Toward Developing an Objective 4DVAR BDA Scheme for Hurricane Initialization Based on TPC Observed Parameters

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1 2054 MONTHLY WEATHER REVIEW VOLUME 132 Toward Developing an Objective 4DVAR BDA Scheme for Hurricane Initialization Based on TPC Observed Parameters KYUNGJEEN PARK AND X. ZOU Department of Meteorology, The Florida State University, Tallahassee, Florida (Manuscript received 10 April 2003, in final form 2 March 2004) ABSTRACT This study aims to make the four-dimensional variational (4DVAR) bogus data assimilation (BDA) scheme for hurricane initialization first proposed by Zou and Xiao more objective. The BDA scheme consists of two steps: (i) specifying a bogus sea level pressure (SLP) field based on parameters observed by the Tropical Prediction Center (TPC) and (ii) assimilating the bogus SLP field under a forecast model constraint adjusting all model variables. In previous studies, specification of the bogus SLP was based on Fujita s formula, requiring the central SLP pressure (P c ), the radius of the outermost closed isobar (R out ), and the radius of the maximum SLP gradient (R 0 ) as inputs. Although the parameters P c and R out are provided directly by the TPC, R 0 is not. In this research, an empirical linear model designed to determine the value of R 0 (the size of the bogus vortex) from the TPC observed radius of 34-kt wind (R 34kt ) is developed. Numerical experiments are carried out for the initialization and prediction of Hurricane Bonnie (1998) over the Atlantic Ocean. The Pennsylvania State University NCAR nonhydrostatic mesoscale adjoint modeling system (Zou et al. 1997) is used for both the data assimilation and prediction components of the forecast. In order to study the sensitivity of hurricane initialization and prediction to the radial profile specification of the bogus vortex, the same experiment is conducted using Fujita s formula with R 0 R max (the radius of the TPC observed maximum wind) and another formula, Holland s formula, for the specification of the bogus SLP. The track prediction is less sensitive to the specification of the bogus SLP than the intensity prediction. The maximum track error is less than 110 km during the entire 3-day forecast for any of the three experiments using different bogus SLP specifications. However, the experiment using the linear model for the size specification required by Fujita s formula considerably outperforms the other two formulations for the intensity prediction of Hurricane Bonnie. Diagnosis of model output indicates that the 4DVAR BDA generated an initial hurricane, which allows for larger amounts of surface fluxes of heat and moisture, angular momentum, and latent heat of condensation, supporting a stronger and more realistic hurricane with more realistic intensity changes than experiments using the other two formulations. 1. Introduction The prediction of hurricane tracks and intensities has always been an important forecasting problem given that hurricanes strike regions of high population and property growth, resulting in large natural disasters. Factors inhibiting the accuracy of hurricane forecast position and intensity include inadequate coverage of observations, difficulties in assimilating satellite and radar data, limitations in physical parameterizations of convection and planetary boundary layer (PBL) processes, and omission of ocean coupling in model integration. An investigation of improvement trends in hurricane track forecasting in the Atlantic basin from 1970 to 1998 indicated that current track forecast errors are still about 50% higher than errors regarded as the limit of predictability (McAdie and Lawrence 2000). The skill level Corresponding author address: Dr. Xiaolei Zou, Dept. of Meteorology, The Florida State University, 404 Love Bldg., Tallahassee, FL zou@met.fsu.edu of hurricane intensity forecasts [the forecasted maximum low-level wind and the minimum sea level pressure (SLP)] is even lower. Improvements in both track and intensity prediction may be obtained by making effective use of observational information, enhancing forecast model resolution and physical parameterizations, as well as including hurricane ocean interactions. In this study, we focus on the effective and innovative use of Tropical Prediction Center (TPC) observed parameters for hurricane initialization. Since initial vortices of storms in large-scale analyses are often too weak and misplaced, hurricane initialization, a procedure using limited observational data to generate a conceptually correct initial bogus vortex, is often needed to improve the description of the initial storm vortex in all model variables (Kurihara et al. 1990; Lord 1991; Trinh and Krishnamurti 1992; Kurihara et al. 1993). Even with the rapid increase in the amount of satellite and radar data, hurricane initialization may still be needed when observations are insufficient to determine all model state variables describing the entire 2004 American Meteorological Society

