We present a conservative formulation and a numerical algorithm for thereduced-gravity shallow-water equations on a beta plane, subjected to a constantwind forcing that leads to the formation of double-gyre circulation in a closedocean basin. The novelty of the paper is that we reformulate the governingequations into a nonlinear hyperbolic conservation law plus source terms. Asecond-order fractional-step algorithm is used to solve the reformulatedequations. In the first step of the fractional-step algorithm, we solve thehomogeneous hyperbolic shallow-water equations by the wave-propagation finitevolume method. The resulting intermediate solution is then used as the initialcondition for the initial-boundary value problem in the second step. As aresult, the proposed method is not sensitive to the choice of viscosity andgives high-resolution results for coarse grids, as long as the Rossbydeformation radius is resolved. We discuss the boundary conditions in each step,when no-slip boundary conditions are imposed to the problem. We validate thealgorithm by a periodic flow on an f-plane with exactsolutions. The order-of-accuracy for the proposed algorithm is testednumerically. We illustrate a quasi-steady-state solution of the double-gyremodel via the height anomaly and the contour of stream function for theformation of double-gyre circulation in a closed basin. Our calculations arehighly consistent with the results reported in the literature. Finally, wepresent an application, in which the double-gyre model is coupled with theadvection equation for modeling transport of a pollutant in a closed oceanbasin.

We consider a two-layer fluid of finite depth with a free surface and, in particular, the surface tension at the free surface and the interface. The usual assumptions of a linearized theory are considered. The objective of this work is to analyse the effect of surface tension on trapped modes, when a horizontal circular cylinder is submerged in either of the layers of a two-layer fluid. By setting up boundary value problems for both of the layers, we find the frequencies for which trapped waves exist. Then, we numerically analyse the effect of variation of surface tension parameters on the trapped modes, and conclude that realistic changes in surface tension do not have a significant effect on the frequencies of these.

The problem of oblique wave scattering by a rectangular submarine trench is investigated assuming a linearized theory of water waves. Due to the geometrical symmetry of the rectangular trench about the central line $x=0$, the boundary value problem is split into two separate problems involving the symmetric and antisymmetric potential functions. A multi-term Galerkin approximation involving ultra-spherical Gegenbauer polynomials is employed to solve the first-kind integral equations arising in the mathematical analysis of the problem. The reflection and transmission coefficients are computed numerically for various values of different parameters and different angles of incidence of the wave train. The coefficients are depicted graphically against the wave number for different situations. Some curves for these coefficients available in the literature and obtained by different methods are recovered.

Many boundary value problems occur in a natural way while studying fluid flow problems in a channel. The solutions of two such boundary value problems are obtained and analysed in the context of flow problems involving three layers of fluids of different constant densities in a channel, associated with an impermeable bottom that has a small undulation. The top surface of the channel is either bounded by a rigid lid or free to the atmosphere. The fluid in each layer is assumed to be inviscid and incompressible, and the flow is irrotational and two-dimensional. Only waves that are stationary with respect to the bottom profile are considered in this paper. The effect of surface tension is neglected. In the process of obtaining solutions for both the problems, regular perturbation analysis along with a Fourier transform technique is employed to derive the first-order corrections of some important physical quantities. Two types of bottom topography, such as concave and convex, are considered to derive the profiles of the interfaces. We observe that the profiles are oscillatory in nature, representing waves of variable amplitude with distinct wave numbers propagating downstream and with no wave upstream. The observations are presented in tabular and graphical forms.

In models of fluid outflows from point or line sources, an interface is present, and it is forced outwards as time progresses. Although various types of fluid instabilities are possible at the interface, it is nevertheless of interest to know the development of its overall shape with time. If the fluids on either side are of nearly equal densities, it is possible to derive a single nonlinear partial differential equation that describes the interfacial shape with time. Although nonlinear, this equation admits a simple transformation that renders it linear, so that closed-form solutions are possible. Two such solutions are illustrated; for a line source in a planar straining flow and a point source in an axisymmetric background flow. Possible applications in astrophysics are discussed.

Problems of wave interaction with a body with arbitrary shape floating or submerged in water are of immense importance in the literature on the linearized theory of water waves. Wave-free potentials are used to construct solutions to these problems involving bodies with circular geometry, such as a submerged or half-immersed long horizontal circular cylinder (in two dimensions) or sphere (in three dimensions). These are singular solutions of Laplace’s equation satisfying the free surface condition and decaying rapidly away from the point of singularity. Wave-free potentials in two and three dimensions for infinitely deep water as well as water of uniform finite depth with a free surface are known in the literature. The method of constructing wave-free potentials in three dimensions is presented here in a systematic manner, neglecting or taking into account the effect of surface tension at the free surface or for water with an ice cover modelled as a thin elastic plate floating on the water. The forms of the wave motion at the upper surface (free surface or ice-covered surface) related to these wave-free potentials are depicted graphically in a number of figures for all the cases considered.

We introduce the entropy of a family of planar curves in terms of the number of intersections of the family with a random line, calculate it for key examples, and discuss the entropy of a pattern of rings produced by an impulse on the surface of still water.

The time-dependent motion of water waves with a parametrically defined free surface is mapped to a fixed time-independent rectangle by an arbitrary transformation. The emphasis is on the general properties of transformations. Special cases are algebraic transformations based on transfinite interpolation, conformal mappings, and transformations generated by nonlinear elliptic partial differential equations. The aim is to study the effect of transformation on variational principles for water waves such as Luke’s Lagrangian formulation, Zakharov’s Hamiltonian formulation, and the Benjamin–Olver Hamiltonian formulation. Several novel features are exposed using this approach: a conservation law for the Jacobian, an explicit form for surface re-parameterization, inner versus outer variations and their role in the generation of hidden conservation laws of the Laplacian. Also some of the differential geometry of water waves becomes explicit. The paper is restricted to the case of planar motion, with a preliminary discussion of the extension to three-dimensional water waves.

