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Internal Tide Generation by Seamounts

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Internal Tide Generation by Seamounts ARTICLE IN PRESS Deep-Sea Research I ] (]]]]) ]]]–]]] www.elsevier.com/locate/dsri Internal tide generation by seamounts Peter G. Bainesa,b,Ã aDepartment of Civil and Environmental Engineering, University of Melbourne, Melbourne 3010, Australia bQUEST, 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 by modelling 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 polarised barotropic tidal flow, the resulting flow field consists of conical beams radiating from the region above the seamount, with largest velocities aligned with the barotropic flow. These beams vary with azimuthal angle but resemble the corresponding beams from two-dimensional steep topography, particularly in the barotropic flow direction. They are primarily forced by the barotropic flow over the seamount, which is amplified by the topography and is independent of the stratification if the radius of the seamount is sufficiently large. In a barotropic tidal flow of 1 cm/s amplitude, energy fluxes from individual seamounts are of order 106 W. Summing this over all seamounts higher than 1 km gives baroclinic energy generation of order 5.109 W, a number that is less than estimates of baroclinic energy flux from the continental slopes and the Hawaiian ridge, but is comparable with them. r 2007 Elsevier Ltd. All rights reserved. 1. Introduction (Munk and Wunsch, 1998; Egbert & Ray, 2000). Internal tides are generated by the tidal flow of the The generation of internal tides in the deep ocean is density-stratified ocean over bottom topography, and of considerable current interest for several reasons, as the first stage in understanding their effects is to follows. Recent observations have shown them to understand and be able to calculate this generation have larger amplitudes than previously expected process. To date, most theoretical studies have focused (Morozov, 1995), most notably near the Hawaiian on generation from two-dimensional topography, and ridge (Ray and Mitchum, 1997, Rudnick et al., 2003). for decades much attention has been focused on Further, in conjunction with the presence of topo- generation from coastal topography, beginning with graphy, they are thought to be important in providing Rattray (1960). The relatively small number of studies an energy source for vertical mixing in the deep ocean of generation by three-dimensional topography have been with linearised bottom topography (Bell, 1975; Balmforth et al., 2002; Llewellyn Smith and Young, ÃDepartment of Civil and Environmental Engineering, Uni- versity of Melbourne, Melbourne 3010, Australia. 2002) or numerical models (Holloway and Merrifield, Tel.: +61 3 83447548; fax: +61 3 83444616. 1999; Munroe and Lamb, 2005). A recent review of E-mail address: [email protected] the field is given by Garrett and Kunze (2007). 0967-0637/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.dsr.2007.05.009 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 ARTICLE IN PRESS 2 P.G. Baines / Deep-Sea Research I ] (]]]]) ]]]–]]] For topography of finite height, the mathematical Accordingly, within the vicinity of an individual problem is very different depending on whether the seamount situated on an otherwise flat ocean floor, topography is ‘‘flat’’, in which the slope of the rays the background barotropic tide may be assumed to (group velocity vectors) of the internal tide is always be spatially uniform. Barotropic tidal motion steeper than the topography that they encounter generally has the form of an ellipse for the velocity (Baines, 1973), or ‘‘steep’’, where the topography is vectors and the particle displacement, for each tidal steeper in some locations (e.g. Baines, 1974). In the component. This may be represented as the sum of latter case, the problem can be sometimes simplified two rectilinear motions at right angles. If the by assuming that the topography has vertical sides. resulting baroclinic motions are small enough to If one wishes to apply such a model to topography be regarded as linear, the response to the elliptic with non-vertical sides, the solution obtained should barotropic tide may be represented as the sum of the have small amplitude in the region of difference, responses to these two rectilinear motions. Hence, it and this constitutes a check on this approximation. suffices to consider the baroclinic generation from a In the case of two-dimensional topography, it is single rectilinear barotropic tidal motion, and