a. Hurricane Jeanne (2004)

Jeanne (2004) was a tropical storm with winds less than 35 m s⁻¹ before 17 September. It then gradually strengthened to a hurricane with 42 m s⁻¹ winds by 23 September. Jeanne crossed cooler waters and it decayed to 35 m s⁻¹ by 0000 UTC 24 September. At the time of Fig. 2a (∼1800 UTC 24 September) Jeanne was in the middle of a reintensification period with the winds increasing to 50 m s⁻¹ by 1200 UTC 25 September. Jeanne made landfall on the east coast of Florida at 0400 UTC on 26 September.

b. Hurricane Katrina (2005)

Katrina (2005) was one of the most devastating natural disasters in the history of the United States. Katrina formed around 1200 UTC 24 August and reached hurricane status around 0000 UTC on 26 August. It nearly doubled in size on 27 August and strengthened from a category 3 hurricane to a category 5 in less than 12 h reaching an intensity of 72 m s⁻¹ by 1200 UTC 28 August. Figure 2b shows the wind field/radar reflectivity structure at ∼1800 UTC on 27 August shortly before Katrina entered its rapid intensification stage and about 40 h before landfall.

c. Hurricane Rita (2005)

Rita (2005) was an intense hurricane that reached category 5 strength over the central Gulf of Mexico and had the fourth lowest central pressure on record for the Atlantic. Rita reached hurricane status ∼1200 UTC on 20 September about 150 km east-southeast of Key West, Florida. Once over the warm waters of the Loop Current it rapidly intensified to 75 m s⁻¹ by 0000 UTC 22 September. Rita completed the transition from a tropical storm to a category 5 hurricane in less than 36 h. Figure 2c shows Rita at ∼1800 UTC on 20 September as it was passing 60 km south of Key West and just before it began its rapid intensification.

a. WRF 3DVAR and its assimilation of airborne Doppler radar data

The configuration of WRF 3DVAR is based on the incremental formulation of Courtier et al. (1994), producing a multivariate incremental analysis in model space. The minimization is performed in preconditioned control variable space. The preconditioned control variables are designed based on the characteristics of the ARW model. They are streamfunction, unbalanced velocity potential, unbalanced temperature, pseudo-relative humidity, and unbalanced surface pressure. The unbalanced control variables are constructed in WRF 3DVAR relative to the geostrophic and hydrostatic relations (Barker et al. 2004; Skamarock et al. 2005). A key to 3DVAR is the background error covariance matrix. In this study, the background errors (BEs) were generated using the 30-day 12- and 24-h forecast around the Hurricane Jeanne case. We separately generated the BE for 12- and 4-km resolution domain. Empirical orthogonal functions (EOFs) are applied for the vertical component background error covariance matrix 𝗕. Recursive filtering is performed for the horizontal component of 𝗕. Eigenvectors/eigenvalues of the vertical component are estimated using the National Meteorological Center (NMC) method (Parrish and Derber 1992). Estimates of the recursive filter’s characteristic length scales depend on the variable and its vertical mode. Regression coefficients to calculate the total part of variables from unbalanced variables are also estimated via the NMC method.

The Doppler radar data assimilations in WRF 3DVAR were described in Xiao et al. (2005, 2007) and Xiao and Sun (2007). The capability has been tested in operational forecasting in Korea (Xiao et al. 2008). Assimilating ADR data for hurricane initialization is a new application of WRF 3DVAR. To extract the most useful information from the data, high-resolution assimilation is necessary. In this study, the ADR data assimilation occurs on a domain of 4-km horizontal grid spacing, nested within an outer domain with 12-km spacing. Only conventional Global Telecommunications System (GTS) data are assimilated on the outer domain. The observations, time–space corrected following the vortex, are assumed to be simultaneous. For the airborne Doppler winds, the data from NOAA/HRD are wind components. Assimilation of the winds is straightforward, similar to the assimilation of conventional sounding winds but with a prespecified error (2 m s⁻¹). For the ADR reflectivity data, we follow the same procedure as in Xiao et al. (2007), in that a warm-rain microphysics scheme is used to bridge the relationship among rainwater, cloud water, moisture, and temperature. When rainwater information (from reflectivity) enters into the minimization iteration procedure, the forward microphysical process and its backward adjoint distribute this information to the increments of other variables (under the constraint of the warm-rain microphysics scheme). The warm-rain microphysics is used in WRF 3DVAR because it captures the major hydrometeor process and is relatively easy for the tangent linear and adjoint development. However, it misses the important roles the ice particles play in the development of a hurricane. There have been many studies that show the impact of ice-phased precipitation on hurricane development (Lord et al. 1984; McFarquhar and Black 2004; McFarquhar et al. 2006; Marks et al. 2008; Rogers et al. 2007). We will discuss the drawbacks of using warm-rain microphysics scheme for hurricane initialization in the last section. The observational error for the ADR wind and reflectivity are empirically set as 2.0 m s⁻¹ and 2 dBZ, respectively.

