Climate change

Author: Jean-Louis Pinault (version française: ici)

This site has the audacity to attempt a rigorous approach to climatology. This project is not trivial at a time when climatologists are closing in their certainties that have been forged from approximations, sometimes crude, in the interpretation of observed phenomena. So much so that it is often difficult to distinguish scientific advances from fantasies, this confusion being maintained by media coverage involving the most prestigious scientific journals. The different scenarios oscillate between a hot or cold future according to the good will of the Gulf Stream. This warm current would threaten to stop due to the melting of the pack ice that would compromise the thermohaline circulation resulting from an increase in the density of seawater at the approach of the polar caps. But what about the Kuroshio, the equivalent of the Gulf Stream that circulates off the coast of Japan while the Pacific is virtually closed to the north by the Bering Strait? It is because forces acting on a planetary scale come into play under the combined effect of gravity and rotation of the earth. Climatology is a young discipline, indeed, the essence of which remains to be discovered.

The warming observed since the beginning of the industrial era is a reality, but the part attributable to human activities, to the emission of greenhouse gases in particular, is poorly known. For the climate has always varied over time, which is confirmed by the archives obtained from ice or sediments cores, the first collected in the polar caps and the second in the ocean trenches. Moreover, the evolution of the global surface temperature of the earth since the beginning of the industrial era is difficult to grasp, with the required precision. The estimates made a few years ago, which showed a slowdown in global temperature growth starting from the end of the 20th century, the famous ‘hiatus’ which seemed to call into question any causal relationship between the growth of greenhouse gases and its assumed effects have been invalidated and replaced by more representative measures. The CRU (Climatic Research Unit, University of East Anglia) has made a retrospective correction by integrating more data, including the Russian Arctic, now showing a continuous increase in temperature (Jones et al., 2012).

These uncertainties, not to mention the images of pure communication that proved to be untrue, fuel a certain skepticism aimed at calling into question the methodological approach of the Intergovernmental Panel on Climate Change (IPCC). What was only a precautionary principle became a reality. Supporters of catastrophic scenarios now speak of consensus to support their assumptions, which goes against any objective approach: the term global warming has given way to climate change following the disappointments of forecasters.

On the other hand, denying or minimizing by pure egocentrism or on the basis of fallacious pseudo-scientific arguments, the anthropogenic impact on the increase of CO2 in the atmosphere as well as its climatic incidence would raise an irresponsible arrogance (Pinault, 2018e). Because the impact of human activities on global warming is undeniable and can now be evaluated objectively thanks to recent work on the resonance of ocean gyres under the influence of solar and orbital forcing. More than half of the warming observed since the beginning of the industrial era is attributable to humans. The average temperature of the earth’s surface has increased by 0.8 ° C in 50 years, linearly. No inflection is perceptible, which suggests the inexorable continuation of this rise over the next decades if the production of greenhouse gases continues to increase at the frenetic pace we are currently experiencing.

This apparently low temperature rise has however an important climate impact because of the increase in energy available to supply cyclonic and anticyclonic systems, leading to more extreme events. Nothing like this has been observed so quickly during the Holocene covering the last 12,000 years of our history.

The purpose of this article is not to make shattering revelations, let alone feed new controversies. It reinforces the growing awareness of global stakes in the light of the author’s scientific publications on medium and long-term climate variability. For it is here that knowledge is most lacking to unambiguously separate natural variations in climate from those related to human activity.

A long adventure has led the author to patiently and methodically dissect the mechanisms involved. Using satellite data on the variation of the height and temperature of the oceans, isotopic analyzes of ice and sediment cores, new concepts on ocean dynamics and climate variability have been established.

Climate is only partly subject to human activities

Our ignorance of the mechanisms controlling climate variability results from the fact that research has mainly focused on atmospheric phenomena over the last few decades, whether they refer to human activities or solar cycles. But the driver of climate change is not the atmosphere but oceans, the atmosphere playing only the role of vector between oceans and continents, like the El Niño phenomenon (Pinault, 2016). The oceans resonate with solar and orbital cycles, storing or on the contrary restoring heat: the resonance called “gyral” because occurring around the five subtropical gyres, closely affects the energy balance of our planet. The amplifying effect of solar and orbital forcing comes from the positive feedback exerted by the geostrophic current (resulting from the combination of gravity and inertial forces due to the rotation of the earth) of the gyral Rossby wave: the oscillation of the thermocline is amplified by the geostrophic current which warms or cools, as the western boundary current accelerates or slows down.

Thus, the modulated response of subtropical gyres helps explain, through observations and based on irrefutable physical basis, climate change at different time scales (Pinault, 2018d). Based on this new phenomenon, this article discusses climate variability with fresh eyes while solving some puzzles on ocean circulation.

The natural variability of climate

Global temperature and carbon dioxide concentration in the last four glacial-interglacial periods derived from the analysis of ice cores (Vostok, Antarctica). The offsets in time observed between the two curves are measurement artifacts.
Global temperature and carbon dioxide concentration in the last four glacial-interglacial periods derived from the analysis of ice cores (Vostok, Antarctica). The offsets in time observed between the two curves are measurement artifacts.

The correlation observed from analysis of Vostok ice cores between global temperatures and atmospheric CO2 over the last four glacial-interglacial periods shows that it is the rise in temperature which increases the CO2 in the atmosphere ( by degassing oceans mainly) and not the reverse, since the observed cycles are the result of orbital forcing (Milankovitch cycles).

This process is still valid today but not noticeable yet because changing equilibriums of the mixed layer of the ocean requires several hundred years. By contrast the emission of combustion gases has a faster effect on the global temperature by radiative forcing.

Total solar irradiance (TSI) obtained from 14C in tree rings and 10Be in ice cores (Steinhilber et al. 2012)
Total solar irradiance (TSI) obtained from 14C in tree rings and 10Be in ice cores (Steinhilber et al. 2012)

Solar activity over the past 11,400 years has been reconstructed by analyzing jointly the concentration of radiocarbon in tree rings and isotopic abundance of beryllium-10 in ice cores. 14C and 10Be isotopes, which are produced by cosmic rays in the upper atmosphere, indeed reflect solar activity because in periods of high activity cosmic rays are deflected in the solar system and therefore produce less cosmogenic isotopes.

The reconstruction of solar activity shows that it varies continuously. During the last millennium, humanitarian disasters generated during periods of low activity suggests that the global temperature has dropped, as was the case during the last Little Ice Age: there is undoubtedly a causal relationship between solar activity and global temperature.

However the solar irradiance variability has low direct impact on climate

How can the 11-year cycle, during which solar activity varies of a few tenths of a percent, be so quiet? Although low, this amplitude is of the same order as that of longer cycles. But this 11-year cycle has little influence on the climate compared to Milankovitch cycles, of very long-periods, which reflect the variations in the orbital parameters of the earth around the sun. Those have a significant impact, setting periods of glaciation, in particular. This selectivity and the variability in the efficiency of solar and orbital forcing during glacial-interglacial periods highlighted from the analysis of ice cores originating from Greenland and Antarctica suggest that resonance phenomena occur, filtering out some frequencies in favor of others while amplifying or mitigating according to the extension of the polar caps.

A novel approach: the modulated response of subtropical gyres (Pinault, 2018d)

The direct impact of solar irradiance does not explain climate variability at different time scales. Such an assumption would come to imagine a submissive climatic system while it has its own frequencies, a considerable inertia, and manifests many whims, harbingers of resonant phenomena. The dynamics of climatic phenomena suggests a leading role of oceans whose influence, though long recognized, remains poorly understood. The oceans offer indeed a field of investigation whose scope is considerable, and that concerns the resonance of the oceanic planetary waves: it not only allows explaining and reproducing the warming of our planet, more precisely the middle and long-term climate variability, but also the El Niño phenomenon, the succession of wet or dry years observed in Western Europe since the 70’s…

Is that the tropical belt of the oceans produces long-waves, whose wavelength is several thousand kilometers. Trapped by the equator due to the Coriolis[i] force resulting from the earth’s rotation, they are deflected at the approach of the continents to form off-equatorial waves. These long tropical waves resonate with the forcing exerted by trade winds, whose period is annual, to produce sub-harmonic whose period is multi-annual. These long-waves that oceanographers appoint of the name of baroclinic[ii] result of the oscillation of the thermocline one hundred meters deep or more that separates the warm surface waters from deep cold waters, denser.

This tropical oceanic resonance is one of the drivers of ocean surface circulation, i.e. above the thermocline, and contributes to the formation of strong western boundary currents[i] such as the Gulf Stream in the North Atlantic and the Kuroshio in the northeast Pacific, by introducing a sequence of warm or cold waters according to the oscillation of the thermocline. Around latitude 40°N or S, those western boundary currents, which flow poleward in both hemispheres, leave the boundary of the continents to join each of the five oceanic subtropical gyres, giant vortexes at the north and south Atlantic and Pacific oceans and the southern Indian ocean. These resonantly forced baroclinic waves then become gyral, suiting their wavelength to the period inherited from the tropical oceans.

