Journalof Marine Research, 59, 167–191, 2001 Journal of MARINE RESEARCH

Volume59, Number 2

NorthPaciŽ c internaltides fromthe Aleutian Ridge: Altimeterobservations andmodeling

byPatrickF. Cummins 1,JosefY. Cherniawsky 1 andMichael G. G.Foreman 1

ABSTRACT Internaltides radiating into the North PaciŽ c fromthe Aleutian Ridge near Amukta Pass are examinedusing 7 yearsof Topex/ Poseidonaltimeter data. The observations show coherent southwardphase propagation at the M2 frequencyover a distanceof atleast1100 km intothe central PaciŽc. Barotropicand baroclinic models are applied to study this internal tidal signal. Results from thebarotropic model show that the strongest cross-slope volume and energy  uxesoccur in the vicinityof Amukta Pass, helping to establish this region as an important site for baroclinic energy conversionalong the eastern half of theridge. Atwo-dimensionalversion of the Princeton Ocean Model is used to simulate internal tide generationand propagation. A comparisonbetween the altimeter data south of the ridge and the sea-surfacesignature of theinternal tide signal of the model shows good agreement for the phase, bothclose to thesourceand well into the far Ž eld.Comparison of thephase between model and data alsoprovides evidence for wave refraction. This occurs due to the slow modulation of wavelength associatedwith the variation in the Coriolis parameter encountered as the internal tide propagates southward.The model results suggest that the net rate of conversion of barotropic to baroclinic energyis about 1.8 GW inthe vicinity of Amukta Pass. This represents about 6% of the local barotropicenergy  uxacrossthe ridge and perhaps 1% ofglobalbaroclinic conversion.

1.Introduction Prior tothelaunch of the Topex-Poseidon (T/ P)missionin August,1992, it was widely heldthat oceanic internal tides generally did not propagate more than short distance away from asourceregion before becoming ‘ incoherent,’that is, before losing their phase relationwith the generating surface tide. For example,in observations from moored

1.Institute of Ocean Sciences, Sidney,BC, Canada V8L4B2. email:cumminsp@ pac.dfo-mpo.gc.ca 167 168 Journalof MarineResearch [59, 2 sensors,internal tides often appear intermittently (Wunsch, 1975), due perhaps to spatial andtemporal variability of thestratiŽ ed medium,but perhaps also to auctuatingsignal to noiselevel. However, data since acquired from theT/Paltimeterhave clearly demonstrated thatenergetic, phase-locked internal tides are indeed present in the open ocean and may propagateover thousands of kilometersbefore being dissipated. The best studied case is theinternal tide generated along the Hawaiian Ridge. Using three years of T/ Paltimetric data,Ray and Mitchum (1996) showed that coherent propagation occurs from bothsides of theridgewith an e-foldingdecay scale of about 1000 km. The netenergy transfer rate from thebarotropic to baroclinic tides along the ridge was estimatedto be15GW,representing perhaps7– 8% ofthe200 GW suggestedby Munk(1997) as the global rate of baroclinic conversionfor the M2 tide. Kang et al. (2000)and MerriŽ eld et al. (2001)have presented resultsfrom numericalmodels of internaltide generation at theHawaiian Ridge. Thesatellite altimetry has also indicated the existence of numerousadditional genera- tionregions for coherentbaroclinic tides. Kantha and Tierney (1997) present a global surveyusing data smoothed over 2° 3 2°bins and show that phase-locked signals typically appearin thevicinity of undersea ridges. On the other hand, there are certain regions where largeinternal tides have been documented (e.g., southwest of theEuropean shelf) that fail toshow a signiŽcant signal in the T/ Pdata.Ray and Mitchum (1997) suggest loss of temporalcoherence with the barotropic tide as a plausibleexplanation. Cherniawsky et al. (2001)recently discussed a techniquefor harmonicanalysis of the T/Pdatausing singular value decomposition (SVD) andapplied the method to over 5 years (1992–97) of T/Paltimetricdata collected over the northeast PaciŽ c. Theirresults permit identiŽcation of small-scalefeatures and, in particular,they mention an internal tide signal apparentlyemanating from theAleutian Island chain. Evidence for thiswas givenin the residualamplitude and phase along T/ Ptrack117. A localizedsource near the Aleutian Islandsmay also be discerned in thesmoothed results of Kanthaand Tierney (1997). Thispaper further discusses the internal tide propagating south from theAleutian Ridge intothe central North PaciŽ c usingan updated analysis of the T/ Pdata,combined with resultsof numericalsimulations from barotropicand baroclinic models. Results from the T/Pdatashow that the internal tide from theAleutian Ridge appears as adirectedbeam of elevatedamplitudes with a robustphase that can be tracedfor over10° of latitudeto the south.In the barotropic model, Amukta Pass isthedominantlocation for semi-diurnaltidal energyto crossthe eastern half of theAleutian Ridge and enter into the Bering Sea. The strongestcross-slope barotropic tidal volume  uxesalong the eastern Aleutian Ridge occur inthe vicinity of thePass, providing the forcing for internaltides. Ahighresolution, two-dimensional ( x2z)baroclinicmodel is applied to simulate the

M2 baroclinictide generated at AmuktaPass. As expected,the model response includes an energeticinternal tide propagating to the south away from theridge topography. A comparisonof thebaroclinic component of themodel sea-level response with along-track altimetricdata shows that the phase of the sea-surface signature is well simulated both closeto thesource and in the far Želd.The results also demonstrate that the observedphase 2001] Cumminset al.:North PaciŽ c internaltides 169

Figure1. Map of the eastern Aleutian Island chain with place names and contours of bottom topography. issigniŽcantly affected by the varying rotation rate encountered by the internal tide as it propagatessouthward into the central PaciŽ c. Inthe next section, the T/ Paltimeterobservations of internal tides from theAleutian Ridgearepresented. Sections 3 and4 discussresults of barotropicand baroclinic numerical models,respectively, including a comparisonwith the altimeter observations. The last sectionsummarizes. For reference,an area map of the eastern Aleutian Island chain is includedas Figure 1.

