The Ages Of Large Amplitude Coastal Seiches On The .
View metadata, citation and similar papers at core.ac.ukbrought to you byCOREprovided by Caltech Authors1 AUGUST 2000HUANG ET AL.2001The Ages of Large Amplitude Coastal Seiches on the Caribbean Coast of Puerto RicoNORDEN E. HUANGEngineering Science, California Institute of Technology, Pasadena, California, and Laboratory for Hydrospheric Processes,Oceans and Ice Branch, NASA Goddard Space Flight Center, Greenbelt, MarylandHSING H. SHIHNOAA/National Ocean Service, Silver Spring, MarylandZHENG SHENEngineering Science, California Institute of Technology, Pasadena, California, andDepartment of Civil and Environmental Engineering, University of California, Irvine, Irvine, CaliforniaSTEVEN R. LONGLaboratory for Hydrospheric Processes, Observational Science Branch, NASA GSFC/Wallops Flight Facility,Wallops Island, VirginiaKUANG L. FANInstitute of Oceanography, National Taiwan University, Taipei, Taiwan(Manuscript received 3 November 1998, in final form 22 September 1999)ABSTRACTUsing a process denoted here as the empirical mode decomposition and the Hilbert spectral analysis, the agesof the seiches on the Caribbean coast of Puerto Rico are determined from their dispersion characteristics withrespect to time. The ages deduced from this method are less than a day; therefore, the seiches could be locallygenerated.1. IntroductionLarge amplitude coastal seiches on the CaribbeanCoast of Puerto Rico have been observed routinely. Thephenomenon has been measured, and the data analyzedby Giese et al. (1990). Later, Chapman and Giese (1990)and Giese and Chapman (1993) have also modeled thegenerating mechanism of the seiches. They concludedthat the seiches are the results of internal waves rushingon the shelf, which then trigger normal modes of theshelf water with the coastal water level change ensuing.Similar phenomena have been observed in many otherareas of the world such as Stigebrandt (1976) at Oslo,Norway; Melville and Buchwald (1976) in the Gulf ofCarpentaria, Australia; Lamy et al. (1981) in the Gulfof Lions by the Lop-Museum; Giese et al. (1982) andCorresponding author address: Steven R. Long, NASA GSFC/WFF Code 972, Building n-159, Room W-134, Wallops Island, VA23337.E-mail: steve@airsea.wff.nasa.govq 2000 American Meteorological SocietyGiese and Hollander (1987) at Palawan Island, the Philippines; Rabinovich and Levyant (1992) near the southern Kuriles; Okihiro et al. (1993) at Barbers Point Harbor, Oahu, Hawaii; Gomis et al. (1993) in the inlets ofBalearic Islands; and Giese and Chapman (1998) in theWestern Mediterranean Sea. The oscillation covers theamplitude range of water level change from a few tensof centimeters to a few meters, and the period from afew minutes to a couple of hours. Both the amplitudeand the period are highly variable depending on thegeometry of the water body and the triggering mechanisms. When the geometry is just right, the oscillationcan be so severe that the water in a small harbor canbe emptied and create great navigation hazards (Gieseand Chapman 1993; Okihiro et al. 1993). In this paper,we will use the data from the new tide gauges installedat Magueyes Island, shown in Fig. 1, to determine thesource of these seiches through the dispersion relationship. The method we used is the newly developed empirical mode decomposition and the associated Hilbertspectral analysis (Huang et al. 1998).
2002JOURNAL OF PHYSICAL OCEANOGRAPHYVOLUME 30FIG. 1. The geographic location of the Next Generation Water Level Measurement System at Puerto Rico.2. The new tide gauge and the dataThe National Ocean Service (NOS) of the NationalOceanic and Atmospheric Administration operates 189new tide gauges known as the Next Generation WaterLevel Measurement System (NGWLMS) in the NationalWater Level Observation Network. The typical installation at these measurement stations is shown in Fig. 2.Basically, the tide measurement apparatus consists ofacoustic (primary) and pressure (secondary) water-levelsensors as described by Mero and Stoney (1988) andGill et al. (1995). The primary sensor is capable ofsampling at rates of up to four Hertz. It measures waterlevel inside a 1.27-cm sounding tube, which is inside a15.24- or 30.48-cm protective well. The protective wellis used to protect the sounding tube from damage causedby waves and surface debris, and to minimize the temperature and hydrodynamic effects as reported by Shihand Baer (1991) and Porter and Shih (1996). It has awell-to-orifice ratio of 3 and parallel end plates to mitigate flow-induced measurement errors.The system normally reports tidal information froman average of 181 one-second samples (3-min average)at 6-min intervals. Also reported is the water-level standard deviation of each 3-min record, which was designed for data quality control purposes. Up to 11 ancillary environmental sensors such as wind, temperature, barometric pressure, and humidity can also be connected to the system. Data are transmitted to the centralcomputer facility via NOAA GOES satellites and dedicated telephone lines. A detailed description of the system and capability is given by Mero and Stoney (1988).Typically, data are relayed to the NOS central officevia satellites every three hours. The systems also havethe capability of random reporting triggered by tsunami(on the West Coast) or storm surge (on the East Coast)events. The data reporting of these events during therandom reporting periods can be set at any desirableintervals. At present the data rate is set at 1 minute and6 minutes for the tsunami and storm surge, respectively.The data used here, as shown in Fig. 3, were collectedat the 6-min rate for five days in September 1994. Although we have conducted an exhaustive search, theonly data showing the seiche events are very localized.This is why only five days of data were used here. Thewater level variation shows typical tidal cycles. Superimposed on the tides are the higher frequency oscillations at the range of 10 cm.3. Method of data analysis: The Hilbert spectralanalysisTraditionally, the tidal data are analyzed with Fourierbased methods from which only an averaged frequencycan be determined. Recently, Huang et al. (1998) havedeveloped a new data analysis method, the empiricalmode decomposition (EMD) and Hilbert spectral analysis (HSA). The key part of this approach is the EMDmethod with which any complicated dataset can be decomposed into a finite and often small number of intrinsic mode functions (IMF). An IMF is defined as anyfunction having the same numbers of zero-crossings andextrema, and also having symmetric envelopes defined
