The functioning of long-period gyral waves is obtained by solving the equations of motion of the quasi-stationary baroclinic waves of very long wavelength in middle latitudes. The solution, that is to say the antinodes as well as the polar and radial modulated currents of gyral waves is obtained by applying the β-cone approximation (Pinault, 2016).

The efficiency of solar and orbital forcing results from the imbalance between incoming and outgoing fluxes through the surface of the ocean. In the absence of resonance of gyral waves the inflow is balanced mainly by evaporation. On the other hand lowering of the thermocline and acceleration of the convergent radial current while the intensity of forcing is increasing makes that the gyral wave behaves as a heat sink because of downwelling. Half a period later, uprising of the thermocline and acceleration of the divergent radial flow while forcing intensity is decreasing makes the gyral wave returns the accumulated heat: it then acts as a heat source due to upwelling. The imbalance between incoming and outgoing fluxes is all the higher as the oscillation of the thermocline is broader, reinforcing the **positive feedback** acting on the speed of the polar current while reducing the temperature gradient between low and high latitudes of the gyre.

A quarter period before the maximum of the solar irradiance, the polar current is anticyclonic. The heat flux transported poleward by the boundary current favors the deepening of the thermocline, therefore the acceleration of the polar current. The polar current speed reaches its maximum nearly at the same time as the maximum solar irradiance. This **positive feedback loop** considerably increases the effectiveness of solar and orbital forcing. Depending itself on the temperature gradient between low and high latitudes of the gyre it is strongly controlled by the extension of the polar caps.

When gyral resonance occurs, it takes precedence over the non-resonant phenomena. As happens for shorter wavelengths, resonance determines the geostrophic forces throughout the basin and occult non-resonant phenomena whose impact is less, this because of the competition between resonant and non-resonant phenomena throughout the gyre. Resonant phenomena absorb maximum solar energy as the wave and forcing are synchronized. Otherwise, during its evolution the wave is necessarily in phase opposition with forcing and therefore its amplitude is less. Since only one mode can remain around the gyre due to the long wavelength of gyral waves that enforce geostrophic forces on the basin scale, it is necessarily resonant.

Acceleration of the polar current of the gyral wave cools the tropical currents by reducing their residence time at low latitudes. But it follows an increase in radiative forcing, so global warming of the planet, due to the reduction in the temperature difference between low and high latitudes of the gyre. This has the effect of reducing the emitted radiative power which is proportional to the fourth power of the temperature according to the Stefan’s law. Conversely, slow-down of the polar current causes a cooling of the planet by increasing the temperature difference between high and low latitudes.

**Simulation of 128-year period gyral waves**

The physical foundations of gyral waves can only be established in solving the equations of motion. However these do not include the positive feedback loop of the oscillation of the thermocline and the speed of the polar current, which leads to underestimate the magnitude of the gyral wave: lowering the thermocline accelerates the polar current when it flows anti-cyclonically, so the western boundary current. It follows an increase in heat flow transported from the tropics towards the poles, which lowers further the thermocline.

In the three videos that follow, the antinodes of gyral waves are represented as polar and radial currents: the thermal energy is stored in the antinode of the gyral wave over a half apparent wavelength (the wavelength seen by a stationary observer).

As shown in the dynamic simulation of the gyral wave, its winding around the gyre corresponds to a half apparent wavelength. The following antinode, in phase opposition relative to the previous one, is outside the gyre and is poleward. In this way, antinodes and nodes are common for all gyral waves, whatever their period, both inside and outside the gyre.

The antinode, which is formed by two bulges on either side of the midline of the gyre remains almost uniform, while the polar current accelerates at high latitudes, due to the thinning of the mixed layer. The radial current has the same property. The antinode is in quadrature with respect to forcing, being a quarter period late. The polar current is in phase with the antinode as it flows cyclonically and the radial currents are in phase with forcing when they converge to the midline of the gyre.