## 1. Introduction

It is reasonable to use a regular grid of equal latitudinal and longitudinal spacing for global gridpoint modeling as a simple and straightforward numerical method of discretization. In this system, the number of grid points is the same for all given latitudes. However, the equal latitudinal/longitudinal spacing is not an equal physical spacing. The physical grid spacing over polar areas is smaller than that over low latitudes, and it will require a smaller time step to satisfy numerical stability. Thus the consequence of using this uniform grid of equal degree spacing in global modeling is the well-known pole problem. In order to integrate with a larger time step, one of the common methods is to apply a polar filter to global gridpoint models.

The spectral transformation for global spectral models (Bourke 1972) avoids the necessity of applying a polar filter. The intrinsic nature of the spectral transformation in global spectral models provides the same effect as a polar filter in global gridpoint models. In the spectral transformation, there is an equal number of grid points for all latitudes with equal grid spacing along any given latitude and a Gaussian distribution along each longitude. This is referred to as the traditional full grid spectral method. Even though the time step is larger, the over specification of grid points over polar areas requires more computations for the transformation and the nonlinear forcing in physical gridpoint space.

Research on reducing the number of polar grid points has been ongoing for more than 40 years, first in global gridpoint modeling (Gates and Riegel 1962; Kurihara 1965) and then in global spectral modeling (Jarraud and Simmons 1983; Hortal 1991; Hortal and Simmons 1991). All results indicate that it may not be necessary to use an equal number of grid points for all given latitudes in global models, especially in spectral modeling. There are already several existing global models constructed with a reduced Gaussian grid, such as weather and climate models at the European Centre for Medium-Range Weather Forecasts (ECMWF) and a weather forecast model at the National Centers for Environmental Prediction (NCEP).

From the previous discussion, we see that it is practical to use a reduced Gaussian grid (instead of a full Gaussian grid) in the NCEP seasonal forecast model. Another reason to apply a reduced Gaussian grid to the operational implementation is to meet the NCEP operational suite wall-time requirements, with a higher resolution if possible. There are two classes of methods to determine the amount of grid reduction in spectral models; the first is based on the geometric considerations of physical grid spacing, and the second is based on the mathematical considerations of spectral transformation. The geometric concept of a reduced grid based on longitudinal physical grid spacing was examined and implemented by Hortal and Simmons (1991). The mathematical concept of a reduced grid based on the accuracy of the associated Legendre functions was proposed by Courtier and Naughton (1994) and investigated and implemented by Williamson and Rosinski (2000).

In this paper, an alternative and more computationally efficient reduced-spectral transformation, also based on the accuracy of associated Legendre functions but using a somewhat different approach, is introduced. The methodology to convert from a full grid spectral transform to a reduced spectral transform is described in section 2. The implementation with run-time options for reduced Legendre transforms and reduced-Gaussian-grid transforms is discussed in section 3, where message- passing interface (MPI) and surface field preparation for reproducibility are introduced. Experimental results, with dynamical code only, are shown in section 4. Results from the full model with dynamics and physics are shown in section 5. Conclusions and future concerns are discussed in the final section.

## 2. Methodology

In this section, the original full grid spectral transform with isosceles right triangular truncation will be shown. We will investigate the behavior of the associated Legendre function and describe a method of selecting an optimal value for the associated Legendre function (in terms of accuracy). Finally, we will introduce a reduced spectral transformation with reduced Gaussian grid and reduced associated Legendre transform and discuss the differences and efficiency of the reduced spectral transform when compared with full grid and reduced grid transforms.

### a. Standard, full grid, spectral transformation with isosceles-right-triangular truncation

The definition of a spherical spectral transformation with an isosceles-right-triangular transformation can be found in Haltiner and Williams (1980). The purpose in paraphrasing it here is twofold. First, the transformation description emphasizes the computational process used in the numerical model, which is described in section 3. Second, in the following subsections this transformation is used in a comparison with the reduced spectral transformation.

