Nonlinear effects in internal waves

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Presentation transcript:

Nonlinear effects in internal waves

Wave steepness Linear wave: h0<1/m 2h0 l/2 2h0 Steep wave: h0m>1. Regions of overturned isopycnals

Connection between high Froude number and wave breaking for surface waves Uw > C Initial interface height C + Uw C - Uw Subsequent interface height If Uw/C is large, then wave shape is modified so that wave surface overturns.

Nonrotating critical level example Vertical wavenumber Velocity profile Z(m) Wave characteristic U m/s when U = Cp, i.e. U=0. Z(m) X(m)

Overturning isopycnals at a critical level (U = 0) Dornbrack and Durbeck, 1998, numerical simulation

Large amplitude topography Fr = U/(Nh) h

Effect of finite amplitude topography: Fr = 1 Blue region shows reverse flow in region of overturned isentropes.

Hydraulic control Topographic obstacle accelerates flow, reduces d, so that transition to supercritical flow occurs. Mixing occurs in supercritical region. d (Numerical simulations, Legg)

Parametric Subharmonic Instability Initial conditions Winters et al, 2004 numerical simulations Subsequent evolution – generation of waves at twice initial frequency

PSI of internal tide at 21 degrees KE spectra as function of Horizontal wavenumber frequency Vertical wavenumber Red = 2days, green = 22 days, blue = 65 days MacKinnon and Winters, 2006. Numerical simulations.

Garrett-Munk spectrum Frequency spectrum Vertical wavenumber spectrum (taken from Lvov et al, 2005)

Comparison of observations and wave-turbulence model for spectra (Lvov et al, 2005)

What have we learned? Nonlinearity in gravity waves can be characterised by wave steepness s=hm, or wave froude number Fr=U/C. Highly nonlinear waves are associated with overturned isopycnals, leading to mixing. Breaking waves can be produced at critical layers, and large amplitude topography. Nonlinear wave-wave interactions allow waves at new k,w to be generated. Parametric Subharmonic Instability is a resonant wave-wave interaction where the new wave has half the frequency of the original wave, especially important when w0=2f. Nonlinear wave-wave interactions are responsible for the continuous Garrett-Munk spectrum observed in the ocean.