Sub-harmonic mode locking of coupled oscillators with inertia
When baroclinic quasi-stationary waves share the output or input modulated currents, depending on it is a tropical or subtropical wave, a sub-harmonic mode locking occurs. In this case the periods of the quasi-stationary waves, which can be assimilated to coupled oscillators, are a multiple of the period of the fundamental wave, i.e. one year because the annual waves have the same period as trade winds that are the main driver of forcing of tropical and subtropical baroclinic quasi-stationary waves.
The coupling arises from the lateral eddy viscosity of seawater because of the speed of the resulting modulated current, which is the sum of the modulated currents of different frequencies, to the nodes of quasi-stationary waves. In this case, the sub-harmonic mode causes the speed of the resulting current, so the average energy dissipated in this dynamic system, is minimal: zero when the resulting current is zero, this energy does indeed increase with the current speed.
The main natural periods of tropical waves are 4 and 8 years, those of gyral waves coupled with solar cycles are 64 y (considered as a harmonic of 128 y), 2×64=128 y, 2×128=256 y, 3×256=768 y, finally those of gyral waves coupled with orbital cycles are 32×768=24,6 Ky, 2×24,6=49,2 Ky, 2×49,2=98,3 Ky and 4×98,3=393,2 Ky. The bands used to calculate the intensity of forcing frame these natural periods by fixing the lower limit to 0.75 x period and the upper limit to 1.5 x period.
Pinault JL (2017), Modulated response of sub-tropical gyres: positive feedback, sub-harmonic modes, resonant solar and orbital forcing, Ocean Dynamics, submitted