Application of a Large Eddy Simulation Database to Optimization of First Order Closures for Neutral and Stably Stratified Boundary Layers
Igor N. Esau, Oyvind Byrkjedal
备注：
The submission to a special issue of the Boundary-Layer Meteorology devoted to the NATO advanced research workshop Atmospheric Boundary Layers: Modelling and Applications for Environmental Security
Large-eddy simulation (LES) is a well-established numerical technique, which resolves the most energetic turbulent fluctuations in the planetary boundary layers. Averaging these fluctuations, high-quality profiles of mean values and turbulence statistics can be obtained in experiments with well-defined initial and boundary conditions. Hence, the LES data can be beneficial for assessment and optimization of turbulence closure schemes. A database of 80 LES runs for neutral and stably stratified PBLs (DATABASE64) is applied in this study to optimize the first-order turbulence closure (FOC). Approximations for the mixing length scale and stability correction functions have been tuned to minimise a relative root-mean square error over the entire DATABASE64. New stability functions have correct asymptotes describing regimes of strong and week mixing found in theoretical exercises, atmospheric data and LES. The correct asymptotes exclude the need for a critical Richardson number in the FOC formulation. Further, we analysed the FOC quality as functions of the integral PBL stability and the vertical model resolution. We show that the FOC is never perfect because the turbulence in the upper half of the PBL is not generated by the local gradients. Accordingly, the parameterized and LES fluxes decorrelate in the upper PBL. With this imperfection in mind, we show that there is no systematic quality deterioration of the FOC in strongly stable PBLs provided the vertical model resolution is better than 10 levels within the PBL. In agreement with previous studies, we found that the quality improves slowly with the vertical resolution refinement. It is generally wise to not overstretch the mesh in the lowest 500 m of the atmosphere where majority of the observed, simulated and theoretically predicted stably stratified PBLs are located.
Revisiting the Local Scaling Hypothesis in Stably Stratified Atmospheric Boundary Layer Turbulence: an Integration of Field and Laboratory Measurements with Large-eddy Simulations
We use a database of direct numerical simulations to evaluate parametrizations for energy dissipation rate in stably stratified flows. We show that shear-based formulations are more appropriate for stable boundary layers than commonly used buoyancy-based formulations. As part of the derivations, we explore several length scales of turbulence and investigate their dependence on local stability.
In this study, we explore several integral and outer length scales of turbulence which can be formulated by using the dissipation of temperature fluctuations ($χ$) and other relevant variables. Our analyses directly lead to simple yet non-trivial parameterizations for both $χ$ and the structure parameter of temperature ($C_T^2$). For our purposes, we make use of high-fidelity data from direct numerical simulations of stratified channel flows.
We present a laboratory experiment of the growth of a turbulent patch in a stably stratified fluid, due to a localized source of turbulence, generated by an oscillating grid. Synchronized and overlapping particle image velocimetry and planar laser induced fluorescence measurements have been conducted capturing the evolution of the patch through its initial growth until it reached a maximum size, followed by its collapse. The simultaneous measurements of density and velocity fields allow for a direct quantification of the degree of mixing within the patch, the propagation speed of the turbulent…
Inertial particles in stably stratified flows play a fundamental role in geophysics, from the dynamics of nutrients in the ocean to the dispersion of pollutants in the atmosphere. We consider the Maxey-Riley equation for small neutrally buoyant inertial particles in the Boussinesq approximation, and discuss its limits of validity. We show that particles behave as forced damped oscillators, with different regimes depending on the particles Stokes number and the fluid Brunt-Väisälä frequency. Using direct numerical simulations we study the particles dynamics and we show that small neutrally buoy…
