1. Introduction

Large-eddy simulation (LES) is a useful numerical approach for simulations of geophysical turbulence at the small-scale end of the atmospheric mesoscale, oceanic submesoscale, and smaller scales, where there is forward kinetic energy transfer from large to small horizontal scales. These scale ranges are broadly characterized by strong stratification and weak rotation (i.e., stratified turbulence) (e.g., Riley and Lindborg 2008). It has been shown that fundamental characteristics of stratified turbulence that are seen in direct numerical simulation (DNS), such as a −5/3 spectral slope in the horizontal wavenumber energy spectra (Waite and Bartello 2004; Lindborg 2006; Brethouwer et al. 2007), layered structures with Kelvin–Helmholtz (KH) instabilities (Bartello and Tobias 2013; Khani and Waite 2016), nonlocal horizontal energy transfer from large scales to the buoyancy scale (Waite 2011; Khani and Waite 2013), and small or negative local Richardson number associated with overturning (Waite and Bartello 2004; Bartello and Tobias 2013), can be captured by LES if the buoyancy scale L_b = 2πu_rms/N is sufficiently well resolved (i.e., Δ < L_b; see Khani and Waite 2014, 2015). Here, u_rms, N, and Δ are the root-mean-square (rms) velocity, buoyancy frequency, and grid spacing, respectively. These and other LES (e.g., Siegel and Domaradzki 1994; Kang et al. 2003; Paoli et al. 2014) used isotropic grid spacing in the horizontal and vertical directions (i.e., Δ_h = Δ_z). However, coarser and anisotropic grid spacings are usually employed in atmosphere and ocean simulations, which require a different subgrid approach.

In large-scale atmosphere and ocean models, it is not possible to explicitly resolve the buoyancy scale L_b in the horizontal direction, due to limits on computation. For example, the horizontal grid spacing Δ_h in global weather prediction models is often around 10 km or larger, while L_b is on the order of 1 km in the atmosphere (see e.g., Augier and Lindborg 2013; Brune and Becker 2013; Schaefer-Rolffs and Becker 2018). Therefore, it is not computationally feasible to use isotropic grid spacing to resolve L_b, and therefore different grid spacings in the horizontal and vertical directions (i.e., anisotropic grids) are usually employed. Atmosphere and ocean models typically use finer grid spacing in the vertical direction than in the horizontal (i.e., Δ_z < Δ_h), and the question of sufficient vertical resolution for capturing the −5/3 power law in the horizontal wavenumber energy spectrum has been an active area of discussion (see e.g., Brune and Becker 2013; Augier and Lindborg 2013; Waite 2016; Schaefer-Rolffs and Becker 2018; Skamarock et al. 2019). Generally, most atmosphere and ocean models use different dissipation schemes in the horizontal and vertical directions because of the large difference in horizontal and vertical grid spacings. These dissipation schemes are typically independent of one another: for example, models may use the horizontal Smagorinsky subgrid-scale (SGS) model for horizontal mixing and a vertical stability-dependent eddy viscosity, possibly as part of the boundary layer scheme, for vertical mixing [see e.g., Griffies and Hallberg 2000; Skamarock et al. 2008, for the Weather Research and Forecasting (WRF) Model and the Modular Ocean Model (MOM), respectively].

In this paper, we use homogeneous stratified turbulence as an idealized problem in which to investigate the consequences of using decoupled horizontal and vertical SGS dissipation schemes in the limit of high vertical resolution. We develop an anisotropic scheme for LES of stratified turbulence based on a scale analysis of the SGS momentum and potential temperature fluxes in stratified turbulence. Initially, we set Δ_z very small (as in DNS) in our LES runs to evaluate the dependence of our new anisotropic scheme on the horizontal grid spacing Δ_h by comparison with a more typical SGS scheme, in which the horizontal and vertical dissipations are treated separately. Next, we study the effects of vertical resolutions in our new scheme. The rest of this paper is composed as follows: the governing equations and mathematical formulations are given in section 2. Section 3 presents the methodology and numerical setup. Results are shown and discussed in section 4, followed by conclusions in section 5.

