3.2. Coupling of Resonantly Forced Waves in the Tropical Oceans
The 1-year period off-equatorial QSW in the Indian Ocean being excluded, QSWs are formed in the three tropical basins, resulting from equatorial Rossby and Kelvin waves equatorially trapped and off-equatorial Rossby waves in the western part of the basins. Each QSW behaves as a single dynamical phenomenon within a characteristic bandwidth. Tuning of the natural period of QSWs to the forcing period results from the off-equatorial Rossby waves [
18,
19]. Thus, tropical basin modes have the propensity to adjust to forcing effects to produce RFW because the equatorial westward-propagating Rossby wave is not really reflected at the western boundary but diverted to form off-equatorial Rossby waves carried by countercurrents. Owing to the geostrophic forces acting at the tropical basin scale, these off-equatorial Rossby waves recede to the western boundary to form an eastward-propagating Kelvin wave. Thus, off-equatorial waves act as “tuning slides”, like what occurs in sound pipes, in which an air column comes into resonance to produce a particular tone, which gives the tropical basins a resonator function. Here the period of the QSWs is not fixed by the length of a tuning slide but by the latitude of off-equatorial Rossby waves on which the phase velocity depends. Therefore, the limits of the basins are not involved in setting the parameters of resonance, hence the ubiquity of tropical waves resonantly forced.
Resonant forcing finds its physical justification by solving the equations of motion, taking into account the equatorial and off-equatorial waves that result from the superposition of first or higher baroclinic mode Kelvin and Rossby waves [
18,
19]. The solution exhibits singularities when the natural period of RFWs coincides with the forcing period, that is, when the respective wavelengths of Kelvin and Rossby waves are adjusted to fulfill the dispersion relation of free waves. The solution is regularized by taking into account the Rayleigh friction. In this way neither the zonal modulated current velocity
nor the pycnocline depth oscillation
vanish at the boundaries of the basin: resonant forcing would be compromised if the wavelength were imposed by the width of the basin. This would jeopardize observations, which is what has led some authors to introduce the concept of no-normal flow at the boundaries of the Indian Ocean to explain the resonant forcing by monsoon winds of semi-annual equatorial baroclinic waves, but this notion is devoid of any physical sense.
Resulting from the superposition of first or higher baroclinic mode Kelvin and Rossby waves, all QSWs are coupled since they share the same node along the equator. In the Atlantic and Pacific oceans, the nodes merge west of the basin, before leaving it to contribute to the western boundary current flowing poleward. In the tropical Pacific in addition to the baroclinic modes, two meridional modes are superposed. In the Indian Ocean, the Equatorial Counter Current feeds the South Equatorial Current and the nodes stretch practically along the whole equatorial band of the basin.
Consequently, coupling of multi-frequency QSWs occurs in the three tropical basins where the equatorial nodes merge before leaving the basins. The mean flow at the outlet of the basin imposes conditions on each of the contributing oscillators, which results in the adjustment of the ocean under geostrophic forces.
3.3. Baroclinic Modes in Tropical Oceans
The first and second baroclinic mode Rossby waves observed in the tropical Indian Ocean as responses to equatorial wind anomalies have been attributed to an adequacy between the wavelength of the baroclinic waves and the width of the tropical basin [
13,
14,
15]. However, what seemed to be a peculiarity of the Indian Ocean occurs in the three tropical oceans. First and second baroclinic modes are observable in the tropical Pacific, first baroclinic mode Rossby and Kelvin waves forming the 4-year period RFW [
18] while second baroclinic mode Rossby and Kelvin waves form the 8-year period RFW, the phase speed being nearly half of that of the first mode. In the Atlantic Ocean, only the first baroclinic mode is highlighted in
Figure 1 and
Figure 2, with the higher modes being of lower amplitude [
19].
Baroclinic modes of equatorial waves result from the density profile along the equator. The solutions of the equations of motion for normal modes of oscillation may be unstable, which means the depth of interfaces may vary significantly when the pycnocline undergoes small modifications, as occurs along the equator [
15]. This difficulty may be overcome by taking advantage of the orthogonality of eigenvectors as shown in
Appendix A.
Taking account of 5 layers within a 4 & 1/2 layer model with ‘motion-less lower layer and active upper layers’, and the boundary conditions
at the surface and the bottom of the ocean (
is the vertical perturbation), phase speeds are calculated by recurrence from the bottom to the top, the upper layer being split into two sub-layers at each step. They are represented in
Table 1 as well as the depths of the interfaces. Depths of the interfaces and density profiles are shown in
Figure 9.