2.Observations from the T/Paltimeter Theresults presented here are derived from sea-leveldata obtained over the northeast PaciŽc Oceanby theT/ Paltimeterduring cycles 1– 250, representing nearly seven years of datafrom September,1992 to July,1999. Standard instrumental and geophysical correc- tionsto thedata, except for tidalanalysis, were madeat theGoddard Space Flight Center. Detailson these algorithms may be foundin Koblinsky et al. (1999).Tidal constants for 21 constituentswere determinedby aleast-squaresharmonic analysis method. This leads to anoverdeterminedmatrix system which is solvedusing SVD (Cherniawsky et al., 2001). Themethodalso provides an estimate of theexpectederror, whichscales with the standard deviationof thenontidal residual. Sincethe T/Psamplingrate of onceevery 9.9156 days is longer than the semidiurnal and diurnalperiods, aliasing occurs between tidal constituents. A revisedRayleigh criterion (Parke et al., 1987)gives the length of time series required to separate individual tidal 170 Journalof MarineResearch [59, 2 constituents.The time series of nearly2500 days used in thisstudy is easilyof sufŽ cient lengthto decouple M2 (theonly constituent considered below) from othertidal constitu- ents.For example,separation of M2 from S2 requiresabout 1100 days of data. The principalsource of aliasing error thusarises from broadbandmesoscale eddies. Such variabilityis relatively weak over the central north PaciŽ c, except perhaps immediately southof the Aleutian chain, where the Alaskan Stream, a swift (100cm s 21), narrow (50km width), western boundary current is found. Even there mesoscale activity in the Streamis relatively weak in comparison to other western boundary currents like the Kuroshioor Gulf Stream (Reed et al., 1991).A preprocessingof the data involving selectivedata trimming helps to reduce the projection of strongsea-level anomalies into thetidal signal. The estimated expected error for M2 isabout1.8 cm in the vicinity of the AlaskanStream, decreasing sharply to about0.8 cm overmost of theregion of interest.

From resultsof theSVD analysis,a residualfor M2 isobtained by takingthe difference betweenthe observed harmonic and a smoothedversion with high wavenumber variability removed.This procedure is fairly standard, but for completeness,a concisesummary is givenin the Appendix. The smoothed Ž eld,taken to represent the barotropic tide, is obtainedfrom aneighth order polynomial Ž ttothe alongtrack T/ Pdata.The residual is interpretedas an internaltide signal plus background noise. Figure2 displaysthe amplitude of the resulting residual over the northeast PaciŽ c. Exceptfor afew distinctsignals, amplitudes over most of theregionare near a background levelof 0.5–0.6 cm. This is theeffective noise level and it appearsto besomewhatlower thanthe formal estimate provided by theSVD analysis.Before turning our attentionto the signalemanating from theAleutianRidge, we Žrst notethe other major sources of coherent internaltides over theregion. The broad area of elevatedamplitudes near the lower left side ofFigure2 isthe far Želdexpression of theinternal tide originating from thenorthern ank oftheHawaiianRidge (Ray and Mitchum, 1996). Also noteworthy are internal tide signals evidenton either side of theMendocino escarpment (125– 130W, 30 –43N),south of the Alaskanshelf (145– 160W, 55– 59N) and in the vicinity of theQueen Charlotte Islands off northernBritish Columbia (130 – 135W,50 – 55N).Except for thelatter,these signals could bediscernedin the results of Kanthaand Tierney(1997); the one occurringoff thenorthern B.C.shelfmay have been obscured by theirspatial smoothing. However, the occurrence of acoherentinternal tide in this region is consistent with the numerical modeling results of Cumminsand Oey (1997) who obtained an energeticbaroclinic tide radiating into the deep oceanfrom theshelf break to the north and south of theQueen Charlotte Islands. Directly westof theseislands they showed a shadowzone, for whichthere is also evidence in the satellitedata. Figure3 showsa close-upfor theinternal tide that apparent lyoriginat esfrom the AleutianIsland chain in the vicinity of Amukta Pass. In additio ntothe amplitu de Želd,the Ž gureidentiŽ es (with an arrowhead symbol) the position swherethe phase is90° (deg UT) for regionswhere the amplitud eisabove the effective backgrou nd noiselevel (0.6 cm). Phaseincreasi ngto the north or south is indicat edby the 2001] Cumminset al.:North PaciŽ c internaltides 171

Figure2. Plan view of theresidual M2 tidalamplitude after removal of the barotropic tide from the along-trackT/ Paltimeterdata. Residual amplitudes above background levels ( ’0.6 cm) are identiŽed ascoherentinternal tide signals. directionofthe arrowhead s.In the North PaciŽ c, the observat ionsindicate coherent , directedpropagat ionfor aconsiderabledistance ( .1000km) almostdirectly due south ofthe Aleutian Ridge into the central PaciŽ c. The signal becomes difŽ cult to distinguishsouth of 42N as it merges with the one originat ingfrom theHawaiian Ridge.The data show relative lylittle geometri calspreadin gofthe Aleutian Ridge internaltide, althoug hsuchspreading could be obscuredto someextent by alowsignal tonoise level off themain axis of the beam. Neverthel essthe amplitu dedecay associatedwithgeometric alspreadin gmustnot be signiŽ cant, else the signal would attenuaterapidlydown to the noise level. This suggests that the generati ngregion alongthe ridge extends over a sufŽciently large distance compared to a wavelength (nominally150km) sothatit moreclosely resembles a linesource, rather than a point source. Thephase signal shown in Figure3 alsoshows propagat ionto thenorth of theridge intothe Bering Sea over a distanceof about two waveleng ths.Inspecti onof this northernsignal shows that it is not as clearly deŽ ned as the one found to the south. SpeciŽcally, there is less regularit yinthe phase variatio nalongT/ Ptracks,and, as well,less consiste ncybetween tracks crossing this signal. A possiblereason that the 172 Journalof MarineResearch [59, 2

