1 INTRODUCTION

A typhoon’s swell propagates far from where it was generated, and basically follows the principles of geometrical optics(Ardhuin et al., 2009). Chen et al.(2002)used a statistical method to partition sea wind and swell, and found that the swell probability was greater than 80% for most areas of the world. Because the swell energy can travel long distances to coastlines with little dissipation, it is a potential and uncertain hazard to coastal infrastructures. Meanwhile, the swell of a typhoon is also regarded as an important early-warning signal for forthcoming disaster. The fundamental work of tracing the swell across the Pacifi c Ocean was first implemented by Munk et al.(1963) and Snodgrass et al.(1966)using wave instruments distributed across the Pacific Ocean from New Zeal and to Alaska.

Satellites can provide accurate data that cover the global ocean, so they can be used to observe the swell evolution over a large range. Using ERS-1 SAR imagery and buoys, Holt et al.(1998)confi rmed that SAR can be used to track the swell originating from an intense storm, using a case study from the Northeast Pacifi c Ocean over several days. Collard et al.(2009)projected the SAR-derived peak wave period both forward and backward along the great circle to trace the swell across the Pacifi c Ocean basin. They assumed that no more waves would be generated after the initial time of the tracking. By excluding the effects of dispersive and angular spreading, they estimated a swell dissipation rate of 3.1×10^-7＜μ＜4.0×10 ^-7 for a peak wave period of T =15 s. Together with the pervious mechanism study on swell dissipation and this observed swell dissipation rate, the wave prediction model was improved by incorporating the swell dissipation as a suitable source term. This made it feasible to investigate the spatial and temporal structure of ocean swell fields(Ardhuin et al., 2006, 2009, 2010; Delpey et al., 2010; Hanafin et al., 2012). More recently, using the altimeter database from transects across the Southern Ocean, the swell decay rate was surveyed by Young et al.(2013), and applied to the swell attenuation model developed from waveturbulence interactions(Babanin, 2012).

In this paper, we propose a new scheme for tracking a swell that was generated by a moving typhoon, using altimeter data and best-track information. We studied a typhoon in the western North Pacific Ocean. Using the concept of the swell traveling along the great circle path and the dispersion relationship, we could relocate the swell source for each altimeter measurement to estimate the swell decay coefficient.

Section 2 gives a description of the data used, an altimeter-based typhoon swell identifi cation scheme to relocate the swell source, and the decay model used to calculate the observed swell decay coefficient. Section 3 presents a case study of typhoon Tingting, and discusses our results. We give our conclusions in Section 4.

2 DATA AND METHOD 2.1 Altimeter dataset and typhoon best-trackinformation

We used altimeter-derived wind speed and signifi cant wave height(SWH)data from the Geophysical Data Record(GDR)of the Jason-1 mission to calculate the wind and wave conditions with a sample frequency of 1 s and a repeat cycle of 10 days. In this study, we chose the wind speed and SWH measurements from the Ku-b and , which is more accurate than the C-b and . The wind speed was accurate to approximately 1.7 m/s, and SWH was approximately accurate to 0.5 m(or 10%).

The sea state is typically a combination of sea wind and swell. We can consider the cases where the atmospheric input is low as swell conditions. We eliminated cases of more than one intense windy area in the North Pacific Ocean to avoid data contamination from other wave systems(we used the European Centre for Medium-Range Weather Forecasts Interim Reanalysis(ERA-interim)wind data to determine the overall wind condition in the North Pacific Ocean). With Cycle 91 of the altimeter data, the principles of selecting data are as follows:

(1)The wind speed derived from the altimeter is lower than 10 m/s, which ensures that the atmospheric input is low;

(2)The distances between each altimeter measurement and its relocated wave source must be less than 4 000 km and the distances between each altimeter measurement and the simultaneous typhoon center should be more than 1 000 km. The other wave systems’ contamination and small wave heights mean that the altimeter cannot be used to detect swell from far away. The altimeter should also not be too close to the typhoon center, where there are nonlinear evolutions;

(3)Signifi cant wave heights must be continuous along the distance, as this model is very sensitive to distraction. Large discrepancies caused by other nearby wave systems were previously eliminated to avoid distracting the characteristic of the swell decay.

The typhoon track information was provided by the China Meteorological Administration(CMA), which archived the best-track information for the western North Pacific Ocean within 0°–55°N and 105°E–180° from 1948 to 2011. The CMA data can be divided into two stages. The first is the reanalysis stage from 1949 to 1971, which includes historical atlases of tropical cyclone tracks, station observations and ship weather reports, automated surface observations, synoptic charts, radiosonde data, aircraft reconnaissance, and real-time tropical cyclone warning advice from various agencies. The second is the annual postseason analysis stage from 1972 to the present day, which includes the observations from the first reanalysis stage and satellite and coastal radar observations. More details can be found in Ying et al.(2013).

