Subharmonic modes of the climate system

While most climate transitions have been recognized as resulting from solar and orbital forcing due to their synchronism,[1] understanding the underlying physical mechanisms is encountering considerable problems. Changes in the forcing are too small to explain the observed climate variations as simple linear responses.[2] Difficulties reach their culmination when the Mid-Pleistocene Transition (MPT) is considered, that is a fundamental change in the behavior of glacial cycles during the Quaternary glaciations. The transition happened approximately 1.2 million years ago, in the Pleistocene epoch. Before the MPT, the glacial cycles were dominated by a 41-year periodicity coherent with the Milankovitch forcing from axial tilt. After the MPT the cycle durations have increased, with an average length of approximately 100,000 years coherent with the Milankovitch forcing from eccentricity. However, the intensity of the forcing resulting from the eccentricity is much lower than that induced by the axial tilt.

Milankovitch theory revisited.

To strictly apply the Milankovitch theory, a mediator involving positive feedbacks must be found, endowing the climate response with a resonant feature. Supported by both observational and theoretical considerations, recent work shows that long-period Rossby waves winding around subtropical ocean gyres meet the requirements of the sought mediator.[3]

Resonantly forced Rossby waves in the oceans

The idea that interannual oceanic Rossby waves are resonantly forced stems from the seminal work of Warren B. White[4] who observed the resonant response of Rossby waves to wind forcing. Outside the equatorial band, Rossby waves are observed where the western boundary currents leave the continents to re-enter the subtropical gyres. Propagating towards the west against the direction of the western boundary current, they are observable thanks to the measurement of the sea surface height from satellites.[5]

Being approximately non-dispersive, the period of Rossby waves increases proportionally to their wavelength. So, the existence of multi-decadal Rossby waves can be inferred, provided they wind around the subtropical gyres as the wavelength increases. Effectively, 64- and 128-year-period Rossby waves are observed around the North Atlantic gyre from sea surface temperature anomalies, synchronized with solar forcing [6]

Propagating cyclonically around the subtropical gyres, the so-called Gyral Rossby waves (GRWs) owe their origin to the gradient β of the Coriolis parameter relative to the mean radius of the gyres. They are ruled by the forced version of linearized equations of motion, that is, momentum, continuity, and conservation of the potential vorticity equations. Whereas the β-plane approximation is used for solving equations relating to the short period Rossby waves,[7] a β-cone approximation is required in the case of GRWs so that the equations are solved in polar coordinates.[6]

The resulting modulated current, whose velocity is added to that of the steady anticyclonic wind-driven current, causes the western boundary current to accelerate/decelerate according to the phase of GRWs. This amplifies the oscillation of the thermocline because more/less heat is transported from the equator to the poles which, in turn accelerates/decelerates the western boundary current.  This powerful positive feedback loop causes the GRWs to strongly impact the climate system nearly in phase with the forcing. This is the reason why long-period Rossby waves can be observed in climate archives from ice and sediment cores.

Dispersion relation of GRWs

In the case of GRWs, the dispersion relation is virtually the same as that referring to short period Rossby waves[7] propagating zonally, that is, approximated by:

${\displaystyle \omega /k=-c^{2}\cos \varphi /2\pi \Omega R\sin ^{2}\varphi }$

where ${\displaystyle c=-0.86}$ m/s is the phase velocity of a first baroclinic mode Rossby wave (resulting from the oscillation of the main pycnocline) propagating westward along the equator; ${\displaystyle R}$ is the radius of the Earth, ${\displaystyle \Omega }$  its rotation rate, ${\displaystyle \varphi }$  the latitude of the centroid of the gyre; ${\displaystyle \omega }$ is the pulsation of the Rossby wave propagating cyclonically around the gyre and ${\displaystyle k}$ its wave number so that ${\displaystyle \omega /k}$ is its phase velocity.

The dispersion relation shows that the phase velocity depends only on the latitude of the centroid of the gyre, and not on the mean radius.

Prototype of Coupled Oscillator Systems

Multi-frequency GRWs are superimposed around the gyres. Since they share the same node, that is, the same polar and radial modulated currents around the gyres, they are coupled. Starting from the equation of coupled oscillators, we will see that GRWs are resonantly forced in subharmonic modes. The issue can be formulated as follows: when the periodic driving with pulsation ${\displaystyle \Omega }$ is ${\displaystyle I_{i}\cos(\Omega t)}$  where ${\displaystyle I_{i}}$  is the amplitude of the periodic components of the forcing on the i^th oscillator, then the average period of the i^th oscillator is a multiple of the period of the fundamental wave.

