Abstract

Numerical simulations of atmospheric circulation models are limited by their finite spatial resolution, so large-eddy simulation (LES) is a preferred approach to study these models. In LES, a low-pass filter is applied to the flow field to separate the large- and small-scale motions. In implicitly filtered LES, the computational mesh and discretization schemes are considered to be the low-pass filter, while in the explicitly filtered LES approach, the filtering procedure is separated from the grid and discretization operators and allows for better control of the numerical errors. The aim of this paper is to study and compare implicitly filtered and explicitly filtered LES of atmospheric circulation models in spectral space. To achieve this goal, the results of implicitly filtered and explicitly filtered LES of a barotropic atmosphere circulation model on a sphere in spectral space are presented and compared with the results obtained from direct numerical simulation (DNS). Different numerical experiments are performed to investigate the efficiency of explicit filtering over implicit filtering under different dissipation terms and rotation rates. The study shows that explicit filtering increases the accuracy of the computations and improves the results, particularly where the location of coherent structures is concerned, a topic of particular importance in LES of atmospheric flows for climate and weather applications.

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.

Fig. 1.

Comparison of implicit and explicit filtering and description of different flow structures: (a) implicit filtering and (b) explicit filtering.

Fig. 1.

Comparison of implicit and explicit filtering and description of different flow structures: (a) implicit filtering and (b) explicit filtering.

Fig. 2.

A schematic illustration of the RS, RSFS and SGS.

Fig. 2.

A schematic illustration of the RS, RSFS and SGS.

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 . Therefore, an extra term , which is called commutation error, needs to be added to the LES equations and modeled in addition to the subgrid-scale term. Vasilyev et al. (1998) have developed a general theory for constructing continuous and discrete filters that commute with the differentiation up to any desired order in the filter width.

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

The BVE describes the motion of a two-dimensional, nondivergent, incompressible fluid on a rotating sphere and is given by

 
formula
 
formula

where is the vertical component of the vorticity, is the streamfunction, is the Coriolis parameter, is the hyperviscosity coefficient, p is the order of the hyperviscosity term, and J is the horizontal Jacobian operator on the sphere, which is defined as

 
formula

where R is radius of the sphere, and are the longitude and latitude, , and Ω is the rotation rate of the sphere.

We nondimensionalize Eqs. (1) and (2) by using the radius of the sphere R as the length scale, U as the characteristic velocity scale, and as the advection time scale. Nondimensionalizing Eq. (1) introduces the Rossby number, an important physical parameter in a rotating system, which is defined as

 
formula

and is a measure of the ratio of inertial forces to Coriolis forces. The Reynolds number also appears as the coefficient of the regular viscosity term when . In addition to the Rossby and Reynolds numbers, the Rhines scale is another important parameter that is associated with the cascade arrest (Rhines 1975). When the flow structures become larger than the Rhines scale, Rossby waves dominate turbulent motions and arrest the inverse energy cascade. The Rhines wavenumber is defined as

 
formula

The BVE is solved under periodic boundary conditions in the λ direction. In the μ direction, ζ should be zero on the poles so

 
formula

and

 
formula

The initial conditions we use are based on the following initial energy spectrum (Cho and Polvani 1996):

 
formula

where A is a normalization constant, is the peak wavenumber of the energy spectrum, and γ is used to control the width of the spectrum. Here, and γ are set to 40 and 20, respectively.

3. Numerical method

The nondimensionalized forms of Eqs. (1) and (2) are solved in spectral space by expanding each variable into a series of spherical harmonics as

 
formula

where are the complex spectral coefficients of , n is the total wavenumber, m is the zonal wavenumber, denotes the truncation wavenumber, and are spherical harmonics defined by

 
formula

where are the normalized associated Legendre polynomials and . Equation (7) is truncated using triangular truncation, which, unlike rhomboidal truncation, is rotationally symmetric. After nondimensionalizing and substituting the spectral representatives of ζ and ψ into Eqs. (1) and (2), the BVE in spectral space is given by

 
formula
 
formula

where is the nondimensional hyperviscosity coefficient, , and is the nonlinear Jacobian term, which is computed using the pseudospectral method.

