Current computing resources tend to limit global spectral weather modeling to about T213 to T255 resolution, where Tn indicates triangular truncation to wave n of the spherical harmonic basis. The National Centers for Environmental Prediction (NCEP) operational model currently runs at T126 resolution, and the European Centre for Medium-Range Weather Forecasts (ECMWF) operates at T213 resolution. At T255 resolution, the research model at The Florida State University (FSU) is fairly expensive to run, and not fast enough for operational use on a Cray YMP or equivalent supercomputer. T255 resolution corresponds to a transform grid spacing of about 50 km at equator. With increasing availability of massively parallel computers, higher resolutions will eventually be obtainable. However, as we go to higher resolutions, modeling of physical processes tend to become more complex as we try to make them more realistic, increasing the demand on computing resources. Furthermore, a fast Legendre transform has yet to be discovered, which for global spectral models becomes a performance bottleneck at high resolutions. Hence, high-resolution regional modeling will remain an important research and forecasting tool in the foreseeable future. Most regional models use gridpoint methods, particularly due to the difficulty in specifying the lateral boundary conditions. However, a number of researchers (Juang and Kanamitsu 1994; Hoyer 1987; Tatsumi 1986) have successfully developed regional spectral models. Spectral models have a number of advantages, such as absence of phase and aliasing errors and ease of implementation of semi-implicit time integration and implicit horizontal diffusion. The superiority of spectral methods over gridpoint methods has been demonstrated (Jarraud et al. 1981; Girard and Jarraud 1982; Tatsumi 1986). With this in mind, we have set forth to develop a nested regional spectral model at FSU. The regional model predicts small-scale perturbations to the FSU Global Spectral Model (FSUGSM) results. The perturbation technique is similar to that used in the NCEP (Juang and Kananitsu 1994) and ECMWF (Hoyer 1987) nested regional models, though there are a number of significant differences, which we point out throughout the paper. The perturbations are represented by π-periodic trigonometric series and are relaxed to zero at the lateral boundaries. Only the perturbations, not the regional (global plus perturbation) fields themselves, are forced to satisfy the mirror (free-slip) boundary conditions. By relaxing the perturbations to zero at the lateral boundary we ensure that the regional model smoothly connects to the global model result. We currently use only one-way nesting from the global model to the regional model. The regional model was designed to be fully compatible with the FSUGSM. The same vertical σ-coordinate system is used, avoiding the necessity of vertical interpolation between the global and regional models.
We present the first case studies of the FSU Nested Regional Spectral Model (FSUNRSM) in this paper. In section 2 we introduce the FSUNRSM and discuss the model equations and some technical details. In section 3 we present a case study of Hurricane Erin from the 1995 storm season. Erin was remarkable in that it made landfall twice on both coasts of Florida, each time as a minimal hurricane. We see definite improvement in the track and intensity forecast due to higher resolution of the regional model. We make some concluding remarks in section 4. An outline of the FSUGSM, with which the regional model has much in common, is given in the appendix.
2. Nested regional spectral model
The nested regional spectral model developed at FSU utilizes a perturbation method that is closer to that of the NCEP and ECMWF nested regional models than that of the Japan Meteorological Agency (Tatsumi 1986). Regional fields are composed of a base field, typically obtained from a low-resolution global model, plus a high-resolution perturbation field. The regional model was designed to be compatible with the FSUGSM, and it shares the same vertical structure and physics. The FSUGSM uses spherical coordinates in the horizontal direction, while the regional model uses a Mercator coordinate system. Thus the spectral basis functions for the global model are spherical harmonics, whereas we use double Fourier sine and cosine series for the regional perturbations. The regional perturbation basis functions are π periodic, and satisfy a free-slip boundary condition. A relaxation scheme, described below, is applied to absorb buildup of perturbations near the boundary to minimize spurious reflections and Gibbs phenomena and, of course, to ensure that the lateral boundary of the regional model approaches the global model result.