2. Governing equations

The governing equations of motion under the Boussinesq approximation with uniform stratification can be written in the following nondimensional form (as in e.g., Khani and Waite 2013):

where u = (u, υ, w) is the velocity vector; θ and p are the potential temperature and pressure perturbations, respectively; and ${\mathrm{Re}}_{\ell }=\mathcal{U}\ell /\nu$, ${\text{Fr}}_{\ell }=\mathcal{U}/\left(\ell N\right)$, and $\text{Pr}=\nu /\kappa$ are the Reynolds, Froude, and Prandtl numbers, respectively. Here, the velocity and length scales are taken to be unity (i.e., $\mathcal{U}\equiv 1$ and $\ell \equiv 1$), and ν and κ are the molecular viscosity and diffusivity, respectively. In LES, the flow variables are filtered using a filtering operator G. For example, the filtered velocity field $\overline{\mathbf{u}}$ is defined as follows (e.g., Pope 2000):

where x = (x, y, z) are the Cartesian coordinates and D is the spatial domain. Applying the filtering operator G to the equations of motions [Eqs. (1)–(3)] is straightforward except for the nonlinear terms, which lead to the subgrid-scale (SGS) momentum stress:

and SGS potential temperature flux:

which are not known in terms of the filtered variables and must be parameterized in LES. In summary, the filtered Navier–Stokes equations under the Boussinesq approximation can be written as

Before introducing SGS models for the momentum and potential temperature fluxes, it is useful to investigate the SGS momentum stress τ_ij using Taylor series and the definition of the filtering operator. A similar procedure is also done for the potential temperature SGS flux h_j.

We can expand the velocity field u(x + r) using a Taylor series at a given point x in r, which is on the order of the filter width Δ (see e.g., Pope 2000; Meneveau and Katz 2000; Khani and Porté-Agel 2017a,b):

Using this expansion, we can find the following nonlinear approximation for the SGS stress tensor (see appendix A for details):

The SGS stress τ_ij depends on the filtering function G and the integral over the associated domain. For example, if G is an isotropic Gaussian function with variance Δ²/12, Eq. (11) yields

Horizontal SGS mixing parameterizes the effects of small unresolved horizontal scales. As a result, it can be investigated by applying a filter to horizontal scales only [i.e., G = G(r_x, r_y)]. In this case, the dummy indices l and k in Eq. (11) will span {1, 2} and Eq. (11) will not include z derivatives. Therefore, using an isotropic horizontal Gaussian filter, Eq. (12) becomes

The vertical components of the SGS stress τ_ij (i.e., τ₁₃, τ₂₃, and τ₃₃) are not zero because the horizontal derivatives of vertical motions (i.e., ∂w/∂x and ∂w/∂y) are nonzero.

Similarly, for the SGS flux term $h_j\left(\mathbf{x}\right)=\overline{\theta \left(\mathbf{x}\right)u_j\left(\mathbf{x}\right)}-\overline{\theta }\left(\mathbf{x}\right){\overline{u}}_j\left(\mathbf{x}\right)$, we can write (see appendix B for more information):

Again, assuming a horizontal Gaussian filter function G, we obtain

where the vertical component of the SGS potential temperature flux h_j is nonzero because ∂w/∂x and ∂w/∂y are nonzero [similar to Eq. (13)]. Note that we do not use Eqs. (13) and (15) as a parameterization, but rather as a guide to determine what terms in τ_ij and h_j should be retained and parameterized. Overall, from Eqs. (13) and (15) it is clear that vertical components of SGS fluxes are not zero even when the focus is on only unresolved horizontal scales (i.e., with a purely horizontal filtering operator G).

We use scale analysis to estimate the size of the various terms in Eqs. (13) and (15) in geophysical simulations. Let l_h and l_z be the horizontal and vertical scales, respectively, where l_z ≪ l_h. In this case, the horizontal and vertical components of the SGS tensor τ_ij can be scaled as (recall that τ_ij is symmetric)

where tilde (~) denotes order of magnitude, U is the horizontal velocity scale, and we have used the continuity equation to scale the vertical velocity as l_zU/l_h (as in e.g., Riley and Lelong 2000). Using a similar scale analysis, the SGS stress tensor divergence in Eq. (7) can be scaled as

The horizontal derivatives of horizontal stress (i, j = 1, 2) and vertical derivatives of SGS stresses with i = 1, 2 and j = 3 are of the same order of magnitude, and therefore the latter terms are not negligible in comparison with the former when a horizontal filter function is employed. Yet these terms, ∂τ₁₃/∂x₃ and ∂τ₂₃/∂x₃, are not included in purely horizontal mixing schemes (e.g., horizontal Smagorinsky in WRF; see Skamarock et al. 2008). Similarly, we can scale the SGS potential temperature flux divergence ∂h_j/∂x_j as

where Θ is the potential temperature scale. Again, both horizontal and vertical derivatives of h_j are of the same order of magnitude.

The SGS term ∂τ_ij/∂x_j includes the following terms in the x, y, and z directions, respectively,

where only the term ∂τ₃₃/∂zx is negligible [see Eq. (17)]. Also, the potential temperature flux ∂h_j/∂x_j includes the following terms:

where all terms are important [see Eq. (18)]. In later sections, we will perform LES runs with anisotropic dissipation following Eqs. (19) to (22), and compare the results with DNS, and classic LES where the horizontal and vertical dissipation schemes are not connected (i.e., the vertical components τ₁₃, τ₂₃, and h₃ are omitted).