2 AUGUST 2004 PARK AND ZOU 2055 structure of the hurricane (Zou et al. 2001). Furthermore, it is important for a bogus hurricane initialization procedure to also include actual observations when they are available. In this regard, a four-dimensional variational (4DVAR) bogus data assimilation (BDA) scheme was developed (Zou and Xiao 2000; Xiao et al. 2000; Zou et al. 2001). This scheme is attractive because the initial fields generated by the scheme are consistent with the forecast model and with observations such as water vapor winds, scatterometer winds, and Special Sensor Microwave Imager (SSM/I) microwave radiance measurements that can be incorporated into the procedure simultaneously if available. For the BDA scheme to work well, it is important to specify an initial bogus vortex that will represent the main features of the corresponding storm, such as the size and intensity, at the initial time of the forecast. In Zou and Xiao (2000), the Fujita (1952) formula was used to formulate an axisymmetric SLP pattern. Input parameters include the central SLP pressure (P c ), the pressure at an infinite distance (P ), the radius of the maximum SLP gradient (R 0 ), and the radius of the outermost closed isobar (R out ). The central SLP pressure P c and the radius of the outermost closed isobar (R out ) can be obtained directly from TPC observed parameters. The pressure at an infinite distance P can either be taken from the observed pressure of the outermost closed isobar (P out ), or found from R 0, R out, and P out by solving Fujita s formula (see section 2 for details). However, direct information on the radius of the maximum SLP gradient (R 0 ) is not available from TPC observed parameters. Therefore, the values of R 0 were specified arbitrarily in previous studies. However, a large degree of variability exists in hurricane size. According to the National Hurricane Center (NHC) climatology of all Atlantic hurricanes during the period , the radius of the outermost closed isobar of these hurricanes can vary from 185 km [Joan (1998), 1800 UTC 20 October] to 1028 km [Gilbert (1988), 1200 UTC 12 September], while the 34-kt wind radius can vary from 85 km [Erin (1995), 1800 UTC 3 August] to 509 km [Grace (1991), 1800 UTC 28 October] (Fig. 1). Such a large variability in hurricane size implies the need to use a case-dependent R 0 for the specification of a bogus vortex. The radius of the maximum SLP gradient (R 0 ) characterizes the horizontal size of the hurricane. It would be ideal to obtain values of all input values of the BDA scheme from TPC observed parameters. However, there is no direct dynamic link between R 0 and the observed radii of various winds and pressure provided by TPC, which include the maximum radii of the 34- (R 34kt ), 50- (R 50kt ), and 64-kt (R 64kt ) winds, the radius of the maximum wind (R max ), and the radius of the outermost closed isobar (R out ). In this study, we seek a statistical relationship between R 0 and either the TPC observed radii of the maximum wind (R max ), the 34-kt wind (R 34kt ), or the 50-kt wind (R 50kt ). FIG. 1. The TPC-observed radius of the 34-kt wind (km) and radius of the outermost closed isobar (km) for all hurricanes during the period A statistical relationship is found to exist between R 0 and R 34kt within BDA-generated vortices. This relationship is then used to construct a bogus SLP field for the 4DVAR BDA procedure in a real case simulation. The results from the forecast using the analysis created by this procedure are then compared with two other forecasts using different 4DVAR BDA procedures to create the analysis fields. In one, the bogus SLP field was created by setting the value of R 0 to the value of R max, while in the other the bogus SLP field was created using the Holland (1980) formula. The paper is organized as follows. In section 2, Fujita s and Holland s formulas are described. A simple statistical model determining the size of the bogus vortex is developed in section 3. A case study is carried out in section 4 to test the proposed simple model, and conclusions are given in section Fujita s and Holland s formulas for specifying a bogus SLP field In the Fujita (1952) formula, the SLP at a particular model grid point is specified according to the distance from the grid point to the vortex center by (P P c) P bogus(r) P, r R out, (1) 2 1/2 r [ 1 ] 2R 0 where P c and P are the values of the central SLP of the hurricane and an estimation of the SLP at an infinite distance, respectively; R out is the radius of the outermost closed isobar; and R 0 is the radius of the maximum gradient of the SLP. The bogus data are confined within a circular region specified by the radius of the outermost closed isobar (R out ). Four input parameters (P c, P, R out, and R 0 ) are required in Fujita s formula (1), with P c defining the in-

3 2056 MONTHLY WEATHER REVIEW VOLUME 132 TABLE 1. TPC observed parameters (representing surface measurements). Wind radii are observed in four quadrants: NE, SE, SW, and NW. Symbol (units) P c (hpa) P out (hpa) R out (n mi) V max (kt) R max (n mi) R 34kt (n mi) R 50kt (n mi) R 64kt (n mi) Description Central SLP Pressure of the outermost closed isobar Radius of the outermost closed isobar Max wind speed Radius of the max wind speed Max radius of the 34-kt wind speed Max radius of the 50-kt wind speed Max radius of the 64-kt wind speed tensity and R 0 defining the size of the initial vortex. Their values should be chosen such that the initial vortex approximates these characteristics of the actual hurricane and results in a good intensity and track prediction. The observed parameters, which are currently provided operationally by the TPC, include those listed in Table 1. Obviously, two of the input parameters required by (1), P c and R out, can be obtained directly from TPC observed parameters. If R 0 is determined, P can be calculated by solving (1) for P and setting r to R out and P bogus to P out, that is, 2 1/2 P out(r out)[1 (R out/ 2R 0)] Pc P. (2) 2 1/2 [1 (R / 2R )] 1 out 0 Therefore, we are left with only one input parameter, R 0, undetermined. In section 3, an empirical linear statistical model is developed to determine R 0 from R 34kt. Holland (1980) proposed another formula to describe the radial profile of SLP of a mature hurricane: ( A/r B) P bogus(r) Pc (P P c )e, (3) where A and B are two scaling parameters that determine the shape of a particular profile and e is the base of the natural logarithm. The scaling parameters A and B in (3) can be determined from TPC observed parameters. First, the cyclostrophic wind is derived from Holland s pressure distribution: ( A/r B) B 1/2 V c(r) [AB(P P c )e / r ], (4) where ( 1.15 kg m 3 ) is the air density. The maximum cyclostrophic wind speed (V cm ) and its radius (R cm ) are given by 1/B Rcm A, (5) 1/2 B 1/2 cm c V (P P ). (6) e If the maximum cyclostrophic wind speed and its radial distance are assumed to be equal to those of the observed maximum wind, A and B can be obtained by solving (5). Holland (1980) showed that distributions of P bogus (r) derived from (3) (5) under these assumptions compared favorably with observed SLP radial profiles. In the numerical experiments that follow, the same assumption is used when Holland s formula is used. TPC TPC TPC Setting P bogus ( Rout ) Pout in (3), R cm Rmax in (5), TPC TPC and V cm ( Rmax) Vmax in (6), we obtain three equations for solving three unknown values: A, B, and P. Specifically, from (6), we obtain TPC 2 e(v max ) TPC P P c. (7) B From (5) using the TPC-observed radius of the maximum wind, we obtain TPC B A (R max ). (8) TPC Substituting (7) and (8) into (3) evaluated at R out, we obtain TPC P out(b) P bogus(r out ) (V TPC 2 e max ) TPC [ (R TPC max ) B/(R TPC) B out ] P c e. (9) B The function P out (B) decreases monotonically as B increases. Holland (1980) also indicated that the value of B ranges from 1 to 2.5. Therefore, a value of B for the 4DVAR BDA scheme using Holland s formula can be obtained by dividing the interval [1, 2.5] for B into several intervals and evaluating the value of TPC P bogus ( R out ) for each B. The value of B for which P out (B) TPC P out will be chosen to specify a bogus SLP. Once B is obtained, A and P can be determined from (7) and TPC (8), respectively. Here, R out is used as it was with Fujita s formula to define the circular area in which bogus data are specified. 3. A linear regression model determining the size of the initial vortex In using Fujita s formula, we seek a quantitative link between R 0 and any of the observed wind-related radii listed in Table 1. Since there is no observed information of R 0, estimation of R 0 is carried out using model fields. Sixteen BDA-generated hurricane fields (Table 2) are used to examine the relationship between the radius of the maximum SLP pressure gradient and the radii of various winds. The wind data are first divided into n subsets for four quadrants. The nth subset includes data whose radial distance to the storm center is greater than (n 1) ds and smaller than n ds, where ds 2 x, and x 18 km is the model s horizontal grid spacing. The wind data are then averaged for each subset, producing an averaged radial wind profile. An example of such a profile is shown in Fig. 2a for the 6-h forecast from experiment EF2 (the experiment is described in section 4). Dots represent all the wind speeds with radial distances less than 700 km. The averaged wind profile is represented by the thick solid line. Finally, the radius of a particular wind speed can be obtained by finding the radius where the averaged wind speed drops below

