Long Wave Resonance in Tropical Oceans and Implications on Climate: The Pacific Ocean

Abstract

The dynamics of the tropical Pacific can be understood satisfactorily by invoking the coupling between the basin modes of 1-, 4- and 8-year average periods. The annual quasi-stationary wave (QSW) is a first baroclinic-mode, fourth meridional-mode Rossby wave resonantly forced by easterlies. The quadrennial QSW is built up from a first baroclinic-mode Kelvin wave and a first baroclinic-mode, first meridional-mode Rossby wave equatorially trapped and two off-equatorial Rossby waves, their dovetailing forming a resonantly forced wave (RFW). The 8-year period QSW is a replica of the quadrennial QSW for the second-baroclinic mode. The coupling between basin modes results from the merging of modulated currents both in the western part of the North Equatorial Counter Current and along the South Equatorial Current. Consequently, a sub-harmonic mode locking occurs, which means that the average period of QSWs is 1-, 4- and 8-year exactly. The quadrennial sub-harmonic is subject to two modes of forcing. One results from coupling with the annual QSW that produces a Kelvin wave at the origin of transfer of the warm waters from the western part of the basin to the central-eastern Pacific. The other is induced by El Niño and La Niña that self-sustain the sub-harmonic by stimulating the Rossby wave accompanying the westward recession of the QSW at a critical stage of its evolution. The interpretation of ENSO from the coupling of different basin modes allows predicting and estimating the amplitude of El Niño events a few months before they become mature from the accelerations of the geostrophic component of the North Equatorial Counter Current.

Appendixes

1.1 Appendix 1: The Equations of Motion of Resonantly Forced Waves Under the Approximation of $\beta$ Plane

In the following equations $x$ and $y$ represent the longitude and latitude relative to the central point of the $\beta$ plane tangent to the terrestrial sphere. The representation of movement in this plane rather than on the sphere simplifies the equations of motion. Their linearized version with forcing terms includes the momentum Eqs. (1), (2), the equation of continuity (3) and the equation of potential vorticity (4). Assuming to simplify writings two superimposed fluids, the components $u$ and $v$ of the velocity vector ($u$ is the longitudinal or zonal component, $v$ is the latitudinal or meridional component) and the perturbation $\eta$ of the height of the ocean surface are solutions of the system of equations (Gill 1982)

$$\partial u/\partial t - fv = g\partial n/\partial x + X/\rho_{1} H_{1}$$

(1)

$$\partial v/\partial t + fu = - g\partial n/\partial y + Y/\rho_{1} H_{1}$$

(2)

$$\partial \eta /\partial t + H_{1} (\partial u/\partial x + \partial v/\partial y) = - E/\rho_{1}$$

(3)

$$\frac{\partial }{\partial t}\left( {\xi - f\eta /H_{1} } \right) + \beta v = \frac{1}{{\rho_{1} H_{1} }}\left( {\partial Y/\partial x - \partial X/\partial y + fE} \right)$$

(4)

$\mu = \eta /h = - g'H_{2} /gH$ $g' = (1 - \rho_{1} /\rho_{2} ),$ where $h$ is the displacement of the interface up, which is resolved in vertical mode: $f$ is the Coriolis parameter, β the gradient of the Coriolis parameter and $\xi = \partial v/\partial x - \partial u/\partial y$ the potential vorticity. $H_{1}$ is the depth of the upper layer, $H_{2}$ that of the lower layer and $H$ the total depth. $\rho_{1}$ and $\rho_{2}$ are the density of the upper and lower layer. The forcing terms $X$ and $Y$ represent the surface stress and $E$ the evaporation rate. As usual we assume that $\varOmega \ll N$ where $N$ is the buoyancy frequency, which is supposed to be constant ($\varOmega$ is the speed of rotation of the earth) and $\beta c \ll N^{2}$.

