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Deep-Sea Research I ] (]]]]) ]]]–]]]

Internal tide generation by seamounts

Peter G. Bainesa,b,!

aDepartment of Civil and Environmental Engineering, University of Melbourne, Melbourne 3010, AustraliabQUEST, Department of Earth Sciences, Bristol BS8 1RJ, UK

Received 12 September 2006; received in revised form 15 May 2007; accepted 18 May 2007

Abstract

The generation of internal tidal wave fields by barotropic tidal flow past a representative seamount is computed bymodelling the seamount as a pillbox, and linearising the equations for internal wave dynamics. This is justifiable for mid-ocean seamounts, which constitute steep topography for internal waves of tidal frequency. For linearly polarisedbarotropic tidal flow, the resulting flow field consists of conical beams radiating from the region above the seamount, withlargest velocities aligned with the barotropic flow. These beams vary with azimuthal angle but resemble the correspondingbeams from two-dimensional steep topography, particularly in the barotropic flow direction. They are primarily forced bythe barotropic flow over the seamount, which is amplified by the topography and is independent of the stratification if theradius of the seamount is sufficiently large. In a barotropic tidal flow of 1 cm/s amplitude, energy fluxes from individualseamounts are of order 106W. Summing this over all seamounts higher than 1 km gives baroclinic energy generation oforder 5.109W, a number that is less than estimates of baroclinic energy flux from the continental slopes and the Hawaiianridge, but is comparable with them.r 2007 Elsevier Ltd. All rights reserved.

1. Introduction

The generation of internal tides in the deep ocean isof considerable current interest for several reasons, asfollows. Recent observations have shown them tohave larger amplitudes than previously expected(Morozov, 1995), most notably near the Hawaiianridge (Ray and Mitchum, 1997, Rudnick et al., 2003).Further, in conjunction with the presence of topo-graphy, they are thought to be important in providingan energy source for vertical mixing in the deep ocean

(Munk and Wunsch, 1998; Egbert & Ray, 2000).Internal tides are generated by the tidal flow of thedensity-stratified ocean over bottom topography, andthe first stage in understanding their effects is tounderstand and be able to calculate this generationprocess. To date, most theoretical studies have focusedon generation from two-dimensional topography, andfor decades much attention has been focused ongeneration from coastal topography, beginning withRattray (1960). The relatively small number of studiesof generation by three-dimensional topography havebeen with linearised bottom topography (Bell, 1975;Balmforth et al., 2002; Llewellyn Smith and Young,2002) or numerical models (Holloway and Merrifield,1999; Munroe and Lamb, 2005). A recent review ofthe field is given by Garrett and Kunze (2007).

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!Department of Civil and Environmental Engineering, Uni-versity of Melbourne, Melbourne 3010, Australia.Tel.: +613 83447548; fax: +61 3 83444616.

E-mail address: [email protected]

Please cite this article as: Baines, P.G., Internal tide generation by seamounts. Deep-Sea Research I (2007), doi:10.1016/j.dsr.2007.05.009

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http://dx.doi.org/10.1016/j.dsr.2007.05.009

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For topography of finite height, the mathematicalproblem is very different depending on whether thetopography is ‘‘flat’’, in which the slope of the rays(group velocity vectors) of the internal tide is alwayssteeper than the topography that they encounter(Baines, 1973), or ‘‘steep’’, where the topography issteeper in some locations (e.g. Baines, 1974). In thelatter case, the problem can be sometimes simplifiedby assuming that the topography has vertical sides.If one wishes to apply such a model to topographywith non-vertical sides, the solution obtained shouldhave small amplitude in the region of difference,and this constitutes a check on this approximation.In the case of two-dimensional topography, it isseen to work quite well (Prinsenberg and Rattray,1975). The non-horizontal terms of the Coriolisforce may also be included in linear models (Bainesand Miles, 2000); these cause changes to thelocations of the rays, but the extra complexity isnot justified for present purposes and they are notincluded here.

This paper presents a study of baroclinic tidalgeneration for steep-sided axisymmetric topographythat approximates typical seamounts. The equationsare linearised, and the seamount is assumed to havea flat top or summit, with vertical sides or flanks. Inspite of these approximations, this model is believedto provide a good description of the overallproperties of the flow, and to give a soundly basedestimate of the energy fluxes produced. The modeland mathematical details are described in Section 2,and the results for individual seamounts arepresented in Section 3. This includes a descriptionof the flow over and around the seamount, thedependence of the energy flux on the relevantdimensionless parameters, and results for differentupper level stratification. In Section 4, these resultsare used to integrate the energy flux over (modelrepresentations of) the whole seamount populationin the deep ocean. These results are compared withcalculations by others (Llewellyn Smith and Young,2002, 2003; St. Laurent et al., 2003) for linearisedor two-dimensional topography in section 5, andthe conclusions are summarised and discussed inSection 6.

2. The model and equations

Seamounts are small topographic features in theocean compared with the scale of ocean basins, andwith the scale of variations of the barotropic tide.Horizontal scales are generally less than 100 km.

Accordingly, within the vicinity of an individualseamount situated on an otherwise flat ocean floor,the background barotropic tide may be assumed tobe spatially uniform. Barotropic tidal motiongenerally has the form of an ellipse for the velocityvectors and the particle displacement, for each tidalcomponent. This may be represented as the sum oftwo rectilinear motions at right angles. If theresulting baroclinic motions are small enough tobe regarded as linear, the response to the ellipticbarotropic tide may be represented as the sum of theresponses to these two rectilinear motions. Hence, itsuffices to consider the baroclinic generation from asingle rectilinear barotropic tidal motion, and sumthe results from the two orthogonal constituents toobtain the net results for forcing by the ellipticalmotion.

One conceptual approach to internal tide genera-tion is to first obtain the flow pattern that wouldexist if the ocean were unstratified, which could betermed the virtual barotropic flow. The advection ofdensity surfaces by this flow constitutes the forcingfor the baroclinic tide (Baines, 1973). In the case ofa non-rotating ocean, small-amplitude periodic flowpast a seamount has the form of three-dimensionalpotential flow in homogeneous fluid of finite depth.In a rotating homogeneous ocean in which potentialvorticity is conserved, the situation is more complex,and this computational procedure is unattractive.A better way of obtaining the internal tide is toconsider the full stratified problem in which onlythe background barotropic flow is specified. Thebarotropic flow over and in the vicinity of theseamount is then determined as part of the solution,and is dependent on the stratification, as seen below.

We therefore assume that the background baro-tropic velocity ub has the form

ub ! U x cos ot, (2.1)

where U is the amplitude of the rectilinear motion,o is the tidal frequency, and x is the unit vector inthe x-direction. The equations for the motion of anincompressible density-stratified ocean, assumingthat these are small enough to be linear, are

quqt

" f xu ! #1

r0rp#

gr0

r0, (2.2)

qr0

qt" w

dr0dz

! 0; r $ u ! 0, (2.3)

where r0 is the perturbation density with r0 amean density, p pressure, g gravity, and

ARTICLE IN PRESSP.G. Baines / Deep-Sea Research I ] (]]]]) ]]]–]]]2




f ! 2O sin(latitude)z, where O is the angular velocityof the rotation of the Earth. The virtual barotropictide mentioned above is governed by these equa-tions with the buoyancy term in (2.2) omitted. Herewe consider plane polar coordinates (r,y), where r isradial horizontal distance from an origin, and y isangular distance from the positive direction of thex- axis. From (2.2) the pressure field for theunidirectional background barotropic flow (2.1) isthen given by

p ! pb ! r0oUr i cos y#f

osin y

! "e#iot, (2.4)

where the real part is taken.For linearised baroclinic motion with pressure p

and vertical velocity w and time dependence e#iot,Eqs. (2.2) and (2.3) give

r2hw#

o2 # f 2

N z% &2 # o2wzz ! 0, (2.5)

r2hp# o2 # f 2

# $ qqz

pzN z% &2 # o2

! "! 0, (2.6)

where r2h is the horizontal Laplacian, and N(z) is the

buoyancy frequency defined by

N z% &2 ! #g

r0

dr0dz

. (2.7a)