Figure3. Plan view similar to Figure2, butfocusing on internaltides originating from the Aleutian Ridgenear 170W. Locations of 90°phase (deg UT) forthose regions where the amplitude of the residualis greater than 0.6 cm are indicated with the arrowhead symbol. The sense of the arrowheadindicates the direction of increasing phase. Bars placed across tracks 117, 79 and41 indicatethe portionof thesetracks that are plotted in Figures9 and10. 2001] Cumminset al.:North PaciŽ c internaltides 173 northwardsignalis apparentlymorecomplica tedthan the one to thesouth may be the irregular, bowl-shapedtopograp hyfound on the northern side of the ridge, with the presencenearby of theBering Sea shelf. This could lead to multiplegenerationsitesin thisregion, producin ginterferingsignals and a complexstructure .Inspite of this uncertainty,the phase propagat ionevident in the T/ Pdatadoes suggest that coherent internaltides in theeastern Bering Sea are predomin antlygenerate dalongthe northern ankof theAleutian Ridge, rather than, for example,at themargin of theBering Sea shelf.

3.Barotropic model Thenext two sections discuss results from barotropicand baroclinic tidal models. The mainpurpose of usinga barotropicmodel is to helpexplain why the region near Amukta Pass shouldbe a sitefor generationof baroclinic tides and to estimate the forcing for a baroclinicmodel. The domain of the barotropic model includes the eastern part of the AleutianIsland chain and the southeastern Bering Sea. Barotropic tides in the Bering Sea havebeen described previously by Mofjeld (1986) and a recentreview is available in Kowalik(1999). However, there has been little work done on tidal currents and energy uxesthrough the passes in the Aleutian Island chain. Here we briey describeresults for the M2 tide;a fullerdiscussion of barotropicmodel results will be presentedelsewhere. Thespherical-coordinate, diagnostic, Ž niteelement model FUNDY5SP (Greenberg et al., 1998)is used to compute tidal harmonics associated with sea-surface elevations and three-dimensionalvelocities over an areaspanning the easternAleutian Islands and Bering

Sea.Eight constituents ( M2, S2, N2, K2, K1, O1, P1, and Q1)areincluded in iterative simulationswith boundary conditions taken from theEgbert (1997) world tidal model. Tidalpotential, earth tide, self-attraction and loading are applied as in Foreman et al. (2000).The model is pseudo-nonlinearthrough its inclusion of advectionand an iterative approximationto quadratic bottom friction in the momentum equations. The numerical solutionis obtainedon atriangulargrid with a resolutionvarying from 40kmoverthe deep oceanto 3kmnearshoreand in the passages between the Aleutian Islands. Bathymetry for themodel was takenfrom theETOPO5, NOAA GEODAS, andSmith and Sandwell (1997)databases. No attempt has been made to represent the seasonal ice cover which occurson theinner Bering Sea shelf.

Anevaluation of the M2 tidein thebarotropic model was madeby comparingthe results withaltimeter observations of thesurface displacement at the crossover points of theT/ P tracks.Based on modeledand observed harmonic constants, the rms differenceover a tidal cycle, D,was computedas, 1 1/2 D 5 ~A2 1 A2! 2 A A cos ~f 2 f ! . (1) H 2 m o m o m o J

Here Ao, Am, fo, and fm denotethe observed and modeled elevation amplitudes and phases,respectively. The error measure, D,was determinedat the93 crossoverlocations 174 Journalof MarineResearch [59, 2 distributedover the model domain where good rates of datareturn ( .2/3ofthetime) were obtained.Some points, located mostly over the northern Bering shelf, were notconsidered becauseof theirpoorer rates of datareturn, due presumably to the seasonal ice cover in the region.The average value of D atthe93 crossoverpoints is 3.4cm, while the average of therelative error ( D/Ao)is10.3%. The largest errors occurnear Bristol Bay, probably reecting a needfor betterlocal bathymetry and perhaps improvements in the parameter- izationof bottomfriction. Overall, this comparison suggests that the model is reasonably accuratein itsrepresentation of the M2 barotropictide over the region.

