The functioning of long-period gyral Rossby waves is obtained by solving the equations of motion of the baroclinic quasi-stationary 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 Rossby waves is obtained by applying the β-cone approximation (Pinault, 2016).
- Solar and orbital forcing
- Simulation of 128-year period gyral Rossby waves
- Modulated response of sub-tropical gyres
- Change of vorticity
- Stability of gyral Rossby waves
- Sub-harmonic mode locking of coupled oscillators with inertia
- The resonance of long-period gyral Rossby waves
- Sea surface temperature anomalies during the warming and cooling phases
- Summary: properties of gyral Rossby waves
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 Rossby 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 Rossby 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 Rossby 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 larger, 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 efficiency 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 Rossby waves that enforce geostrophic forces on the basin scale, it is necessarily resonant.
Acceleration of the polar current of the gyral Rossby 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.
The physical foundations of gyral Rossby 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 Rossby 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 Rossby waves are represented as polar and radial currents: the thermal energy is stored in the antinode of the gyral Rossby wave over a half apparent wavelength (the wavelength seen by a stationary observer).
As shown in the dynamic simulation of the gyral Rossby 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 Rossby 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.
Few studies have examined the modulated response of subtropical gyres. It’s that the subtropical gyres are generally considered in a steady state, with western intensification, in conformity with the theories of Sverdrup (1947), Stommel (1948) and Munk (1950). According to this theory, the subtropical gyres result from the wind-driven recirculation (accelerated by Coriolis force alone), westerly winds at mid-latitudes and the trade winds at low latitudes, i.e. they only owe their existence to wind-driven currents. The center of a subtropical gyre is indeed a high pressure area. The circulation around the high pressure occurs clockwise in the northern hemisphere and counter-clockwise in the southern hemisphere, due to the Coriolis force. Upwelling in the center of the gyre creates a flow toward the equator, then poleward when it merges with the western boundary current (see Ekman transport).
However the observations of short-period Rossby waves from where the western boundary currents leave the continents to enter the gyre, which are part of the Meridional Overturning Circulation, as well as long-period Rossby waves wrapping around the gyre and resulting from the variation in the Coriolis effect with the mean radius of the gyre suggest a key role of modulated responses of the gyres in the general ocean circulation (Pinault, 2018d).
Subtropical gyres respond to varying forcing effects that are of two types according to the period with which they alter the Sverdrup balance. Short period forcing effects result from the variation in warm water masses carried by the western boundary currents, which gives rise to the oscillation of the thermocline at mid-latitudes of the gyres. Resulting westward-propagating baroclinic Rossby waves are formed, dragged by the wind-driven current. Those forcing effects are inherited from the tropical oceans. Long period forcing effects result from solar and orbital cycles. Thermal perturbations, however very weak, are strongly amplified owing to the positive feedback on the baroclinic Rossby waves.
The speed of the steady anti-cyclonically wind-driven circulation being higher than the phase velocity of cyclonically propagating Rossby waves, amplified forcing effects occur, caused by a positive feedback on the baroclinic waves: warming of surface water at mid-latitudes deepens the thermocline around the gyre, which accelerates the modulated geostrophic current – this current, which results from the rotation of the earth and the forces of gravity, is proportional and in phase with the thermocline oscillation – thus increasing the poleward heat advection by the western boundary current. So, the thermocline deepens more, hence a positive feedback loop.
Several gyral Rossby waves of different periods may cohabit, which give subtropical gyres a remarkable property that distinguishes them from smaller vortices. Sharing the same modulated current where the western boundary currents leave the continents to re-enter the interior flow of the subtropical gyres, these gyral Rossby waves are coupled. Consequently, as occurs in the general case of coupled oscillators with inertia they are subject to a subharmonic mode locking giving the dynamic system an optimal stability (Pinault, 2018c). Resonance phenomena may then occur when the period of long-wavelength gyral Rossby waves is close to the period of solar and orbital cycles.
The periods of gyral Rossby waves are integer numbers of years which are deduced by recurrence (see Properties of Rossby gyral waves). To the long periods corresponds an integer number of turns made by the gyral Rossby wave around the gyre (anticyclonically) during half a period. This number of turns is the subharmonic mode. For the 128 year period the gyral Rossby wave travels 2 turns except in the South Pacific where it is 1 and the south of the Indian Ocean where it is 3/2.