3. Methodology

We consider a domain with periodic boundary conditions. The horizontal side length is ${\mathcal{L}}_h=2\pi$, and the vertical side length, depending on the simulation, is ${\mathcal{L}}_{\upsilon }=2\pi$ or π. Decaying stratified turbulence is simulated: we have scaled the velocity amplitude to set an initial energy of 0.13 with random phases over a spherical wavevector shell k_i − 0.5 < k ≤ k_i + 0.5, where $k=\sqrt{k_x^2+k_y^2+k_z^2}$ is the total wavenumber and k_i = 3 is the initial wavenumber (a similar initial condition is used in Bartello and Tobias 2013). The initial potential temperature fluctuation is zero. The buoyancy frequency is N = 2.1 or 4.2, both of which ensure small initial Froude numbers in our simulations (i.e., Fr_l = 1/N). Grid sizes vary from n_x = n_y = 240 to 960. For the DNS, n_z = n_x (for ${\mathcal{L}}_{\upsilon }=2\pi$) or n_z = n_x/2 (for ${\mathcal{L}}_{\upsilon }=\pi$), where n_i is the number of grid points in the i = x, y, and z directions. Spatial derivatives are discretized using the spectral transform method, and the two-thirds rule (Orszag 1971) is employed to eliminate aliasing errors, which leads to an effective grid spacing ${\text{Δ}}_h=3{\mathcal{L}}_h/\left(2n_x\right)$ and ${\text{Δ}}_z=3{\mathcal{L}}_{\upsilon }/\left(2n_z\right)$. The third-order Adams–Bashforth scheme is employed for the time stepping of all terms in the transport equations, except the molecular dissipation terms, which are treated with a Crank–Nicolson approach (see e.g., Durran 2010).

A new anisotropic¹ LES method, in which the vertical derivatives of the SGS stress and flux are retained as shown in Eqs. (19)–(22), is tested. We employ the dynamic Smagorinsky SGS model because it has the best overall performance in comparison with other SGS parameterizations in LES of stratified turbulence (see Khani and Waite 2014, 2015). The eddy viscosity and diffusivity terms in the anisotropic dynamic Smagorinsky model are given as follows:

and

where

is the rate-of-strain tensor, $\overline{S}=-c_s{\text{Δ}}_h^2{\left(2{\overline{s}}_{ij}{\overline{s}}_{ij}\right)}^{1/2}$ is proportional to the total strain rate, c_s is the time and space dependent Smagorinsky coefficient, which is calculated using an isotropic explicit coarse filter scale $\tilde{\mathrm{\Delta }}=2{\text{Δ}}_h$ (negative c_s values are clipped, see Khani and Waite 2015); the turbulent Prandtl number Pr_t = 1. Since the anisotropic LES approach is based on horizontal-only filtering, these simulations use high (DNS) resolution in the vertical.

For comparison, classic LES and DNS runs are also performed. In the classic LES runs, the vertical dissipation scale is resolved with high (DNS) vertical resolution, and therefore terms including vertical derivatives in Eqs. (23)–(26) are neglected in the classic horizontal SGS parameterization. The spatial resolution of DNS runs is high in all directions and no SGS model is included. The DNS resolution of the Kolmogorov scale L_d is $k_{\max }/k_d\gtrsim 0.67$, which is in line with the criterion given by Moin and Mahesh (1998). Here, k_max is the maximum wavenumber and k_d is the Kolmogorov wavenumber, calculated over the time interval of maximum kinetic energy dissipation. Our setup with small Δ_z (DNS) is not standard in atmosphere and ocean simulations—indeed, the vertical grids in such models do not ever resolve the Kolmogorov scale—but it is a sensible experiment in which to investigate the parameterization of subgrid horizontal scales. In current SGS models, which we call here classic LES, the limit of Δ_z → 0 would shut off the requirement for a vertical mixing scheme; here we show that simulations with the classic LES model will fail in such a limit, for example, the kinetic energy spectra are underdissipated (see below), thereby illustrating a problem with the classic approach.

Simulation results are averaged over a time interval around which the kinetic energy dissipation rate ϵ is maximum. The rms velocity $u_{\text{rms}}=\sqrt{\langle E\left(t\right)\rangle }$, where the angle bracket ⟨⋅⟩ denotes time averaging and E(t) is the domain-averaged kinetic energy. The buoyancy scale L_b = 2πu_rms/N, the Ozmidov scale L_o = 2π(ε/N³)^1/2, and the Kolmogorov scale L_d = 2π(υ³/ε)^1/4. In our LES, unless there is no eddy viscosity/diffusivity terms (i.e., in cases with high vertical resolution Δ_z), molecular viscosity and diffusivity are switched off. Table 1 shows a list of averaged variables and parameters for the DNS and anisotropic LES runs.² In the rest of this paper, simulation names start with “D” for DNS, “LA” for anisotropic LES with the same vertical resolution as DNS but coarser horizontal resolution, and “LAV” for anisotropic LES with half the vertical resolution compared with “LA” runs (see Table 1). For LAV runs, the Kolmogorov scale is not resolved in the vertical direction, and there is no extra vertical SGS parameterization (only the anisotropic SGS model is employed). The classic LES simulations, which are labeled with “LC” (see the caption of Fig. 5), are not reported in Table 1 because these runs are underdissipated (see below).