4 AUGUST 2004 PARK AND ZOU 2057 TABLE 2. The 16 BDA experiments used to construct a linear regression model for R 0. All experiments contained 27 vertical levels Hurricane name Time Grid dimension Resolution (km) R 0 (km) Opal Felix Erika Bonnie Bonnie Bonnie Bonnie Bonnie Bonnie Bonnie Bonnie Floyd Floyd Gordon Erin Erin 1200 UTC 2 Oct UTC 16 Aug UTC 9 Sep UTC 24 Aug UTC 23 Aug UTC 23 Aug UTC 23 Aug UTC 23 Aug UTC 23 Aug UTC 23 Aug UTC 23 Aug UTC 13 Sep UTC 13 Sep UTC 17 Sep UTC 9 Sep UTC 11 Sep that speed (see examples for the 34-kt wind, thin solid line in Fig. 2a). The averaged wind speed as a function of radial distance may not have an inverse function; that is, a single wind speed may correspond to multiple wind radii. If this is the case, the largest value of the radial distance is taken. The radius of the maximum SLP gradient in a BDAgenerated initial condition is equal to R 0. The value of R 0 can be estimated from the radial profile of the averaged SLP values, defined in a similar manner as the radial profile of the averaged wind; R 0 can then be found where the difference between two adjacent averaged SLP values is largest (Fig. 2b). Figure 3 shows the relationship between R 0 and R 34kt for the 16 BDA experiments listed in Table 2. The correlation coefficient (0.86) between these two parameters is higher than the correlation coefficient between R 0 and any of the other wind radii. The solid line in Fig. 3 is the regression line, satisfying the following equation: FIG. 2. Radial profile of (a) wind speed and (b) SLP for EF2 at 6 h in the NE quadrant. Dots represent wind speed in (a) and SLP in (b) with respect to the radial distance. The averaged SLP and wind in each subset are represented by circles. The solid horizontal line in (a) represents 34-kt (17.5 m s 1 ) wind speed, while the dotted lines in (b) show the SLP gradient. FIG. 3. The relationship between R 0 and R 34kt for a total of 16 BDA experiments (listed in Table 3). The solid line is the linear regression line.

5 2058 MONTHLY WEATHER REVIEW VOLUME 132 FIG. 4. Time evolution of the observed (a) track, (b) maximum wind, and (c) minimum SLP of Hurricane Bonnie for the 12-day period starting from 1800 UTC 19 Aug The initial time for the initialization and forecast experiments is indicated by an X. R0 0.38R34kt 3.8, (10) where both R 0 and R 34kt are expressed in kilometers. Since the best linear fit was found between R 0 and R 34kt the above equation will be used to specify R 0 to calculate the bogus SLP field using Fujita s formula. The forecast from the analysis created by this procedure is compared in the next section with two other forecasts that used analyses that were created from different specifications of the bogus SLP. 4. A case study a. Case: Hurricane Bonnie The case chosen for hurricane initialization and forecast experiments in this study is Hurricane Bonnie (1998). Figure 4 provides the actual track and observed intensity based on TPC data. Hurricane Bonnie originated from a tropical wave that moved over Dakar, Senegal, on 14 August It turned into a tropical depression on 19 August One day later, it became a tropical cyclone. Bonnie became a category 1 hurricane at 0600 UTC 22 August and a category 3 hurricane at 1200 UTC 23 August whereupon Bonnie obtained a maximum wind of 100 kt and a minimum pressure of 954 hpa. It was located about 150 n mi east of San Salvador in the Bahamas at 0300 UTC 24 August. After weakening slightly, Bonnie made landfall at 0330 UTC 27 August near Wilmington, North Carolina, as a category 2 hurricane. Bonnie directly impacted the coast of North Carolina, causing three deaths and an estimated 720 million dollars in total damage. b. Experiment design Numerical experiments of hurricane initializations are conducted beginning at 1200 UTC 23 August Three-day forecasts are made from analyses created from the BDA procedure. The fifth-generation Pennsylvania State University National Center for Atmospheric Research (Penn State NCAR) Mesoscale Model (MM5; Dudhia, 1989) and its adjoint modeling system (Zou et al. 1997) are used for this study. The 4DVAR BDA experiments are carried out on a domain of grid points (the BDA domain) with a horizontal resolution of 18 km at 27 vertical levels. Three-day forecasts are performed using a larger (forecast) domain that includes the BDA domain as a subdomain. The forecast domain has grid points for 27 levels, also utilizing an 18-km horizontal resolution. The Kuo cumulus parameterization, stable precipitation, and Medium-Range Forecast (MRF) model PBL schemes are used in the BDA procedure, while the Grell et al. (1994) cumulus parameterization, Dudhia s (1989) explicit microphysics, and the MRF PBL schemes are used for the hurricane forecasts.

