Nonlinear harmonic generation by internal waves in a density staircase

Recent work has explored linear internal wave transmission and reflection across a density staircase. Here, weakly nonlinear theory is used to extend previous linear internal wave solutions to include nonlinear effects at density interfaces within the staircase. Near-resonant forcing of freely propagating modes of the staircase by weakly nonlinear incident internal waves is shown to occur for some wavelength-frequency combinations. In some cases, this results in the staircase radiating double-frequency, double-wave-number harmonic waves. In others, incident waves excite one or more interfacial waves which propagate along the density interfaces within the staircase. This nonlinear effect is expected to be significant for internal waves of realistic amplitude incident on oceanic thermohaline staircases.

Figure 1

Example stratification profile (8) and linear solution (10) for $J=3$.

Figure 2

Stream function profiles for the steady-state harmonic perturbation $\delta {\psi }_s\left(z\right)$ (solid lines) and primary (linear) wave ${\psi }_0\left(z\right)$ (dotted lines) in a staircase with $J=4$ and incident wave amplitude $|A_{\mathrm{inc}}|=0.05N_0{L}^2$. The harmonic perturbation is trapped in the staircase for this incident frequency $\left(\omega =0.6N_0\right)$. There is no rotation $\left(f=0\right)$.

Figure 3

Stream function profiles for the steady-state harmonic perturbation $\delta {\psi }_s\left(z\right)$ (solid lines) and linear solution ${\psi }_0\left(z\right)$ (dotted lines) in a staircase with $J=4$ and incident wave amplitude $|A_{\mathrm{inc}}|=0.05N_0{L}^2$. The harmonic perturbation radiates away from the staircase for this incident frequency $\left(\omega =0.3N_0\right)$. There is no rotation $\left(f=0\right)$.

Figure 4

Dimensionless steady-state harmonic amplitude at the staircase top $\left(\omega |\delta A_0|/|k{A}_{\mathrm{inc}}{|}^2\right)$ on a logarithmic (base 10) scale as a function of incident wave frequency $\omega /N_0$ and wave number $kL$ for staircases with one, two, four, and ten steps. The dashed black line shoes the approximate linear wave transmission boundary (reflection above, transmission below), while the dotted black line shows the pycnocline interfacial harmonic resonance criteria from [27].

Figure 5

Dimensionless steady-state staircase kinetic energy ratio $R_K$ as a function of incident wave frequency $\omega /N_0$ and wave number $kL$ for staircases with one, two, four, and ten steps. The dashed black line shoes the approximate linear wave transmission boundary (reflection above, transmission below), while the dotted black line shows the pycnocline interfacial harmonic resonance criteria from [27].

Figure 6

Dimensionless steady-state harmonic amplitude at the staircase top $\left(\omega |\delta A_0|/|k{A}_{\mathrm{inc}}{|}^2\right)$ on a logarithmic (base 10) scale as a function of incident wave frequency $\omega /N_0$ and wave number $kL$ for four different rotation frequencies $f/N_0$ in a staircase with $J=5$ steps.