Thus, modulated response of subtropical gyres raises considerable interest since it opens up a new field to be investigated: long-period baroclinic waves have a key role in the formation and stability of the subtropical gyres, in total volume transport and abrupt change in potential vorticity of western boundary currents, medium and long-term climate variability (see Holocene, Glacial-interglacial era), as well as the global temperature Tmg decomposed into a ‘natural’ and anthropogenic component.
From a conceptual point of view the modulated response of subtropical gyres is of considerable importance because it helps explain the abrupt change in the vorticity of the western boundary current off the coasts (Pinault, 2018d). This phenomenon, which will be confirmed for longer periods, allows to solve a mystery that has persisted since the work of the Norwegian oceanographer and meteorologist Harald Ulrik Sverdrup (1888-1957). Sverdrup has been appointed director of the prestigious Scripps Institution of Oceanography (SIO) in California from 1936 to 1948. During this period he developed his theory of the circulation of the oceans. As we will see, this theory based on the formation of steady surface currents driven by wind stress only partially explains the functioning of the subtropical gyres. In particular, it does not apply to the change in vorticity of western boundary currents when they merge with the subtropical gyres or, conversely, when they deviate to migrate poleward as do the drift currents in the northern hemisphere, or the circumpolar currents in the southern hemisphere. Indeed, any reasoning implying a purely inertial behavior (accelerated by the Coriolis force alone) of the gyre is incomplete because it must involve the modulated response of subtropical gyres and the associated geostrophic forces (combination of Coriolis force and gravity) around the gyres.
Sverdrup (1947) and Stommel (1948) theories do not explain more the total transport of western boundary currents, which can be understood only by considering the modulated response of subtropical gyres. Whether the change in vorticity or the magnitude of the flow of western boundary currents, attempts at explanation have been given since the pioneering work of Sverdrup, but unconvincingly.
The gyral resonance can occur only if the steady wind-driven current speed (anticyclonic) is higher than the phase velocity of the Rossby Rossby wave (cyclonic). As concerns the short-period gyral waves, the length of Rossby waves adapts so that their natural period coincides with the forcing period. The problem of stability is posed for long-period gyral Rossby waves for which a sub-harmonic mode locking occurs (Pinault, 2018c). Their natural period being thus imposed, generally it does not coincide with the forcing period.
Phenomena of regulation, mainly involving the average latitude of the gyre whose role in the tuning between the Rossby wave and forcing is essential (a decrease in the average latitude of 0.1°, i.e. 11.1 km, generates an increase in the circumference of the gyre of the North Atlantic of 122 km), maintain the conditions of resonance despite the wind changes (whose impact on the wind-driven movement is weak owing to the long-periods involved) but mostly changes in the period of the oscillations of solar irradiance.
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 (Pinault, 2018c). 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.
For periods exceeding 8 years, the length of resonantly forced Rossby waves exceeds the width of the oceans at mid-latitudes, so that the long-period waves necessarily develop around the subtropical gyres. The gyre of the North Atlantic allows the estimation of anomalies for periods extending up to 128 years relatively accurately because sea surface temperature measurements were already performed in 1870 in a systematic way, which is not true for the other gyres.
Like for short periods the baroclinic Rossby wave follows the subtropical gyre from the western boundary of the basin while changing the potential vorticity of the western boundary current to allow it to enter the gyre. Again, the gyral resonance of Rossby waves requires the speed of the steady wind-driven current of the gyre, which is anticyclonic, be higher than the phase velocity of the Rossby wave, which itself is cyclonic. The latter remains constant around the gyre as only depending on the mean latitude of the gyre.
The modulated geostrophic current is non-divergent. This can be ascertained by observing that the current lines remain substantially parallel around the gyre. Multiple turns may overlap, which implies that the Rossby wavelength has not upper limit. In other words, first baroclinic mode, first radial mode Rossby waves of long-period can resonate at mid-latitudes, tuning to long-period solar cycles.
Suppose that the number of coils corresponding to a half apparent wavelength (the wavelength seen by a stationary observer) is N. Within a period a warming phase occurs during which warm water is accumulated along the overlapping turns, followed by a cooling phase during which the warm water leaves the gyre. The gyral resonance can occur indeed only if an antinode develops outside the gyre, in phase opposition with the antinode around the gyre, as this happens for short periods. It follows that, to resonate, the gyral Rossby wave must be such that an integer number N of turns corresponds to a half apparent wavelength. For the North Atlantic gyre, N = 2 for the period of 128 years, which allows the excitation of the harmonic of 64 yrs period (a single winding).