b. AHW model and domain configurations

The numerical model used in this study is the AHW, a derivative of ARW version 2¹ (Skamarock et al. 2005). It is a compressible, three-dimensional, nonhydrostatic model using terrain-following coordinates and its governing equations are written in flux form. The Runge–Kutta third-order time scheme is employed and fifth- and third-order advection schemes are chosen for the horizontal and vertical directions, respectively.

For the study of Hurricane Jeanne, a spatially fixed, inner domain of 4-km grid spacing was nested interactively within a 12-km outer domain for all simulations. The grid dimensions were 400 × 301 × 35 for domain 1 and 502 × 451 × 35 for domain 2 in the east–west, north–south, and vertical directions, respectively. The following parameterizations were activated for both domains: WRF Single-Moment 3 classes (WSM-3) microphysics scheme (Dudhia 1989); the new Kain–Fritch cumulus parameterization (Kain 2004), which includes deep and shallow convection (only on the outer domain); the Yonsei University (YSU) boundary layer parameterization, which accounts for local and nonlocal mixing (Hong and Noh 2006); the Dudhia shortwave parameterization (Dudhia 1989); and the Rapid Radiative Transfer Model (RRTM) longwave parameterization (Mlawer et al. 1997).

For Hurricanes Katrina and Rita, two moving nests were nested interactively within the outer domain. The innermost nest (domain 3), with 1.33-km grid spacing, was centered within the nest of 4-km grid spacing (domain 2), and the outer domain (domain 1) had a 12-km grid spacing as before. The dimensions were 460 × 351 × 35 for domain 1, 202 × 202 × 35 for domain 2, and 241 × 241 × 35 for domain 3, where domain 2 and 3 are moving nested following the hurricane tracks. The location of the nests was determined by the minimum geopotential height at 500 hPa and was repositioned every 15 min. The physics parameterizations are the same as for the simulations of Hurricane Jeanne, except the microphysics was changed to the WRF Single-Moment 5 classes (WSM-5) scheme (Hong et al. 2004).

The model was integrated for 48 h with a time step of 60, 20, and 6.7 s for domains 1, 2, and 3, respectively.

c. Experimental design

The NCEP/GFS analysis with a spatial resolution of 1° × 1° was used to produce the first guess for all data assimilation. Four sets of experiments for each case were conducted (Table 1): the control run (CTL), which used the NCEP/GFS analysis as the initial condition; the second experiment (GTS), which assimilated only the conventional GTS data in all domains; the third run (GRV), which assimilated GTS in the outer domain (12 km) and the GTS plus airborne radar wind data on domains 2 and 3; and the fourth experiment (GVZ), which used the same strategy as GRV except both the ADR wind and reflectivity data were assimilated on domains 2 and 3.

