Abstract

Identifying the condition(s) of how tropical cyclones intensify, in particular rapid intensification, is challenging, because of the complexity of the problem involving internal dynamics, environments, and mutual interactions; yet the benefit to improved forecasts may be rewarding. To make the analysis more tractable, an attempt is made here focusing near the sea surface, by examining 23-yr global observations comprising over 16 000 cases of tropical cyclone intensity change, together with upper-ocean features, surface waves, and low-level atmospheric moisture convergence. Contrary to the popular misconception, we found no statistically significant evidence that thicker upper-ocean layers and/or warmer temperatures are conducive to rapid intensification. Instead, we found in storms undergoing rapid intensification significantly higher coincidence of low-level moisture convergence and a dimensionless air–sea exchange coefficient closely related to the youth of the surface waves under the storm. This finding is consistent with the previous modeling results, verified here using ensemble experiments, that higher coincidence of moisture and surface fluxes tends to correlate with intensification, through greater precipitation and heat release. The young waves grow to saturation in the right-front quadrant as a result of trapped-wave resonance for a group of Goldilocks cyclones that translate neither too slowly nor too quickly, which 70% of rapidly intensifying storms belong. Young waves in rapidly intensifying storms also produce relatively less (as percentage of the wind input) Stokes-induced mixing and cooling in the cyclone core. A reinforcing coupling between tropical cyclone wind and waves leading to rapid intensification is proposed.

1. Introduction

In part because of the complex dynamics involved, forecasts of tropical cyclone intensity have not improved significantly in the past several decades (Rogers et al. 2006; Houze et al. 2007; Rappaport et al. 2009; Gall et al. 2013; Rios-Berrios et al. 2014). Yet understanding and identifying the conditions of how a tropical cyclone changes intensity can lead to improved prediction and storm preparedness, which may save lives and properties. Tropical cyclone intensity depends on various often interrelated factors such as the following:

  1. the large-scale environment including the ocean and cyclone translation (Schade and Emanuel 1999; Dunion and Velden 2004; Wang and Wu 2004; Zeng et al. 2007; Lin et al. 2008; Hendricks et al. 2010; Sun and Oey 2015; Sun et al. 2017);

  2. the inner-core dynamics (Emanuel 1986; Hendricks et al. 2004; Wang and Wu 2004; Montgomery et al. 2009, 2015);

  3. the air–sea energy exchanges (Donelan et al. 2004);

  4. the upper-tropospheric outflow and inertial instability (Merrill 1988; Rappin et al. 2011); and

  5. their mutual influences (e.g., Wang and Wu 2004; Evans et al. 2010).

The physics of tropical cyclone intensification thus involves a wide range of spatial and temporal scales (Wang and Wu 2004). The literature may be very broadly grouped into one that examines the influence of the environment [e.g., Hendricks et al. (2010); overviews are given in Evans et al. (2010) and Sun et al. (2017)], and the other one that studies internal dynamics (Riehl 1963; Emanuel 1991; Montgomery and Smith 2014). There is, however, a considerable overlap of the internal-dynamical and environmental processes, and how a tropical cyclone can change intensity likely depends both on the inner-core dynamics and the environments (Houze et al. 2007; Evans et al. 2010). One proposed mechanism of tropical cyclone intensification by internal dynamics is the wind-induced surface heat exchange (WISHE) (Emanuel 1986; Yano and Emanuel 1991), in which there exists a positive-feedback reinforcement between tropical cyclone winds and wind speed–dependent surface enthalpy fluxes. An alternative proposed mechanism does not depend on the reinforcing feedback (Montgomery et al. 2009); instead, the intensification process is fundamentally nonaxisymmetric and involves the growths of highly localized deep convective structures that amplify the local vorticity, which then form vortical hot towers (VHTs). In both mechanisms, boundary layer processes near the air–sea interface play an important role. Although the wind speed dependence of the surface enthalpy flux is not strictly necessary for sustained intensification, the WISHE mechanism can alter the rate of tropical cyclone intensification [Montgomery and Smith (2014); Zhang and Emanuel (2016, see their Figs. 4 and 5)].

Tropical cyclones that undergo rapid intensity change, defined as the change of maximum 1-min winds at 10-m altitude of more than 15 m s−1 in a 24-h period (Kaplan and DeMaria 2003), are of particular interest to forecasters and researchers. They often develop into extremely powerful and dangerous storms, and attract the attention of the general public, as they should. Recent well-known examples are Hurricane Katrina (2005), Typhoon Haiyan (2013), Hurricane Patricia (2015), and Hurricane Irma (2017). Some studies have suggested that reduced surface friction at high wind speeds (Powell et al. 2003; Donelan et al. 2004; Holthuijsen et al. 2012; Takagaki et al. 2016) may favor rapid intensification (Soloviev et al. 2014, 2017; Donelan 2018), which in turn may explain the observed bimodal distribution of lifetime maximum intensity (Kossin et al. 2013; Lee et al. 2016). Barenblatt et al. (2005) modeled the reduced friction by incorporating spray into the marine boundary layer: the suspension of spray droplets depletes turbulence energy production, making the flow more “laminar,” resulting in flow acceleration. Holthuijsen et al. (2012; see also Donelan et al. 2004) observed that as hurricane wind speed increases, large amount of streaks of foam and spray merge into a “white out” over the sea surface, effectively smoothing out the surface roughness. Soloviev et al. (2014) showed that the white out can be caused by the breakup of Kelvin–Helmholtz waves (because in very high wind speeds the pressure drop over the wave overcomes gravity and surface tension), which disrupts the air–sea interface and rips droplets from it to form spume, resulting in drag reduction as short gravity-capillary waves are suppressed. On the other hand, spray may also change the surface enthalpy flux that in turn can modify intensification (Lighthill 1999; Andreas and Emanuel 2001). Soloviev et al. (2014, see their Fig. 4) found that the bimodal distribution of lifetime maximum intensity may also be explained by a similar bimodal distribution of the environmental potential intensity. Kowch and Emanuel (2015) have suggested that rapid intensification may be primarily controlled by random environmental and internal variability. These studies further reiterate the complexity of the tropical cyclone intensity problem: while surface momentum and enthalpy fluxes are important, they alone are probably insufficient in explaining intensity changes.

Asymmetry in the structure of a tropical cyclone plays an important role in determining its intensity (Emanuel 1991; Wang and Wu 2004). Numerical experiments have shown that an idealized symmetrical vortex in a uniformly rotating Earth (the f plane) can achieve high intensity (Peng et al. 1999). A wavenumber-1 azimuthal asymmetry (i.e., the lowest possible asymmetric Fourier wave) begins to develop in the core of the vortex under more realistic conditions such as the latitudinal variation of Earth’s Coriolis parameter (the β plane), the storm’s translation, and the large-scale steering flow and vertical wind shear. Peng et al. (1999) (see also Frank and Ritchie 2001; Wu and Braun 2004) then found that an azimuthal mismatch between the locations of the maximum turbulent air–sea exchange surface flux and low-level moisture flux convergence (MFC) weakens the cyclone. As a working hypothesis, we posit that a high areal coincidence of the surface flux and low-level moisture convergence around the cyclone center favors intensification. (The term “coincidence” is used in a geometric sense, from medieval Latin “coincidere,” in the sense of “occupying the same space.”) The idea is that larger coincidence would favor intensification through greater precipitation and heat release.

