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  • View in gallery

    A snapshot of the 200-hPa horizontal winds at 0600 UTC 14 Sep 2004. Question marks show the downstream sides of upper-level troughs and propose the question of which area (which trough or even which grid point) is most favorable for TC intensification.

  • View in gallery

    Schematic showing how the horizontal map of ULFI (or its REFC, PEFC, and absolute vorticity components) is constructed. The black crosses represent the latitude–longitude grid points. Arrows show the translating vector of a TC (red TC symbol). A cylindrical coordinate system whose origin overlaps with a latitude–longitude grid point is established for the evaluation of ULFI with respect to that point. Then, ULFI is averaged radially between two specific radii (pink shaded). Finally, every grid point should have only one averaged value, forming a continuous horizontal map of ULFI.

  • View in gallery

    The 200-hPa winds (vector; m s−1) and horizontal maps of 300–600-km-averaged (a) REFC (m s−1 day−1), (b) PEFC (m s−1 day−1), (c) absolute vorticity (10−5 s−1), and (d) ULFI (m s−1) valid at 0600 UTC 14 Sep 2004. (e)–(h) As in (a)–(d), but for the 500–800-km average.

  • View in gallery

    The 200-hPa storm-relative winds (vector; m s−1) and 300–600-km-averaged horizontal map of REFC (shaded; m s−1 day−1) valid at 0600 UTC 14 Sep 2004. The cylindrical coordinate established at each grid point moves (a) northward, (b) southward, (c) eastward, and (d) westward at a constant speed of 10 m s−1. (e) As in (a), but the cylindrical coordinates are fixed.

  • View in gallery

    Along-track quantities for Hurricane Elena (1985). (a) Minimum sea level pressure (hPa; black line), SST (°C; red line), VWS (m s−1; blue line), and ULFI (m s−1; green line). SST and VWS are averaged within a 300-km radius, and ULFI is averaged over a 300–600-km radial band. Note that SST is undefined after 1200 UTC 2 Sep because of Elena’s landfall. The black arrow highlights the intensification of interest here, which is associated with an upper-level environmental influence. (b) The 200-hPa REFC (m s−1 day−1; black line) and PEFC (m s−1 day−1; green line) averaged over a 300–600-km radial band. The horizontal dashed line is the threshold for identifying interaction using the REFC diagnostic process. (c) The 200-hPa mean absolute vorticity (10−5 s−1; black line) and its two components, mean relative vorticity (10−5 s−1; green line) and mean planetary vorticity (10−5 s−1; blue line), all of which are averaged over a 300–600-km radial band. The vertical solid line in each panel indicates the time when Elena was over land.

  • View in gallery

    Synoptic evolution of 200-hPa winds (vectors; m s−1) and horizontal maps of ULFI (shaded; m s−1) defined in Eq. (6) from 1200 UTC 29 Aug to 1200 UTC 2 Sep 1985 at 12-h intervals. The black line and dot, respectively, show the track and current position of Hurricane Elena (1985).

  • View in gallery

    As in Fig. 5, but for Tropical Storm Haima (2004).

  • View in gallery

    As in Fig. 6, but for Tropical Storm Haima (2004) from 1200 UTC 11 Sep to 1200 UTC 15 Sep 2004.

  • View in gallery

    As in Fig. 5, but for Typhoon Omar (1992).

  • View in gallery

    As in Fig. 6, but for Typhoon Omar (1992) from 0600 UTC 29 Aug to 0600 UTC 2 Sep 1992.

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A Horizontal Index for the Influence of Upper-Level Environmental Flow on Tropical Cyclone Intensity

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  • 1 State Key Laboratory of Tropical Oceanography, South China Sea Institute of Oceanology, Chinese Academy of Sciences, and Center for Monsoon and Environmental Research, Department of Atmospheric Science, Sun Yat-sen University, Guangzhou, China
  • 2 South China Sea Marine Prediction Center, State Oceanic Administrative, Guangzhou, China
  • 3 State Key Laboratory of Tropical Oceanography, South China Sea Institute of Oceanology, Chinese Academy of Sciences, Guangzhou, and School of Oceanography, Qinzhou University, Qinzhou, China
  • 4 State Key Laboratory of Tropical Oceanography, South China Sea Institute of Oceanology, Chinese Academy of Sciences, Guangzhou, China
  • 5 Luogang District Meteorological Bureau, Guangzhou, China
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Abstract

A horizontal map of the upper-level forcing index (ULFI) is constructed to show the possible influence of upper-level large-scale environmental flow on the intensity change of tropical cyclones (TCs). The ULFI includes three commonly used diagnostics, that is, 200-hPa eddy flux convergences of both relative (REFC) and planetary angular momentum (PEFC), as well as axisymmetric absolute vorticity as a denominator that rescales the strength of the eddy forcings similar to the outflow-layer inertial stability. A simple procedure is adopted to convert these storm-relative components and the ULFI into Eulerian horizontal maps. Applications of this index map to three selected TC cases clearly demonstrate the process of upper-level TC–environment interaction: when a TC moves into a region of high (low) index, significant upper-level asymmetric forcing is exerted on the TC, leading to the strengthening (weakening) of the TC’s axisymmetric outflow and then possibly its intensity. As such, the horizontal map of ULFI not only provides a quantitative way of estimating the strength of upper-level asymmetric forcing at each grid point, but also serves as an indicator showing where the possible intensity change of a TC would occur under the influence of upper-level environmental flow. The index is thus recommended to be used in future studies of TC–environment interaction.

Corresponding author address: Shiqiu Peng, South China Sea Institute of Oceanology, State Key Laboratory of Tropical Oceanography, West Xingang Road 164, Guangzhou 510301, China. E-mail: speng@scsio.ac.cn

Abstract

A horizontal map of the upper-level forcing index (ULFI) is constructed to show the possible influence of upper-level large-scale environmental flow on the intensity change of tropical cyclones (TCs). The ULFI includes three commonly used diagnostics, that is, 200-hPa eddy flux convergences of both relative (REFC) and planetary angular momentum (PEFC), as well as axisymmetric absolute vorticity as a denominator that rescales the strength of the eddy forcings similar to the outflow-layer inertial stability. A simple procedure is adopted to convert these storm-relative components and the ULFI into Eulerian horizontal maps. Applications of this index map to three selected TC cases clearly demonstrate the process of upper-level TC–environment interaction: when a TC moves into a region of high (low) index, significant upper-level asymmetric forcing is exerted on the TC, leading to the strengthening (weakening) of the TC’s axisymmetric outflow and then possibly its intensity. As such, the horizontal map of ULFI not only provides a quantitative way of estimating the strength of upper-level asymmetric forcing at each grid point, but also serves as an indicator showing where the possible intensity change of a TC would occur under the influence of upper-level environmental flow. The index is thus recommended to be used in future studies of TC–environment interaction.