1 AUGUST 2000HUANG ET AL.2003FIG. 2. The schematic diagram of the NOAA NGWLMS.by the local maxima and minima respectively. The IMFadmits well-behaved Hilbert transforms. This decomposition method is adaptive and, therefore, highly efficient. Since the decomposition is based on the localcharacteristic timescale of the data, it is applicable tononlinear and nonstationary processes. With the Hilberttransform, the intrinsic mode functions yield instantaneous frequencies by differentiation of the phase function; therefore, the frequency is a function of time,which gives sharp identifications of embedded structures. The final presentation of the results is an energy–frequency–time distribution, designated as the Hilbertspectrum. As the frequency can be defined very precisely, we can use the dispersive properties of the waterFIG. 3. Selected data from the Puerto Rico tidal station for fivedays in Sep 1994.waves to determine their source as Snodgrass et al.(1966) did for the ocean swell.The data were collected by the new NOS tidal systemfor September 1994. The sampling rate is 1 Hz. Whenthe data is subjected to the EMD method, they give ninecomponents as shown in Fig. 4. The IMF componentsmost noticeable are c6 and c7, which are clearly thesemidiurnal and diurnal tides. Yet their amplitude andfrequencies are ever changing. This is very differentfrom the Fourier expansion, which would require tensof modes to represent the whole data. With these components, one can reconstruct the original data as follows:In Fig. 5a, we plot the original data in dotted line andthe last component, c9, in solid line. The trend of thedata is faithfully depicted by the data. Now we will addthe components back successively as in Figs. 5b,c. Notein Fig. 5c: the record is dominated by the diurnal tide.Adding the next component, we get the sum as shownin Fig. 5d. At this point, we have practically recoveredall the tidal energy. The physical meaning for the nextthree components are not immediately clear. Fortunately, their magnitudes are all small. When we add all thecomponents up to c2 together, we have the result in Fig.5f in which the sum up to c3 is given as a solid lineand the sum up to c2 as a dotted line. Here the contribution of c2 is clearly illustrated. It essentially represents all the energy in the seiches. When all the components are added back together, we note that all theenergy is recovered, as shown for all the cases in Huanget al. (1998).With this IMF expansion, we can next construct theHilbert spectrum. In order to show the details of thehigher-frequency components, we decided to separatethe spectrum into the low (0–5 cycle/day) and the high
2004JOURNAL OF PHYSICAL OCEANOGRAPHYVOLUME 30FIG. 4. The intrinsic mode function (IMF) components derived from the data by the empirical mode decompositionmethod.(5–60 cycle/day) frequency parts. The low-frequencyMorlet wavelet spectrum is shown in Fig. 6, while thecorresponding Hilbert spectrum is given in Fig. 7. Adetailed calibration of the Hilbert spectrum is given inHuang et al. (1998). Here, both the wavelet and theHilbert spectrum show a similar time–energy–frequencydistribution, but the Hilbert spectrum is more quantitative. There are some end effects showing in the Hilbertspectrum that is inherited from the implementation ofthe Hilbert transform via the fast Fourier transform routine. Usually, these end effects can be masked by awindow, but we did not invoke such a window here.According to both these spectra, the fluctuations of thefrequency indicate that the tide is actually nonlinear atthe measurement station. This is reasonable, for all thetidal stations are situated in shallow waters. The propagation of the tide over the shallow shelf made theirprofile nonlinear, even if the generating forces and thegoverning equations are linear.The high frequency part of the Hilbert spectrum isgiven in Fig. 9, while the corresponding Morlet waveletspectrum is given in Fig. 8. If we integrate the Hilbertspectrum with respect to time, we have the marginalspectrum as shown in Fig. 10 in which we have alsoplotted the marginal spectrum of the wavelet analysisand the Fourier spectrum, with their magnitude staggered by two decades. Here the comparison should beread with care. To begin with, the marginal spectra donot have the same information content as the full Hilbertor wavelet spectra, for they are merely the projectionsand not the real substance. Second, the integration ofthe spectrum with respect to time totally eliminates thetime variation. As a result, the advantage of the presentmethod and the wavelet over the Fourier is lost. Evenwith these understandings, the difference is clear: Thesmoothly smeared energy contour in the wavelet spectrum was even amplified in this marginal presentation.It really betrays the lack of resolution in the waveletresults as a consequence of the conflicting requirementsof localization and frequency resolution or the uncertainty principle. To preserve the localization, the compromise in the Morlet wavelet analysis is to select 5.5waves in the ‘‘mother’’ wavelet. With only 5.5 wavesin a window, the frequency resolution is certainly verypoor as shown here. The marginal Hilbert spectrum,though not as informative as the full spectrum, still provides a more precise frequency definition than the Fourier spectrum. It indicates that the high-frequency component is in a cleanly cut region around 30 cycle/day,not a smooth but wide peak as in the Fourier spectrum.