It appears that two main baroclinic modes coexist in each of the tropical oceans, the corresponding interfaces are located at the bottom and at the top of the pycnoclines. However, intermediate modes are stimulated in the Atlantic and the Indian Oceans, probably due to the coupling between the 1-year period RFW and sub-harmonics. In the Pacific Ocean, comparison of
Figure 5 and
Figure 7 shows that the equatorial antinode of the 8-year period RFW is shorter than that of the 4-year period RFW, the former not extending beyond 100° W whereas the latter stretches to the South American coast, which confirms the phase speed of the second mode is lower than half the phase speed of the first mode (
Table 1).
3.5. The Equations of Motion under Resonant Forcing at Mid-Latitudes
At mid-latitudes, the formation of baroclinic Rossby waves relies on the stratification of the oceans, and their adjustments to the periodic change in warm water mass carried by the western boundary currents from the tropics, as will be shown, by solving the equations of motion with relevant boundary conditions. As a consequence of multi-frequency QSWs in the tropical oceans, western boundary currents have the potential ability to convey a succession of warm water masses at particular frequencies, thus producing the deepening or the rising of the thermocline where they leave the continents to enter the subtropical gyres. Indeed, QSWs exhibit western and eastern antinodes in opposite phases, as well as western and eastern nodes also in opposite phases: the western antinode follows the gyre whereas the eastern antinode is outside the gyre, escaping poleward [
17]. On the other hand, the phases of the western antinode and node are opposite. In such condition, the western antinode takes the place of the eastern antinode after half a cycle, while a new western antinode is formed, in opposite phase versus the previous one. This leads to the transfer poleward and outside the gyres of successive warm water masses.
The formation of westward-propagating Rossby waves at mid-latitudes, as observed by many authors (e.g., [
12]), requires the westward phase velocity of the Rossby wave be lower than the eastward velocity of the wind-driven circulation in which it is embedded (
Figure 10a). The resulting wave propagates eastward with an apparent phase velocity −ω/k’ where −k’ is the apparent wavenumber; ω = 2π/T where T is the period. The phase velocity of progressive Rossby waves is determined from the dispersion relation of free waves. Rossby waves being approximately non-dispersive, the natural period of embedded waves is proportional to their apparent wavelength. A phenomenon of resonance occurs when the natural period of the embedded Rossby waves coincides with the forcing period, which supposes that their apparent wavelength is adjusted to achieve the tuning.
Here again the equations are solved using the
β-plane approximation for boundless waves, but taking into account forcing effects at the western boundary. To remove the inconsistency that would arise if forcing terms resulted in nonzero divergence, the potential vorticity equation is added to the momentum equations. Therefore, resonantly forced Rossby waves at mid-latitudes are governed by the same equations of motion as those applied to the tropical oceans [
18].
Assuming two superimposed fluids, the components
and
of the velocity vector (
is the zonal component,
is the meridional component) and the perturbation
of the height of the ocean surface are solutions of the system of equations [
35]:
,
where
is the displacement of the interface up, which is resolved in vertical mode:
is the Coriolis parameter,
β the gradient of the Coriolis parameter and
the potential vorticity.
is the depth of the upper layer,
that of the lower layer and
the total depth.
and
are the density of the upper and lower layer. The forcing terms
and
represent the surface stress.
Under the approximation of the plane, and are the components of the velocity vector expressed relative to the longitude and latitude. Similarly, and are the forcing terms relative to the surface stress, expressed in relation to the longitude and the latitude. is the evaporation rate.
At mid-latitudes the gradient of the Coriolis parameter
and the Coriolis parameter
at the central latitude (
is the rotation rate of earth,
is the radius of the earth and
is the central latitude in the local Mercator projection). The Coriolis parameter
is expanded with respect to
such that
. Thus, the dispersion relation is, in the case of long planetary waves, approximated by:
where
is the phase velocity along the equator for the first baroclinic mode [
35]. The first equality is deduced from the leading-order approximation of the potential vorticity equation, which is the same as that relating to the tropics. This expression of the phase velocity of long waves amounts to consider only the
part of the ageostrophic motion, ignoring both the isallobaric part and the nonlinear part of ageostrophic velocity (the Rossby radius is small in comparison with the wavelength of the waves). For an observer moving at the speed of the wind-driven circulation, the wavelength is 2780 km in latitude 40° for the period of 8 years, whereas it is only 174 km for the biannual wave.
