Near-inertial internal wave interactions with mesoscale. Craig M. Lee. fronts: Observations and models

Transcription

1 JOURNAL OF GEOPHYSCAL RESEARCH, VOL. 102, NO. C2, PAGES , FEBRUARY 15, 1997 Near-nertal nternal wave nteractons wth mesoscale fronts: Observatons and models Crag M. Lee Department of Physcal Oceanography, Woods Hole Oceanographc nsttuton, Woods Hole, Massachusetts Charles C. Erksen School of Oceanography, Unversty of Washngton, Seattle Abstract. Moored observatons of near-nertal currents and buoyancy fluctuatons from the Frontal Ar Sea nteracton Experment (FASNEX) may be understood as nearnertal nternal waves reflectng off sheared background currents. The FASNEX moored array measured upper ocean currents, temperature, and conductvty between January and June 1986 n a regon of the Sargasso Sea heavly populated by strong upper ocean fronts. The background jets assocated wth these features have typcal cross-front scales of 20 km, maxmum currents of 0.5 rn s -, and relatve vortctes of +_0.3f. n the vcnty of fronts, observed near-nertal currents have short horzontal scales, show no clear horzontal phase structure, and trace out elongated current ellpses wth elevated acrossfront over along-front varance. Durng these perods, nertal band-pass buoyancy fluctuatons often correlate wth major-axs currents. WKB ray-tracng models of propagatng, monochromatc plane waves exhbt horzontal phase propagaton and correlatons between mnor-axs currents and buoyancy and thus cannot explan these results. Models of near-nertal nternal waves reflected or trapped by a barotropc frontal jet produce mode-lke structures wth short horzontal scales, lttle horzontal phase propagaton, and ansotropc current ellpses algned prmarly cross front. Model buoyancy may correlate wth one or both current ellpse components, dependng on the ncdence angle of the wave and the degree to whch t reflects. Completely reflected waves wth horzontal scales smlar to those of the front and near-normal ncdence angle produce current ellpses and correlaton patterns consstent wth the observatons. 1. ntroducton Mesoscale shear can strongly modfy near-nertal nternal wave propagaton and thus the transfer of nertal energy nto the ocean nteror. Numerous studes examne the generaton of surface-forced near-nertal motons and descrbe the sub- sequent decay of near-nertal energy by the propagaton of near-nertal nternal waves out of the mxed layer [Pollard, 1970; Prce, 1983; Kundu and Thomson, 1985; Erksen, 1988a; Rubensten, 1994; Zervaks and Levne, 1995]. n the absence of background shear, horzontal wavenumber largely governs near-nertal wave propagaton and energy transport. Characterstcs of the appled wnd stress feld, partcularly ts propagaton speed and horzontal scale, determne the wavelengths of the nertal motons generated [Kundu and Thomson, 1985; DMsaro, 1987; Erksen, 1988a; Rubensten, 1994]. Typcal atmospherc systems force near-nertal waves wth wavelengths of the order of km, mplyng very slow vertcal propagaton. Several factors may modfy how near-nertal waves propagate away from the surface, ncludng the ntroducton of smaller scales by small-scale features n the appled stress feld [DMsaro, 1985], scale reducton over tme due to /3 effect [DMsaro, 1989, 1995; DMsaro et al., 1995], and nteracton wth the mesoscale flow feld [Kunze and Sanford, 1984]. Ths study examnes the effects of horzontally sheared background cur- Copyrght 1997 by the Amercan Geophyscal Unon. Paper number 96JC /97/96J C rents on near-nertal nternal wave propagaton and presents observatonal evdence of near-nertal wave reflecton by shear assocated wth upper ocean fronts. Near-nertal waves propagatng n a background flow experence both Doppler shftng and the effects of subnertal shearng [Kunze, 1985]. For a monochromatc plane wave of wavenumber vector (hereafter referred to as wave vector) K - (k, l, m) (eastward, northward, upward) propagatng n a background current 15 = (U, V, W), the Doppler shft may be expressed as = u), (1) where the Euleran frequency o-s nvarant n a steady flow and the ntrnsc (Lagrangan) frequency to vares as the wave propagates through the background current. Dynamcally, the ntrnsc frequency governs the wave's behavor and (1) descrbes advecton of the wave by the background flow. Near-nertal nternal waves respond to varatons n background relatve vortcty, dvergence, and buoyancy [Jones, 1967; Mooers, 1975a, b; Olbers, 1981]. Weller [1982] reformulates the model presented by Pollard [1970] to nclude horzontally sheared background flows. He fnds that the relatve vortcty of the background flow shfts the frequency of the nertal oscllatons, whle dvergent flows cause exponental decay and convergent flows cause exponental growth. Kunze [1985] derves an approxmate dsperson relaton for nearnertal waves propagatng n a shear flow,

