1. Introduction

Numerical prediction of tropical cyclone tracks has improved enormously over the past few decades, but there is little skill in forecasting the storm’s intensity change (Elsberry et al. 1992) and inner-core structures (Liu et al. 1997). In fact, hurricane intensity change is closely related to the evolving three-dimensional structures of the hurricane. The difficulties in the prediction of hurricane intensity and inner-core structure are associated with insufficient observations over the oceans and with the limitations of forecast models, such as low-resolution, hydrostatic dynamics, crude physical parameterization, and the inability to treat multiscale interactions. Recently, hurricane forecast models at high resolution have greatly improved as a result of advances in computer architecture. More sophisticated models have been developed and used for hurricane study and forecast. The Florida State University atmospheric global circulation model was run at a resolution as high as T213, which can rebuild some mesoscale signatures of the observed hurricane features with the help of satellite data using physical initialization (Krishnamurti et al. 1995). The well-known Geophysical Fluid Dynamics Laboratory (GFDL) hurricane model has been running operationally with a multiply nested movable mesh system at a grid resolution as high as 6 km, and a very sophisticated model initialization scheme (Bender et al. 1993; Kurihara et al. 1995; Ross and Kurihara 1995). The Pennsylvania State University–National Center for Atmospheric Research (Penn State–NCAR Mesoscale Model version 5 MM5) was also employed to simulate the development of Hurricane Andrew of 1992 (Liu et al. 1997) and Hurricane Felix of 1995 (Zou and Xiao 2000). It was run at a grid spacing of less than 10 km with explicit microphysics.

By contrast, advances in hurricane initialization have been relatively slow. With the sparse data coverage over the oceans, numerical analyses cannot adequately represent the initial circulations of tropical cyclones (Leslie and Holland 1995). Hurricane initialization becomes a necessary step before making a prediction. There are several approaches for hurricane initialization: (i) substitute a specified vortex circulation defined by an analytical expression for the analyzed vortex into the initial conditions (Mathur 1991; Ueno 1989; Serrano and Unden 1994; Leslie and Holland 1995), (ii) implant a“spinup” vortex generated by the same forecast model into the initial conditions (Kurihara and Ross 1993; Kurihara et al. 1995; Peng et al. 1993; Liu et al. 1997); and (iii) improve the initial conditions by making use of satellite–rain gauge based measurements of rainfall through a physical initialization procedure (Krishnamurti and Ross 1993, 1995, 1997, 1998). Although bogusing schemes are employed by many NWP centers, they do not always work well for a large range of tropical cyclones (Wang 1998). In order to minimize the effects of the “initial shock” resulting from the ad hoc assumptions in these bogus schemes, a procedure is usually needed to balance the model initial state. For example, the National Centers for Environmental Prediction (NCEP) global model generates bogus observations, which are run through the multivariate analysis scheme. This ensures that the initial state will be balanced. In the GFDL model, a quasi-balanced initial state is generated using a simplified version of the model in the bogus procedure (Kurihara and Ross 1993).

In the recent work of Zou and Xiao (2000), a variational bogus data assimilation (BDA) scheme was proposed to generate the structure of a tropical hurricane in the initial condition of a high-resolution mesoscale model. In the scheme, a specified surface low based on a few observed parameters was incorporated into the model’s initial conditions through a variational approach in which the forecasting model serves as a strong constraint. The initial vortex obtained by the proposed method contains a more realistic hurricane structure in all model fields and is also well adapted to the prediction model. Such an initialization procedure is not only objective, but also flexible for the inclusion of any available observations due to its variational formulation.

The BDA scheme worked well for the numerical simulation of Hurricane Felix (1995), a hurricane that made a recurvature prior to its landfall (Zou and Xiao 2000). In this paper, we further test the performance of the BDA scheme for the numerical simulations of Hurricane Fran (1996) during its landfall, which occurred from 0000 UTC 3 September to 0600 UTC 6 September 1996. We extend the work of Zou and Xiao (2000) to examine the BDA scheme’s sensitivity to model resolution, to the size of the specified vortex, and to the specification of bogused sea level pressure (SLP) versus wind. The advantage of studying this case is that there were extensive observational analyses after 0000 UTC 3 September 1996 that are available from the National Oceanic and Atmospheric Administration’s Hurricane Research Division (NOAA/HRD) for forecast verification, such as Weather Surveillance Radar-1988 Doppler (WSR-88D) reflectivities and surface wind analysis from before the landfall of Fran. These data are not available for Hurricane Felix (1995).