The spectral method used in global spectral models comprises two transforms: a Fourier transform and an associated Legendre transform. The method requires equal grid lengths at any given latitude for the Fourier transform and Gaussian-type grid spacing through a Gauss–Legendre integration formulation along longitudes for the associated Legendre transform. This spectral transformation can also be described in both directions, from spectral coefficients in spectral space to gridpoint values in physical space and vice versa.

*q*(

*λ*

_{i},

*φ*

_{j}), where

*λ*is longitude and

*φ*is latitude, for

*i*= 1, … ,

*I*[for

*λ*from 0 to (2

*π*− Δ

*λ*)], and

*j*= ±1, … , ±

*J*(for

*φ*from the first row off the equator to the pole). Any global field can be transformed from physical values to spectral coefficients with maximum wavenumber

*N,*using an isosceles right triangular truncation. First, the Fourier transform at any given latitude is applied to the transformation from gridpoint values to Fourier coefficients aswhere

*q*

^{m}(

*μ*

_{j}) is the Fourier coefficient at latitude

*μ*

_{j}(=sin

*φ*

_{j}), and |

*m*| = 0, … ,

*N.*There is a relationship between the number of grid points and spectral coefficients when we consider an alias-free representation of quadratic terms in a nonlinear numerical model, so

*I*≥ (3

*N*+ 1). After Fourier coefficients are obtained for all latitudes, the associated Legendre transformation is applied to construct all spectral coefficients aswhere

*q*

^{m}

_{n}

*n,*Fourier wavenumber

*m,*and |

*m*| = 0, … ,

*N*and

*n*=

*m,*… ,

*N,*again for an isosceles right triangular truncation

*N.*Here

*P*

^{m}

_{n}

*μ*

_{j}) is the associated Legendre function at

*μ*

_{j}for wave (

*m,*

*n*), and

*w*

_{j}is the weighting function at latitude

*μ*

_{j}. For an aliasing-free representation of quadratic term,

*J*has to be larger than or equal to (3

*N*+ 1)/2.

*q*

^{m}

_{n}

*m*| = 0, … ,

*N*and

*n*=

*m,*… ,

*N*) can be transformed to physical values by the following. First, the associated Legendre transformation is applied to obtain Fourier coefficients at any given latitude usingand then the Fourier transformation is applied to transform Fourier coefficients into gridpoint values usingThus, both directions of spectral transformations are described. Combining Eqs. (1) and (2) results in a single formula to transform from gridpoint values to spectral coefficients, and combining Eqs. (3) and (4) results in a single formula for the reversed transformation.

### b. Accuracy of the associated Legendre transformation

The amplitude of the associated Legendre function is the major key to this application of a reduced spectral transformation. Figure 1 gives three examples of magnitude plots for the associated Legendre function (*P*^{m}_{n}_{10}|*P*^{m}_{n}*P*^{m}_{n}*m*) near the pole to a higher zonal wavenumber near the equator, with an axis at the zero wavenumber, *m* = 0 and *n* = 0. Second, the amplitude at any given latitude shows two regions separated by the rightmost white line, with significant values of |*P*^{m}_{n}*P*^{m}_{n}^{−44}, 10^{−24}, and 10^{−18} at 80°, 70°, and 60°, respectively. The contribution of *P*^{m}_{n}

*q*

^{m}

_{n}

*P*

^{m}

_{n}

*m, n,*and

*j*) are scanned to find the maximum magnitude of the associated Legendre coefficient, which can be expressed as max|

*P*

^{m}

_{n}

*μ*

_{j})|. Then an optimal value of the Legendre coefficient,

*P**, is defined by multiplying 10

^{−d}with the maximum magnitude of the associated Legendre coefficient as

*P*

^{*}

^{−d}

*P*

^{m}

_{n}

*μ*

_{j}

*d*is an integer used to specify the number of digits of accuracy. All absolute values of

*P*

^{m}

_{n}

*P** will be truncated at any given latitude, because their contribution to the spectral transformation could be neglected for a given

*d*in Eq. (5). The summation in the Legendre transform where all |

*P*

^{m}

_{n}

*P** will then be eliminated. For example, in Fig. 1b the dashed line indicates the boundary when

*d*= 4. The coefficients located to the right of the dashed line are neglected. Thus, at this latitude, the spectral summation for

*n*at given

*m*does not start from

*m*but from

*n*

_{mj}(indicated by the dashed line). The maximum longitudinal wavenumber will be

*Mj,*which is the maximum

*m*along the dashed line less than or equal to

*N.*Then the related Fourier transform at this latitude will require only

*Ij*≥ (3

*Mj*+ 1).

