Oblique Interactions between Internal Solitary Waves

Abstract

In this paper, we study the oblique interaction of weakly, nonlinear, long internal gravity waves in both shallow and deep fluids. The interaction is classified as weak when where Δ 1 =| c m /c n −cosδ|, Δ 2 =| c n /c m −cosδ|,c m,n , are the linear, long wave speeds for waves with mode numbers m, n, δ is the angle between the respective propagation directions, and α measures the wave amplitude. In this case, each wave is governed by its own Kortweg‐de Vries (KdV) equation for a shallow fluid, or intermediate long‐wave (ILW) equation for a deep fluid, and the main effect of the interaction is an 0(α) phase shift. A strong interaction (I) occurs when Δ 1,2 are 0(α), and this case is governed by two coupled Kadomtsev‐Petviashvili (KP) equations for a shallow fluid, or two coupled two‐dimensional ILW equations for deep fluids. A strong interaction (II) occurs when Δ 1 is 0(α), and (or vice versa), and in this case, each wave is governed by its own KdV equation for a shallow fluid, or ILW equation for a deep fluid. The main effect of the interaction is that the phase shift associated with Δ 1 leads to a local distortion of the wave speed of the mode n . When the interacting waves belong to the same mode (i.e., m = n ) the general results simplify and we show that for a weak interaction the phase shift for obliquely interacting waves is always negative (positive) for (1/2+cosδ)>0(<0), while the interaction term always has the same polarity as the interacting waves.