Weakening of the Geostrophic Component of the Gulf Stream: A Positive Feedback Loop on the Melting of the Arctic Ice Sheet

: The North Atlantic gyre experiences both a signiﬁcant temperature rise at high latitudes and a considerable weakening of the geostrophic component of the Gulf Stream, which is reﬂected by the 64-year fundamental gyral Rossby wave (GRW). This singular behavior compared to the South Atlantic and South Indian Ocean gyres highlights a feedback loop of Arctic ice sheet melting on mid-latitude Atlantic Ocean temperature. The warming of the northern oceanic gyre at high latitudes due to the retreat of Arctic ice sheet via the Labrador Current decreases the thermal gradient between the high and low latitudes of the north Atlantic gyre. This results in a weakening of the geostrophic forces at the basin scale and a reduction in the amplitude of the GRWs. Reducing the amplitude of the variation of the upward and downward movement of the pycnocline modiﬁes air–sea interactions, weakening vertical mixing as well as the evaporation processes and the departure of latent heat when the pycnocline rises. The resulting thermal anomaly stretching along the Gulf Stream from where it leaves the American continent is partly transferred to the Arctic sea ice via the drift current and thermohaline circulation, which contributes to the retreat of the ice sheet, and the closing of the feedback loop. The 64-year-period GRW should disappear around 2050 if its damping continues linearly, favoring an increasingly rapid warming of the ocean at mid-latitudes. These interactions are less acute in the southern hemisphere due to the circumpolar current.


Introduction
The Arctic exhibits the most dramatic warming in recent decades, known as the Arctic amplification of climate change [1,2].Accelerating ice mass loss from the Arctic ice sheet has multiple effects on ocean circulation and climate, as shown by numerous studies (e.g., [3]).The relationship between Arctic sea ice melting and climate change involves sea-atmosphere interactions and positive feedbacks.This concerns the incidence of ice sheet melting on atmospheric circulation, the Atlantic meridional overturning circulation (AMOC), and the sea subsurface temperature from where the Gulf Stream leaves the north American continent to re-enter the north Atlantic gyre.In this case, the sea-atmosphere interactions and positive feedbacks are attributed to long-period Rossby waves winding around the subtropical gyres, i.e., the Gyral Rossby Waves (GRWs) [4].This approach is based on the evolution of the geostrophic forces of the North Atlantic basin, the thermohaline circulation appearing as an overflow of the North Atlantic drift current.Within this context, the thermohaline circulation loses its driving role.

Incidence of Ice Sheet Melting on Atmospheric Circulation
The loss of Arctic sea ice can change the atmospheric circulation, enhancing the sea-air heat flux exchange.This affects climate through a positive sea ice feedback [5,6], atmospheric temperature feedback [7], and cloud and water vapor feedback [8].However, nothing has been quantitatively established on the impact of these feedbacks on the atmospheric circulation and on the climate at middle and high latitudes [9].
The loss of sea ice over the Barents-Kara Seas can induce positive geopotential height anomalies over the Arctic region and negative anomalies over Eurasia, similar to a wave train [10].Significant changes in the blocking highs [11,12] or the Siberian high may lead to Eurasian climate anomalies [13,14].

Incidence of Ice Sheet Melting on the Atlantic Meridional Overturning Circulation
The transient response of global ocean circulation to increased freshwater forcing associated with the melting of the Greenland Ice Sheet has been studied [15].Increased freshwater runoff from Greenland results in a basin-wide response of the North Atlantic on timescales of a few years via eastward propagating equatorial Kelvin waves, and westward propagating Rossby waves.In addition, modified air-sea interactions play a fundamental role in the Atlantic circulation in its subpolar and subtropical gyres.
Most of the studies on the impact of melting Arctic sea ice on ocean circulation are focused on the AMOC, which is the outlet of the branch of the Gulf Stream flowing north, i.e., the North Atlantic current.It results from differences in temperature and salt content.This global process contributes to the world's oceans mixing, distributing heat and energy around the earth.
The AMOC's hysteresis response to freshwater perturbations may endanger its stability properties as a result of the modification of the water's density [16].However, long-term trends are debated, although there were large and rapid changes in the AMOC towards the end of the last ice age [17][18][19][20].Sensitivity experiments suggest that Greenland discharge has implications beyond the AMOC, heavily impacting subsurface temperatures in the Gulf of Saint Lawrence [21].On the other hand, the interannual sea surface salinity variability observed in the subpolar gyre is attributed both to excess ice melt and to AMOC variability [22].Mass redistribution resulting from ice melt and land hydrology causes both land uplift and sea-level rise along the northwestern coast of the North Atlantic Ocean, which is likely caused by variability in the Labrador Sea spreading south [23].
The Beaufort Gyre circulation system, which is the dominant circulation in the Canadian Basin, intensified and widened while accumulating freshwater during the first part of the 2000s.A recent work shows that over the past decade the Beaufort Gyre has transitioned into a quasi-stable state.However, continued thinning of the cold halocline layer could modulate the present stable state, impacting the AMOC [24].