Acontourplot of theamplitude of themajor semi-axis for the M2 transportellipses is shownin Figure 4a. The Aleutian Ridge generally presents an obstacle to cross-slope motionswhich are relatively weak along most of its length. However, in the vicinity of AmuktaPass, owing to the greater depth of this passage, the pattern is disrupted and signiŽcant cross-slope volume transports are obtained. The vertically-integrated barotropic energy uxvectors are given in Figure4b. Thepredominant pattern in theGulfof Alaskais acyclonicgyre associated with propagation of alarge-scaleKelvin wave around the basin. SigniŽcant leakage of tidal energy from theNorth PaciŽ c intothe eastern Bering Sea occursin the region of the Amukta Pass. In particular, the model yields a totalpower crossingthe ridgeof 29 GWthroughAmukta Pass andthe threeother smaller passes found betweenAmlia and Umnak Islands. In contrast, relatively little M2 tidalenergy crosses UnimakPass. (Locations are shown in Fig. 1.) Theseresults from thebarotropic simulations indicate that for theeastern half of the AleutianRidge, the most pronounced cross-slope transport occurs in the vicinity of AmuktaPass. In two-dimensional models of internaltides (e.g., Baines, 1982; Morozov, 1995)the forcingis proportional to the the cross-slopevolume  ux,whichis often the most importantfactor determining internal tide generation. On this basis, we mayexpect that the AmuktaPass regionis theprincipal site along the eastern side of the AleutianRidge for the conversionof barotropictidal energy into baroclinic tides. This corroborates the altimeter observationsof Figure 3 whichshow an apparent internal tide signal emanating from this region.SigniŽ cant cross-slope motions through Amukta Pass andother nearby passes are distributedalong a 275km portion of the ridge. Although the extent of the generation regionis not particularly long compared to a wavelengthof 150km, it may be sufŽcient to accountfor therelative small degree of geometricalspreading evident in the observations.

4.Baroclinic model Atwo-dimensionalversion of thesigma-coordinate Princeton Ocean Model (POM) is usedto simulate internal tide generation at Amukta Pass andpropagation into the North PaciŽc. Thetwo-dimensional approach is usefulbecause a largelatitudinal range can be coveredwhile maintaining good resolution in both horizontal and vertical coordinates. Representationof three-dimensional effects such as geometrical spreading is not at- tempted.Nevertheless, as shown below, the 2-D approachis able to capture the main 2001] Cumminset al.:North PaciŽ c internaltides 175

Figure4. Results from the three-dimensional, Ž niteelement barotropic model. (a) Amplitude of the 2 21 majorsemi-axis of the M2 transportellipses (m s ),and(b) depth-integrated barotropic energy

 ux for the M2 constituent.The 1000 m bathymetriccontour is indicated by the dashed line in (b). 176 Journalof MarineResearch [59, 2 featuresof propagation along the axis of theinternal tidal beam, as observed in the T/ P data. Twotopographic sections were considered,both crossing the Aleutian Ridge at Amukta Pass.One is alignedN-S alongthe 171.6W meridian; the second is inclinedby about12° counterclockwisewith respect to north, so as tolie approximately normal to theAleutian Ridgeat thePass. The results shown below are from thelatter case; however, there was littleto distinguish the model response with the two topographies. In particular the comparisonwith altimeter data was nearlyunchanged. Topographic data are from the Smithand Sandwell (1997) bathymetry, smoothed with a 15kmwide boxcar Ž lter.A uniformgrid resolution of 1kmisspeciŽed over an inner region of variabletopography spanningabout 7° of latitude across the ridge. Depths are constant north and south of this innerregion, and the grid resolution increases linearly to a maximumof 5 kmat the boundaries.Since we areconcerned only with the internal tide in the North PaciŽ c, no attempthas been made to representthe shallow shelf bathymetry of the Bering Sea. There are1201 grid points in thehorizontal direction and 101 sigma levels in the vertical. These levelsare distributed uniformly within the interior, but with enhanced resolution near the surfaceand the seabed. The model includes a realisticvariation in the Coriolis parameter andcomparison is madeto acasewith a uniformrotation rate. Themodel is initializedwith a horizontallyhomogeneous stratiŽ cation approximating theannual mean density Ž eldof the North PaciŽ c nearthe Aleutian Ridge. Figure 5 comparesthe initial vertical density proŽ le andassociated buoyancy frequency with those derivedfrom theWorldOcean Atlas climatology (Antonov et al., 1998; Boyer et al., 1998; henceforthWOA98) at51.5N, 171.5W. Experiments were alsoconducted in which the initialstratiŽ cation was modiŽed to allow for a100m surfacemixed layer, but the comparisonwith altimeter observations again showed little sensitivity to this change. The horizontallyuniform stratiŽ cation is only a Žrst approximation,and anumberof permanent featuresof the region (for example,the subpolar front) are not represented. Some justiŽcation for neglectingthe in uence of horizontal variations in the stratiŽ cation on wavepropagation is given at the end of Section 4b. The Alaskan Stream is also not included.The Stream is asufŽciently narrow current that the internal tide will propagate throughit in a waveperiod or less,so that its in uence is likelyto beonly local. Tocomplete the description of the model, we notethat the model is driven on the southernopen boundary through the north-south component of barotropiccurrent oscillat- ing at the M2 frequency.Open boundary conditions are similar to thoseused by Oeyand Chen(1992). In particular, a gravitywave radiation condition (Flather, 1976) is applied for thenormal component of barotropic velocity and the surface elevation. The tangential componentof thebarotropic current at the boundaries is speciŽ ed through a zerogradient condition.For thebaroclinic velocity Ž elds,an Orlanski radiation condition is applied, whilea lateraladvection condition is used to determine temperature and salinity at the boundaries(Oey andChen, 1992). To facilitatelong integrations, a ‘sponge’layer is also implemented.Near theboundaries, temperature and salinity Ž eldsare relaxed back to 2001] Cumminset al.:North PaciŽ c internaltides 177

