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Several Earth system components are at a high risk of undergoing rapid, irreversible qualitative changes or “tipping” with increasing climate warming. It is therefore necessary to investigate the feasibility of arresting or even reversing the crossing of tipping thresholds. Here, we study feedback control of an idealized energy balance model (EBM) for Earth’s climate, which exhibits a “small icecap” instability responsible for a rapid transition to an ice-free climate under increasing greenhouse gas forcing. We develop an optimal control strategy for the EBM under different forcing scenarios to reverse sea-ice loss while minimizing costs. Control is achievable for this system, but the cost nearly quadruples once the system tips. While thermal inertia may delay tipping, leading to an overshoot of the critical forcing threshold, this leeway comes with a steep rise in requisite control once tipping occurs. Additionally, we find that the optimal control is localized in the polar region.

Potential vorticity (PV) is one of the most important quantities in atmospheric science. In the absence of dissipative processes, the PV of each fluid parcel is known to be conserved, for a dry atmosphere. However, a parcel's PV is not conserved if clouds or phase changes of water occur. Recently, PV conservation laws were derived for a cloudy atmosphere, where each parcel's PV is not conserved but parcel‐integrated PV is conserved, for integrals over certain volumes that move with the flow. Hence a variety of different statements are now possible for moist PV conservation and non‐conservation, and in comparison to the case of a dry atmosphere, the situation for moist PV is more complex. Here, in light of this complexity, several different definitions of moist PV are compared for a cloudy atmosphere. Numerical simulations are shown for a rising thermal, both before and after the formation of a cloud. These simulations include the first computational illustration of the parcel‐integrated, moist PV conservation laws. The comparisons, both theoretical and numerical, serve to clarify and highlight the different statements of conservation and non‐conservation that arise for different definitions of moist PV. Potential vorticity (PV) of each fluid parcel is known to be not conserved if clouds and phase changes are present. Numerical simulations here illustrate a new moist PV conservation principle that is valid even if clouds and phase changes occur. Moist PV, when integrated over certain volumes that move with the flow, will remain conserved (and, in this case, zero), even with the formation and presence of clouds.

With shifting environmental trends, many Earth system elements may be poised to undergo critical transitions or ‘tipping’. Reliable anticipation of these tipping elements is vital to inform policy decisions. Many of the current methods for tipping point detection are based on loss of resilience or ‘critical slowdown’ of the system as it approaches a tipping point. However, these methods are prone to false alarms; the detected slowdown may be an artifact of nonstationary noise unrelated to tipping behavior. Here, we explore the efficacy of early warning signs based on a nonequilibrium thermodynamics framework. The model-free detection method relies on the increased intrinsic time-irreversibility due to detailed balance breaking, preceding the onset of tipping or instabilities. We demonstrate that these EWSs are effective for tipping point detection and robust against false alarms due to nonstationary noise, using idealized models for two key elements of the Earth system that are prone to tipping: the Atlantic Meridional Overturning Circulation and Arctic sea-ice loss. Increased intrinsic time-irreversibility provides a robust early warning for Earth system tipping points, outperforming traditional methods in the presence of nonstationary noise, according to a study based on a nonequilibrium thermodynamics framework.

One of the most important conservation laws in atmospheric and oceanic science is conservation of potential vorticity. The original derivation is approximately a century old, in the work of Rossby and Ertel, and it is related to the celebrated circulation theorems of Kelvin and Bjerknes. However, the laws apply to idealized fluids, and extensions to more realistic scenarios have been problematic. Here, these laws are extended to hold with additional fundamental complexities, including salinity in the ocean, or moisture and clouds in the atmosphere. In the absence of these additional complexities, it is known that potential vorticity is conserved following each fluid parcel; here, for a salty ocean or cloudy atmosphere, the general conserved quantity is potential vorticity integrated over certain pancake-shaped volumes. Furthermore, the conservation laws are also related to a symmetry in the Lagrangian, which brings a connection to the symmetry-conservation relationships seen in other areas of physics.

