Using a Scale-Selective Filter for Dynamical Downscaling with the Conformal Cubic Atmospheric Model

Marcus Thatcher CSIRO Marine and Atmospheric Research, Aspendale, Victoria, Australia

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John L. McGregor CSIRO Marine and Atmospheric Research, Aspendale, Victoria, Australia

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Abstract

This article examines dynamical downscaling with a scale-selective filter in the Conformal Cubic Atmospheric Model (CCAM). In this study, 1D and 2D scale-selective filters have been implemented using a convolution-based scheme, since a convolution can be readily evaluated in terms of CCAM’s native conformal cubic coordinates. The downscaling accuracy of 1D and 2D scale-selective filters is evaluated after downscaling NCEP Global Forecast System analyses for 2006 from 200-km resolution to 60-km resolution over Australia. The 1D scale-selective filter scheme was found to downscale the analyses with similar accuracy to a 2D filter but required significantly fewer computations. The 1D and 2D scale-selective filters were also found to downscale the analyses more accurately than a far-field nudging scheme (i.e., analogous to a boundary-value nudging approach). It is concluded that when the model is required to reproduce the host model behavior above a specified length scale then the use of an appropriately designed 1D scale-selective filter can be a computationally efficient approach to dynamical downscaling for models having a cube-based geometry.

Corresponding author address: Marcus Thatcher, CSIRO Marine and Atmospheric Research, 107 Station St., Aspendale, VIC 3195, Australia. Email: marcus.thatcher@csiro.au

Abstract

This article examines dynamical downscaling with a scale-selective filter in the Conformal Cubic Atmospheric Model (CCAM). In this study, 1D and 2D scale-selective filters have been implemented using a convolution-based scheme, since a convolution can be readily evaluated in terms of CCAM’s native conformal cubic coordinates. The downscaling accuracy of 1D and 2D scale-selective filters is evaluated after downscaling NCEP Global Forecast System analyses for 2006 from 200-km resolution to 60-km resolution over Australia. The 1D scale-selective filter scheme was found to downscale the analyses with similar accuracy to a 2D filter but required significantly fewer computations. The 1D and 2D scale-selective filters were also found to downscale the analyses more accurately than a far-field nudging scheme (i.e., analogous to a boundary-value nudging approach). It is concluded that when the model is required to reproduce the host model behavior above a specified length scale then the use of an appropriately designed 1D scale-selective filter can be a computationally efficient approach to dynamical downscaling for models having a cube-based geometry.

Corresponding author address: Marcus Thatcher, CSIRO Marine and Atmospheric Research, 107 Station St., Aspendale, VIC 3195, Australia. Email: marcus.thatcher@csiro.au

1. Introduction

Regional atmospheric models are widely used for dynamical downscaling from global analyses or simulations to finer-resolution regional length scales (e.g., Wang et al. 2004) to take into account the influence of local orography, land use, and appropriately parameterized physics. Regional downscaling models also need to account for the influence of atmospheric conditions outside their regional focus. For this reason, various nudging or boundary nesting techniques have been developed to efficiently assimilate the relevant large-scale information from the coarse-resolution analysis or simulation to the regional atmospheric model.

Most of the approaches in the literature for atmospheric nudging can be broadly grouped under either boundary condition (BC) nudging methods (e.g., Davies 1976; Wang et al. 2004), or scale-selective filter methods (e.g., Kida et al. 1991; Waldron et al. 1996; von Storch et al. 2000; Denis et al. 2002; Misra 2007). The BC nudging methods allow the interior of the regional model to evolve freely while lateral boundary conditions are provided by the host model. In this way, BC nudging represents dynamical downscaling as a lateral boundary value problem. An attraction of BC nudging is that it only requires O(N) computations for N2 grid points. However, BC nudging schemes require special treatment near the simulation boundaries to avoid reflections (Davies 1976), and the results of the regional downscaling can depend on the size and position of the regional model domain (Jones et al. 1995).

Scale-selective downscaling (also known as spectral nudging) replaces lateral boundaries with a cutoff in the spectral domain at a particular length scale. In this way, the state of the regional atmosphere at large length scales is specified by a host model while the atmosphere at small length scales is allowed to evolve freely. As a consequence, the regional downscaling results are, in principle, independent of the domain size. Scale-selective downscaling is sometimes referred to as poor-man’s data assimilation (von Storch et al. 2000; Kanamaru and Kanamitsu 2007). A disadvantage of scale-selective downscaling methods is that they are usually more computationally intensive than BC nudging methods, typically O(N2 log2N) computations for schemes based on fast Fourier transforms (FFTs).

In this paper we propose using 1D scale-selective filters for dynamical downscaling in the Commonwealth Scientific and Industrial Research Organisation (CSIRO) Conformal Cubic Atmospheric Model (CCAM). In contrast to limited-area models, CCAM can simulate the regional atmosphere using a stretched conformal cubic grid where the grid is focused on the region of interest (McGregor 2005). Since the stretched conformal cubic grid has no lateral boundaries, we can implement scale-selective downscaling in CCAM without the need for any special treatment of simulation boundaries. Using CCAM, we show that a 2D scale-selective filter can be successfully approximated by a sequence of 1D filters when downscaling for a regional area. As a consequence, the number of required computations is substantially reduced. The approach is illustrated using CCAM to downscale from National Centers for Environmental Prediction Global Forecast System (NCEP GFS) analyses over Australia, although the downscaling technique can also be applied to regional climate change projections or numerical weather prediction (NWP) simulations (e.g., for downscaling ensembles). To ensure that the results of this paper are applicable to a wide range of applications, we concentrate our analysis on the development of the 1D and 2D scale-selective filters and their properties. Recent examples of CCAM’s climate and weather modeling capabilities can be obtained from Nunez and McGregor (2007) and Thatcher and McGregor (2007, 2008).

