The Generation of Internal Tides at Abrupt Topography

ARTICLE IN PRESS

Deep-Sea Research I 50 (2003) 987–1003

The generation of internal tides at abrupt topography

Louis St. Laurenta,*, Steven Stringerb, Chris Garrettc, Dominique Perrault-Joncasd a Department of Oceanography, Florida State University, Tallahassee, FL 32306, USA b School of Earth and Ocean Sciences, University of Victoria, Victoria, BC, Canada c Department of Physics and Astronomy, University of Victoria, Victoria, BC, Canada d Department of Physics, McGill University, Montreal, QC, Canada

Received23 July 2002; receivedin revisedform 2 April 2003; accepted2 April 2003

Abstract

Internal tide generation is examined for a knife-edge ridge and an abrupt step. The energy flux from a knife-edge ridge with a height much less than the water depth is shown to be twice that from a Witch of Agnesi ridge with the same height but a small slope. In contrast, the energy flux from an abrupt step with an infinitesimal depth change compared to the water depth is the same as from a small slope with the same depth change. For larger topographic heights in both cases, the energy flux from the abrupt topography can significantly exceedthat from gentle topography. The energy flux generated at a top-hat ridge and top-hat trench is also calculated. A top-hat ridge generates more energy flux than a knife edge of equivalent height, though the increase is large only for ridges whose height is small compared to the total depth. Additionally, the energy flux produced by a top-hat ridge is found to be rather insensitive to the ridge width. In contrast, the energy flux generated at a top-hat trench is strongly dependent on width. A knife-edge ridge of moderate height has much of its energy flux in mode 1. For a height to depth ratio comparable to that of the Hawaiian Ridge this fraction is 75%, consistent with observations. We also show that energy flux estimates basedon representing general topography as a number of independent steps are flawed. r 2003 Elsevier Ltd. All rights reserved.

1. Introduction Here, hðx; yÞ is the bottom topography relative to a deep constant-depth level z ¼ÀH; U is the In the stratiﬁedocean, internal tidesare gener- barotropic tidal velocity for the region of depth atedat regions where the barotropic tidalcurrent H; and uðx; y; z; tÞ and wðx; y; z; tÞ are the lateral encounters variations in bottom topography. The andvertical components of the baroclinic wave problem for the wave response involves the response. These waves obey the conventional solution to the momentum, continuity, andbuoy- dispersion relation of internal waves which can ancy equations satisfying the surface ðz ¼ 0Þ and be expressedin terms of wave slope, bottom ðz ¼ÀH þ hðx; yÞÞ boundary conditions 2 2 1=2 k o À f a ¼ ¼ ; ð2Þ wð0Þ¼0; wðÀH þ hÞ¼U Árh þ u Árh: ð1Þ m N2 À o2

*Corresponding author. where a is the wave slope, k and m are the lateral E-mail address: [email protected] (L. St. Laurent). andvertical wavenumbers, and f and N are the

988 L. St. Laurent et al. / Deep-Sea Research I 50 (2003) 987–1003 inertial andbuoyancy frequencies. We will take the kU0=o51is tidal frequency o satisfying the condition XN À2 * *n f oooN so that internal tides are freely radiating. Flinear ¼ F0H knhðknÞh ðknÞDk; ð3Þ Several nondimensional parameters are needed n¼1 to characterize the physical regime of wave where kn ¼ nDk ¼ anp=H denote the wavenum- * generation. One parameter, kU0=o; measures the bers of the resonant modes and h is the Fourier ratio of the tidal excursion length scale U0=o to transform of the topography. Here, the length scale of the topography kÀ1: This 2 2 2 2 1=2 parameter is discussed by Bell (1975) andothers, 1 ððN À o Þðo À f ÞÞ 2 2 F0 ¼ r U0 H ð4Þ and distinguishes a wave response dominated by 2p o the fundamental tidal frequency ðkU0=oo1Þ from is a convenient metric for the energy flux a lee-wave response involving higher tidal harmo- amplitude, with the rest of (3) being nondimen- nics ðkU0=o > 1Þ: A secondparameter, d ¼ h0=H; sional. measures the ratio of the topographic amplitude h0 Internal tide generation at topography of finite to the total depth H: A thirdparameter, e ¼ s=a; steepness was considered by Baines (1982). In that measures the ratio of the maximum topographic study, ray tracing methods were developed to slope s ¼jrhj to the ray slope given by (2). This permit the full range of d and e: The integral parameter also distinguishes two regimes. In the equations derived by Baines (1982) are difficult to case of eo1; the topographic slopes are less steep apply to arbitrary topography, andBaines didnot than the radiated tidal beam, and internal wave examine the sensitivity of the conversion rate to generation is termedsubcritical. In the case of e > the steepness of the topography. Using an 1; the topographic slopes exceedthe steepness of approach similar to Baines, Craig (1987) specifi- the radiated beam and the internal wave genera- cally considered a continental slope of finite tion is termedsupercritical. The critical generation steepness, but only considered ep2: Taking a condition is met when the radiated tidal beam is different approach, St. Laurent andGarrett (2002) alignedwith the slope of the topography. examinedthe first-ordercorrection to linear theory The subcritical generation of internal tides was estimates of energy flux for sinusoidal topography first considered by Cox andSandstrom(1962) , of subcritical steepness ðeo1Þ; finding that Baines (1973), and Bell (1975). These studies 1 examinedsubcritical topography in the limit of F ¼ F 1 þ e2 þ ? : ð5Þ linear 4 d51ande51; for which the bottom boundary condition can be linearized to wðÀHÞ¼U Árh: In Full series expansions for the full corrections for this case, the internal tide generation problem can sinusoidal and gaussian topographies were derived be solvedfor topography of arbitrary shape by by Balmforth et al. (2002) for increasing steepness Fourier decomposition. Later studies have exam- up to the critical condition ðe ¼ 1Þ: They find inedthe initial transient wave response, finite increasedlevels of conversion at critical slopes, depth effects, depth varying stratification, and though the increase varies from only 14% for the spatial variations in topography andtidalforcing gaussian ridge to 56% for the sinusoidal case. (Hibiya, 1986; Khatiwala, 2003; Li, in press; Numerical simulations have allowedcalcula- Llewellyn Smith andYoung, 2002 ; St. Laurent tions of internal tide generation over the full andGarrett, 2002 ). Of central interest in all studies variation of the steepness parameter. Studies by is the tidal conversion rate F; the production of Khatiwala (2003) and Li (in press) also show baroclinic energy as the barotropic tide responds increasedenergy flux production at critical topo- to changes in topography. In the limit of small graphies to levels comparable with those reported tidal excursion ðkU0=o51Þ; the conversion by Balmforth et al. (2002). Both Li (in press) and rate is equal to the energy flux away from the Khatiwala (2003) finda reductionin energy flux topography. Llewellyn Smith andYoung (2002) for supercritical sinusoidal topography. Khatiwala show that the energy flux in the limits of e51 and (2003) also explores isolated ridges, and finds that ARTICLE IN PRESS