From 0000 UTC 3 September through 0600 UTC 6 September 1996, Hurricane Fran moved in a northwest direction in the western Atlantic, north of the Caribbean, and made landfall on the North Carolina coast around 0030 UTC 6 September (Fig. 1). At landfall, the minimum central SLP of Fran was estimated as 954 hPa and the maximum sustained surface winds were 51.7 m s⁻¹. The strongest winds occurred in streaks within the deep convective areas north and northeast of the center. Hurricane Fran was responsible for 34 deaths. The storm surge on the North Carolina coast destroyed or seriously damaged numerous beachfront houses. Extensive flooding caused additional damage in the Carolinas, Virginia, West Virginia, Maryland, Ohio, and Pennsylvania. Fran was responsible for an estimated $3.2 billion in damage to the United States (Pasch and Avila 1999).

This paper is organized as follows: The next section describes the experimental design, including a brief description of the hurricane variational initialization scheme. The BDA generated initial hurricane structure and the subsequent forecast are discussed in sections 3 and 4. Section 5 presents the sensitivity of the forecasted hurricane’s track and intensity to the size of initial vortex. The importance of bogused SLP versus wind fields in the BDA scheme is also tested in section 5. Finally, a summary and conclusions are provided in the last section.

2. Experimental design

b. The BDA scheme

The variational initialization scheme includes two parts. The first step is vortex specification. The second step is a minimization procedure forcing the forecast model to generate the hurricane vortex. The vortex specification applies to a few model variables and the minimization procedure produces fields for all model variables, objectively satisfying the dynamic and physical constraint for the atmosphere.

1) Vortex specification

The SLP of the hurricane vortex is specified according to Fujita’s formula (1952). The formula is expressed as a function of r (radial distance from the cyclone center) as follows:

View Expanded

where P_c is the hurricane’s central pressure, ΔP is a parameter related to the hurricane SLP gradient information (which will be determined by the maximum wind as shown below), and R is the estimated radius of maximum SLP gradient. The hurricane location and central pressure are specified according to the National Hurricane Center’s (NHC) observational report.

The wind can be derived from Eq. (1) in terms of the gradient wind relation, which gives

View Expanded

Therefore, the hurricane maximum wind V_max can be calculated according to Eq. (2), and ΔP in Eq. (1) can be adjusted so as to match the observed maximum wind. Here, ρ is air density (assumed constant). For Hurricane Fran (1996) with P_c = 977 hPa and V_max = 75 kt (38.8 m s⁻¹) at 0000 UTC 3 September 1996, the relation between ΔP and R is calculated and shown in Fig. 2. In addition, there were extensive observational analyses at NOAA/HRD after 0000 UTC 3 September 1996. The radius of maximum wind (RMW) at 0000 UTC 3 September 1996 is around 80 km (marked in Fig. 2). In the vertical, the wind has a weighting profile of 1.0, 1.0, 0.95, 0.85, 0.65, and 0.35 at the corresponding 1000-, 850-, 700-, 500-, 400-, and 300-hPa levels. This vertical weighting function is empirically determined. The gradient wind balances are not necessarily maintained at high levels before the minimization. During the minimization, model-constraint fields are then gradually produced while the BDA scheme fits the specified wind step by step.

The assigned pressure P_o and wind fields V_o values in the domain Ω are used as “observations” data to modify the MM5 analysis through a variational procedure, which is described below.

2) Minimization procedure

In this procedure, a cost function is defined as

JXJ_bJ_pJ_υ(3)

where

View Expanded

where X = (u, υ, w, p′, T, q)^T represents all the model variables at the initial time. The model variables u and υ are horizontal velocities, w is the vertical velocity, p′ is the pressure perturbation from a constant reference state, T is the temperature, and q is the specific humidity. In addition, X is gradually adjusted via an iteration procedures, X_b is the background analysis obtained from standard MM5 analysis fields with a crudely estimated diagonal error covariance matrix B, J_b represents the background term of the cost function, and J_p is the pressure term, in which P(r) represents the SLP of the model atmosphere. The hydrostatic relation is included in the operation to obtain the SLP from the surface pressure of the model. The wind term is J_υ, in which V(r, k) includes model wind velocity at sea level, 1000-, 850-, 700-, 500-, 400-, and 300-hPa level within the domain Ω. Also, Ω is a two-dimensional domain in the vicinity of the hurricane center. The rim of Ω is defined as the limit line while SLP calculated by Eq. (1) is larger than that of the MM5 analysis. It is obvious that the area of Ω is related to R in Eq. (1). Larger R is associated with the larger area of Ω. Here, Ω is not necessarily a circle and the average radius of Ω is about 2.5–3.5 times of R. Summation over t_i is carried out over half-hour windows at every 5 min.

