Dynamics of Benjamin–Ono Solitons in a Two-Layer Ocean with a Shear Flow

Abstract

The results of a theoretical study on Benjamin–Ono (BO) soliton evolution are presented in a simple model of a two-layer ocean with a shear flow and viscosity. The upper layer is assumed to move with a constant speed relative to the lower layer with a tangential discontinuity in the flow profile. It is shown that in the long-wave approximation, such a model can be appropriate. If the flow is supercritical, i.e., its speed (U) exceeds the speed of long linear waves (c1), then BO solitons experience “explosive-type” enhancement due to viscosity, such that their amplitudes increase to infinity in a finite time. In the subcritical regime, when Uc1, BO solitons experience very slow decay due to viscosity. Soliton amplitude decays with time as A∼t−1/2 or A∼t−1/3, depending on whether both layers are weakly viscous (the former case) or only the lower layer is viscous (the latter case). Estimates of "explosion time" are presented for real oceanic parameters.

Keywords

• ocean waves
• Benjamin–Ono equation
• soliton
• explosive instability
• stratified ocean
• viscous fluid