For convenience we have chosen to implement scale-selective downscaling in CCAM using a convolution rather than using a spherical harmonic FFT. This is because a convolution can be expressed in terms of CCAM’s native conformal cubic coordinates, making it easier to describe and implement. Two-dimensional convolutions are usually avoided in numerical schemes since they are relatively computationally expensive because they require O(N4) computations, where N2 is the number of points. This can be compared with spherical harmonic FFTs that in the past required O(N3) computations, although more recently spherical harmonic FFTs have been devised that take O(N2 log2N) computations (e.g., Mohlenkamp 1999). Nevertheless, the use of a 1D scale-selective filter reduces the number of computations from O(N4) to O(N3) and therefore mitigates the computational speed disadvantage of the convolution approach.

In section 2 we briefly review the CCAM model, describing how its stretched conformal cubic grid is used for regional atmospheric modeling. Section 3 describes how 1D scale-selective filters can be designed to approximate a 2D filter for dynamical downscaling. The performance of scale-selective filters is compared in section 4 for the case of dynamical downscaling from 200-km-resolution analyses to 60-km resolution over Australia. The results of this paper are summarized in section 5.

2. The Conformal Cubic Atmospheric Model

The Conformal Cubic Atmospheric Model is a global atmospheric model principally developed for atmospheric climate modeling and regional climate downscaling (McGregor 2005; McGregor and Dix 2008). CCAM employs a conformal cubic grid to model the atmosphere without requiring lateral boundaries (see Fig. 1). The relationship between conformal cubic coordinates (A, B, C) on the surface of a unit cube and Cartesian coordinates (X′, Y′, Z′) on the surface of a sphere is described by
i1520-0493-137-6-1742-e1
where A, B, and C each vary between −1 and 1 and R is the radius of the sphere (e.g., the radius of the earth). The six cubic panels used by CCAM to describe the surface of a sphere are then indicated by (A, B, ±1), (A, ±1, C), and (±1, B, C). In CCAM, the degree of grid stretching is controlled by the Schmidt (1977) transformation. The stretched (X, Y, Z) Cartesian grid coordinates are related to the unstretched coordinates (X′, Y′, Z′) according to
i1520-0493-137-6-1742-e2
i1520-0493-137-6-1742-e3
i1520-0493-137-6-1742-e4
where S is the Schmidt factor. A Schmidt factor of greater than 1 indicates a focusing of the grid points, whereas a Schmidt factor of less than 1 describes a dilution of grid points. Since for S > 1 the grid is focused at the geometric North Pole, that is, (X, Y, Z) = (0, 0, R), the coordinate system is typically rotated so that the grid is focused at the center of the region of interest. The map factor, m(A, B, C), in the primitive equations is related to the Jacobian as described by Rancic et al. (1996), although in CCAM the map factors are numerically calculated according to a scheme described by McGregor (2005).

CCAM is a hydrostatic atmospheric model with two-time-level semi-implicit time differencing. It uses semi-Lagrangian advection associated with bicubic horizontal interpolation and total-variation-diminishing vertical advection. In a typical case, CCAM runs with 18 vertical levels and employs a C48 grid on which there are 48 × 48 grid points for each of the six cubic panels (i.e., N2 = 6 × 48 × 48 grid points in the horizontal). A detailed description of the CCAM model dynamics can be obtained from McGregor (2005).

3. Scale-selective downscaling for a conformal cubic grid

Scale-selective downscaling schemes for various models have been previously described by several authors, including Kida et al. (1991), Waldron et al. (1996), von Storch et al. (2000), Denis et al. (2002), and Misra (2007). The basic principle is to perturb the atmospheric model fields (i.e., wind components, temperature, surface pressure, and moisture) at large length scales by the host model. For example,
i1520-0493-137-6-1742-e5
where x is a vector indicating the physical position of grid points corresponding to the atmosphere at some model level. In this paper we use pressure-sigma levels σ for the vertical coordinate; η(σ) is a weighting factor between 0 and 1 (see, e.g., von Storch et al. 2000). Here Ψold(x) is an intermediate scalar atmospheric field produced by the model dynamical calculations, Ψnew(x) is a perturbed scalar atmospheric field, and Ψhost(x) is the host-model scalar atmospheric field. Here, W(x) is a low-bandpass filter that is designed to ensure that only large length scales are perturbed by the host model. In this article, we have chosen W(x) to have a Gaussian form, as defined by
i1520-0493-137-6-1742-e6
where ϵ is a constant that determines the length scale of the filter, with a larger value of ϵ indicating perturbing at a smaller length scale. We note that a Gaussian low-bandpass filter such as (6) has been used in the literature by many authors for scale-separation of atmospheric fields, with an important example being the Barnes (1964) objective analysis filter. The asterisk denotes a convolution which is defined in the usual way; that is,
i1520-0493-137-6-1742-e7
where S indicates that the integral is performed over the surface of a sphere and dSp is the area element associated with the position vector p. Last, for this paper, we define the vertical weighting factor as
i1520-0493-137-6-1742-e8
so that the atmosphere in the boundary layer evolves freely, whereas the atmosphere above the boundary layer is strongly perturbed by the host model at large length scales, although applied only once every 6 h and using a large filter length scale in our examples (see section 4).
Equation (5) differs from the usual approach to scale-selective downscaling in that it is based on a convolution, instead of a Fourier transform. Of course the results obtained using (5) for continuous functions are equivalent to those obtained using a Fourier transform–based scheme, because of the well-known convolution theorem (e.g., Kreyszig 1993)
i1520-0493-137-6-1742-e9
where F {} denotes a Fourier transform; that is,
i1520-0493-137-6-1742-e10
where k is a vector describing the wavenumber coordinate. Equation (5) can then be equivalently written as
i1520-0493-137-6-1742-e11
where ω(k) is defined as
i1520-0493-137-6-1742-e12
In this way, the scale-selective filter described in (5) using a convolution is the same as using a Fourier transform if we neglect differences introduced by the numerical discretization (discussed at the end of this section).
For CCAM we have chosen to use a convolution approach (5) instead of employing a Fourier transform (11), because the convolution can be readily expressed in terms of CCAM’s conformal cubic coordinates (A, B, C). Since x describes the Cartesian coordinate on the surface of a sphere with radius R, the great circle distance between two points x(A, B, C) and x(a, b, c) can be calculated as
i1520-0493-137-6-1742-e13
As a consequence, we have replaced the low-bandpass filter W(x) in (6) by
i1520-0493-137-6-1742-e14
Instead of including a constant normalization factor for W ′(x) we normalize the convolution by replacing (5) with
i1520-0493-137-6-1742-e15
The advantage with (15) is that the low-bandpass filter is still correctly normalized when applied to a discrete, conformal cubic grid as is used by CCAM. Using the revised definition for the low-bandpass filter W ′(x), the 2D convolution (7) can then be defined in terms of conformal cubic coordinates for x(A, B, C) as
i1520-0493-137-6-1742-e16
where a, b, and c are integrated along the conformal cubic coordinates and m(a, b, c) is the map factor. Note that (16) has been separated into the six contributions corresponding to each cubic panel. The normalization factors 1 * W′(x) are calculated by replacing Ψ(x) with 1 in (16).

