Phase baroclinic in a differentially experiment subject to periodic forcing with a variable duty cycle spatially extended

A series of laboratory experiments in a thermally driven, rotating ﬂuid annulus are presented that investigate the onset and characteristics of phase synchronization and frequency entrainment between the intrinsic, chaotic, oscillatory amplitude modulation of travelling baroclinic waves and a periodic modulation of the (axisymmetric) thermal boundary conditions, subject to time-dependent coupling. The time-dependence is in the form of a prescribed duty cycle in which the periodic forcing of the boundary conditions is applied for only a fraction d of each oscillation. For the rest of the oscillation, the boundary conditions are held ﬁxed. Two proﬁles of forcing were investigated that capture different parts of the sinusoidal variation and d was varied over the range 0 : 1 (cid:2) d (cid:2) 1. Reducing d was found to act in a similar way to a reduction in a constant coupling coefﬁcient in reducing the width of the interval in forcing frequency or period over which complete synchronization was observed (the “Arnol’d tongue”) with respect to the detuning, although for the strongest pulse-like forcing proﬁle some degree of synchronization was discernible even at d ¼ 0 : 1. Complete phase synchronization was obtained within the Arnol’d tongue itself, although the strength of the amplitude modulation of the baroclinic wave was not signiﬁcantly affected. These experiments demonstrate a possible mechanism for intraseasonal and/or interannual “teleconnections” within the climate system of the Earth and other planets that does not rely on Rossby wave propagation across the planet along great circles.

Synchronization is commonly discussed in the context of discrete, coupled oscillators which may be periodic or chaotic. But under some circumstances, extended nonlinear systems which are formally infinite-dimensional, such as fluid flows, may exhibit discrete oscillations and behave as if they consisted of discrete oscillating components. We study such an example in the laboratory, consisting of amplitude-modulated, azimuthally travelling baroclinic waves in a thermally driven, rotating annulus experiment. Earlier experiments showed that, under conditions in which the unperturbed travelling waves are spontaneously (either periodically or weakly chaotically) modulated in time, the amplitude modulation of the waves can be phase-locked and synchronized with periodic perturbations of the applied (axisymmetric) temperature difference between the inner and outer cylinders of the experiment. We use this potentially synchronized system to explore what happens if the periodic perturbations are applied for only a given fraction of each cycle-i.e., for a duty cycle <100%. Our systematic exploration of variations in both the duty cycle and detuning (difference in period between the natural oscillation period and that of the imposed boundary temperature variations) shows that the duty cycle acts in the same way as a variable coupling coefficient, demonstrating a narrowing of the frequency interval over which complete phase synchronization is observed as the duty cycle is reduced. We also report a new form of scaling of residual phase fluctuations within the synchronized state with the duty cycle parameter, which may apply more generally with respect to coupling coefficients. These results are relevant, among other things, to our understanding of possible mechanisms coupling different components of the climate system of Earth and other planets, and their impact leading to potentially correlated behaviour in different parts of the world during different seasons, even if the coupling between these regions is not present throughout the year.

I. INTRODUCTION
Synchronization is generally taken to imply a correlated behavior between two or more oscillating systems, in which the oscillations of one system follow that of another. It takes a number of forms depending on the degree of coherence between the systems, ranging from complete synchronization (where every feature of the behavior of the systems behaves a) coherently) through weaker forms, such as phase synchronization [which is the form most commonly discussed, e.g., Rosenblum et al. (1996) and Pikovsky et al. (2001)], partial synchronization, and the so-called generalized synchronization (Rulkov et al. 1995). The essential elements required for synchronization typically include some form of coupling between components of the oscillating systems which can exert an influence on the phase of the oscillations, enabling the frequency of one system to be entrained by another into a constant ratio (on average), typically of simple integers. Such systems have been widely studied for many years with diverse applications in both discrete and continuum systems, such as in networks of physical or biological systems (e.g., Watts andStrogatz, 1998 andArenas et al., 2008), ensembles of chemical oscillators (e.g., Taylor et al., 2009), lasers (e.g., Van Wiggeren and Roy 1998), cardiac and circadian rhythms (Glass 2001), reaction-diffusion systems, and in mechanically oscillating systems among many others (e.g., see Pikovsky et al., 2001).
