Article

Linear water waves with vorticity: Rotational features and particle paths

Department of Mathematics, Lund University, PO Box 118, 221 00 Lund, Sweden; Dipartimento di Matematica, Viale Morgagni 67/A, 50134 Firenze, Italy
Journal of Differential Equations (Impact Factor: 1.48). 01/2008; DOI: 10.1016/j.jde.2008.01.012
Source: arXiv

ABSTRACT We study steady linear gravity waves of small amplitude travelling on a current of constant vorticity. For positive vorticity the situation resembles that of Stokes waves, but if the vorticity is large enough the particle trajectories are affected. For negative vorticity we show that there may appear internal waves and vortices, wherein the particle trajectories are not ellipses.

1 Bookmark
 · 
62 Views
  • [Show abstract] [Hide abstract]
    ABSTRACT: We use bifurcation theory to construct small periodic gravity stratified water waves with density which depends linearly upon the pseudostream function. As a special feature the density may also decrease with depth and the waves we obtain may posses two different critical layers with catʼs eye vortices. Within the vortex, the density of the fluid has an extremum at the stagnation point.
    Journal of Differential Equations 01/2011; 251(10):2932-2949. · 1.48 Impact Factor
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: Steady, free-surface, vortical flows of an inviscid, incompressible, heavy fluid over a horizontal, rigid bottom are considered. All flows of constant depth are described for any Lipschitz vorticity distribution. It is shown that the values of Bernoulli’s constant, for which such flows exist, are greater than or equal to some critical value depending on the vorticity. For the critical value, only one flow exists and it is unidirectional. Supercritical flows exist for all values of Bernoulli’s constant greater than the critical one; every such flow is also unidirectional and its depth is smaller than that of the critical flow. Furthermore, at least one flow other than supercritical does exist for every value of Bernoulli’s constant greater than the critical one. It is found that for some vorticity distributions, the number of constant depth flows increases unrestrictedly as Bernoulli’s constant tends to infinity. However, all these flows, except for one or two, have counter-currents; their number depends on Bernoulli’s constant and increases by at least two every time when this constant becomes greater than a critical value (the above mentioned is the smallest of them), belonging to a sequence defined by the vorticity. A classification of vorticity distributions is presented; it divides all of them into three classes in accordance with the behaviour of some integral of the distribution on the interval [0, 1]. For distributions in the first class, a unidirectional subcritical flow exists for all admissible values of Bernoulli’s constant. For vorticity distributions belonging to the other two classes such a flow exists only when Bernoulli’s constant is less than a certain value. If Bernoulli’s constant is greater than this value, then at least one flow with counter-currents does exist along with the unidirectional supercritical flow. The second and third classes of vorticity distributions are distinguished from one another by the character of the counter-currents. If a distribution is in the second class, then a near-bottom counter-current is always present for sufficiently large values of Bernoulli’s constant. For distributions in the third class, a near-surface counter-current is always present for such values of the constant. Several examples illustrating the results are considered.
    The Quarterly Journal of Mechanics and Applied Mathematics 08/2011; 64(3). · 1.27 Impact Factor
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: We construct three-dimensional families of small-amplitude gravity-driven rotational steady water waves on finite depth. The solutions contain counter-currents and multiple crests in each minimal period. Each such wave generically is a combination of three different Fourier modes, giving rise to a rich and complex variety of wave patterns. The bifurcation argument is based on a blow-up technique, taking advantage of three parameters associated with the vorticity distribution, the strength of the background stream, and the period of the wave.
    Archive for Rational Mechanics and Analysis 10/2013; · 2.29 Impact Factor

Full-text (3 Sources)

Download
46 Downloads
Available from
May 28, 2014