Corresponding author address: Shiqiu Peng, South China Sea Institute of Oceanology, State Key Laboratory of Tropical Oceanography, West Xingang Road 164, Guangzhou 510301, China. E-mail: speng@scsio.ac.cn

1. Introduction

Although improvement in intensity forecast skill for tropical cyclones (TCs) has recently been shown to be statistically significant (DeMaria et al. 2014), as compared to the increase in the skill of TC track predictions, there is still a substantial lag between intensity forecasts in the past decades (Elsberry et al. 1992; Emanuel 1999; Fitzpatrick 1997; Knaff et al. 2005; Montgomery and Smith 2014). Therefore, forecasting TC intensity is still more of a challenge for meteorologists, requiring more effort in the future.

It has been known that there are multiscale factors (from the convective scale to the synoptic scale) controlling TC intensity (e.g., Wang and Wu 2004). One of these factors is the upper-tropospheric (typically 200 hPa) environmental flow. The emphasis on its role may be traced back to Riehl (1950), who noted that in a hurricane’s development, some forcings were needed to ensure that upper-level outflow did not sink in the immediate storm environment and ultimately destroy the radial temperature gradient required to intensify the storm. Later, modulation of upper-level flow on TC intensity was confirmed by early observational studies (Colón 1963; Fett 1966; Ramage 1974; Rodgers et al. 1990; Sadler 1976, 1978). Case studies have also been extensive in the ensuing decades (e.g., Bosart et al. 2000; Erickson 1967; Hanley 2002; Leroux et al. 2013; Molinari et al. 1998; Molinari and Vollaro 1989; Qian et al. 2011; Rodgers et al. 1991; Rodgers and Olson 1998; Shi et al. 1997; Wu and Cheng 1999; Yu and Kwon 2005). Nowadays, this branch of research is also identified as upper-level TC–environment (or TC–trough) interaction. Intensification induced by interaction can be found during all stages of a TC’s lifetime, such as tropical depression formation (Bracken and Bosart 2000), tropical depression to hurricane transition (Bosart and Bartlo 1991), tropical storm to hurricane transition (Molinari et al. 1998; Shi et al. 1997), rapid intensification to a category 5 hurricane (Bosart et al. 2000), development into a supertyphoon (Titley and Elsberry 2000; Wu and Cheng 1999), and tropical storm (Qian et al. 2011) or typhoon (Zhu et al. 2005) to extratropical cyclone transition.

Two factors distinguish the special role of upper-level flow on affecting TC intensity. One is that in the upper troposphere, a TC does not retain its axisymmetric structure and thus environmental asymmetric forcing is generally stronger than that found in the middle or lower troposphere (e.g., Pfeffer and Challa 1981). Another is that the inertial stability in the upper levels is much smaller than those in the middle and lower levels (e.g., Holland and Merrill 1984). Therefore, upper-level forcing may excite vortex responses of large radial extent so that the forcing effects can easily penetrate to the inner-core region and then affect a TC’s intensity.

Eddy flux convergence of relative angular momentum (REFC) is usually adopted as a diagnostic for estimating the strength of upper-level asymmetric forcing and identifying TC–environment interaction. DeMaria et al. (1993) investigated the relation between 200-hPa REFC and Atlantic named hurricanes from 1989 to 1991 and found that about ⅓ of the storms intensified just after the enhanced REFC. Hanley et al. (2001) investigated the 12-yr records of hurricanes from 1985 to 1996. After excluding the records that occurred over cold water or close to land, they found that 78% of superposition and 61% distant interaction cases deepened. Both studies confirmed that interaction would favor TC intensification in a statistical sense. As for a TC’s rapid intensification (RI), however, Kaplan and DeMaria (2003) and Hanley et al. (2001) suggested that RI is less likely to occur for higher values of REFC. Only under the right circumstances then can TC–environment interaction result in a TC’s RI. Therefore, the “bad trough–good trough” issue proposed by Hanley et al. (2001) is still one of the biggest challenges in the context of RI (e.g., Leroux et al. 2013).

Nevertheless, many studies, from either individual or statistical perspectives, have argued for the utility of REFC as a predictor in TC intensity forecast schemes, for example, in the Statistical Hurricane Intensity Prediction Scheme (SHIPS; DeMaria and Kaplan 1994, 1999), the Typhoon Intensity Prediction Scheme (TIPS; Fitzpatrick 1997), and the Statistical Typhoon Intensity Prediction Scheme (STIPS; Knaff et al. 2005). In these schemes, REFC is computed in storm-relative cylindrical coordinates along TC tracks (i.e., an along-track diagnostic). This is not intuitive as compared to a map of sea surface temperature (SST) or vertical wind shear (VWS). That is, given a horizontal map of SST or VWS, one could easily point out the areas of high SST or low VWS over which a TC is likely to be maintained or intensify. In contrast, one cannot easily identify those areas favorable for TC intensification given a map of the upper-level horizontal wind field (e.g., Fig. 1). Although the downstream areas of troughs (denoted by three question marks in Fig. 1) are usually accompanied by ascending motion according to quasigeostrophic theory (e.g., Holton 2004) and thus may favor TC strengthening, it is only a qualitative way of identification. We still do not know which of the three downstream areas is most favorable for TC intensification. Even for a single trough, it is still unclear which part or even which grid point is the best place for TC intensification. To answer these questions, one needs a quantitative method to construct a corresponding horizontal map of an index similar to that of SST or VWS. Good guidance for constructing this index is to incorporate upper-level REFC forcing and inertial stability, both of which are important quantities emphasized by previous studies (e.g., Holland and Merrill 1984; Pfeffer and Challa 1981).

Fig. 1.
Fig. 1.

A snapshot of the 200-hPa horizontal winds at 0600 UTC 14 Sep 2004. Question marks show the downstream sides of upper-level troughs and propose the question of which area (which trough or even which grid point) is most favorable for TC intensification.

Citation: Weather and Forecasting 31, 1; 10.1175/WAF-D-15-0091.1

The present study is going to construct such a horizontal index and then demonstrates its usefulness in selected cases of TC–environment interaction. Section 2 describes the data used in the present study. Section 3 shows how to construct the index. Three TC cases are selected to demonstrate the index in section 4. Conclusions and discussion are given in section 5.