An unfortunate drawback with the 2D convolution described in (16) is that it is relatively computationally expensive, requiring O(N4) computations for N2 grid points. Nevertheless, it is possible to improve the computational efficiency of the scale-selective filter by approximating the 2D convolution with a sequence of 1D convolutions that only require O(N3) computations. Such a scheme was conceptually proposed for a conformal cubic grid by McGregor (2006), in which a 1D convolution is performed around the A cubic axis, then around the B axis, and finally around the C axis. However, a drawback with this approach is that the three 1D convolutions cannot always act orthogonally to each other since the three 1D convolutions are performed on a two-dimensional surface (i.e., an S2 spherical surface embedded in an R3 space). As a consequence of this redundancy, grid points can make multiple contributions to the filter and therefore have a disproportionate influence on the result near the vertices.

In this paper we present an alternative 1D convolution scheme in which the low-bandpass filter W ′(x) is approximated by a function that is separable with respect to the conformal cubic coordinates. This approximation and the associated error terms are written as
i1520-0493-137-6-1742-e17
where the leading error terms have been derived from a Taylor expansion of the filter with respect to Aa, Bb, and Cc. In section 4 we find that these error terms do not lead to any significant degradation in the performance of the 1D filter when the filter length scale is approximately the size of the fine-resolution panel.
To derive a set of 1D convolutions, we substitute (17) into the 2D convolution (16) and neglect error terms. To explain the resultant set of 1D filters, consider the first term of the 2D filter for the B = −1 panel in (16), which in the following discussion we label as Ψ′1(A, B, C). After substituting (17) we obtain
i1520-0493-137-6-1742-e18
giving
i1520-0493-137-6-1742-e19
where Ψ(a, C) is defined as
i1520-0493-137-6-1742-e20
Since Ψ′1(A, B, C) is only defined on the surface of the sphere, we substitute A = ±1, B = ±1, and C = ±1 into (19) to obtain
i1520-0493-137-6-1742-e21
i1520-0493-137-6-1742-e22
i1520-0493-137-6-1742-e23
Equations (20)(23) describe how the contribution from the B = −1 panel in (16) can be approximated by a set of 1D scale-selective filters. The above derivation can be repeated for the other terms (i.e., panels) in the 2D convolution (16) to obtain the functions Ψ′2(A, B, C), Ψ′3(A, B, C), Ψ′4(A, B, C), Ψ′5(A, B, C), and Ψ′6(A, B, C), respectively. The scale-selective filter is then written as
i1520-0493-137-6-1742-e24
A schematic for implementing (24) is shown in Fig. 2. As for the 2D filter in (16), the normalization factors 1 * W ′(x) are again calculated by replacing Ψ(x) with 1 in (24).
For the above discussion, we have considered the convolution on the conformal cubic grid for continuous functions. However, to implement the convolution numerically it is necessary to discretize the above equations. A convolution is normally discretized according to (see, e.g., Smith 2007)
i1520-0493-137-6-1742-e25
where xn is the coordinate of the nth grid point (n = 0, 1, … , M − 1), M is the number of grid points, Δx is the grid spacing, and we require that Ψ(xn) is periodic [i.e., Ψ(xn+M) = Ψ(xn)]. An important property of the discrete convolution is that it satisfies the convolution theorem (9) for discrete FFTs (e.g., for spherical coordinates). However, in this paper we wish to apply the convolution to discrete conformal cubic coordinates. As a consequence of the different discrete coordinate systems, we expect the numerical results for discrete conformal cubic coordinates to differ slightly from those obtained using a discrete spherical harmonic FFT, within truncation errors.

The results of this section show that it is possible to use 1D scale-selective filters as an approximation to the 2D filter for dynamical downscaling. However, it is necessary to test the 1D filter scheme proposed in this section in a realistic context. To this end, in the next section we compare the downscaling accuracy of the 1D scale-selective filter scheme with that of the 2D filter when downscaling NCEP GFS analyses over Australia.

4. Evaluation of the 1D scale-selective filter

In this section we evaluate the 1D scale-selective filter scheme when downscaling from NCEP GFS analyses. To be specific, we interpolate a sequence of NCEP GFS analyses to a quasi-uniform 200-km-resolution conformal cubic grid (i.e., a C48 grid with S = 1). We then downscale from the 200-km-resolution host analyses to 60-km resolution over Australia using a C48 grid but with S = 3.3 (see Fig. 1). The results of the dynamical atmospheric downscaling are then compared with NCEP GFS 0.5° analyses so as to compare the behavior of the 2D filter, 1D filter, and the BC nudging schemes. For the comparison we use an output domain defined from 95° to 175°E and from 60° to 10°S, although CCAM still simulates the atmosphere outside this domain. RMS errors are calculated in the usual way, and the pattern correlation ρ is derived according to the standard definition
i1520-0493-137-6-1742-e26
where Ψ(x) is an atmospheric scalar field, Ψave is the spatial mean value of Ψ(x), Ψa(x) is the analysis scalar field, and is the spatial mean value of Ψa(x).