Incidence of Ice Sheet Melting on the Geostrophic Component of the Gulf Stream
The velocity of the modulated geostrophic current induced by GRWs is added or subtracted from the velocity of the anticyclonic steady current of the subtropical gyres induced by wind curl.The amplitude of variation of the speed of the geostrophic current is closely dependent on the temperature gradient between the high and low latitudes of the subtropical gyres.Particularly in the north Atlantic, the polar regions are warming faster than other regions located at mid-latitudes, which reduces the temperature difference between the low and high latitudes of the north Atlantic gyre under the influence of the Labrador Current at mid-latitudes.The fact that this temperature difference is no longer as great as before weakens the circulation of the modulated geostrophic component of the polar and radial currents of the gyre.The same applies to the western boundary current, i.e., the Gulf Stream.

Very-Long-Period Gyral Rossby Waves
Very-long-period GRWs wind around subtropical gyres for half a wavelength, then propagate poleward.Because of their very long wavelength, these waves are approximately non-dispersive.Propagating cyclonically, their phase velocity is lower than the velocity of the stationary anticyclonic wind-driven circulation in which they are embedded in such a way that they appear to be propagating anticyclonically to a fix observer [4].
GRWs are governed by the linearized equations of motion in stratified oceans, i.e., the equation of continuity, the momentum equations, and the equation of the conservation of the potential vorticity.Instead of using the β-plane approximation, as is used for shortperiod baroclinic waves, these equations are solved by using a β-cone approximation so that the circle in contact with the terrestrial sphere and the cone, which is tangential to it, come as close as possible to the gyre.In this way, Rossby waves no longer owe their existence to the gradient β of the Coriolis parameter relative to the latitude, but to the gradient β of the Coriolis parameter relative to the radius of the circle, i.e., the mean radius of the gyre.
Multifrequency gyral Rossby waves wind around the gyres.They are coupled since they share the same modulated geostrophic currents around the gyres.Solving the equations of vertical motion for the normal modes of oscillation, it can be seen that GRWs have the property of resonating in subharmonic modes [25,26].This is the sine qua non condition to ensure their stability so that on average each of these coupled waves receives as much energy as it yields.This stability condition applies to any dynamic system of interacting entities.It can be established from the Caldirola-Kanai equation of coupled oscillators [27].The period of the fundamental GRW is 64 years in the North Atlantic gyre; therefore, it makes a revolution in 64 years in average.This is the propagation time required to travel one half of the apparent wavelength.
The oscillation of the pycnocline results in a variation in the temperature of the subsurface waters of the subtropical gyres and even beyond the gyres, due to the modulated radial currents.Consequently, GRWs have a driving role on the global temperature of the surface of the earth because, under the effect of vertical motions of the pycnocline, they stimulate ocean-atmosphere exchanges.

