Long Wave Phase Research Articles

Excitation of Surface Waves Due to Thermocapillary Effects on a Stably Stratified Fluid Layer

In chemical engineering applications, the operation of condensers and evaporators can be made more efficient by exploiting the transport properties of interfacial waves excited on the interface between a hot vapor overlying a colder liquid. Linear theory for the onset of instabilities due to heating a thin layer from above is computed for the Marangoni–Bénard problem. Symbolic computation in the long wave asymptotic limit shows three stationary, non-growing modes. Intersection of two decaying branches occurs at a crossover long wavelength; two other modes co-exist at the crossover point—propagating modes on nascent, shorter wavelength branches. The dispersion relation is then mapped numerically by Newton continuation methods. A neutral stability method is used to map the space of critical stability for a physically meaningful range of capillary, Prandtl, and Galileo numbers. The existence of a cut-off wavenumber for the long wave instability was verified. It was found that the effect of applying a no-slip lower boundary condition was to render all long waves stationary. This has the implication that any propagating modes, if they exist, must occur at finite wavelengths. The computation of 8000 different parameter sets shows that the group velocity always lies within 1 2 to 2 3 of the longwave phase velocity.

Seiche excitation in a highly stratified fjord of southern Chile: the Reloncaví fjord

Abstract. We describe a seiche process based on current, temperature, and sea-level data obtained from the Reloncaví fjord (41.6° S, 72.5° W) in southern Chile. We combined 4 months of acoustic Doppler current profiler (ADCP) data with sea-level, temperature, and wind time series to analyze the dynamics of low-frequency (periods > 1 day) internal oscillations in the fjord. Additionally, seasonal conductivity, temperature, and depth (CTD) data from 19 along-fjord stations were used to characterize the seasonality of the density field. The density profiles were used to estimate the internal long-wave phase speed (c) using two approximations: (1) a simple reduced gravity model (RGM) and (2) a continuously stratified model (CSM). No major seasonal changes in c were observed using either approximation (e.g., the CSM yielded 0.73 < c < 0.87 m s−1 for mode 1). The natural internal periods (TN) were estimated using Merian's formula for a simple fjord-like basin and the above phase speeds. Estimated values of TN varied between 2.9 and 3.5 days and were highly consistent with spectral peaks observed in the along-fjord currents and temperature time series. We conclude that these oscillations were forced by the wind stress, despite the moderate wind energy. Wind conditions at the end of winter gave us an excellent opportunity to explore the damping process. The observed damping time (Td) was relatively long (Td = 9.1 days).

DARTs and CORK in Cascadia Basin: High-resolution observations of the 2004 Sumatra tsunami in the northeast Pacific

[1] We report unique deep-sea recordings of the Sumatra tsunami of December 2004 by high-resolution DART® (Deep-ocean Assessment and Reporting of Tsunami) and CORK (Circulation Obviation Retrofit Kit) bottom pressure sensors deployed at depths of ∼1500–3500 m in Cascadia Basin in the northeast Pacific. The simultaneous records from these sites establish the first-ever regional-scale tsunami detection array for the open ocean, enabling us to resolve both seafloor and crustal tsunami signals and to determine fundamental properties of the waves following their 22,000 km journey from the source region. Waves reaching the basin had mean amplitudes of ∼5 mm with energy spread over a broad frequency band from 0.4 to 7 cph. Peak tsunami energy was in the 0.8 to 2 cph (75 to 30 min) band. Leading waves from the event arrived 34–35 h after the earthquake, roughly 7 h later than expected, suggesting that the tsunami mainly propagated by the “most economic” (minimum energy loss) path along mid-ocean ridge wave-guides rather than taking the direct and fastest path across the ocean. Motions within the peak energy band comprise roughly 50% coherent progressive waves propagating from the south at longwave phase speeds of ∼150 ms−1 and 50% random waves scattered from the coast and bottom irregularities. Tsunami amplitudes in the borehole were one-third those on the seafloor. The extensive “ringing” and anomalously slow (3.5-day) e-folding decay time of the tsunami wave energy indicates long duration energy flux radiating from the Indian Ocean via the Southern Pacific Ocean.

