Editorial Type:
Article
Article Type:
Research Article

A Gradient Tensor–Based Subgrid-Scale Parameterization for Large-Eddy Simulations of Stratified Shear Layers Using the Weather Research and Forecasting Model

Sina Khani aOden Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, Texas

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Fernando Porté-Agel bWind Engineering and Renewable Energy Laboratory (WiRE), École Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland

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Online Publication:
07 Sep 2022
Print Publication:
01 Sep 2022
DOI:
https://doi.org/10.1175/MWR-D-21-0217.1
Page(s):
2279–2298
Open access
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Abstract

The transition process from laminar stratified shear layer to fully developed turbulence is usually captured using direct numerical simulations, in which the computational cost is extremely high and the numerical domain size is limited. In this work, we introduce a scale-aware subgrid-scale (SGS) parameterization, based on the gradient tensor of resolved variables, which is implemented in the Weather Research and Forecasting (WRF) Model. With this new SGS model, we can skillfully resolve the characteristics of transition process, including formation of vortex cores, merging vorticity billows, breaking waves into smaller scales, and developing secondary instability in the stratified shear layer even at coarse-resolution simulations. Our new model is developed such that the time scales of the eddy viscosity and diffusivity terms are represented using the tensor of the gradient and not that of the rate-of-strain, which is commonly used in the parameterization of turbulent-viscosity models. We show that time scales of unresolved transition processes in our new model are correlated with those of vorticity fields. At early times, the power-law slopes in the kinetic and available potential energy spectra are consistent with the process of formation and merging waves with an upscale energy transfer. At later times, the power-law slopes are in line with the process of breaking waves into small-scale motions with a downscale transfer. More importantly, the efficiency of turbulent mixing is mainly high at the edge of vortex filaments and not at the vortices’ eyes. These findings can improve our understanding of turbulent mixing process in large-scale wind-induced events, such as tropical cyclones, using the WRF Model.

Significance Statement

The evolution of instabilities in stratified shear layers has significant impacts on the structure of large-scale geophysical flows and also on the energy pathway to smaller-scale motions in internal waves and turbulence. Resolving transition processes in stratified shear layers requires very high-resolution simulations in climate models. We propose a new subgrid-scale parameterization that is implemented in the Weather Research and Forecasting Model to capture the dynamics of transition process from laminar to three-dimensional turbulence in stratified shear layers at coarse-resolution simulations. Our new scale-aware parameterization can reduce biases in climate models by skillfully representing unresolved fluxes, leading to higher accuracy in weather predictions of temperature, precipitation, and surface fluxes with an affordable computational cost.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Sina Khani, sina.khani@austin.utexas.edu

Abstract

The transition process from laminar stratified shear layer to fully developed turbulence is usually captured using direct numerical simulations, in which the computational cost is extremely high and the numerical domain size is limited. In this work, we introduce a scale-aware subgrid-scale (SGS) parameterization, based on the gradient tensor of resolved variables, which is implemented in the Weather Research and Forecasting (WRF) Model. With this new SGS model, we can skillfully resolve the characteristics of transition process, including formation of vortex cores, merging vorticity billows, breaking waves into smaller scales, and developing secondary instability in the stratified shear layer even at coarse-resolution simulations. Our new model is developed such that the time scales of the eddy viscosity and diffusivity terms are represented using the tensor of the gradient and not that of the rate-of-strain, which is commonly used in the parameterization of turbulent-viscosity models. We show that time scales of unresolved transition processes in our new model are correlated with those of vorticity fields. At early times, the power-law slopes in the kinetic and available potential energy spectra are consistent with the process of formation and merging waves with an upscale energy transfer. At later times, the power-law slopes are in line with the process of breaking waves into small-scale motions with a downscale transfer. More importantly, the efficiency of turbulent mixing is mainly high at the edge of vortex filaments and not at the vortices’ eyes. These findings can improve our understanding of turbulent mixing process in large-scale wind-induced events, such as tropical cyclones, using the WRF Model.

Significance Statement

The evolution of instabilities in stratified shear layers has significant impacts on the structure of large-scale geophysical flows and also on the energy pathway to smaller-scale motions in internal waves and turbulence. Resolving transition processes in stratified shear layers requires very high-resolution simulations in climate models. We propose a new subgrid-scale parameterization that is implemented in the Weather Research and Forecasting Model to capture the dynamics of transition process from laminar to three-dimensional turbulence in stratified shear layers at coarse-resolution simulations. Our new scale-aware parameterization can reduce biases in climate models by skillfully representing unresolved fluxes, leading to higher accuracy in weather predictions of temperature, precipitation, and surface fluxes with an affordable computational cost.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Sina Khani, sina.khani@austin.utexas.edu
Keywords: Kelvin-Helmholtz instabilities; Mesoscale processes; Shear structure/flows; Turbulence; Wave breaking; Large eddy simulations; Mesoscale models; Numerical analysis/modeling; Subgrid-scale processes

1. Introduction

Stratified shear layers are usually considered to provide the transition from laminar to turbulent flows in the presence of velocity shears and density gradients (Smyth and Peltier 1990, 1993; Caulfield and Peltier 2000). The evolution of stratified shear layers includes forming Kelvin–Helmholtz (KH) billows and Holmboe instability, and developing secondary instabilities on the edge of KH billows due to the increase of potential energy by irreversible mixing. For example, turbulent KH billows with horizontal wavelength of 4–6 km are observed in the middle and upper atmosphere above the midlevel cloud base (Luce et al. 2018). Also, coherent Gulf Stream rings with vorticity of diameter ∼20 km are affected by large-scale velocity shear (Leaman and Molinari 1987). A pathway to smaller-scale turbulence in internal waves at the stratified ocean interior is also connected to the evolution of instabilities in the stratified shear layers (Caulfield and Peltier 2000; Mashayek and Peltier 2012a; Smyth and Moum 2012).

Most numerical simulations of stratified shear layers and KH (or Holmboe) instabilities are performed using the direct numerical simulation (DNS) approach (Caulfield and Peltier 1994; Scinocca 1995; Caulfield and Peltier 2000; Mashayek and Peltier 2012a; VanDine et al. 2021), where a huge amount of computational resources are invested on resolving small-scale motions around the molecular (viscous) dissipation scale (see e.g., Pope 2000). However, these non-energetic small-scale motions around the viscous dissipation scale are not important in the transition process and large-scale energy interactions in stratified turbulence (see e.g., Armenio and Sarkar 2002; Khani and Waite 2014, 2015; Khani 2018).