For periods between half a year and eight years, the forcing of these gyral Rossby waves is induced from the sequence of warm and cold waters conveyed by western boundary currents, and now cause the oscillation of the thermocline of the gyre. But these gigantic gyral Rossby waves also have the ability to tune with the long-period solar cycles of one to up to several centuries, as well as Milankovitch cycles that affect the occurrence of glacial and interglacial periods, which reflect the changes in terrestrial astronomical parameters throughout tens of thousands of years.

These resonant baroclinic waves have the ability to ‘hide’ the thermal energy that drives them by lowering the thermocline; due to a positive feedback those warm deep-waters favor the acceleration of the western boundary current and the development of sea surface temperature anomalies, supporting the heat exchanges between the ocean surface and the atmosphere: thermal surface anomalies induce atmospheric instabilities say again baroclinic, depressions or cyclones, which, carried by the jet-streams[ii] at high altitude, travel throughout the continents.

In this way, surface temperature of continents respond to thermal anomalies of subtropical gyres. Positive or negative according to the motion of the thermocline, these thermal anomalies of sea surface, which result from the remanence of the vertical thermal gradient, tend to produce the same anomalies at the surface of continents. This because of the cyclonic or anticyclonic activity of the atmosphere stimulated at mid-latitudes. This thermal balancing internal to our planet, which occurs over the years, smooth the climate variations we observe daily at mid-latitudes. The imbalance between the energy received and re-emitted by the earth mainly depends on the depth of the thermocline of gyral Rossby waves.

The resonance of tropical waves

To understand the modulated response of subtropical gyres, driver of mid- and long-term climate variability and its coupling with the solar and orbital cycles, first we have to focus on the resonance of oceanic tropical waves from which are inherited periods of resonance. The study of the quasi-stationary waves in the three tropical oceans consists in bringing out sea surface height anomalies and modulated currents in characteristic bands such as the 8-16 month band for annual waves. From measurements of sea surface height are deduced the amplitude and phase of sea surface height anomalies, but also the speed and phase of the modulated geostrophic currents from the slope of the sea surface[iii], thanks to the use of cross wavelets.

Quasi-Stationary-Waves are formed, representing a single dynamic phenomenon within a characteristic bandwidth. Geostrophic forces closely constrain the behavior of the baroclinic waves at the limits of the basin, forming antinodes at the place of Sea Surface Height anomalies and nodes where modulated geostrophic currents ensure the transfer of warm water from an antinode to another. Modulated geostrophic currents that change direction twice per cycle, are superimposed on the background current. The observed current is the resultant of the modulated geostrophic current and the background current. Although these terms node and antinode are abusive because the phase of Quasi-Stationary-Waves is not uniform, which may involve overlapping of nodes and antinodes, they explicitly reflect the evolution of the wave during a cycle.

The Atlantic Ocean



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[i].

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[i] 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[i] 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.

The Pacific Ocean

As the Atlantic Ocean, the Pacific is subject to the resonance of equatorial waves formed from the first baroclinic mode (vertical) Kelvin waves, and the first baroclinic mode, first meridional mode Rossby waves. However, while under the effect of forcing resulting from wind stress the resonance occurs at a frequency of one cycle per year in the Atlantic Ocean, due to the width of the basin, 17,760 km instead of 6,500 km, the period of tropical waves in the Pacific is necessarily multiyear. The resonant basin mode produces the El Niño phenomenon well known for its meteorological effects on a global scale (Pinault, 2015).

The annual wave



On the other hand, an annual quasi-stationary wave is observable north of the equator between latitudes 0°N and 12°N. The dynamic representation of the quasi-stationary wave shows two long-crested waves, almost in opposite phase, which extend from the eastern coasts of Southeast Asia to Central America, suggesting that they result from resonant forcing of the first baroclinic mode, fourth meridional mode Rossby wave. Meridional modes indeed show an increasing number of zonal energetic strips as the mode increases (one for the first mode, two strips for the second and so on).

The nodes are the two zonal surface currents flowing north and south of the equator, i.e. between 4.5°N and 7.5°N, forming the North Equatorial Counter-Current, and 0°N, 4.5°S, forming the South Equatorial Current. The period of the modulated currents is one year, too. The modulated South Equatorial Current feeds the western boundary currents, the outlet being located near the equator, i.e. off North Maluku and north Sulawesi islands in the Indonesian archipelago.

Evolution of the quasi-stationary wave

Only the stimulation of a high meridional mode can indeed explain the structure in strips of the wave and the two nodes whose analysis reveals that they are only one, being highly correlated. The antinodes are mainly visible in the northern hemisphere when they should appear antisymmetric in the southern hemisphere: only the southernmost antinode is visible west of 160°W, in phase with the southernmost antinode in the northern hemisphere, as the trade winds are weaker south of the equator.

The wavelength, which is 9400 km, is less than the width of the basin. The phase of the quasi-stationary wave thus reverse at both ends, which may explain the large variability from one cycle to the other of the observed waveform. In the illustrated realization, the speed of the modulated component of the Counter-Current peaks in September-October in the northern hemisphere, as it flows eastward, in phase with the South Equatorial Current in the southern hemisphere. This occurs while warm water has been transferred to the southernmost antinode in each hemisphere, then a few months later at the northernmost antinode.

The quadrennial wave



The pattern of antinodes and nodes of the quadrennial wave recalls what is observed in the tropical Atlantic (the frequency representation of the sea surface height and the speed of the geostrophic current near the equator west of the basin show that the period is about 4 years with a high variability). As in the Atlantic basin this mode shows a main node where the modulated equatorial current extends into the western half of the basin, two antinodes on both sides of the equator west of the basin and an equatorial antinode: the central-eastern antinode results from the superposition of the first baroclinic mode, first meridional mode Rossby wave and a Kelvin wave, both trapped by the equator, but propagating in opposite directions. Western antinodes on the one hand and the central-eastern antinode secondly separate the Pacific into two parts where the thermocline oscillates almost in phase opposition.

The north-western antinode forms a curl joining the eastern coast of central and southern Philippines, overlapping the North Equatorial Current in its northern part, and riding the North Equatorial Counter-Current and the South Equatorial Current in the Southern part. A Rossby wave, which is reflected against the eastern coast of the Indonesian archipelago, spreads along the curl, driven by the North Equatorial Counter-Current.

The south-western antinode forms a tongue extending to the coast of north-eastern New Guinea through the Solomon Islands. It also proceeds from a Rossby wave that is reflected against New Guinea, driven by the South Equatorial Current.

The main node is the modulated component of the South Equatorial Current flowing westward along the equator between latitudes 0°N and 8°S.

Evolution of the quasi-stationary wave

The evolution of quasi-stationary waves during a cycle can be expressed relative to the ENSO event that occurred during the cycle (Pinault, 2018b). The westward phase propagation of the quasi-stationary wave along the equator begins when the ridge is reflected against the South American coast, during the maturation stage of the ENSO event. It lasts almost two years during which the thermocline rises along the central-eastern antinode. At the end of the westward phase propagation, the main modulated current, which then flows to the west, reaches its maximum speed and partially leaves the equatorial belt to feed the western boundary currents. In the absence of a powerful counter-current south of the equator, which prevents any reflection of the equatorial ridge to the south-western antinode, the resonant wave is partly reflected to the curl forming the north-western antinode. The westward phase propagation of the resonant wave along the equator stimulates upwelling at the eastern limit of the basin, i.e. the South American coast, while the trough of the wave is formed along the central-eastern antinode. Thus, in the central and eastern part of the basin cold water gradually replaces the warm water leaving the equatorial belt.

When the phase is ± 2 years relative to the ENSO event, under the effect of winds western antinodes form a ridge with the deepening of the thermocline in the “warm water pool” down to 250 m. The ridge of the north-western antinode as well as the speed of the North Equatorial Counter-Current flowing eastward reach their maximum at the same time as the south-western antinode. Meanwhile, the trough deepens at the central-eastern antinode, stimulating the migration of warm water from the western antinodes, replacing cold water while upwelling weakens off the South American coast.

As in the Atlantic Ocean, the north-western antinode plays the role of “tuning slide”, but the propagation time along the curl is short compared to the period of the quasi-stationary wave, so that only a fine tuning of the period occurs. Again, the south-western antinode acts as a heat sink.

