = IB(TA)/IB(TB), which measures the lasting impact of the elevated KHBs on the BL (Table 1). The RBA values suggest that, while the elevated KH billows have substantial impact on the BL for all the cases, their lasting impact is more dramatic in CONTROL and LOWER, and least significant in CBL and FWSHR. This is consistent with the variation of the responsive ratio from simulation to simulation.

It is instructive to examine the pressure and u-wind fluctuations in the x direction at a given level in the BL. According to the linear wave theory, for a two-dimensional internal wave in a uniform atmosphere, the density normalized pressure and u-wind perturbations are related as p′=(c−u¯)u′ (see appendix). If the wave is neutral and inviscid, c−u¯ is real and the wave-induced fluctuations along the x direction appear as a straight line in a u(x)–p′(x) diagram with a slope of c−u¯. For a growing or decaying wave, c−u¯ is complex and the wave manifested itself as an ellipse. Shown in Fig. 13 are such u(x) –p′(x) phase diagrams for z = 50 and 100 m before the breakdown of the KHBs. At z = 50 m, the lower half of the trajectory for CONTROL corresponding to larger mean u winds in wave troughs nearly collapses onto a straight line with a negative slope of around 4.5 (Fig. 13a), which is in reasonable agreement with linear theory (i.e., c−u¯≈−4.8 m s−1). The upper half of the trajectories, corresponding to weaker u winds in the wave crests, manifest as loops implying stronger nonlinearity in wave crests. The slopes of the major axes are between −3.5 and −4.0, in reasonable agreement with the u¯ for the upper half of the loops. The u(x)–p′(x) phase diagrams for HIGHER and LOWER at z = 50 m exhibit similar looping patterns as in CONTROL with a mean negative slope comparable to that in CONTROL. However, the KHB-induced pressure undulations are weaker in LOWER and HIGHER, resulting in smaller perturbations in the u winds than in CONTROL.

Fig. 13.
Fig. 13.

The variation of perturbation pressure (p′) along the x direction is plotted against the u-wind variation for y = Ly/2 at level (a) z = 50 m and (b) z = 100 m, valid at T = 45 min, approximately 10 min before breakdown occurs.

Citation: Journal of the Atmospheric Sciences 78, 12; 10.1175/JAS-D-21-0062.1

According to the linear theory, the amplitude of KH induced pressure fluctuations tends to decay exponentially with the distance from the shear layer [appendix: (A4)]. Using parameters estimated from CONTROL (i.e., c = 0, U ~ 6 m s−1, N2 ~ 2.2 × 10−4 s−2 and k = 2π/1490 m−1), Eq. (A3) yields l ~ 0.0034 m−1. Using (A4), the amplitude of KHB oscillations decreases by about a half for every 200 m increase in the vertical distance. If the KH wave is short enough [i.e., N2/(c − U)2 ≪ k2], Eq. (A4) becomes ϕ1e−k|zc−D|. Hence we can define a nondimensional shear level as h^=k⁡(zc−D)and the maximum BL response occurs for h^ ~1 (Table 1). In CBL, the u and p′ ranges are comparable to those in HIGHER with much stronger chaotic perturbations. For FWSHR, the p′ and u oscillations are positively correlated and approximately collapsed onto a straight line with a mean slope around 7, which is consistent with the linear theory prediction (i.e., c ~ 12 m s−1, u¯~5 m s−1, and c−u¯≈7 m s−1). Therefore, for a given pressure oscillation, the corresponding u oscillation is smaller in FWSHR than in the rest due to its larger c−u¯ near the surface. For a quantitative comparison, we define the dimensionless u fluctuation as sz = U0Δu/Δp, where Δp and Δu denote the oscillation ranges of p′ and u at level z, which provides a convenient measure of the u-wind response to a given pressure forcing. The s50 values (Table 1) indicate that the strongest response of the BL winds occur in CONTROL and LOWER while the weakest occurs in FWSHR.

