Editorial Type:
Article
Article Type:
Research Article

Near-Inertial Surface Currents around Islands

Mika N. Siegelman aUniversity of Hawai‘i at Mānoa, Honolulu, Hawaii
bScripps Institution of Oceanography, La Jolla, California

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https://orcid.org/0000-0001-5781-0061
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Eric Firing aUniversity of Hawai‘i at Mānoa, Honolulu, Hawaii

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Mark A. Merrifield bScripps Institution of Oceanography, La Jolla, California

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Janet M. Becker bScripps Institution of Oceanography, La Jolla, California

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Ruth C. Musgrave cDalhousie University, Halifax, Nova Scotia, Canada

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Online Publication:
19 Jan 2023
Print Publication:
01 Feb 2023
DOI:
https://doi.org/10.1175/JPO-D-21-0310.1
Page(s):
433–455
Free access

Abstract

Motivated by observations of enhanced near-inertial currents at the island chain of Palau, the modification of wind-generated near-inertial oscillations (NIOs) by the presence of an island is examined using the analytic solutions of Longuet-Higgins and a linear, inviscid, 1.5-layer reduced-gravity model. The analytic solution for oscillations at the inertial frequency f provides insights into flow adjustment near the island but excludes wave dynamics. To account for wave motion, the numerical model initially is forced by a large-scale wind field rotating at f, where the forcing is increased then decreased to zero. Numerical simulations are carried out over a range of island radii and the ocean response detailed. Near the island, wind energy in the frequency band near f can excite subinertial island-trapped waves and superinertial Poincaré waves. In the small-island limit, both the Poincaré waves and the island-trapped waves are very near f, and their sum resembles the Longuet-Higgins analytic solution but with increased amplitude near the island. The flow field can be viewed as primarily a far-field NIO locally deflected by the island plus an island-trapped contribution, leading to enhanced near-inertial currents near the island, on the scale of the island radius. As the island size is increased, the island-trapped wave frequency deviates further from f and its amplitude depends strongly on the frequency bandwidth and wavenumber structure of the wind forcing. In the large-island limit, the island-trapped wave resembles a Kelvin wave, and the sum of incident and reflected Poincaré waves suppresses the near-inertial current amplitude near the island.

Significance Statement

Strong, impulsive winds over the ocean excite currents that rotate in the opposite direction to Earth’s rotation. This work examines how these wind-generated currents, known as near-inertial oscillations (NIOs), are modified by the presence of an island. Around small islands, the primary response is locally enhanced near-inertial currents. Alternatively, around large islands, near-inertial currents are weaker. Understanding how these currents behave should provide insight into the physical processes that drive current variability near islands and spur local mixing.

© 2023 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

This article is included in the Oceanic Flow–Topography Interations Special Collection.

Corresponding author: Mika Siegelman, msiegelm@hawaii.edu

Abstract

Motivated by observations of enhanced near-inertial currents at the island chain of Palau, the modification of wind-generated near-inertial oscillations (NIOs) by the presence of an island is examined using the analytic solutions of Longuet-Higgins and a linear, inviscid, 1.5-layer reduced-gravity model. The analytic solution for oscillations at the inertial frequency f provides insights into flow adjustment near the island but excludes wave dynamics. To account for wave motion, the numerical model initially is forced by a large-scale wind field rotating at f, where the forcing is increased then decreased to zero. Numerical simulations are carried out over a range of island radii and the ocean response detailed. Near the island, wind energy in the frequency band near f can excite subinertial island-trapped waves and superinertial Poincaré waves. In the small-island limit, both the Poincaré waves and the island-trapped waves are very near f, and their sum resembles the Longuet-Higgins analytic solution but with increased amplitude near the island. The flow field can be viewed as primarily a far-field NIO locally deflected by the island plus an island-trapped contribution, leading to enhanced near-inertial currents near the island, on the scale of the island radius. As the island size is increased, the island-trapped wave frequency deviates further from f and its amplitude depends strongly on the frequency bandwidth and wavenumber structure of the wind forcing. In the large-island limit, the island-trapped wave resembles a Kelvin wave, and the sum of incident and reflected Poincaré waves suppresses the near-inertial current amplitude near the island.

Significance Statement

Strong, impulsive winds over the ocean excite currents that rotate in the opposite direction to Earth’s rotation. This work examines how these wind-generated currents, known as near-inertial oscillations (NIOs), are modified by the presence of an island. Around small islands, the primary response is locally enhanced near-inertial currents. Alternatively, around large islands, near-inertial currents are weaker. Understanding how these currents behave should provide insight into the physical processes that drive current variability near islands and spur local mixing.

© 2023 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

This article is included in the Oceanic Flow–Topography Interations Special Collection.

Corresponding author: Mika Siegelman, msiegelm@hawaii.edu
Keywords: Coastal flows; Currents; Internal waves

1. Introduction

The initial response of the ocean mixed layer to any change in the wind stress may be approximated using a one-dimensional slab model (Pollard and Millard 1970; Pollard 1980) in which momentum is uniform within the mixed layer. The momentum balance may be split into an Ekman contribution, with the wind stress balanced by part of the Coriolis force, and an inertial oscillation (IO), in which local acceleration balances the remainder of the Coriolis force. In the IO, the velocity vector rotates anticyclonically at the frequency f (the Coriolis parameter). The slab model may be improved by taking into account a finite domain, horizontal variations in the wind, and the variation of f with latitude, any of which introduces horizontal divergence and horizontal pressure gradients (D’Asaro 1989). The inertial momentum balance is modified by the addition of the horizontal pressure gradient to yield a rotationally modified gravity wave, or Poincaré wave, with frequency f as the lower limit. Hence, the rotation frequency of the velocity vector shifts to slightly higher frequencies (i.e., it is blue shifted), and the motion can be described as a near-inertial oscillation (NIO). When the propagation of these waves is important, we may alternatively call them near-inertial waves (NIWs). In a model with continuous stratification below the mixed layer, NIW propagation is fully three-dimensional.

Kundu et al. (1983) explored the behavior of wind-generated currents along an infinite, vertical wall, with continuous stratification below a surface mixed layer. Their solution to the linear problem was expressed as the sum of Ekman and NIO responses in the mixed layer, plus NIW energy propagating both offshore and downward. The net effect is a solution for which the amplitude of the mixed layer NIO tends to zero at the coast, consistent with the no-normal-flow boundary condition. This phenomenon, known as coastal inhibition, has been well documented and observed (Schahinger 1988; Tintoré et al. 1995; Shearman 2005; Fontán and Cornuelle 2015).

The occurrence of coastal inhibition near islands is unclear, as islands have both finite lateral scale and complex bathymetric profiles. Few studies have explored the interaction of NIOs with islands, though in a recent paper, Brink (2021) modeled the island response to large-scale currents oscillating at tidal frequencies. He found that superinertial “leaky” modes can be generated; they resemble Kelvin waves propagating around the island, but they slowly lose energy to internal waves that radiate away.

Longuet-Higgins analyzed trapped waves and inertial currents around circular and elliptical islands in two papers. In the first, Longuet-Higgins (1969, hereafter LH69) found the trapped analytical solutions to the barotropic, long-wave equation:
(∇2+σ2−f2gH)η=0,
where ∇2 is the horizontal Laplace operator, σ is the radian frequency, H is the ocean depth, g is the acceleration of gravity, f is the Coriolis frequency, and η is the surface elevation. Assuming a horizontally periodic solution, imposing the boundary condition that η tends to zero as the radial distance r goes to infinity to exclude radiating waves, and requiring no flow through the island boundary at r = a, he found solutions for subinertial trapped modes when the island radius a exceeds a critical radius [n(n − 1)gH]1/2/f, where n is the azimuthal wavenumber. Additionally, he determined that a trapped wave precisely at f could exist if the island radius is identical to the critical radius.

In the second paper, Longuet-Higgins (1970, hereafter LH70) focused on inertial motions around circular and elliptical islands. By setting σ = −f and applying the boundary condition that the velocity tends to an IO at infinity, the long-wave equation (1) is reduced to the Laplace equation. In contrast to the coastal inhibition observed along an infinite coastal boundary, Longuet-Higgins found that inertial currents strengthen near the boundary of a circular island, reaching speeds twice that of the far-field IOs. With an elliptical island, inertial currents are further enhanced near the narrow tips of the island. While LH69 considered the trapped wave modes of an island, the solutions of LH70 can be thought of as the adjustment to an IO near an island boundary and are relevant at the inertial frequency around an island with a radius much smaller than the Rossby radius of deformation.

In an observational study, Siegelman et al. (2019) examined near-inertial currents around the island chain of Palau. Near-inertial motions coherent between the northern and southern tips of the island chain were observed, despite over 170 km separating the two sites. The generation of these coherent motions was attributed to wind stress, which had decorrelation length scales of 380 km. Spatial variability of the near-inertial currents around the island chain also was observed, with stronger currents near the narrow northern and southern tips of the island than in the far field, which is qualitatively consistent with the behavior of IOs around an elliptical island presented in LH70 and in contrast to coastal inhibition.

The work of Longuet-Higgins provides an initial, though limited, framework to understand the behavior of wind-generated NIOs around island topography similar to Palau. LH69 found the theoretical island-trapped modes that could be excited by a wind forcing around an island. This solution is valid for frequencies equal to f if the island radius is precisely equal to the critical radius, as defined by LH69. LH70 describes linear solutions for IOs near small islands, enabling a simplification of the governing equations by eliminating free waves from the dynamics and only considering motions precisely at f. As such, the solutions are limited in their applicability to the ocean where forcings are broadband, and stratification results in a smaller internal Rossby radius. Furthermore, neither paper considers forcing mechanisms, such as the wind, that excite the trapped or near-inertial motions.

Here, we revisit the work of LH70 and explore the assumptions, dynamics, and energetics of the analytical solution to gain theoretical understanding of IOs around circular and elliptical islands in an infinite domain. Then, to relax some of the constraints of the analytic solutions, we use a wind-forced 1.5-layer reduced-gravity numerical model on an f plane. For simplicity the model dynamics are linear and inviscid. A spatially uniform wind forcing is used to excite NIOs; the periodic boundary conditions we use would admit spatially uniform solutions in the absence of an island. Last, we use a wind patch east of the island to excite near-inertial Poincaré waves (i.e., NIWs) that are partly reflected and partly diffracted as they encounter the island.

This paper is organized as follows: Section 2 explores the analytical solution of rotary, inertial currents around a circular island and an elliptical island, section 3 presents results from a 1.5-layer reduced-gravity model, section 4 summarizes the results and provides concluding remarks.

2. Analytical solution

a. Rotary flow around a circular island forced by boundary oscillations

The analytical solution of LH70, extended here to a 1.5-layer ocean, provides a useful starting point for assessing the 1.5-layer reduced-gravity model results (section 3). LH70 studied analytically the behavior of IOs in an infinite domain with a circular island. Replacing his barotropic formulation with the reduced-gravity equivalent, the linearized momentum and continuity equations are
∂u∂t+fk^×u=−g′∇h, and
∂h∂t+H(∂u∂x+∂υ∂y)=0,
where g′ is the reduced gravity, f is the Coriolis frequency, H is the layer thickness, and h is the layer thickness perturbation (i.e., the negative of the interface displacement, hi). The interface displacement may be converted to the surface elevation using the relationship: η=−(g′/g)hi, making the total interface depth H − hi. Here, g′ = 0.09 m s−1, f = 2.024 × 10−5 s−1 for 8°N, a nominal latitude for Palau, and H = 100 m. Assuming an oscillatory solution of the form u, h ∝ e−iσt in Eqs. (2) and (3), we obtain the long-wave equation and the polarization relation between the velocity and layer thickness perturbation:
(∇2+σ2−f2g′H)h=0, and
u=−g′σ2−f2(iσ∇h+f×∇h).
At σ = f, the apparent singularity in (5) does not exist because h yields a factor of σ2 − f 2 in the numerator that cancels the factor in the denominator (see appendix B). If σ = f or in the limit that the island radius a is much less than the baroclinic Rossby radius of deformation, Ld=g′H/f, the long-wave Eq. (4) reduces to the Laplace equation:
∇2h=0.
More precisely, we define an island size parameter, ϵL=a2f2/(g′H), and a frequency parameter, ϵσ=(σ/f)−1. Then the Laplace equation approximation is valid in the limit that the product ϵLϵσ ≪ 1, and the horizontal divergence goes to zero in the small-island limit, ϵL ≪ 1 (appendix A). The LH70 solution of (6) has motions restricted to f; the role of waves, that is, σ ≠ f, is explored using the linear, reduced-gravity model in section 3.
Focusing on IOs (i.e., σ = f), following LH70, Eq. (6) may be solved in polar coordinates by imposing the no-normal-flow condition on the island boundary, while requiring spatially uniform IOs in the far field:
ur=Cei(θ+ft),uθ=iCei(θ+ft) as r→∞,
where ur is the radial velocity, uθ is the tangential velocity, and C is a constant. Applying those conditions to the reduced-gravity equations, the solution follows that in LH70 with g replaced by g′:
ur=(1−a2r2)Cei(θ+ft),
uθ=i(1+a2r2)Cei(θ+ft), and
g′h=−2if(a2r)Cei(θ+ft).
(See appendix B for details of the derivation.) The radial (8) and tangential (9) velocity fields are, at any instant, those of steady potential flow around a cylinder having a background magnitude of C (Fig. 1). At the upstream and downstream edges of the cylinder, where the incident flow is normal to the island boundary, there are stagnation points. At the top and bottom of the cylinder, ±90° from the upstream stagnation point, the flow reaches maximal speed. The flow field and surface elevation simply rotate anticyclonically around the cylinder at the inertial frequency as the direction of the incident flow changes.
Fig. 1.
Fig. 1.