Coupling of basin modes

The functioning of the quadrennial quasi-stationary wave cannot be dissociated from the ENSO. Indeed, El Niño events are triggered at a critical time in the cycle of the wave when, at the end of the eastward phase propagation, the ridge reaches the west coast of South America. These El Niño events stimulate evaporation from the surface of the central-eastern equatorial anomaly, which cools the mixed layer, and thus raises the thermocline. ENSO is also a way of forcing of the resonant wave because it stimulates the propagation of the ridge to the west. More La Niña, which announces the resumption of the Walker circulation[i] with increased surface stress from easterlies, is also a way of forcing because it becomes effective after El Niño, so during the westward phase propagation of the ridge.

However, the equations of motion show that forcing associated with ENSO is not sufficient to explain the amplitude of antinodes and speed of modulated currents. Thus, a coupling between annual and quadrennial basin modes has to be invoked. The modulated components of the North Equatorial Counter-Current and South Equatorial Current, which form an integral part of the annual quasi-stationary wave, indeed merge with the main node of the quadrennial quasi-stationary wave along a narrow equatorial strip west of 150°W. The special location of the island of Papua New Guinea, near the equator, modifies the current lines of zonal flows and instabilities are exacerbated along a narrow line between 136°E and 141°E longitude and between 0 N° and 2°N latitude.

These instabilities result from the North Equatorial Counter-Current that flows closer to the equator and in the amplification of the modulated current acceleration common to both basin modes. Off the Cape d’Urville 137.5°E 0.5°N the North Equatorial Counter-Current may accelerate from 0 to 1.5 m/s within one month, which is considerable. Some of these accelerations are harbingers of an ENSO event. In this case the current accelerates rapidly eastward and its speed decreases before increasing again, reaching a maximum two months later. The geostrophic current off the Cape d’Urville is perturbed at an early stage of the development of an ENSO event, and then returns to its original speed. This occurs 7 times late 1992 to mid-2015: the corresponding ENSO events occur in 08/1994, 11/1997, 12/2002, 12/2004, 12/2006, 11/2009 and 09/2012 (an event is maturing in June 2015).

a) The geostrophic current observed off the Cape d'Urville 137.5°E, 0.5°N. The dotted lines indicate the different ENSO events to their maturity. The arrows show the acceleration of the current associated with each of the ENSO events – b) Gross signal –SOI and filtered in the band 1.5-15 years. ENSO events occur to the maxima of the filtered signal.
a) The geostrophic current observed off the Cape d’Urville 137.5°E, 0.5°N. The dotted lines indicate the different ENSO events to their maturity. The arrows show the acceleration of the current associated with each of the ENSO events – b) Gross signal –SOI and filtered in the band 1.5-15 years. ENSO events occur to the maxima of the filtered signal.

On the other hand, these instabilities anticipate the maturation stage of ENSO events of 4 to 6 months, this time depending on how the surface temperature anomalies develop during the evolution of ENSO. The acceleration of the North Equatorial Counter-Current that flows closer to the equator at the western part of the basin stimulates a baroclinic Kelvin wave, crossing the basin from west to east in two months and causing a deepening of the thermocline in the central-eastern part of the basin. Geostrophic forces in the tropical basin make that two Kelvin waves cannot succeed in less than one year and a half. This is the minimum time required for a complete cycle of the quasi-stationary wave to occur, the tilt along the equator of the sea surface having to promote the propagation of Kelvin waves to the east. Therefore all the current accelerations, whose average period is one year, do not produce a Kelvin wave. This is why some accelerations, even of large amplitude as that of 2003, does not produce ENSO event. In this case geostrophic forces remain confined to the west of the basin.

The periods of coupled oscillators

Applied to the case of coupled ocean waves, the theory of Sub-harmonic mode locking in coupled oscillators with inertia indicates that the average periods of the coupled waves are multiples of the average period of the fundamental wave[i], i.e. one year here when we consider that the trade wind cycle is the temporal reference of the tropical basin. The quadrennial wave is indeed subject to a sub-harmonic mode locking of the annual mode (Pinault, 2018c). This holds true irrespective of the variability of the period from a cycle to another. As a result, the average periods of the two basin modes are precisely 1 and 4 years. This latter period is unambiguously determined from the distribution of ENSO events.

The 8-year period sub-harmonic



A sub-harmonic whose average period is 8 years exhibits three antinodes and a main node like the quadrennial quasi-stationary wave. The central-eastern antinode extends between 160°E and 130°W. The north-western antinode is located off the Philippines against which Rossby waves are reflected. The south-western antinode, parallel to the equator, stretches from the eastern Australian coast to 130°W between latitudes 25°S and 20°S. The functioning of this basin mode is reminiscent of the quadrennial mode. The central-eastern antinode is formed in the waveguide formed by the equator, the north-western antinode plays the role of “tuning slide” and the south-western antinode is a heat sink.

This basin mode involves Rossby and Kelvin waves whose phase velocities are necessarily lower than these of the quadrennial basin mode: the second baroclinic mode Rossby and Kelvin waves have to be invoked. The phase velocity is nearly 1 m/s, i.e. less than half that of the first baroclinic mode. This second baroclinic mode (or vertical) not correspond to the oscillation of the thermocline, more precisely the interface at the basis of the pycnocline, depth of 200 to 250 m, but the oscillation of the interface atop the pycnocline at the depth of 125 m.

The main node coincides with that of the quadrennial basin mode west of 170°E, so that this mode is coupled to the previous basin modes of 1 and 4-year period: the average period is 8 years since a sub-harmonic mode locking occurs. This basin mode contributes to the ENSO less than the quadrennial basin mode because the central-eastern antinode interferes little with the cold currents in the eastern basin.

In this way, the outlet of the tropical Pacific, where the modulated currents leave the basin to supply the western boundary currents, is common to the three basin modes. Thus the Pacific Ocean differs from the Atlantic Ocean, due to the superposition of strong modulated currents at the output of the tropical basin whose average periods are 1, 4 and 8 years.

The Indian Ocean

As well as the Atlantic and Pacific oceans, the functioning of the tropical Indian Ocean is subject to resonant forcing of long waves, leading to introduction in the only western boundary current, the Agulhas, of alternately warm and cold water at a characteristic frequency. The tropical Indian Ocean is involved in the same way as the other oceans in the resonance of long waves at mid-latitudes leading to subtropical oceanic gyre forcing. But it has two features, its closing north, which prevents any western boundary current from flowing northward, and its openness to the Pacific Ocean to the east, which produces the Throughflow, i.e. an Indonesian system of surface currents flowing from the Pacific to the Indian Ocean through the Indonesian seas. In this way, the Pacific Ocean influences the Indian Ocean, in the region extending from 17.5°S to 7.5°S in particular, due to the propagation of Rossby waves from the western Pacific to the Indian Ocean. The Throughflow plays an important role in the transport of heat in the climate system, linking the warm waters of the Western Pacific pool and the cold waters of the South Equatorial Current of the Indian Ocean.

Another feature of the tropical basin is that its width, which is 6,300 km from the east coast of Africa to the west coast of Sumatra, is close to half the wavelength of a Rossby wave of biannual frequency, i.e. 12,100 km. Considering the first baroclinic mode, the propagation time of a Rossby wave, then a Kelvin wave in return is nearly two thirds of the period to make a round trip along the equator, which enables the tuning of natural and forcing periods due to the delayed response of the Quasi-Stationary-Wave at the western antinode.

The Indian Ocean is subject to a phenomenon comparable to El Niño, which is an irregular oscillation of surface temperatures in which the western part of the ocean becomes alternately warmer and colder than the eastern part, forming the Indian Ocean Dipole (IOD). When the dipole is positive temperature anomalies are observed in the western part of the basin, which induces an increase in rainfall in East Africa and a higher than normal Indian monsoon. Cooling then occurs in the eastern part of the basin, which tends to cause drought in Indonesia and Australia. When the dipole is negative conditions are reversed, with warm water and an increase in precipitation in the eastern Indian Ocean, and cooler and drier conditions west.

However, again, the functioning of the tropical Indian Ocean cannot be understood adequately without involving the resonant forcing of long ocean waves. This concerns the IOD, of course, but also the Somali, a modulated current along the eastern coast of Africa in front of Somalia. Two quasi-stationary long-waves can be identified, a biannual equatorial wave which is the superimposition of a Kelvin and a Rossby wave, and an annual off-equatorial Rossby wave. It spreads across the Indian Ocean from the outlet of the Timor passage at 120°E along the South Equatorial Current. It is deflected northward approaching the western boundary of the Indian Ocean, then follows the Somali and the monsoon drift, avoids the Indian subcontinent south of Sri Lanka to go along the coast of the Bay of Bengal.

The biannual wave



The biannual quasi-stationary wave seesaws from the western part of the basin to its eastern part, the equator acting as a waveguide, as shown by the amplitude and phase of the cross-wavelet of sea surface height. The geostrophic component of the modulated zonal current, which is the Equatorial Counter-Current, preferably flows east between longitudes 50°E and 90°E, but may reverse. This inversion indicates that during a cycle exchanges occur between western and eastern antinodes.