Caldirola–Kanai Oscillators

Major properties of GRWs can be found by solving the equation of the Caldirola–Kanai (CK) oscillator,[8] which is a fundamental model of dissipative systems.[9] Note that this formulation is very general and can be applied to any planetary wave system sharing a same node. So, the oscillators can just as easily replace GRWs or equatorial Rossby and Kelvin waves that tightly control the genesis of El Niño events in the Pacific Ocean.

Consider the system of CK equations governing the motion for a system of ${\displaystyle N}$ coupled oscillators corresponding to the ${\displaystyle N}$ resonance frequencies:[10]

${\displaystyle M_{i}\phi _{i}''+\gamma M_{i}\phi _{i}'+\sum _{k=1}^{N}J_{i,k}(\phi _{i}-\phi _{k})=I_{i}\cos(\Omega t)}$

where ${\displaystyle \phi _{i}}$ represents the phase of the i^th oscillator, ${\displaystyle M_{i}}$  the inertia parameter, i.e., the mass of water displaced during a cycle resulting from the quasi-geostrophic motion of the i^th oscillator, ${\displaystyle \gamma }$ the damping parameter, i.e., the Rayleigh friction and ${\displaystyle J_{i,k}}$ measures the coupling strength between the oscillators ${\displaystyle i}$ and ${\displaystyle k}$. The phase ${\displaystyle \phi _{i}}$ represents the variation in displacement of the pycnocline upwardly resolved in vertical mode associated with the i^th oscillator.

Since the solution of the momentum equations is such that the polar current ${\displaystyle u_{i}}$ and the vertical motion ${\displaystyle \Delta h_{i}}$ of the pycnocline are proportional and coherent,[6] every oscillator is characterized by a constant ${\displaystyle A_{i}}$ such that ${\displaystyle \Delta h_{i}(t)=A_{i}u_{i}(t)}$. By appropriately subdividing the total cross section of the common node around the gyre into partial cross sections allocated to each free oscillator proportionally to its contribution to the resultant rate of flow, it can be written: ${\displaystyle \Delta h_{i}(t)=Au_{i}(t)}$ ${\displaystyle i=1,...,N}$, the same constant ${\displaystyle A}$ applying to all oscillators.

The restoring force, which depends on the phase difference between the oscillators ${\displaystyle i}$ and ${\displaystyle k}$, vanishes when the phases are equal: ${\displaystyle \phi _{i}=\phi _{k}}$ because this condition involves ${\displaystyle u_{i}=u_{k}}$ which, in the absence of friction, removes any interaction between the oscillators ${\displaystyle i}$ and ${\displaystyle k}$. On the other hand, the interaction is all the stronger as the difference in polar current velocities is higher (a linear approximation is used for the CK oscillator). Without having to detail the dynamics of the coupling between the oscillators that involves geostrophic forces at the basin scale, suppose that the polar velocity ${\displaystyle u_{i}}$ of a free oscillator is lower than the resultant velocity of the flow where the nodes of all oscillators are merging. Then, the polar current ${\displaystyle u_{i}}$ of the coupled oscillator at the common node is faster than that of the free oscillator. The opposite happens when the polar velocity ${\displaystyle u_{i}}$ of a free oscillator is higher than the resultant velocity of the flow at the common node. This property applies to all oscillators, which results from the adjustment under gravity of the ocean.

Since the steady background current is not involved in the motion, only the modulated polar currents are considered. Consequently, the average of the left-hand side vanishes as the time elapses and this property also applies to the right-hand side that represents both forcing and geostrophic forces, that is, Coriolis and pressure gradient forces.

Conditions to ensure the durability of the resonant dynamical system.

Subharmonic modes appear when the oscillatory system is periodic, each oscillator transferring on average as much interaction energy to all the others that it receives from them, a required condition to ensure the durability of the resonant dynamical system.The interaction energy ${\displaystyle E_{k}(t)}$ of the k^th oscillator in the resonant dynamical system includes a transient term which tends towards 0 as time increases.[8] In the stationary state (${\displaystyle t\rightarrow \infty }$), the inertia parameter ${\displaystyle M_{i}}$, the damping parameter ${\displaystyle \gamma }$, and the amplitude of the forcing ${\displaystyle I_{i}}$ disappear from ${\displaystyle E_{k}(t)}$ so that:

${\displaystyle E_{k}(t)=-\sum _{i=1}^{k-1}J_{i,k}\cos(\phi _{i}-\phi _{k})}$

The durability of the oscillatory system is proven when the periods of coupled oscillators are commensurable with the fundamental period that coincides with the forcing period ${\displaystyle T=2\pi /\Omega }$, that is, ${\displaystyle T_{i}=n_{i}2\pi /\Omega }$  where ${\displaystyle n_{i}}$ is an integer. In such condition, the oscillatory system is itself periodic. Its period ${\displaystyle mT}$ is such that ${\displaystyle m}$ is the smallest common multiple of ${\displaystyle n_{1}...n_{N}}$. Then, the interaction energy time-averaged over the period ${\displaystyle mT}$ is zero for every oscillator:

${\displaystyle E_{k}^{m}={1 \over mT}\textstyle \int _{0}^{mT}\displaystyle E_{k}(t)dt}$ = ${\displaystyle -{1 \over mT}\sum _{i=1}^{k-1}{J_{i,k}\sin(\omega _{i}mT-\omega _{k}mT) \over \omega _{i}-\omega _{k}}}$=0 ${\displaystyle k=1...N}$

where ${\displaystyle \omega _{i}={\Omega /n_{i}}}$. Consequently, in a stationary state, the periodicity of the total energy of the coupled oscillator system ensures its durability.

Actually, the period of the oscillators may deviate significantly from the average value during each cycle without jeopardizing the sustainability of the coupled oscillator system when the periods ${\displaystyle \tau _{i}=2\pi n_{i}/\Omega }$ time-averaged over the period ${\displaystyle mT}$ converge towards their average value ${\displaystyle {\overline {\tau _{i}}}}$. In this way, the subharmonic modes of the oscillatory system ensure its durability in all circumstances, even though the forcing period ${\displaystyle T}$ deviates from its mean value.

In this case:

${\displaystyle {\overline {E_{k}^{m}}}=-{1 \over mT}\sum _{i=1}^{k-1}{J_{i,k}{\overline {\sin(\omega _{i}mT-\omega _{k}mT)}} \over {\overline {\omega _{i}}}-{\overline {\omega _{k}}}}}$${\displaystyle \approx }$${\displaystyle -{1 \over mT}\sum _{i=1}^{k-1}{{J_{i,k}\sin({\overline {\omega _{i}}}m{\overline {T}}-{\overline {\omega _{k}}}m{\overline {T}})} \over {\overline {\omega _{i}}}-{\overline {\omega _{k}}}}}$=0 ${\displaystyle k=1,...,N}$

The stability of the complete system requires the stability of the different subsystems whose preponderance can vary considerably over time so that the resonance conditions are to be defined recursively:

${\displaystyle \tau _{i}=m_{i}\tau _{i-1}}$ with ${\displaystyle \tau _{0}=T}$

The stability of the subsystem formed by the oscillators ${\displaystyle k=1,...,i-1}$ ensures that of the subsystem of oscillators ${\displaystyle k=1,...,i}$ when ${\displaystyle m_{i}}$=2 or 3. Indeed, the smaller the integer ${\displaystyle m_{i}}$ the more stable the system so that the optimal conditions are reached.

Thus, the functioning of the coupled oscillator system in subharmonic modes is a sufficient condition to ensure the durability of that system robustly. But it is also a necessary condition to ensure the resonant forcing of the system. Indeed, when the mean periods of the oscillators are commensurable with the mean forcing period, the energy the oscillation absorbs is maximized when the forcing is periodically ‘in phase’ with the oscillations, while the oscillation’s energy is more extracted when it is never in phase with the forcing. In this way any non-resonant oscillatory system being less efficient than a resonant system disappears in favor of the resonant system.

Properties of GRWs

The way in which GRWs mediate the solar and orbital forcing of the climate system is demonstrated from their properties.[3][6]

1.       GRWs wind an integer number of turns around the gyre, over a half-wavelength. Thus, the modulated currents allow the warm water to accumulate around the gyre for a half-period, then to evacuate to the pole via the drift current (the circumpolar current in the southern hemisphere) during the following half-period.

2.       GRWs resonate in subharmonic modes; resonance periods are obtained recursively, any period being deduced from the previous one by multiplying it by 2 or 3, which corresponds to an optimum stability of the gyral Rossby waves. This multiplying factor is deduced from long time series of climate (ice and sediment cores). See the table.

3.       GRWs respond selectively according to the period of the forcing: the efficiency is maximum when the latter coincides with a natural period of the Rossby waves.

4.       When the forcing period is close to the natural period of a GRW, a fine tuning occurs, resulting from the latitudinal drift of the centroid of the gyre in accordance with the dispersion relation of GRWs.

.5.       GRWs do not dampen as their period increases because Rayleigh friction is compensated by the lengthening of the forcing duration. So, in theory, the periods of GRWs have no upper limit.

6.       The pycnocline depth and the modulated current velocity around the gyres are nearly in phase with the forcing but some time lags may occur because of the own dynamics of GRWs.