The slope of the rays, or group velocity vectors ofinternal waves of frequency o in fluid with buoy-ancy frequency N is c, where

c2 !o2 # f 2

N2 # o2. (2.7b)

Horizontal velocity components (ur, uy) in theradial and y-directions are then given by

ur !1

r0 o2 # f 2# $ #io

qpqr

"f

r

qpqy

! ",

uy ! #1

r0 o2 # f 2# $ f

qpqr

"ior

qpqy

! ", %2:8&

and the vertical velocity by

w !1

r0

ioN z% &2 # o2# $ qp

qz. (2.9)

We consider seamounts with axisymmetric geo-metry, and take the coordinate origin at the centreof the seamount as shown in Fig. 1. Each seamountis assumed to have the shape of a ‘‘pillbox’’ —axisymmetric with a flat top and vertical sides. Thisis justified because seamounts are generally steep

(typical slope angle 171, Wessel, 2001) and are muchsteeper than the slope c of the rays, which aretypically 3–61. Hence they may be assumed to beaxisymmetric ‘‘steep topography’’ in the language ofprevious two-dimensional studies (e.g. Baines, 1973,1974). The region on the top of the seamount hassmall slope, and is adequately represented by ahorizontal surface. The geometry of the rays for theinternal tide, therefore, is well-represented by thatfor a pillbox, for a typical seamount. This assump-tion is assessed below when the solutions areobtained. The pillbox has radius a and height hmabove the horizontal ocean bottom, in an ocean oftotal depth De. The coordinate origin is taken at thecentre of the seamount, at the ocean surface.

The mathematical problem for the internal tidethen takes the following form. The total tidalvelocity u is equal to

u ! ub " ui, (2.10)

where the background barotropic flow ub ishorizontal and given by (2.1) and ui denotes theforced response due to the seamount.ui ! (uir, uiy,w) in polar coordinates, with theboundary conditions

w ! 0 on z ! 0;

z ! #%De # hm&; 0oroa;

z ! #De; r4a;

(2.11)

and

ur ! 0; uir ! #ubr on r ! a,

#Deozo# %De # hm&. %2:12&

ARTICLE IN PRESS

De

z

x

hm

r

!

Ocean surface

Ocean bottoma

Fig. 1. The pillbox model of a seamount, with coordinate system.

P.G. Baines / Deep-Sea Research I ] (]]]]) ]]]–]]] 3




Note that ui will contain a non-zero verticallyaveraged (barotropic) component, to be determinedas part of the solution. We also require radiationconditions for the far field (i.e. large r) which implythat there is no incoming internal wave energy frominfinity.

We obtain solutions in terms of the pressure, andwrite p ! pb+pi. It is then convenient to write pbfrom (2.4) in the form

pb ! r0Uoar

a

i

21"

f

o

! "eiy

%

" 1#f

o

! "e#iy

&e#iot, %2:13&

so that pi may be expressed

pi ! r0Uoa'pp r; z% &eiy " pm r; z% &e#iy(e#iot, (2.14)

the subscripts p and m notionally denoting ‘‘plus’’and ‘‘minus’’ the iy exponent. The conditions on wthen imply that

qppqz ! 0; qpm

qz ! 0 on z ! 0;

z ! #%De # hm&; 0oroa;

z ! #De; r4a;

(2.15)

and

qppqr

#f

oapp ! #

i 1# f 2

o2

' (

2aon r ! a,

#Deozo# %De# hm&, %2:16&

and similarly for pm with (#f) replacing f.We will beconsidering different forms of the solution in theregions r4a, and 0oroa above the seamount, andthe conditions of continuity of ur and p imply thatpp and pm and their derivatives must be continuousacross r ! a in the range – (De#hm)ozo0. Hencepp and pm are calculated separately.

For the motion above the summit of theseamount, there are two possibilities. The first isto assume that the wave field there is essentiallyinviscid, and may be regarded as a sum of discretemodes that do not contain singularities at the origin.This is mathematically simple, but as will be seen,the accurate description of the flow requires manyvertical modes (typically, about 100 or more), andfor many of these modes, the wavelengths are veryshort if the depth of the top of the seamount isshallow. Consequently, dissipation is likely to besignificant, and it seems doubtful that in practice all

of these modes will be coherent over the wholeseamount summit, particularly if the seamount isnot exactly circular. A more realistic option may beto assume that the baroclinic motion over theseamount is generated at the seamount edge,propagates inward and is dissipated without sig-nificant internal reflection occurring from thesummit rim. Both of these cases are computed here.The real situation for a given seamount may be amixture of the two—a small number of modes (oneor two?) may have little dissipation and havestanding—wave character over the seamount, withthe rest having inward energy propagation from theouter edge, or summit rim, that is dissipated beforereaching the far side.

From (2.6), both pp and pm satisfy

1

r

qqr

rqppqr

! "#

ppr2

# o2 # f 2# $ q

qzppz

N z% &2 # o2

! "! 0.

(2.17)

This may be solved by separation of variables:pp ! R(r)Z(z), giving

Rrr "1

rRr " m2 #

1

r2

! "R ! 0,

1

N z% &2 # o2Zz

! "

z

"m2

o2 # f 2Z ! 0, %2:18&

where the subscripts denote derivatives and

Zz ! 0 on z ! 0; #De for the deep water; and

Zz ! 0 on z ! 0; #%De # hm&for the shallow water over the seamount. %2:19&

These conditions on Z determine the eigenvaluesmsj on the shallow side, and mdj on the deep side,where j ! 1, 2, 3,y.m ! 0 is also an eigenvalue,and corresponds to the barotropic mode. For agiven value of m, the solutions for R(r) are Besselfunctions, except for m ! 0, when they are R(r) ! rand 1/r.

Hence, if we assume that the dynamics areinviscid and the internal wave motion over theseamount consists of standing waves (i.e. ‘‘sloshingmodes’’), we may express the solution in the form

pp r; z% & ! C0r

a"X1

j!1

CjJ1 msjr# $

Zsj z% &; 0oroa,

! D0a

r"X1

j!1

DjH%1&1 mdjr# $

Zdj z% &; r4a,

%2:20&





and similarly for pm(r,z) but with Sj, Ej replacing Cj,Dj, respectively, for j ! 0, 1, 2,y. Here J1 denotesthe familiar Bessel function, and H %1&

1 the Hankelfunction of the first kind. The latter functionguarantees that each internal wave mode in thedeep water propagates energy away from theseamount to infinity.

The above formulation of the problem is labelledas Case J. If we instead take the alternativeformulation where the waves over the seamountare assumed to propagate inward and dissipate, thecorresponding equations are:

pp r; z% & ! C0r

a"X1

j!1

CjH2% &1 msjr# $

Zsj z% &; 0oroa,

! D0a

r"X1

j!1

DjH1% &1 mdjr# $

Zdj z% &; roa,

%2:21&

and similarly for pm(r,z), with Sj ; Ej replacingCj ; Dj, respectively, for j ! 0, 1, 2,y. This secondcase is known as Case H(2) (where these labels referto the Bessel functions involved over the summit).Note that H %2&

1 becomes singular at the origin r ! 0,and hence does not give a realistic description of thebaroclinic motion over the seamount. However, thisbaroclinic motion is found to be small and relativelyinconsequential, so that this consideration is notsignificant. This H(2) model is preferred for thecalculation of the baroclinic tide in the deep water.

