The (real) butterfly effect: the impact of resolving the mesoscale range


What does the ‘butterfly effect’ exactly mean? Many people would attribute the butterfly effect to the famous 3-dimensional non-linear model of Lorenz (1963) whose attractor looks like a butterfly when viewed from a particular angle. While it serves as an important foundation to chaos theory (by establishing that 3 dimensions are not only necessary for chaos as mandated in the Poincaré-Bendixson Theorem, but are also sufficient), the term ‘butterfly effect’ was not coined until 1972 (Palmer et al. 2014) based on a scientific presentation that Lorenz gave on a more radical, more recent work (Lorenz 1969) on the predictability barrier in multi-scale fluid systems. In this work, Lorenz demonstrated that under certain conditions, small-scale errors grow faster than large-scale errors in such a way that the predictability horizon cannot be extended beyond an absolute limit by reducing the initial error (unless the initial error is perfectly zero). Such limited predictability, or the butterfly effect as understood in this context, has now become a ‘canon in dynamical meteorology’ (Rotunno and Snyder 2008). Recent studies with advanced numerical weather prediction (NWP) models estimate this predictability horizon to be on the order of 2 to 3 weeks (Buizza and Leutbecher 2015; Judt 2018), in agreement with Lorenz’s original result.

The predictability properties of a fluid system primarily depend on the energy spectrum, whereas the nature of the dynamics per se only plays a secondary role (Rotunno and Snyder 2008). It is well-known that a slope shallower than (equal to or steeper than) -3 in the energy spectrum is associated with limited (unlimited) predictability (Lorenz 1969; Rotunno and Snyder 2008), which could be understood through analysing the characteristics of the energy spectrum of the error field. As shown in Figure 1, the error appears to grow uniformly across scales when predictability is indefinite, and appears to ‘cascade’ upscale when predictability is limited. In the latter case, the error spectra peak at the small scale and the growth rate is faster there.

Figure 1: Growth of error energy spectra (red, bottom to top) in the Lorenz (1969) model under the influence of a control spectrum (blue) of slope (left) -3 and (right) -\frac{5}{3}.

The Earth’s atmospheric energy spectrum consists of a -3 range in the synoptic scale and a -\frac{5}{3} range in the mesoscale (Nastrom and Gage 1985). While the limited predictability of the atmosphere arises from mesoscale physical processes, it would be of interest to understand how errors grow under this hybrid spectrum, and to what extent do global numerical weather prediction (NWP) models, which are just beginning to resolve the mesoscale -\frac{5}{3} range, demonstrate the fast error growth proper to the limited predictability associated with this range.

We use the Lorenz (1969) model at two different resolutions: K_{max}=11, corresponding to a maximal wavenumber of 2^{11}=2048, and K_{max}=21. The former represents the approximate resolution of global NWP models (~ 20 km), and the latter represents a resolution about 1000 times finer so that the shallower mesoscale range is much better resolved. Figure 2 shows the growth of a small-scale, small-amplitude initial error under these model settings.

Figure 2: As in Figure 1, except that the control spectrum is a hybrid spectrum with a -3 range in the synoptic scale and a -\frac{5}{3} range in the mesoscale, truncating at (left) K_{max}=11 and (right) K_{max}=21. The colours red and blue are reversed compared to Figure 1.

In the K_{max}=11 case where the -\frac{5}{3} range is not so much resolved, the error growth remains more or less up-magnitude, and the upscale cascade is not visible. The error is still much influenced by the synoptic-scale -3 range. Such behaviour largely agrees with the results of a recent study using a full-physics global NWP model (Judt 2018). In contrast, with the higher resolution K_{max}=21, the upscale propagation of error in the mesoscale is clearly visible. As the error spreads to the synoptic scale, its growth becomes more up-magnitude.

To understand the dependence of the error growth rate on scales, we use the parametric model of Žagar et al. (2017) by fitting the error-versus-time curve for every wavenumber / scale to the equation E\left ( t \right )=A\tanh\left (  at+b\right )+B, so that the parameters A, B, a and b are functions of the wavenumber / scale. Among the parameters, a describes the rate of error growth, the larger the quicker. A dimensional argument suggests that a \sim (k^3 E(k))^{1/2}, so that a should be constant for a -3 range (E(k) \sim k^{-3}), and should grow 10^{2/3}>4.5-fold for every decade of wavenumbers in the case of a -\frac{5}{3} range. These scalings are indeed observed in the model simulations, except that the sharp increase pertaining to the -\frac{5}{3} range only kicks in at K \sim 15 (1 to 2 km), much smaller in scale than the transition between the -3 and -\frac{5}{3} ranges at K \sim 7 (300 to 600 km). See Figure 3 for details.

Figure 3: The parameter a as a function of the scale K, for truncations (left) K_{max}=8,9,10,11 and (right) K_{max}=11,13,15,17,19,21.

This explains the absence of the upscale cascade in the K_{max}=11 simulation. As models go into very high resolution in the future, the strong predictability constraints proper to the mesoscale -\frac{5}{3} range will emerge, but only when it is sufficiently resolved. Our idealised study with the Lorenz model shows that this will happen only if K_{max} >15. In other words, motions at 1 to 2 km have to be fully resolved in order for error growth in the small scales be correctly represented. This would mean a grid resolution of ~ 250 m after accounting for the need of a dissipation range in a numerical model (Skamarock 2004).