Although initially restricted to systems of simple, periodic oscillators, the concept of synchronization is now also widely applied in the context of chaotic oscillators, in which spontaneously aperiodic behavior can also be entrained to follow an external signal or the behavior of another component of the network under some circumstances (e.g., Pecora et al., 1997 andBoccalletti et al., 2002). This has broadened the application of synchronization concepts much more widely to problems in engineering, cryptography, biology, medicine, and physics [such as in nonlinear optics-e.g., see Van Wiggeren and Roy (1998)]-see e.g., Bocalletti et al. (2002) for a review. Most applications, however, have generally focused on systems of discrete oscillators (periodic or chaotic) though potentially in very large and complex interconnected networks (Watts andStrogatz, 1998 andArenas et al., 2008). But continuous (formally infinite dimensional) systems, such as fluids or plasmas, can also behave as if they consist of a finite (countable) set of discrete component oscillators under certain circumstances, given appropriate forcing and dissipation (e.g., Williamson and Govardhan, 2004;Pereira et al., 2013;and Goldstein et al., 2016), especially for flows confined within finite-sized containers. This raises the possibility of observing discrete, synchronized behavior within fluid dynamical systems that may be analogous to synchronization phenomena in networks of discrete oscillators.
Such behavior has been explored in the context of theoretical and numerical models, laboratory experiments, and even in observations of large-scale natural systems such as the Earth's atmosphere and climate system, which Lorenz (1991) suggested could be regarded as a complex network of weakly coupled (regional) sub-systems. Early explorations of synchronization phenomena in meteorology and climate were prompted by an interest in accounting for the so-called "teleconnections" in the atmosphere, in which geographically widely separated events may be causally related by various coupling mechanisms. Mechanistic studies of these phenomena were initially based on discretized (often highly truncated) numerical models of atmospheric flows in distinct regions of the atmosphere (mid-latitude channels or whole hemispheres) in which ad hoc coupling of all (or most) flow components between models was imposed (e.g., Duane, 1997 andLunkheit, 2001). Later work, however, showed that only weak coupling of a subset of flow components (usually non-zonal eddies) was needed to achieve discernible synchronization (e.g., Kocarev et al., 1997;Duane et al., 1999;Duane and Tribbia, 2001;2004;and Hiemstra et al., 2012). In a precursor to the present work, Duane et al. (1999) also conducted some simulations in which the cross-hemispheric Rossby wave coupling they invoked to account for synchronized atmospheric blocking events between the north and south varied with the time of year, although this was not investigated in much detail.
Meteorological data assimilation, in which observations are introduced into a numerical simulation of the atmosphere or oceans, has also been reinterpreted in terms of the synchronization of chaotic systems through weak coupling of a subset of simulated flow components with sequences of observations (e.g., Duane et al., 2006;Yang et al., 2006;and Abarbanel et al., 2009). This approach has been further developed most recently in the context of multi-model ensembles of climate simulations as a means of accounting for systematic errors between different models through mutual assimilation of model variables, forming the socalled "super-model" (van den Berge et al., 2011;Duane, 2015;and Shen et al., 2016).
Evidence for actual identifiable synchronization phenomena within the Earth's climate system has also been sought through analysis of the time series of climate observations [e.g., see Read and Castrej on-Pita (2010) for a review]. Duane et al. (1999), for example, noted a weak tendency for blocking events to occur simultaneously in both the northern and southern hemispheres under certain conditions/at certain times of the year. Maraun and Kurths (2005) found evidence for phase synchronization between the El Niño-Southern Oscillation (ENSO) index and variations in the Indian Monsoon, while Feliks et al. (2010) detected phase synchronization between various climate indicators in the middle East and the North Atlantic Oscillation (Hurrell et al., 2003) on timescales of 7-8 years. On annual/seasonal timescales, Gruzdev and Bezverkhny (2000), Kuai et al. (2009), and Read and Castrej on-Pita (2012), showed that the stratospheric Quasi-Biennial Oscillation (QBO) was phasesynchronized with the annual cycle in that its period was found to be entrained to a rational ratio of the annual period, although this ratio evidently fluctuated chaotically on timescales of several years. Evidence of partial phase synchronization of the ENSO index to the annual cycle was also found by Stein et al. (2011).
