Intraseasonal oscillations in sea surface temperature, wind stress, and sea level off the central California coast

Continental Shelf Research 21 (2001) 727–750

Intraseasonal oscillations in sea surface temperature, wind stress, and sea level off the central California coast Laurence C. Breakera,*, Paul C. Liub, Christopher Torrencec a

National Centers for Environmental Prediction, Washington, DC 20233-9910, USA Great Lakes Environmental Research Laboratory, Ann Arbor, MI 48105-1593, USA c National Center for Atmospheric Research/Advanced Study Program, P.O. Box 3000, Boulder, CO 80307-3000, USA b

Received 14 October 1999; accepted 27 April 2000

Abstract The wavelet transform is used to conduct spectral and cross-spectral analysis of daily time series of sea surface temperature (SST), surface wind stress, and sea level off the central California coast for an 18-year period from 1974 through 1991. The spectral band of primary interest is given by intraseasonal time scales ranging from 30 to 70 days. Using the wavelet transform, we examine the evolutionary behavior of the frequently observed 40–50 day oscillation originally discovered in the tropics by Madden and Julian, and explore the relative importance of atmospheric vs oceanic forcing for a range of periods where both could be important. Wavelet power spectra of each variable reveal the event-like, nonstationary nature of the intraseasonal band. Peaks in wavelet power typically last for 3–4 months and occur, on average, approximately once every 18 months. Thus, their occurrence and/or duration off central California is somewhat reduced in comparison to their presence in the tropics. Although peaks in wind stress often coincide with peaks in SST and/or sea level, no consistent relationships between the variables was initially apparent. The spectra suggest, however, that relationships between the variables, if and where they do exist, are event-dependent and thus have time scales of the same order. Cross-wavelet spectra between wind stress and SST indicate that periods of high coherence (>0.90) occur on at least six occasions over the 18-year period of record. Phase differences tend to be positive, consistent with wind forcing. For wind stress vs sea level, the cross-wavelet spectra indicate that periods of high coherence, which tend to correlate with lags close to zero, also occur, but are less frequent. As with SST, the periods of high coherence usually coincide with events in the wavelet power spectra. The somewhat weaker relationship between wind stress and sea level may be due to an independent contribution to sea level through remote forcing by the ocean originating in the tropics. Finally, simple dynamical arguments regarding the lag relationships between the variables appear to be consistent with the cross-wavelet results. # 2001 Elsevier Science Ltd. All rights reserved. Keywords: Wind stress, Sea surface temperature, California coast

*Corresponding author. 0278-4343/01/$ - see front matter # 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 2 7 8 - 4 3 4 3 ( 0 0 ) 0 0 0 8 0 - 7

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1. Introduction Quasi-periodic oscillations in zonal winds with periods in the neighborhood of 40–50 days were first observed by Madden and Julian in the lower troposphere in the western tropical Pacific (Madden and Julian, 1971). These oscillations are usually attributed to equatorially trapped Kelvin waves that are generated by convective disturbances in the tropical Indian and tropical western Pacific Oceans (e.g., Chang, 1977). These oscillations propagate eastward around the earth at speeds of 5–15 m/s and are usually confined to within
208 of the equator. Anderson and Rosen (1983) showed that this tropical oscillation is related to a 50 day oscillation in the relative angular momentum of the atmosphere. Spectral analysis of length-of-day observations also indicates increased variability in the neighborhood of 50 days (e.g., Langley et al., 1981). Anderson et al. (1984) showed that these quasi-periodic oscillations have amplitudes and frequencies which vary widely, with frequencies (periods) ranging from approximately 0.1 (63) to 0.2 (31) rad/day (days). Subsequent observations by Parker (1973) and Weickmann (1983) have shown that the range of frequencies associated with these oscillations can extend from as low as 0.07 rad/day (90 days) to as high as 0.25 rad/day (25 days). Madden and Julian (1994) conclude that the 40–50 day oscillation is a relatively broadband phenomenon with a central frequency which is not fixed but varies with time. Lau and Chan (1988) have speculated that El Nino episodes may, in fact, be triggered by 40–50 day oscillations which occasionally amplify through coupled air-sea interactions. Estimates of the oscillation’s presence, or ‘‘duty cycle’’, vary from 58% (Madden and Julian, 1994) to 75% (Knutson et al., 1986) of the time. The amplitude of the 40–50 day oscillation is seasonally dependent. It is largest between December and February and smallest between June and August and is associated with the seasonal migration of convection associated with the Intertropical Convergence Zone (Madden, 1986). Indications of the 40–50 day tropospheric oscillation are also observed at mid-latitudes; however, it is not yet clear whether such oscillations have a separate source or are linked to the tropical oscillation, perhaps through atmospheric teleconnections. According to Madden and Julian (1994), because of the magnitude of the tropical disturbance it is not unreasonable to expect some impact at higher latitudes, but in their experience, ‘‘robust mid-latitude responses are hard to find’’. Alternatively, Ghil (1987), Ghil and Childress (1987), Dickey et al. (1991), and Marcus et al. (1994), for example, indicate the possibility of a separate nontropical source for these oscillations, possibly through the interaction of nonzonal westerly flow with mountain topography. Indications of an oceanic response to these intraseasonal oscillations (IOs) have been observed at many locations in the Indian and Pacific Oceans extending from the tropics to mid-latitudes. In one case, IOs in sea surface temperature (SST) were also reported in the Gulf of Guinea in the tropical Atlantic (Picaut and Verstraete, 1976). Away from coastal boundaries and in the region where equatorial Kelvin waves are constrained to travel, i.e. the equatorial waveguide, the oceanic oscillations often appear to be forced by the local winds (e.g., McPhaden, 1982; Mysak and Mertz, 1984; Mertz and Mysak, 1984; Krishnamurti et al., 1988; Shetye et al., 1991). In some cases, oceanic IOs have been observed where no clear connection to the 40–50 day atmospheric oscillation could be made (Picaut and Verstraete, 1976; Quadfasel and Swallow, 1986; Schott et al., 1988).