2. Data

“Best track” datasets are obtained from the Regional Specialized Meteorological Centre (RSMC) Tokyo for TCs over the western North Pacific and from the National Hurricane Center (NHC) for those over the North Atlantic. These datasets contain maximum surface wind speed and minimum sea level pressure (SLP) at least every 6 h for each TC valid at 0000, 0600, 1200, and 1800 UTC. Here, the minimum SLP will be used to describe the evolution of TC intensity.

A new version of the European Centre for Medium-Range Weather Forecasts (ECMWF; Dee et al. 2011) interim reanalysis dataset (ERA-Interim) is adopted here to describe 200-hPa environmental flows. The benefits of using the ECMWF reanalysis for studying the upper-level environment of TCs have been documented previously (Molinari and Vollaro 1990; Molinari et al. 1992). In the present study, horizontal wind data on a 0.75° × 0.75° latitude–longitude grid with 6-h interval are used.

3. Methodology

a. Construction of the horizontal index

According to the Sawyer–Eliassen balance (SEB) vortex theory (Eliassen 1952; Sawyer 1956), the axisymmetric secondary circulation of a TC is proportional to the magnitude of the REFC and inversely proportional to the inertial stability. To show this, we start from the axisymmetric tangential wind tendency equation in storm-relative cylindrical-pressure coordinates, which can be written as
e1
In Eq. (1), , , and are the azimuthal , radial , and vertical components of storm-relative velocity, respectively. Primes denote deviations from the azimuthal mean (overbar) quantities. In addition, is the radial component of the coordinates’ motion, is the Coriolis parameter, and is the tangential component of friction. Through incorporating the continuity equation of deviated velocities,
e2
Eq. (1) can be rewritten in flux form as
e3
where is the absolute vorticity. At the upper levels, friction generally vanishes. Neglecting those terms involving vertical motion that are large near the core regions, and assuming a steady state of the tangential wind, then Eq. (3) consists of only three terms:
e4
The first and second terms on the rhs are the REFC and the eddy flux convergence of planetary angular momentum (PEFC), respectively (e.g., DeMaria and Kaplan 1994). Following the conventions used in previous studies (e.g., Molinari and Vollaro 1989), both terms are in unit of meters per second per day (m s−1 day−1). Consistent with SEB theory, Eq. (4) clearly shows that the TC axisymmetric radial flow at upper levels is proportional to the magnitudes of REFC and PEFC, and inversely proportional to the axisymmetric absolute vorticity, defined as
e5
where is relative vorticity. Thus, the index, termed the upper-level forcing index (ULFI) in the present study, can be directly defined as
e6
The value of ULFI will be close to the observed axisymmetric outflow if the upper-level vortex is in steady state. Otherwise, ULFI could be viewed as the portion of the observed outflow that is exactly balanced by eddy momentum forcings. It is the imbalance between ULFI and the observed outflow that causes the adjustment of the tangential wind.
If the numerator and denominator in Eq. (6) are all multiplied by a factor of , then ULFI can also be written as
e7
where is the inertial stability commonly defined in SEB theory (e.g., Eliassen 1952). From Eq. (7), it is clear that the axisymmetric outflow (or ULFI) is proportional to the sum of REFC and PEFC and inversely proportional to the inertial stability I. Note that Eq. (7) is quite similar to the SEB vortex equation [e.g., see Eqs. (2), (3), and (6) in Molinari and Vollaro (1990)], and thus it can be deduced alternatively from the SEB vortex equation with similar assumptions. Now, we have obtained ULFI as defined in Eq. (7), and ULFI indeed has incorporated the two important factors (i.e., REFC and inertial stability, as emphasized in the introduction). But in the following practical calculation, we choose the form of Eq. (6) because the factor of in Eq. (7) can be eliminated. In the upper troposphere ( is not large) and not too close to the TC center ( is large), we may have and thus the temporal variation, as well as the sign of the absolute vorticity, are quite similar to those of the inertial stability in real TCs (figure not shown). In this sense, the absolute vorticity in Eq. (6) can be viewed as a proxy of inertial stability although their magnitudes are not even close.

The ULFI defined in Eq. (6), as well as its components REFC, PEFC, and absolute vorticity, are defined in storm-relative (quasi Lagrangian) cylindrical coordinates and thus cannot be used to construct a horizontal (Eulerian) map. It is noticed that these quantities are azimuthally averaged and are only functions of radius from the TC’s center. Similar to previous studies (e.g., DeMaria et al. 1993; Hanley et al. 2001), they can be averaged along the radial direction and result in a single mean value that depends only on the location of the coordinates’ origin (i.e., TC center), forming along-track diagnostics. What is needed for computing these along-track diagnostics is essentially a location of an origin and its translating velocity. In fact, at lower levels there could be TC-like vortices located at any positions moving at any speeds. Therefore, to form a horizontal map (see Fig. 2), one can simply assume that there is a vortex at every latitude–longitude grid point moving at the same speed as a real TC and establish cylindrical coordinates at each grid point. Then, horizontal maps can be constructed by computing the along-track quantities at each grid point.

Fig. 2.
Fig. 2.

Schematic showing how the horizontal map of ULFI (or its REFC, PEFC, and absolute vorticity components) is constructed. The black crosses represent the latitude–longitude grid points. Arrows show the translating vector of a TC (red TC symbol). A cylindrical coordinate system whose origin overlaps with a latitude–longitude grid point is established for the evaluation of ULFI with respect to that point. Then, ULFI is averaged radially between two specific radii (pink shaded). Finally, every grid point should have only one averaged value, forming a continuous horizontal map of ULFI.

Citation: Weather and Forecasting 31, 1; 10.1175/WAF-D-15-0091.1

Following this idea, cylindrical coordinates, containing 72 azimuthal grid points with 5° intervals and 19 radial grid points with 0.3° intervals (about 33.3 km), are established at each latitude–longitude grid point. Latitude–longitude gridded wind data are interpolated onto the cylindrical coordinates in 16-point bicubic polynomial fashion and then reprojected to azimuthal and radial components. Finally, those along-track quantities defined in Eq. (6) are computed in cylindrical coordinates for every grid point, forming horizontal maps.