In order to complete the solutions, it remains todetermine the constants Cj, Dj, etc. from theboundary conditions at r ! a. It is readily shownthat, for arbitrary N(z), Zdj(z) and Zsj(z) have thefollowing properties (see for example LlewellynSmith and Young, 2002):

Z 0

#De

Zdj z% &dz ! 0; i ! 1; 2; 3; . . . ,

Z 0

#De

Zdi z% &Zdj z% &dz ! 0; if iaj,

m2dj

Z 0

#De

Z2dj dz

! o2 # f 2# $ Z 0

#De

1

N z% &2 # o2

dZdj

dz

! "2

dz,

%2:22&

with the same applying to Zsj but with De#hmreplacing De in the above. Continuity of pp across

r ! a above the seamount then gives, for Case J,

C0 "X1

j!1

CjJ1 msja# $

Zsj z% &

! D0 "X1

j!1

DjH1% &1 mdja# $

Zdj z% &,

# %De # hm&ozo0. %2:23&

Integrating this equation over the range–(De#hm)ozo0 gives

C0 ! D0 "1

De # hm

X1

j!1

g0jDj . (2.24)

Multiplying (2.23) by Zsk(z) and integrating overthe same range gives

Ck !1

bk

X1

j!1

gkjDj ; k ! 1; 2; 3; . . . , (2.25)

where

bk ! J1 mska% &Z 0

# De#hm% &Z2

sk dz,

g0j ! H 1% &1 mdja# $ Z 0

# De#hm% &Zdj dz,

gkj ! H 1% &1 mdja# $ Z 0

# De#hm% &ZskZdj dz. %2:26&

The normal velocity condition ((2.15), (2.16))gives

#D0 1"f

o

! ""X1

j!1

) )mdjaH0 1% &1 mdja# $%

#f

oH 1% &

1 mdja# $&

DjZdj z% &

! #i

21#

f 2

o2

! "; #Deozo# %De # hm&,

! C0 1#f

o

! ""X1

j!1

) )msjaJ01 msja# $%

#f

oJ1 msja# $&

CjZsj z% &; #%De # hm&ozo0,

%2:27&

where primes denote derivatives of Bessel functions.Multiplying this equation by Zdk(z) and integratingfrom #De to 0 gives, for the barotropic motion withk ! 0, Zd0 ! 1,

D0 !i

2

hmDe

1#f

o

! "#

1# fo

1" fo

1#hmDe

! "C0, (2.28)

ARTICLE IN PRESSP.G. Baines / Deep-Sea Research I ] (]]]]) ]]]–]]] 5




and for k ! 1, 2, 3,y,

Dk mdkaH%1&01 mdka% & #

f

oH 1% &

1 mdka% &% & Z 0

#De

Z2dk dz

! #i

21#

f 2

o2

! " Z # De#hm% &

#De

Zdk dz

" C0 1#f

o

! " Z 0

# De#hm% &Zdk dz

"X1

j!1

msjaJ01 msja# $

#f

oJ1 msja# $% &

* Cj

Z 0

# De#hm% &ZsjZdk dz: %2:29&

Substituting C0 and Cj from (2.24) and (2.25) inthese equations gives

XMd

k!1

djkAdj # gd0jg0k #XMs

i!1

gdijgik

" #

Dk ! Qdj ,

j ! 1; 2; . . . ; Md , %2:30&

where djk is the Kroneker delta: djk ! 1 if j ! k,djk ! 0 otherwise, and

Adj ! mdjaH1% &01 mdja

# $#

f

oH 1% &

1 mdja# $% & Z 0

#De

Z2dj dz,

(2.31)

Qdj !i 1# f 2

o2

' (

2# hmDe

1# fo

' (Z 0

# De#hm% &Zdj dz,

gd0j !1# f 2

o2

2# hmDe

1# fo

' ( 1

De # hm

Z 0

# De#hm% &Zdj dz,

%2:32&

gdij !1

bimsiaJ1 msia% & #

f

oJ1 msia% &

% & Z 0

# De#hm% &ZsiZdj dz.

(2.33)

Here, instead of being taken from 1 to infinity,from the viewpoint of summing these seriesnumerically the sums are taken from 1 to Md orMs, depending on whether the sum is over modes inthe deep water, or in the shallow water over theseamount. From both practice and common sense,Md and Ms need to reflect the same resolution, orsame spatial scale, so that, in general,

Ms + Md%1# hm=De&. (2.34)

Eq. (2.30) is then solved for the Dj at the requiredresolution, and then D0, C0 are obtained from (2.24)and (2.28), and Cj from (2.25). For most purposes in

this paper, Md is taken as 150, and then Ms from(2.34).

The above equations describe the method ofsolution only for the function pp for Case J. Toobtain pm for Case J, one takes the same equationsbut with (#f) replacing f, obtaining Sj, Ej in place ofCj, Dj. For CaseH(2), one obtains Cj ; Dj for pp fromthe same procedure as above, but with J1(msja)replaced by H1

(2)(msja). For pm for Case H(2), oneobtains Sj ; Ejfrom the same equations as for pp butwith f replaced by (#f).

The energy flux radiating away from the sea-mount for case J is given by

Eflux !Z 2p

0

Z 0

#De

Re pi# $

Re ur% &) *

:rdzdy, (2.35)

where the integral is taken at some arbitrary radiusr4a, and

Eflux ! r0oa2U2De

2

1# f 2

o2

' ( 1

De

*X1

j!1

Z 0

#De

Z2dj dz

1

2p

Z 2p

0Term dy, %2:36&

where

Term ! DjD)j " EjE

)j " %DjE

)j "D)

j Ej& cos 2y

" i %DjE)j #D)

j Ej& sin 2y, %2:37&

and the asterisks denote the complex conjugate.This Term gives the angular dependence of theenergy flux, and the sin 2y and cos 2y terms sum tozero when integrated over all azimuths. In general,the direction of maximum flux is aligned with thatof the barotropic tide (y ! 0), with minima aty ! 7p/2, for all modes. However, as f/o-1 wherethe flow is close to the inertial resonance, for thelowest order modes this direction moves to positiveangles, and for mode 1 it may reach values in excessof 1 rad close to the limit. For the correspondingexpression for case H(2), one obtains (2.35)–(2.37)but with Dj ; Ej in place of Dj, Ej. In this case there isalso energy propagating inward from the boundaryof the seamount, and the total of this flux is given by

Eflux ! r0oa2U2 De # hm% &

2

1# f 2

o2

' (

*X1

j!1

CjCn

j " SjSn

j

h i 1

De # hm% &

Z 0

# De#hm% &Z2

sj dz,

%2:38&

This flux is relatively small, as discussed below.





3. Results for individual seamounts

3.1. Flow field properties

As shown above, the barotropic tidal pressurefield that coexists with unidirectional backgroundbarotropic flow (2.1) may be written

pb ! r0Uoar

a

i

21"

f

o

! "eiy " 1#

f

o

! "e#iy

% &e#iot,

which consists of a cum sole component (e#i(y+ot) inthe northern hemisphere) of magnitude propor-tional to (1#|f|/o), and a contra sole componentwith magnitude proportional to (1+|f|/o). Therelative magnitudes are equal at the equator, butthe contra sole component for the pressure is thelarger in each hemisphere, and becomes increasinglydominant with increasing latitude. From (2.20) and(2.21), the barotropic component of ui over theseamount has a form similar to (2.13), and in thedeep water it resembles potential flow aroundthe seamount. The corresponding internal tide pimay be expressed as

pi ! rUoa'pp r; z% &eiy " pm r; z% &e#iy(e#iot,

where the pp component is forced by the ei(y#ot)

term in (2.15), and the pm component by thee#i(y+ot) term. Accordingly, we may expect thebaroclinic contra sole pressure component todominate over the cum sole component except nearthe equator because the respective forcing termsare larger. However, as shown below this is notnormally the case.

Two types of stratification are considered here.First, N is assumed uniform everywhere, to give thesimplest description of the flow types and a base forthe energy flux results. For this case, the eigenfunc-tions for the vertical structure are given by

Zdj z% & ! cosjpzDe

! "; Zsj z% & ! cos

jpzDe # hm

! ",

j ! 1; 2; 3; . . . . %3:1&

Second, stratification with uniform N but with asurface mixed layer and shallow thermocline isdiscussed. For the uniform N case we may identifyfour main dimensionless parameters for individualseamounts: f/o, o/N, hm/De, and oa/NDe, wherethe last may be identified as a form of Burgernumber for this system. In general, the quantities o,N, and De have reasonably well-defined values forthe ocean, but f, hm, and a depend on the location

and shape of the seamount. We take De ! 4 km, ando ! 1.41 $ 10#4 rad/s, focusing on the semi-diurnaltide, with N ! 10#3 rad/s for definiteness. Henceo/N is chosen, or deemed to have a narrow range ofvalues. We consider seamounts that have a max-imum radius of about 100 km, unusually large for aseamount. Hence, for the other parameters thecomputations need to cover the ranges of values asfollows:

0of =oo1; 0ohm=Deo1; and 0ooa=NDeo5.