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Oceanic IOs have also been observed in the eastern tropical Pacific. They have been traced to equatorial Kelvin waves which are generated by low-level winds in the western and central Pacific (Luther, 1980; Erickson et al., 1983; Enfield, 1987). At one location along the coast of Peru, Enfield and Lukas (1984) find a 45-day oscillation in sea level that is correlated at large lags with similar oscillations in zonal wind in the western tropical Pacific. Relevant to this study, Spillane et al. (1987) and Enfield (1987) both find high coherence in sea level variability at intraseasonal frequencies along the coasts of North and South America extending from central Peru to at least as far north as northern California for two different periods, 1971–1975 (Spillane et al., 1987) and 1979–1984 (Enfield, 1987). According to Enfield (1987), IOs along the coasts of North and South America are forced by equatorial Kelvin waves in the atmosphere in the western and central Pacific. These oscillations in zonal wind excite internal waves in the ocean that propagate eastward along the equatorial waveguide with propagation speeds which are consistent with the lowest baroclinic-mode Kelvin wave. When these remotely forced Kelvin waves arrive at the eastern boundary along the coast of South America, the energy associated with these disturbances is redirected north and south along the coastal boundaries as internal Kelvin waves. According to Clarke (1992), the poleward-propagating energy flux transmitted along the coastal boundaries is greater in the northern Hemisphere than in the southern Hemisphere due to the non-meridional or asymmetric nature of the coastline where the wave/boundary interaction takes place. Due to the influence of the continental shelf and slope, these poleward-propagating disturbances take on the characteristics of coastal-trapped waves. In the northern Hemisphere, these waves propagate poleward with phase speeds of 150–200 km/day (Spillane et al., 1987). Also, it is found that there is little coherence between the local wind and sea level at most coastal locations, implying that the IOs in sea level are remotely forced and thus travel as freely-propagating waves along the coast. This result is also consistent with the results of Breaker and Lewis (l988) who found no significant coherence between wind stress and sea level off the central coast of California. The work of Spillane et al. and Enfield have brought to light and clarified the large-scale, spatially coherent nature of these poleward-propagating IOs along the coasts of North and South America. Although the results of each study were spatially extensive, the lengths of record employed were relatively short (5 years for Spillane et al., and 6 years for Enfield). Consequently, we have taken records of surface wind stress, SST, and sea level along the central California coast which are 18 years in length to examine the temporal behavior of these IOs. The intraseasonal band is of interest, however, not only because of the occurrence of the well-known 40–50 oscillations but also because the time scales associated with this band lie between synoptic and seasonal, and so it is not necessarily clear whether the atmosphere drives the ocean or vice versa in this case. To examine both the IOs per se as well as the relationships between the variables employed in this study, we have drawn upon the continuous wavelet transform (e.g., Daubechies, 1990). The wavelet transform, because of its unique ability to provide localization in time and thus to better resolve discontinuous or transient events, is well-suited to the task at hand. We have also calculated cross-wavelet spectra in order to estimate the phase and coherence between wind stress and SST, and wind stress and sea level with the possibility of shedding some light on the relationship between atmospheric (oceanic) forcing and the oceanic (atmospheric) response on intraseasonal time scales. Calculation of the coherence and phase from the cross-wavelet transform is relatively new since these quantities have only recently been adequately defined (Torrence and Webster, 1999).

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The paper is divided into six sections (Section 1 is the introduction), Section 2 describes the observations, Section 3 presents the mathematics of the wavelet and cross-wavelet transforms and a description of the analysis procedures employed, Section 4 contains the results of the wavelet and cross-wavelet analyses, Section 5 contains a discussion of several related issues including dynamical considerations, and, finally, Section 6 contains a summary and conclusions.

2. The observations Time series of surface wind stress, SST, and sea level are available on a daily basis for the 18year period from 1/l/74 through 12/31/91 along the central California coast between 36.08 and 36.68N. The locations where these data were acquired are shown in Fig. 1. Geostrophic winds at 1800 UTC were calculated from Fleet Numerical Meteorology and Oceanography Center’s surface atmospheric pressure fields for an ocean point located just off the coast at 368N, 1228W, 35 km SSW of Pt. Sur, California (Bakun, 1975). The geostrophic winds were then used to calculate the surface wind stress using a simple square law relationship with a constant drag coefficient of 1.3  103. In this study, only the alongshore (v) component of the wind stress was used. The SST data were acquired at approximately 1600 UTC each day at a well-exposed location on the coast at Granite Canyon (also called Rocky Point), 23 km south of Monterey. The accuracy of these data is
0.28C (SIO Ref. 81–30, 1981). The sea level data were acquired hourly at the tide gage maintained by the National Ocean Service in Monterey Harbor. The hourly sea levels were daily-averaged and then low-pass filtered to remove the tides(Godin, 1972). In all cases, leap days have been removed from the data to create time series with 365 days per year. The time series of surface wind, SST, and sea level are shown in Fig. 2.