From the above method, the ideas on an accuracy of the associated Legendre function were adopted from Courtier and Naughton (1994) and on the selection of an optimal value of the associated Legendre function from Williamson and Rosinski (2000). But the precise method used here is somewhat different from theirs. We will describe the precise method of the spectral transformation in section 2c and compare this method to others in section 2d.

### c. The reduced spectral transformation

*m*| = 0, … ,

*Mj,*where

*Mj*is the maximum longitudinal wavenumber according to the accuracy of the associated Legendre coefficient at latitude

*φ*

_{j};

*I*

_{j}must satisfy

*Ij*≥ (3

*Mj*+ 1) and be suitable for a specific fast Fourier transform (FFT). The model uses not only factors of 2, 3, and 5 (with at least one factor of 2) but also larger prime numbers, such as 7 and 11, if the specific FFT package allows the use of higher prime numbers.

*m*and

*n,*

*J*

_{mn}is the polewardmost latitude before we start neglecting the

*P*

^{m}

_{n}

*P*

^{m}

_{n}

*μ*

_{Jmn+1})| <

*P** and |

*P*

^{m}

_{n}

*μ*

_{Jmn})| ≥

*P** for positive and negative

*μ*

_{Jmn}and can be described in a formula asfor each

*m*and

*n*. And

*J*

_{mn}=

*J,*while all

*j*satisfies |

*P*

^{m}

_{n}

*μ*

_{j})| ≥

*P** for any given

*m*and

*n.*

*n*

_{mj}is the smallest latitudinal wavenumber at given longitudinal wavenumber

*m*and latitude

*j*and satisfies the condition for positive and negative

*μ*

_{j}byfor each

*m*and

*j,*and

*n*

_{mj}=

*m,*while all

*n*satisfies |

*P*

^{m}

_{n}

*μ*

_{j})| ≥

*P** for any given

*m*and

*j.*In the final transformation, Eq. (4), used to transform Fourier coefficients into gridpoint values, should be modified toIn this transformation, Gaussian grids have been reduced in computing the Fourier transform, and the number of waves retained is progressively reduced for Gaussian latitudes approaching the poles. Thus, this transformation comprises reduced Gaussian grids and reduced spectral summations (based on the accuracy of the associated Legendre coefficients) and is referred to as a reduced spectral transformation.

The reduced spectral summations save computational time on the Legendre transformation, but reduced Gaussian grids save time not only on the Fourier transformation but also on all nonlinear computation (because of the reduced physical grids). However, the total spectral wavenumbers with *N* isosceles right triangular truncation have not been changed.

### d. Comparisons and efficiency of the reduced spectral transformation

As mentioned in section 2b, the reduced method proposed here has two major differences from Courtier and Naughton (1994) and Williamson and Rosinski (2000). First is the method of selecting the optimal value for accuracy consideration, and second is the reduced method of computing Legendre summation.

In the Courtier and Naughton (1994) method, ε and ε′ are based on absolute but arbitrary values for accuracy (say, 10^{−6}). The *l*-digit accuracy in Williamson and Rosinski (2000) is based on relative values among a group of *P*_{mn} at the same longitudinal wave [shown in their Eq. (2.6)]. My selection, in Eq. (5), is based on a global maximum value of all associated Legendre coefficients at all latitudes. Even though my approach is different from Williamson and Rosinski (2000), we both follow the ε-reduced method of Courtier and Naughton (1994), and we have the same approach in using the associated Legendre function magnitude for selecting the optimal value of ε. According to the analysis from Courtier and Naughton (1994), ε-reduced and ε′-reduced have the same behavior for triangular truncation ranging from 21 to 213 waves, which covers the resolution considered in seasonal forecast models. Furthermore, it was found that one could reduce the grids more by using global selection than Williamson and Rosinski (2000) could by using local selection within the same wavenumber.