The basic operation of the regional model is as follows. First, a global model is run to provide a base field for the regional model at every nest interval. The nest interval may range from every time step to several hours. To avoid interpolation error, the global model variables and their derivatives are spectrally transformed directly to the regional grid, rather than interpolated from the global Gaussian grid. These base fields are then linearly interpolated in time so that they are available at every time step of the regional model. The perturbation variables and derivatives are then Fourier transformed to the regional grid and added to the base variables and derivatives. Once the full regional variables and derivatives are thus obtained, the nonlinear dynamical and physical time tendencies can be computed. The perturbation time tendencies are obtained by subtracting the global time tendencies from the regional time tendencies. The perturbation time tendencies are then spectrally analyzed, and a semi-implicit time integration is performed to get the perturbations at the next time step.
While the perturbation method used in the FSUNRSM is very similar to the above-mentioned models, there are some differences. Unlike NCEP, we use a Mercator projection rather than a polar stereographic projection. The Mercator projection is nearly regularly spaced at the equator, which is preferable for the study of tropical systems. For regional domains at higher latitudes, a rotation of the coordinate system can be used (Hoyer 1987). The Mercator coordinate system has other advantages as well. Since the longitudinal coordinate is the same as in the global model, the fast Fourier–Legendre transform can be used to transform the global variables and derivatives to the regional grid. This can result in significant computational savings if frequent nesting, say every time step, is used. Another consequence of sharing the same longitudinal coordinate is that the map factor in the regional model depends only on one coordinate, rather than two. Thus there is an absence of aliasing in the east–west direction due to the map factor. The presence of the map factor in the primitive equations results in some cubic products, which are not performed alias free. There will be some aliasing in the north–south direction, which can be minimized by rotating the coordinate system so that the center of the regional grid is at the equator, as is done in the ECMWF regional model. To perform alias-free calculations, one may spectrally truncate the map factor and account for the presence of the map factor in determining the size of the spectral basis needed to compute alias-free terms, as is done by Tatsumi (1986). We have not done that here. If the regional domain is near the equator, the map factor is nearly constant and we neglect any aliasing in the present studies. In Juang and Kanamitsu (1994) aliasing due to the map factor was not discussed.
We include in the FSUNRSM a perturbation orography. This allows the regional model to have a more detailed orography than the global base field. It is not clear from Juang and Kanamitsu (1994) how the regional orography differed from the base field. In at least some cases, the global base fields used were truncated fields of a higher-resolution global model. In the results presented here, the global model was run at low resolution, and only the initial perturbations were obtained from a higher-resolution global model dataset, which included a finer orography.
The FSUGSM and FSUNRSM may either run concurrently or consecutively. Running the models concurrently obviates the need for intermediate storage. Running the models consecutively, or in parallel on separate processors or machines, allows each model to take full advantage of computing resources. Also, more than one regional model can be run, covering different domains.
a. Model equations
b. Spectral representation of perturbations
c. Perturbation orography
A perturbation orography is included via the variable P′ = P − Pg = Φ′ + RT∗q′, where Φ′ =
The relaxation technique used here is rather crude, but appears to perform acceptably. One problem with this type of relaxation scheme is that the perturbations are not relaxed to the desired values instantaneously, thus giving rise to Gibbs phenomena. Another approach is to directly force the perturbation tendencies to satisfy the boundary condition at every time step. This is essentially what the blending technique of Juang and Kanamitsu (1994) does. However, with a perturbation orography this will require some modification of the basic equations. Also one has to be concerned about possible aliasing effects, since the tendencies contain quadratic terms and smoothing them to zero at the boundaries will result in higher than quadratic terms. Also of concern are boundary reflections caused by the zero boundary condition. In Juang and Kanamitsu (1994) it was suggested, based on a number of forecasts, that blending may not be necessary. We have taken that suggestion here for the time being.
e. Truncation and diffusion
We have included an implicit ∇4 horizontal diffusion that is very similar to that used in the FSUGSM. A weak time filter as described in Asselin (1972) is used to suppress some computational modes due to the leapfrog scheme.