The quasi-stationary wave is the superposition of a westward propagating Rossby wave and a Kelvin wave in the opposite direction, both reflecting on the limits of the basin that are the coast of Eastern Equatorial Africa on the one hand, Malacca and Sumatra secondly. The western antinode forms a ridge in March and September, and the Eastern antinode in May and November, whence a slight asymmetry in the duration of transfers between the eastern and western tropical basins due to the difference in phase velocity of the Rossby and Kelvin waves. The speed of the modulated current is maximal in May and November when directed eastward, i.e. it is in phase with the eastern antinode. The resulting basin mode is tuned to the monsoon winds.

Evolution of the quasi-stationary wave

Quasi-stationary equatorial waves are the superposition of the first baroclinic mode Kelvin wave and the first baroclinic mode, first meridional mode Rossby wave. During the eastward phase propagation, warm water is transferred from the western antinode to the eastern antinode where it partially leaves the tropical basin to join the eastern boundary current while cold water replaces warm water to the west of the basin by stimulating upwelling off the eastern coast of Africa, leading to the rise of the thermocline. This phase, during which upwelling off the coast of Sumatra is reduced, ends in spring or autumn.

The speed of the Equatorial Counter-Current, which flows preferentially to the east, increases in spring and autumn: the Kelvin wave is reflected against the west coast of the Indonesian archipelago, forming coastal waves that propagate poleward.

During the summer and winter, the modulated current vanishes or reverses. Warm water replaces cold water at the western antinode while upwelling is reinforced along the coast of Sumatra, causing the rise of the thermocline.

During a period the mixed layer, warm, is advected from the western antinode where it is formed to the eastern antinode. According to the geostrophy of the tropical ocean, advection may also be performed back, when the modulated current reverses. Thus, the biannual basin mode induces heat transfer between the western and eastern parts of the tropical Indian Ocean while stimulating or reducing upwelling at the boundaries of the basin.

Due to the seasonal reversal of monsoon winds, forcing mainly occurs at the eastern antinode and southern India. Northwest winds reach their maximum in April-May and October-November, and are reversed in March and September, in phase with the eastern antinode. Thus, the biannual basin mode turns out to be the response of the tropical ocean to resonant forcing induced by the seasonal reversal of the monsoon winds.

The annual wave



Unlike the biannual equatorial wave and its sub-harmonics the annual quasi-stationary wave has a leading role in the circulation of the western boundary current, which here is the Agulhas propagating southward. In this way it is involved in the long-term climate variability like the tropical quasi-stationary waves in the Atlantic and Pacific oceans.

Two main antinodes are visible in both hemispheres. The southernmost antinode extends westward from the Timor passage, longitude 80°E, following first the Indonesian Throughflow then the South Equatorial Current. The northernmost antinode follows the southwest monsoon drift off the east coast of Africa, south of the Arabian Sea, to the southern tip of the Indian subcontinent. Less extended, antinodes develop along the coast of the Bay of Bengal. To the east they are formed from the coastal Kelvin waves, as evidenced by the phase change north of the bay.

Three main nodes are recognizable. To the south is the South Equatorial Current between the Timor passage and longitude 60°E, to the west the Somali, a current that follows the eastern African coast, to the north the northeast monsoon drift that is mostly visible south of the Indian subcontinent. South of the coast of Java the South Equatorial Current flows mainly to the west, vanishing periodically, while the Somali and monsoon drift are reversing currents.

Evolution of the quasi-stationary wave

The annual Rossby wave is formed at the outlet of the Timor passage to cross the Indian Ocean; once deflected by the East African coast, the Rossby wave propagates eastward into the northern hemisphere to join the monsoon drift, the southern coasts of India and Sri Lanka acting as a waveguide. The wave propagation in the northern hemisphere results from the Doppler Effect when the speed of the current flowing eastward is higher than the phase velocity of the Rossby wave flowing westward. When the phase is reversed, the Rossby wave propagates westward, and the Somali along the coast of Somalia is reversed, too, a part of this current leaving the tropical ocean to feed the western boundary current along the eastern coast of Madagascar and the coast of southeast Africa to form the Mozambique current.

Antinodes show a north-south seesaw of warm waters of the tropical ocean. From the Pacific they accumulate during the boreal summer to form the southern antinode whereas, due to upwelling that is stimulated in the Bay of Bengal and the Arabian Sea, cold water overruns the northern part of the basin. In spring the phenomenon is reversed, warm water accumulating in the north of the basin. Upwelling weakens as well as the South Equatorial Current; reversing of monsoon drift promotes seesaw of warm waters.

Thus, the thermal energy is transferred from the western basin in the Pacific, which acts as a heat sink, to the Indian Ocean via the Timor passage. Then, heat exchange occurs between the two hemispheres via the Somali and the monsoon drift, each reversing periodically in phase. The annual wave feeds the western boundary current, i.e. the Agulhas, through a succession of warm and cold waters.

In contrast the Equatorial Counter-Current is not part of this system, being out of phase with the two nodes that are the Somali and the monsoon drift. The biannual equatorial wave and sub-harmonics thus behave independently of the annual wave, which itself propagates out of the equator. These two systems have no node in common, the first producing the Equatorial Counter-Current and the second the South Equatorial Current, then the monsoon drift. This situation, which is unprecedented in the functioning of tropical oceans, highlights two independent basin modes in the Indian Ocean.

The resonance of gyral Rossby waves

To the 5 subtropical gyres correspond 5 western boundary currents that are the Gulf Stream and the Brazil Current in the North and South Atlantic, the Kuroshio and the eastern Australian current in the North and South Pacific, the Agulhas in the South Indian Ocean. Under the influence of resonantly forced baroclinic waves, the three tropical oceans feed those western boundary currents through a sequence of warm and cold waters to one cycle every 1/2, 1, 4 and 8 years. Tropical oceans behave indeed as “resonators” under the effect of forcing due to surface stress and ENSO as regards the Pacific.

In fact, during these cycles the temperature of the water carried by the western boundary currents does not change, or very little. The wavelet analysis of the sea surface temperature does not show anomalies in the different characteristic frequency bands. It is the depth of the thermocline that varies, thus the warm water mass transported poleward, without generating the formation of baroclinic waves that, facing west, would inevitably be wiped out against the coasts.

This is no longer true when the western boundary current reaches a latitude nearby 35° to 40°N or S. At high latitudes, the velocity of the western boundary current increases as the phase velocity of the baroclinic waves decreases: baroclinic waves are formed when the velocity of the western boundary current becomes higher than their phase velocity.

More precisely, the western boundary current becomes unstable when its speed is higher than the phase velocity of Rossby waves, this condition inducing a resonance. Any obstacle causing the current to move away from the coast leads to the formation of quasi-stationary Rossby waves, either because of the line of the coast or the collision with a current flowing in the opposite direction along the coast: the western boundary current orientates gradually eastwards while Rossby waves propagate in the opposite direction.



Rossby waves being non-dispersive, for a given latitude their phase velocity does not depend on frequency. In other words their wavelength is directly proportional to the period. Thus, for the period of 8 years, the wavelength is 2,780 km in latitude 40°, whereas it is only 174 km for the biannual wave.

Where the resonance occurs the cross-wavelet analysis of sea surface height reveals two antinodes in opposite phase (sometimes more), as occurs in the North Atlantic for the 8-year period Rossby wave. Off the Cape Hatteras, the Gulf Stream leaves the eastern North American coast around 35°N. The westernmost anomaly faces east, along the subtropical gyre, followed by the second anomaly headed north-east along the north Atlantic drift. The phase change occurs at the longitude 50°W. These antinodes are always associated with a modulated geostrophic current at the node of the quasi-stationary wave.

The most important consequence of the modulated response of subtropical gyres concerns the long-term variability of climate. The modulated response of subtropical gyres occurs indeed at very specific frequencies that are inherited either of tropical waves or of the oscillations of solar irradiance for longer periods. Direct observation of the modulated response of subtropical gyres can be done from sea surface height for short periods, as has been done for the North Atlantic gyre using the available data sets covering a period twenty years. For longer periods one is interested in sea surface temperature anomalies using data sets this time covering nearly a century and a half. In practice, using sea surface height series jointly with sea surface temperature series, quasi-stationary waves can be observed at frequencies ranging from 2 cycles per year to one cycle for 128 years.

Resonant forcing

The resonance occurs when the speed, facing east, of the steady wind-driven current is higher than the phase velocity, facing west, of the Rossby wave. In this case, the length of the Rossby wave adapts so that its natural period coincides with the forcing period. The ridge of the western antinode is advected of a half apparent wavelength (the wavelength seen by a stationary observer) to the eastern antinode during a half-cycle, the troughs are then translated in the next half-cycle. This leads to eastward transfer of a sequence of warm and cold waters.