a. ADR wind assimilation

Figure 3 shows the sea level pressure (SLP) and 10-m winds with and without ADR wind assimilation for Hurricane Jeanne (2004) at 1800 UTC 24 September (the initialization time of the forecast). The cyclonic circulation is strengthened with a maximum surface wind (MSW) speed increase to 38 m s⁻¹ (Fig. 3c), compared with 23.4 m s⁻¹ in GTS (Fig. 3b) and 23.7 m s⁻¹ in CTRL (Fig. 3a). The observed MSW speed is 44 m s⁻¹. Its central sea level pressure (CSLP) is decreased in experiment GRV after assimilating ADR wind data. Along with the intensity enhancement in wind speeds, the CSLP is correspondingly decreased about 5 hPa through the multivariate constraint in WRF 3DVAR. Figure 4 shows the H*WIND analysis at 2100 UTC 24 September, the nearest in time to 1800 UTC. H*WIND combines data from reconnaissance aircraft, dropsondes, satellite-derived winds, in situ observations, and stepped-frequency microwave radiometer retrievals (Powell et al. 1998; Uhlhorn and Black 2003), and produces a gridded storm-centered 10-m, 1-min, marine exposure sustained wind field. The GRV wind speed distribution in Fig. 3c is much closer to the H*WIND than those in Figs. 3a,b. There are two maximum wind centers in both northwest and southeast quadrants of the vortex in Fig. 3c, due to the data coverage in WRF 3DVAR assimilation. The wind distribution in the assimilated one flight leg data (Fig. 2a) presents the major wind innovations in WRF 3DVAR assimilation are in the northwest and southeast quadrants.

Vertical cross sections of horizontal wind along the line AB in Fig. 3c for GRV and GTS experiments are presented in Fig. 5. A strong and clear maximum wind band with a contracted radius in Fig. 5b is produced after assimilating the ADR wind data. Some asymmetry is apparent, and the maximum wind speed is located at around 900 hPa (Fig. 5b). The pattern and the maximum value concentrate in a small scale within the vortex. However, in the GTS data assimilation experiment, the hurricane vortex circulation is very weak and the radius of maximum wind is too large (Figs. 5a). The CTL experiment has a similar structure as GTS, so it is omitted in the discussion. To our knowledge, the vortex feature in Fig. 5b has not been produced in 3DVAR-based hurricane initialization with any other kind of real data before, except for methods that prescribe a synthetic vortex (Xiao et al. 2009). The recovery of the vortex feature is attributed to assimilating the very high horizontal and vertical resolution ADR data in the inner-core region of the storm. The assimilated airborne Doppler radar data extend up to the top of troposphere. Both the data vertical coverage and the WRF 3DVAR structure function are the reasons of the coherent deep vortex structure in Fig. 5b.

The azimuthally mean tangential wind in Fig. 6 shows again the maximum wind located at the lower level, consistent with the cross sections in Fig. 5. The radius of maximum winds with the ADR wind assimilation is much closer to the hurricane vortex message (74 km) than the GTS experiment. The storm with the ADR wind data assimilation is enhanced with a 40 m s⁻¹ maximum azimuth mean of tangential wind in the boundary layer, whereas peak winds are quite weak without ADR data assimilation, about 24 m s⁻¹ in both the GTS and CTL experiments.

WRF 3DVAR also produces temperature increments using ADR data due to the multivariate incremental structure in the system (Fig. 7). At 300 hPa, both temperature and wind vector increments in the vortex region in GTS experiment are small (Fig. 7a) because no GTS observations exist in the vortex at high levels. The small increments for both the wind vector and temperature in the hurricane vortex region are due to the covariance with the hurricane environment. However, once assimilating the ADR wind data in the vortex region, positive temperature increments occur associated with the strong wind vector increments and the maximum positive temperature increment is 1.117°C at 300 hPa in the vortex inner region (Fig. 7b). In the vertical cross sections above the hurricane vortex, GTS shows only slight increments of temperature above 850 hPa (Fig. 7c). On the contrary, GRV produces a vertical incremental structure with the largest positive temperature increments in the high troposphere around 300 hPa. In the middle troposphere around 500 hPa, there is a negative layer of temperature increments. Notice that both GTS and GRV produce a negative layer of temperature increments above the vortex near the surface due to the assimilation of winds from buoys and ships. It is clear that the ADR data assimilation not only improves the three-dimensional inner vortex wind structure, but also contributes to the vortex’s warm-core structure as well. It should be mentioned that, the temperature response to the wind increments is rather small due to the use of more “climatological” error covariance from 1-month forecasts as opposed to more flow-dependant error covariance that would know about the presence of the hurricane. The vortex region is highly imbalanced; the climatological error covariance can only construct a small part of the warm core, a highly unbalanced structure.