Figures 3a–c show the 200-hPa horizontal maps of REFC, PEFC, absolute vorticity, and finally ULFI defined in Eqs. (4)(6), computed using the above procedure. An average across a 300–600-km radial band is taken following previous studies (e.g., Hanley et al. 2001). As shown in Fig. 3a, on an REFC map a local maximum center represents the strongest 300–600-km asymmetric environmental forcing among the neighboring grid points. The traditional along-track REFC value (computed in storm-relative cylindrical coordinates) can be obtained just by sampling the values on this REFC map along any trajectory (assuming slow translating speed along the trajectory). From the map, it is found that large REFC values occur in midlatitude westerlies north of ~30°N. Maximum REFC centers are located downstream of troughs accompanied by southwest winds while minimum centers are associated with areas upstream of troughs accompanied by northwest winds. The maximum magnitude of REFC can reach above 140 m s−1 day−1, an order of magnitude larger than the threshold (10 m s−1 day−1) used for identifying TC–environment interaction (e.g., DeMaria et al. 1993; Hanley et al. 2001). The value of 140 m s−1 day−1 is close to those of extreme interaction cases found by Qian et al. (2016) using 25-yr-long TC records, indicating that when TCs move into midlatitude westerlies they would experience extremely large REFC conditions and possibly be transformed into extratropical cyclones in such environments. For the PEFC map shown in Fig. 3b, as the Coriolis parameter does not have zonal deviations, PEFC is affected only by full (not storm relative) meridional winds. In contrast to REFC, the maximum PEFC centers occur in upstream areas of troughs while the minimum centers occur along the downstream sides. Therefore, the effects of the REFC and PEFC tend to be canceled out when they are summed up in Eq. (6). As the magnitude of PEFC is only one-tenth of that of REFC, summing PEFC will not strongly alter the maximum/minimum centers on the REFC map. Similar to DeMaria and Kaplan (1994), who have incorporated PEFC as a predictor into their intensity forecast scheme, we keep PEFC in Eq. (6) to account for more factors but still retain the minimal requirement of wind data (i.e., only one single-layer wind data source).

Fig. 3.
Fig. 3.

The 200-hPa winds (vector; m s−1) and horizontal maps of 300–600-km-averaged (a) REFC (m s−1 day−1), (b) PEFC (m s−1 day−1), (c) absolute vorticity (10−5 s−1), and (d) ULFI (m s−1) valid at 0600 UTC 14 Sep 2004. (e)–(h) As in (a)–(d), but for the 500–800-km average.

Citation: Weather and Forecasting 31, 1; 10.1175/WAF-D-15-0091.1

Figure 3c shows the 300–600-km-averaged absolute vorticity . As in the denominator of Eq. (6), it will scale up or down the REFC and PEFC forcings. This is similar to the role of inertial stability, which exerts some resistant effects on environmental forcings (Holland and Merrill 1984; Schubert and Hack 1982). Absolute vorticity is invariant to the coordinate transform, and thus the result computed in storm-relative cylindrical coordinates is the same as that computed in spherical coordinates. Therefore, the 300–600-km average only tends to blur some details of its structure in traditional latitude–longitude grids without averaging (figure not shown) while the general north–south gradient is still obvious in Fig. 3c. Note that there are some blank regions near the equator where , indicating inertial instability and the possible failure of the SEB theory. Thus, it is suggested to compute the index several degrees away from the equator in order to avoid the instability and small-denominator problem in Eq. (6).

Figure 3d shows the horizontal map of ULFI defined in Eq. (6). We can see that the maximum/minimum REFC centers embedded in the westerlies have been scaled down by the larger absolute vorticity at higher latitudes. The magnitudes of midlatitude centers are comparable to those in the subtropics or even the tropics. However, one should notice the extremely large index near the equator where inertial instability occurs (blank regions in Fig. 3c). To avoid the problems of inertial instability and a small denominator, we suggest computing the index away from the equator or masking out regions where the absolute vorticity is small (e.g., < 1 × 10−5 s−1). Such a map provides direct and quantitative information to forecasters. Based on the REFC map (Fig. 3a), we can determine which specific area will exert the strongest environmental asymmetric eddy forcing when a TC moves in; based on the ULFI map (Fig. 3d), we can see how strongly a TC could feel the environmental forcing. The difference is that the axisymmetric absolute vorticity will scale up or down the asymmetric forcing effects. If eddy forcings are the dominant process controlling TC intensity [i.e., Eq. (4) holds well], the maximum ULFI center is also the place where the TC axisymmetric outflow response reaches its maximum. Thus, if a TC moves from a low index area into a high index area, it is likely to intensify from the perspective of upper-level environmental interaction. This is as simple as reading a horizontal map of SST and VWS.

In the present study, a threshold value of ULFI should be determined as a high index value. This ULFI threshold can also be used to identify interaction. A direct choice is to use a value corresponding to the 10 m s−1 day−1 REFC threshold proposed by previous studies (e.g., DeMaria et al. 1993) for identifying interactions. If is approximated by planetary vorticity at 20°N, then that REFC threshold corresponds to a value of 2.3 m s−1 for ULFI. Taking into account that will be smaller if upper-level flow is anticyclonic, a value of 3 m s−1 will be a good choice, which will be used in the present study (see Fig. 3d). The following cases will show that both thresholds of ULFI and REFC give very similar results in identifying interactions.

b. Possible influence of some parameters

There are several issues related to the index defined in Eq. (6), and these should be discussed before applying this tool to real TCs. The first is the choice of radial band for averaging quantities that may have possible influence on their magnitudes and patterns. Previous studies (e.g., DeMaria et al. 1993) have tried different radial bands from 100 to above 1000 km, which actually reflects different distances between environment-induced forcings and the TC’s inner core. Molinari and Vollaro (1989) estimated the errors of REFC as a function of radius and have shown that within 300 km the errors (or uncertainty) could be larger than 40%. This indicates that the start radius for averaging should not be too small. On the other hand, if large radii are chosen, the effects of flow forcing may not reach the TC center and thus have no direct and immediate impact on the TC’s intensity. The second issue is that the translating velocity should be subtracted first before calculating REFC. However, if we want to plot the index at any given time, there could be no TCs or more than a single TC over the basin of interest so that the translating velocity may not be properly defined. For simplicity, it is better not to take into account the translating velocity in the calculation. But the possible influence should be estimated first.

To examine the influence of different averaging radial bands on the index, we choose 300–600 and 500–800 km for comparison. Here, the translation speed is not accounted for as no specific TC is selected so that the cylindrical coordinates are all fixed at latitude–longitude grid points. Figures 3e–h show the horizontal maps averaged using the 500–800-km radial band corresponding to Figs. 3a–d. For REFC (Fig. 3e), maximum and minimum centers embedded in westerlies, as well as their patterns, are not strongly altered because of their large magnitudes. Those centers in the subtropics tend to be blurred out in the 500–800-km average that resembles a larger-window running mean. This situation is similar to that of the absolute vorticity so that the outer-radius average (Fig. 3g) tends to reduce the area of inertial instability where . In contrast to REFC, the magnitude of the PEFC center has been enhanced in the 500–800-km average because the meridional asymmetry increases as the radius increases. Taking a comparison between ULFI shown in Figs. 3d and 3h, we can conclude that the outer-radius average only tends to smooth the index centers slightly without strongly altering their pattern.