Thermal Gradient between the High and Low Latitudes of the Subtropical Gyres
The geostrophic forces exerted on the GRWs are all the stronger as the thermal gradient between the low and high latitudes of the gyre is high.Indeed, the oscillation amplitude of the pycnocline is all the higher as the thermal gradient is steeper where the western boundary current flows from the tropics towards the high latitudes.As shown by the linearized equations of motion, positive feedback occurs from the modulated component of the polar current of the gyre, which is proportional and in phase with the oscillation of the pycnocline, whereas the radial current is in quadrature.The more the modulated component of the polar current accelerates while flowing anticyclonically, the more the pycnocline lowers due to the influx of warm waters from the tropics.In return, the lower the pycnocline, the more the western boundary current accelerates, hence the positive feedback loop.This positive feedback loop responds all the more strongly as the thermal gradient is steeper because this increases the thermal amplitude at high latitudes of the gyre (the positive feedback would disappear if the thermal gradient were zero).The opposite occurs when the modulated component of the polar current flows cyclonically in such a way that the resultant velocity of the polar current of the gyre, i.e., the sum of the geostrophic and the wind-driven currents, decreases.

Incidence of the Gyral Rossby Waves on Climate: Some Examples
The mixing of the oceans influences many elements of the Earth system, including circulation, planetary-scale climate, and the ecosystem.In particular, the efficiency of mixing has a direct effect on the temperature of subsurface waters.While the seasonal evolution of ocean mixing results from stirring by mechanical forces (wind and tides), destabilizing buoyancy forces (cooling or ice freezing at sea surface), and the resulting long-term variations in the efficiency of mixing resulting from the driving processes of the modulated component of the western boundary currents, that is, the amplitude of the pycnocline oscillation, has a strong impact on climate.
The Atlantic multi-decadal Oscillation (AMO) is a North Atlantic basin-wide Sea Surface Temperature fluctuation correlating with the 64-year-period fundamental wave of the North Atlantic gyre [4].The AMO has wide-ranging climate impacts and is linked to variability of Sahel rainfall, North and South American hydroclimate, and the Indian monsoon [28,29].
The climate system responds resonantly to solar and orbital forcing in subharmonic modes, leading to new hypotheses on the evolution of the past climate being advocated.This implicates the deviation between forcing periods and natural periods according to the subharmonic modes, and the polar ice caps [26].The glacial-interglacial cycles can thus be explained, as well as the modification of their duration as occurred during the Mid-Pleistocene Transition (MPT), when GRWs were considered to mediate the climate system in response to external forcing.
By breaking down speleothems used as archives of past climate into subharmonic modes, the following were shown: (1) how the weakening of ENSO activity occurred in the mid-Holocene and (2) the quasi-resonance of the equatorward migration of the summer Inter-Tropical Convergence Zone (ITCZ) during the Holocene as a result of the progressive decrease in the thermal gradient between the low and high latitudes of the gyres [30].
The resonant forcing of GRWs according to high subharmonic modes helps explain the change in atmospheric circulation that occurred in the North Atlantic during the middle Holocene.High-pressure systems prevailed over the North Atlantic in the first half of the Holocene while low-pressure systems resulting from the baroclinic instabilities of the atmosphere dominated during the second half, favoring the growth of glaciers in Scandinavia [31].
The paper is organized as follows.In addition to the data (Section 2.1), Section 2 discusses the method used to separate the contribution of anthropogenic warming from the natural evolution of the Sea Surface Temperature (SST).For this, the time series are detrended by using polynomial approximations of data (Section 2.2), and filtered in the period range of 48-96 years, from which the evolution of the 64-year period SST, that is, a proxy of GRWs, has been deduced since 1900 (Section 2.3).The results are displayed in Section 3.They relate to the comparative study of the evolution of the SST of 3 subtropical gyres at mid-latitudes, that is, the North and South Atlantic and the Indian Ocean.In Section 4, the discussion relates to the particularity of the North Atlantic gyre compared to the other two (Section 4.1), which is attributed to a positive feedback loop closely related to the evolution of the temperature gradient between the high and low latitudes of the gyres (Section 4.2).Section 5 is devoted to the conclusion.

