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Humio Mitsudera and Roger Grimshaw

Abstract

The resonant interaction of a longshore baroclinic current with a topographic feature is investigated, using a quasi-geostrophic two-layer model, where the lower layer is assumed to be deep but is not stagnant. In this model the current may be baroclinically unstable. When a long-wave phase speed is close to zero (in a fixed reference frame), which is found to be realized when the current has almost zero velocity at the coast, there is an enhanced generation of mesoscale variability due to a combination of resonant topographic forcing and baroclinic instability. A forced evolution equation of the KdV-type, which includes an additional coupling term with the lower-layer equation, describes the behavior of the upper layer. On the other hand, the lower-layer motion is governed by a linear vorticity equation, which in turn is coupled to the upper-layer equation.

A stability analysis shows that a solitary wave is unstable when a parameter Γ (the phase speed in the absence of any coupling between the two layers) takes values in a certain range determined by considering a linear stability problem. A variety of numerical solutions are presented, covering stable and unstable cases, characterized by the property of the baroclinic current and the forcing mechanism, which is due either to a coastline perturbation or to bottom topography. It is found that upstream and downstream nonlinear waves are generated due to resonant forcing and may be further amplified by baroclinic instability if the wave parameter Γ meets the instability criterion. These destabilized nonlinear waves show very complicated interactive behavior.

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Georg Gottwald and Roger Grimshaw

Abstract

It is shown that the influence of topography on the interaction of long, weakly nonlinear, quasigeostrophic baroclinic waves can be described by a pair of linearly coupled Korteweg–de Vries equations, with a forcing term in one of the equations. This system exhibits a rich dynamics that is suggestive of atmospheric blocking such as stable stationary solutions, transient quasi-steady-state solutions, multiple equilibria, and baroclinic instability. Topography is shown to favor the formation of blocking systems. This system is investigated both analytically, using techniques from asymptotic perturbation theory, and through numerical simulations.

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Humio Mitsudera and Roger Grimshaw

Abstract

The authors have demonstrated that a large amplitude, nearly stationary solitary wave can be induced either by direct resonant forcing or by the capture of a traveling wave over the forcing region, using a two-layer model in a weakly nonlinear, long-wave limit. This two-layer model consists of a thin upper layer (where the motion is relatively strong) and a deep lower layer. From this system, an evolution equation of the KdV-type is derived to describe the upper-layer motion, while the deep lower-layer motion is described by a linear long-wave vorticity equation. The authors are particularly interested in the role of baroclinic instability in the evolution of solitary waves, as well as the effects of topographic forcing and frictional dissipation.

Resonant forcing occurs within a bandwidth of a detuning parameter that scales with the square root of the (nondimensional) forcing amplitude. On the other hand, the capture of traveling waves, whose amplitude is larger than a critical value, occurs when the detuning parameter is outside the resonant band, and it is in this range that multiple equilibria (coexistence of the large and small amplitude stationary states for a given parameter set) can be realized. Whether the large amplitude stationary state appears upstream or downstream from the forcing region depends on the relative importance of baroclinic energy conversion, topographic forcing, and frictional dissipation. Further, a topographic feature can trigger baroclinic instability, which can then induce not only large amplitude stationary waves but also large amplitude traveling waves going away from the forcing region. The model results are suggestive of the bimodality of Kuroshio upstream from the Izu Ridge.

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Georg Gottwald and Roger Grimshaw

Abstract

It is shown that the interaction of long, weakly nonlinear, quasigeostrophic baroclinic waves can be described by a pair of linearly coupled Korteweg–de Vries equations. Baroclinic energy conversion is investigated as the interaction of interacting upper- and lower-layer structures, which are represented by solitary waves. This system exhibits a rich dynamics that is suggestive of atmospheric blocking features, such as stable stationary solutions, transient quasi-steady-state solutions, multiple equilibria, and baroclinic instability. This system is investigated both analytically, using techniques from asymptotic perturbation theory, and through numerical simulations.

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Fernando Viera and Roger Grimshaw

Abstract

Using the methodology of contour dynamics on a quasigeostrophic model, the nonlinear evolution of a coastal potential vorticity front over a Gaussian topographic feature in the presence of an overlying linearly stable basic flow is investigated. The simulations show that increasing the amplitude of the forcing leads to four different qualitative regimes: 1) small amplitude wavelike disturbances are formed, 2) a primary (trapped) disturbance breaks and forms filaments, 3) a secondary (moving) disturbance breaks and forms filaments, and 4) the primary filament winds around the topographic feature until an eddy with considerable internal mixing finally detaches. Other parameters such as the topographic width, the position of the undisturbed front relative to the topography, and the potential vorticity in the ocean region are also shown to be important in controlling (either enhancing or inhibiting) the process of filamentation and vortex formation. The main conclusion is that nonlinear dynamics alone may be responsible for the formation of meanders and eddies without the necessary presence of instabilities in the basic flow.