Another important factor affecting the index is the translating velocity of the cylindrical coordinates (or TC). Skubis and Molinari (1987) have shown that in order to get a more accurate estimate, one needs to consider the effects of a storm-relative framework on fast-moving TCs in a nonuniform environment. Thus, in principle, quantities defined in Eq. (6) should be evaluated within a storm-relative framework. If there are two or more TCs simultaneously over the same ocean basin, constructing the index could be problematic because the translating velocity is ambiguous. The best solution would be to compute the index with respect to different TCs of different translating velocities. Here, the possible influence of the translating velocity on the index is also examined.

Among the three components (e.g., REFC, PEFC, and absolute vorticity) used to define the ULFI, the absolute vorticity (vertical component of the vorticity vector) can be viewed as a common scalar (similar to temperature) and thus will not affected by the storm-relative framework. Nor will PEFC be affected because it uses the full velocity rather than the storm-relative velocity. Therefore, only REFC is examined here. Figure 4 shows the maps of REFC with respect to translating velocities of the same magnitude (10 m s−1) but of different directions. A TC moving at a speed of 10 m s−1 can be viewed as a fast-translating TC. For TCs moving at this speed, the storm-relative winds will grow to as large as twice the magnitude of the translating speed. If the full winds are not strong enough (such as in the subtropical regions), their signs can be easily altered within a storm-relative framework so that both the pattern and the magnitude of REFC would change slightly, especially when the translating velocities are in opposite directions (cf. Figs. 4a and 4b or Figs. 4c and 4d). Therefore, in order to accurately estimate the eddy momentum forcing, a storm-relative framework should be taken into account. If there is more than one TC appearing at the same time, one should compute the index with respect to different TCs so that the storm-relative effect can be considered correctly.

Fig. 4.
Fig. 4.

The 200-hPa storm-relative winds (vector; m s−1) and 300–600-km-averaged horizontal map of REFC (shaded; m s−1 day−1) valid at 0600 UTC 14 Sep 2004. The cylindrical coordinate established at each grid point moves (a) northward, (b) southward, (c) eastward, and (d) westward at a constant speed of 10 m s−1. (e) As in (a), but the cylindrical coordinates are fixed.

Citation: Weather and Forecasting 31, 1; 10.1175/WAF-D-15-0091.1

4. Application in TCs

In this section, the ULFI will be applied to real TC cases to demonstrate its utility. Three TC cases (Hurricane Elena 1985; Tropical Storm Haima 2004; and Typhoon Omar 1992) are selected here. The first two cases have already been well documented by previous studies (e.g., Molinari and Vollaro 1990; Qian et al. 2011). Through solving the SEB vortex model, both cases have shown that their intensifications during specific periods resulted from interacting with the upper-level environment. The third case, Omar (1992), is selected as a countercase, as it did not intensify when experiencing interaction, indicating a possible failure of the index and highlighting the unresolved “good trough–bad trough” problem in the area of TC–environment interaction. Although these cases cover only two ocean basins (Elena for the North Atlantic and Haima and Omar for the western North Pacific), the applicability of the index can be extended to the Southern Hemisphere. High-index areas in the Southern Hemisphere still stand for regions favorable for TC strengthening since REFC and PEFC, as well as absolute vorticity, change sign simultaneously in the Southern Hemisphere.

a. Hurricane Elena (1985)

Hurricane Elena (1985) is selected here as a favorable interaction case during the mature hurricane stage. It is studied by Molinari and his coauthors in detail (Molinari et al. 1995; Molinari and Vollaro 1989, 1990), and its historical evolution can also be found in Molinari and Vollaro (1989). We emphasize the RI process that started at 0000 UTC 1 September 1985 (Fig. 5a) when Elena hovered around the Gulf of Mexico just before making landfall. This intensification is highly correlated to outer-radius eddy momentum fluxes 27–33 h before (Molinari and Vollaro 1989). Through solving the SEB vortex equation, Molinari and Vollaro (1990) found that the inward propagation of eddy momentum fluxes excited a contracting upper-level outflow maximum that was related to the RI process.

Fig. 5.
Fig. 5.

Along-track quantities for Hurricane Elena (1985). (a) Minimum sea level pressure (hPa; black line), SST (°C; red line), VWS (m s−1; blue line), and ULFI (m s−1; green line). SST and VWS are averaged within a 300-km radius, and ULFI is averaged over a 300–600-km radial band. Note that SST is undefined after 1200 UTC 2 Sep because of Elena’s landfall. The black arrow highlights the intensification of interest here, which is associated with an upper-level environmental influence. (b) The 200-hPa REFC (m s−1 day−1; black line) and PEFC (m s−1 day−1; green line) averaged over a 300–600-km radial band. The horizontal dashed line is the threshold for identifying interaction using the REFC diagnostic process. (c) The 200-hPa mean absolute vorticity (10−5 s−1; black line) and its two components, mean relative vorticity (10−5 s−1; green line) and mean planetary vorticity (10−5 s−1; blue line), all of which are averaged over a 300–600-km radial band. The vertical solid line in each panel indicates the time when Elena was over land.

Citation: Weather and Forecasting 31, 1; 10.1175/WAF-D-15-0091.1

To provide a clear description here about the large-scale environment of Elena, two commonly used along-track diagnostics, SST and VWS, as well as ULFI proposed in the present study, are shown along with Elena’s minimum sea level pressure (Fig. 5a). Additionally, three components (i.e., REFC, PEFC, and absolute vorticity) in ULFI are also calculated separately to show their relative contributions to the index (Figs. 5b,c). Here, VWS is defined as the magnitude of the difference between the 850- and 200-hPa wind vectors, and both SST and VWS are averaged within a 300-km radius.

First of all, we briefly review the temporal evolutions of SST, VWS, and REFC, three commonly used environmental diagnostics employed in many previous studies (e.g., Wu and Cheng 1999; Yu and Kwon 2005; Chen et al. 2015). Figure 5a shows that starting from 31 August and before Elena’s landfall, SSTs within 300 km of Elena’s center fluctuated slightly between 28° and 28.5°C, which provided a favorable background for TC maintenance. However, the triggering of Elena’s RI could not be attributed to the SST environment because the SSTs remained roughly constant. On the other hand, as noted by Molinari and Vollaro (1989), 1 day before RI, there was a significant increase in 200-hPa REFC, which remained above 10 m s−1 day−1 (dashed line in Fig. 5b), indicating Elena’s interaction with upper-level environmental flows according to the 10 m s−1 day−1 threshold proposed by previous studies (e.g., DeMaria et al. 1993). Such interaction is also accompanied by an enhanced VWS (Fig. 5a) whose peak reached about 10 m s−1. The subsequent weakening of VWS at 0000 UTC 1 September, although transient, provided another favorable factor and coincided well with the start of Elena’s RI.