L. St. Laurent et al. / Deep-Sea Research I 50 (2003) 987–1003 989 the reduction in energy flux does not occur. He discuss implications of these abrupt topography suggests that the downward propagating energy models. Internal tide energy flux for the Hawaiian from supercritical sinusoidal topography will Ridge is considered using the knife-edge ridge accumulate in valleys as trappedmodes.While model in Section 5a. Finally, the model used by numerical simulations have provided some in- Sjoberg. andStigebrandt(1992) and Gustafsson sights, the properties of internal tides generated at (2001), in which arbitrary topography is repre- supercritical topography are still poorly under- sented as a number of independent steps, is stood. examinedin Section 5b. Conclusions are presented In contrast to studies that explore wave genera- in Section 6. tion for incremental increases in e; models for generation at abrupt depth discontinuities ðe ¼ NÞ have also been formulated. Rattray (1960) con- 2. Internal tide generation at a knife-edge ridge sidered the first-mode baroclinic response to barotropic flow over shelf topography. Stigeb- We consider a knife-edge (zero width) ridge of randt (1980) also considered a topographic step height h0 at x ¼ 0 in an ocean of uniform depth H andexaminedthe full spectrum of baroclinic (Fig. 1). A barotropic tidal current is given by modes. However, both of these studies focus on UðtÞ¼U0 cos ot; taken perpendicular to the ridge. the coastal generation problem, where dC1: Past Previous studies by Larsen (1969) and Robinson studies have not compared the energy flux (1969) have examinedthe scattering of waves, production at abrupt topography with estimates rather than wave generation, from a knife-edge made using linear theory. ridge. While their solutions could be adapted to In the present study, we examine the internal tide the barotropic to baroclinic conversion of the tide, generation problem for a knife-edge ridge and a we will use a straightforwardapproach basedon step. For the case of the knife-edge ridge (Section 2), matching conditions for the modal solutions: we compare estimates of energy flux to linear XN npz theory estimates for the ‘‘Witch of Agnesi’’ ridge. u1 ¼ U0 an cos cos ðknx þ otÞ; H For the case of the step (Section 3), we compare n¼1 estimates of energy flux to linear theory estimates XN npz u ¼ U b cos cos ðÀk x þ otÞ; ð6Þ for a slope. In both cases, we examine the energy 2 0 n H n n¼1 flux over the full range of d; paying particular attention to the limit of d-0 applicable to small XN npz w ¼ aU a sin sin ðk x þ otÞ; amplitude topography in the deep ocean. We 1 0 n H n n¼1 extendthe modelfor the step to examine energy XN npz flux generatedby a top-hat ridge(Section 4.1) and w ¼ÀaU b sin sin ðÀk x þ otÞ; ð7Þ 2 0 n H n a top-hat trench (Section 4.2). In Section 5, we n¼1

z=0

ω ω U = U o cos t u u U = U o cos t

z=-H x=0

Fig. 1. Sketch showing a knife-edge ridge of height h0 in an ocean of depth H: The barotropic tidal current and baroclinic response are denoted by U and u; respectively. ARTICLE IN PRESS

990 L. St. Laurent et al. / Deep-Sea Research I 50 (2003) 987–1003 Z XN ÀHþh where an and bn are the nondimensional coeffi- 0 npz mpz cients of the modal series, np=H are vertical an cos cos dz H H n¼1 ÀH wavenumbers, and kn are the horizontal wave- Z numbers given by (2) as k ¼ anp=H: The sub- 0 npz mpz n þ sin cos dz scripts 1 and2 denote the baroclinic response on ÀHþh0 H H the x 0 and x > 0 sides of the ridge, respectively. Z o ÀHþh0 mpz Here, the modal forms have been chosen to satisfy ¼À cos dz: ð15Þ the boundary condition w ¼ 0atz ¼ 0 and z ¼ ÀH H ÀH: The modal coefficients are obtained through In practice, the summation in (15) is done over a matching conditions at x ¼ 0; finite number of modes, n ¼ 1; 2; y; N: Thus, for any integer choice of m; (15) gives an equation w1 ¼ w2; ÀH þ h0pzp0; ð8Þ with N unknown an coefficients. A coupledset of N equations in N unknowns can be produced by u þ U ¼ u þ U; ÀH þ h z 0; ð9Þ 1 2 0p p taking m ¼ 0; 1; 2; y; N À 1: These can be written in matrix form as u1 þ U ¼ 0; ÀHpzo À H þ h0; ð10Þ Amnan ¼ cm; ð16Þ u2 þ U ¼ 0; ÀHpzo À H þ h0: ð11Þ where

n sin npð1 À dÞ cos mpð1 À dÞÀm cos npð1 À dÞ sin mpð1 À dÞ A ¼ mn ðm2 À n2Þ n À n cos npð1 À dÞ cos mpð1 À dÞÀm sin npð1 À dÞ sin mpð1 À dÞ þ ð17Þ ðm2 À n2Þ