Figure5. Comparison of the initial model proŽ les of potential density ( st 5 r 2 1000) and buoyancyfrequency, N,(solidlines) with observations (dashed lines) from the annual mean WOA98climatology at 51.5N, 171.5W. initialconditions on a timescale that varies from about17 hours at adistanceof 100km to only40 minutes directly at the boundary. This provides an absorbing region that effectivelydamps incident internal wave energy and also any re ected baroclinic motions dueto inadequacies in the radiation conditions. The sponge layer has no effect on the barotropicmode. Resultsdiscussed below are based on harmonic analyses of 24 hourly Ž eldsstored followinga spin-upperiod of 20daysduration. Tests were conductedin whichthe spin-up periodwas variedbetween 8 and80 days.While the comparison with the T/ Pdatashows littlesensitivity to thesevariations, some dependence is evident in thecharacteristics of the internalresponse. SpeciŽ cally, appreciable differences were apparentin the depth- integratedbaroclinic energy  uxbetween 8-day and 20-day spin-up periods. This is presumablybecause sufŽ cient time must be allowed for moreslowly propagating higher modesto advance into the far Želd.A furtherextension to 40 and 80 days leads to only minorquantitative changes. 178 Journalof MarineResearch [59, 2 a. Results Experimentswere conductedin which the amplitude of theforcing at the southern open boundarywas varied.A volume uxper unit length of 100m 2 s21 was foundto yield a surfaceinternal tide signal of comparable strength to the observed one. Most results presentedbelow are from thisbasic case. By comparison, in the barotropic model the amplitudeof the M2 volume uxacross the ridge near Amukta Pass is19 Svwith about 60%occurring in the Amukta Pass itselfand the rest through three smaller openings nearby.Averaging this  uxover the approximately 200 km distance spanned by these openingsyields a similarvalue of 95 m 2 s21 for thelaterally averaged  ux. Figures6a andb presentthe amplitude and phase of thesea-surface displacement from thebaroclinic model. There is a generalprogression of phase from southto north associatedwith propagation of thebarotropic tide. Wiggles in theamplitude and phase are thesurfaceexpression of internal tides propagating away from eitherside of the ridge. Also shownare smooth curves, obtained from apolynomialŽ ttothe model surface displace- ment(see AppendixA), whichare taken to represent the barotropic tide. Theinternal response of themodel is characterized by the rms barocliniccurrent Ž elds shownin Figure 7a, b. These Ž eldswere determinedfrom aharmonicanalysis of the horizontalvelocity of themodel. Also shown on theŽ gureare the internal tide characteris- tics(rays) originatingfrom 52.1N,a criticalpoint where the magnitude of the bottomslope isequalto thatof therays. For hydrostaticwaves ( N2 @ v2),theray slopes are given by dz/dx 5 6=v2 2 f 2/N, where f isthe Coriolisparameter, v theangular frequency of the 21 M2 tide, and N 5 =2gr0 dr/dz,thebuoyancyfrequency with r thepotentialdensity. As acriticalslope is resonant for thebaroclinic tide, this point is apparently the source for muchof thevariability found on thesouthern side of theridge. The baroclinic current is mostintense along this ray, which re ects off thebottom near 51.7N, and then off the surfaceat about51N. Theray structure gradually becomes incoherent and is lost south of about49N. In the far Želd(Fig. 7b), the baroclinic current is intensiŽ ed over the upper 500 m ofthe water columnwith maximum rms valuesof 10 to 12 cm s 21 occurringnear the surface. The modaldistribution of horizontalkinetic energy was determinedby projecting the horizon- talcurrent onto the vertical modes (see AppendixB ofCummins and Oey, 1997). The resultsshowed that beyond the immediate source region, the gravest internal mode dominatesthe baroclinic current. Accordingly, at 51N,the Ž rst modeaccounts for halfof thevariance in thebaroclinic current, increasing to 70–90%south of 46N. Asimilarsituation prevails on thenorthern side of theridge (not shown) with an intense internalresponse occurring along the ray emanating downward from acriticalpoint on the northern ank.Since the ridge is steeperon itsnorthern side, the critical point is foundat a shallowerdepth (370 m) thanon the southern side. Away from thesource, there is a comparableloss of theray structure after a coupleof re ections. Thedepth-integrated baroclinic energy  uxwas determinedfrom theharmonically analyzedŽ elds(Cummins and Oey, 1997) and the results are shown in Figure8. Jumpsin 2001] Cumminset al.:North PaciŽ c internaltides 179

Figure6. Solid lines give the phase (a) and amplitude (b) of thesea-surface displacement from the basicexperiment with the baroclinic model. Phase and amplitude from an 8thorderpolynomial Ž t

to the M2 harmonicare also given (dashed lines). The lower panel (c) gives the bottom topography ofthemodel. theenergy uxoccurat the pointson eitherside of theridgecrest where the bottom slope is critical.For thebasic case (maximum volume  uxof 100 m 2 s21), thepeak of the southwardsenergy  uxis about3000 W m 21,decreasingto about2000 W m 21 near 51N andthen remaining almost constant. These values scale as the square of the cross-slope volume uxand are, therefore, sensitive to the magnitude of the barotropic forcing. A meanbaroclinic conversion rate of 3000 W m 21 integratedover the approximately200 km lengthof ridge with signiŽ cant cross-slope barotropic forcing (Fig. 4a, b) implies a net baroclinic uxofabout0.6 GW intothe N. PaciŽc. Themodel response on the northern side of the ridge is associated with a baroclinic Figure7. Rms M2 barocliniccurrent on the southern  ankof the ridge (a) and in the far Želd(b). The baroclinicvelocity is computed as the residual horizontal velocity once the depth-averaged componentis removed. Contours are drawn at 2 cms 21 intervalsfrom 2 to20 cms 2 1 . The dashed-dottedline indicates the path of arayemanating from the critical point occurring at adepth of740m onthesouthern  ankof theridge. 2001] Cumminset al.:North PaciŽ c internaltides 181