Concerning laboratory fluid flow experiments, Maza et al. (2000), for example, demonstrated synchronization phenomena within a chaotic convective flow in a rectangular box to which oscillatory thermal perturbations were applied locally to a small region of the upper surface. Maestrello (2004) studied synchronized interactions between a turbulent aerodynamic boundary layer and an adjacent flexible boundary and other sustained resonant interactions between mechanical structures and periodic vortex shedding were reviewed by Williamson and Govardhan (2004). Most relevant to the present study, Eccles et al. (2009) investigated synchronization effects in a rotating, thermally driven annulus experiment in which the basic (unperturbed) flow consisted of a rotating, temporally modulated travelling baroclinic wave that propagated azimuthally around the annular channel. Clear synchronization was observed between oscillations of the wave amplitude and cyclic perturbations of the imposed (axisymmetric) temperature contrast, provided the period of the perturbations was not too far removed from that of the natural period of the wave amplitude modulation. Phase synchronization with the periodically perturbed thermal boundary conditions was observed in both quasi-periodic and chaotic regimes. This was taken one stage further by Castrej on-Pita and , who coupled two such experimental chambers via their (azimuthally symmetric) boundary conditions in a master-slave configuration, such that fluctuations in the heat transport in one annulus could influence the boundary conditions of the other. They were able to demonstrate phase synchronization of the wave modulation in the slave system in both quasi-periodic and chaotic regimes.
In the present work, we extend the experiments of Eccles et al. (2009) and Castrej on-Pita and  to explore the impact of time-dependent coupling on the tendency of the baroclinic wave system to synchronize with periodic modulations in the thermal boundary conditions, by imposing a duty cycle on the applied perturbations. In many respects, this forms a laboratory analogue of the hemispheric coupling model simulations of Duane et al. (1999), with the important difference that, in our experiments, the coupling between the applied perturbation and the experimental flow is imposed axisymmetrically, via the "annular mode" in the parlance of climate dynamics. One possible motivation was to explore the potential for such annular mode coupling (essentially via modulations in the effective hemispheric heat transfer) to result in significant cross-hemispheric synchronization, even if the coupling is only effective during certain seasons of the year. Such a configuration may be seen as the prototype of a whole new class of "climate teleconnections" which does not rely on Rossby wave propagation from one region to another (cf. Hoskins and Karoly 1981).
Section II describes the experimental configuration, introducing the key variables to be explored and the methods of analysis. The main results are presented in Sec. III and the overall conclusions are discussed in Sec. IV.

II. EXPERIMENTAL CONFIGURATION
The apparatus was modified from the thermally driven, rotating annulus experiment used in earlier studies by Hignett et al. (1985), Read et al. (1992), Fr€ uh and Read (1997), Read (2003), Eccles et al. (2009), andCastrejon-Pita et al. (2010). It comprised two upright, coaxial brass cylinders, mounted along the rotation axis of a horizontal rotating table with flat, thermally insulating (Perspex) horizontal boundaries, both of which were in contact with the working fluid in the annular channel. Water was circulated in two separate, independently controlled channels at two well-controlled temperatures in thermal contact with each sidewall to maintain quasiisothermal boundaries. The temperature of the water in each circuit was maintained to a precision of 60.01 K by a combination of inline chillers, heaters, and platinum resistance thermal sensors, servo-controlled by a commercial process controller (Eurotherm 2704). Pumps and chillers were located in the stationary laboratory frame and the coolant was circulated onto and off of the rotating frame via rotating fluid couplers. The whole apparatus in the rotating frame was placed inside a temperature controlled enclosure, whose air temperature was maintained at 60.5 K by a thermostatically controlled heater, within the laboratory whose temperature was maintained at around 61.5 K. The dimensions and parameters for the experiments described herein are listed in Table I, and a schematic diagram and photograph of the apparatus are depicted in Fig. 1.
In common with other studies using rotating annulus experiments (e.g., see Mason, 1975 andRead et al., 2015), the parameter space is determined with reference to two principal dimensionless parameters, the thermal Rossby or Hide number defined as where g is the acceleration due to gravity and other parameters are shown in Table I, and a form of Taylor number scaled by the aspect ratio, is the radial width of the channel, following the study by Fowlis and Hide (1965). The measurement system for the annulus involved acquiring data from copper-constantan thermocouples in a ring with 32 equally spaced sensors in the azimuthal plane, positioned at midheight and mid-radius, together with additional thermocouple sensors embedded in the sidewall boundaries and others to sample the ambient air temperature. These temperature measurements were digitized by an on-board Agilent 30904A Data Acquisition system in the rotating frame at a sample interval of 4 s, and sent in real time to a computer (in the stationary laboratory frame, connected via electrical slip rings) for further analysis. The rotation of the table was maintained to a precision of a few parts in 10 5 through analogue servo-control of the DC direct-drive motor via a tachometer.

