PUBLICATIONS. Journal of Geophysical Research: Oceans

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1 PUBLICATIONS Jurnal f Gephysical Research: Oceans RESEARCH ARTICLE 1.1/1JC1591 Key Pints: With hrizntally unifrm stratificatin, semidiurnal tidal activity peaked in the diurnal critical latitude range Diffusivities peaked at the K 1 critical latitude, as determined in the mdel using the Nakanishi-Niin vertical mixing parameterizatin Energy transfers were directin dependent, with mre transfers ccurring in the acrss-slpe directin at lwer frequencies, particularly the semidiurnal tides Supprting Infrmatin: Supprting Infrmatin S1 Crrespndence t: R. Rbertsn, r.rbertsn_nmad@yah.cm Citatin: Rbertsn, R., Dng, J., & Hartlipp, P. (17). Diurnal Critical latitude and the latitude dependence f internal tides, internal waves, and mixing based n Barc seamunt. Jurnal f Gephysical Research: Oceans, 1, JC1591 Received 8 NOV 1 Accepted 3 SEP 17 Accepted article nline 1 SEP 17 Published nline 1 OCT 17 Diurnal Critical Latitude and the Latitude Dependence f Internal Tides, Internal Waves, and Mixing Based n Barc Seamunt Rbin Rbertsn 1,,3, Jihai Dng, and Paul Hartlipp 1 1 Schl f Physical, Envirnmental, and Mathematical Sciences, University f New Suth Wales, Canberra BC, ACT, Australia, Schl f Marine Science, Nanjing University f Infrmatin Science and Technlgy, Nanjing, Jiangsu, China, 3 Nw at Xiamen University Malaysia, Department f Marine Bitechnlgy, Sepang, Selangr, Malaysia Abstract Vertical mixing is a key issue in cean circulatin mdeling tday. Mixing, particularly tidal mixing, is prly represented in cean and climate mdels, which generally ignre critical latitude effects. Critical latitude is the latitude where the inertial frequency equals the tidal frequency and differs fr each tidal cnstituent. Critical latitudes strngly influence generatin and prpagatin f internal tides. Using a mdel, latitude effects n tidal interactins with a seamunt were examined by varying the latitude frm 8 t 388, thrugh the range f the diurnal critical latitudes. The diurnal critical latitudes were fund t strngly influence prpagatin f the diurnal internal tides, the magnitude f the semidiurnal tides, the energy in the harmnic and higher frequencies, the bartrpic mean flw, and the diffusivities. The strngest effects ccurred between the K 1 and O 1 critical latitudes. Here the semidiurnal tides, harmnics, and high frequencies were enhanced, bartrpic mean velcities weakened, energy at the harmnics and higher frequencies increased, and diffusivities increased. Spectral techniques indicate that mst f these impacts are the result f nnlinear wave-wave interactins and resnant phenmena with the prminent mechanism harmnic transfers. There was n evidence f parametric subharmnic instabilities. The semidiurnal tides indicated a resnant respnse at 8S, which is near the latitude fr the cmbined M and K 1 tidal perid, 198S. Plain Language Summary This study investigates hw the cean respnse t the tides changes with latitude using a series f mdel simulatins fr a seamunt. It fcuses n the latitudes which resnate with the daily tides, the diurnal critical latitudes, where small changes are amplified. The effects n mixing and the transfer f energy t ther frequencies is a key aspect. It was fund that even small tides significantly increase mixing near the K1 critical latitude. The transfer f energy increases near the diurnal critical latitudes. Energy transfers in currents frm the daily tide t the twice daily tide were primarily arund the seamunt, whereas thse frm twice daily tides t their harmnics and high frequencies were arund the seamunt twards and away frm t the seamunt. VC 17. The Authrs. This is an pen access article under the terms f the Creative Cmmns Attributin-NnCmmercial-NDerivs License, which permits use and distributin in any medium, prvided the riginal wrk is prperly cited, the use is nn-cmmercial and n mdificatins r adaptatins are made. 1. Intrductin Presently, vertical cean mixing is ne f the mst prminent prblems in cean and climate mdeling. Vertical mixing plays a key rle in maintaining the stratificatin, driving the glbal verturning circulatin (Munk & Wunsch, 1998), distributing nutrients and larvae fr bilgical prductivity and fisheries (Stevens et al., 1; Wilsn, 11), redistributing heat and salt, and influencing climate dynamics. Much f the cean s mixing ccurs in specific ht spts near tpgraphic features, resulting frm interactins f currents and/r tides with tpgraphy (Garrett, 3). A significant amunt f this mixing, TW(1TW5 1 1 W), is attributed t tides (Egbert & Ray, ; Munk & Wunsch, 1998), with rughly half assciated with internal tides. Observatin f tidal mixing is difficult bth t its lcalized nature and its time dependence, i.e., tidal mixing fllws the spring-neap and daily tidal cycles. Until recently, cean circulatin and climate mdels ignred tides and vertical tidal mixing, due t cmputatinal requirements. Reslutin requirements fr internal tides (5 km; Rbertsn, ) make them difficult t simulate and impractical fr mst glbal mdels. T accunt fr vertical tidal mixing, Lee et al. ROBERTSON ET AL. CRITICAL LATITUDE, TIDES, AND MIXING 7838