Figure8. Depth-integrated baroclinic energy  uxin W m 21 forthe basic case. Negative values denotesouthward radiation of energy.The vertical dotted lines indicate the points near the ridge crestwhere the bottom slope is critical. energy uxofabouttwice the magnitude of theone radiating southward. The strength of theforcingfor internaltides is sensitive to the shapeof thetopography and thestratiŽ cation nearthe generation region (Baines, 1982). The enhanced response to thenorth is dueto the asymmetricalshape of the topography (steeper on the northern side) and the increased stratiŽcation near the shallower critical point. (By comparison, in anumericalexperiment withan idealizedsymmetrical topography of comparable scale to theAleutian Ridge, the modelyielded similar energy  uxeson both sides of theridge.) Sincethe northern signal is less well deŽ ned in the altimetry data and the local three dimensionalityof the topography likely more important, we areless conŽ dent that the modelresponse is realistic to the north of the Ridge. If thiscomponent is nevertheless included,then the model yields an estimate of about 1.8 GW for thenet baroclinic conversionnear Amukta Pass. This value is 6%ofthebarotropic energy  ux(Section2) crossingthe ridge in this region, and it represents about 1% of the global baroclinic conversionestimated by Munk (1997). Thus while the region near Amukta Pass isan 182 Journalof MarineResearch [59, 2 importantsource of coherentinternal tides for theN. PaciŽc (e.g.,Fig. 2), its signiŽ cance in termsof globalenergy conversion is apparently quite limited. b.Comparison with observations and evidence for refraction Figure9 presentsa comparisonbetween the sea-surface signature of the internal tide obtainedfrom themodel (black curves) and from thealtimeter observations for three descendingtracks in theNorth PaciŽ c (T/Ptracks117, 79 and41; colored curves). The top andmiddle panels of the Ž gureshow the latitudinal variation of phase, fi,anddisplace- mentamplitude, hi,deŽned by Eqs. (A3) and(A4) oftheAppendix, respectively. In the lowerpanel, the sea-surface displacement, at a Žxedtime ( t 5 0), hi cos (fi),isplotted againstlatitude. Each of theT/ Ptrackstraverse the Aleutian Ridge internal tide obliquely overa rangeof latitudes. Accordingly, the data for eachtrack are plotted over the latitudinalrange in whichthe central region of theinternal tide beam is traversed by the altimeter,with some overlap allowed between adjacent tracks. The portions of the three tracksplotted in Figure9 areindicated in theplan view of Figure3. Theupper panel of Figure 9 indicatesthat the phase of the internal tide in the model comparesfavorably with the observedphase. Although there are some discrepancies, these arerelatively minor. In the far Želd,over nearly 10° of latitude ( ’1100km) from the sourceregion, the model phase continues to agree with the observations. Phase from the overlappingtracks of datatends to beinmutualagreement, except perhaps in the vicinity of48.5N and near 45N. At theserespective latitudes, tracks 117 and 79 lienear the outer edgeof the detectable beam of internal tidal energy and the small phase advance with respectto thecenter of thebeam may be dueto somegeometrical spreading of thesignal. Themiddle panel (Fig. 9b) indicates that the amplitude of theinternal tidal signal from the modelis comparable to that of the observations, especially in the far Želd.There is, however,little correspondence between model and observations with respect to  uctua- tionsin theamplitude, hi,whichare generally below the noiselevel of thedata.The largest discrepanciesin amplitude occur close to theridge. However, owing to the presence of the AlaskanStream, this is also the region of largest measurement uncertainty. Overall, the modelsea-surface displacement (Fig. 9c, lower panel) appears to be in generally good agreementwith observations well into the far Želd. Thephase signal shown in Figure 9a is signiŽ cantly affected by the variation in the Coriolisparameter. This is demonstrated straightforwardly by comparing the observed phasewith that obtained from anumericalrun in whichthe Coriolis parameter is givena constantvalue appropriate to 52N, the approximate latitude of the source region. The comparisonis shown in Figure10; for conveniencethe phase comparison from Figure9a is reproducedin the Ž gure.Close to the source, results are similar to the previous case. However,large systematic differences develop within 4° – 5°south of theridgebetween the observationsand the model with uniform rotation rate, which fails to reproduce the observedphase signal. The variable rotation rate appears essential for accuratesimulation ofpropagation into the far Želd. 2001] Cumminset al.:North PaciŽ c internaltides 183

Figure9. Comparison of thesea-surface signature of theinternal tide from the model (black lines) withT/ Pdataalong tracks 117 (red lines), 79 (blue lines) and 41 (greenlines). The plotted portions

ofthesetracks are identiŽ ed inFigure 3. The upper panel (a) shows the phase, fi ,(degUT) the

middlepanel (b) the amplitude, hi ,andthe lower panel (c), the sea surface displacement at aŽxed

time (t 5 0), hi cos (fi ). 184 Journalof MarineResearch [59, 2

Figure10. Comparison of theinternal tidal phase from the model(black lines) with observed phases alongT/ Ptracks(colored lines). The upper panel is from a numericalexperiment with uniform rotationrate ( f 5 2V sin52° ), whilethe lower panel is withvariable f (andidentical to that in Fig. 9a).

