How Meteorologists Get Rid of Lorenz Chaos

In this presentation we summarize what meteorologists have always done to get rid of the deleterious effects of Lorenz's deterministic chaos. First of all, it is necessary to understand the difference between the “deterministic chaos” discovered by Lorenz and an alleged “atmospheric chaos” commonly cited by the literature but which probably does not even exist. Lorenz's discovery of chaos occurred as he was simulating the atmospheric behavior using a non-linear numerical model. He discovered that the predictions changed a lot after some time-steps, only due a very small and inadvertent change in the accuracy of the input data when there was a change of machines. Lorenz did not discover a physical fact that could be tested. It’s not possible to keep the atmosphere totally unchanged twice, and make a slight change in the initial condition. It will never be possible to guarantee the repetition of a physical experiment in the atmosphere by modifying only a detail similar to the fluttering of a butterfly's wings.

Certainly, there is a widespread misconception of what Lorenz really discovered. The butterfly effect refers to the exponential growth of any small perturbation that is inserted into nonlinear weather models, either by erroneous data or by incompatible scales involved in integration. However, this exponential growth only continues as long as the perturbations remain confined to the phase space defined by the volume occupied by the attractor, because the solution evolving in time always returns towards the attractor. Unfortunately, most researchers do not take this latter part into account and find that the disturbance continues to grow until it is so large that it makes the prediction useless. Due to Lorenz's deterministic chaos, one should not attempt to obtain detailed predictions on scales smaller than appropriated one, although this chaos has no significant effect on larger scales. In short, one should put aside the misconception about the “butterfly effect”. To the question “could the fluttering of a butterfly's wings in Brazil set off a tornado in Texas", the answer is absolutely no.

While worrying about atmospheric chaos is unnecessary, Lorenz's deterministic chaos exists and should be cause for concern, because the models used in numerical weather forecasting show uncertainties that grow during integration time, every time small changes are introduced into the input data. These small changes in the input data may be associated with inaccuracies in those data, or may result from datasets with different scales being concomitantly assimilated by the model, and so, the effects of deterministic chaos appear and should be minimized.

Early weather forecasts using numerical models came shortly after World War II due to strong demand for good wind forecasts at altitude and tropical latitudes, primarily for aviation applications. At the same time radiosonde data has been incorporated into the input datasets of the prediction models. Initially numerical predictions were made with linear models, the so-called filtered models, because they did not include gravity waves as part of their solution. However, gravity waves are one of the most important aspects of tropical convection prediction, where geostrophism does not work. This forced meteorologists to use nonlinear models, but with this came some totally artificial gravity waves, mainly due to the inclusion of advective terms in the prediction model equations. Spurious shortwaves not only appear but also amplify, making it necessary to apply procedures to keep these unwanted gravity waves under control, preventing them from contaminating predictions, which in some cases could become entirely useless.

These unwanted spurious waves arising due to the nonlinearity of the models, we are calling butterflies to make a joke with the Lorenz butterfly effect. But seriously, by replacing the first filtered models with nonlinear ones, and by introducing data from the most diverse sources and characteristics into the models, not only spurious shortwaves appear but they grow as the prediction horizons widen.

It is estimated that the number of degrees of freedom in a modern numerical weather prediction model is of the order of 10⁷, while the total number of conventional observations is of the order of 10⁴ to 10⁵. To supply this gap, the modern meteorology uses unconventional data such as satellite and radar data.

In order to avoid deterioration of the weather forecasting, taking advantage of both the prediction ability of the most complete models and the richness of information derived from the expansion of the data sources, several initialization techniques have been proposed to assimilate the various types of input data.

The global operational centers for numerical weather prediction produce their initial conditions through a statistical combination of observations and short-term prediction, usually 6-hour forecasts. In the Variational Assimilation approach (3D-Var), one defines a cost function proportional to the square of the distance between the analysis with both the background (a prediction of 6 hours) and the observations. This cost function is minimized in order to obtain the analysis. Recently, the variational approach has been extended to four dimensions (4D-Var), including in the cost function the time interval between observations and the moment of analysis, or assimilation window. More recently, but for the same purpose, elegant and efficient digital filtering methods have been introduced into models to dampen unwanted gravitational waves.

References:

Kalnay, E., 2003: Atmospheric Modeling – Data, Assimilation and Predictability. Cambridge, Cambridge University Press, U.K. 276 p.

Lorenz, E. N., 1963: Deterministic nonperiodic flow. J. Atmos. Sci., 20, 130-141.

Lorenz, E. N., 1969: Predictability of a flow which possesses many scales of motion. Tellus, 21, 289-307.

Santos, I. A., & J. Buchmann, 2013: Would be the atmosphere chaotic? Anuário do Instituto de Geociências, 36, 40-44.

Santos, I. A., & J. Buchmann, 2015: Revisiting the Question of Atmospheric Predictability. Journal of Geography and Geology, 7, 18-31.