LH70 solution for inertial oscillations in the vicinity of a small island. (left) Snapshot of the velocity (blue vectors) and surface elevation (color) from Eqs. (8)–(10). At any instant the velocity field is identical to potential flow around a cylinder. The surface elevation, which has a mode-1 spatial structure, adjusts instantaneously with the velocity field, rotating anticyclonically at f. (right) Terms of the x and y momentum balance equations. The sum of the local acceleration ∂u/∂t and the Coriolis acceleration fk^×u equals the pressure gradient −g∇η.

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-21-0310.1

The pressure pattern in Fig. 1 superficially resembles an azimuthal mode-1 island-trapped wave, but the approximation (6) excludes wave dynamics. The velocity field is horizontally nondivergent, as in the limit of a rigid lid and infinite gravity wave speed.

Despite the similarities of the IO velocity field to that of steady potential flow, the IO and potential flow momentum balances differ, as is evident in the relationship between velocity and the surface elevation (Fig. 1). In steady potential flow, the pressure gradient force balances the advection of momentum. This results in an azimuthal mode-2 structure with pressure maxima at stagnation points around the island, and pressure minima at velocity maxima. For the IO analytical solution, the far-field balance is between the local and Coriolis accelerations, but near the cylinder the balance includes the pressure gradient. At the upstream and downstream stagnation points, local accelerations balance pressure gradients, whereas at the top and bottom edges of the island, the Coriolis acceleration balances the pressure gradient in a Geostrophic balance. These balances also result in an azimuthal mode-2 pattern, but the pressure and velocity extrema now coincide (Fig. 1), instead of being 180° out of phase as in the potential flow example.

The structure of the solution (8), (9), and (10) may also be shown via the normalized polar velocity components and pressure as functions of r*, where r*=r/a (Fig. 2). On the island boundary, the radial velocity must go to zero, while the tangential velocity is twice that of the far-field inertial current. Both velocity components rapidly approach the far-field velocity as r*−2. In contrast, the surface elevation decays only as r*−1.

Fig. 2.
Fig. 2.

Radial (blue) and tangential (orange) velocity and surface elevation (green) normalized by far-field velocity. This figure was reconstructed based on Fig. 1 from LH70, which was missing a factor of −2 in the pressure term. The radial velocity goes to zero on the island boundary to satisfy the boundary condition, while the tangential velocity is twice that of the far-field IO.

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-21-0310.1

b. Rotary flow around an elliptical island

Continuing to follow LH70, the solution for IOs around a circular island may be extended to an elliptical island, more representative of Palau, using conformal mapping (further details in appendix B). Consider a circle of radius a centered at the origin on the λ plane (λ = μ + iν). This circle, where a=(1/2)(Smaj+Smin), can be transformed into an ellipse centered at the origin in the z plane (z = x + iy) with semimajor and semiminor axes lengths of Smaj and Smin, respectively, using the Joukowski transformation:
z=λ+c24λ,
where c2=Smaj2−Smin2 and Smaj ≥ Smin. The inverse of Eq. (11),
λ±=12[z±(z2−c2)1/2],
transforms an ellipse in the z plane to a circle in the λ plane. Thus, to find the solution for rotary flow around an ellipse in the z plane, the positive inverse transformation (12) is substituted into the solution for the flow around a circle in the λ plane (8), (9), and (10):
u=(∂λ∂z−a2λ*2∂λ∂z*)Ceift,
υ=i(∂λ∂z+a2λ*2∂λ∂z*)Ceift, and
g′h=−2ifCa2λ*eift,
where u and υ are in elliptical coordinates. The negative inverse of Eq. (12) is neglected as it maps into the interior of the circle.

Rotary flow around an elliptical island exhibits many of the same characteristics as flow around a circular island. Regardless of flow direction, there are two stagnation points and two speed maxima on the island boundary. The flow field near the island is largely in geostrophic balance, with high pressure on the left side of the island and low pressure on the right when facing downstream. Once again, the azimuthal wavelength of the surface elevation is equal to the circumference of the island, and the rotation of the far-field velocity forces this pattern to rotate anticyclonically around the ellipse at the inertial frequency.

For an elliptical, rather than circular, island, the rotary flow is blocked by an obstacle of varying shape as the angle of incidence changes. Figure 3 shows the velocity field and surface elevation over a quarter of an inertial period for an ellipse with eccentricity (c = 83) similar to Palau. When the incident flow is exactly normal to the minor axis (Fig. 3a), island blocking and flow acceleration are minimal and the maximum speed is low. A quarter of an inertial period later the incident flow is blocked by the broad side of the island (Fig. 3d), maximizing the flow acceleration around the island tips. Figure 4a shows, for an ellipse of eccentricity c = 83, the azimuthal velocity maximum on the island boundary is approximately 6.4 times greater when the flow is blocked by the major axis than when blocked by the minor axis. The amplitude of the surface elevation on the island boundary is not dependent on the eccentricity (Fig. 4b). The radial pressure gradient force, however, is a function of the eccentricity and increases for a narrower island (larger c), resulting in high flow speeds around the tips of the ellipse (Fig. 4c).

Fig. 3.
Fig. 3.

(a)–(d) Snapshots of the velocity (red vectors) and surface elevation (color) from the analytical solution of rotary flow around an elliptical island, when σ = f, over a quarter of an inertial period. When the incident flow is normal to the minor axis as in (a), flow acceleration is minimal. After a quarter of an inertial period as seen in (d), the IO is normal to the major axis resulting in enhanced inertial currents at the narrow ends of the ellipse due to maximal flow blocking.

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-21-0310.1

Fig. 4.
Fig. 4.

Cross-shore transects from the σ = f solution of the (a) normalized azimuthal velocity (contours mark 2.0, 3.0, 4.0, and 7.0), (b) normalized surface elevation (contours mark 0.5, 1.0, and 1.5), and (c) normalized pressure gradient force (contours mark 0.5, 1.0, 1.5, 2.0, and 7.0) extending distance L, normalized by the ellipse radius, a=(1/2)(Smaj+Smin) offshore from the northern most point on the island for a range of ellipse eccentricity values, c=Smaj2−Smin2. Above the black line (c > 0), the upstream velocity is purely zonal and blocked by the broad side of the island. Below the black line (c < 0), the purely zonal flow is blocked by the narrow side of the island. When the incident flow is normal to the major axis (c > 0), there is flow enhancement and large pressure gradients near the narrow tips of the elliptical island. Alternatively, when the incident flow is normal to the minor axis (c < 0), there is marginal flow blocking and flow acceleration. As the eccentricity increases, the energy enhancement increases around the narrow tips of the ellipse and decreases along the flanks of the ellipse.

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-21-0310.1

c. Energetics

The presence of an island alters the spatial distribution of kinetic and potential energy (Fig. 5). The amount of additional energy around the island is identical to that in a homogeneous “background” IO contained within an area of equal size to the island in the absence of topography, and one may regard the IO energy as having been displaced by the island. This finding is reflected in the equivalence of the area integral of the kinetic energy of the circular island solution [Eqs. (8) and (9)] minus the background IO [Eq. (7)] from r = a to r = ∞ and the area integral of the background IO [Eq. (7)] from r = 0 to r = a in the absence of the island:
∫02π∫a∞12ρ(ur′2+uθ′2)rdrdθ=∫02π∫0a12ρ(ur2+uθ2)rdrdθ=12ρC2a2π,
where ur′=−(a2/r2)Ccos⁡(θ−ft),uθ′=(a2/r2)Csin⁡(θ−ft), ur = C cos(θ − ft) and uθ = C sin(θ − ft). The azimuthal velocity (9) and pressure (10) are exactly 180° out of phase, which indicates an anticyclonic energy flux around the island.
Fig. 5.
Fig. 5.

Mean kinetic energy of the σ = f solution [Eqs. (8)–(10)], which satisfy [Eqs. (5) and (6)] over an inertial period in the presence of an (a) circular and (b) elliptical island. The mean energy distribution is uniformly distributed around the circular island, but focused around the narrow tips of the elliptical island, similar to observations around the island chain of Palau.

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-21-0310.1

Kinetic energy is enhanced around the elliptical island tips and reduced along the flanks (Fig. 5b). Additionally, the mean kinetic energy along the major axis of the ellipse is less than in the far field because of the stagnation point that travels along the ellipse boundary. Despite the assumptions of the analytical solution, the distribution of kinetic energy around the ellipse shows many similarities to the distribution of kinetic energy observed around Palau by Siegelman et al. (2019), most notably the observed flow enhancement along the narrow tips of the island chain and weaker inertial currents along the longer island boundaries.

3. Numerical model

In this section, a linear 1.5-layer reduced-gravity model is used to relax the limitations of the analytical solution, providing insight into the more realistic behavior of wind-generated NIOs and NIWs around islands. The model is forced by wind stress, rather than assuming a solution that is periodic at the inertial frequency and that matches an IO in the far field. The idealized wind forcing, with anticyclonic rotation and a Blackman modulation window, is chosen to efficiently excite NIOs. After the forcing has stopped, the flow in the model is free to evolve in time; unlike the analytic solution, it is not constrained to oscillate at a single frequency. In a finite domain and in the presence of a circular island, which introduces a finite length scale, the free solutions to the governing reduced gravity [Eqs. (17)–(19)] consist of Poincaré waves, manifest as NIOs or NIWs for frequencies slightly above f, and island-trapped waves, which resemble Kelvin waves as the island size increases. The model allows us to vary the island size without constraint, to see how the solution evolves as the small-island condition is relaxed. In a final set of numerical experiments, the wind forcing is moved away from the island, showing the response around the island to incident NIWs.

The inviscid, linear 1.5-layer reduced-gravity model assumes motion is confined to the surface layer and the lower layer is immobile and infinitely deep. The governing equations are
∂u∂t−fυ=g′∂hi∂x+τxρoH,
∂υ∂t+fu=g′∂hi∂y+τyρoH, and
−∂hi∂t+H(∂u∂x+∂υ∂y)=0.
The surface elevation is η=−(g′/g)hi, where g is the gravitational acceleration and g′ is the reduced gravity. Finally, τ is the wind stress, H is the undisturbed layer thickness, and f is the Coriolis parameter. We use H = 100 m, g′ = 0.09 m s−2, and f = 2.024 × 10−5 s−1 for 8°N, a nominal latitude for Palau. The baroclinic Rossby radius of deformation, Ld=g′H/f, is 148.2 km. The island-size parameter, ϵL=a2f2/(g′H), is varied by changing the island radius a over a range of 50–300 km.

The governing [Eqs. (17)–(19)] are explicitly time integrated using the third-order Adams–Bashforth time-stepping method (Durran 1991). This finite-difference model is solved on a 6000 km × 6000 km, Arakawa C grid with 5-km resolution. Periodic open boundary conditions are used in the far field, with a free-slip condition on the island boundary.

a. Spatially uniform winds

To compare with the LH70 solution, which describes an island in an infinite IO field, a spatially uniform wind forcing is used to excite NIOs. While the theory described in LH70 requires that the island is small relative to the Rossby radius of deformation (ϵL ≪ 1), this condition is removed in numerical simulations where the island radius is varied, but the Rossby radius of deformation is kept constant. In most of the experiments the wind forcing is modulated by an envelope of one inertial cycle. An additional experiment is run for a large-island case with a three-cycle forcing envelope to show the effect of reduced forcing bandwidth.