The Coriolis parameter $f$ is expanded with respect to $y$ such that $f = f_{0} + \beta y,$ where $f_{0}$ is the Coriolis parameter at the central point of the $\beta$ plane whose latitude is $\phi_{0}$, with $\beta = (2\varOmega /R)\cos \phi_{0}$ (in the case of equatorial waves $f_{0} = 0$).

Under the approximation of the $\beta$ plane, $u$ and $v$ are the components of the velocity vector $\bf u$ expressed relative to the longitude and latitude. Similarly, $X$ and $Y$ are the forcing terms relative to the surface stress, expressed in relation to the longitude and the latitude.

In order to link $v$ to the other variables, new variables $q$ and $r$ are defined and replace $u$ and $v$ (Gill 1982), such that $q = g\eta /c + u$, $r = g\eta /c - u$ where $c$ is the phase velocity for the baroclinic mode considered. Combining (1) and (4), three new independent equations are obtained:

$$\frac{\partial q}{\partial t} + c\frac{\partial q}{\partial x} + c\frac{\partial v}{\partial y} - fv = \frac{1}{{\rho_{1} H_{1} }}(X - cE)$$

(5)

$$\frac{\partial r}{\partial t} - c\frac{\partial q}{\partial x} + c\frac{\partial v}{\partial y} + fv = - \frac{1}{{\rho_{1} H_{1} }}(X + cE)$$

(6)

$$\frac{\partial }{\partial t}\left( {c\frac{\partial r}{\partial y} - fr + \frac{\partial v}{\partial t} + c\frac{\partial v}{\partial x}} \right) + \beta cv = \frac{1}{{\rho_{1} H_{1} }}\left\{ {\frac{\partial Y}{\partial t} + c\left( {\frac{\partial Y}{\partial x} - \frac{\partial X}{\partial y} + fE} \right)} \right\}$$

(7)

In a shallow water model, inviscid, equations are solved by expanding $r$, $q$, $v$, $X$, $Y$ and $E$ in terms corresponding to the meridional modes, i.e. parabolic cylinder functions:

$$(v,q,r) = \sum\limits_{n = 0}^{\infty } {(v_{n} ,q_{n} ,r_{n} )D_{n} \left( {(2\beta /c)^{1/2} y} \right)}$$

(8)

$$(X,Y,E) = \rho_{1} H_{1} \sum\limits_{n = 0}^{\infty } {(X_{n} ,Y_{n} ,E_{n} )D_{n} \left( {(2\beta /c)^{1/2} y} \right)}$$

(9)

Then, using the property of the parabolic cylinder functions

$$\left( {\frac{\text{d}}{{{\text{d}}\xi }} + \xi /2} \right)D_{m} = mD_{m - 1} \quad {\text{and}}\quad \left( {\frac{\text{d}}{{{\text{d}}\xi }} - \xi /2} \right)D_{m} = - D_{m + 1} ,$$

$\xi = (2\beta /c)^{1/2} y$, the equations of the coefficients become for the modes $n \ge 0$ (coefficients with a negative index are zero; $n = 0$ corresponds to mixed Rossby-gravity waves).

$$(\partial /\partial t - c\partial /\partial x)r_{n - 1} + f_{0} v_{n - 1} + (2\beta c)^{1/2} nv_{n} = - (X_{n - 1} + cE_{n - 1} )$$

(10)

$$(\partial /\partial t + c\partial /\partial x)q_{n + 1} - f_{0} v_{n + 1} - (2\beta c)^{1/2} v_{n} = X_{n + 1} - cE_{n + 1}$$

(11)

$$\begin{aligned} &\frac{\partial }{\partial t}\left\{ { - f_{0} r_{n} - (2\beta c)^{1/2} r_{n - 1} + \left( {\frac{\partial }{\partial t} + c\frac{\partial }{\partial x}} \right)v_{n} } \right\} + \beta cv_{n} \hfill \\ &\quad= \left( {\frac{\partial }{\partial t} + c\frac{\partial }{\partial x}} \right)Y_{n} - (\beta c/2)^{1/2} \left[ {\left( {n + 1} \right)X_{n + 1} - X_{n - 1} } \right] + fE_{n} \hfill \\ \end{aligned}$$

