Narrowband Oscillations in the Upper Equatorial Ocean. Part I: Interpretation as Shear Instabilities

Transcription

1 VOLUME 41 J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y MARCH 2011 Narrowban Oscillations in the Upper Equatorial Ocean. Part I: Interpretation as Shear Instabilities J. N. MOUM, J.D.NASH, AND W. D. SMYTH College of Oceanic an Atmospheric Sciences, Oregon State University, Corvallis, Oregon (Manuscript receive 17 February 2010, in final form 12 November 2010) ABSTRACT Extene measurements of temperature fluctuations that inclue the turbulence wavenumber ban have now been mae using rapily sample fast thermistors at multiple epths above the core of the Equatorial Unercurrent on the Tropical Atmosphere Ocean (TAO) mooring at 08, 1408W. These measurements inclue the signature of narrowban oscillations as well as turbulence, from which the temperature variance issipation rate x T an the turbulence energy issipation rate x are estimate. The narrowban oscillations are characterize by the following: groupiness packets consist of O(10) oscillations; spectral peaks of up to two orers of magnitue above backgroun; a clear ay night cycle with more intensive activity at night; enhance mixing rates; frequencies of Hz, close to both the local buoyancy an shear frequencies, N/2p an S/2p, which vary slowly in time; high vertical coherence over at least 30-m scales; an abrupt vertical phase change (p/2 over,20 m). The abrupt vertical phase change is consistent with instabilities forme in stratifie shear flows. Linear stability analysis applie to measure velocity an ensity profiles leas to preicte frequencies that match those of the observe oscillations. This corresponence suggests that the observe oscillation frequencies are set by the phase spee an wavelength of instabilities as oppose to the Doppler shifting of internal gravity waves with intrinsic frequency set by the local stratification N. 1. Introuction Because of the importance of turbulent heat an momentum transports to the ynamics of the equatorial current system (Crawfor an Osborn 1981; McCreary 1981), experiments to examine the nature of mixing in the central equatorial Pacific were conucte in 1984 (Tropic Heat), 1987 (Tropic Heat 2), 1991 [Tropical Instability Wave Experiment (TIWE)], an autumn From these experiments, we gaine etaile glimpses into the complexity of small-scale processes within the equatorial current system. The intense iurnal cycle of mixing extening beneath the mixe layer was first observe in 1984 (Moum an Calwell 1985; Gregg et al. 1985) an Corresponing author aress: J. N. Moum, College of Oceanic an Atmospheric Sciences, Oregon State University, COAS Amin. Blg. 104, Corvallis, OR moum@coas.oregonstate.eu later foun (in 1987) to be associate with intermittently occurring bursts of oscillations with frequencies near the local buoyancy frequency (McPhaen an Peters 1992) an horizontal wavelengths of m (Moum et al. 1992a). Recently, surprisingly intense subsurface mixing was observe above the core of the Equatorial Unercurrent (EUC) in association with tropical instability waves (Moum et al. 2009); this mixing oes not cycle on a aily basis. There are several ways that narrowban oscillations (frequencies near N, 200-m wavelengths) may be generate; caniates inclue shear instability in the stratifie flui above the EUC core (Sun et al. 1998; Mack an Hebert 1997; Smyth an Moum 2002), shear instabilities forme in the mixe layer when shear is enhance by easterly win intensifications (Skyllingsta an Denbo 1994), convectively riven eies in the mixe layer (Gregg et al. 1985), an an obstacle effect associate with sheare flow over a perturbe mixe layer base DOI: /2010JPO Ó 2011 American Meteorological Society 397