Mean flow in hexagonal convection: stability and nonlinear dynamics

Weakly nonlinear hexagon convection patterns coupled to mean flow are investigated within the framework of coupled Ginzburg–Landau equations. The equations are in particular relevant for non-Boussinesq Rayleigh–Bénard convection at low Prandtl numbers. The mean flow is found to: (1) affect only one of the two long-wave phase modes of the hexagons, and (2) suppress the mixing between the two phase modes. As a consequence, for small Prandtl numbers the transverse and the longitudinal phase instability are expected to occur in sufficiently distinct parameter regimes that they can be studied separately. Through the formation of penta–hepta defects, they lead to different types of transient disordered states. The results for the dynamics of the penta–hepta defects shed light on the persistence of grain boundaries in such disordered states.

Solitary Rossby Waves in Zonally Varying Jet Flows

The dynamics of solitary Rossby waves (SRWs) embedded in a meridionally sheared, zonally varying background flow are examined using a non-divergent barotropic model centered on a midlatitude g -plane. The zonally varying background flow, which is produced by an external potential vorticity (PV) forcing, yields a modified Korteweg-de Vries (K-dV) equation that governs the spatial-temporal evolution of a disturbance field that contains both Rossby wave packets and SRWs. The modified K-dV equation differs from the classical equation in that the zonally varying background flow, which varies on the same scale as the disturbance field, directly affects the disturbance linear translation speed and linear growth characteristics. In the limit of a locally parallel background flow, equations governing the amplitude and propagation characteristics of SRWs are derived analytically. These equations show, for example, that a sufficiently large (small) translation speed and/or a sufficiently weak (strong) background zonal shear favor transmission (reflection) of the SRW through (from) the jet. Conservation equations are derived showing that time changes in the domain averaged amplitude ("mass") or squared amplitude ("momentum") are due to zonal variation in both the linear, long-wave phase speed and linear growth; dispersion and nonlinearity do not affect the "mass" or "momentum". Provided (1) the background PV forcing is sufficiently small, or (2) the background PV forcing is meridionally symmetric and the disturbance is a SRW, the dynamics of the disturbance field is Hamiltonian and mass and energy are thus conserved. Numerical solutions of the K-dV equation show that the zonally varying background flow yields three general classes of behavior: reflection, transmission, or trapping. Within each class there exists SRWs and Rossby wave packets. SRWs that become trapped within the zonally localized jet region may exhibit the following behaviors: (1) an oscillatory decay to a steady state at the jet center, (2) the creation of additional SRWs within the jet region, or (3) a steady-state wherein the solution has a smoothed step-like structure located downstream along the jet axis.

Continuous Dynamical Modes in Straits Having Arbitrary Cross Sections, with Applications to the Bab al Mandab*

The continuous dynamical modes of the exchange flow in the Bab al Mandab are computed in an attempt to assess the hydraulic character of the flow at the sill. First, an extended version of the Taylor–Goldstein equation for long waves that accounts for cross-channel topographic variations, is developed. A series of calculations using idealized background velocity U(z) and buoyancy frequency N(z) are presented to illustrate the effects of simple topographic cross sections on the internal modes and their speeds. Next, hydrographic and direct velocity measurements from April to November 1996 using moored CTDs and a bottom-mounted ADCP are utilized to construct monthly mean vertical profiles of N2(z) and at the U(z) sill. An analytical approximation of the true topography across the strait is also constructed. The observed monthly mean profiles are then used to solve for the phase speeds of the first and second internal modes. Additional calculations are carried out using a selection of “instantaneous” (2-h average) profiles measured during extremes of the semidiurnal tide. The results are compared with a three-layer analysis of data from the previous year. Many of the authors’ conclusions follow from an intriguing observation concerning the long-wave phase speeds. Specifically, it was nearly always observed that the calculated speeds c−1 and c1 of the two waves belonging to the first internal mode obey c−1 < Umin < Umax < c1, where Umin and Umax are the minimum and maximum of the velocity profile. An immediate consequence is that neither wave has a critical level. For monthly mean profiles, each of which have Umin < 0 < Umax, the flow is therefore subcritical (the phase speeds of the two waves have opposite signs). For instantaneous profiles this relationship continues to hold, although the velocity profile can be unidirectional. Thus the flow can be critical (c−1 = 0 and/or c1 = 0) or even supercritical (c−1 and c1 have the same sign) with respect to the first mode. Similar findings follow for the second baroclinic mode phase speeds (c−2 and c2). The authors conclude that hydraulically critical flow is an intermittent feature, influenced to a great extent by the tides. It is noted that the phase speed pairs for each mode lie very close to Umin and Umax. As suggested by the analysis of idealized profiles, this behavior is characteristic of flows that are marginally stable, perhaps as a result of prior mixing. This suggestion is supported by Richardson number (Ri) profiles calculated from the monthly mean and instantaneous data. Middepth values of Ri were frequently found to be O(1) and sometimes <1/4, a result consistent with the presence of mixing over portions of the water column.