This study attempts to identify intensification condition(s) due to physical processes in the atmosphere and at the sea surface. We seek observational evidence if rapidly intensifying tropical cyclones have higher areal coincidence of surface flux and horizontal moisture convergence in their cores than slower and nonintensifying storms. However, instead of “surface flux,” we seek a greater generality by using a dimensionless surface wave-induced air–sea exchange coefficient. The complex intensity-change identification problem is thus reduced to examining two parameters, and their coincidences, which, relatively speaking, can be more easily deduced from observations. However, as our analysis and discussion (last section) will indicate, this should not be misunderstood as an oversimplistic view of a complex problem.

It is well established that the ocean’s surface waves play an indispensable role in characterizing and modulating the air–sea exchange flux (Andreas and Emanuel 2001; Donelan et al. 2004; Holthuijsen et al. 2012; Soloviev et al. 2014; Hwang and Walsh 2016; Takagaki et al. 2016). To study the coincidence of surface flux and low-level moisture flux convergence, we therefore use observations of tropical cyclone–induced surface waves to estimate the patterns of a dimensionless wave-induced air–sea exchange flux coefficient, and compare them with the patterns of the moisture flux convergence for global tropical cyclones under different states of the intensification process. The global data yield a total of over 16 000 cases, from which we may obtain stable composites with a high confidence level. Our analysis will show that rapidly intensifying tropical cyclones have younger surface waves, as well as significantly higher coincidence and larger low-level moisture flux convergence and ocean wave-induced air–sea exchange coefficient in the core of the cyclone than slower-intensifying and weakening cyclones.

For completeness in examining the ocean influence, we also consider the potential impact of warm ocean features on tropical cyclone rapid intensification. Several studies found that in some cases warmer ocean features: thicker upper layers and/or higher surface temperatures may induce rapid intensification (e.g., Shay et al. 2000; Lin et al. 2008). Using the global data, we will show that there is no statistically significant difference in the upper-ocean thickness and/or sea surface temperature under rapidly and nonrapidly intensifying tropical cyclones.

The manuscript is organized as follows. The “methods” section describes in details how the wave and atmospheric observations were prepared, so that the air–sea exchange coefficients and moisture flux convergences could be calculated for each of the 16 000 cases of intensity change, and their coincidences and degrees of asymmetry estimated for different intensity-change groups. In the “results” section, we first describe the patterns of the observed air–sea exchange coefficient and moisture flux convergence for different intensity-change groups. We then discuss their significances focusing on distinguishing between the results of rapidly and nonrapidly intensifying tropical cyclones. The section then analyzes the potential effects of warm ocean features on intensity change, and ends with a brief summary of the overall finding and implication. The final section proposes a plausible mechanistic feedback between wave-induced enthalpy flux and intensity change; there, we also discuss how ocean mixing by the Stokes’s currents (Stokes 1880) contributes in explaining why young ocean waves could favor the rapid intensification of tropical cyclones.

2. Methods

a. Momentum exchange coefficient

Tropical cyclone winds impart energy to ocean surface waves. In return, surface waves determine air–sea exchanges of momentum and energy by producing roughness and sea spray at the air–sea interface (Lighthill 1999; Donelan et al. 2004; Black et al. 2007; Holthuijsen et al. 2012; Soloviev et al. 2014), which affect cyclone intensity (Riehl 1963; Emanuel 1986, 1991; Lighthill 1999). We estimate air–sea exchange through knowledge of surface wave significant wave height Hs obtained from satellite altimeter data in tropical cyclones, as follows.

Excluding the eye region, surface waves under a tropical cyclone cannot in general propagate faster than the strong cyclone’s wind; they are constantly under the influence of the strong wind and may be classified as active wind seas. The inverse wave age ωn = U10/Cp generally exceeds one; here U10 is the magnitude of wind vector U10 at 10-m altitude, Cp = (g/ω) is deep-water wave phase speed, g is acceleration due to gravity, and ω is wave frequency. Hwang and Walsh (2016) (see also Young 2017) show that the dimensionless wave growth functions are then comparable to those in field experiments collected under more ideal quasi-steady fetch-limited conditions (Toba 1972; Donelan et al. 1992; Hwang and Wang 2004; Mellor et al. 2008). We use the wave growth function of Hwang and Walsh (2016), Eq. (1a), to express the air–sea momentum exchange (i.e., wind input to generate surface wave momentum) Mt, Eq. (1b), in terms of U10 and significant wave height Hs:

 
formula
 
formula

Here, ρa is the air density, ηn is the dimensionless significant wave height, and αM is the dimensionless momentum exchange coefficient. The ηn is also the dimensionless spectral wave energy: (Longuet-Higgins 1952). Using the wave growth function in Eq. (1a), we obtain

 
formula

The momentum exchange coefficient αM is therefore proportional to the ratio of local wind speed to , and is directly related to the inverse wave age ωn: young waves have larger αM. Hwang and Walsh (2016) showed that the αM is closely related to the surface drag coefficient CD, and found good agreements between the two coefficients using observations in Hurricane Bonnie (1998) for wind speeds up to 42 m s−1. At high wind (30 < U10 < 50 m s−1), the CD has been found to saturate and/or decrease in laboratory and field experiments (Powell et al. 2003; Donelan et al. 2004; Jarosz et al. 2007; Holthuijsen et al. 2012; Takagaki et al. 2012; Soloviev et al. 2014). Donelan et al. (2012; and others) showed that the CD too depends on waves through the Jeffreys’s (1925) sheltering coefficient A1 modulating the wind energy input (Sin) to the waves. Using laboratory and field observations, Donelan (2018) deduced that the saturation and/or decrease of CD is due to a Reynolds number–dependent A1, which is analogous to the Reynolds number–dependent behavior of form drag for flow past a solid cylinder of fixed diameter (Batchelor 1967, see their section 5.11). The A1 would decrease and increase gradually as the Reynolds number increases, and would drop precipitously as the Reynolds number exceeds a critical value. For completeness, we recalculated A1, following Donelan (2018) (see the  appendix),1 and used it in the wind energy input Sin (Hwang and Walsh 2016) to obtain a modified αM (i.e., αMM) so that it too is Reynolds number dependent and saturates/decreases near U10 ≈ 30–50 m s−1. The coincidence analysis (below) is insensitive to either αM, however, and both give very similar results.