1) A case study: Dynamics near a small island (ϵL = 0.11)

The motivation for this case study is to assess the behavior of spatially uniform wind-generated NIOs around a dynamically small island after wind forcing has stopped and motion is not constrained to σ = f. This case is compared with LH70’s analytical solution, which requires a small island and imposes spatially uniform IOs. The island radius is 50 km and ϵL = 0.11.

The numerical model starts from rest with a spatially uniform wind event that gradually grows and decays over an inertial period, at which point the wind shuts off (Fig. 6a). To efficiently excite NIOs, the wind forcing τ rotates anticyclonically at the inertial frequency:
τ=τx+iτy=w(t)τ′e−i2πft,
where the wind amplitude τ′ is 0.05 N m−2, which is typical for wind events near Palau. A Blackman window smoothly modulates the wind amplitude over a single inertial period:
w(t)=0.42−0.5cos 2πft+0.08cos 4πft,(0≤t≤1/f)
=0,(t>1/f).
As the wind strengthens, the currents accelerate and begin to rotate anticyclonically. Unlike the inherently horizontally nondivergent analytical solution, the flow field near the island immediately becomes divergent (Fig. 6b) as free Poincaré waves are generated to satisfy the boundary condition at the island. The Hovmöller diagram of a zonal transect to the east of the island shows a wave radiating away from the island boundary (Fig. 6c). The estimated wavelength and frequency of the initial spin-up wave, estimated from the Hovmöller diagram (Fig. 6c) and denoted by the red star in Fig. 6d, falls directly on the Poincaré wave dispersion relationship (blue line).
Fig. 6.
Fig. 6.

(a) Time series of the applied spatially uniform, anticyclonically rotating wind stress until it shuts off after 1 inertial period. (b) Horizontal divergence of the velocity at the eastern boundary of the island. (c) Hovmöller diagram of the surface elevation from a zonal transect beginning at the eastern most end of the island and extending 1000 km offshore. The phase speed, estimated from the slope of the blue dashed line, is 3.64 m s−1. (d) Poincaré wave dispersion relation. Red star indicates estimated wavenumber and frequency of the first wave to radiate away from the island. Black dashed line marks the inertial frequency.

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-21-0310.1

Once the wind forcing stops, NIWs continue to radiate outward, superimposed on the far-field NIOs. Qualitatively, similarities exist between the numerical and analytic solutions of the flow field and surface elevation around a circular island (Fig. 7). As in Fig. 1, the numerical solution has two stagnation points where the flow is normal to the island boundary, and two speed maxima as the flow accelerates around the island after reaching the initial stagnation point. This velocity pattern rotates anticyclonically around the island at approximately the inertial frequency. Furthermore, two surface elevation extrema (one minimum and one maximum) associated with the strongest currents around the island exist.

Fig. 7.
Fig. 7.

(top) Snapshot of the surface elevation (color) with the corresponding velocity (cyan vectors) (left) within 10 and (right) zoomed into just 5 radii away from the island at 1 inertial period. The spiral pattern (which is not evident in the analytical solution) is a superinertial Poincaré wave radiating away from the island. (middle),(bottom) The x- and y-momentum balance terms at one inertial period after the model is initialized, i.e., immediately after the wind forcing is shut off. This snapshot highlights the predominant far-field balance between the local acceleration and Coriolis term suggesting NIOs, the Poincaré waves radiating away from the island, and the phase lag due to the island-trapped wave propagating clockwise around the island.

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-21-0310.1

Unsurprisingly, the analytical and numerical model solutions differ. First, an outward propagating superinertial Poincaré wave is generated at the island boundary even after the wind forcing is shut off. Second, there is a phase lag between the far-field NIO, which is purely zonal at the instant shown in Fig. 7, and the near-field currents and surface elevation. According to the analytical solution (9) and (10), given a far-field, zonal flow, the surface elevation and velocity extrema would be located at the north and south ends of the island (Fig. 1), but here, the extrema have yet to reach these locations. The phase lag is consistent with a slightly subinertial island-trapped wave propagating anticyclonically around the island.

Consistent with the analytical solution, the model is linear and inviscid. Figure 7 shows snapshots of the terms from the numerical model x- and y- momentum balance Eqs. (17) and (18) at one inertial period, just after the wind stops. The far field is primarily a balance between the local acceleration ∂υ/∂t and the Coriolis acceleration fu which is consistent with NIOs at an instant when the far field is purely zonal. In the y-momentum equation, the pressure gradient is balanced by the local acceleration where the flow is normal to the boundary and by the Coriolis acceleration where the flow skirts the island. The region corresponding to the surface elevation spiral is a Poincaré wave–like balance of all three terms in both the x- and y-momentum equations. The phase lag between the near and far-field currents and the surface elevation is also evident in the momentum terms.

Quantitative comparisons between the model and analytic solutions of LH70 illustrate some key differences. At one inertial period, as the wind forcing ends, the analytical and modeled solutions are in best agreement (Figs. 8b–d, dashed black line compared to pale red line). Consistent with LH70, the azimuthal velocity at the island boundary is approximately twice that of the far field, and the normalized surface elevation is approximately −2 at the island boundary. Over time, the surface elevation and velocity of the numerical solution amplify, resulting in increasingly larger values compared to the analytical solution (Figs. 8b–d, dashed black line compared to increasingly dark red lines). This enhancement is also seen in the Hovmöller diagrams of these fields along the island boundary (Figs. 9a,c,e). Over time, the amplified region (between the sloping red dashed lines) slowly spreads outward from the island. Beyond this region there is a subtle dip in velocity amplitude (Figs. 9c,e between red and blue dashed lines). After one inertial period there is no wind forcing, so these structures result from the superposition of free waves, including incident and reflected Poincaré waves and the subinertial trapped wave. The energy enhancement will be further discussed in section 3a(2).

Fig. 8.
Fig. 8.

(a) Time series of the azimuthal velocity near the northern extent of the model domain. Extrema (dots) occur when the tangential speed at the furthest extent north of the island is at a maximum, thus the velocity of the far-field NIO is almost purely zonal. Given purely zonal far-field flow, (b) the radial velocity along a zonal transect and (c) the azimuthal velocity and (d) surface elevation along a meridional transect should compare well with the analytical solution presented in Fig. 2 (black dashed lines). The color of the transects corresponds to the color of the dots and denotes the time in inertial periods when the snapshot occurs. The best agreement between the analytical and numerical models occurs at one inertial period. As the model run progresses, energy continues to accumulate near the island, which is explained in section 3.

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-21-0310.1

Fig. 9.
Fig. 9.

(left) Hovmöller diagrams of (a) surface elevation, (c) zonal, and (e) meridional velocity along a zonal transect that bisects the island. Magenta dashed line marks the instant that the wind forcing stops. Between the island boundary and red dashed line, the surface elevation increases over time. The velocity Hovmöller diagrams show enhanced velocities near the island (between the island boundary and red dashed line) and weakened velocities beyond those regions (between red and blue dashed lines). (right) Power spectral density of (b) surface elevation, (d) zonal, and (f) meridional velocity along the same transect shown in the left panels. The inertial frequency f is indicated by the cyan dashed line. Near the island, the surface elevation bandwidth is relatively broad, with energy ranging from sub- to superinertial. Away from the island, the PSD peak frequency is blue shifted (i.e., shifted toward higher frequencies) from the inertial frequency. Across the transect, the velocity PSD has a narrower peak, with most energy concentrated at the inertial frequency. The υ component of the velocity has a wider peak near the island that is slightly red shifted from the inertial frequency.

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-21-0310.1

The Hovmöller diagram of surface elevation shows slanting phase lines during the entire run corresponding to Poincaré waves radiating away from the island (Fig. 9a). The decreasing slope of the phase lines over time reflects the increase in phase speed of the waves as their frequency approaches, but never reaches, the inertial frequency.

The power spectral density of the surface elevation along the zonal transect emphasizes the island-trapped wave and the energy radiating away from the island (Fig. 9b). Near the island boundary, the spectral peak is wide with a peak frequency at f, indicating energetic sub- and superinertial motions. The subinertial energy is the signature of the island-trapped wave (LH69). Away from the island, consistent with radiating Poincaré waves, the spectral peak is blue shifted (i.e., above f), with little energy near the inertial frequency. The surface elevation expression of NIOs and NIWs vanishes as the frequency approaches f.

Compared to the surface elevation spectrum, the velocity spectrum is more narrow banded and concentrated near f and relatively uniform with distance from the island (Figs. 9d,f). The radial component (u) goes to zero at the boundary over an island radius scale. The variance of the tangential component (υ) increases toward the boundary, consistent with the free slip boundary condition.

2) ϵL dependence

To move away from the small-island case, we increase ϵL by increasing the island radius, a. Immediately after the winds shut off, the phase difference between the far-field NIOs and the currents near the island is greater as ϵL increases (Fig. 10). The angular frequency dθ/dt, estimated from a Hovmöller diagram of the surface elevation along the island boundary, decreases as ϵL increases (Fig. 11, blue dots), qualitatively consistent with the theoretical mode-1 island-trapped wave solutions derived by LH69. A quantitative comparison in Fig. 11, however, shows the angular frequency decreasing with ϵL more gradually in the numerical results than in the LH69 theory.

Fig. 10.
Fig. 10.

Snapshot of surface elevation (color) with the corresponding velocity (cyan vectors) after 1 inertial period when ϵL is (a) 0.11, (b) 1.02, and (c) 1.82. As ϵL increases, the phase lag between the trapped wave and far-field NIOs grows more rapidly due to the decreasing frequency of the trapped wave (LH69) (Fig. 11).

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-21-0310.1

Fig. 11.
Fig. 11.

Comparison between the estimated azimuthal phase speed normalized by f for model runs of varying ϵL (light blue dots) and the mode-1 island-trapped wave theory presented by LH69 (solid blue line). The frequency difference between the numerical model and the analytical solution for the mode-1 trapped wave frequencies is due to the inertial and subinertial motions in the numerical model.

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-21-0310.1

The deviation between the numerical results and LH69 is due to the superposition of superinertial, near-inertial, and subinertial motions in the model, while the theory only reflects the subinertial trapped modes (LH69). Evidence of the superposition is seen in the increasing bandwidth of the spectral peak as ϵL increases (Fig. 12). When ϵL = 0.11, the trapped wave frequency is 0.99f, which is consistent with the slight phase lag near the island boundary, while maintaining a single spectral peak (spectral resolution = 0.05 cpd). When ϵL = 1.82, there are two distinct spectral peaks, one at the inertial frequency and the other at the trapped wave frequency, 0.154 cpd. In the far field, there is a slight increase in the variance of η at superinertial frequencies with increasing ϵL, suggesting more energetic Poincaré waves with larger ϵL.

Fig. 12.
Fig. 12.

PSD of (a),(b) u and (c),(d) η (bottom row) at the (left) northern island boundary and (right) in the far field. When ϵL is small (0.11, 0.26), the PSD of u and η have a single peak at the inertial frequency. As ϵL increases (0.46–1.82), the single peak broadens until two peaks emerge, one at the inertial and the other at the trapped wave frequency from LH69.

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-21-0310.1

The superposition of the incident and reflected Poincaré waves and the island-trapped wave results in local fluctuations of energy due to constructive and destructive interference patterns. The temporal and spatial variability of the energy are seen by examining the azimuthal mean of kinetic energy around a circle at radius r (Fig. 13, left column):
KE¯=Hρ2π∫02π12(u2+υ2)dθ,
and potential energy (Fig. 13, middle column):
PE¯=ρg2π∫02π∫0ηzdzdθ=ρg4π∫02πη2dθ.
The radially outward flux across a circle of radius r (Fig. 13, right column):
Er=∫02πpurrdθ,
shows the radial direction of energy propagation due to the interference among the excited waves.
Fig. 13.
Fig. 13.

The azimuthal mean of (left) kinetic [Eq. (23)] and (center) potential [Eq. (24)] energy and (right) the radial energy flux [Eq. (25)] for a range of ϵL. The radial energy flux is positive in the direction away from the island. Near the island, the energy fluctuates at the beat period between the NIOs and the trapped wave. At the island boundary, periods of increasing energy correspond with negative radial energy flux, suggesting that the increase is due to offshore energy moving toward the island. In the far field, fluctuations of the radial energy flux are the result of Poincaré waves superposed on the background NIOs.