A. Coupling and synchronization
As employed by Castrej on-Pita et al. (2010), cyclic, controlled perturbations to the sidewall boundary temperatures were applied through an additional in-line heater, placed after the Eurotherm control sensor, whose heating rate was controlled in real time by the computer in the stationary laboratory frame. The applied boundary temperature was then monitored by thermistor probes. In the cases investigated by Eccles et al. (2009), the perturbations applied to the thermal boundary conditions consisted of periodic (sinusoidal) variations in T a and T b such that DT ¼ DT 0 þ sinct: The amplitude of the forcing, ; was kept fixed and the frequency, c; was chosen to be close to that of the natural vacillation frequency of the baroclinic wave flow.
For the purposes of this experiment, the effect of intermittent coupling was achieved by switching off the periodic forcing for a certain fraction of each cycle (i.e., imposing a duty cycle d) to vary the degree of forcing. This was achieved using one of two different methods, as illustrated in Figs. 2 and 3. Upon imposing a duty cycle on the periodic forcing using profile 1, the forcing closely followed the sinusoidal oscillation in DT for times centred around the peak or trough of each oscillation, deviating back to the nominal reference value around the time of the zero-crossing. When imposing a duty cycle on the periodic forcing using profile 2, however, the forcing was applied only for part of the positive-going phase of each cycle, thereby acting more like a pulse to try to "kick" the system back into synchronization during each period. Schematic of the rotating annulus showing inner and outer cylinders and the circular array of thermocouples at mid-height and mid-radius. The working fluid is contained within the annular channel between the cylinders. The outer cylinder is heated from the outside, and the inner cylinder is cooled from the inside. X is the rotation rate, and DT ¼ jT a À T b j is the applied temperature difference.

B. Experimental design and procedures
All experiments reported here were focused around a flow dominated by an azimuthal wavenumber 3 travelling baroclinic wave exhibiting a "modulated amplitude vacillation" (Read et al., 1992). In this regime, the amplitude of the dominant wavenumber (its harmonics and, at least intermittently, its sidebands) was subject to a relatively fast, nearly periodic modulation (with a period of around 200 s) whose strength varied more slowly and cyclically (though aperiodically) on a timescale of around 1100 s. An example of a time series taken from an unperturbed experiment is shown in Fig. 4, which shows azimuth-time contour maps (or Hovm€ oller diagrams) of the variations of temperature at mid-radius and mid-height. The slow azimuthal drift of the wavenumber 3 pattern is apparent in the diagonal patterns of positive and negative temperature anomalies, while the fast oscillations of wave amplitude ("amplitude vacillations") are apparent in the break-up of the diagonal stripes into a braided or pulsating pattern. The depth of the modulation by the fast vacillation is seen to vary slowly in time, indicating that this is a "modulated amplitude vacillation regime." Modulations in the dominant wave amplitude were also accompanied by fluctuations in the advective heat transfer between the inner and outer sidewall (Read et al. 1992), which could also be affected by changes in DT, forming the basis for coupling the system to an oscillating heat source or sink (Castrej on-Pita et al., 2010).
Intermittent synchronization experiments were typically initialized by setting the required mean temperature difference, DT 0 ; and rotation rate, X, and allowing the system to come to equilibrium, typically for 1-3 h. The modulation to DT was then applied with given values of ; c, and duty cycle d held constant for intervals of at least 1 h (if synchronization was actually observed), or for up to 3 h if synchronization was not clearly evident. The value of c was then incremented automatically in order to scan a range of frequencies, starting with a value at which permanent synchronization was not observed so that the onset of phase synchronization and subsequent phase-locking was clear. The experimental sequence was then repeated at a different duty cycle level (expressed as a percentage of each cycle) in order to map out where phase-locking was observed and then eventually lost, thereby delineating the "Arnol'd tongue" for that particular flow.