2 398 J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y VOLUME 41 (Wijesekera an Dillon 1991). It is possible that ifferent instability mechanisms occur at ifferent times (or even simultaneously). It is important for us to etermine exactly how these instabilities occur to make the link to the next larger temporal an spatial scales, which is the way we will improve mixing parameterizations. We have recently emonstrate a new means of obtaining moore measurements of mixing (Moum an Nash 2009) that consists of a pair of fast-response thermistors protruing from a pressure case attache to a mooring cable. The instrument, which we call xpo, is vane to steer the thermistors into the flow, thus permitting measurement of unisturbe turbulence. This has proven successful at current spees above 0.05 m s 21.Thetechnique is especially useful in the upper equatorial current system, where strong currents prevail. The purpose of these measurements is to obtain recors of mixing that are sufficiently long to evaluate the interplay between mixing, waves, an the larger circulation scales that affect interannual phenomena such as El Niño an La Niña. Long time series are require because attempts to parameterize mixing from the intensive but short equatorial turbulence profiling experiments to ate have not been particularly successful (Zaron an Moum 2009). We suggest that this is partly because our limite experiments have not measure the full range of equatorial mixing states. A complementary set of objectives inclues obtaining a useful measure of the turbulent heat flux over long time scales to allow estimates of the vertical ivergence an thus heating rates through the water column. In this way, we anticipate etermining the role that variations in mixing have on the large-scale equatorial current structure an particularly on the sea surface temperature (SST). Here we examine an 8-month recor from four epths above the core of the EUC with the intention of pointing out the variety of states of mixing an how these are relate to narrowban oscillations, which are ubiquitous in these time series. We are hesitant to refer to these a priori as internal waves, because we wish to istinguish freely propagating internal gravity waves from shear instability waves an hence use the term oscillation to cover both phenomena. We begin with a short escription of the ata (section 2) an the backgroun oceanographic conitions uring the eployment (section 3). We then take a closer look at a 52-ay portion of the recor that most clearly contains the narrowban signal in orer to examine the temperature variability an properties of the narrowban oscillations (section 4) an their relationship to turbulence (section 5). A iscussion follows in which we argue that these observations are consistent with shear instabilities, which prompts a linear stability analysis (LSA) using observe currents an stratification, resulting in the fining that the observe peak frequencies of narrowban oscillations are etermine by the phase spees an wavelengths of shear instabilities forme above the EUC core (section 6). This is followe by a short iscussion (section 7) an a summary an conclusions (section 8). In the companion paper (Smyth et al. 2011, hereafter Part II), the computational metho for the LSA is presente, the compute instabilities are escribe in greater etail, an a theory is avance to explain the narrowban character of the oscillations. 2. Details of the eployment Five xpos were eploye on the Tropical Atmosphere Ocean (TAO) mooring at 08, 1408W on 18 September 2006 at 29-, 39-, 49-, 59-, an 84-m epths (xpo epths are shown on the image plots in Fig. 1). These were recovere on 30 May 2007, an a refurbishe set of six xpos were eploye on the replacement mooring. The unit targete for 84 m was amage prior to eployment an no ata were recore. All of the other units provie continuous ata for the uration of the eployment. The xpos provie temperature at 10-Hz sample rate an its erivative at 120 Hz. Using currents measure nearby an the cable motion as sense by linear accelerometers on boar the xpos, the flow spee past the sensors is etermine. The temperature variance issipation rate x T is then compute as escribe by Moum an Nash (2009). From this, we estimate the turbulence kinetic energy issipation rate x 1 an the turbulence iffusion coefficient for heat K T. The backgroun temperature structure in the upper 100 m at 08, 1408W uring the eployment was erive from temperature measurements on the TAO mooring at 1, 5, 10, 13, 20, 25, 28, 40, 45, 48, 60, 80, an 100 m (Fig. 1b). From these, an estimate of mixe layer epth was mae as the epth at which the temperature eviate from its value at 1 m by 0.15 K (aily maxima of mixe layer epths are shown in Fig. 1b). Win stress (Fig. 1a) was erive from win spee measurements on the surface float using bulk formulas (Fairall et al. 1996). A nearby subsurface mooring with 150-kHz narrowban acoustic Doppler current profiler (ADCP) at 270-m epth provie a measure of ambient currents in the upper 250 m (zonal u an meriional y currents are shown in Figs. 1c,, an square shear compute from these in Fig. 1e), augmente by iscrete measurements on the surface mooring 1 The subscript x is use to inicate that is compute from microscale temperature, not shear.

3 MARCH 2011 M O U M E T A L. 399 FIG. 1. (a) Win stress t w at 08, 1408W from mi-september 2006 to en of May (b) Temperature in upper 100 m (image color); aily maximum mixe layer epth (efine as the epth where temperature eviates from SST by 0.15 K) is the white line. (c) Zonal velocity, () meriional velocity, an (e) square shear. The xpo epths are enote by the short horizontal lines at both abscissa extrema of (b) (e). The bar at the top of (a) refers to the time perio of further focus in the present analysis. in the upper 40 m, where sielobe contamination affects the ADCP measurement. The subsurface mooring is part of the stanar TAO project array at this location. Data are provie as 15-min averages at 1-h intervals with 8-m vertical bins. The local buoyancy frequency N is compute in two ways. A large-scale estimate is erive from temperature an conuctivity measurements on the TAO mooring. The spacing of temperature sensors is given in the previous paragraph. Conuctivity sensors were space at 20-m intervals in the m epth range that inclues our xpos. These measurements were interpolate to 5-m vertical spacing an ensity r was compute; from this, N 2 52gr z /r, where r z is the local vertical ensity graient an g is the acceleration ue to gravity. For the purpose of this analysis, this estimate of N is use for time-average, vertical profiles. An estimate of N that is local in both time an space to our xpos an oes not require ifferencing measurements from inepenent sensors (an operation that amplifies uncertainties in graients) is erive irectly from xpo measurements of T. Vertical pumping of the TAO mooring by surface gravity waves acting on the surface float causes the xpos to move vertically (nominally, 61.5 m) about their mean epth on the mooring (Moum an Nash 2009). Here T is fit to z over 2-min intervals from which the local vertical graient T z is erive. From this follows N T 2 5 gat z, where a is the thermal expansion coefficient of seawater. The terms N an N T are compare in section 4. Because it turns out that the contribution from salinity graients is small uring the perios of greatest interest an because N T provies a local estimate uncompromise by interpolation, we employ this value in combination with wave isplacement in section 4 to estimate wave energy. Here N T is also use in the xpo estimate of x T an x,as prescribe by Moum an Nash (2009). Again, uring the perios of interest, x T an x are foun not to be biase by our choice of N. Uses of N an N T are clearly istinguishe in the figures an text to follow. 3. Backgroun conitions The variability in Fig. 1 reflects the great range of equatorial motions. Deep mixe layers precee the