Table 1.

List of numerical simulations with DNS and LES.

View Table

4. Results and discussion

a. Overview of simulations

Figures 1a–d show the time evolution of the total, kinetic, and potential energy, respectively, for the DNS and anisotropic LES runs. The total energy is almost constant up to approximately t = 5, and then it decays due to the onset of turbulence (see below). The time series of the kinetic and potential energy (KE and PE) show oscillations, mainly before turbulence decay, due to buoyancy exchanges between KE and PE, since only KE is present at t = 0 (Figs. 1c,d). The oscillation time scale is related to the frequency of the gravity waves excited by the initial conditions, and therefore becomes smaller in the case with stronger stratification (Figs. 1c,d). The anisotropic LES runs correctly capture the energy oscillations and onset of dissipation from the DNS runs, although the anisotropic LES cases with coarser horizontal resolution (i.e., cases LA18N2b and LA22N4b) slightly underestimate the total energy level after the occurrence of turbulence, which is due to larger eddy dissipation in these coarse LES simulations. Moreover, the onset of turbulence happens earlier in the anisotropic LES cases with the lowest horizontal resolution (see green and magenta dash–dot lines in Figs. 1a,b).

Time series of (top) total energy and (bottom) the kinetic and potential energy for DNS and anisotropic LES runs with (a),(c) weak and (b),(d) strong stratification. Potential energy curves in (c),(d) are those that start from zero.

Citation: Monthly Weather Review 148, 10; 10.1175/MWR-D-19-0351.1

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Time series of (top) total energy and (bottom) the kinetic and potential energy for DNS and anisotropic LES runs with (a),(c) weak and (b),(d) strong stratification. Potential energy curves in (c),(d) are those that start from zero.

Citation: Monthly Weather Review 148, 10; 10.1175/MWR-D-19-0351.1

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Time series of (top) total energy and (bottom) the kinetic and potential energy for DNS and anisotropic LES runs with (a),(c) weak and (b),(d) strong stratification. Potential energy curves in (c),(d) are those that start from zero.

Citation: Monthly Weather Review 148, 10; 10.1175/MWR-D-19-0351.1

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The kinetic energy dissipation rate ε for cases with the initial Reynolds number Re_l = 18 000, buoyancy frequency N = 2.1, and Re_l = 22 200, N = 4.2 are displayed in Figs. 2a and 2b, respectively. The solid black line in Fig. 2a shows ε for the DNS run, to which we compare the ε in the anisotropic LES cases (red dash and green dash–dot lines). A similar comparison is provided in Fig. 2b, in which the solid gray line shows ε for the DNS case, and the blue dash and magenta dash–dot lines show ε in the anisotropic LES runs. The kinetic energy dissipation rate has a maximum around t = 7 and t = 8 in Figs. 2a and 2b, respectively, for the DNS runs. These maxima give the approximate time at which turbulence onset occurs. The onset time for turbulence is relatively well estimated by the anisotropic LES cases with finer horizontal resolutions (i.e., cases LA18N2a and LA22N4a), although the magnitudes of ε at the maximum times are a little higher in these LES cases in comparison with the DNS. If the horizontal grid spacing decreases further in the anisotropic LES cases (LA18N2b and LA22N4b), turbulence onset occurs earlier while the maximum ε values would be around or a little smaller than the corresponding DNS runs (solid versus dash–dot line in Fig. 2). Overall, the anisotropic LES runs show larger ε at early times compared to the DNS cases, but differences between the kinetic energy dissipation rates in the DNS and anisotropic LES cases are smaller after the onset of turbulence (Fig. 2).

Time series of the kinetic energy dissipation rate for DNS and anisotropic LES runs with (a) weak and (b) strong stratification.

Citation: Monthly Weather Review 148, 10; 10.1175/MWR-D-19-0351.1

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Time series of the kinetic energy dissipation rate for DNS and anisotropic LES runs with (a) weak and (b) strong stratification.

Citation: Monthly Weather Review 148, 10; 10.1175/MWR-D-19-0351.1

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Time series of the kinetic energy dissipation rate for DNS and anisotropic LES runs with (a) weak and (b) strong stratification.

