Space-Time scales of internal waves

We have contrived a model E(αω) α μ−1ω−p+1(ω 2−ω i 2)−+ for the distribution of internal wave energy in horizontal wavenumber, frequency-space, with wavenumber α extending to some upper limit μ(ω) α ω r-1 (ω 2−ω i 2)½, and frequency ω extending from the inertial frequency ω i to the local Väisälä frequency n(y). The spectrum is portrayed as an equivalent continuum to which the modal structure (if it exists) is not vital. We assume horizontal isotropy, E(α, ω) = 2παE(α1, α2, ω), with α1, α2 designating components of α. Certain moments of E(α1, α2, ω) can be derived from observations. (i) Moored (or freely floating) devices measuring horizontal current u(t), vertical displacement η(t),…, yield the frequency spectra F (u,η,…)(ω) = ∫∫ (U 2, Z 2,…)E(α1, ∞2, ω) dα1 dα2, where U, Z,… are the appropriate wave functions. (ii) Similarly towed measurements give the wavenumber spectrum F (…)(α1) = ∫∫… dα2 dω. (iii) Moored measurements horizontally separated by X yield the coherence spectrum R(X, ω) which is related to the horizontal cosine transform ∫∫ E(α1, α2 ω) cos α1 Xdα1 dα1. (iv) Moored measurements vertically separated by Y yield R(Y, ω) and (v) towed measurements vertically separated yield R(Y, α1), and these are related to similar vertical Fourier transforms. Away from inertial frequencies, our model E(α, ω) α ω −p-r for α ≦ μ ω ω r, yields F(ω) ∞ ω −p, F(α1) ∞ α1 −q, with q = (p + r − 1)/r. The observed moored and towed spectra suggest p and q between 5/3 and 2, yielding r between 2/3 and 3/2, inconsistent with a value of r = 2 derived from Webster's measurements of moored vertical coherence. We ascribe Webster's result to the oceanic fine-structure. Our choice (p, q, r) = (2, 2, 1) is then not inconsistent with existing evidence. The spectrum is E(∞, ω) ∞ ω −1(ω 2−ω i 2 −1, and the α-bandwith μ ∞ (ω 2−ω i 2)+ is equivalent to about 20 modes. Finally, we consider the frequency-of-encounter spectra F(sgrave) at any towing speed S, approaching F(ω) as S ≦ S o, and F(α1) for α1 = sgrave/S as S ≧ S o, where S o = 0(1 km/h) is the relevant Doppler velocity scale.