In the course of its evolution the gyre is subject to radial transformations. During the warming phase the two edges of the gyre converge towards the median current line, while the Rossby wave is retained around the gyre. In contrast, during the cooling phase the movement reverses when the wave leaves the gyre.
In this way the resonance of Rossby waves of long-period is similar to that of 4 or 8-year period for which antinodes are separated by a half-wavelength, because of the adequacy between the length of Rossby wave and the period. Gyral Rossby waves sharing the same node where the western boundary current leaves the coast to merge with the subtropical gyre, a sub-harmonic mode locking occurs, so that the average periods of the coupled waves are multiple of short periods (Pinault, 2018c).
Referring to thermal anomalies of ocean surface, ocean-atmosphere exchanges mainly result from the latent heat flux. The impact on climate of these sea surface temperature anomalies, which either stimulate or, on the contrary, reduce evaporation, is substantial because they generate baroclinic instabilities that may lead to the formation of cyclonic or, on the contrary, anticyclonic systems of the atmosphere.
The direct impact of variations in solar irradiance on the sea surface temperature would be low if the gyral Rossby waves did not come into resonance. In this case, the heat budget would be balanced, i.e. the input and output heat fluxes through the surface of the ocean would be equal (as a first approximation, if we ignore fluxes carried by ocean currents), and in the absence of sea surface temperature anomaly, the forcing efficiency would be of the order of 0.1 °C(W/m2)-1. But it is much higher, around 1.0 °C(W/m2)-1 in the conditions that have been prevailing for the last few thousand years.
As shown in the dynamic representation of the North Atlantic, surface temperature anomalies observed in the band 96-144 years are indicative of an imbalance between incoming and outgoing fluxes through the surface of the ocean. According to the equations of motion the oscillation of the thermocline of the baroclinic wave is in quadrature with respect to forcing. Lowering the thermocline accelerates the western boundary current which thereby reduces the temperature gradient between low and high latitudes. In turn, the increased heat flux from the equator to the poles tends to further lower the thermocline. This positive feedback induces a phenomenon of amplification of the oscillation of the thermocline. Acceleration of the polar current stimulates upwelling off the eastern boundary of the basin where the current lines tighten, i.e. the Canary Current in the North Atlantic, the Benguela Current in the South Atlantic, the West Australian Current in the South Indian Ocean, the California Current in the North Pacific, and the Peru (Humboldt) Current in the South Pacific, by vertical pumping effect without changing vorticity significantly. Cooling the polar current compensates warming the western boundary current due to acceleration, which prevents from runaway effect resulting from the positive feedback.
Due to these effects the observed thermal anomaly may be delayed relative to forcing, which occurs in the northern and southern part of the gyre. Besides, the surface temperature anomaly out the gyre, which reaches 0.10°C, is in phase opposition with respect to the anomaly around the gyre.
The functioning of long-period gyral Rossby waves is derived from the equations of motion.
The gyral resonance helps explain precisely climate variability at different time scales. The vagaries of climate find indeed a convincing explanation from the properties of gyral Rossby waves of very long wavelength, resonantly forced from variations in solar irradiance.
To resume the main properties of gyral Rossby waves:
- Cyclonic non-dispersive gyral Rossby waves resonantly forced by solar and orbital variations with zero wind stress require their phase velocity is lower than the speed of the anticyclonic wind-driven current in which they are embedded, at the critical latitude where the western boundary current leaves the continent to enter the subtropical gyre. The speeds of the modulated geostrophic polar and radial currents are proportional to h, that is, the magnitude of the oscillation of the thermocline that results from the first baroclinic mode, first radial mode Rossby waves around the gyres.
- The oscillation of the thermocline is subject to a positive feedback loop in which the effects of a small disturbance induces an increase in the magnitude of the oscillation. This is because the acceleration of the polar current enhances the warming of the western boundary current that transfers more rapidly warm water from low to high latitudes. Thus, an amplification occurs, limited by the ability of sea water to warm up at low latitudes and by cooling resulting from upwelling along the eastern boundary currents of the gyres.