Citation: Monthly Weather Review 148, 10; 10.1175/MWR-D-19-0351.1

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Figures 3 and 4 show the y component of vorticity on the x–z plane at y = 0.25 and t = 15 for weak and strong stratification cases, respectively (Figs. 3a and 4a for DNS and Figs. 3b,c and 4b,c for anisotropic LES). In the DNS, the vorticity snapshot shows layers, KH instabilities, and regions of more isotropic small-scale turbulence (see e.g., regions around z = 2 and x = [0–3], or z ≈ 0.8 and x = [5–6] in Fig. 4a). Layering is more pronounced in the simulation with larger stratification, which has smaller Re_b and is therefore more influenced by dissipation (as shown in Fig. 4a, and also see Brethouwer et al. 2007; Bartello and Tobias 2013; Khani and Waite 2014). For example, Fig. 3b depicts many regions with small-scale isotropic turbulence, while layered structures are more visible in Fig. 4b where stratification is increased. If we further decrease the horizontal resolution in the anisotropic LES, similar large-scale structures are generally seen in both weak and strong stratification cases. Interestingly, the anisotropic LES runs with larger grid spacing (i.e., coarser resolution compared to DNS) reproduce similar structures that are seen in the DNS runs with smaller grid spacing. The horizontal layers are much more pronounced in these low-resolution simulations due to significantly larger horizontal dissipation because Δ_h is much larger here, which reduces the transition to small-scale isotropy (Figs. 3c and 4c).

Vorticity field in y direction on the x–z plane at y = 0.25 and t = 15 for the case with ${\mathrm{Re}}_{\ell }=18\hspace{0.17em}000$ and N = 2.1: (a) DNS, (b) high-horizontal-resolution anisotropic LES, and (c) low-horizontal-resolution anisotropic LES.

Citation: Monthly Weather Review 148, 10; 10.1175/MWR-D-19-0351.1

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Vorticity field in y direction on the x–z plane at y = 0.25 and t = 15 for the case with ${\mathrm{Re}}_{\ell }=18\hspace{0.17em}000$ and N = 2.1: (a) DNS, (b) high-horizontal-resolution anisotropic LES, and (c) low-horizontal-resolution anisotropic LES.

Citation: Monthly Weather Review 148, 10; 10.1175/MWR-D-19-0351.1

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Vorticity field in y direction on the x–z plane at y = 0.25 and t = 15 for the case with ${\mathrm{Re}}_{\ell }=18\hspace{0.17em}000$ and N = 2.1: (a) DNS, (b) high-horizontal-resolution anisotropic LES, and (c) low-horizontal-resolution anisotropic LES.

Citation: Monthly Weather Review 148, 10; 10.1175/MWR-D-19-0351.1

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Vorticity field in y direction on the x–z plane at y = 0.25 and t = 15 for the case with ${\mathrm{Re}}_{\ell }=22\hspace{0.17em}200$ and N = 4.2: (a) DNS, (b) high-horizontal-resolution anisotropic LES, and (c) low-horizontal-resolution anisotropic LES.

Citation: Monthly Weather Review 148, 10; 10.1175/MWR-D-19-0351.1

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Vorticity field in y direction on the x–z plane at y = 0.25 and t = 15 for the case with ${\mathrm{Re}}_{\ell }=22\hspace{0.17em}200$ and N = 4.2: (a) DNS, (b) high-horizontal-resolution anisotropic LES, and (c) low-horizontal-resolution anisotropic LES.

Citation: Monthly Weather Review 148, 10; 10.1175/MWR-D-19-0351.1

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Vorticity field in y direction on the x–z plane at y = 0.25 and t = 15 for the case with ${\mathrm{Re}}_{\ell }=22\hspace{0.17em}200$ and N = 4.2: (a) DNS, (b) high-horizontal-resolution anisotropic LES, and (c) low-horizontal-resolution anisotropic LES.

