Numerical Modeling of the Internal Dispersive Shock Wave in the Ocean

Numerical modeling of dispersive shock waves called solibore in a stratified fluid is conducted. The theoretical model is based on extended version of the Korteweg-de Vries equation which takes into account the effects of cubic nonlinearity and Earth rotation. This model is now very popular in the physical oceanography. Initial conditions for simulations correspond to the real observed internal waves of shock-like shape in the Pechora Sea, the Arctic. It is shown that a sharp drop (like kink in the soliton theory) in the depth of the thermocline is conserved at a distance of one–three kilometers, and then it is transformed into dispersive shock waves (shock wave with undulations)

Internal solitary waves: propagation, deformation and disintegration

In coastal seas and straits, the interaction of barotropic tidal currents with the continental shelf, seamounts or sills is often observed to generate large-amplitude, horizontally propagating internal solitary waves. Typically these waves occur in regions of variable bottom topography, with the consequence that they are often modeled by nonlinear evolution equations of the Korteweg-de Vries type with variable coefficients. We shall review how these models are used to describe the propagation, deformation and disintegration of internal solitary waves as they propagate over the continental shelf and slope.

Vertical structure of the velocity field induced by mode-I and mode-II solitary waves in a stratified fluid

The structure of the velocity field induced by internal solitary waves of the first and second modes is determined. The contribution from second-order terms in asymptotic expansion into the horizontal velocity is estimated for the models of almost two- and three-layer fluid for solitons of positive and negative polarity. The influence of the nonlinear correction manifests itself firstly in the shape of the lines of zero horizontal velocity: they are curved and the shape depends on the soliton amplitude and polarity, while for the leading-order wave field they are horizontal. Also the wave field accounting for the nonlinear correction for mode I has smaller maximal absolute values of negative velocities (near-surface for the soliton of elevation, and near-bottom for the soliton of depression) and larger maximums of positive velocities. For solitary waves of negative polarity, which are the most typical for hydrological conditions in the ocean for low and middle latitudes, the situation is the opposite. The velocity field of the mode-II soliton in a smoothed two-layer fluid reaches its maximal absolute values in a middle layer instead of near-bottom and near-surface maximums for mode-I solitons

Nonlinear Models of Finite Amplitude Interfacial Waves in Shallow Two-Layer Fluid

We present an example of systematic use of asymptotic methods for the description of long waves at the interface between two fluids of different densities. The governing equations for weakly nonlinear long interfacial waves are consistently derived for a general case of nonpotential flow and taking into account surface tension between two layers. Particular attention is given to the situations when some terms in the resulting fifth-order KdV-type evolution equation become small, vanish, or change their sign

The impact of seasonal changes in stratification on the dynamics of internal waves in the Sea of Okhotsk

The properties and dynamics of internal waves in the ocean crucially depend on the vertical structure of water masses. We present detailed analysis of the impact of spatial and seasonal variations in the density-driven stratification in the Sea of Okhotsk on the properties of the classic kinematic and nonlinear parameters of internal waves in this water body. The resulting maps of the phase speed of long internal waves and coefficients at various terms of the underlying Gardner’s equation make it possible to rapidly determine the main properties of internal solitary waves in the region and to choose an adequate set of parameters of the relevant numerical models. It is shown that the phase speed of long internal waves almost does not depend on the particular season. The coefficient at the quadratic term of the underlying evolution equation is predominantly negative in summer and winter and therefore internal solitons usually have negative polarity. Numerical simulations of the formation of internal solitons and solibores indicate that seasonal variations in the coefficient at the cubic term of Gardner’s equation lead to substantial variations in the shape of solibores