Relations (9)–(11) can be combinedto show and

XN XN sin mpð1 À dÞ npz npz cm ¼ : ð18Þ an cos ¼ bn cos ; ÀHpzp0: ð12Þ m H H n¼1 n¼1 Amn is singular when m ¼ n; so those elements are Through the orthogonality of cosines, (12) implies replacedwith those foundby re-evaluating the an ¼ bn; a result which is also impliedby the integrals on the left-handsideof Eq. (15) after symmetry of the topography. This allows us to setting m ¼ n; giving write all the matching conditions as two state- npd À sin npð1 À dÞ cos npð1 À dÞÀsin2 npð1 À dÞ A ¼ : ments, nn 2n N ð19Þ X npz an sin ¼ 0; ÀH þ h0pzp0; ð13Þ H Also, c is singular at m ¼ 0: This element is n¼1 m replacedby setting m ¼ 0 in the integral on the right-hand side of (15), yielding XN npz an cos ¼À1; ÀHpzo À H þ h0: ð14Þ c0 ¼Àpd: ð20Þ H n¼1 The matrix problem (16) has been solvedfor The coefﬁcients an can now be determined by a 2000 baroclinic modes. The cosine expansion was Fourier series expansion of the terms in (13) and also done for 2000 terms, m ¼ 0; 1; 2; y; 1999: (14). Multiplying by cos mpz=H andvertically This gives a square matrix for Amn; andthe integrating gives coefﬁcients an are given by the matrix inversion ARTICLE IN PRESS

L. St. Laurent et al. / Deep-Sea Research I 50 (2003) 987–1003 991

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 u’ / U0 0

-0.25

-0.5 z / H

-0.75

-1 -1 -0.5 0 0.5 1 x / α -1H

Fig. 2. A snapshot of instantaneous baroclinic velocity (u1 and u2 at ot ¼ 0; 2p; 4p; y; from (6)) normalizedby the barotropic current amplitude, for the knife-edge ridge with d ¼ 0:75: The ridge at x ¼ 0 is shown, and the distance coordinate is normalized by aÀ1H: Dark gray shading indicates regions where u ¼ÀU0:

0 10 0 -5 / F 10 knife F

-10 10

1 0.75 0.5 0.25 0 cumulative F 0 1 2 3 10 10 10 10 mode number Fig. 3. Nondimensional energy ﬂux for 1000 modes generated from the knife ridge with d ¼ 0:75: In the lower portion of the panel, the cumulative sum of the energy ﬂux is shown, normalizedby the total sum in (21).

À1 an ¼ Amncm: In general, only Oð300Þ terms of the For this finite depth system, the conversion rate series expansions are needed to give solutions that is equal to the energy flux away from the comply within 1% of the matching conditions (8)– topography. The depth-integrated energy flux is (11). The higher mode contributions become given by important when d-1: Fig. 2 shows the horizontal Z velocity field( u1 and u2) for the case of d ¼ 0:75: 0 XN / 0 0S À1 2 The rays emanating from the critical generation Fknife ¼ p u dz ¼ F0 n an; ð21Þ point at the ridge tip are clearly visible. For the ÀH n¼1 constant stratification usedin this calculation, the wave energy is concentratedalong the character- where F0 is given by (4) and / Á S denotes an istic linear paths given by dx=dz ¼ 7aÀ1: The average over all components of wave phase. Fig. 3 current amplitudes along the upward and down- shows the nondimensional energy flux for 1000 wardpaths are equal. modes for the case of d ¼ 0:75: The spectrum is ARTICLE IN PRESS

992 L. St. Laurent et al. / Deep-Sea Research I 50 (2003) 987–1003 red, with the energy flux in mode 1 accounting for given by 75% of the total sum in (21). 33=2 h We can compare (21) to the linear theory e ¼ aÀ1 0: ð23Þ 8 b solution for baroclinic energy flux causedby tidal flow over a two-dimensional ridge of varying In the limit of d=e51; (22) reduces to the result for steepness. Specifically, we examine a topographic an infinitely deep ocean presented by Bell (1975), ridge with the ‘‘Witch of Agnesi’’ profile, hðxÞ¼ C 2 2 2 2 À1 Fwitch F0ðp =4Þd : ð24Þ h0ð1 þðx =b ÞÞ (the story behindthe name of this curve is given in Chapter 3 of Singh (1997)). As discussed by Llewellyn Smith andYoung Bell (1975) examinedthe internal tidegeneration (2002), (24) is independent of the width of the by this in an infinitely deep ocean, and Llewellyn sloping region. Smith andYoung (2002) examinedthe modal Fig. 4 shows the ratio of Fknife from (21) to Fwitch response to wave generation in an ocean of finite from (22) for the case of a witch ridge with critical depth. Using their formulation, it is possible to slope, e ¼ 1: Here, we have stretchedthe applic- express the conversion rate for the witch as ability of the linear theory estimate, since (22) is formally validonly for e51: However, this

2 XN estimate for critical generation provides the basis p 2 2 Ànc Fwitch ¼ F0 d nc e for a useful comparison with (21). In the limit of 4 n¼1 d51; the knife produces twice the energy flux 2 2 Àc predicted for the witch. Specifically, we find p 2 c e ¼ F0 d ; ð22Þ F =F is equal to 2:00070:0004 for 10 4 ð1 À eÀcÞ2 knife witch calculations in the parameter range 0:003o do0:03: Thus, it seems that Fknife=Fwitch is exactly where c ¼ð33=2p=4Þðd=eÞ; andthe slope parameter 2 for d51; as Llewellyn Smith andYoung (in for the maximum topographic steepness is press) have now foundanalytically. Since (24) is

14 =1) ε , 12 δ (

witch 10 ) / F δ ( 8 knife F 6

0 0 0.2 0.4 0.6 0.8 1 δ

Fig. 4. Ratio of the knife-edge ridge energy flux to the linear theory estimate of energy flux for a Witch of Agnesi ridge. The witch energy flux was computedfrom (22), with e ¼ 1: ARTICLE IN PRESS