2 1.1/1JC1591 Table 1 Latitudes f Resnance and Perids fr the Fur Majr Tidal Cnstituents (M,S,K 1, and O 1 ) and Their Cmbinatins O 1 K 1 1 O 1 K 1 M 1 O 1 S 1 O 1 M 1 K 1 K 1 1 S O 1 M M 1 M 1 S S S 1 K 1 M S Perid (h) Latitude (8) Critical latitude a a b a a b a The critical latitudes using the half frequency definitin. b The critical latitudes using ur definitin. (), Jayne and St Laurent (1), and Simmns et al. () develped parameterizatins f vertical tidal mixing t be used in cean and climate mdels. Inclusin f these tidal mixing parameterizatins in cean mdels imprved perfrmance bth glbally (Jayne, 8; Simmns et al., ) and reginally (Indnesian Seas: Kch-Larruy et al., 8; Nrth Atlantic: Lee et al., ; Australia and the Indnesian Seas: Lee et al., ; Schiller et al., 13). Althugh these parameterizatins are an imprtant imprvement, they ignre critical latitude effects. Here the critical latitude is defined as the latitude where the inertial frequency equals the tidal frequency (e.g., Furevik & Fldvik, 199; Middletn & Denniss, 1993; Rbertsn, 1), althugh thers define it as the latitude with half the tidal frequency (e.g., MacKinnn et al., 13b; see Table 1). Critical latitudes affect cean dynamics in several ways. They have been bserved t be a key factr in energy transfer and dissipatin (Hibiya et al., 1). They strngly influence generatin and prpagatin f internal tides (e.g., Middletn & Denniss, 1993; Rbertsn, 1). Tides becme resnant near critical latitude, increasing generatin and energy transfers t harmnics. Parametric subharmnic instabilities (psi) increase energy transfers frm the semidiurnal t the diurnal tidal cnstituents near the diurnal critical latitudes (MacKinnn et al., 13b). Critical latitudes act as turning latitudes fr Pincare waves. Accrding t linear internal wave thery, as internal tides apprach the critical latitude, their hrizntal wave number increases and their prpagatin angle Figure 1. (a) Lcatin f Barc seamunt with the dmain utlined by a yellw bx and (b) the bathymetry in the dmain. ROBERTSON ET AL. CRITICAL LATITUDE, TIDES, AND MIXING 7839

3 1.1/1JC1591 flattens, becming infinite and hrizntal, respectively, at critical latitude. This implies that internal tides generated pleward f the critical latitude are trapped. Near critical latitude, the benthic bundary layer thickens (Furevik & Fldvik, 199). Critical latitude has been linked t anmalusly high bilgical prductivity (Wilsn, 11) and t increases in ice shelf melt rates f 5% (Rbertsn, 13). Observatins and mdels have shwn that the critical latitude can be shifted by the relative vrticity f mean currents t an effective critical latitude with shifts f several degrees pssible (Kunze & Tle, 1997; Rbertsn,, 13). And bservatinal studies, such as MacKinnn et al. (13a, 13b), shw diffusivities increase near the diurnal critical latitudes. The dependence f internal tides, waves, and mixing n latitude was investigated using a numerical mdel. The latitude range extended 88 n either side f the K 1 and O 1 critical latitudes, 8S 388S. The simple case f a seamunt was chsen t reduce interactins frm ther tpgraphic features. Additinally, an area withut strng tides was chsen t be representative f typical glbal tides. Barc seamunt (Figure 1) met these criteria (supprting infrmatin Figure S-1) and is nly a few degrees pleward f the diurnal critical latitude. Additinally accurate bathymetric and hydrgraphic data fr the area were available. The mdel and methdlgy used fr the simulatins are described in sectin and Appendix B. A discussin f wave-wave interactins is given in sectin 3 fr reference in the later sectins. Sectin utlines the latitudinal dependencies f the generatin and prpagatin f internal tides and waves, mean velcities, energy transfers, and mixing as represented by the mdel diffusivities. Thickening f the bundary layer was nt discussed, bth fr brevity and an extensive discussin was dne fr the M tide by Furevik and Fldvik (199). A discussin highlighting the cmmnalities in the results fllws in sectin 5. Sectin gives the verall cnclusins. Additinal figures have been included in supprting infrmatin, which are referred t as S-#.. Mdeling Simulatins were perfrmed with the Reginal Ocean Mdelling System (ROMS) (Shchepetkin & McWilliams, ) versin 3. (dwnladed in late 1) with a few mdificatins, which are utlined in Appendix A. The bathymetry fr the regin was taken frm multibeam ech sunder data cllected during RV Suthern Surveyr vyage SS9 in Octber 9 (Figure 1) and kindly prvided by CSIRO Marine and Atmspheric Research staff. The mdel grid is 3 grid cells with a hrizntal reslutin f 1 km and vertical levels. A small amunt f smthing was required at a steep area ver the crest f the seamunt; hwever, elsewhere the tpgraphy was nt smthed. The initial temperature and salinity fields were adapted frm Cnductivity, Temperature, and Depth (CTD) cast data cllected during SS9 (supprting infrmatin Figure S-). A single averaged prfile was used ver the entire dmain, resulting in hrizntally unifrm hydrgraphy and simplifying the dynamics (supprting infrmatin Figure S-). Frcing was limited t fur tidal cnstituents (M,S,K 1, and O 1 ), with n wind r net slar radiatin frcing. Tidal frcing was set by specifying the elevatin and -D velcities. Mre infrmatin n the simulatins and their analysis can be fund in Appendix B. T investigate latitude dependence f the internal tides and mixing, additinal grids were created by mdifying the latitude s the center latitude ranged frm.8 t 38.8 in 8 intervals with tw extra dmains near the diurnal critical latitudes, 7.8 and 9.8. The same initial hydrgraphy was used fr all simulatins. Althugh this is nt realistic, since the stratificatin will change with latitude, it was dne in rder t islate latitude effects. 3. Resnant Interactins Waves interact mre readily when they are resnant with each ther, i.e., their frequencies are integer multiples f each ther. These resnant interactins are generally cmpsed f tw parent waves generating a daughter wave accrding t the fllwing relatins: k 1 k 5k 3 m 1 m 5m 3 x 1 x 5x 3 (1) where k is the hrizntal wave number, m the vertical wave number, x the frequency, and the subscripts refer t the different waves invlved, representing either the parent r daughters depending upn the ROBERTSON ET AL. CRITICAL LATITUDE, TIDES, AND MIXING 78

