OBLIQUE INTERACTIONS BETWEEN INTERNAL SOLITARY WAVES

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 DELTA1,2 much greater than alpha where DELTA1 = \c(m)/c(n) - cos delta\, DELTA2 = \c(n)/c(m) - cos delta\, c(m,n) are the linear, long wave speeds for waves with mode numbers m, n, delta is the angle between the respective propagation directions, and alpha 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(alpha) phase shift. A strong interaction (I) occurs when DELTA1,2 are 0(alpha), 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 DELTA1 is 0(alpha), and DELTA2 much greater than a (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 DELTA1 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 (\1 - cos delta\ much greater than alpha) the phase shift for obliquely interacting waves always negative (positive) for (1/2 + cos delta) > 0(< 0), while the interaction term always has the same polarity as the interacting waves.