The tropical oceans are subject to the resonance of Rossby and Kelvin waves.
Contents
The quasi-stationary Rossby and Kelvin waves
The dynamic representation of the nodes and antinodes of the tropical quasi-stationary wave allows distinguishing (Pinault, 2013):
- Three antinodes of very unequal amplitude, two antinodes on either side of the equator and one antinode along the equator. The northern antinode has the highest amplitude, around 10 cm in November. It extends from the Brazilian coast to the west coast of Africa that it follows northwards. The equatorial antinode, which extends from South America to the West African coast, forms a ridge in January. The antinode south of the equator, barely perceptible, is divided into two branches: extending from the coast of South America, it forms a ridge in April.
- Two main nodes, both coinciding with the western part of antinodes, to the north and along the equator. The modulated current located north of the equator flows mainly to the west, its velocity reaching 30 cm/s in October-November (in February in the northern portion). It straddles the North Equatorial Current and the North Equatorial Counter-Current. The maximum velocity of the modulated equatorial current is, on the other hand, reached in May.
Evolution of the quasi-stationary wave
Following the geostrophic forces acting in the tropical basin, which result from the conjugate effects of rotation and gravity, Rossby waves are formed both north and south of the equator. Under the effect of wind stress these waves partially reflect against the eastern coast of South America to join the equatorial wave. A part leaves the tropical basin to merge with the western boundary currents flowing north and south. The other part forms the Kelvin wave trapped by the equator which propagates eastward to the coast of Africa, producing a coastal Kelvin wave. It propagates mainly southward from the Gulf of Guinea. Under the effect of recession, some of these Kelvin waves are reflected against the African coast to form the equatorial Rossby wave which, trapped by the equator and stimulated by trade winds, propagates westward to the south American coast where it is diverted to the north along the North Equatorial Counter-Current. This is due to the Doppler Effect: Rossby waves propagate apparently eastward when the counter-current that drags them is faster than their phase velocity.
Thus, geostrophic forces resulting from antinodes control the motion of Rossby and Kelvin waves, allowing their reflection against the eastern and western limits of the basin, or otherwise promoting their leaving as this occurs when the northern Rossby wave flows westward. In order that the resonance occurs, the average period of the complete cycle must coincide with that of forcing, i.e. one year. The adjustment of the basin to forcing results from the northern wave which owes its existence to the North Atlantic Counter-Current. It actually plays the role of a « tuning slide » so that the sea surface height of the tropical basin adjusts to allow the forced wave to achieve its cycle in an average time of one year exactly.
The evolution of tropical waves is subject to resonant forcing due to trade winds. The entire tropical basin adjusts to resonate, allowing it to capture the maximum energy. This resonant basin mode outweighs the non-resonant modes that are not synchronized with forcing. In this case the waves are damped very quickly as inevitably opposed to forcing during their evolution. Under these conditions, for the Atlantic Ocean the Rossby wavelength obtained from the dispersion relation is 24,700 km at the equator for the period of one year, which corresponds to the cycle of winds. It is 12,350 km for the Kelvin wave, faster, whose period is 2 months.
Resonant forcing, which involves the transfer of warm water between the two hemispheres, takes advantage of trade wind rocking from an hemisphere to another as does the inter-tropical convergence zone (ITCZ), what sailors call « doldrums ». The inter-tropical convergence zone is a belt of low pressure areas around the Earth, close to the equator. Its location oscillates on both sides of the equator from one hemisphere to the other according to an annual pace, following the declination of the Sun. During the austral winter the ITCZ migrates to the northern hemisphere while trade winds blow in the southern hemisphere, forcing the southern annual wave. During the boreal winter the ITCZ migrates to the southern hemisphere when trade winds blow in the northern hemisphere, forcing the northern wave.
Seasonal upwelling in the Gulf of Guinea is impeded during the eastward phase propagation of the resonant equatorial wave whereas it is stimulated during the westward phase propagation, whence cold water replaces warm water in the mixed layer. In boreal winter, the ridge is formed along the equator while the northern anomaly deepens, cold water gradually replacing the warm water that has just left the tropical basin to supply the western boundary currents. At the end of the boreal winter the northern antinode forms a trough, which promotes migration to the northern hemisphere of warm water accumulated during the austral summer in the southern hemisphere. This is due to the sea surface height of the tropical ocean: geostrophic currents flow along the steepest lines (from positive to negative antinodes). Furthermore, the equatorial Rossby wave is deflected by the western boundary of the basin, merging with the North Atlantic counter-current. Six months later the northern antinode is reversed to form a ridge. This ridge, which is associated with the deepening of the thermocline, goes with the recession of the wave during which the western boundary currents are fed with warm water, and the cycle can start again…
During a cycle, the warm waters migrate from the southern antinode to the northern antinode via the equatorial antinode. At each step the volume of warm water increases by cumulating to that already in place, in formation, which is displayed by the amplitude of antinodes. These warm waters leave the northern antinode to join the two western boundary currents that are the Gulf Stream north and the North Brazil Current south, with an annual periodicity.
Glossary
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.
Upwelling. Here, this term indicates an upward flow of deep water, therefore cold. The upwelling is associated with the functioning of tropical resonant waves.
In 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.
The dispersion relation, connecting the pulsation (or frequency) of a free wave (unconstrained) ω = 2π/T to its wavelength, takes a very simple form when the waves are non-dispersive, as is the case of the Kelvin and Rossby waves of long wavelength. In the first case, ω/k = c where k is the wave number (reciprocal wavelength) and in the second case ω/k = -c/(n+1), the sign – indicating that the wave propagates westward. c is the phase velocity of the first baroclinic mode, n is the rank (order) of the meridional mode.
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.