At z = 100 m (i.e., near the top of the SBL, Fig. 13b), the diagrams are qualitatively consistent with their counterparts at the 50 m level with two noticeable differences. First, the minor axes of the trajectory ovals are shorter than their counterparts at the 50 m level, suggesting that the KHB-induced undulations at z = 100 m are more consistent with the linear neutral wave assumption. Second, in general, the BL response is weaker than at z = 50 m, presumably because the mean u winds at 100 m are larger than at the 50-m level.

5. Discussion and conclusions

The impact of elevated KHBs on the underlying atmospheric BL is examined utilizing a group of large-eddy simulations. In these simulations, KHBs develop in an ESL, characterized by a hyperbolic-tangent wind profile. The KH event lasts for about 90 min and its life cycle can be divided into three stages, namely, the growth, saturation, and decay stages. In the growth stage, quasi-two-dimensional KHBs rapidly grow with the total PKE increasing exponentially with time by extracting energy from the sheared flow. In the saturation stage, the total PKE slowly evolves while the growth is partially counterbalanced by turbulent dissipation, emerging SI, and breakdown of KHBs. The turbulent breakdown propagates from the billow cores outward, with the total PKE exponentially decaying with time. The heat and momentum fluxes associated with KH billows are in general downgradient and their magnitudes grow with the KH wave amplitude during the growth stage and decrease sharply after the turbulent breakdown occurs. The simulated KHB life cycle and general features are consistent with previous studies of KHI in high-Reynolds-number fluids.

This study finds that elevated KH billows could exert a significant impact on the underlying BL by inducing vertical undulations of isentropic surfaces, wavelike oscillations in the horizontal winds, and intermittent turbulence in the BL. A few aspects of this impact are worth mentioning. First, the wavy motion in the BL grows simultaneously with the elevated KHBs. The PKE maximum in the BL is comparable or even larger than its counterpart aloft. The total PKE in the BL experiences similar exponential growth, saturation, and exponential decay stages, approximately in pace with its counterpart aloft, suggestive of tight coupling between the elevated KHBs and the BL waves. Second, the undulations in the BL bear some characteristics of both KH waves and trapped internal waves. While the roll up of the surface shear layer and the exponential growth of the PKE in the BL resemble KH billows, the solid surface suppress vigorous vertical motion and the formation of well-defined KH billow cores in the BL. On the other hand, the wavy patterns of the BL top inversion and rotors underneath wave crests resemble typical trapped waves and rotors observed in the lee of mountains (e.g., Doyle et al. 2009). In accordance with the breakdown of the elevated KHBs, SI and breakdown occur in the BL, and consequently the BL flow underneath the BL top inversion becomes less stratified and more turbulent. Turbulent dissipation in the BL exhibits a pronounced maximum, approximately at the time when the PKE peaks in the BL and ESL. Third, the interaction between waves and turbulence contributes to the turbulence enhancement in the BL even before the breakdown occurs, as evidenced by the power spectral analysis. Furthermore, the BL depth increases sizably after the KH event with reduced wind shear and stratification even in the presence of surface cooling. Finally, the KH event also leads to enhanced air–land interaction, including substantial increase in the surface friction velocity and the sensible heat flux.

The impact of the elevated KH billows on the BL varies with the ESL and BL characteristics. The interaction between KHBs and the underlying BL is sensitive to the nondimensional ESL height, h^=k⁡(zc−D). The maximum impact occurs when h^ is near unity. If the shear layer is too high (i.e., h^>1), the ESL and the BL are less coupled and the influence of the KHBs on the BL weakens with increasing height. This is largely because KHB-induced undulations (e.g., pressure) tend to weaken with the vertical distance from the ESL. On the other hand, if the ESL is too low (h^<1), while the ESL and BL are closely coupled, the solid surface and the turbulent BL limit the full development of KHBs. For given KHB forcing, its impact on a convective BL is less significant than on a SBL, due to the typically deeper BL depth and stronger turbulent mixing. The stronger vertical momentum mixing tends to reduce the horizontal wind fluctuations and suppress the u-wind reversal near the surface. In addition, the upper portion of a convective BL could effectively dampen the KHB-induced forcing on the surface layer. As a result, while there are pronounced PKE maxima during an KH event, rotors and breakdown of BL waves are absent in the convective BL. Turbulent dissipation in a convective BL is still significantly enhanced during a KH event, presumably through the interaction between turbulence and KHB-induced oscillations in the wind shear. Finally, the sign of the shear layer is important in driving the BL response. In general, KHBs in an ESL with a backward shear have substantial stronger impact on the BL than a forward shear layer with the same shear strength.