3. The wavelet transform 3.1. Wavelet power spectra Following Daubechies (1990), for example, the continuous wavelet transform of a time series, XðtÞ, is defined as 0  Z 1

t t 1=2 0 Xðt Þc ð1Þ dt0 ; Wðs; tÞ ¼ s s 1 which is the convolution of XðtÞ with a family of functions, cst ðt0 Þ, given by 0  t t 0 1=2 cst ðt Þ ¼ s c s

ð2Þ

where the asterisk in Eq. (1) indicates the complex conjugate, s > 0 represents dilation and corresponds to frequency, 15t51 represents translation in time, and cðtÞ; the so-called mother wavelet, must satisfy the following admissibility condition: Z 1 cðtÞ dt ¼ 0: ð3Þ 1

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Fig. 1. A map showing the exact locations off central California where the wind stress (368000 N, 1228000 W), sea surface temperature (Rocky Point } also called Granite Canyon), and sea level (Monterey) time series were acquired.

In this study, we use the Morlet wavelet for cðtÞ. This basis function is complex and nonorthogonal and consists of a damped sine and cosine wave, 2 cðtÞ ¼ p1=4 eðimtÞ eðt =2Þ

ð4Þ

where m ¼ 6 has been chosen to satisfy the admissibility condition. Because the primary objective is to calculate wavelet power spectra, defined as WXX ðs; tÞ ¼ jWX ðs; tÞj2 ;

ð5Þ

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Fig. 2. Raw time series of the (a) meridional component of surface wind stress, (b) sea surface temperature, and (c), sea level. The data are daily and extend from 1/1/74 through 12/31/91. Leap days (1976, 1980, 1984, and 1988) have been removed.

the choice of a particular wavelet basis function is not critical, since one function will produce results which are qualitatively similar to another (Torrence and Compo, 1998). The wavelet transform, in essence, takes a one-dimensional function of time (in our case) and expands it into a two-dimensional space consisting of time and frequency. 3.2. Wavelet coherence and phase Following Liu (1994), the cross-wavelet transform of two time series XðtÞ and YðtÞ, with wavelet transforms WX ðs; tÞ and WY ðs; tÞ, is given by the product, WXY ðs; tÞ ¼ WX ðs; tÞWY ðs; tÞ: ð6Þ

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This cross-wavelet transform is essentially the decomposition of the Fourier co- and quadraturespectra into time-scale space. In the spirit of the Fourier squared coherency, one can define the wavelet coherence as the smoothed cross-wavelet spectrum normalized by the smoothed individual wavelet power spectra, where  1   s WXY ðs; tÞ 2 2 ED E; ð7Þ R ðs; tÞ ¼ D s1 jWX ðs; tÞj2 s1 jWY ðs; tÞj2 where h i indicates smoothing in time and scale and the factor s1 converts wavelet power to energy density. The wavelet–coherence phase difference is given by  1    1 Imf s WXY ðs; tÞ g : ð8Þ Fðs; tÞ ¼ tan Refhs1 WXY ðs; tÞig The smoothing in Eqs. (7) and (8) is done over both time and scale using running weighted averages. Over time, the smoothing is performed using the absolute value of the Morlet wavelet function at the particular scale, normalized to unity. In the scale direction, the smoothing uses a boxcar filter of width equal to the scale-decorrelation length of the Morlet wavelet, given approximately by one-half of a power-of-two period band. Additional information on the smoothing procedure can be found in Torrence and Webster (1999). The wavelet coherence ranges from 0 to 1 and gives a measure of the correlation between the two time series as a function of both scale (or period) and time. The wavelet phase measures the phase difference between the complex wavelet transforms and indicates the presence of any lag or lead relationship between the two time series, also as a function of scale and time. 3.3. Analysis procedures The wavelet transforms were computed following the methods of Liu (1994) and Torrence and Compo (1998). Significance levels for wavelet power spectra were computed using the chi-square distribution with two degrees-of-freedom (Torrence and Compo, 1998). The Morlet wavelet was used as the wavelet basis function, where the wavelet scale, s, is almost equivalent to the corresponding peak-to-peak Fourier period, and the terms ‘‘scale’’ and ‘‘period’’ are taken to be synonymous. For Figs. 4–6, 8, and 9, the Morlet parameter with m ¼ 5:336 was used, while for Figs. 3, 7, 10, and 11, a value of m ¼ 6:0 was used. However, the results are insensitive to the choice of this parameter. The wavelet spectra presented in Fig. 3 are shown in two ways. The standard, time-dependent wavelet spectrum is shown to the left in each case followed by a global wavelet spectrum (GWS) for that series to the right obtained by taking a global average over time. As indicated above, the original spectra have been normalized by taking the original wavelet power spectra and dividing them by the average wavelet power, i.e., the global wavelet spectra, for the entire time series. Thus, the normalized wavelet spectra show the ratio of the current wavelet power to the ‘‘normal’’ or expected wavelet power. This procedure emphasizes (1) wavelet scales where the variability is relatively high (i.e. intraseasonal), and (2), periods when there are major peaks in power (i.e., during particular oscillations). Conversely, this procedure tends to deemphasize wavelet