Since the phase velocity of the Rossby waves only depends on latitude, the resonance that supposes the adequacy of the frequency and the wavelength occurs at all frequencies. Otherwise, the lack of synchronization between the waves of different frequencies and forcing would inexorably lead to their destruction.

Evolution of short-period gyral Rossby waves

 The observation of quasi-stationary waves for different periods teaches us how the sequence of warm and cold waters is transferred from the western boundary to the east. This is particularly clear for the wave of 8-year period of the North Atlantic. A ridge is formed at the western antinode and a trough at the eastern antinode. The speed of the western modulated zonal current is maximum, facing west. The resulting modulated current, sum of the modulated geostrophic current and the steady wind-driven current, vanishes or reverses. At this time the speed of the eastern modulated current, facing east, is maximum. Thus warm water flows from the western antinode to the eastern antinode while the transfer from the western antinode to the western boundary is low due to geostrophic forces that prevent it.

Half a cycle later, a trough is formed at the western antinode and a ridge at the eastern antinode. The velocity of the resulting current at the western boundary is maximum, being the superposition of two eastward propagating currents. Gradually the trough of the western antinode gives way to a ridge while the speed of the resulting modulated current vanishes or reverses in the east, being the superposition of two currents flowing in opposite directions: this locks out any transfer from the eastern antinode to the western antinode. The same mode of transfer occurs for different frequencies of gyral Rossby waves due to the proportionality between the wavelength and the period.

Growth of the western antinode induces a horizontal pumping effect, as evidenced by the modulated current when it is oriented to the east, west of the basin, being the superposition of the steady wind-driven current and the modulated geostrophic current. This pumping effect drastically increases the flow of the western boundary current. Another consequence is the rapid change in potential vorticity of the western boundary current when it leaves the coast to flow eastward.

Another important aspect of the resonance relates to the simultaneity of the phases of western antinodes. Synchronism of resonances in the five subtropical gyres reflects a common mode of forcing. This is particularly evident for the resonance at the frequency of 2 cycles per year which occurs in April and October, an indicator in the three tropical oceans of forcing by trade wind stress, active during the boreal winter in the northern hemisphere and during the austral winter in the southern hemisphere. The transfer time of the mixed layer from the equator to mid-latitudes is short compared to the period of gyral Rossby waves. Thus, the synchronism persists at mid-latitudes regardless of the period. A final aspect of the resonance involves the coupling of gyral Rossby waves of different frequencies that share the same node at the western boundary of the basin, which again leads to a sub-harmonic mode locking in accordance with the periods 1/2, 1, 4 and 8 years (Pinault, 2018c).

How the gyral Rossby wave moves poleward after leaving the gyre reflects the thermohaline circulation which is generated by differences in temperature and salinity of water masses acting on their density. Thermohaline circulation occurs when the currents approaching the ice cap, water cooled and salty sinks to depths between 1 and 3 km to participate in the deep ocean circulation.

The resonance of long-period gyral Rossby waves


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 ( download.html).

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).

Sea surface temperature anomalies during the warming and cooling phases

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.

Where the earth is warming… or cooling

The earth does not warm evenly: these are the regions impacted by the resonant thermal anomalies of oceanic origin which warm or cool the first.

Baroclinic atmospheric instabilities

The gyral and tropical resonance producing positive or negative surface temperature anomalies these can induce high and low atmospheric pressure systems that affect the climate globally. To quantify the energy transfer from the thermal anomaly to the continents, the unperturbed state of the system in the absence of resonant thermal anomaly (produced by gyral Rossby waves) has to be considered at first, which implies that the average energy captured by the earth is completely re-emitted in space. This is only true if the energy transfers are averaged over one or even several years to remove fingerprints of non-resonant phenomena that cause an imbalance in energy budget during the annual cycle: this is the case, for example, of the formation of sea ice during the winter and its melting during the summer.

Then oceanic thermal anomaly is considered as a perturbation and the perturbed system tends to a new steady state. In the perturbed state the resonant thermal anomalies act either as a heat source, or on the contrary as a heat sink. The perturbation behaves as an isolated thermodynamic system because heat transfers between oceans and continents mainly involve latent heat with low shortwave radiative forcing. Minor effects are induced by a variation of humidity as a result of high or low pressure systems. A slight amplification effect may occur because energy gain resulting from downwelling longwave radiations outweighs the albedo effect due to low clouds when humidity increases.

In such a quasi-isolated thermodynamic system thermal transfers between oceans and impacted continental areas occur until a thermal equilibrium is established between the oceanic and continental anomalies. Processes that lead to this balance, i.e. how high and low pressure are formed from resonant thermal anomalies and move to the continents, result from baroclinic instabilities of the atmosphere. Due to the large heat capacity of seawater relative to the continents and to the feeding or to the draining of warm water at the antinode, resonant thermal anomalies warm or otherwise cool the impacted land areas without weakening significantly. Then, more global processes take over to warm or cool the continents globally as a result of long-period SST anomalies. Thus everything happens as if the perturbed state were deduced from the unperturbed by equalizing both the resonant ocean and land surface temperature anomalies, considered as perturbations.

Although the areas covered by oceanic thermal anomalies are small compared to the ocean surface, they generate atmospheric baroclinic instabilities that have a key role in the transfer of heat between the oceans and continents. However, the mechanisms involved differ depending on whether one considers the tropics or mid-latitudes. As we will see by referring to the rainfall oscillation in the band 5-10 years, thermal transfer, positive or negative, between the resonant oceanic anomalies and impacted continental regions is performed in two main ways (Pinault, 2018a). On the one hand the oceanic temperature anomalies at mid-latitudes deflect tropical cyclones to mid-latitudes or otherwise confine them within the tropical belt according to the sign of the anomalies. On the other hand they promote depressions, anticyclones and troughs at mid-latitudes, these atmospheric phenomena being aroused under the effect of the polar or sub-tropical jet-stream. In all cases, atmospheric baroclinic instabilities may generate heat transfers to the synoptic scale[i], mainly in the form of latent heat.

Rainfall in the band 5-10 years allows to highlight how certain land areas are affected by atmospheric baroclinic instabilities induced by resonant thermal anomalies of oceanic origin. Indeed, the transfer of heat from the oceans to the continents mainly resulting from evaporation and condensation processes according to what has been seen previously, how rainfall varies over time characterizes the impacted regions.

Rainfall oscillation

To highlight how some land areas are affected by atmospheric baroclinic instabilities induced by resonant thermal anomalies of oceanic origin, it is convenient to use monthly rainfall height data which are known since 1901 on the terrestrial scale. Indeed, heat transfer from the oceans to the continents mainly resulting from processes of evaporation and condensation according to what has been seen previously, how rainfall varies over time characterizes the impacted areas.

The resonant oscillation of precipitation, i.e. linked to the resonant origin of oceanic thermal anomalies, is recognizable by its large amplitude in the characteristic bands of the periods 1/2, 4 and 8 years with a low amplitude of the annual oscillation. Resulting from depressions formed or guided by resonant thermal anomalies, rainfall in the impacted areas is distributed evenly between seasons due to the moderator effect of the oceans. Indeed, the thermal anomalies produced by annual waves reach their maximum during the boreal / austral winter.

Amplitude (top) and phase (bottom) of oscillation of reduced precipitation height averaged in the band 8-16 months and over the observation period 1901-2009. The areas subject to strong annual rainfall oscillation are little affected by sea surface temperature anomalies of baroclinic waves.
Amplitude (top) and phase (bottom) of oscillation of reduced precipitation height averaged in the band 8-16 months and over the observation period 1901-2009. The areas subject to strong annual rainfall oscillation are little affected by sea surface temperature anomalies of baroclinic waves.

In contrast, non-resonant precipitation exhibits strong seasonality, whether due to tropical cyclones, tropical and extra-tropical depressions or monsoon, i.e. seasonal changes in atmospheric circulation and precipitation associated with the asymmetric heating of land and sea.

The dynamic of sea surface temperature anomalies and the oscillation of rainfall in a characteristic frequency band reveals the mechanisms leading to atmospheric baroclinic instabilities and the formation of high and low pressure systems to cause oscillation of precipitation. In particular, the analysis of rainfall in the band 5-10 years allows to connect resonant oceanic and atmospheric phenomena unambiguously because sea surface temperature anomalies of 8-year period are well identified, resulting from the highest baroclinic mode in the three tropical oceans, while not being sensitive to ENSO. In addition, the 8-year period is close to the time required to balance the oceanic and terrestrial thermal anomalies of the perturbed state, which reduces their phase shift.