Many definitions of moist potential vorticity (PV) have been proposed to extend the dry theory of Ertel PV. None of the moist PV definitions seem to have all of the desirable properties of the dry Ertel PV. For instance, dry PV is not only a globally conserved quantity, but also a material invariant that is conserved along fluid parcel trajectories. Therefore, an open question remains: is there a moist PV that is a material invariant, if clouds and phase changes of water are present? In prior studies, definitions of moist PV have been proposed based on physical and mathematical intuition. Here, a systematic approach is used. In particular, a particle relabeling symmetry is devised for a moist atmosphere and then Noether's theorem is employed to arrive at the associated conservation laws for moist PV. A priori, it is not clear whether this systematic approach will be viable, since it relies on variational derivatives in Hamilton's principle, and phase changes introduce singularities that could potentially prevent derivatives at cloud edge. However, it is shown that the energy and the Lagrangian density are sufficiently smooth to allow variational derivatives, in a moist Boussinesq system with reversible phase transitions between water vapor and liquid cloud water.. From the particle relabeling symmetry, a moist Kelvin circulation theorem is found, along with a moist PV conservation law that applies not for each individual parcel but for parcel-integrated PV, integrated over certain local volumes.

For two-dimensional Rayleigh-Bénard convection, classes of unstable, steady solutions were previously computed using numerical continuation (Waleffe, 2015; Sondak, 2015). The `primary' steady solution bifurcates from the conduction state at \\(Ra 1708\\), and has a characteristic aspect ratio (length/height) of approximately \\(2\\). The primary solution corresponds to one pair of counterclockwise-clockwise convection rolls with a temperature updraft in between and an adjacent downdraft on the sides. By adjusting the horizontal length of the domain, (Waleffe, 2015; Sondak, 2015) also found steady, maximal heat transport solutions, with characteristic aspect ratio less than \\(2\\) and decreasing with increasing \\(Ra\\). Compared to the primary solutions, optimal heat transport solutions have modifications to boundary layer thickness, the horizontal length scale of the plume, and the structure of the downdrafts. The current study establishes a direct link between these (unstable) steady solutions and transition to turbulence for \\(Pr = 7\\) and \\(Pr = 100\\). For transitional values of \\(Ra\\), the primary and optimal heat transport solutions both appear prominently in appropriately-sized sub-fields of the time-evolving temperature fields. For \\(Ra\\) beyond transitional, our data analysis shows persistence of the primary solution for \\(Pr = 7\\), while the optimal heat transport solutions are more easily detectable for \\(Pr = 100\\). In both cases \\(Pr = 7\\) and \\(Pr = 100\\), the relative prevalence of primary and optimal solutions is consistent with the \\(Nu\\) vs. \\(Ra\\) scalings for the numerical data and the steady solutions.