4 400 J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y VOLUME 41 FIG. 2. Temperature recors at 29, 39, 49, an 59 m from 18 Sep 2006 to 30 May 2007 as measure by xpos on the 08, 1408W TAO mooring. The bar at the top refers to the time perio of further focus in the present analysis. surfacing of the EUC in early 2007; shallower mixe layer epths followe. Accompanying the surfacing of the EUC is the shallowing of a eeper sustaine shear layer above the EUC core into the epth range of the xpos (enote in Figs. 1b e). The temperature recor at each of the four xpos is shown in Fig. 2. The xpo temperatures inicate goo agreement with nearby (1-m offset) temperature sensors on the TAO mooring, typically within a simple offset of 0.18C. 4. Signatures of the narrowban wave fiel The temperature signals shown in Fig. 2 inclue a highly energetic narrowban signal of varying intensity. This signal is particularly clear in a 52-ay perio from late December to early February 2007, as illustrate using variance-preserving spectrograms of T (Fig. 3). For this reason an because the longer recor obscures aily variations in images such as Fig. 3, we restrict our FIG. 3. (a) Win stress t, (b) hourly temperature an spectrograms of temperature in variancepreserving form at (c) 29, () 39, (e) 49, an (f) 59 m from 20 Dec 2006 to 10 Feb The bars an roman numerals at the top of (a) represent 12 h examples reprouce in etail in Fig. 8. In these images, the narrowban spectral peaks appear as bright blue streaks in the vicinity of f Hz.

5 MARCH 2011 M O U M E T A L. 401 effect of salinity graient on N 2 is reasonably small. We believe the estimate of N 2 from the coarsely space an slowly sample moore sensors is not particularly relevant to local turbulence ynamics, making N T 2 the relevant stratification for use in the computation of x T. The istinctive peak in temperature spectra suggests that we can fairly reliably extract the narrowban signal by filtering. We efine the perturbation temperature signal T9 by high-pass filtering the temperature recor at Hz. A relative measure of the narrowban isplacement is then z 5 T9 T z, (1) FIG. 4. Temperature spectra (from 1 min ata) at 29, 39, 49, an 59 m for the two-ay perio shown in Fig. 6. Vertical ashe lines represent local buoyancy frequency, N/2p, at each epth. attention to this 52-ay perio for the remainer of this analysis. The narrowban signal is seen as a peak at approximately Hz at the beginning of January 2007, slightly greater than N/2p. The narrowban spectral peak pulses at the iurnal perio an ecreases in frequency throughout the month of February to approximately Hz. Average spectral amplitues over a two-ay perio (1 3 January 2007) are about two orers of magnitue greater at frequencies between 1.5 an Hz than at backgroun frequencies (Fig. 4). Note that the frequencies of elevate variance are the same for each epth an are not specifically linke to the local buoyancy frequency, which varies consierably in epth (enote by the vertical ashe lines in Fig. 4). The frequency at the amplitue maxima in Fig. 3 is ientifie by fining the peak in each spectrum an requiring it to have a minimum amplitue of K 2 ; we refer to the frequency at this peak as f NB. The result for the xpoat29misshowninfig.5a.theshort-term variability in f NB is significantly smaller than that in N T /2p, an the two frequencies are practically uncorrelate (actually weakly correlate in a negative sense, r ). AcomparisonbetweenN an N T is also seen in Fig. 5a. Salinity graients apparently occur with either sign (interpolate to 29 m) in this recor, estabilizing early in the recor when stratification is weak an stabilizing at the en of the recor. However, uring the perios of interest for this stuy (when f NB is well efine), the from which we estimate N T 2 z 2 as twice the available potential energy associate with the narrowban oscillations, following Moum et al. (1992a). A two-ay example that shows the variability of the narrowban signal an its relationship to the turbulence is represente in Fig. 6. Unfortunately, the solar raiation signal on the TAO mooring was incomplete for this eployment. To inicate the time of the solar ay, an estimate of clear sky, shortwave raiation is shown in Fig. 6a, as etermine from a simple moel. 2 The uration of narrowban bursts is several hours, most intensely at night. These are typically accompanie by high x. Generally, x is lower when narrowban oscillations are absent. In the first energetic wave group (just past miay on 1 January), the oscillations appear at all epths at close to the same time. However, high x is observe first at the upper xpo an thence progressively later with epth. The other oscillation groups seem to show that both oscillations an turbulence appear at the upper xpo first, following which they appear progressively later with increasing epth. During the perio of the measurements shown in Fig. 6, the core of the EUC was about 125 m eep (Fig. 7a). The four xpos were above the EUC core, in the westwarflowing South Equatorial Current (SEC). The spectral peak at f NB Hz (Fig. 7) is similar to the local value of N/2p (6 cph) at m (Fig. 7b), but f NB. N/2p at 29 m. Oscillations are vertically coherent at all measurement epths (Fig. 7e), with a strong vertical phase epenence (Fig. 7f). The phase lag at 39 m is approximately p/4 relative to 29 m, an p/2 at 49 an 59 m (also relative to 29 m), suggesting that the p/2 phase shift occurs over a istance of D z, 20 m. These results are similar to 2 J q SW 5 I 0 sinv 0 (t 2 t 0 ), when sinv 0 (t 2 t 0 ). 0, where I W m 22, v 0 5 2p raians ay 21, an t / /4 ays; otherwise, J q SW 5 0.