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Fig. 3. Wavelet power spectra of (a) wind stress, (b) sea surface temperature, and (c) sea level, using the Morlet wavelet. The x-axis is time in years while the y-axis is the Fourier period that corresponds to the wavelet scale. The tick marks correspond to the beginning (1 January) of each year. Eight sub-periods are used within each power-of-two. The wavelet power for each day is normalized by the global wavelet spectrum (GWS), shown in the plot at right. The contour levels are at 0.5 (gray) and 2.0 (black), i.e. half and twice the GWS power, respectively. The contour level at 2.0 corresponds to the 13.5% significance level above the GWS. The hatched region indicates the cone-of-influence, where edge effects (due to zero-padding the ends) have reduced the power. The dashed lines show periods of 30, 45, and 70 days.

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scales where there is not much variability (i.e., constant power), such as the annual cycle. It thus becomes easier to identify regions that are significantly above the background GWS. The hatched regions in the lower right- and left-hand corners of each plot indicate the cones-of-influence, where edge effects (due to zero-padding the ends of the time series) have reduced the power. Since the time series for SST actually extends back to 1972, the entire series was used in calculating the wavelet spectra in this case and so no cone-of-influence was required for the left-hand side of this figure. 4. Results 4.1. The wavelet spectra For the purposes of this study, we have defined that band of frequencies with periods between 30 and 70 days as ‘‘intraseasonal’’, close to, but not identical with, the definition of 36–73 days adopted by Spillane et al. (1987). The normalized wavelet spectra to the left, together with the corresponding global1 wavelet spectra (GWS) to the right, are shown for each variable in Fig. 3. The dominance of the annual cycle is clearly apparent in the GWS in each case. A secondary peak also appears in the neighborhood of 182 days which corresponds to the first harmonic of the annual cycle. The wavelet power spectra span a range of frequencies with periods from 8 to 512 days. The three horizontal dashed lines correspond to periods of 30, 45, and 70 days, which cover the region of primary interest. The contour levels correspond to half (gray) and twice (black) the GWS power, to which they are referenced. Note that after normalizing by the GWS, the lack of structure in the spectral region which contains the annual cycle suggests that the annual cycle is essentially stationary. Within the intraseasonal band, the regions of high variability are generally well-resolved and appear as isolated peaks. The exact form of these spectral peaks is partly governed by the wavelet ‘‘uncertainty principle’’ (e.g., Chui, 1992) which precludes the possibility of achieving high localization in time and frequency simultaneously, and thus depends somewhat on our choice of a wavelet basis function. Nevertheless, the event-like, intermittent, pulsating nature of the variability in the intraseasonal frequency band is one attribute of these data that stands out. The occurrence and duration of these spectral peaks is irregular, but roughly, a major event or pulse occurs, on average, once every 18 months.2 With respect to frequency, these peaks are often not restricted to the intraseasonal frequency band per se, but extend to frequencies well above and below this region. Between 1977 and 1980, for example, two periods of increased variability are particularly apparent in the sea level spectrum which extend up to periods of 200 days or more. Major spectral peaks in wind stress occur in 1980 and 1983, with lesser peaks occurring in 1975, 1978, 1988, and 1990. For SST, periods of increased variability occur in 1974, 1975, 1980, 1983, 1984, 1987, and 1988. Major spectral peaks for sea level occur in 1975, 1977–1978, 1979, 1980, 1981, 1983 and 1987. The peaks in wind stress often coincide with the 1

The term ‘‘global’’ is used here to indicate a temporal average taken over the entire length of record, in this case, 18 years. This estimate is very approximate and possibly misleading to the extent that it implies intervals of constant duration when, in fact, the intervals between events varied from approximately one year to more than two years depending on the method of estimation used. 2