4 1.1/1JC1591 interactin. Resnant interactins are knwn t exchange energy between the three wave numbers and frequencies. There are many resnant interactins, but three distinct classes are prminent in cean dynamics: parametric subharmnic instabilities, induced diffusin, and elastic scattering (McCmas & Brethertn, 1977). Mst f these interactins ccur between waves with similar frequencies r between waves that have half the frequency. This is basically the cnditin fr the diurnal and semidiurnal tides. Additinally, they ften transfer energy t near-inertial frequencies, which is the case fr the diurnal tides near their critical latitudes. Therefre, the cnditins fr these mechanisms t ccur are present in the simulatins, particularly near and pleward f the diurnal critical latitudes. Many f the interactins ccur between tw waves with nearly the same frequency, such as between the tw diurnal r semidiurnal tides. These interactins will transfer energy t lw frequencies r a mean velcity. One f the resnant mechanisms that exchanges energy between frequencies is parametric subharmnic instabilities (psi). In psi, there is a nnlinear interactin between a wave triad, where tw waves have near the same frequency (x 1 x 3 ), but nearly ppsite wave numbers (k 1 k 3 ) and the third wave has twice the frequency f the ther waves (x x 1 ) and a higher wave number (k > k 1 ) (McCmas, 1977; McCmas & M uller, 1981; M uller et al., 198). Psi has been fund t transfer significant energy in midlatitudes (Hibiya et al., ) and t be assciated with tidal mixing, particularly near the diurnal critical latitudes, with the M tide giving energy t the diurnal tides (MacKinnn et al., 13a, 13b). Induced diffusin describes the interactin between a wave with high wave number and frequency (k 1, x 1 ) with a high energy wave with a lw wave number and frequency (k, x ) generating a wave similar t the high frequency wave (k 3 k 1, x 3 x 1 ). If the energy spectrum decreases rapidly fr wave numbers exceeding k 3, energy frm the high energy, lw frequency wave will be transferred t the higher wave number wave 1, diminishing wave. The net effect f this is t fill in higher wave numbers beynd that f k 3. In elastic scattering, the prpagatin directin f the energy is switched. A dwnward prpagating wave (k 3, m 3, x 3 ) interacts with a near-inertial wave with half the vertical wave number f wave 3 (k, m m 3, x ), shifting energy int anther wave prpagating in the ppsite directin (k 1 k 3, m 1 m 3, x 1 x 3 ). These energy transfers balance the upward and dwnward prpagatin at that frequency t be mre equal.. Latitude Dependence The tidal respnse is cmprised f many factrs. Here diurnal and semidiurnal barclinic tidal behavir, tidally generated bartrpic mean velcities, and temperature diffusivities are examined. Additinally, exchanges f energy between frequencies are discussed..1. Diurnal Tides (1 cpd) The diurnal respnse was dminated by K 1. The strngest bartrpic K 1 and O 1 tidal velcities, as represented by the majr axes f the tidal ellipses, ccurred ver a small patch at the crest f the seamunt and in patches in the deep basin n the nrthern and suthern sides (supprting infrmatin Figures S-3 and S- ). The bartrpic K 1 velcities (supprting infrmatin Figure S-3) were rughly twice the O 1 velcities, 3 cm s 1 cmpared t 1 cm s 1 (supprting infrmatin Figure S-). The largest, bartrpic diurnal velcities develped pleward f the O 1 critical latitude (supprting infrmatin Figures S-3 and S-) with the maximum ver the rim, pleward f the respective critical latitudes. The barclinic respnse fr bth diurnal cnstituents exceeded their respective bartrpic respnses by mre than a factr f, but fllwed the same trends (Figure cmpared t supprting infrmatin Figure S-3 fr K 1 and Figure 3 cmpared t supprting infrmatin Figure S- fr O 1 ). The peak diurnal respnse ccurred just pleward f their critical latitudes, at 3.8S fr K 1 and 8.8S fr O 1 (Figures i and a and c). There, internal tides reaching 8 cm s 1 fr K 1 and cm s 1 fr O 1 were generated ver the crest f the seamunt (Figures and 3). Generatin at the crest increased with increasing latitude as critical latitude was apprached and prpagatin ceased after their critical latitudes (Figures and 3) in agreement with the theretical generatin sites accrding t Baines criteria (supprting infrmatin Figures S-5 and S-; see Appendix C fr the Baines criteria descriptin.) The strngest K 1 barclinic velcities ccurred just pleward f its critical latitude and clung t the bttm ver the upper prtin f the seamunt (Figure ). The O 1 ROBERTSON ET AL. CRITICAL LATITUDE, TIDES, AND MIXING 781

5 1.1/1JC1591 Figure. Barclinic majr axes fr K 1 at each latitude frm.8s t 38.8Sin8 increments with tw extra dmains near the diurnal critical latitudes. barclinic tides were similar, with the peak generatin at the crest and rim f the seamunt, reaching cm s 1 just pleward f the critical latitude (Figure 3). The O 1 velcities remained strng at the crest pleward f critical latitude and the patch f higher velcities extended further dwn the slpe, indicating bth ROBERTSON ET AL. CRITICAL LATITUDE, TIDES, AND MIXING 78

6 1.1/1JC1591 a) e) i) Depth (m) S 7. S 7.1 S O 1 critical latitude 3. S actual latitude b) f) j) Depth (m) Depth (m) S S c) g) k) 3. S S. S 3. S (cm s -1 ) d) h) l) Depth (m) S Latitude 3. S 3 S K 1 critical latitude Latitude 38. S Latitude Figure 3. Barclinic majr axes fr O 1 at each latitude frm.8s t 38.8Sin8 increments with tw extra dmains near the diurnal critical latitudes. a benthic respnse and trapping (Figure 3). The peaking f the respnse just pleward f the critical latitude and the trapping agreed with linear internal wave thery. ROBERTSON ET AL. CRITICAL LATITUDE, TIDES, AND MIXING 783