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-21-0310.1

First, consider the small island case, where ϵL = 0.11 (Fig. 13, first row). On the island boundary, energy continues to increase in the absence of wind forcing after one inertial period (also evident in Figs. 9c,d). The negative radial energy flux near the boundary shows that energy from offshore is moving toward the island; energy increases adjacent to the island and decreases farther from the shore. With increasing distance from shore, additional fluctuations in kinetic energy correspond to alternating positive and negative radial energy flux.

The flux of energy within the domain is the result of the changing interference pattern due to the superposition of the incident, radiating and trapped waves. In all simulations, the far field (>1000 km from the island boundary) is dominated by the NIOs and Poincaré waves, which superpose to create regions of increased and decreased kinetic energy that correspond to positive and negative radial energy flux. While the far-field NIOs are identical in all runs, the radial energy flux increases with the island size. A larger island intercepts more near-inertial energy from the large-scale NIO, and therefore reflects more energy away from its boundary in the form of Poincaré waves. A smaller island allows a larger fraction of incident energy to diffract rather than reflect.

To understand the interference pattern near the island (within 1000 km from the island boundary), consider the linear superposition of two waves of differing frequencies (f1 and f2). Due to constructive and destructive interference, the energy fluctuates at the beat frequency, that is, |f1 − f2|. In these simulations, near the island, the two waves are the near-inertial motions and the trapped wave propagating anticyclonically around the island. The near-inertial motions consist of the large-scale NIOs plus the outward-radiating NIWs that are necessary to satisfy the boundary condition at the island. Initially, the energy near the island increases as the trapped wave and the near-inertial motions become more in phase and interfere constructively. When ϵL is small, the trapped wave and the near-inertial motions have nearly the same frequency; thus, the beat period is extremely long and only a small fraction of it occurs during a 20-day run. As a result, for small values of ϵL (0.11, 0.26), less than half the beat period is resolved and only a period of energy increase near the island is seen.

As ϵL increases, the trapped wave frequency decreases and the beat period becomes shorter. The three runs that resolve at least half the beat period (ϵL = 0.46, 1.02, 1.82) show both constructive and destructive interference near the island. As with ϵL = 0.11, energy initially increases on the island boundary. However, after half a beat period, energy decreases on the island boundary and moves offshore, indicated by the positive radial energy flux. Given a long enough run and a large enough island, we see multiple cycles.

3) A large island (ϵL = 4.10): Sensitivity to forcing bandwidth

The previous section demonstrates that the superposition of the large-scale NIOs, radiating Poincaré waves, and subinertial island-trapped waves accounts for the net ocean response near an island to an episode of spatially uniform near-inertial wind forcing. For small islands the trapped wave frequency is too close to f for simple data analysis techniques to isolate that component of the motion in a numerical run lasting a few inertial cycles. The mode-1 trapped wave frequency decreases with increasing island size, though, so in this section we use a large enough island (ϵL = 4.10) to easily distinguish the trapped wave from the remainder of the motion. Additionally, we use a second run with a longer forcing interval, hence a narrower forcing bandwidth (Fig. 14), to show the sensitivity of the response to this parameter. In this run the amplitude is reduced by a third and the modulation envelope spans 3 inertial periods, so the work done by the wind in forcing NIOs is unchanged. For both runs the island radius is 300 km, the domain is expanded to 25 000 km × 25 000 km, and the resolution is reduced to 10 km.

Fig. 14.
Fig. 14.

Rotary spectrum of the wind stress τ when the forcing is applied over 1 inertial period (magenta) and 3 inertial periods (blue). The clockwise component is denoted by a solid line, and the counterclockwise component is denoted by a dashed line.

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-21-0310.1

To isolate the island-trapped wave, the time series at each point is bandpass filtered via complex demodulation at the theoretical island-trapped wave frequency, 0.11 cpd, from LH69. The first and last trapped wave periods are subject to filtering end effects and are therefore omitted from the analysis. The residual elevation shows the expected radiating Poincaré wave signal (Fig. 15a), while the residual velocity is dominated by NIO energy that is suppressed near the island (Figs. 15c,e). The trapped wave (Fig. 15, right column) has a Kelvin wave character with energy in the surface elevation and the alongshore velocity component (υ in this zonal transect). The modeled surface elevation as a function of distance from the island agrees well with the LH69 theory, and both are close to the exponential behavior of a Kelvin wave (Fig. S1 in the online supplemental material).

Fig. 15.
Fig. 15.

Hovmöller diagrams from a zonal transect that bisects the island when a 1 inertial period forcing is applied. Residual (a) surface elevation, (c) u, and (e) υ after removing the trapped wave component. Different ranges of horizontal distance (x axis) are used to highlight the far-field radiating waves in the surface elevation (a) and the near-field velocities (c) and (e). The residual is predominantly near inertial. Slanted phase lines in surface elevation (a) show the outward radiation of Poincaré waves with frequencies successively closer to f as time increases. In the far field, the residual currents (c) and (e) are NIOs. Approaching the island, the normal velocity component, u, decays to zero, satisfying the boundary condition. The tangential velocity υ reaches a null approximately 100 km from the island boundary and reverses closer to shore. (right) The trapped wave component of (b) surface elevation, (d) u, and (f) υ. The surface elevation has an azimuthal mode-1 structure, but with radial decay rather than propagation. The cross-shore velocity in (d) is negligible, consistent with Kelvin wave dynamics.

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-21-0310.1

The sensitivity of the NIOs and Poincaré waves to the bandwidth of the wind stress is reflected in comparisons between the 1 and 3 inertial period forcing (1IPF and 3IPF, respectively) runs 3000 km east of the island boundary, beyond the radial extent of the trapped wave. The rotary spectra show that NIOs dominate the far field with anticyclonic, inertial peaks of identical magnitude in each of the numerical runs (Fig. 16a). In the time domain, the inertial peaks manifest as the plateau of kinetic energy at 0.002 m2 s−2 for both runs (Fig. 17a). The kinetic energy plateau is disrupted by the arrival of dispersive Poincaré waves generated at the island boundary. The superposition of these waves on the background NIO field causes fluctuations of the total kinetic energy that correspond to the positive and negative radial energy flux patterns observed in the far field of Fig. 13. Compared to the 3IPF run, the broader forcing bandwidth in the 1IPF run generates higher-frequency waves that travel faster and arrive sooner (Fig. 17a, magenta arrow). The power spectral density of η in the far field reflects only the Poincaré waves because the background NIOs have negligible surface expression. Increasing the forcing period from 1 to 3 inertial periods slightly reduces the near-inertial spectral peak height and decreases its bandwidth. (Fig. 16c).

Fig. 16.
Fig. 16.

The rotary spectrum of the (a),(b) velocity and the (c),(d) PSD of the surface elevation when the forcing is applied over 1 inertial period (magenta) and 3 inertial periods (blue) in the (left) far field and (right) at the eastern most point on the island boundary. In (a) and (b), the solid and dashed lines are the CW and CCW rotary components, respectively. The PSD of u and η on the island boundary show the sensitivity of the trapped wave to the bandwidth of the wind forcing. In the far field, the PSD of η has more energy at higher frequencies for the 1IPF run.

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-21-0310.1

Fig. 17.
Fig. 17.

Time series of kinetic energy when the forcing is applied over 1 inertial period (magenta) and 3 inertial periods (blue) 3000 km east of the island boundary, beyond the radial extent of the trapped wave. The arrival of the Poincaré waves (denoted by the arrows) is delayed in the 3IPF runs. (b) The residual tangential velocity at the eastern most point on the island boundary initially oscillates at a superinertial frequency and has a larger initial amplitude for the 1IPF run. Within approximately 4 inertial periods, the currents oscillate at a near-inertial frequency and the amplitude is the same regardless of the forcing. (c) The tangential velocity of the trapped wave component at the eastern boundary, which was removed from the currents shown in (b), has a larger amplitude for the 1IPF than the 3IPF. The gray boxes in (c) and (b) mark the period contaminated by end effects.

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-21-0310.1

At the eastern most point on the island boundary, the currents are rectilinear (alongshore), and the clockwise and counterclockwise rotary components are identical (Fig. 16b, dashed line overlies the solid line). The two distinct peaks are at the trapped wave frequency ftw (0.11 cpd, consistent with LH69) and the inertial frequency. Once again, the power spectral density (PSD) at the inertial frequency is identical in the two runs (seen also in the time domain, Fig. 17b). The near-inertial peak is much lower than it is in the far field, consistent with the reduction in residual current near the island seen in Figs. 15c and 15e.

The dynamics driving the attenuation of near-inertial currents toward the island are similar to those along an infinite, straight coastal boundary, known as coastal inhibition (Kundu et al. 1983). After high-frequency Poincaré waves quickly radiate away, the more slowly propagating NIWs remain. These NIWs superpose with the NIOs reducing the amplitude of the near-inertial motions near the island. The residual of the tangential velocity reflects the superposition of the NIWs and NIOs (Fig. 17b). One important difference in the work of Kundu et al. (1983), is that their continuously stratified model allows for downward propagation of near-inertial energy as opposed to our model that only allows for horizontal energy propagation. Despite this difference, both models admit near-inertial waves, which reduce energy in the NIOs near the coastal boundary.

As expected, the PSD in the trapped wave band is lower in the 3IPF run (Figs. 16b,d) because the forcing contains less energy at subinertial frequencies (Fig. 14). The amplitude of the tangential velocity does not decay over time (Fig. 17c), confirming that the energy at this frequency is in fact trapped. (For a view of the spectra as functions of distance from the coast see Fig. S2.)

b. Nonuniform winds

Although comparable to the LH70 analytical solution, the applicability of the results from the previous section is limited by the assumption of uniform wind stress across the entire domain. The generation of subinertial island-trapped waves by local winds has been well documented (Pizarro and Shaffer 1998; Brink 1999; Merrifield et al. 2002), but the excitation of slightly superinertial oscillations by NIWs impinging on an island has not been previously considered. A related problem is the model of Dale et al. (2001) for a straight coast with continuous stratification, a continental shelf, and a spur. They found that a superinertial current along the coast caused radiation from the spur of leaky coastally trapped waves at the same frequency. Similarly, we might expect NIWs impinging on an island to excite leaky island-trapped waves—that is, motions with structure very similar to pure island-trapped modes, but at slightly higher frequency that gradually lose energy by radiating NIWs. Therefore, we conduct simulations in which there is no local wind forcing, but a remote wind patch forces a spectrum of radiating Poincaré waves that impinge on the island. In these simulations, subinertial island-trapped waves cannot be excited because only energy from superinertial Poincaré waves reaches the island.

To examine this case, the original wind forcing, still varying in time as in Fig. 6a, is multiplied by a spatial Blackman window (21) that is nonzero for 1000 < x < 2000 km (blue rectangle in Fig. 18c). The domain extends farther in the east–west direction than the north–south direction (8000 km × 6000 km, 5-km resolution) to increase the distance between the island and the waves that propagate eastward from the source region and reenter in the west through the periodic boundary. Time periods contaminated by this energy reentry are excluded from the analysis.

Fig. 18.
Fig. 18.

Surface elevation (color) and velocity (cyan vectors) snapshots at (a) 3.50 and (b) 3.75 inertial periods after the start of the run where ϵL = 0.11. NIWs excited by divergent wind stress applied between 1000 and 2000 km along the x axis, propagate westward and eastward. (c) Hovmöller diagram of the surface elevation along a zonal transect that bisects the island. Cyan shading delimits the wind forcing region. The black dashed line marks 1 inertial period, after which the wind stops. NIWs radiate eastward and westward from the wind forcing region. Poincaré waves are reflected eastward away from the island boundary as a result of incident westward propagating NIWs. The superposition of the incident and reflected waves results in a null region that moves eastward over time between 0 and 1000 km from the island. A trapped wave propagates clockwise around the island, traveling approximately a quarter of the circumference in a quarter of an inertial period.

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-21-0310.1

The zonal structure in the wind excites Poincaré waves that propagate eastward and westward, with the higher wavenumber (higher frequency) components spreading the fastest (Figs. 18a,b). A slowly decaying NIO remains at the axis of symmetry, x = 1500 km. The sloping phase lines in the Hovmöller diagram of surface elevation also show the radiation of Poincaré waves from the axis of wind forcing, with frequency declining toward f over time (Fig. 18c).

