Long Wave Speed Research Articles

Resonant generation of a solitary wave in a thermocline

The resonant generation of a second-mode internal solitary wave, resulting from a ship internal waves system damping in a thermocline, is studied experimentally. The source of the stationary internal waves was provided by an oblong ellipsoid of revolution towed horizontally and uniformly at the depth of the thermocline center. The ranges of the Reynolds and Froude numbers were 500⩽Re=Ul/v⩽ 15000 and 0.3⩽Fi=U/NmaxD⩽1.0, respectively. When the body's speed and the linear long-wave second-mode phase speed were equal, an internal solitary wave of the “bulge” type was observed. The shape of the wave satisfied the Korteweg-de Vries equation. The Urcell parameter was equal to 10.2.

Simple Internal Waves and Bores

A simple surface wave develops, for example, after a dam break in a reservoir, or when water is suddenly withdrawn from one end of a long open channel. The present contribution is on simple internal waves, which can develop at the thermocline of a stratified reservoir when a sudden change of the inflow rate occurs due to man-made or natural causes. Use is made of the fact that the interface along a simple wave advances with the local long-wave speed, and of flow continuity. General analytical solutions for the velocity difference between the upper and lower layers along simple waves, and for their shape, are derived for Boussinesq flows. Also, the conditions are outlined under which a simple wave becomes unstable, and an internal bore develops. An example is provided for the shape and extent of a simple wave, which is due to a sudden dense inflow into a stratified reservoir. Such a wave is stable.

A simple model of Rossby-wave hydraulic behaviour

This paper considers hydraulic control and upstream influence in systems where the only wave propagation mechanism arises from the variation of vorticity or potential vorticity. These systems include two-dimensional shear flows as well as many simple paradigms for large-scale geophysical flows. The simplest is a flow in which the vorticity or potential vorticity is piecewise constant. We consider such a flow confined to a rotating channel and disturbed by a topographic perturbation. We analyse the behaviour of the system using steady nonlinear long-wave theory and demonstrate that it exhibits behaviour analogous to open-channel hydraulics, with the possibility of different upstream and downstream states. The manner by which the system achieves such states is examined using time-dependent long-wave theory via integration along characteristics and using full numerical solution via the contour-dynamics technique.The full integrations agree well with the hydraulic interpretation of the steady-state theory. One aspect of the behaviour of the system that is not seen in open-channel hydraulics is that for strong subcritical flows there is a critical topographic amplitude beyond which information from the control cannot propagate far upstream. Instead flow upstream of the topographic perturbation adjusts until the long-wave speed is zero, the control moves to the leading edge of the obstacle and flow downstream of the control is supercritical, with a transition from one supercritical branch to another on the downstream slope of the obstacle.

Unglazed Solar Collector Performance Characteristics

A modified form of the Hottel-Whillier-Bliss solar collector efficiency function is investigated for unglazed solar collector characterisation. The modified function includes allowance for long-wave sky radiation and wind speed on collector performance. The effect of uncertainties in long-wave radiation and wind speed measurement on the evaluation of the collector efficiency is discussed. The use of four coefficient and three coefficient forms of the modified Hottel-Whillier-Bliss function are considered from the point of view of test data reliability.

On hydraulic control in a stratified fluid

The conditions for hydraulic control to occur in a continuously stratified fluid are discussed, using density as a vertical coordinate in place of height. A suitable definition of Froude number, which varies with depth, is given. Three conditions for control emerge. One is that the flow be everywhere well-behaved; another is that control occurs when the local long-wave speed vanishes. These are shown to be equivalent. The location of the control is determined indirectly by the Froude number, which occurs as the coefficient in an ordinary differential equation; the Froude number must be somewhere less than a critical value for control to occur. The third condition requires the coalescence of two different solutions for the same boundary conditions at the point of control. It is shown that this requirement is non-trivial: examples given include a simple control by topography, a virtual control, and a control by a constriction. A direct connection with layered theory is produced. Brief discussions of bidirectional flow (where the isopycnal surface of zero velocity must be flat) and weak shocks are given.