Peng et al. (1999) found in their tropical cyclone model that the moisture and heat fluxes in the core have basically the same pattern as the momentum flux. Guided by their results, we therefore also assume that the pattern of momentum flux Mt is similar to the pattern of surface fluxes of heat and moisture. However, instead of Mt, it is more desirable to examine αM, which is dimensionless and is much less dependent on the details of the tropical cyclone wind distribution in the core; we use it to present our results.

b. Storm-following maps

We used 6-hourly tropical cyclone data from the IBTrACS archive (Knapp et al. 2010) from 1992 to 2015 (Fig. 1a). We followed each cyclone and constructed 800 km × 800 km storm-centered, 6-hourly maps with 10-km grid resolution. The maps were rotated with a positive y axis in the instantaneous direction of the storm, and a positive x axis at 90° to the right. An example is shown in Fig. 1b for such a map centered at the center of Typhoon Haiyan on 7 November 2013. Variables were gridded on these maps and were then composited (i.e., arithmetically averaged) in four 24-h intensity-change (ΔV) groups: rapid intensification (RI), ΔV > 15 m s−1; slow to medium intensification (SMI), 15 ≥ ΔV > 5 m s−1; neutral (N), 5 ≥ ΔV > −5 m s−1; and weakening (W), −5 m s−1 ≥ ΔV (Fig. 1a). Here ΔV is the change in the peak 1-min wind speeds Vm at 10-m altitude. We found that weaker, tropical storms with Vm < 33 m s−1 generally have a large percentage of older waves (see Figs. S1a–d in the online supplemental material), for which the use of the wave growth function may be problematic. We therefore chose only track points with Vm > 33 m s−1, which included 98% of all rapidly intensifying storms. To avoid near-coast regions where intensity changes may be influenced by terrain, we used the ETOPO2 2-min resolution dataset to choose only those tracks over the open ocean where the water depth is deeper than 1000 m.

Fig. 1.

IBTrACS data from 1992 to 2015. (a) Locations of rapidly intensifying (RI, red), slow to medium intensifying (SMI, green), neutral (N, cyan), and weakening (W, gray) tropical cyclones of category 1 and stronger. Legends show the mean occurrence per year for each case. (b) An example [Typhoon Haiyan wind vector and speed (color shading) on 7 Nov 2013] of a storm-following, storm-centered 800 km × 800 km box within which wind, waves, and other analyzed parameters such as the moisture flux convergence MFC and air–sea exchange coefficient αM are calculated and composited.

Fig. 1.

IBTrACS data from 1992 to 2015. (a) Locations of rapidly intensifying (RI, red), slow to medium intensifying (SMI, green), neutral (N, cyan), and weakening (W, gray) tropical cyclones of category 1 and stronger. Legends show the mean occurrence per year for each case. (b) An example [Typhoon Haiyan wind vector and speed (color shading) on 7 Nov 2013] of a storm-following, storm-centered 800 km × 800 km box within which wind, waves, and other analyzed parameters such as the moisture flux convergence MFC and air–sea exchange coefficient αM are calculated and composited.

c. Significant wave height

The 6-hourly tracks of Hs from satellite altimeter (AVISO) were gridded without interpolation onto the above storm-following, storm-centered 6-hourly maps. The satellite Hs was checked against NDBC buoy data. These buoys were mostly located near the coast and over the continental shelves and slopes of the United States. The comparison yielded a regression slope = 1.1 (satellite underestimates) and r2 = 0.97.

d. Wind

The 10-m wind U10 was calculated using a parametric tropical cyclone model (Holland et al. 2010) using 6-hourly center pressure, location, and Vm from the IBTrACS, the radius of maximum wind estimated from Knaff et al. (2007) with the cyclone-moving component added following Jakobsen and Madsen (2004). The tropical cyclone wind was then merged with the cross-calibrated multi-platform (CCMP) reanalysis wind (Atlas et al. 2011) at radial distances >~350 km from the cyclone center. The merged wind was then gridded onto the storm-centered maps using bilinear interpolation. The data were checked for self-consistency against observed NDBC buoy winds and maximum wind speeds for tropical cyclones from 1992 to 2015, yielding a regression slope = 1.02 and r2 = 0.93.

The U10 was bilinearly interpolated onto the storm-following maps. Then together with the Hs, other derived variables such as ωn, αM, etc., from Eqs. (1) and (2) were calculated on the maps.

e. Moisture flux

Low-level column-integrated moisture flux convergence (MFC) and the column-integrated moisture flux (Iuq) were computed using wind, specific humidity (calculated from temperature and relative humidity) data from the 6-hourly National Centers for Environmental Prediction Final Operational Global Analysis [(NCEP-FNL) from 2000 to 2015] analysis at 1° resolution:

 
formula
 
formula

The upper-bound pressure for the integration was 700 hPa, and the lower bound was the surface pressure. The 1° × 1° NCEP data are too coarse to resolve the tropical cyclone core and the wind magnitude is also underestimated. However, observations have been assimilated in the NCEP data, and we focus only on the asymmetric pattern of the MFC, which is caused primarily by the interaction of the cyclone with the larger-scale environment (Peng et al. 1999). We expect that the MFC pattern and its relative distribution with respect to the pattern of αM are little affected by the magnitude of MFC. The MFC and Iuq were gridded onto the storm-centered maps using bilinear interpolation.

f. Degree of asymmetry (DOA)

The degree of asymmetry of the moisture flux Iuq is , subscript “a” indicates the asymmetric component of the flux (i.e., total minus the axis-symmetric part), and the summation is over grid points within the circle of radius 200 km about the storm center. The DOAs for scalars αM and MFC [i.e., DOA(αM) and DOA(MFC)] are similarly defined by partitioning each into asymmetric and axis-symmetric components. The DOA(MF) and DOA(MFC) are very similar; only DOA(MF) is shown below.

g. Composites

We calculated composites of variables for all selected tracks in the global ocean (Fig. 1a), and separately also in each of the six tropical cyclone basins, as well as the Northern and Southern Hemispheric basins. Southern Hemispheric storms were mapped into Northern Hemispheric–equivalent storms by reflecting the field in the storm-following maps into the y axis: Field(x, …) = Field(−x, …). Since rapid intensification occurs mostly near the tropics (Fig. 1a), the atmospheric and SST environments are different from those at higher latitudes where many nonrapidly intensifying changes could occur. It is necessary therefore to check for possible biases in the composites due to the different geographic environments. We therefore repeated the composite calculations of moisture flux vectors and convergences using only those storm tracks within the 30°S–30°N latitude band. The wave-related composites (e.g., ωn) are unchanged however. (Tests using waves also within 30°S–30°N yielded virtually identical ωn patterns.) Standard statistical tests (e.g., Press et al. 1992) were applied to calculate the significances of the composites and differences. The nonparametric Wilcox–Mann–Whitney test (Chu 2002) was also used to calculate the significance of differences, yielding similar results. Because of the relatively large samples used for global and individual basins, except the north Indian basin, the significance tests were well satisfied at the 95% confidence level.

h. Ensemble model experiments

As stated previously in the introduction, an objective of the present study is to use observations to test the hypothesis that high areal coincidence of the surface flux and low-level moisture convergence around the tropical cyclone center favors intensification (Peng et al. 1999). Peng et al. (1999) conducted single-member experiments. Before presenting the results of the observational analyses, it is desirable to place the hypothesis on a statistically firmer ground. We therefore repeated their β-plane model experiment of the translation and intensification of an initial vortex; but instead of one experiment, we conducted a suite of experiments with the initial vortex placed at different tropical latitudes, with the model initial fields randomly perturbed, and with different boundary layer and microphysics schemes (see supplemental material for more information). The experiments yielded different cyclones with different trajectories and translation speeds, and over 120 samples of intensity change δV. Together with the corresponding surface fluxes and MFCs, the change in their coincidence, δ(Coincidence), was calculated and regressed against δV (see Fig. S2). There is a close (r2 ≈ 0.65), nearly linear relation with positive slope between the two variables: higher coincidences tend to be associated with stronger intensifications, consistent with the Peng’s hypothesis.