(12)

Without forcing the solutions are $r_{n - 1} = R_{n - 1} \sin (kx - \omega t)$, $v_{n} = V_{n} \cos (kx - \omega t)$ and $q_{n + 1} = Q_{n + 1} \sin (kx - \omega t)$ whose amplitudes $R_{n - 1}$ and $Q_{n + 1}$ are (supposing $V_{n} = 1$)$R_{n - 1} = (2\beta c)^{1/2} n/(ck + \omega )$, $Q_{n + 1} = (2\beta c)^{1/2} /(ck - \omega )$ and the relation between the wave number $k$ and the pulsation $\omega$ is (for long-period long-wavelength waves)

$$k \approx - (2n + 1)(\omega /c)$$

(13)

Considering Kelvin waves for which $v = 0$ and $\omega = kc$, the coefficients are

$$\partial r_{0} /\partial t - c\partial r_{0} /\partial x = - (X_{0} + cE_{0} )$$

(14)

$$\partial q_{0} /\partial t + c\partial q_{0} /\partial x = X_{0} - cE_{0}$$

(15)

While Fourier or Laplace transforms are typically used to calculate the coefficients $r_{j} (x,t)$, $q_{j} (x,t)$ and $v_{j} (x,t)$ (which assume that the waves vanish at infinity), here the equations are solved by expanding in Fourier series versus $x$ and $t$ both the coefficients and the forcing terms $r_{i}$, $q_{i}$, $v_{i}$, $X_{i}$, $Y_{i}$ and $E_{i}$

$$\left( {v_{j} ,q_{j} ,r_{j} } \right) = \sum\limits_{l = 0}^{\infty } {\sum\limits_{m = 0}^{\infty } {\left( {v_{j,m,l} ,q_{j,m,l} ,r_{j,m,l} } \right)} } \exp \left[ {i\left( {mkx - l\omega t} \right)} \right]$$

(16)

$$\left( {X_{j} ,Y_{j} ,Z_{j} } \right) = \sum\limits_{l = 0}^{\infty } {\sum\limits_{m = 0}^{\infty } {\left( {X_{j,m,l} ,Y_{j,m,l} ,Z_{j,m,l} } \right)} } \exp \left[ {i\left( {mkx - l\omega t} \right)} \right]$$

(17)

The coefficients $v_{n,m,l}$, $r_{n - 1,m,l}$ and $q_{n + 1,m,l}$ can be expressed in terms of forcing terms $X_{n + 1,m,l}$, $X_{n - 1,m,l}$, $Y_{n,m,l}$, $E_{n - 1,m,l}$ $E_{n,m,l}$ and $E_{n + 1,m,l}$: the terms of the Fourier series can be considered as normal modes of forced waves, the solution for the free waves corresponding to the mode $m = 1$ and $l = 1$.

$$v_{n,m,l} = \frac{{\left( {l\omega + cmk} \right)\varPhi - l\omega \left( {2\beta c} \right)^{1/2} \left( {X_{n - 1} + cE_{n - 1} } \right)}}{{\beta c\left( {l\omega \left( {2n + 1} \right) + cmk} \right) + l\omega \left( {\left( {cmk} \right)^{2} - \left( {l\omega } \right)^{2} } \right)}}$$

(18)

$$\varPhi = \left( {cmk - l\omega } \right)Y_{n} - \left( {\beta c/2} \right)^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} \left( {\left( {n + 1} \right)X_{n + 1} - X_{n - 1} } \right) + fE_{n}$$

(19)

$$r_{n - 1,m,l} = \frac{ - i}{cmk + l\omega }\left[ {X_{n + 1} + cE_{n - 1} + \left( {2\beta c} \right)^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} nv_{n,m,l} } \right]$$

(20)

$$q_{n + 1,m,l} = \frac{ - i}{cmk - l\omega }\left[ {X_{n + 1} - cE_{n + 1} + \left( {2\beta c} \right)^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} v_{n,m,l} } \right]$$

(21)

Forced equatorial and off-equatorial planetary waves are governed by Eqs. (18)–(21). The coefficient $v_{n,m,l}$ is in quadrature relative to the coefficients $r_{n - 1,m,l}$ and $q_{n + 1,m,l}$.