The first waves to reach the island are high frequency and short wavelength (Fig. 18). Generally, these waves pass by the island unmodified. As the frequency of the waves decreases toward f and the wavelength increases, a leaky trapped wave is excited that propagates anticyclonically around the island (Figs. 18a,b). The wave has mode-1 structure and a near-inertial peak frequency.

Snapshots for these simulations show NIWs reflected and diffracted by the island superimposed on low wavenumber energy incident from the east and higher-wavenumber energy that has passed the island (Figs. 19a–c). In addition, leaky trapped waves are evident. Their azimuthal modal structure is dependent on ϵL. When the island size is dynamically small (ϵL = 0.11, r = 50 km), the NIWs excite a mode-1 leaky trapped wave. When ϵL = 1.02, a mode-2 wave almost develops, but does not maintain its structure. Finally, when ϵL = 1.82, a mode-2 leaky trapped wave is clearly excited. When the wind is spatially uniform (section 3a), the surface elevation associated with the adjustment around the island projects onto azimuthal mode 1. Here, however, the spatial structure of the incident NIWs can project onto higher modes.

Fig. 19.
Fig. 19.

(a)–(c) Snapshots at 4.6 inertial periods of the surface elevation (color) and velocity (cyan vectors) when ϵL = 0.11, 1.02, and 1.82. Poincaré waves propagate westward, reflecting off the eastern boundary of the island and refracting around it regardless of ϵL. When ϵL = 1.82, a mode-2 leaky trapped wave is excited. (d)–(f) PSD of surface elevation (red line) and rotary spectra of velocity (blue, clockwise rotational is solid and counterclockwise rotational is dashed) on the eastern boundary of the island when ϵL = 0.11, 1.02, and 1.82 in (d)–(f), respectively. The black dashed line marks the inertial period. The PSD shows that the mode-2 trapped wave has a near-inertial frequency.

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-21-0310.1

The rotary spectra of the currents at the easternmost point on the island boundary have a broad, slightly blue shifted near-inertial peak (Figs. 19d–f). The wider near-inertial peak, which is higher when ϵL is small, reflects the bandwidth of the NIWs impinging on the island. The total variance of the currents is 3 times larger when ϵL = 0.11 than ϵL = 1.82, indicating stronger near-inertial currents occur near a dynamically small island, which is consistent with the uniform wind forcing runs.

Similar to the velocity spectrum, the spectrum of η on the eastern boundary of the island also shows one slightly blue shifted, near-inertial peak regardless of ϵL (Figs. 19d–f). When ϵL = 1.82, this result may seem surprising because of the distinct subinertial trapped wave peak that arises in response to a spatially uniform wind forcing (Fig. 12c); however, this result is consistent with the theory of LH69. As noted in the introduction, LH69 found that a trapped wave could exist at the inertial frequency under the condition that the island radius is precisely equal to the critical radius [n(n − 1)g′H]1/2/f. Given a Rossby radius of deformation of 148.2 km, the critical radius is 209 km when n = 2. The island radius is 200 km when ϵL = 1.82, so the incident NIWs excite a mode-2 leaky trapped wave. Note the slight blue shift of the near-inertial peak (Fig. 19f), consistent with the fact that only slightly superinertial energy can reach the island in this model. When the island radius is too far from the critical radius for a mode, for example, the radius is 150 km (ϵL = 1.02), a trapped wave is not excited.

4. Summary and conclusions

Results from the 1.5-layer reduced-gravity model show how an episode of spatially uniform wind forcing can generate far-field NIOs, island-trapped waves, and NIWs radiating outward from the island. Similarly, when Poincaré waves generated by a remote wind forcing encounter an island, the dominant response is reflected NIWs, diffraction of incident wave fronts, and the generation of leaky island-trapped waves. In the small-island case (ϵL ≪ 1) with very large-scale forcing, the superposition of the incident, reflected, and trapped waves results in locally enhanced near-inertial motions similar to the analytical solution (section 2). Despite this similarity, there are fundamental differences between the analytic solution and the numerical experiments. The analytic solution, by design, is in a straitjacket: it requires that all motion in the infinite domain is harmonic at a single frequency. That frequency can only be f, since the far-field solution must be an inertial oscillation—the limit of a Poincaré wave as its horizontal scale goes to infinity. In contrast, after the forcing has been turned off, the numerical experiments can be viewed as initial value problems with the initial state being whatever fields of velocity and elevation have been generated by the forcing. Thereafter, the solution is simply the superposition of free waves, which can be island trapped (at discrete frequencies) or propagating to infinity (with a spectrum of superinertial frequencies).

As the island size increases from zero, the mode-1 trapped wave frequency decreases from f (LH69). For ϵL < ∼1 the frequency is close to f, so the relative phases of the trapped wave and the NIWs (i.e., Poincaré waves) change only slowly. With a much larger island, for example, ϵL = 4.1, the frequency difference is obvious, and we can easily separate the trapped wave, with its discrete subinertial frequency, from all the NIWs.

The large-island case differs from the small-island in another important way: the net effect of the large island is to reduce the near-inertial energy near the island, but the inertial frequency band, which includes the slightly subinertial trapped wave, is enhanced near the small island. As ϵL increases, the island reflects more near-inertial energy. As a result, the dominant near-inertial response to the island is not diffraction, but rather reflection. Large amplitude NIWs radiate energy away from the island. What remains locally is a subinertial island-trapped wave and weak near-inertial motions. In the limit of an infinite coastal wall, the island-trapped wave would be replaced by a Kelvin wave and the phenomenon of weak near-inertial motions could be described as coastal inhibition as in the stratified model of Kundu et al. (1983).

The present study has moved beyond the LH70 analytical solution for IOs around idealized island topography by relaxing the small-island limit and by looking at the response to an initial interval of forcing. Additional steps toward understanding near-inertial motion around islands such as Palau include considering more realistic topography and continuous stratification. Brink (2021) found nearly trapped superinertial waves at sloping island topography. While the topography around Palau is steep, and nearly vertical in some locations, it would be interesting to consider how the solution changes once a slope is introduced. Additionally, the influence of nonlinearity and viscosity should be explored to examine generation of vertical vorticity by NIOs shearing against topography. Siegelman et al. (2019) observed a period of enhanced near-inertial currents that corresponded to a large scale wind event near the narrow, northern tip of Velasco Reef, Palau. During this period, the vorticity estimated from a mooring array deployed near the reef boundary had energy in the near-inertial band. While Siegelman et al. (2019) hypothesized that near-inertial vorticity was generated due to enhanced NIOs shearing against the reef wall, this has not yet been supported by a numerical or theoretical study.

Acknowledgments.

We thank the two anonymous peer reviewers for their thoughtful comments. We acknowledge Brian Powell for helping to lay the foundation for the numerical modelling used in this paper. Pat and Lori Colin at the Coral Reef Research Foundation were essential to the fieldwork campaign at Palau, which motivated our efforts. This work was funded by the Office of Naval Research through Grant N00014-16-1-2671. Mika Siegelman was partially funded by the Denise B. Evans Fellowship, and Ruth C. Musgrave was supported, in part, thanks to funding from the Canada Research Chairs Program.

Data availability statement.

The model output analyzed in this paper is available from https://doi.org/10.6075/j0j67h3p.

APPENDIX A

Nondimensionalization of the Long-Wave Equation

Assuming the horizontal scale of the response in the small-island case is the island scale, we use the island radius a to define a nondimensional Laplacian operator as ∇*2=a2∇2. By “small island” we mean small compared to the Rossby radius of deformation, Ld=g′H/f, so ϵL=f2a2/(g′H)≪1. For motions with frequency near f, we define an ϵσ such that σ = f(1 + ϵσ). Substituting these definitions into (4) and dropping terms of O(ϵσ2) yields
(∇*2+2ϵσϵL)h=0,
and hence, the long-wave equation is approximated by the Laplace equation in the limit of small ϵσϵL, having already assumed that each of the two parameters is individually small.
The small-island approximation also leads to a nondivergent velocity field as we can see by taking the divergence of the polarization relation (5):
∇⋅[u=−g′σ2−f2(iσ∇h+f×∇h)],∇⋅u=−g′σ2−f2(iσ∇2h)⋅
Use of the Laplacian equation then implies zero divergence. To see how this limit is approached, nondimensionalize by defining an arbitrary amplitude for height h0 so that h*=h/h0. Then the momentum balance (2) yields a scaled velocity, u∗=fa/(g′h0). Substitution of (A2) together with (A1) leaves
∇*⋅u∗=iϵLh*.
Hence, the divergence goes to zero with ϵL.

APPENDIX B

Analytical Solution

Expanding on sections 2a and 2b, we present the detailed solution to the analytical work of LH70. We revisit this work to correct errors and clarify steps in his original publication.

a. Circular island

We begin with the baroclinic, long-wave equation:
(∇2+σ2−f2g′H)h=0,
and the polarization relation:
u=−g′σ2−f2(iσ∇h+f×∇h).
In LH70, the second term of the polarization, f × ∇h, has the wrong sign [LH70, Eq. (2.2)]. As described in section 2, given a small island (ϵL ≪ 1), the long-wave Eq. (B1) is reduced to the Laplace equation:
∇2h=0.
Next, we assume a far-field velocity that is spatially uniform with both components oscillating at the same frequency:
u=Ae−iσt,υ=Be−iσt,
where A and B are complex constants. To match boundary conditions on the circular island, (B4) can be converted to polar coordinates
ur=ucos⁡θ+υsin⁡θ=Cei(θ−σt)+De−i(θ+σt)uθ=−usin⁡θ+υcos⁡θ=i(Cei(θ−σt)−De−i(θ+σt)),
where C=(1/2)(A−Bi) and D=(1/2)(A+Bi). We will be interested in inertial oscillations, σ = ±f, but for reasons that will become apparent in the algebra, we do not want to set this equality until later. Note that we could choose either sign for σ; the solution for inertial oscillations requires C = 0 for positive σ or D = 0 for negative σ. Although we could make the choice of C versus D now to simplify the development, we will follow Longuet-Higgins and retain both C and D terms until the last step.
The polarization relation (B2) can also be transformed to polar coordinates:
uθ=−g′σ2−f2(iσr∂h∂θ+f∂h∂r), and
ur=−g′σ2−f2(iσ∂h∂r−fr∂h∂θ).
Equations (B6) and (B7) are combined and rearranged as follows:
g′r∂h∂θ=iσuθ−fur, and
g′∂h∂r=fuθ+iσur.
Plugging (B5) into (B9) and requiring h = 0 at r = 0, which would be the case with open ocean inertial oscillations, we find the surface perturbation in the absence of an island is
g′h0=iCr(σ+f)ei(θ−σt)+iDr(σ−f)e−i(θ+σt).
Next, we account for the island by adding
g′h1=Pa2rei(θ−σt)+Qa2re−i(θ+σt)
to (B10). Because this system is linear, the total layer thickness perturbation is h = h0 + h1. The constants P and Q are determined by the boundary condition, ur = 0 at r = a:
(iσ∂h∂r−fr∂h∂θ)r=a=0.
Plugging h into (B12) produces
[−σ(σ+f)C−σPia2r2+f(σ+f)C−fiPa2r2]ei(θ−σt)+[−σ(σ−f)D−σQia2b2−f(σ−f)D+fiQa2r2]e−i(θ+σt)=0,
which has a solution if the coefficients of the linearly independent complex exponentials equal zero:
[−σ(σ+f)C−(σ+f)Pi+f(σ+f)C]=0,[−σ(σ−f)D−(σ−f)Qi−f(σ−f)D]=0,
when r = a. Finally, from (B14):
P=i(σ−f)C,Q=i(σ+f)D⋅
Substituting these constants into (B11) and summing (B10) and (B11), we find
g′h=i[(σ+f)r+(σ−f)a2r]Cei(θ−σt)+i[(σ−f)r+(σ+f)a2r]De−i(θ+σt).
In LH70, P and Q (B15) are incorrectly found to be negative [LH70, Eq. (2.11)]. This error then propagates through [LH70, (2.12)–(2.14)] to the final solution of h [LH70, (2.15)], which is plotted incorrectly in Fig. 1 of LH70, but reconstructed here (Fig. 2). The solution for the radial and azimuthal velocity is found by plugging (B16) into (B6) and (B7):
ur=−1σ2−f2{[−(σ2−f2)+(σ2−f2)a2r]Cei(θ−σt)+[−(σ2−f2)+(σ2−f2)a2r]De−i(θ+σt)}=(1−a2r2)(Cei(θ−σt)+De−i(θ+σt)),
uθ=−1σ2−f2{[i(σ2−f2)−i(σ2−f2)a2r2]Cei(θ−σt)+[i(σ2−f2)+i(σ2−f2)a2r2]De−i(θ+σt)}=i(1+a2r2)(Cei(θ−σt)−De−i(θ+σt)).
Note that because the Laplace equation (B3) is solved rather than the long-wave equation (B1), there is no singularity in (B17) or (B18) because the solution for h provides a factor of (σ2 − f2) in the numerator to cancel the factor in the denominator of (B6) and (B7). Assuming the far field goes to inertial oscillations:
ur=Cei(θ+ft),uθ=iCei(θ+ft) as r→∞,
we require D = 0 and σ = −f, resulting in the following for the solution of velocity and layer thickness perturbation:
ur=(1−a2r2)Cei(θ+ft),
uθ=i(1+a2r2)Cei(θ+ft), and
g′h=−2if(a2r)Cei(θ+ft).
Note that inertial oscillations are the only nontrivial spatially uniform motion on the infinite f plane for which h is bounded; as seen in (B16), h grows linearly with r in the far field unless σ = −f with D = 0 or σ = f with C = 0.