An elevated shear layer may develop in the lower troposphere across a BL top inversion, over complex terrain, over the top of a nocturnal BL, and associated with a variety of synoptic or mesoscale processes such as cold fronts and sea-breeze circulations. KHI occurs in elevated shear layers likely more frequently than observed due to its episodic nature and being invisible to naked eyes in a cloud-free environment. According to this study, elevated KH billows may exert substantial impact on the underlying BL. Especially over the SBL, elevated KHB induced oscillations are a plausible source for “dirty waves” and intermittent turbulence documented in stable boundary layers (e.g., Sun et al. 2015).

Finally, a few limitations of this study are worth noting. First, the KH waves examined here are mostly stationary (except for in FWSHR), while propagating KH waves are likely more common in the real world (e.g., Sun et al. 2002; Belušić et al. 2007). A few remarks can be made regarding the impact of propagating KH waves on the underlying BL utilizing the linear wave relation between pressure and velocity perturbations, p′=(c−u¯)u′. If elevated KH waves propagate against the mean BL wind [i.e., real(c) and u¯ have opposite signs], the BL response tends to be weaker for a given KH wave, and if real(c) and u¯ share the same sign, stronger BL response is expected. However, if c−u¯≈0, this linear relationship breaks down and a critical level is present in the BL. Further study is needed to fully understand the interaction between propagating KH waves and the BL, especially when the wave phase speed matches the mean wind in the BL. In theory, the above linear pressure–wind relation is valid for a PBW as well, and therefore the same can be said about the influence of a PBW on the underneath BL. Second, KH waves observed in the field were often more transient with a finite number of waves. The use of periodic boundary condition in our idealized LES study implies an infinite number of waves, although the KH waves are transient with a life cycle of about 90 min. Consequently, the simulated BL wave response is likely stronger than observed under more transient conditions. This issue will be addressed in a future study. Furthermore, the wavelengths of KH waves documented in the atmospheric BL vary from a few hundreds of meters (e.g., Petenko et al. 2016) to several kilometers (e.g., Cronenwett et al. 1972; Kjelaas et al. 1974). In general, shorter waves have shorter life cycles and their influence decays faster with the vertical distance away from the shear layer.

Acknowledgments

This research is supported by the Chief of Naval Research through the NRL Base Program, PE 0601153N. The author would like to thank Dr. Peter Sullivan at the National Center for the Atmospheric Research for kindly providing us his LES code. Computational resources were supported by a grant of HPC time from the Department of Defense Major Shared Resource Centers.

APPENDIX

Linear Theory Review

Small-amplitude two-dimensional perturbations in a continuously stratified unidirectional inviscid Boussinesq flow are governed by the following linear equations:
⁡(∂∂t+U∂∂x)u′+w′dUdz=−∂p′∂x,
⁡(∂∂t+U∂∂x)w′=−∂p′∂z+gθ0θ′,
⁡(∂∂t+U∂∂x)θ′+w′dθ¯dz=0,
∂u′∂x+dw′dz=0.
Here U and θ¯ are the mean u wind and potential temperature. Applying normal mode expansion to the perturbation variables, namely, ⁡(u′,w′,θ′,p′)=(u^,w^,θ^,p^) exp[ik⁡(x−c)], and using the streamfunction ϕ^=iw^/k, Eqs. (A1) can be combined into a single Taylor–Goldstein equation (e.g., Tanaka 1975):
d2ϕ^∂z2−l2ϕ^=0,
where l2 = k2 − N2/(c − U)2 − Uzz/( c − U). The wave phase speed c is complex for a KH wave and real for a neutral internal gravity wave. KH modes and their growth rate can be obtained by numerically solving (A2) with proper boundary conditions. Outside of the shear layer, assuming that U and N2 are vertically uniform, we have
l2=k2−N2/⁡(c−U)2,
ϕ^=ϕ^1e−l|z−z1|,
p^=(c−U)u^,
w^=ik⁡(c−U)θ^.
In (A4), ϕ^=ϕ^1 at reference level z1. For l2 > 0, ϕ^ decays exponentially with the vertical distance from the shear layer. In (A6), if c is real, w^ and θ^ are in quadrature and the sensible flux is zero. Letting c = cr + ici, we have w^θ^=−kciθ^2, which implies that the heat flux is negative for a growth mode (i.e., ci > 0) and positive for a decay mode.