Transcritical, shallow-water flow over compact topography

The flow of a shallow layer of fluid over isolated topography is considered when the velocity is near the critical, long-wave speed. The response of the free surface is obtained when the length-scale of the topographic variation is comparable to the fluid depth (compact topography). The weakly nonlinear response is described by a forced Korteweg-de Vries equation. When the topography is compact, an inner and outer matching is required to obtain a uniform description of the free-surface deformation for all streamwise positions. The important classes of compact forcing are defined and general results for the shape of the free surface directly over the topographic forcing are obtained. A variety of stationary, outer solutions are presented including solitary and lee-wave responses.

Upstream-advancing nonlinear waves in an axisymmetric resonant flow of rotating fluid past an obstacle

A numerical study of rotating flow past an axisymmetric obstacle in a cylindrical tube is described. Long nonlinear inertial waves propagate upstream of the obstacle even when the theoretical speed of long waves as given by the linearized equations is equal to the incoming axial flow speed. This occurs when the upstream swirling velocity field is of the Brugers vortex type and the quadratic nonlinear term in the forced KdV equation is not zero. If the flow has a rigid-body rotation, so that the quadratic nonlinear term vanishes, the nonlinear correction of the wave speed is small and cannot be identified even after large times of evolution.

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.

Resonant internal waves in fluid flow

UNDERSTANDING the flow of a stratified fluid over an uneven bed is an important problem in hydrodynamics. Previous field experiments using acoustic imagery1–3 have enhanced our understanding of internal-wave generation mechanisms. These studies have generally concentrated on isolated topographic features that show large-amplitude long internal waves and hydraulic jumps as the flow becomes critical defined in the hydraulic sense such that a long-wave speed vanishes with respect to a fixed coordinate system. We present field observations of stratified fluid flow over a nearly periodic topographic relief which show a unique series of resonant internal waves. The observations, carried out in the Rotterdam Waterway, show the presence of internal waves with wavelengths commensurate with those of the topography of the waterway bed and amplitudes that are three to four times larger than the relief of the topography generating them. Resonant internal waves may be important in understanding the flow dynamics of stratified coastal environments and may have consequences for atmospheric flows over mountain ranges.

Baroclinic wave generation by barotropic waves passing over a shelf

In a 10×2×0.6 m channel we studied baroclinic wave generation by barotropic periodic waves passing over a shelf. Firstly, for fixed experimental conditions, we varied the period, and noticed: (1) that the Froude number defined by F=UsbC2s is suited tharacterize the flow near the edge of the shelf (Usb is the maximum value of the barometric current near the edge of the shelf and C2s is the baroclinic longwave speed over the shelf); (2) that for F<1, the baroclinic wave recorded near the edge of the shelf is quasi-linear and then evolves into non-linear waves as it propagates over the shelf; (3) that, for F>1, the waves are non-linear everywhere. Secondly, for a given period the longitudinal profiles of the interface at various times reveal that: (i) over the slope there is lee-wave formation during part of the period; (ii) when H1<H2s (H1 and H2s upper and lower thickness, respectively), such a wave propagates over the shelf where it evolves to trains of solitary waves of depression; (iii) when H1>H2s, the phenomena are much more complex, but, under some conditions, there may appear trains of solitary waves of elevation over the shelf.

Preface

Observations of internal tide generation over continental slopes in a laboratory experiment have been carried out, with the objectives of making comparisons with linear generation theory and investigating its limitations. Both continuous and layered stratification have been considered. A measure of the amplitude of the barotropic tidal forcing (and hence of non-linearity) is given by the Froude Number F = usb/cw, where usb is the maximum barotropic tidal velocity at the shelf break, and cw is the long-wave speed of the lowest internal mode.For continuous stratification, good agreement was obtained for “steep” slopes (α/c > 1), where α is the slope at the continental slope and c is the slope of the internal wave rays of tidal frequency), even for quite large amplitude motions (F < 1.6), and the upper limit of its quantitative usefulness was not reached. For “flat” slopes (α/c < 1) reasonable agreement was also obtained, even up to quite amplitudes (F < 3.1), although some departure from linear theory was apparent.For two-layer flows the applicability of linear theory was much more restricted. For F ∼ 0.5 there was only qualitative agreement and for larger F (>1) significant differences were observed. The latter were due to the substantial advection and associated hydraulic jumps which occured seaward of the shelf-break during the ebb-phase of the barotropic tide. Shelf-break values of F > 1 are common in the ocean.