In the North Atlantic, the apparent half-wavelength of the 8-year period Rossby wave, which is about 3400 km (
Figure 7), involves the apparent eastward phase velocity is 0.027 m/s relative to the geoid of reference (3400 kilometers traveled in 4 years). Consequently, considering the westward phase velocity of the Rossby wave at 40° N given by (7) is 0.011 m/s, the speed of the wind-driven circulation is 0.011 + 0.027 = 0.038 m/s. The same reasoning applies to the five subtropical gyres (
Table 2). The speed of the wind-driven circulation is the lower in the South Pacific gyre, that is, 0.028 m/s, and the higher in the North Pacific and the South Indian gyres where it reaches 0.055 m/s.
The speed of the wind-driven circulation measured from the drift of the 8-year period Rossby wave, which is dragged along this wind-driven current, is far from the velocity of a material element in the flow of the gyre. This because the geostrophic current, which may be nearly ten times faster than the wind-driven current, is superimposed on the latter (
Figure 10a). The same measurement performed from a shorter period Rossby wave (so shorter wavelength) would have resulted in a higher speed of the wind-driven circulation because of its deceleration as soon as the western boundary current enters the gyre.
The geostrophic current induced by the baroclinic waves is superimposed on the wind-driven circulation (
Figure 10a).
3.5.1. Forcing Terms
Consider the
-plane approximation for the quasi-geostrophic momentum equations of waves propagating along parallels. The forcing term
in (3), which is referring to longitudinal stress, is replaced by
.
in (4) that is referring to meridional stress is set to zero.
in (5), which is referring to evaporation, is replaced by
.
and
represent the speed of the zonal forcing current and the forcing pycnocline depth at the western boundary, there value being zero everywhere else (neither surface stress forcing nor evaporation are taken into account explicitly into the present model). The solution of the equations of motion exhibits singularities, which emphasizes the resonant character of Rossby waves produced at mid-latitude whose natural period is adjusted to the forcing period resulting from the warm water masses transported periodically by the western boundary layers. As this was done in the tropics [
18], the solution is regularized by taking into account the Rayleigh friction into the equations of motion.
3.5.2. Representation of the Solution
The equations of motion solved for the tropical and the subtropical 8-year period RFWs in the North Atlantic are represented in
Figure 11, assuming the amount of warm water transferred from the tropical oceans to the mid-latitudes is conserved and the wind-driven circulation velocity at 40° N is 0.038 m/s. In the tropical ocean the forcing modulated current
is oriented to the west at the western boundary whereas it is oriented to the east at mid-latitudes. Thus, forcing modulated currents at the frontier are in the opposite phase, the transfer time of warm water masses from the equator to the mid-latitudes being small in comparison with the period. The phase of the forcing pycnocline depth is such that
at the western boundary of the equator and
where the western boundary current leaves the coast at mid-latitudes. In addition, the following relation is used:
where
is the thermocline depth and
is the apparent phase speed associated with the first baroclinic mode. This relationship simply indicates that the displacement of the interface up would be equal to the thermocline depth if the velocity
of the forcing current was equal to the apparent phase speed of the progressive wave.
3.5.3. Evolution of the RFW
As displayed in
Figure 11, both the amplitude of the perturbation of SSH and the velocity of the modulated zonal current are comparable to what is observed. The phases of the subtropical antinode and the node are opposite. In such condition, the western antinode takes the place of the eastern antinode after half a cycle, while a new western antinode is formed, in opposite phase versus the previous one. This leads to the transfer toward the east and outside the gyres of successive warm water masses.
Growing of the western antinode results from the meridional currents that flow in quadrature relative to the antinode. Growing is maximum when the convergent meridional currents are maximum, half a period before the ridge is formed at the antinode (
Figure 11e,f). At this time, the meridional currents vanish. On the contrary, deepening of the antinode is maximum when the divergent meridional currents are maximum, half a period before the trough is formed at the antinode. At this time, the meridional currents vanish again.
