## 1. Introduction

Atmospheric and oceanic general circulation models are key components of global climate models. The full forms of the general circulation models are computationally expensive to solve numerically. Therefore, different approximations are employed to simplify the full models and allow for detailed investigation of some specific effects. The barotropic vorticity equation (BVE) represents the simplest nontrivial model of the atmosphere that describes the evolution of a two-dimensional, nondivergent flow on the surface of a sphere. The BVE contains the nonlinear interactions of atmospheric motions and has been used extensively in the study of large-scale atmospheric dynamics. Charney et al. (1950) performed the first successful numerical weather prediction based on the BVE.

Atmospheric flows have a wide range of time and length scales, which can vary from seconds to decades and from micrometers to several thousand kilometers. Because of limited computational resources, resolving all of these scales numerically is not feasible in numerical simulations. Large-eddy simulation (LES), in which the large-scale motions are resolved explicitly and the effects of small-scale motions are modeled, is a preferred method to solve these kinds of flows.

In LES, a low-pass filter is applied to separate the flow field into large- and small-scale motions. The filtering operation in LES can be implicit, or an explicit filter can be applied in addition to the implicit filter. In implicit filtering the computational grid and discretization schemes are considered to be the low-pass filter that divides the flow field into resolved-scale (RS) and subgrid-scale (SGS) motions. In explicit filtering, the filtering procedure is separate from the grid and discretization operations. The explicit filter width is usually larger than the grid spacing (implicit filter width), so in explicit filtering, the flow field is divided into three portions: RS, resolvable subfilter scale (RSFS), and unresolvable subfilter-scale (USFS) or SGS motions. Resolved scales are scales larger than the explicit filter width, the contributions of which are computed numerically. Resolvable subfilter scales are those with a size between the explicit filter width and the implicit filter width. These scales can be reconstructed theoretically by using an inverse filtering operation. Unresolvable subfilter scales are scales smaller than the grid spacing and are known as subgrid scales, the effects of which are typically modeled using an eddy viscosity model (Zhou et al. 2001). A schematic illustration of the implicit and explicit filtering and RS, RSFS, and SGS structures is shown in Fig. 1. Figure 2 also shows a schematic of the RS, RSFS, and SGS structures similar to that of Zhou et al. (2001) for a typical energy spectrum of a turbulent flow.

Implicit filtering is the most commonly used technique in LES of turbulent flows because it is computationally less expensive and less complicated than explicit filtering. However, implicit filtering is associated with some numerical issues (Lund 2003). Explicit filtering overcomes some of the difficulties associated with implicit filtering and therefore has received increasing attention over the last few years. The accuracy of the explicitly filtered LES results depend on three key factors: the filtering operation, the reconstruction model, and the subgrid-scale model.

The filtering operation is performed by convolving a flow variable with the filter kernel. The most commonly used filter functions in LES of turbulent flows are the sharp cutoff filter, the Gaussian filter, and the top-hat filter. If the filter width is constant, the differentiation and filtering operations commute. For inhomogeneous turbulent flows where the smallest turbulence scales vary with time and space, a filter with a variable width is required. In general, filters with variable widths do not commute with the differentiation operator

The RSFS motions can be recovered by applying a reconstruction model. In physical space, reconstruction models are typically based on series expansion methods. The first reconstruction model was proposed by Leonard (1974), who provided an analytical expression based on Taylor series expansions of the filtering operator to reconstruct the filtered scales due to explicit filtering. The method was then improved by Clark et al. (1979) and is known as the gradient or nonlinear or tensor-diffusivity model. Bardina et al. (1983) presented the scale similarity model, which assumes that the smallest resolved scales are similar to the largest unresolved scales. Thus, the unknown unfiltered quantities can be approximated by the filtered quantities. The velocity estimation model was proposed by Domaradzki and Saiki (1997). In this model, the unfiltered velocity field is estimated by expanding the resolved velocity field to subgrid scales 2 times smaller than the grid scale. The approximate deconvolution model (ADM) of Stolz and Adams (1999) is the most popular method for reconstructing the resolvable subfilter scales. In this model, the unfiltered flow quantities are approximated based on repeated application of an inverse filter to the filtered quantities. In spectral space, RSFS motions can be exactly recovered by convolving the inverse filter kernel with the filtered flow field.

