1 Introduction

Among all hazardous weather systems, tropical and extratropical cyclones possess the most extensive spatial scale. These synoptic-scale systems interact with the ocean, giving rise to swells and storm surges that pose a threat to coastal areas. Consequently, comprehending the impact of tropical and extratropical cyclones on the ocean is a crucial matter that warrants thorough investigation. Conducting a comparative study of TCs and ECs can provide valuable insights into this subject.

When considering the oceanic responses to the atmosphere, two evident factors come into play: the ocean current, which can be effectively represented by the drift ratio, and the wave activity induced by the wind, which can be adequately represented by the significant wave height \({H}_{s}\). The drift ratio is defined by U/W, with U the ocean current speed and W the surface wind speed. Chang et al. (2012) found that the drift ratio is approximately 2% at high wind speed (W > 26 m s−1) in the tropical and temperate zones of the Northwestern Pacific (0° N–50° N, 100° E–170° E). Oruba et al. (2017) indicated that inertial currents under strong winds are nearly aligned with the wind, with a drift ratio of 2%, consistent with \({\rho }_{\mathrm{air}}{W}^{2}\sim {\rho }_{\mathrm{water}}{U}^{2}\). Here, \({\rho }_{\mathrm{air}}\) and \({\rho }_{\mathrm{water}}\) are densities of air and seawater, respectively. Fan et al. (2022) showed that the mean drift ratio under TCs is around 2% in the left-front and left-rear quadrants with relatively small variability, compared to between 2 and 4% in the right-front and right-rear quadrants, with much higher variation. This implies that the drift ratio is different between an EC and a TC.

The wave activity represented by the significant wave height \({H}_{s}\) is another important oceanic response to the atmosphere. Sverdrup and Munk (1947) established the quantitative relationship among \(W\), significant wave height (\({H}_{s}\)), fetch (\(F\)), and duration (\(t\)) from the observed data. Bretschneider (1952) further improved the method of Sverdrup and Munk (1947), and applied it to forecasting waves in deep water, called the Sverdrup–Munk–Bretschneider (SMB) method. The SMB wind and wave forecast formula is as follows:

$$\frac{g{H}_{s}}{{W}^{2}}=1.6\times {10}^{-3}{(\frac{gF}{{W}^{2}})}^{1/2} \left(\mathrm{fetch}-\mathrm{limited\;\; sea}\right),$$
(1)
$$\frac{g{H}_{s}}{{W}^{2}}=8.3\times {10}^{-5}{(\frac{gt}{W})}^{5/7} (\mathrm{duration}-\mathrm{limited\;\; sea}),$$
(2)
$$\frac{g{H}_{s}}{{W}^{2}}=2.4\times {10}^{-1}\, (\mathrm{fully}-\mathrm{developed \;\;sea}),$$
(3)

where \(g=\) 9.81 m s−1 is the gravitational acceleration.

At wind speed (\(W\)) of 7–30 m s−1, Ochi (1993, 2003) provided the empirical relationships between \({H}_{s}\) and \(W\),

$${H}_{s}=0.24W$$
(4)

for the relatively calm sea, and

$${H}_{s}=0.078{W}^{1.57}$$
(5)

for the disturbed sea. Young (2017) reviewed the parametric descriptions of tropical cyclone wind wave generation and provided an updated formula

$${H}_{s}=8.1\times {10}^{-4}{W}^{1.19}{F}^{0.405}.$$
(6)

Key questions arise after summarizing the existing studies: Is the drift ratio different between a TC in low latitudes and an EC in mid-latitudes? Are the relationships between W and (Hs, U) different between a TC and an EC? If yes, what are the mechanisms causing such differences? To answer these questions, the wind, wave, current data for the northern Pacific are used in this study to identify the characteristics of winds, currents, and waves under the TCs and ECs.

2 Data and method

The wind data are from the NASA QuikSCAT reprocessed wind records (the Ku-2011 winds) (Ricciardulli and Wentz 2011) and ECMWF Reanalysis v5 (ERA5). The extensive validations of the Ku-2011 winds (Ricciardulli and Wentz 2015) were performed to improve the consistency of aircraft winds, satellite winds from different sensors, in situ observations, and numerical models at all wind speed regimes. ERA5 provides hourly estimates of atmospheric climate variables on a 30 km grid after assimilating a large number of historical observations into global estimates.