3. Results

a. Patterns and coincidences of αM and MFC

Excluding the eye region, the RI group (Fig. 2a) has the largest momentum exchange coefficient αM and therefore youngest and most active waves [please see Eq. (2) showing αM nearly linearly related to the inverse wave age ωn]. The large αM wraps clockwise around the storm’s center, nearly reaching the left-front quadrant, thus making the αM pattern nearly symmetrical. By contrast, young waves in the weakening (W) group (Fig. 2d) are mostly confined to the right side, while older waves may be seen in the left-front quadrant. The SMI (Fig. 2b) and N (Fig. 2c) groups have intermediate αM patterns. All the groups display a common feature that the αM is the largest and most widespread in the right-back quadrant, in agreement with the field measurements and modeling that also show youngest waves in that quadrant (Holthuijsen et al. 2012; Hwang and Walsh 2016). The asymmetric moisture flux enters in the back and exits in the front “through” the cyclone generally in the direction of the translation; it contributes to an MFC that is “front heavy” (Fiorino and Elsberry 1989; Peng et al. 1999). The RI group has the most symmetrical MFC pattern (Fig. 2a), with the region of large MFC shifted toward the right-front quadrant. By contrast, the other three non-RI groups (Figs. 2b–d) show a more widespread asymmetric pattern with larger MFC tilted toward the left-front quadrant. For these non-RI groups, the left-front MFC asymmetry is therefore azimuthally 180° out of phase with their right-back αM asymmetry, while the phase shift for the RI group is less. These MFC and αM patterns generally hold also for individual basins (Figs. S3–S8) and Northern and Southern Hemispheric composites (Figs. S9–S10); the exceptions are in the north Indian and the South Pacific basins (Figs. S6–S7) where the samples are too few (Fig. S12). The composite for storms within the 30°S–30°N latitude band show very similar patterns (Fig. S11). Because of the large sample size, the composite results are relatively insensitive to the different geographic locations of the storms.

Fig. 2.

(a)–(d) Global composites of αM (shading) and MFC (blue contours) each normalized by their maxima shown across the top, and the asymmetric component of the moisture flux (vectors) for each of the four ΔV groups: RI, SMI, N, and W; values > 0.3 are significant at the 95% confidence level. (e)–(h) Corresponding coincidence maps = product of anomalies of αM and MFC; stippling shows positive points where both anomalies are positive.

Fig. 2.

(a)–(d) Global composites of αM (shading) and MFC (blue contours) each normalized by their maxima shown across the top, and the asymmetric component of the moisture flux (vectors) for each of the four ΔV groups: RI, SMI, N, and W; values > 0.3 are significant at the 95% confidence level. (e)–(h) Corresponding coincidence maps = product of anomalies of αM and MFC; stippling shows positive points where both anomalies are positive.

We quantify the extent with which the αM and MFC coincide, as shown in Figs. 2e–h. Positive coincidence points indicate where strong surface and moisture convergence fluxes coincide, while negative coincidence indicates where they are out of phase. The former favors storm intensification while the latter does not (Peng et al. 1999; Wu and Braun 2004). Both the MFC (blue contours on Fig. 2, left column) and αM (color shading) contribute to the coincidence patterns. For the RI (Figs. 2a,e) and W tropical cyclone groups (Figs. 2d,h), the largest αM is on the right-back quadrant (Figs. 2a,d). But because the largest MFC leans toward the right-front quadrant where the αM while not the largest is quite substantial, the resulting coincidence is controlled more by MFC and therefore its positive region also leans more to the right-front quadrant (Figs. 2e,h). The distinguishing difference between the “RI” and “W” groups is that the former (Figs. 2a,e) has nearly symmetrical MFC and αM (Fig. 2a), thus making their positive coincidence not only larger (as mentioned above), but also more symmetrical (Fig. 2e). For the SMI (Figs. 2b,f) and N groups (Figs. 2c,g), both the MFC and αM contribute, but the αM has a slightly stronger control over their coincidence patterns.

We summarize these results by plotting the summed positive coincidence (COIN) and the degrees of asymmetry for moisture flux DOA(MF) and αM DOA(αM) (Fig. 3; see Fig. S13 for the corresponding plots for individual basins). For the RI group, the positive coincidence is distributed nearly symmetrically around the core of the storm (Fig. 2e), and the corresponding COIN is significantly higher than the other three non-RI groups (Fig. 3). By contrast, the W group (Fig. 2h) shows large areas of negative coincidence in the left-front quadrant. The SMI and N groups have intermediate distributions (Figs. 2f,g). The summed positive coincidences for these two groups are nearly indistinguishable from each other but both remain significantly larger than the COIN of the W group (Fig. 3). For the degrees of asymmetry, both DOA(αM) and DOA(MF), are smallest for the RI group, and are significantly smaller than those for the W group. The analyses for coincidences and degrees of asymmetry for storms confined within the 30°S–30°N latitude band show similar stark differences between RI and non-RI groups (Fig. S14). However, the DOAs for the RI group are generally indistinguishable from those of the SMI and N groups. These results suggest that the coincidence of moisture flux convergence and αM is a more relevant determinant than the axis-symmetry properties in distinguishing rapidly from non-RI tropical cyclones.

Fig. 3.

Summed positive coincidence COIN of MFC and αM (gold bars), the degree of asymmetries of the moisture flux [DOA(MF), gray bars], and αM [DOA(αM), purple bars] for the four ΔV groups: RI, SMI, N, and W for the global ocean. Error bars show ±1 standard deviation from the mean for that group. The COIN for each group is arbitrarily normalized by the sum (same value for all four groups) of grid points within a 200-km radius surrounding the cyclone.

Fig. 3.

Summed positive coincidence COIN of MFC and αM (gold bars), the degree of asymmetries of the moisture flux [DOA(MF), gray bars], and αM [DOA(αM), purple bars] for the four ΔV groups: RI, SMI, N, and W for the global ocean. Error bars show ±1 standard deviation from the mean for that group. The COIN for each group is arbitrarily normalized by the sum (same value for all four groups) of grid points within a 200-km radius surrounding the cyclone.