7 1.1/1JC1591 Table Dispersin Relatins, Hrizntal Wavelengths, and Hrizntal and Vertical Grup Speeds, c gh and c gv, Respectively fr Different Types f Waves Internal tide in cntinuus stratificatin Barclinic kelvin wave Barclinic Pincare wave p Dispersin relatin x j 5 f m j 1N k x5 ffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g H k x5 f 1c ðk 1l Þ k 1m j Hrizntal grup km j N f p speed relatin, c c gh 5 gh f m j 1N k 1= k 1m 3= m j 5 jp c gh 5 ffiffiffiffiffiffiffi g H c gh 5 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c k f H 1c ðk 1l Þ j Vertical grup speed, c gv k m j N f c gv 5 c c gv 5 f m j 1N k 1= k c gv 5 mj pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3= f 1m 1c ðk 1l Þ j The theretical prpagatin rays fr internal tidal beams fr the diurnal (and semidiurnal) cnstituents were determined accrding t linear internal wave thery (supprting infrmatin Figure S-7). The rays riginated alng the slpe at the depth f Baines value clsest t ne within the upper 1,5 m. Near critical latitude, a default lcatin ver the rim was used fr illustratin. Ray slpes fr the cnstituents varied with depth due t stratificatin differences at the rigins. N internal wave characteristics are shwn pleward f the respective critical latitudes (see Appendix C). The diurnal cnstituents were strngly latitude dependent, as expected. Fr the K 1 cnstituent, the nrthernmst fur dmains indicated rays which prpagated slightly upward frm the hrizntal (Figures a d) in agreement with linear thery (red lines in supprting infrmatin Figures S-7a S-7d). The internal wave beams at these latitudes reached well acrss the dmain. As critical latitude was apprached, the beams bradened, their slpe flattened, and their prpagatin did nt extend as far frm the seamunt (Figures e g). Since the grup speed decreases near the critical latitude (supprting infrmatin Figures S-8 and S-9), the rays d nt prpagate as far frm the seamunt. At critical latitude, the beams were the strngest with the highest velcities, reaching 8 cm s 1 (Figure h). Pleward f critical latitude, the prpagatin f the beams was reduced, and the majr axes were mre unifrm vertically (Figures i l). The diurnal tides became much mre bartrpic, unifrm with depth, at these latitudes (Figures i l). The O 1 respnse was similar, resulting in large, nearly vertical patches f the largest majr axes (Figure 3). Again, the respnse at the seamunt crest peaked just pleward f the O 1 critical latitude, became mre bartrpic, and remained strng at higher latitudes (Figure 3). Hrizntal wavelengths and hrizntal and vertical grup speeds, L h, c gh, and c gv, respectively, were determined fllwing linear thery and the equatins in Table. Diurnal hrizntal wavelengths were much lnger and changed drastically with latitude (supprting infrmatin Figures S-8 and S-9 fr mdes 1 and ). At.8S, they ranged frm 5 km fr the 5 m depth t km fr the 1, m depth. Fr the dmains further pleward, the diurnal hrizntal wavelength increased expnentially within 8 f critical latitude, where it apprached infinity (supprting infrmatin Figures S-8c and S-8d and S-9c and S-9d fr mdes 1 and ). Further pleward f critical latitude, the hrizntal wavelength was imaginary. There were als drastic changes in the hrizntal and vertical grup speeds. At 8S, the hrizntal grup speed ranged frm.5 t 1. m s 1 fr K 1 and. t m s 1 fr O 1, fr the depths range f 5 t, m, respectively, with mde 1 faster than mde. The hrizntal grup speeds drpped t m s 1 at the respective critical latitudes, as did the vertical grup speeds (supprting infrmatin Figures S-8 and S-9 fr mdes 1 and ). The diurnal vertical grup speeds were negative, indicating a dwnward flux f energy. They were significantly smaller than the hrizntal grup speeds, with maximum magnitudes in deep water f.5 ms 1 fr K 1 and. m s 1 fr O 1. S wave energy prpagatin fr the diurnal cnstituents slwed as their critical latitudes were apprached. The minr axes f the diurnal cnstituents shwed little latitude dependence (supprting infrmatin Figures S-1 and S-11 fr K 1 and O 1, respectively). Equatrward f the critical latitude, larger minr axes did appear in beam-like patterns emanating frm the crest f the seamunt; whereas pleward f the critical latitudes, the minr axes appeared in patches. The latitude dependence f the prpagatin can be seen in the phases (supprting infrmatin Figures S-1 and S-13 fr K 1 and O 1, respectively). Nte, that phases are ROBERTSON ET AL. CRITICAL LATITUDE, TIDES, AND MIXING 78

8 1.1/1JC1591 a) e) i) Depth (m) S 7. S 7.1 S O 1 critical latitude 3. S actual latitude b) f) j) Depth (m) Depth (m) S S c) g) k) 3. S S. S 3. S (cm s -1 ) d) h) l) Depth (m) S Latitude 3. S 3 S K 1 critical latitude Latitude 38. S Latitude Figure. Barclinic majr axes fr M at each latitude frm.8s t 38.8Sin8 increments with tw extra dmains near the diurnal critical latitudes. nt shwn where the majr axes are less than 1. cm s 1. Equatrward f the critical latitude, the K 1 phases spread frm the seamunt with beams emanating at several depths alng the seamunt flank, prpagating bth away frm the seamunt and alng the bttm. Interference between internal tidal beams appeared ROBERTSON ET AL. CRITICAL LATITUDE, TIDES, AND MIXING 785