6 AUGUST 2004 PARK AND ZOU 2059 FIG. 5. Radial profile of SLP generated by Holland s and Fujita s formulas for experiments EH (solid line), EF1 (dotted line), and EF2 (dot dash line). A surface low of the initial hurricane is specified based on TPC parameters. Three 4DVAR BDA experiments are conducted. The first two experiments use TPC Fujita s formula with R 0 R max (EF1) and R 0 TPC 0.38R 34kt 3.8 (EF2), respectively. The third experiment uses Holland s formula (EH). The value of the TPC observed central SLP P c 958 hpa at 1200 UTC 23 August is used in all three experiments. At the initialization time, the radius of the maximum wind speed is R max 25 n mi. This value is used as R 0 in EF1. The observed 34-kt wind radii for Hurricane Bonnie are 150, 150, 100, and 150 n mi for the NE, SE, SW, and NW quadrants, respectively (see Table 1). The averaged 34- kt wind radius is thus 255 km. Using (10), we obtain R 0 93 km for EF2. The values P 1000 hpa and P 1020 hpa, derived from (2), are used for EF1 and EF2, respectively. In the third experiment (EH), the TPC observed maximum wind speed (100 kt) and the radius of the maximum wind speed (25 n mi) are used to determine the two parameters (A and B) required by Holland s formula. The values of A and B are found to be A 397 and B The radial profiles of Fujita s and Holland s formulas are shown in Fig. 5. Holland s formula produces a radial profile that is much steeper than that of Fujita s formula at small radial distances (less than 100 km). Furthermore, the profile for EF2 is more linear in radial distance than the profile for EF1 (which is slightly curved), while the profile for EH possesses the most curvature. c. Numerical results 1) HURRICANE INITIALIZATION RESULTS The 4DVAR BDA scheme imposes a forecast model constraint into a hurricane initialization procedure, which assimilates a bogus SLP field and adjusts all model fields to fit the bogus SLP under the constraint of the model. The 4DVAR BDA experiments are performed during a 30-min assimilation window. The bogus SLP is assumed invariant during the assimilation window and is fitted at 3-min intervals. More information on the 4DVAR procedure used for the BDA scheme including information on the cost function, which is minimized during the assimilation procedure, can be found in Zou and Xiao (2000). Vertical east west cross sections of the differences in a control forecast (one in which only a standard analysis without a hurricane initialization is used to produce the forecast) and the forecast that results from EF2 (EF2 control forecast) are shown in Figs. 6 8 for different model fields. Negative pressure perturbation differences, with a magnitude of about 40 hpa, are observed in the lower troposphere (Fig. 6a) between the EF2 and control forecasts at all times during the first 30 min of the forecast period. The EF2 forecast is also 20 K warmer than the control forecast in the middle troposphere during this same time period (Fig. 6b) at all times. However, no significant change in these difference fields occur during the first 30 min of the forecast (the same time period as the assimilation window). What dynamical processes are responsible for these adjustments in pressure and temperature fields? A careful examination of the divergence difference field (Fig. 7a) reveals that at the initial time, strong divergence exists in the lower troposphere and convergence exists in the upper levels for EF2 when compared to the control. Strong downward motion is also observed within the simulated hurricane through all levels (Fig. 8a). These adjustments in the model variables at the initial time suggest that the hydrostatic constraint plays a dominant role in generating the desired low pressure system. The air within the hurricane is rarefied through the downward motion. The downward motion, in the meantime, produces adiabatic warming (Fig. 6b), which reduces the low-level pressure further. As the forecast model is integrated forward in time, the pressure gradient force associated with the low pressure system starts to play a dominant role. Strong convergence develops in the lower troposphere after 24 min (Fig. 7a). Adiabatic cooling is found in the lower troposphere, accompanying the upward motion associated with the low-level convergence. In the last 6 min of the first 30- min forecast period, for example, a 3.6-K temperature cooling is found in the lower troposphere (figure omitted). The dynamic adjustment continues beyond the first 30 min (Fig. 7a). The low-level convergence intensifies and extends farther into the midtroposphere. The lowlevel divergence seen at the beginning of the assimilation weakens and is pushed farther upward. A new divergence center in the upper atmosphere develops, intensifies, and connects to the middle-level divergence system. By 48 min, the upper-level troposphere is characterized by a single divergence flow that extends from 800 to 100 hpa. At 60 min, a nearly balanced hurricane forms. The eyewall structure is clearly seen in the ver-