Amplitude (top) and phase (bottom) of oscillation of reduced precipitation height averaged in the band 5-10 years and over the observation period 1901-2009. The regions subjected to a strong rainfall oscillation are heavily affected by sea surface temperature anomalies of baroclinic waves. The phase is expressed relative to the signal –SOI (Southern Oscillation Index). The regions whose phase is delayed with respect to -SOI signal are in phase with the sea surface temperature anomalies. Those whose phase is ahead are in phase opposition with the sea surface temperature anomalies.
Amplitude (top) and phase (bottom) of oscillation of reduced precipitation height averaged in the band 5-10 years and over the observation period 1901-2009. The regions subject to a strong rainfall oscillation are heavily affected by sea surface temperature anomalies of baroclinic waves. The phase is expressed relative to the signal –SOI (Southern Oscillation Index). The regions whose phase is delayed with respect to -SOI signal are in phase with the sea surface temperature anomalies. Those whose phase is ahead are in phase opposition with the sea surface temperature anomalies.

This analysis emphasizes the regions impacted rapidly by the resonant thermal anomalies of oceanic origin where baroclinic instability of the atmosphere originates preferentially. These baroclinic instabilities are most active when the resulting systems of high or low pressure are stimulated and guided by the jet-streams, these ribbons along a winding path through which flows from west to east a strong and rapid airflow at high altitude. The strongest are the polar jet-streams, around latitude 60°, while the subtropical jet-streams are located between 20° and 40° latitude, which explains the key role of subtropical gyres on climate variability.

Since the average annual rainfall height is not relevant in our analysis, rainfall is reduced, i.e. divided by the mean rainfall height. Being dimensionless, the temporal variations of reduced rainfall are homogenized this way on global scale and a causal relationship can be established between sea surface temperature and rainfall anomalies. The amplitude of reduced rainfall anomalies highlights areas primarily impacted by oceanic thermal anomalies. Phase allows to discern transfer mechanisms.

The main areas subject to rainfall oscillation, that is to say those that warm or cool at first, are a) The south-west of North America, b) Texas, c) The south-east of North America, d) The north-east of North America, e) The southern Greenland, f) Europe and Central and Western Asia, g) Region of the Río de la Plata, h) Southwestern and southeastern Australia, i) The Southeast Asia.

The climate at different time scales

State of the art

In many cases paleoclimatology is still at the stage of speculating what may have been the underlying causes of rapid climate transitions, cycles and forcing effects. This leads us to how the climate system responds to external stimuli with its own dynamics. When the internal dynamics of the climate system is consistent with an external stimulus, a resonance phenomenon occurs. The study of such resonances therefore tells us about the internal dynamics of the Earth system, spearheading our understanding of the mechanisms involved in vagaries of climate.

Understand the vagaries of climate becomes possible from the archives of past climate. We have, in fact, over the last decade data of exceptional quality for tracing the climate up to several million years before present (BP), with a resolution of a few years. This technological feat was made possible through the analysis of stable isotopes[ii] in ice cores from the Arctic and Antarctic ice caps and in sediment cores from the ocean trenches.

Total solar irradiance: gross data (TSI) and data filtered into the band 576-1152 years.
Total solar irradiance: gross data (TSI) and data filtered into the band 576-1152 years.

The analysis of ice cores plays a key role in understanding the different mechanisms involved in the natural evolution of climate over the last major cycles of glacial and interglacial periods. The oldest records obtained to date cover 800,000 years, the second half of the Quaternary. Deuterium data 2H obtained from Antarctica Dome C ice core (European Project for Ice Coring in Antarctica EPICA) are used for global mean temperature estimate in the southern hemisphere considering for calibration 5.53‰ 2H/°C [Jouzel et al, 2007]. 18O data obtained from Greenland Summit Ice Cores GISP2 (Greenland Ice Sheet Project 2 Ice Core), Grootes and Stuiver, [1997] are used as proxies of global mean temperature in the northern Atlantic. 18O data are calibrated considering a variation of 0.67‰ 18O/°C [Jouzel and Merlivat, 1984].

Frequency representation of total solar irradiance. The spectrum is broad, with a peak centered on 935 years. High frequencies (short periods) are filtered out.
Frequency representation of total solar irradiance. The spectrum is broad, with a peak centered on 935 years. High frequencies (short periods) are filtered out.

Sediment cores allow the study of the composition of the different layers of sediment accumulated over time on the ocean floor. There are fossil microorganisms composed of calcium carbonate. By studying the “abundance ratio” of 18O and carbon isotope 13C, past climates can be reconstructed going back millions of years. They establish how the oceans have evolved in different climatic periods (temperature, salinity, nutrients, …).

Understanding the temporal variation of cosmic radiation and solar activity during the Holocene allows specifying the solar-terrestrial relationship. 10Be and 14C which are stored in polar ice cores and tree rings, offer the opportunity to reconstruct the history of cosmic radiation and solar activity. In series obtained by Steinhilber et al., [2012] different 10Be ice core records from Greenland and Antarctica are combined with the global 14C tree ring record to provide total solar irradiance variations (W/m2).

The last few million years have been punctuated by many abrupt climate transitions. Many of them occur on time-scales of centuries or even decades. The ability of climate to change abruptly has been one of the most surprising outcomes of the study of Earth history [e.g., Jouzel et al 1987, Taylor et al 1993; Petit et al 1999, Dansgaard et al 1993; Alley, 2000, Jouzel et al 2007].

Climate changes at the planetary scale are responses to external forcing mechanisms, that’s what we strive to show in this chapter. The role of the Sun in climate variability, and specifically solar irradiance fluctuations which reflect the internal dynamics of the Sun but also orbital forcing that alter the net radiation budget of the Earth are frequently referred [e.g. Magny, 1993, Karlén and Kuylenstierna, 1996, Chambers et al., 1999, Bond et al., 2001, Gavin et al., 2011]. Nevertheless, the internal mechanisms involved in long-term climate variability are poorly understood. The idea often mentioned that the deep ocean is the only candidate for driving and sustaining long-term climate change (of hundreds to thousands years) because of its volume, heat capacity, and inertia [e.g. Maslin et al, 2001], can easily be counteracted in the glow of the present results. Indeed, variations in the flow and temperature of deep water that is known to have a direct effect on global climate is a consequence of a mechanism of much greater scope involving the resonance of sub-tropical oceanic gyres.

The Holocene

The climate during Holocene, which began with the interglacial period about 12500 years ago, can be studied from proxies of solar irradiance and Earth’s average temperature in both hemispheres. From the coupling between solar irradiance and global mean temperature will be deduced information about the internal dynamics of the climate system. Furthermore, superimposed on oscillations are several distinct climate steps which appear to be of widespread significance, the most prominent being observed 8.2 Kyr, 5.5-5.3 Kyr and 2.5 Kyr (Kyr=103 years) BP. These events, which are recognized as part of the millennial scale quasi-periodic climate changes, alongside the Dansgaard–Oeschger (D-O)[i] cycles, and are characteristic of the Holocene [O’Brien et al, 1996; Bond et al, 1997; Bianchi and McCave, 1999; de Menocal et al, 2000; Giraudeau et al, 2000].

The glacial-interglacial period

Although the study of Holocene alone brings key elements necessary for understanding current climate variability, new properties of gyral resonance emerge when looking at the glacial-interglacial era, properties that will allow us to raise some mysteries surrounding the climate of the past few million years.

As for the Holocene, the comparative study of Earth’s average temperature and solar irradiance can be carried out to deduce the efficiency of orbital forcing onto the gyral resonance. The Earth’s average temperature can be inferred from the sediment core records after they have been calibrated relative to ice core records. Orbital forcing is calculated from the Milankovitch parameters (Berger, 1992).

Orbital forcing

A perspective on climate variability and orbital forcing, i.e. the precession, the obliquity and the eccentricity, what are commonly named Milankovitch parameters, arises huge problems and there is currently no consensus on the responsible mechanism. On one hand the impact of orbital variations on climate seems not proportional to the amplitude of solar irradiance variations. On the other hand, over the past 800,000 years, the period of oscillation of glacial-interglacial that dominated is 100,000 years, showing it is mainly subject to eccentricity parameter. During the interval from 3.0 to 0.8 million years before our era, the period of 41,000 years prevailed, corresponding to changes in the obliquity of the Earth, what is named the transition problem.

Fourier power spectrum of the three components of orbital forcing – a) Each component is normalized – b) Real spectrum.
Fourier power spectrum of the three components of orbital forcing – a) Each component is normalized – b) Real Spectrum.

The observations suggest that a resonance phenomenon occurs, filtering out some frequencies in favor of others. This hypothesis, issued for several decades, has so far not found plausible physical explanation. This suggests a priori that the gyral resonance is the missing link to solve this riddle. Effectively, there exists a link between orbital forcing and the amplitude of the long baroclinic waves around the gyres.