For simplicity, the background error covariance B contains only the diagonal elements. In this study, background error variances are estimated from the differences between the 12-h MM5 model forecast and the MM5 analysis at the same time. The weightings in Eq. (4), W_p and W_υ, are treated as constants and determined empirically. We take W_p = 1.6 1/hPa² and W_υ = 0.185 s² m⁻² for all experiments (corresponding to 0.8 hPa pressure error and 2.325 m s⁻¹ wind error assumed).

In order to minimize (3), a forward model and its adjoint are required. In this paper, the MM5 adjoint system (Zou et al. 1997) is used as a tool to minimize the cost function J for the purpose of obtaining the optimum hurricane initial conditions. The hurricane variational initialization is carried out on the coarsest 54-km domain (domain A) and/or the intermediate 18-km domain (domain B). The model initial conditions for the 6-km domains (domain C_i) are interpolated from that of the 18-km domain.

The dynamic structure of the MM5 adjoint system is the same as the MM5 forecast model described above. However, the physical processes that were employed in the minimization procedure are different. These processes are the bulk aerodynamic planetary boundary layer parameterization, surface friction, surface fluxes, dry convective adjustment, Kuo-type cumulus parameterization, and large-scale precipitation process. The forward MM5 model and the backward adjoint version use the same set of model physics. The gradient is checked with a procedure suggested by Navon et al. (1992).

The limited-memory quasi-Newton method of Liu and Nocedal (1989) is used to minimize the objective function in this study. The background X_b (the standard MM5 analysis X⁽⁰⁾) is used as the first guess of X for the minimization procedure. In all experiments, 30 iterations are performed to minimize the cost function J. After the minimization, the model fields, not only pressure and wind fields, but also temperature and humidity fields, are adjusted. The new adjusted fields are taken as the model initial conditions for the subsequent simulation.

3. Results from the BDA scheme

4. Experimental results from the simulation of Hurricane Fran (1996)

According to the real-time surface wind analyses prepared by NOAA/HRD Atlantic Oceanographic and Meteorological Laboratory (AOML), Hurricane Fran had an RMW of close to 80 km at 0000 UTC 3 September 1996. Two experiments, A80 and B80, which both had an RMW of 80 km (close to the observed one), were carried out to examine the performance of the BDA scheme on the initialization and simulation of the landfall of Hurricane Fran while studying the sensitivities of the numerical results to the horizontal resolution of the assimilation model used in BDA.

Figure 5 shows the predicted hurricane tracks of Fran (1996) from CTL, A80, and B80, starting from 0000 UTC 3 September 1996. The NHC observed best track (OBS) is also plotted for comparison. Starting from 0000 UTC 3 September, the observed storm moved north-northwestward and landed at about 0030 UTC 6 September on the North Carolina coast. The CTL-simulated hurricane track is on the west side of the observed track and the position error at 78 h is about 352 km (see also Table 3). The landing time predicted by CTL had a 6-h delay. On the contrary, the hurricane tracks from A80 and B80 were on the east side of the observed track. The errors in both the movement and landing time were much reduced (Fig. 5 and Table 3). At 78 h of model integration, the track forecast made by B80 has about a 96.5-km position error. The forecasted landing time was about 1 h earlier than the observed one. This demonstrates the positive impact of the BDA scheme on the hurricane track prediction.

The hurricane track predicted by A80 with the BDA scheme carried out on 54-km resolution (domain A) was slightly degraded. The simulated track was farther to the east and slightly faster than that of B80.