9 1.1/1JC1591 Table 3 Latitude Dependence f the Diurnal and Semidiurnal Barclinic Tides, Tidally Generated Mean Bartrpic Velcities, and Mmentum and Temperature Diffusivities Diurnal barclinic tides Equatrward f the O 1 critical latitude 8 8 Tidal beams emanate frm upper flank and prpagate hrizntally In the diurnal critical latitude range Tidal beams fr K 1 emanate frm upper flank and crwn with diminished hrizntal prpagatin Semidiurnal barclinic tides Tidal beams emanate frm crwn Enhanced tidal beams emanate frm crwn; increased benthic activity at crwn Pleward f the K 1 critical latitude Tidal beams attached t bttm alng flank and crwn, mre unifrm vertically Tidal beams emanate frm crwn; decreased magnitudes Harmnic frequencies Lw kinetic energy Increased kinetic energy Lw kinetic energy Mean bartrpic velcities Centered ver crwn, strngest Centered ver crwn, but weakest Centered ver crwn Mmentum and temperature diffusivities Variable; strngest just equatrward f O 1 critical latitude Strngest just equatrward f and at K 1 critical latitude and in critical latitude range Lwest and trapped near seamunt; increases pleward f 38S as phase jumps. Pleward f the critical latitudes, the phases were trapped t the seamunt and prpagated nly alng the bttm. Inclinatins were cncentrated near the seamunt pleward f their critical latitudes, althugh there was less latitude dependence (supprting infrmatin Figures S-1 and S-15)... Semidiurnal Tides ( cpd) M dminated the semidiurnal respnse with bartrpic majr axes reaching cm s 1 (supprting infrmatin Figures S-1 and S-17), twice thse f the diurnals. The semidiurnal bartrpic tidal currents peaked ver the tp f the seamunt and were primarily cnfined in the upper 1, m fr bth cnstituents (supprting infrmatin Figures S-1 and S-17 fr M and S, respectively). The strngest M respnse ccurred at.8s and between the diurnal critical latitudes, 7.8S 3.8S (supprting infrmatin Figure S-1). The M barclinic velcities (Figure ) emanated frm the crwn f the seamunt spreading in the upper 5 m. The S barclinic velcity respnse was similar, but weaker (supprting infrmatin Figure S-18). The M barclinic respnse was strngest at.8s and between 7.8S and 3.8S (Figure ) and S between 7.8S and 3.8S (supprting infrmatin Figure S-18). The mdel generated internal tides ver the seamunt crwn as predicted by the Baines criteria (supprting infrmatin Figures S-19 and S- fr M and S, respectively). The strngest barclinic semidiurnal tides rughly fllwed internal wave paths accrding t linear wave thery (blue fr M and cyan fr S in supprting infrmatin Figure S-7). Bth the mdel and theretical ray paths varied very little with latitude fr the semidiurnal cnstituents. Nte the rays in supprting infrmatin Figure S-7 are sample rays and the mdel generates mre internal wave beams, the strngest f which are higher n the seamunt. The prpagatin f the semidiurnal tides did nt have an appreciable latitude dependence. The phases f the semidiurnal barclinic tides were relatively latitude independent (supprting infrmatin Figures S-1 and S-). Due t interference between the multiple beams emanating frm the seamunt, the phases ften changed in jumps. Phases als ccurred in large patches, indicating prpagatin nrmal t the transect. The minr axes f the M tidal ellipses were larger at.8s and in the diurnal critical latitude range (supprting infrmatin Figures S-3 and S-). Generally, their magnitude is largest where the majr axes were largest and away frm the seamunt, where tpgraphic restrictins are reduced. The semidiurnal inclinatins were relatively independent f latitude (supprting infrmatin Figures S-5 and S-). Hrizntal wavelengths fr the semidiurnal cnstituents, M and S, ranged frm t 18 km dependent n the water depth, the mde, and the latitude (Table 3 and supprting infrmatin Figures S-8 and S-9 fr mdes 1 and ). (Deeper water depths had lnger wavelengths and higher mdes had shrter wavelengths.) There was a slight increase in wavelength at higher latitudes fr the semidiurnal cnstituents. Hrizntal grup speeds were faster fr deeper depths and lwer mde numbers (supprting infrmatin Figures S-8 and S-9 fr mdes 1 and ). Hrizntal grup speeds decreased by..5 cm s 1 as latitude increased fr the semidiurnal cnstituents. The magnitudes f the vertical grup speed increased by.. cm s 1 as latitude increased. Vertical grup speeds were negative, indicating a dwnward energy flux (upward prpagating phase), and were slwer in shallwer water and faster in deeper water. ROBERTSON ET AL. CRITICAL LATITUDE, TIDES, AND MIXING 78

10 1.1/1JC1591 a) Spectral Density N-S Velcity ((cm s -1 ) cpd -1 ) S Crwn Rim Flank Shulder Basin.1 b) Spectral Density N-S Velcity ((cm s -1 ) cpd -1 ) S c) Spectral Density N-S Velcity ((cm s -1 ) cpd -1 ) S Frequency (cpd) 1 Figure 5. Nrth-Suth velcity spectra at the crwn, rim, flank, shulder, and basin at level 5 (near the surface) fr three different latitudes,.8s, 3.8S, and 3.8S, representing equatrward, at, and pleward f the diurnal critical latitudes. The water depths and latitude distance frm the center at the crwn, rim, flank, shulder, and basin are 88 m.38, 39 m.38, 158 m.818, 535 m.188, and 583 m.1538, respectively. ROBERTSON ET AL. CRITICAL LATITUDE, TIDES, AND MIXING 787

11 1.1/1JC1591 a) c) Integrated Spectral Energy ((cm/s)) b) Nrmalized Energy Bands (-) E-W Velcity Latitude O 1 critical latitude K 1 critical latitude Latitude Integrated Spectral Energy ((cm/s) ) d) Nrmalized Energy Bands (-) N-S Velcity O 1 critical latitude Ttal Diurnal (1 cpd) Semidiurnal ( cpd) Harmnics (3-1 cpd) High Frequencies (>1 cpd) K 1 critical latitude beta (lw frequencies) alpha (high frequencies) Latitude Figure. Integrated velcity spectra versus latitude fr the frequency bands: all frequencies, black; diurnal, red; semidiurnal, blue; harmnics 3 1 cpd, cyan; high frequencies > 1 cpd, green fr the (a) East-West and (c) Nrth-Suth velcities. Integrated velcity spectra fr the a and b frequency bands defined as between f and N fr a (black) and between f and f fr b (magenta) fr the (b) East-West and (d) Nrth-Suth velcities..3. Spectral Distributin: Harmnics and Higher Frequencies (3 1 and >1 cpd) Spectral analysis was perfrmed n the barclinic anmalies at each depth and each grid cell. The largest spectral peaks in the Nrth-Suth velcities ccurred at the diurnal and semidiurnal frequencies fr all latitudes and all lcatins (Figure 5): the crwn (red), rim (pink), flank (green), shulder f the seamunt (blue), and the deep basin (black). The diurnal peak in the Nrth-Suth velcities at the crwn and rim increased frm.8 t 3.8S, but decreased pleward f the diurnal critical latitudes at the flank, shulder, and deep basin (Figure 5). The semidiurnal peak was greatly enhanced ver the flank (green) and shulder (blue) f the seamunt equatrward f the diurnal critical latitudes near the surface (Figure 5). Deeper in the water clumn, the semidiurnal peak was decreased at the rim pleward f the diurnal critical latitude range fr the East-West velcities (supprting infrmatin Figure S-7). The respnse at the harmnic and higher frequencies was als enhanced bth in and pleward f the diurnal critical latitude range, particularly near the bttm (supprting infrmatin Figures S-7 and S-8). Fr a mre quantitative cmparisn, the spectra were integrated ver different frequency bands alng a transect thrugh the center f the dmain, t prvide an estimate f the ttal energy within each frequency band. The diurnal band, represented frequencies near 1 cpd, the semidiurnal near cpd, the harmnics and cmpund tides between 3 and 1 cpd, and the high frequencies the band > 1 cpd. The integrated energy fr the diurnal band fr the East-West velcities, alng-slpe, increased tward the O 1 critical latitude, decreased at the O 1 critical latitude, increased thrugh the diurnal critical latitude range, peaked at 3.8S, then decreased slightly (red line in Figure a). The integrated energy fr the diurnal band fr the Nrth- Suth velcities, acrss-slpe, was significantly smaller than in the alng-slpe directin and peaked at the diurnal critical latitudes and 3.8S (red line in Figure c). The integrated kinetic energy in the semidiurnal band peaked in the critical latitude range and at.8s fr the East-West and Nrth-Suth velcities (blue ROBERTSON ET AL. CRITICAL LATITUDE, TIDES, AND MIXING 788

