Direct Numerical Simulation of Turbulence Collapse and Rebirth in Stably Stratified Ekman Flow

Abstract

Direct numerical simulations of an Ekman layer are performed to study flow evolution during the response of an initially neutral boundary layer to stable stratification. The Obukhov length, L, is varied among cases by imposing a range of stable buoyancy fluxes at the surface to mimic ground cooling. The imposition of constant surface buoyancy flux , i.e. constant-flux stability, leads to a buoyancy difference between the ground and background that tends to increase with time, unlike the constant-temperature stability case where a constant surface temperature is imposed. The initial collapse of turbulence in the surface layer owing to surface cooling that occurs over a time scale proportional to $L/u_*$, where $u_*$ is the friction velocity, is followed by turbulence recovery. The flow accelerates, and a “low-level jet” (LLJ) with inertial oscillations forms during the turbulence collapse. Turbulence statistics and budgets are examined to understand the recovery of turbulence. Vertical turbulence exchange, primarily by pressure transport, is found to initiate fluctuations in the surface layer and there is rebirth of turbulence through enhanced turbulence production as the LLJ shear increases. The turbulence recovery is not monotonic and exhibits temporal intermittency with several collapse/rebirth episodes. The boundary layer adjusts to an increase in the surface buoyancy flux by increased super-geostrophic velocity and surface stress such that the Obukhov length becomes similar among the cases and sufficiently large to allow fluctuations with sustained momentum and heat fluxes. The eventual state of fluctuations, achieved after about two inertial periods ($ft \approx 4\pi$), corresponds to global intermittency with turbulent patches in an otherwise quiescent background. Our simplified configuration is sufficient to identify turbulence collapse and rebirth, global and temporal intermittency, as well as formation of low-level jets, as in observations of the stratified atmospheric boundary layer.

Appendix: Comparison of Constant-Temperature and Constant-Flux Stability

We have simulated two additional cases with constant-temperature stability, i.e. constant surface temperature (constant $Ri_b$), to illustrate their differences with the constant surface buoyancy flux cases (where $Ri_b$ continuously changes). The time evolution of $Ri_b$, plotted previously in Fig. 3b, shows that the $\textit{SLD}1$ case, that has an initially low value of $Ri_b \approx 0.17$, has a large value of $Ri_b=0.62$ at $ft\approx 8$. This motivates our choice of $Ri_b=0.62$ and $Ri_b=0.17$ as the two constant-temperature stability cases to compare with case $\textit{SLD}1$. The comparison is elaborated upon in the following paragraphs.

Fig. 13

Profiles of normalized statistics as a function of normalized height: a velocity components; (b) TKE. Here plots are presented for two cases with constant-temperature stability ($Ri_b = 0.17$ and $Ri_b = 0.62$ lines) and constant-flux stability ($L^-=1$ lines)

Profiles of mean velocity and TKE are shown in Fig. 13a, b, respectively, and compared to long-time, quasi-steady profiles of the constant $Ri_b$ cases. The $ft \approx 8$ (when $Ri_b (t) \approx 0.62$) velocity profiles of case $\textit{SLD}1$ in Fig. 13a show a stronger LLJ in the streamwise velocity component than the constant $Ri_b = 0.62$ case, and the spanwise velocity component is larger in magnitude. These differences in the mean velocity persist at a longer time of $ft \approx 30$. It is worth noting that case $\textit{SLD}1$ exhibits inertial oscillations in the outer layer during its evolution unlike the cases with constant-temperature stability that achieve a quasi-steady state. At $ft \approx 8$, the $\textit{SLD}1$ case is in a state just before turbulence collapse in the near-surface layer and Fig.  13b shows that the TKE is low relative to the $Ri_b = 0.62$ case. However, the TKE recovers later during the evolution of case $\textit{SLD}1$ and, at $ft \approx 30$, TKE is comparable to the constant $Ri_b = 0.62$ case and slightly larger near the wall.

Fig. 14

The time evolution of normalized friction velocity ($u_*/U_\infty$) for the simulated cases with constant-temperature stability ($Ri_b = 0.17$ and $Ri_b = 0.62$ lines) and constant-flux stability ($L^-=1$ line)

Figure 14 shows differences in the time evolution of normalized friction velocity among the different cases. Case $\textit{SLD}1$ has an initial $Ri_b=0.17$ at $t=0$ and exhibits a similar evolution of $u_*$ as the constant $Ri_b=0.17$ case until $ft\approx 3$. Later in time, $u_*$ increases in case $\textit{SLD}1$ to values larger than the neutral case due to the strong LLJ in the $\textit{SLD}1$ case (Fig. 13a) that increases both streamwise and spanwise shear in the surface layer. The constant-temperature stability case with $Ri_b = 0.62$ shows a large initial reduction of $u_*$ followed by an increase. However, unlike the $\textit{SLD}1$ case, the value of $u_*$ in the $Ri_b = 0.62$ case remains smaller than that in the neutral case, consistent with studies of constant-temperature stability by other investigators.