2 3238 LEE AND ERKSEN: NEAR-NERTAL NTERNAL WAVE NTERACTONS 0) Lff-']- meffkh N2K 2fm 2 m (OB - x-x k + - y-y OB) l =Lff--]- 22 where the effectve Corols frequency 2fm2 (2) Lff= f +5 Ox 3 (3) wave packet refractng as t propagates through a front. They measure the wave's ntrnsc frequency and demonstrate that t s strongly nfluenced by Doppler shftng. The wave turns nto the across-front drecton, whle ts vertcal wavenumber and ntrnsc frequency shrnk, consstent wth ray-tracng predctons. Weller et al. [1991] fnd enhancement of across- over along-front near-nertal current varance, whle Med et al. and the effectve buoyancy frequency [Mooers, 1975a] [1990] report a preference for nertal waves to propagate perpendcular to the background current n a cold-core rng. N2 ff=n2-2- x u-2 -. Med et al. [1990] argue that waves propagatng aganst the (4) mean flow are Doppler shfted to the local buoyancy frequency and absorbed n horzontal crtcal layers, whle those propa- Here gu = V'k s the horzontal wavenumber, f s the gatng wth the flow undergo refracton and are ether too weak Corols frequency, N s the buoyancy frequency, and B s the or have wavelengths too long to be resolved by the observabackground buoyancy. Buoyancy relates to densty p as b = tons. These mechansms act as a flter admttng only those -#9/9o, where g s gravty and 9o s the mean densty. A wave waves propagatng cross stream. feels the rotaton of the flud as well as planetary rotaton (3). Erksen [1988b] reports currents wth rr --- f dsplayng hghly Varatons n relatve vortcty alter the lower bound of the ansotrop current ellpses. For a monochromatc plane wave, nternal wave band, forcng the wave vector to evolve and modulatng wave ampltude [Mooers, 1975a, b; Kunze, 1985]. Dependng on the orentaton of the wave, horzontal densty gradents (vertcal shears of geostrophc velocty) can shft the = - -u' (5) upper bound of the nternal wave band (4) by modfyng the densty contrast seen by a partcle traversng an nclned nternal wave current ellpse. Wave turnng and crtcal layers [Jones, 1967; Booker and Bretherton, 1967; Mooers, 1975a, b; Kunze, 1985] arse from Doppler shftng and nteracton wth background shear. As a near-nertal wave propagates horzontally through a nonunform background flow, ts wave vector changes accordng to Dk/Dt = -Vrr to satsfy dsperson relaton (2), where Dk/Dt s the change n wavenumber followng a ray. Only the wavenumber components n the drecton of background nhomowhere u' s the velocty component parallel to the drecton of the wave vector and v' s the velocty component perpendcular to the drecton of the wave vector. Doppler shftng can ncrease a wave's ntrnsc frequency, makng t dynamcally more supernertal and elongatng the current ellpse. Nearnertal waves propagatng through regons of negatve relatve vortcty experence a smlar effect due to a reducton n effectve Corols frequency. All the studes mentoned above nterpret the nternal wave feld as dstnct, monochromatc wave packets spectrally separated n tme and space. Kunze et al. [1995] suggesthat supergenety evolve. Ths produces wave refracton and, as fef postons of waves formng standng modes mght offer a better due to Doppler shftng or fef due to lateral changes n the descrpton n the vcnty of geostrophc currents. They suceffectve Corols frequency, total or partal reflecton. Kunze cessfully descrbe near-nertal wave crtcal layer nteracton n and Sanford [1984] and Kunze [1985, 1986] consder nertal a warm-core rng and n a vortex cap atop a seamount [Kunze wave trappng n wells of negatve relatve vortcty assocated et al., 1995; E. Kunze and J. M. Toole, Tdally forced vortcty, wth warm-core rngs and the warm sdes of upper ocean durnal shear and turbulence atop Feberlng Seamount, subfronts. Waves generated n regons where f f < f encounter mtted to Journal of Physcal Oceanography, 1996] usng a turnng ponts near the edges of the negatve vortcty well and model combnng a modal descrpton n the horzontal wth cannot propagate outsde t. Ths may lead to enhanced nearray tracng to descrbe vertcal propagaton. nertal energy levels n the trappng regon and, n baroclnc Our goal s to understand how near-nertal waves nteract background flows, enhanced dsspaton where the trapped wth the strong jets and densty gradents assocated wth mewaves encounter vertcal crtcal layers [Kunze et al., 1995]. soscale upper ocean fronts. Secton 2 detals observatons from Near-nertal waves encounter vertcal crtcal layers when the Frontal Ar-Sea nteracton Experment (FASNEX). Sec f f by Doppler shftng or f f --> 60 due to vertcal ton 3 descrbes the strong upper ocean fronts whch domnate varatons n relatve vortcty. As a wave approaches ts crtcal subnertal current varance, followed by observatons of neardepth, (2) ndcates that ts vertcal wavenumber grows and ts vertcal group velocty, am/am, shrnks. Conservaton of acton nertal waves nteractng wth the background flow. Nearnertal varance s ntermttent and shows some enhancement flux requres the wave's energy densty to ncrease as t slows. Eventually, amplfcaton and sharpenng leads to nstablty, n regons of negatve relatve vortcty. The observatons are and the wave may lose energy through transfer to the mean not well descrbed as packets of horzontally propagatng, flow, wave-wave nteractons, or turbulent dsspaton [Booker monochromatc plane waves but often show ansotropy and and Bretherton, 1967; Gregg et al., 1986; Jones, 1967; Kunze et component correlatons more consstent wth standng modes al., 1990a, b, 1995; Marmorno et al., 1987]. Observatons provde evdence of near-nertal wave nteracton wth background shear flow. The negatve vortcty reformed by superpostons of waves. To understand the observed patterns of current and buoyancy varance, we develop a smple dynamcal model for near-nertal waves refractng gons assocated wth fronts and rngs often dsplay enhanced and reflectng off dealzed fronts. Secton 4 presents these near-nertal knetc energy [Kunze and Sanford, 1984; Weller, 1985; Kunze, 1986; Med et al., 1986], consstent wth predcresults and uses them to explan ansotropy and small horzontal scales and to dentfy near-nertal wave reflecton n the tons of nertal wave trappng by horzontal turnng ponts and vertcal crtcal layers. Med et al. [1987] observe an nertal observatons. secton 5. We dscuss the results and offer conclusons n

3 LEE AND ERKSEN: NEAR-NERTAL NTERNAL WAVE NTERACTONS Data 38,5 km We use moored observatons made n the North Atlantc Subtropcal Convergence Zone, near 27øN, 70øW, from wnter to late sprng of 1986 [Weller et al., 1990a, h. The FASNEX array (Fgure 1) was desgned to resolve cross-frontemperature scales, wth typcal horzontal separatons of 20 kn. Three proflng current meters (PCMs) provde observatons of velocty, temperature, and conductvty at 5-m ntervals between 40 and 180 m. The PCMs collected profles at 4-hour ntervals for the duraton of the experment. Surface moorngs F2, F4, F6, FS, and F10 sampled velocty and temperature at depths of 10, 20, 30, 40, 80, 120, and 160 m usng a combnaton of vector averagng (VACM) and vector measurng (VMCM) current meters [Weller et al., 1990a]. Hourly averaged surface moorng data are further averaged nto 4-hour bns correspondng to PCM profle tmes when performng calculatons nvolvng the entre array. We correct for dfferences n the response characterstcs of the varous velocty sensors followng Lee and Erksen [ 1996]. 3. Observatons 12.2 km l PCM moorng Surface moorng We solate near-nertal nternal waves nteractng wth strongly sheared frontal jets and quantfy ther current var- Fgure 1. Plan vew of the FASNEX moored array [after ance, ansotropy, and velocty-buoyancy correlatons for com- Weller et al., 1991]. Squares mark proflng current meters parson wth a smple analytcal model of near-nertal wave (PCM) (F3, F5, and F7) and crcles mark surface moorngs reflecton n secton 4. Due to the epsodc nature of nertal (F2, F4, F6, FS, and F10). Separatons between moorngs are wave generaton [Pollard and Mllard, 1970; DMsaro, 1984] and the ntermttent passage of mesoscale fronts, examples of nearnertal waves nteractng wth the background flow occur as noted n klometers. solated events assocated wth some, but not all, of the ob- Large changes n relatve vortcty occur as fronts pass served fronts. We begn wth an overvew of the low-frequency through the moored array. As motons wth nadequately reflow feld, focusng on two specfc frontal events to llustrate the background envronment n whch the waves propagate. A band-pass flter solates current varance assocated wth the near-nertal spectral peak. We show that these currents exhbt ansotropy, wth enhanced varance assocated wth the crossfront drecton. Correlaton coe cents between current ellpse components and buoyancy fluctuatons often exhbt behavor consstent wth lateral nternal wave reflecton durng the passage of fronts. solved spatal scales wll be alased nto horzontal velocty gradent estmates, we flter the velocty feld as descrbed by Lee and Erksen [1996]. An objectve map of the resultng felds yelds an estmate of relatve vortcty. Near a front, resolved relatve vortctes can reach _ 0.3f, wth negatve relatve vortcty assocated wth the warm sde of the front and postve relatve vortcty assocated wth the cold sde of the front. Rudnck and Weller [1993] fnd the low-frequency heat balance to be between rate of change of heat and horzontal advecton The Background Flow Feld Thus, durng a frontal event, temporal varablty at a sngle moorng may be prmarly attrbuted to horzontal advecton of Mesoscale upper ocean fronts domnate the observed lowfrequency current varance, creatng a hghly sheared background envronment. A 48-hour low-pass flter separates lowfrequency varablty assocated wth fronts and other mesoscale features from nternal wave varance. Large vertcal excursons 0(50 m) n sopycnal depths and currents of up to 0.5 m s- (Fgures 2 and 3) mark dstnct frontal passages n the front past the array (Fgure 3). Temporal evoluton of the fronts and other processes whch contrbute current varance wll complcate ths pcture, but the conceptual model remans useful. Postve vortcty rdges and negatve vortcty troughs, such as those observed n early March, can cause nertal wave refracton, whle surface-ntensfed regons of negatve relatve vortcty may result n crtcal layer trappng. early February, March, Aprl, and May Horzontal temperature gradents strengthen wth depth and have typcal val Near-nertal Motons ues of øc per 20 km. On average, FASNEX fronts were algned NE-SW [Erksen et al., 1991], though ndvdual events can show consderable curvature and arbtrary algnment. n February a northwestward frontal jet translates through the array, turnng westward over tme (Fgure 3). Warm water and negatve relatve vortcty are consstently on the north sde of the front. The March record contans two events, wth the frontal jet reversng from northeast to southwest around March 11, 1986 (year day 70) (Fgure 3). Thus the warm sde of the front s to the south for the frst half of the perod and to the north for the remander. dentfyng nertal varablty requres care, as we wsh to solate motons wth ntrnsc frequences o - f usng Euleran observatons. Strong background currents may result n large Doppler shfts (1), smearng the Euleran nertal peak. Varance-preservng rotary current spectra (Fgure 4) show energetc peaks at the clockwse nertal and durnal frequences, wth a smaller peak n the clockwse semdurnal band and elevated low-frequency energy at 40 m due to the fronts. The nertal peak rolls off sharply, returnng to levels ndstngushable from the background wthn _+20% of f. Both Doppler smearng and energy leakage from the durnal band contrbute 12.4 km 11.8 km