The second difference is in the proposed reduced Legendre transform. If *m,* instead of *n*_{mj}, is used as the starting point for *n* at a given *m* and *j* [see Eq. (9)], the transformation (when, e.g., *d* = 4) will be at an accuracy given by selecting the coefficients to the left of the black solid line in Fig. 1b. It can be referred to as a trapezoidal summation because of its shape and will be the same as in Williamson and Rosinski (2000) and others. However, if we use *n*_{mj} as the starting point for *n* at a given *m* and *j* [as in Eq. (9)] the transformation will be at an accuracy given by selecting the computation region to the left of the black dashed line in Fig. 1b. In the case of *d* = 1, the selected computational region is shaped like a scalene right triangle. To conveniently distinguish it from a trapezoidal summation, we refer to this kind of summation as “scalenelike” (scalene “like” because it is not a scalene right triangle when *d* > 1). As it is a reduced spectral transform, we can refer to it as a reduced spectral summation when it is not being compared with a trapezoidal summation. Even though Eqs. (6) and (11) are the same as Eqs. (2.7) and (2.11) in Williamson and Rosinski (2000), Eqs. (7) and (9) are different from their Eqs. (2.9) and (2.10), respectively. The computation of our Legendre transform ends at *J*_{mn} (not *J*) for *j* in Eq. (7) and starts at *n*_{mj} (not *m*) for *n* in Eq. (9). The scalenelike summation used here has the advantage of saving more computation time than a trapezoidal summation. In fact, our summation is similar to the method by Jarraud and Simmons (1983) except that they used a full Gaussian grid for testing.

Figure 2a shows the ratio of total grid numbers of a reduced Gaussian grid over a full Gaussian grid for T42 (light dashed and solid curves) and T62 (bold dashed and solid curves) for *d* ranging from 1 to 12. The dashed curves are for percentages independent of any particular FFT, and the solid lines are for percentages using a particular FFT (which required that the grid numbers along any latitude be factors of only 2, 3, and 5). From Fig. 2a, we can conclude that the computational savings increase with resolution and that the FFT degrades savings by less than 2.5%. Since an FFT is considerably faster than a nonlinear computation done in gridpoint space, it is worthwhile to use an FFT that allows two more factors (7 and 11), which results in only a 1% grid reduction. Figure 2a shows a slight grid reduction over the results from Williamson and Rosinski (2000) (in their Fig. 1b, for T42) because of the different methods used in selecting optimal values (ours is through all associated Legendre functions and theirs is limited locally).

The grid number ratio in Fig. 2b shows the percentage of computational time needed by the reduced Legendre transform with respect to a full transform, for T62 truncation. The trapezoidal summation (dashed line in Fig. 2b) shows an approximate 20% savings with *d* = 1 accuracy, decreasing to a 7% savings when *d* = 12. The scalenelike summation proposed here (the solid line in Fig. 2b) saved about 50% more than the trapezoidal summation at any given value of *d.* These two transformations use the same reduced Gaussian grid, but different Legendre computations. Hereafter, when I refer to a full grid transform I mean the traditional spectral transformation with uniform Gaussian grids, a reduced grid transform is the reduced Gaussian grid with trapezoidal spectral summation, and a reduced spectral transform is the reduced Gaussian grid with scalenelike spectral summation.

## 3. Implementation

This section describes how to implement the reduced spectral transformation into the NCEP seasonal forecast model (SFM). Included is a short description of the seasonal forecast model, a dynamical implementation of the transform with run-time options, a noninterpolation method for surface fields (without constructing pre- and postprocessors), and a consideration of massively parallel computation.

### a. Description of seasonal forecast model

The atmospheric portion of NCEP SFM is a modified version of the previous NCEP operational atmospheric global spectral model (GSM) used in medium-range forecasting on a CRAY machine. The modifications are mainly for the purpose of improving seasonal forecasts. They include some dynamical, physical, and code structure changes, which are described in Kanamitsu et al. (2002).