6 402 J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y VOLUME 41 FIG. 5. (a) Time series of N T /2p (black line), f NB (symbols), an N/2p (gray line) at 29 m. Here f NB correspons to the spectral peaks in Fig. 3c. (b) N T /2p vs f NB. those erive from towe thermistor chain measurements nearby an along the equator in 1987 (Moum et al. 1992a). The etaile structure of the narrowban oscillations is iverse. Five 12-h examples from the perio shown in Fig. 3 are reprouce in Fig. 8 (each example is enote by a Roman numeral at left in Fig. 8, which is echoe in Fig. 3). Over this time perio, the mixe layer shallowe from about 50 to 15 m an the zero-crossing point of the zonal velocity (which separates the EUC from the SEC) shallowe from about 70 to 35 m (Fig. 1). The shallowing of the zero-crossing point of the zonal velocitycanbeseeninfig.8a. These examples were chosen because each is istinguishe by a narrowban spectrum but with unique characteristics that we note in turn. Case I: The xpos were all locate in the westwarflowing SEC. At m, f NB N/2p;at29m,f NB. N/2p. For reference, the spectrum at 59 m (Fig. 8, case I, panel e) is reprouce in gray in Fig. 8, cases II V, FIG. 6. (a) Incoming shortwave raiation as etermine from a clear-sky solar raiation moel, (b) z (1-min means), an (c) x (1-min averages of 1-s spectral estimates) compute from the four xpos for the first two ays of January In (c), the otte baselines represent x m 2 s 23. In this an all plots, times are in UTC.

7 MARCH 2011 M O U M E T A L. 403 FIG. 7. Average profiles for the perio shown in Fig. 6. (a) Zonal (u; blue) an meriional (y; re) velocities; (b) buoyancy perio (N; black) an shear magnitue (re); an (c) 1/Ri, where the vertical ashe line correspons to Ri 5 1 /4, the critical value for shear instability. Spectral characteristics of the wave fiel are shown: () variance-preserving isplacement spectra, where the spectral peak is highlighte in gray, which is echoe in (b); (e) square coherence between isplacements at successive epths (95% confience shown as otte line); an (f) phase ifference relative to 29 m (shown only where coherence is significant an f ffi f NB ). The 95% confience limits are inicate. The sign of the phase is such that 0, phase, p when the signal at 29-m leas. panel e. The narrowban signal is most intense in the first half of the recor, with oscillations graually growing in amplitue over the first 12 cycles an then graually ecreasing in amplitue over the final 12 cycles. This is accompanie by a graual increase an ecrease in x. Case II: The zero-crossing point of the zonal velocity has shallowe to the epth of the thir xpo so that the upper two xpos are in westwar-flowing water an the lowest in eastwar-flowing water. At all xpo epths, f NB, N/2p. The ominant signal near the mile of the recor consists of oscillations growing in amplitue over approximately 12 cycles, after which they vanish. This abrupt ening is accompanie by an equally abrupt an large increase in at the epths of the lowest three xpos. Case III: Here, f NB is close to that in cases I an II. As in case II, there is significantly greater broaban energy at f, f NB (Fig. 8e, case III) than there is in case I. Case IV: This particularly strong event with f NB. N/2p, at 29 an 39 m, has the signature of the time inverse of case II. That is, initially large amplitue oscillations ecrease in amplitue over about 12 cycles. High x accompanies the leaing, largest oscillations. Case V: In this example of low-frequency oscillations, f NB, N/2p at all epths. 5. Relationship to mixing Two trens are apparent in the 52-ay recor (20 December February 2007) uring which narrowban waves were consistently observe (Fig. 9a). First, there is a clear iurnal cycle in both x an N T 2 z 2 (Figs. 9b,c). There is also a moulation of both x an N T 2 z 2 on the time scales of the narrowban waves seen in (Fig. 9a). The ay night ifferences in N T 2 z 2 can be seen in histograms of 1-h values sorte into aytime an nighttime perios (Fig. 10). These ifferences are clear at m an marginal eeper. Because of covariation on both iurnal an longer time scales, N T 2 z 2 is highly correlate to x (r , 95% confience limits [0.60, 0.69]; Fig. 11). For comparison,