4 LEE AND ERKSEN: NEAR-NERTAL NTERNAL WAVE NTERACTONS F3 c e (48 h low-pass), ,50 Yeor Doy: /,5...,, :: ,,,, 1 t 1,5 JAN 30 JAN ]a FEB ] MAR 16 MAR 31 MAR APR 30 APR 15 MAY 30 MAY 14 JUN 1986 Fgure 2. Gray scale plot of F3 48-hour low pass cr o (klograms per cubc meter). Profles are plotted as a functon of depth (vertcal) and tme (horzontal) and shaded accordng to the bar dsplayed across the top of the fgure. Closely spaced changes n shadng ndcate strong stratfcaton. Rapdly rsng and dvng sopycnals n early February, March, Aprl, and May mark the passage of upper ocean fronts. to the wdth of the nertal peak. We band-pass flter the observatons (1 _+ 0.05) f, excludng the tals of the nertal peak but retanng the majorty of the varance. The narrow passband excludes durnal varablty, whch would obscure the nertal motons of nterest. nertal motons generated n regons of weakly sheared flow, away from the nfluence of fronts, wll have Euleran frequences cr wthn a few percent of f. As cr remans constant for an nternal wave propagatng through a steady background flow, the band-pass flter effectvely solates near-nertal waves propagatng nto the fronts from the outsde. nertal motons generated wthn a front may have Euleran frequences close to fee, whch wll fall outsde the flter's passband when relatve vortcty s large. Near-nertal currents vary on scales comparable to the moorng separatons, wth large-ampltude changes and no clear sense of horzontal phase propagaton [Weller et al., 1991]. Attempts to ft sets of supermposed plane waves to the observatons faled to explan a sgnfcant fracton of the varance, perhaps due to the falure of any sngle wavenumber to domnate the varablty or to the presence of unresolved, small-scale near-nertal motons. n the followng sectons we quantfy near-nertal velocty and buoyancy varablty and demonstrate that standng modes produced by near-nertal nternal waves propagatng n sheared background flows provde a consstent descrpton of the observatons Near-nertal Waves n a Front Durng the passage of upper ocean fronts, observed nearnertal currents exhbt sgnfcant ansotropy wth enhanced across-front varance. Ray-tracng results [Kunze, 1985] suggest that fronts wll focus an ntally sotropc nertal wave feld nto a narrow, ansotropc beam, whle prevous observatons of nertal waves n sheared background currents report a preference for across- over along-front propagaton. We test for frontal-related ansotropy by rotatng nertal band-pass currents nto coordnates defned by the vertcally averaged, objectvely mapped low-pass temperature gradent. A 4-day runnng wndow estmates tme seres of RMS along- and acrossgradent veloctes (Fgure 5). Durng the March and Aprl frontal events, nertal currents at 40 m show statstcally sgnfcant ansotropy, wth greater across- than along-front varance. Shallow near-nertal currents show lttle temperature gradent-related ansotropy durng other perods. nertal currents at 160 m, n the seasonal pycnoclne, reman sotropc throughout the record. n the vcnty of strong upper ocean fronts the observed near-nertal veloctes and buoyancy perturbatons often ex- hbt behavor nconsstent wth that expected from monochromatc nternal waves. From (5) we expect waves wth ntrnsc frequency to --> f to trace out nearly crcular current ellpses.

5 LEE AND ERKSEN: NEAR-NERTAL NTERNAL WAVE NTERACTONS 3241 Year Day 1986 o a o? o o o o Jlbl LLL 1, lb,j V4J t l la.kj &J ll llll tll llll Ubp,V m Hl, " t--... ljl l d , q Ulp 0 U p Ulp..._ q U p r-j'f -w., -'";x V o.o -0.3 ll El f llllll lllll lllllllllll,, 11111,,,1 J 15 J 30 FEB14 M M 16 R31 R15 R30 MAY15 MAY30 JUN14 = U p 0.25 FASNEX: F3 U, Ub, VxU s No hw d = U p 0.10 s Noahward Fgure 3. Low-pass (Ulp) and nertal band-pass (Ubp) currents and relatve vortcty /f for F3. Tme seres of velocty stck vectors and emprcal orthogonal functon (EOF)-fltered, objectvely mapped relatve vortcty are plotted at depths of 40, 80, 120, and 160 m. Note the dfferent velocty scales for low-pass and band-pass current vectors. Durng frontal passages, low-pass currents are strong and typcally algn NE-SW. Relatve vortctes reach values of up to +_ 0.3f wth negatve values on the warm sde of the front and postve values on the cold sde of the front. Perods when < 0 are often accompaned by enhanced near-nertal current varance. Smlarly, lnear monochromatc nternal waves satsfy the consstency relatonshp between near-nertal buoyancy perturbatons and currents. Contrary to predctons that buoyancy should vary n phase wth transverse currents v' for monochromatc plane waves N 2 k N 2 k b = u' = v' (6) (6), buoyancy fluctuatons are n phase wth currents along the (.OFF/ f /T/ ' major axs of the ellpses. Near-nertal currents above 80 m where buoyancy perturbatons are n quadrature wth currents durng the early March front (Plate 2) provde a partcularly u' parallel to the drecton of the wave vector. The currents clear example of ths phase relatonshp. Followng current assocated wth a sngle, propagatng plane wave trace out an vectors at a constant depth for several days, the polarty of the ellpse elongated n the drecton of the wave vector wth max- buoyancy fluctuatons reverses. Durng March the shft from red to blue n the 40-m northward current vectors llustrates mum buoyancy fluctuatons orented wth the mnor axs (Plate 1). We estmate near-nertal buoyancy fluctuatons by ths behavor. Band-pass flter wdth restrcts the modulaton band-pass flterng b = -gp'/po at ( )f, where tmescale of the observatons to perods longer than 10 days. perturbaton densty p' s calculated by removng the 48-hour Tme seres of zero-lag correlaton coeffcents between prnlow-passtratfcaton from the densty feld. Two examples cpal axs current ellpse components and buoyancy perturbacorrespondng to the February and March frontal events llus- tons also produce evdence of nertal wave nteracton wth trate typcal behavor durng perods of strong background flow the background flow. Usng a 4-day runnng wndow, we cal- (Plate 2). Velocty vectors colored by perturbaton buoyancy culate prncpal axes and eccentrctes, e = X/u'2 _ v'2/u' trace out near-nertal current ellpses. At depths shallower for the nertal band-pass currents, where u' and v' are taken than 120 m the observed currents dsplay nearly lnear polar- as the RMS velocty components along the semmajor and zaton, consstent wth waves Doppler shfted to supernertal semmnor axes. Standard errors n the varance estmates ntrnsc frequences. Deeper, where the background flow (Fg- [Bendat and Persol, 1986] ndcate that only ellpses wth e > ure 3) and Doppler shfts weaken, currents are essentally 0.8 are dstngushable from crcularly polarzed motons. We crcularly polarzed. More strkng s the phase relatonshp can estmate only the algnment of the semmajor axs and