Owing to the sub-harmonic mode locking of coupled oscillators with inertia, periods of gyral Rossby waves forced by the Milankovitch cycles are a multiple of shorter periods that meet the cycles of solar radiation (Pinault, 2018c). Thus, the sub-harmonics form a sequence whose average periods are multiples of 768 years, the dominant period of the gyral resonance during the Holocene. But, contrarily to what occurs during the Holocene, the narrow frequency bands of orbital forcing complicate the interpretation of the coupling because of the deviation between the frequencies of forcing and the natural frequencies of gyral Rossby waves.

To tune the natural frequency of gyral Rossby waves to the forcing frequency while keeping the wavenumber, the latitude of the centroid has to be shifted. Because several components tuned to their own orbital frequency coexist, instabilities occur around the gyre but the resonance conditions are no longer fulfilled accurately. In this way, the efficiency of forcing depends on the latitude of the centroid of the gyral Rossby waves, which oscillate on either side of their mean value.

Ice and sediment core records allow a detailed study of the glacial-interglacial era in the various characteristic bands.

The current climate (Pinault, 2018e)

The current climate is the subject of intense debate because of the economic and societal implications of global warming. Greenhouse gases, mainly carbon dioxide but also other gases that are rarer but have a high absorption capacity of long-wave radiation re-emitted by the Earth, are accused of playing an important role in anthropogenic climate change. But the real anthropogenic impact on global temperature is difficult to assess accurately because of possible feedback loops and non-linear effects.

Yet the real anthropogenic contribution to climate change can be reliably known by taking into account the oceanic signature in the observed global surface temperature. This can be achieved by subtracting to the instrumental surface temperature the SST anomalies located on the internal antinodes of the Gyral Rossby Waves (GRW), that is, where the depth of the thermocline oscillates according to subharmonic modes. The oceanic contribution in the global temperature can be formally identified after 1870, the date from which the data is available, and before the anthropogenic impact becomes noticeable.

Areas representative of the oceanic signature in the global temperature are to be selected at high latitudes of the five subtropical gyres such that the perturbation ΔT represented by the SST anomalies, averaged over each area, can be deduced from the land surface temperature. The resulting oceanic signature of the global temperature is obtained by doing the weighted average of the SST anomalies in the five subtropical gyres. The weights are indicative of the incidence on the global temperature of the corresponding gyres, that is, they are approximately proportional to the areas of the continents impacted by each of the gyres.

Some continental regions are directly impacted by heat exchanges between the oceans and the continents. To these continental regions can be associated particular SST anomalies by jointly analyzing, both in space and time, the SST and the rainfall oscillations in the 5–10 year band. This method allows representing the inland areas subject to such rainfall oscillation, from which the signatures of the SST and rainfall height anomalies can be unambiguously associated. This is made possible because of the selectivity of both SST and rainfall height anomalies within this band. The active SST anomalies are located on the internal antinodes of the GRWs for the relevant subharmonic modes. The main areas subject to rainfall oscillation at mid-latitudes where condensation / precipitation of water vapor releases the latent heat, are Southwest North America, Texas, Southeastern and Northeastern North America, Southern Greenland, Central and Western Europe and Western Asia, the region of the Río de la Plata, Southwestern and Southeastern Australia, and Southeast Asia.

However, SST anomalies in the 5–10 year band are representative of short-term exchanges between the oceans and the continents resulting from the resonance of GRWs for low subharmonic modes and their inland thermal imprints are evanescent. For these reasons, representative areas of SST anomalies on the long-period internal antinodes, which correspond to higher subharmonic modes, must be judiciously selected to accurately represent the persistence of continental thermal footprints. Accuracy of the SST anomalies averaged over such areas requires the latter are as small as possible not to integrate short-term exchanges the signature of which is masked by long-term exchanges. Short-term and long-term exchanges are governed by short-wavelength and long-wavelength GRWs, respectively. Internal antinodes of short-wavelength GRWs extend from where the western boundary current leaves the coast to the bifurcation of the re-circulating wind-driven current of the gyre and the drift current leaving the gyre. Internal antinodes of long-wavelength GRWs extend all around the gyres so that areas representative of persistent exchanges are necessarily located to the east of the short-period internal antinodes.


Wavelet power of SST in 1958, scale-averaged over the band 48-96 years (64 year average period). Areas that are representative of the oceanic signature of the global temperature are displayed.

Preselected areas are considered as representative of thermal exchanges in the perturbed state of the global climate system when the oceanic perturbation ΔT, that is, the weighted average of SST anomalies over the five subtropical gyres, is a replica of the instrumental global temperature. This can be done before the global temperature is subject to anthropogenic warming. Then, the net anthropogenic contribution in the global temperature can be estimated by subtracting from the latter the weighted sum of the SST anomalies from the global temperature. Actually, the contributions of the SST anomalies are estimated by using the least squares method, that is, by minimizing the sum of the squares of the differences between the instrumental global temperature and the weighted sum of the SST anomalies within a relevant interval of time, the sum of the weighting factors being one.

a) The instrumental temperature Tinst and the weighted sum of SST anomalies (SST Gl). The Mobile Average (MA) over 5 years is displayed – b) Mobile average over 13 years of the SST in the Northern Hemisphere (NA=North Atlantic, NP=North Pacific) – c) Mobile average over 13 years of the SST in the Southern Hemisphere (SA=South Atlantic, SP=South Pacific, SI=South Indian Ocean). Signals are centered.

Oceanic signatures exhibit particular behaviors according to the gyres. In the Figure the instrumental surface temperature is compared to the weighted sum of SST anomalies wNANA+wNPNP+wSASA+wSPSP+wSISI where the weighting factors are wNA=0.50, wNP=0.17, wSA=0.15, wSP=0.13, wSI=0.05. Systematic differences are observed. Beyond 1970, the discrepancies highlight the contribution of the anthropogenic warming. Before 1900 they reflect systematic errors on measurements.

The contribution of the component in the band 48-96 years, whose amplitude of variation is 0.3°C, is significant as well as that in the band 192-576 years, which varies between ±0.1°C. The latter can be considered as a rebound following the little ice age although this subharmonic mode is weakly exogenously forced and behaves as a harmonic of lower frequency GRWs. The 256-year average period GRWs are indeed coupled to those of 128-year average period, which are forced by the Gleissberg cycle of the Sun.

The TSI reconstructed is decomposed into the frequency bands representative of the subharmonic modes, which allows the accurate estimation of the forcing efficiency in the 96-192 year band. Calculated from the maximum oscillation occurring in 1976 for both the TSI and the global temperature, it is 0.21 °C(W/m2)-1. This estimate is very low compared to what happens during the Holocene and corresponds to a low amplitude of the Gleissberg cycle that occurs after 700 years BP and more particularly 300 years BP. To compare, this forcing efficiency is close to the value deduced from the greenhouse effect resulting from the increase in atmospheric water vapor following an increase in the global temperature, that is, 0.22 °C(W/m2)-1.

Components of the weighted sum of the SST anomalies into the bands characteristic of subharmonic modes (surface temperature in the northern hemisphere).
a) The Total Solar Irradiance (TSI) – b) The components within the bands characteristic of subharmonic modes (Coddington et al., BAMS, 2015 doi: 10.1175/BAMS-D-14-00265.1)

Spatial pattern of anthropogenic and natural temperature responses

The weights associated with the SST anomalies that represent at best the gridded surface temperatures Ts are estimated by using the same least squares method as that explained previously. Here, the time interval from which the fitting is performed is 1940-1970, for which the weights are the most precise and the most representative of natural forcing when the surface temperatures are considered individually in the 5°×5° grid. This choice allows us to minimize the noise in the spatial pattern of the natural temperature. However the estimation of the part of the anthropogenic response within Ts by subtracting from the latter the weighted sum of the SST anomalies very little depends on the time interval, 1900-1970 or 1940-1970.

a) The part of the anthropogenic response within the surface temperature Ts in 2015 – b) the natural temperature response in the northern hemisphere and in the southern hemisphere. 1970 is the reference year for temperatures (both natural and anthropogenic responses are supposed to be zero). Areas without dots have no data. Data are provided by the Climatic Research Unit (CRU) at the University of East Anglia,

The temperature response to the natural radiative forcing exhibits a low spatial variability in both hemispheres. In the northern hemisphere it is because the temperature response of the Atlantic and the Pacific oceans in 2015 are close (the temperature increase since 1970 in the Pacific is slightly lower than in the Atlantic). The influence of the Pacific can be seen in the Central Asia whereas North America and Europe are rather influenced by the Atlantic. The southern hemisphere reflects the influence of the warmer Indian Ocean than the other two oceans. Everywhere the natural temperature response is positive because all oceanic signatures increase since 1970s. The increase is most noticeable in the North America and north of 60°N, where it reaches 0.6 ° C.

