## Wave breaking in long wave models and undular bores

##### Type

Master thesis###### Not peer reviewed

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##### Date

2015-12-18##### Author

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Show full item record##### Abstract

Bores are a well known phenomena in fluid mechanics, although
their occurrence in nature is relatively rare. The circumstances in
which they occur is usually when a tidal swell causes a difference
in surface elevation in the mouth of a river, or narrow bay, causing
long waves to propagate upstream. The term 'tidal bore' is also
frequently used in this context. Depending on the conditions the
bore may take on various forms, ranging from a smooth wavefront
followed by a smaller wave train, to one singe breaking wavefront.
Some noteworthy locations where tidal bores can be found include
the River Seine in France, the Petitcodiac River in Canada, and the
Qiantang River in China. Common for all these locations is a large
tidal range. Bores, when powerful enough, can produce particularly
unsafe environments for shipping, but at the same time popular
opportunities for river surfing.
As found by Favre in 1935 by wave tank experiments, the strength
of the bore can be determined by the ratio of the incident water
level above the undisturbed water depth to the undisturbed water
depth. Denoting this ratio by $\alpha$, bores can occur in one of
three categories: If $\alpha$ is less than 0.28 the bore is purely
undular, and will feature oscillations downstream of the bore front.
If $\alpha$ is between 0.28 and 0.75 the bore will continue to
feature oscillations, but one or more waves behind the bore front
will start to break. If $\alpha$ is greater than 0.75 the bore is
completely turbulent, and can no longer be described by the
standard potential flow theory.
The goal of this report is to simulate the time evolution of an
undular bore through numerical experiments, using a dispersive
nonlinear shallow water theory, in particular the Korteweg-deVries
(KdV) equation. This is a third order nonlinear partial differential
equation, where the dependent variable describes the
displacement of the free surface. When deriving this equation, an
expression for the velocity field of the flow is also available. This
can be calculated at any point in the fluid, as long as the
displacement of the surface is known. Thus, solving the KdV
equation also yields the fluid particle velocity field, which can be
used to calculate fluid particle trajectories, as done by Bjørkvåg
and Kalisch, but also to formulate a breaking criterion. By applying
this breaking criterion to the undular bore, the onset of breaking,
and thus also a maximum allowable wave height (due to the
nonlinearity of the model equation), can be computed numerically.
This criterion can also be applied to the exact traveling wave
solutions of the KdV equation, namely the 'solitary wave' solution
and the 'cnoidal wave' solution. These are waves of constant shape
traveling at constant velocity, thus applying the breaking criterion
yields a maximum height for which they can exist.
The theory leading to the formulation of the KdV equation is also
included, in addition to formulation of the linearized and shallow
water equations. These, however, serve only as 'stepping stones'
towards the higher order Boussinesq equations, and are not used
in any further calculations.

##### Publisher

The University of Bergen##### Collections

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