6 _ -- _ LEE AND ERKSEN: NEAR-NERTAL NTERNAL WAVE NTERACTONS (a) F3 U 40m _,,m Clockwse : : Antclockwse '"e...,. ß ph cph 10-1 lo 0 (b) F3 U 160m _...,..., /1 cph... Clockwse Antclockwse l ph o.oo 0.000,,, -,,....,-... _:_¾ cph Fgure 4. Varance-preservng rotary current spectra at (a) 40 and (b) 160 m for F3. Sold lnes ndcate antclockwse motons, and dashed lnes ndcate clockwse motons. Promnent peaks mark the clockwse nertal, durnal, and semdurnal frequences. Passng fronts enhance low-frequency varance at shallow depths where frontal currents are strongest. Gray bars mark 95% confdence ntervals on selected estmates. choose the drecton ( ) along ths axs that s closesto the temperature gradent drecton. We use the runnng wndow to calculate zero-lag correlaton coeffcents between (u', v' ) and b whenever there s a dstngushablellpse. Results from F3 (Fgure 6) typfy those at the other moorngs and may be compared wth the tme seres n Plate 2. where the sgn of r<u,t,> changes wth tme. The two events consdered n Plate 2 provde strkng examples of ths pattern. n early February the 40- and 120-m records (Fgure 6) show hgh r<u,t,> whch changesgn over a few days, whle r<v,t,> s not sgnfcantly dfferent from zero. At 80 and 160 m the ellpse s not eccentrc enough to relably estmate a prncpal Correlaton coeffcents r<u,t,> and r< v,t,> (Fgure 6) exhbt axs durng ths perod. Durng the frst half of March a smlar patterns of temporal varaton resemblng the cross-front pat- pattern s apparent both n the correlaton coeffcents and n terns assocated wth near-nertal wave reflecton n secton 4 the color stck vectors (Plate 2) shallower than 120 m. Prncpal (Fgure 7 and Plate 5). Monochromatc near-nertal waves axs orentaton tends to concde wth the drecton of the obey (6) and thus have large r<v,t, > and small r<u,t,>, as seen at 120 m and shallower durng the latter half of March and early Aprl and 120 m and deeper n md May. The lnearly polarzed vertcally averaged temperature gradent durng these two perods. n secton 4 we show that mode-lke structures resultng from near-nertal waves reflectng off a front produce corremotons emphaszed n ths analyss frequently show strong laton patterns consstent wth the observatons. Correlaton correlatons between major-axs currents and buoyancy, r<u,t,>, patterns nvolvng nonzero along- and across-wave vector co-

7 ,,,= LEE AND ERKSEN: NEAR-NERTAL NTERNAL WAVE NTERACTONS 3243 Year Day / :.. \ ',,:,.. \..' JAN 15 JAN 30 FEB 14 MAR1 MAR 16 MAR 31 APR 15 APR 30 MAY 15 MAY 30 JUN 14 F3 < u w >, <Uñw> Fgure 5. F3 across-front (llv T, thck sold lnes) and along-front (_L V T, thn dotted lnes) nertal bandpass RMS veloctes. Lght gray shadng marks standard random errors for each curve. Gray bars runnng along the tme axs ndcate frontal events, wth thck bars markng the early February and early March fronts detaled n Plate 2. Durng the March and Aprl frontal events, 40-m across-front exceeds along-front current varance, suggestng ansotropc nertal currents wth a preferred across-front orentaton. n the absence of strong fronts and n the seasonal pycnoclne (160 m), nertal currents are sotropc and show no tendency to algn wth the background flow. etfcents may be attrbuted to modes resultng from partal reflecton or random superposton of propagatng wave packets. K H 4. Near-nertal nternal Wave Reflecton by a Barotropc Front A smple model of near-nertal nternal waves propagatng nto a barotropc frontal jet suggests explanatons for three features of the observatons: (1) ansotropy of the nertal wave feld, (2) small horzontal wave scales, and (3) correlaton patterns between current ellpse components and buoyancy per- 0.1 rn s- turbatons. Prevous modelng studes employ WKB ray tracng to examne nertal wave propagaton n more realstc, baroclnc shear flows. Ray tracng effectvely llustrates nertal wave turnng and reflecton by followng ndvdual wave packets but allows only complete reflecton or transmsson at potental barrers such as rdges of postve relatve vortcty. As the observatons lack dscernble horzontal phase propagaton and have component correlatons nconsstent wth those expected for monochromatc nternal waves, ray-tracng solutons cannot explan them. The superposton of ncdent and re buo ancy (ms- ) flected waves can produce mode-lke structures whch ether stand or propagate n the horzontal, dependng on how com- (o- 1.5f, N = 5x10-3 S '1, h = 40 km pletely ncdent waves reflect. We examne the nature of these modes by solvng for the across-front velocty and buoyancy Plate 1. A plan vew current ellpse for a monochromatc nternal wave, colored by perturbaton buoyancy accordng to structure produced by near-nertal waves propagatng n a the scale at the bottom of the plate. The drecton of the barotropc jet. horzontal wave vector s plotted above the ellpse. Mnmum We model the background flow assocated wth the front as buoyancy fluctuatons algn wth the mnor axs of the ellpse a barotropc, merdonal jet l/(x). Neglectng wave-wave n- and are marked by red shadng, and maxmum buoyancy flucteractons, lnearzed equatons of moton for nertal waves propagatng n the mean flow are tuatons algn wth the mnor axs of the ellpse and are marked wth red shadng.

8 3244 LEE AND ERKSEN: NEAR-NERTAL NTERNAL WAVE NTERACTONS February: F3 Ubv, bbp buoyancy Year Day a a = 0.1 rn ' northward J 80 ' March: F3 Ubp, bbp FEB 6 FEB buoyancy (ms' ).o.ooo3.o.ooo2.o.oom o.ooo o.ooo o.ooo3 Year Day = 0.1 m-* northward!:.11.,.,11..tl t F m * * Plate 2. F3 nertal band-pass current vectors colored by band-pass buoyancy at 40, 80, 120, and 160 m durng the (top) February and (bottom) March frontal events. Shallow current ellpses durng both events exhbt nearly lnear polarzaton, whle deep currents are more crcularly polarzed. Buoyancy fluctuatons are n quadrature wth mnor-axs currents, contrary to theoretcal expectatons for lnear, monochromatc progressve plane waves. Buoyancy fluctuatons reverse polarty n md-february (all depths) and md-march (40 and 80 m) [ 2 - w2m 2 dv]du - xx + [m2(w2 N 2 jre2 ) _/ 2 Ou Ou Op d2u 2wm V - fv = Ox (7) dx 2 N2 Ov ov Op + V + u - +fu = ay (8) Ob ap 0= - +b (9) Ou Ov Ow O = 0 (10) Ob --+ V + N2w = 0. (11) at Here u = (u, v, w) are wave veloctes and pressure p --> P/Po s normalzed by the mean densty As the background flow vares only across the front, we assume plane wave solutons exp [(ly + mz - trt)] and derve the cross-front structure equaton where N2l 2 w2m 2 dx f + -ff + -- dx2 ] tt = O, (12) [ 2 N2l _ w2m 2 art Ox wm -/02m 2 ( OV) ] v = N2l -- + N212 and 2 f + - u (13) b = N212 _ w2m2 w - + l f + u. (14) Equaton (12) dffers from equaton (9.1) of Kunze [1985], whch ncludes an erroneous term n the coeffcent of the frst dervatve. To the extent that temporal varablty can be nterpreted as horzontal advecton of a structure past fxed lo-