b. ADR reflectivity assimilation

Figure 8 shows the ADR reflectivity assimilation results of Hurricane Jeanne at 1800 UTC 24 September 2004 in the GVZ experiment. The CSLP (987 hPa) and 10-m wind analysis in GVZ (Fig. 8a) are similar to the results of GRV (Fig. 3c). However, GVZ produces cloud water and rainwater analyses after assimilating reflectivity data. Compared with the observation (Fig. 2a), the reflectivity information is only partially recovered to the analysis. Because the microphysics involved in the reflectivity assimilation has many “on–off” switches, the linearity assumption used in the WRF 3DVAR for the reflectivity assimilation is compromised. In addition, the warm-rain scheme in WRF 3DVAR does not represent well the microphysical process above the melting level (∼500 hPa) for hurricanes. Nevertheless, GVZ contains some hydrometeor representation in the initialization, compared to no hydrometeors in GRV.

HH Figures 8b–d illustrates the changes occurring in variables not directly assimilated owing to the addition of reflectivity data. These changes arise from the multivariate structure in the analysis. The temperature and water vapor mixing ratio in the mid to lower levels above hurricane vortex are decreased, while in upper levels are increased (Figs. 8b,c). The storm center’s stability ∂θ_e/∂z increases (Fig. 8d), while at about 120-km radius, the stability between 900 and 500 hPa decreases, with a maximum difference of θ_e at around 900 hPa. GVZ also produces a warmer upper-level core. In general, ADR reflectivity assimilation increases the atmospheric stability in the vortex center, while decreasing stability outside the eye region.

a. Hurricane structure

In general, the ADR data assimilation improves the forecast of hurricane structures for all of the three cases during the entire 48-h forecast period. In this section, we take Hurricane Jeanne (2004) as an example to illustrate these improvements by comparing the forecast structures with and without ADR data assimilation.

Figure 9 presents SLP and 10-m wind vectors and wind speed at 24-h from the GTS, GRV, and GVZ experiments. The 24-h distribution of the predicted SLP shows quite different structure in the three experiments. The CSLP is only 986 hPa in GTS (Fig. 9a), 972 hPa in GRV (Fig. 9b), and 965 hPa in GVZ (Fig. 9c). The observed CSLP at the time is 952 hPa (Fig. 1). Although all do not attain the observed SLP minimum, the assimilation of ADR data results in better SLP forecast relative to the experiment of GTS. The predicted wind structures also exhibit significant differences. GRV and GVZ both increase the 10-m MSW speed to 42.4 and 47.6 m s⁻¹, respectively, up from 40.3 m s⁻¹ in GTS. The MSW speed of 47.6 m s⁻¹ in GVZ is very close to the observed (48.8 m s⁻¹). Comparing the wind features among Figs. 9a–c, another obvious difference is that ADR data assimilation experiments (GRV and GVZ) predict structures with smaller radius of MSW than experiment GTS (cf. Figs. 9b,c with Fig. 9a). The forecast in experiment GTS is effectively missing an inner core.

Cross sections of equivalent potential temperature (θ_e) and horizontal wind speed through Hurricane Jeanne (2004) for GTS, GRV, and GVZ experiments at 1800 UTC 25 September (24-h forecast) are shown in Fig. 10. Consistent with results shown in Fig. 9, the inner core is effectively absent in experiment GTS. The GFS analysis, which is the background state, is too coarse to resolve any inner-core structure; the GTS data cannot enhance the inner core, and apparently the model is unable to contract the inner core within the first 24 h of the forecast. In experiment GRV (Fig. 10b), there is an inner core showing evidence of a frontlike structure in θ_e with vertical plume of high θ_e values radially inward from the strongest winds. With the reflectivity also assimilated (Fig. 10c), there arises a more obvious outward slope of the tangential wind contours (and therefore angular momentum surfaces—not shown), and a further, modest increase in intensity. The structures in both Figs. 10b,c are clearly more indicative of an intensifying hurricane near category 3 intensity than is the structure in Fig. 10a (e.g., Eliassen 1959; Emanuel 1995).