L. St. Laurent et al. / Deep-Sea Research I 50 (2003) 987–1003 993 independent of e; this result is generally applicable horizontal andvertical wavenumbers are given by to the linear theory solution for witch topography kn ¼ anp=H and np=H; respectively. The velocity with d51 of arbitrary steepness. components on the shallow side ðx > 0Þ of the step 0 The energy ﬂux production from the knife are ðu2; w2Þ; with wavenumbers kn ¼ anp=ðH À h0Þ exceeds the linear prediction by more than a factor and np=ðH À h0Þ: The matching conditions at of 4 for d > 0:6(Fig. 4). x ¼ 0 are similar to (8)–(10), with the barotropic velocity on the shallow side of the step scaled by 3. Internal tide generation at a step H=ðH À h0Þ: These can be statedas XN npz XN npz We now consider a topographic step at x ¼ 0; a sin ¼ b sin ; n H n ðH À h Þ where the depth is ÀH on the deep side of the step n¼1 n¼1 0 and ÀH þ h0 on the shallow side of the step À H þ h0pzp0; ð27Þ (Fig. 5). A barotropic tidal current given by UðtÞ¼ XN U cos ot is taken perpendicular to the barrier. npz H 0 1 þ a cos ¼ This problem was examinedby Stigebrandt (1980), n H ðH À h Þ n¼1 0 who assumedthat the wave response occurred XN npz only on the deep side of the step. We allow internal þ b cos ; n ðH À h Þ tides to radiate away from the step in both n¼1 0 directions. In this case, the modal solutions take À H þ h0pzp0; ð28Þ the form: XN XN npz npz 1 þ an cos ¼ 0; ÀHpzo À H þ h0: ð29Þ u1 ¼ U0 an cos cos ðknx þ otÞ; H H n¼1 n¼1 XN npz 0 The terms in (28) and(29) can be multipliedby u2 ¼ U0 bn cos cos ðÀknx þ otÞ; H À h0 cos mpz=H andintegratedvertically. For n ¼ n¼1 ð25Þ 1; y; N modes and m ¼ 1; y; N; this gives a XN npz coupledset of equations for the coefﬁcients am w1 ¼ aU0 an sin sin ðknx þ otÞ; and b which can be written as the matrix problem: H n n¼1 am ¼ Amnbn þ cm: ð30Þ XN npz w2 ¼ÀaU0 bn sin Additionally, multiplying the terms in (27) by H À h0 n¼1 sin lpz=ðH À h0Þ andvertically integrating the set 0 of equations generatedby taking l ¼ 1; y; N gives Â sin ðÀknx þ otÞ: ð26Þ the matrix problem: Here, ðu1; w1Þ are the baroclinic velocity components on the deep side ðxo0Þ of the step where the bn ¼ Bnlal: ð31Þ

z=0

ω H ω U = Uo cos t u u U = U o cos t H-ho

z=-H x=0

Fig. 5. Sketch showing a topographic step of height h0 above a deep ocean of depth H: The barotropic tidal current and baroclinic response are denoted by U and u; respectively. ARTICLE IN PRESS

994 L. St. Laurent et al. / Deep-Sea Research I 50 (2003) 987–1003

The matrices for the two coupledproblems are where / Á S denotes an average over all compo- given by nents of wave phase. Fig. 7 shows the nondimen- 2mð1 À dÞ2ðÀ1Þn sin mpð1 À dÞ sional energy flux for 1000 modes for the case of Amn ¼ ; ð32Þ d ¼ 0:75: As in the case of the knife-edge spectrum, p½m2ð1 À dÞ2 À n2 the step spectrum is redwith mode1 accounting n 2nðÀ1Þ sin lpð1 À dÞ for 75% of the total sum in (36). Fig. 8 shows the Bnl ¼ ; ð33Þ p½n2 À l2ð1 À dÞ2 fraction of energy flux radiated to the shallow side of the step relative to the total flux. In the limit of 2 sin mpð1 À dÞ c ¼ : ð34Þ infinitesimal topography, the energy flux radiated m mpð1 À dÞ to each side is equal, as expected from symmetry. The coefficients for the baroclinic velocity series However, the energy flux radiated to the shallow are solvedby combining (30) and(31) into a single side of the step drops to less than 10% for dX0:25: matrix inversion, The asymptotic behavior of (36) as d-0 can be À1 al ¼ðI À AmnBnlÞ cm; ð35Þ derived through analysis of terms in (32)–(34) and the use of (30) and(31). It can be shown that for where I is the identity matrix. Using (31) and (35), À modes with noOðd 1Þ; the modal coefficients were computed for 2000 n lim an ¼Àlim bn ¼ÀðÀ1Þ d; ð37Þ baroclinic modes. Fig. 6 shows the horizontal d-0 d-0 velocity field( u and u ) for the case of d ¼ 0:75: 1 2 which implies The amplitudes of the baroclinic velocities are largest along characteristic paths, with slope OXðdÀ1Þ 2 À1 7 À1 lim Fstep ¼ F0d n dx=dz ¼ a emanating from the edge of the d-0 step. On the deep-side of the step, the current n¼0 2 À1 amplitudesalong the upwardanddownwardpaths ¼ F0d lnðad Þþsmaller terms: ð38Þ are equal. The leading term in (38) provides an excellent The depth-integrated energy flux is given by approximation to F in the limit of small d when Z step 0 aC3:4: We compare (36) to the linear theory / 0 0S Fstep ¼ p u dz solution for energy flux by tidal flow over a two- ÀH XN dimensional slope up to a shelf. Specifically, we 1 À1 2 2 2 will examine a topographic slope with the ¼ F0 n ½an þð1 À dÞ bn; ð36Þ 2 À1 À1 n¼1 profile hðxÞ¼p h0 tan ðx=bÞ: We note that this

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 u’ / U 0 0

-0.25

-0.5 z / H

-0.75

-1 -1 -0.5 0 0.5 1 x / α-1 H

Fig. 6. A snapshot of instantaneous baroclinic velocity (u1 and u2 at ot ¼ 0; 2p; 4p; y; from (25)), normalizedby the barotropic current amplitude, for a topographic step with d ¼ 0:75: Dark gray shading on the deep side of the step indicates regions where u ¼ÀU0: ARTICLE IN PRESS

L. St. Laurent et al. / Deep-Sea Research I 50 (2003) 987–1003 995

0 10

-4 0 10 / F step F

-8 10 1 0.75 0.5 0.25

0 cumulative F 0 1 2 3 10 10 10 10 mode number Fig. 7. Nondimensional energy ﬂux for 1000 modes generated from the step with d ¼ 0:75: In the lower portion of the panel, the cumulative sum of the energy ﬂux is shown, normalizedby the total sum in (36).