sum seen to work quite well (Prinsenberg and Rattray, the results from the two orthogonal constituents to 1975). The non-horizontal terms of the Coriolis obtain the net results for forcing by the elliptical force may also be included in linear models (Baines motion. and Miles, 2000); these cause changes to the One conceptual approach to internal tide genera- locations of the rays, but the extra complexity is tion is to first obtain the flow pattern that would not justified for present purposes and they are not exist if the ocean were unstratified, which could be included here. termed the virtual barotropic flow. The advection of This paper presents a study of baroclinic tidal density surfaces by this flow constitutes the forcing generation for steep-sided axisymmetric topography for the baroclinic tide (Baines, 1973). In the case of that approximates typical seamounts. The equations a non-rotating ocean, small-amplitude periodic flow are linearised, and the seamount is assumed to have past a seamount has the form of three-dimensional a flat top or summit, with vertical sides or flanks. In potential flow in homogeneous fluid of finite depth. spite of these approximations, this model is believed In a rotating homogeneous ocean in which potential to provide a good description of the overall vorticity is conserved, the situation is more complex, properties of the flow, and to give a soundly based and this computational procedure is unattractive. estimate of the energy fluxes produced. The model A better way of obtaining the internal tide is to and mathematical details are described in Section 2, consider the full stratified problem in which only and the results for individual seamounts are the background barotropic flow is specified. The presented in Section 3. This includes a description barotropic flow over and in the vicinity of the of the flow over and around the seamount, the seamount is then determined as part of the solution, dependence of the energy flux on the relevant and is dependent on the stratification, as seen below. dimensionless parameters, and results for different We therefore assume that the background baro- upper level stratification. In Section 4, these results tropic velocity ub has the form are used to integrate the energy flux over (model ub Ux^ cos ot, (2.1) representations of) the whole seamount population ¼ in the deep ocean. These results are compared with where U is the amplitude of the rectilinear motion, calculations by others (Llewellyn Smith and Young, o is the tidal frequency, and x^ is the unit vector in 2002, 2003; St. Laurent et al., 2003) for linearised the x-direction. The equations for the motion of an or two-dimensional topography in section 5, and incompressible density-stratified ocean, assuming the conclusions are summarised and discussed in that these are small enough to be linear, are Section 6. qu 1 gr0 f xu p , (2.2) 2. The model and equations qt þ ¼ À r0 r À r0 qr0 dr Seamounts are small topographic features in the w 0 0; u 0, (2.3) ocean compared with the scale of ocean basins, and qt þ dz ¼ r Á ¼ with the scale of variations of the barotropic tide. where r0 is the perturbation density with r0 a Horizontal scales are generally less than 100 km. mean density, p pressure, g gravity, and 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 ARTICLE IN PRESS P.G. Baines / Deep-Sea Research I ] (]]]]) ]]]–]]] 3 f 2O sin(latitude)z^, where O is the angular velocity z of¼the rotation of the Earth. The virtual barotropic Ocean surface tide mentioned above is governed by these equa- r tions with the buoyancy term in (2.2) omitted. Here ! we consider plane polar coordinates (r,y), where r is xˆ radial horizontal distance from an origin, and y is De angular distance from the positive direction of the hm x- axis. From (2.2) the pressure field for the unidirectional background barotropic flow (2.1) is then given by a Ocean bottom f iot p p r oUr i cos y sin y eÀ , (2.4) ¼ b ¼ 0 À o where the real part is taken. For linearised baroclinic motion with pressure p Fig. 1. The pillbox model of a seamount, with coordinate system. iot and vertical velocity w and time dependence eÀ , Eqs. (2.2) and (2.3) give (typical slope angle 171, Wessel, 2001) and are much 2 f 2 2 o steeper than the slope c of the rays, which are hw 2À wzz 0, (2.5) r À N z o2 ¼ typically 3–6 . Hence they may be assumed to be ð Þ À 1 axisymmetric ‘‘steep topography’’ in the language of 2 2 2 q pz previous two-dimensional studies (e.g. Baines, 1973, hp o f 2 0, (2.6) r À À qz N z o2 ¼ 1974).
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