Modulation of three resonating gravity–capillary waves by a long gravity wave

We consider a resonant triad of gravity–capillary waves, riding on top of a much longer gravity wave. The long-wave phase is assumed to vary on the same scale as the slow modulation of the short waves. Envelope equations are first deduced in the Lagrangian description. By perturbation analysis for a weak long wave, we then find that the long wave can resonate the natural modulation oscillations of the triad envelope, giving rise to various bifurcations in the Poincaré map. Numerical integration for a stronger long wave reveals that chaos can emerge from these bifurcations. The bifurcation criterion of Chen &amp; Saffman (1979) for collinear Wilton's ripples is generalized to arbitrary non-collinear triads, and is found to play an important role as a criterion for the onset of chaotic behaviour.

The Logic of World Capitalism

The “world-system” theory, in which the development pattern of each “nation-state” is determined by its location in the system, has been criticized of its holist framework. In this paper, by contrast, “national societies” are patternized in accordance with their ways of transition to capitalism and endogenous class relations. And then changes of world capitalism are tried to suggest as results of those of “national societies” relations.In order to attain this aim, the renewal of whole solid capital as a system is combined with industrial relations and the long wave theory is recomposed, factors of class struggles, state intervention and ideology being taken into consideration.As a result, it is theoretically explained that at the “down” phase of long wave a “national society” which has the initiative is likely to be replaced and the capitalist developments of societies in the “semi-periphery” and/or “periphery” may be influenced.

Generation of Mesoscale Variability by Resonant Interaction between a Baroclinic Current and Localized Topography

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 t...

The extended Korteweg-de Vries equation and the resonant flow of a fluid over topography

The extended Korteweg-de Vries equation which includes nonlinear and dispersive terms cubic in the wave amplitude is derived from the water-wave equations and the Lagrangian for the water-wave equations. For the special case in which only the higher-order nonlinear term is retained, the extended Korteweg-de Vries equation is transformed into the Korteweg-de Vries equation. Modulation equations for this equation are then derived from the modulation equations for the Korteweg-de Vries equation and the undular bore solution of the extended Korteweg-de Vries equation is found as a simple wave solution of these modulation equations. The modulation equations are also used to extend the solution for the resonant flow of a fluid over topography. This resonant flow occurs when, in the weakly nonlinear, long-wave limit, the basic flow speed is close to a linear long-wave phase speed for one of the long-wave modes. In addition to the effect of higher-order terms, the effect of boundary-layer viscosity is also considered. These solutions (with and without viscosity) are compared with recent experimental and numerical results.

Resonant forcing of coastally trapped waves in a continuously stratified ocean

In a previous paper (Grimshaw, 1987) the resonant forcing of coastally trapped waves was discussed in the barotropic case. In order to extend that theory to more realistic situations, we have considered the analogous theory whereby a longshore current interacts with a longshore topographic feature, or the forcing is due to longshore wind stress, for the case of the continuously stratified ocean. As in the previous theory, near resonance, when a long-wave phase speed is close to zero (in the reference frame of the forcing), the wave motion is governed by a forced evolution equation of the KdV-type. The behaviour of the wave field is characterized by three parameters representing the bandwidth for resonance, the forcing amplitude and the dissipation. We have evaluated these parameters in various practical cases, and found that the bandwidths, which scale with α1/2 when the forcing has dimensionless amplitude α, can often be quite broad. Typically the second, third, or higher, modes may be resonant. Concurrently, the dissipation is also usually significant, leading to a steady state balance between the forcing, dissipation and nonlinear terms.