0.5

0.4 (total) 0.3 step

(shallow) / F 0.2 step F

0.1

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 δ

Fig. 8. Fraction of energy flux radiated to the shallow side of the step relative to the total flux. arctan slope is relatedto the spatial derivative (2002) as of the Witch of Agnesi profile, just as the XN 2 À1 À4nd=eð2ÀdÞ step relates to the knife-edge. The linear conver- Fslope ¼ F0d n e sion rate for this slope can be calculatedusing n¼1 2 À4d=eð2ÀdÞ the theory of Llewellyn Smith andYoung ¼ÀF0d lnð1 À e Þ; ð39Þ ARTICLE IN PRESS

996 L. St. Laurent et al. / Deep-Sea Research I 50 (2003) 987–1003

4.5

3.5 =1) ε

, 3 δ (

slope 2.5 ) / F δ ( 2 step F 1.5

0.5

0 0 0.05 0.1 0.15 0.2 0.25 0.3 δ

Fig. 9. Ratio of step generatedenergy ﬂux to the linear theory estimate of energy ﬂux for a slope.

where the slope parameter for the maximum by Fslope with e ¼ 6:8 produces a curve that topographic steepness is given by approaches unity smoothly in the limit of d ¼ 0: The energy flux from the topographic step exceeds h e ¼ aÀ1pÀ1 0: ð40Þ the linear prediction by more than a factor of 4 for b d > 0:3: In the limit of d=e51; the conversion rate is approximatedby 2 À d e 4. Internal tide generation at topography of finite F C À F d2 ln : ð41Þ slope 0 4 d width Furthermore, if both d51 and d=e51; 4.1. A top-hat ridge F CF d2 lnð1 e=dÞ; ð42Þ slope 0 2 The formulation andsolution technique for the which to lowest order in d is the same as (38) for step is readily adapted to the problem of internal the step energy flux. Unlike the case of d51 and tide generation by a top hat, rather than knife- d=e51 for a ridge as in (24), (42) remains edge ridge. As shown in Fig. 10, we still consider a dependent on e=d; andhence the widthof the ridge of height h0 in an ocean of depth H; but now sloping region, even in the limit of an infinitely give it a width 2L: The barotropic tide is, as before, deep ocean. U0 cos ot in regions 1 and3, and H=ðH À h0Þ Fig. 9 shows the ratio of Fstep to Fslope for the times this in region 2. The baroclinic part of the case of a critical slope. In the limit of d51; the step solution in regions 1 and3 is now composedof andthe slope producethe same energy flux, as internal modes propagating away from the ridge, expectedfrom (38) and(42). However, this ratio but in region 2 the baroclinic solution must be increases rapidly as d is increased, with standing waves. Symmetry conditions require that Fstep=FslopeC1:3 for d ¼ 0:001: Normalizing Fstep w ¼ 0atx ¼ 0; so that we may write the baroclinic ARTICLE IN PRESS

L. St. Laurent et al. / Deep-Sea Research I 50 (2003) 987–1003 997

z=0 132

U = U cos ω t H ω U = U cos ω t o u U = U o cos t u o H-ho

z=-H x= -L x=0 x=L

Fig. 10. Schematic of a top-hat ridge of height h0 andwidth2 L: solution in regions 1 and2 as that from the knife edge, as a function of x andfor () various values of the depth ratio d ¼ h =H: As XN npz 0 u ¼ Re U a cos eiðknxþotÞ ; required, the energy flux ratio tends to 1 for an 1 0 n H n¼1 infinitely narrow ridge with x ¼ 0: It is also easily () XN seen from the structure of the problem that the npz 0 iot u2 ¼ Re U0 bn cos cos k xe ; solution is periodic in x; with a periodof 2 p: The H À h n n¼1 0 energy flux is, in fact, periodic with a period of p ð43Þ andsymmetric about x ¼ p=2: It may seem a bit ()physically surprising that for x ¼ p; or any integer XN npz multiple of this, the radiated energy is the same as w ¼ Re ÀiaU a sin eiðknxþotÞ ; 1 0 n H from a knife edge, but the most interesting result n¼1 () is the weak dependence of the flux on the depth XN npz w ¼ Re aU b sin sin k0 xeiot : ratio d: Even for d as small as 0.1, the maximum 2 0 n H À h n n¼1 0 increase in energy flux, over that for the knife edge, is only about 70%, andis much less for a ridge ð44Þ that occupies a significant fraction of the water The solution in region 3 is as in region 1 with a column. The knife edge thus seems to be a good change of sign in kn and w: As for the step, the model for many tall ridges in the ocean. For very total horizontal velocity must match across x ¼ small values of d; the internal tide is as for two ÀL for ÀH þ h0ozo0 andbe zero for steps. However, if x is an integer multiple of p; the ÀHozo À H þ h0; and w must also be contin- waves generatedat each edgeof the ridge uous at x ¼ÀL for ÀH þ h0ozo0: As for the destructively interfere. For x well away from step, this leads to two coupled matrix equations multiples of p; andgiven the asymptotic fluxes for the coefficients an and bn; though these (24) and(38), the energy flux is much greater than coefficients may now be complex. Because of the for a knife edge of the same height. symmetry of the problem, the matching conditions at x ¼ 7L leadto exactly the same matrix 4.2. A top-hat trench equations as for the step. However the total energy fluxP away from the ridge is now given by It is also straightforwardto extendthe above N À1 2 F0 n¼1 n janj : We proceedmuch as before by problem to a trench of depth h0 in an ocean of solving the problem for 2000 modes for a variety depth H elsewhere (Fig. 12). The energy flux of values of the ratio d ¼ h0=H andfor various clearly vanishes for x ¼ 0 at which the trench has values of the width parameter x ¼ apL=H: collapsedto zero width.Interestingly, as a Fig. 11 shows the internal tide energy flux consequence of the periodicity of the problem, radiated from the top-hat ridge compared with this result of zero flux also applies if x is an integer ARTICLE IN PRESS