Citation: Monthly Weather Review 148, 10; 10.1175/MWR-D-19-0351.1

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b. Kinetic energy spectra

The horizontal and vertical wavenumber kinetic energy spectra for DNS, anisotropic LES, and classic LES runs are shown in Fig. 5 (Figs. 5a,b show simulations with Re_l = 18 000 and N = 2.1, and Figs. 5c,d show simulations with Re_l = 22 200 and N = 4.2). The spectra are averaged over a time interval Δt = 4 around the maximum kinetic energy dissipation rate. The high-resolution anisotropic LES (LA18N2a and LA22N4a cases) show almost identical vertical wavenumber kinetic energy spectra to those for DNS (red and blue dashed versus black and gray solid lines in Figs. 5b,d). This trend may not be unexpected since both the DNS and high-resolution anisotropic LES have the same vertical resolution, but the horizontal resolutions are different. The horizontal wavenumber kinetic energy spectra of the anisotropic LES and DNS are also very similar with higher horizontal resolution in the LES (with ${\text{Δ}}_h^{\text{LA}}=2{\text{Δ}}_h^D$; red and blue dashed versus black and gray solid lines in Figs. 5a,c). If we further reduce the horizontal resolution in the anisotropic LES to $4{\text{Δ}}_h^D$, we still can obtain reasonable results at large scales, although both the vertical and horizontal wavenumber kinetic energy spectra are less energetic at smaller scales due to the larger eddy dissipation at small horizontal scales (see green and magenta dash–dotted lines in Fig. 5). Overall, the anisotropic LES model is able to capture the inertial subrange of stratified turbulence similar to the DNS, with significantly less computational cost. Nevertheless, the kinetic energy spectra in the coarse horizontal-resolution cases (LA18N2b and LA22N4b) are more steep at small resolved scales.

The time-averaged (left) horizontal and (right) vertical wavenumber kinetic energy spectra for (top) weak and (bottom) strong stratification cases. Simulations labeled with “LC” are the same as LA, but the horizontal derivatives of vertical motions are omitted. Time averaging is performed over a window (Δt = 4) around the maximum kinetic energy dissipation rate. The solid black line segments show −5/3 and −3 slopes.

Citation: Monthly Weather Review 148, 10; 10.1175/MWR-D-19-0351.1

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The time-averaged (left) horizontal and (right) vertical wavenumber kinetic energy spectra for (top) weak and (bottom) strong stratification cases. Simulations labeled with “LC” are the same as LA, but the horizontal derivatives of vertical motions are omitted. Time averaging is performed over a window (Δt = 4) around the maximum kinetic energy dissipation rate. The solid black line segments show −5/3 and −3 slopes.

Citation: Monthly Weather Review 148, 10; 10.1175/MWR-D-19-0351.1

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The time-averaged (left) horizontal and (right) vertical wavenumber kinetic energy spectra for (top) weak and (bottom) strong stratification cases. Simulations labeled with “LC” are the same as LA, but the horizontal derivatives of vertical motions are omitted. Time averaging is performed over a window (Δt = 4) around the maximum kinetic energy dissipation rate. The solid black line segments show −5/3 and −3 slopes.

Citation: Monthly Weather Review 148, 10; 10.1175/MWR-D-19-0351.1

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The coarser resolution anisotropic LES cases have more dissipation at small vertical scales, as evidenced by the steeper vertical spectra, in comparison with DNS (or high-resolution anisotropic LES), despite the fact that they have the same Δ_z. This behavior suggests that horizontal resolution can have a significant impact on the resolution of small vertical scales in LES of stratified turbulence, and that the dissipation mechanisms in the horizontal and vertical directions are actually connected.

If the vertical derivatives of the SGS stress and flux are omitted in our LES runs (i.e., classic LES), the impact on the kinetic energy spectra are significant. Indeed, the spectra are underdissipated; there is insufficient small-scale dissipation and, as a result, energy accumulates around the smallest resolved scales in both horizontal and vertical wavenumber spectra (see cyan lines in Fig. 5). The vertical derivatives of SGS fluxes, which are missing in the classic LES runs, therefore play an important role in removing energy from small horizontal and vertical scales. Neglecting these terms can lead to unrealistic results, even with fine (DNS) grid spacings in the vertical. Overall, the results of this section show that the scale analyses in Eqs. (17) and (18), which lead to the anisotropic LES parameterizations that are shown by Eqs. (19)–(22), are confirmed using numerical simulations.

If we further reduce the horizontal resolution in the anisotropic LES model, the results become underdissipated when Δ_h/Δ_z > 4 (not shown). This trend suggests that the ratio Δ_h/Δ_z can also play a role in dissipation terms of anisotropic LES runs. To investigate this point further, we consider a series of additional anisotropic LES runs in the case with Re_l = 18 000 and N = 2.1, for which the vertical grid spacing is double the vertical grid spacing of DNS runs, with different horizontal resolution (these runs are labeled with “LAV”). Figure 6 shows the horizontal and vertical wavenumber kinetic energy spectra with ${\text{Δ}}_z^{\text{LAV}}=2{\text{Δ}}_z^D$, where cases b, c, and d show simulations with different horizontal resolutions: that is, ${\text{Δ}}_{h,b}^{\text{LAV}}=1/2{\text{Δ}}_{h,c}^{\text{LAV}}=2/3{\text{Δ}}_{h,d}^{\text{LAV}}={\text{Δ}}_{h,b}^{\text{LA}}$ (see Table 1). We should mention here that the coarse horizontal resolution in cases c and d gave underdissipated results with high vertical resolution ${\text{Δ}}_z^D$ (not shown), but interestingly, these cases are not underdissipated when coarser vertical grid spacing $2{\text{Δ}}_z^D$ is used (Fig. 6). Anisotropic LES with coarser vertical resolution (LAV) and DNS curves show very similar kinetic energy spectra when k_h and k_υ are $\lesssim 20$. For larger wavenumbers, however, LAV runs are much more dissipative than the DNS. This behavior is expected because both horizontal and vertical grids are larger than those in DNS; for example, the curve with dash–dot green line in Fig. 6 is for a simulation with ${\text{Δ}}_z^{\text{LAV}}=2{\text{Δ}}_z^D$ and ${\text{Δ}}_h^{\text{LAV}}=8{\text{Δ}}_h^D$. In this case, we can significantly save on computational resources since this anisotropic LES run is ≈300 times cheaper than the DNS run because of having larger grid spacing and time step for running simulations. From DNS to anisotropic LES, we can successfully reduce computational costs remarkably while the accuracy of results are still high.