Such interaction features can also be clearly seen from the synoptic evolution of the 200-hPa winds and the corresponding horizontal map of ULFI constructed with respect to Elena (Fig. 6). After Elena straightforwardly headed into the Gulf of Mexico, a westerly trough started to deepen over the U.S. mainland at 1200 UTC 30 August. The bottom of the trough was located just north of Elena. Areas with ULFI larger than 3 m s−1 (red areas in Fig. 6) are associated with the area downstream of the trough and span ~25° in latitude from the midlatitudes to the subtropics, which is to the east of Elena. At 0000 UTC 31 August, Elena started to move east and circle around very close to Florida, just inside the red areas in Fig. 6, and continued this motion for about 1 day. The overlapping of Elena with the red index area is in accordance with the enhancement of the 200-hPa eddy momentum flux shown in Fig. 5b. One day later at 0000 UTC 1 September, Elena detached from the bottom of the trough (as indicated by the separation of the TC and the red index area) and started to move westward with the RI highlighted in Fig. 5a.

Fig. 6.
Fig. 6.

Synoptic evolution of 200-hPa winds (vectors; m s−1) and horizontal maps of ULFI (shaded; m s−1) defined in Eq. (6) from 1200 UTC 29 Aug to 1200 UTC 2 Sep 1985 at 12-h intervals. The black line and dot, respectively, show the track and current position of Hurricane Elena (1985).

Citation: Weather and Forecasting 31, 1; 10.1175/WAF-D-15-0091.1

From the along-track evolutions of the three components of ULFI (Figs. 5b,c), it is clear that ULFI is largely determined by REFC, especially when the magnitude of REFC is larger than 10 m s−1 day−1. The contribution of PEFC to the index is negligible except when REFC is very small. However, the role of absolute vorticity (similar to that of inertial stability) cannot be ignored. According to Eq. (3), large REFC may cause a significant cyclonic spinup () of the upper-level vortex. Such spinup then leads to an increase in the azimuthally averaged relative vorticity (green line in Fig. 5c). Therefore, the role of REFC is twofold: the positive one is to exert environmental asymmetric forcing on an axisymmetric vortex while the negative one is to spinup and stabilize the upper-level vortex so that the vortex responds less to the forcing. These two effects are opposite and the net effect is illustrated by ULFI as defined here. For example, the along-track REFC increased dramatically from 1200 UTC 30 August to 1800 UTC 31 August, with a nearly constant rate of increase (see REFC slope in Fig. 5b). ULFI also increased during this period (green line in Fig. 5a) but the rate of increase after 0000 UTC 31 August greatly slowed down, which is attributed to the increase in the azimuthal-mean absolute vorticity (Fig. 5c). As the REFC peak (Fig. 5b) is greatly smoothed in ULFI (Fig. 5a), we can say that during a persistent interaction, the positive contribution of REFC to ULFI will be gradually scaled down so that the eddy-induced portion of the outflow will get smaller.

There are several things that make Hurricane Elena a special interaction (or typical “good” trough) case. First, Elena experienced a moderate REFC (~25 m s−1 day−1) in the 300–600-km radial band (Molinari and Vollaro 1990) on 31 August rather than extremely large REFC (>40 m s−1 day−1) that is generally associated with strong VWS cases. Although the red index area extends over a wide range from the midlatitudes to the subtropics (Fig. 6), the part over the Gulf of Mexico is more favorable for Elena’s intensification when taking into account the effect of VWS. Elena never went into the westerly trough, and the trough never deepened dramatically to enclose Elena. It is just the moderate environmental forcing with not-too-large VWS that caused Elena’s unexpected RI very close to the shore rather than destroying it. Second, there is a lead/lag correlation between the enhanced environmental forcing and subsequent RI so that the force–response explanation based on SEB theory may be improper. Molinari et al. (1995) speculated that the environmental forcing triggered the wind-induced surface heat exchange (WISHE; Emanuel 1986) process, and thus there would be a time lag for Elena’s RI. However, this cannot be seen or inferred directly from the ULFI map. We note the possible lead/lag correlation here to draw forecasters’ attention to this pattern.

b. Tropical Storm Haima (2004)

A second example, Tropical Storm Haima (2004), has been documented by Qian et al. (2011). During its life cycle, it only reached tropical-storm strength according to the RSMC best-track data. After moving around Taiwan Island, Haima made landfall along the eastern coast of China and dissipated with its minimum central pressure increasing to 1008 hPa at 1200 UTC 13 September (Fig. 7a). It is selected here because of its unusual reintensification after 14 September with its pressure continuously falling, down to 995 hPa, even over land as it moved north. Thus, the ULFI can be applied to those cases close to land or even overland, not just to those over open water. Through solving the SEB vortex model, Qian et al. (2011) show that the dominant contributor to Haima’s secondary circulation at this stage is the 200-hPa eddy absolute angular momentum flux convergence associated with an upper-level S-shaped flow pattern. Therefore, its unusual reintensification is attributed to the interaction with the S-shaped flow. Another reason for choosing this case is that this interaction occurred during Haima’s extratropical cyclone stage, rather than during the tropical storm (or typhoon) stage, as was the case with Elena. Thus, the applicability of ULFI can also be examined in such extratropical transition cases.

Fig. 7.
Fig. 7.

As in Fig. 5, but for Tropical Storm Haima (2004).

Citation: Weather and Forecasting 31, 1; 10.1175/WAF-D-15-0091.1

Figure 7 shows the along-track environmental diagnostics for Haima. Haima’s filling during 13 September is mainly attributed to its landfall, as well as the dropping SST. Beginning at 1800 UTC 13 September, Haima started to reintensify significantly while interacting with 200-hPa environmental flow identified by REFC exceeding 10 m s−1 day−1 (Fig. 7b). This interaction is characterized by large REFC (Fig. 7b) and VWS (Fig. 7a) values whose peaks are almost 3 times as large as those in the case of Elena. In such a strongly sheared environment, Haima could not maintain its warm-core structure and was transformed into an extratropical cyclone (EC) at 1800 UTC 13 September as identified by RSMC. This case is quite different from that of Elena as interaction occurred during Haima’s EC stage and thus is selected on purpose to evaluate the index during a TC’s different evolutionary stages.