The effect of subgrid-scale motions on the resolved scales is considered by applying a subgrid-scale model. The first subgrid-scale model for two-dimensional turbulence was probably developed by Leith (1971), wherein he derived an eddy viscosity parameterization of the effects of unresolved scales on resolved scales using the eddy-damped Markovian approximation. Leith (1971) applied the model to the simulation of isotropic homogeneous large-scale atmospheric turbulence in Cartesian coordinates in Fourier spectral space and found a good agreement between experimental and numerical results. Kraichnan (1976) used the direct-interaction approximation (DIA) to develop an eddy viscosity subgrid-scale model for two- and three-dimensional turbulence in Fourier spectral space. Unlike the eddy-damped Markovian approximation, the DIA is applied to inhomogeneous and anisotropic flows, as well as homogeneous and isotropic flows. Boer and Shepherd (1983) used the method proposed by Leith (1971) to develop a SGS model for computing large-scale atmospheric flows on a sphere in spectral space. Koshyk and Boer (1995) modeled the nonlinear interactions between the resolved and unresolved scales in the simulation of general circulation models on a sphere in spectral space by using an empirical interaction function (EIF). The EIF function they used in their simulations was obtained based on a high-resolution computation. Frederiksen and Davies (1997) derived eddy viscosity and stochastic backscatter parameterizations for the simulation of a two-dimensional atmospheric circulation model on a sphere in spectral space. They used both eddy-damped quasi-normal Markovian (EDQNM) and DIA closures in spherical geometry to derive some equations for eddy viscosity and stochastic backscatter for the forced-dissipative BVE. Frederiksen and Kepert (2006) then improved the dynamic version of the Frederiksen and Davies’s model by including time–history effects in the stochastic modeling. Gelb and Gleeson (2001) developed a spectral eddy viscosity model for computing shallow-water flows in spherical geometry using spherical harmonics. Their model is based not on physical arguments, but rather on a mathematical approach to the problems exhibited at large wavenumbers in truncated spectral methods.

Explicit filtering in LES of geophysical flows has been studied by few researchers (Chow et al. 2005; San et al. 2011, 2013). Chow et al. (2005) used ADM to reconstruct the RSFS motions and the dynamic Smagorinsky model (Germano et al. 1991) to parameterize the effects of SGS motions in the computations of the atmospheric boundary layer in physical space. They found significant improvements in the accuracy of the results obtained from explicit filtering over the results obtained from implicit filtering. San et al. (2011) and San et al. (2013) used the ADM for recovering the RSFS term in the computations of the one-layer and two-layer wind-driven circulation in a shallow ocean basin in the plane in physical space without applying any SGS models. Results obtained using explicit filtering and ADM showed the correct four-gyre circulation structure predicted by the direct numerical simulation (DNS) results in simulation of the forced BVE while the results obtained from implicit filtering yielded a two-gyre structure, which is not consistent with the DNS results.

This paper aims to study the effects of explicit filtering on LES of two-dimensional atmospheric flows in spherical coordinates in spectral space, because of the high accuracy of spectral methods and because the unfiltered flow fields can be reconstructed exactly without resorting to reconstruction models, such as ADM. We use a spectral method based on spherical harmonic transforms to solve the BVE in spectral space. A differential filter is applied to separate the flow field into resolved and subfilter scales. To reconstruct the unfiltered flow variables, we use exact deconvolution by applying the exact inverse filter to the filtered flow field. The effects of the subgrid-scale term are taken into account by applying a spectral eddy viscosity term.

The organization of the rest of this paper is as follows. The governing equations are presented in section 2. In section 3, the numerical method is discussed. Reconstruction procedure and the LES equations are explained in section 4. The SGS model is presented in section 5. The results and discussion are given in section 6, and conclusions are made in section 7.