f. Model physics and vertical structure
The regional model has the same physics and vertical structure as the FSUGSM. Both models use the same vertical σ-coordinate system where σ = P/Ps (P is pressure and Ps is surface pressure). In the present study, there are 14 sigma levels in the vertical, except for the moisture variable for which there are 11. The discrete sigma levels used are 0.05, 0.07, 0.10, 0.20, 0.30, 0.40, 0.50, 0.60, 0.70, 0.80, 0.90, 0.95, and 0.99. Model physics include long- and shortwave radiation, including the effect of clouds. The radiation calculations are performed every 3 h in the global model and every hour in the regional model. The model uses Kuo-type parameterization of deep cumulus convection, and includes large-scale condensation. Boundary layer processes such as sensible heat flux, water vapor flux, and surface stress are included. Further details can be found in the appendix.
3. Erin case study
Erin became a named storm at about 0000 UTC 31 July, located at 22.3°N, 73.2°W with a central low pressure of 1004 mb and winds of 45 kt. Within 24 h it gained minimal hurricane strength, and at 0615 UTC 2 August Erin made its first landfall near Vero Beach, Florida, as a category 1 hurricane with winds of 75 kt and a central low pressure of 984 mb. The storm weakened as it crossed central Florida and entered the Gulf of Mexico, where it regained strength. Erin became a category 2 storm while in the gulf, took a northward turn, and made its second landfall near Pensacola, Florida, on 3 August 1600 UTC with sustained winds near 80 kt and low pressure of 973 mb. We perform two regional forecasts. In the first run we use a T106 global model as a base and obtain initial perturbations from a T170 dataset. No bogussing of the storm was done. In the second run we use physical initialization (Krishnamurti et al. 1984, 1991). Since a T170 physically initialized dataset was not available at the time of this study, we used a T126 physically initialized global model output as a base field and set the initial perturbations to zero. A storm bogus was used.
a. T106 base case study
For this study we used a T106 global model output for a base field with a nest interval of 3 h. The initial regional perturbations were obtained from a subdomain of a T170 global model dataset. Both the T106 and T170 global models were initialized with 1.125° ECMWF analyses. The regional model transform grid has 0.5° spacing, effectively giving it an equivalent resolution of a T240 global model. Since the initial perturbations were obtained from a T170 model, the regional model was not initialized to its highest resolution. We used rectangular truncation in the regional model, designated as R(85, 42), with 85 waves in the east–west direction and 42 waves in north–south direction. The time step of the T106 global model is 450 s, and 240 s for the regional model.
We began our forecast of Erin at 1200 UTC 31 July. At that time, Erin was a tropical storm located at 22.8°N, 73.9°W with winds of 55 kt and a central low pressure of 999 mb. The initial state of the regional model showed a closed circulation near the observed storm center, but with weaker winds (near surface) of about 40 kt. Both the T106 global model and the regional model showed Erin developing to hurricane strength within 12 h of the forecast start time, with somewhat stronger than observed maximum winds of about 78 kt and low pressures of 987–988 mb.
In Fig. 1 we show wind speeds near the surface (lowest sigma level) predicted by the regional and T106 global models after 30 h. The regional model produces a more realistic looking storm due to its higher resolution. The maximum wind speeds are much closer to the center of circulation, as would be expected in an actual hurricane. The coarser T106 model tends to expand the size of the storm, due to its inability to fully resolve it. Such expansion of storm features due to coarse resolution was noted in a previous study (Krishnamurti et al. 1995). Corresponding streamlines are shown in Fig. 2. The circulation appears better organized in the regional model, and the center of the circulation indicates a more northerly track predicted by the regional model. In Fig. 3 we show the surface pressures. The regional model showed a deeper low near the center of circulation. Note the lower pressures over the mountains in Cuba in the regional model due to the increased mountain heights. Thus after 30 h, the regional model showed the hurricane continuing to strengthen, with top winds of 80 kt and a low pressure of 981 mb. This compares very well with the observed wind speeds of 75 kt and low pressure of 980 mb. The T106 global model showed Erin had weakened, with winds of 65 kt and a low pressure of 993 mb. Both models predicted first landfall in southeastern Florida at approximately 0000–0300 UTC 2 August, a few hours ahead of the observed time (0615 UTC 2 August). The regional model predicted a somewhat closer landfall to the observed, but was still about 200 km to the south.