Dissipative effects on the resonant flow of a stratified fluid over topography

The effect of dissipation on the flow of a stratified fluid over topography is considered in the weakly nonlinear, long-wave limit for the case when the flow is near resonance, i.e. the basic flow speed is close to a linear long-wave speed for one of the long-wave modes. The two types of dissipation considered are the dissipation due to viscosity acting in boundary layers and/or interfaces and the dissipation due to viscosity acting in the fluid as a whole. The effect of changing bottom topography on the flow produced by a force moving at a resonant velocity is also considered. In this case, the resonant condition is that the force velocity is close to a linear long-wave velocity for one of the long-wave modes. It is found that in most cases, these extra effects result in the formation of a steady state, in contrast to the flow without these effects, which remains unsteady for all time. The flow resulting under the action of boundary-layer dissipation is compared with recent experimental results.

Multiple solutions and flow limitation for steady flow through a collapsible tube held open at the ends

Equations for the steady flow of an incompressible, inviscid fluid through a collapsible tube under longitudinal tension are derived by treating the tube longitudinally as a membrane, and taking the collapsibility of the tube into account in an approximate way by replacing in the equation for an axisymmetric membrane a term representing the resistance of the tube to area change by the tube law for collapsible tubes. The flow is assumed to be uniform in a cross-section. A nonlinear differential equation is obtained for the shape of the tube for given values of total pressure p0, flow rate q, longitudinal tension τ and tube law P = P(ρ); where ρ = (A/πR2)½ is the equivalent radius of the tube (A = area of a cross-section, R = radius of the unloaded, then circular tube). The equation can be integrated and analysed in the phase plane. Equilibrium points correspond to uniform flow through cylindrical tubes; saddle points correspond to subcritical flow (S &lt; 1), centrepoints to supercritical (S &gt; 1) and a higher-order point to critical flow (S = 1). Here S is the speed index, the ratio of the flow speed to the speed of long waves. Near centrepoints there are solutions, that represent area-periodic tubes. For a finite tube, held open at the ends, the steady flow is formulated as a two-point boundary-value problem. On the basis of numerical calculations, and a bifurcation analysis using the method of Lyapunov–Schmidt, the existence and multiplicity of the solutions of this problem are discussed and the process of flow limitation studied. For negative total pressures two collapsed solutions are found that disappear at the flow-limitation value of the flow rate. For positive total pressures a distinction is made between subcritical, critical and supercritical total pressures. In all these cases there is a multiplicity, proportional to the ratio of the tube length to [Lscr ]1(0), the wavelength of the collapsed periodic solution for vanishing flow rate, and having maximum radius ρ = 1. For subcritical total pressures increase of the flow rate leads to a gradual loss of all solutions in higher-order flow limitations until final flow limitation occurs by the mergence of two collapsed solutions. For supercritical total pressures increase of the flow rate also leads to a gradual loss of all solutions in higher-order flow limitations in a process which now also depends upon the ratio of the tube length to the wavelength L of periodic solutions with vanishing amplitude and ρ ≡ 1.

Resonant Forcing of Barotropic Coastally Trapped Waves

Abstract The interaction of a longshore current with a longshore topographic feature is investigated in the barotropic case. It is shown that near resonance, when a long-wave speed is close to zero (in a fixed reference frame), there is enhanced generation of upstream and downstream coastally trapped waves. An evolution equation of the the KdV-type is derived to describe the resonant behavior, and numerical solutions are discussed for a range of parameters describing the forcing terms, the detuning term and dissipation. The analogous situation of resonant generation due to wind stress is developed in an appendix.