On the equator, forced Kelvin waves are governed by (14) and (15), propagating eastward. The coefficients $r_{0,m,l}$ and $q_{0,m,l}$ are

$$r_{0,m,l} = \frac{ - i}{cmk + l\omega }\left[ {X_{0} + cE_{0} } \right]$$

(22)

$$q_{0,m,l} = \frac{ - i}{cmk - l\omega }\left[ {X_{0} + cE_{0} } \right]$$

(23)

According to the dispersion relation (13) $r_{n - 1,m,l}$ is infinite if $l = m\left( {2n + 1} \right)$ for the westward propagating planetary waves. Similarly, $q_{0,m,l}$ is infinite if $l = m$ for eastward propagating Kelvin waves. These singularities highlight the resonances. This so-called ‘ill-posed’ problem as having singularities has to be regularized to be resolved, which is done by considering the Rayleigh friction. From a physical point of view the Rayleigh friction moderates variations in wave amplitude in the vicinity of resonances, assigning a finite value in all circumstances.

To introduce Rayleigh friction terms in (10)–(12), (14), (15) ${\partial \mathord{\left/ {\vphantom {\partial {\partial t}}} \right. \kern-0pt} {\partial t}}$ is replaced by $r + {\partial \mathord{\left/ {\vphantom {\partial {\partial t}}} \right. \kern-0pt} {\partial t}}$ everywhere, $r$ being the decay rate of the friction $r = \alpha \omega$ where $0 < \alpha < 1$ (the case $\alpha = 1$ is singular). In this way, $l\omega$ has to be replaced by $- r + l\omega$ in (18)–(23).

1.2 Appendix 2: Forcing Terms of the Annual RFW

The solution of the equations of motion is $v_{n,m,l}$, $r_{n - 1,m,l}$ and $q_{n + 1,m,l}$, where the meridional mode $n = 4$ indicates the rank of the parabolic cylinder functions with respect to the latitude $y$; $m$ and $l$ are the ranks of the terms of the Fourier series with respect to the longitude $x$ and the time $t$ (e.g. Pinault 2013). The forcing terms corresponding to surface stress are $X_{3,m,l}$, $X_{5,m,l}$ and $Y_{4,m,l}$, and those relating to evaporation are $E_{3,m,l}$ and $E_{5,m,l}$ ($f = 0$ on the equator). As in the tropical Atlantic, additional forcing terms are necessary to model the modulated components of the NECC and the SEC.

Thus the forcing term $X_{3,m,l}$ is replaced by

$$X_{3,m,l} + \rho_{1} H_{1} \omega u_{3} \sin \left( {\omega t - \varphi_{u3} } \right)$$

(24)

and $E_{3,m,l}$ by:

$$\rho_{1} \omega \eta_{3} \sin \left( {\omega t - \varphi_{\eta 3} } \right)/\mu ,$$

(25)

where $\omega = {{2\pi } \mathord{\left/ {\vphantom {{2\pi } T}} \right. \kern-0pt} T}$ (the period $T$ is 1 year); $u_{3}$ and ${{\eta_{3} } \mathord{\left/ {\vphantom {{\eta_{3} } \mu }} \right. \kern-0pt} \mu }$ represent the amplitude of the third term of the zonal modulated current and thermocline depth (which does not depend on longitude) expanded in series of parabolic cylinder functions (evaporation is not explicitly taken into account in the model). The positive forcing term (25) indicates that the current carries warm water. Only additional forcing terms of rank 3 are important to inform about the two modulated forcing currents both sides of the equator.

