Resonance phenomena at the long wave run-up on the coast

Run-up of long waves on a beach consisting of three pieces of constant but different slopes is studied. Linear shallow-water theory is used for incoming impulse evolution, and nonlinear corrections are obtained for the run-up stage. It is demonstrated that bottom profile influences the run-up characteristics and can lead to resonance effects: increase of wave height, particle velocity, and number of oscillations. Simple parameterization of tsunami source through an earthquake magnitude is used to calculate the run-up height versus earthquake magnitude. It is shown that resonance effects lead to the sufficient increase of run-up heights for the weakest earthquakes, and a tsunami wave does not break on chosen bottom relief if the earthquake magnitude does not exceed 7.8.

Resonance phenomena at the long wave run-up on the coast

Run-up of long wave on a beach consisting of three pieces of constant but different slopes is studied. Linear shallow-water theory is used for incoming impulse evolution and non-linear corrections are obtained for the run-up stage. It is demonstrated that bottom profile influences the run-up characteristics and can lead to the resonance effects: increasing of wave height, particle velocity, and number of oscillations. Simple parameterization of tsunami source through an earthquake magnitude is used to calculate the run-up height versus earthquake magnitude. It is shown that resonance effects lead to the sufficient increasing of run-up heights for weakest earthquakes and tsunami wave does not break on chosen bottom relief if the earthquake magnitude does not exceed 7.8.

Abrupt Transition to Strong Superrotation Driven by Equatorial Wave Resonance in an Idealized GCM

Persistent superrotation is seen in the atmospheres of other terrestrial bodies (Venus, Titan) but not in that of present Earth, which is distinguished by equatorial easterlies. Nevertheless, superrotation has appeared in numerical simulations of Earth’s atmosphere, from two-layer models to multilevel comprehensive GCMs. Simulations of warm climates that generate enhanced tropical convective variability seem particularly prone to superrotation, which has led to hypotheses that the warmer atmospheres of the early Pliocene and Eocene may have been superrotating, and that the phenomenon may be relevant to future climate projections. This paper considers a positive feedback leading to superrotation based on an equatorial wave resonance that occurs in a westerly background flow. The authors present simulations with an idealized multilevel GCM forced with a zonally varying equatorial heating, which show abrupt transitions to strongly superrotating states. Linear shallow water theory is used to show that these transitions occur as the superrotating jet velocity approaches the phase speed of free equatorial Rossby wave modes, leading to a resonant amplification of the response to eddy heating and its associated equatorward momentum flux. The resonance and transition are most prominent in simulations where the meridional temperature gradient has been reduced, and hysteresis behavior is seen when the gradient is eliminated completely. No evidence is found in these simulations for the midlatitude wave feedback believed to drive abrupt transitions in two-layer models, and there is only a minor role for the axisymmetric feedback based on vertical advection by the Hadley circulation.

TRANSFORMATION, BREAKING AND RUN-UP OF A LONG WAVE OF FINITE HEIGHT

On studying the transformation, breaking and run-up of a relatively steep wave of a short period, the theory for waves of permanent type has given us many fruitful results. However, the theory gradually loses its applicability as a wave becomes flat, since a considerable deformation of the wave profile is inevitable in its propagation. In § 1, a discussion concerning the transformation of a long wave in a channel of variable section is presented based on the non-linear shallow water theory. Approximate solutions obtained by G. B. Whitham's method (1958) are shown. Further, some brief considerations are given to the effects of bottom friction on wave transformation. In § 2, breaking of a long wave is discussed. Breakings on a uniformly sloping beach and on a beach of parabolic profile are considered and the effects of beach profile on breaking are clarified. Finally in § 3, experimental results on wave run-up over l/30 slope are described in comparing with the Kaplan's results.

Trapping of water waves above a round sill

The three-dimensional problem of wave trapping above a submerged round sill was first analysed by Longuet-Higgins on the basis of a linear shallow-water theory. The large responses predicted by the theory were, however, not well borne out by the experiments of Barnard, Pritchard & Provis, and this has motivated a more detailed study of the problem. A full linear theory for both inviscid and weakly viscous fluid, without any shallow-water assumptions, is presented here. It reveals important limitations on the use of shallow-water theory and the reasons for them. In particular, while the qualitative features of wave trapping are similar to those of shallow-water theory, the nearly resonant frequencies differ significantly, and, since the resonances are narrow, the observed amplitudes at a given frequency differ greatly. The geometry is strongly indicative of long waves, and the dispersion relation appears quite consistent with that, but the part of the motion at wavenumbers that are not small has, despite the small amplitude, a substantial effect on the response to excitation.

Long waves generated by ground motion

The initial-value problem for waves generated by ground motion near a shore is solved using linear shallow water theory and an exponential bottom profile. It is found that long waves can be trapped along the coast and travel with the deep water wave speed, (gh)½. The energy in these waves decays with x−½ instead of x−1 so that more energy would be observed on this coast than expected on the basis of deep water wave amplitudes.

The downstream effect of closing a barrier across an estuary with particular reference to the Thames

The theory given in this paper considers the reflected surge experienced downstream due to closing a tidal barrier across an estuary during a rising tide or storm surge. The estuary considered is exponential in width and of constant depth, and the discussion is based on linear shallow-water theory with friction taken proportional to the velocity in accord with Lorentz’s linearization of Chezy’s law. In the calculation of the surge, the initial non-steadiness of the motion of the water in the estuary, due to the original tidal action, is neglected. The theory finds the surge initiated when a steady velocity is brought impulsively or otherwise to rest at the barrier. Shortly after closure the resultant water level is obtained by adding the transient surge, obtained in this way, to the tidal curve for the unobstructed estuary. Expressions are obtained for the reflected surge which occurs seawards of the barrier for the three cases, namely, complete instantaneous closure, a state approximating partial instantaneous closure and gradual closure. For example, in the case of instantaneous closure, the surge at any point downstream of the barrier begins with a bore followed by a more gradual change of the water surface level. The height of the bore is shown to fall off seawards as (width of estuary) -1/2 x (an exponential damping factor dependent on the friction). The theory is applied to the proposed Thames Tidal Barrier and the results are in reasonable agreement with model tests.