In this paper we study the kinetic and potential energy transfers in a stratified shear layer using the large-eddy simulation (LES) approach, in which the small-scale motions are not directly resolved but the effects of small scales on large-scale motions are parameterized using subgrid-scale (SGS) models (see e.g., Meneveau and Katz 2000). One of the main focuses of this work is to introduce a new eddy viscosity model, in which the eddy time scale of eddy viscosity (or diffusivity) coefficients is set by the gradient tensor:
Gij=∂u¯i∂xk∂u¯j∂xk,
rather than the strain rate tensor sij=1/2(∂u¯i/∂xj+∂u¯j/∂xi), where u¯i is the resolved velocity field (i, j = 1, 2 and 3; the overbar denotes resolved or filtered variable in the LES approach), and subscript k is a dummy variable (according to the Einstein’s summation convention). The actual SGS momentum stress is proportional to the gradient tensor Gij based on the Taylor series expansion of the resolved velocity field u¯ (Meneveau and Katz 2000; Khani and Porté-Agel 2017a,b; Khani and Waite 2020); therefore, we shall demonstrate that setting the eddy time scale based on Gij is more relevant to the structure of SGS eddies in comparison with setting an eddy time scale based on the strain rate tensor sij, as in the Smagorinsky (1963) model. The rest of this paper is organized as follows: the governing equations for the stratified shear layer simulations, along with our new SGS model are introduced in section 2. Section 3 gives the methodology and simulations setup. Results and discussion are given in section 4, followed by conclusions in section 5.

2. Governing equations

The nondimensional governing equations of motion under the Boussinesq approximation are given as follows:
∂u¯i∂t+∂∂xj(u¯iu¯j)=−∂p¯∂xi+Riθ¯ez+1Re∇2u¯i−∂τij∂xj,
∂u¯i∂xi=0,
∂θ¯∂t+∂∂xj(θ¯u¯j)+w¯=1RePr∇2θ¯−∂hj∂xj,
where u¯=(u¯,υ¯,w¯), θ¯, p¯ are the filtered (resolved) velocity, potential temperature and pressure; Ri=gdδθ/(U2θ0) is the bulk Richardson number, Re=Ud/ν is the Reynolds number, and Pr = ν/D is the Prandtl number. Here, d, δθ, U, θ0, ν and D are the initial thickness of shear layer, potential temperature deviation, velocity scale, reference potential temperature, molecular viscosity, and diffusivity coefficients, respectively. The SGS momentum flux τij and potential temperature flux hj need to be parameterized using SGS models. In LES, we mainly resolve scales much larger than the Kolmogorov (Batchelor) scale, leading to neglect the molecular viscosity (diffusivity) dissipation in comparison with the SGS dissipation by the eddy viscosity (diffusivity).

Parameterizing eddy viscosity and diffusivity terms

The SGS momentum flux τij can be approximated by a Taylor series expansion of the resolved velocity field u¯ at a point x over a distance r, where |r| ∼ Δ, which is the filter width (grid spacing) in the LES approach (Pope 2000; Meneveau and Katz 2000; Khani and Porté-Agel 2017b; Khani and Waite 2020):
τij≈Δ212Gij=Δ212∂u¯i∂xk∂u¯j∂xk.
However, an SGS model based on the Eq. (5) cannot provide enough dissipation at scales around the grid spacing Δ (as discussed in Vreman et al. 1997; Khani and Porté-Agel 2017a,b). To take advantage of the high correlation between the gradient tensors and SGS fluxes, we suggest an eddy viscosity model to parameterize τij as follows:
τij=−2νgsij,
where νg uses the structure of Gij instead of sij that is used in the Smagorinsky model. In our new parameterization, the eddy viscosity coefficient νg is proposed as
νg=cgΔ2(GijGij)1/4,
which is different from the eddy viscosity coefficient in the Smagorinsky parameterization with νg=csΔ2(sijsij)1/2. Here, cg and cs are model coefficients. The novelty of our new model is that we employ the gradient tensor, which provides a theoretical approximation of the actual SGS tensor, in the definition of the eddy viscosity coefficient νg. The advantage of our new SGS model to the Smagorinsky is that the eddy viscosity coefficient in our model includes the structure of the gradient tensor, which has shown to be highly correlated with actual SGS motions (see e.g., Clark et al. 1979; Pope 2000; Porté-Agel et al. 2001; Khani and Porté-Agel 2017a). Overall, our new model is as simple as the Smagorinsky model but it sets the eddy time scale based on a theoretical approximation of the gradient tensor Gij, and not the rate of strain tensor sij.

In theory, our SGS eddy viscosity coefficient in Eq. (7) sets the eddy turnover time scale using the gradient tensor, which mathematically approximates the actual SGS motions using the Taylor series expansion of the resolved velocity field [see Eq. (5)]. We propose this new model as an alternative to the Smagorinsky model because the Smagorinsky model is too dissipative to simulate overturning and transition to turbulence (Vreman et al. 1997; Khani and Waite 2015). A theoretical value is obtained for the constant coefficient cg using an isotropic inertial subrange for the kinetic energy spectrum (as shown in the appendix).

The eddy diffusivity term hj in our new model is parameterized as follows:
hj=−Dg∂θ¯∂xj,
where Dg = νg/Prt, and Prt is the SGS Prandtl number.

3. Methodology

We consider idealized simulations using the Weather Research and Forecasting (WRF-LES) model to study stratified shear layers. The WRF Model uses a hydrostatic-pressure vertical coordinate, and also includes a rotating term with the Coriolis parameter fc = 10−4 s−1 (in our simulations, the effects of Coriolis parameter on the evolution and transition of shear layers are negligible since the computational domain is small enough to have a fairly large Rossby number). Free-slip boundaries are imposed at the top and bottom surfaces and lateral directions have periodic boundary conditions. The third-order Runge–Kutta scheme is employed for the time advancement; fifth- and third-order upwind biased finite difference methods are also employed for the spatial discretization in the horizontal and vertical directions, respectively. The initial streamwise velocity and potential temperature profiles are given as follows (similar initialization is used in Smyth and Peltier 1990, 1993; Caulfield and Peltier 2000; Mashayek and Peltier 2012a):
u¯(z,t=0)=Utanh⁡(z−Hd),
θ¯(z,t=0)−θ0=δθtanh⁡(z−Hd),
where H is the half-height of the computational domain.

Eight simulations are performed using our newly developed SGS parameterization. Simulations are characterized with different initial bulk Richardson number Ri, half shear layer thickness d, and velocity scale U; simulations are also run at different resolutions with grid points nx × ny × nz = 80 × 80 × 80, 160 × 160 × 160, 230 × 230 × 320, and 320 × 320 × 320 over a cubic domain where Lx and Ly are integer multiples of the most unstable KH modes (Hazel 1972), and Lz=2 km. The SGS Prandtl number is set to 1/3 (similar to Skamarock et al. 2008). Table 1 shows a list of parameters and variables in our LES runs. Flow structures and dynamics at coarser resolution simulations are similar to those at high-resolution runs, but there is less clarity on capturing fundamental processes in coarse-resolution simulations with nx,y,z = 1603 and 803 grid points; hence, in this work we mainly focus on the results of high-resolution cases with nx,y,z = 3203 and n{x,y,z} = 2302 × 320 grid points.