8 404 J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y VOLUME 41 FIG. 8. (a) Average zonal (u; blue) an meriional (y; re) velocity profiles. (b) Average buoyancy frequency (N/2p; black) an shear (S/2p; re) profiles. Averages were compute over (c) the 12-h time perios, which shows temperature at 29- (re), 39- (cyan), 49- (blue), an 59-m (black) epths from the xpos, following the color convention introuce in Fig. 2. () x. (e) Temperature spectra normalize by the mean value of N at that epth. The gray spectral line reprouce in all of the plots is the 59-m spectrum from the first (upper) time perio. The gray vertical bars in (e) represent the peaks in the spectra an are reprouce in (b).

9 MARCH 2011 M O U M E T A L. 405 FIG. 9. (a) Temperature spectrogram (variance preserving), (b) N T 2 z 2, an (c) x at 29 m for the perio 20 Dec Feb The time perios epicte in Fig. 8 are note at the top of (a). () The histogram correspons to frequencies estimate from 155 unstable moes as etermine from LSA applie to observe profiles of currents an stratification as iscusse in section 6a. The soli line in () is the temperature spectrum (variance-preserving form) for the full perio shown in (a). the ata points plotte in Fig. 9 of Moum et al. (1992a) are also shown (r ). The current xpo estimates overlap the parameter space of the 1987 measurements an exten it. Lines of unity slope in Fig. 11 represent constant time scales relate to the ecay of energy N T 2 z 2 by turbulence issipation, t N T 2 z 2 /. These suggest that the narrowban oscillations ecay rapily, on a time scale of O(1 h) or a few oscillation perios. This woul seem to be consistent with the relatively few oscillations that compose a group as well as the patchiness of the narrowban signal. 6. Mechanistic interpretation The narrowban signal that we have escribe from these measurements has several istinctive characteristics. The ominant frequency, f NB Hz, varies remarkably slowly in time (Fig. 3) an is similar in magnitue to both N/2p an S/2p. At times, f NB. N/2p, although it is not generally observe to be greater than the maximum value of N above the EUC core (Fig. 8). The oscillations typically appear in groups of about 10 before isappearing from the time series (Fig. 8). The groups are associate with elevate mixing (Fig. 11). Packets appear as both symmetric groups with largest amplitues an most-elevate mixing in the mile of the group (Fig. 8, case I) an as highly asymmetric groups, with largest amplitues an most-elevate mixing at either trailing or leaing ege (Fig. 8, cases II an IV). If we interpret the oscillations as propagating wave packets, it is impossible to istinguish trailing from leaing ege in the time series ata. Signals are vertically coherent over the full epth range of the measurements (29 59 m) an exhibit a change in phase of p/2 over a vertical span #20 m. Both of these characteristics were also observe in the towe thermistor chain measurements of Moum et al. (1992a); only the high coherence was etecte in the moore observations of McPhaen an Peters (1992). Because of a lack of confience in relative timing between sensors, McPhaen an Peters (1992) were unable to etermine whether vertical ifferences in phase at the frequencies of the narrowban signal iffere from 0.

10 406 J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y VOLUME 41 FIG. 11. Comparison of N T 2 z 2 an x at 29 m base on 1-h averages over the 52-ay perio shown in Fig. 9. Yellow/re points plotte represent the two-imensional histogram; reer points occur more frequently. The black ots represent estimates from measurements in 1987 of using the turbulence profiler, Chameleon, an a thermistor chain towe behin the ship (Moum et al. 1992a). The correlation for the xpo ata is note at the top left, with 95% confience limits. The correlation for the 1987 ata is FIG. 10. Histograms of N T 2 z 2 at (a) 29-, (b) 39-, (c) 49-, an () 59-m epths. These were sorte into aytime ( UTC) an nighttime perios ( UTC), which are represente by light an ark gray shaing, respectively. In an introuction to the original set of papers escribing the narrowban phenomenon beneath the equatorial mixe layer, Moum et al. (1992b) mae two significant points. They acknowlege the ifficulty in making the istinction between freely propagating internal gravity waves an force shear instabilities from the observations. They also recognize the importance of oing so, primarily because of the potential for vertical reistribution of momentum by vertically propagating internal waves that woul not exist in the case of short-live shear instabilities. A number of hypotheses have been propose to explain this narrowban signal beneath the equatorial mixe layer in the form of internal waves. Wijesekera an Dillon (1991) suggeste that the interaction of turbulent overturns in the nighttime mixe layer with the mixe layer base in the highly sheare equatorial current system might prouce an internal wave fiel. From two-imensional simulations, Skyllingsta an Denbo (1994) showe generation of internal waves in the equatorial surface layer uring perios of easterly win intensifications by shear instabilities generate there. For the ata presente here, the shear is generate by easterly wins. In fact, our ata show instances of enhancement of narrowban wave activity when easterly wins increase (cf. Figs. 3a,c). However, there are enough counterexamples to inicate that, if easterly win intensification is a factor, it oes not work in isolation. More recently, Pham an Sarkar (2010) have use threeimensional simulations to show the generation an growth of internal waves by energetic vortex tubes ejecte from Kelvin Helmholtz instabilities into the sheare flui above the core of the EUC. At the same time, arguments have been mae that the narrowban signal is irectly relate to shear instabilities. LSA using observe profiles of currents an ensity (Sun et al. 1998) has shown the generation of shear instabilities with wavelengths close to those observe by Moum et al. (1992a). Most relevant may be the observation of the ;p/2 change in phase across the waves, which echoes the results of Moum et al. (1992a) an is expecte for growing Kelvin Helmholtz instabilities (Smyth an Peltier 1989; Baines an Mitsuera 1994). We procee with this argument by applying LSA to the present ata an comparing LSA results with the properties of these observations.