The wave data are from the Global Ocean Waves Reanalysis (WAVERYS), which is based on the version 4 of Météo-France WAve Model (MFWAM), a third-generation numerical weather forecast model provided by the French national meteorological service. WAVERYS is the multi-year wave reanalysis that provides global 3-h integrated wave parameters with a grid resolution of 1/5°. WAVERYS includes the wave–current interactions and assimilation of altimeter wave data provided by the ocean reanalysis GLORYS and Sentinel-1 Synthetic Aperture Radar. GLORYS reanalysis project is carried out in the framework the European Copernicus Marine Environment Monitoring Service (CMEMS). The validation of WAVERYS has been performed with independent altimeter and buoy wave data (Law-Chune et al. 2021).

The current data are from the NOAA Surface Velocity Program (SVP) drifter. HYbrid Coordinate Ocean Model (HYCOM) and Navy Coupled Ocean Data Assimilation (NCODA) 1/12° Global Reanalysis are used in this study. The SVP was established during the Tropical Ocean Global Atmosphere experiment in 1988 (Hansen and Poulain 1996) and led to the creation of the Global Drifter Program (Niiler 2001). The HYCOM (Bleck 2002) is a primitive equation, general circulation model. The NCODA is a multivariate optimal interpolation scheme to assimilate surface observations from satellites, expendable bathythermographs (XBTs), Argo floats, conductivity temperature depth (CTDs), and moored buoys (Cummings 2005).

Selecting example TCs/ECs is crucial for the current comparative study. TCs have been extensively documented, and the widely accepted Saffir–Simpson hurricane wind scale provides an intensity classification system. Consequently, the process of choosing example TCs is straightforward: opting for a powerful TC located far from the continent in the open ocean, ensuring a uniform environment devoid of topographic influences.

However, the selection of an example EC is more challenging due to the absence of a systematic data bank for ECs. Typically, ECs have a larger horizontal length scale compared to TCs, and their wind speed intensity is lower. Consequently, ECs are inevitably influenced by the presence of the continent. Besides, it is difficult to find an EC with similar wind speed to the chosen TC. This difficulty in finding comparable wind speeds with example TCs is the reason why only one EC is being analyzed. To provide a comparison with this EC, only one example TC is discussed as well.

The wind, wave, and current data are identified as the result of a TC or EC based on the time of data acquisition. That is, if the data are recorded during the period of time the TC or EC existed, it is considered as the result of that TC or EC.

Before a detailed analysis, it is necessary to show the wind, the flow, and the wave height are physically consistent. It can be noted that both the flow speed roughly obeys the existing relationship with the wind speed (U ~ 0.02 W) of Chang et al. (2012), and the wave height is approximately ruled by the wind–wave relationship (Hs = 0.24 W) of Ochi (1993). That is, the flow speed and the wave height are in general proportional to the wind speed with the proportional constant close to the average value. Therefore, the flow and the wave basically were generated by the wind of the EC/TC and the physical consistency is insured.

3 Results

On 21–23 August 2004, Chaba was the strongest TC in the western Pacific during that year, and peaked as a TC with maximum sustained winds equivalent to that of a Category 5 on the Saffir–Simpson hurricane wind scale (Fig. 1). The wind speeds of a powerful EC on 14–16 January 2004 were comparable (Fig. 2). The size of the powerful EC far exceeds that of the Category 5 TC. Two NOAA SVP drifters (Argos ID: 39606 and 41143) measured the current speeds of the oceanic mixed layer (OML) under the category-5 TC on 20–28 August 2004, and the powerful EC on 10–18 January 2004 in Figs. 1, 2, and 3. The maximum wind speeds of the TC (Fig. 1) and EC (Fig. 2) are similar, but their wind directions are very different. The wind directions of the TC change rapidly, but the wind directions of the EC are quite stable. Due to its larger size and more consistent wind direction, an EC possesses a broader fetch compared to a TC. The presence of strong winds in a stable direction, combined with the extensive fetch of ECs, leads to the superposition of wind waves and swells over a long distance, resulting in large waves (Fig. 2). The rapidly changing wind directions of TCs cause wind waves and swells to radiate in different directions (Fig. 1). Upon comparing the top 9 wind images and the bottom 9 wave images in Fig. 1, it is evident that the region of highest wave heights does not correspond to the area of maximum wind speeds. The presence of maximum waves in the right-front quadrant can be attributed to a combination of previously generated swells that propagate at a faster pace than the TC’s center, as well as local wind waves.