998 L. St. Laurent et al. / Deep-Sea Research I 50 (2003) 987–1003

3.5

δ=0.001

δ=0.01 2.5 knife /F ridge F 2

δ=0.1

1.5 δ=0.25

δ=0.75 1 0 π/4 π/2 3π/4 π ξ

Fig. 11. Energy ﬂux generated by the top-hat ridge normalized by the energy ﬂux generated by the knife-edge ridge. The nondimensional ridge width parameter is given by x ¼ apL=H:

z=0 1 2 3

ω H ω U = U o cos t ω U = U o cos t U = U o cos t H+ho u u

z=-H

x=-L x=0 x=L

Fig. 12. Sketch showing a top-hat trench of depth h0 (relative to z ¼ÀH) andwidth2 L: multiple of p: Fig. 13 shows the energy flux for a trench, unlike the situation for a ridge, the energy trench comparedwith that for a ridgeof the same flux generatedis a rather sensitive function of the width as a function of the ratio d ¼ h0=H; where h0 width. is the depth of the trench or height of the ridge relative to z ¼ÀH: For small values of d the trench, like the ridge, generates internal tides as 5. Discussion from two steps. For x away from integer multiples of p; the energy flux from the trench is almost the 5.1. The Hawaiian Ridge same as from the ridge. As d increases, the range in x of this near equality is reduced and only holds Our model of internal tide generation at a knife- exactly at x ¼ p=2: The results show that for a edge ridge can serve as a crude model for internal ARTICLE IN PRESS

L. St. Laurent et al. / Deep-Sea Research I 50 (2003) 987–1003 999

0.9 δ=0.01

0.8 δ=0.1 0.7

0.6 ridge

/F 0.5 δ=0.25 trench

F 0.4

0.3 δ=0.75 0.2

0.1 δ=0.99

0 0 π/4 π/2 3π/4 π ξ

Fig. 13. Energy flux generatedby the top-hat trench normalizedby the energy flux generatedby the top-hat ridgeof equal height amplitude. tide generation at the Hawaiian Ridge, or other depth of the Hawaiian Ridge cannot realistically tall ridges. The Hawaiian Ridge has been the focus be characterizedby a single value of d: Also, the of numerous internal tide studies (Ray and barotropic current amplitude varies significantly Mitchum, 1996, 1997; Kang et al., 2000; Pinkel over the Hawaiian Ridge, and a 40% change in the et al., 2000; Merrifieldet al., 2001 ), and Althaus value of U0 results in a factor of two change in et al. (in press) have observedinternal tidesfrom Fknife: Additionally, if WKB stretching of depth is the knife-edge like Mendocino Escarpment. Here, considered to account for varying stratification we approximate the Hawaiian Ridge as a knife- with depth, a more appropriate value of d is edge ridge with d ¼ 0:75 (h0 ¼ 3000 m and given by the ratio of stretchedvariables,R dWKB ¼ # d # #D 0 0 0 H ¼ 4000 m). The nondimensional energy flux ðH ÀðH À h0ÞÞ=H; where z z ðNðz Þ=Nref Þ dz spectrum for this case is shown in Fig. 3.We (Leaman andSanford1975 ). Here, Nref is a now seek a dimensional estimate of the energy reference value of the stratification, which may flux. We use N ¼ 5 Â 10À3 sÀ1 as a rough scale for be taken as the stratification value at the ridge the stratification at 1000 m depth, near the critical crest where the critical internal tide generation generation point on the ridge. We take other occurs. Using an exponential stratification, NðzÞ¼ À4 À1 z=b À1 parameters as o ¼ 1:4 Â 10 s ; f ¼ 5 Â N0e with N0 ¼ 0:00524 s and b ¼ 1300 m; we À5 À1 À1 10 s ; and U0 ¼ 0:02 m s to estimate Fknife ¼ estimate dWKB ¼ 0:44 for the Hawaiian Ridge. À1 À1 11; 000 W m : Taking 2000 km as the Hawaiian With U0 ¼ 0:02 m s ; this implies Fknife ¼ À1 Ridge length, this implies 21 GW of M2 internal 3000 W m ; or roughly 6 GW summedaround tide production. This estimate is close to the the ridge. 20 GW estimatedby Egbert andRay (2000, 2001) While the total energy flux is sensitive to the for the total barotropic tidal conversion occurring choice of parameters, the ratio of the mode-1 at Hawaii, but larger than the model result by energy flux to the total energy flux is not. Fig. 14 Merrifieldet al. (2001) of 9 GW for the energy flux shows this ratio for general d: For the case of d ¼ radiated away from the ridge. We note that our 0:75; mode 1 carries roughly 75% of the total flux. parameter values are somewhat subjective. The Ray andMitchum (1997) have usedsea-surface ARTICLE IN PRESS

1000 L. St. Laurent et al. / Deep-Sea Research I 50 (2003) 987–1003

0.8

(total) 0.6 knife

0.4 (mode 1) / F knife F

0.2

0 0 0.2 0.4 0.6 0.8 1 δ

Fig. 14. Ratio of mode-1 energy flux to the total energy flux for a knife-edge ridge. altimetry data to estimate that 15 GW of mode-1 depth. Furthermore, these studies treat each topo- internal tide energy is radiated by the Hawaiian graphic step as an independent generator of internal Ridge. This is indeed 75% of the Egbert andRay tides, ignoring the interference between internal (2000, 2001) estimate for the total internal-tide tides produced at two or more nearby steps. This energy flux. We note that a knife-edge ridge of simplification is clearly applicable to fjords, where height d ¼ 0:65 generates a mode-1 baroclinic one topographic step is usedto represent a sill. response containing roughly 80% of the total flux. However, multiple steps are needed to represent general topography in the deep ocean, and the 5.2. Internal tide generation at multiple steps neglect of interference phenomena makes the usefulness of the resulting estimates questionable. Following Stigebrandt (1980), several studies We have examinedthe Sjoberg. andStigebrandt have appliedthe result for internal tidegeneration (1992) energy flux estimate for the Witch of Agnesi at a single step to arrangements of multiple steps topography. As discussed previously, the linear (Sjoberg. andStigebrandt,1992 ; Gustafsson, 2001). theory conversion rate for the witch is given by These models attempt to represent arbitrary (22). Here, we will focus on the limit of d=e51; 2 2 topography as a series of vertical pillars, with with Fwitch ¼ F0ðp =4Þd : The witch topography internal tide generation occurring at the steps can be representedas a series of vertical pillars, between them. Stigebrandt (1980) assumedthat and Sjoberg. andStigebrandt(1992) estimate the internal tide energy was radiated to the deep ocean internal tide energy flux from each pillar as side of the step only. He applied this simplification XN 2 in fjords where the radiation of the internal tide À1 sin ðjp½1 À dÞ Fpillar ¼ 2F0 j ; ð45Þ 2 2 2 onto a shallow sill was assumedto be negligible. j¼1 p j ð1 À dÞ However, Sjoberg. andStigebrandt(1992) and Gustafsson (2001) apply this simplification to where the sum is over modes with index j: For each topographic steps in the deep ocean where the pillar, d is taken as the difference in height from deep and shallow sides of the step are of similar the neighboring pillar normalizedby the water ARTICLE IN PRESS