b. Elliptical island

LH70 presents the elliptical island case using conformal mapping; however, the final solution for the velocities around the ellipse [LH70, (2.26)] is incorrect in stating that the complex velocities, u − iυ, equal the real component of an expression. Furthermore, it is unclear to us how that equation was derived. Here, we reexamine the elliptical island case. We use a different transformation from LH70 (2.23). The Joukowski transformation of the form
z=λ+c24λ,
transforms a circle of radius a centered at the origin in the λ plane (λ = μ + iν):
μ2+ν2=a2,
into an ellipse centered at the origin in the z plane (z = x + iy):
x2A2+y2B2=1,
where A is the semimajor axis, B is the semiminor axis, a=(1/2)(A+B), and c2 = A2 − B2 (A ≥ B). The inverse of Eq. (B23):
λ±=12[z±(z2−c2)1/2],
inverts the transformation, taking an ellipse in the z plane and transforming it to a circle in the λ plane. Here, only the positive inverse transformation (B26) is considered because the negative inverse maps into the interior of the circle. Thus, to find the solution of h given an elliptical island in the z plane, the positive inverse transformation (B26) is substituted into the solution to the flow around a circle in the λ plane. To execute this substitution, the equation for h (B22) is rearranged as follows:
g′h=−2if(a2re−iθ)Ceift, and
g′h=−2if(a2λ*)Ceift,
where λ*=(1/2)(z*+z*2−c2) and z* = x − iy. Thus, (B28) is the layer thickness perturbation around an elliptical island on the z plane.
Next, the velocity field is found by considering the polarization relation between the surface interface and the velocity in Cartesian coordinates (B2). The spatial gradients of h, ∂h/∂x and ∂h/∂y, are converted to gradients of z or z* by applying the chain rule to the derivatives ∂h/∂z and ∂h/∂z∗:
∂h∂z=∂h∂x∂x∂z+∂h∂y∂y∂z,∂h∂z*=∂h∂x∂x∂z*+∂h∂y∂y∂z*⋅
Rearranging (B29) and using the relationship between (x, y) and (z,z*):
x=12(z+z*),y=12i(z*−z),
results in the spatial gradients as a function of z and z*:
∂h∂x=∂h∂z+∂h∂z*,∂h∂y=i(∂h∂z−∂h∂z*)⋅
Substituting (B31) in for the spatial gradients, (B28) becomes
u=g′iσ2−f2[(f−σ)∂h∂z−(f+σ)∂h∂z*],υ=−g′σ2−f2[(f+σ)∂h∂z*+(f−σ)∂h∂z].
The chain rule is applied once again to evaluate ∂h/∂z and ∂h/∂z∗ in (B32):
∂h∂z=∂h∂λ∂λ∂z,∂h∂z*=∂h∂λ*∂λ*∂z*.
We evaluate ∂h/∂z and ∂h/∂z* using h before assuming D = 0 and σ = f:
g′h=i[(σ+f)λ+(σ−f)a2λ*]Ce−iσt+i[(σ−f)λ*+(σ+f)a2λ]De−iσt,
which is identical to (B16) from the circular island case, but substitutes λ and λ∗ for reiθ and re−iθ, respectively. Using h before making assumptions about D and σ ensures the apparent singularity in (B32) is removed and results in the solution for the velocity field around an elliptical island:
u=(∂λ∂z−a2λ*2∂λ∂z*)Ce−iσt, and
υ=i(∂λ∂z+a2λ*2∂λ∂z*)Ce−iσt.

REFERENCES

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    • Search Google Scholar
    • Export Citation

  • Brink, K. H., 2021: Near-resonances of superinertial and tidal fluctuations at islands. J. Phys. Oceanogr., 51, 2721–2733, https://doi.org/10.1175/JPO-D-20-0298.1.

    • Search Google Scholar
    • Export Citation

  • Dale, A. C., J. M. Huthnance, and T. J. Sherwin, 2001: Coastal-trapped waves and tides at near-inertial frequencies. J. Phys. Oceanogr., 31, 2958–2970, https://doi.org/10.1175/1520-0485(2001)031<2958:CTWATA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • D’Asaro, E. A., 1989: The decay of wind-forced mixed layer inertial oscillations due to the β effect. J. Geophys. Res., 94, 2045–2056, https://doi.org/10.1029/JC094iC02p02045.

    • Search Google Scholar
    • Export Citation

  • Durran, D. R., 1991: The third-order Adams-Bashforth method: An attractive alternative to leapfrog time differencing. Mon. Wea. Rev., 119, 702–720, https://doi.org/10.1175/1520-0493(1991)119<0702:TTOABM>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Fontán, A., and B. Cornuelle, 2015: Anisotropic response of surface circulation to wind forcing, as inferred from high-frequency radar currents in the southeastern Bay of Biscay. J. Geophys. Res. Oceans, 120, 2945–2957, https://doi.org/10.1002/2014JC010671.

    • Search Google Scholar
    • Export Citation

  • Kundu, P. K., S.-Y. Chao, and J. P. McCreary, 1983: Transient coastal currents and inertio-gravity waves. Deep-Sea Res., 30A, 1059–1082, https://doi.org/10.1016/0198-0149(83)90061-4.

    • Search Google Scholar
    • Export Citation

  • Longuet-Higgins, M. S., 1969: On the trapping of long-period waves round islands. J. Fluid Mech., 37, 773–784, https://doi.org/10.1017/S0022112069000875.

    • Search Google Scholar
    • Export Citation

  • Longuet-Higgins, M. S., 1970: Steady currents induced by oscillations round islands. J. Fluid Mech., 42, 701–720, https://doi.org/10.1017/S0022112070001568.

    • Search Google Scholar
    • Export Citation

  • Merrifield, M. A., L. Yang, and D. S. Luther, 2002: Numerical simulations of a storm-generated island-trapped wave event at the Hawaiian Islands. J. Geophys. Res., 107, 3169, https://doi.org/10.1029/2001JC001134.

    • Search Google Scholar
    • Export Citation

  • Pizarro, O., and G. Shaffer, 1998: Wind-driven, coastal-trapped waves off the island of Gotland, Baltic Sea. J. Phys. Oceanogr., 28, 2117–2129, https://doi.org/10.1175/1520-0485(1998)028<2117:WDCTWO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Pollard, R. T., 1980: Properties of near-surface inertial oscillations. J. Phys. Oceanogr., 10, 385–398, https://doi.org/10.1175/1520-0485(1980)010<0385:PONSIO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Pollard, R. T., and R. C. Millard, 1970: Comparison between observed and simulated wind-generated inertial oscillations. Deep-Sea Res. Oceanogr. Abstr., 17, 813–821, https://doi.org/10.1016/0011-7471(70)90043-4.

    • Search Google Scholar
    • Export Citation

  • Schahinger, R. B., 1988: Near-inertial motion on the South Australian shelf. J. Phys. Oceanogr., 18, 492–504, https://doi.org/10.1175/1520-0485(1988)018<0492:NIMOTS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Shearman, R. K., 2005: Observations of near-inertial current variability on the New England shelf. J. Geophys. Res., 110, C02012, https://doi.org/10.1029/2004JC002341.

    • Search Google Scholar
    • Export Citation

  • Siegelman, M., and Coauthors, 2019: Observations of near-inertial surface currents at Palau. Oceanography, 32, 74–83, https://doi.org/10.5670/oceanog.2019.413.

    • Search Google Scholar
    • Export Citation

  • Tintoré, J., D.-P. Wang, E. Garćia, and A. Viúdez, 1995: Near-inertial motions in the coastal ocean. J. Mar. Syst., 6, 301–312, https://doi.org/10.1016/0924-7963(94)00030-F.

    • Search Google Scholar
    • Export Citation

Supplementary Materials

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Cover Journal of Physical Oceanography
Journal of Physical Oceanography

  • Brink, K. H., 1999: Island-trapped waves, with application to observations of Bermuda. Dyn. Atmos. Oceans, 29, 93–118, https://doi.org/10.1016/S0377-0265(99)00003-2.

    • Search Google Scholar
    • Export Citation

  • Brink, K. H., 2021: Near-resonances of superinertial and tidal fluctuations at islands. J. Phys. Oceanogr., 51, 2721–2733, https://doi.org/10.1175/JPO-D-20-0298.1.

    • Search Google Scholar
    • Export Citation

  • Dale, A. C., J. M. Huthnance, and T. J. Sherwin, 2001: Coastal-trapped waves and tides at near-inertial frequencies. J. Phys. Oceanogr., 31, 2958–2970, https://doi.org/10.1175/1520-0485(2001)031<2958:CTWATA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • D’Asaro, E. A., 1989: The decay of wind-forced mixed layer inertial oscillations due to the β effect. J. Geophys. Res., 94, 2045–2056, https://doi.org/10.1029/JC094iC02p02045.

    • Search Google Scholar
    • Export Citation

  • Durran, D. R., 1991: The third-order Adams-Bashforth method: An attractive alternative to leapfrog time differencing. Mon. Wea. Rev., 119, 702–720, https://doi.org/10.1175/1520-0493(1991)119<0702:TTOABM>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Fontán, A., and B. Cornuelle, 2015: Anisotropic response of surface circulation to wind forcing, as inferred from high-frequency radar currents in the southeastern Bay of Biscay. J. Geophys. Res. Oceans, 120, 2945–2957, https://doi.org/10.1002/2014JC010671.

    • Search Google Scholar
    • Export Citation

  • Kundu, P. K., S.-Y. Chao, and J. P. McCreary, 1983: Transient coastal currents and inertio-gravity waves. Deep-Sea Res., 30A, 1059–1082, https://doi.org/10.1016/0198-0149(83)90061-4.

    • Search Google Scholar
    • Export Citation

  • Longuet-Higgins, M. S., 1969: On the trapping of long-period waves round islands. J. Fluid Mech., 37, 773–784, https://doi.org/10.1017/S0022112069000875.

    • Search Google Scholar
    • Export Citation

  • Longuet-Higgins, M. S., 1970: Steady currents induced by oscillations round islands. J. Fluid Mech., 42, 701–720, https://doi.org/10.1017/S0022112070001568.

    • Search Google Scholar
    • Export Citation

  • Merrifield, M. A., L. Yang, and D. S. Luther, 2002: Numerical simulations of a storm-generated island-trapped wave event at the Hawaiian Islands. J. Geophys. Res., 107, 3169, https://doi.org/10.1029/2001JC001134.

    • Search Google Scholar
    • Export Citation

  • Pizarro, O., and G. Shaffer, 1998: Wind-driven, coastal-trapped waves off the island of Gotland, Baltic Sea. J. Phys. Oceanogr., 28, 2117–2129, https://doi.org/10.1175/1520-0485(1998)028<2117:WDCTWO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Pollard, R. T., 1980: Properties of near-surface inertial oscillations. J. Phys. Oceanogr., 10, 385–398, https://doi.org/10.1175/1520-0485(1980)010<0385:PONSIO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Pollard, R. T., and R. C. Millard, 1970: Comparison between observed and simulated wind-generated inertial oscillations. Deep-Sea Res. Oceanogr. Abstr., 17, 813–821, https://doi.org/10.1016/0011-7471(70)90043-4.