12 1.1/1JC1591 lines in Figures a and c, respectively). The increase in the critical latitude range was much strnger in the acrss-slpe directin than alng-slpe, althugh there was rughly half f the semidiurnal energy in the acrss-slpe directin as in the alng-slpe directin utside the critical latitude range (blue lines in Figures a and c). Energy at the harmnics and high frequencies were significantly smaller than the tidal frequencies fr bth hrizntal velcities (cyan and green lines in Figures a and c, respectively). The energy at the high frequencies in the acrss-slpe directin was mre than twice that in the alng-slpe directin and the harmnic energy was als slightly higher in the acrss-slpe directin. Bth the harmnic and high frequency energy peaked in the diurnal critical latitude range (cyan and green lines in Figures a and c, fr the harmnics and high frequencies, respectively). Fllwing Hibiya et al. (1), spectra were als integrated ver the alpha and beta bands, where alpha, a, represents frequencies, x, frm f < x < N and beta, b, represents frequencies between f < x < f. With bth a and b, the width f the integratin frequency band changes with latitude, narrwing at higher latitudes 1.78 times the width at.8 as at Furthermre, when interpreting b, cnsideratin needs t be taken that the diurnal frequencies will be included in b equatrward f their critical latitudes, but will drp ut at their respective critical latitudes. This resulted in big drps in b at the critical latitudes and little energy in the b band pleward f the diurnal critical latitudes. Energy at the lw frequencies, b, decreased as the dmain mves pleward until the diurnal critical latitudes are reached, where it peaks at the O 1 critical latitude in the alng-slpe directin and 8.8S in the acrss-slpe (magenta lines in Figures b and d). Energy fr the high frequency, a, band peaked at the K 1 critical latitude in bth the alng-slpe and acrssslpe directins (black lines in Figures b and d)... Spectral Distributin: Lw Frequencies (..8 cpd) The lw frequency respnse includes frequencies belw the O 1 tidal frequency. Fr the latitudes frm.8s t.8s, the lw frequencies include bth the subinertial and inertial frequencies. Hwever, between 7.8S and 38.8S, the subinertial range increased, eventually including the tidal diurnal tidal frequencies. The lw frequency respnse shwed n cnsistent, significant latitude dependence (Figures 5 and supprting infrmatin Figures S-7 and S-8). Althugh this mdel has previusly simulated diurnal tpgraphically trapped Rssby waves (Rbertsn, 5a, 5b), there was n evidence f them in the elevatins. There was an apparent wave-like pattern with tw wavelengths arund the seamunt in the bartrpic majr axes f the tidal ellipses fr K 1,O 1, and S, which was cnstant with latitude (supprting infrmatin Figures S-3, S-, and S-17). Semidiurnal Rssby waves are undefined at these latitudes and diurnal Rssby waves are undefined equatrward f their critical latitudes. Barclinic tpgraphically trapped waves wuld change wavelengths and prpagatin thrugh this latitude range. This did nt happen. Therefre, this wave-like pattern was nt a tpgraphically trapped wave but was mre likely assciated with the tpgraphy, specifically the elngated shape f the seamunt..5. Mean Velcities The tides generated bth bartrpic and barclinic mean velcities. Mean currents are ften frmed due t tidal interactins with seamunts and have been nted previusly fr Fieberling Guyt bth in bservatins (Kunze & Tle, 1997) and in mdel simulatins (Rbertsn, ). The magnitudes f the bartrpic mean currents were latitude dependent, with weaker respnses in the diurnal critical latitude range (Figure 7). The bartrpic mean currents peaked ver the tp f the seamunt, likely due t the shallwer water depths, and were strngest in the dmains.8s.8s, and 38.8S (Figure 7). Nte, that the entire dmain is nt shwn in these figures. The edges f the dmain,.18, were ignred. The barclinic means did nt shw a clear latitude dependence (nt shwn)... Kinetic Energy and Energy Transfers The spatial distributin f kinetic energy varied with latitude mre than the spectra in Figure 5 and supprting infrmatin Figures S-7 and S-8 indicate. The highest spectral energy at the diurnal frequencies (1 cpd) mved deeper in the water clumn with prgressin pleward frm.8s tward the diurnal critical latitudes (Figure 8 and supprting infrmatin Figure S-3 fr the East-West and Nrth-Suth velcities, respectively) based n the barclinic anmalies. The high diurnal energy mrphed frm rays emanating frm the crwn f the seamunt and traveling in accrdance t theretical ray paths equatrward f the ROBERTSON ET AL. CRITICAL LATITUDE, TIDES, AND MIXING 789

13 1.1/1JC1591 Figure 7. Bartrpic mean velcities at each latitude frm.8s t 38.8Sin8 increments with tw extra dmains near the diurnal critical latitudes. diurnal critical latitudes t vertical patches in the lwer water clumn with a benthic peak (Figure 8 and supprting infrmatin Figure S-9). (Nte, in Figures 8 and 9 and supprting infrmatin Figures S-9 S-38 that the y axis is the mdel level, which changes with water depth.) Energy levels were much higher in the East-West, alng-slpe directin (Figure 8) than the Nrth-Suth, acrss-slpe, directin (supprting infrmatin Figure S-9). The spectral energy near the semidiurnal frequencies ( cpd) decreased ROBERTSON ET AL. CRITICAL LATITUDE, TIDES, AND MIXING 785

14 1.1/1JC1591 Figure 8. Latitude-level cuts f the lg 1 f the spectral energy fr the East-West velcity at the diurnal frequency at each latitude frm.8s t 38.8Sin8 increments with tw extra dmains near the diurnal critical latitudes. dramatically in the alng-slpe directin pleward f the K 1 critical latitude, althugh the lcatin f the energy in the water clumn remained cnsistent (Figure 9). In the acrss-slpe directin, the semidiurnal energy peaked at.8s and in the diurnal critical latitude range (supprting infrmatin Figure S-3). Bth ROBERTSON ET AL. CRITICAL LATITUDE, TIDES, AND MIXING 7851