We repeated the above analyses (Figs. 2 and 3) to account for a Reynolds number–dependent αM (i.e., αMM) such that its value peaks at U10 ≈ 30–50 m s−1 and then decreases at higher wind speeds (see the methods section and the  appendix). (Maps of αMM and MFC and their coincidences, as well as the corresponding summary bar plots are shown in Figs. S15 and S16.) As may be expected, the αMM values are generally smaller than the αMs and the corresponding coincidences are weaker for all intensity change groups, but the patterns are very similar (cf. Figs. 2 and S15). As a result, the contrast in COIN between the RI group and other, non-RI intensity-change groups, while slightly weaker, remains highly significant (cf. Figs. 3 and S16).

b. The potential influence of warm ocean features on intensity change

Previous studies have noted that some tropical cyclones intensified as they passed over warm ocean features (e.g., eddies with diameters of a few hundred kilometers with deepened 26°C isotherm in the ocean’s upper layer). Shay et al. (2000) observed that Hurricane Opal (1995) rapidly intensified when passing over a warm-core ring in the Gulf of Mexico. Lin et al. (2008) noted eight cases of intensification, from 1993 to 2005, when typhoons passed over warm features in the western North Pacific. Sun et al. (2017) found some correspondence between intensification and warm ocean features for the western North Pacific typhoons, but only obtained high and significant correlations after incorporating other parameters, such as the maximum potential intensity and the wind shear. Here we wish to check if the higher COIN identified above for rapid intensification could also have occurred when the tropical cyclones passed over warm ocean features. Specifically, we ask the following question: Are rapid intensification events more likely to occur, compared to slower intensification events, when tropical cyclones pass over warm ocean features? To answer this question, we compared the depth of the ocean upper layer and the sea surface temperature at locations where RI occurred against the same ocean variables when SMI and/or N intensification occurred. The answer would then be “YES” if the difference (RI minus SMI or RI minus N) is statistically significantly positive; otherwise it is “NO.”

As before, we again followed each storm, but this time recorded the sea surface height anomaly (SSHA; http://www.aviso.altimetry.fr/en/data.html) from satellite altimetry data under the storm. We use the depth of the 26°C isotherm, Z26, as a measure of the upper-ocean depth (e.g., Goni et al. 1996; Shay et al. 2000; Lin et al. 2008; Sun et al. 2017). Using the reduced gravity approximation, changes in the depth of the 26°C isotherm, , due to, for example, eddies, can be related to η′ = SSHA: ηρ0ρ (Goni et al. 1996), where ρ0 is a reference (seawater) density and Δρ is the density difference between the upper and lower layer, from the WOA climatology (Stephens et al. 2001). Since Δρ/ρ0 varies much slower than η′ both spatially and temporally, ~ η′. This formulation was used by Shay et al. (2000), Lin et al. (2008), Sun et al. (2017), and others; as pointed out by these authors, larger (smaller) SSHAs generally collocate well with thicker (thinner) upper-ocean warm layers. Note that the ocean heat content correlates well with the 26°C isotherm depth (e.g., Oey et al. 2007). Two methods were used to record the resulting SSHA “seen” by the moving storm. The first method simply recorded the SSHA averaged over a small circle of radius 50 km around the storm center. The second, more stringent method additionally removed a spatially slowly varying SSHA, taken as the averaged SSHA over a 500-km-radius circle around the moving cyclone center, and then averaged the resulting residual or relative SSHA over the 50-km circular area. Similar analyses were applied to the sea surface temperature (SST; https://www.ghrsst.org/). (More details are provided in the supplemental material.) We then compared the SSHA and SST under the storms that underwent RI against the SSHA and SST under the storms that underwent SMI and N intensifications, by displaying their differences on a 5° × 5° grid over the global ocean (Figs. 4a,b). The means and standard deviations of these differences for individual and global ocean basins are given as bar plots in Figs. 4c,d. (For completeness, the differences between the RI and W cases are also shown in Figs. 4c,d). [The result from the less stringent method 1 (i.e., more favorable to the argument of a connection between SSHA/SST and intensification) is shown in Fig. 4, while the corresponding result from the more stringent method 2 is shown in Fig. S17.] Positive (negative) differences indicate that, on average, the RI storms passed over warmer (cooler) features than the SMI and N storms. If RI occurred consistently more frequently over warmer eddies compared to SMI and N intensifications, the plots in Figs. 4a,b would then show mostly “warm colored” grids that are also statistically significant. Globally, this is only nearly (i.e., “warm” but insignificant) true for the RI storms in the eastern Pacific for the SSHA (Figs. 4a,c), and additionally in the north and south Indian Oceans for SST (Figs. 4b,d). Indeed, the numbers of negative and positive grids are nearly the same (see the legends in Figs. 4a,b). Over most of the global ocean, it is therefore nearly equally likely for all three intensification groups RI, SMI, and N to occur, regardless of whether or not the storm passes over a warm eddy. The global means for “RI minus (SMI+N)” and “RI minus W” SSHAs are in fact not significant (Fig. 4c; “significant” in the sense of mean exceeding plus or minus one standard deviation). The “RI minus W” SSTs (Fig. 4d, red bars) are significantly positive, however, except for the eastern Pacific and north Indian Oceans. However, that is to be expected as cooler SST can more likely weaken a storm. Similar inferences are obtained using the more stringent method 2 (see Fig. S17 and descriptions). It is interesting that, comparing Fig. 4 with the corresponding plots for method 2 (in Fig. S17), many of the warm-colored grids in the former plots had either become “less warm” or had turned “cool” in the latter plots, which suggests that some of the warm grids in Fig. 4 were induced by large-scale warming that has less to do with localized warmer ocean features.

Fig. 4.

The RI minus “SMI+N” (a) SSHA and (b) SST; dots indicate significant points at the 90% confidence level according to the Wilcox–Mann–Whitney test. Legends show global mean, standard deviation (sd), and numbers of positive and negative significant grids. (c),(d) The means (bars) and sd (lines) for individual and global ocean basins, for SSHA and SST, for RI minus W (red) and RI minus “SMI+N” (yellow) storms, averaged from grid points where they are significant [e.g., in (a) and (b) for RI minus “SMI+N”]. The ×s in (d) for the eastern Pacific and north Indian Oceans indicate that the “means” were only for ONE value (i.e., sd = ∞), and the NaNs in (c) for the north Indian Ocean indicate no significant points.

Fig. 4.

The RI minus “SMI+N” (a) SSHA and (b) SST; dots indicate significant points at the 90% confidence level according to the Wilcox–Mann–Whitney test. Legends show global mean, standard deviation (sd), and numbers of positive and negative significant grids. (c),(d) The means (bars) and sd (lines) for individual and global ocean basins, for SSHA and SST, for RI minus W (red) and RI minus “SMI+N” (yellow) storms, averaged from grid points where they are significant [e.g., in (a) and (b) for RI minus “SMI+N”]. The ×s in (d) for the eastern Pacific and north Indian Oceans indicate that the “means” were only for ONE value (i.e., sd = ∞), and the NaNs in (c) for the north Indian Ocean indicate no significant points.

In conclusion, rapid intensification is neither more nor less likely to occur over ocean warm features than other, less-rapid intensifications. We have thus precluded the possibility that the significantly high COIN identified for rapidly intensifying tropical cyclones (Fig. 3) was due to some happenstance when the analyzed storms passed over ocean warm eddies. Following the logical steps used in our analysis, that began with the four intensity groups of tropical cyclones and ended with COIN, one may write the following:

 
formula

where “⇒” denotes the logical symbol “if … then.” While this observational analysis result does not constitute an absolute mathematical truth, the significantly high COIN for the RI group compared to other intensity-change groups suggests that high COIN may likely be a necessary condition (one of several) for a tropical cyclone to rapidly intensify.