When downscaling using the 1D and 2D scale-selective filters, we perturb the surface pressure, air temperature, and winds once every 6 h using a strong perturbation weighting η(σ), as described by (8). No filtering is performed at other time steps. The winds and air temperature are perturbed at all levels above 900 hPa while the fields below 900 hPa are allowed to evolve freely so as to be consistent with surface forcings, similar to the treatment of Waldron et al. (1996) and von Storch et al. (2000). Also, to preserve basic thermodynamic consistencies, we vertically interpolate the host atmospheric fields to CCAM’s sigma vertical coordinates. For convenience the winds are perturbed with respect to their Cartesian components [i.e., VX(x), VY(x), and VZ(x)], which makes it straightforward to combine the contributions from different conformal cubic panels in the discrete convolution. Unless otherwise stated, the Gaussian filter W ′(x) was defined by ϵ = π4R2/(18S2), which corresponds to a cutoff length-scale radius of approximately 12° (i.e., when the Gaussian low-bandpass filter has dropped to 50% of its magnitude at its center). No adverse effects were observed after applying the 1D and 2D filters with the strong perturbation weighting (8) when the filter length-scale radius is 3° or greater, with an example of the 1D filter applied to surface pressure shown in Fig. 3. However, an artificial oscillation in the surface pressure (approximately 1 hPa in amplitude) is observed if the filter length scale is too small, as demonstrated in Fig. 3 by a filter with a length-scale radius of 0.4°. Similar results were also obtained for the wind and temperature perturbations (not shown). Note that the size of the perturbation increases as the filter length scale is decreased (i.e., as the filter is applied over an increasingly smaller area). Last, we have designed the scale-selective filters to take advantage of the Message Passing Interface so that the time to calculate a convolution was further reduced by running CCAM on six processors.

To provide a further comparison, we also downscale NCEP GFS analyses using a type of BC nudging in CCAM, which is performed using a far-field nudging scheme designed specifically for the conformal cubic grid (McGregor 2005). To explain this scheme, we first define far-field nudging according to
i1520-0493-137-6-1742-e27
where Δt is the model time step (chosen here to be 1200 s) and τ(x) is an e-folding time, defined as a function of position on the conformal cubic grid. For far-field nudging there is no nudging on the front conformal cubic panel (panel 1, which typically has higher resolution for the region of interest) or on the inner half of the adjacent four side panels (panels 0, 2, 3, and 5). Nudging is only applied with τ(x) = 1 day on the farthest rear panel (panel 4) and with the inverse of τ(x) linearly decreasing to zero at the half panel boundaries (i.e., on panels 0, 2, 3, and 5). This linear change in the inverse of τ(x) is designed to minimize reflections [analogous to Davies (1976)]. For consistency with the scale-selective filters, we apply far-field nudging in this paper to the surface pressure, air temperature, and winds, with the air temperature and winds nudged above 900 hPa. For these experiments, the mixing ratio is not nudged.

In the following numerical experiments, we first wish to determine whether the downscaling accuracy of the 1D scale-selective filter is similar to that of the 2D filter scheme. For our study we concentrate on the 500-hPa winds, 500-hPa air temperature, and mean sea level pressure (MSLP). Downscaling errors are measured up to the 28th simulation day, and we start the CCAM simulation for each day in 2006, resulting in n = 365 downscaling simulations. A simulation length of 28 days was chosen to be sufficient time for the average pattern correlation and RMS errors to reach values that can be considered representative of the downscaling accuracy (i.e., values that do not change if the simulation length is increased, see below). The experiment was repeated for n = 365 times so that any differences found in the performance of the 1D filter, 2D filter, and far-field downscaling techniques are statistically significant. CCAM soil data are initialized using a climatological dataset so that soil temperatures and moisture realistically account for seasonal variability.

Figures 4 –6 show the pattern correlations between the NCEP GFS 0.5° analyses and the downscaling results of the 2D filter, 1D filter, and far-field nudging schemes as a function of simulation time for MSLP, 500-hPa meridional wind component, and 500-hPa temperature, respectively. Figures 4 –6 indicate that the average pattern correlation for both the 1D and 2D scale-selective filters and for the far-field nudging does not change significantly after the 16th simulation day. The scale-selective filter RMS errors are also found to not change after the 16th simulation day (not shown), which suggests that the pattern correlation and RMS errors on the 28th simulation day can be considered to be representative of the downscaling technique (i.e., the correlations and RMS errors do not change if the simulation time is further increased). Tables 1 and 2 compare the pattern correlation and RMS errors for the different nudging schemes on the 28th simulation day for each season. Figures 4 –6 and Tables 1 –2 show no significant difference in the downscaling accuracy of the 1D and 2D scale-selective filter schemes. In particular, Tables 1 and 2 show that the 1D and 2D scale-selective filters still produce equivalent results for all seasons, despite the seasonal dependence of the average pattern correlation and RMS errors. Based on these results, we conclude from the downscaling experiments that the 1D filter scheme produces equally valid downscaling of the NCEP GFS 0.5° analyses on average when compared with that obtained using a 2D scale-selective filter, but with much greater computational efficiency. Tables 1 and 2 also show that the 1D and 2D scale-selective filters provide a higher pattern correlation and lower RMS errors on average than the far-field nudging scheme. Similar conclusions have been obtained by other authors who compared scale-selective methods with BC nudging schemes (e.g., von Storch et al. 2000; Kanamaru and Kanamitsu 2007) and also demonstrated that scale-selective filters are more effective at assimilating information from the host model than are BC-based methods. Typical wall times for CCAM using a 1D filter, 2D filter, and far-field nudging are described in Table 3, which shows that increased wall time due to the 1D filter (i.e., relative to far-field nudging) is relatively small in comparison with the increased wall time due to the 2D filter. Note that as the number of processors is increased the additional overhead due to message passing results in an increasing wall-time ratio between the far-field and 1D filter as well as the far-field and 2D filter nudging schemes. Nevertheless, the 1D filter scheme is always faster than the 2D scheme, because both have the same message passing overhead. Note that the differences in the typical wall times shown in Table 3 will increase as the number of horizontal grid points is increased; that is, proportional to O(N), O(N3), and O(N4) for far-field nudging, the 1D filter, and 2D filter, respectively, when applied to N2 horizontal grid points.

