Turbulent transport mechanisms in long-lived stable Ekman layers

Abstract

Direct numerical simulations (DNS) are conducted for long-lived stable atmospheric boundary layers (SABLs) maintained by constant ambient stratification and surface cooling. The study examines how stratification mechanism and strength influence turbulence within a four-dimensional parameter space defined by the stratification perturbation parameter ($Π_s$), wind-forcing parameter ($Π_w$), Rossby-radius factor ($Π_f$), and Prandtl number. A regime map in the $Π_s-Π_w$ plane identifies linearly stable, very stable, and weakly stable regimes, showing that turbulence sustenance is inherently multi-parametric. At low $Π_s$ and high $Π_w$, a weakly stable regime with persistent turbulence emerges. The SABL exhibits a multilayered thermal structure-comprising a near-surface stable layer, an intermediate unstable layer, and an overlying inversion-that strengthens with decreasing $Π_s$, indicating enhanced downward heat transport. Turbulence statistics reveal weak dependence of the momentum field and turbulent kinetic energy (TKE) on stratification, while buoyancy and turbulent potential energy (TPE) show strong sensitivity due to additional production from turbulent heat flux interacting with ambient stratification. Energy transfer occurs among mean gradients, momentum-buoyancy flux, and TKE-TPE exchange. The turbulent Prandtl number exhibits strong vertical variation, exceeding values typical of nocturnal SABLs, underscoring the limits of constant eddy-diffusivity models. Barycentric anisotropy maps indicate weak near-surface effects but enhanced isotropy aloft. These results motivate improved parameterizations for ambient-stratification-dominated SABLs.

Figures (26)

• Figure 1: Illustration of the long-lived stable Ekman boundary layer with key parameters. $U_g$ is the geostrophic wind, $B_s$ is the imposed surface buoyancy flux, and $N_a^2$ is the strength of the ambient stratification that is imposed independent of $B_s$ in the equations.
• Figure 2: Comparison of (a) dimensionless velocity $M^+-y^+$ and (b) dimensionless velocity gradient $(\phi_m)$ for neutrally stratified conditions at $Re_D=400$ and $900$ with DNS data of shah_direct_2014. The log-law in frame (a) refers to $M^+=\kappa^{-1}~log(y^+)+B$, where $\kappa=0.41$ and $B=5.5$. The dashed line in frame (b) shows $\phi_m=1$.
• Figure 3: Dimensionless spectra of vertical velocity $(E_{vv}$) for cases XIX-XXII computed at $y^+=32$, where the $y^+$ is computed using the friction velocity from the neutral case. Turbulence dissipation rate $\epsilon = 2 \nu \langle \partial_j u_i^\prime \partial_j u_i^\prime \rangle$, and Kolmogorov length scale $\eta = (\nu^3/\epsilon)^{0.25}$ at $y^+=32$ are used for normalization. These four cases are later used for budget analysis.
• Figure 4: Effect of $\varPi_s$ and $\varPi_w$ parameters on (a) time series of vertical velocity sampled at the geometric center of a horizontal plane at $y/D(y^+)=0.5(32.70)$ and (b) PDF of the vertical velocity fluctuations. The gray shaded region in frame (a) indicates turbulence collapse, which is a measure of distinction between very stable and weakly stable. Markers $P_1-P_3$ denote time instances before turbulence collapse, during turbulence collapse and after turbulence resurgence, respectively. Note that the red , blue and black curves are shifted vertically by $0.1,-0.1$ and $-0.2$, respectively, for a better representation in frame (a), and analytical Gaussian distribution with zero mean and unit standard deviation is shown with cyan colored line in panel (b).
• Figure 5: Contour visualisations of instantaneous vertical (a-c) and horizontal velocity (d-f) fields from Case 2 with $\varPi_w = 8100$ and $\varPi_s=1$ in $y-z$ and $x-z$ planes, respectively, at time instances $P_1 (a,d), P_2 (b,e)$ and $P_3(c,f)$, as marked in Fig. \ref{['fig:Ri_timeseries']}(a). The arrow indicates the geostrophic wind direction.