Thein uence of a variableCoriolis parameter on phasemay be understood in termsof a Žrst modeplane wave whose local wavelength is gradually modulated as it propagates from northto south. As phaseis thelatitudinal integral of wavenumber,we have

u0 kRd f~u! 5 f~u0! 2 E u9, (2) u 2001] Cumminset al.:North PaciŽ c internaltides 185 where u isa latitudecoordinate, u0,thereference latitude of thesource, k thewavenumber, and R,earth’s radius.The wavenumber for verticalmode n isobtained from thelocal dispersionrelation,

2 2 2 2 v 5 cnkn 1 f , (3) wherethe Coriolis parameter f 5 2V sin u, with V theangular velocity of the earth. The parameter, cn,isthe eigenvalue of the Sturm Liouville equation governing the vertical modes, Pn( z),for hydrostaticinternal waves (LeBlond and Mysak, 1978),

d 1 dPn Pn 2 1 2 5 0, (4) dz X N dz D cn withboundary conditions, dPn/dz 5 0 at z 5 0, 2H. Takingthe negative root in (3) for southwardpropagation ( kn , 0)andsubstituting into (2) we have

u0 1 2 g2 sin2 u9 f~u! 5 f~u ! 1 v Rdu9, (5) 0 c2 E Î n u where g 5 2V/v. If cn isassumed independent of latitude, (5) hasan exact integral (Gradshteynand Ryzhik, 2000; p. 198)given by

Rv f~u! 5 f~u0! 1 ~I~u0! 2 I~u!!, (6) cn where

~g2 2 1! I~u! 5 Î 1 2 g2 sin2 udu 5 gE~a, g21! 2 F~a, g21! @g2 . 1#, E g a 5 arcsin (g sin u) and

a dj F~a, p! 5 @p2 , 1#, E Î 1 2 p2 sin2 j 0

a E~a, p! 5 E Î 1 2 p2 sin2 jdj @p2 , 1# 0 areelliptic integrals of theŽ rst andsecond kind, respectively.

Setting f(u0)arbitrarilyto zeroand taking u0 5 52N, f(u)was determinedfrom (6) with thealgorithm given in Press et al. (1992)used to compute the elliptic integrals. The buoyancyfrequency, N( z),was basedon themean annual stratiŽ cation from theWOA98 21 climatologyat 51.5N, 171.5W. Eq. (4) thenyields c1 5 2.08 m s for thedominant gravestinternal mode. 186 Journalof MarineResearch [59, 2

Figure11. (a) Comparison of phase, f(u),givenby Eq. (6) (solid line) to a casewith constant

Coriolisparameter (dotted line). The eigenvalue, c1 ,isconstant for both cases. (b) The solid line

repeatsthe solution of (6)given in (a), while the dashed line is anumericalsolution of (5)with c1 nowdependent on latitude. The dash-dotted line in (a) and (b) gives the cumulative phase differencefrom the sourcebetween the respective cases.

Figure11a compares the phase from (6) witha casein which the phase varies linearly withthe Coriolis parameter held constant at thevalue appropriate to 52N.The two cases graduallydiverge away from thesource with the phase difference becoming appreciable overa few degreesof latitude, much as in Figure 10a. The cumulative phase difference over10° of latitude is 320°. Thus,the slow variation in theCoriolisparameter encountered 2001] Cumminset al.:North PaciŽ c internaltides 187 bytheinternal tide as itpropagates to thesouth may be expectedto leadto a progressive decreasein wavelength,signiŽ cantly affecting the phase of thesignal. This is essentiallya refractionphenomena and it largely accounts for theresults of Figure 10, where the cumulativephase difference at the southern boundary between the two numerical experi- mentsis about 300° . Inthe altimeter observations, small variations in wavelength are difŽcult to discern. However, the integrated effect on the phase appears to be readily detectable. As acheckon the in uence of horizontal variations in stratiŽ cation, f(u) was recalculatedfrom (5) withlatitudinal variations in c1 retained.Data from theWOA98 climatologywere usedto compute c1 along171.5W, from 51.5Nto 42.5Nat 1°intervals. Thesevalues were linearlyinterpolated onto a Žneresolution grid and (5) was integrated numerically.The results are given in Figure 11b and compared to the previous case with constant c1.For mostof the latitude range, f(u)isnearly indistinguishable between the twocases. Only toward the southern end of theregion do variationsin stratiŽcation begin tohavea noticeable,yet still comparatively small, in uence on thephase. For thelatitude rangeconsidered, Coriolis parameter variations appear to have considerably greater inuence on refraction of theinternal tide from AleutianRidge than lateral inhomogene- itiesin the stratiŽ cation. This provides some justiŽ cation for theinitial horizontally homogeneousstratiŽ cation of themodel. c.Vertical mixing Asidefrom theintrinsic interest that such a phenomenamay hold, internal tides are potentiallyimportant because they may make a signiŽcant contribution to vertical mixing processesin theocean.The numericalmodel suggests that baroclinic energy conversion on thesouthern  ankof theridge occurs at a rateof about3000 W m 21.Followingarguments givenby Munkand Wunsch (1998, § 3.3),an internal tide carrying an energy  uxofthis magnitudewith e-foldingdissipation scale of O(1000km)will lead to a verticaldiffusiv- itywithin the interior of O(102 5 m2 s21).Thisvalue corresponds to the ‘ pelagic’ diffusivityof Munk and Wunsch (1998) and is consistent with observations taken over muchof the ocean interior (e.g., Kunze and Sanford, 1996). However, it is usually consideredinsufŽ cient to maintain the abyssal stratiŽ cation of the ocean. Exceptfor thebottom logarithmic layer, the vertical diffusivity over almost the entire modeldomain is dominatedby theweak background constant (2 3 1025 m2 s21) that is addedto the mixing coefŽ cient determined from thelevel— 2.5 turbulence closure scheme ofMellorand Yamada (1982). However, the internal tide produces strong shears near the generationregion and the turbulence closure responds with large mixing coefŽ cients in localizedareas. Figure 12 shows the mean vertical diffusivity produced by the closure schemenear the crest of the ridge. Large mixing coefŽ cients ( .102 2 m2 s21)occurwithin theinterior in localized patches on either side of the ridge crest. The turbulent kinetic energyŽ eld(not shown) has a similarpattern. Superposed on the Ž guresare the internal tidecharacteristics emanating from thetwo points nearest the crest where the bottom slope 188 Journalof MarineResearch [59, 2