15 1.1/1JC1591 Figure 9. Latitude-level cuts f the lg 1 f the spectral energy fr the East-West velcity at the semidiurnal frequency at each latitude frm.8s t 38.8Sin8 increments with tw extra dmains near the diurnal critical latitudes. respnses ccurred primarily in the upper half f the water clumn ff the sides f the seamunt (Figure 9 and supprting infrmatin Figure S-3). Like the diurnal frequency, semidiurnal energy levels were much higher in the alng-slpe directin (Figure 9) than in the acrss-slpe directin (supprting infrmatin ROBERTSON ET AL. CRITICAL LATITUDE, TIDES, AND MIXING 785

16 1.1/1JC1591 Figure S-9). The spectral energies assciated with the lw frequencies/mean velcities were als higher in the alng-slpe directin than the crss-slpe directin (supprting infrmatin Figures S-31 and S-3). They were als fcused ver the crwn f the seamunt and in the upper water clumn (supprting infrmatin Figures S-31 and S-3). There was n clear latitude dependence f the lw frequency/mean velcities, althugh they did decrease just pleward f the K 1 critical latitude. Bth alng-slpe and acrss-slpe velcities indicated increased energy at and cpd in the diurnal critical latitude range; hwever, there was n significant increase at 3 cpd (supprting infrmatin Figures S-33 S-38 at 3,, and cpd, fr the East-West and Nrth-Suth velcities respectively). At and cpd, the increased energy in the diurnal critical latitude range ccurred ver the crest f the seamunt and in the upper water clumn with mre energy in the acrss-slpe than alng-slpe directin (supprting infrmatin Figures S-3 S-38). The energy at the harmnics was cncentrated near the bttm at the crest f the seamunt and in the upper cean (supprting infrmatin Figures S-33 S-38). The strnger respnse at the harmnics in the critical latitude range near the surface became apparent when spectral density is shwn with levels, such as ver the rim (supprting infrmatin Figure S-39). Frequency-vertical wave number spectra f the barclinic velcity anmalies added insight int the spectral distributin between high and lw mdes. The first vertical mde, with a half f a wavelength thrugh the water clumn is unreslvable using Furier techniques, making mde the lwest reslvable mde. Bth the vertical wave number range and the lwest reslvable wave number increased as water depth decreased frm the basin t the rim f the seamunt. Cnsequently, the wave number range in shallwer water (Figure 1) had higher wave numbers/shrter wavelengths than in deeper water (supprting infrmatin Figure S-). Mst f the spectral energy ccurred in discrete peaks at the lw frequency/mean, diurnal, and semidiurnal frequencies and the lwest mdes up t wavelengths f m ver the rim and t 5 m ver the flank (Figures 1 and supprting infrmatin Figure S- fr sites ver the rim and flank, respectively). Energy increased at the lw frequency/mean, diurnal, and semidiurnal frequencies in the diurnal critical latitude range and decreased pleward f the K 1 critical latitude (Figure 1). Energy at the rim als increased at the harmnic frequencies in the diurnal critical latitude range. And energy appeared at higher wave numbers, reaching 15 cpkm (shrter wavelengths 7 m) in the diurnal critical latitude range at the rim (Figure 1g). (Nte: The 1 cpd and half f the cpkm prtins f the spectra are nt shwn in rder t fcus n the lwer frequency and wave numbers.) There is n apparent respnse at the inertial frequencies at any f these latitudes (arrws n left axis in Figure 1 and supprting infrmatin Figure S-) r ther sites investigated (nt shwn). T illuminate the transfer f energy between frequencies, bispectra were perfrmed n the velcity data. Bispectra indicated strnger energy transfers in the Nrth-Suth, acrss-slpe velcities than in the East-West, alng-slpe velcities. At mst latitudes, interactins near the bttm ccurred between the diurnal, semidiurnal, and mean frequencies (Figures 11b 11l); hwever, at.8s, the primary exchanges ccurred at the semidiurnal and mean frequencies with the diurnals inactive (Figure 11a). Near the bttm, the strngest energy exchanges ccurred at the semidiurnal frequencies between the diurnal critical latitudes (Figure 11). The largest transfers entailed a wide range f frequencies. Higher in the water clumn, strng energy transfers ccurred at.8s,.8s, and 9.8S with energy spreading frm the diurnal and semidiurnal frequencies and mean t a wide range f frequencies (supprting infrmatin Figure S-1). Since this is cut alng levels, slight changes in the vertical extent f the high respnse patches (supprting infrmatin Figure S-) result in larger changes in the energy transfers at that level (supprting infrmatin Figure S-1). T investigate the spatial distributin f the bispectra energy, cuts were made thrugh the bispectra at bth the diurnal (1 cpd) (Figure 1) and semidiurnal ( cpd) (Figure 13) frequencies. The diurnal energy transfers were quite high at all latitudes and peaked at 9.8S arund level fr a site ver the rim (Figure 1). Energy transfers frm the diurnal frequencies spread t all frequencies alng the levels, but shifted frm the upper levels t the lwer levels at and pleward f the diurnal critical latitudes (Figure 1). The semidiurnal energy transfers indicated a clear peak at 9.8S, just equatrward f the K 1 critical latitude, transferring energy t all frequencies fr this site ver the rim (Figure 13). Pleward f the K 1 critical latitude, energy exchanges frm the semidiurnal frequencies were reduced. Interactins were much weaker with the East-West, alng-slpe, velcities than with the Nrth-Suth, acrss-slpe, velcities (supprting infrmatin Figures S- S-51), and were mre cnfined t the lw frequency/mean, diurnal, semidiurnal, and harmnic frequencies, with less energy ging t high frequencies. ROBERTSON ET AL. CRITICAL LATITUDE, TIDES, AND MIXING 7853

17 Jurnal f Gephysical Research: Oceans 1.1/1JC1591 Figure 1. Frequency-vertical wave number cuts f the lg1 f the -D spectral energy fr the Nrth-Suth velcity at level 5 fr a site ver the rim f the seamunt at each latitude frm.8s t 38.8S in 8 increments with tw extra dmains near the diurnal critical latitudes. The inertial frequency fr the latitude is indicated by an arrw n the left side in each figure. ROBERTSON ET AL. CRITICAL LATITUDE, TIDES, AND MIXING 785