In this study, the stability dependence of turbulent Prandtl number ($Pr_t$) is quantified via a novel and simple analytical approach. Based on the variance and flux budget equations, a hybrid length scale formulation is first proposed and its functional relationships to well-known length scales are established. Next, the ratios of these length scales are utilized to derive an explicit relationship between $Pr_t$ and gradient Richardson number. In addition, theoretical predictions are made for several key turbulence variables (e.g., dissipation rates, normalized fluxes). The results from our p…
Turbulent fluxes in the atmospheric surface layer are key input for the prediction of weather, hydrology, and carbon dioxide concentration. In numerical modelling of turbulent fluxes, a -7/3 power-law scaling in turbulence cospectra is usually assumed at high wavenumbers. In eddy-covariance (EC) measurements of turbulent fluxes, an assumed shape of turbulence cospectra is typically required for high-frequency spectral corrections, typically assuming a -7/3 power law. The derivation of -7/3 power-law scaling is based primarily on dimensional analysis, and other cospectral scaling has also been …
We present a turbulent kinetic energy (TKE) closure scheme for the stably stratified atmosphere in which the mixing lengths for momentum and heat are not parameterized in the same manner. The key difference is that, while the mixing length for heat tends towards the stability independent mixing length for momentum in neutrally stratified conditions, it tends towards one based on the Brunt-Väisälä time scale and square root of the TKE in the limit of large stability. This enables a unique steady-state solution for TKE to be obtained, which we demonstrate would otherwise be impossible if the mix…
We investigate the effects of condensation and liquid water loading on the stably stratified surface layer, with an eye towards understanding the influence of turbulent mixing on fog formation. Direct numerical simulations (DNS) of dry and moist open channel flows are conducted, where in both a constant cooling rate is applied at the ground to mimic longwave radiative cooling. Depending on the cooling rate, it can lead to either turbulent (weakly stable) or laminar (very stable) flows. Compared to the completely dry case, the condensation of liquid water in the moist case enables slightly high…
We extend the scaling relations of stably stratified turbulence from the geophysical regime of unity Prandtl number to the astrophysical regime of extremely small Prandtl number applicable to stably stratified regions of stars and gas giants. A transition to a new turbulent regime is found to occur when the Prandtl number drops below the inverse of the buoyancy Reynolds number, i.e. $PrRb<1$, which signals a shift of the dominant balance in the buoyancy equation. Application of critical balance arguments then derives new predictions for the anisotropic energy spectrum and dominant balance of t…
The energy and flux budget (EFB) closure theory for a passive scalar (non-buoyant and non-inertial particles or gaseous admixtures) is developed for stably stratified turbulence. The physical background of the EFB turbulence closures is based on the budget equations for the turbulent kinetic and potential energies and turbulent fluxes of momentum and buoyancy, as well as the turbulent flux of particles. The EFB turbulence closure is designed for stratified geophysical flows from neutral to very stable stratification and it implies that turbulence is maintained by the velocity shear at any stra…
Machine-Learned Closure of URANS for Stably Stratified Turbulence: Connecting Physical Timescales & Data Hyperparameters of Deep Time-Series Models
National Center for Computational Sciences, Oak Ridge National Laboratory（橡树岭国家实验室国家计算科学中心）；University of Massachusetts Amherst（马萨诸塞大学阿默斯特分校）；University of Washington（华盛顿大学）；DoD HPCMP PET/GDIT（美国国防部高性能计算现代化计划项目执行办公室/通用动力信息技术公司）
We have advanced the energy and flux budget (EFB) turbulence closure theory that takes into account a two-way coupling between internal gravity waves (IGW) and the shear-free stably stratified turbulence. This theory is based on the budget equation for the total (kinetic plus potential) energy of IGW, the budget equations for the kinetic and potential energies of fluid turbulence, and turbulent fluxes of potential temperature for waves and fluid flow. The waves emitted at a certain level, propagate upward, and the losses of wave energy cause the production of turbulence energy. We demonstrate …