Although the results in Figs. 4 –6 show the performance of the 1D and 2D scale-selective filters to be equivalent on average with respect to the NCEP GFS 0.5° analyses, nevertheless there are differences in the downscaling results produced by the two schemes. As discussed in section 3, differences between the 1D and 2D filter schemes arise mainly as a consequence of treating the filter as separable with respect to the conformal cubic coordinates. These differences are quantified in Fig. 7 which plots the average 500-hPa meridional wind pattern correlation between the two scale-selective filter schemes as a function of simulation time. In particular, an average pattern correlation of 0.964 ± 0.003 is achieved after 28 simulation days despite the various approximations made by the 1D scheme. Higher pattern correlations are achieved for the 500-hPa zonal wind, 500-hPa air temperature, and MSLP (not shown).

Figure 8 shows the sensitivity of the downscaled results to the length scale of the scale-selective filter. We specifically plot the pattern correlation of the 500-hPa meridional winds with respect to the NCEP GFS 0.5° analyses after 28 days for the 1D scale-selective filter. As expected, the results show that the correlation increases as the filter length scale is reduced (i.e., by decreasing ϵ), because more information is assimilated from the host model. However, the results also show that the strength of the pattern correlation decreases rapidly once the length-scale radius is increased above approximately 12° [i.e., ϵ > π4R2/(18S2)], which corresponds to a filter length-scale diameter that is roughly the width of the fine-resolution panel (C = 1). Similar conclusions are obtained for the MSLP and zonal wind pattern correlation (not shown). It can be argued that setting ϵ = π4R2/(18S2) is a reasonable compromise between effectively assimilating information from the host model (i.e., as indicated by a high pattern correlation in Fig. 8) while also maximizing the range of length scales that CCAM simulates independent of the host model (i.e., smaller than the width of the fine-resolution cubic panel). For this reason we have used ϵ = π4R2/(18S2) for all results shown in this paper unless stated otherwise.

The scale-selective filter produces improved pattern correlation and RMS errors when downscaling from NCEP GFS analyses (i.e., it is beneficial for reproducing patterns of the host model above the specified length scale). However, it is important to note that the scale-selective filter will not necessary produce superior downscaling results for all applications. For example, we have also compared the relative performance of the scale-selective filters and the far-field nudging when the host model is replaced with a 4-day CCAM NWP simulation for Australia in 2006, where 4 days was chosen to be a realistic forecasting time scale for a deterministic forecasting model to have a high level of forecasting skill. For this NWP experiment we found that the scale-selective filters and far-field nudging both produced downscaled results that are equivalent. In specific terms, average pattern correlation and RMS errors of 0.917 ± 0.005 and 5.1 ± 0.1 hPa for MSLP, 0.64 ± 0.02 and 7.9 ± 0.2 m s−1 for 500-hPa meridional winds, and 0.962 ± 0.002 and 2.64 ± 0.06 K for 500-hPa temperature were found after 4 simulation days for both the 1D and 2D scale-selective filters and for far-field nudging (i.e., including forecast errors). Therefore, for this particular NWP experiment, there is no advantage using a scale-selective filter relative to far-field nudging, since the additional information assimilated by the scale-selective filter has not resulted in any improvement in forecast skill. Furthermore, as discussed by Kanamaru and Kanamitsu (2007), it is possible that errors in the host model can degrade the forecast skill of the regional model if these errors are also assimilated more effectively by the scale-selective filter. Nevertheless, the scale-selective filter and spectral nudging methods should be effective in applications for which the regional atmospheric model is required to reproduce the behavior of the host model above a specified length scale.

5. Conclusions

In this article we have investigated the use of a combination of 1D scale-selective filters for dynamical downscaling with CCAM. Since CCAM uses a stretched global conformal cubic grid, we can implement scale-selective downscaling in CCAM without any lateral boundaries. Our scheme differs from the conceptual proposal by McGregor (2006) in that our scheme approximates a filter as being separable with respect to the conformal cubic coordinates. The resultant 1D scale-selective filters therefore provide a reasonable approximation of a 2D filter but require significantly fewer computations. We have estimated the leading-order error terms in this approximation and found that they do not lead to any significant degradation in the performance of the 1D filter as long as the filter length scale is not too large. In particular, we recommend using ϵ = π4R2/(18S2) as a reasonably optimal choice (i.e., the filter diameter is approximately the size of the finest-resolution panel).

To demonstrate the proposed 1D scale-selective filters in CCAM, we have found it convenient to base our scheme on a convolution instead of a spherical harmonic FFT. This is because the convolution can be readily evaluated in terms of CCAM’s native conformal cubic grid, making the scheme and its approximations easier to describe and implement. Convolution-based schemes are traditionally avoided since they are relatively computationally expensive. However, since the required number of computations is reduced from O(N4) to O(N3) by using a combination of 1D filters, the computational speed disadvantage of the convolution is largely mitigated for practical applications.

