**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.

Subharmonic modes of RFWs can be deduced from the equation of the Caldirola-Kanai oscillator, which is a fundamental model of dissipative systems for the damped harmonic oscillator. The Hamiltonian (the energy) of a CK oscillator being time-dependent, subharmonic modes appear when the oscillatory system is periodic, each oscillator transferring as much interaction energy to all the others that it receives during a period. This is a required condition to ensure the durability of the dynamic system in stationary state.

The main natural periods of tropical waves are 4 and 8 years, those of gyral Rossby 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 Rossby 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