The time-averaged (left) horizontal and (right) vertical wavenumber kinetic energy spectra for DNS and vertically reduced resolution anisotropic LES for the weak stratification case. Time averaging is performed over a window (Δt = 4) around the maximum kinetic energy dissipation rate. The solid black line segments show −5/3 and −3 slopes.

Citation: Monthly Weather Review 148, 10; 10.1175/MWR-D-19-0351.1

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The time-averaged (left) horizontal and (right) vertical wavenumber kinetic energy spectra for DNS and vertically reduced resolution anisotropic LES for the weak stratification case. Time averaging is performed over a window (Δt = 4) around the maximum kinetic energy dissipation rate. The solid black line segments show −5/3 and −3 slopes.

Citation: Monthly Weather Review 148, 10; 10.1175/MWR-D-19-0351.1

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The time-averaged (left) horizontal and (right) vertical wavenumber kinetic energy spectra for DNS and vertically reduced resolution anisotropic LES for the weak stratification case. Time averaging is performed over a window (Δt = 4) around the maximum kinetic energy dissipation rate. The solid black line segments show −5/3 and −3 slopes.

Citation: Monthly Weather Review 148, 10; 10.1175/MWR-D-19-0351.1

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Overall, we found that retaining the vertical derivatives of SGS fluxes as given by Eqs. (19)–(22) is a key in our anisotropic LES scheme to reproduce DNS results. Also, our results show that by increasing the ratio of horizontal to vertical grid spacing in the anisotropic LES model, we can help to prohibit underdissipative LES results. Nevertheless, this latter parameter setup needs to be further investigated in the realistic atmosphere and ocean models with our new horizontal dissipation scheme along with an appropriate vertical mixing scheme. Implementing this parameterization in atmosphere and ocean models would require geometrical adjustments for spherical coordinates (some geometrical modifications for diffusion coefficients in spherical geometry have been introduced, as in e.g., Gordon and Stern 1982; Smagorinsky 1993; Becker and Burkhardt 2007).

c. Mixing efficiency

Mixing efficiency is a key parameter in atmospheric sciences and physical oceanography, where breaking internal waves in stratified shear layers and diapycnal mixing in the upper ocean are significantly influenced by the efficiency of turbulent mixing (see e.g., Riley and Lelong 2000; Gregg et al. 2018). The irreversible mixing efficiency γ_i is defined as the ratio of the molecular potential energy dissipation to the total molecular dissipation rates ε_p/(ε + ε_p) (Winters and D’Asaro 1996; Caulfield and Peltier 2000). This quantity has been extended to be used in LES with SGS eddy dissipation rates (Khani 2018). The SGS mixing efficiency γ_i depends on the turbulent Prandtl number Pr_t as follows:

${\gamma }_i~{\displaystyle \frac{1}{1+2{\text{Pr}}_t}},$(28)

implying γ_i ≈ 1/3 in stratified turbulence with Pr_t = 1 (Khani 2018).

Figure 7 shows the irreversible mixing efficiency γ_i versus the resolution of the Ozmidov scale L_o in the horizontal direction, for DNS and anisotropic LES approaches. As expected, the ratio L_o/Δ_h is larger in DNS compared to that in LES. Nevertheless, values of γ_i in LES overlap well with those from DNS and are in line with the theoretical estimate of 1/3 for LES of stratified turbulence (Fig. 7). This agreement is due to the resolution of the Ozmidov scale L_o in LES runs (see Khani (2018) for more information). Noteworthy, unlike in the simulations in Khani (2018), where only large-scale vortical modes were initially excited, here we excite large horizontal and vertical motions, which results in more efficient energy exchange between KE and PE through the buoyancy fluxes. In this case, γ_i is slightly larger than 1/3 for both DNS and LES runs (Fig. 7).