Table 1

List of numerical simulations with LES.
Table 1

4. Results and discussion

a. Vorticity fields

Figure 1 shows time evolution of the y-direction vorticity field in the x–z plane for the case with the initial bulk Richardson number Ri = 0.041 and shear layer thickness d = 25 m (i.e., run 1 in Table 1). At t = 0, the shear layer is thin and approximately steady (not shown). As time evolves, intermittency grows, and we can see the onset of KH instability at t = 75 s. At t = 105 s, a train of four KH vorticity cores (billows) is formed, and these vortex cores are connected by vorticity filaments (braids). The vorticity cores start merging at t = 135 s; hence, two vorticity billows are formed and breaking KH waves into smaller-scale structures (turbulence) occurs at t = 195 s. At later times, the process of merging vorticity cores and breaking waves into turbulence, along with developing secondary instability that enhances turbulence collapse in the mixing layer, are seen (Figs. 1e and 1f). The occurrence of secondary instability mostly emerges along braids when the vorticity cores merge, and also at the edge of KH billows (as shown in Figs. 1e–g). At t = 615 s, the process of transition into turbulence has mostly completed and small-scale vorticity structures and filaments are mainly visible. These structures and processes are in line with those reported in the literature on the formation of KH waves and three-dimensional evolution into turbulence (Corcos and Sherman 1976; Smyth and Peltier 1991, 1993; Staquet 1995, 2000; Caulfield and Peltier 2000; Smyth 2003, 2004; Mashayek and Peltier 2012a, b; Fritts et al. 2014).

Fig. 1.
Fig. 1.

Time evolution of vorticity field in the x–z plane at y ≈ 500 m for the case with the initial bulk Richardson number Ri = 0.041 and shear layer thickness d = 25 m.

Citation: Monthly Weather Review 150, 9; 10.1175/MWR-D-21-0217.1

Next, we increase the initial shear layer thickness to d = 50 m, thereby the bulk Richardson number Ri = 0.082 is doubled (run 2 in Table 1). In this case, the processes of KH billow formation, breaking waves, and transition into turbulence delay over time in comparison with those in run 1. In the case with the initial bulk Richardson number Ri = 0.082 and shear layer thickness d = 50 m, a train of three vorticity cores has emerged by t = 195 s (Fig. 2a). At t = 255 s, vorticity cores merge into two billows while KH waves start breaking into small-scale structures and the secondary instability develops across the braids of KH vortices (as shown in Fig. 2b; see also Fig. 2c, where breaking KH waves and secondary instabilities are well captured at t = 285 s). Ultimately, small-scale vorticity filaments and enlarged shear layer thickness are seen at t = 465 s (Fig. 2d). It is also important to note that the intensity of vorticity filaments attenuates at later time due to the effect of SGS eddy dissipation on smaller scales (as shown in Fig. 2d). Time evolution of the y-direction vorticity field ωy in the x–y and y–z planes is shown in Fig. 3. Small-scale instabilities emerge in the y direction at a time around t = 195 s (Figs. 3a and 3b). These instabilities grow over time during the waves breaking process in three dimensions (Figs. 3c and 3d at t = 255 s). As time evolves, small-scale instabilities further grow during the process of merging billows and formation of secondary instability at time t = 285 s (Figs. 3e and 3f). Overall, these results demonstrate that the transition from laminar stratified shear layer to fully developed turbulence includes the wave breaking, merging billows and formation of secondary instability through a 3D process (Figs. 2 and 3).

Fig. 2.
Fig. 2.

Time evolution of vorticity field in the x–z plane at y ≈ 500 m for the case with the initial bulk Richardson number Ri = 0.082 and shear layer thickness d = 50 m. See also Fig. 3 for plots in the x–y and y–z planes.

Citation: Monthly Weather Review 150, 9; 10.1175/MWR-D-21-0217.1

Fig. 3.
Fig. 3.

Time evolution of vorticity field ωy in the (left) x–y plane at z ≈ 1200 m and (right) y–z plane x ≈ 500 m for the case with the initial bulk Richardson number Ri = 0.082 and shear layer thickness d = 50 m. Plots in the x–z plane are shown in Fig. 2.

Citation: Monthly Weather Review 150, 9; 10.1175/MWR-D-21-0217.1

For a case with the initial bulk Richardson number Ri = 0.164 and shear layer thickness d = 100 m, breaking KH waves and formation of secondary instabilities along vorticity braids are seen at t = 585 s (Fig. 4a). It is noteworthy that secondary KH billows are formed at scales that are mainly smaller than the scale of the first-generation KH waves (see Fig. 4b). Interestingly, secondary KH billows may evolve over time toward breaking secondary waves, turbulence occurrence, and even merging with larger-scale KH billows (these processes are similar to those are captured in breaking the first generation KH waves; see Figs. 4b and 4c). For the case with the initial bulk Richardson number Ri = 0.197 and layer thickness d = 100 m, the shear layer evolves to form Holmboe-like waves at t = 765 s (Fig. 5a; similar wave structures are reported in DNS of Smyth and Winters 2003). In this case, the thickness of the mixing layer is shallower than previous cases in which the initial bulk Richardson number and shear layer thickness were smaller. Furthermore, the formation of Holmboe waves delayed in the case with Ri = 0.235 and d = 100 m (Fig. 5b). As elaborated in Salehipour et al. (2016), Holmboe waves grow slowly over time and they scour (versus KH waves that overturn) as time evolves by exhibiting a −5/3 power-law in the kinetic energy spectra with an energy level that is relatively smaller than that in KH waves (see Fig. 9 in section 4b). If we further increase the initial bulk Richardson number to Ri =0.327, the emergence of instability and waves is suppressed (not shown) because Ri becomes larger than the critical bulk Richardson number Ricr = 1/4, which is a necessary condition for instability growth in stratified shear layers (Miles 1961; Peltier and Caulfield 2003).

Fig. 4.
Fig. 4.

Time evolution of vorticity field in the x–z plane at y ≈ 500 m for the case with the initial bulk Richardson number Ri = 0.164 and shear layer thickness d = 100 m.

Citation: Monthly Weather Review 150, 9; 10.1175/MWR-D-21-0217.1

Fig. 5.
Fig. 5.

The vorticity field in the x–z plane at y ≈ 500 m for cases with the initial bulk Richardson number (a) Ri = 0.197 and (b) Ri = 0.235, and shear layer thickness d = 100 m.

Citation: Monthly Weather Review 150, 9; 10.1175/MWR-D-21-0217.1

b. Energy spectra

Figure 6 shows the time evolution of vertically averaged horizontal wavenumber kinetic and available potential energy (KE and APE) spectra within the stratified shear layer for the case with Ri = 0.041 and d = 25 m. At t = 105 and 135 s, when KH waves are formed and the merging process occurs, KE and APE spectra are dominated by peaks at the wavenumbers of KH harmonics. The APE spectra are more energetic at early times in comparison with KE spectra, and the spectral slope of kinetic energy is close to −3 (a sign of upscale energy transfer because of formation and merging KH waves; see Figs. 6a and 6b). As time evolves, at t = 195 s a time which three-dimensionalization occurs where KH waves start breaking, the KE and APE spectra show similar energy levels at large horizontal scales (small wavenumbers), and the spectral slopes become closer to −5/3 over the range of intermediate horizontal wavenumbers (a sign of downscale energy transfer because of breaking waves into smaller structures; see Figs. 6c and 6d). As time further evolves, the KE spectra become more energetic in comparison with those of APE and a clear −5/3 spectral power law is visible (downscale cascade of energy by onset of turbulence; see Figs. 6e and 6f). At later times, SGS eddy dissipation increases due to the onset of turbulence (see section 4c); hence, the energy levels of KE and APE spectra are remarkably decreased (Fig. 6g).

Fig. 6.
Fig. 6.