Before describing the results of computations forthese ranges of variables, we first examine the natureof the baroclinic tide generated by two representa-tive seamounts where f/o ! 0.5, oa/NDe ! 0.5,and hm/De ! 0.8 and 0.2. In all cases with thistopography, the flow is dominated by an axisym-metric pattern of conical rays with slope c thatoriginate from the tangent point of these rayswith the summit rim, as illustrated schematicallyin Fig. 2. Although the flow is computed as a sum ofmodes consisting of Hankel (Bessel) functionsradially and cosines vertically, they sum to give thisdistinct conical pattern (with apex above theseamount) with rapid transitions in flow across theboundaries of the cone. The more modes that areincluded in this sum, the sharper are the transitionsacross the boundaries of this cone and its continua-tions by reflection at the top and bottom boundariesof the fluid. These conical regions contiguous withthe region over the seamount tend to contain largervelocities, and show phase propagation in thevelocity that is consistent with energy flux awayfrom the seamount, at all azimuthal positions. Thisis consistent with two-dimensional solutions forsteep topography. The solution is calculated interms of two distinct functions of r and z: onecontra-sole (up, pp if f positive), the other cum sole(um, pm). Here the velocities up and um have similarmagnitudes (though they are not identical), but the

ARTICLE IN PRESS

z

x

Fig. 2. The pattern of conical rays for the baroclinic tide for arepresentative seamount.





cum-sole pressure pp has larger amplitude than thecontra-sole. This difference increases with increas-ing hm/De. This is the opposite situation for thepressure function for the barotropic tide in Eq.(2.13), and occurs because the cum-sole componentis closer to the inertial resonance at o ! f.

Fig. 3 shows a sample of the flow for the case withhm/De ! 0.8, for a vertical section along theseamount axis aligned with the barotropic tidalflow direction on the downstream side (y ! 0), withFig. 3a showing the amplitude of the total velocityand Fig. 3b the phase. The flow has obvioussimilarities to the two-dimensional case, withupward phase propagation in the downward-slop-ing beam, and smaller velocities in the broad regionbelow it. Fig. 4 shows the amplitude (Fig. 4a) andthe phase (Fig. 4b) of the radial velocity for thesame flow but at azimuthal longitude y ! 901. Herethe velocities are much weaker, but the raysemanating from the summit rim are quite conspic-uous, and show virtual discontinuities in phase andamplitude across them. Upward phase propagation,

implying downward energy propagation from linearwave theory, is again apparent in the beamemanating from the region above the summit. Incontrast, Figs. 5 and 6 show the correspondingflow patterns for the same parameters but withhm/De ! 0.2. Here the narrow ‘‘beam’’ containingmost of the structure emanates from the topogra-phy, rather than the region above the summit.Inside it the amplitude of the radial velocity is low,and its phase propagation is opposite to thatexpected from linear wave theory. In the broaderregions between, the radial velocity is much larger,and has outward phase propagation as expected.The constituent modes of these flows each haveoutward energy propagation, but when these aresummed to give the overall spatial pattern, thephase propagation in the various regions delineatedby the rays from the upper seamount edge maydiffer from what one might expect, and also varywith the variable (p, u, or v) considered. Theamplitude of the baroclinic motion is small in the‘‘shadow region’’ adjacent to the steep sides of

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th Z

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th Z

a

Amplitude ofradial velocity

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r

Fig. 3. (a) Amplitude and (b) phase of the total velocity u/U in the x-direction (y ! 0) for the conditions hm/De ! 0.8, f/o ! 0.5,ao/NDe ! 0.5, calculated using theH(2) model with 150 modes. The contour interval is 0.2 for (a), and 0.2 rad for (b). The ‘‘fuzzy’’ natureof some of the contours is a consequence of summing a large number of modes.

P.G. Baines / Deep-Sea Research I ] (]]]]) ]]]–]]]8




the topography, in each part of Fig. 4. If most ofthese regions beneath the downward-sloping rays fromthe summit rim were filled in with solid topography,little change would be expected in the remainingsolution elsewhere. This provides good justification forthe steep-sided pillbox model for seamounts.

3.2. Energy fluxes

We consider primarily the H(2) case (with inward-propagating energy flux over the summit), focusing onthe baroclinic energy flux away from the seamount,and present the energy flux in terms of thedimensionless variables hm/De, ao/NDe, and f/o ascalculated from Eq. (2.36). Although the results forthe energy flux were computed in terms of r0oa2U2De,it turns out to be more appropriate to present themin terms of r0NaU2De

2. Fluxes in the limit of largeao/NDe are shown in Fig. 7. These results werecomputed for ao/NDe ! 20, but are a good approxi-mation for ao/NDe41, as comparison with Fig. 8and 9 shows. The curve for small f/o is approxi-

mately described by the function [sin(p(1#hm/De))/(2#hm/De)]

2, which is contained in the equations ofSection 2, but there does not seem to be an obviousanalytic extension that includes f/o. As f/o increasesabove about 0.4, the maximum of the curves movesto larger values of hm/De, shrinking to small values asf/o approaches unity. This energy flux must necessa-rily be zero when either f/o or hm/De ! 1.

The three-dimensional structure of this energyflux with the same scaling is shown for ao/NDeo1in Figs. 8 and 9 as a function of the seamount shapeparameters hm/De, ao/NDe. These curves level offin the direction of increasing ao/NDe, so that thedimensional energy flux increases linearly with a forao/NDe40.5, approximately. For small ao/NDe,these curves increase quadratically with ao/NDe, sothat the energy flux increases with a3 for ao/NDe

oo1, up to about 0.3. As a function of f/o, thesecurves have their largest amplitude near f/o ! 1, asshown in Fig. 9 (where all curves have the samescale), though they descend rapidly to zero as thislimit is approached. This is clearly associated with

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r

Fig. 4. The same as Fig. 3, but for the velocity in the y-direction (y ! p/2). The contour interval is 0.05 for (a), 0.4 rad for (b).





inertial resonance, and is discussed further below.Some computations in which N and De were eachvaried by an order of magnitude gave essentially thesame dimensionless magnitudes as given in thesefigures, so that this scaling is robust for thesevariables, as expected.

The above results have been compared with thosefrom the J-model, in which the motion over theseamount is composed of stationary oscillations forall modes. The energy flux results are very similar,with magnitudes differing by a few percent. Thereare, however, small systematic differences which areshown in Fig. 10, a representative example withf/o ! 0.3; as in Fig. 8 for the H(2) model,corresponding figures for f/o ranging from 0 to0.5 are similar to this one. Fig. 10 may be comparedwith Fig. 8b. The radiated energy flux plot in theJ-model contains small ridges and valleys that areapproximately straight on this diagram. Theselinear ridges approximately correspond with thezeroes of the Bessel function J1(msia), the firstzero with the first ridge, the second zero with thesecond, and so on. Hence these ridges correspondwith near-resonances associated with the forcing of

the stationary modes over the seamount, thestructure of which has the form of this Besselfunction. In the inviscid theory these are not fullresonances, as the energy is always able to leakaway across the open boundary at the edge of theseamount, but as the computations show, a smallmaximum in the energy flux results.

Fig. 11 shows a representative distribution of thetotal energy flux across the internal wave modes. Herehm,/De ! 0.8, f/o ! 0.5, ao/NDe ! 1, model H(2) isused, and 150 modes are taken. This ‘‘energyspectrum’’ has character similar to those for a two-dimensional knife-edge ridge (St. Laurent et al.,2003). For this tall seamount, much of the energy isin mode 1, with a power law decrease with increasingmode number. As for the 2D ridge there is also anoscillation in this amplitude, with a period of severalmodes (10 in this case) that remains approximatelyconstant as the mode number increases.