9 LEE AND ERKSEN: NEAR-NERTAL NTERNAL WAVE NTERACTONS 3245 Year Day 1õ 30 4õ 60 7õ 90 10õ õ 1õ0 16õ W llll,11tlll, ' l l llllllllll lllll! lllllllllllll 11 l ' &lllllllllll' tllllllllll 0.0 ' 270 ',-/Z, o ' ll 1 ] t l l[,lll ll *ll lll ltl ll q tl ll, l[ f*l llll J t,, q [llll,, &l tl t ll Cl,ll JAN 15 JAN 30 FEB 14 MAR 1 MAR 16 MAR 31 APR 15 APR 30 MAY 15 MAY 30 JUN F3: r<u,b>, r<v,b> &; 0pa, 0VT Fgure 6. nertal wave component correlatons, prncpal axs orentatons, and temperature gradent drecton for F3 at 40, 80, 120, and 160 m. n the top panel for each depth, the heavy trace represents zero-lag correlaton coeffcent r<,,,> between the semmajor axs ellpse component and buoyancy fluctuatons, and the lghtrace represents zero-lag correlaton coeffcent r<v,,> between the semmnor axs ellpse component and buoyancy fluctuatons. Grey bars runnng along the tme axs ndcate frontal events as n Fgure 5. Dashed lnes mark the level of zero sgnfcance at 95% confdence. n the bottom panel of each depth the heavy trace represents the prncpal axs drecton n degrees, antclockwse from east, and the lght trace represents the vertcally averaged temperature gradent drecton n degrees, antclockwse from east. Data are plotted only when ellpseccentrcty e > 0.8. Mode-lke structures resultng from near-nertal wave reflecton off frontal jets can produce patterns of hgh r<,,,> consstent wth those seen n early February and March. Prncpal axs and temperature gradent drectons tend to concde durng these perods. catons, solutons to (12) may be compared wth the observed tme seres (Plate 2 and Fgure 6). Near-nertal waves ncdent from arbtrary drectons on a barotropc, Gaussan frontal jet (Plate 3, bottom panel) provde a model of current and buoyancy fluctuatons for comparson wth the observatons. Both Doppler shfts and changes n fee modfy propagaton. We assume a constant buoyancy frequency N = s- and a Corols frequencyf = x 10 -s s - (27øN) and choose a jet magntude of 0.5 m s - wth a decay scale of 10 km to represent an dealzed front. The negatve (antcyclonc) and postve (cyclonc) vortcty sdes of the jet correspond to the warm and cold sdes of a baroclnc front. Solutons to the cross-front structure equaton (12) predct the sze and prncpal axs algnment of nternal wave current ellpses resultng from nteracton wth the frontal jet. We also consder phase relatonshps between nternal wave buoyancy fluctuatons and current ellpse components for comparson wth observatons. For waves propagatng nto the jet

10 3246 LEE AND ERKSEN: NEAR-NERTAL NTERNAL WAVE NTERACTONS from the outsde, we solve (12) by numercally ntegratng the transmtted wave back through the front to an endpont on the ncdent sde. We solve for trapped modes generated wthn the negatve relatve vortcty pool [Kunze, 1985] by ntegratng (12) from both sdes of the front and matchng the soluton and ts gradent near the center of the trappng regon. To facltate comparson wth the observatons (Plate 2), we normalze model currents to have maxmum ampltudes of 0.1 m s - and calculate buoyancy fluctuatons accordng to (14). Near-nertal waves propagatng nto a front after generaton n a regon of weak background currents exhbt a combnaton of turnng and refracton whch s hghly dependent on the wave's horzontal wavenumber and orentaton. Prevous ob- servatonal studes of near-nertal waves propagatng n fronts and rngs [Kunze and Sanford, 1984; Kunze, 1986; Med et al., 1986, 1987] fnd short horzontal wavelengths, smlar to the scales of the background currents. n contrast, the horzontal wavelengths expected from atmospherc forcng are an order of magntude larger [D54saro, 1987]. We choose horzontal wavelengths X h of 40 km and 400 km and solve for a varety of ncdence angles to llustrate a range of plausble solutons. nternal wave current ellpses, colored by perturbaton buoyancy (Plates 3 and 4), summarze the model results and should be compared wth the observed nertal band-pas stck vectors (Plate 2). We plot plan-vew ellpses at 2-km ntervals spannng the regon near the jet for sx dfferent ncdent wave angles. For each row the hollow vector to the rght of the ellpses represents the frontal jet drecton, and the smaller, sold vector ndcates the orentaton of the ncdent wave vector. El- lpse magntude and orentaton ndcate the sze and drecton of current fluctuatons at the cross-front poston marked by the center of the ellpse. Lnes drawn outward from the ellpse centers ndcate phase relatve to the other ellpses n the row. Antclockwse turnng of phase wth ncreasng x ndcates propagaton n the postve x drecton, and clockwse turnng of phase wth ncreasng x ndcates propagaton n the negatve x drecton, whle phase lnes whch undergo 180 ø changes but show no other sgns of turnng ndcate standng modes. A sngle, propagatng plane wave would appear as an ellpse elongated n the drecton of the wavevector as n Plate 1. Short waves (,k/, = 40 km, cr = 1.05f) undergo ether total or partal reflecton as they encounter the frontal jet (Plate 3), resultng n substantally dfferent current ellpses and buoyancy fluctuatons from those predcted for monochromatc, near-nertal plane waves. The jet completely reflects waves normally ncdent on the cyclonc sde of the front (0 ø) and oblquely ncdent on ether sde when the wave vector s algned wth the background flow (45 ø and 135 ø ) (Plate 3, top three rows). The normally ncdent wave penetrates farthest nto the jet, whle Doppler shftng lowers the ntrnsc frequency of waves propagatng wth the background jet (45 ø and 135ø), forcng them to turn earler when encounterng the postve vortcty rdge. The superposton of ncdent and reflected waves forms a standng mode on the ncdent sde, marked by the lack of horzontal phase propagaton and a snusodal mod- (Plate 3). Here the effectve Corols frequency fee decreases, makng the waves more supernertal and ther current ellpses more elongated. The ellpses algn n the cross-front drecton, wth mnmum and maxmum buoyancy fluctuatons along the semmajor axs. Waves propagatng aganst the background current (225 ø and 315 ø ) are Doppler shfted to hgher ntrnsc frequences, makng the front a less effectve barrer and allowng a porton of the ncdent energy to pass through the jet. The superposton of the ncdent wave and a weaker reflected wave generates a propagatng soluton modulated by a modelke horzontal structure (Fgure 7, last two groups), whle a sngle propagatng plane wave exts the opposte sde of the front. Doppler shftng combned wth decreasng fee makes waves propagatng over the negatve vortcty trough hghly supernertal and leads to lnearly polarzed current ellpses algned prmarly n the cross-front drecton. On the ncdent sde, buoyancy exhbts coherence wth both semmajor and semmnor current ellpse components, whle buoyancy fluctuatons only occur along the semmnor axs for the transmtted wave. Long waves (400 km, rr = 1.05f) partally reflect off the frontal jet for all angles of ncdence (Plate 4). Ths produces propagatng solutons on both sdes of the front, wth ncdentsde ampltude modulated by a mode-lke structure n the cross-front drecton. n the regon of negatve relatve vortc- ty, current ellpses elongate and algn across the front, whle ellpses n the rdge of postve vortcty algn along the front. ncdent-sde buoyancy correlates wth both semmajor and semmnor axs currents, whle the transmtted wave obeys (6). The superposton of ncdent and reflected waves n the X h = 40 km soluton produces cross-front patterns of correlaton coeffcents r<,,,>, r(,,,>, ellpse magntude lu'l, v', and orentaton Op, (Fgure 7) that we may compare wth the temporal patterns seen n the observatons (Fgure 6). The superposton of ncdent and reflected waves generates a snusodal modulaton of ellpse ampltude wth a length scale of half the ncdent cross-front wavelength. Standng modes have nodes n the cross-front drecton, whle propagatng modes have ampltude mnmums and sngle propagatng waves show no ampltude modulaton. For ncdence angles of 180 ø, 225 ø, and 315 ø, tu' > v' and Op, algns across the front n the negatve vortcty trough. Elsewhere, current ellpses are crcularly polarzed. For all angles of ncdent wave vector, transmtted waves have r<,,,> = 0.0 and r<v,, > = - 1.0, as expected for sngle, lnear nternal waves (6). The standng mode produced by normally ncdent waves (0 ø and 180 ø) yelds r(,,,> = 0.0 and r(,,,> = 1.0 wth a length scale of half the ncdent wavelength, shrnkng as the wave approaches the front. Supermposng the ncdent wave wth a weaker reflected wave (225 ø and 335 ø) produces r(,,,> near 1.0 and r<,,,> whch osclates over some nonzero range wth a length scale smlar to that n the normally ncdent case. Perodc 90 ø shfts n prncpal axs orentaton produce complex cross-front patterns n both r(,,,> and r(,,,> for oblquely ncdent waves whch are completely reflected (45 ø and 135ø). Correlaton coeffcents ulaton of ellpse ampltudes lu'l and 'l (Fgure 7). Contrary are defned usng prncpal axs coordnates, so the shft n axs to (6), whch predcts buoyancy fluctuatons correlated wth orentaton also alters r<,,,> and r<,,,>. These 90 ø shfts occur cross-wave vector currents, standng mode buoyancy fluctua- only when the ellpses are essentally crcular and do not affect tons can be correlated wth currents algned along the wave our results for more lnearly polarzed motons. vector. Short waves normally ncdent on the antcyclonc sde of the Waves normally ncdent on the antcyclonc sde of the front front or propagatng agansthe background current (Fgure 7, (180 ø ) also undergo complete reflecton but have ansotropc 180 ø, 225 ø and 315 ø) undergo a shortenng of cross-front wavecurrent ellpses n the regon of negatve relatve vortcty length n the negatve vortcty trough. As the waves enter ths