7 2060 MONTHLY WEATHER REVIEW VOLUME 132 FIG. 6. East west cross sections through the center of the hurricane for the forecast differences between experiment EF2 and a control forecast experiment initialized with the NCEP large-scale analysis at a 6- min interval. (a) Pressure perturbation (Pa) from the initial time (leftmost) to 30 min (rightmost). (b) Temperature (K) from the initial time (leftmost) to 30 min (rightmost). The horizontal distance of each cross section is about 700 km. tical velocity field (Fig. 8b), characterized by two strong upward motions near the hurricane center. Figure 7b shows the time evolution of moisture difference fields for the first 60 min of the forecast. At the initial time the forecast produced in EF2 is moister than the control forecast; however, the downward motion at the beginning of the model simulation brings drier air from the upper atmosphere into the simulated hurricane center. Then, as the low-level convergence develops, a significant increase of moisture content is observed, first near the surface and later in the middle troposphere. By 1 h, a conceptually correct hurricane structure has developed, including low-level convergence, moist lower levels, a warm core, and upper-level divergence. In other words, the spinup time for hurricane simulation based on the 4DVAR BDA generated initial vortex presented here is about 1 h. 2) FORECAST RESULTS The predicted track and track error of Hurricane Bonnie from 1200 UTC 23 August to 1200 UTC 26 August for the three experiments previously discussed are shown in Fig. 9. The location of the hurricane is defined by the grid point at which the value of the SLP is minimal. No significant differences are found amongst the three experiments in track prediction. The track error experiences a rapid increase during the initial 12-h forecast period but levels off at subsequent forecast times. The maximum track error is below 110 km during the entire 3-day forecast period, and the averaged track error for the 3-day period is about 60 km in all three cases. All three cases show initial track errors: hurricanes move toward the west while the observed track heads north. The 4DVAR BDA scheme assimilates only SLP (not steering flow), so the steering flow could be damaged during assimilation processes. In order to reduce the initial track error, steering flow needs to be considered. The intensity forecast is more sensitive to the specification of the initial hurricane than to the track prediction. The time evolution of the minimum SLP and the maximum low-level winds (Figs. 10a and 10b) reveals that EF2 outperforms both EF1 and EH. The 3-day average P c errors of EH, EF1, and EF2 are 17.5, 9.6, and 3.5 hpa, respectively. The average errors of the maximum low-level wind for the 3-day period are 12.2, 8.6, and 6.6ms 1 for EH, EF1, and EF2, respectively. The maximum wind speed errors for EF2 (the best case) are 4.2,

8 AUGUST 2004 PARK AND ZOU 2061 FIG. 7. East west cross sections through the center of the hurricane for the forecast differences of divergence field (s 1 ) between experiment EF2 and a control forecast experiment initialized with the NCEP large-scale analysis at a 12-min interval. (a) Divergence field (s 1 ) from the initial time (leftmost) to 60 min (rightmost). (b) Mixing ratio (kg kg 1 ) from the initial time (leftmost) to 60 min (rightmost). The horizontal distance of each cross section is about 700 km. 5.7, 2.5, 5.2, and 6.5 m s 1 at the 12-, 24-, 36-, 48-, and 72-h forecast times, respectively. Compared with the official forecast errors for Bonnie (Table 3), track and intensity predictions from EF2 are quite promising. During the initial 6hofthemodel forecast, a weakening in intensity measured by SLP is seen for all three cases (Fig. 10a). However, such a weakening in intensity is less dramatic in experiment EF2 than in the other two experiments. In order to understand differences in the intensity change among these three experiments, we examine hourly variations of the central SLP during the initial 6 h (Fig. 11). A rapid weakening is found during the first hour of all experiments. This occurred due to the fact that during the first 20 min, the hurricane center becomes drier due to the continuous downward motion within the simulated hurricane and the suppression of the condensation of water vapor and upward surface heat fluxes. Without a latent heat supply from both the surface and condensation, the hurricane weakens until the surface heat and water vapor is transported upward by low-level convergent winds and the associated upward vertical motion. After 1 h, a slight deepening is observed in the next hour for EH, followed by a gradual weakening again. In EF1, the SLP also deepens during the second forecast hour. In contrast, the simulated hurricane in EF2 experiences a continuous deepening for a much longer period of time (2 24 h) (see also Fig. 10a). Therefore, the intensity change is closely related to the size specification of the initial vortex. In the following, we show that the size of the initial vortex affects the total amount of surface flux of heat and moisture, the angular momentum, and the latent heat of condensation within the hurricane. The amount of heat and water vapor transported into the hurricane depends on the surface wind, which is derived mainly by the pressure gradient force. Different SLP fields were used to generate the initial vortices of EH, EF1, and EF2, so the surface flux patterns are dissimilar. The smallest hurricane (EH) shows a rapid increase of SLP near the center but a smoother increase in the outer regions. As a result, smaller hurricanes produced by this BDA procedure have stronger winds near the hurricane center and weaker winds in the outer regions compared to larger hurricanes. The surface latent heat fluxes at 2 h are shown in Figs. 12 and 13, and the time evolution of domain-averaged (860 km 860 km) latent heat fluxes are shown in Fig. 14 for all three