3.3. The nature of the flow over the seamount

For the H(2) model, where inward propagationfrom the seamount edge is assumed, the baroclinic

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motion over the seamount is not realistic, but it isnot large. The inward energy fluxes from (2.36) aremuch smaller than the external ones. As Figs. 3–6indicate, the baroclinic motion over the summit issmall compared with the barotropic—there is littlestructure evident inside the conical beams radiatingfrom the shallow region above the summit. Hencethe main feature of interest is the structure of thebarotropic motion over the seamount, forced bythe external barotropic tide, and determined by thecomplex coefficients C0 and S0 in the expressions forpp and pm of Eq. (2.20). Three factors of practicalinterest are considered here: the magnitude of thisbarotropic perturbation, the orientation of thelargest motion, and its phase relative to the originalbarotropic forcing.

Here it is appropriate to use Cartesian coordi-nates (x,y), with corresponding velocities (u,v). Theperturbation barotropic velocity is spatially uniform

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Fig. 6. As for Fig. 4, but with hm/De ! 0.2. Contour interval is 0.01 for (a), 0.4 rad for (b).

0 0.4 0.5 0.6 0.7 0.8 0.9 10

0.05

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En

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y F

lux U

nit

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aN

U 2

De

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0.95f/$ = 0

0.4

Fig. 7. Energy flux radiating away from the seamount in units ofr0NaU2De

2 in the limit of large ao/NDe as a function of hm/De

for various values of f/o. These values were calculated forao/NDe ! 20, but are a good approximation for ao/NDe41.





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hm /D

ehm /D

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Eflux/#0NaU 2De2

hm /D

ea$/NDe

Fig. 9. Energy flux as for Fig. 8, but with values close to the inertial cut-off. (a) f/o ! 9, (b) 0.95, (c) 0.99; (a) is the same as Fig. 8(d) and isrepeated here for comparison with (b) and (c). The curves are all plotted on the same scale to display the variation when the tidal frequencyis close to the inertial frequency.

0 0.2 0.4 0.6 0.81

0

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10

0 0.2 0.4 0.6 0.8 1

0

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10

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0.15

0.1

0.05

hm /D

e a$/NDe

hm /D

e a$/NDe

hm /D

e a$/NDe

hm /D

e a$/NDe

Fig. 8. Energy flux in units of r0NaU2De2 as a function of hm/De, ao/NDe for various values of f/o, calculated with the model H(2):

(a) f/o ! 0, (b) 0.3, (c) 0.6, (d) 0.9. Note the change in scale for (d) compared with (a)–(c).





over the flat top of the seamount, and may beexpressed as

uiU

! #i

1# f 2=o2# $ C0 1#

f

o

! "" S0 1"

f

o

! "% &e#iot,

(3.2)

viU

!1

1# f 2=o2# $ C0 1#

f

o

! "# S0 1"

f

o

! "% &e#iot,

for the velocity components in the x and y direc-tions, with magnitudes

C0

1" f =o"

S0

1# f =o

++++

++++;C0

1" f =o#

S0

1# f =o

++++

++++,

(3.3)

respectively. We first note the magnitude of thisquantity for the case where there is no stratification(i.e. N ! 0). Here (2.24) and (2.28) give D0 ! C0,

E0 ! S0, and

C0 !i2hmDe

1# f 2

o2

' (

2# hmDe

1# fo

' ( ; S0 !i2hmDe

1# f 2

o2

' (

2# hmDe

1" fo

' ( .

(3.4)

Adding the initial barotropic velocity, the totalvelocity over the seamount for this unstratified caseis, for this model,

u ! U cos ot 1"hmDe

2# hmDe

' (21# f 2

o2

' (

4 1# hmDe

' (" hm

De

' (21# f 2

o2

' (

2

64

3

75,

v ! #U sin othmDe

f

o2

4 1# hmDe

' (" hm

De

' (21# f 2

o2

' ( .

%3:5&

These expressions are independent of the sea-mount radius a, and give cum-sole elliptical motionover the seamount, with the major axis aligned withthe external barotropic flow. This barotropic flow,with potential flow in the deep water, is illustratedin Fig. 12 for the parameter values hm/De ! 0.8,f/o ! 0.5. For tall seamounts where hm/De ap-proaches unity so that the depth of the seamounttop is a small fraction of the total depth, this isapproximated by

u + U cos ot 1"1" f 2=o2

1# f 2=o2

" #

,

v + #U sin ot2f =o

1# f 2=o2

" #

. %3:6&

The perturbation increases dramatically as thecut-off latitude (where o ! f) is approached, be-cause of tidal forcing at the inertial resonance. Thisimplies that, even when there is no stratification, thediurnal tide should be large over seamounts near the

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Efl

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Fig. 11. The contribution to the total energy flux from variousinternal wave modes for a seamount with parametershm/De ! 0.8, f/o ! 0.5, ao/NDe ! 1, computed with the H(2)model, shown on a log–log scale. Most of the energy flux iscarried by a small number of low-order modes, and there is aperiodic variation with period ,10 modes.

0

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1

00.20.40.60.810

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0.2

Eflux in units: "NaU2De2, f/# = 0.3

hm /Dea$

/ND

e

Fig. 10. Energy flux as for Fig. 8 but for the J-model withstationary modes over the seamount, for f/o ! 0.3. Results forf/oo0.5 are similar. Compared with Fig. 8b, the additional smallridges and valleys are due to partial resonances of the internalwave modes over the seamount.





cut-off latitude of 301. Large amplification of thediurnal tide (up to a factor of 10) has been observedover the Great Meteor Seamount (summit width31 km), centred at latitude 301S in the SouthAtlantic (Mohn and Beckmann, 2002), and overthe smaller Fieberling Guyot (summit width 10 km,latitude 321 270N) in the North Pacific (Brink, 1995).

For seamounts of small radius, the presence ofstratification and the forcing of the internal modesradiating energy away from the seamount can causechanges to the barotropic motion over it. Themagnitude of the x-component perturbation ui/U isshown in Fig. 13 for a range of values of f/o: 0, 0.3,0.6, and 0.95. For most of the parameter range, (3.5)gives a good description of this variable, theexceptions being when the seamount radius is smalland, approximately,

hm=Deo2ao=NDe. (3.7)

In this range, the velocity is smaller than thehomogeneous value given by (3.5). Computations ofthe phase of this maximum (not shown) show that itis significantly negative, particularly for smallao/NDe where it reaches values near #p/2, implyingthat it occurs 1

4-cycle earlier. The amplitude of thev-velocities are shown in Fig. 14, for the valuesf/o ! 0.3, 0.6, and 0.95 (the value for f/o ! 0 beingzero). As for the u-velocities, the values for v arewell-described by (3.5), except when (3.7) applies,when they are generally smaller as seen in Fig. 14.The phases are approximately p/2 from those for ui.Hence, the velocities over the seamount are gen-erally well-described (apart from phase) by (3.5),except when hm/De is large and ao/NDe is small.