C. Analysis procedures
During the whole experiment, temperature data and the rotation rate were logged for later analysis. Variations in the wave amplitude were determined from Fourier analyses in azimuth of the thermal measurements from the thermocouple ring at mid-height and mid-radius of the annulus at a given timestep. A fast Fourier transform of the 32 sensors in the ring at a given timestep enabled a decomposition of the thermal structure into azimuthal harmonics, from which the amplitudes and phases of the different components could be determined. An example is illustrated in Fig. 5, showing the complex spatio-temporal amplitude variations of wavenumbers 1, 2, and 3 which reveal the modulation of the dominant wave mode, its sideband, and long wave component, much as found by Read et al. (1992) and Fr€ uh and Read (1997).
In the same way as investigated by Eccles et al.
where X m ðtÞ is the measured timeseries of the amplitude of wavenumber m and X H m ðtÞ is its Hilbert Transform. The phase was then determined from u m ðtÞ is restricted by construction to the range [0,2p], but accumulating the phases such that, following every cycle, u m ðtÞ increases or decreases by 2p (termed "unwrapping" of the phase), its increasing or decreasing value in time could be observed and compared with the phase of the forcing, given (without loss of generality) by Finally, by calculating the instantaneous phase difference, phase synchronization signatures (typically related to nearstationary values of Du) could be identified and investigated. These included compiling histograms of Duðmod2pÞ and the synchronization index, as a measure of phase coherence, where R takes the value of unity for perfect phase synchronization and zero for a uniform statistical distribution of phases (Mormann et al., 2000). The observed frequencies or periods were also computed to study the frequency entrainment between forcing and response (e.g., Pikovsky et al., 2001).

III. RESULTS
As described above, an unperturbed flow was set up at a nominal point in ðH; T Þ parameter space at ð0:544; 6:81 Â10 6 Þ, corresponding to a freely evolving wavenumber 3 MAV flow pattern which was weakly chaotic (e.g., Fr€ uh and Read, 1997). Upon application of full sinusoidal forcing of the boundary conditions at frequency c with an amplitude e % 0:05DT, the vacillation phase was recorded and compared with the phase of the forcing to establish the extent to which it was phase-synchronized.
A. Duty cycle d 5 100% Figure 6 shows some examples of phase difference timeseries Du t ð Þ and histograms of Duðmod2pÞ for three different forcing periods at the full 100% duty cycle that illustrate both fully synchronized [(c) and (d)] and partially synchronized [(a), (b), (e), and (f)] cases.
In cases where the detuning (difference between the forcing and natural vacillation periods) is too large for sustained, continuous synchronization, Du t ð Þ is seen either to increase [ Fig. 6(a)] or decrease [ Fig. 6(e)] monotonically with occasional phase slips, representing a partially synchronized state. In the fully synchronized case, however, [ Fig. 6(c)] Du t ð Þ oscillates gently about a constant mean value. Even for the unsynchronized cases, however, the phase exhibits plateaux where Du t ð Þ appears to "pause" and remains nearly constant for a short while before resuming a drift. This is indicative of a nonlinear interaction between the forcing and vacillation in which the forcing partially affects the oscillation but not sufficiently to achieve a permanent synchronization, so is classified as a partial synchronization. This is also evident in the histograms of Duðmod2pÞ, which are not uniform across all phases, even for the unsynchronized cases shown in Fig. 6, also indicating a partially  Read et al. Chaos 27, 127001 (2017) synchronized state. The histogram for the fully synchronized state in Fig. 6(d) is sharply peaked, consistent with a roughly constant phase difference between forcing and response. For the other cases, however, all phases appear in the histograms in Figs. 6(b) and 6(f) but with peaks at certain favoured values of Du. This indicates some degree of partial synchronization. Other measures of the degree of synchronization for these cases are illustrated in Fig. 7, showing (a) the mean difference in the oscillation period between observed vacillation and the forcing, and (b) the synchronization index, R [cf. Eq. (7)] over a range of forcing periods, s F ; from $158 s to 174 s. Figure 7(a) clearly shows a plateau in Ds ¼ s R À s F from s F ¼ 160 s to 172 s around a natural vacillation period of approximately 164.5 s, with a markedly different s R outside this range, more or less consistent with the difference in period between s F and the natural period. The loss of synchronization outside this interval is also clearly seen in the profile of R in Fig. 7(b), which takes values close to 1 within the synchronized interval and exhibits a sharp drop-off towards smaller values beyond it. The value of R at s F ¼ 158 s and 174 s, however, does not drop immediately to zero but remains finite, indicative of partial but incomplete phase synchronization in which the monotonic progression of Du t ð Þ with time pauses temporarily in metastable synchronized states, followed by more rapid phase slips as synchronization is lost [e.g., see Figs. 6(b) and 6(f)].