Figure 8 shows the synoptic evolution of 200-hPa winds and the horizontal map of ULFI. At 0000 UTC 13 September, a southwest–northeast-tilted westerly trough came into the domain from the west and swept over mainland China. As the trough moved eastward, it gradually became southeast–northwest tilted and finally encountered the north-moving Haima at 0000 UTC 14 September. At this time, Haima had almost died out 1 day after landfall. However, as its remnants moved into the red ULFI area associated with the downstream region of the trough, it reintensified continuously over the following 2 days. During these 2 days, under the advection of the downstream southerly flow of the trough, Haima moved northward and remained inside the red index region all of the time (Fig. 7), with maximum ULFI close to 6 m s−1 during 15 September.

Fig. 8.
Fig. 8.

As in Fig. 6, but for Tropical Storm Haima (2004) from 1200 UTC 11 Sep to 1200 UTC 15 Sep 2004.

Citation: Weather and Forecasting 31, 1; 10.1175/WAF-D-15-0091.1

From the evolutionary patterns of the components of ULFI (Figs. 7b,c), we can see that although REFC reached ~70 m s−1 day−1, almost 3 times as large as that in the case of Elena, the corresponding ULFI (~5 m s−1) is only slightly larger than that of Elena (~4.5 m s−1). The relatively large negative contribution of PEFC, which arises from the meridional asymmetry of the radial wind associated with the S-shaped flow (see the snapshot from 0000 UTC 15 September in Fig. 8), may partially account for this. More importantly, as a result of the very large REFC forcing as well as Haima’s fast northward movement, the absolute vorticity increased dramatically so that the upper-level vortex is stabilized significantly and thus the forcing effect is greatly reduced, as shown in the ULFI (Fig. 7a).

It is also worth noting some features of this case that are different from those of Elena. First, Haima penetrated the disturbed westerlies and eventually became embedded inside the trough (last snapshot in Fig. 8). Thus, from the start, strong VWS was brought in by environmental interaction. However, the interaction occurred during Haima’s EC stage, which is generally different from a typical TC, and thus the accompanying VWS may not be destructive to Haima’s reintensification. Second, there is no obvious lead/lag relationship between the environmental forcing and Haima’s intensity change. Therefore, Haima’s reintensification can be explained quite well as an axisymmetric vortex response to asymmetric eddy momentum forcing based on the SEB theory (Qian et al. 2011).

c. Typhoon Omar (1992)

The two cases discussed previously are typical examples illustrating how ULFI can indicate TC intensification when interacting with upper-level flows. There are of course countercases, for example, TCs that do not intensify or even weaken when interaction occurs. In reality, these cases are not rare, and the reason why they weakened could be explained by the decreasing of SST or increasing of VWS during interactions (e.g., DeMaria et al. 1993; Qian et al. 2016). Such cases generally occur during the tropical storm and typhoon stages rather than the EC stage because they are more vulnerable to low SST and high VWS. Typhoon Omar (1992) is selected here as an “unusual” countercase as compared to those more common countercases. Unusual in this sense means that its weakening during interaction cannot be attributed to decreasing SST or increasing VWS. Through this case, the possible failure of ULFI in indicating the TC intensity change during interaction is clearly shown. More importantly, the problem of favorable interaction [or equivalently the “good” trough problem proposed by Hanley et al. (2001)], which is still an unresolved one in the field of TC–environment interaction, is highlighted for forecasters.

Typhoon Omar is characterized by its high level of intensity (Fig. 9a) and its straight-line track (Fig. 10). Figure 9a shows that Omar’s minimum sea level pressure deepened to 920 hPa at 1800 UTC 29 August and remained unchanged for 1 day. During this time, there was a significant increase in REFC (Fig. 9b) from −20 to 20 m s−1 day−1 the following day. According to the REFC criterion, Omar interacted with an upper-level environment for only half a day on 31 August. As compared to Elena (see Fig. 5), Omar experienced slightly smaller REFC forcing while its upper-level mean absolute vorticity is also smaller (Fig. 9c), leading to a peak ULFI value of 4 m s−1, comparable to that of Elena. At the same time, its underlying SST remained close to 28.5°C and the VWS dropped from above 12 to below 6 m s−1, both of which seem to be favorable (at least not hostile) for TC intensification. In such favorable environments, the upper-level interaction did not initiate Omar’s intensification. Instead, Omar weakened slowly from 31 August and lasted for about 2 days with its pressure increasing step by step (arrow in Fig. 9a).

Fig. 9.
Fig. 9.

As in Fig. 5, but for Typhoon Omar (1992).

Citation: Weather and Forecasting 31, 1; 10.1175/WAF-D-15-0091.1

Fig. 10.
Fig. 10.

As in Fig. 6, but for Typhoon Omar (1992) from 0600 UTC 29 Aug to 0600 UTC 2 Sep 1992.

Citation: Weather and Forecasting 31, 1; 10.1175/WAF-D-15-0091.1

Figure 10 shows more details about Omar’s interaction and associated upper-level wind flows. Before its weakening on 31 August, Omar’s outflow was clear and located equatorward. In addition, there was a developing anticyclone to the northwest of Omar, with the strong southward wind on its east flank flushing across the TC’s center (see snapshots from 30 August in Fig. 10), leading to a VWS larger than 10 m s−1 during this period. This southward flow curved anticyclonically and merged with Omar’s outflow, resulting in small or even negative absolute vorticity. Therefore, there are some discontinuities in the map of ULFI. During the subsequent day (31 August), Omar started to overlap with red ULFI areas. Such overlap indicates its interaction with the upper-level environment, which is consistent with REFC criterion. The red ULFI area around Omar’s center is associated with an upper-level inverted trough (easterly wave), in contrast with traditional westerly trough interaction. This flow feature was transient and the interaction only lasted for half a day. Starting at 1200 UTC 30 August, Omar moved into regions of strong easterly wind associated with the fully developed anticyclone that occupied the north half of the domain. Because of its strong advective effect, Omar continued to move northwestward with a dramatic increasing in VWS (Fig. 9a).

