‘Does the flap of a butterfly’s wing in Brazil stir up a tornado in Texas?’ These words reflect the so-called ‘butterfly effect’ typically ascribed to Edward Lorenz (MIT), who was a pioneer in the field of numerical weather prediction (NWP). (Interestingly, Lorenz himself first referred to the ‘flap of a seagulls’ wing’; the butterfly swap out was the work of a creative conference session convener. Somewhat coincidentally, the shape of the ‘Lorenz attractor’ is also reminiscent of a butterfly, see Figure 1). Motivated by the very salient challenge in the post-WWII era to improve weather prediction capabilities for strategic purposes, he developed a theoretical framework for the predictability of non-linear dynamical systems in a seminal paper, ‘Deterministic nonperiodic flow’ (Lorenz, 1963), that would come to be known as ‘chaos theory’ or ‘chaotic dynamics’. This theory was foundational to the development of ensemble forecasting (and a whole host of other things), and still underpins our current theoretical understanding of the inherent predictability of complex dynamical systems like the atmosphere (Lorenz, 2006).
Figure 1. Book cover (left panel) and the characteristic ‘butterfly wing’ shape of the Lorenz attractor (right panel) from his seminal 1963 work. (Palmer, 1993)
In actuality, the whole ‘flap of a butterfly’s wing creating a tornado in Texas’ relates specifically to only one aspect of the theoretical framework developed by Lorenz. In this post, I will attempt to give a slightly fuller treatment of this theoretical framework and its development. My hope is that this post will be a theoretical primer for two follow-on practically oriented posts: 1) a discussion of the state-of-the-science of hydrometeorological ensemble forecasting and its application in emerging water resources management practices like forecast informed reservoir operations, and 2) a deeper dive into ensemble verification techniques. While the techniques for (2) are grounded in the theoretical statistical properties of stochastic-dynamic predictive ensembles, they have broad applications to any scenario where one needs to diagnose the performance of an ensemble.
The Lorenz model and ‘chaos’ theory
When most people hear the word ‘chaos’, they tend to think of the dictionary definition: ‘complete disorder and confusion’ (Oxford), which in a scientific context might aptly describe some sort of white noise process. As you will see, this is somewhat of a far cry from the ‘chaos’ described in Lorenz’s theory. The ‘chaos’ terminology was actually coined by a later paper building on Lorenz’s work (Li & Yorke, 1975) and as noted by Wilks (2019): ‘…this label is somewhat unfortunate in that it is not really descriptive of the sensitive-dependence phenomenon’. The sensitive-dependence phenomenon is one of a set of 4 key properties (to be discussed later) that characterize the behaviors of the sort of non-linear, non-periodic deterministic systems that Lorenz argued were most representative of the atmosphere. In contrast to ‘disorder’ and ‘confusion’, these properties, in fact, lend some sense of order to the evolution of these dynamical systems, although the nature of that order is highly dependent on the system state.
A deterministic, non-periodic systems model
First, let’s dive into a few details of Lorenz’s 1963 work, using some figures and descriptions from a later paper (Palmer, 1993) that are easier to follow. While the 1963 paper is quite dense, much of the mathematical logic is dedicated to justifying the use of a particular system of equations that forms the basis of the study. This system of 3 equations and 3 variables (X, Y, Z) describes a non-linear, dissipative model of convection in a fluid of uniform depth, where there is a temperature difference between the upper and lower surfaces. Lorenz derived the set of 3 equations shown in the upper left panel of Figure 2 from earlier work by Rayleigh (1916) on this particular problem. In short, X relates to the intensity of convective motion, Y relates to the temperature difference between ascending and descending currents, and Z relates to the distortion of the vertical temperature profile from linearity; the details of these variables are not actually all that important to the study. What is important is that this system of equations has no general solutions (aside from the steady state solution) and must be numerically integrated in the time dimension to determine the convective flow dynamics. The ‘trajectories’ of these integrations shown in the phase space diagrams in Figure 2 exhibit the sorts of unstable, non-periodic behaviors that Lorenz thought were the most useful analog to atmospheric dynamics, albeit in a much simpler system. (‘Much’ is an understatement here; modern NWP models have a phase space on the order of 109 in contrast to the 3-dimensional phase space of this problem, Wilks, 2019. Of note, the use of phase-space diagrams (i.e. plots where each axis corresponds to a dependent variable in the system of equations) preceded Lorenz, but his use of them is perhaps one of the best-known early instances of this kind of visualization. Other uses of phase space relationships can be found in Rohini’s post or Dave’s post)
Figure 2. a) Lorenz equations and the XY phase space diagram of their integrations. b-c) X variable timeseries of two separate integrations of the Lorenz equations from very similar initial conditions. (Palmer, 1993)
Regime structure and multiple distinct timescales
What ‘behaviors’ then, are we talking about? Referring again to Figure 2a, we see the projection of a long series of model integration trajectories onto the XY-plane of the phase space diagram, where each dot is a timestep in the integration. What is immediately apparent in the form of these integrations is the two lobed ‘butterfly’ shape of the diagram. Each lobe has a central attractor where the trajectories often reside for multiple revolutions, but can then transition to the other attractor when passing near the center of the diagram. These so-called ‘Lorenz attractors’ comprise one of the key properties of chaotic dynamics, namely regime structure, which is tendency of the trajectories to reside around phase space attractors for some period of time. This residence time in a regime is generally quite a bit longer than the timescales of the trajectories’ individual revolutions around an attractor. This attribute is referred to as multiple distinct timescales and is evidenced in Figure 2b-c, where smaller amplitude sections of the timeseries show residence in one or the other regime and large amplitude sections describe transitions between regimes. Often, but not always, the residence in these regimes occurs for multiple revolutions, suggesting that there are shorter timescale evolutions of the system that take place within these regimes, while infrequent, random shifts to the other regimes occur at longer timescales.