The model's main features are summarized here. It is an atmospheric model with primitive equations in hydrostatic approximation on sigma coordinates. Spectral computations with isosceles right triangular truncation are used for linear computations, which include a semi- implicit time scheme, second-order high-wavenumber horizontal diffusion, time filtering, and zonal damping. The spectral transformation (Orszag 1970) is used to get gridpoint values for all nonlinear computations, which include 1) nonlinear dynamics such as pressure gradient force and Coriolis force and 2) model physics such as radiation, surface energy budget, surface layer and planetary boundary layer, gravity wave drag, shallow and deep cumulus convection, large-scale precipitation, and some hydrologic calculations. Isosceles right triangular truncation at 62 wavenumbers (T62) with 28 vertical layers is used for all the tests in this paper. The time step for all tests is 1800 s.

### b. Dynamical implementation

In order to have flexibility in testing different spectral transforms and different levels of accuracy for Legendre coefficients, it is better to implement spectral transforms in a way that can run at any transformation and accuracy without recompilation. We use a run-time name list to control the accuracy and type of spectral transform in the experiments, with options to run full grid, reduced grid or reduced spectral transformations.

The model initial and restart files of atmospheric data are stored as spectral coefficients and, for a given truncation, are independent of the transformation. Thus atmospheric data in spectral form can be used for all experiments with the options mentioned above. In the model surface data file, the initial file will be different from the restart file for the experiments with different Gaussian grid options. All options have the same number and locations for latitudinal grid points but differ in the number and locations of longitudinal grids. In order to solve this problem, a nearest point replacement (shown by solid arrows in Fig. 3) is used. The remaining points (shown by dashed arrows in Fig. 3) are discarded when going from a full to reduced Gaussian grid, and the value to the immediate west of the remaining points is used when going from a reduced to full Gaussian grid.

### c. MPI considerations

In order to meet the NCEP operational model suite wall-time requirements, one needs to run the model on a massively parallel machine using a message-passing interface (MPI). This requires an additional modification: a two-dimensional decomposition (with a one-dimensional option) to slice the model domain (Juang and Kanamitsu 2001) for massively parallel computing.

The reduced Gaussian grid and spectral transformation are irregular grids. They can be load balanced by a local symmetric distribution, which can be described in the following example, using six MPI tasks. The first and longest bar in Fig. 4a is assigned to MPI task 1, the second to task 2, … , the sixth to task 6, the seventh to task 6, the eighth to task 5, … , the twelfth to task 1, and the thirteenth to task 1, … , until all are assigned. Figure 4 shows the coefficient length of regular non- MPI Legendre computations at any given latitude for (a) reduced spectral (scalenelike) and (b) reduced grid (trapezoidal) transformations (see, e.g., Fig. 1b), and MPI-balanced computations using six tasks for (c) reduced spectral and (d) reduced grid transformations. It is clear that the reduced spectral transform (c) has better computational load balancing. Thus, there is an advantage in using a reduced spectral transformation with MPI under the aforementioned algorithm of local symmetric distribution for decomposition, which is used by most global spectral models, such as in ECMWF and Japan Meteorological Agency (JMA). Furthermore, it is not easy to have good load balancing with the reduced Gaussian grid; nevertheless, a local symmetric distribution is a simple way to get load balancing because each group will have low, middle, and high latitudes to compute. Even though we use only six MPI tasks and a scalene shape (*d* = 1) as an example in Fig. 4, the conclusion is the same for different numbers of tasks and for scalenelike shapes (*d* > 1) with regard to load balancing.

## 4. Results from dynamical code without physics

The main theme of this paper is to select an optimal function for a reduced spectral method and to emphasize the differences between the proposed and existing transformations. Since a reduced spectral transform has not been seen in previous publications, its behavior should be examined. Williamson and Rosinski (2000) pointed out that there is little reason to choose any accuracy higher than four significant digits. Courtier and Naughton (1994) proposed 10^{−4} as a threshold, based on experiments performed with an aquaplanet model. So an accuracy of four digits is used [i.e., *d* = 4 in Eq. (5)] to compare the performance between reduced spectral and reduced grid transforms and between reduced spectral and full grid transforms.