As shown in Fig. 21 the solution varies significantly, even when the forcing terms are small; the phases $\varphi_{u3}$ and $\varphi_{\eta 3}$ are set such that the forcing terms (24) and (25) reach a maximum in August, as observed ($\varphi_{u3} = \varphi_{\eta 3}$). Under these conditions, the period of the solution is 1 year and the phase of the antinode and the node is uniform in the central Pacific, but is reversed at both ends of the basin (Fig. 21c, d), the wavelength of the annual RFW being close to 9800 km (Fig. 5).

Fig. 21

Representation of the solution of the equations of motion. The blue curves are obtained by introducing the surface stress alone, the red curves by introducing forcing terms (24) and (25): a antinode at 160°W, 3°N, b node at 160°W, 3°N, c antinode in 10/1998, d node in 10/1998

In Fig. 5 the average depth of the thermocline is assumed to be 255 m, the phase velocity for the first-baroclinic mode 2.8 m/s. The length of the partial sums of the Fourier series is 20 with respect to the longitude and 20 with respect to time. T = 12 months. $\omega u_{3}$ = 3.5E−07 m/s² ${{\omega \eta_{3} } \mathord{\left/ {\vphantom {{\omega \eta_{3} } \mu }} \right. \kern-0pt} \mu } = \left( {{{H_{1} } \mathord{\left/ {\vphantom {{H_{1} } c}} \right. \kern-0pt} c}} \right)\omega u_{3}$ = 3.1E−05 m/s.

1.3 Appendix 3: The Forcing Terms of the Quadrennial RFW

Like for the Atlantic Ocean (e.g. Pinault 2013) three sets of equations of motion are solved, in relation with the north-western, the south-western and the equatorial antinodes, i.e. the two off-equatorial Rossby waves and the equatorially trapped Kelvin wave that reflect off the eastern boundary to produce an equatorially trapped Rossby wave. Additional forcing terms are to be introduced in the equations of motion to account for geostrophic forces at the basin scale. Geostrophic forces result from the merging of the nodes of the annual and quadrennial QSWs, which form the North Equatorial Counter-Current. By stimulating a Kelvin wave from the extreme western part of the basin, geostrophic forces initiate the eastward phase propagation of the quadrennial RFW. The forcing term X is replaced by X ± ρ ₁ H ₁ ωu ₀ and E by ±ρ ₁ ωη ₀ /μ where u ₀ and η ₀/μ refer to the forcing of the zonal modulated current and the thermocline depth (the direct impact of evaporation due to El Niño events is ignored). These forcing terms are not dependent on the longitude: u ₀ is proportional to the dimensionless current u _r  = u _b /max(|u _b |) where u _b is the zonal current observed at the western boundary 0.5°N, 137.5°E (Fig. 8c) and ωη ₀ /μ = (H ₁ /c)ωu ₀. The signs depend on the direction of forcing of the current and the up or down movement of the thermocline. When the forcing current accelerates at the western boundary, either it produces a Kelvin wave propagating eastward when ∂η/∂x < 0 along the equator, or it leaves the basin to join the western boundary currents when ∂η/∂x > 0 (implying ∂u/∂t < 0) that occurs when two accelerations of the forcing current are too close.

In Fig. 9 the length of partial sums of Fourier series is 10 versus the longitude and 20 versus the time. The Rayleigh friction parameter α is set equal to 0.4 for Rossby waves and 0.06 for Kelvin waves so that the decay rate of friction r = αω are 2E−08 s⁻¹ and 2.3E−08 s⁻¹, respectively: ω = 2π/T, T = 4 years for the Rossby waves and T = 0.5 year for the Kelvin wave. The forcing terms are u ₀ = −2.4E−04 u _r m/s for the northern antinode, u ₀ = −7.8E−05 u _r m/s for the southern antinode and u ₀ = 1.2E−06 u _r m/s for the equatorial antinode.

