DECEMBER 2010 K E A N D Y A N K O V S K Y 2757 The Hybrid Kelvin–Edge Wave and Its Role in Tidal Dynamics ZIMING KE Marine Science Program, University of South Carolina, Columbia, South Carolina ALEXANDER E. YANKOVSKY Department of Earth and Ocean Sciences, University of South Carolina, Columbia, South Carolina (Manuscript received 13 January 2010, in final form 23 July 2010) ABSTRACT A full set of long waves trapped in the coastal ocean over a variable topography includes a zero (funda- mental) mode propagating with the coast on its right (left) in the Northern (Southern) Hemisphere. This zero mode resembles a Kelvin wave at lower frequencies and an edge wave (Stokes mode) at higher frequencies. At the intermediate frequencies this mode becomes a hybrid Kelvin–edge wave (HKEW), as both rotational effects and the variable depth become important. Furthermore, the group velocity of this hybrid mode be- comes very small or even zero depending on shelf width. It is found that in midlatitudes a zero group velocity occurs at semidiurnal (tidal) frequencies over wide (;300 km), gently sloping shelves. This notion motivated numerical experiments using the Regional Ocean Modeling System in which the incident HKEW with a semidiurnal period propagates over a wide shelf and encounters a narrowing shelf so that the group velocity becomes zero at some alongshore location. The numerical experiments have demonstrated that the wave energy increases upstream of this location as a result of the energy flux convergence while farther downstream the wave amplitude is substantially reduced. Instead of propagating alongshore, the wave energy radiates offshore in the form of Poincare´ modes. Thus, it is concluded that the shelf areas where the group velocity of the HKEW becomes zero are characterized by an increased tidal amplitude and (consequently) high tidal energy dissipation, and by offshore wave energy radiation. This behavior is qualitatively consistent with the dynamics of semidiurnal tides on wide shelves narrowing in the direction of tidal wave propagation, including the Patagonia shelf and the South China Sea. 1. Introduction that if the shelf is wide enough the group velocity of this hybrid mode becomes zero. A zero group velocity implies A full set of barotropic long waves trapped in the that the wave energy does not propagate along the coastal ocean over a variable topography includes a zero coastline. (fundamental) mode that exists at both subinertial and Semidiurnal tides represent the strongest signal in the superinertial frequencies and propagates with the coast intermediate superinertial frequency band in the sub- on its right (left) in the Northern (Southern) Hemisphere tropics and midlatitudes and often propagate along the (hereinafter, we refer to this direction as downstream) continental margins in the form of a fundamental trapped (e.g., Huthnance 1975). This zero mode resembles a mode, possibly as a HKEW (Munk et al. 1970). The Kelvin wave at lower, near- or subinertial frequencies and transmission of tidal energy along the shelf can be im- an edge wave (Stokes mode) at high frequencies (Mysak paired by a reduction of the group velocity, especially as it 1980). At the intermediate superinertial frequencies this approaches zero. This effect was previously discussed for mode becomes a hybrid Kelvin–edge wave (HKEW), as the diurnal tides off the Scottish west coast (Smith 1975). both rotational effects and the variable depth become In this area, diurnal tides propagate as subinertial conti- important (Munk et al. 1970). Yankovsky (2009) showed nental shelf waves (Cartwright et al. 1980), and the re- duction of their group speed can explain the observed amplification of tidal currents, especially at the shelf Corresponding author address: Alexander Yankovsky, Dept. of Earth and Ocean Sciences, University of South Carolina, 701 break. The goal of this paper is twofold: 1) to determine Sumter St. EWS-617, Columbia, SC 29208. characteristics of shelf geometry when the HKEW group E-mail: [email protected] velocity becomes zero at a semidiurnal frequency and DOI: 10.1175/2010JPO4430.1 Ó 2010 American Meteorological Society Unauthenticated | Downloaded 09/24/21 06:02 AM UTC 2758 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 40 2) to model the evolution of a semidiurnal HKEW when it encounters topographic variations that cause the wave group velocity to become zero. The rest of the paper is organized as follows: Section 2 describes the HKEW dis- persion characteristics and relates zero group velocity at a semidiurnal frequency with the shelf geometry. Section 3 presents a numerical experiment in which the incident wave encounters a variable shelf width and its down- stream propagation is essentially blocked because of a zero group velocity. Section 4 compares these results with World Ocean tides exhibiting similar patterns of alongshore propagation and concludes the paper. 