While this seems to be a pessimistic statement, we have observed that the sensitivity of the error growth behaviour to the model resolution is itself sensitive to the initial error profile. The results presented above are for an initial error confined to a single small scale. When the initial error distribution is changed, the qualitative picture of error growth may not present such a contrast between the two resolutions. Thus, we highlight the need of further research to assess the potential gains of resolving more scales in the mesoscale, especially for the case of a realistic distribution of error that initiates the integrations of operational NWP models.

A manuscript on this work has been submitted and is currently under review.

This work is supported by a PhD scholarship awarded by the EPSRC Centre for Doctoral Training in the Mathematics of Planet Earth, with additional funding support from the ERC Advanced Grant ‘Understanding the Atmospheric Circulation Response to Climate Change’ and the Deutsche Forschungsgemeinschaft (DFG) Grant ‘Scaling Cascades in Complex Systems’.


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Leung, T. Y., Leutbecher, M., Reich, S. and Shepherd, T. G. (2019). Impact of the mesoscale range on error growth and the limits to atmospheric predictability. Submitted.

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Palmer, T. N., Döring, A. and Seregin, G. (2014). The real butterfly effect. Nonlinearity 27, R123—R141.

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Žagar, N., Horvat, M., Zaplotnik, Ž. and Magnusson, L. (2017). Scale-dependent estimates of the growth of forecast uncertainties in a global prediction system. Tellus A 69:1, 1287492.

The 27th General Assembly of the International Union of Geodesy and Geophysics (IUGG) in Montréal, Canada

Earlier this month (9th – 17th July, 2019), Elena Saggioro and I from the Mathematics of Planet Earth Centre of Doctoral Training (MPE CDT) were in Montréal for the General Assembly of the IUGG, a quadrennial gathering of nearly 4000 geoscientists from all over the world sharing their latest scientific advances.

At the conference centre.

The IUGG, which celebrates its centenary this year, is an international organisation ‘dedicated to advancing, promoting, and communicating knowledge of the Earth system, its space environment, and the dynamical processes causing change’ (from the Mission Statement on its website).  The IUGG consists of eight constituent associations, among which the International Association of Meteorology and Atmospheric Sciences (IAMAS) and the International Association for the Physical Sciences of the Oceans (IAPSO) are of the most relevance to meteorology students here in Reading.  Other fields under the IUGG umbrella include hydrology, cryospheric sciences, seismology, volcanology, geodesy and geomagnetism.

In the General Assembly I presented a poster on my own PhD research, revisiting and proposing a new argument for the finite-time barrier of weather predictability. The poster turned out to be popular, with a good number of scientists visiting and discussing in depth. It is great to know these people, especially those who work in the relatively small field of predictability. Earlier that day, Elena gave an interesting talk on studying southern-hemisphere stratosphere-troposphere coupling using casual network. A member in the audience came to her after the talk for a follow-up chat which lasted for hours! In addition, our supervisor Ted Shepherd gave a solicited talk advocating his storylines approach to the construction of regional climate-change information.

Elena Saggioro’s oral presentation.
With my poster.

For the variety of subjects covered, the General Assembly was also an excellent opportunity for us to interact with geoscientists of other fields and to get an idea of their research. I did this primarily through the poster sessions, as there’s already so much going on in the oral-presentation sessions of the IAMAS symposia (just a matter of fact: the IAMAS, at 21%, was by far the association with the most attendees), and because it’s easier for a beginner to learn through interacting with a poster presenter than listening to short talks that usually presume some background knowledge in the field. The outcome of visiting posters in such an international conference could be somewhat unexpected. This time, I gave a little more focus on posters from remote parts of the world and learnt how research is being done in these places. To give an example, I saw how hydrologists in French Polynesia use analogue techniques to forecast rainfall and flood on the island of Tahiti which has a complex geography of drainage basins (poster by Lydie Sichoix, University of French Polynesia). This is a very challenging problem, and I think their commitment to protecting the public’s safety during floods is clear, yet there’s only so much they can do as they don’t have the money to buy even a single RADAR instrument for nowcasting. The situation in underprivileged places like this definitely deserves more attention.

Aside from the scientific programme, Elena and I spent some time as a tourist in Montréal. We are delighted to learn how committed Montréal is to sustainability and climate-change adaptation. The Biosphère Museum of the Environment nicely outlines the resilient city’s master plan 50 years ahead: new space reserved for nature in the city centre, green alleyways throughout the city, and harvesting storm and rain water are just a few examples in their long-term plan.

The Biosphère Museum.

Montréal is also rich in history, culture and diversity. Churches and museums are everywhere. There were also a multi-cultural festival and a series of fireworks depicting different national themes during our stay, and we went to some of them. Situated along St Lawrence’s River, the city is also home to a range of water sports, including white-water rafting which was a fun experience. Before coming home, Elena and I went up to Mount Royal for an exhilarating view of Montréal, a city that we much enjoyed!

A panoramic view from the Mount Royal Lookout.