If the above quantities are computed with theJ-model, incorporating standing baroclinic oscilla-tions over the seamount, the curves obtained arevery similar to those of Figs. 13 and 14, with some

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0

-1

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3

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Fig. 12. Schematic plan view of the stream field of barotropic motion in the absence of stratification, for the parameter valueshm/De ! 0.8, f/o ! 0.5, at four phases of the tide: (a) t ! 0; (b) t ! p/2o; (c) t ! p/o; (d) t ! 3p/2o. The slab-like motion of the fluid overthe seamount is shown by the central arrows, with potential flow in the deep water. The amplitude of the external barotropic velocity isshown by the arrows at the top of (a) and (c); for these parameters the relative amplitudes over the seamount are 1.95 in (a) and (c), and0.31 in (b) and (d). With stratification present, this barotropic field is essentially unchanged if hm/Deo2ao/NDe. Otherwise, the motionover the seamount is reduced in amplitude. The internal tide is generated by the mismatch at the summit rim between the slab-like motionover the seamount and the potential flow around it.





small oscillations superimposed, much like thecomparison of Fig. 10 with Fig. 8. Hence theseresults for this barotropic flow may be regarded as

robust. For the J-model it is possible to calculate thebaroclinic modes, and (for the pressure variable)these are smaller than the barotropic. The largest is

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a$/NDehm /D

e

hm /D

e

hm /D

e

hm /D

e

f/$

= 0

f/$

= 0

.3

f/$

= 0

.6

f/$

= 0

.95

2

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1

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01

0.5

0 0 0.2 0.4 0.6 0.8 1

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1

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01

0.50 0 0.2 0.4

0.6 0.81

8

6

4

2

01

0.5

0 0 0.20.4 0.6

0.8 1

Fig. 13. The x-component of the maximum perturbation barotropic velocity ui/U over the seamount as calculated for the H(2) model as afunction of the seamount shape parameters and f/o ! (a) 0, (b) 0.3, (c) 0.6, (d) 0.95.

1.5

1

0.5

01

0.5

0 0 0.2 0.40.6 0.8

1

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a$/NDe

a$/NDehm /D

e

hm /D

e

hm /D

e

f/$

= 0

.3

f/$

= 0

.6

f/$

= 0

.95

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0.6

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0.2

01

0.5

0 00.2

0.40.50.8 1

8

6

4

2

01

0.5

0 00.2 0.4 0.6 0.8 1

Fig. 14. As for Fig. 13 but for the velocity vi/U in the (transverse) y-direction. The case f/o ! 0 is omitted because the values are all zero.





usually the first baroclinic mode with coefficient S1,which may become noticeable for small values ofao/NDe, implying that the gravest baroclinic modemay be significant for small seamounts, but notothers. For wide seamounts (large a), the presenceof dissipation will degrade the possibility forcoherent modes in any case.

3.4. Variable stratification

Some computations were carried out with adensity profile that incorporated a surface mixedlayer, and a shallow thermocline modelled as aninterface at the bottom of the mixed layer. Here therelevant parameters are d/(De#hm), where d is themixed layer thickness, and G ! g0/o2De, whereg0 ! gDr/r0 and Dr is the density difference acrossthe interface. These were done with a seamount witha summit 500m below the surface; the depth of theinterface was varied from zero to a level just abovethe seamount top, and the interface strength G wasvaried from 0 to 20. For these variables, a surfacemixed layer that is created by mechanical stirringwithout surface buoyancy flux satisfies d/De !(o/N)2G, which lies within these ranges of values.

The results were initially surprising. Althoughthese variations caused expected changes in theform of the internal tide field from the changes inthe density structure and the associated modes, thetotal energy flux varied by only a few percent. Thisinsensitivity of the radiated energy flux to thestratification above the seamount is consistent withthe above result that the main forcing for theinternal tides is associated with the barotropicmotion there.

4. Seamount distributions and integrated results

A global distribution of seamounts as inferredfrom satellite altimetry has been given by Wessel(2001). Various studies of seamount morphologyhave concluded that a typical seamount is closelyapproximated by a flat-topped cone with a flankslope of 171, and a ratio of height hm to radius a ofthe flat top of

hm=a ! 0:69, (4.1)

which may be termed the ‘‘Wessel relation’’. Wesselidentifies nearly 15,000 seamounts by satellitealtimetry, and they have a distribution in height asshown in Fig. 15. They are also consistent with apower law proposed by Wessel and Lyons (1997) of

the form

Npl ! 6:104=h4:16m , (4.2)

where Npl denotes the number of seamounts in250m bands in height. This relationship is alsoshown in Fig. 15, and fits the data for hm42 km. Itsextrapolation to the range hm ! 1–2 km, whereexisting satellite altimetry resolution is inadequatefor detecting seamounts, suggests that the totalnumber is closer to 100,000, mostly made up byseamounts with heights smaller than 2 km.

Since (4.1) approximates many seamounts in theocean, it is appropriate to examine the properties ofthe internal tides generated by these seamounts inmore detail. They are too small to conform tothe wide seamount limit (ao/NDe41), and arerepresented by a section across the diagrams ofFigs. 8–10. Fig. 16 shows the total energy flux forthese seamounts as functions of hm/De and f/o; thecharacter is similar to the wide-seamount limit ofFig. 7. With realistic values of the parameters, theseresults show that individual seamounts generatefluxes of order 106W for linearly polarised tidalcurrents of 1 cm/s. As Fig. 11 indicates, most of theenergy flux radiated from a seamount is containedin a small number of the lowest modes. Results arepresented here for the three lowest modes. Fig. 17ashows the total energy flux radiated by thefirst mode (vertical structure given by (3.1)), andthis is clearly a substantial fraction of the total formost values of the parameters. This radiation isnot uniform with azimuthal position around the

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Seamounts inferred fromsatellite alimetry

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Fig. 15. Numbers of seamounts in 250m height ranges, inferredfrom satellite altimetry (Wessel, 2001), and the extension of thesedata to the range hm ! 1–2 km by a power law relationship(Wessel and Lyons, 1997).





seamount, and has cos 2(y#a1) dependence on y.The angle a1 of maximum energy flux is shown inFig. 17b, and the maximum and minimum values ofthis elliptical distribution are plotted in Fig. 17c.Corresponding results for modes 2 and 3 are shownin Figs. 18 and 19 respectively. The shapes of themode 1 energy flux curves in Fig. 17a are generallysimilar to those for the total energy flux shown inFig. 16. For most situations, the direction a1 of themaximum of this flux is slightly positive (i.e rotatedanti-clockwise) relative to the x-axis by up to 151. Iff/o ! 0, this energy flux is bi-directional along thex-axis, but as f/o increases the ellipticity of thedistribution decreases and it becomes more azi-muthally uniform, as shown in Fig. 17c. For modes2 and 3, some of the energy flux curves show asecond maximum near hm/De ! 0.5; the values of a2and a3 are also somewhat larger, and may benegative (i.e. clockwise relative to the barotropictide direction) for f/o near unity. These curves forthe three modes give a good indication of the natureand location of most of the internal wave field andassociated energy flux around these seamounts.

In order to estimate the total baroclinic energyflux generated by the sum of these seamounts, weconsider the height ranges given by Wessel, andemploy the model of the previous sections. For alinearly polarised barotropic tidal flow with velocityof magnitude U, the energy flux radiated from theseamount may be expressed as

Eflux ! FhmDe

;aoNDe

;f

o

! "r0aN UDe% &2. (4.3)

Here we take U ! 1 cm/s, and De ! 4.5 km. Mostseamounts lie in latitudes of less than 601, and asFig. 16 shows, the dependence of the energy flux onf is not large unless f/o is close to unity, so weassume that f/o ! 0.5 for all seamounts for thesecomputations, giving the F curve shown in Fig. 20a.Eq. (4.3) then gives energy fluxes for individualseamounts, as also shown in Fig. 20a. Multiplyingthese fluxes by the number of seamounts in the250m height ranges shown in Fig. 15, according tothe two seamount number-models, gives the totalsum of the energy flux in each range box, which isshown in Fig. 20b. Summing over all of the boxesthen gives the total energy flux, for each number-model. The results are 1.8 $ 109W for the satellitealtimeter seamount count, and 2.4 $ 109W for thepower law model.

If the tide were circularly polarized with thesame strength, the fluxes would be double thesevalues. Also, this velocity is at the lower end ofranges of mid-ocean barotropic tidal velocities,and the flux will be larger in regions with largerU. The result is not as sensitive to the chosenvalue of De as Eq. (4.3) suggests. A seamountsituated on a mid-ocean ridge where the effectivedepth is less than over an abyssal plane willhave a correspondingly larger value of U, whichwill compensate for the depth decrease—thebarotropic tidal volume flux UDe will remaineffectively constant, and vary only with the large-scale pattern of the barotropic tide. Accordingly,the above expression is seen to be reasonablyrobust.