Figure 6 depicts the time variation of the minimum central pressure (Fig. 6a) and the maximum low-level winds (Fig. 6b) from CTL, B80, and A80. The observed minimum central pressures and the maximum low-level winds from observations are also shown as a reference in the figure. The simulated maximum low-level wind is the highest value of the averaged 25th and 26th model level winds calculated at each grid point. Without the variational initialization procedure, model simulation (CTL) did not capture the intensity of Hurricane Fran, either in the central pressure or the maximum low-level winds. The errors from the B80 and A80 simulations initialized with the BDA technique were significantly smaller compared to CTL. The spinup problem seen during the first 6-h integration of A80 and B80, reflected in the central pressure jump during that interval in Fig. 6a, is probably due to the use of triply nested domains for predication and a single domain for initialization.

According to the HRD real-time surface wind analysis and NHC aircraft reconnaissance (Pasch and Avila 1999), Hurricane Fran had its maximum wind streak from the east to the north-northeast of the hurricane center. Figure 7 shows the surface analyses of the streamlines and isotachs (kt) at 0100 and 0700 UTC 5 September 1996 using the available observations and the aircraft reconnaissance by NOAA/HRD. The maximum winds are observed to the northeast of the hurricane center. The maximum surface wind at 0100 UTC is 95 kt (49.1 m s⁻¹). It is 90 kt (46.5 m s⁻¹) at 0700 UTC 5 September 1996. The simulations of the hurricane surface wind fields at 0100 and 0700 UTC 5 September 1996 are shown in Figs. 8a and 8b, respectively. We can see that the simulated maximum wind streaks were located in the northeast quadrant. The maximum surface wind was 49.6 m s⁻¹ at 0000 UTC 5 September and 56.3 m s⁻¹ at 0600 UTC 5 September 1999. The simulated wind features resemble the HRD real-time surface wind analysis. However, the simulated RMW is slightly smaller than the observations.

To show the structures of the simulated Hurricane Fran, the vertical cross sections of temperature, specific humidity, horizontal wind, and vertical velocity of the B80 simulated 49-h hurricane along line AB in Fig. 8a are given in Fig. 9. The temperature field had an evident warm core above the hurricane (Fig. 9a) because of compensating descent inside the hurricane eye. The maximum value of specific humidity occurred near the surface in the hurricane center (Fig. 9b). The regions of maximum moisture were in the eyewalls from lower to middle troposphere. The hurricane center area was not as moist as the eyewall regions from 600 to 950 hPa. As seen in Fig. 8a, the cross sections are along the direction of the most asymmetric wind. Therefore, the wind fields in Fig. 9c also depict the asymmetric characteristics in the vertical distribution. The radius of the maximum wind increases with the height. It is noted that the ascending motion (Fig. 9d) was also not axisymmetric. The axes of the ascending motion in the eyewall and the compensating descent in the hurricane eye were tilted. The maximum ascending vertical motion occurred between 400 and 500 hPa in the eyewall.

From the simulated results we can summarize the following characteristics of the hurricane’s flow and its thermodynamic structure. 1) The tangential or swirling wind of the hurricane is strongly asymmetric. The maximum wind speed occurred around 900 hPa. 2) The ascending vertical motion around the eye increases the moisture in that region, while the descent inside the eye makes the area of the hurricane center drier in the lower to middle troposphere. Near the surface at the hurricane center, the specific humidity reaches its maximum value. 3) The compensating descent inside the hurricane eye is the main reason for the formation of the warm core. Because the temperature increase does not occur in the eyewall where the convective latent heating release takes place, we suspect that convective latent heating release is not the main reason for the formation of the hurricane’s warm core.

At 66 h of model integration, the 6-km grid-spacing domain C was activated. Therefore, we obtain a detailed hurricane forecast structure that can be compared with the onshore radar reflectivity distribution observed during the landfall of Hurricane Fran. The radar reflectivity was estimated from the model cloud water, rainwater, and other ice particles such as graupel. The method for calculation of the model reflectivity was the same as that in Liu et al. (1997) and it was based on the R–Z relation of Jorgenson and Wills (1982) and Fujiyoshi et al. (1990). Contributions of cloud water to reflectivity were small and therefore neglected. For Hurricane Fran, the structure of the storm at landfall was well captured by the WSR-88D radar onshore at North Carolina (Fig. 10b). The radar-observed stronger reflectivity represents the deep convection that occurred in the north-northwest area of the hurricane center. The simulated reflectivities of Hurricane Fran immediately after its landfall (Fig. 10a) showed that the simulated strong reflectivities were also in the north-northwest section of the hurricane center. The model reproduced very well the intense precipitation in the north-northwestern direction of the hurricane center, in accordance with the radar observations onshore. The horizontal structure of radar reflectivities were highly asymmetric from both the simulation and observation. The simulated higher radar reflectivities were more intense over land, a feature consistent with observations from the onshore radar at 0028 UTC 6 September 1996.