2.2 An altimeter-based typhoon swell identifi cation scheme

Unlike the situation investigated by Young et al.(2013), the typhoon investigated in this paper can be considered a point source that continuously moved during its life span. Because the swell source was not static, the swells generated at different times with different peak frequencies were superimposed on each other according to the dispersion relationship principle. The close range and the continuity and perpetual movement properties of the typhoon offset the effects of the dispersive and angular spreading, to some degree.

In this study, the range of the observations was more than 1 000 km, which is a long distance for the swell to traverse. For example, it takes waves with a frequency of 0.08 Hz nearly a day to traverse 1 000 km. The typhoon was actually moving during the observation process. We assumed that the typhoon was moving smoothly, and could be approximately considered as a continuous wave source with the same intensity along the best track. Instead of tracking the swell along the great circle path in the same way as SAR, altimeters can provide the instantaneous transect profi le of the swell radiating from its source. In the cases considered by Young et al.(2013), the altimeter track fortunately followed the direction of the swell propagation. However, this kind of situation is rare in the North Pacific Ocean. Therefore, we reprocessed the altimeter dataset to give the distribution of the SWH with k²x, where k is the wavenumber of the peak wave period and xis the spherical distance from the source. k depends on peak wave period T, that is

Considering the movement of the typhoon, we used the peak wave period along the altimeter track to determine the source of swell at each altimeter measurement. This peak wave period was derived from the ERA-interim with the constant ratio T/T_z =1.29±0.14, where T represents the peak wave period, and T_z represents the mean wave period, as introduced by

where T denotes the peak wave period. We must calculate the spherical distances(X_i)between the typhoon center(S_i(φ_i, λ_i))of the best track and each specifi c altimeter measurement(P(φ_p, λ_p)), to determine the time(t_i)it takes for waves with this group velocity to traverse the distance. where n is the number of typhoon centers along the best track. By comparing the traverse time with the actual time difference between each typhoon center and altimeter measurement, we can relocate the swell source, as illustrated in Fig. 1.

2.3 Swell decay model

The swell decay rate presented by Babanin(2012)is the result of turbulent interactions with water. Young et al.(2013)based their work on this, and proposed that the form of the swell decay coeffi cient b₁ is

where H₀ represents the initial swell height, H_s is the swell height along the swell track, x is the distance from the initial point, and k is the wave number of the peak wave period. For a point swell source, the frequency and angular spreading of the energy packet has a 1/ αsin(α)decay along the propagation(where α is the radian spherical distance from the source). To eliminate the effects of the frequency and angular dispersion, we set the highest signifi cant wave height as the reference point with a radian spherical distance from the source α ' . Additionally, the signifi cant wave height measured by an altimeter is fi rst multiplied by a factor of .

Given that this track is no longer following the swell propagation, we defi ne H₀ as the swell wave height at the source and x as the spherical distance from each relocated source. Assuming that the intensity is steady, H₀ remains the same for every altimeter measurement from one specifi c case. However, H₀ does not represent the actual source wave height, as in windy areas it is not swell. Before the wave departs a windy area, the nonlinear wave energy transfer redistributes the wave energy so that the swell decay coeffi cient is not applicable. H₀ is an imaginary source wave height derived by assuming that the swell dissipates from the source with a leastsquare fi t. The actual wave height at the source will be much higher than the derived H₀.

3 CASE STUDY

We selected the North Pacifi c Ocean because it is a vast ocean, does not have many isl and s that block the swell propagation, and has deep water(so we can ignore bottom friction). It appears to be an ideal region to observe the decay rate of the swell generated by a typhoon over a long range. In this study, the altimeter data from Cycle 91 were used to address the decay rate of the swell generated by typhoon Tingting in the western North Pacifi c Ocean.

Tingting formed on 25 June 2004 and ended on 4 July 2004, after me and ering on the western fringe of the North Pacifi c Ocean. We selected three altimeter passes using the previously mentioned selection criteria. Figure 2 shows Jason-1 passes in the North Pacifi c Ocean during Tingting.

Using the method introduced in Section 2, Fig. 3 shows the features of the swell decay along the parameter k₂x . Details of these three cases are given in Table 1, where the odd passes are ascending and the even passes are descending. We retrieved the initial swell heights(H₀)emanating from the source for each case, and the swell decay coefficient(b₁)derived from the swell decay model.

In the case of typhoon Tingting, the data from passes that were almost along the swell propagation have well-defined features of the wave’s declination at distances near to the source. The passes that were far from the typhoon source may have other significant wave heights from windy areas. However, the wave’s declination with the distance is hard to identify in the passes that were nearly parallel to the best track of the typhoon, in which the moving source disturbs the previously generated swell. Besides the altimeter track’s position relative to the typhoon’s best track, another important factor that affects the precision of the swell decay is the accuracy of the peak wave period, which is used to determine the propagation distance and relocate the swell source. Here, the peak wave period is not directly observed, and inaccuracies in the peak wave period are the main error source when deriving swell decay.