The energy and enstrophy spectra are defined as

 
formula
 
formula

and the total kinetic energy and enstrophy are given by

 
formula
 
formula

4. Explicit filtering and LES equations

a. Explicit filtering

In LES, a spatial filter is applied to the fluid field to separate flow motions into large and small scales. The filtering operation is mathematically represented as a convolution product:

 
formula

where u is a typical flow variable, is the filtered variable, G is the filter kernel, and the subfilter flow variable is defined by

 
formula

Deconvolution is a process for obtaining the unfiltered variable by applying an inverse filter:

 
formula

In Eq. (17), is the inverse filter and can be defined by the Neumann series:

 
formula

where I is the identity operator. This nonconvergent Neumann series can be approximated by the van Cittert (van Cittert 1931) equation as

 
formula

where,

 
formula

An approximate deconvolution of u can now be obtained by (Stolz and Adams 1999)

 
formula

In spectral methods, the convolution is turned into a multiplication:

 
formula

where and are spectral representatives of and u, respectively, and is the filter kernel in spectral space. The unfiltered flow fields can be exactly reconstructed by applying the exact inverse filter to the filtered fields, so the exact deconvolution of filtered flow variables is given by

 
formula

b. Explicit filter

We use the differential filter as an explicit filter, which in physical space is defined by

 
formula

where is related to the filter width. The differential filters were first proposed and their properties were investigated by Germano (1986a,b) for LES of turbulent flows. Bose (2012) applied the differential filters to LES of turbulent flows on structured grids. Here, we use a differential filter with a constant width, so Eq. (24) becomes

 
formula

Applying the definition of the Laplacian in spectral space

 
formula

and substituting this into Eq. (25), the differential filter kernel in spectral space is defined by

 
formula

We use in our computations. The mesh size in physical space is not constant. The mesh is large on the equator and smaller on the poles. However, the width of the differential filter is constant and consistent with the smallest mesh size.

c. LES equations

Applying the implicit filter due to the discretization scheme, along with the explicit filter to Eqs. (9) and (10), the governing equations of the LES are obtained as follows:

 
formula
 
formula

where the tilde indicates implicit filtering, and the bar indicates explicit filtering. The quantity is the subfilter-scale term and is defined as

 
formula

where

 
formula

is the SGS term to be addressed in section 5, and

 
formula

is the RSFS term.

Considering the effect of implicit filtering, the exact deconvolution of filtered flow variables is given by

 
formula
 
formula

5. Subgrid-scale model

SGS models have an important impact on the accuracy of the LES results. Following the work done by Gelb and Gleeson (2001) on shallow-water computations, we use the following equation to model the SGS term:

 
formula

where

 
formula

ϵ is the spectral eddy viscosity amplitude, and is the cutoff wavenumber. The values of ϵ and depend on the truncation wavenumber . To overcome the failure of convergence, Maday et al. (1993) suggested the following dependencies of ϵ and on :

 
formula

We found that the 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 can produce LES results that agree better with the DNS results in our computations:

 
formula

All implicitly and explicitly filtered LES results are obtained based on the above chosen values of 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 Eq. (9); in the other set, an eighth-order hyperviscosity term is applied, which corresponds to in Eq. (9). Each set of experiments is performed at two Rossby numbers: a small Rossby number, where the rotation rate of the sphere is large and turbulence is weak compared to jets, and at a large Rossby number, when the rotation rate is small and turbulence is dominant. To demonstrate the efficiency of the LES method (both implicitly filtered and explicitly filtered), the nonlinear term represented by the Jacobian in Eq. (1) should be comparable to the linear terms. At large Rossby numbers, the nonlinear term is larger than the linear term due to the rotation and turbulence dominates flow motions. However, at small Rossby numbers, the rotation term is greater than the nonlinear term, and jets dominate the flow behavior. Here, we perform computations at and . At , the nonlinear term and the rotation term are comparable, and both turbulence and jets are present in the flow. At , turbulence is the dominant term; however, jets are also present, and their interaction with turbulence causes oscillations of some integral quantities, such as total kinetic energy, that can be seen in the following figures.

In all experiments, the spectral resolution of the DNS computations is . In physical space, 1000 (longitude) × 500 (latitude) grid points are used to compute nonlinear terms in accordance with the dealiasing rule (Canuto et al. 2006). The resolution in the implicitly and explicitly filtered LES runs is or 200 (longitude) × 100 (latitude). In this paper, we refer to the computation with high resolution in which the coherent structures are properly resolved as DNS; it does not signify direct numerical simulation, in which all the turbulent structures are resolved numerically. The fourth-order Runge–Kutta scheme is used for the time integration of the governing equations.

a. Experiment 1

The first experiment is performed at and with the regular viscosity term, , and . The evolution of flow structures with time is shown in Fig. 3. This figure shows the variation of the streamfunction field with time. It can be seen that following the nature of decaying two-dimensional turbulence, coherent structures are smaller at initial times and get larger with time. The change in size of flow structures can be attributed to the vortex merging mechanism. In the vortex merging phenomenon, two neighboring eddies with the same rotational sense merge to form a single larger eddy. After a certain time, the size of coherent structures no longer changes with time. The structures only move within the flow and change their location with time.