Time evolution of vertically averaged horizontal kinetic and available potential energy spectra for the case with the initial bulk Richardson number Ri = 0.041 and shear layer thickness d = 25 m. Spectra are averaged over the mixed layer height (between z ≈ 710 and 1460 m).

Citation: Monthly Weather Review 150, 9; 10.1175/MWR-D-21-0217.1

Similar trends are seen in the case with Ri = 0.082 and d = 50 m, for which the energy spectra at t = 195 s include wavenumber peaks of KH waves, and show domination of APE because of the merging process and formation of KH billows with a spectral slope ∼ −3 (see Fig. 7a). At later times, however, KE levels become gradually larger than APE levels with a spectral slope ∼ −5/3 (Figs. 7b–e). Also, KE and APE levels significantly decrease due to the SGS eddy dissipation on smaller scales at later times (see Fig. 7e when t = 735 s). For the case with the initial bulk Richardson number Ri = 0.164 and shear layer thickness d = 100 m, spectral shapes are similar to those we reported in Figs. 6 and 7 (see Figs. 8a and 8b; consistent with the time delay on formation and merging KH waves, and turbulence onset that are discussed in section 4a). In the case with the initial bulk Richardson number Ri = 0.197 and d = 100 m, the KE and APE spectra show (less-energetic) wavenumber peaks and harmonics at t = 585 s when Holmboe like waves start forming (Fig. 9a). At the later time t = 885 s, energy spectra are more smooth and depict a −5/3 spectra power law due to turbulence occurrence and breaking Holmboe waves into smaller-scale structures (Fig. 9b). For the case with Ri = 0.235 and d = 100 m, the KE and APE spectra show the wave harmonics at very late times (e.g., Holmboe instability in this case occurs at around t = 825 s; not shown). Moreover, the process of breaking waves into turbulence, with a–5/3 power spectra, are also seen at t = 1005 s in this case (delayed instability formation and downscale transfer of energy; not shown).

Fig. 7.
Fig. 7.

Time evolution of vertically averaged horizontal kinetic and available potential energy spectra for the case with the initial bulk Richardson number Ri = 0.082 and shear layer thickness d = 50 m. Spectra are averaged over the mixed layer height.

Citation: Monthly Weather Review 150, 9; 10.1175/MWR-D-21-0217.1

Fig. 8.
Fig. 8.

Vertically averaged horizontal kinetic and available potential energy spectra at time (a) t = 525 and (b) t = 795 s, for the case with the initial bulk Richardson number Ri = 0.164 and shear layer thickness d = 100 m. Spectra are averaged over the mixed layer height.

Citation: Monthly Weather Review 150, 9; 10.1175/MWR-D-21-0217.1

Fig. 9.
Fig. 9.

Vertically averaged horizontal kinetic and available potential energy spectra at time (a) t = 585 and (b) t = 885 s, for the case with the initial bulk Richardson number Ri = 0.197 and shear layer thickness d = 100 m. Spectra are averaged over the mixed layer height.

Citation: Monthly Weather Review 150, 9; 10.1175/MWR-D-21-0217.1

c. Shear characteristics and dissipation rates

Figure 10 shows the time evolution of resolved shear (filtered shear characteristics) S¯=(GijGij)1/4 in the x–z plane for the case with the initial bulk Richardson number Ri = 0.041 and shear layer thickness d = 25 m. From Figs. 1 and 10, it is suggestive that the resolved shear S¯ and vorticity field ωy are highly correlated. Figure 11 shows a scatter diagram of ωy versus S¯ at different times for the case with Ri = 0.041 and d = 25 m. At early time, when primary KH billows start emerging, ωy is well approximated by S¯ with a correlation coefficient around 1 (ωy≈S¯ as shown in Fig. 11a). As time evolves, where waves breaking occurs and secondary instability starts forming, the nonlinearity becomes dominant due to the onset of turbulence; hence the correlation coefficient between ωy and S¯ varies from −1 to 1 (−S¯ ≲ ωy ≲ S¯, see Figs. 11b–d). Time scales of KH billows, merging processes, breaking waves and turbulence onset in vorticity field are better correlated with those of filtered shear characteristics because the time scale of SGS stress τij is proportional to S¯ and sij, and S¯ is computed from Gij in the gradient model. Moreover, our work shows that the eddy viscosity coefficient in this new definition of S¯ is less-dissipative in the region with high shears in comparison with the regular Smagorinsky model (see section 4e).

Fig. 10.
Fig. 10.

Time evolution of resolved shear characteristics S¯=(GijGij)1/4 in the x–z plane at y ≈ 500 m for the case with the initial bulk Richardson number Ri = 0.041 and shear layer thickness d = 25 m.

Citation: Monthly Weather Review 150, 9; 10.1175/MWR-D-21-0217.1

Fig. 11.
Fig. 11.

Time evolution of the scatter diagram of vorticity field ωy vs the resolved shear S¯ for the case with the initial bulk Richardson number Ri = 0.041 and shear layer thickness d = 25 m. Lines ωy=±S¯ are shown for references.

Citation: Monthly Weather Review 150, 9; 10.1175/MWR-D-21-0217.1

Figures 12 and 13 show the time evolution of kinetic and potential SGS eddy dissipation rates ϵ and ϵp, respectively, in the x–z plane. When turbulence onsets at t ≈ 195 s, SGS eddy dissipation rates depict maximum values at vorticity filaments and smaller-scale structures. It is shown that the SGS potential eddy dissipation rate ϵp is much larger than the SGS kinetic eddy dissipation rate ϵ at vorticity braids and wave edges, while ϵ is mostly dominant at vorticity cores (as shown in Figs. 12 and 13; in particular, see Figs. 12d–g and Figs. 13d–g). Similar trends are also shown in the case with Ri = 0.082 and d = 50 m, where the shear characteristics S¯ can correctly predict the transition of stratified shear layer from KH billow formation to breaking waves and the occurrence of turbulence (Figs. 14a–c). The time evolution of kinetic and potential eddy dissipation rates shows the transition process where maximum values of ϵ occur at vorticity cores while ϵp shows maximum values at vorticity braids and filaments (see Figs. 15a–d).

Fig. 12.
Fig. 12.

Time evolution of kinetic energy eddy dissipation rate ϵ = 2νgsijsij in the x–z plane at y ≈ 500 m for the case with the initial bulk Richardson number Ri = 0.041 and shear layer thickness d = 25 m.

Citation: Monthly Weather Review 150, 9; 10.1175/MWR-D-21-0217.1

Fig. 13.
Fig. 13.

Time evolution of potential energy eddy dissipation rate ϵp=RiDg(∂θ¯/∂xj)(∂θ¯/∂xj) in the x–z plane at y ≈ 500 m for the case with the initial bulk Richardson number Ri = 0.041 and shear layer thickness d = 25 m. Note that Dg = νg/Prt.

Citation: Monthly Weather Review 150, 9; 10.1175/MWR-D-21-0217.1

Fig. 14.
Fig. 14.

Time evolution of resolved shear characteristics S¯=(GijGij)1/4 in the x–z plane at y ≈ 500 m for the case with the initial bulk Richardson number Ri = 0.082 and shear layer thickness d = 50 m.

Citation: Monthly Weather Review 150, 9; 10.1175/MWR-D-21-0217.1

Fig. 15.
Fig. 15.

