Colloqium
To exhange research ideas about the mathematics of climate both within and beyond the network, MCRN runs an online Colloquium series Mondays noon-1pm PDT. Speakers aim for 45 min talks, with the rest of the time devoted to discussion.
MCRN members receive Colloquium invitations by email. To join MCRN, please fill out this form. For inquiries about the Colloquium, please email Dr. Christian Sampson at csampson-at-ucar.edu.
Our 2024 schedule includes:
Aug. 12 Dynamics of glacial cycles
Hans Kaper, Georgetown U.
Abstract: The geological record shows great variability of Earth’s climate. During the Pleistocene Epoch (from approximately 2.6 Myr before present (BP) until approximately 11.7 Kyr BP), continental ice sheets expanded and contracted over significant areas, especially in the Northern Hemisphere. This happened in a more or less cyclical fashion, with periods of approximately 41 Kyr during the early Pleistocene and approximately 100 Kyr during the late Pleistocene. In this talk I will discuss a conceptual model first presented by Maasch and Saltzman (1990) to explain this persistence of glacial cycles as the result of the interaction of atmospheric carbon dioxide and the strength of the North Atlantic overturning circulation. The model consists of a system of three ordinary differential equations with a rich bifurcation structure.
Aug. 19 An overview of mathematical results regarding atmospheric convection
Roland Welter, U. Hamburg
Abstract: In the study of climate, general circulation models aim to accurately represent the motions of the atmosphere and ocean using equations from fluid dynamics. However, vertical accelerations are often set equal to zero in primitive equation models. This assumption is partially justified from a physical viewpoint, since the vertical accelerations should be small compared to more dominant forces (gravity, etc) which are in an approximate hydrostatic balance. Furthermore, this yields a significant mathematical benefit since the well-posedness theory for such equations is then much more satisfactory. On the other hand, such an assumption cannot be entirely physically accurate, and hence often in the climate community additional terms are included to account for the discrepancy. This procedure is known as parameterization, and there is not a consensus about a correct or optimal way to parameterize convection.
In this presentation, I will present recent results which develop a mathematically rigorous framework for studying vertical heat transport in turbulent convection. Starting from the paradigmatic Boussinesq-Oberbeck equations, heat transport is investigated via the HKC hierarchy of Galerkin truncated ODE models of increasing dimension. The dynamics of these models are studied, and particular attention is given to stable values of heat transport, as well as the convergence across models where the models accurately represent the PDE. Implications for energetically consistent parameterization of convection will then be discussed.
Aug. 26 Tipping and EWS in some nonautonomous scalar ODEs
Iacopo Longo, Imperial College London
Abstract: Unlike their autonomous counterparts, nonautonomous bifurcations are inherently complicated already in dimension one. Not only equilibria and periodic orbits become extremely rare, but also new phenomena appear which are not possible in the autonomous case. In this talk, we characterize the dynamical scenarios for nonautonomous scalar concave and d-concave differential equations and study the occurrence of nonautonomous saddle-node bifurcations of hyperbolic solutions. Concave and d-concave problems are largely used in theoretical ecology but also in theoretical climate science. Additionally, we investigate the use of finite-time Lyapunov Exponents as Early Warning signs of a critical transition and show how they can be successfully employed to avoid the collapse of a certain desirable state.
Sep. 9 The pending collapse of the Atlantic Overturning Circulation: a false alarm or an Early Warning Signal of a tipping point?
Ka-Kit Tung, University of Washington
Abstract: There has recently been intense publicity in news media about the pending collapse of AMOC, as early as 2025, with severe global climate consequences. AMOC is just one of many possible tipping points of the global climate system; there is urgent need to understand the possible Early Warning Signals (EWS) that tell us how close we are to any tipping point. We examine the mathematics and the data analyses of EWS. There does not appear to be any EWS in the deterministic system, but the behavior of noise in the stochastic system may yield some clues. Approaching a saddle-node bifurcation in a stochastic dynamical system, variance of the noise and its autocorrelations may increase and these are taken as classic examples of EWS, but they could also increase due to other external reasons. More sophisticated methods involve using results of Ornstein-Uhlenbeck process, and empirical generalized regression analyses. The need for deep mathematics has fostered close collaborations between climate scientists and mathematicians/statisticians, but going for ever deeper mathematics may not necessarily be the right approach. As EWS depends sensitively on the character of the “noise” in the time series, how data are collected, reanalyzed or assimilated also need to be carefully considered. Because of the large natural oscillations in AMOC, with a quasi-period of about 60-100 years, data lengths need to be at least 120 years. Our conclusion is that the existing prediction on tipping points are false alarms, and furthermore, the prospect of finding EWS may need to wait another hundred years.
Sep. 16 Transit times for nonautonomous compartmental systems and applications to ecology (Modified schedule: 8am PDT)
Martin Rasmussen, Imperial College London
Abstract: We develop a theory for transit times and mean ages for nonautonomous (linear) compartmental systems. We show that the mean ages of mass in a compartmental system satisfy a linear nonautonomous ordinary differential equation that is exponentially stable. We then define a nonautonomous version of transit time as the mean age of mass leaving the compartmental system at a particular time. We apply these results two nonautonomous compartmental systems, modeling the terrestrial carbon cycle and breeder states of the Southern Fulmar seabird. Joint work with Alan Hastings, George Chappelle, Matthew Smith, Yiqi Luo, Folashade Agusto, Benito Chen-Charpentier, Forrest Hoffman, Jiang Jiang, Katherine Todd-Brown, Ying Wang, Ying-Ping Wang.
Sep. 23 The Maslov index in dynamical systems and PDEs: recent applications and connections to stability
Emmanuel Fleurantin, George Mason University and RENCI
Abstract: This talk introduces the Maslov index and its application to analyzing dynamical systems, particularly in the context of wave stability. The Maslov index, a topological invariant associated with classical oscillation theorems, has been generalized to detect instabilities in solutions of evolutionary partial differential equations. We will explore computational and analytical tools for stability analysis, presenting a method that represents solutions as angles fluctuating within the phase space of the eigenvalue problem. As a case study, we will examine the cubic nonlinear Schrödinger equation (NLSE) with a decaying potential. Although the NLSE is not widely used in large-scale climate modeling, it has applications in climate science and oceanography, providing an excellent model for our analysis. We will demonstrate how to count eigenstates in the gap on the imaginary axis beneath the essential spectrum, extending Sturm-Liouville theory via the Maslov index. This approach has broad implications across applied mathematics and physics, offering insights into complex phenomena in both natural and engineered systems
Oct. 7 No need to reinvent the wheel: data assimilation in the age of AI
Ivo Pasmans, University of Reading
Abstract: TBD
Oct. 21 Stability of floating objects
Dan Anderson, George Mason University
Abstract: TBD
Nov. 11 An investigation of tipping mechanisms in a carbon cycle model
Katherine Slyman, Boston College
Abstract: TBD
Dec. 2 Rate induced tipping in a habitat PDE model
Blake Barker, Brigham Young University
Abstract: TBD
Our 2025 schedule includes:
Jan. 27 Computing Lyapunov exponents efficiently
Evelyn Sander, George Mason University
Abstract: TBD