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Sub-Neptune type exoplanets are abundant in our galaxy yet have no solar system analogs. They exist in a broad range of stellar forcing and rotational regimes that are distinctly different from solar system planets and more commonly studied hot Jupiters. Here we present simulations that explore global atmospheric circulation of sub-Neptunes generated with a two-dimensional shallow-water model, SWAMPE. We explore the circulation regimes of synchronously rotating sub-Neptunes with a focus on the interaction of planetary rotation rate and radiative timescale in a variety of stellar insolations. In highly irradiated, short-timescale regimes, our models exhibit high day-night geopotential contrasts. As the timescales become longer, the geopotential contrasts and longitudinal variability decrease, while temporal variability increases. The transition from day-to-night flow to jet-dominated flow is primarily driven by the radiative timescale. Strong- and medium-forcing regimes exhibit transitions between day-to-night flow and jet-dominated flow at similar points in the parameter space. Weak-forcing regime differs due to comparatively stronger rotational effects. Planetary rotation period dominates in determining equator-to-pole geopotential contrast. Our simulations exhibit higher time variability when either radiative timescale or rotation period is long., Comment: Accepted for publication in PSJ on 5/12/2023. 25 pages, 1 table, 11 figures

Using a combination of theory, experiment, and simulation, we revisit the dynamics of two coupled metronomes on a moving platform. Our experiments show that the platform's motion is damped by a dry friction force of Coulomb type, not the viscous linear friction force that has often been assumed in the past. Prompted by this result, we develop a new mathematical model that builds on previously introduced models, but departs from them in its treatment of the friction on the platform. We analyze the model by a two-timescale analysis and derive the slow-flow equations that determine its long-term dynamics. The derivation of the slow flow is challenging, due to the stick-slip motion of the platform in some parameter regimes. Simulations of the slow flow reveal various kinds of long-term behavior including in-phase and antiphase synchronization of identical metronomes, phase locking and phase drift of non-identical metronomes, and metronome suppression and death. In these latter two states, one or both of the metronomes come to swing at such low amplitude that they no longer engage their escapement mechanisms. We find good agreement between our theory, simulations, and experiments, but stress that our exploration is far from exhaustive. Indeed, much still remains to be learned about the dynamics of coupled metronomes, despite their simplicity and familiarity., Comment: 25 pages, 19 figures

We introduce a new method for analyzing the carbon budget using box models and a mass balance approach. The method describes the net flow of carbon between the atmosphere and other reservoirs. The method assumes that the data can be explained by having the carbon move in and out of the atmosphere through two reservoirs, one consisting of isotopically light carbon (biotic) and the other consisting of isotopically heavy carbon (abiotic). The systems are underdetermined from the data, so the Occam's razor approach is to assume that the least amount of carbon moves between the atmosphere and the reservoirs. As data from known sources and sinks are added to the model, one can determine the constraints on the unknown sources and sinks. To illustrate the method, we analyze data from the Mauna Loa Observatory and data from Antarctic ice cores., Comment: 28 pages, 15 figures

The current definition of rate-induced tipping is tied to the idea of a pullback attractor limiting in forward and backward time to a stable quasi-static equilibrium. Here we propose a new definition that encompasses the standard definition in the literature for certain scalar systems and includes previously excluded $N$-dimensional systems that exhibit rate-dependent critical transitions., Comment: 11 pages, 5 figures

We present a new conceptual model of the Earth's glacial-interglacial cycles, one leading to governing equations for which the vector field has a hyperplane of discontinuities. This work extends the classic Budyko- and Sellers-type conceptual energy balance models of temperature-albedo feedback by removing the standard assumption of planetary symmetry about the equator. The dynamics of separate Northern and Southern Hemisphere ice caps are coupled to an equation representing the annual global mean surface temperature. The system has a discontinuous switching mechanism based on mass balance principles for the Northern Hemisphere ice sheet. We show the associated Filippov system admits a unique nonsmooth and attracting limit cycle that represents the cycling between glacial and interglacial states. Due to the vastly different time scales involved, the model presents a nonsmooth geometric perturbation problem, for which we use ad hoc mathematical techniques to produce the periodic orbit. We find climatic changes in the Northern Hemisphere drive synchronous changes in the Southern Hemisphere, as is observed for the Earth on orbital time scales., Comment: 21 pages, 5 figures