De-Trending and Wavelet-Filtering of Data
De-trending of yearly SST is performed by fitting the data with a 4th-degree polynomial.The degree k of the polynomials is optimized using the Akaike information criterion (AIC), consisting of minimizing the AIC applied to least squares model fitting: where n is the number of observations, σ2 = RSS/(n − k), and the RSS being the residual sum of squares [33].
Applying the AIC allows for the optimization of the number of degrees of freedom of the polynomial that best fits the yearly SST in the North Atlantic (averaged over the region ), which deals with both the risk of overfitting and the risk of underfitting (Figure 1).Here, degree 4 of the polynomial corresponds to the inflection point of the AIC (Figure 1a).Choosing the point of inflection rather than the minimum amounts to favoring the principle, according to which it is better to underestimate the slope of the temperature at the end of the series rather than the contrary (Figure 1b).In the latter case, there would be a risk of correlation between the result of the adjustment of the polynomial and the wavelet-filtering of the series in the period range of 48-96 years (Figure 1c) [34].
where n is the number of observations,  /   , and the RSS being the residual sum of squares [33].
Applying the AIC allows for the optimization of the number of degrees of freedom of the polynomial that best fits the yearly SST in the North Atlantic (averaged over the region [76°W, 65°W] × [37°N, 41°N]), which deals with both the risk of overfitting and the risk of underfitting (Figure 1).Here, degree 4 of the polynomial corresponds to the inflection point of the AIC (Figure 1a).Choosing the point of inflection rather than the minimum amounts to favoring the principle, according to which it is better to underestimate the slope of the temperature at the end of the series rather than the contrary (Figure 1b).In the latter case, there would be a risk of correlation between the result of the adjustment of the polynomial and the wavelet-filtering of the series in the period range of 48-96 years (Figure 1c) [34].

Estimation of the Weakening of the GRW Amplitude
GRWs open a new field of investigation on the interconnections between the acceleration of the loss of ice mass of the Arctic ice sheet subject to the North Atlantic Drift Current and the increase in SST along the North Atlantic gyre from where the western boundary current leaves the north American continent.This warming is accompanied by a weakening of the geostrophic component of the polar and radial currents of the gyre.To highlight this weakening, the 64-year period of the GRW, that is, the fundamental GRW in the North Atlantic, is represented through the SST anomaly in the period range of 48−96 years (Figure 2).Indeed, this SST anomaly results from the up and down motion of the pycnocline along the gyre, which modifies the air-sea interactions.On the other hand, the modulation of the polar and radial geostrophic currents of the gyre contributes to the mixing of subsurface waters along the gyre.
In Figure 2, the maxima correspond to the anticyclonic circulation of the modulated polar current of the GRW, whereas the minima corresponds to the cyclonic circulation.Only the anticyclonic circulation, for which the velocity of the modulated polar current is added to the velocity of the steady wind-driven current, is considered (the resulting velocity of the polar current is much less reproducible when its modulated component propagates cyclonically because the velocities of the wind-driven and the geostrophic currents subtract each other).

Estimation of the Weakening of the GRW Amplitude
GRWs open a new field of investigation on the interconnections between the acceleration of the loss of ice mass of the Arctic ice sheet subject to the North Atlantic Drift Current and the increase in SST along the North Atlantic gyre from where the western boundary current leaves the north American continent.This warming is accompanied by a weakening of the geostrophic component of the polar and radial currents of the gyre.To highlight this weakening, the 64-year period of the GRW, that is, the fundamental GRW in the North Atlantic, is represented through the SST anomaly in the period range of 48-96 years (Figure 2).Indeed, this SST anomaly results from the up and down motion of the pycnocline along the gyre, which modifies the air-sea interactions.On the other hand, the modulation of the polar and radial geostrophic currents of the gyre contributes to the mixing of subsurface waters along the gyre.

Results
In order to better highlight the weakening of the modulated components of the Gulf Stream, the crest of the SST anomaly with a period of 64 years is first represented.This is achieved by using the Morlet wavelet analysis of the SST in the period range of 48 to 96 years (Figure 3) [34].
The main part of the Gulf Stream concerned is located from where it leaves the North American continent to where its split between the branch continuing eastward following the North Atlantic gyre and the branch migrating northwards, which gives rise to the North Atlantic Drift Current (Figure 3a).In Figure 2, the maxima correspond to the anticyclonic circulation of the modulated polar current of the GRW, whereas the minima corresponds to the cyclonic circulation.Only the anticyclonic circulation, for which the velocity of the modulated polar current is added to the velocity of the steady wind-driven current, is considered (the resulting velocity of the polar current is much less reproducible when its modulated component propagates cyclonically because the velocities of the wind-driven and the geostrophic currents subtract each other).