These numbers may be compared with theestimates of internal tide generation from continen-tal slopes, and from the Hawaiian ridge. Internaltidal energy flux from continental slopes has beenestimated as 1.45 $ 1010W for the M2 tide, and2.73 $ 109W for the S2 tide (Baines, 1982). For thewhole Hawaiian ridge, the corresponding flux forthe M2 tide is estimated to be 2 $ 1010W (Rudnicket al., 2003), a number very similar to thatfrom the sum of continental slopes. For the nettotal from all the seamounts, we cannot make adirect comparison without a more careful estimateof the magnitudes of the background baro-tropic tide in the locations of the seamounts, butif we assume a circularly polarised barotropic M2

tide, the above numbers (from this section) give5 $ 109W, which is less than those of the continentalshelves and Hawaiian ridge, but is comparable withthem.

ARTICLE IN PRESS

0.2

0.18

0.16

0.14

0.12

0.1

0.08

0.06

0.04

0.02

00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

hm/De

hm/a = 0.69f/$ = 0.99

0.95

0.9

0.80.7

0.5

0

Efl

ux

## 0N

aU2 D

2 e

Fig. 16. Energy flux radiating away from the seamount in unitsof r0NaU2De

2 for seamounts satisfying the ‘‘Wessell relation’’hm/a ! 0.69 (c.f. Fig. 7).





5. Comparison with previous models

The energy flux results of Section 4 are presentedin the most appropriate form for seamounts, butthis differs from that which arises naturally fromlinear perturbation theory. In particular, LlewellynSmith and Young (2002) derive the energy fluxesfrom linearised seamounts of Gaussian shape

h ! hme#%x2"y2&=2a2 , (5.1)

and for the same external conditions as used abovefor the pillbox model, obtain

Eflux ! Gr0NaU2h2m 1#f

o

! "2 !1=2

, (5.2)

where a is now the Gaussian width of an axisym-metric seamount, and G is a function of a parameterA defined as

A ! paoNDe

1#f

o

! "2 !1=2

. (5.3)

For present purposes, G is given approximatelyby

G ! 0:696; 0oAo1,

+p2A3 e#A2 " 4e#4A2

' (A41. %5:4&

G decreases rapidly from 0.696 as A increasesbeyond unity, and is effectively zero for A43.

ARTICLE IN PRESS

0.14

0.12

0.1

0.08

0.06

0.04

0.02

00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

hm/De hm/De

hm/De

f/$ = 0.99

f/$ = 0.99

f/$ = 0.9

0.95

0.950.9

0.70.5

0.30

25

20

15

10

5

0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.9

0.70.5

0.3 0.1Angle

of M

axi

mum

Energ

y F

lux

% 1in

mode 1

(degre

es)

0.01

0.009

0.008

0.007

0.006

0.005

0.004

0.003

0.002

0.001

00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.7

0.50.3

00.5

0.3

Maxima and minima inangular distribution

Eflux, M

ode 1

## 0N

aU

2D

2 e

Eflux, M

ode 1

# 0N

aU

2D

2 e

Fig. 17. Properties of the energy flux in mode 1 for seamounts satisfying the Wessel relation. (a) Total energy flux away from the seamountin mode 1, as in Fig. 16. (b) Orientation a1 of the direction of maximum energy flux in mode 1, measured anti-clockwise from the real axis.a1 ! 0 for f/o ! 0. (c) Maximum (solid lines) and minimum (dashed) amplitude of the elliptical azimuthal distribution of energy flux. Theenergy flux minimum is zero for f/o ! 0. These curves come closer together as f/o increases, implying that the azimuthal distributionchanges from unidirectional to nearly zonally uniform. Contour 0.1 is plotted for (b), but not for (a) and (c).





The corresponding results from two-dimensionalridges are often presented in the same manner. Inparticular, for a 2D knife-edge ridge, for a lengthof ridge 2a (to be compared with seamounts ofthis diameter) the energy flux is given by (5.2)but with G replaced by a function Gkn, whichdepends only on hm/De and increases monotonicallyfrom p/2 (St. Laurent et al., 2003, LlewellynSmith and Young, 2003). Gkn is shown plotted inFigs. 21a and b.

Fig. 21a shows the results for the pillbox model,presented in the same manner, for a range of valuesof f/o and for ao/NDe41 (wide seamounts). Thefunction Gkn and the maximum value of G areshown dashed, for comparison. It is clear that thisform of scaling is not particularly appropriate forseamounts, as these curves vary substantially withhm/De, though they do tend to coalesce for f/osmall. There is a weak singularity at the origin withthe approximate form 1/hm

0.1. Compared with the

knife-edge ridge, these curves decrease with increas-ing height because the fluid can pass around theseamount rather than over it. Interestingly, thepillbox values are generally larger than the Gaussianlinear ones for small f/o, and this may also be thecase for larger f/o for wide seamounts, dependingon Eq. (5.4).

Fig. 21b shows the corresponding curves forWessel’s observed ratio hm/a ! 0.69 (i.e. theparameter ao=NDe ! %o=0:69N&hm=De (for theseparameter values, the plotted value of G is thevalid one). In these cases both the 2D and line-arised models give substantially larger energyfluxes than the pillbox model, for all seamountheights except for large, high seamounts when thesystem is close to the inertial resonance. This is,again, a reflection of the fact that steep-sidedseamounts tend to deflect the barotropic tidearound them, an effect not fully captured by line-arised topography.

ARTICLE IN PRESS

0.025

0.02

0.015

0.01

0.005

00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

hm/De

80

60

40

20

0

-20

-40

-60

-80

-100

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70.8 0.9 1

hm/De

3.5

3

2.5

2

1.5

1

0.5

00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

hm/De

Angula

r m

axi

mum

and m

inim

um

valu

es,

Mode 2

x 10-3

Angle

of M

axi

mum

Energ

y F

lux

$2

in m

ode 2

(degre

es)

f/$ = 0.99

f/$ = 0.99

f/$ = 0

0.95

0.9

0 0.9 0.95

0

0.1

0.3

0.50.7

0.30.5

0.7

0.90.9

0.7

0.5

0.3

Eflux, M

ode 2

# 0N

aU

2D

2 e

Fig. 18. As for Fig. 17, but for mode 2.





6. Summary and discussion

I have described a model of internal tidalgeneration over steep-sided axisymmetric topogra-phy that is applicable to seamounts. In this modelthe seamounts are pillbox-shaped, with flat top andvertical sides. Two versions of the model have beencompared: one in which the generated internalwaves over the seamount form standing modes, andone in which they are assumed to propagate inwardsfrom the summit rim and decay. The latter ispreferred for most purposes here, but the twomodels give similar answers for the structure andamplitude of the waves radiating away from theseamount, and the associated energy flux. Thereason for this is due, at least in part, to the factthat the waves radiating away in the deep water areforced primarily by the forced barotropic motionover the seamount. For this simple topography, this

barotropic motion is spatially uniform over theseamount. For hm/Deo2ao/NDe, it is independentof the stratification, and is given by the simpleexpressions for zero stratification (3.5). However,this relationship is not satisfied for most seamountsin the ocean, where the radiated energy loss due tothe stratification causes a reduction in the baro-tropic motion over the seamount. This occursbecause the radiated energy loss from a seamounttends to reduce the amplitude of the barotropicmotion over the seamount that (primarily) forces it,and this effect becomes more significant as theradius of the seamount becomes smaller. Thismotion over the seamount becomes large for diurnaltides near 301 latitude, and this is consistent withobservations.