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peaks in sea level (particularly for the years leading up to, and including, 1983), and occasionally with the peaks in SST. Finally, in some cases these periods of increased variability coincide, at least approximately, with tropical El Nino events. Such is the case in 1977 and 1983, for example. To look at the variability associated with the IOs in greater detail, we have calculated separate wavelet spectra for each of the 18 years of record. In this case, we restrict the range of frequencies to periods of 20 to 100 days and no normalization with respect a global mean spectrum has been performed. In Figs. 4–6, we show yearly wavelet spectra for 1975, 1980, and 1989. In 1975 (Fig. 4), peaks in the yearly wavelet spectra for wind stress and sea level coincide, suggesting a possible cause and effect relationship. A maximum in SST also occurs during 1975 but the primary maximum is delayed by at least three months with respect to the other variables. However, a weaker peak in SST occurs in March, following rather closely behind the peaks in wind stress and sea level (10–15 days) suggesting a possible connection in this case (see Section 6 for additional information concerning phase relationships among the variables). Although the maxima in wavelet power for IOs frequently tend to coincide with tropical El Nino events, this was not the case in 1975 since no El Nino activity occurred at that time (Quinn et al., 1978). In 1980 (Fig. 5), a relatively weak El Nino episode by tropical standards occurred (Donguy et al., 1982) which produced an unusually strong response at mid-latitudes along the California coast (e.g., Breaker, 1989). In this case, all three variables display major peaks during February and March and are essentially in phase, suggesting a cause and effect relationship between the local wind and both sea level and SST. Over the 18-year period of study, the co-occurrence of maxima for all three variables is unusual (only in 1983 and 1985 were similar situations observed but these cases were more ambiguous). In 1989 (Fig. 6), maxima in wind stress and SST occur between May and July and so are approximately in phase, with a peak in sea level preceding the peaks in wind stress and SST by 3–4 months. There were no major tropical El Ninos during the latter half of 1988 or in 1989.3 In some years, no peaks were observed and in some years when peaks did occur, no obvious connection between the variables was suggested. From the yearly spectra we estimate the time scales for these IO-related events to range from about 2–5 months with the majority of events lasting 3–4 months. Duty cycles4 were also estimated from the yearly spectra using a simple objective criterion, yielding values of 21.5% for wind stress, 33.6% for SST, and 27.0% for sea level. We note that these values are much lower than the duty cycles estimated by Madden and Julian (1994) (58%) and Knutson et al. (1986) (75%), in the tropics. We do not know, however, how much of this difference in the presence of the oscillation between the tropics and mid-latitudes is due to differences in the duration of the events vs. differences in their frequency of occurrence. Overall, although consistent relationships between the variables or connections to tropical El Nino events may exist, they are not readily apparent from the results presented here. However, the wavelet spectra do suggest that possible relationships between the variables, when and where they exist, are most likely event-dependent and thus have time scales which are commensurate. A different view of the wavelet spectra shown in Fig. 3 can be taken by creating separate ‘‘time slices’’ or wavelet power time series for specific periods. In our case we have chosen periods of 30, 3

Based on statistics provided by NOAA’s Climate Prediction Center on their web site, most of 1988 and 1989 were dominated by La Nina, or cold conditions in the tropical Pacific. 4 The term ‘‘duty cycle’’ is taken from electrical engineering and indicates the fraction of time a signal is present.

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Fig. 4. Wavelet power spectra for (a) wind stress, (b) sea surface temperature (SST), and (c) sea level, calculated for a single year } 1975. The range of periods, in this case, has been restricted to between 20 and 100 days (and the direction has been reversed from the previous figure), and there has been no normalization relative to a global mean spectrum applied. Dashed lines are again shown for periods of 30, 45, and 70 days.

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Fig. 5. Same as Fig. 4 except for the year 1980.

45 and 70 days (Fig. 7). This method of display accentuates the differences in response as a function of frequency within the intraseasonal band. Perhaps the most striking feature of these plots is the apparent difference in response at 30 days vs. that at 45 and 70 days. We emphasize ‘‘apparent’’ because although the response at 30 days does appear to be less coherent (i.e., noisier) than the responses at the two lower frequencies, this is not necessarily the case as one looks more

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Fig. 6. Same as Fig. 4 except for the year 1989.

closely at the relationships between the variables for this period. Clear correspondences between at least two of the three variables can be seen, for example, early in 1975, from 1977 into 1978, and in 1985. Conversely, the clear correspondence between all three variables that occurs in 1980 at 45 and 70 days, is not apparent at 30 days. Also, peaks in wavelet power co-occur in 1983 and 1990 for all three variables at 45 days which do not show up clearly at either of the other two

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Fig. 7. Time series of wavelet power at periods of (a) 30, (b) 45, and (c) 70 days taken as ‘‘time slices’’ through the wavelet spectra shown in Fig. 3. The dotted trace represents wind stress, the solid trace, sea surface temperature, and the dashed trace, sea level.

frequencies. This last observation, together with the fact that more peaks with higher power occur at 45 days than at 30 or 70 days, suggests that the maximum response is band-limited to periods of 40–50 days. Previous studies have shown that IOs in the tropics are strongest between December and February (Madden and Julian, 1994). Due to the facts that atmospheric teleconnections between the tropics and mid-latitudes are well-established during the colder half of the year (e.g., Wallace and Gutzler, 1981) and that the strongest IO forcing occurs between December and February (Madden and Julian, 1994), it may be reasonable to expect a seasonal dependence in the intensity of IOs at mid-latitudes as well. In order to examine the seasonal behavior of our data, we have calculated mean wavelet spectra over the year to show seasonal variations in wavelet power for wind stress, SST, and sea level, using the 18 years of record (i.e., 1974–1991) as a basis. The maximum in wind stress (Fig. 8a) occurs between January and May for periods between about 25 and 55 days. For SST (Fig. 8b), the maximum values occur later in the year, starting in June and extending into November with maximum values occurring in September and October. The range of frequencies in this case extends from 30 days up to 80 days. A smaller, less intense peak for SST also occurs in April with periods in the range of 35–45 days, generally coincident with the peak in wind stress. For sea level (Fig. 8c), the maximum wavelet power spans the period from November through March, over periods from roughly

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Fig. 8. Mean annual wavelet spectra for the entire period from 1974 through 1991 showing the seasonal dependence of wavelet power for (a) wind stress, (b) sea surface temperature, and (c) sea level.