Figure12. Time-averaged vertical diffusivity near the crest of theridge from the basic experiment. TheŽ lledcontour delineates regions where the diffusivity is greaterthan 10 22 m2 s21 . The broken linesindicate ray paths emanating from critical points on eitherside of theridge. iscritical(cf. Fig.7). Although the regions of largeturbulent kinetic energy and mixing are ratherirregularly distributed, they appear to beassociatedwith the internal tide characteris- tics.Strong shears between regions of intenseinternal tidal motions along these ray paths andadjacent quiescent regions apparently produce the patches of turbulentmixing in the model. Thereare, of course, no measurements to verify this aspect of the model response. Nevertheless,these results bear a qualitativesimilarity to therecent dissipation measure- mentsof Gregg et al. (1999)taken over a submarinefan on thenorthside of MontereyBay. Theyshowed intense turbulence and mixing in a 50mwidebeam extending 5– 6 km offshoreand lying along an internal tide characteristic. Figure 12 suggests that the Mellor-Yamadaparameterization is able to capture aspects of intense turbulent mixing associatedwith the tidal rays, and that this may be a generalfeature of the near-Ž eld response.

5. Summary Satellitealtimeter measurements have demonstrated the existence of coherentinternal tidesin theocean whose phase propagation may be tracedover large distances. The global 2001] Cumminset al.:North PaciŽ c internaltides 189 surveyof Kanthaand Tierney (1997) shows that these signals occur throughout the world oceanand originate principally from largesubmerged ridge topography. This paper has consideredthe North PaciŽ c internaltide signal originating from theAleutian Ridge. In thiscase, over mostof thelength the ridge,the large topography and associated complex of islandsblock the cross-slope barotropic motions required to generate internal tides. However,as demonstrated in simulations with a barotropictidal model, strong cross-slope motionsdo occur in and around Amukta Pass, a relativelydeep gap in the ridge. This permitsthe generation of adirected,coherent internal tide that propagates into the central NorthPaciŽ c. There is also a similarbut less clearly deŽ ned signal which propagates to the northinto the southern Bering Sea. Ahighresolution, two-dimensional version of the Princeton Ocean Model was applied toexamine generation and propagation of the M2 internaltide emanating from Amukta Pass.Comparison with the T/ Pobservationsshows that the model is able to capture the surfacesignature of the internal tide as it propagates over 1100 km southward into the centralNorth PaciŽ c. The phase of thissignal is the most robust aspect of theobservations andthe comparison shows that it is sensitive to the latitudinal variation of the Coriolis parameter.This sensitivity is consistent with an expectedphase modulation due to gradual reductionin the wavelength of aŽrst modeinternal plane wave propagating to the south. Over almostthe entire model domain, the vertical mixing coefŽ cient is dominatedby the weakbackground constant that supplements the parameterization of turbulence in the model.There are, however, isolated interior regions close to the ridge crest where strong shearsobtain, and the turbulence closure scheme yields large mixing coefŽ cients. These regionsoccur near the internal tide characteristics emanating from locationswhere the bottomslope is critical. This result appears to be qualitatively consistent with recent observationsby Gregg et al. (1999)of intenseturbulent dissipation along an internal tide ray. Thenumerical model yields a netbaroclinic energy conversion rate near Amukta Pass of about9000 W m 21,withone third of this total radiating southward into the central North PaciŽc. Assuming that this is theaverage rate of baroclinic energy production along the 200km regionof cross-slopebarotropic  owimplies an integratedbaroclinic  uxofabout 1.8GW. Inglobal terms this represents perhaps 1% of the total baroclinic energy conversionfor the M2 tide. Animproved estimate of baroclinicconversion at theAleutian Ridge will likely require applicationof a three-dimensionalbaroclinic model. Such a modelwould permit an assessmentof thein uence of smaller scale transverse topographic irregularities along the ridgein scattering additional barotropic energy into baroclinic motions. Further problems thatcould be addressed with a three-dimensionalmodel include an assessment of the inuence of the Alaskan Stream, information on propagation and dissipation of internal tidesin theBering Sea, and a determinationof theextent of geometricalspreading of the southwardradiating signal. 190 Journalof MarineResearch [59, 2

APPENDIX Separatingthe surfacesignature of the internaltide

We assumethat tidal oscillations of seasurface, ho,arethe sum of contributionsfrom surface, hs,andinternal, hi tides,

ho 5 hs 1 hi. (A1)

Expressedin harmonicform we have

hi 5 hi cos ~vt 2 fi! 5 ai cos ~vt! 1 bi sin ~vt!, (A2) where v is the M2 frequencyand ( ai, bi) 5 (hi cos(fi), hi sin (fi)).Similarforms are assumed for ho and hs.Theinternal tidal phase, fi,andamplitude, hi,arethen given by

21 fi 5 tan ~B/A!, (A3)

hi 5 A/cos ~fi!, (A4) where (A, B) 5 (ao 2 as, bo 2 bs).Thebarotropic tidal component, ( as, bs),isdetermined from aleast-squarespolynomial Ž tto( ao, bo).Theorder of thispolynomial was chosenso astoprovide a smoothinglength scale of about300 km. The results shown are based on an 8thorder polynomial Ž ttotheT/ Pdata.The model sea surface was treatedsimilarly, and it was veriŽed that results are essentially unchanged using a 6thorder Ž t.

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Received: 5October,2000; revised: 20February,2001.