The pattern of internal waves produced by thelinearly polarised barotropic tidal flow past aseamount consists of an axisymmetric conical beam

ARTICLE IN PRESS

0.014

0.012

0.01

0.008

0.006

0.004

0.002

00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

100

80

60

40

20

0

-20

-40

-60

-80

-1000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.6 0.9 1

hm/De

Angle

of M

axi

mum

Energ

y F

lux

in m

ode 3

(degre

es)

1.5

1

0.5

0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

hm/De

hm/De

Angula

r m

axi

mum

and m

inim

um

valu

es,

Mode 3

f/$=0

0.9

0.9

0.7

0.50.3

f/$=0.9 0.95

0.99

0

0.10.30.5

0.7

0

0.9

0.95

f/$ = 0.99

Eflux, M

ode 3

# 0N

aU

2D

2 e

Fig. 19. As for Fig. 17, but for mode 3.





radiating along ‘‘characteristics’’ or wave rays fromthe region over the seamount. A vertical sectionthrough this cone shows that the flow (examples ofwhich are shown in Figs. 3–6) resembles the familiarpattern from two-dimensional steep topography.There is variation in amplitude and phase withazimuthal angle, the flow having larger magnitudein the x-direction, aligned with the initial barotropicflow, than in the transverse y-direction. In thex-direction in particular, if hm/De40.5, the ampli-tude of u in the beam emanating from above theseamount is approximately uniform across thebeam, with phase variation of less than 0.3 rad.

This is consistent with the beam being generatedprimarily by the forced barotropic motion over theseamount.

Most seamounts in the ocean conform to therelationship hm/a,0.69, which generally implies thatthe Burger-number parameter ao/NDeo0.2. Forthese cases, energy fluxes from individual seamountsare of order 106W. Integrating over all of theseseamounts gives a total estimate of 2.4 $ 109W forlinearised background barotropic flow of 1 cm/s. Ifwe assume a circularly polarised M2 tide of thismagnitude, the estimate is doubled, but is lessthan the estimates for energy flux from the sum of

ARTICLE IN PRESS

0.45

0.4

0.35

0.3

0.25

0.2

0.15

0.1

0.05

0

1 1.5 2 2.5 3 3.5 4 4.5

Energy flux/seamount

10F

0.25

0.2

0.15

0.1

0.05

01 1.5 2 2.5 3 3.5 4 4.5

hm km

Wessel

power law

Energy flux/seamountunits : 106 Watt

Total energy fluxper height bandunits : 109 Watt

Fig. 20. (a) The function F(hm/De, ao/NDe, f/o) (plotted as 10F) for the relation (4.1) with f/o ! 0.5 and De ! 4.5 km, and energy fluxper seamount ( by 10#6 on the same scale), with the same units as in Fig. 16. (b) Total energy flux obtained by summing results from (a)over all seamounts in each 250m height band, for the numbers shown in Fig. 15.





continental slopes and from the Hawaiian ridge forM2, but is comparable with both. The precise valuefor the seamounts depends on how they are

counted, the magnitude and polarisation of localbarotropic tidal currents, latitude of seamounts,etc., but the total order of magnitude seems robust.

ARTICLE IN PRESS

2.5

2

1.5

1

0.5

00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

hm/De

2

1.8

1.6

1.4

1.2

1

0.8

0.6

0.4

0.2

00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

hm/De

Eflu

x

# # 0N

aU

2h

2

1-

f $

21

/2

2D knife-edgeridge

3D Gaussianseamount-linear

f/$ = 0

f/$ = 0.99

0.99

0.95

0.90.8

2D knife-edgeridge

3D Gaussianseamount-linear

0.95

0.9

0.8

Eflu

x

f $

21

/2

m# 0

Na

U2h

2

1-

m

Fig. 21. (a) Energy flux for seamounts of radius a, expressed in terms of the factor r0NaU2h2m 1# fo

' (2! "1=2

, which arises from linear

perturbation theory. The solid curves are for seamounts with ao/NDe41, and f/o ! 0, 0.3, 0.5, 0.7, 0.8, 0.9, 0.95, 0.99 in sequence. Upperdashed curve: two-dimensional knife-edge ridge, length 2a (Gkn in text). Lower dashed line: linearised Gaussian seamount with

aoNDe

1# fo

' (2! "1=2

o 1p. (b) As for (a), but for seamounts with hm/a ! 0.69. Here f/o ! 0, 0.3, 0.6, 0.8, 0.9, 0.95, 0.99.





Acknowledgements

This work has benefited from discussions withseveral colleagues, but I would like to specificallymention conversations with Bill Young, which ledto improvements in the formulation of the model.Comments from two anonymous referees werehelpful in improving the manuscript. This workhas been funded in part by the QUEST Programmeof the Natural Environmental Research Council ofthe UK.

References

Baines, P.G., 1973. The generation of internal tides by flat-bumptopography. Deep-Sea Research 20, 179–205.

Baines, P.G., 1974. The generation of internal tides over steepcontinental slopes. Philosophical Transactions of the RoyalSociety of London 277, 27–58.

Baines, P.G., 1982. On internal tide generation models. Deep-SeaResearch 29, 307–338.

Baines, P.G., Miles, J.W., 2000. On topographic coupling ofsurface and internal tides. Deep-Sea Research part I 47,2395–2403.

Bell, T.H., 1975. Topographically generated internal waves in theopen ocean. Journal of Geophysical Research 80, 320–327.

Balmforth, N.J., Ierley, G.R., Young, W.R., 2002. Tidalconversion by subcritical topography. Journal of PhysicalOceanography 32, 2900–2914.

Brink, K., 1995. Tidal and lower frequency currents aboveFieberling Guyot. Journal of Geophysical Research 100 (C6),10817–10832.

Egbert, G.D., Ray, R.D., 2000. Significant dissipation of tidalenergy in the deep ocean inferred from satellite altimeter data.Nature 405, 775–778.

Garrett, C.J.R., Kunze, E., 2007. Internal tide generationin the deep ocean. Annual Review of Fluid Mechanics 39,57–87.

Holloway, P., Merrifield, M., 1999. Internal tide generation byseamounts, ridges and islands. Jounal of GeophysicalResearch 104, 25937–25951.

Llewellyn Smith, S.G., Young, W.R., 2002. Conversion of thebarotropic tide. Journal of Physical Oceanography 32,1554–1566.

Llewellyn Smith, S.G., Young, W.R., 2003. Tidal conversion at avery steep ridge. Journal of Fluid Mechanics 495, 175–191.

Mohn, C., Beckmann, A., 2002. The upper ocean circulation atGreat Meteor Seamount, Part I: structure and density of flowfields. Ocean Dynamics 52, 179–193.

Morozov, E.G., 1995. Semidiurnal internal tidal wave field.Deep-Sea Research Part I 42, 135–148.

Munk, W., Wunsch, C., 1998. Abyssal recipes II: energeticsof tidal and wind mixing. Deep-Sea Research Part I 45,1977–2100.

Munroe, J.R., Lamb, K.G., 2005. Topographic amplitudedependence of internal wave generation by tidal forcing overidealized three-dimensional topography. Journal of Geophy-sical Research 110.

Prinsenberg, S.J., Rattray Jr., M., 1975. Effects of continentalslope and variable Brunt-Vaisala frequency on the coastalgeneration of internal tides. Deep-Sea Research 22, 251–265.

Rattray, M., 1960. On the coastal generation of internal tides,Tellus 12, 54–62.

Ray, R.D., Mitchum, G.T., 1997. Surface manifestationsof internal tides in the deep ocean: observations fromaltimetry and island gauges. Progress in Oceanography 40,135–162.

Rudnick, D.L., et al., 2003. From tides to mixing along theHawaiian ridge. Science 301, 355–357.

St. Laurent, L., Stringer, S., Garrett, C., Perrault-Joncas, D.,2003. The generation of internal tides at abrupt topography.Deep-Sea Research Part I 50, 987–1003.

Wessel, P., 2001. Global distribution of seamounts inferred fromGeosat/ERS-1 altimetry. Journal of Geophysical Research106, 19431–19441.

Wessel, P., Lyons, S., 1997. Distribution of large Pacificseamounts from Geosat/ERS-1: implications for the historyof intraplate volcanism. Journal of Geophysical Research 102,22459–22476.

ARTICLE IN PRESSP.G. Baines / Deep-Sea Research I ] (]]]]) ]]]–]]] 23




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