Irreversible mixing efficiency γ_i vs the ratio L_o/Δ_h for DNS and anisotropic LES runs. LES and LES vert. refer to those anisotropic LES runs with high and low vertical resolutions (LA and LAV), respectively.

Citation: Monthly Weather Review 148, 10; 10.1175/MWR-D-19-0351.1

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Irreversible mixing efficiency γ_i vs the ratio L_o/Δ_h for DNS and anisotropic LES runs. LES and LES vert. refer to those anisotropic LES runs with high and low vertical resolutions (LA and LAV), respectively.

Citation: Monthly Weather Review 148, 10; 10.1175/MWR-D-19-0351.1

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Irreversible mixing efficiency γ_i vs the ratio L_o/Δ_h for DNS and anisotropic LES runs. LES and LES vert. refer to those anisotropic LES runs with high and low vertical resolutions (LA and LAV), respectively.

Citation: Monthly Weather Review 148, 10; 10.1175/MWR-D-19-0351.1

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5. Conclusions

A new anisotropic SGS model in LES of stratified turbulence is introduced. The new scheme uses coarse grid spacing in the horizontal direction, and also retains the vertical derivatives of horizontal motions in the eddy dissipation terms, which are omitted in the classic LES approach for horizontal dissipation. Therefore, our new model maintains anisotropy in the resolution, and the connection between the horizontal and vertical motions in the eddy dissipation. The new anisotropic SGS parameterization is tested in LES of decaying stratified turbulence, and the results are compared with those from DNS: the time series of total energy and kinetic energy dissipation rate, vorticity field, horizontal and vertical wavenumber spectra, and mixing efficiency are fairly well reproduced in the new LES scheme similar to those in DNS, while the computational cost is largely decreased in LES.

It has been shown that if we neglect the vertical derivatives of SGS motions in our eddy dissipation terms, our results will be underdissipated at small scales. We think a similar story should exist in atmosphere and ocean models (e.g., Griffies and Hallberg 2000; Griffies et al. 2004; Skamarock et al. 2008), where the vertical derivatives of SGS fluxes are neglected by horizontal mixing schemes. As a result, we hypothesize that the horizontal eddy dissipation parameters may sometimes be artificially increased in atmosphere and ocean models to ensure model convergence since the zonal and meridional SGS eddy fluxes do not include fluxes from vertical motions. This unrealistically enhanced horizontal eddy dissipation can affect the results of atmosphere and ocean models, and may be compensated by adding an energizing term in the form of a stochastic or negative Laplacian backscatter (as in Mana and Zanna 2014; Jansen and Held 2014) to the equations of motion, in order to improve the performance of these models. Our work suggests that if we keep the neglected terms in the horizontal eddy dissipation scheme, the model performance may be improved without adding any additional energizing terms to the zonal and meridional momentum equations. Nevertheless, this suggestion has to be tested in large-scale atmosphere and ocean models since the flow regime in such models, even at the grid scale, is affected by rotating, unlike the stratified turbulence considered here. In addition, as model resolutions continue to increase, gridscale motions in such models will become closer to the stratified turbulence regime, and our findings will become increasingly relevant.

In atmosphere and ocean models, different types of SGS eddy viscosity and diffusivity parameterizations can be used. In addition to the Smagorinsky model, a common SGS model in atmosphere and ocean simulations is the turbulent kinetic energy (TKE) model, where the term $\overline{S}$ in Eqs. (23)–(26) is replaced by $K_{\text{tke}}=c_k{\mathcal{L}}_m{\overline{e}}^{1/2}$. Here, c_k is a constant coefficient, ${\mathcal{L}}_m$ is a mixing length that is usually proportional to the grid spacing, and $\overline{e}$ is the SGS turbulence kinetic energy that is computed by solving a transport equation for the TKE budget (see e.g., Sommeria 1976; Deardorff 1980; Schumann 1991; Kaltenbach et al. 1994). Our anisotropic horizontal-filtering framework can be easily adapted to such models. Furthermore, our approach can also be used in more complicated atmosphere and ocean models where, for example, water vapor or other scalars are considered. For these cases, we would need to perform scale analysis for the SGS fluxes, which are proportional to the gradient of corresponding quantities, and include the vertical derivatives of quantities in the horizontally filtered transport equations. Meanwhile, we should evaluate the performance of our anisotropic SGS parameterization in atmosphere and ocean simulations, which have much lower vertical resolution than what we consider here, with a hierarchy of vertical subgrid schemes. We will consider simulations with and without current vertical SGS parameterizations in atmosphere and ocean models. Also, depending on the ratio of Δ_h/Δ_z, we can develop a new SGS scheme based on a vertical-filtering approach. For future work, we plan to test the performance of our new anisotropic SGS parameterization in atmosphere and ocean models considering these modifications and complexities.