4. Summary and discussion

We have shown from multidecadal global observations that the coincidence (COIN) of the atmospheric low-level moisture flux convergence (MFC) and oceanic wave–induced air–sea exchange coefficient (αM) in the core of a tropical cyclone can serve to differentiate rapid from non–rapid intensity change of the storm. In non–rapidly intensifying storms, the asymmetric component of the MFC is “front heavy” with larger values most likely found in the front of the moving cyclone, while active young waves that contribute to the large αM generally reside in the back. By contrast, in rapidly intensifying storms, the MFC becomes more symmetrical; at the same time, young waves appear in the right-front quadrant. These changes in the patterns of the MFC and αM result in their strong coincidence. Our finding is consistent with the model results of Peng et al. (1999), which we also verified through ensemble experiments, that COIN and intensity change ΔV are positively correlated. The physical basis, given by Peng et al. (1999), among others, is that an intensifying cyclone tends to be associated with a small wavenumber-1 asymmetry that allows a more symmetric distribution of the surface fluxes and low-level moisture convergence around the center, while a mismatch between the surface flux maximum and the lateral moisture convergence reduces precipitation and prevents intensification.

To understand the high COIN associated with a rapid intensity change, it is therefore necessary to explain why more young waves shift into the right-front quadrant. The shifting pattern was previously described in the plot for αM ~ (Figs. 2a–d), but is also clear from the plot of the inverse wave age ωn itself (see Fig. S18). Suppose a tropical cyclone, of relatively weak or moderate intensity (tropical storm or category 1), intensifies (as indicated by the U10 “up-arrow” symbol near the 7 o’clock position of the clockface in Fig. 5). Finite waves (Hs ≈ 2 m or more) grow according to the Jeffreys’s (1925) idea that the sheltering of the wave troughs due to flow separation at the wave crests gives rise to form drag, transferring energy from wind to wave (Donelan et al. 2012) (Fig. 5, near the 4 o’clock position). Assuming that the tropical cyclone translates at a speed Uh that is near the wave group speed Cg (to be shown below), resonant trapped-fetch waves grow to large amplitudes on the right-front side of the cyclone (Moon et al. 2003; Bowyer and MacAfee 2005). As the wind speed U10 increases, the inverse wave age ωn ≈ 0.47 × [U10/(gHs)1/2]1.2 [see Eq. (2)], which is above one, increases further with U10 if U10 varies like , k > ½ (Fig. 5, near the 3 o’clock position). Figure 6a graphs the probabilities of tropical cyclones in three Uh categories: slow, medium, and fast, in bins of the intensity-change ΔV. Figure 6b shows the Cg as a function of U10 and Hs, derived by Zhang and Oey (2018), and multiplied by cos(35°) to adjust for an average misalignment angle of 35° between waves and wind, the latter being aligned on average with the tropical cyclone's translation on the right side of the storm (Moon et al. 2003; Zhang and Oey 2018). Figures 6c and 6d show the 50th and 90th percentile Hs binned according to the maximum wind speed Vm, in the western Pacific and North Atlantic. (There were insufficient samples in the other basins to derive meaningful statistics.) Figure 6a shows that in the rapid intensity change ΔV > 15 m s−1 (i.e., RI) range, the so-called “Goldilocks tropical cyclones” that translate neither too slowly nor too quickly at the medium speeds Uh between 3 and 7 m s−1, significantly outnumber (70%) both the slow (Uh < 3 m s−1; 16%) and fast (Uh > 7 m s−1; 14%) storms.2 In general, the Goldilocks cyclones have a higher probability of intensifying than weakening (the green line slopes up with ΔV), while the fast storms have a higher probability of weakening (the red line slopes down), and the slow storms have nearly equal probabilities of intensifying and weakening (the blue line remains mostly flat). Figure 6b shows that in the range of Hs considered, Cg varies between 3 and 8 m s−1, approximately in the range of Uh of the Goldilocks tropical cyclones. Finally, Figs. 6c and 6d show that as Vm increases, the Hs also increases, nearly linearly up to about 45 m s−1, and then Hs plateaus or saturates at ~10–12 m for stronger wind speeds 50 m s−1 and larger.3 The best fit of the 90th percentile curve is Vm ~ (i.e., k is likely to be > ½). Instead of Vm, estimated U10 also gives a similar variation U10 ~ (Zhang and Oey 2018). The high-wave saturation guarantees that, under the Goldilocks condition, more young waves appear in the right-front quadrant where ωn, hence αM, becomes large. The “spilling” of high αM into the right-front quadrant (Fig. 5, near the 2 o’clock position) allows a high coincidence of the αM with the MFC, further intensifying the cyclone (Fig. 5, back to the 7 o’clock position). We suggest that this positive reinforcing loop between the tropical cyclone wind and young waves in the right-front quadrant favors rapid intensification.

Fig. 5.

A sketch of the positive reinforcing loop (big red circle, like a clock, with anticlockwise-directed red arrows indicating positive reinforcement of the physical processes) between tropical cyclone wind and waves that favors rapid intensification as discussed in the text. The tropical cyclone wind is represented by U10, wave height is represented by Hs, Uh is the cyclone’s translation speed, Cg is the wave’s group velocity, ωn is the inverse wave age, αM is the wave-induced air–sea exchange coefficient, MFC is the moisture flux convergence (anomaly), and COIN is the coincidence between MFC and αM. The up arrow ↑ indicates an increase of the variable preceding it. The pink-shaded dashed circle indicates the area of increased COIN anomaly as a result of the “spilling” of young waves into the right-front quadrant. Effects of the Goldilocks translation speeds, Uh ~ Cg, and the spray droplets due to breaking waves, both favoring intensification, are discussed in the text.

Fig. 5.

A sketch of the positive reinforcing loop (big red circle, like a clock, with anticlockwise-directed red arrows indicating positive reinforcement of the physical processes) between tropical cyclone wind and waves that favors rapid intensification as discussed in the text. The tropical cyclone wind is represented by U10, wave height is represented by Hs, Uh is the cyclone’s translation speed, Cg is the wave’s group velocity, ωn is the inverse wave age, αM is the wave-induced air–sea exchange coefficient, MFC is the moisture flux convergence (anomaly), and COIN is the coincidence between MFC and αM. The up arrow ↑ indicates an increase of the variable preceding it. The pink-shaded dashed circle indicates the area of increased COIN anomaly as a result of the “spilling” of young waves into the right-front quadrant. Effects of the Goldilocks translation speeds, Uh ~ Cg, and the spray droplets due to breaking waves, both favoring intensification, are discussed in the text.

Fig. 6.

(a) Percentages of the three indicated storms’ translating speed groups in western Pacific and North Atlantic plotted as a function of intensity-change ΔV in 5 m s−1 bins. Vertical lines show ±1 sd. (b) Group speeds from Zhang and Oey (2018): , multiplied by cos(35°) (see text), plotted as a function of the observed significant wave height Hs and reference wind speed U10. (c),(d) The mean 90th percentile Hs (filled circles with ±1 sd lines) observed in the right-front quadrant of tropical cyclones in the (c) western Pacific and (d) North Atlantic, plotted as a function of the maximum tropical cyclone wind Vm in 5 m s−1 bins. Dashed lines show the mean 50th percentile Hs (without the ±1 sd lines for clarity).