9 2062 MONTHLY WEATHER REVIEW VOLUME 132 FIG. 9. Time evolution of the simulated hurricane (a) tracks and (b) averaged track errors of EH (solid line with empty circle), EF1 (dotted line with square), and EF2 (dot dash line with triangle) during the 3-day forecast period. The line with black circles in (a) represents the observed best track. FIG. 8. East west cross sections through the center of the hurricane for the forecast differences of vertical velocity (m s 1 ) at (a) 0 and (b) 1 h. The contour intervals are 1 m s 1 for (a) and 0.5 m s 1 for (b). Solid (dotted) lines represent upward (downward) motion. forecasts. Since the latent heat flux is dependent on lowlevel winds, the radial profiles of latent heat flux of all three cases show a pattern corresponding to a typical hurricane wind distribution; that is, the latent heat flux is minimum at the hurricane center, increases rapidly from the center, and then decreases gradually as the radial distance increases, but there are differences between the experiments. The area of hurricane-induced surface latent heat flux in EF2 is larger than in both EF1 and EH. As a result, the total domain-averaged latent heat flux in EF2 is consistently larger than EF1 and EH (Fig. 14). At 2 h, the domain-averaged latent heat fluxes, for example, are 454, 539, and 688 W m 2 for EH, EF1, and EF2, respectively. With more latent heat supply from the surface, the hurricane simulated in EF2 experiences less weakening. The sensible heat flux is also affected by the size of the hurricane. The sensible heat fluxes averaged over the same domain are 23.0, 22.8, and 19.4 W m 2 for EH, EF1, and EF2, respectively. While the latent heat flux is always positive, the sensible heat flux can be negative. The air is not cooler than the ocean everywhere. In fact, the air is warmer in the center of the hurricane and cooler in

10 AUGUST 2004 PARK AND ZOU 2063 TABLE 3. Averaged official NHC forecast errors for Hurricane Bonnie (1998). Forecast interval Track error (km) V max error (m s 1 ) FIG. 10. Same as Fig. 9b, except for (a) the central pressure (hpa) and (b) the maximum wind speed (m s 1 ). the outer regions. Stronger winds near the hurricane center and weaker winds in the outer regions in EH increase the overall sensible heat transport from the ocean than in the other two experiments. However, the sensible heat flux in EH is only slightly larger than in EF2, so that the differences in the sensible heat flux among the three experiments have a negligible impact on the resulting hurricane intensity. The effect of the simulated hurricane size is not limited only to surface fluxes. The angular momentum distribution is sensitive to the size of the simulated hurricane. The absolute angular momentum (AAM) of the atmosphere is 1 2 AAM m fr r, (11) 2 where m is the mass of the atmosphere, r is radial distance from the hurricane s center, f is the Coriolis parameter, and is the tangential wind component. In a hurricane, the differential mass between the radial distance r and r r can be roughly estimated as 2 2 m H [(r r) r )] 2 H (2 r r ) 2 2H r r, (12) where is the air density and H is the height of the atmospheric column. The r 2 term has been omitted in the far right-hand side of (12). Therefore, the mass is proportional to the square of distance. From (11) and (12), we find that the second term in the AAM is proportional to the square of the radial distance multiplied by the tangential wind component. The angular momentum is dependent on the tangential wind component, which is related to the distribution of pressure. The radial profile of the 30-min forecasted low-level tangential wind for all three experiments is shown in Fig. 15. The main difference among EF1, EF2, and EH is that the maximum tangential wind in EF2 appears at a larger distance from the hurricane center than in EF1 and EH. Since the mass increases with the square of the radial distance, the angular momentum of the simulated hurricane in EF2 is much larger than in EF1 and EH. The larger angular momentum within the EF2 hurricane allows the model to maintain a stronger hurricane. By 12 h, the radial profiles of SLP among the three experiments (Fig. 16a) are not as different as at the initial time (Fig. 5). The radial profile of SLP in EF2 FIG. 11. Same as Fig. 9b, except for central pressure (hpa) during the first 12 h of the simulation.

11 2064 MONTHLY WEATHER REVIEW VOLUME 132 FIG. 12. The latent heat flux (W m 2 ) from (a) EH, (b) EF1, and (c) EF2 at t 2h. FIG. 13. East west cross section through the hurricane s center for EH (solid line with circle), EF1 (dotted line with square), and EF2 (dot dash line with triangle) at t 2 h. The contour interval is 200 Wm 2. FIG. 14. The latent heat flux (W m 2 ) averaged over a domain centered at the hurricane s eye. The domain size is 860 km 860 km.