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.

Joint Vortices, Eastward Propagating Eddies and Migratory Taylor Columns

The behavior of an isolated pair of vortices consisting of two eddies situated on top of each other in a three-layer ocean is examined analytically. The amplitudes of both eddies are high and, consequently, the two eddies behave as one unit and migrate together in the ocean. For this reason, it is proposed to call the system joint vortices. The eddies are of equal or opposite sign; each vortex is situated in a different layer so that there are two active layers and one passive layer. Attention is focused on the behavior of joint vortices on a sloping bottom in the deep ocean and on a β plane in the upper ocean. That is, we consider deep joint eddies situated on an inclined floor in the lowest two layers of a three-layer ocean and upper joint eddies in the upper two layers. Special attention is given to the cases where one of the vortices is a lens-like eddy. Approximate solutions for slope (or β) induced drifts in the east-west direction are obtained. It is found that because of the high amplitudes and the resulting nonlinear coupling, the joint eddies have a mutual drift which is very different from the drift that each individual vortex would have. For example, while each individual vortex translates to the west in the absence of a conjugate vortex, the combined vortices may drift steadily to the east. This bizarre behavior stems from the presence of a “planetary lift” which is the oceanic equivalent of the side pressure force associated with the so-called Magnus effect. It is directed at 90° to the left of the drifting eddies. Other results of interest are: (i) Under some conditions, the westward drift of joint eddies consisting of two cyclonic vortices it much faster than the long-wave speed. Such fag drift contradict previously held contentions that the speed of cyclonic eddies cannot exceed the long wave speed. (ii) As it translates westward, an anticyclonic lens-like eddy can carry a Taylor column on top of it. Possible application of this theory to various eddies in the ocean is discussed.

A separated-flow model for collapsible-tube oscillations

A new model is presented to describe flow in segments of collapsible tube mounted between two rigid tubes and surrounded by a pressurized container. The new features of the model are the inclusion of (a) longitudinal wall tension and (b) energy loss in the separated flow downstream of the time-dependent constriction in a collapsing tube, in a manner which is consistent with the one-dimensional equations of motion. As well as accurately simulating steady-state collapse, the model predicts self-excited oscillations whose amplitude is large enough to be observable only if the flow in the collapsible tube becomes supercritical somewhere (fluid speed exceeding long-wave propagation speed). The dynamics of the oscillations is dominated by longitudinal movement of the point of flow separation, in response to the adverse pressure gradient associated with waves propagating backwards and forwards between the (moving) narrowest point of the constriction and the tube outlet.

Internal tide generation at a continental shelf/slope junction: A comparison between theory and a laboratory experiment

Observations of internal tide generation over continental slopes in a laboratory experiment have been carried out, with the objectives of making comparisons with linear generation theory and investigating its limitations. Both continuous and layered stratification have been considered. A measure of the amplitude of the barotropic tidal forcing (and hence of non-linearity) is given by the Froude Number F = usb/cw, where usb is the maximum barotropic tidal velocity at the shelf break, and cw is the long-wave speed of the lowest internal mode. For continuous stratification, good agreement was obtained for “steep” slopes (α/c > 1), where α is the slope at the continental slope and c is the slope of the internal wave rays of tidal frequency), even for quite large amplitude motions (F < 1.6), and the upper limit of its quantitative usefulness was not reached. For “flat” slopes (α/c < 1) reasonable agreement was also obtained, even up to quite amplitudes (F < 3.1), although some departure from linear theory was apparent. For two-layer flows the applicability of linear theory was much more restricted. For F ∼ 0.5 there was only qualitative agreement and for larger F (>1) significant differences were observed. The latter were due to the substantial advection and associated hydraulic jumps which occured seaward of the shelf-break during the ebb-phase of the barotropic tide. Shelf-break values of F > 1 are common in the ocean.