1.4 Appendix 4: Sub-Harmonic Mode Locking in Coupled Oscillators with Inertia

The following is deducted from the works of Choi and Thouless in the particular case of quasi-stationary waves that share the same modulated current at the node, the coupling resulting from processes involving viscosity μ of seawater. Indeed, viscosity is a means of removing mechanical energy from the system by giving rise to stresses on the surface of material volume elements related to the rate of strain. In this way, the terms (μ/ρ ₁) × (∂² u/∂x ² + ∂² u/∂y ²) and (μ/ρ ₁) × (∂² v/∂x ² + ∂² v/∂y ²) have to be added to the right-hand side of the two momentum Eqs. (1) and (2) in Appendix 1 to introduce the coupling terms in the equations of motion: u and v are the zonal and meridional currents; μ/ρ ₁ is the kinematic viscosity (Gill 1982).

The following results are deduced from the general equations of coupled oscillators with inertia without having to solve the particular equations of motion of barocline waves. Let us suppose the set of equations of motion for a system of N coupled oscillators:

$$\sum\limits_{j = 1,N} {M_{ij} \ddot{\varphi }_{j} + \gamma M_{ij} \dot{\varphi }_{j} + J_{ij} \sin \left( {\varphi_{i} - \varphi_{j} } \right) = I_{i} + \varepsilon } ,$$

(26)

where $\varphi_{i}$ represents the phase of the $i{\rm th}$ oscillator, i.e. the coordinate of the crest of the quasi-stationary wave ($\varphi_{i}$ increases by $2\pi$ every time the crest covers a complete cycle along the equatorial and the off-equatorial waves in the case of tropical waves), ${\dot\varphi}_{i}$ the first derivative versus time, i.e. the phase velocity, ${\ddot\varphi}_{i}$ the second derivative, $M_{ij} = M\delta_{ij}$ the inertia matrix, $\gamma$ the damping parameter, and $J_{ij}$ the coupling strength between the oscillators i and j. The right-hand side describes the periodic forcing with frequency Ω: $I_{i} \equiv I_{i,a} \cos \varOmega t$ where $I_{i,a}$ is the amplitude of forcing on the ith oscillator. Each of the coupled oscillator possesses inertia $M$ and suffers from dissipation of strength $\gamma M$. $\varepsilon$ is a white noise of zero mean and variance $\langle \varepsilon^{2} \rangle$ that reflects the variability of resonant forcing from cycle to cycle.

An element of simplification of the theory relies on the fact that forcing terms in (26) have a zero mean. Indeed, at any positive antinode of a quasi-stationary wave corresponds a negative antinode half a period later, and vice versa, offsetting the effects over time. This also applies to the modulated zonal and meridional currents at the nodes of the quasi-stationary waves, the sign of the velocity of which is reversed during each half period.

Using the canonical transformation ${\dot\varphi}_{i} = \partial H/\partial p_{i}$ and ${\dot p}_{i} = - \partial H/\partial \phi_{i}$ ($H$ is the Hamiltonian), then Eq. (1) can be written in the following form where ($\phi_{i}$,$p_{i}$) are conjugate variables:

$${\dot\theta}_{i} = M^{ - 1} p_{i} e^{\gamma t}$$

(27)

$${\dot p}_{i} = - \sum\limits_{j} {J_{ij} e^{\gamma t} \sin (\theta_{i} - \theta_{j} - a_{ij} ) + \varepsilon e^{\gamma t} } ,$$

(28)

where $\theta_{i} \equiv \phi_{i} + a_{i}$, $M\,{\dot a}_{i} \equiv e^{ - \gamma t} Q_{i}$,$Q_{i}$ is defined so that ${\dot Q}_{i} e - \gamma t = I_{i}$ and

$$a_{ij} = a_{i} - a_{j} = \frac{1}{M}\left[ {\frac{{I_{a} }}{{\gamma^{2} + \varOmega^{2} }}(\frac{\gamma }{\varOmega }\sin \varOmega t - \cos \varOmega t)} \right]\,\,\,\,I_{a} = I_{i,a} - I_{j,a}$$

(29)

apart from an arbitrary constant.