Sensitive-dependence and state-dependent variation in predictability
Figure 3. a-c) Different trajectories through the XY phase space dependent on the initial condition state-space (black circle). (Palmer, 1993)
Returning now to the ‘butterfly effect’; what, then, is this whole sensitive-dependence thing mentioned earlier? Figure 3a-c provide a nice graphical representation of this phenomenon. In each panel, a set of nearby initial states are chosen at different location in the phase space diagram and then are followed through their set of trajectories. In 3a, these trajectories neatly transition from one regime to the other, remaining relatively close to each other at the trajectories’ end. In 3b, a set of initial states not so far from 3a are chosen and instead of neatly transitioning to the other regime, they diverge strongly near the center of the diagram, with some trajectories remaining in the original regime, and some transitioning. However, for about half of the timespan, the trajectories remain very close to one another. Finally, an initial state chosen near the center of the diagram (3c) diverges quickly into both regimes, with trajectories ending up at nearly opposite ends of the phase space (black tails at upper right and left of diagram). Figures b-c, in particular, showcase the sensitive-dependence on initial conditions attributes of the system. In other words, from a set of very close-by initial states, the final trajectories in phase space can yield strongly divergent results. Importantly, this divergence in trajectories over some period of time can occur right away (3c), at some intermediate time (3b), or not at all (3a).
This is the basic idea behind the last core property of these systems, state-dependent variation in predictability. Clearly, a forecast initialized from the starting point in 3a could be a little bit uncertain about the exact starting state and still end up in about the right spot for an accurate future prediction at the end of the forecast horizon. At medium ranges, this is also the case for 3b, but the longer range forecast is highly uncertain. For 3c, all forecast ranges are highly uncertain; in other words, the flap of a butterfly’s wing can mean the difference between one or the other trajectory! Importantly, one can imagine in this representation that an average value of 3c’s trajectories (i.e. the ensemble mean) would fall somewhere in the middle of the phase space and be representative of none of the two physically plausible trajectories into the right or left regimes. This is an important point that we’ll return to at the end of this post.
From the Lorenz model to ensemble forecasting
The previous sections have highlighted this idealized dynamical system (Lorenz model) that theoretically has properties of the actual atmosphere. Those 4 properties (the big 4!) were: 1) sensitive-dependence on initial conditions, 2) regime structure, 3) multiple distinct timescales, and 4) state-dependent variation in predictability. In this last section, I will tie these concepts into a theoretical discussion of ensemble forecasting. Notably, in the previous sections, I discussed ‘timescales’ without any reference to the real atmosphere. The dynamics of the Lorenz system prove to be quite illuminating about the actual atmosphere when timescales are attached to the system that are roughly on the order of the evolution of synoptic scale weather systems. If one assumes that a lap around the Lorenz attractor equates to a synoptic timescale of ~5 days, then the Lorenz model can be conceptualized in terms of relatively homogenous synoptic weather progressions occurring within two distinct weather regimes that typically persist on the order of weeks to months (see Figure 3b-c). This theoretical foundation jives nicely with the empirically derived weather-regime based approach (see Rohini’s post or Nasser’s post) where incidentally, 4 or more weather regimes are commonly found (The real atmosphere is quite a bit more complex than the Lorenz model after all). This discussion of the Lorenz model has hopefully given you some intuition that conceptualizing real world weather progressions outside the context of an appropriate regime structure could lead to some very unphysical representations of the climate system.
