Assessing Turbulence Strength via Lyaponuv Exponents

In this paper we study the link between ‘turbulence strength’ in a flow and the leading Lyaponuv exponent that characterize it. To this end we use two approaches. The first, analytical, considers the truncated convection equations in 2-dimensions with three (Lorenz model) and six components and study their leading Lyaponuv exponent as a function of the Rayleigh number. For the second approach we analyze fifteen time series of measurements taken by a plane flying at constant height in the upper troposphere. For each of these time series we estimate the leading Lyaponuv exponent which we then correlate with the structure constant for the temperature. Turbulence is still considered as one of the major unsolved problems of modern science. Usually one attempts to characterize the turbulent state of the system in terms of various numbers such as the Reynolds number, Grashof number, Rayleigh number, etc. How a combination of these numbers actually characterize the strength of turbulence in the system is not answered easily. The situation is even more complex when the flow is represented only by a time series. Furthermore is there any relation between this ‘strength’ and chaos theory? Although chaos owes it modern origins to the work of Lorenz who investigated the onset of convection, a characterization of turbulence in terms of dynamical invariants such as Lyapunov exponents, fractal dimension and so on is still an open question which needs further investigation. This paper attempts to present a modest contribution in this direction.