The equation modelling the evolution of a foam (a complex porous medium consisting of a set of gas bubbles surrounded by liquid films) is solved numerically. This model is described by the reaction–diffusion differential equation with a free boundary. Two numerical methods, namely the fixed-point and the averaging in time and forward differences in space (the Crank–Nicolson scheme), both in combination with Newton’s method, are proposed for solving the governing equations. The solution of Burgers’ equation is considered as a special case. We present the Crank–Nicolson scheme combined with Newton’s method for the reaction–diffusion differential equation appearing in a foam breaking phenomenon.

The stochastic Lagrange wave model is a realistic alternative to the Gaussian linear wave model, which has been successfully used in ocean engineering for more than half a century. In this paper we present the slope distributions and other characteristic distributions at level crossings for asymmetric Lagrange time waves, i.e. what can be observed at a fixed measuring station, thereby extending results previously given for space waves. The distributions are given as expectations in a multivariate normal distribution, and they have to be evaluated by simulation or numerical integration. Interesting characteristic variables are the slope in time, the slope in space, and the vertical particle velocity when the waves are observed close to instances when the water level crosses a predetermined level. The theory has been made possible by recent generalizations of Rice's formula for the expected number of marked crossings in random fields.

Gravity and surface-tension effects are examined for inviscid–inviscid interactions between two fluids close to a wall. The ratios of density and viscosity of the two fluids are taken to be small. A nonlinear integro–differential equation is found to govern the near-wall flow velocity, interface shape and pressure; analysis, computation and comparisons are then applied. Travelling-state solutions are of particular interest.

Quintic B-spline collocation schemes for numerical solution of the regularized long wave (RLW) equation have been proposed. The schemes are based on the Crank–Nicolson formulation for time integration and quintic B-spline functions for space integration. The quintic B-spline collocation method over finite intervals is also applied to the time-split RLW equation and space-split RLW equation. After stability analysis is applied to all the schemes, the results of the three algorithms are compared by studying the propagation of the solitary wave, interaction of two solitary waves and wave undulation.

Families of vortex equilibria, with constant vorticity, in steady flow past a flat plate are computed numerically. An equilibrium configuration, which can be thought of as a desingularized point vortex, involves a single symmetric vortex patch located wholly on one side of the plate. Given that the outermost edge of the vortex is unit distance from the plate, the equilibria depend on three parameters: the length of the plate, circulation about the plate, and the distance of the innermost edge of the vortex from the plate. Families in which there is zero circulation about the plate and for which the Kutta condition at the plate ends is satisfied are both considered. Properties such as vortex area, lift and free-stream speed are computed. Time-dependent numerical simulations are used to investigate the stability of the computed steady solutions.

Consider a Navier-Stokes incompressible turbulent fluid in R 2. Let x(t) denote the position coordinate of a moving vortex with initial circulation Γ0 > 0 in the fluid, subject to a force F. Define x(t) as a stochastic process with continuous sample paths described by a stochastic differential equation. Assuming a suitable notion of weak rotationality, it is shown that the stochastic equation is equivalent to a linear partial differential equation for the complex function ψ, i∂ψ/∂t = [-Γ0Δ + F] ψ, where |ψ|2 = ρ(x,t), ρ being the probability density function of finding the vortex centre in position x at time t.

In this paper time-harmonic surface wave motion for progressive waves incident normally on and scattered by a partially immersed fixed vertical barrier in water of infinite depth is considered in the presence of surface tension. The problem for the velocity potential is solved, as others have been previously, by first supposing that the free-surface slopes at the barrier are prescribed and the formal solution in terms of these is obtained explicitly by complex-variable methods. To simplify the calculation the known solution corresponding to zero free-surface slopes at the barrier is subtracted out first and emphasis is placed on determining the residual potential. Finally, an appropriate dynamical edge condition is imposed on the formal solution to determine the required values of the edge-slope constants and hence fully solve the transmission problem. The problem was first examined some time ago using a complex-variable reduction procedure before the advent of this condition, although an explicit formal solution was not obtained, that earlier work forms a basis for the present investigation. It is noted in conclusion how the solution of the problem for waves generated by a partially immersed non-uniform heaving vertical plate may easily be obtained in a similar manner, since the formal solution required is just the residual potential determined in our main problem.

In this paper two expansions are obtained by contour integration methods for the velocity potential describing two-dimensional time-harmonic surface waves due to a free-surface wave source on water of infinite depth in the presence of surface tension. First the series expansion at r = 0 is found and then the asymptotic expansion as Kr®¥, where K is the wave number for progressive waves and r the radial distance from the source. The corresponding expansions for the more important submerged wave source in terms of the radial distance from the image source in the free surface may then easily be deduced. The latter are required in a number of surface wave problems, particularly those of a short-wave asymptotic nature, and are also relevant in obtaining expansions for finite constant depth.

Attempts to extend known two-dimensional results (Ursell, 1947) to the fully three-dimensional case can lead to unpredictable results. We show how the use of a variational approximation for a finite plane vertical barrier leads to apparently different results when different formulations are used. The reason for this is not so much that the method is wrong, but rather that several different limits are taken in the process, which are hard to control. We suggest an alternative matching scheme, based on Ayad and Leppington (1977), which holds for the case ka → ∞, l/a → ∞, kl → ∞, where / is the length of the barrier, a its depth and k the wavelength of the incident wave. The method is applied to a channel with impeding side walls, as a model of French's (1977) wave-energy device.