11 x (km) '" 'o.o o E o.o o..._. o.ooo œ o.5 ', ', -o.5 -? ' :... : -1.0 O. oo 1... '["' o E '... '.... '... E 90 ¾o.o o...'/,.....,... o ß : oooo -- 8o,? ' o.o... o.o.o. ; t -o.,_v "" E ß 90 o.o5o o Z Co - = o,ooo ' o.o '; -, ' -... o.o ' ' ', ' -0.5 "" -1.0 [... '... '... 1,0 E q....:.,,, 'X ' 90 o.o o '. -'J x f' t' \ / ß -[o -- o.o - o CO : o. ; ' -o. "" ' " E _ O,, o,ooo ,1 -- 1,0 : 1.0 o.5 o o.o _ ' -, -, L - ø'ø -0.5 ' : , 'O. loo, [ 180 E f '-.._ ' ' 90 ¾o.o o : _ -_.,... -'..:.,...,...,:::.....:..:..:..:.:.:.:..:...-..:..-..:..:.:..:..:..: o o - o.o25 : oooo '. '... '".' '- o - o CO M 10 c ,5 d ' o.o-l----- '"" ' - ' ' ' = cyclonc ({ > O)...--'""- ><;,,, '. antcyclonc ( < o) / --,- ['- 0,25...., o.oo...,... "" o x (km) / h = 40 km, /f = 1.05 Fgure 7. Current ellpse component magntudes, prncpal axs orentaton, and correlaton coeffcents between veloctes and buoyancy as a functon of cross-front dstance for a wave wth X/, = 40 km and normalzed Euleran frequency o-/f = ncdence angles are specfed followng Plate 3. The top panel of each par dsplays the magntude of semmajor (1 '1, sold curves) and semmnor ( v' l, dashed curves) axs currents and prncpal axs orentaton 0,, (dotted curves). The bottom panel of each par shows correlaton coeffcents between semmajor axs currents and buoyancy r<,,/,> (sold curves) and between semmnor axs currents and buoyancy r<,,,/,> (dashed curves). The frontal jet (sold curve) and normalzed effectve Corols frequencyf ff/f (dashed curve) are shown at the bottom of the fgure. The horzontal scale has been expanded relatve to that n Plate 3. Note the square wave pattern of r<,,/,> produced by waves normally ncdent on the front (0 ø and 180ø).,.?

12 3248 LEE AND ERKSEN: NEAR-NERTAL NTERNAL WAVE NTERACTONS regon, effectve Corols frequency decreases and ntrnsc frequences ether ncrease or reman unchanged. To satsfy the dsperson relaton (2), the cross-front wavenumber ncreases, turnng oblquely propagatng waves nto the cross-front drecton [Kunze, 1985] and producng short-scale current varance wthn the front. n the postve vortcty rdge, Doppler shft and ncreasng effectve Corols frequency compete and the cross-front wavenumber may shft n ether drecton dependng on the relatve strengths of the two effects. A smlar scenaro holds for waves propagatng wth the jet n the negatve vortcty trough, whle the cross-front wavenumber decreases for such waves n the postve vortcty rdge. A general decrease n ellpse ampltude accompaneshortenng of the cross-front wavelength. Conservaton of wave acton flux requres O[Cax(EKEq-EpE) co =O (15) Ox where the cross-front group velocty s O co Cox = Ok - fm N2k (16) 2 --wth and E CE and EpE are the wave knetc and potental energy. To keep the term nsde the square brackets of (15) constant, ncreased cross-front group velocty (cross-front wavenumber) must be compensated by decreased knetc and potental energy and/or ncreased ntrnsc frequency. As near-nertal waves have lttle potental energy, decreases n EpE have lmted value for compensatng changes n group velocty. Furthermore, ncreasng ntrnsc frequency also results n an ncrease n potental energy. Thus knetc energy must decrease ncreasng group velocty to satsfy conservaton of wave acton flux (15). Trapped near-nertal waves (rr = 0.95f) form standng modes n the negatve vortcty regon of the front. A low-mode soluton for waves generated wthn the negatve vortcty trough (Plate 5) shows cross-front orented current ellpses n the trappng regon. Outsde the trough, no propagatng solutons exst and energy decays rapdly wth cross-front dstance. nsde the trappng regon, r< v',> = 0.0 and r<,,,> = _+ 1.0, changng sgn once across the trough. Trapped modes have the same characterstcs as ther untrapped counterparts, except the sze of the trappng regon sets the cross-front mode scale. Hgher modes exhbt smlar behavor but are less lkely to be excted by large-scale atmospherc forcng. 5. Dscusson and Conclusons The mode-lke structures produced by models of nearnertal waves nteractng wth barotropc frontal jets provde a better descrpton of the observatons than sngle plane waves. Observatons show: 1. Near-nertal currents have small horzontal scales and no dstnct phase structure. 2. n the vcnty of upper ocean fronts, near-nertal currents dsplay enhanced across- over along-front varance (Fgure 5). 3. nertal band-pass veloctes (Plate 2) trace out ansotropc ellpses n whch buoyancy vares wth currents along the prncpal axs. Contrary to theoretcal expectatons for monochromatc, propagatng plane waves (6), major-axs currents

13 . LEE AND ERKSEN: NEAR-NERTAL NTERNAL WAVE NTERACTONS 3249 ¾1 o% /.!. \... \ o : (.stu) A

14 3250 LEE AND ERKSEN: NEAR-NERTAL NTERNAL WAVE NTERACTONS o o A Plate 4. Current ellpses for near-nertal nternal waves propagatng nto the front wth horzontal wavelength Xh = 400 km and normalzed Euleran frequency o'/f = 1.05, plotted as n Plate 3. Long waves undergo partal reflecton at all ncdence angles. n the negatve vortcty trough, current ellpses elongate n the cross-front drecton, whle they algn along front where relatve vortcty s postve.