Fig. 6.

(a) Percentages of the three indicated storms’ translating speed groups in western Pacific and North Atlantic plotted as a function of intensity-change ΔV in 5 m s−1 bins. Vertical lines show ±1 sd. (b) Group speeds from Zhang and Oey (2018): , multiplied by cos(35°) (see text), plotted as a function of the observed significant wave height Hs and reference wind speed U10. (c),(d) The mean 90th percentile Hs (filled circles with ±1 sd lines) observed in the right-front quadrant of tropical cyclones in the (c) western Pacific and (d) North Atlantic, plotted as a function of the maximum tropical cyclone wind Vm in 5 m s−1 bins. Dashed lines show the mean 50th percentile Hs (without the ±1 sd lines for clarity).

Indicated in Fig. 5 is also the potential intensity-enhancing effect of spray as large amounts are spewed into “the near-surface droplet evaporation layer” (Andreas 2011) at high wind speeds (Andreas and Emanuel 2001; Donelan et al. 2004; Holthuijsen et al. 2012; Soloviev et al. 2014). In addition to potentially reducing the surface drag (Soloviev et al. 2014), discussed below, the spray induces spray-mediated enthalpy flux (in addition to the customary interfacial flux), which at high winds, U10 ~35 m s−1 and higher, scales like , with γ between ~2 and 4 (Andreas 2010; Richter and Stern 2014). Despite this robust, increasing trend of the enthalpy flux with U10 from field measurements, the corresponding trend in the graph of the enthalpy flux coefficient Ck versus U10 is statistically insignificant because of large scatter in the field data (Richter and Stern 2014), in part due to the very difficult conditions under which field measurements were often taken (Jarosz et al. 2007; Andreas 2010; Holthuijsen et al. 2012). A recent laboratory experiment by Komori et al. (2018) provides some evidence of enhanced enthalpy flux at high winds. They showed that Ck is approximately constant for wind speeds below ~35 m s−1, but increases linearly with the wind speed for stronger wind. The result appears to be consistent with Andreas’s (2011) analysis, which shows a significant increase in Ck with U10 once the spray-mediated fluxes begin to dominate over the interfacial fluxes. It is interesting that, since αM ~ U10/(gHs)1/2, the exchange coefficient we have used (i.e. Hwang and Walsh 2016) also scales linearly with U10 when waves become saturated. Komori et al.’s result is consistent with Andreas (2010) and Richter and Stern (2014), but is in contrast to what has customarily been assumed based on data at lower wind speeds (e.g., Drennan et al. 2007; Jeong et al. 2012), that the enthalpy coefficient remains constant at high winds (e.g., Soloviev et al. 2014, 2017; Donelan 2018). However, conditions in laboratory experiments are not the same as those in the field (Richter and Stern 2014). How spray may affect tropical cyclone intensity is complex, and remains a topic of intense debate and research (e.g., Wang et al. 2001; Gall et al. 2008; Garg et al. 2018).

The proposed wind–wave reinforcing mechanism of Fig. 5 appears similar to the WISHE evaporation–wind feedback mechanism, previewed in the introduction. However, WISHE relies on the feedback between the wind and surface enthalpy flux (Qk) that is a function of the wind speed |U10|:

 
formula

where ρa is the air density; Ck is the dimensionless exchange coefficient for enthalpy; and ks and ka are the specific enthalpies of the air at saturation at the ocean surface and ambient boundary layer air, respectively. On the other hand, the wind–wave reinforcing mechanism depends on the increased wave-induced air–sea exchange coefficient αM and its coincidence with the MFC. Unlike WISHE, the wind–wave reinforcing mechanism is nonaxisymmetric. While one might anticipate a functional relation between Ck and αM (~ |U10| as waves saturate; in which case WISHE can augment intensification; Zhang and Emanuel 2016), the presence of young waves in the wind–wave reinforcing mechanism injects new processes (e.g., spray) as outlined above; thus, further research is needed.

In Emanuel’s (1986, 1999) heat engine model, the maximum steady intensity a storm can achieve is

 
formula

where Vpot is the maximum wind speed, and Ts and To are the absolute temperatures of the sea surface and storm top. Soloviev et al. (2014, 2017) and Donelan (2018) proposed that CD peaks around a wind speed of ~30–40 m s−1, but reaches some minimum around ~50–60 m s−1 before increasing again. Thus, if Ck is assumed constant, then Vpot may amplify, sufficiently rapidly to explain the observed bimodal distribution of the lifetime maximum intensity of tropical cyclones (Kossin et al. 2013). In view of the above discussion that Ck is not constant at high winds, but varies like ~ with the power k most likely ≥ 1, the lone effect of CD in explaining rapid intensification is unclear. In Soloviev et al.’s theory, the large spray droplets (spume) that contribute to smoothing the sea surface and drag reduction are also those that play a central role in the spray-mediated enthalpy fluxes that potentially can enhance Ck (Andreas and Emanuel 2001; Andreas 2011). It is likely that both a decrease in CD and an increase in Ck at high wind speeds contribute to TC intensification process. Hence, more research is needed.

Instead of examining the lifetime maximum intensity, our study focuses on actual intensity changes (ΔV), and our finding suggests a close coupling between waves and the MFC for the rapid intensification of the tropical cyclone. Young waves play a central role in our results. Waves drive ocean Stokes current, us (Stokes 1880), which contributes to mixing that cools the sea surface (Tamura et al. 2012). A formula for the Stokes speed at the sea surface can be derived from the second form of Eq. (2) relating the significant wave height with inverse wave age; thus, |us|/U10 = 2.94 × 10−3. For a given cyclone wind, the Stokes speed is least in the right-back quadrant where waves are youngest (Fig. 7). This prediction is supported by full-physics numerical experiments that showed also that the least amount of sea surface temperature cooling was in the right-back quadrant, caused by wave-induced mixing (Fan et al. 2009). For young waves, since a fraction of wind energy is spent in growing them, there is less available for mixing and cooling the sea surface. Compared to non–rapidly intensifying storms, a rapidly intensifying storm with abundant young waves in its core has therefore relatively weaker Stokes current (i.e., for the same wind speed) and is therefore more immune to intensity-inhibiting surface cooling. Our results suggest that a new surface layer parameterization including young waves may be developed to improve tropical cyclone intensity forecast.

Fig. 7.

The Stokes speed divided by wind speed |us|/U10 for the four different tropical cyclone intensity change groups: (a) RI, (b) SMI, (c) N, and (d) W. Vector at center shows tropical cyclone heading and its length is the averaged translation speed with the scale at top right. The plots show that the |us|/U10 hence also the ocean mixing and therefore cooling caused by the Stokes current is least in the right-back quadrant where waves are youngest (Figs. S1a–d). This agrees with numerical experiments showing also the least amount of sea surface temperature cooling in the right-back quadrant (Fan et al. 2009). We see then that the RI group in (a) with the youngest waves among the four groups has the weakest Stokes current relative to the wind, and therefore also the least fractional cooling relative to the wind. In other words, for the same wind speed, the RI tropical cyclones are more immune to intensity-inhibiting surface cooling because of the widespread presence of young waves in its core.