Ensemble forecasting as a discrete approximation of forecast uncertainty
In terms of weather prediction, though, the big 4 make things really tough. While there are certainly conditions in the atmosphere that can lead to excellent long range predictability (i.e. ‘forecasts of opportunity’, see Figure 3a), the ‘typical’ dynamics of the atmosphere yield the potential for multiple regimes and associated transitions within the short-to-medium range timeframe (1-15 days) where synoptic-scale hydro-meteorological forecasting is theoretically possible. (Note, by ‘synoptic-scale’ here, I am referring to the capability to predict actual weather events, not prevailing conditions. Current science puts the theoretical dynamical limit to predictability at ~2 weeks with current NWP technology achieving ‘usable’ skill out to ~7 days, give or take a few dependent on the region)
Early efforts to bring the Lorenz model into weather prediction sought to develop analytical methods to propagate initial condition uncertainty into a useful probabilistic approximation of the forecast uncertainty at various lead times. This can work for simplified representation of the atmosphere like the Lorenz model, but quickly becomes intractable as the complexity and scale of the governing equations and boundary conditions increases.
Figure 4. Conceptual depiction of ensemble forecasting including sampling of initial condition uncertainty, forecasts at different lead times, regime structure, and ensemble mean comparison to individual ensemble members. Solid arrow represents a ‘control’ member of the ensemble. Histograms represent an approximate empirical distribution of the ensemble forecast. (Wilks, 2019)
Thankfully, rapid advances in computing power led to an alternate approach, ensemble forecasting! Ensemble forecasting is a stochastic-dynamic approach that couples a discrete, probabilistic sampling of the initial condition uncertainty (Figure 4 ‘Initial time’) and propagates each of those initial condition state vectors through a dynamical NWP model. Each of these NWP integrations is a unique trajectory through state space of the dynamical model that yields a discrete approximation of the forecast uncertainty (Figure 4 ‘Intermediate/Final forecast lead time’). This discrete forecast uncertainty distribution theoretically encompasses the full space of potential hydro-meteorologic trajectories and allows a probabilistic representation of forecast outcomes through analysis of the empirical forecast ensemble distribution. These forecast trajectories highlight many of the big 4 properties discussed in previous sections, including regime structure and state-dependent predictability (Figure 4 trajectories are analogous to the Figure 3b trajectories for the Lorenz model). The ensemble mean prediction is an accurate and dynamically consistent prediction at the intermediate lead time, but at the final lead time where distinct regime partitioning has occurred, it is no longer dynamically consistent and delineates a region of low probability in the full ensemble distribution. I will explore properties of ensemble averaging, both good and bad, in a future post.
Lastly, I will note that the ensemble forecasting approach is a type of Monte Carlo procedure. Like other Monte Carlo approaches with complex systems, the methodology for sampling the initial condition uncertainty has a profound effect on the uncertainty quantification contained within the final NWP ensemble output, especially when considering the high degree of spatiotemporal relationships within the observed climatic variables that form the initial state vector. This is a key area of continued research and improvement in ensemble forecasting models.
Final thoughts
I hope that you find this discussion of the theoretical underpinnings of chaotic dynamics and ensemble forecasting to be useful. I have always found these foundational concepts to be particularly fascinating. Moreover, the basic theory has strong connections outside of ensemble forecasting, including the ties to weather regime theory mentioned in this post. I also think these foundational concepts are necessary to understand how much actual ensemble forecasting techniques can diverge from the theoretical stochastic-dynamic framework. This will be the subject of some future posts that will delve into more practical and applied aspects of ensemble forecasting with a focus on water resources management.
Reference
Lorenz, E. N. (1963). Deterministic nonperiodic flow. Journal of the Atmospheric Sciences, 20, 130–141.
Lorenz, E. N. (2006). Reflections on the Conception , Birth , and Childhood of Numerical Weather Prediction. Annual Review of Earth and Planetary Science, 34, 37–45. https://doi.org/10.1146/annurev.earth.34.083105.102317
Palmer, T. N. (1993). Extended-range atmospheric prediction and the Lorenz model. Bulletin – American Meteorological Society, 74(1), 49–65. https://doi.org/10.1175/1520-0477(1993)074<0049:ERAPAT>2.0.CO;2
Rayleigh, L. (1916). On convective currents in a horizontal layer when the higher temperature is on the underside. Phil. Mag., 32, 529-546.
Wilks, D. S., (2019). Statistical Methods in the Atmospheric Sciences, 4th ed. Cambridge, MA: Elsevier.
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