15 LEE AND ERKSEN: NEAR-NERTAL NTERNAL WAVE NTERACTONS 3251 k.! ß ß ß ß ß ß! -..,-,, C:::) -,, o o ß v % -- _ x '- o -, q c?, "!! <q,a>j,<q,n>j A

16 3252 LEE AND ERKSEN: NEAR-NERTAL NTERNAL WAVE NTERACTONS correlate wth near-nertal buoyancy durng the February and March frontal events (Fgure 6). Models of near-nertal wave reflecton and trappng at barotropc frontal jets produce results consstent wth the observa- tons: array cannot accommodate both modal structures and propagatng waves. Consderng a barotropc frontal jet solates near-nertal wave turnng from vertcal crtcal layer processes but fals to accurately represent upper ocean fronts, whch may have sgnfcant horzontal densty gradents and thus vertcal shear. Baroclnc effects become mportant when the vertcal wavelength s greater than the vertcal scale of the background 1. On the ncdent sde of a front, reflectng near-nertal waves produce mode-lke structures wth short horzontal scales, current ellpses elongated n the cross-front drecton, and lttle horzontal phase propagaton (Plates 3 and 4). 2. Buoyancy fluctuatons correlate wth major-axs currents (Fgure 7), consstent wth observatons durng the February and March frontal events (Fgure 6). 3. Trapped modes generated n regons of negatve relatve currents. Horzontal densty contrasts act on dsperson relaton (2), shftng the wave vector nto the plane of the sopycnals and alterng the ntrnsc frequency [Mooers, 1975a; Kunze, 1985]. Of the two cases consdered n secton 4 the 40-km wave has vertcal scales (X z = 170 m) smlar to those of the frontal vortcty produce current ellpses and buoyancy fluctuatons jet, whle the vertcal wavelength of the 400-km wave s much smlar to those of completely reflected waves (Plate 5). longer (X z = 1700 m), suggestng that a baroclnc model 4. Model correlatons depend strongly on ncdence angle would be more approprate for the longwave case. ntutvely, a front wth vertcal scales shorter than the ncdent wave's and degree of reflecton. Both observatons (Fgure 6) and model correlatons (Fgure 7) dsplay a varety of patterns n vertcal wavelength presents a less effectve barrer, and bawhch buoyancy may be correlated wth one or both ellpse roclnc effects should yeld more effcent longwave transmscomponents. son than predcted by the barotropc model (Plate 4). Both the observatons and models show correlatons be- 5. Near-nertal waves propagatng upstream through a negatve vortcty trough change horzontal wavelength to sat- tween perturbaton buoyancy and cross-front currents, suggestng sfy dsperson relaton (2), producng horzontal scale shorter wave-drven acceleraton of the mean flow. n the barotropc than the ncdent wavelength (Fgure 7) and turnng the jet model (secton 4) the Charney-Drazn nonacceleraton thewavevector nto the cross-front drecton. orem [Andrews et al., 1987] suggests that reflecton of near- Perhaps the most strkng comparson s between the square nertal waves by the frontal jet cannot alter the background flow. n a more realstc case, near-nertal nternal waves propwave patterns n r(,,b> generated by normally ncdent short waves (Fgure 7, 0 ø and 180 ø) and the observed correlatons n agatng n a baroclnc background flow are subjecto dsspatve processesuch as crtcal layer nteractons and may nduce early February (40 and 120 m) and early March (40 and 80 m) acceleratons. t s thus nterestng to consder what nforma- (Fgure 6), when the observed prncpal axes are generally ton would be requred to dagnose the nfluence of wavealgned across the front. Weather systems typcally translated drven acceleraton n the observatons. To demonstrate how eastward to the northwest of the moored array, generatng cross-front eddy fluxes of buoyancy are balanced, consder the near-nertal waves ncdent from the north. Although fronts transformed Euleran mean (TEM) equatons [Andrews et al., were generally orented wth ther warm sdes to the south, the 1987] and, specfcally, the Elassen-Palm flux February front and the latter half of the March event had warm water to the north. The barotropc model shows that waves ncdent on the antcyclonc ("warm") sde of the front produce ansotrop current ellpses and short horzontal scales (Plates v = ov0 - o, 3 and 4 and Fgure 7), consstent wth the observatons. Alternatvely, trapped modes produce smlar condtons (Plate 5) N2 f + ] (u'b') - (v'w') (17) and may also explan the observatons durng perods of negfor nternal waves ncdent on a baroclnc, merdonal backatve relatve vortcty. n both cases, observed correlaton patground flow. Here the angle brackets ( ) ndcate along-front terns suggest complete reflecton and short horzontal scales, averagng, the prme symbol ndcates wave quanttes, and we as longer waves would only partally reflect and produce patassume along-front perodcff. For lnear, steady, nondsspaterns wth sgnfcant r( v,t,>. tve waves, Elassen and Palm [1961] demonstrate that The barotropc jet models demonstrate mechansms whch may explan the short horzontal scales and lack of clear hor- V.F=0, ( 8) zontal phase propagaton n the observatons. Wthn the where eddy fluxes of momentum and buoyan do not act model front, Doppler shftng and propagaton through regons ndependently but must compensate to sats (18). Essentally, of low effectve Corols frequency produce varance at hor- gradents of cross-front buoyancy flux are balanced by dverzontal scales consderably smaller than the ncdent wavelength gence of along-front momentum fl. By generalzng the El- (Fgure 7). Reflectng waves produce dstnctly dfferent phase assen-palm theorem, Andrews and Mcln m [1976, 1978] llusstructures on ether sde of the front, where ncdent-sde neartrate how tme-va ng and dsspatv effects alter (18) and nertal currents show snusodal ampltude modulaton and no allow acceleraton of the background flow. Sgnfcantly, the phase propagaton whle transmtted-sde currents dsplay con- TEM equatons and Elassen-Palm theorem demonstrate the stant ampltude and smooth phase varaton. Trapped waves (Plate 5) have mode-lke phase structure n the trappng regon wth varance decayng rapdly outsde t. Thus a smple superposton of plane waves cannot smultaneously descrbe the need for estmates of all the eddy-flux terms and ther gradents f we are to predct the nfluence of waves on the background flow; nferences drawn from pont estmates of an ndvdual term may produce msleadng results. phase structure on both sdes of the model front. n partcular, attempts to ft plane waves to moored observatonspannng a front may produce poor results, as a sngle ft usng the entre Acknowledgments. We thank Robert Weller for generously provdng the surface moorng observatons used n ths study. Nel Bogue