(a),(b) Kinetic energy eddy dissipation rate ϵ = 2νgsijsij and (c),(d) potential energy eddy dissipation rate ϵp=RiDg(∂θ¯/∂xj)(∂θ¯/∂xj) in the x–z plane at y ≈ 500 m for the case with the initial bulk Richardson number Ri = 0.082 and shear layer thickness d = 50 m. Note that Dg = νg/Prt.

Citation: Monthly Weather Review 150, 9; 10.1175/MWR-D-21-0217.1

For the case with the initial bulk Richardson number Ri = 0.164 and shear layer thickness d = 100 m, the SGS kinetic and potential eddy dissipation rates show peak values where both primary KH wave and secondary instability are formed. As shown in Figs. 16a and 16b, ϵ and ϵp are mainly dominant at vortex eyes and braids of the first (and also secondary) wave(s), respectively. The SGS potential eddy dissipation rate ϵp is more dominant at braids of Holmboe vorticities in the case with Ri = 0.197 and d = 100 m (Figs. 17a and 17b). These trends are consistent with the time evolution of ϵp over x–z plane for KH waves we discussed above (see Fig. 13).

Fig. 16.
Fig. 16.

(a) Kinetic energy eddy dissipation rate ϵ = 2νgsijsij and (b) potential energy eddy dissipation rate ϵp=RiDg(∂θ¯/∂xj)(∂θ¯/∂xj) in the x–z plane at y ≈ 500 m for the case with the initial bulk Richardson number Ri = 0.164 and shear layer thickness d = 100 m. Note that Dg = νg/Prt.

Citation: Monthly Weather Review 150, 9; 10.1175/MWR-D-21-0217.1

Fig. 17.
Fig. 17.

Potential energy eddy dissipation rate ϵp=RiDg(∂θ¯/∂xj)(∂θ¯/∂xj) in the x–z plane at y ≈ 500 m for the case with the initial bulk Richardson number Ri = 0.197 and shear layer thickness d = 100 m. Note that Dg = νg/Prt.

Citation: Monthly Weather Review 150, 9; 10.1175/MWR-D-21-0217.1

d. Mixing efficiency

Mixing efficiency, which is defined as a measure of the effectiveness of turbulence in mixing process, is an important microscale parameter that influences breaking gravity waves and vertical transport of heat and tracers in atmospheric shear layers (Lilly et al. 1974; Weinstock 1978; Caulfield and Peltier 2000). In the LES approach, the irreversible mixing efficiency γi is defined as the ratio of SGS potential eddy dissipation ϵp to the total SGS dissipation (ϵp + ϵ) as follows (Khani 2018):
γi=ϵpϵp+ϵ.

Figure 18 shows the time evolution of γi in the x–z plane. Interestingly, the irreversible mixing efficiency is more effective at the edge of vorticity filaments and braids where the SGS potential dissipation rate is maximum, while the efficiency of turbulent mixing is weaker at vorticity cores (see Figs. 18a–d). Similar structures are also seen in our cases with different initial Richardson numbers and shear layer thicknesses where KH and Holmboe waves are formed (not shown). These results demonstrate that the irreversible turbulent mixing is mainly influential at vorticity filaments and braids rather than vorticity cores (eyes). Small values of mixing efficiency at vorticity cores, where waves breaking occurs, is due to small ratio of potential to kinetic energy eddy dissipation (i.e., ϵp/ϵ ≪ 1) because in this scenario γi ≈ ϵp/ϵ [see Eq. (11)]. Our results are in line with those from Gassmann (2018) that the thermal dissipation is much smaller than the kinetic energy dissipation in numerical experiments of breaking gravity waves. These findings can be useful for better understanding the interactive mixing in large-scale motions of tropical cyclones, in which mixing is mainly carried by transient wind-driven events at low local Richardson numbers (i.e., turbulent mixing) rather than mixing by diffusion (as discussed in Raymond et al. 2004; Korty et al. 2008).

Fig. 18.
Fig. 18.

Time evolution of irreversible mixing efficiency γi in the x–z plane at y ≈ 500 m for the case with the initial bulk Richardson number Ri = 0.041 and shear layer thickness d = 25 m.

Citation: Monthly Weather Review 150, 9; 10.1175/MWR-D-21-0217.1

e. Comparison with the Smagorinsky parameterization

We have performed an extra LES run with the Smagorinsky parameterization in order to compare the results of our newly implemented SGS model in WRF-LES code with those from the existing SGS model. We consider the case with the initial bulk Richardson number Ri = 0.164 and shear layer thickness 100 m in the LES run with the Smagorinsky model. The stratified shear layer is stable until t = 945 s, a time at which instabilities start growing, and also the formation of KH instabilities emerges at t ≈ 1125 s (Figs. 19a and 19b). With the gradient model, however, this case shows breaking waves and formation of secondary instabilities at t ≈ 585 s (as was shown in Fig. 4a). As a result, the evolution time of stratified shear layer and transition processes are significantly postponed because, unlike the gradient model, the Smagorinsky model is too dissipative and excessive eddy dissipation does not allow the growth of instabilities and formation of primary and secondary instabilities at an early time. Similar trends are also shown in the kinetic and potential energy dissipation rates ϵ and ϵp, where the onset of turbulence is suppressed at early time in the LES run with the Smagorinsky parameterization. In fact, the formation of KH billows and merging processes in this case are delayed until t ≈ 1185 s when the Smagorinsky model is employed (see Figs. 19c and 19d).

Fig. 19.
Fig. 19.

Time evolution of vorticity field in the x–z plane at y ≈ 500 m when (a) t = 945 and (b) t = 1125 s for the case with the initial bulk Richardson number Ri = 0.164 and shear layer thickness d = 100 m, when the Smagorinsky parameterization is employed. (c) Kinetic and (d) potential eddy dissipation rates at t = 1185 s.

Citation: Monthly Weather Review 150, 9; 10.1175/MWR-D-21-0217.1

5. Conclusions

In this work we implement a new SGS parameterization, in which the eddy time scale of unresolved fluxes is set by the gradient tensor Gij, for LES of stratified shear layers in the WRF Model. We test our new model in cases with different initial bulk Richardson number and layer thicknesses. Using coarse-resolution WRF-LES runs, we could capture the formation of vorticity cores, merging vorticity billows, breaking waves into small-scale structures and development of secondary instability. Usually, very high-resolution DNS runs are required to capture the transition process from laminar shear layer to fully developed three-dimensional turbulence (as are shown in e.g., Smyth 2003; Smyth and Winters 2003; Smyth 2004; Mashayek and Peltier 2012a,b). Here, we have shown that these fundamental dynamics of stratified shear layers can be skillfully captured in coarse-resolution LES runs when a scale-aware SGS parameterization based on the gradient tensor is employed.

