Space-Time scales of internal waves

Abstract

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.