Figure 11 shows the composite reflectivity at 24- and 36-h forecasts for the GTS, GRV, and GVZ experiments. Experiment GTS does not produce an eyewall, indicating that the vortex is not well organized (Figs. 11a,d). However, GRV and GVZ produce well-organized hurricane structures in radar reflectivity with compact eyewalls embedded in the vortex. Comparison of the observed reflectivity at landfall (Fig. 12) with these forecasts suggests that both GRV and GVZ produce a realistic distribution of reflectivity, but the heavier rainband over the east coast of Florida and the suggestion of a break in reflectivity on the east side of the eyewall in GVZ matches the observations somewhat better.

b. Hurricane track and intensity verification

Verification of track and intensity for all three hurricanes is discussed in this subsection for experiments CTL, GTS, GRV, and GVZ. ADR data assimilation improves the intensity forecast for all three hurricanes (Jeanne, Katrina, and Rita). It seems that the forecast improvement in track is not as significant as in intensity, but it is noticeable. It is consistent with the idea that the hurricane track is mostly influenced by the environment, instead of the inner structure of the hurricane. In this subsection, verification results of Hurricane Jeanne (2004) are presented in detail, consistent with previous sections. The average forecast errors of track, CSLP, and MSW for all three cases are then shown. The forecast error is calculated at a 6-h interval using the best-track data as truth.

Figure 13 shows the track forecasts by experiments CTL, GTS, GRV, and GVZ for Jeanne (2004). There is an initial position deviation in CTL (Fig. 13a) with a position error of 40 km (Fig. 13b). The initial position is adjusted closer to the right position in all data assimilation experiments (GTS, GRV, and GVZ). In the subsequent forecast, all data assimilation experiments have less track deviation than CTL, which is biased to the south of the best track (Fig. 13a). GTS follows the best track in the first 18-h forecasts, and then turns to the south of best track until 36 h. With ADR data assimilation, GRV and GVZ improve the track forecast, especially during the first 24-h forecast. However, reflectivity does not add much benefit to the track forecast (Fig. 13b). In general, all experiments predict a slightly slower-moving speed than observed, with the predicted landfall times about 4–5 h later than best track.

The 48-h evolution of Hurricane Jeanne’s (2004) CSLP and MSW from the best track and the four experiments (CTL, GTS, GRV, and GVZ) are shown in Fig. 14. GTS does not improve the CSLP and MSW in the whole 48-h period. Figure 14a shows that ADR data assimilation experiments (GRV and GVZ) significantly improve the CSLP forecast compared with CTL and GTS, although the storm is still not as deep as observed. The CSLP error of GRV (GVZ) is reduced from 37 hPa in CTL to 23 hPa (12 hPa) at 24 h, and from 20 to 8 hPa (<1 hPa) at 48 h. Similarly, the MSW time series also shows the improvement due to ADR data assimilation. At 24 h, GRV (GVZ) predicts MSW of 92 kt (98 kt), compared with the observed MSW of 105 kt. At the same time, experiments CTL and GTS predict 86 and 82 kt, respectively. Overall, the improved intensity forecast for Jeanne offered by assimilating ADR data lasts between 24 and 48 h and is better maintained for CSLP than for maximum wind.

Similar experiments for Hurricanes Katrina and Rita (2005) were also conducted. The airborne Doppler radar data are at 1800 UTC 27 August 2005 for Hurricane Katrina and at 1800 UTC 20 September 2005 for Hurricane Rita. The hurricane forecasts that initialize from the times of data undergo rapid intensifications and follow weakening periods for the two hurricanes. Instead of describing the verifications for each case, the average mean absolute errors for hurricane position, CSLP, and MSW are calculated for the three cases (Fig. 15). The average mean absolute track errors of CTL at 24- and 48-h forecasts are 58 and 125 km, respectively (Fig. 15a). The GTS run improves the track forecast with position errors at 24 and 48 h reduced to 50 and 83 km, respectively. ADR data assimilation further improves the track forecasts, with position errors at 24 and 48 h for GRV (GVZ) reduced to 28 km (32 km) and 90 km (91 km), respectively. These results demonstrate that ADR data assimilation benefits the hurricane track forecast.