12 AUGUST 2004 PARK AND ZOU 2065 FIG. 15. Radial profile of tangential wind (m s 1 ) of EH (solid line with circles), EF1 (dotted line with squares), and EF2 (dot dash line with triangles) at the end of the assimilation window. at 12 h becomes much sharper than that at the initial time. We have also calculated three radial profiles of SLP based on Holland s and Fujita s formulas using model forecast parameters (P c, R max, R 34kt, V max, R out, and P out ) from EF2 at 12 h (Fig. 16b). Compared with these radial profiles, we find that the predicted SLP profile at 12 h in EF2 resembles Holland s profile. The contraction of the hurricane s size during the model forecast in EF2 is also reflected in the domain-averaged latent heat flux (Fig. 14). Starting at 5 h, a decrease of domain-averaged surface latent heat flux is observed. In other words, the larger initial bogus vortex generated by Fujita s formula using the linear regression model (10) for the size determination allows for a sufficient supply of angular momentum and surface fluxes. The radial profile of SLP becomes more realistic during the subsequent model forecast. Besides surface fluxes and angular momentum, differences are also observed in latent heat release due to condensation. Figure 17 shows the mean vertical velocity averaged over an area of km 2 centered at the hurricane s eye during the initial 6 h. The mean vertical velocity in EF2 is consistently stronger than in EF1 and EH, implying a larger amount of diabatic heating in EF2. The latent heat release associated with precipitation provides a strong positive feedback to the intensification of the simulation of Hurricane Bonnie in EF2. d. Impact of horizontal grid spacing It is well known that hurricane intensity prediction is sensitive to model resolution. Rosenthal (1970) investigated the resolution effect on hurricane simulation by comparing experiments performed on 10- and 20-km horizontal grid spacing and showed that a hurricane simulated by 10-km resolution is slightly stronger and more FIG. 16. The SLP radial profiles of (a) the simulated hurricane and (b) SLP profiles calculated by Holland s and Fujita s formulas using the parameters obtained from the EF2 simulation result at 12 h. realistic than one simulated by 20-km resolution. Tenerelli and Chen (2001) used the MM5 with a vortexfollowing mesh refinement scheme to simulate Hurricane Floyd (1999). Starting from a 45-km mesh, they added finer meshes successively: a 15-, 5-, and km mesh. They found that the track was insensitive to resolution, but the maximum wind speed became stronger and compared more favorably with observations when a higher resolution was used. Liu et al. (1997) used a triply nested MM5 to simulate Hurricane Andrew (1992) with a 6-km resolution and emphasized the importance of high grid resolution in hurricane prediction. As shown in Fig. 16b, the radial profile of modelsimulated SLP is better represented by Holland s formula than by Fujita s formula. However, Holland s formula fails to generate an initial vortex that leads to a good intensity forecast. Since Holland s radial profiles of SLP are in general sharper than those of Fujita, the poor intensity forecast of Holland s case (EH) may be attributed to the use of an 18-km model resolution in-

13 2066 MONTHLY WEATHER REVIEW VOLUME 132 FIG. 17. The vertical profile of vertical wind speed (m s 1 ) averaged over a domain sized 860 km 860 km, at 1 to 6 h. sufficient for resolving properly Holland s initial vortex. It would be interesting to examine if the model forecasts initialized with Holland s formula may be improved using a higher model resolution. In order to examine the impact of model resolution, the EH case was repeated using a 6-km horizontal resolution (instead of 18 km) for both the BDA and forecast experiments. As expected, the 6-km resolution resolves the structures of Holland s vortex much better than those at the 18-km resolution, as seen in cross sections of

14 AUGUST 2004 PARK AND ZOU 2067 FIG. 18. East west cross section through the hurricane s center of (a) pressure perturbation (Pa), (b) temperature (K), and (c) vertical velocity (m s 1 ) at 1 h, taken from the EH case performed at an 18- km resolution. Solid (dotted) lines are for positive (negative) values, and the contour intervals are (a) 4 Pa, (b) 5 K, and (c) 2 m s 1, respectively. pressure perturbation, temperature, and vertical velocity from west to east through the hurricane vortex center of the 1-h forecast (Figs. 18 and 19). Both the 6- and 18-km experiments show a bell-shaped negative pressure perturbation pattern in the lower troposphere, with the pressure perturbation in the 6-km experiment being stronger and sharper than that of the 18-km experiment. A clear warm core is observed in the 6-km experiment. At 700 hpa, the temperature near the hurricane center is about 15 K higher than its surrounding air at the same height. In contrast, the 18-km experiment produces only a few degrees of temperature increase near the hurricane center. The eyewall structure is also better represented in the 6-km experiment. A downward motion is found between two columns of strong upward motion, defining a hurricane eye of about km in diameter. The abrupt weakening of the simulated hurricane at the first few hours of model integration seen in the 18-km experiment is greatly reduced in the 6-km experiment (Fig. 20). However, the simulated hurricane in the 6-km forecast experiences a steady weakening. By 7 h, the central SLP is as weak as that of the 18-km run (Fig. 20). Therefore, compared with the 18-km experiment, the 6- km experiment better captures the initial vortex structure and produces a smoother start of the simulated hurricane evolution during the beginning few hours of model forecast. However, increasing model resolution does not improve the hurricane intensity forecast beyond 7 h in this case study. 5. Summary and conclusions High quality satellite observations or their retrievals are often least available over hurricane regions where precipitation occurs. Assimilation of the raw measurements, such as SSM/I microwave radiance, may require the prediction of hydrometeor variables (such as cloud and ice) to be greatly improved (Amerault and

15 2068 MONTHLY WEATHER REVIEW VOLUME 132 FIG. 19. Same as Fig. 18, except for the EH case performed at 6- km resolution. Zou 2003). On the other hand, improvement of hurricane prediction in terms of both track and intensity depends critically on the initial vortex. Therefore, hurricane initialization has been and might continue to be required to improve track and intensity forecasts. The 4DVAR BDA scheme incorporating the forecast model constraint was found to be a very successful and efficient method to obtain the initial structures of a storm (Zou and Xiao 2000; Xiao et al. 2000; Zou et al. 2001). It specifies only the SLP of the bogused vortex and installs the forecast model constraint to generate modelconsistent initial fields of all variables that fit the specified bogus SLP. In order to apply the BDA scheme in an operational environment, input parameters of Fujita s formula need to be determined objectively. These input parameters are the minimum SLP (P c ), the pressure at an infinite distance (P ), the radius of the maximum SLP gradient (R 0 ), and the radius of the outermost closed isobar (R out ). We propose to determine all these input parameters based on TPC-observed parameters. Both P c and R out are direct TPC parameters, and the pressure at an infinite distance can be calculated by solving Fujita s formula to obtain P at r R out. The only remaining parameter to be determined is R 0, the maximum radial gradient of SLP. Based on the results of many BDA-initialized hurricanes, a linear relationship between R 0 and R 34kt is found. A linear model that provides an actual value of R 0 based on the TPC-observed parameter, R 34kt, is therefore developed. One of the advantages of using model-generated storms rather than observed storms to derive the statistical relation between the radius of the maximum SLP gradient and the 34-kt wind radii is that the model biases can be removed. Because of model limitations, such as resolution and parameterization of physical processes, the relationship between R 0 and R 34kt from model-generated storms is different from that of observed storms. Since R 0 is determined from observed R 34kt data, the wind-