## 2. Governing equations

*p*is the order of the hyperviscosity term, and

*J*is the horizontal Jacobian operator on the sphere, which is defined aswhere

*R*is radius of the sphere,

*R*as the length scale,

*U*as the characteristic velocity scale, and

*λ*direction. In the

*μ*direction,

*ζ*should be zero on the poles soand

*A*is a normalization constant,

*γ*is used to control the width of the spectrum. Here,

*γ*are set to 40 and 20, respectively.

## 3. Numerical method

*n*is the total wavenumber,

*m*is the zonal wavenumber,

*ζ*and

*ψ*into Eqs. (1) and (2), the BVE in spectral space is given bywhere

## 4. Explicit filtering and LES equations

### a. Explicit filtering

*u*is a typical flow variable,

*G*is the filter kernel, and the subfilter flow variable

*I*is the identity operator. This nonconvergent Neumann series can be approximated by the van Cittert (van Cittert 1931) equation aswhere,An approximate deconvolution of

*u*can now be obtained by (Stolz and Adams 1999)In spectral methods, the convolution is turned into a multiplication:where

*u*, respectively, and

### b. Explicit filter

### c. LES equations

## 5. Subgrid-scale model

*ϵ*is the spectral eddy viscosity amplitude, and

*ϵ*and

*ϵ*and

*ϵ*obtained from the above relation is too large and makes the model developed here too dissipative. Comparison of the LES results obtained with the above SGS model with the results obtained from DNS showed that the following relations for

*ϵ*and

*ϵ*and

## 6. Results and discussion

We performed four numerical experiments. In one set of experiments, the regular viscosity term is used in the BVE, which corresponds to

In all experiments, the spectral resolution of the DNS computations is

### a. Experiment 1

The first experiment is performed at

Contour plots of absolute vorticity,

The variation of the total kinetic energy with time is presented in Fig. 5. The total kinetic energy is the sum of the kinetic energy of the RS, RSFS, and SGS terms. Figure 5 shows that the total kinetic energy decays and gets to about

Figure 6 shows the temporal variation of the total enstrophy. The total enstrophy is also the sum of the enstrophy of the RS, RSFS, and SGS terms. It can be seen that the results obtained from explicitly filtered LES agree well with the DNS results and converge approximately to the same value as the DNS results while the results obtained from implicitly filtered LES converge to a final value, which is a little smaller than that of the DNS results.

The variation of the energy spectrum with wavenumber at early time

### b. Experiment 2

Experiment 2 is performed with the same regular viscosity term as experiment 1 at Ro = 5. The general evolution of coherent structures with time for this case is similar to that in Fig. 3. The size of coherent structures increases with time, and, after some time, all vortices merge to only four vortices, similar to the DNS results in Fig. 9. The vortices move in the flow, but the number of vortices does not change with time.

The absolute vorticity and streamfunction fields are shown in Fig. 9. Long-time integration of the BVE at high Rossby numbers produces a vortical quadrupole state (Cho and Polvani (1996)), which can be seen in Fig. 9. It can be seen that the results obtained from the explicitly filtered LES are a nearly perfect match with the DNS results, while the implicitly filtered LES results show incorrect vortical structures.

The temporal variation of the total kinetic energy for this case is shown in Fig. 10. The results obtained from implicitly and explicitly filtered LES both agree very well with the DNS results. However, the oscillatory behavior of the total kinetic energy, which is caused by the interaction of the turbulence and jets, is different for DNS and implicitly and explicitly filtered LES. At initial times, the phase difference between DNS and implicitly and explicitly filtered LES is small, but the difference increases with time.

The variation of the total enstrophy with time is shown in Fig. 11. As in Fig. 10, the results obtained from the implicitly and explicitly filtered LES both show good agreement with the DNS results.

Figures 12 and 13 show the decay of the energy spectrum with wavenumber at

### c. Experiment 3

*p*= 4, to the BVE. The hyperviscosity coefficient

The evolution of coherent structures with time is again similar to Fig. 3. The size of coherent structures increases up to a given time and then does not change any more.