In Fig. 4 we show a plot of the central low pressure of both models compared with the observed for the first 78 h of the forecast. Both models showed some weakening as Erin crossed central Florida and headed into the Gulf of Mexico. The T106 model showed Erin remaining as a strong tropical storm in the gulf, while the regional model showed Erin weakening but maintaining hurricane strength as it crossed Florida and continued into the gulf. One reason the regional model did not show Erin losing much strength while crossing Florida is probably due to the fact that the southern half of the Florida peninsula is not well resolved in the model, with the land–sea matrix indicating that Erin is moving mostly over ocean rather than land. This could be remedied by initializing the regional model to its fullest resolution. The regional model did show Erin strengthening in the gulf to a category 2 hurricane as was observed, with 80-kt winds and a low of 979 mb compared to 974 mb observed just prior to landfall. The T106 model continued to show only tropical storm strength.
The intensity forecast of the regional model was much better than the T106 model, and remarkably close to the observed. The predicted low pressure by the regional model never deviated by more than 6 mb from the observed 6-h report during the forecast and has an average deviation (per 6-h report) of less than 2.5 mb. This compares to a maximum deviation of 19 mb and average deviation of 7.3 mb for the T106 model. The forecast tracks of both models were consistently 100–200 km due south of the observed best track. The error in the track is likely due to an upper-level low near Florida, which slowed and steered Erin more northerly, that was not present in the initial analysis.
b. T126 base physical initialization case study
In this case study we used a T126 global model output base that was physically initialized over a 24-h integration period. A storm bogus was included prior to initialization. The forecast start time was again 1200 UTC 31 July. Unlike the previous case study, the initial perturbations were set to zero. The storm bogus profile was altered somewhat during the initialization process. As a result, the initial storm was again weaker than the observed with a central pressure of about 1005 mb and 40-kt winds. However, the initial vorticity appeared much improved compared to the previous case study.
The T126 global and regional models showed Erin reaching hurricane strength within 12–18 h of the forecast. The regional model showed Erin becoming a much stronger storm than is observed, briefly dropping to 963 mb before rising back to 977 mb at first landfall. We believe this overintensification may be due to the storm bogus profile, rather than initialization. The profile used was more appropriate for a T126 resolution model than the T240-equivalent resolution of the regional model. We observed a similar strong intensification using a T126 global model base with the same bogus but with normal mode initialization rather than physical initialization. When no bogus is used, as in the previous case study, the storm does not overintensify nearly as much. Further studies using a high-resolution storm bogus profile are clearly needed. The global model showed a central pressure of 980 mb at first landfall, but with stronger than observed winds of 85–90 kt. The regional model predicted landfall about 100 km south of the observed, compared to about 200 km south for the global model.
We show the surface winds after 60 h of forecast in Fig. 5 for both the regional and global models. The regional model predicted maximum winds of around 70 kt, compared to about 65 kt for the observed. The global model forecast weaker winds of around 40 kt, and the radius of maximum winds are well into land and even extends back into the Atlantic.
In Fig. 6 we show a plot of the surface pressure after a 66-h forecast. At this time, both models show the storm in nearly the same position. The regional model showed Erin reintensifying in the gulf, with the central pressure dropping to 978–979 mb, in agreement with the observed 979 mb. The global model showed Erin to be mainly of tropical storm to minimal hurricane strength with a low pressure of 987 mb. On the whole, the regional model produced a better intensity forecast in the gulf.