Figure 4 shows the three altimeter passes with 885 data measurements. The solid line is a least-square fit to all the data measurements, which was used to calculate a swell decay coeffi cient of b₁ =1.3×10^-4 . This result is smaller than the decay coeffi cients of b₁ =0.001 4 derived by Young et al.(2013)from Great Australian Bight, and b₁ =1.7×10^-4 derived by Ardhuin et al.(2009). It resembles the same signifi cant scatter as Young et al.(2013), which could possibly be induced by contamination from other wave systems(including the contamination from the moving typhoon itself), the intensity change of the source, and inaccurate locations for the swell source(the lower right quadrant of the track of the storm had more intense wind and waves). Without more specific wave structure details, the observation scheme that only uses altimeters should be considered a rough estimate.

We used the NOAA 46035 buoy at 57.4°N 177.45°W to test the swell decay coeffi cient. A swell peak of 0.675 Hz appeared at 1800 UTC July 4, generated by the typhoon source at 1200 UTC June 29, and traversed about 5 000 km over 5 days. We chose a wave height of 3.2 m and a distance of 2 000 km from the source, derived from pass 186. After considering the frequency and angular dispersion, and the derived swell decay, this was reduced to approximately 1.3 m traversing 3 700 km. The swell height of buoy 6035 was approximately 1.6 m at 1800 UTC July 4.

Assuming that the swell wave height was 3.2 m 2 000 km away from the source, we applied the derived swell decay coeffi cient(b₁ =1.3×10^-4)in the swell decay relationship(Eq.4)to determine the decrease in the swell wave height according to the frequency, for different propagation times and distances(Fig. 5). The frequency range of the swell waves is due to the frequency range of the swell waves of the conspicuous energy observed by the NOAA 46035 buoy. We also considered the frequency and angular dispersions by multiplying the wave energy by αsin(α). The group velocity for a certain wave period is given in Eq.2, and the dispersive relationship for deep water is

As Collard et al.(2009)proposed, the angular spreading effects are asymptotically close to 1/sin(α) and the frequency dispersion is 1/α for a point source. Close to a point fi eld, the angular spreading effects have a huge influence on the energy redistribution. The angular and frequency spreading effects may be counteracted by the movement of the typhoon, if it is close. However, it is difficult to define the extent of this effect, which depends on the specific conditions of the typhoon and the altimeter tracking profile. It is worth noting that a typhoon is not typically a regular point source in a snapshot, especially close to the source. However, because altimeters have limitations when discriminating waves from different systems and with different frequencies, it is not practical to use them far away from the source. Unlike at large distances, the shape of the typhoon center and consequent asymmetry of the swell field divert the frequency and angular dispersion from 1/ αsin(α)at close distances. A better frequency and angular dispersion model that adapts to different typhoon shapes at close distances should be researched in the future.

4 CONCLUSION

Typhoons have important influences on the wave structure in the North Pacific Ocean. In this paper, we studied wave characteristics and the wave energy budget in the North Pacific Ocean, using a typhoon swell decay rate derived from altimeters.

We focused on a moving point source at close distances, as opposed to the distant point source in Collard et al.(2009) and the belt-like homogenous source in Young et al.(2013). Assuming a steady moving typhoon and relocating the swell source for each altimeter measurement, we considered three selected altimeter passes during typhoon Tingting. We investigated features of the swell decay along the parameter k²x . Combining all the cases using a leastsquare fit, we determined a resultant decay coeffi cient of b₁ =1.3×10^-4 . This demonstrates that this method can be used to illustrate part of the swell evolution during the typhoon, which may last several days.

Considering the complicated wind and wave conditions at close distances, the accuracy could be significantly improved if we could simulate more explicit typhoon information. As shown in our results, the dispersive and angular spreading significantly contribute to the energy redistribution for a moving typhoon at close distances, and improvements are needed. Further investigations should analyze detailed wave structures close to typhoons. The observed swell evolution may also potentially be used to reconstruct the specific typhoon conditions and present an overview of the infl uence of typhoons on ocean dynamics.

Altimeters cannot distinguish different wave systems, because they lack direction information. However, an altimeter has an enormous dataset to choose from when investigating a particular synoptic process. Previous work has shown that SAR can track swell over long distances in the unique “wave mode”, but its sampling resolution and data capacity is not ideal. In future investigations, multi-sensor observations may be applied to track swell over the full range of distances(both near and far).

5 ACKNOWLEDGMENT

The wind and SWH data pairs of Jasson-1 used in this study were obtained from the Jet Propulsion Laboratory. Track information data for the typhoon were provided by the China Meteorological Administration tropical cyclone database. The bathymetry data used in this study are ETOPO5 from the National Geophysical Data Center of the National Oceanic and Atmospheric Administration. The authors would also like to thank JIANG Haoyu for his insightful advice.