Comparisons of simulated and actual synthetic aperture radar gravity wave images

The synthetic aperture radar (SAR) images obtained with the Jet Propulsion Laboratory's L band system in the Tower Ocean Wave and Radar Dependence (TOWARD) oceanographic experiment are compared with images generated by a simulation program implementing a SAR ocean imaging model based on two‐scale hydrodynamic and electromagnetic scattering approximate models. A critically regarded test of a theory of SAR ocean imaging is its prediction of the best image focus dependence on surface motion. The primary means of comparison here is an estimate of the best focus parameter that uses a subimage cross‐correlation technique. The focus parameter estimates for both the actual and simulated images show (1) a reversal in sign with reversal in dominant longwave direction relative to the SAR direction, (2) a magnitude increase with an increase in magnitude of the angle between the dominent long wave and SAR axes, and (3) an independence of the altitude and also the range‐to‐velocity ratio. All equivalent velocity estimates are of the order of the dominant longwave phase velocity, normalized by the SAR vehicle velocity, and agree well qualitatively though there are some quantitative differences. These behaviors are in agreement with a related analytical prediction. The “visibility” of the long waves' SAR image artifacts, for this TOWARD data set, increased with increased altitude and also with increased range‐to‐velocity ratio for both the actual and simulated images, as was predicted by a related analysis. Agreement of spectral density estimates of the actual and simulated SAR images generally required an order of magnitude increase in the strength of the hydrodynamic interaction of the long and short waves. This may indicate, as have surface measurements in TOWARD and other experiments, that the two‐scale model does not fully describe this nonlinear interaction. With this interaction adjustment, a comparison of actual and simulated images using an alternative focus criterion showed very close agreement in some cases.

Resonant flow of a stratified fluid over topography

The flow of a stratified fluid over topography is considered in the long-wavelength weakly nonlinear limit for the case when the flow is near resonance; that is, the basic flow speed is close to a linear long-wave phase speed for one of the long-wave modes. It is shown that the amplitude of this mode is governed by a forced Korteweg-de Vries equation. This equation is discussed both analytically and numerically for a variety of different cases, covering subcritical and supercritical flow, resonant or non-resonant, and for localized forcing that has either the same, or opposite, polarity to the solitary waves that would exist in the absence of forcing. In many cases a significant upstream disturbance is generated which consists of a train of solitary waves. The usefulness of internal hydraulic theory in interpreting the results is also demonstrated.

On the Interaction Between Long and Short Surface Waves

Abstract Short, dissipative, surface waves superposed on longer waves cause a growth of the long wave momentum Ml at a ratewhere kl, al are the amplitude and wavenumber of the long waves, so that klal is their steepness; Sa is the radiation stress of the short waves and τs, the rate of transfer of momentum to the short waves by the wind; and the angle braces denote an average over the long-wave phase θ = klx−ωlt. The first term in the above equation is the radiation stress interaction (Phillips, 1963; Hasselmann, 1971) and is generally negligible compared with the second term, neglected by Hasselmann (1971), which shows that long waves can grow if short wave generation (rather than dissipation) is correlated with the long wave orbital velocity. Even if the modulation of τs is only O(klal) times 〈τs〉, this mechanism can contribute a significant fraction of long wave momentum. However, even a substantially greater modulation of τs, perhaps due to varying exposure of short waves to the wind, is unlikely to a...

The Pamir Earthquake of 1911 February 18, in Relation to the Depths of Earthquake Foci.

SUMMARY Formulae have been derived for computing the energy in an earth-quake wave from seismographic records. These differ considerably from Galitzin's formula, especially with regard to the long-wave phase. An estimate is made of the fraction of the energy of a falling body that reappears as the energy of the seismic wave generated, the body being supposed broken up by the impact. With regard to the earth-quake under consideration, Galitzin computed the energy actually in the earthquake wave from seismological observations, and also found the energy set free by the impact of the falling mass on the ground. The two agreed. It is here shown that his estimate of the energy in the wave was about 250 times too high, but that only 1/300 of the energy of the impact would go into the wave : accordingly theory and observation still agree, and we can infer that the wave was actually produced by the landslip. The observations of this earthquake, whose focus must have been in the surface, are compared with those derived from ordinary earth-quakes, which have occurred at various depths. No systematic difference is found of the kind that would correspond to a difference in the depth of focus. Accordingly it is inferred that the foci of the earth-quakes used in the ordinary tables were not at depths greater than 120 km.

VII. Surface reflexion of earthquake waves

A seismogram, or instrumental time record of the earth movement produced by an earthquake at some distance, may be divided into three main parts, known as the first phase, the second phase, and the long-wave phase respectively. As a rule, the experienced observer has little difficulty in distinguishing these phases, and although difficult cases do arise, there is general agreement that the three divisions are characteristic and correspond to definite properties of the earth. The beginning of the-first phase is denoted by P and the beginning of the second phase by S, and these are interpreted as representing the arrival of longitudinal and transversal waves which start simultaneously at the focus of the earthquake and reach the observing station at different times. By general convention P and S are also used to represent the time interval between the occurrence of the earthquake and the arrival of the corresponding waves at the observing station.