    • Search Google Scholar
    • Export Citation

  • Schahinger, R. B., 1988: Near-inertial motion on the South Australian shelf. J. Phys. Oceanogr., 18, 492–504, https://doi.org/10.1175/1520-0485(1988)018<0492:NIMOTS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Shearman, R. K., 2005: Observations of near-inertial current variability on the New England shelf. J. Geophys. Res., 110, C02012, https://doi.org/10.1029/2004JC002341.

    • Search Google Scholar
    • Export Citation

  • Siegelman, M., and Coauthors, 2019: Observations of near-inertial surface currents at Palau. Oceanography, 32, 74–83, https://doi.org/10.5670/oceanog.2019.413.

    • Search Google Scholar
    • Export Citation

  • Tintoré, J., D.-P. Wang, E. Garćia, and A. Viúdez, 1995: Near-inertial motions in the coastal ocean. J. Mar. Syst., 6, 301–312, https://doi.org/10.1016/0924-7963(94)00030-F.

    • Search Google Scholar
    • Export Citation
  • <sc>Fig</sc>. 1.
    View in gallery
    Fig. 1.

    LH70 solution for inertial oscillations in the vicinity of a small island. (left) Snapshot of the velocity (blue vectors) and surface elevation (color) from Eqs. (8)–(10). At any instant the velocity field is identical to potential flow around a cylinder. The surface elevation, which has a mode-1 spatial structure, adjusts instantaneously with the velocity field, rotating anticyclonically at f. (right) Terms of the x and y momentum balance equations. The sum of the local acceleration ∂u/∂t and the Coriolis acceleration fk^×u equals the pressure gradient −g∇η.

  • <sc>Fig</sc>. 2.
    View in gallery
    Fig. 2.

    Radial (blue) and tangential (orange) velocity and surface elevation (green) normalized by far-field velocity. This figure was reconstructed based on Fig. 1 from LH70, which was missing a factor of −2 in the pressure term. The radial velocity goes to zero on the island boundary to satisfy the boundary condition, while the tangential velocity is twice that of the far-field IO.

  • <sc>Fig</sc>. 3.
    View in gallery
    Fig. 3.

    (a)–(d) Snapshots of the velocity (red vectors) and surface elevation (color) from the analytical solution of rotary flow around an elliptical island, when σ = f, over a quarter of an inertial period. When the incident flow is normal to the minor axis as in (a), flow acceleration is minimal. After a quarter of an inertial period as seen in (d), the IO is normal to the major axis resulting in enhanced inertial currents at the narrow ends of the ellipse due to maximal flow blocking.

  • <sc>Fig</sc>. 4.
    View in gallery
    Fig. 4.

    Cross-shore transects from the σ = f solution of the (a) normalized azimuthal velocity (contours mark 2.0, 3.0, 4.0, and 7.0), (b) normalized surface elevation (contours mark 0.5, 1.0, and 1.5), and (c) normalized pressure gradient force (contours mark 0.5, 1.0, 1.5, 2.0, and 7.0) extending distance L, normalized by the ellipse radius, a=(1/2)(Smaj+Smin) offshore from the northern most point on the island for a range of ellipse eccentricity values, c=Smaj2−Smin2. Above the black line (c > 0), the upstream velocity is purely zonal and blocked by the broad side of the island. Below the black line (c < 0), the purely zonal flow is blocked by the narrow side of the island. When the incident flow is normal to the major axis (c > 0), there is flow enhancement and large pressure gradients near the narrow tips of the elliptical island. Alternatively, when the incident flow is normal to the minor axis (c < 0), there is marginal flow blocking and flow acceleration. As the eccentricity increases, the energy enhancement increases around the narrow tips of the ellipse and decreases along the flanks of the ellipse.

  • <sc>Fig</sc>. 5.
    View in gallery
    Fig. 5.

    Mean kinetic energy of the σ = f solution [Eqs. (8)–(10)], which satisfy [Eqs. (5) and (6)] over an inertial period in the presence of an (a) circular and (b) elliptical island. The mean energy distribution is uniformly distributed around the circular island, but focused around the narrow tips of the elliptical island, similar to observations around the island chain of Palau.

  • <sc>Fig</sc>. 6.
    View in gallery
    Fig. 6.

    (a) Time series of the applied spatially uniform, anticyclonically rotating wind stress until it shuts off after 1 inertial period. (b) Horizontal divergence of the velocity at the eastern boundary of the island. (c) Hovmöller diagram of the surface elevation from a zonal transect beginning at the eastern most end of the island and extending 1000 km offshore. The phase speed, estimated from the slope of the blue dashed line, is 3.64 m s−1. (d) Poincaré wave dispersion relation. Red star indicates estimated wavenumber and frequency of the first wave to radiate away from the island. Black dashed line marks the inertial frequency.

  • <sc>Fig</sc>. 7.
    View in gallery
    Fig. 7.

    (top) Snapshot of the surface elevation (color) with the corresponding velocity (cyan vectors) (left) within 10 and (right) zoomed into just 5 radii away from the island at 1 inertial period. The spiral pattern (which is not evident in the analytical solution) is a superinertial Poincaré wave radiating away from the island. (middle),(bottom) The x- and y-momentum balance terms at one inertial period after the model is initialized, i.e., immediately after the wind forcing is shut off. This snapshot highlights the predominant far-field balance between the local acceleration and Coriolis term suggesting NIOs, the Poincaré waves radiating away from the island, and the phase lag due to the island-trapped wave propagating clockwise around the island.

  • <sc>Fig</sc>. 8.
    View in gallery
    Fig. 8.

    (a) Time series of the azimuthal velocity near the northern extent of the model domain. Extrema (dots) occur when the tangential speed at the furthest extent north of the island is at a maximum, thus the velocity of the far-field NIO is almost purely zonal. Given purely zonal far-field flow, (b) the radial velocity along a zonal transect and (c) the azimuthal velocity and (d) surface elevation along a meridional transect should compare well with the analytical solution presented in Fig. 2 (black dashed lines). The color of the transects corresponds to the color of the dots and denotes the time in inertial periods when the snapshot occurs. The best agreement between the analytical and numerical models occurs at one inertial period. As the model run progresses, energy continues to accumulate near the island, which is explained in section 3.

  • <sc>Fig</sc>. 9.
    View in gallery
    Fig. 9.

    (left) Hovmöller diagrams of (a) surface elevation, (c) zonal, and (e) meridional velocity along a zonal transect that bisects the island. Magenta dashed line marks the instant that the wind forcing stops. Between the island boundary and red dashed line, the surface elevation increases over time. The velocity Hovmöller diagrams show enhanced velocities near the island (between the island boundary and red dashed line) and weakened velocities beyond those regions (between red and blue dashed lines). (right) Power spectral density of (b) surface elevation, (d) zonal, and (f) meridional velocity along the same transect shown in the left panels. The inertial frequency f is indicated by the cyan dashed line. Near the island, the surface elevation bandwidth is relatively broad, with energy ranging from sub- to superinertial. Away from the island, the PSD peak frequency is blue shifted (i.e., shifted toward higher frequencies) from the inertial frequency. Across the transect, the velocity PSD has a narrower peak, with most energy concentrated at the inertial frequency. The υ component of the velocity has a wider peak near the island that is slightly red shifted from the inertial frequency.

  • <sc>Fig</sc>. 10.
    View in gallery
    Fig. 10.

    Snapshot of surface elevation (color) with the corresponding velocity (cyan vectors) after 1 inertial period when ϵL is (a) 0.11, (b) 1.02, and (c) 1.82. As ϵL increases, the phase lag between the trapped wave and far-field NIOs grows more rapidly due to the decreasing frequency of the trapped wave (LH69) (Fig. 11).

  • <sc>Fig</sc>. 11.
    View in gallery
    Fig. 11.

    Comparison between the estimated azimuthal phase speed normalized by f for model runs of varying ϵL (light blue dots) and the mode-1 island-trapped wave theory presented by LH69 (solid blue line). The frequency difference between the numerical model and the analytical solution for the mode-1 trapped wave frequencies is due to the inertial and subinertial motions in the numerical model.

  • <sc>Fig</sc>. 12.
    View in gallery
    Fig. 12.

    PSD of (a),(b) u and (c),(d) η (bottom row) at the (left) northern island boundary and (right) in the far field. When ϵL is small (0.11, 0.26), the PSD of u and η have a single peak at the inertial frequency. As ϵL increases (0.46–1.82), the single peak broadens until two peaks emerge, one at the inertial and the other at the trapped wave frequency from LH69.

  • <sc>Fig</sc>. 13.
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    Fig. 13.

    The azimuthal mean of (left) kinetic [Eq. (23)] and (center) potential [Eq. (24)] energy and (right) the radial energy flux [Eq. (25)] for a range of ϵL. The radial energy flux is positive in the direction away from the island. Near the island, the energy fluctuates at the beat period between the NIOs and the trapped wave. At the island boundary, periods of increasing energy correspond with negative radial energy flux, suggesting that the increase is due to offshore energy moving toward the island. In the far field, fluctuations of the radial energy flux are the result of Poincaré waves superposed on the background NIOs.

  • <sc>Fig</sc>. 14.
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    Fig. 14.

    Rotary spectrum of the wind stress τ when the forcing is applied over 1 inertial period (magenta) and 3 inertial periods (blue). The clockwise component is denoted by a solid line, and the counterclockwise component is denoted by a dashed line.

  • <sc>Fig</sc>. 15.
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    Fig. 15.

    Hovmöller diagrams from a zonal transect that bisects the island when a 1 inertial period forcing is applied. Residual (a) surface elevation, (c) u, and (e) υ after removing the trapped wave component. Different ranges of horizontal distance (x axis) are used to highlight the far-field radiating waves in the surface elevation (a) and the near-field velocities (c) and (e). The residual is predominantly near inertial. Slanted phase lines in surface elevation (a) show the outward radiation of Poincaré waves with frequencies successively closer to f as time increases. In the far field, the residual currents (c) and (e) are NIOs. Approaching the island, the normal velocity component, u, decays to zero, satisfying the boundary condition. The tangential velocity υ reaches a null approximately 100 km from the island boundary and reverses closer to shore. (right) The trapped wave component of (b) surface elevation, (d) u, and (f) υ. The surface elevation has an azimuthal mode-1 structure, but with radial decay rather than propagation. The cross-shore velocity in (d) is negligible, consistent with Kelvin wave dynamics.

  • <sc>Fig</sc>. 16.
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    Fig. 16.

    The rotary spectrum of the (a),(b) velocity and the (c),(d) PSD of the surface elevation when the forcing is applied over 1 inertial period (magenta) and 3 inertial periods (blue) in the (left) far field and (right) at the eastern most point on the island boundary. In (a) and (b), the solid and dashed lines are the CW and CCW rotary components, respectively. The PSD of u and η on the island boundary show the sensitivity of the trapped wave to the bandwidth of the wind forcing. In the far field, the PSD of η has more energy at higher frequencies for the 1IPF run.

  • <sc>Fig</sc>. 17.
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    Fig. 17.

    Time series of kinetic energy when the forcing is applied over 1 inertial period (magenta) and 3 inertial periods (blue) 3000 km east of the island boundary, beyond the radial extent of the trapped wave. The arrival of the Poincaré waves (denoted by the arrows) is delayed in the 3IPF runs. (b) The residual tangential velocity at the eastern most point on the island boundary initially oscillates at a superinertial frequency and has a larger initial amplitude for the 1IPF run. Within approximately 4 inertial periods, the currents oscillate at a near-inertial frequency and the amplitude is the same regardless of the forcing. (c) The tangential velocity of the trapped wave component at the eastern boundary, which was removed from the currents shown in (b), has a larger amplitude for the 1IPF than the 3IPF. The gray boxes in (c) and (b) mark the period contaminated by end effects.

  • <sc>Fig</sc>. 18.
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    Fig. 18.

    Surface elevation (color) and velocity (cyan vectors) snapshots at (a) 3.50 and (b) 3.75 inertial periods after the start of the run where ϵL = 0.11. NIWs excited by divergent wind stress applied between 1000 and 2000 km along the x axis, propagate westward and eastward. (c) Hovmöller diagram of the surface elevation along a zonal transect that bisects the island. Cyan shading delimits the wind forcing region. The black dashed line marks 1 inertial period, after which the wind stops. NIWs radiate eastward and westward from the wind forcing region. Poincaré waves are reflected eastward away from the island boundary as a result of incident westward propagating NIWs. The superposition of the incident and reflected waves results in a null region that moves eastward over time between 0 and 1000 km from the island. A trapped wave propagates clockwise around the island, traveling approximately a quarter of the circumference in a quarter of an inertial period.