In Fig. 7 we show the tracks of Erin predicted by the T126 global and regional models as well as the observed best track for the first 3 days of the forecast. We determined the track by the position of minimum central pressure. The regional model produced a better track throughout the entire forecast. Both models show nearly the same position of final landfall near the Alabama–Florida border and very close to the observed landfall, though a few hours ahead of time. The storm tracks of both models were much better than those of the T106 global and regional models of the previous case study. In Fig. 8 we show the tracks of Erin predicted by the regional model with and without physical initialization. Apparently, physical initialization was able to bring about some component of the upper-level low that helped to steer Erin more northerly.
The case studies presented here show that the FSU nested regional model performs well. The intensity and track forecasts of the regional model for Hurricane Erin are generally better than that predicted by the global model, and in very good agreement with the observed. In the first case study, where no storm bogussing was done, the predicted central pressure deviated from the observed by about 2.5 mb on average. In the second case study, with storm bogussing and physical initialization, the storm track (neglecting phase differences) rarely deviated by more than 100 km from the observed. We do not, by any means, assert that the regional model is as accurate as these results suggest. Many case studies will be needed to determine the performance of the model. Rather these case studies show the capabilities of the regional model afforded by the increased resolution. The regional model was able to produce lower central pressures of the storm. The storm cyclone was generally more intense and smaller than that produced by the coarser global model, and closer in size to that expected for a hurricane such as Erin. In neither of these studies was the regional model initialized to its fullest resolution. We hope to increase the resolution of the initial state in future studies.
The primary motivation for developing a regional model is computational economy, allowing us to go to horizontal resolutions that are too expensive or impractical in a global model. In Table 1 we show CPU time on a Cray YMP for a 1-day forecast for the regional model and T106 and T255 global models. The computational advantage of the regional model is clearly evident. The regional model is more than 24 times faster than a T255 global model, which has about the same resolution. The regional model, however, can be run at even higher resolutions. Experiments with T300-equivalent or better resolutions are currently under way.
I want to thank T. N. Krishnamurti for getting me started on this project. I am grateful to Dr. H. S. Bedi for his invaluable advice on the FSU Global Spectral Model and general words of wisdom. I am indebted to C. E. Williford for providing me with the datasets used in these experiments and useful discussions.
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FSU Global Spectral Model
Below is an outline of the FSU Global Spectral Model:
(a) Independent variables: (x, y, σ, t).
(b) Dependent variables: vorticity, divergence, surface pressure, vertical velocity, temperature, and humidity.
(c) Horizontal resolution: triangular 170 waves.
(d) Vertical resolution: 14 layers between roughly 10 and 1000 mb.
(e) Semi-implicit time integration scheme.
(f) Envelope orography (Wallace et al. 1983).
(g) Centered differences in the vertical for all variables except humidity, which is handled by an upstream differencing scheme.
(h) Fourth-order horizontal diffusion (Kanamitsu et al. 1983).
(i) Kuo-type cumulus parameterization (Krishnamurti et al. 1983).
(j) Shallow convection (Tiedke 1984).
(k) Dry convective adjustment.
(l) Large-scale condensation (Kanamitsu 1975).
(m) Surface fluxes via similarity theory (Businger et al. 1971).
(n) Vertical distribution of fluxes utilizing diffusive formulation where the exchange coefficients are functions of the Richardson number (Louis 1979).
(o) Long- and shortwave radiative fluxes based on a band model (Harshvardan and Corsetti 1984; Lacis and Hansen 1974).
(p) Diurnal cycle (day/night by variation of zenith angle).
(q) Parameterization of low, middle, and high clouds based on threshold relative humidity for radiative transfer calculations.
(r) Surface energy balance coupled to the similarity theory (Krishnamurti et al. 1991)
(s) Nonlinear normal mode initialization—five vertical modes (Kitade 1983).
(t) Physical initialization (Krishnamurti et al. 1991).
Model performance—CPU time (Cray YMP) for 1-day forecast.