30 to 70 days. The annual average spectra for SST and sea level differ significantly, with the maxima in wavelet power for each variable being approximately six months apart. However, both variables have at least some overlap with wind stress (February–April). The seasonal maximum for sea level clearly occurs during the winter season, generally consistent with a source

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in the tropics.5 Because the annual peak in wind stress occurs between February and April, it is delayed by at least a month or two compared to the seasonal maximum which occurs in the tropics. 4.2. The cross-wavelet spectra Previous results using global (in time) analysis techniques have indicated that along the US West Coast, intraseasonal oscillations in alongshore wind stress and sea level are not correlated (e.g., Spillane et al., 1987) but that intraseasonal oscillations in alongshore wind stress and SST are correlated at least off central California (Breaker and Lewis, 1988). According to Enfield (1987), the intraseasonal oscillations in sea level are primarily due to remote forcing by equatorial Kelvin waves that originate in the western tropical Pacific. According to Breaker and Lewis (1988), intraseasonal oscillations in SST are consistent with forcing by the local winds which may be related to intraseasonal oscillations in the tropical troposphere through atmospheric teleconnections. In comparing the occurrences of the intraseasonal peaks in wind stress, SST, and sea level in the previous section, certain events tended to coincide. For example, major peaks in sea level in 1975, 1980, and 1983 coincide approximately with major peaks in wind stress. Weaker peaks also occur in SST which may correspond to the peaks in wind stress and sea level; however, there are lags of at least several months in 1975 and 1983. Major peaks also occur in sea level and SST in 1987, but not in wind stress. As pointed out earlier (Breaker and Lewis, 1988), the peaks which occur in 1980 and 1983 coincide with, and apparently correspond to, El Nino events which occurred in those years.6 In order to examine the relationships between wind stress and the other two variables in more detail we have calculated cross-wavelet spectra from which wavelet coherence and phase can be derived, analogous to the coherence and phase which can be obtained from conventional cross-spectral analysis (Torrence and Webster, 1999). In Fig. 9, we plot coherence and phase for wind stress vs. SST, and for wind stress vs. sea level for the entire 18-year period. We have plotted coherence and phase separately as functions of time at a period of 45 days (Fig. 9a–d). For wind stress vs. SST, periods of high coherence (>0.90; Fig. 9a) occur on six occasions and most often correspond to periods when the wavelet power is relatively high for both variables. High values occur during 1976, 1978–1979, 1980, 1984, 1988, and 1989. Whereas the values of high coherence are often sustained for periods of up to six months, values of low coherence (50.5), usually persist for periods of a month or less. Overall, the phase appears to be positive in most cases, varying primarily between zero and p=2 (Fig. 9b). The positive phase relationship between wind stress and SST indicates that the fluctuations in wind stress lead those of SST, consistent with atmospheric forcing. A further discussion of the expected phase relationship between these variables based on simple dynamics is given in Section 5.

5

Since only a month or two is required for waves to propagate across the central Pacific and up the coast of North America to 368N (Enfield, 1987). 6 Actually the 1983 El Nino started in 1982 in the tropics but its full impact was not experienced along the California coast until 1983 (McGowan et al., 1998).

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Fig. 9. (a) Wavelet squared coherence (0.0–1.0) and (b) phase (p to þp radians) between wind stress and sea surface temperature (SST) for the entire period from 1974 through 1991 at a period of 45 days. (c) Wavelet coherence and (d) phase, between wind stress and sea level at a period of 45 days for the same period.

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Two periods of very low coherence (90.2) also occur, first in 1982 and then in 1985. The very low coherence in 1982 is accompanied by an abrupt change in phase. In each case, these periods of very low coherence precede, by at least several months, the major El Nino episodes which occurred along the California coast in 1983 and 1987. Coherence and phase for wind stress vs. sea level at 45 days are shown in Figs. 9c and d, respectively. In this case, values of coherence which clearly exceed 0.9 occur in 1978, 1980, and 1983, again usually when the wavelet power is high for both variables. Periods of high coherence in this case often, but not always, coincide with periods of high coherence between wind stress and SST. In 1983, however, the major extended maximum in coherence that occurs for wind stress vs. sea level does not appear the same for wind stress vs. SST where two peaks of shorter duration and somewhat lesser amplitude occur. Also, the phase now appears to be more symmetrically distributed about zero. In several cases the periods of high coherence clearly correspond to phase differences close to zero as in 1980 and 1983, for example. Thus, intraseasonal oscillations in wind stress may not necessarily precede similar oscillations in sea level as they did in the case of SST. This result is again consistent with dynamical arguments presented in Section 5. We also note that the five brief periods of extremely low coherence (1976–1977, 1980, 1982, 1985, and 1990) coincide with sudden changes in phase.7 Additional analyses were performed to compare the relative strengths of the relationships between wind stress and SST, and between wind stress and sea level. In each case, the relationship between wind stress and SST was found to be stronger than the relationship between wind stress and sea level. Finally, in Fig. 10 we present a histogram (i.e., density) plot of the wavelet phase difference as a function of period. In this case, the 18-year time series of wavelet phase at each period is converted to a normalized histogram. Since phase at the longer periods is highly correlated in time, the bin size has been made to increase linearly with period. The phase density for wind stress vs. SST (Fig. 10a) shows that wind stress leads SST by approximately 3 days at 30 days, and by 6 days at 70 days.8 The corresponding histogram for wind stress and sea level (Fig. 10b) does not show any consistent lead–lag relationship, but appears to be in phase at all periods shorter than about 60 days, consistent with our previous results.