Results
In order to better highlight the weakening of the modulated components of the Gulf Stream, the crest of the SST anomaly with a period of 64 years is first represented.This is achieved by using the Morlet wavelet analysis of the SST in the period range of 48 to 96 years (Figure 3) [34].
the region [76°W, 65°W] × [37°N, 41°N]).The attenuation of the wave is estimated from the slope of the red line joining the two successive maxima of the gyral Rossby wave within the period range of 48−96 years.

Results
In order to better highlight the weakening of the modulated components of the Gulf Stream, the crest of the SST anomaly with a period of 64 years is first represented.This is achieved by using the Morlet wavelet analysis of the SST in the period range of 48 to 96 years (Figure 3) [34].
The main part of the Gulf Stream concerned is located from where it leaves the North American continent to where its split between the branch continuing eastward following the North Atlantic gyre and the branch migrating northwards, which gives rise to the North Atlantic Drift Current (Figure 3a).In Figure 4a, the slope of the SST in 2020 is represented according to the longitude by using the first derivative in 2020 of the fourth-degree polynomial approximation fitted from 1900 to 2022.Regarding the North Atlantic, the seven adjacent regions (Figure 3b) along the crest represented in Figure 3a are used.The slope of the SST increases up to almost 0.12 °C/year between where the Gulf Stream leaves the continent and its ramification nearly 50°W.Then, the slope decreases from 0.06 to 0.04 °C/year until longitude 26.5°W, from where the gyre curves towards the south off the Iberian coast.
In Figure 4b, the relative amplitude variation of the 64-year SST anomaly, which is interpreted as representative of the amplitude of the fundamental GRW in the North Atlantic, increases between 1943 and 2010 from −1.14%/year at 70.5°W to −0.96%/year at The main part of the Gulf Stream concerned is located from where it leaves the North American continent to where its split between the branch continuing eastward following the North Atlantic gyre and the branch migrating northwards, which gives rise to the North Atlantic Drift Current (Figure 3a).
In Figure 4a, the slope of the SST in 2020 is represented according to the longitude by using the first derivative in 2020 of the fourth-degree polynomial approximation fitted from 1900 to 2022.Regarding the North Atlantic, the seven adjacent regions (Figure 3b) along the crest represented in Figure 3a are used.The slope of the SST increases up to almost 0.12 • C/year between where the Gulf Stream leaves the continent and its ramification nearly 50 • W.Then, the slope decreases from 0.06 to 0.04 • C/year until longitude 26.5 • W, from where the gyre curves towards the south off the Iberian coast.
54.5°W.Note that 64 years is the average period of the fundamental wave.Here, the time elapsed between two maxima is around 67 years.After the ramification, the relative variation stabilizes close to −0.80%/year.
In Figure 4c, the relative amplitude variation of the 64-year SST anomaly in the North Atlantic is represented versus the slope of the SST in 2020.It appears that the attenuation of the fundamental wave is not very sensitive to the temperature increase observed in 2020, as it is around −1.0%/year.In addition to showing that the attenuation is characteristic of the North Atlantic gyre, this plateau demonstrates the de-correlation of the estimation of the slope of the SST in 2020, and the estimation of the amplitude of the 64-yearperiod fundamental GRW.The same approach is used for the South Atlantic and South Indian Ocean gyres, whose mean periods of the fundamental GRW are also 64 years.The crests of the SST anomalies in the period range 48 to 96 years are represented in Figures 5 and 6, as well as the selected regions for the estimation of the slope of SST in 2020 and the attenuation of the 64-year-period SST anomalies.In Figure 4b, the relative amplitude variation of the 64-year SST anomaly, which is interpreted as representative of the amplitude of the fundamental GRW in the North Atlantic, increases between 1943 and 2010 from −1.14%/year at 70.5 • W to −0.96%/year at 54.5 • W. Note that 64 years is the average period of the fundamental wave.Here, the time elapsed between two maxima is around 67 years.After the ramification, the relative variation stabilizes close to −0.80%/year.
In Figure 4c, the relative amplitude variation of the 64-year SST anomaly in the North Atlantic is represented versus the slope of the SST in 2020.It appears that the attenuation of the fundamental wave is not very sensitive to the temperature increase observed in 2020, as it is around −1.0%/year.In addition to showing that the attenuation is characteristic of the North Atlantic gyre, this plateau demonstrates the de-correlation of the estimation of the slope of the SST in 2020, and the estimation of the amplitude of the 64-year-period fundamental GRW.
The same approach is used for the South Atlantic and South Indian Ocean gyres, whose mean periods of the fundamental GRW are also 64 years.The crests of the SST anomalies in the period range 48 to 96 years are represented in Figures 5 and 6, as well as the selected regions for the estimation of the slope of SST in 2020 and the attenuation of the 64-year-period SST anomalies.The same approach is used for the South Atlantic and South Indian Ocean gyres, whose mean periods of the fundamental GRW are also 64 years.The crests of the SST anomalies in the period range 48 to 96 years are represented in Figures 5 and 6, as well as the selected regions for the estimation of the slope of SST in 2020 and the attenuation of the 64-year-period SST anomalies.Only three regions can be selected for the South Atlantic gyre given the rapid damping of the 64-year-period SST anomaly while the south-propagating Brazil Current meets the north-propagating Malvinas Current (Figure 5a).The SST slope is nearly 0.03 °C/year (Figure 4a), and the weakening of the 64-year period GRW is nearly −0.4%/year between 1923 and 1990 (Figure 4b,c).
Regarding the South Indian Ocean gyre, five bevel regions are defined (Figure 6b) from the crest of the 64-year-period SST anomaly displayed in Figure 6a.Located halfway between Madagascar and Australia, this thermal anomaly is of low amplitude since it barely exceeds 0.5 °C against 2.54 °C for the North Atlantic gyre.The slope of the SST in 2020 was low, varying between 0.00 and 0.03 °C/year (Figure 4a).In Figure 4b,c, the relative amplitude variation of the 64-year SST anomaly is positive, between 0.47 and Only three regions can be selected for the South Atlantic gyre given the rapid damping of the 64-year-period SST anomaly while the south-propagating Brazil Current meets the north-propagating Malvinas Current (Figure 5a).The SST slope is nearly 0.03 • C/year (Figure 4a), and the weakening of the 64-year period GRW is nearly −0.4%/year between 1923 and 1990 (Figure 4b,c).
Regarding the South Indian Ocean gyre, five bevel regions are defined (Figure 6b) from the crest of the 64-year-period SST anomaly displayed in Figure 6a.Located halfway between Madagascar and Australia, this thermal anomaly is of low amplitude since it barely exceeds 0.5 • C against 2.54 • C for the North Atlantic gyre.The slope of the SST in 2020 was low, varying between 0.00 and 0.03 • C/year (Figure 4a).In Figure 4b,c, the relative amplitude variation of the 64-year SST anomaly is positive, between 0.47 and 0.04%/year, leading to the amplitude of the 64-year period GRW increases slightly between 1938 and 1996.
The study of the behavior of the fundamental GRW in the North and South Pacific cannot be carried out because its average period is 128 years, which would require much longer SST data.The slope of the SST in 2020 is included between 0.031 and 0.041 • C/year in the North Pacific, and between 0.025 and 0.045 • C/year in the South Pacific.