Fig. 3.

Evolution of the streamfunction field with time for DNS of experiment 1 ( and ).

Fig. 3.

Evolution of the streamfunction field with time for DNS of experiment 1 ( and ).

Contour plots of absolute vorticity, , and streamfunction for the DNS, explicitly filtered LES, and implicitly filtered LES are shown in Fig. 4. These plots show flow structures at . At small Rossby numbers, the planetary vorticity is the dominant term in the definition of absolute vorticity. Therefore, the absolute vorticity plots represent mainly the planetary vorticity, not the actual variation of flow structures. The streamfunction plots are more informative in this case and show that the explicitly filtered LES results are better in prediction of coherent structures.

Fig. 4.

(a) The absolute vorticity field and (b) the streamfunction field for (left to right) the DNS, explicitly filtered LES (EFLES), and implicitly filtered LES (IFLES) at for experiment 1 (, ).

Fig. 4.

(a) The absolute vorticity field and (b) the streamfunction field for (left to right) the DNS, explicitly filtered LES (EFLES), and implicitly filtered LES (IFLES) at for experiment 1 (, ).

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 of the initial value at in the DNS run. The rate of the decay of the total kinetic energy is greater for the implicitly and explicitly filtered LES runs. However, the total kinetic energy obtained from explicitly filtered LES decays slower and is in better agreement with the DNS results.

Fig. 5.

Comparison of the total kinetic energy for the DNS, EFLES, and IFLES for experiment 1 (, ).

Fig. 5.

Comparison of the total kinetic energy for the DNS, EFLES, and IFLES for experiment 1 (, ).

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.

Fig. 6.

Decay of the total enstrophy for the DNS, EFLES, and IFLES for experiment 1 (, ).

Fig. 6.

Decay of the total enstrophy for the DNS, EFLES, and IFLES for experiment 1 (, ).

The variation of the energy spectrum with wavenumber at early time and at later is shown in Figs. 7 and 8, respectively. Figure 7 shows that at initial times both implicitly and explicitly filtered LES results are too dissipative. However, the explicitly filtered LES results are in better agreement with the DNS results at high wavenumbers. At larger times, as can be seen in Fig. 8, implicitly filtered LES results are still too dissipative, while explicitly filtered LES results become closer to the DNS results and overlap the DNS results at large wavenumbers. Comparison of Figs. 7 and 8 shows the evolution of the energy spectrum with time. The energy spectrum decays with time resulting in smaller values with time. Also, the dissipative scales move to larger scales with time.

Fig. 7.

Variation of the energy spectrum with wavenumber for the DNS, EFLES, and IFLES for experiment 1 at (, ).

Fig. 7.

Variation of the energy spectrum with wavenumber for the DNS, EFLES, and IFLES for experiment 1 at (, ).

Fig. 8.

Variation of the energy spectrum with wavenumber for the DNS, EFLES, and IFLES for experiment 1 at (, ).

Fig. 8.

Variation of the energy spectrum with wavenumber for the DNS, EFLES, and IFLES for experiment 1 at (, ).

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.

Fig. 9.

As in Fig. 4, but for experiment 2 ().

Fig. 9.

As in Fig. 4, but for experiment 2 ().

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.

Fig. 10.

As in Fig. 5, but for experiment 2 ().

Fig. 10.

As in Fig. 5, but for experiment 2 ().

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.

Fig. 11.

As in Fig. 6, but for experiment 2 ().

Fig. 11.

As in Fig. 6, but for experiment 2 ().

Figures 12 and 13 show the decay of the energy spectrum with wavenumber at and . It can be seen that at both the implicitly and explicitly filtered LES results are too dissipative and predict low values for the energy spectra, while at the energy spectra predicted by the implicitly and explicitly filtered LES results are higher than the DNS energy spectrum at large wavenumbers. However, at both and , the explicitly filtered LES results are in better agreement with the DNS results.

Fig. 12.