The time evolution of the probability distribution $P\left( {\left\{ {\theta_{i} } \right\},\left\{ {p_{i} } \right\},t} \right)$ of phases and momenta at time $t$ is described by the Fokker–Planck equation (Risken 1989) for the inertial behavior of the system:

$$\frac{\partial p}{\partial t} = - \sum\limits_{i} {\frac{{p_{i} }}{M}e^{\gamma t} } \frac{\partial P}{{\partial \theta_{i} }} + \sum\limits_{ij} {J_{ij} e^{\gamma t} \sin (\theta_{i} - \theta_{j} - a_{ij} )\frac{\partial P}{{\partial p_{i} }} + \frac{{\langle \varepsilon^{2} \rangle }}{2}\sum\limits_{i} {e^{2\gamma t} } \frac{{\partial^{2} P}}{{\partial p_{i}^{2} }}}$$

(30)

Equation (30) yields the stationary solution valid in the limit $t \to \infty$

$$P\left( {\left\{ {\theta_{i} } \right\},\left\{ {p_{i} } \right\},t} \right) = A{\text{e}}^{{ - 2\gamma MH/\langle \varepsilon^{2} \rangle )}} ,$$

(31)

where $A$ is the normalization constant; the Hamiltonian is given by

$$H = \frac{{{\text{e}}^{ - 2\gamma t} }}{2M}\sum\limits_{i} {p_{i}^{2} - \sum\limits_{i < j} {J_{ij} \cos (\theta_{i} - \theta_{j} - a_{ij} )} }$$

(32)

so that (30) gives

$$\frac{\partial p}{\partial t} = \frac{{\langle \varepsilon^{2} \rangle }}{2}\sum\limits_{i} {{\text{e}}^{2\gamma t} \frac{{\partial^{2} P}}{{\partial p_{i}^{2} }} = A\gamma {\text{e}}^{{ - 2\gamma MH/\langle \varepsilon^{2} \rangle }} \sum\limits_{i} {\left( {2\gamma p_{i}^{2} {\text{e}}^{ - 2\gamma t} /\langle \varepsilon^{2} \rangle - 1} \right)} }$$

The first term of (32) becomes vanishingly small in the stationary state (t → ∞). One thus obtains the Hamiltonian

$$H = - \sum\limits_{i < j} {J_{i,j} \cos (\theta_{i} - \theta_{j} - a_{ij} ) = - \sum\limits_{i} {J_{i,j} \cos (\varphi_{i} - \varphi_{j} )} }$$

(33)

Equation (29) shows that $a_{ij}$, defined modulo 2π, is periodic in time with period $\tau = 2\pi /\varOmega$. Since the Hamiltonian as well as $a_{ij}$ have the periodicity $\tau$ the Floquet theorem is applicable to the corresponding Schrödinger equation for the wave function $\varPsi_{i} (t + \tau ) - e^{ - iE\tau } \varPsi_{i} (t)$ with the quasi-energy $E$ (e.g. Whittaker and Watson 1952). This imposes that, apart from the dynamical contribution $E\tau$, the corresponding change in the phase of the wave function $\varPsi_{i}$ should be a whole number of $2\pi$ $\theta_{i} (t + \tau ) - \theta_{i} (t) = 2n_{i} \pi - E\tau ,$ where $n_{i}$ is an integer depending on the form of $a_{ij}$, i.e. of the driving $I_{i}$, and $E$ has been assumed to be real. One thus has the average change rate of the phase

$$\left\langle {\dot{\varphi }_{i} } \right\rangle = \left\langle {\dot{\theta }_{i} } \right\rangle = \frac{1}{\tau }\int_{0}^{\tau } {\dot{\theta }_{i} {\text{d}}t = n_{i} \varOmega - E} ,$$

(34)

where $n_{i}$ is the winding number. So, the difference in phase velocities $\left\langle {\dot{\varphi }_{i} } \right\rangle - \left\langle {\dot{\varphi }_{j} } \right\rangle = (n_{i} - n_{j} )\varOmega$ that is a multiple of the frequency $\varOmega$ displays a particular sub-harmonic mode locking.