5. Discussion A number of studies in the North Pacific have examined the forcing and response relationships between the atmosphere and ocean over a range of time scales (Namais, 1975; Davis, 1976, 1979). From the results of these studies it is clear that these relationships are rather complicated and time scale dependent. For periods of 30–70 days, and under the conditions of this study, it is not clear in which direction forcing between the atmosphere and the ocean takes place. In order to gain 7

It is possible that these brief periods of very low coherence (a few days at most) between wind stress and sea level correspond to periods when coastal trapped waves (possibly of tropical origin) were present along the central California coast. The periods of coastal trapped waves are similarly of the order of several days. However, to verify this conjecture, an analysis of sea level data at several locations along the California coast would be required. 8 It is interesting to note that for a constant phase shift of p=2 between wind stress and SST, the amount by which wind stress leads SST would increase by 10 days between periods of 30 and 70 days, indicating that the observed phase shift (i.e., 3 days between 30 and 70 days) actually decreases slightly (based on period) over the intraseasonal band.

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Fig. 10. Density plots of wavelet phase difference as a function of period for (a) wind stress vs. sea surface temperature (SST), and (b) for wind stress vs. sea level. The histogram of wavelet phase difference (converted from radians to days) was computed at each period using the bin sizes shown by the horizontal lines in the middle. The histograms were then normalized (separately at each scale) as a percentage. The dashed lines correspond to periods of 30, 45, and 70 days.

some insight into this problem we look at the governing dynamical equations. Following Breaker and Lewis (1988), we first look at the relationship between wind stress and SST. Considering the equation for the conservation of heat, we can write (ignoring heat exchange across the ocean surface and using standard notation) @T=@t ¼ u@T=@x;

ð9Þ

where x is positive in the onshore direction next to a meridional (i.e., alongshore) boundary. The geostrophic balance can be expressed in terms of the alongshore wind stress o^y (e.g., Bowden, 1983), as fu ¼ ty =rH;

ð10Þ

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where H is the water depth. Substituting the second equation into the first yields @T=@tffi ty =ðrfHÞ @T=@x:

ð11Þ

The solutions for T indicate that ty and T are in quadrature with ty leading T as one choice. Thus, for a period of 45 days, wind stress would lead SST by approximately 11 days. Because the thermal response of the ocean to wind forcing is essentially baroclinic, a significant lag (i.e., 10 days) appears to be reasonable. This result is also generally consistent with results presented in the previous section where lags between wind stress and SST of roughly 5–10 days were observed when the coherence was high. Finally, these results agree with those of Breaker and Lewis who found lags of about 8 days between the same variables using conventional cross-spectrum analysis applied to 12 years of data from the same locations employed in this study. Next, we consider the relationship between wind stress and sea level. Consider the following vertically integrated momentum equation in the x (onshore–offshore) direction, @u=@t ¼ g@Z=@x þ tx =H;

ð12Þ

where u is the x-component of velocity, tx is the x-component of the surface wind stress, Z is the surface elevation and, as before, H is the water depth. Thus, u is in quadrature with Z and tx , implying that surface elevation and wind stress are in phase. In reality, due to spin-up time and dissipation processes, there may be a small delay between wind and water levels initially but subsequently both fields will come into phase. Also, we note that during periods when the upper ocean is highly stratified, the assumption of a purely barotropic response to fluctuations in surface wind stress may not be valid. Previous work in the upwelling region off the coast of Oregon has clearly shown that the sea level response to forcing by the alongshore component of the wind is primarily barotropic (e.g., Smith, 1974; Huyer et al., 1979). Huyer et al. found, for example, that coastal sea level responded to impulsive wind forcing within about 2 h. Based on this argument then, we might expect during periods of high coherence between wind stress and sea level that variations in surface wind drive the variations in sea level with zero lag. Our results from the previous section agree with this conclusion since the observed lags during periods of high coherence between wind stress and sea level were generally close to zero. A question worth asking at this point is why we have found strong, albeit intermittent, correlations between wind stress and sea level locally, when previous studies along the US West Coast have found no correlation between these variables (e.g., Spillane et al., 1987)? By calculating wavelet and cross-wavelet spectra, we have been able to follow the relationships between the variables employed in this study over time. During periods when peaks in wind stress and sea level in the intraseasonal band occurred together, correlations between these variables were usually high. However, there were also periods when the correlations were low, particularly between events or maxima in wavelet power. Duty cycles ranged from approximately 21–34%, and so for a majority of the time, wavelet power within the intraseasonal band was relatively low. Thus, it would not be unreasonable to expect that global statistics would most likely indicate relatively low correlations, overall. Although the same argument can be used for the relationship between wind stress and SST, we have gone to some effort to show that the relationship between wind stress and SST is clearly stronger than the one between wind stress and sea level. In our view, herein lies the power of the wavelet transform } its ability to resolve spectral characteristics and

