Abstract
There is no theoretical underpinning that successfully explains how turbulent mixing is fed by wave breaking associated with nonlinear wave-wave interactions in the background oceanic internal wavefield. We address this conundrum using one-dimensional ray tracing simulations to investigate interactions between high frequency internal waves and inertial oscillations in the extreme scale separated limit known as “Induced Diffusion”. Here, estimates of phase locking are used to define a resonant process (a resonant well) and a non-resonant process that results in stochastic jumps. The small amplitude limit consists of jumps that are small compared to the scale of the resonant well. The ray tracing simulations are used to estimate the first and second moments of a wave packet’s vertical wavenumber as it evolves from an initial condition. These moments are compared with predictions obtained from the diffusive approximation to a self-consistent kinetic equation derived in the ‘Direct Interaction Approximation’. Results indicate that the first and second moments of the two systems evolve in a nearly identical manner when the inertial field has amplitudes an order of magnitude smaller than oceanic values. At realistic (oceanic) amplitudes, though, the second moment estimated from the ray tracing simulations is inhibited. The transition is explained by the stochastic jumps obtaining the characteristic size of the resonant well. We interpret this transition as an adiabatic ‘saturation’ process which changes the nominal background wavefield from supporting no mixing to the point where that background wavefield defines the normalization for oceanic mixing models.
Highlights
One of the stories that gets told in Physical Oceanography is that, at the turn of the preceding century, there was one major unsolved problem in Physics, turbulence, and some niggling questions concerning an ultraviolet catastrophe
The ray tracing results contain, crudely, two processes: a resonant one with an interaction time scale giving rise to the ‘resonant well’ concept that is made apparent by the phase locking distributions, and off resonant, stochastic jumps at a shorter time scale which are apparent as differences in the first and second passage estimates of the first moment (Figure 4) and as approximate half-power points of the Lagrangian shear spectra
The decorrelation time scales are shorter than that associated with the resonant well; by itself this is too short to explain the inhibition of the second moment at oceanic amplitudes
Summary
One of the stories that gets told in Physical Oceanography is that, at the turn of the preceding century, there was one major unsolved problem in Physics, turbulence, and some niggling questions concerning an ultraviolet catastrophe. We present a necessarily brief summary of equations for the time evolution of the wave action spectrum, n( p), in which action is energy density e( p) divided by frequency σ ( p), with the intent of divining whether a breakdown in diffusion can be inferred from a self-consistent kinetic equation at finite amplitude Such equations are built around perturbations to plane wave solutions of ei(r · p−σt) having wavenumber p and Eulerian frequency σ related by the dispersion relation (4) and represent the coupled oscillator paradigm. Many wave systems (e.g., internal waves, Rossby waves, acoustic waves, capillary-gravity waves) have dispersion relations that permit triadic interactions, so our findings may well be generally applicable Interactions in this particular internal wave system are dominated by extreme scale separations such that the kinetic equation does not converge unless a lower bound in vertical wavenumber is inserted, which is accomplished by including rotation [9]
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