In order to compare the results of these three spectral transformations, an arbitrary initial condition of 0000 UTC on 9 March 1990 was selected. First, a 1-month integration of each spectral transformation without model physics is performed, then the reduced spectral and the full grid transforms are integrated up to 4 months. For a 1-month integration, there are 1440 time steps of integration. From past experience, we know that over 1000 time steps are more than enough to test the stability of a new dynamical scheme. The purpose of the experiment without physics is to eliminate any possible influence by strong internal variability due to physical parameterizations, such as different responses by the surface physics caused by different sea–land locations among the different transformations. Thus, results from different transformations will show only the dynamical response from each transformation. Furthermore, since a reduced grid transform has been examined fully and used in global spectral models (as in Williamson and Rosinski 2000), it may not be necessary to compare results between the reduced grid and full grid transforms. Instead, the differences between reduced spectral and reduced grid transforms should be investigated. Thus, most of the figures comparing reduced spectral to full grid transforms and reduced spectral to reduced grid transforms are shown, except Fig. 5b (also Fig. 8b), plotted between reduced grid and full grid transforms, which are included for comparison.

Figure 5 shows the zonal mean of root-mean-square difference (RMSD) in mean sea level pressure (MSLP) in hectopascals between (a) reduced spectral and full grid transforms, (b) reduced grid and full grid transforms, and (c) reduced spectral and reduced grid transforms. It indicates that RMSD grows to a maximum of about 3.5 hPa at around 50°N in Fig. 5a, but to only about 2.5 hPa in Fig. 5b at 60°N after a 30-day integration. Nevertheless, the proximity of the RMSD patterns between Figs. 5a and 5b implies that the behavior of reduced spectral transforms is similar to reduced grid transforms. And the difference between reduced spectral and reduced grid transforms in Fig. 5c is much smaller than that between reduced spectral and full grid transforms in Fig. 5a.

Next, let us examine the differences in MSLP between reduced spectral and full grid transforms, as in Fig. 5a, for day 30 (3.5 hPa localized maximum RMSD). Figure 6a shows MSLP after a 30-day integration. The shaded plots with white contours are for the reduced spectral transform, and black contours are for the full grid transform. It is clear that the white and black contours coincide with each other. Both sets of contours show proximity in terms of high/low center locations and in the patterns of curvature and gradient. Note that the white contours in Fig. 6a are mostly invisible because they were plotted before the black contours and because both contours are so close to each other.

Figure 6b shows the latitudinal cross section of zonal mean RMSD of temperature for reduced spectral and full grid transforms after a 1-month integration. The maximum RMSD, approximately 2 K, is at 1000 hPa over the Tibetan Plateau, where this pressure level is below surface height. Extrapolation error could be contributing to the size of the maximum RMSD. Thus, it can be concluded that RMSD for temperature is around 1 K at low levels, but otherwise at or below 0.5 K. This result implies that a reduced spectral transform is an acceptable substitute for a full grid transform, even up to a 30-day integration without model physics, although there is a localized maximum RMSD of 3.5 hPa.

Because of the lack of model physics, the only forcings are from terrain and Coriolis effects. Thus, after more than 1 month of integration, we can expect the model states will be close to zonal symmetry, except at those latitudes dominated by high terrain and at high latitudes where the Coriolis effect is larger. Figure 7 shows expected results in terms of (a) zonal mean of RMSD for MSLP up to a 4-month integration and (b) MSLP after 4-month integration for reduced spectral (white contours) and full grid (black contours) transforms. It implies that the behavior of the reduced spectral method is similar to that of full grid transform beyond a 1-month integration. Nevertheless, it is not practical to use a model without physics; thus, it is necessary to fully examine the reduced spectral method with model physics.

## 5. Results from full model dynamics and physics

In this section, the full model dynamics and physics are used to perform integrations with full grid, reduced grid, and reduced spectral transforms. Again, only the results with *d* = 4 accuracy are shown here, and 1- month integrations are conducted first.