- Multi-frequency gyral Rossby waves are superimposed. Sharing the same polar currents they are coupled. Consequently, a subharmonic mode locking occurs, which means that the average periods of gyral Rossby waves are deduced by recurrence, the period of order n being a multiple of the period of order n-1 (Pinault, 2018c). Therefore, gyral Rossby waves are characterized by a number of revolutions, that is, their subharmonic mode, which corresponds to half their apparent wavelength.
- When the period of gyral Rossby waves increases, the Rayleigh friction is offset by the increase in the duration of heating of the mixed layer so that the amplitude of gyral Rossby waves does not vanish. This remarkable property enables the gyral Rossby waves to be tuned to forcing over very long periods as is the case for orbital variations of the Earth, without mitigation.
- The efficiency of the positive feedback loop is enhanced as the temperature lowers at high latitudes of the gyre, which strongly depends on the location of the front of the polar pack.
- Resonant forcing of gyral Rossby waves requires the natural frequencies have the ability to be finely tuned to the frequencies of solar and orbital forcing by shifting the centroid of the gyres along a meridian. Several natural periods of gyral Rossby waves may be tuned to different forcing periods when they are sufficiently close.
Thus, the interest aroused by the modulated responses of subtropical gyres (Pinault, 2018d), which has been ignored so far while being supported both by the observations and the theory, is promising in physical oceanography (the total volume transport of western boundary currents, their abrupt change in potential vorticity at critical latitudes, the global ocean circulation) and long and very long-term climate variability while complementing the current theories.
Munk, W. H. On the wind-driven ocean circulation, J. Meteorol., Vol. 7,1950
Stommel, H., The westward intensification of wind-driven ocean currents, Trans. Amer. Geophys. Union, 1948, 29, 202-206.
Sverdrup, H.U., Wind-Driven Currents in a Baroclinic Ocean; with Application to the Equatorial Currents of the Eastern Pacific, Proc. Natl. Acad. Sci., 1947, 33, 318-326.
Pinault, J.-L. Resonantly Forced Baroclinic Waves in the Oceans: Subharmonic Modes, J. Mar. Sci. Eng. 2018, 6, 78; doi:10.3390/jmse6030078
1) The Hadley area that lies between the equator and 30 degrees N and S where there are regular winds blowing from the northeast in the northern hemisphere and south-east in the southern hemisphere: the trade winds
2) mid-latitudes are characterized by transient low pressure systems in circulation in altitude generally from west, it is the Ferrel cell
3) polar cells are found respectively north and south of parallel 60-th north and south with a surface circulation generally from east
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 wind-driven current results from wind stress, i.e. from the Ekman Transport. It was formulated in 1902 by the Swedish oceanographer Vagn Walfrid Ekman (1874-1954) after he observed with Fridtjof Nansen the icebergs do not drift with the wind but at an angle of 20°-40° thereof. The Ekman transport moves layers of surface waters horizontally. But the Coriolis force deflects the movement to the right in the northern hemisphere and to the left in the southern hemisphere. This movement propagates downward due to viscosity and material is conveyed in a direction different from the axis of the wind. According to the path of winds, there is divergence or convergence of material, which creates two situations, pumping and ventilation.
The Ekman pumping is the upward transport of seawater as a result of depression. Under the action of wind, water of the mixed layer is set in motion and deflected by Coriolis force outwardly of the depression. This creates divergence. In contrast, in a high pressure, the Ekman transport occurs to the center of the system, creating convergence and transport of material downwardly.
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.
Baroclinic wave. In contrast with barotropic waves that move parallel to isotherms, baroclinic Rossby or Kelvin waves cause a vertical displacement of the thermocline, often of the order of several tens of meters. The seconds are usually slower than the first.
The Coriolis parameter f is equal to twice the speed of rotation Ω of the earth multiplied by the sine of the latitude φ: f = 2Ωsin φ. The Coriolis force, on the other hand, is perpendicular to the direction of movement of the moving body. It is proportional to the velocity of the body and the speed of rotation of the medium.
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.
Positive feedback loops amplify changes in a dynamical system; this tends to move the system away from its equilibrium state and make it more unstable. Negative feedbacks tend to dampen changes; this tends to hold the system to some equilibrium state making it more stable.