In this paper we advance physical background of the energy- and flux-budget turbulence closure based on the budget equations for the turbulent kinetic and potential energies and turbulent fluxes of momentum and buoyancy, and a new relaxation equation for the turbulent dissipation time-scale. The closure is designed for stratified geophysical flows from neutral to very stable and accounts for the Earth rotation. In accordance to modern experimental evidence, the closure implies maintaining of turbulence by the velocity shear at any gradient Richardson number Ri, and distinguishes between the tw…
Rotation is one of the key physical mechanisms that deeply impact the evolution of stars. Helio- and asteroseismology reveal a strong extraction of angular momentum from stellar radiation zones over the whole Hertzsprung-Russell diagram. Turbulent transport in differentially rotating stably stratified stellar radiation zones should be carefully modeled and its strength evaluated. Stratification and rotation imply that this turbulent transport is anisotropic. Only phenomenological prescriptions have been proposed for the transport in the horizontal direction, which however constitutes a corners…
In the stably stratified ocean, small-scale turbulence is important for vertical exchange and hence for the mixing of water masses and suspended matter. To observationally study turbulent motions and the buoyancy- and shear-generators behind them, a 100-m tall array of high-resolution temperature (T) sensors was moored at a 2480-m deep seafloor near the steep continental slope of the Western Mediterranean Sea. The area is dominated by boundary flow, (sub-)mesoscale eddies, internal waves, wintertime dense water formation, and very weak vertical density stratification. Various physical oceanogr…
Freshwater from rivers influences Indian summer monsoon rainfall and regional tropical cyclones by shallowing the upper layer and warming the subsurface ocean in the Bay of Bengal. Here, we use in situ and satellite data with reanalysis products to showcase how river water can experience a significant increase in salinity on subseasonal timescales. This involves the trapping and homogenization of freshwater by a cyclonic eddy in the Bay. Specifically, in October 2015, river water is shown to enter a particularly long-lived eddy along with its attracting manifolds within a period of two weeks. …
Convection over a wavy heated bottom wall in the air flow has been studied in experiments with the Rayleigh number $\sim 10^8$. It is shown that the mean temperature gradient in the flow core inside a large-scale circulation is directed upward, that corresponds to the stably stratified flow. In the experiments with a wavy heated bottom wall, we detect large-scale standing internal gravity waves excited in the regions with the stably stratified flow. The wavelength and the period of these waves are much larger than the turbulent spatial and time scales, respectively. In particular, the frequenc…
We study forced, rapildy rotating and stably stratified turbulence in an elongated domain using an asymptotic expansion at simultaneously low Rossby number $Ro\ll1$ and large domain height compared to the energy injection scale, $h=H/\ell_{in}\gg1$. The resulting equations depend on the parameter $λ=(h Ro )^{-1}$ and the Froude number $Fr$. An extensive set of direct numerical simulations (DNS) is performed to explore the parameter space $(λ,Fr)$. We show that a forward energy cascade occurs in one region of this space, and a split energy cascade outside it. At weak stratification (large $Fr$)…
We derive equations for the mean entropy and the mean internal energy in the low-Mach-number temperature stratified turbulence (i.e., for turbulent convection or stably stratified turbulence), and show that turbulent flux of entropy is given by ${\bf F}_s=\overlineρ \, \overline{{\bf u} s}$, where $\overlineρ$ is the mean fluid density, $s$ are fluctuations of entropy and overbars denote averaging over an ensemble of turbulent velocity field, ${\bf u}$. We demonstrate that the turbulent flux of entropy is different from the turbulent convective flux, ${\bf F}_c=\overline{T} \, \overlineρ \, \o…
Planetary rotation organizes fluid motions into coherent, long-lived swirls, known as large-scale vortices (LSVs), which play an important role in the dynamics and long-term evolution of geophysical and astrophysical fluids. Here, using direct numerical simulations, we show that LSVs in rapidly rotating mixed convective and stably stratified fluids, which approximates the two-layer, turbulent-stratified dynamics of many geophysical and astrophysical fluids, have a generic shape and that their size can be predicted. We show that LSVs emerge in the convection zone from upscale energy transfers a…