18 1.1/1JC1591 Figure 11. Frequency-frequency cuts f the lg 1 f the bispectral energy fr the Nrth-Suth velcity at level 5 fr a site ver the flank f the seamunt at each latitude frm.8s t 38.8Sin8 increments with tw extra dmains near the diurnal critical latitudes. ROBERTSON ET AL. CRITICAL LATITUDE, TIDES, AND MIXING 7855

19 Jurnal f Gephysical Research: Oceans a). S 7. S e) 1.1/1JC S i) 5 Level 3 1 b). S 8. S f) 3. S j) 5 Level c) g) S. S 3. S - k) 5-5 Level (lg1. S d) S h) -1 ((cm s ) -1 S cph ) ) l) Level Frequency (cpd) Frequency (cpd) Frequency (cpd) Figure 1. Level-frequency cuts f the lg1 f the bispectral energy fr the Nrth-Suth velcity at the diurnal frequency at a site ver the rim f the seamunt at each latitude frm.8s t 38.8S in 8 increments with tw extra dmains near the diurnal critical latitudes. ROBERTSON ET AL. CRITICAL LATITUDE, TIDES, AND MIXING 785

20 Jurnal f Gephysical Research: Oceans a). S 7. S e) 1.1/1JC S i) 5 Level 3 1 b). S f) 8. S 3. S j) Level c) g) S. S 3. S -. k) 5-5. Level (lg1. S d) 3. S h) ((cm s ) -1 S cph ) ) l) Level Frequency (cpd) Frequency (cpd) Frequency (cpd) Figure 13. Level-frequency cuts f the lg1 f the bispectral energy fr the Nrth-Suth velcity at the semidiurnal frequency at a site ver the rim f the seamunt at each latitude frm.8s t 38.8S in 8 increments with tw extra dmains near the diurnal critical latitudes. ROBERTSON ET AL. CRITICAL LATITUDE, TIDES, AND MIXING 7857

21 1.1/1JC Vertical Tidal Mixing as Represented by Diffusivities Of primary interest is hw the critical latitude influences mixing. Here mixing is represented by the diffusivity f temperature, K T, as determined by the mdel, althugh the diffusivity f mmentum, K V, was als calculated. Since the diffusivity f temperature and salinity are equivalent in the mdel, K T can be used as a prxy fr the density diffusivity, K q. Mixing ccurs episdically and lcally resulting in a large number f very lw diffusivity values with a smaller number f large values. It is the large values that were f interest. The mean diffusivity at any site is biased by the large number f nearly zer values. Cnsequently, a significant diffusivity, defined as the average f the highest ne-third f the diffusivity values, will be used t characterize diffusivity here. Significant K T were small, less than 1 5 m s 1, away frm the seamunt at mst latitudes (the lg 1 is shwn Figure 1). Higher temperature diffusivities, exceeding m s 1, ccurred ver the crest f the seamunt, in patches alng the sides, and in the bttm 5 7 m. At the K 1 critical latitude, temperature diffusivities exceeded 1 m s 1 ver the entire dmain (Figure 1h). High significant diffusivities ccurred in the midwater clumn between 1, and, m, radiating away frm the seamunt, particularly between.8s and the K 1 critical latitude. An anmalus patch f high diffusivity ccurred where this beam met the edge f the dmain, near the K 1 critical latitude. The highest diffusivities ccurred at the K 1 critical latitude. Pleward f the K 1 critical latitude, diffusivities were smaller and primarily trapped adjacent t the seamunt. The psitin and shape f the high diffusivity patches cincided with velcity shears belw 5 m in the diurnal cnstituents (Figures and 3). Average K T had a similar pattern with smaller magnitudes (supprting infrmatin Figure S-5). These trends als held in the standard deviatins f the diffusivities, indicating that the high diffusivities fluctuate in time, which is expected f tidally driven mixing (supprting infrmatin Figure S-53). Spectra f the diffusivity indicate they respnded t the diurnal and semidiurnal tides bth equatrward and pleward f the diurnal critical latitudes, particularly near the bttm (Figure 15). Additinally, spectral energy ccurred at the harmnics pleward f the critical latitudes, especially ver the rim (Figure 15c). At the K 1 critical latitude, the diffusivity spectra were essentially white, particularly at the rim and crwn, indicating a respnse at all frequencies (Figure 15). T quantify the latitude dependency f the diffusivity, temperature diffusivities were integrated alng a transect acrss the seamunt. The integrated temperature diffusivity peaked at the K 1 critical latitude (red line in Figure 1). Pleward f the critical latitude, the integrated temperature diffusivity drpped ff sharply, then increased with higher latitude after 3.8S. The mmentum diffusivity fllwed the same pattern, but with a higher value (black line in Figure 1). This behavir was partially influenced by the patch f high diffusivity next t the bundary; hwever, it is clear frm Figure 1 that the temperature diffusivity peaks at the K 1 critical latitude. 5. Discussin The behavirs f many parameters were latitude dependent. Linear internal wave thery predicts that the latitudinal dependence shuld fall int three bands: equatrward f critical latitude, at critical latitude, and pleward f critical latitude and these results substantiate this behavir. A summary f the behavir in each f the latitude bands is given in Table 3. Additinally, within the bands, the behavir was spatially dependent bth with distance frm the seamunt and vertically within the water clumn. Detailed analysis f all the spatial dependencies wuld exceed the page limitatins fr this jurnal, s the discussin will primarily fcus n the verall picture. First, whether the diurnal critical latitudes were at their exact latitudes r were shifted will be addressed Shifting f the Diurnal Critical Latitudes An bservatinal study at Fieberling Guyt (383 N) fund prpagating barclinic diurnal tides mre than 8 pleward f the K 1 critical latitude (Kunze & Tle, 1997). They determined that a tidally generated mean/residual rtatinal velcity added enugh relative vrticity t the dynamics t have shifted the K 1 critical latitude pleward f the seamunt s latitude, enabling prpagatin f barclinic diurnal tides. At Barc seamunt, mean velcities were f sufficient magnitude t shift the diurnal critical latitude pleward; hwever, inspectin shwed the means were generally asymmetrical and nt well rganized int a rtatinal flw, particularly in the diurnal critical latitude range (Figure 7). This is attributed t the elliptical ROBERTSON ET AL. CRITICAL LATITUDE, TIDES, AND MIXING 7858