7.       The forcing efficiency of GRWs increases with the temperature difference between the low and high latitudes of the gyres.

8.       Since the amplitudes ${\displaystyle I_{i}}$ of the different forcing cycles do not appear explicitly in the interaction energy of the coupled oscillators, it is almost impossible to distinguish, only from the amplitudes of the oscillations, the oscillators which are under the influence of a direct external forcing from those which are under the effect of a collective driving.

Subharmonic modes

Subharmonic modes of Gyral Rossby Waves. Their natural periods are obtained by multiplying the n^th subharmonic mode (number of turns performed by the Gyral Rossby Waves around the North and South Atlantic gyres, and the Indian Ocean gyre) by 64 years (48 years before the Mid-Pleistocene Transition). The number of turns must be divided by 2 in the North and South Pacific gyres.

Period of Resonance (yr)	Subharmonic Mode	Characteristic band (yr)	Forcing Mode
64	n1=2^0	48-96	No external forcing
128	n2=2^1	96-192	Solar forcing (Gleissberg)
256	n3=2^2	192-576	No external forcing
768	n4=3×2^2	576-1152	Solar forcing
1,536	n5=3×2^3	1152-2304	No external forcing
3,072	n6=3×2^4	2304-4608	No external forcing
6,144	n7=3×2^5	4608-9216	No external forcing
12288	n8=3×2^6	9216-18432	No external forcing
24,576	n9=3×2^7	18432-36864	Orbital forcing (precession)
49,152	n10=3×2^8	36864-73728	Orbital forcing (obliquity)
98,304	n11=3×2^9	73728-147456	Orbital forcing (eccentricity)
196608	n12=3×2^10	147456-294912	No external forcing
393,216	n13=3×2^11	294912-589824	Orbital forcing (eccentricity)
1,179,648	n14=3^2×2^11	589824-1769472	Orbital forcing (eccentricity)
2,359,296	n15=3^2×2^12	1769472-3538944	No external forcing

References

1. Hays, J. D.; Imbrie, John; Shackleton, N. J. (1976-12-10). "Variations in the Earth's Orbit: Pacemaker of the Ice Ages". Science. 194 (4270): 1121–1132. doi:10.1126/science.194.4270.1121. ISSN 0036-8075. PMID 17790893.
2. Rial, J (2000-12-01). "Understanding nonlinear responses of the climate system to orbital forcing". Quaternary Science Reviews. 19 (17–18): 1709–1722. doi:10.1016/S0277-3791(00)00087-1.
3. Pinault, Jean-Louis (2021-01-01). "Resonantly Forced Baroclinic Waves in the Oceans: A New Approach to Climate Variability". Journal of Marine Science and Engineering. 9 (1): 13. doi:10.3390/jmse9010013.
4. White, Warren B. (1985-04-01). "The Resonant Response of Interannual Baroclinic Rossby Waves to Wind Forcing in the Eastern Midlatitude North Pacific". Journal of Physical Oceanography. 15 (4): 403–415. doi:10.1175/1520-0485(1985)0152.0.CO;2. ISSN 0022-3670.
5. Chelton, D. B.; Schlax, M. G. (1996-04-12). "Global Observations of Oceanic Rossby Waves". Science. 272 (5259): 234–238. doi:10.1126/science.272.5259.234. ISSN 0036-8075.
6. Pinault, Jean-Louis (2018-09-19). "Modulated Response of Subtropical Gyres: Positive Feedback Loop, Subharmonic Modes, Resonant Solar and Orbital Forcing". Journal of Marine Science and Engineering. 6 (3): 107. doi:10.3390/jmse6030107. ISSN 2077-1312.
7. Gill, Adrian E. (1982). Atmosphere-ocean dynamics. New York: Academic Press. ISBN 978-0-08-057052-5. OCLC 277664880.
8. Pinault, Jean-Louis (2018-07-02). "Resonantly Forced Baroclinic Waves in the Oceans: Subharmonic Modes". Journal of Marine Science and Engineering. 6 (3): 78. doi:10.3390/jmse6030078. ISSN 2077-1312. Text was copied from this source, which is available under a Creative Commons Attribution 4.0 International License.
9. Um, Chung-In; Yeon, Kyu-Hwang; George, Thomas F. (2002-05-01). "The quantum damped harmonic oscillator". Physics Reports. 362 (2–3): 63–192. doi:10.1016/s0370-1573(01)00077-1. ISSN 0370-1573.
10. Choi, M. Y.; Thouless, D. J. (2001-06-14). "Topological interpretation of subharmonic mode locking in coupled oscillators with inertia". Physical Review B. 64 (1). doi:10.1103/physrevb.64.014305. ISSN 0163-1829.