### a. Accuracy of short-range integration

Figure 8 shows zonal mean MSLP RMSD in hectopascals for (a) reduced spectral and full grid transforms, (b) reduced grid and full grid transforms, and (c) reduced spectral and reduced grid transforms, after a 1- month integration. RMSD grows from 1 to 8 hPa between days 8–15 in Figs. 8a and 8b and days 10–18 in Fig. 8c. Then RMSD levels off at 8 hPa after day 15 in Figs. 8a and 8b and day 18 in Fig. 8c. This is very similar to the global mean results obtained by Williamson and Rosinski (2000) in their Fig. 6 for a 2-month integration, where their globally averaged results show a level-off in maximum RMSD after 20 days.

RMSD may give us an indication of the difference between two fields, but not the proximity of features when comparing two similar fields. Again, let us investigate MSLP results from these three different transforms. Figure 9 shows MSLP fields after a 7-day integration. In the shaded plots, white contours are for reduced spectral transforms, and black contours are for (a) full grid transform and (b) reduced grid transform. The white contours (reduced spectral transform) are slightly more visible in Fig. 9a than 9b, which again shows that the best match is between reduced spectral transform and reduced grid transform (Fig. 9b). Nevertheless, the reduced spectral transform is acceptable, in terms of 1-hPa RMSD, when compared to the full grid transform in Fig. 9a after a 7-day integration.

Figure 10 shows the MSLP fields after a 30-day integration. It is easy to distinguish the white and black contours here, because the differences between reduced spectral and full grid transforms or between the reduced spectral and reduced grid transforms are much greater than in Fig. 9. A look at the black contours in the two plots shows that the full grid, in Fig. 10a, and the reduced grid transforms, in Fig. 10b, have also diverged from each other. The locations of pressure centers and the pattern of curvature and gradient have diverged among these three transforms. This implies that zonal mean RMSD of 8 hPa over most latitudes (in Fig. 8) after 30 days is an indicator of dramatic differences between the two MSLP fields all over the globe. These features mirror the well-known phenomenon of model internal variability with physical parameterization, once the time scales are long enough (say, 1 month). The resulting differences in Fig. 10a (between reduced spectral and full grid transforms) are expected because of different physical grids and different spectral transformations, and the equally large differences in Fig. 10b are also expected through internal variability with physical parameterization, though there is only a small difference between the scalenelike and trapezoidal spectral summations of reduced spectral and reduced grid transforms, respectively. This implies that the instantaneous results after a monthly integration with model physics can dramatically diverge from even a very small truncation difference, as, for example, between reduced spectral and reduced grid transforms.

Figure 11 shows zonal mean RMSD for temperature (in K) from reduced spectral and full grid transforms, after (a) 7- and (b) 30-day integrations. The magnitude of RMSD is below 1 K, and in most areas below 0.5 K, for the 7-day integration, and as expected, in the 30- day integration, maximum RMSD is above 13 K, and above 2 K in all areas except the Tropics. After a 7-day integration, the reduced spectral and full grid RMSD is small in terms of mass (MSLP, as in Fig. 9a) and in terms of upper-air fields (temperature, as in Fig. 11a). Other fields examined after a 7-day integration show the same conclusion. This implies that a reduced spectral transform can be used instead of a full grid transform in short-range integrations.

### b. Climatologic responses

In the previous subsection, we saw that the reduced spectral transform behaves about the same as a full grid transform after a 7-day integration, producing results similar to a full grid transform over short periods of time. After 7 days, the internal variability of the model plays a role in diverging results among these different spectral transformations. So we cannot expect the results of a reduced spectral transform to be similar to those of a full-grid transform, and ensemble integrations should be performed. An alternative to ensemble integration is to use single long-range integrations to examine the climatology of reduced spectral transforms to see if they can substitute for a full grid transform.

