By Matthew R. Francis
For many years, scientists have warned that the Atlantic meridional overturning circulation (AMOC)—the thermal cycle that drives currents in the Atlantic Ocean—is getting weaker [1]. Among other effects, the AMOC carries warm water to Ireland and the U.K. and returns cooler water from the north to southern regions. Instability in this circulation cycle could result in its complete collapse and cause widespread disruptions in temperature, changes in rain and snowfall patterns, and other natural disasters.
The potential loss of the AMOC represents a possible tipping point due to human-driven climate change. Global increases in temperature lead to warmer ocean water and melting polar ice, both of which decrease water density (see Figure 1). The subsequent lower-density water does not sink as much as it cools, thus disrupting the thermal cycle. When the AMOC collapsed in the prehistoric past, it jolted Earth’s climate and affected every ecosystem.
Figure 1. Sea ice off the coast of Greenland, which has been diminished by warming Arctic temperatures. The infusion of cold water into the ocean risks disrupting the Atlantic meridional overturning circulation. Public domain image courtesy of NASA.
Many researchers, including Christopher K. Jones of the University of North Carolina at Chapel Hill, have searched for mathematical models to describe processes that resemble the AMOC collapse. Jones delivered an invited talk about this subject at the
2021 SIAM Conference on Applications of Dynamical Systems (DS21), which took place virtually in May. “Climate change is all about the
rate at which something is changing,” Jones said. “The impacts of climate change may be triggered not just because the state of the system you’re looking at is reaching a certain threshold, but because the
rate at which the state of the system is changing is triggering the event.”
In other words, the rapidity of climate change might be influencing the likelihood of AMOC collapse, severe droughts, extreme hurricane seasons, and other phenomena more than a static temperature or greenhouse-gas concentration threshold. For instance, some studies suggest that a fairly rapid infusion of cold freshwater into the ocean could have triggered a 1,000-year cold snap known as the Younger Dryas [2]. According to this theory, a slower introduction of the same amount of water would not have created such a large disruption; it was the rapidity of the cold water infusion that made the difference.
Jones and his graduate student Katherine Slyman are therefore examining a particular class of dynamical system models that involve rate-induced tipping (R-tipping), which differs from the better-known bifurcation-based tipping (B-tipping). In B-tipping, the dynamical system has different steady-state configurations; the one in which the system resides depends upon the value of state parameters, not the rate at which those parameters change. “There are identifiable different stable states that are quite distinct from each other,” Jones said. “The transition from one to the other can be abrupt.” It need not be a small perturbation either; the trigger for the transition could be large in some cases.
In her own minisymposium presentation at DS21, Slyman used the analogy of a well-known magic trick wherein a magician sets dishes on a tablecloth and quickly pulls the cloth away. If the magician is too slow in yanking the cloth, the dishes fall on the floor and break; with sufficient speed they remain on the table. The rate of change makes all the difference.
Time Is (Not) On Our Side
Sebastian Wieczorek of the University of Cork and his colleagues were the first researchers to work out much of the R-tipping formalism. They were examining the “compost bomb instability,” a phenomenon that causes peatlands to spontaneously catch fire. “They’re still below their ignition temperature,” Jones said. “But the rate at which they’ve been warmed causes them to catch fire. So, the rate has reached a certain threshold and not the state itself.”
One challenge of R-tipping compared to B-tipping is that the systems are non-autonomous, which means that the equations include explicit time dependence. A generic way to write a system of \(n\) equations with a single time-dependent parameter \(\Lambda\) is \(\dot x = f(x, \Lambda(t)) x \in \mathbb{R}^n\), \(\Lambda \in \mathbb{R}\). “When you introduce rate-induced tipping, you have this parameter that now changes in time,” Slyman said. “That makes this problem hairy to solve and kind of unpleasant.”
The particular behavior of some non-autonomous R-tipping systems allows for simplifications that exploit the essential two-state nature of tipping points. As is common with other non-autonomous systems, one must first define a new variable \(s=rt\) with a constant rate parameter \(r>0\):
\[\dot x = f(x, \Lambda(s))\] \[\dot s =r.\]
This formulation makes the system autonomous, albeit by increasing the model’s dimension from \(n\) to \(n+1\). However, the choice of the function \(\Lambda\) also plays an important role in the model’s manageability. “It’s important to understand that this is not an instantaneous thing and not a very long-term thing,” Jones said. “It’s kind of an intermediate time thing. The rate is changing rapidly but it’s not changing instantaneously. And it’s not not changing at all; it’s somewhere in the middle.”