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  • Jiang, Q., Q. Wang, S. Wang, and S. Gaberšek, 2020: Turbulence adjustment and scaling in an offshore convective internal boundary layer: A CASPER case study. J. Atmos. Sci., 77, 1661–1681, https://doi.org/10.1175/JAS-D-19-0189.1.

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    • Search Google Scholar
    • Export Citation

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    • Search Google Scholar
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    • Crossref
    • Search Google Scholar
    • Export Citation

  • Jiang, Q., J. D. Doyle, V. Grubišić, and R. B. Smith, 2010: Turbulence characteristics in an elevated shear layer over Owens Valley. J. Atmos. Sci., 67, 2355–2371, https://doi.org/10.1175/2010JAS3156.1.

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    • Search Google Scholar
    • Export Citation

  • Jiang, Q., Q. Wang, S. Wang, and S. Gaberšek, 2020: Turbulence adjustment and scaling in an offshore convective internal boundary layer: A CASPER case study. J. Atmos. Sci., 77, 1661–1681, https://doi.org/10.1175/JAS-D-19-0189.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Kelvin, L., 1871: Hydrokinetic solutions and observations. Philos. Mag., 42, 362–377, https://doi.org/10.1080/14786447108640585.

  • Kjelaas, A. G., D. W. Beran, W. H. Hooke, and B. R. Bean, 1974: Waves observed in the planetary boundary layer using an array of acoustic sounders. J. Atmos. Sci., 31, 2040–2045, https://doi.org/10.1175/1520-0469(1974)031<2040:WOITPB>2.0.CO;2.

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    • Search Google Scholar
    • Export Citation

  • Klemp, J. B., and D. R. Durran, 1983: An upper boundary condition permitting internal gravity wave radiation in numerical mesoscale models. Mon. Wea. Rev., 111, 430–444, https://doi.org/10.1175/1520-0493(1983)111<0430:AUBCPI>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Luce, H., L. Kantha, M. Yabuki, and H. Hashiguchi, 2018: Atmospheric Kelvin–Helmholtz billows captured by the MU radar, lidars, and a fish-eye camera. Earth Planets Space, 70, 162, https://doi.org/10.1186/s40623-018-0935-0.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Lyulyukin, V., and Coauthors, 2019: Sodar observation of the ABL structure and waves over the Black Sea offshore site. Atmosphere, 10, 811, https://doi.org/10.3390/atmos10120811.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Medina, S., and R. A. Houze Jr., 2016: Kelvin–Helmholtz waves in extratropical cyclones passing over mountain ranges. Quart. J. Roy. Meteor. Soc., 142, 1311–1319, https://doi.org/10.1002/qj.2734.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Miles, J., 1961: On the stability of heterogeneous shear flows. J. Fluid Mech., 17, 496–508, https://doi.org/10.1017/S0022112061000305.

  • Moeng, C. H., 1984: A large-eddy-simulation model for the study of planetary boundary-layer turbulence. J. Atmos. Sci., 41, 2052–2062, https://doi.org/10.1175/1520-0469(1984)041<2052:ALESMF>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Mondal, S., and Coauthors, 2019: On the long lasting “C-type” structures in the sodium lidargram: The lifetime of Kelvin–Helmholtz billows in the mesosphere and lower thermosphere region. J. Geophys. Res. Space Phys., 124, 3110–3124, https://doi.org/10.1029/2019JA026630.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Muschinski, A., 1996: Possible effect of Kelvin–Helmholtz instability on VHF radar observations of the mean vertical wind. J. Appl. Meteor., 35, 2210–2217, https://doi.org/10.1175/1520-0450(1996)035<2210:PEOKHI>2.0.CO;2.

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