Fig. 1
figure 1

Six-hourly wind speeds and wind directions (top 9 pictures), significant wave heights (Hs), (bottom 9 pictures), TC’s track, and drifter’s track (Argos ID: 39606) under TC on 21–23 Aug 2004

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Fig. 2
figure 2

Six-hourly wind speeds and wind directions (top 9 pictures), significant wave heights (Hs), (bottom 9 pictures), and drifter’s track (Argos ID: 41143) under EC on 14–16 Jan 2004

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Fig. 3
figure 3

a, b The drifter’s tracks, c, d observed current speeds (U) of SVP drifter (ID: 39606 and 41143), e, f Hs of WAVERYS, g, h observed wind speeds (W) of QuikSCAT data during the TC (on 20–28 Aug 2004) and EC (on 10–18 Jan 2004)

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Drifter-measured U of 1.5 m s−1 (0.8 m s−1) and Hs of 6 m (10 m) from WAVERYS data are shown in Fig. 3, respectively, under the TC (EC) at an observed W of 27 m s−1 from QuikSCAT data. Chang et al. (2013) indicated that the maximum U of a TC occurs at approximately 1–2 Rmax to the right of the storm center. Due to the faster propagation of the swell that forms the maximum wave, the maximum Hs of the TC is observed earlier than the maximum U (see Fig. 3).

One may wonder why only one drifter is used for each example of a TC and EC. During the brief storm period, there are very few drifter data and very limited Argos floats located in the storm. As in this study, only one drifter is found in the EC area. For TC Chaba, two argos drifters were found, but only one drifter (Argos ID: 39606) was in the area of strong wind.

4 Discussion and conclusion

Different from Ochi’s wind–wave relationship (Ochi 1993) under a TC and an EC, our findings reveal a consistent pattern: the Hs of ECs is consistently larger than that of TCs for equivalent wind speeds, and Hs saturates when W is 32–34 m s−1 (Fig. 4a) which seems to be related to the reduced drag coefficient (Cd) when the wind speed is approximately 32 m s−1 at high wind speed in previous studies (Powell et al. 2003; Jarosz et al. 2007). If the observed W of QuikSCAT data is 27 m s−1 (in Fig. 3g and h), \({H}_{s}\) should be 17.8 m for a fully developed sea, based on the SMB formula (Equation-3). However, at the same W of 27 m s−1 in Fig. 3g and h, Hs of Fig. 3e and f are only approximately 6 m (10 m) under TC (EC), and therefore this does not represent a fully developed sea. Substitution of (W = 27 m s−1, Hs = 6 m) under TC and (W = 27 m s−1, Hs = 10 m) under EC into Eq. 1 (for a fetch-limited sea) leads to F = 189 km under TC and 525 km under EC, respectively. These estimates of 189 km and 525 km for W of 27 m s−1 from the SMB formula match well with the fetches under W of 26–28 m s−1 in Figs. 1 and 2 (see contours of 26–28 m s−1) of this study. With F = 189 km and W = 27 m s−1 under TC, Eq. 6 (Young 2017) gives Hs = 5.6 m. With W = 27 m s−1, Equation-5 (Ochi 1993) gives Hs = 6.5 m for relatively calm sea, which is also similar to our result of 6 m in Fig. 3.