To support the analytic arguments made in the text as to the validity of using 1D scale-selective filters, we have used CCAM to downscale NCEP GFS analyses (interpolated onto a C48 conformal cubic grid at 200-km resolution) to 60-km resolution over Australia for 28-day forecasts for each day in 2006. The results obtained using the 1D filter method were validated against NCEP GFS 0.5° analyses, as well as using a 2D filter and a far-field nudging scheme. The 1D filter scheme produced equally valid downscaled predictions of NCEP GFS 0.5° analyses when compared with those of the 2D filter scheme, indicating that the 1D filter scheme provides a computationally efficient method for practical applications of scale-selective downscaling on the surface of a sphere. We also observed that, when scale-selective filters are compared with CCAM’s existing far-field nudging scheme (i.e., a BC-style approach), the scale-selective methods achieve statistically significant improvements in average pattern correlation and RMS errors when downscaling from analyses. These results are consistent with previous findings (e.g., Waldron et al. 1996; von Storch et al. 2000; Misra 2007; Kanamaru and Kanamitsu 2007), concerning the effectiveness of scale-selective methods to assimilate large-length-scale information from the host model.

Although the above results show that the scale-selective filter is more effective at downscaling analyses than is the far-field approach, it has been noted previously by Kanamaru and Kanamitsu (2007) that there are situations in which the scale-selective filter’s more effective assimilation of host-model data into the regional model can be a disadvantage. For example, when considering NWP and general circulation model downscaling applications it is likely that the host model will contain errors and biases. These host errors will also be effectively assimilated into the regional model by the scale-selective filter, which can degrade the performance of the regional simulation. A more subtle problem is illustrated by replacing the host model with a 4-day CCAM NWP forecast as considered in this paper. Under these circumstances we found that the scale-selective filter and the far-field nudging methods both gave equally valid downscaled output, demonstrating that the additional information assimilated into the regional model by the scale-selective filter does not necessarily result in an improved downscaled NWP forecast. The purpose of this paper is to develop a scale-selective filter for atmospheric models with a cube-based geometry and to demonstrate the filter’s effectiveness at assimilating information from the host model into the regional model. Evaluating the usefulness of the scale-selective filter for different downscaling applications (NWP, climate simulations, regional climate change projections, etc.) is the subject of ongoing work.

Acknowledgments

The authors thank Debbie Abbs, Martin Dix, Jorgen Frederiksen, Jack Katzfey, Kim Nguyen, and Terry O’Kane for their advice during the writing of this paper. The authors also thank the two anonymous reviewers for their constructive comments. This work was partially funded under the South Eastern Australian Climate Initiative (SEACI).

REFERENCES

  • Barnes, S., 1964: A technique for maximizing details in numerical weather map analysis. J. Appl. Meteor., 3 , 396409.

  • Davies, H., 1976: A lateral boundary formulation for multi-level prediction models. Quart. J. Roy. Meteor. Soc., 102 , 405418.

  • Denis, B., J. Coté, and R. Laprise, 2002: Spectral decomposition of two-dimensional atmospheric fields on limited-area domains using the discrete cosine transform (DCT). Mon. Wea. Rev., 130 , 18121829.

    • Search Google Scholar
    • Export Citation
  • Jones, R., J. Murphy, and M. Noguer, 1995: Simulation of climate change of Europe using a nested regional-climate model. I: Assessment of control climate, including sensitivity to location of lateral boundaries. Quart. J. Roy. Meteor. Soc., 121 , 14131449.

    • Search Google Scholar
    • Export Citation
  • Kanamaru, H., and M. Kanamitsu, 2007: Scale-selective bias correction in a downscaling of global analysis using a regional model. Mon. Wea. Rev., 135 , 334350.

    • Search Google Scholar
    • Export Citation
  • Kida, H., T. Koide, H. Sasaki, and M. Chiba, 1991: A new approach for coupling a limited area model to a GCM for regional climate simulations. J. Meteor. Soc. Japan, 69 , 723728.

    • Search Google Scholar
    • Export Citation
  • Kreyszig, E., 1993: Advanced Engineering Mathematics. 7th ed. John Wiley and Sons, 1271 pp.

  • McGregor, J. L., 2005: C-CAM: Geometric aspects and dynamical formulation. CSIRO Marine and Atmospheric Research Tech. Paper 70, 43 pp.

  • McGregor, J. L., 2006: Regional climate modelling using CCAM. BMRC Research Rep. 123, 118–122.

  • McGregor, J. L., and M. R. Dix, 2008: An updated description of the Conformal Cubic Atmospheric Model. High Resolution Simulation of the Atmosphere and Ocean, K. Hamilton and W. Ohfuchi, Eds., Springer, 51–76.

    • Search Google Scholar
    • Export Citation
  • Misra, V., 2007: Addressing the issue of systematic errors in a regional climate model. J. Climate, 20 , 801818.

  • Mohlenkamp, M., 1999: A fast transform for spherical harmonics. J. Fourier Anal. Appl., 5 , 159184.

  • Nunez, M., and J. L. McGregor, 2007: Modelling future water environments of Tasmania, Australia. Climate Res., 34 , 2537.

  • Rancic, M., R. Purser, and F. Mesinger, 1996: A global shallow-water model using an expanded spherical cube: Gnomonic versus conformal coordinates. Quart. J. Roy. Meteor. Soc., 122 , 959982.

    • Search Google Scholar
    • Export Citation
  • Schmidt, F., 1977: Variable fine mesh in spectral global models. Beitr. Phys. Atmos., 50 , 211217.

  • Smith, J., 2007: Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications. 2nd ed. W3K Publishing, 320 pp.

  • Thatcher, M., and J. L. McGregor, 2007: Regional climate downscaling for the Marine and Tropical Sciences Research Facility (MTSRF) between 1971 and 2000. Marine and Tropical Sciences Research Facility Rep., 12 pp. [Available online at www.rrrc.org.au/publications/downloads/25ii1-CSIRO-Thatcher-2007-Climate-Downscaling-1971-2000.pdf].