1 AUGUST 20002005HUANG ET AL.FIG. 5. (a)–(f ) The component-by-component reconstruction of the data from the IMF components starting from thelowest frequency components. It is clear in (e) that the tidal oscillation is effectively removed.Now, let us return to the Hilbert spectrum. In it, wesee clear trends of a frequency modulation of the mainenergy concentration regions on the time axis aroundthe 0.5 and 1.25 day marks. To examine these frequencychanges with time, we present two detailed drawings ofthe Hilbert spectral contour and the IMF component c2in Figs. 11a and 11b, which represent the main wavepacket of the seiche events extracted by the EMD method. In Fig. 11a, the group of waves has their frequencyincreasing gradually from 21 to 29 cycle/day over thetime span of about a quarter of a day. In Fig. 11b, thegroup of waves near the center of the time span againshows a similar frequency shift. Now we will use thisinformation to determine the source of the seiches.4. The source of the seiches: Their agesThe age of the seiches can be computed easily fromthe dispersive properties of the waves as follows. Letus imagine that the propagation of the wave in spacealong the horizontal axis representing the distance fromthe source (S) to the measuring station (L), and thevertical axis representing the time. Any waves that initiate from the common source will reach the measuringstation at different times according to the dispersiverelationship. Therefore,L 5 c1 t 5 c 2 (t 1 dt),(1)where the dispersive relationship (see, e.g., Phillips1977) is given bydrgk[cothkd 1 cothk(D 2 d)]21;ro(2)sdr g5[cothkd 1 cothk(D 2 d)]21kro s(3)s2 5thusc5in which the group velocities of wavenumber 1 andwavenumber 2 are indicated by c1 and c 2 ; s and k arethe frequency and wavenumber; dr and r are the densityjump and density of the seawater; D and d are the totalwater depth and the surface layer depth respectively. Asthe waves are generated at the same place and time butare propagated at different group velocities, they willarrive at different times, at t and t 1 dt. Since we donot know the exact density structure of the ocean at thetime the event took place, we can only be certain that
2006JOURNAL OF PHYSICAL OCEANOGRAPHYFIG. 6. The low-frequency part of the Morlet wavelet spectrum of the data.FIG. 7. The low-frequency part of the Hilbert spectrum from the data.VOLUME 30
1 AUGUST 20002007HUANG ET AL.FIG. 8. The high-frequency part of the Morlet wavelet spectrum of the data.the dispersion relationship given in Eq. (3) describesthe group velocity varying inversely with its frequency.Thus,c1 5k,sc2 5ks 1 ds(4)in which k is a function of the density structure of theocean and the wavenumber. Combining Eqs. (1) and(4), one immediately obtainsdtdsd ts5or t 5.tsds(5)Thus we can use the quantitative information gleanedfrom the Hilbert spectrum to determine the ages of theseiches. This method of using the dispersive propertiesof the water waves to determine their source was firstapplied by Snodgrass et al. (1966) for ocean swell. Itis a method well founded on the mathematics and physics of wave motions. Based on our best estimate, theages of the seiches are less than one day. This is verydifferent from the one given by Giese et al. (1990).The above computation is based on the linear dispersion relationship. As the waves here all appeared ingroups, the nonlinear evolution of the groups could bedifferent from that of an infinite train of waves as dis-cussed in Whitham (1975) and Infeld and Rowlands(1990). Let us also discuss the implication of the nonlinear dispersion of the groups. For weakly nonlinearwaves as we studied here, the dispersion relationshipaccording to Fornberg and Whitham (1978) should beexpressed as11 ]2 as 5 s0 (k) 1 s2 (k)a 2 2 s00(k)1 ···,2a ]x 2(6)in which a is the amplitude, s 2 (k) is the nonlinear correction to the dispersion rel
sensors as described by Mero and Stoney (1988) and Gill et al. (1995). The primary sensor is capable of sampling at rates of up to four Hertz. It measures water level inside a 1.27-cm sounding tube, which is inside a
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