  • <sc>Fig</sc>. 19.
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    Fig. 19.

    (a)–(c) Snapshots at 4.6 inertial periods of the surface elevation (color) and velocity (cyan vectors) when ϵL = 0.11, 1.02, and 1.82. Poincaré waves propagate westward, reflecting off the eastern boundary of the island and refracting around it regardless of ϵL. When ϵL = 1.82, a mode-2 leaky trapped wave is excited. (d)–(f) PSD of surface elevation (red line) and rotary spectra of velocity (blue, clockwise rotational is solid and counterclockwise rotational is dashed) on the eastern boundary of the island when ϵL = 0.11, 1.02, and 1.82 in (d)–(f), respectively. The black dashed line marks the inertial period. The PSD shows that the mode-2 trapped wave has a near-inertial frequency.

Fig. 1.

LH70 solution for inertial oscillations in the vicinity of a small island. (left) Snapshot of the velocity (blue vectors) and surface elevation (color) from Eqs. (8)–(10). At any instant the velocity field is identical to potential flow around a cylinder. The surface elevation, which has a mode-1 spatial structure, adjusts instantaneously with the velocity field, rotating anticyclonically at f. (right) Terms of the x and y momentum balance equations. The sum of the local acceleration ∂u/∂t and the Coriolis acceleration fk^×u equals the pressure gradient −g∇η.
Fig. 2.

Radial (blue) and tangential (orange) velocity and surface elevation (green) normalized by far-field velocity. This figure was reconstructed based on Fig. 1 from LH70, which was missing a factor of −2 in the pressure term. The radial velocity goes to zero on the island boundary to satisfy the boundary condition, while the tangential velocity is twice that of the far-field IO.
Fig. 3.

(a)–(d) Snapshots of the velocity (red vectors) and surface elevation (color) from the analytical solution of rotary flow around an elliptical island, when σ = f, over a quarter of an inertial period. When the incident flow is normal to the minor axis as in (a), flow acceleration is minimal. After a quarter of an inertial period as seen in (d), the IO is normal to the major axis resulting in enhanced inertial currents at the narrow ends of the ellipse due to maximal flow blocking.
Fig. 4.

Cross-shore transects from the σ = f solution of the (a) normalized azimuthal velocity (contours mark 2.0, 3.0, 4.0, and 7.0), (b) normalized surface elevation (contours mark 0.5, 1.0, and 1.5), and (c) normalized pressure gradient force (contours mark 0.5, 1.0, 1.5, 2.0, and 7.0) extending distance L, normalized by the ellipse radius, a=(1/2)(Smaj+Smin) offshore from the northern most point on the island for a range of ellipse eccentricity values, c=Smaj2−Smin2. Above the black line (c > 0), the upstream velocity is purely zonal and blocked by the broad side of the island. Below the black line (c < 0), the purely zonal flow is blocked by the narrow side of the island. When the incident flow is normal to the major axis (c > 0), there is flow enhancement and large pressure gradients near the narrow tips of the elliptical island. Alternatively, when the incident flow is normal to the minor axis (c < 0), there is marginal flow blocking and flow acceleration. As the eccentricity increases, the energy enhancement increases around the narrow tips of the ellipse and decreases along the flanks of the ellipse.
Fig. 5.

Mean kinetic energy of the σ = f solution [Eqs. (8)–(10)], which satisfy [Eqs. (5) and (6)] over an inertial period in the presence of an (a) circular and (b) elliptical island. The mean energy distribution is uniformly distributed around the circular island, but focused around the narrow tips of the elliptical island, similar to observations around the island chain of Palau.
Fig. 6.

(a) Time series of the applied spatially uniform, anticyclonically rotating wind stress until it shuts off after 1 inertial period. (b) Horizontal divergence of the velocity at the eastern boundary of the island. (c) Hovmöller diagram of the surface elevation from a zonal transect beginning at the eastern most end of the island and extending 1000 km offshore. The phase speed, estimated from the slope of the blue dashed line, is 3.64 m s−1. (d) Poincaré wave dispersion relation. Red star indicates estimated wavenumber and frequency of the first wave to radiate away from the island. Black dashed line marks the inertial frequency.
Fig. 7.

(top) Snapshot of the surface elevation (color) with the corresponding velocity (cyan vectors) (left) within 10 and (right) zoomed into just 5 radii away from the island at 1 inertial period. The spiral pattern (which is not evident in the analytical solution) is a superinertial Poincaré wave radiating away from the island. (middle),(bottom) The x- and y-momentum balance terms at one inertial period after the model is initialized, i.e., immediately after the wind forcing is shut off. This snapshot highlights the predominant far-field balance between the local acceleration and Coriolis term suggesting NIOs, the Poincaré waves radiating away from the island, and the phase lag due to the island-trapped wave propagating clockwise around the island.
Fig. 8.

(a) Time series of the azimuthal velocity near the northern extent of the model domain. Extrema (dots) occur when the tangential speed at the furthest extent north of the island is at a maximum, thus the velocity of the far-field NIO is almost purely zonal. Given purely zonal far-field flow, (b) the radial velocity along a zonal transect and (c) the azimuthal velocity and (d) surface elevation along a meridional transect should compare well with the analytical solution presented in Fig. 2 (black dashed lines). The color of the transects corresponds to the color of the dots and denotes the time in inertial periods when the snapshot occurs. The best agreement between the analytical and numerical models occurs at one inertial period. As the model run progresses, energy continues to accumulate near the island, which is explained in section 3.
Fig. 9.

(left) Hovmöller diagrams of (a) surface elevation, (c) zonal, and (e) meridional velocity along a zonal transect that bisects the island. Magenta dashed line marks the instant that the wind forcing stops. Between the island boundary and red dashed line, the surface elevation increases over time. The velocity Hovmöller diagrams show enhanced velocities near the island (between the island boundary and red dashed line) and weakened velocities beyond those regions (between red and blue dashed lines). (right) Power spectral density of (b) surface elevation, (d) zonal, and (f) meridional velocity along the same transect shown in the left panels. The inertial frequency f is indicated by the cyan dashed line. Near the island, the surface elevation bandwidth is relatively broad, with energy ranging from sub- to superinertial. Away from the island, the PSD peak frequency is blue shifted (i.e., shifted toward higher frequencies) from the inertial frequency. Across the transect, the velocity PSD has a narrower peak, with most energy concentrated at the inertial frequency. The υ component of the velocity has a wider peak near the island that is slightly red shifted from the inertial frequency.
Fig. 10.

Snapshot of surface elevation (color) with the corresponding velocity (cyan vectors) after 1 inertial period when ϵL is (a) 0.11, (b) 1.02, and (c) 1.82. As ϵL increases, the phase lag between the trapped wave and far-field NIOs grows more rapidly due to the decreasing frequency of the trapped wave (LH69) (Fig. 11).
Fig. 11.

Comparison between the estimated azimuthal phase speed normalized by f for model runs of varying ϵL (light blue dots) and the mode-1 island-trapped wave theory presented by LH69 (solid blue line). The frequency difference between the numerical model and the analytical solution for the mode-1 trapped wave frequencies is due to the inertial and subinertial motions in the numerical model.
Fig. 12.

PSD of (a),(b) u and (c),(d) η (bottom row) at the (left) northern island boundary and (right) in the far field. When ϵL is small (0.11, 0.26), the PSD of u and η have a single peak at the inertial frequency. As ϵL increases (0.46–1.82), the single peak broadens until two peaks emerge, one at the inertial and the other at the trapped wave frequency from LH69.
Fig. 13.

The azimuthal mean of (left) kinetic [Eq. (23)] and (center) potential [Eq. (24)] energy and (right) the radial energy flux [Eq. (25)] for a range of ϵL. The radial energy flux is positive in the direction away from the island. Near the island, the energy fluctuates at the beat period between the NIOs and the trapped wave. At the island boundary, periods of increasing energy correspond with negative radial energy flux, suggesting that the increase is due to offshore energy moving toward the island. In the far field, fluctuations of the radial energy flux are the result of Poincaré waves superposed on the background NIOs.
Fig. 14.

Rotary spectrum of the wind stress τ when the forcing is applied over 1 inertial period (magenta) and 3 inertial periods (blue). The clockwise component is denoted by a solid line, and the counterclockwise component is denoted by a dashed line.
Fig. 15.

Hovmöller diagrams from a zonal transect that bisects the island when a 1 inertial period forcing is applied. Residual (a) surface elevation, (c) u, and (e) υ after removing the trapped wave component. Different ranges of horizontal distance (x axis) are used to highlight the far-field radiating waves in the surface elevation (a) and the near-field velocities (c) and (e). The residual is predominantly near inertial. Slanted phase lines in surface elevation (a) show the outward radiation of Poincaré waves with frequencies successively closer to f as time increases. In the far field, the residual currents (c) and (e) are NIOs. Approaching the island, the normal velocity component, u, decays to zero, satisfying the boundary condition. The tangential velocity υ reaches a null approximately 100 km from the island boundary and reverses closer to shore. (right) The trapped wave component of (b) surface elevation, (d) u, and (f) υ. The surface elevation has an azimuthal mode-1 structure, but with radial decay rather than propagation. The cross-shore velocity in (d) is negligible, consistent with Kelvin wave dynamics.
Fig. 16.

The rotary spectrum of the (a),(b) velocity and the (c),(d) PSD of the surface elevation when the forcing is applied over 1 inertial period (magenta) and 3 inertial periods (blue) in the (left) far field and (right) at the eastern most point on the island boundary. In (a) and (b), the solid and dashed lines are the CW and CCW rotary components, respectively. The PSD of u and η on the island boundary show the sensitivity of the trapped wave to the bandwidth of the wind forcing. In the far field, the PSD of η has more energy at higher frequencies for the 1IPF run.
Fig. 17.

Time series of kinetic energy when the forcing is applied over 1 inertial period (magenta) and 3 inertial periods (blue) 3000 km east of the island boundary, beyond the radial extent of the trapped wave. The arrival of the Poincaré waves (denoted by the arrows) is delayed in the 3IPF runs. (b) The residual tangential velocity at the eastern most point on the island boundary initially oscillates at a superinertial frequency and has a larger initial amplitude for the 1IPF run. Within approximately 4 inertial periods, the currents oscillate at a near-inertial frequency and the amplitude is the same regardless of the forcing. (c) The tangential velocity of the trapped wave component at the eastern boundary, which was removed from the currents shown in (b), has a larger amplitude for the 1IPF than the 3IPF. The gray boxes in (c) and (b) mark the period contaminated by end effects.
Fig. 18.

Surface elevation (color) and velocity (cyan vectors) snapshots at (a) 3.50 and (b) 3.75 inertial periods after the start of the run where ϵL = 0.11. NIWs excited by divergent wind stress applied between 1000 and 2000 km along the x axis, propagate westward and eastward. (c) Hovmöller diagram of the surface elevation along a zonal transect that bisects the island. Cyan shading delimits the wind forcing region. The black dashed line marks 1 inertial period, after which the wind stops. NIWs radiate eastward and westward from the wind forcing region. Poincaré waves are reflected eastward away from the island boundary as a result of incident westward propagating NIWs. The superposition of the incident and reflected waves results in a null region that moves eastward over time between 0 and 1000 km from the island. A trapped wave propagates clockwise around the island, traveling approximately a quarter of the circumference in a quarter of an inertial period.
Fig. 19.

(a)–(c) Snapshots at 4.6 inertial periods of the surface elevation (color) and velocity (cyan vectors) when ϵL = 0.11, 1.02, and 1.82. Poincaré waves propagate westward, reflecting off the eastern boundary of the island and refracting around it regardless of ϵL. When ϵL = 1.82, a mode-2 leaky trapped wave is excited. (d)–(f) PSD of surface elevation (red line) and rotary spectra of velocity (blue, clockwise rotational is solid and counterclockwise rotational is dashed) on the eastern boundary of the island when ϵL = 0.11, 1.02, and 1.82 in (d)–(f), respectively. The black dashed line marks the inertial period. The PSD shows that the mode-2 trapped wave has a near-inertial frequency.
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