McGill.CA / Science / Department of Physics

Nonlinear Physics
(S. Lovejoy)

The weather and climate as problems in physics

Since the 1980's, the nonlinear physics and atmospheric physics group has worked on a series of new geophysical paradigms. A particularly exciting one is the idea that atmospheric dynamics repeat scale after scale from large to small scales in a cascade-like way. The key is recognizing that as the scales get smaller, the horizontal gets “squashed” much more than the vertical so that the stratification which starts out being extreme (structures very flat at planetary scales) become rounder and rounder at small scales. This allows the scaling (and the stratified cascades) to occur over huge ranges of scale. The cascade mechanism implies that the variability builds up scale by scale; the resulting “intermittency” is huge and is a consequence of the large range of scales. It has nonclassical statistical features, the result is a multifractal process. The physical implications are that atmospheric fluxes of energy, moisture etc. are far from uniform, they are concentrated in storms, and even in the centre of storms.

[Dynamics and types of scaling variability]
Dynamics and types of scaling variability»
Representative temperature series from weather, macroweather and climate (bottom to top respectively). Each sample is 720 points long and was normalized by its standard deviation (bottom to top: 0.35o C, 4.48o C, 2.59 o C, 1.39 o C, dashed lines indicate means). The resolutions are 0.066 s, 1 hour, 20 days and 1 century, the data are from, Montreal Canada (the roof of the physics building at McGill), Lander Wyoming, the 20th Century reanalysis and Vostok (Deuterium paleotemperatures, Antarctic) adapted from [Lovejoy, 2013].
The classical laws of turbulence (Kolmogorov, Bolgiano, Obukhov, Corrsin...) are based on strong assumptions of isotropy and homogeneity of the fluxes and fields, they are expected to be high level laws “emergent” from the lower level laws of continuum mechanics and thermodynamics. When the nonlinearity (Reynolds number) is strong enough one obtains “fully developed turbulence”. However, due to the atmosphere's strong stratification and heterogeneity, it was believed that they would only apply over very narrow ranges of scale (several hundred meters at most). The developments of anisotropic scale invariance and multifractals effectively generalize them allowing them to hold up to planetary scales.

In the last five years, with the help of massive amounts of in situ, aircraft, satellite data, and reanalysis data, these emergent laws have been extensively verified up to planetary scales. An overall in depth review (for atmospheric scientists) has recently been published: “The weather and climate: emergent laws and multifractal cascades” Lovejoy and Schertzer 2013; a powerpoint presentation is available here. It was also found that over almost all of their ranges, numerical models of the atmosphere (and reanalyses) also have cascade structures, so that cascades do indeed provide stochastic models of deterministic Global Circulation Models (GCM's) and can be used to understand and improve the latter (for example by “stochastic” subgrid paramertrisations).

[Typical (Haar) fluctuations]
Typical (Haar) fluctuations»
This shows the typical (Haar) fluctuations for the Lander data (upper left), and paleo Antarctic data (from the EPICA core, near and similar to the Vostok core). Also shown is the globally averaged (at 2o resolution) fluctuations from the Twentieth century reanalysis. The various regimes are shown, along with power law (straight) reference lines with the slopes as indicated.
When these scaling ideas are applied to the temporal structure of the atmosphere, they predict that there is a fundamental change in behavior at about ten days; this is the lifetime of planetary sized structures; it is determined by the solar (turbulent) forcing of about one milliwatt per kilogram. Indeed, all the atmospheric fields show qualitative changes in their statistics at about 10 days. It turns out that whereas fluctuations grow with scale in the weather regime, over scales longer than this, the fluctuations tend to cancel out - the signs of the fluctuation exponents change from positive to negative. Averaging over longer and longer periods thus gives smaller and smaller fluctuations, apparently converging to a well-defined “climate”. However, this turns out to be an illusion: at scales of 10- 30 years (industrial period, 50 -100 years, preindustrial), the exponents again change sign, with fluctuations again increasing with scale. The intermediate “macroweather” regime is dominated by weather dynamics, the longer regime is the true climate; it is the focus of much of our research in the last few years (see e.g. chapters 10, 11 of Lovejoy and Schertzer 2013, but also Lovejoy and Schertzer 2012. Scaling techniques (including the much neglected Haar fluctuations) are transforming our view of the climate by allowing us to compare scale by scale instrumental, paleo (proxy) data and outputs of numerical models.

Solid earth Geophysics

In the area of solid earth studies, we have shown that the surface topography is a universal multifractal, showing that - contrary to prevailing wisdom - scaling surfaces cannot generally be regarded as self-affine fractals (Gagnon et al 2006). Similarly, the variability of the earth's surface magnetic field can be explained by a similar scaling stratification of the rock susceptibility (Lovejoy et al 2001). Interestingly, the lithospheric stratification is opposite to that of the atmosphere becoming stronger at smaller rather than larger scales. Analogous results apply to the rock density and geogravity fields (Lovejoy et al 2008), see Lovejoy and Schertzer 2007 for a review.

Other applications

Other applications include the analysis and simulation of scaling properties of ocean and ice surfaces, chemical pollution, low frequency human speech, hadron jets and the large scale structure of the universe. As disparate as some of these applications may seem, they are linked by the common theme of (nonlinear, dynamical) scale invariance, a symmetry principle whose generality and significance is great.

Nonlinear Geophysics

This work is part of a family of approaches to geophysics collectively termed “nonlinear geophysics”. For more information on this and on its place in the American Geophysical Union and the European Geosciences Union, see “Nonlinear geophysics: why we need it”.
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