A Peculiarity of the North Atlantic
The examination of Figure 4c highlights the responses of the three oceans, i.e., the north and south Atlantic and the Indian oceans, to the warming and weakening of the 64-year-period GRW.In the southern hemisphere, the two oceans are moderately affected because the warming of subsurface waters estimated from the slope of the SST in 2020 does not exceed 0.04 • C/year.On the other hand, the North Atlantic Ocean is very affected, as the slope of the SST in 2020 exceeds 0.1 • C/year between where the Gulf Stream leaves the American continent and its ramification at nearly 50 • W.
This difference in behavior can be interpreted from the positive feedback of the amplitude of the fundamental GRW to the thermal gradient between the high and low latitudes of the three subtropical gyres.In peculiar, the melting of the Greenland ice sheet reflects a significant warming, which decreases the thermal gradient between the high and low latitudes of the northern oceanic gyre due to the influence of the Labrador Current at mid-latitudes.This results in a considerable weakening of the fundamental GRW, almost −1%/year between 1943 and 2010, which means that the GRW will disappear in 2050.On the other hand, since the Antarctic Sea ice being less affected by the global warming, the thermal gradient between the high and low latitudes of the two gyres in the southern hemisphere is also less affected.The feedback of the amplitude of the fundamental GRWs is therefore less.Regarding the two gyres in the southern hemisphere, the difference in the responses of the amplitude of the fundamental GRW to global warming is explained by the difference in the average SST between the two gyres at high latitudes, 12.1 • C in the South Atlantic and 20.4 • C in the south Indian ocean.Thus, the response of GRWs to positive feedback is less in the southern Indian Ocean than in the southern Atlantic Ocean (mean SST is 16.6 • C in the northern Atlantic Ocean).

