Internal hydraulic jumps and mixing in two-layer flows

Abstract

Internal hydraulic jumps in two-layer flows are studied, with particular emphasis on their role in entrainment and mixing. For highly entraining internal jumps, a new closure is proposed for the jump conditions. The closure is based on two main assumptions: (i) most of the energy dissipated at the jump goes into turbulence, and (ii) the amount of turbulent energy that a stably stratified flow may contain without immediately mixing further is bounded by a measure of the stratification. As a consequence of this closure, surprising bounds emerge, for example on the amount of entrainment that may take place at the location of the jump. These bounds are probably almost achieved by highly entraining internal jumps, such as those likely to develop in dense oceanic over flows. The values obtained here are in good agreement with the existing observations of the spatial development of oceanic downslope currents, which play a crucial role in the formation of abyssal and intermediate waters in the global ocean.

Are cascading flows stable?

“…In the past, this problem has generally been avoided by taking two-layer flows with uniform densities and velocities in the layers both up-and down-stream of a jump, and with the density in each layer remaining unchanged through the transition (e.g. see Baines 1995) or changing in one of the two layers as a consequence of entrainment at the jump (Holland et al, 2002). Even if the flow approaching a transition is layered (with Kelvin-Helmholtz instability at the interfaces between layers somehow suppressed), in miscible fluids, a jump to another layered flow with discontinuous density and velocities is unrealistic.…”

Fourier spectrum and shape evolution of an internal Riemann wave of moderate amplitude

“…The breaking of internal waves contributes significantly to the mixing processes in the ocean and has been frequently observed in real nature (Smyth and Holloway, 1988;Henyey and Hoering, 1997;Afanasyev and Pelter, 2001;Chant and Wilson, 2000;Horn et al, 2001;Konyaev and Filonov, 2006). Strongly nonlinear deformation and breaking of internal waves is described theoretically in the framework of twolayer flows with no dispersion (Baines, 1995;Klemp et al, 1997;Lyapidevsky and Teshukov, 2000;Jiang and Smith, 2001;Holland et al, 2002;Fringer and Street, 2003;Milewski et al, 2004;Sakai and Redekopp, 2007;Zahibo et al, 2007). In fact, the long interfacial waves in a twolayer fluid as described in the Boussinesq approximation with rigid-lid approximation are described by almost the same equations of motion as surface shallow-water waves, allowing to introduce the Riemann invariants and use the classic theory of nonlinear hyperbolic equations.…”

Strongly nonlinear steepening of long interfacial waves

“…Breaking internal waves contribute to the mixing of salt, heat and tracers in water (Henyey and Hoering, 1997;Munk and Wunsh, 1998;Chant and Wilson, 2000;Horn et al, 2001;Fringer and Street, 2003). There are many observations and theoretical models of bore-like shapes of long internal waves on ocean shelves, and in fjords and lakes (Klemp et al, 1997;Afanasyev and Pelter, 2000;Armi and Farmer, 2001;Holland et al, 2002). In many cases these strongly nonlinear internal waves are described in the framework of two-layer flows; see, for instance, the books by Baines (1995) and Lyapidevsky and Teshukov (2000).…”