Fig. 4
figure 4

Dependence of a significant wave height, Hs, and b current speed, U, on surface wind speed with the error bars showing one standard deviation, and c dependence of pair numbers of concurrent wind and surface current (or significant wave height) data on surface wind speed during the TC and EC. Relationships among wind speeds, latitudes, d significant wave height (Hs), and e current speeds (U)

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Dependence of U on W from this study (Fig. 4b) agrees with earlier studies (Ardhuin et al. 2009; Chang et al. 2012). U in low latitudes (mid-latitudes) are relatively high (low) measured around W \(\le\) 26 m s−1. Parametric formulation of Price et al. (1994) can be expressed as

$$U=\frac{\tau {R}_{max}}{{h}_{OML}{U}_{h}},$$
(7)

where Uh is the moving speed of a TC center. Based on previous studies (de Boyer Monte´gut et al. 2004; Chu and Fan 2019), the hOML of ~ 30–40 m in low latitudes of the North Pacific during summer for the TC is thinner than hOML of ~ 125–150 m in mid-latitudes during winter for the EC. Therefore, U in low latitudes is higher than that in mid-latitudes as W < 26 m s−1 in Fig. 4b. W of 26 m s−1 is approximately equal to the maximum wind speed (VMAX) of a tropical storm (TS), which ranges from 18 to 32 m s−1. However, VMAX of a category-1 TC ranges from 33 to 42 m s−1. Previous studies (Price 1981; Foltz et al. 2015) indicated that TCs will cause upwelling and mixing and thicken hOML. When W increases to over 26 m s−1 (\(\ge\) VMAX of TS), OML becomes deeper. Then, a larger body of water needs to be driven by the same wind stress. Therefore, U becomes lower because of larger \({h}_{\mathrm{OML}}\) (see Eq. 10 and Fig. 4b). To sum up, under a TC, it takes less time for U to reach saturation than for \({H}_{s}\) in Fig. 4a and b.

In the earlier study, Chang et al. (2012) used the SVP drifter and NASA QuikSCAT wind data (1999–2009) for the northwestern Pacific Ocean to obtain the relationship. To obtain a statistical relationship between (Hs, U) and [W, latitude (\(\mathrm{\varnothing }\))] during the EC (00:00 UTC 14 January 2004–00:00 UTC 16 January 2004) and TC (18:00 UTC 21 August 2004–18:00 UTC 23 August 2004) in this study (Fig. 5), the wind data of ERA5, wave data of WAVERY, and current data of HYCOM/NCODA in the area of EC (25° N–55° N, 140° E–170° N; 00:00 UTC 14 January 2004–00:00 UTC 16 January 2004) and TC (0° N–30° N, 130° E–160° N; 18:00 UTC 21 August 2004–18:00 UTC 23 August 2004) are processed by the ensemble average method (Freeland et al. 1975; Centurioni and Niiler 2003; Centurioni et al. 2004; Lee and Niiler 2005; Chang et al. 2013) in Fig. 4d and e. The concept of a stationary ensemble is crucial to applications of statistical ensembles. A statistical ensemble does not change over time. Thus, it can be said to be in statistical equilibrium. For important physical cases, calculating averages directly over the whole of the dynamic ensemble can obtain explicit formulas. Maximum Hs of approximately 9 m (7 m) occur at W of 32 m s−1 and \(\mathrm{\varnothing }\) of 35° N–40° N (15° N–20° N) in mid-latitudes (low latitudes) in Fig. 4d. Maximum U of approximately 0.8 m s−1 (0.7 m s−1) occurs at W of 32 m s−1 (27 m s−1) and \(\mathrm{\varnothing }\) of 32° N–36° N (15° N–20° N) in Fig. 4e. EC (TC) prevails in mid-latitudes (low latitudes) of 35° N–40° N (15° N–20° N), and their mean Hs are 5 (3), 6 (4), 7 (5), 8 (6), 9 (7), 7 (6) m at limits of W of 10–15, 15–20, 20–25, 25–30, 30–35, 35–40 m s−1, respectively (Table 1). The Beaufort wind force scale (Met Office) and our results from this study are listed in Table 2. The Probable wave heights of the Beaufort wind force scale are 3.5, 5.5, 7.0, 9.0, 11.5, and > 14 m at limits of W of 11–17, 17–21, 21–24, 25–28, 29–32, and > 33 m s−1, respectively. The Beaufort wind force scale should improve overestimated wave heights because of the reduced Cd at high wind speeds over 32 m s−1 (Powell et al. 2003; Jarosz et al. 2007). EC (TC) prevails in mid-latitudes (low latitudes) of 35° N–40° N (15° N–20° N), and their mean U are 0.4 (0.3), 0.5 (0.4), 0.6 (0.6), 0.7 (0.7), 0.8 (0.7), 0.6 (0.6) ms-1at limits of W of 10–15, 15–20, 20–25, 25–30, 30–35, 35–40 m s−1, respectively (Table 3). For W of 10–35 m s−1, this study provides the empirical relationships during cyclonic storms in mid-latitudes and low latitudes between \({H}_{s}\) and \(W\), and between \(U\) and \(W\),