16 AUGUST 2004 PARK AND ZOU 2069 related features of initial vortices can be improved. For example, the model errors, such as underestimated latent heat flux, which is very sensitive to the surface winds, can be compensated for by specifying larger hurricanes. Numerical results for the simulation of Hurricane Bonnie demonstrate that the forecast model experiences only 1 h of spinup time using an analysis created by the 4DVAR BDA procedure. The track prediction is excellent and not sensitive to the details of the specified bogused low. However, the intensity forecasts are extremely sensitive to the specification of the radial profile of SLP. A numerical experiment using the linear model for the size specification required by Fujita s formula outperforms those experiments using the other two formulations, one using Fujita s formula with R 0 R max and the other using Holland s formula to calculate the bogus SLP. The size of the simulated hurricane was found to control the amount of surface fluxes of heat and moisture, angular momentum, and latent heat of condensation, and therefore controlled the intensity change. We realize that this is a single case study. A systematic evaluation of this linear model, as well as 4DVAR BDA in general, is required. Further studies of hurricane initialization that are being conducted also include developing a conceptual model of hurricane structures, with a few parameters to be determined by TPC-observed parameters without the use of a complex primitive equation model such as the MM5, as well as introducing scatterometer vector wind and SSM/I microwave radiance data into the BDA procedure. Eventually, we wish to use as little bogus data as possible for hurricane initialization with simple dynamical and physical constraints. Acknowledgments. This research is supported by the National Science Foundation under Project ATM REFERENCES FIG. 20. Time evolution of the simulated central minimum pressure during the first 12-h forecast for the EH case with 18-km resolution (solid line with square), EH case with 6-km resolution (dotted line with triangle), and the observations (solid line with black circle). Amerault, C., and X. Zou, 2003: Preliminary steps in assimilating SSM/I brightness temperatures in a hurricane prediction scheme. J. Atmos. Oceanic Technol., 20, Dudhia, J., 1989: Numerical study of convection observed during the Winter Monsoon Experiment using a mesoscale two-dimensional model. J. Atmos. Sci., 46, Fujita, T., 1952: Pressure distribution within a typhoon. Geophys. Mag., 23, Grell, G. A., J. Dudhia, and D. R. Stauffer, 1994: A description of the fifth-generation Penn State/NCAR Mesoscale Model (MM5). NCAR Tech. Note NCAR/TN-398 STR, 117 pp. Holland, G. J., 1980: An analytic model of the wind and pressure profiles in hurricanes. Mon. Wea. Rev., 108, Kurihara, Y., M. A. Bender, R. E. Tuleya, and R. J. Ross, 1990: Prediction experiments of Hurricane Gloria (1985) using a multiply nested movable mesh model. Mon. Wea. Rev., 118, , M. A. Bender, and R. J. Ross, 1993: An initialization scheme of hurricane models by vortex specification. Mon. Wea. Rev., 121, Liu, Y., D.-L. Zhang, and M. K. Yau, 1997: A multiscale numerical study of Hurricane Andrew (1992). Part I: Explicit simulation and verification. Mon. Wea. Rev., 125, Lord, S. J., 1991: A bogusing system for vortex circulations in the National Meteorological Center global forecast model. Preprints, 19th Conf. on Hurricanes and Tropical Meteorology, Miami, FL, Amer. Meteor. Soc., McAdie, C. J., and M. B. Lawrence, 2000: Improvements in tropical cyclone track forecasting in the Atlantic basin, Bull. Amer. Meteor. Soc., 81, Rosenthal, S. L., 1970: Experiments with a numerical model of tropical cyclone development: Some effects of radial resolution. Mon. Wea. Rev., 98, Tenerelli, J. E., and S. S. Chen, 2001: High-resolution simulations of Hurricane Floyd using MM5 with vortex-following mesh refinement. Preprints, 18th Conf. on Weather Analysis and Forecasting/14th Conf. on Numerical Weather Prediction, Fort Lauderdale, FL, Amer. Meteor. Soc., CD-ROM, JP1.11. Trinh, V. T., and T. N. Krishnamurti, 1992: Vortex initialization for typhoon track prediction. Meteor. Atmos. Phys., 47, Xiao, Q., X. Zou, and B. Wang, 2000: On the initialization and prediction of a landfalling hurricane using a variational bogus data assimilation scheme. Mon. Wea. Rev., 128, Zou, X., and Q. Xiao, 2000: Studies on the initialization and simulation of a mature hurricane using a variational bogus data assimilation scheme. J. Atmos. Sci., 57, , F. Vandenberghe, M. Pondeca, and Y.-H. Kuo, 1997: Introduction to adjoint techniques and the MM5 adjoint modeling system. NCAR Tech. Note NCAR/TN-435-STR, 110 pp., Q. Xiao, A. E. Lipton, and G. D. Modica, 2001: A numerical study of the effect of GOES sounder cloud-cleared brightness temperatures on the prediction of Hurricane Felix. J. Appl. Meteor., 40,