Positive Feedback Loop and the Arctic Amplification of Climate Change
The influence of the longitudinal limit of the sea ice on the amplitude of the oscillation of the pycnocline at the high latitudes of the gyre can therefore be assessed from the amplitude of the oscillation of the temperature of the subsurface water.Benefiting from reliable SST data from 1900, it is the reason why the influence of the longitudinal limit of sea ice on the amplitude of the pycnocline oscillation, i.e., on the efficiency of long-term ocean mixing, can be obtained by measuring the amplitude of the fundamental GRW along the Gulf Stream.This is made possible by the fact that the fundamental wave of the North Atlantic gyre has a period of 64 years, which is compatible with the length of available SST data.
The particular behavior of the North Atlantic compared to the two oceans of the southern hemisphere suggests that the response of the North Atlantic gyre to the melting of the Arctic Sea ice reflects powerful feedback.Indeed, the weakening of the geostrophic component of the Gulf Stream has modified air-sea interactions.Therefore, this has resulted in a weakening of the vertical mixing of subsurface waters as well as of the exchanges between the ocean and the atmosphere, as they are essentially dominated by evaporative processes and the departure of latent heat during the ascending phase of the pycnocline [4].This is the reason why the weakening of GRWs leads to warming of the ocean, which, in turn, contributes to the further attenuation of the fundamental GRW by decreasing the thermal gradient between the high and low latitudes of the gyre under the influence of the Labrador Current at mid-latitudes.This process, leading to positive feedback, does not seem to be triggered in the two oceans of the southern hemisphere in the presence of the circumpolar current.On the contrary, the warming of the Gulf Stream at high latitudes of the gyre has a direct impact on the melting of the Arctic Sea ice via the North Atlantic Drift Current and the thermohaline circulation.Apart from seasonal cycles, the thermal gradient between low and high latitudes of the North Atlantic gyre is essentially governed by the latitude of the edge of the sea ice.This positive feedback contributes to the Arctic amplification of climate change.Indeed, underlying physical mechanisms that can give rise to Arctic amplification include both local feedbacks and changes in poleward energy transport [35].

Conclusions
The North Atlantic gyre experiences both a significant temperature rise at high latitudes and a considerable attenuation of the 64-year fundamental GRW.This singular behavior, compared to the South Atlantic and South Indian Ocean gyres, highlights a feedback loop of Arctic ice sheet melting as mid-latitude Atlantic Ocean temperatures rise, which is probably a substantial contributor to the Arctic amplification of climate change, which is strengthened by local feedbacks.
By reducing the thermal gradient between the high and low latitudes of the subtropical gyres, the warming of the mid-latitude oceans weakens the amplitude of the GRWs and, consequently, the vertical mixing of subsurface waters.The modification of air-sea interactions leads to a reduction in the cooling of the subsurface waters resulting from evaporation during the ascending phase of the pycnocline, hence the net increase in the temperature of sea water above the pycnocline at mid-latitudes.
In the north Atlantic, warming subsurface waters of the Gulf Stream interacts with Arctic Sea ice via the drift current and the thermohaline circulation, accelerating sea ice melting and promoting ice sheet retreat.In turn, this contributes to the warming of the gyre at high latitudes via the Labrador Current, which reduces the thermal gradient between the high and low latitudes of the north Atlantic gyre.The 64-year-period GRW should disappear around 2050 if its damping continues linearly, favoring an increasingly rapid warming of the ocean at mid-latitudes.These interactions are less acute in the southern hemisphere due to the circumpolar current.