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the correlative properties between two variables which change with time, i.e., to identify and track nonstationary processes. One interpretation of the results obtained from this study which is consistent with the discussion above is the following. During periods when the surface winds are active locally within the intraseasonal band, i.e., during ‘‘events’’, SST and sea level may be expected to respond in accordance with this forcing. However, IOs in sea level can also arise locally due to remote forcing through the ocean, initially in the form of equatorially trapped waves in the tropics, and then as coastal-trapped waves along the coasts of north and south America (Enfield, 1987), a sequence which our results suggest, could be particularly important during winter. During such periods, it might be reasonable to expect low correlations between local wind stress and sea level. Thus, the additional impact of remote forcing in the case of sea level would be expected to contribute to lower correlations between wind stress and sea level, overall, than to the correlations between wind stress and SST. We believe that this is the most reasonable explanation for the weaker relationship between wind stress and sea level. It is important to realize that the results from the cross-wavelet analyses are ‘‘necessary’’ but clearly not ‘‘sufficient’’ to establish cause and effect. In reality, an ocean–atmosphere model with full, two-way coupling which includes the fluxes of heat, moisture, and momentum is needed to explore the forcing/response relationships involved over the time scales of interest here. However, even with access to such a model, it would not necessarily be a simple task to isolate these relationships. Lending some additional credence to these results is the fact that they appear to be generally consistent with certain simplified dynamical arguments. One of the several unanswered questions that arises from this work relates to the origin of the intraseasonal oscillations in surface wind off central California. Breaker and Lewis speculated that these oscillations might be traceable to the tropics and linked to mid-latitudes along the California coast through atmospheric teleconnections. More recent work suggests that an independent source for these oscillations may arise at mid-latitudes through the interaction of non-zonal westerly flow with mountain topography (e.g., Marcus et al.,1994). We have initiated a follow-on study that will address this question.

6. Concluding remarks The wavelet transform has been used to conduct spectral and cross-spectral analysis of time series data acquired from 1974 to 1992 off the central California coast. These data include surface wind stress (alongshore component), sea surface temperature, and sea level. By using wavelet transform methods, we have been able to examine the evolution of intraseasonal oscillations in the range of 30 to 70 days over an 18-year period and to witness the event-like, pulsating, nonstationary nature of these oscillations. The peaks in wavelet power which characterize these events typically last for 3–4 months and occur, on average, approximately once every 18 months. Overall, however, their occurrence off central California is reduced considerably from their occurrence in the tropics. In some cases, peaks in wind stress coincide with peaks in sea level and/ or SST. Although consistent relationships between the variables are not readily apparent, such relationships are clearly suggested in specific cases. In 1980, for example, the co-occurrence of major peaks in wavelet power for all three variables was striking. In some cases, these peaks occur

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during El Nino episodes. Such cases occurred in 1980 and 1983. Time slices of the wavelet spectra at several frequencies suggest that the maximum response does indeed occur near the center of the band in the neighborhood of 40 to 50 days. The seasonal dependence of the IOs off central California is somewhat different from the tropics where they are strongest between December and February. Although sea level has a seasonal maximum during winter (November to March), the maximum in wind stress occurs between January and May, and for SST, it occurs between May and November. The seasonal maxima for sea level and SST are clearly different but both overlap wind stress, at least partially. Finally, the wavelet spectra strongly suggest that where relationships between the variables do exist, they are event-dependent and thus have time scales which are of the same order. To further examine the nature of the relationships between wind stress, and SST and sea level, cross-wavelet spectra were calculated, providing estimates of wavelet phase and coherence. For wind stress vs. SST, six periods where the coherence exceeded 0.90 occur over the 18 years of record and generally correspond to periods when the wavelet power is relatively high for both variables. In most cases, the phase difference between wind stress and SST is positive, indicating that the IOs in wind stress lead those of SST, consistent with wind forcing. A simple dynamical argument indicates that wind stress should lead SST by a quarter wavelength or approximately by 11 days at a period of 45 days. For wind stress vs. sea level, periods of high coherence (>0.90) also occur but their occurrence is somewhat less frequent than in the previous case. Periods of high coherence in some cases tend to correlate with lags closer to zero, consistent with an in-phase relationship between these variables. This result is again consistent with dynamical arguments. Previous results have consistently shown that IOs in wind stress are not correlated with those in sea level along the US West Coast using standard global analysis techniques (Spillane et al., 1987; Breaker and Lewis, 1988). By using the wavelet transform, we have been able to show that wind stress is correlated with sea level during certain events where the coherence between these variables is high, a result which is not in obvious agreement with previous findings. The relationship between wind stress and sea level is not as strong as the relationship between wind stress and SST off the central California coast, and may be explained in part by the oceanic contribution to sea level by remote forcing from the tropics. Such independent contributions to local variations in sea level will almost certainly act to weaken the relationship between the local winds and sea level. Finally, because the results described above apply to a single location only, similar analyses should be conducted at other locations along the west coast of North America in order to determine whether or not the results presented in this study are representative of other locations along the West Coast.

Acknowledgements The authors would like to acknowledge the assistance of Roland Madden of the National Center for Atmospheric Research (NCAR) in Boulder, and Jerrold Norton, of the Pacific Fisheries Environmental Group in Monterey, in completing this study. We also thank Roland Madden, Dave Enfield, and Jerry Norton for critical reviews of the manuscript. We thank D.B. Rao both for reviewing the manuscript and for providing input to Section 5 of this paper. Finally, we thank an anonymous reviewer for providing a number of helpful comments.

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