11 MARCH 2011 M O U M E T A L. 407 a. Comparison with textbook shear instability Three rules of thumb arise from stuies of the iealize shear instability case of the Kelvin Helmholtz form growing on a shear layer (e.g., Hazel 1972; Turner 1973): 1) the wavelength is about 7 times the thickness of the shear layer; 2) the phase ifference across the shear layer is near p/2; an 3) the phase spee equals the spee of the backgroun flow where the shear is a maximum an the phase shift is most rapi. The observe p/2 phase shift (Fig. 7f) is consistent with property 2 above. The thickness of the layer over which this phase shift occurs provies an estimate for the thickness of the shear layer that generate the instability. In this example, the thickness is #20 m. For the purpose of rough scaling, we use 20 m. If property 1 above is vali, the resulting wavelength woul be 7 times this, or l ffi 140 m. This is within the range of wavelengths observe by Moum et al. (1992a), although slightly smaller than the m wavelengths that ominate the spectrum. The phase velocity is likely to be negative because the p/2 phase change of the isturbance occurs over the epth range of the xpos, which is within the westwarflowing SEC (Figs. 7a,b). To estimate this phase velocity, we use the observe frequency f NB Hz (Fig. 7). As efine above, f NB is an absolute value; the actual frequency is negative. We therefore compute the phase velocity as c 52lf NB m s 21. This is within the range of the backgroun velocity (Fig. 7a) in agreement with the semicircle theorem (Howar 1961). It is, however, significantly slower than the backgroun current in the epth range of the xpos where the phase change is observe, contrary to property 3 above. This iscrepancy is aresse in section 6b. In summary, the observe oscillations are consistent with at least some of the well-known properties of Kelvin Helmholtz instability on a shear layer. b. LSA of observe mean states To expan on the results of section 6a, we have carrie out stability analyses of hourly average profiles of velocity an stratification. The linear stability problem for a viscous, iffusive, stratifie, parallel shear flow was solve in matrix form, as escribe in Part II. Vertical profiles of current shear an stratification N 2 were compute from the relatively coarsely resolve measurements available from the TAO mooring (Fig. 1). Unerresolution of both shear an stratification means that Ri is likely overestimate, as high-resolution measurements show small Ri on vertical scales smaller than the 5-m vertical samples an time scales shorter than the 1-h samples available (Moum et al. 2009). We assume that the hourly average profiles represent a stationary, parallel flow perturbe by infinitesimal isturbances, whereas in fact the flow is continuously subjecte to large-amplitue perturbations. Despite these caveats, the normal moe spectrum yiels a testable preiction of the fastest-growing isturbance. The LSA yiels growth rates an phase spees as functions of wavenumber. The moe corresponing to the wavenumber where the growth rate is a maximum is the one most likely to reach observable amplitue. The preicte frequency is f 5 c/l, where c an l are the phase velocity an wavelength of the fastest-growing moe. This analysis was repeate for all hourly average profiles uring the time perio shown in Figs. 3 an 9, an moes were selecte accoring to criteria given in Part II. The sum of these analyses elivere an ensemble of 155 instability events. It was usually not possible to correlate a specific instability with an observe increase in narrowban signal. This is not surprising for several reasons. First of all, as mentione earlier, the input ata are low resolution. Secon, the velocity an ensity measurements are not collocate. The temperature an conuctivity measurements use to compute the stratification were obtaine from the TAO surface mooring, which was eploye typically at least 1 km from the subsurface mooring that houses the acoustic Doppler current profiler from which velocity profiles were obtaine. Lateral inhomogeneities in currents an stratification between these two locations will contribute to both false positive an false negative inications of instabilities. Thir, any observe instability must have ha sufficient time to grow to observable amplitue. During this time, it will have propagate at spee c, which is typically 6¼ 0, meaning that the instability was forme elsewhere. Again, lateral inhomogeneities in currents an stratification will contribute to lack of agreement between observe an preicte instabilities. In view of these unavoiable limitations, we focus not on iniviual events but rather on the statistical properties of the ensemble. Properties of the 155 unstable moes are iscusse in etail in Part II an briefly reviewe here. Growth rates range from 1 to 11 e-folings per hour (for comparison, a close look at Fig. 6b suggests that the growth rate of the first large packet at miay on 1 January 2007 was approximately 2 e-folings per hour). As expecte from the ominance of zonal shear, unstable moes were generally zonally propagating, with phase velocities ranging from 21 ms 21 in the SEC to 11 ms 21 in the EUC. Negative values were consierably more common, but values close to the peak SEC velocity were rare because moes with critical levels above 37.5-m epth (where the quality

