Meridional modes

If the vertical baroclinic modes are resulting from the stratification of the oceans, meridional (latitudinal) modes of Rossby waves reflect the effects of the variation of the Coriolis parameter ƒ. For this reason these modes result from what is called the dispersion β where β represents the variation of the Coriolis parameter with respect to the latitude. The Coriolis parameter or Coriolis frequency ƒ is defined from the speed of rotation on the surface of the terrestrial sphere, and increases with latitude. This parameter is zero at the equator and reaches its maximum at the poles.

There are different modes of propagation of Rossby waves, which differ in their latitudinal structure. This is why they are called meridional modes. For the first baroclinic mode, the phase velocity[i] of the first meridional mode equals c/3. In other words, the first meridional mode Rossby waves propagate three times slower than the Kelvin waves (and in the opposite direction). On the other hand the waveform is symmetrical about the propagating direction.

The phase velocity of the second meridional mode equals c/5 and the wave is anti-symmetric with respect to the direction of propagation. When the meridional mode increases the phase velocities decrease from c/7 to c/9… The waves whose mode is odd are symmetrical, those whose mode is even are anti-symmetric. The meridional structure becomes complex when the mode increases as the amplitude of the wave vanishes and changes sign periodically. Unlike baroclinic modes whose amplitude decreases with the mode, interfaces becoming shallower, meridional modes of high order may be of large amplitude when they are resonantly forced, as is the case in the tropical Pacific. In solving the equations of motion, the forcing terms indeed depend on the meridional mode considered. Wind stress, for example, does not act the same way on the first meridional mode, which is symmetrical about the equator, and the second meridional mode, which is anti-symmetric.

Glossary

[i] 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.

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