Two different types of atmospheric flows: flows with clouds and phase changes and Rayleigh–Bénard convection were studied in this thesis. Our efforts led to the identification of coherent structures associated with potential vorticity conservation in the moist atmosphere and heat transport in transitional Rayleigh–Bénard convection.Moist atmospheric flows [Kooloth et al., 2022b,a]: One of the most important conservation laws in atmospheric and oceanic science is conservation of potential vorticity. The original derivation is approximately a century old, in the work of Rossby and Ertel, and it is related to the celebrated circulation theorems of Kelvin and Bjerknes. However, the laws apply to idealized fluids, and extensions to more realistic scenarios have been problematic. Here, these laws are extended to hold with additional fundamental complexities, including salinity in the ocean, or moisture and clouds in the atmosphere. In the absence of these additional complexities, it is known that potential vorticity is conserved following each fluid parcel; here, for a salty ocean or cloudy atmosphere, the general conserved quantity is potential vorticity integrated over certain pancake-shaped volumes. Furthermore, the conservation laws are also related to a symmetry in the Lagrangian, which brings a connection to the symmetry-conservation relationships seen in other areas of physics.Rayleigh–Bénard convection [Kooloth et al., 2021]: For two-dimensional (2D) Rayleigh-Bénard convection, classes of unstable, steady solutions were previously computed using numerical continuation [Waleffe et al. [2015], Sondak et al. [2015]]. The ‘primary’ steady solution bifurcates from the conduction state at Ra ≈ 1708, and has a characteristic aspect ratio (length/height) of approximately 2. The primary solution corresponds to one pair of counterclockwise-clockwise convection rolls with a temperature updraft in between and an adjacent downdraft on the sides. By adjusting the horizontal length of the domain, Waleffe et al. [2015], Sondak et al. [2015] also found steady, maximal heat transport solutions, with characteristic aspect ratio less than 2 and decreasing with increasing Ra. Compared to the primary solutions, optimal heat transport solutions have modifications to boundary layer thickness, the horizontal length scale of the plume, and the structure of the downdrafts. The current study establishes a direct link between these (unstable) steady solutions and transition to turbulence for Pr = 7 and Pr = 100. For transitional values of Ra, the primary and optimal-heat-transport solutions both appear prominently in appropriately-sized sub-fields of the time-evolving temperature fields. For Ra beyond transitional, our data analysis shows persistence of the primary solution for Pr = 7, while the optimal heat transport solutions are more easily detectable for Pr = 100. In both cases Pr = 7 and Pr = 100, the relative prevalence of primary and optimal solutions is consistent with the Nu vs. Ra scalings for the numerical data and the steady solutions.

Symmetry breaking is a central organizing principle of nonlinear dynamics, yet its information-theoretic signatures remain poorly understood. In this work, we develop an entropic framework for analysing spontaneous symmetry breaking in equivariant dynamical systems, that distinguishes between local and global mechanisms of symmetry loss and shows that each requires a different analytical lens. For local spontaneous symmetry breaking, where a symmetric equilibrium loses stability through bifurcation, we show that critical slowing down broadens the regularized invariant density, increases Shannon entropy, and sharply amplifies steady-state directional information transfer. For global spontaneous symmetry breaking, where the invariant measure itself undergoes qualitative reorganization, we derive a general entropy law for invariant-density restructuring. The broken-phase entropy decomposes into an internal sector entropy and a symmetry-label information term, with classical invariant-set splitting arising as the disjoint-sector limit. Consequently, global symmetry breaking may either increase or decrease entropy depending on how probability is redistributed across symmetry-related sectors. Analytical results are illustrated through canonical examples, providing a quantitative bridge between symmetry, bifurcation structure, invariant measures, and information theory in dynamical systems.

We develop an entropy based framework for analyzing symmetry breaking in dynamical systems. Information transfer, which measures the directional exchange of entropy between observables, provides a quantitative early indicator of symmetry loss. For local spontaneous symmetry breaking (SSB), we show that as a symmetric equilibrium approaches instability, trajectories exhibit pronounced critical slowing down accompanied by a rise in Shannon entropy. This establishes a direct link between symmetry loss, slowing down, and entropy growth. We further characterize the entropy discontinuity associated with global symmetry breaking (GSSB) through an ergodic decomposition viewpoint. Numerical examples illustrate that entropy and information transfer measures serve as reliable precursors and diagnostics of symmetry breaking transitions.

We propose a novel framework to analyze symmetry breaking in dynamical systems through the lens of entropy and information transfer. Information transfer quantifies the directional exchange of entropy between observables, allowing us to anticipate the onset of symmetry breaking. For local symmetry breakings, namely, local Spontaneous Symmetry Breaking (SSB) and Dynamical Symmetry Breaking (DSB), we show that as a system loses symmetry, its trajectories exhibit a pronounced slowdown accompanied by an increase in Shannon entropy. This establishes a direct link between symmetry loss, dynamical slowing down, and entropy growth. We also extend the analysis to global symmetry breaking and characterize its associated entropy change. Finally, we demonstrate the efficacy of the proposed framework using representative examples, showing that information theoretic quantities can serve as reliable precursors and diagnostics of symmetry breaking transitions.