Figures 15b,c indicate that the intensity forecast is more significantly improved by ADR data assimilation than track. While GTS data shows no universal benefits for the intensity forecast, the CSLP average mean absolute errors with ADR data initialization are reduced at each forecast lead time within 48 h. The improvement in MSW is maintained for roughly 30 h, echoing the results for Jeanne alone. Significant reduction of MSW mean absolute errors occurs at the initial time. On average, the error reduction is nearly 29 kt (25 kt) by the ADR data initialization in GRV (GVZ). In response to the vortex wind correction, the CSLP is decreased. The average decrease from the three cases is about 7 hPa, which is not as significant as the MSW increase. The results indicate that the current correlation between CSLP and MSW is relatively weak in the WRF 3DVAR system. The increments in vortex dynamical fields obtained by assimilating ADR wind do not result in a correspondingly large pressure response. Because the background error statistics used in this study were based on statistics averaged over an entire month, they are not flow dependent and therefore do not reflect the vortex structure among variables in the background covariance. We anticipate further improvements by using specific error covariance that recognizes the hurricane vortex structure.

Nevertheless, the hurricane-intensity forecasts are improved with ADR data assimilation using the relatively computationally inexpensive 3DVAR approach. The largest error reduction of CSLP and MSW occurs at 24 h. At 48 h, the CSLPs of both GRV and GVZ experiments still show less error than CTL and GTS. However, the MSW errors at 48 h are larger than CTL and GTS experiments. Comparing GVZ with GRV, we notice that adding ADR reflectivity in hurricane initialization (GVZ) further reduces the hurricane-intensity forecast errors from GRV. Assimilating reflectivity has added value for hurricane-intensity forecast (Figs. 15b,c), even though it does not show much benefit in track forecast (Fig. 15a).

In order for the readers to know the current status of WRF 3DVAR hurricane initialization compared with other schemes, such as the GFDL hurricane initialization scheme, we also calculated the average mean absolute errors of the operational GFDL results for the three cases (dashed line with circle in Fig. 15). The experiment using WRF 3DVAR with ADR data (GVZ) produces better track and intensity forecasts than the GFDL results beyond 20 h. Although the short-term (<20 h) hurricane forecasts (track, CSLP, and WSM) show significant improvement from CTL; however, WRF 3DVAR hurricane initialization still needs further development for short-term hurricane forecasts compared with GFDL scheme. Using previous cycling WRF forecast (with the spunup vortex) as the background for 3DVAR hurricane initialization can further improve the short-term hurricane forecast. As we know, variational data assimilation is a weighted combination of the background and observation information. Because the vortex position in the background filed from previous WRF forecast is not necessarily consistent with the observation, some kind of relocation may be necessary. We have started this work and will report the results in the future.

Examining the maximum errors in our experiments for the three cases, we found that the ability to reduce intensity errors by ADR data assimilation decreases as hurricane intensity increases. CTL has the maximum CSLP error of 35 hPa for Jeanne, 48 hPa for Katrina, and 52 hPa for Rita at initial time. The improvement from ADR data initialization is larger for Hurricane Jeanne than for Katrina and Rita. There are two reasons for this behavior. First, only a single set of background error statistics are computed and these represent Jeanne. The background error covariance used in the experiments for Hurricanes Katrina and Rita (2005) are interpolated from the covariance used for the experiments for Hurricane Jeanne (2004). Second, Hurricanes Katrina and Rita (2005) are much stronger than Jeanne. Hurricane Jeanne (2004) is a category 3 hurricane with a strongest intensity of 951 hPa, whereas both Katrina and Rita (2005) are category 5 hurricanes with strongest intensities of 902 and 897 hPa, respectively. Because of the intensity of Katrina and Rita, and in particular the fact that both storms were entering a rapid intensification stage, the use of monthly mean covariance statistics should be a worse approximation to the true statistics than for Jeanne. It is possible that the approach used herein will be most effective for initialization of larger tropical cyclones or systems not undergoing rapid intensification at initialization time. It seems logical that time-dependent and flow-dependent assimilation strategies will be essential for properly initializing small and rapidly intensifying storms.