17 LEE AND ERKSEN: NEAR-NERTAL NTERNAL WAVE NTERACTONS 3253 helped wth data preparaton and provded nvaluable computer assstance. Emmanuel Boss, Dave Hebert, Erc Kunze, and an anonymous revewer made helpful comments on the manuscrpt. We gratefully acknowledge the support of the Offce of Naval Research through grant N J References Andrews, D. G., and M. E. Mcntyre, Planetary waves n vertcal and horzontal shear: The generalzed Elassen-Palm relaton and the zonal mean crculaton, J. Atmos. Sc., 33, , Andrews, D. G., and M. E. Mcntyre, Generalzed Elassen-Palm and Charney-Drazn theorems for waves on axsymmetrc mean flows n compressble atmospheres, J. Atmos. Sc., 35, , Andrews, D. G., J. R. Holton, and C. B. Leovy, Mddle Atmosphere Dynamcs, 1st ed., 489 pp., Academc, San Dego, Calf., Bendat, J. S., and A. G. Persol, Random Data: Analyss and Measurement Procedures, 2nd ed., 566 pp., John Wley, New York, Booker, J. R., and F. P. Bretherton, The crtcal layer for nternal gravty waves n a shear flow, J. Flud Mech., 27, , D'Asaro, E. A., Wnd forced nternal waves n the North Pacfc and Sargasso Sea, J. Phys. Oceanogr., 14, , D'Asaro, E. A., Upper ocean temperature structure, nertal currents, and Rchardson numbers observed durng strong meteorologcal forcng, J. Phys. Oceanogr., 15, , D'Asaro, E. A., Horzontal scales of wnd forced nertal motons, n Hawa Wnter Workshop: Dynamcs of the Oceanc Surface Mxed Layer, edted by P. Muller and D. Henderson, pp , Unv. of Hawa at Manoa, D'Asaro, E. A., The decay of wnd-forced mxed layer nertal oscllatons due to the/3 effect, J. Geophys. Res., 94, , D'Asaro, E. A., Upper-ocean nertal currents forced by a strong storm,, Modelng, J. Phys. Oceanogr., 25, , D'Asaro, E. A., C. E. Erksen, M.D. Levne, P. Nler, C. A. Paulson, and P. Van Meurs, Upper-ocean nertal currents forced by a strong storm,, Data and comparsons wth lnear theory, J. Phys. Oceanogr., 25, , Elassen, A., and E. Palm, On the transfer of energy n statonary mountan waves, Geophys. Norv., 22, 1-23, Erksen, C. C., On wnd forcng and the observed oceanc wavenumber spectra, J. Geophys. Res., 93, , 1988a. Erksen, C. C., Varablty n the upper ocean nternal wave feld at a Sargasso Sea ste, J. Phys. Oceanogr., 18, , 1988b. Erksen, C. C., R. A. Weller, D. L. Rudnck, R. T. Pollard, and L. A. Reger, Ocean frontal varablty n the Frontal Ar-Sea nteracton Experment, J. Geophys. Res., 96, , Gregg, M. C., E. A. D'Asaro, T. J. Shay, and N. Larson, Observatons of persstent mxng and near-nertal nternal waves, J. Phys. Oceanogr., 16, , Jones, W. L., Propagaton of nternal gravty waves n fluds wth shear flow and rotaton, J. Flud Mech., 30, , Kundu, P. K., and R. E. Thomson, nertal oscllatons due to a movng front, J. Phys. Oceanogr., 15, , Kunze, E., Near-nertal wave propagaton n geostrophc shear, Ph.D. dssertaton, 90 pp., Unv. of Wash., Seattle, Kunze, E., The mean and near-nertal velocty felds n a warm-core rng, J. Phys. Oceanogr., 8, , Kunze, E., and T. B. Sanford, Observatons of near-nertal waves n a front, J. Phys. Oceanogr., 14, , Kunze, E., and J. M. Toole, Fne and mcrostructure observatons of trapped durnal oscllatons atop Feberlng Seamount, n Hawa Wnter Workshop, n press, Kunze, E., M. G. Brscoe, and A. J.. Wllams, nterpretng shear and fne structure from a neutrally buoyant float, J. Geophys. Res., 95, 18,111-18,125, 1990a. Kunze, E., A. J.. Wllams, and M. G. Brscoe, Observatons of shear and vertcal stablty from a neutrally buoyancy float, J. Geophys. Res., 95, 18,127-18,142, 1990b. Kunze, E., R. W. Schmtt, and J. M. Toole, The energy balance n a warm-core rng's near-nertal crtcal layer, J. Phys. Oceanogr., 25, , Lee, C. M., and C. C. Erksen, The subnertal momentum balance of the North Atlantc Subtropcal Convergence Zone, J. Phys. Oceanogr., 26, , Marmorno, G. O., L. J. Rosenblum, and C. L. Trump, Fne-scale temperature varablty: The nfluence of near-nertal waves, J. Geophys. Res., 92, 13,049-13,062, Med, R. P., C. Y. Shen, C. L. Trump, and G. J. Lndemann, nternal- nertal waves n a Sargasso Sea front, J. Phys. Oceanogr., 16, , Med, R. P., G. J. Lndemann, and C. L. Trump, nertal wave dynamcs n the North Atlantc Subtropcal Zone, J. Geophys. Res., 92, 13,063-13,074, Med, R. P., C. Y. Shen, and M. J. Kdd, Current shear-nertal wave nteracton n the Sargasso Sea, J. Phys. Oceanogr., 20, 81-96, Mooers, C. N. K., Several effects of a baroclnc current on the crossstream propagaton of nertal-nternal waves, Geophys. Flud Dyn., 6, , 1975a. Mooers, C. N. K., Several effects of a baroclnc current on the threedmensonal propagaton of nertal-nternal waves, Geophys. Flud Dyn., 6, , 1975b. Olbers, D. J., The propagaton of nternal waves n a geostrophc current, J. Phys. Oceanogr., 11, , Pollard, R. T., On the generaton of wnds by nertal waves n the ocean, Deep Sea Res., 17, , Pollard, R. T., and R. C. J. Mllard, Comparson between observed and smulated wnd-generated nertal oscllatons, Deep Sea Res., 17, , Prce, J. F., nternal wave wake of a movng storm,, Scales, energy budget and observatons, J. Phys. Oceanogr., 13, , Rubensten, D. M., A spectral model of wnd-forced nternal waves, J. Phys. Oceanogr., 24, , Rudnck, D. L., and R. A. Weller, The heat budget of the North Atlantc Subtropcal Frontal Zone, J. Geophys. Res., 98, , Weller, R. A., The relaton of near-nertal motons observed n the mxed layer durng the JASN (1978) Experment to the local wnd stress and to the quas-geostrophc flow feld, J. Phys. Oceanogr., 12, , Weller, R. A., Near-surface velocty varablty at nertal and subnertal frequences n the vcnty of the Calforna Current, J. Phys. Oceanogr., 15, , Weller, R. A., D. L. Rudnck, R. E. Payne, J.P. Dean, N.J. Pennngton, and R. P. Trask, Measurng near-surface meteorology over the ocean from an array of surface moorngs n the Subtropcal Convergence Zone, J. Atmos. Oceanc Technol., 7, , 1990a. Weller, R. A., D. L. Rudnck, N.J. Pennngton, R. P. Trask, and J. R. Valdes, Measurng upper ocean varablty from an array of surface moorngs n the Subtropcal Convergence Zone, J. Atmos. Oceanc Technol., 7, 68-84, 1990b. Weller, R. A., D. L. Rudnck, C. C. Erksen, K. L. Polzn, N. S. Oakey, J. W. Toole, R. W. Schmtt, and R. T. Pollard, Forced ocean response durng the Frontal Ar-Sea nteracton Experment, J. Geophys. Res., 96, , Zervaks, V., and M.D. Levne, Near-nertal energy propagaton n the mxed layer: Theoretcal consderatons, J. Phys. Oceanogr., 25, , C. C. Erksen, School of Oceanography, Unversty of Washngton, Box , Seattle, WA C. M. Lee (correspondng author), Department of Physcal Oceanography, Woods Hole Oceanographc nsttuton, MS-21, Woods Hole, MA (e-mal: clee@who.edu) (Receved November 28, 1995; revsed June 14, 1996; accepted September 10, 1996.)