Several aspects of this countercase are summarized here. First, Omar’s interaction with upper-level flow is not a typical TC–westerly trough interaction. Thus, the interaction is not strong (in terms of REFC) and the duration is not long. Second, under two favorable large-scale environments (i.e., high SST and weak VWS), interaction with an inverted trough did not initiate Omar’s intensification. This is not too surprising because the TC’s intensity change is controlled not only by large-scale environmental factors but also by small-scale inner-core dynamics such as vortex Rossby waves, eyewall replacement cycles, etc. Here, it is speculated that after reaching 920 hPa, Omar slowed down its movement and thus induced strong upwelling and mixing of cooler water to the surface, which in turn lead to Omar’s gradual weakening. Finally, this case (among others) shows the possible failure of the ULFI in indicating the TC intensity change. Although many of the countercases could be attributed to low SST or high VWS, there are also cases similar to Typhoon Omar that weakened in favorable interaction environments. This horizontal map of ULFI will succeed in indicating the TC intensity change only when the intensity change is largely controlled by upper-level environments rather than other factors. This is also the basic assumption under which Eq. (4) holds.

5. Summary and discussion

The present study defines an index, named the upper-level forcing index (ULFI), in storm-relative cylindrical coordinates [Eq. (6)] to represent the possible influence of upper-level environmental forcing on TC intensity. This index takes into account three dynamic factors: asymmetric eddy forcings that arise from both relative and planetary angular momentum, as well as outflow-layer absolute vorticity, whose effect is similar to that of inertial stability, which provides some resistance in the vortex response to the eddy forcings. All of these are important dynamical factors that have been emphasized by previous studies (e.g., Challa and Pfeffer 1980; DeMaria et al. 1993; DeMaria and Kaplan 1994; Holland and Merrill 1984; Molinari and Vollaro 1989; Rappin et al. 2011; Schubert and Hack 1982). A simple procedure is then applied to transform the cylindrical-coordinate-based ULFI into an Eulerian map for forecasters’ convenience. On such maps of ULFI, high index areas represent favorable upper-level environments for TC intensification. Therefore, one could easily point out which area, or even which grid point, is most favorable for TC intensification, only requiring 200-hPa synoptic wind fields. Finally, this ULFI map is applied to three selected interaction cases. Results show that this index succeeds in identifying upper-level TC–environment interaction and indicating the associated intensity change for the first two cases but failed in the third countercase. Characteristics of each case are analyzed and reasons for the failure in the third case are also discussed.

Interaction of a TC with its upper-tropospheric environment has been a branch of TC intensity research for more than half a century, and REFC has become a commonly used diagnostic for measuring the strength as well as identifying the occurrence of interactions. The ULFI defined here may also be used for such purposes and the corresponding results, as has already been shown in the three TC cases, agreed well with those obtained by the REFC diagnostic. This is because ULFI is largely determined by the REFC component. However, there is an important difference between ULFI and REFC: ULFI incorporates the axisymmetric absolute vorticity as a denominator. This denominator rescales the strength of upper-level eddy forcings, indicating that vortex rotation would resist the secondary response to the eddy forcings. Such a resistant effect of absolute vorticity is similar to that of inertial stability emphasized by many studies (e.g., Holland and Merrill 1984; Rappin et al. 2011; Schubert and Hack 1982), and their connection is shown by Eq. (7). Without the vorticity denominator, the maximum centers of the index map will be embedded in the midlatitude westerlies (Fig. 3a). However, if absolute vorticity is included as a denominator, the index maxima inside the westerlies will be significantly reduced and then will be comparable to those in the subtropics (Fig. 3d). In this sense, ULFI should be used as a more complete diagnostic as compared to REFC in the study of upper-level TC–environment interactions.

There are several advantages of the horizontal map of ULFI in practice. First of all, it is very easy to understand what this index means. According to Eq. (6), this index stands for the portion of TC axisymmetric radial flow that is exactly balanced by eddy angular momentum forcings. When a TC moves into high index area, its outflow is likely to be enhanced. But when a TC moves into low index area, an upper-level inflow will be generated to balance the eddy forcing. If the forcing and the excited outflow affect the TC core in an optimized manner, for example, through the wind-induced surface heat exchange (WISHE) mechanism (Molinari and Vollaro 1990) or by exciting an eyewall replacement cycle (Leroux et al. 2013), the upper-level eddy forcing will cause TC intensification. Second, reading the ULFI map is just as easy as reading a map of SST and VWS. A higher index area, similar to the areas of high SST and low VWS, is favorable for TC intensification. This is applicable to both hemispheres without caring about the sign of the index except in tropical regions where inertial instability occurs and SEB theory may fail. If REFC alone is chosen as the index, one should be aware that in the Southern Hemisphere a negative index area would be favorable for TC intensification. Finally, it is very easy to compute ULFI because it requires only a single layer (usually 200 hPa) of wind data. Along with the index, its three components (i.e., REFC, PEFC, and absolute vorticity) can also be illustrated separately (as shown in Fig. 3) to see their relative contributions to the index. Utilizing predicted winds and TC position from operational forecasts, one can also gain predictive information straightforwardly from this index map.

There are also some limitations. The most important one is that this index shows the effect of only one large-scale environmental factor (i.e., upper-level flow). It becomes useful or accurate only when the upper-level environment plays the dominant role in affecting the TC intensity. However, there are also many cases that experienced interactions but still weakened, as documented by case studies (e.g., Lewis and Jorgensen 1978; Yu and Kwon 2005) or inferred from statistical studies (e.g., DeMaria and Kaplan 1994; Hanley et al. 2001; Qian et al. 2016), primarily because of the influence of other large-scale environmental factors such as low SST or strong VWS. Therefore, one cannot ignore the effects of SST and VWS. Interactions with strong VWS may favor the strengthening of TCs that undergo extratropical transitions (see the case of Haima). However, such interactions are likely to weaken typical TCs (such as Elena) by destroying their warm-core structures.

Therefore, including SST and VWS when constructing a more complete horizontal index map will be much better than using upper-level dynamic factors alone. The present study has provided a simple way to combine quasi-Lagrangian factors (e.g., REFC and PEFC defined in cylindrical coordinates) with Eulerian factors (e.g., SST and VWS). Such combinations can be made simply through regression processes similar to SHIPS (DeMaria and Kaplan 1994, 1999) but implemented grid by grid to form a horizontal map of a multipredictor index. Such a horizontal index will definitely be more indicative for forecasters.

Acknowledgments

This study is jointly supported by the MOST of China (Grant 2014CB953904), the China Special Fund for Meteorological Research in the Public Interest (Grant GYHY201406008), the National Natural Science Foundation of China (Grants 41205032, 41405048, and 41376021), and the Guangdong Marine’s Disaster Emergency Response Technology Research Center (2012A032100004). Discussion with Prof. Zhuojian Yuan from Sun Yat-sen University is much appreciated. The authors gratefully acknowledge the use of the HPCC at the South China Sea Institute of Oceanology, Chinese Academy of Sciences.

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