It must be pointed out that the simulation of B80 has some shortcomings. The predicted hurricane intensity is still weak compared with observation. The spinup problem is not eliminated during the first 6-h integration. The simulated winds in the hurricane environment are weaker than the analysis from observations (cf. Fig. 7 with Fig. 8). It is necessary to conduct more experiments and case studies to improve the BDA scheme in the future.

5. Sensitivities of the BDA results to the model resolution, RMW, and bogus variable specification

The specification of the bogus vortex [see Eq. (1)] requires knowledge of the RMW. The RMW is not always known, nor can it be estimated exactly. Previous studies, such as Fiorino and Elsberry (1989), addressed the sensitivity of hurricane prediction to the RMW using barotropic models. In our study, however, the baroclinic effects were included through the use of the primitive-equation mesoscale model MM5. We will examine the influence of the specified RMW upon the prediction of hurricane track and intensity using BDA. In addition, we will also test the relative importance of specifying the bogused SLP versus wind in the BDA scheme for the hurricane initialization and prediction.

6. Summary and conclusions

One of the most challenging problems for the tropical cyclone forecaster and researcher is defining the structure of the tropical cyclone and the adjacent synoptic features with insufficient observations over the ocean. In this paper, the BDA scheme for hurricane initialization, proposed by Zou and Xiao (2000), is tested for the landfalling simulation of Hurricane Fran (1996). We studied the sensitivity of the BDA scheme to the model resolution, the radii of maximum SLP gradient for the specification of the bogus vortex, and the type of bogus data assimilated in the BDA. Numerical simulations of MM5 using the BDA scheme showed a remarkable forecast skill in reproducing the movement, intensity, and structure of Hurricane Fran near landfall. Sensitivity studies reveal many interesting features. The most important results are summarized as follows.

• The BDA scheme is very efficient in recovering the initial structure of the hurricane using very little observational information. Because the hurricane forecast model was used as a strong constraint in the hurricane BDA scheme, all model fields were adjusted to fit the bogus data of the SLP low and the derived balance wind fields. For instance, a moist warm-core hurricane structure was found in the initial vortex after BDA. These are adjustments in the model variables not directly bogused. An automatic result of BDA is that the initial conditions were consistent with the hurricane forecast model.

• The simulation of the hurricane track was sensitive to the model resolution on which the BDA scheme was performed. The predicted hurricane track was improved from the initialization with higher resolution. But the improvement from using higher resolution does not occur in the intensity forecast.

• In the present case with the current model configuration, the RMW used in the BDA scheme is a sensitive parameter for the hurricane track and intensity forecast. If the selected domain grid mesh can resolve the specified bogus vortex, hurricanes with larger RMW have a larger component of westward motion and are usually weaker.

• The sensitivity study shows that assimilation with the use of the bogused SLP field in the BDA scheme is more effective than the wind fields in recovering hurricane structure in all model variables. The hurricane simulation from the initial conditions produced by the use of bogused SLP low data is closer to observation than the use of bogused wind data only.

• Despite the differences resulting from variations in bogus data and model resolution, all the forecast experiments initialized with the BDA scheme obtained reasonably good hurricane track forecasts, which were reflected in the landfall location and the landfalling time. It is of particular importance that the BDA scheme showed great capabilities in reproducing the intensity and intensity changes of Hurricane Fran.

• The hurricane flow structure is simulated well using BDA and is in conformity with the structures illustrated by many authors such as Willoughby (1995). During the landing period of Fran, the model-simulated wind and reflectivity exhibited many structures similar to the observed wind and radar reflectivity distributions onshore.

This study is a natural extension of the work by Zou and Xiao (2000) where the BDA scheme was tested for the prediction of Hurricane Felix (1995). It is demonstrated that the BDA scheme works well, not only for a developing hurricane, but also for a mature hurricane simulation. The studies of the sensitivity of the BDA scheme to the model resolution, the RMW of the specified bogused SLP low, and the bogus variables in BDA increased our understanding of the performance of the BDA scheme in many aspects. Further experiments will be conducted to assess the capability, propriety, and feasibility of the proposed scheme for operational application.