    • Search Google Scholar
    • Export Citation
  • Thatcher, M., and J. L. McGregor, 2008: Dynamically downscaled Mk3.0 climate projection for the Marine and Tropical Sciences Research Facility (MTSRF) from 2060–2090. CSIRO Marine and Atmospheric Research Rep., 23 pp. [Available online at www.cmar.csiro.au/e-print/open/2008/thatchermj_b.pdf].

    • Search Google Scholar
    • Export Citation
  • von Storch, H., H. Langenberg, and F. Feser, 2000: A spectral nudging technique for dynamical downscaling purposes. Mon. Wea. Rev., 128 , 36643673.

    • Search Google Scholar
    • Export Citation
  • Waldron, K., J. Paegle, and J. Horel, 1996: Sensitivity of a spectrally filtered and nudged limited-area model to outer model options. Mon. Wea. Rev., 124 , 529547.

    • Search Google Scholar
    • Export Citation
  • Wang, Y., L. R. Leung, J. L. McGregor, D-K. Lee, W-C. Wang, Y. Ding, and F. Kimura, 2004: Regional climate modelling: Progress, challenges and prospects. J. Meteor. Soc. Japan, 82 , 15991628.

    • Search Google Scholar
    • Export Citation

Fig. 1.
Fig. 1.

Example of a C48 global conformal cubic grid that has been stretched according to the Schmidt transformation (S = 3.3), producing a resolution of approximately 200 km on average and 60 km over Australia.

Citation: Monthly Weather Review 137, 6; 10.1175/2008MWR2599.1

Fig. 2.
Fig. 2.

Schematic representation of the 1D scale-selective filter described in (24). This example shows the contribution from the B = −1 panel, as described by the Ψ1(A, C) and Ψ′1(A, B, C) convolutions outlined in (20)(23). The Ψ1(A, C) convolutions are performed first and are represented by short dashed lines, and the Ψ′1(A, B, C) convolutions are performed second as indicated by long dashed lines.

Citation: Monthly Weather Review 137, 6; 10.1175/2008MWR2599.1

Fig. 3.
Fig. 3.

Plot of the surface pressure as a function of simulation time after applying the 1D scale-selective filter. Four choices of the filter length-scale radius are shown as solid lines for 0.4°, 3°, 12°, and 48° (i.e., the smaller the radius is, the greater is the perturbation by the filter), as well as when no filter is applied, which is indicated by the dashed line.

Citation: Monthly Weather Review 137, 6; 10.1175/2008MWR2599.1

Fig. 4.
Fig. 4.

Average pattern correlations for MSLP after downscaling to 60-km resolution over Australia (see Fig. 1) from NCEP GFS analyses interpolated to a 200-km-resolution conformal cubic grid. Pattern correlation is measured relative to NCEP GFS 0.5° analyses. Note that error bars for the 1D and 2D scale-selective filters are smaller than the size of plot markers and therefore are not visible on this plot.

Citation: Monthly Weather Review 137, 6; 10.1175/2008MWR2599.1

Fig. 5.
Fig. 5.

Average pattern correlations as for Fig. 4, but for the 500-hPa meridional wind component.

Citation: Monthly Weather Review 137, 6; 10.1175/2008MWR2599.1

Fig. 6.
Fig. 6.

Average pattern correlations as for Fig. 4, but for the 500-hPa temperature.

Citation: Monthly Weather Review 137, 6; 10.1175/2008MWR2599.1

Fig. 7.
Fig. 7.

Average pattern correlations of 500-hPa meridional wind component as for Fig. 5, but measuring the similarity between the 1D scale-selective filter and the 2D filter (i.e., instead of validating against the NCEP GFS 0.5° analyses).

Citation: Monthly Weather Review 137, 6; 10.1175/2008MWR2599.1

Fig. 8.
Fig. 8.

Plot of the average 500-hPa meridional wind pattern correlation after 28 simulation days as a function of the scale-selective filter length-scale radius. The pattern correlation is measured relative to NCEP GFS 0.5° analyses.

Citation: Monthly Weather Review 137, 6; 10.1175/2008MWR2599.1

Table 1.

Average pattern correlation for each season using the 2D scale-selective filter, 1D scale-selective filter, and far-field nudging at 28 simulation days while downscaling from the NCEP GFS analyses to 60-km resolution over Australia in 2006 (n = 365). Errors are measured relative to NCEP GFS 0.5° analyses, and ± indicates the 95% confidence interval measured using Student’s t distribution; the seasons are December–February (DJF), March–May (MAM), June–August (JJA), and September–November (SON).

Table 1.
Table 2.

Average RMS errors for each season using the 2D scale-selective filter, 1D scale-selective filter, and far-field nudging at 28 simulation days while downscaling from the NCEP GFS analyses to 60-km resolution over Australia in 2006 (n = 365). Errors are measured relative to NCEP GFS 0.5° analyses, and ± indicates the 95% confidence interval measured using Student’s t distribution.

Table 2.
Table 3.

Comparison of the CCAM wall times per simulation day using the 1D scale-selective filter, the 2D scale-selective filter, and far-field nudging for a C48 grid with 18 vertical levels (where the filter is applied from levels 4 to 18). The CCAM simulation timings are for Xeon 3.2-GHz processors.

Table 3.
Save
  • Barnes, S., 1964: A technique for maximizing details in numerical weather map analysis. J. Appl. Meteor., 3 , 396409.

  • Davies, H., 1976: A lateral boundary formulation for multi-level prediction models. Quart. J. Roy. Meteor. Soc., 102 , 405418.

  • Denis, B., J. Coté, and R. Laprise, 2002: Spectral decomposition of two-dimensional atmospheric fields on limited-area domains using the discrete cosine transform (DCT). Mon. Wea. Rev., 130 , 18121829.

    • Search Google Scholar
    • Export Citation
  • Jones, R., J. Murphy, and M. Noguer, 1995: Simulation of climate change of Europe using a nested regional-climate model. I: Assessment of control climate, including sensitivity to location of lateral boundaries. Quart. J. Roy. Meteor. Soc., 121 , 14131449.