B. Other duty cycles
Other duty cycles were investigated for both forcing profiles in sequences in which the forcing period was incremented successively over the same range as for the 100% duty cycle. The vacillation response was recorded as above for intervals of 1-3 h before incrementing the forcing period to the next value. The experiment was typically run continuously without stopping throughout each sequence at a given duty cycle and forcing profile.
The degree of synchronization was determined as for the 100% duty cycle presented in Sec. III A and Fig. 8 illustrates the complete results in terms of the period difference Ds for (a) forcing profile 1 and (b) forcing profile 2. For both forcing profiles, the interval in s F over which frequency entrainment and synchronization occur is seen to be a strong function of duty cycle, with the interval decreasing towards zero as the duty cycle was reduced. The boundary of the synchronized region is shown with bold lines in Fig. 8, showing   FIG. 7. Variation of (a) difference in period between wave amplitude vacillation and forcing, and (b) synchronization index, R, for wavenumber m ¼ 3 with a duty cycle d of 100%, as a function of forcing period s F . The dashed line in (a) indicates the mean natural period of the unperturbed vacillation.
FIG. 8. 3D plot of the period difference between forcing and response for fixed duty cycles using forcing profile 1 (left) and forcing profile 2 (right). These graphs show the boundaries of the phase synchronized regions (Arnol'd Tongues) as heavy black lines in each case. The results for forcing profile 2 show that the system became identifiably synchronized with a duty cycle as low as 10%, although a similar degree of synchronization required a duty cycle of $30% for forcing profile 1.
the convergence of the boundaries as d is decreased. This was more pronounced for forcing profile 1, however, for which the synchronization interval was found to shrink to almost zero by d ¼ 10%. For forcing profile 2, however, a significant interval of synchronization was evident even at d ¼ 10%. In fact, for forcing profile 2, the synchronized interval in s F at d ¼ 70% actually seemed to increase slightly over the value at d ¼ 100%, although this may be only a marginal effect. The corresponding variations of R for these two forcing profiles as a function of duty cycle d are shown in Fig. 9, for comparison with Fig. 8. These both show a plateau in R close to R ¼ 1, indicating complete phase synchronization, with steep sides that demonstrate the rapid loss of synchronization as the detuning in s F becomes too large. The sides are not completely vertical, however, indicating that some weak partial synchronization is retained even outside the Arnol'd tongue, although this decays rapidly as the detuning becomes larger. As with the profiles of Ds, the width of the plateaux becomes steadily narrower as d is reduced, but with a greater effect notable for forcing profile 1 compared with forcing profile 2.
Some typical timeseries of the wave amplitude and Du t ð Þ (normalized by 2p) are illustrated in Fig. 10 for a range of duty cycle d from 10% to 100% using forcing profile 2 with a forcing period of s F ¼ 162 s. Together with Fig. 6(d), this clearly shows that, even when fully synchronized, Du t ð Þ oscillates about its mean value at a period of s F , though does not drift. The amplitude modulation index and intensity of the irregular fluctuations of the spatial sidebands are relatively unaffected by phase-synchronizing with the periodic forcing, with sporadic fluctuations in the modulation index of m ¼ 3 even at large values of d. The azimuthal propagation of the baroclinic waves (not shown) is also not much affected by the synchronization, which is largely confined to entraining the phase of the main oscillation in wave amplitude and associated variations in heat transfer.
As d is reduced, however, the amplitude of phase fluctuations is seen to increase, especially at the lowest values of d. This is clearly shown in Table II, which shows the standard deviation of phase difference fluctuations r Du (normalized by 2p) as a function of d for two values of s F within the fully synchronized region, where the subscript number indicates the forcing profile. Even though the flow remains synchronized down to the smallest values of d in both cases, the amplitude of the oscillations in Du t ð Þ nearly doubles in size from their value at d ¼ 100% by d ¼ 10%, roughly consistent with a scaling that follows the form where r 0 ð¼ 0:039960:0013Þ is the value of r Du at d ¼ 100% and l is an exponent with a fitted value around l ¼ 0.247 6 0.027 for forcing profile 1 at s F ¼ 164 s. The data for both forcing profiles and the corresponding fit to profile 1 are shown in Fig. 11.

