Higher-order resonant instabilities of internal gravity waves

We investigate the three-dimensional characteristics of a general class of resonant temporal instabilities of internal gravity waves, in which the disturbance comprises two infinitesimal internal gravity waves satisfying the conditions w(I)+w(II) = n (w) over tilde, k(I)+k(II) = n (k) over tilde, with growth rates of order N(\(u) over tilde\\(k) over tilde\/N)(n), assuming small dimensional primary-wave velocity amplitude \(u) over tilde\. We derive simple equations for their wave-numbers, frequency, growth rate, and energy budget. Interactions involving more than two disturbance components are shown not to represent distinct families of solutions, but rather to comprise order transitions linking together two or more 'two-disturbance-component' solutions of different order n. Unlike n = 1 resonant instabilities, those with n greater than or equal to 3 can align with the wave shear flow. We calculate the peak growth rate, spanwise wavenumber, and energy budget of shear-aligned resonance, as functions of wave frequency; they extract energy from both the wave shear and buoyancy fields.