Two Atmospheric Model Intercomparison Project (AMIP)-type multiyear integrations were conducted, one with a full grid transform and the other with the reduced spectral transform. Seasonal results with a 10-, 30-, or 50-yr mean trended to the same conclusion, so only the 10-yr mean is shown here. Figure 12 shows 10-yr averaged MSLP for (a) winter [December–January–February (DJF)], (b) spring [March–April–May (MAM)], (c) summer [June–July–August (JJA)], and (d) fall [September–October–November (SON)]. Solid contours are for the reduced spectral transform and dotted contours for the full grid transform. It is clear that the locations and intensities of MSLP centers and the patterns of curvature and gradient are all similar between these two transforms for all four seasons. Figure 13 shows RMSD of 10-yr- averaged zonal mean temperature between these two transforms for (a) winter (DJF), (b) spring (MAM), (c) summer (JJA), and (d) fall (SON) with contour intervals of 0.2 K. The maximum is around 1 K, and many areas have RMSDs below 0.5 K. From Figs. 12 and 13, we can conclude that the reduced spectral and full grid transforms have a similar climatology, so the proposed reduced spectral transform can be used for climatic integration.

## 6. Conclusions

A reduced spectral transform is implemented into the NCEP atmospheric seasonal forecast model. This implementation allows different accuracies to be selected, as well as an option of choosing to use a reduced spectral (scalenelike) transform, a reduced grid (trapezoidal) transform, or a full grid transform. The magnitude of Legendre coefficients is the basis in constructing the reduced spectral transform. We use an accuracy of the maximum absolute value of Legendre coefficients in selecting significant coefficients for the spherical spectral transform. The patterns of local significant values of Legendre coefficients form a scalene-right-triangular shape for any given latitude in the reduced spectral transform when *d* = 1 and form a “scalenelike” shape when *d* > 1. The scalenelike reduced Legendre transform needs less computation compared to the trapezoidal reduced Legendre transform and has an extra 50% computational saving on the Legendre transformation.

To simplify the automation process, this implementation modified only the model itself, not any pre- or postprocessor utility package. We reach reproducibility in case of restart by applying a nearest-point-replacement method to the preparation of surface fields when transitioning between full Gaussian and reduced Gaussian grids. Our results show that the reduced spectral transform has better load balancing for massively parallel computation than does the reduced grid transform under a specific algorithm of 1D decomposition.

Monthly integration without model physics shows that there is no significant difference among full grid transform, reduced grid, and reduced spectral transforms, though the reduced spectral and reduced grid transforms are closer to each other than to the full grid transform. And the reduced spectral transform echoes the same behavior as a full grid transform beyond a 1- month integration. This implies that the reduced spectral transform, with *d* = 4 accuracy, is good enough for spherical transforms in comparison to a full grid transform. Because of the enhanced model's internal variability of a chaotic nature with model physical parameterizations, the differences among these three transformations become significant after a 30-day integration. The differences in MSLP synoptic patterns highlight the fact that it is impossible to compare simultaneous results after a 1-month integration even with small truncation errors. Nevertheless, over the short- range (say, a 7-day integration), there are negligible differences among these three transformations with model physics. We conducted 10-yr AMIP integrations of reduced spectral and full grid transforms, and their 10-yr means were used to compare the climatology between them. Similarities in mean MSLP patterns and small values of RMSD in zonal temperature over the four seasons demonstrate the possibility of using reduced spectral transforms for climatic integrations.

In November of 2001, the reduced spectral transform was implemented into the NCEP seasonal forecast model, with a resolution of T62 and 28 layers, for operational seasonal forecasts. The wall time for a 1-day integration with 60 processors running on an IBM SP is 18 s. The 18-s wall time meets the operational computing resource requirements when running a 20-member ensemble for a 7-month forecast, a 210-member ensemble for a 7- month hindcast, model pre- and postprocessing runs for all members, and computing a monthly average for the ensemble mean.

## Acknowledgments

The author would like to thank Dr. Masao Kanamitsu for his encouragement during the initial stage of development, Mary Hart and Dr. Steve Lord for final proofreading, and Drs. Joe Sela and Mark Iredell for internal reviews. The author extends his appreciation to two anonymous reviewers for their instructive and invaluable suggestions and comments that improved the original manuscript.

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