12 408 J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y VOLUME 41 FIG. 12. (a) Amplitue an (b) phase of the vertical isplacement eigenfunction corresponing to the 69 fastestgrowing moes with frequencies near to the peak frequency in Fig. 9, as etermine from a LSA using observe currents an stratification from TAO mooring ata at 08, 1408W. Amplitue is nonimensional an is efine up to an arbitrary scaling factor. The vertical axis is epth reference to the critical layer at which the fastest-growing moe was foun an nonimensionalize by its wavelength. Horizontal ashe lines ientify a layer of thickness l/7 about z c. Thick lines are mean values, an thin lines encompass 50% of the eigenfunctions. of the TAO velocity measurements was iminishe) were rejecte. Critical level epths extene nearly to the epth of the EUC core. Although unstable moes coul in theory have wavelengths up to 7 times the epth of the EUC core, or 700 m, compute moes all ha l # 340 m. The compute range of wavelengths is consistent with existing observations (Moum et al. 1992a; Lien et al. 1996). For the purpose of irect comparison of the properties of observe an preicte instabilities, we compare f NB with the apparent frequency f 5 c/l ue to the propagation of compute instabilities past our observation point; the vertical phase epenence of the compute instabilities with those observe; an preicte wavelengths to estimate shear layer thicknesses, to assess consistency with our first rule of thumb. 1) FREQUENCIES OF SHEAR INSTABILITIES The ensemble of frequencies etermine for the 155 instabilities over the 52-ay recor shown in Fig. 9a is represente by the histogram in Fig. 9. Despite the fact that the instabilities arise from a range of critical epths z c above the core of the EUC, encompassing a range of both c an l (Part II), the peak frequency of the istribution is very close to that efine by the spectral peak at frequency f NB Hz (Fig. 9). Peak frequencies of the observations an compute unstable moes are both close in magnitue to N/2p (Fig. 5a) although f NB is not temporally correlate with N (Fig. 5b). 2) VERTICAL PHASE DEPENDENCE To illustrate the general vertical structure of the instabilities, we compute composite amplitue an phase profiles as follows: We compile vertical isplacement eigenfunctions for 69 unstable moes selecte by the requirement that 10 23, j f j, Hz, representing the frequency range of the narrowban instabilities (inicate by the moe of the histogram in Fig. 9). Each vertical isplacement eigenfunction was normalize by its value at the critical level. The eigenfunctions were then converte to polar form, an the amplitue an the tangent of the phase were sorte into bins accoring to the scale vertical coorinate (z 2 z c )/l. The meian an quartiles were compute for each bin. Results are shown in Fig. 12. The amplitue peak clearly coincies with z c in Fig. 12a an the phase changes by close to p/2 raians across the shear layer, in accor with our secon rule of thumb an the observe vertical phase shift (Fig. 7f). Closer inspection of Fig. 12b reveals that the phase shift actually occurs not over the whole