The power-law of KE and APE spectra changes over time in this transition process: a −3 spectral slope for upscale energy transfer is seen at early stages due to the formation and merging KH waves (or Holmboe instability when Ri is large but still ≲Ricr), while a −5/3 spectral slope for downscale energy transfer is captured at later times associated with breaking waves into small-scale motions (Figs. 6 and 7). Furthermore, time scales of SGS transition processes, which are indicated by the filtered shear characteristics S¯=(GijGij)1/4 can lead to set in a less-dissipative correlation coefficient in the eddy viscosity model. It is remarkable that the SGS potential eddy dissipation rate ϵp is large at the leading edges of waves (instabilities) and vorticity braids whereas the SGS kinetic eddy dissipation rate ϵ is mostly effective at vorticity cores (Figs. 12 and 13). As a result, the efficiency of turbulent mixing, which is carried by wind-driven events at low Ri values, is mainly important at vorticity braids and filaments and not at the vortex eyes (Eq. 11 and Fig. 18).

The new SGS parameterization based on the gradient tensor is useful in high-resolution numerical weather prediction models to enhance the performance of eddy viscosity (diffusivity) parameterizations for boundary layer processes, convection, and cloud dynamics. Since SGS fluxes in the gradient-based parameterization are less-diffusive, modeling unresolved processes, potential temperature, precipitation, air-sea fluxes, and other meteorological elements shall be predicted with higher accuracy in high-resolution weather forecasting models. For future work, we will employ our new gradient-based SGS parameterization in numerical simulations of tropical cyclones and hurricanes using the WRF Model (similar work with current microphysics parameterizations has been done in WRF-LES and Cloud Model 1; see Zhu 2008; Nolan et al. 2017).

Acknowledgments.

This work has benefited from comments by Michael Waite and three anonymous reviewers. Computing resources from Scientific IT and Application Support (SCITAS) at École Polytechnique Fédérale de Lausanne (EPFL) are acknowledged. Model data and source code for generating the results can be accessible from the GitHub (https://github.com/sinakhani/Stratified-shear-layer-in-WRF).

APPENDIX

A Theoretical Approach to Determine cg

According to Eqs. (6) and (7), the subgrid stress tensor τij can be written as follows:
τij=−2cgΔ2(GklGkl)1/4sij,
and, therefore, the dissipation rate ϵ = |τijsij| is given in the following form:
ϵ=2cgΔ2(GklGkl)1/4sijsij.
We can write the resolved gradient and strain-rate terms GijGij and sijsij, respectively, in the Fourier space (see e.g., Pope 2000):
GijGij≈4[∫k2G2(k)E(k)dk][∫k2G2(k)E(k)dk],
sijsij=∫k2G2(k)E(k)dk,
where G(k) and E(k) are the filter function and kinetic energy spectrum, respectively. Assuming that G(k) is the cutoff filter function with the cutoff wavenumber kc =π/Δ, and E(k)=Cϵ2/3k−5/3 is the isotropic Kolmogorov inertial subrange, Eqs. (A3)–(A4) can be simplified:
GijGij≈4C2ϵ4/3(34)2(πΔ)8/3,
sijsij≈Cϵ2/3(34)(πΔ)4/3.
Here, C is a constant coefficient (we are aware that turbulent flows are highly anisotropic in atmospheric sciences. The isotropic Kolmogorov subrange is only a primitive assumption to develop a theoretical approach for estimating the correlation coefficient cg in the gradient model). Substituting Eqs. (A5)–(A6) into Eq. (A2) results in
ϵ=cgΔ2[(32)3/2C3/2ϵ(πΔ)2],
from which we can obtain a theoretical estimate for cg in the following form
cg=(2/3)3/2C3/2π2.
If we use C=1.51 (as is suggested in Pope 2000; Meneveau and Katz 2000), cg can be estimated as follows
cg=0.0299,
which is interestingly close to the Smagorinsky coefficient cs2=(0.17)2=0.0289. The similarity of cg and cs2 is left for the future investigation.

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    • Search Google Scholar
    • Export Citation

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  • Armenio, V., and S. Sarkar, 2002: An investigation of stably stratified turbulent channel flow using large-eddy simulation. J. Fluid Mech., 459, 1–42, https://doi.org/10.1017/S0022112002007851.

    • Search Google Scholar
    • Export Citation

  • Caulfield, C., and W. Peltier, 1994: Three dimensionalization of the stratified mixing layer. Phys. Fluids, 6, 3803–3805, https://doi.org/10.1063/1.868370.

    • Search Google Scholar
    • Export Citation

  • Caulfield, C., and W. Peltier, 2000: The anatomy of the mixing transition in homogeneous and stratified free shear layers. J. Fluid Mech., 413, 1–47, https://doi.org/10.1017/S0022112000008284.

    • Search Google Scholar
    • Export Citation

  • Clark, R. A., J. H. Ferziger, and W. C. Reynolds, 1979: Evaluation of subgrid-scale models using an accurately simulated turbulent flow. J. Fluid Mech., 91, 1–16, https://doi.org/10.1017/S002211207900001X.

    • Search Google Scholar
    • Export Citation

  • Corcos, G. M., and F. S. Sherman, 1976: Vorticity concentration and the dynamics of unstable free shear layers. J. Fluid Mech., 73, 241–264, https://doi.org/10.1017/S0022112076001365.

    • Search Google Scholar
    • Export Citation

  • Fritts, D. C., K. Wan, J. Werne, T. Lund, and J. H. Hecht, 2014: Modeling the implications of. Kelvin–Helmholtz instability dynamics for airglow observations. J. Geophys. Res. Atmos., 119, 8858–8871, https://doi.org/10.1002/2014JD021737.

    • Search Google Scholar
    • Export Citation

  • Gassmann, A., 2018: Entropy production due to subgrid-scale thermal fluxes with application to breaking gravity waves. Quart. J. Roy. Meteor. Soc., 144, 499–510, https://doi.org/10.1002/qj.3221.

    • Search Google Scholar
    • Export Citation

  • Hazel, P., 1972: Numerical studies of the stability of inviscid stratified shear flows. J. Fluid Mech., 51, 39–61, https://doi.org/10.1017/S0022112072001065.

    • Search Google Scholar
    • Export Citation

  • Khani, S., 2018: Mixing efficiency in large-eddy simulations of stratified turbulence. J. Fluid Mech., 849, 373–394, https://doi.org/10.1017/jfm.2018.417.

    • Search Google Scholar
    • Export Citation

  • Khani, S., and F. Porté-Agel, 2017a: Evaluation of non-eddy viscosity subgrid-scale models in stratified turbulence using direct numerical simulations. Eur. J. Mech. B Fluids, 65, 168–178, https://doi.org/10.1016/j.euromechflu.2017.03.009.

    • Search Google Scholar
    • Export Citation

  • Khani, S., and F. Porté-Agel, 2017b: A modulated-gradient parametrization for the large-eddy simulation of the atmospheric boundary layer using the Weather Research and Forecasting Model. Bound.-Layer Meteor., 165, 385–404, https://doi.org/10.1007/s10546-017-0287-5.

    • Search Google Scholar
    • Export Citation

  • Khani, S., and M. L. Waite, 2014: Buoyancy scale effects in large-eddy simulations of stratified turbulence. J. Fluid Mech., 754, 75–97, https://doi.org/10.1017/jfm.2014.381.

    • Search Google Scholar
    • Export Citation

  • Khani, S. and M. L. Waite, 2015: Large eddy simulations of stratified turbulence: The dynamic Smagorinsky model. J. Fluid Mech., 773, 327–344, https://doi.org/10.1017/jfm.2015.249.