L. St. Laurent et al. / Deep-Sea Research I 50 (2003) 987–1003 1001

1.5 1.5 ΣF / F =0.81 ΣF / F =1.34 n= 3 pillar witch n=11 pillar witch

o 1 o 1 z/h z/h 0.5 0.5

0 0 -100 0 100 -100 0 100 (a) x/H (b) x/H

1.5 1.5 ΣF / F =0.85 ΣF / F =0.46 n=21 pillar witch n=41 pillar witch

o 1 o 1 z/h z/h 0.5 0.5

0 0 -100 0 100 -100 0 100 (c) x/H (d) x/H 1.5

witch 1 / F

pillar 0.5 F

0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 (e) 1/n Fig. 15. Calculations using the pillar model of Sjoberg. andStigebrandt(1992) , appliedto the Witch of Agnesi ridge.Panels a–dshow successively increasing pillar representations of the same topography. The number of pillars ðnÞ is reportedin the upper left corner of each panel, andthe total pillar energy flux to the witch-topography energy flux is shown in the upper right. Panel e shows the energy flux ratio for pillar representations with n ¼ 3–5000. depth above the pillar. In the case where the for an increasing number of pillars ðnÞ: In each difference in height between pillars is small case, (45) was usedtoP estimate the energy flux for comparedto the total depth ðd51Þ; (45) can be each pillar. The ratio Fpillar=Fwitch is reportedin expressedas upper right of each panel. The witch is coarsely resolvedby 3 pillars in panel a, andpanels b–d OXðdÀ1Þ show successively more numerous pillars with F ¼ 2F d2 jÀ1ð1 þ Oð2dÞÞ: ð46Þ pillar 0 smaller steps which resolve the topography with j¼1 Pincreasedaccuracy. The general trendis that À1 Here, the approximation is validout to jCOðd Þ: Fpillar approaches zero linearly as the number Sjoberg. andStigebrandt(1992) and Gustafsson of steps is increased(panel e). We note that (2001) truncate the summation in (45) at mode 10. Sjoberg. andStigebrandt(1992) and Gustafsson PThe total energy-flux (per unit cross-stream width) (2001) represent an obstacle’s total height ðh0Þ with À1 Fpillar is calculatedby summing the conversion n steps, such that d for each pillar is H ðh0=nÞ: It rates of all the pillars. follows from (46) that Fig. 15 shows the Witch of Agnesi topography X representedas a series of vertical pillars. Four h2 F pn Â HÀ2 0 ln n: ð47Þ different representations of the witch are shown pillar n2 ARTICLE IN PRESS

1002 L. St. Laurent et al. / Deep-Sea Research I 50 (2003) 987–1003

Since Sjoberg. andStigebrandt(1992) and Gus- for large amplitude ridges such as Hawaii, the tafsson (2001) truncate the summation at j ¼ 10; internal tide energy flux is largely determined by their energy flux estimates are proportional to nÀ1 the overall topographic height, andnot by finer À1 ratherP than n ln n: Fig. 15e confirms that scale roughness along the ridge. This contrasts the À1 Fpillarpn ; causing the energy flux estimate internal tide generation occurring at mid-ocean to approach zero as the number of pillars becomes ridges, where the energy flux is produced by large. This is a general problem, not relatedto roughness at all scales (e.g., St. Laurent and truncating the sum in (46), andnot limitedto any Garrett, 2002). specific topographic shape. The case of a topographic step was also Thus, in cases where a series of steps are needed considered. In the limit of infinitesimal topogra- to represent topography, the estimates of Sjoberg. phy, the energy flux of the step equals the linear andStigebrandt(1992) and Gustafsson (2001) will theory prediction for a slope of critical steepness. be sensitive to the number of steps usedto resolve Energy flux estimates for step topography have the bathymetry. While the energy flux estimates been previously employedin calculations by presentedin these studies are plausible, their Sjoberg. andStigebrandt(1992) and Gustafsson results must be viewedas fortuitous. A valid (2001). In those studies, deep ocean topography model for internal tide generation at a series of was representedas pillars of varying height, and steps must include interference phenomena occur- estimates of internal tide energy flux were made at ring between internal tides generated at neighbor- the steps between adjacent pillars. Since Sjoberg. ing steps. Interference effects allow for andStigebrandt(1992) and Gustafsson (2001) convergence of internal tide estimates as the regard each step as an independent generator of number of steps is increasedto better resolve the the tides, their estimates are entirely dependent on topography. the number of pillars usedto represent the topography. In the limit of many pillars for which the topography is finely resolved, the Sjoberg. and 6. Conclusion Stigebrandt (1992) and Gustafsson (2001) calculations predict a vanishing energy flux. A knife-edge ridge may be regarded as a simple For the case of mid-ocean ridges such as the model for an isolated ridge of supercritical Mid Atlantic Ridge, changes in depth over the steepness such as Hawaii. We have foundthat in lateral scale of the mode-1 tidal wavelength the limit of infinitesimal amplitude topography, a ðOð150 kmÞÞ are generally less than 500 m: This knife-edge ridge produces exactly twice the provides a rough estimate for the height h0 of amount of energy flux as the linear prediction for topographic obstacles along the mid-ocean ridge. critical topography. We contrast this to the Furthermore, the mean depth H of a mid-ocean estimate of Balmforth et al. (2002) for a gaussian ridge system can vary from 3000 to 5000 m: Thus, ridge. They find that at critical steepness, the 0:1odo0:2 is a rough estimate of the amplitude energy flux produced by the ridge is 1.14 times range for typical mid-ocean ridge topographies. greater than the linear prediction. Generalizing, we These regions are also characterizedas subcritical conclude that in the infinitesimal topography limit (St. Laurent andGarrett, 2002 ). It therefore seems ðd51Þ; increasing the steepness from e ¼ 1toe ¼ reasonable to use linear theory for the internal N increases the power production by only a tides generated at mid-ocean ridge topographies. further 75%. However, we note that for a knife- The complex topography at these sites can then be edge ridge of finite amplitude, the energy flux can modeled as individual Fourier components, and considerably exceed the linear theory estimates. the total baroclinic response is determined through We have further found that the knife-edge ridge is superposition. The simple models for knife-edge, an efficient generator for the first baroclinic mode. step, andtop-hat topography are not useful for For 0:35odo0:9; mode-1 accounts for 50% or calculating internal tide energy flux at these more of the total energy flux. It seems likely that sites. Additionally, the methods of Sjoberg. and ARTICLE IN PRESS