As in Fig. 7, but for experiment 2 ().

Fig. 12.

As in Fig. 7, but for experiment 2 ().

Fig. 13.

As in Fig. 12, but for .

Fig. 13.

As in Fig. 12, but for .

c. Experiment 3

Experiments 3 and 4 are performed by applying an eighth-order hyperviscosity term, p = 4, to the BVE. The hyperviscosity coefficient is obtained approximately by the following equation, presented by Cho and Polvani (1996):

 
formula

By using this relation, the value of the hyperviscosity coefficient for DNS runs () in experiments 3 and 4 is set to be . The hyperviscosity terms play the role of the SGS models. Therefore, for the computations with a hyperviscosity term, there is no need to add an SGS model. An approximate value of the hyperviscosity coefficient for the implicitly filtered LES runs is obtained by Eq. (39), and then the exact value is empirically obtained in order to get good agreements with the DNS results. The hyperviscosity coefficient for the explicitly filtered LES runs is the same as that for the implicitly filtered LES. Experiment 3 is performed at , , and the hyperviscosity coefficient for implicitly and explicitly filtered LES is set to be .

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.

Fig. 14.

As in Fig. 4, but for experiment 3.

Fig. 14.

As in Fig. 4, but for experiment 3.

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.

Fig. 15.

As in Fig. 5, but for experiment 3.

Fig. 15.

As in Fig. 5, but for experiment 3.

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.

Fig. 16.

As in Fig. 6, but for experiment 3.

Fig. 16.

As in Fig. 6, but for experiment 3.

Figures 17 and 18 show the decay of the energy spectrum with wavenumber at early and late , respectively. Figure 17 shows that at , explicitly filtered LES results are in better agreement with the DNS results at large wavenumbers. At , both the implicitly and explicitly filtered LES results are on top of the DNS results at high wavenumbers, but at intermediate wavenumbers the explicitly filtered LES results are closer to the DNS results.

Fig. 17.

As in Fig. 7, but for experiment 3.

Fig. 17.

As in Fig. 7, but for experiment 3.

Fig. 18.

As in Fig. 8, but for experiment 3.

Fig. 18.

As in Fig. 8, but for experiment 3.

d. Experiment 4

Experiment 4 is performed by applying the same hyperviscosity term of experiment 3 but at . The hyperviscosity coefficient for the implicitly and explicitly filtered LES for this experiment is set to .

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.

Fig. 19.

As in Fig. 4, but for experiment 4 ().

Fig. 19.

As in Fig. 4, but for experiment 4 ().

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.

Fig. 20.

As in Fig. 5, but for experiment 4 ().

Fig. 20.

As in Fig. 5, but for experiment 4 ().

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.

Fig. 21.

As in Fig. 6, but for experiment 4 ().

Fig. 21.

As in Fig. 6, but for experiment 4 ().

The energy spectrum at and is shown in Figs. 22 and 23, respectively. It can be seen that at the implicitly and explicitly filtered LES results show the same behavior, but at the explicitly filtered LES results perform better at intermediate wavenumbers.

Fig. 22.

As in Fig. 7, but for experiment 4 ().

Fig. 22.

As in Fig. 7, but for experiment 4 ().

Fig. 23.

As in Fig. 8, but for experiment 4 ().

Fig. 23.

As in Fig. 8, but for experiment 4 ().

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 and . At , the nonlinear term in the BVE is comparable to the rotation term, and both turbulence and jets are present in the flow. At , turbulence controls the flow behavior while jets also exist and interact with turbulence. Comparison of the results obtained for the total kinetic energy, total enstrophy, and energy spectrum showed that, although for the experiments with regular viscosity implicit filtering and explicit filtering give similarly accurate results, for the experiments with hyperviscosity the explicitly filtered LES results agree better with the results obtained from direct numerical simulation (DNS). The superior performance of explicit filtering over implicit filtering was seen in the contour plots of the vorticity and streamfunction in all experiments. While the explicitly filtered LES results predicted the exact locations of the coherent structures, implicit filtering was not able to capture the correct behavior of the coherent structures and even predicted the wrong number of coherent structures in some cases. Coherent structures play an important role in computations of atmospheric flows and need to be captured correctly for accurate numerical weather prediction and climate modeling. The results obtained in this paper show that applying an explicit filter in addition to an implicit filter in LES of a turbulent barotropic flow in spectral space can improve the results and accurately predict the behavior of the coherent structures in the flow.

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