  • Khani, S., and M. L. Waite, 2020: An anisotropic subgrid-scale parameterization for large-eddy simulations of stratified turbulence. Mon. Wea. Rev., 148, 4299–4311, https://doi.org/10.1175/MWR-D-19-0351.1.

    • Search Google Scholar
    • Export Citation

  • Korty, R. L., K. A. Emanuel, and J. R. Scott, 2008: Tropical cyclone–induced upper-ocean mixing and climate: Application to equable climates. J. Climate, 21, 638–654, https://doi.org/10.1175/2007JCLI1659.1.

    • Search Google Scholar
    • Export Citation

  • Leaman, K. D., and R. L. Molinari, 1987: Topographic modification of the Florida Current by little Bahama and great Bahama banks. J. Phys. Oceanogr., 17, 1724–1736, https://doi.org/10.1175/1520-0485(1987)017<1724:TMOTFC>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Lilly, D., D. Waco, and S. Adelfang, 1974: Stratospheric mixing estimated from high-altitude turbulence measurements. J. Appl. Meteor. Climatol., 13, 488–493, https://doi.org/10.1175/1520-0450(1974)013<0488:SMEFHA>2.0.CO;2.

    • 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, 1–15, https://doi.org/10.1186/s40623-018-0935-0.

    • Search Google Scholar
    • Export Citation

  • Mashayek, A., and W. Peltier, 2012a: The “zoo” of secondary instabilities precursory to stratified shear flow transition. Part 1: Shear aligned convection, pairing, and braid instabilities. J. Fluid Mech., 708, 5–44, https://doi.org/10.1017/jfm.2012.304.

    • Search Google Scholar
    • Export Citation

  • Mashayek, A., and W. Peltier, 2012b: The “zoo” of secondary instabilities precursory to stratified shear flow transition. Part 2: The influence of stratification. J. Fluid Mech., 708, 45–70, https://doi.org/10.1017/jfm.2012.294.

    • Search Google Scholar
    • Export Citation

  • Meneveau, C., and J. Katz, 2000: Scale-invariance and turbulence models for large-eddy simulation. Annu. Rev. Fluid Mech., 32, 1–32, https://doi.org/10.1146/annurev.fluid.32.1.1.

    • Search Google Scholar
    • Export Citation

  • Miles, J. W., 1961: On the stability of heterogeneous shear flows. J. Fluid Mech., 10, 496–508, https://doi.org/10.1017/S0022112061000305.

    • Search Google Scholar
    • Export Citation

  • Nolan, D. S., N. A. Dahl, G. H. Bryan, and R. Rotunno, 2017: Tornado vortex structure, intensity, and surface wind gusts in large-eddy simulations with fully developed turbulence. J. Atmos. Sci., 74, 1573–1597, https://doi.org/10.1175/JAS-D-16-0258.1.

    • Search Google Scholar
    • Export Citation

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  • <sc>Fig</sc>. 1.
    View in gallery
    Fig. 1.

    Time evolution of vorticity field in the x–z plane at y ≈ 500 m for the case with the initial bulk Richardson number Ri = 0.041 and shear layer thickness d = 25 m.

  • <sc>Fig</sc>. 2.
    View in gallery
    Fig. 2.

    Time evolution of vorticity field in the x–z plane at y ≈ 500 m for the case with the initial bulk Richardson number Ri = 0.082 and shear layer thickness d = 50 m. See also Fig. 3 for plots in the x–y and y–z planes.

  • <sc>Fig</sc>. 3.
    View in gallery
    Fig. 3.

    Time evolution of vorticity field ωy in the (left) x–y plane at z ≈ 1200 m and (right) y–z plane x ≈ 500 m for the case with the initial bulk Richardson number Ri = 0.082 and shear layer thickness d = 50 m. Plots in the x–z plane are shown in Fig. 2.

  • <sc>Fig</sc>. 4.
    View in gallery
    Fig. 4.

    Time evolution of vorticity field in the x–z plane at y ≈ 500 m for the case with the initial bulk Richardson number Ri = 0.164 and shear layer thickness d = 100 m.

  • <sc>Fig</sc>. 5.
    View in gallery
    Fig. 5.

    The vorticity field in the x–z plane at y ≈ 500 m for cases with the initial bulk Richardson number (a) Ri = 0.197 and (b) Ri = 0.235, and shear layer thickness d = 100 m.

  • <sc>Fig</sc>. 6.
    View in gallery
    Fig. 6.

    Time evolution of vertically averaged horizontal kinetic and available potential energy spectra for the case with the initial bulk Richardson number Ri = 0.041 and shear layer thickness d = 25 m. Spectra are averaged over the mixed layer height (between z ≈ 710 and 1460 m).

  • <sc>Fig</sc>. 7.
    View in gallery
    Fig. 7.

    Time evolution of vertically averaged horizontal kinetic and available potential energy spectra for the case with the initial bulk Richardson number Ri = 0.082 and shear layer thickness d = 50 m. Spectra are averaged over the mixed layer height.

  • <sc>Fig</sc>. 8.
    View in gallery
    Fig. 8.

    Vertically averaged horizontal kinetic and available potential energy spectra at time (a) t = 525 and (b) t = 795 s, for the case with the initial bulk Richardson number Ri = 0.164 and shear layer thickness d = 100 m. Spectra are averaged over the mixed layer height.

  • <sc>Fig</sc>. 9.
    View in gallery
    Fig. 9.

    Vertically averaged horizontal kinetic and available potential energy spectra at time (a) t = 585 and (b) t = 885 s, for the case with the initial bulk Richardson number Ri = 0.197 and shear layer thickness d = 100 m. Spectra are averaged over the mixed layer height.

  • <sc>Fig</sc>. 10.
    View in gallery
    Fig. 10.

    Time evolution of resolved shear characteristics S¯=(GijGij)1/4 in the x–z plane at y ≈ 500 m for the case with the initial bulk Richardson number Ri = 0.041 and shear layer thickness d = 25 m.

  • <sc>Fig</sc>. 11.
    View in gallery
    Fig. 11.

    Time evolution of the scatter diagram of vorticity field ωy vs the resolved shear S¯ for the case with the initial bulk Richardson number Ri = 0.041 and shear layer thickness d = 25 m. Lines ωy=±S¯ are shown for references.

  • <sc>Fig</sc>. 12.
    View in gallery
    Fig. 12.

    Time evolution of kinetic energy eddy dissipation rate ϵ = 2νgsijsij in the x–z plane at y ≈ 500 m for the case with the initial bulk Richardson number Ri = 0.041 and shear layer thickness d = 25 m.

  • <sc>Fig</sc>. 13.
    View in gallery
    Fig. 13.

    Time evolution of potential energy eddy dissipation rate ϵp=RiDg(∂θ¯/∂xj)(∂θ¯/∂xj) in the x–z plane at y ≈ 500 m for the case with the initial bulk Richardson number Ri = 0.041 and shear layer thickness d = 25 m. Note that Dg = νg/Prt.

  • <sc>Fig</sc>. 14.
    View in gallery
    Fig. 14.

    Time evolution of resolved shear characteristics S¯=(GijGij)1/4 in the x–z plane at y ≈ 500 m for the case with the initial bulk Richardson number Ri = 0.082 and shear layer thickness d = 50 m.