Hyperbolic tangent is a simple function that behaves in this way. It has a finite range \([-1,1]\) for its domain and is invertible, differentiable, and ramp-like in shape:
\[\Lambda(t)=1+\tan h(rt).\]
The rate \(r\) controls the ramp function’s steepness and the limits as \(t\) approaches \(\pm \infty\) define the system’s behavior before and after the transition. Researchers use a topological trick known as compactification—which attaches points at infinity onto the phase space \(\{x, \Lambda\}\)—to extract this asymptotic behavior [3]. Since the derivative of the ramp function is zero at \(\Lambda(\pm \infty)= \lambda_\pm\) in the compactified space, the new task involves identifying places where \(\dot x = 0\) as well; these spots are asymptotic fixed points (see Figure 2).
Figure 2. A toy mathematical model that demonstrates compactification for rate-based tipping. Adjusting the rate at which the system changes leads to qualitatively different asymptotic behaviors with a critical value in between; compactification attaches those asymptotic fixed points to the phase space. Figure courtesy of [3].
Even with a ramp function, the system is not guaranteed to tip. In order for R-tipping to occur, the system must have qualitatively different outcomes based on \(r\)'s value. The critical rate is the value of \(r\) at which there are heteroclinic orbits — trajectories that connect two saddle points in the phase space of the dynamical variable and the rate-dependent parameter. Above and below this critical value, the system variables \(x\) follow different trajectories that are unreachable from each other if \(r\) does not vary. R-tipping does transpire for many model problems, including some that correspond to real-world applications.
Bring in the Noise, Bring in the Tipping
In addition to B- and R-tipping, researchers have also investigated noise-induced tipping (N-tipping). As the name suggests, random fluctuations cause the dynamical system in these models to change state. Consider the standard Wiener process in the general case
\[dx=f(x, \Lambda)dt+\sigma dW.\]
Here, \(\sigma\) and \(W\) quantify the noise. \(\Lambda\) may be fixed for N-tipping alone or time dependent for a system with both noise and R-tipping; the latter case is especially interesting.
“What we see in the model problem is that you don’t need to reach the \(r\) critical value if you put this addition of noise in the system,” Slyman said. “The system can tip, and tip quite often—not even rarely—when you lower that ramp parameter and add noise to it.” But if the ramp function is absent, noise alone cannot tip the model system in a reasonable amount of time. Noise and changing rates collectively produce qualitatively different behavior than either R- or N-tipping on their own.
The study of R-tipping is still a relatively new field, with most results in low-dimension systems. However, realistic climate change models—including those for AMOC—are very complex. Therefore, not all major climate-related changes are describable via R-tipping (with or without noise). For instance, Jones spoke at DS21 about a model for which R-tipping can describe hurricane dissipation but not hurricane genesis. Because climate change involves so many factors, it seems likely that the paradigm will prove successful for certain phenomena — particularly those driven by changes that commence more quickly than anything Earth has ever experienced.
References
[1] Boers, N. (2021). Observation-based early-warning signals for a collapse of the Atlantic meridional overturning circulation. Nat. Clim. Change, 11, 680-688.
[2] Mooney, C. (2018, July 11). Scientists may have solved a huge riddle in Earth’s climate past. It doesn’t bode well for the future. The Washington Post. Retrieved from https://www.washingtonpost.com/energy-environment/2018/07/11/scientists-may-have-solved-huge-riddle-earths-climate-past-it-doesnt-bode-well-future.
[3] Wieczorek, S., Xie, C., & Jones, C.K.R.T. (2021). Compactification for asymptotically autonomous dynamical systems: Theory, applications and invariant manifolds. Nonlin., 34(5), 2970.
Further Reading
Ashwin, P., Perryman, C., & Wieczorek, S. (2017). Parameter shifts for nonautonomous systems in low dimension: Bifurcation- and rate-induced tipping. Nonlin., 30(6), 2185.
Matthew R. Francis is a physicist, science writer, public speaker, educator, and frequent wearer of jaunty hats. His website is BowlerHatScience.org.