Contour plots of absolute vorticity and streamfunction for DNS, implicitly filtered LES, and explicitly filtered LES are shown in Fig. 14 As in Fig. 4a, the dominance of the planetary vorticity is apparent in Fig. 14a. In this case, both implicitly and explicitly filtered LES results show good agreement with the streamfunction field obtained from the DNS results. The emergence of jets is apparent in these plots.

The variation of the total kinetic energy with time is shown in Fig. 15. It can be seen that the total kinetic energy obtained from DNS increases from the initial value of 1 and reaches a nearly constant value. The initial increase of the total kinetic energy can be attributed to the mean shear caused by velocity gradients. At initial times, the mean shear exceeds the dissipation term and increases the kinetic energy, but at larger times, the dissipation term overcomes the mean shear and keeps the kinetic energy in a constant level. Figure 15 shows that although both implicitly and explicitly filtered LES predict lower levels for the total kinetic energy, explicitly filtered LES results are closer to the DNS results.

The better performance of explicitly filtered LES in computation of the total enstrophy is shown in Fig. 16. Both the implicitly and explicitly filtered LES results converge to smaller values of the total enstrophy, but the value for the explicitly filtered LES is closer to that for the DNS results.

Figures 17 and 18 show the decay of the energy spectrum with wavenumber at early

### d. Experiment 4

Experiment 4 is performed by applying the same hyperviscosity term of experiment 3 but at

The evolution of coherent structures with time is again similar to experiment 2. The size of coherent structures increases with time and, after some time, a vortical quadrupole state forms and the number of vortices does not change with time.

The absolute vorticity and streamfunction fields are shown in Fig. 19. The quadrupole state is clear from the streamfunction plots; however, it seems more time is needed to see this state in absolute vorticity plots. The superior performance of the explicitly filtered LES results is obvious from these plots. The implicitly filtered LES results can neither predict the location of the coherent structures, nor the exact number of coherent structures.

The temporal variation of the total kinetic energy is shown in Fig. 20. It can be seen that the explicitly filtered LES is better in predicting the total kinetic energy. The oscillations of the total kinetic energy plots for DNS and implicitly and explicitly filtered LES are random, and there is no clear pattern in the phase differences of these plots.

The variation of the total enstrophy with time is shown in Fig. 21. Although the explicitly filtered LES results are closer to the DNS results, both the implicitly and explicitly filtered LES results are in poor agreement with the DNS results.

The energy spectrum at

Comparison of the results obtained in each experiment shows that at small Rossby numbers the total kinetic energy plot is more oscillatory than large Rossby numbers. This behavior can be attributed to the presence of jets at small Rossby numbers. It can also be seen that in the presence of the regular dissipation term, the total kinetic energy decays, while in the presence of the hyperviscosity term the total kinetic energy increases at initial times and reaches a nearly constant value. The hyperviscosity term keeps the level of the total energy constant and increases the inertial range. Therefore, the dissipative scales move to smaller scales. This behavior can be seen by comparing the energy spectrum plots of experiments 1 and 2 with the energy spectrum plots of experiments 3 and 4.

Although in the computation of some quantities the results obtained from the implicitly filtered LES appear to be as accurate as the results obtained from the explicitly filtered LES, in other cases explicitly filtered LES performs better than implicitly filtered LES. In all comparisons, the explicitly filtered LES results show a better agreement with the DNS results in capturing the behavior of the coherent structures.

## 7. Conclusions

We performed large-eddy simulation (LES) of the turbulent barotropic vorticity equation (BVE) on a sphere in spectral space. Both implicitly and explicitly filtered LES were investigated, and results obtained from implicit filtering were compared with the results obtained from explicit filtering. In explicitly filtered LES, the differential filter was used to separate the large and small scales, and exact deconvolution was applied to reconstruct the unfiltered flow variables from the filtered flow variables. A spectral eddy viscosity model was used to parameterize the effects of the subgrid scales on the resolved scales in both the implicit and explicit filtering computations when regular viscosity was applied while, for the experiments with a hyperviscosity term, the hyperviscosity term itself acted as the subgrid-scale model. We performed different numerical experiments for two different orders of the hyperviscosity term, each at

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