IV. DISCUSSION
In this paper, we have extended the experiments of Eccles et al. (2009) to demonstrate that a high degree of phase synchronization occurs between oscillations in baroclinic wave amplitude and imposed variations in boundary temperatures, even when these variations are only imposed for a fraction of the oscillation cycle as small as 10%. This, and the observation of similar degrees of synchronization for two quite different forcing profiles, clearly illustrates the robustness of this synchronization phenomenon, depending only on the coherence and repeatability of the forcing signal and not its precise waveform. The systematic variation in the width of the synchronized region in s F defines a wedgeshaped region resembling an Arnol'd tongue, indicating that the duty cycle parameter d plays a similar role to a coupling coefficient in a more conventional, steadily forced, masterslave pair of coupled oscillators.
Immediately outside the fully phase-synchronized region, the oscillations in wave amplitude are only partially synchronized but still exhibit noticeable temporary plateaux in Du t ð Þ, indicating that the periodic forcing is still exerting an influence on the amplitude oscillations, though not sufficient to constrain Du t ð Þ to a quasi-stationary value. The interval between such phase plateaux evidently increases as the detuning in s F increases. This time interval would be expected to exhibit scaling behavior as the boundary of the Arnol'd tongue is approached (e.g., Boccalletti et al., 2002), but many more experiments than were possible in the present study would be needed to confirm this.
Some indicative evidence for scaling in the amplitude of fluctuations in Du t ð Þ within the Arnol'd tongue itself was found, however, which appears to exhibit a deterministic periodic component as well as stochastic noise. This would FIG. 9. 3D plot of the synchronization index for fixed duty cycles and variable forcing period using Forcing profile 1 (left) and Forcing profile 2 (right). These graphs show the Arnol'd Tongues (black lines) of the system depending on the Forcing profile. The results for Forcing profile 2 show that the system synchronized with a duty cycle as low as 10%, showing that there are some small differences in the width of the region of synchronization depending on the Forcing profile.
suggest the effects of a higher order nonlinearity in the phase dynamics than is typically assumed (e.g., Boccalletti et al., 2002). The data are roughly consistent with a power law dependence which could suggest a growth of the variance of these fluctuations as d À1=2 which deserves further attention in future theoretical work, especially if such behavior was common to other phase-synchronized systems. It would be of significant interest to explore other aspects of scaling related to this kind of phase synchronization in the laboratory, although the long duration of experiments needed to

127001-10
Read et al. Chaos 27, 127001 (2017) acquire sufficient statistics to identify and characterize such scaling relationships represents a formidable task. As mentioned above, a significant motivation for the present study was a possible analogy between these experiments and models of synchronized teleconnections in the climate system of the Earth and other planets. Recent explanations for observed teleconnections have tended to focus on mechanisms involving the propagation of information across the planet via trains of planetary Rossby waves, which may be launched from anomalies of surface heating or topography (Hoskins and Karoly, 1981). This would tend to imply coupling via non-axisymmetric eddies in model simulations exploring synchronized teleconnections (cf. Duane et al., 1999;Lunkheit, 2001;and Duane and Tribbia, 2001). In the present case, however, and in the earlier work of Eccles et al. (2009) and Castrej on-Pita and Read (2010), coupling was implemented via the axisymmetric boundary conditions, effectively coupling the zonally symmetric "annular modes," leaving the waves and eddies within the system to determine a secondary response to this type of forcing. The results presented here and by Eccles et al. (2009) and Castrej on-Pita and  show that such annular mode forcing can also lead to a more subtle form of synchronized teleconnection that directly affects only the zonally symmetric flow though may influence the overall amplitude of wave activity indirectly. This kind of teleconnection could well play a role in the dynamics of annular modes in the climate system, such as the North Atlantic and Arctic Oscillations (Hurrell et al., 2003).
The present work, moreover, indicates that even if such annular coupling were to be significantly modulated (or even interrupted intermittently), e.g., by circulation changes during the seasonal cycle, such that it might actually decouple remote regions for significant fractions of the year, this would not preclude the possibility of significant coherent interactions that would appear as correlated cyclic behaviours between climatic components in different parts of the globe. Such a propensity for teleconnected synchronization may have significant implications, e.g., for seasonal forecasting of the climate or on longer timescales and should be explored further using more detailed and realistic models.