5 gyral Rossby waves of 1/2, 1, 4 and 8 year-periods are inherited from the tropical oceans. The average period of the yearly fundamental Rossby wave is inherited from easterlies that force quasi-stationary waves in tropical oceans, which is to say from the average period of rocking of the intertropical convergence zone. This results from the variation in warm water masses carried by the western boundary currents, which gives rise to the oscillation of the pycnocline depth at mid-latitudes of the gyres. Resulting westward-propagating baroclinic Rossby waves are formed, embedded in the wind-driven current.
1/2, 4 and 8 years period subharmonics are superimposed on the fundamental wave, as well as waves of longer period.
Gyral Rossby waves are almost in phase, whatever the period. The antinodes of the biannual waves peak in March-April or September-October, and at the same places that the resonances at lower frequencies. This simultaneity is confirmed by modulated currents whose speeds are highest in May-June or November-December when they are eastward, i.e. in phase opposition with the antinodes.
As regards the annual wave antinodes are highest in November-December near the coast, regardless of the gyre. Speeds of modulated currents facing east are highest in July-August near the coast.
The period of 8 years is most appropriate for discerning antinodes due to longer wavelengths. Gyral Rossby waves are almost in phase near the coast where the antinodes are highest in average when the phase is about -2.5 years (in comparison with the signal SOI). Although noisy, the phase of the modulated current shows that the speed is maximum when it is included between 1.8 and 3.6 years, almost in opposite phase with respect to the antinode.
Rossby waves being non-dispersive, for a given latitude their phase Velocity does not depend on frequency. In other words their wavelength is directly proportional to the period. Thus, for the period of 8 years, the wavelength is 2,780 km in latitude 40°, whereas it is only 174 km for the biannual wave.
Where the resonance occurs the cross-wavelet analysis of sea surface height reveals two antinodes in opposite phase (sometimes more), as occurs in the North Atlantic for the 8-year period Rossby wave. Off the Cape Hatteras, the Gulf Stream leaves the eastern North American coast around 35°N. The westernmost anomaly faces east, along the subtropical gyre, followed by the second anomaly headed north-east along the north Atlantic drift. The phase change occurs at the longitude 50°W. These antinodes are always associated with a modulated geostrophic current at the node of the quasi-stationary wave.
The most important consequence of the modulated response of subtropical gyres concerns the long-term variability of climate. The modulated response of subtropical gyres occurs indeed at very specific frequencies that are inherited either of tropical waves or of the oscillations of solar irradiance for longer periods. Direct observation of the modulated response of subtropical gyres can be done from sea surface height for short periods, as has been done for the North Atlantic gyre using the available data sets covering a period twenty years. For longer periods one is interested in sea surface temperature anomalies using data sets this time covering nearly a century and a half. In practice, using sea surface height series jointly with sea surface temperature series, quasi-stationary waves can be observed at frequencies ranging from 2 cycles per year to one cycle for 128 years.
The resonance occurs when the speed, facing east, of the steady wind-driven current is higher than the phase velocity, facing west, of the Rossby wave. In this case, the length of the Rossby wave adapts so that its natural period coincides with the forcing period. The ridge of the western antinode is advected of a half apparent wavelength (the wavelength seen by a stationary observer) to the eastern antinode during a half-cycle, the troughs are then translated in the next half-cycle. This leads to eastward transfer of a sequence of warm and cold waters.
Since the phase velocity of the Rossby waves only depends on latitude, the resonance that supposes the adequacy of the frequency and the wavelength occurs at all frequencies. Otherwise, the lack of synchronization between the waves of different frequencies and forcing would inexorably lead to their destruction.
The observation of quasi-stationary waves for different periods teaches us how the sequence of warm and cold waters is transferred from the western boundary to the east. This is particularly clear for the wave of 8-year period of the North Atlantic. A ridge is formed at the western antinode and a trough at the eastern antinode. The speed of the western modulated zonal current is maximum, facing west. The resulting modulated current, sum of the modulated geostrophic current and the steady wind-driven current, vanishes or reverses. At this time the speed of the eastern modulated current, facing east, is maximum. Thus warm water flows from the western antinode to the eastern antinode while the transfer from the western antinode to the western boundary is low due to geostrophic forces that prevent it.
Half a cycle later, a trough is formed at the western antinode and a ridge at the eastern antinode. The velocity of the resulting current at the western boundary is maximum, being the superposition of two eastward propagating currents. Gradually the trough of the western antinode gives way to a ridge while the speed of the resulting modulated current vanishes or reverses in the east, being the superposition of two currents flowing in opposite directions: this locks out any transfer from the eastern antinode to the western antinode. The same mode of transfer occurs for different frequencies of gyral Rossby waves due to the proportionality between the wavelength and the period.
Growth of the western antinode induces a horizontal pumping effect, as evidenced by the modulated current when it is oriented to the east, west of the basin, being the superposition of the steady wind-driven current and the modulated geostrophic current. This pumping effect drastically increases the flow of the western boundary current. Another consequence is the rapid change in potential vorticity of the western boundary current when it leaves the coast to flow eastward.
Another important aspect of the resonance relates to the simultaneity of the phases of western antinodes. Synchronism of resonances in the five subtropical gyres reflects a common mode of forcing. This is particularly evident for the resonance at the frequency of 2 cycles per year which occurs in April and October, an indicator in the three tropical oceans of forcing by trade wind stress, active during the boreal winter in the northern hemisphere and during the austral winter in the southern hemisphere. The transfer time of the mixed layer from the equator to mid-latitudes is short compared to the period of gyral Rossby waves. Thus, the synchronism persists at mid-latitudes regardless of the period. A final aspect of the resonance involves the coupling of gyral Rossby waves of different frequencies that share the same node at the western boundary of the basin, which again leads to a sub-harmonic mode locking in accordance with the periods 1/2, 1, 4 and 8 years (Pinault, 2018c).
How the gyral Rossby wave moves poleward after leaving the gyre reflects the thermohaline circulation which is generated by differences in temperature and salinity of water masses acting on their density. Thermohaline circulation occurs when the currents approaching the ice cap, water cooled and salty sinks to depths between 1 and 3 km to participate in the deep ocean circulation.
A standing wave is the phenomenon resulting from the simultaneous propagation in different directions of several waves of the same frequency. In a standing wave nodes remain fixed, alternating with antinodes. A quasi-stationary wave acts as a standing wave but the antinodes and nodes may overlap.
The fundamental quasi-stationary wave is in phase with forcing. In sound pipes, strings and vibrating membranes form harmonics whose period is a divisor of that of the fundamental wave. As regards the long ocean waves, sub-harmonics are formed whose period is a multiple of that of the fundamental wave as occurs for high rank baroclinic modes.
a homogeneous medium, propagation in a given direction of a monochromatic wave (or sine) results in a simple translation of the sinusoid at a speed called phase velocity or celerity. In a non-dispersive medium, the speed does not depend on the frequency (or wavelength). In this case every complex wave is the sum of several monochromatic waves that also undergo an overall translation of its profile, this without deformation. In contrast, in a dispersive medium the phase velocity depends on the frequency and the energy transported by the wave moves at a speed lower than the phase velocity, said group velocity.
Geostrophic currents are derived from measurements of wind, temperature and satellite altimetry. The calculation uses a quasi-stationary geostrophic model while incorporating a wind-driven component resulting from wind stress. Geostrophic current thus obtained is averaged over the first 30 meters of the ocean.