  • <sc>Fig</sc>. 15.
    View in gallery
    Fig. 15.

    (a),(b) Kinetic energy eddy dissipation rate ϵ = 2νgsijsij and (c),(d) potential energy eddy dissipation rate ϵp=RiDg(∂θ¯/∂xj)(∂θ¯/∂xj) in the x–z plane at y ≈ 500 m for the case with the initial bulk Richardson number Ri = 0.082 and shear layer thickness d = 50 m. Note that Dg = νg/Prt.

  • <sc>Fig</sc>. 16.
    View in gallery
    Fig. 16.

    (a) Kinetic energy eddy dissipation rate ϵ = 2νgsijsij and (b) potential energy eddy dissipation rate ϵp=RiDg(∂θ¯/∂xj)(∂θ¯/∂xj) in the x–z plane at y ≈ 500 m for the case with the initial bulk Richardson number Ri = 0.164 and shear layer thickness d = 100 m. Note that Dg = νg/Prt.

  • <sc>Fig</sc>. 17.
    View in gallery
    Fig. 17.

    Potential energy eddy dissipation rate ϵp=RiDg(∂θ¯/∂xj)(∂θ¯/∂xj) in the x–z plane at y ≈ 500 m for the case with the initial bulk Richardson number Ri = 0.197 and shear layer thickness d = 100 m. Note that Dg = νg/Prt.

  • <sc>Fig</sc>. 18.
    View in gallery
    Fig. 18.

    Time evolution of irreversible mixing efficiency γi in the x–z plane at y ≈ 500 m for the case with the initial bulk Richardson number Ri = 0.041 and shear layer thickness d = 25 m.

  • <sc>Fig</sc>. 19.
    View in gallery
    Fig. 19.

    Time evolution of vorticity field in the x–z plane at y ≈ 500 m when (a) t = 945 and (b) t = 1125 s for the case with the initial bulk Richardson number Ri = 0.164 and shear layer thickness d = 100 m, when the Smagorinsky parameterization is employed. (c) Kinetic and (d) potential eddy dissipation rates at t = 1185 s.

Fig. 1.

Time evolution of vorticity field in the x–z plane at y ≈ 500 m for the case with the initial bulk Richardson number Ri = 0.041 and shear layer thickness d = 25 m.
Fig. 2.

Time evolution of vorticity field in the x–z plane at y ≈ 500 m for the case with the initial bulk Richardson number Ri = 0.082 and shear layer thickness d = 50 m. See also Fig. 3 for plots in the x–y and y–z planes.
Fig. 3.

Time evolution of vorticity field ωy in the (left) x–y plane at z ≈ 1200 m and (right) y–z plane x ≈ 500 m for the case with the initial bulk Richardson number Ri = 0.082 and shear layer thickness d = 50 m. Plots in the x–z plane are shown in Fig. 2.
Fig. 4.

Time evolution of vorticity field in the x–z plane at y ≈ 500 m for the case with the initial bulk Richardson number Ri = 0.164 and shear layer thickness d = 100 m.
Fig. 5.

The vorticity field in the x–z plane at y ≈ 500 m for cases with the initial bulk Richardson number (a) Ri = 0.197 and (b) Ri = 0.235, and shear layer thickness d = 100 m.
Fig. 6.

Time evolution of vertically averaged horizontal kinetic and available potential energy spectra for the case with the initial bulk Richardson number Ri = 0.041 and shear layer thickness d = 25 m. Spectra are averaged over the mixed layer height (between z ≈ 710 and 1460 m).
Fig. 7.

Time evolution of vertically averaged horizontal kinetic and available potential energy spectra for the case with the initial bulk Richardson number Ri = 0.082 and shear layer thickness d = 50 m. Spectra are averaged over the mixed layer height.
Fig. 8.

Vertically averaged horizontal kinetic and available potential energy spectra at time (a) t = 525 and (b) t = 795 s, for the case with the initial bulk Richardson number Ri = 0.164 and shear layer thickness d = 100 m. Spectra are averaged over the mixed layer height.
Fig. 9.

Vertically averaged horizontal kinetic and available potential energy spectra at time (a) t = 585 and (b) t = 885 s, for the case with the initial bulk Richardson number Ri = 0.197 and shear layer thickness d = 100 m. Spectra are averaged over the mixed layer height.
Fig. 10.

Time evolution of resolved shear characteristics S¯=(GijGij)1/4 in the x–z plane at y ≈ 500 m for the case with the initial bulk Richardson number Ri = 0.041 and shear layer thickness d = 25 m.
Fig. 11.

Time evolution of the scatter diagram of vorticity field ωy vs the resolved shear S¯ for the case with the initial bulk Richardson number Ri = 0.041 and shear layer thickness d = 25 m. Lines ωy=±S¯ are shown for references.
Fig. 12.

Time evolution of kinetic energy eddy dissipation rate ϵ = 2νgsijsij in the x–z plane at y ≈ 500 m for the case with the initial bulk Richardson number Ri = 0.041 and shear layer thickness d = 25 m.
Fig. 13.

Time evolution of potential energy eddy dissipation rate ϵp=RiDg(∂θ¯/∂xj)(∂θ¯/∂xj) in the x–z plane at y ≈ 500 m for the case with the initial bulk Richardson number Ri = 0.041 and shear layer thickness d = 25 m. Note that Dg = νg/Prt.
Fig. 14.

Time evolution of resolved shear characteristics S¯=(GijGij)1/4 in the x–z plane at y ≈ 500 m for the case with the initial bulk Richardson number Ri = 0.082 and shear layer thickness d = 50 m.
Fig. 15.

(a),(b) Kinetic energy eddy dissipation rate ϵ = 2νgsijsij and (c),(d) potential energy eddy dissipation rate ϵp=RiDg(∂θ¯/∂xj)(∂θ¯/∂xj) in the x–z plane at y ≈ 500 m for the case with the initial bulk Richardson number Ri = 0.082 and shear layer thickness d = 50 m. Note that Dg = νg/Prt.
Fig. 16.

(a) Kinetic energy eddy dissipation rate ϵ = 2νgsijsij and (b) potential energy eddy dissipation rate ϵp=RiDg(∂θ¯/∂xj)(∂θ¯/∂xj) in the x–z plane at y ≈ 500 m for the case with the initial bulk Richardson number Ri = 0.164 and shear layer thickness d = 100 m. Note that Dg = νg/Prt.
Fig. 17.

Potential energy eddy dissipation rate ϵp=RiDg(∂θ¯/∂xj)(∂θ¯/∂xj) in the x–z plane at y ≈ 500 m for the case with the initial bulk Richardson number Ri = 0.197 and shear layer thickness d = 100 m. Note that Dg = νg/Prt.
Fig. 18.

Time evolution of irreversible mixing efficiency γi in the x–z plane at y ≈ 500 m for the case with the initial bulk Richardson number Ri = 0.041 and shear layer thickness d = 25 m.
Fig. 19.

Time evolution of vorticity field in the x–z plane at y ≈ 500 m when (a) t = 945 and (b) t = 1125 s for the case with the initial bulk Richardson number Ri = 0.164 and shear layer thickness d = 100 m, when the Smagorinsky parameterization is employed. (c) Kinetic and (d) potential eddy dissipation rates at t = 1185 s.