Some Aspects of the Flow of Stratified Fluids: I. A Theoretical Investigation

Abstract

The following paper is the first of a series of two relating to the problem of internal oscillations of a fluid in a gravity field with vertical gradients of density and velocity. The theoretical analysis in this paper will be supplemented by a report on an experimental investigation along the same lines to be published in a future issue of this journal. The exact, steady‐state equations of motion and continuity of a perfect liquid moving two‐dimensionally, with an arbitrary vertical distribution of density and velocity, are integrated once to yield a second‐order differential equation. This equation is examined with regard to uniqueness and stability of the motion. A criterion is developed giving a sufficient condition for the motion to be uniquely determined by the configuration of the topography over which the fluid moves. It appears, further, that the condition of uniqueness is also a condition that a certain integrated quantity, called the kinetic potential of the motion, be a maximum. The suggestion is offered that this may correspond to a form of fluid instability. A detailed study is made in the special cases of a uniform basic velocity, and a certain type of shearing flow. In either case, it is shown that an internal Froudc number of about 1/3 divides the motion into two states, one of which is called supercritical, the other subcritical. From several viewpoints, these regimes are analogous to the corresponding states of flow of water in a channel. In the subcritical state the flow is in the form of standing wave patterns. When flowing supercritically, conditions may be favorable for the formation of internal “hydraulic jumps”.

An example of enhanced tephra deposition driven by topographically induced atmospheric turbulence

“…The effects of lee waves have been extensively reported through observations (Jiang and Doyle 2004;Doyle et al 2005;Smith et al 2007;Lane et al 2009;Jiang et al 2010). Some early theoretical studies (Long 1953;Miles and Huppert 1968) elucidated the nonlinear mechanisms for an early stage of the wave breaking processes. Understanding of these mechanisms has been significantly enriched through numerical experiments (Laprise and Peltier 1989;Lin and Wang 1996;Jiang et al 2004;McHugh and Sharman 2013).…”

Numerical simulations of shoaling internal solitary waves of elevation

“…In the context of internal wave dynamics, such simplification leads to the Dubreil-Jacotin-Long (DJL) equation (Dubreil-Jacotin, 1932;Long, 1953), a single nonlinear PDE for the isopycnal displacement. The DJL equation is equivalent to the full set of stratified Euler equations, and hence its solutions are exact solitary wave solutions.…”

Instability in internal solitary waves with trapped cores

“…The most common approach has been to model the flow, both inside and outside of the core region, using the Dubriel-Jacotin-Long 26,27 (DJL) equation of models have assumed that the Brunt-Väisälä frequency in the core region takes the same value as that at the core boundary, i.e. the stratification is simply extended into the core region.…”