In this study, we compare two analytic energy balance models with explicit dependence on obliquity to study the likelihood of different stable ice configurations. We compare the results of models with different methods of heat transport and different insolation distributions. We show that stable partial ice cover is possible for any obliquity, provided the insolation distribution is sufficiently accurate. Additionally, we quantify the severity of the transition to the Snowball state as different model parameters are varied. In accordance with an earlier study, transitions to the Snowball state are more severe for higher values of the albedo contrast and energy transport across latitudes in both model; however, we find that the Snowball transition is not equally likely across both models. This work is general enough to apply to any rapidly rotating planet and could be used to study the likelihood of Snowball transitions on planets within the habitable region of other stars., Comment: 14 pages, 6 figures. Accepted for publication in Planetary Science Journal

In 1665, Huygens observed that two pendulum clocks hanging from the same board became synchronized in antiphase after hundreds of swings. On the other hand, modern experiments with metronomes placed on a movable platform show that they often tend to synchronize in phase, not antiphase. Here we study both in-phase and antiphase synchronization in a model of pendulum clocks and metronomes and analyze their long-term dynamics with the tools of perturbation theory. Specifically, we exploit the separation of timescales between the fast oscillations of the individual pendulums and the much slower adjustments of their amplitudes and phases. By scaling the equations appropriately and applying the method of multiple timescales, we derive explicit formulas for the regimes in parameter space where either antiphase or in-phase synchronization are stable, or where both are stable. Although this sort of perturbative analysis is standard in other parts of nonlinear science, it has been applied surprisingly rarely in the context of Huygens's clocks. Unusual features of our approach include its treatment of the escapement mechanism, a small-angle approximation up to cubic order, and both a two- and three-timescale asymptotic analysis., Comment: 15 pages, 8 figures

We derive the Legendre series expansion for the insolation distribution on rapidly rotating planets as a function of sine of the latitude and the planet's obliquity. We give an explicit formula for the coefficients of this series as it depends on the obliquity and approximate the convergence rate. We determine the optimal truncation of the series for use in climate models and compare this approximation to other approximations in the literature., Comment: 29 pages, 8 figures

A wide variety of physical systems ranging from the firing of neurons to eutrophication of lakes to the presence of Arctic summer sea ice exhibit a phenomenon known as tipping. In mathematical models, tipping can be caused by bifurcations, noise, and the rate at which parameters are changing in time [2]. Because traditional methods in dynamical systems are usually concerned with the long-term behavior of the system, these methods are not always able to detect the transient dynamics characteristic of rate-tipping. In this paper, we consider one- and two-dimensional dynamical systems with nonautonomous parameters that exhibit rate-tipping, as defined as not tracking the evolution of stable equilibria (QSEs) in the corresponding autonomous systems. We find that nonautonomous stability spectra in the form of Steklov averages and their derivatives appear to be correlated with transient rate-tipping in systems with unique QSEs or with parameters that change at a constant rate. Furthermore, for systems in two dimensions and higher, comparison of the angle between leading Lyapunov vectors of different trajectories admits a possible criterion for detecting rate-tipping. Our heuristic results add to the body of work dedicated to studying and understanding the phenomenon of rate-tipping., Comment: 20 pages, 7 figures, 2 tables

In this paper, we use a sixth order Legendre series expansion to approximate the mean annual insolation by latitude of a planet with obliquity angle $\beta$, leading to faster computations with little loss in the accuracy of results. We discuss differences between our method and selected computational results for insolation found in the literature., Comment: 5 pages, 2 figures