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Humio Mitsudera and Roger Grimshaw

Abstract

In this paper, the effects of bottom and interfacial friction on localized baroclinic instability are discussed in the weakly nonlinear, long-wave limit. Using a quasigeostrophic two-layer model in which the lower layer is assumed to be deep, we have derived a coupled evolution equation set that consists of a KdV-type equation for the upper layer and a linear long-wave equation for the lower layer. A perturbation theory reveals that there are multiple equilibria in this system, where baroclinic energy conversion and frictional dissipation are in balance; the flow is not forced externally, and multiplicity here refers to the presence or absence of solitary waves propagating steadily on a zonal flow. Further, direct numerical calculations show a rich variety of behavior of solitary waves, including steady, periodic, and complicated interacting evolutions. For a two-layer model to have multiple steady or oscillatory states, both bottom and interfacial friction should be included because if one of these vanishes, friction destabilizes rather than damps the otherwise neutral waves. The localized baroclinic instability is highly suggestive of the dynamics of the Kuroshio large meander.

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Yong Ming Tang and Roger Grimshaw

Abstract

The authors use numerical simulations of the shallow-water equations to study the generation of coastally trapped waves by localized forcing mechanisms. The model parameters are chosen to be typical for coastal ocean situations. The waves are generated by various wind-stress forcing mechanisms typical of atmospheric fronts or tropical cyclones that travel either parallel or normal to the coast, or in a combination of both of these. Coastally trapped waves fall into three classes: superinertial edge waves, Kelvin waves, and subinertial shelf waves. Mode-fitting routines are described, which when applied to the model output enable one to identify the type and modal properties of the waves generated. The authors’ results show that for the typical wind stress forcing considered here, the generated wave field is dominated by low-mode shelf waves.

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Roger Grimshaw, Dave Broutman, and Lauren E. Sahl

Abstract

An analytical model is developed for nondivergent barotropic flow in a rectangular region, with depth contours parallel to the longshore boundary. The model is linear, with flow forced by wind stress and subject to dissipation by bottom stress. The friction due to the bottom stress is parameterized by a single friction coefficient, which represents a gross decay time for the whole region. Analytical solutions are developed in detail for a steady or periodic wind stress, which is independent of the cross-shelf coordinate. Results are presented for a range of values of the region dimension and the friction coefficient. Comparisons are made between the analytical model and recent observations of the wind-driven flow in the northern Great Barrier Reef.

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Karl R. Helfrich and Roger H. J. Grimshaw

Abstract

The disintegration of a first-mode internal tide into shorter solitary-like waves is considered. Since observations frequently show both tides and waves with amplitudes beyond the restrictions of weakly nonlinear theory, the evolution is studied using a fully nonlinear, weakly nonhydrostatic two-layer theory that includes rotation. In the hydrostatic limit, the governing equations have periodic, nonlinear inertia–gravity solutions that are explored as models of the nonlinear internal tide. These long waves are shown to be robust to weak nonhydrostatic effects. Numerical solutions show that the disintegration of an initial sinusoidal linear internal tide is closely linked to the presence of these nonlinear waves. The initial tide steepens due to nonlinearity and sheds energy into short solitary waves. The disintegration is halted as the longwave part of the solution settles onto a state close to one of the nonlinear hydrostatic solutions, with the short solitary waves superimposed. The degree of disintegration is a function of initial amplitude of the tide and the properties of the underlying nonlinear hydrostatic solutions, which, depending on stratification and tidal frequency, exist only for a finite range of amplitudes (or energies). There is a lower threshold below which no short solitary waves are produced. However, for initial amplitudes above another threshold, given approximately by the energy of the limiting nonlinear hydrostatic inertia–gravity wave, most of the initial tidal energy goes into solitary waves. Recent observations in the South China Sea are briefly discussed.

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Yong Ming Tang, Peter Holloway, and Roger Grimshaw

Abstract

In 1983 Tropical Cyclone Jane crossed the North West Coast of Australia generating a storm surge. Currents associated with this storm surge were recorded at two offshore moorings south of the cyclone track. The data from these moorings are suggestive of the propagation of a continental shelf wave between the two stations. This hypothesis is tested by carrying out a numerical simulation of this storm surge based on the depth-integrated shallow-water equations, with wind-wave-enhanced bottom friction. Analysis of the numerical results shows that the storm surge can be interpreted as due to continental shelf waves.

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