13 MARCH 2011 M O U M E T A L. 409 FIG. 13. (a) Histogram of wavelengths nonimensionalize by the thickness of the shear layer h s in which they were foun. Only the 69 moes with absolute frequencies between an Hz were inclue. (b) Scatterplot of l vs h s. shear layer but over approximately the inner half of that layer. We will show that this property resolves the phase spee iscrepancy note in section 6a. 3) RELATIONSHIP BETWEEN WAVELENGTHS AND SHEAR LAYER THICKNESSES The shear layer thickness h s was etermine for each moe by fitting the shear magnitue to a function S 0 sech 2 [(z 2 z c )/h s ], where S 0 is the shear at the critical level. Figure 13 shows statistics of the shear layer thickness an the wavelength for moes with frequencies in the range 10 23, j f j, The meian shear layer thickness is 30 m. Combine with our previous observation that the phase shift occurs over half the shear layer, the phase shift shoul typically occur over 15 m in accorance with the example shown in Fig. 7. The meian wavelength for this subsample of the unstable moes is 245 m, consistent with the observations of Moum et al. (1992a). If we now recompute the phase velocity of the moe shown in Fig. 7 using this wavelength, we fin c m s 21. This is within the range of backgroun flow spees observe in the epth range of the xpos. This improve agreement with linear theory is thanks mainly to the refinement of our previous assumption that the phase shift occupies the entire shear layer. The canonical moel for shear layer instability is the Kelvin Helmholtz form, erive from hyperbolic-tangent profiles of velocity an ensity (Hazel 1972), whence comes the expectation that the wavelength is about 7 times the shear layer thickness. In our compute moes, the mean value of l/h s is 7.4, the meian value is 8.8, an 75% of the values of l/h s lie within the range [6, 11] (Fig. 13). Although it is likely that this ratio is altere by the range of naturally occurring shear profiles that spawn unstable moes (Part II), this result suggests that the Kelvin Helmholtz moel provies a reasonable escription of the observe instabilities. In sum, the apparent frequencies of preicte shear instabilities are close to f NB (an close to N/2p as well), the preicte vertical phase epenence of p/2 across h s echoes the observe phase shifts, an 92% of the compute wavelengths are within a factor of 2 of 7h s. 7. Discussion The LSA oes not prouce a preference for nighttime instabilities over aytime ones. Although this might be consiere to be inconsistent with our interpretation of the narrowban signal (which has a strong aily variation, at least above 40 m, as inicate by Fig. 10) as shear instabilities, it may also be ue to the relatively coarse quality of the ata use to perform the LSA. OneofthegreatbenefitsoftheTAOataisthelong, freely available time series from which we can test theories such as ours. At the same time, we are pushing that ata beyon its intene use as it is not resolve finely enough, either in time or in space, to aequately capture the true variability of shear an stratification in equatorial flows. This can be clearly seen by comparison to the richness of small-scale variability shown in the measurements of Moum et al. (2009). Perhaps of most concern is the lack of high-quality ata near the surface where the ownwar migration of the mixe layer at night inclues both the highly stratifie an highly sheare mixe layer base, which is a potential instability trigger on a aily time scale. Yet, espite the limitations of the ata, we have foun that instabilities are preicte an that the properties of these instabilities are consistent with the properties of the observe narrowban oscillations. In a complementary future analysis, we will examine the well-resolve

14 410 J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y VOLUME 41 measurements escribe by Moum et al. (2009) to etermine if LSA can efine the aily variations. A etaile one-to-one comparison of observe an preicte instabilities is beyon the capability of the ata available from the TAO moorings (which o not offer coincient velocity an ensity profiles). It may also be generally impossible simply because practically all instabilities (those characterize by c 6¼ 0) forme in sheare flows such as these are not observe until they have achieve observable amplitue. Hence, observe shear instabilities must coincie with mean flow conitions ifferent from those in which they were create. 8. Summary an conclusions These measurements provie a epiction of the narrowban oscillations at the equator that has not previously been available. They emonstrate the following characteristics: The oscillations are groupy. They appear as groups of O(10) iniviual oscillations (Fig. 8). Spectral peaks at f NB are approximately two orers of magnitue greater than at backgroun frequencies, even when compute over two-ay perios. f NB is similar in magnitue to N/2p, but these are not correlate in time. Daily variation in oscillation potential energy is greatest at night an concentrate at the surface (Fig. 10). Mixing is enhance (Fig. 11). There is high vertical coherence at f NB. Finally, there exists strong vertical phase epenence at f NB, typifie by the abrupt p/2 phase shift in Fig. 7f. At minimum, the observe abrupt vertical phase change motivates our interpretation of these oscillations as shear instabilities. Together with previous analyses (Sun et al. 1998; Smyth an Moum 2002) that have shown instability wavelengths comparable to observe wavelengths (Moum et al. 1992a), this has prompte the computation of the properties of shear instabilities from the available TAO ata. From this analysis, we have shown that the localize shear layers where instabilities are forme are approximately l/7 in thickness, with apparent frequencies comparable to f NB (Fig. 9), further supporting the case that the observe narrowban signal is ue to shear instability resembling the classic Kelvin Helmholtz form. This scenario iffers from the moel of a Doppler-shifte wave fiel suggeste by McPhaen an Peters (1992). These measurements support previous observations emonstrating the short-live nature of packets of narrowban oscillations forme at night in the upper equatorial thermocline (McPhaen an Peters 1992; Moum et al. 1992a). The ratio N T 2 z 2 / x maybeconsiere a time scale representing the ecay of the energy in the oscillations solely ue to turbulence. This time scale is O(1 h) (Fig. 11), an it correspons roughly to the observe uration of packets of O(10 perios) at 5 cph (in the reference frame of the fixe observer). The suggestion has been that the combination of space time patchiness an high turbulence means these packets are generate locally an ecay locally (Moum et al. 1992a). Our analysis inicates that the properties of these oscillations are consistent with shear instabilities forme from the hourly mean structure of currents an stratification. It seems that the observe narrowban signal in the temperature recor is consistent with locally generate shear instabilities that o not propagate or avect very far before breaking. The fact that f NB is near N/2p has to o with propagation of the spatial structure of the instability characterize by l at c past the measurement point, rather than oscillation of a gravity wave at frequency N (Part II). In Part II, we show that N is a natural scale because 1) the ominant frequency of an ensemble of shear instabilities scales with S an 2) S an N are closely relate in this near-critical Ri flow. 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