In addition to the natural response, the part of the anthropogenic response within the instrumental surface temperature Ts shows considerable spatial variability. Lower than 0.8°C and even 0.5°C in Australia, southern South America, eastern North America, northern and Western Europe, and Southeast Asia it overreaches 2°C in Eastern Europe, Russia, Kazakhstan, Mongolia, east of North America, east of Brazil, eastern Africa, Angola, Namibia, even more than 2.5°C north of 70°N. This great disparity questions the nature of the positive feedback loop responsible for such amplification in some regions, regardless the latitude.

Regions primarily impacted by latent heat fluxes from the oceans

The distribution of extra-tropical regions with low anthropogenic impact coincides with those subject to the rainfall oscillation in the 5-10 year band. In the North American continent, these are mainly the regions of eastern and south-western United States. In South America this concerns the countries of the north, and both eastern and southern Argentina. In Europe, these are the South Greenland, and western and northern countries. In Africa, the oscillation concerns the North of the Maghreb countries and South Africa. In Oceania the oscillation is observable almost everywhere.

At mid-latitudes, the 5–10 year band rainfall oscillation characterizes regions impacted by latent heat transfers from the oceans. Within the intertropical convergence zone, monsoonal regions are weakly impacted by anthropogenic warming, that is, the Central America, Western Africa, India and the South-East Asia.

Regions primarily impacted by sensible heat fluxes from the oceans

The areas that are heavily impacted by anthropogenic warming are characterized by a high amplitude of the rainfall oscillation in the 0.5-1.5 year band, exhibiting a strong seasonality. The annual rainfall pattern displays a peak time in late boreal and austral summer, that is, when the difference between the temperature of the air aloft and the surface temperature is the greatest, leading to the greatest potential for instability. In this way the precipitations occur endogenously within the continents, and mainly sensible heat transfer occurs from the oceans.

Positive feedback

Amplifying effects on the anthropogenic temperature response do not depend on the latitude, but on the way in which the thermal exchanges occur between the oceans and the continents. The south Greenland and the eastern Africa are respectively little and heavily impacted, which is contrary to the general trend. On the other hand, contrary to commonly accepted ideas, amplifying effects do not result from the greenhouse effect resulting from the supposed increased atmospheric water vapor associated with the increased temperature. Regions least affected by anthropogenic warming are such that the increase in water vapor with temperature occurs in the free troposphere.

In regions impacted by latent heat fluxes extra-tropical free tropospheric water vapor (above the boundary layer) is mostly associated with the tropospheric circulation. An increase in the temperature of the atmosphere increases its water-holding capacity, in proportion with the Clausius-Clapeyron relation. In contrast, arising from endogenous primary moisture sources, free tropospheric water vapor in regions impacted by sensible heat fluxes is mainly controlled by the difference between the temperature of the air aloft and the surface temperature.

Those findings reinforce the idea that the climate response is closely linked to the top of atmosphere flux as suggested by the spatial pattern of climate feedback. The only way indeed to explain the spatial distribution of the anthropogenic temperature response is to assign a driving role in the amplification effect of the high troposphere cloud cover, which involves the lapse rate, surface albedo and cloud feedbacks.

What is the future of our planet?

The Response of the ice cap to global warming

One of the most important consequences of global warming is melting the polar ice. This is followed with the greatest attention. In particular, the satellite measurement of the concentration of sea ice by microwaves provides relevant information on the temporal evolution of the polar ice: the amplitude of the variations in the percentage of ice highlights the most impacted areas.

Cumulative effects of gyral Rossby waves and human activity

The warming observed since 1970 is mainly attributable to human activities, the evolution of which will be decisive in the coming decades. No inflection is perceptible on the growth curve of the last 50 years, everything suggests that Tmg will increase by nearly 1 ° C over the next 50 years if the rise in production of greenhouse gases does not weaken.

The natural variability of Tmg initiates a decrease due mainly to the harmonic of gyral Rossby waves of 64 year average period. However it remains low (a few tens of degrees) compared to the anthropogenic component and cannot be enough to reverse the trend.

Inform, persuade, beyond the dogmas and myths

The study of resonantly forced planetary waves of very long wavelength, which has been ignored so far, is promising in physical oceanography and climatology. This may be a step forward in the areas that are still poorly understood, such as the formation and stability of the subtropical gyres and long and very long-term climate variability, citadels that remained invincible for more than half a century and therefore, can be overcame only by using new concepts. This article provides a physical basis for a resonant phenomenon that many researchers have foreseen for a long time, allowing to explain how the efficiency of solar and orbital forcing could vary by a factor of 5 during glacial-interglacial periods, as well as the contribution of the natural variability in the global warming that prevailed since the beginning of the 20th century.

Taking into account the natural variability of the climate in the warming confirms the importance of the anthropic impact and specifies its evolution. A lot of work remains to be done but we hope that this article will help raise awareness and mobilize those in charge of public policies.

To make climatology a science, a fight every moment…


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[i] Heinrich events, first described by marine geologist Hartmut Heinrich, occurred during the last glacial period. During such events, numerous icebergs broke off from glaciers and traversed the North Atlantic. The icebergs contained rock mass eroded by the glaciers, and as they melted, this matter was dropped onto the sea floor as “ice rafted debris”.

[i] Dansgaard–Oeschger events (often abbreviated D–O events) are rapid climate fluctuations that occurred during the last glacial period.

[i] The dew point or dew temperature is the lowest temperature at which an air mass may be subject, with fixed pressure and moisture, without the occurrence of a water liquid formed by saturation.

[ii] The term stable isotopes usually refers to isotopes of the same element. The relative abundance of such stable isotopes can be measured experimentally (isotope analysis), yielding an isotope ratio: relative abundances are affected by isotope fractionation in nature, hence their interest in geochemistry.

[i] In meteorology, synoptic scale phenomena are characterized by a length of several hundred to several thousand kilometers and a period of several days.

[ii] The horse latitudes, between 30 and 35°N or S, define an area highs, the quiet zone created by the subtropical descending column of the Hadley cell. It is said that the term comes from the days when Spanish sailing vessels transported horses to the West Indies. In the absence of wind in these latitudes prolongation of trip resulted in shortages of food and water and the crews were sometimes forced to throw horses overboard or kill them to prevent starvation on board.

[i] 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.

[i] 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.

[i] Upwelling. Here, this term indicates an upward flow of deep water, therefore cold. The upwelling is associated with the functioning of tropical resonant waves.

[i] Western boundary currents. Western boundary currents, warm, deep, narrow and fast are formed along the western boundary of the ocean. They carry warm water from the tropics to the poles, forming the western branch of the subtropical gyres. It is the Gulf Stream (North Atlantic), the Brazil Current (South Atlantic) the Agulhas (South Indian Ocean), the Kuroshio (North Pacific), and the western boundary currents of the subtropical gyre in the South Pacific.

[ii] Jet-streams are fast winds aloft blowing from west to east. Along a curved and sinuous path, they play a major role in atmospheric circulation as they participate in the formation of depressions and anticyclones at middle latitudes, which then move under these powerful atmospheric currents.

[iii] 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.

[i] 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.

[ii] 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.

 [i] SOI (Southern Oscillation Index). The SOI is the amplitude of the Southern Oscillation; it is a measure of the monthly change in the normalized atmospheric pressure difference at sea level between Tahiti and Darwin (Australia).

[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.

[i] 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.

[i] The Walker Circulation, El Niño, La Niña. In the tropics, the direct airflow on the surface towards the equator (called Hadley cell) forms the inter-tropical convergence zone. The Coriolis force is small at these latitudes but enough to divert to the west the circulation, giving the trade winds (north-east in the northern hemisphere and south -east in the south). The Humboldt Current from Antarctica cools the coast of South America. So there is a large temperature difference between the western and eastern Pacific that leads to direct circulation similar to Hadley circulation (air masses rise close to Asia and Australia and down along the coast of South America). If convective activity decreases in the western Pacific, the eastward aloft flow decreases or stops, cutting cold air intake in the eastern Pacific and the return surface flow weakens. The opposite of El Niño is La Niña. Convection in the western Pacific increases in this case which amplifies cell Walker bringing colder air along the coast of America.

[i] The albedo, which is the ratio of solar energy reflected by a surface to the incident solar energy is high on the polar caps (about 60%) and much lower on the oceans (5-10%). For a period of cooling the polar caps extend, increasing the albedo. The planet reflects more solar radiation, absorbs less, which enhances cooling. Warming has the opposite effect: the global warming melts the polar ice, which reduces the albedo and thus increases the temperature of the planet.