2. Dispersion characteristics of the fundamental mode FIG. 1. Model geometry for boundary problems (1)–(2) and (4). A full set of barotropic wave modes trapped over the modes in the stratified ocean (e.g., Dale et al. 2001). continental shelf and slope topography can be obtained However, in our case of a barotropic ocean, only a Kelvin by applying linear shallow-water equations under an wave mode exists at the inertial frequency and its struc- f-plane approximation and seeking the solution that is ture is easily distinguishable from the spurious inertial periodic in time and alongshore coordinate. This yields mode. Nevertheless, we did not calculate a Kelvin wave a well-known boundary problem (e.g., Huthnance 1975), mode in the vicinity of the inertial frequency so as to avoid !a numerical interference of ‘‘real’’ and spurious wave- d2h 1 dh dh kf dh f 2 À v2 1 1 h À À k2 5 0, (1) mode solutions. dy2 h dy dy vh dy gh The topography of the continental shelf and slope is approximated as a two-slope depth profile adjacent to a subject to boundary conditions deep ocean of a constant depth (Fig. 1). The depth profile is characterized by the width of the shelf, slope, and deep y 5 0(y 5 0) and (2) ocean (L1, L2,andL3, respectively) as well as by the depth of the coastal wall, shelf break, and deep ocean (h , h ! 0(y ! ‘), (3) 1 h2,andh3, respectively). The offshore boundary condition 5 1 where x and y are the alongshore and offshore coordinates (4) is applied at y 1.5(L1 L2) km. The spatial reso- D (Fig. 1), u and y are the x-axis and y-axis velocity com- lution y in the finite-difference approximation is set to ponents, f is the Coriolis parameter, h is the free surface 1 km. We conduct a sensitivity study of the fundamental- elevation, h 5 h(y) is the water depth, g is the acceleration mode dispersion characteristics by systematically varying due to gravity, k is the alongshore wavenumber, and v is the shelf and slope widths L1 and L2,aswellasthe the wave frequency. The boundary condition for trapped shelfbreak and deep-ocean depths h2 and h3. The coastal 5 waves (3) implies an exponential decay of free-surface wall h1 5 m represents a nearshore zone boundary at perturbation in the offshore direction. At the finite off- which the wave-breaking and turbulent stresses domi- shore distance at which the water depth becomes constant nate, and it is kept constant for all cases. it can be expressed as Our study indicates that the dispersion curve of a zero mode is far more sensitive to the variations of a shelf 2 2 dh f À v width L1 than of a slope L2. Hence, we first show how the 5 Àlh, where l2 5 1 k2 (h 5 const). dy gh dispersion diagram of a zero mode depends on the shelf (4) width (Fig. 2) while other metrics of the depth profile are held constant: h2 5 100 m, h3 5 2000 m, and L2 5 Boundary problem (1)–(2) and (4) is solved numerically 100 km. Two values are used for the Coriolis parameter: by applying resonant iteration in v for a specified k.Inthis f 5 1024 s21 (representing midlatitudes) and f 5 6.59 3 study we utilize a computational code developed by Brink 1025 s21 (representing the subtropics). and Chapman (1987). The numerical solution contains The fundamental mode has two asymptotic regimes a spurious mode at the inertial frequency (e.g., Pedlosky corresponding to two different mechanisms of the wave 2003). This spurious mode represents a certain difficulty energy trapping. At low frequencies/wavenumbers, it for numerical calculations of near-inertial trapped wave resembles a Kelvin wave trapped by a vertical wall in an Unauthenticated | Downloaded 09/24/21 06:02 AM UTC DECEMBER 2010 K E A N D Y A N K O V S K Y 2759 FIG. 2. Dispersion curves of zero wave modes corresponding to the different values of the shelf width L1 and propagating in the positive x direction for f 5 (left) 1024 and (right) 6.59 3 1025 s21. Thick lines are the continuum of Poincare´ waves and the Kelvin wave in the coastal ocean of constant depth h3. ocean of a constant depth h3 because of the earth’s ro- for different L1 at high frequencies is evident in Fig. 2. At tation described by the dispersion relation low frequencies, the wave extends offshore well beyond the area of the continental shelf and slope, and its phase 1/2 v 5 (gh3) k. (5) speed is close to that of a Kelvin wave. Only at the in- termediate wavenumbers (k ; 1026–1025 m21)isthe At high frequencies/wavenumbers, it resembles a zero offshore extension of the wave structure comparable to edge-wave mode (often called a Stokes mode) formed the continental shelf and slope width, so that the phase by long-wave refraction over a sloping bottom.