Resonant and Forced Internal Waves

The periodicity inherent in weather patterns creates over-lake wind fields that occur at regular frequencies. For example, the winds over Lake Erie have c c

Figure 14 Schematic showing the reflection of internal wave rays from a uniform slope. The lines indicate crests and troughs (lines of constant phase) and cgI and cgR the direction of the incident and reflected group velocity, respectively; cb is the direction of phase propagation on the slope. (a-b) show the supercritical case where a <b and rays continue upslope or downslope. (c-d) show the subctitical case a > b where rays are reflected out to deeper water. The distance between rays is proportional to the wavelength and can increase or decrease upon reflection. As b ! a, the reflected rays become parallel to the slope and their wavelength goes to zero. Adapted from Thorpe SA and Umlauf L (2002) Internal gravity wave frequencies and wavenumbers from single point measurements over a slope. Journal of Marine Research 60: 690-723.

Figure 14 Schematic showing the reflection of internal wave rays from a uniform slope. The lines indicate crests and troughs (lines of constant phase) and cgI and cgR the direction of the incident and reflected group velocity, respectively; cb is the direction of phase propagation on the slope. (a-b) show the supercritical case where a <b and rays continue upslope or downslope. (c-d) show the subctitical case a > b where rays are reflected out to deeper water. The distance between rays is proportional to the wavelength and can increase or decrease upon reflection. As b ! a, the reflected rays become parallel to the slope and their wavelength goes to zero. Adapted from Thorpe SA and Umlauf L (2002) Internal gravity wave frequencies and wavenumbers from single point measurements over a slope. Journal of Marine Research 60: 690-723.

periodicities of 10 d and 24 h, associated with frontal weather systems and diurnal land/sea breeze phenomena, respectively. When the forcing frequency matches one of the natural frequencies of the basin-scale wave modes, resonant amplification will occur. Seasonal heating and deepening of the thermocline will adjust the natural frequencies relative to the forcing frequencies, thus tuning the system into and out of resonance.

Laboratory experiments show that the relative frequencies of the wind forcing fw and H1 internal seiche fH1 can be used to model internal wave response to periodic forcing conditions (Figure 16). When the forcing frequency of the wind stress is less than the natural frequency of the H1 internal seiche (fw < ~ 2/3fH 1), the phase of the basin-scale oscillations are reset with each wind event and a forced internal seiche is generated at the same frequency as the wind forcing. A resonant H1 seiche will occur when the frequency of the wind forcing is near the frequency of the H1 seiche (2/3fH 1 < fw < 2fH 1) and when the forcing frequency is greater than the natural frequency of the H1 seiche (fw >~ 2fH 1), higher-mode horizontal seiches are generated. Resonance appears to be particularly effective in amplifying the response of the second vertical mode (e.g., Lake Alpnach) and the even horizontal modes that are not naturally energized by a wind-induced interfacial setup.

Resonance between the wind and the H1 internal mode leads to increased seiche amplitudes under relatively weak wind forcing conditions. For example, when W—1 > 0.03, nonlinearities become significant (compared to W—1 > 0.3 under non-resonant conditions), favoring the formation of steepened internal wave fronts (e.g., Loch Ness) and progressive NLIWs. If the wind induced tilting increases, W—1 > 0.2 will cause Kelvin-Helmholtz instabilities to form within the progressive NLIWs, leading to significant diapyc-nal mixing within the basin interior. The energy content of the H1 seiche, NLIWs and shear instability modes thus appear to be capped and continued energy input via resonant amplification is transferred between these discrete modes, ultimately being lost to dissipation and mixing at turbulent scales.

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