The zonal currents that are in phase with the antinodes govern warm water transfer between the western and the eastern antinodes. A ridge at the western antinode around the gyre corresponds to a trough at the eastern antinode outside the gyre (both antinodes are in opposite phase). The resulting velocity of the western zonal node (sum of velocities of modulated geostrophic and steady wind-driven currents of opposite direction) vanishes or reverses. At this time, the velocity of the eastern node reaches a maximum, while flowing eastwards. Then, warm water flows from the western to the eastern antinode while transfer from the western antinode to the west is low, which results in extraction of warm water from the gyre. Half a period later, a trough is formed at the western antinode and a ridge at the eastern antinode. The velocity of the western boundary current is maximal, being the sum of velocities of modulated geostrophic and steady wind-driven currents of the same direction. The trough at the western antinode is being replaced progressively by a ridge while the velocity of the resulting current at the eastern node vanishes or reverses, being the superposition of two opposite velocities: transfer from the western to the eastern antinode is low.
The same transfer process applies to all frequencies knowing that this representation is true on average, actual RFWs exhibiting strong variability from one cycle to another (
Figure 10). Consequently, the resonance of Rossby waves at mid-latitudes leads to the formation of an eastward propagating mixed layer represented by the successive warm water masses advected to half a wavelength during half a cycle. Growing of the western antinode induces a horizontal pumping effect evidenced by the eastward western node that results from the conjugation of the steady wind-driven circulation and the zonal modulated geostrophic current of the resonant baroclinic Rossby wave: the slowness of the steady wind-driven circulation compared to the maximum speed of the modulated geostrophic currents that may peak to 0.3 m/s (
Figure 10a) evidences the driving force of Rossby waves. Consequently, the total volume transport of the western boundary current is increased and an abrupt change in its potential vorticity occurs as soon as it leaves the coast: the Rossby waves play a key role in the adjustment of the ocean circulation around and outside the gyre to high latitudes, as well as in the western intensification.
3.7. A Necessary and Sufficient Condition to Ensure the Durability of the Resonant Oscillatory System
The durability of the oscillatory system is proven when the periods of coupled oscillators are commensurable with the fundamental period that coincides with the forcing period
, that is,
where
is an integer. In such condition, the oscillatory system is itself periodic. Its period
is such that
is the smallest common multiple of
. Then, the interaction energy time-averaged over the period
is zero for every oscillator:
where
. Consequently, in a stationary state, the periodicity of the total energy of the coupled oscillator system ensures its durability.
Actually, the period of the baroclinic RFWs deviates significantly from the average value during each cycle. But this variability does not jeopardize the sustainability of the coupled oscillator system when the periods time-averaged over the period converge towards their average value . In this way, the subharmonic modes of the oscillatory system ensure its durability in all circumstances, even though the forcing period deviates from its mean value.
The actual conditions referring to subharmonic modes are more draconian when they are applied to baroclinic RFWs. The observations show that in the oceans the coupled oscillators form oscillatory subsystems so that the resonance conditions are to be defined recursively:
This additional constraint results from the great variability over time of the amplitude of each oscillator. Also, the stability of the complete system requires the stability of the different subsystems whose preponderance can vary considerably over time. Considering the period of the fundamental QSW is year, the average period of coupled oscillators is a multiple of years. Therefore, as already stated, the average periods of coupled RFWs are exactly 1-, 4- and 8-years, even though those periods are subject to a large variability (note that the 2-year period RFW, clearly visible in the Indian Ocean, is very low in the Atlantic and Pacific oceans).
We have seen that the functioning of the coupled oscillator system in subharmonic modes is a sufficient condition to ensure the durability of that system robustly. But it is also a necessary condition to ensure the resonant forcing of the system. Indeed, when the mean periods of the oscillators are commensurable with the mean forcing period, the energy the oscillation absorbs is maximized when the forcing is periodically ‘in phase’ with the oscillations, while the oscillation’s energy is more extracted when it is never in phase with the forcing. In this way any non-resonant oscillatory system being less efficient than a resonant system disappears in favor of the resonant system.
The subharmonic mode locking of multi-frequency RFWs is very general because this property does not involve the actual interaction mechanisms. Since the subharmonic modes condition the resonance of baroclinic waves, they tightly control climate, which happens in the tropical Pacific with ENSO. Since ENSO events are mature at the end of the eastward phase propagation of the quadrennial QSW, their characteristics can be joined to the date of occurrence of the events within a regular 4-year cycle [
21]. On the other hand, ice core records show that solar and orbital forcing exhibit a resonant character that might result from the influence of subharmonic modes of very long period baroclinic waves. As an example, the 100,000-year problem of the Milankovitch theory of orbital forcing, which refers to a discrepancy between the reconstructed geologic temperature record and the reconstructed amount of incoming solar radiation, or insolation over the past 800,000 years [
36], might result from such resonant modes.