Figure 1 .
Figure 1.(a) Akaike information criterion (AIC) applied to polynomials fitting the yearly SST in (b), and wavelet-filtering of the residual (de-trended SST using a polynomial of degree 4 in the period range of 48−96 years in (c).

Figure 1 .
Figure 1.(a) Akaike information criterion (AIC) applied to polynomials fitting the yearly SST in (b), and wavelet-filtering of the residual (de-trended SST using a polynomial of degree 4 in the period range of 48-96 years in (c).

Figure 2 .
Figure 2. Amplitude of the 64-year-period gyral Rossby wave into the North Atlantic (averaged over the region [76°W, 65°W] × [37°N, 41°N]).The attenuation of the wave is estimated from the slope of the red line joining the two successive maxima of the gyral Rossby wave within the period range of 48−96 years.

Figure 2 .
Figure 2. Amplitude of the 64-year-period gyral Rossby wave into the North Atlantic (averaged over the region [76 • W, 65 • W] × [37 • N, 41 • N]).The attenuation of the wave is estimated from the slope of the red line joining the two successive maxima of the gyral Rossby wave within the period range of 48-96 years.

Figure 3 .
Figure 3. Wavelet analysis of the Sea Surface Temperature anomalies in the North Atlantic in the period range of 48−96 years: (a) the amplitude divided into 16 quantiles, (b) the regions from which the slopes of SST in 2020 and the amplitudes of the 64-year period gyral Rossby wave are estimated.

Figure 3 .
Figure 3. Wavelet analysis of the Sea Surface Temperature anomalies in the North Atlantic in the period range of 48-96 years: (a) the amplitude divided into 16 quantiles, (b) the regions from which the slopes of SST in 2020 and the amplitudes of the 64-year period gyral Rossby wave are estimated.

Figure 4 .
Figure 4. Characterization of the North Atlantic, the South Atlantic, and the South Indian oceans from (a) the variation in SST slope in 2020 along the gyre, (b) the variation in the relative amplitude of the 64−year-period GRW along the gyre, (c) the variations in the relative amplitude of the 64−year-period GRW versus the variation in the SST slope in 2020.

Figure 4 .
Figure 4. Characterization of the North Atlantic, the South Atlantic, and the South Indian oceans from (a) the variation in SST slope in 2020 along the gyre, (b) the variation in the relative amplitude of the 64-year-period GRW along the gyre, (c) the variations in the relative amplitude of the 64-year-period GRW versus the variation in the SST slope in 2020.

Figure 4 .
Figure 4. Characterization of the North Atlantic, the South Atlantic, and the South Indian oceans from (a) the variation in SST slope in 2020 along the gyre, (b) the variation in the relative amplitude of the 64−year-period GRW along the gyre, (c) the variations in the relative amplitude of the 64−year-period GRW versus the variation in the SST slope in 2020.

Figure 5 .
Figure 5. Wavelet analysis of the Sea Surface Temperature anomalies in the South Atlantic in the period range of 48−96 years: (a) the amplitude divided into 16 quantiles.(b) the regions from which the amplitudes of the 64-year-period gyral Rossby wave are estimated in the South Atlantic.

Figure 5 .
Figure 5. Wavelet analysis of the Sea Surface Temperature anomalies in the South Atlantic in the period range of 48-96 years: (a) the amplitude divided into 16 quantiles.(b) the regions from which the amplitudes of the 64-year-period gyral Rossby wave are estimated in the South Atlantic.J. Mar.Sci.Eng.2023, 11, x FOR PEER REVIEW 8 of 11

Figure 6 .
Figure 6.Wavelet analysis of the Sea Surface Temperature anomalies in the South Indian Ocean in the period range of 48−96 years: (a) the amplitude divided into 16 quantiles.(b) the regions from which the amplitudes of the 64-year-period gyral Rossby wave are estimated in the South Indian Ocean.

Figure 6 .
Figure 6.Wavelet analysis of the Sea Surface Temperature anomalies in the South Indian Ocean in the period range of 48-96 years: (a) the amplitude divided into 16 quantiles.(b) the regions from which the amplitudes of the 64-year-period gyral Rossby wave are estimated in the South Indian Ocean.