L. St. Laurent et al. / Deep-Sea Research I 50 (2003) 987–1003 1003

Stigebrandt (1992) and Gustafsson (2001) will not Gustafsson, K.E., 2001. Computations of the energy flux to produce reliable estimates of energy flux for mid- mixing processes via baroclinic wave drag on barotropic ocean ridge regions. tides. Deep-Sea Research 48, 2283–2295. Hibiya, T., 1986. Generation mechanism of internal waves by In contrast, ridges associated with oceanic tidal flow over a sill. Journal of Geophysical Research 91, islandandtrench systems, such as the Hawaiian 7696–7708. andAleutian Ridges, are both large amplitude Kang, S.K., Foreman, M.G.G., Crawford, W.R., Cherniawsky, ð0:5odo1Þ andsupercritical. Linear theory for the J.Y., 2000. Numerical modeling of internal tide generation internal tides will significantly under-estimate the along the Hawaiian Ridge. Journal of Physical Oceano- graphy 30, 1083–1098. energy flux produced at these topographies. We Khatiwala, S., 2003. Generation of internal tides in the ocean. propose that the simple model for the knife-edge Deep-Sea Research I 50, 3–21. ridge can serve as the basis for energy flux Larsen, L.H., 1969. Internal waves incident upon a knife edge estimates for these features. barrier. Deep-Sea Research 16, 411–419. Leaman, K.D., Sanford, T.B., 1975. Vertical energy propagation of inertial waves: a vector spectral analysis of velocity profiles. Journal of Geophysical Research 80, Acknowledgements 1975–1978. Li, M. Energetics of internal tides radiated from deep-ocean The authors acknowledge the support of the US topographic features. Journal of Physical Oceanography, Office of Naval Research andthe Natural Science submittedfor publication. Llewellyn Smith, S.G., Young, W.R. 2002. Conversion of the andEngineering Council of Canada. DP-J thanks barotropic tide. Journal of Physical Oceanography 32, NSERC for an Undergraduate Student Research 1554–1566. Award. We thank Eric Kunze, and three anon- Llewellyn Smith, S.G., Young, W.R. Tidal conversion by a very ymous reviewers for helpful comments. steep ridge. Journal of Fluid Mechanics, submitted for publication. Merrifield, M.A., Holloway, P.E., Shaun Johnston, T.M., 2001. The generation of internal tides at the Hawaiian Ridge. References Geophysical Research Letters 28, 559–562. Pinkel, R., Munk, W., Worcester, P., et al., 2000. Ocean mixing Althaus, A.M., Kunze, E., Sanford, T.B. Internal tide radiation studied near Hawaiian Ridge. EOS, Transactions of the from Mendocino Escarpment. Journal of Physical Oceano- American Geophysical Union, 81, pp. 545, 553. graphy, in press. Rattray, M., 1960. On the coastal generation of internal tides. Baines, P.G., 1973. The generation of internal tides by flat- Tellus 12, 54–61. bump topography. Deep-Sea Research 20, 179–205. Ray, R., Mitchum, G.T., 1996. Surface manifestation of Baines, P.G., 1982. On internal tide generation models. Deep- internal tides generated near Hawaii. Geophysical Research Sea Research 29, 307–338. Letters 23, 2101–2104. Balmforth, N.J., Ierley, G.R., Young, W.R., 2002. Tidal Ray, R., Mitchum, G.T., 1997. Surface manifestation of conversion by nearly critical topography. Journal of internal tides in the deep ocean: observations from Physical Oceanography 32, 2900–2914. altimetry andislandgauges. Progress in Oceanography 40, Bell, T.H., 1975. Lee waves in stratifiedflows with simple 135–162. harmonic time dependence. Journal of Fluid Mechanics 67, Robinson, R.M., 1969. The effects of a barrier on internal 705–722. waves. Deep-Sea Research 16, 421–429. Cox, C.S., Sandstrom, H., 1962. Coupling of surface and St. Laurent, L., Garrett, C., 2002. The role of internal tides in internal waves in water of variable depth. Journal of the mixing the deep ocean. Journal of Physical Oceanography Oceanographic Society of Japan 20th Anniversary Volume, 32, 2882–2899. 499–513. Singh, S., 1997. Fermat’s Enigma: The Quest to Solve the Craig, P.D., 1987. Solutions for internal tide generation World’s Greatest Mathematical Problem. Walker and Co., over coastal topography. Journal of Marine Research 45, New York, 288pp. 83–105. Sjoberg,. B., Stigebrandt, A., 1992. Computations of the Egbert, G.D., Ray, R.D., 2000. Significant dissipation of tidal geographical distribution of the energy flux to mixing energy in the deep ocean inferred from satellite altimeter processes via internal tides and the associated vertical data. Nature 405, 775–778. circulation in the ocean. Deep-Sea Research 39, 269–291. Egbert, G.D., Ray, R.D., 2001. Estimates of M2 tidal energy Stigebrandt, A., 1980. Some aspects of tidal interaction with dissipation from TOPEX/POSEIDON altimeter data. Jour- FjordConstrictions. Estuarine andCoastal Marine Science nal of Geophysical Research 106, 22475–22502. 11, 151–166.