Fig. 7.

The Stokes speed divided by wind speed |us|/U10 for the four different tropical cyclone intensity change groups: (a) RI, (b) SMI, (c) N, and (d) W. Vector at center shows tropical cyclone heading and its length is the averaged translation speed with the scale at top right. The plots show that the |us|/U10 hence also the ocean mixing and therefore cooling caused by the Stokes current is least in the right-back quadrant where waves are youngest (Figs. S1a–d). This agrees with numerical experiments showing also the least amount of sea surface temperature cooling in the right-back quadrant (Fan et al. 2009). We see then that the RI group in (a) with the youngest waves among the four groups has the weakest Stokes current relative to the wind, and therefore also the least fractional cooling relative to the wind. In other words, for the same wind speed, the RI tropical cyclones are more immune to intensity-inhibiting surface cooling because of the widespread presence of young waves in its core.

Acknowledgments

The authors are grateful for correspondence with Paul Hwang, whose work (Hwang and Walsh’s 2016 in particular) has inspired this study. We thank Prof. Soloviev and the anonymous reviewer 2 for providing useful comments, which improved the manuscript. Data used here are publically available from AVISO (http://www.aviso.altimetry.fr/en/my-aviso.html), GHRSST (https://www.ghrsst.org/), NCEP (https://rda.ucar.edu/datasets/ds083.2/), and NDBC (http://www.ndbc.noaa.gov/). This research was supported by the Taiwan Ministry of Science and Technology Grants 107-2111-M-008-035 and 107-2611-M-008-003, awarded to the National Central University. Author contributions: LO conceived the idea, designed the study, interpreted the data, and wrote the paper. LZ analyzed the data, conducted the WRF simulations, and plotted the figures, with some contributions from LO. Both authors derived the equations and discussed the research.

APPENDIX

The Sheltering Coefficient A1 and the Reynolds Number–Dependent, Modified αM: αMM

Here we incorporate the Donelan’s (2018) Reynolds number–dependent sheltering coefficient A1 (see Donelan et al. 2012) into the Hwang and Walsh’s (2016) ,αM to obtain the Reynolds number–dependent αMM. We first fitted Donelan’s (2018, his Fig. 5) data to obtain the following for the sheltering coefficient A1:

 
formula

where x = ln(Rew). The septic formula is plotted in Fig. A1, which corrects the late Professor Donelan’s (2018) fitted formula (dark blue; which appears to contain typos). Here, Rew is the Reynolds number:

 
formula

where U20 = 20-m wind speed, νa = 1.48 × 10−5 m2 s−1 is the kinematic viscosity of air, Hs is the significant wave height from satellite data, and Cp is the wave phase speed estimated from U10, Hs, and the wave-growth function (see the main text; Zhang and Oey 2018). For simplicity the 20-m height for the wind speed is used in place of the wind at half-peak wavelength height (Uλ/2) used in Donelan (2018), following the recommendation of Donelan et al. (2012) [the Uλ/2 may be calculated also more precisely using the dispersion relation; Mellor et al. (2008)]. The logarithmic law is used to estimate U20 from U10:

 
formula

where the drag coefficient

 
formula

The first form of the CD formula is shown for completeness only to fit the Large and Pond’s (1981) value, as the U10 in our analysis all exceed 4.5 m s−1. The second form is from Hwang and Walsh (2016); it fits for example, Powell et al. (2003) and others’ data, peaking at around 2.3 × 10−3 for U10 ≈ 30 m s−1 and decreasing for higher wind speeds. The last form uses Holthuijsen et al.’s (2012) value for the minimum CD. The sheltering coefficient is thus completely determined from the given Hs from satellite data.

Fig. A1.

The Donelan’s (2018) sheltering coefficient A1 vs Reynolds number Rew from ocean measurements. The black dots are Donelan’s data points from his Fig. 5. The red line is the fitted septic formula: A1 = 8.2987 × 10−7x7 − 7.4548 × 10−5x6 + 2.7999 × 10−3x5 − 5.7021 × 10−2x4 + 6.7957 × 10−1x3 − 4.7287x2 + 17.722x − 27.392, for 662 ≤ Rew ≤ 4 × 107, where x = ln(Rew). The dark blue line is based on the late Professor Donelan’s fitted formula, Eq. (13) of Donelan (2018), which appears to have typos.

Fig. A1.

The Donelan’s (2018) sheltering coefficient A1 vs Reynolds number Rew from ocean measurements. The black dots are Donelan’s data points from his Fig. 5. The red line is the fitted septic formula: A1 = 8.2987 × 10−7x7 − 7.4548 × 10−5x6 + 2.7999 × 10−3x5 − 5.7021 × 10−2x4 + 6.7957 × 10−1x3 − 4.7287x2 + 17.722x − 27.392, for 662 ≤ Rew ≤ 4 × 107, where x = ln(Rew). The dark blue line is based on the late Professor Donelan’s fitted formula, Eq. (13) of Donelan (2018), which appears to have typos.

The Reynolds number–dependent αMM is calculated by noting that for a sinusoidal motion the wave energy is equal to wave phase speed times the wave momentum (Hwang and Walsh 2016); the αMM can then be related to the wind energy input to the waves, and is directly proportional to A1 (c.f. Donelan et al. 2012). After some algebra to reconcile the two different formulations (Donelan et al. 2012; Hwang and Walsh 2016) we get

 
formula

where “9.36” is the proportionality factor, and the Reynolds number–independent αM was calculated as described in the text. It is interesting that the proportionality factor 9.36 makes αMM = αM at Rew = 106 (roughly corresponding to U10 ≈ 30 m s−1), prior to the A1 maximum near Rew = 6 × 106 and the anticipated drop at higher wind speeds.

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Footnotes

Supplemental information related to this paper is available at the Journals Online website: https://doi.org/10.1175/MWR-D-18-0214.s1.

© 2018 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

1

The late Professor Donelan’s septic formula, Eq. (13) of Donelan (2018), does not fit his Fig. 5. The corrected formula is given in the  appendix.

2

Zeng et al. (2007) had previously noted that western Pacific cyclones moving at the medium-translating speeds are more conducive to intensification. No direct explanation was given; although they suggested that slow storms produce upper-ocean mixing that cools the SST, while fast storms produce large asymmetric structure, both of which are not conducive to intensification.

3

Donelan et al. (2004) vividly described the process: “…a breaking wave, with abrupt change in slope., causes the (air) flow to separate from the surface and reattach near the crest of the preceding wave… The outer flow, unable to follow the wave surface, does not ‘see’ the troughs of the waves and skips from breaking crest to breaking crest…” (p. 4). The upshot is a reduction in the Jeffreys’s (1925) sheltering effect and the form drag (Donelan 2018), and the large waves become saturated.

Supplemental Material