$${H}_{s}=0.2W+2.5\, (\mathrm{mid}-\mathrm{latitudes}, 35^\circ \mathrm{N}-40^\circ \mathrm{N}),$$
(8)
$${H}_{s}=0.2W+0.5\, (\mathrm{low}-\mathrm{latitudes}, 15^\circ \mathrm{N}-20^\circ \mathrm{N}),$$
(9)
$$U=0.020W+0.15\, (\mathrm{mid}-\mathrm{latitudes}, 35^\circ \mathrm{N}-40^\circ \mathrm{N}),$$
(10)
$$U=0.022W+0.05\, (\mathrm{low}-\mathrm{latitudes}, 15^\circ \mathrm{N}-20^\circ \mathrm{N}),$$
(11)

with high R2-values of 0.91–0.99 (0.997, 0.994, 0.967, and 0.917). Their root-mean-square errors from Eqs. (8–11) are 0.113 m, 0.116 m, 0.031 m s−1, and 0.061 m s−1, respectively. Swells can travel long distances. These results imply that the sea condition in mid-latitudes (low latitudes) is the disturbed (relatively calm) sea, which has more (less) swells. Equations (10) and (11) show the current speed and the drift ratio is slightly different for a TC in low latitudes and an EC in mid-latitudes. Different drift ratios can be attributed to the angular velocity of the earth’s rotation which is faster at mid-latitudes than at lower latitudes. When seawater flows, due to the smaller rotation radius at mid-latitudes, the seawater will accelerate forward to maintain the conservation of angular momentum. Besides wind-driven currents, there are Kuroshio geostrophic velocity components in the mid-latitude ocean of the Northwest Pacific Ocean which cause the background current speed in the mid-latitude to be relatively higher. Our results of wind–wave and wind–current relationships are updates from the existing relationships of Hs = 0.24 W by Ochi (1993), and U ~ 0.02 W by Chang et al. (2012).

Fig. 5
figure 5

Six-hourly wind speeds (W) in the area of the TC (top 9 pictures; 0° N–30° N, 130° E–160° N; 2004/8/21 UTC 18:00–2004/8/23 UTC 18:00) and EC (bottom 9 pictures; 25° N–55° N, 140° E–170° N; 2004/1/14 UTC 00:00–2004/1/16 UTC 00:00)

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Table 1 Relationships among significant wave height (Hs, unit: m), wind speeds (W), and latitudes (\(\mathrm{\varnothing }\)) during cyclonic storms
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Table 2 Relationships among current speeds (U, unit: m s−1), wind speeds (W), and latitudes (\(\mathrm{\varnothing }\)) during cyclonic storms
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Table 3 Wave heights of Beaufort wind force scale and our results during cyclonic storms
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This article is an analysis of one case for each phenomenon (e.g., there is a need for the additional analysis of other cases). The example TC was a category-5 super typhoon with a typical moving speed (5 m/s) and direction; the example EC induced the heaviest snow in some towns on the coast of the Okhotsk. Coincidently, these two examples were both in 2004 and the wind data can be retrieved from NASA QuikSCAT which is available only from 1999 to 2009. These data are not very recent but are consistent with the wave and current regression relations (Ochi 1993; Chang et al. 2012) used in the present study.

To see global warming effects and other changes with time, it may be necessary to analyze more recent TC and EC events. However, QuikSCAT wind data are not available since 2009 and hence, the wind field after that time should be retrieved with another device. Newer TC and EC data can be analyzed in a separate study as a future work.