    • Search Google Scholar
    • Export Citation
  • Kanamaru, H., and M. Kanamitsu, 2007: Scale-selective bias correction in a downscaling of global analysis using a regional model. Mon. Wea. Rev., 135 , 334350.

    • Search Google Scholar
    • Export Citation
  • Kida, H., T. Koide, H. Sasaki, and M. Chiba, 1991: A new approach for coupling a limited area model to a GCM for regional climate simulations. J. Meteor. Soc. Japan, 69 , 723728.

    • Search Google Scholar
    • Export Citation
  • Kreyszig, E., 1993: Advanced Engineering Mathematics. 7th ed. John Wiley and Sons, 1271 pp.

  • McGregor, J. L., 2005: C-CAM: Geometric aspects and dynamical formulation. CSIRO Marine and Atmospheric Research Tech. Paper 70, 43 pp.

  • McGregor, J. L., 2006: Regional climate modelling using CCAM. BMRC Research Rep. 123, 118–122.

  • McGregor, J. L., and M. R. Dix, 2008: An updated description of the Conformal Cubic Atmospheric Model. High Resolution Simulation of the Atmosphere and Ocean, K. Hamilton and W. Ohfuchi, Eds., Springer, 51–76.

    • Search Google Scholar
    • Export Citation
  • Misra, V., 2007: Addressing the issue of systematic errors in a regional climate model. J. Climate, 20 , 801818.

  • Mohlenkamp, M., 1999: A fast transform for spherical harmonics. J. Fourier Anal. Appl., 5 , 159184.

  • Nunez, M., and J. L. McGregor, 2007: Modelling future water environments of Tasmania, Australia. Climate Res., 34 , 2537.

  • Rancic, M., R. Purser, and F. Mesinger, 1996: A global shallow-water model using an expanded spherical cube: Gnomonic versus conformal coordinates. Quart. J. Roy. Meteor. Soc., 122 , 959982.

    • Search Google Scholar
    • Export Citation
  • Schmidt, F., 1977: Variable fine mesh in spectral global models. Beitr. Phys. Atmos., 50 , 211217.

  • Smith, J., 2007: Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications. 2nd ed. W3K Publishing, 320 pp.

  • Thatcher, M., and J. L. McGregor, 2007: Regional climate downscaling for the Marine and Tropical Sciences Research Facility (MTSRF) between 1971 and 2000. Marine and Tropical Sciences Research Facility Rep., 12 pp. [Available online at www.rrrc.org.au/publications/downloads/25ii1-CSIRO-Thatcher-2007-Climate-Downscaling-1971-2000.pdf].

    • Search Google Scholar
    • Export Citation
  • Thatcher, M., and J. L. McGregor, 2008: Dynamically downscaled Mk3.0 climate projection for the Marine and Tropical Sciences Research Facility (MTSRF) from 2060–2090. CSIRO Marine and Atmospheric Research Rep., 23 pp. [Available online at www.cmar.csiro.au/e-print/open/2008/thatchermj_b.pdf].

    • Search Google Scholar
    • Export Citation
  • von Storch, H., H. Langenberg, and F. Feser, 2000: A spectral nudging technique for dynamical downscaling purposes. Mon. Wea. Rev., 128 , 36643673.

    • Search Google Scholar
    • Export Citation
  • Waldron, K., J. Paegle, and J. Horel, 1996: Sensitivity of a spectrally filtered and nudged limited-area model to outer model options. Mon. Wea. Rev., 124 , 529547.

    • Search Google Scholar
    • Export Citation
  • Wang, Y., L. R. Leung, J. L. McGregor, D-K. Lee, W-C. Wang, Y. Ding, and F. Kimura, 2004: Regional climate modelling: Progress, challenges and prospects. J. Meteor. Soc. Japan, 82 , 15991628.

    • Search Google Scholar
    • Export Citation
  • Fig. 1.

    Example of a C48 global conformal cubic grid that has been stretched according to the Schmidt transformation (S = 3.3), producing a resolution of approximately 200 km on average and 60 km over Australia.

  • Fig. 2.

    Schematic representation of the 1D scale-selective filter described in (24). This example shows the contribution from the B = −1 panel, as described by the Ψ1(A, C) and Ψ′1(A, B, C) convolutions outlined in (20)(23). The Ψ1(A, C) convolutions are performed first and are represented by short dashed lines, and the Ψ′1(A, B, C) convolutions are performed second as indicated by long dashed lines.

  • Fig. 3.

    Plot of the surface pressure as a function of simulation time after applying the 1D scale-selective filter. Four choices of the filter length-scale radius are shown as solid lines for 0.4°, 3°, 12°, and 48° (i.e., the smaller the radius is, the greater is the perturbation by the filter), as well as when no filter is applied, which is indicated by the dashed line.

  • Fig. 4.

    Average pattern correlations for MSLP after downscaling to 60-km resolution over Australia (see Fig. 1) from NCEP GFS analyses interpolated to a 200-km-resolution conformal cubic grid. Pattern correlation is measured relative to NCEP GFS 0.5° analyses. Note that error bars for the 1D and 2D scale-selective filters are smaller than the size of plot markers and therefore are not visible on this plot.

  • Fig. 5.

    Average pattern correlations as for Fig. 4, but for the 500-hPa meridional wind component.

  • Fig. 6.

    Average pattern correlations as for Fig. 4, but for the 500-hPa temperature.

  • Fig. 7.

    Average pattern correlations of 500-hPa meridional wind component as for Fig. 5, but measuring the similarity between the 1D scale-selective filter and the 2D filter (i.e., instead of validating against the NCEP GFS 0.5° analyses).

  • Fig. 8.

    Plot of the average 500-hPa meridional wind pattern correlation after 28 simulation days as a function of the scale-selective filter length-scale radius. The pattern correlation is measured relative to NCEP GFS 0.5° analyses.

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