Predictability of Turbulent Flows

Abstract

The test-field model of turbulence is used to compute the growth of prediction error for inertial-range turbulence in both three and two dimensions. It is found that initial uncertainty in high wavenumbers spreads through the entire inertial range according to a similarity behavior. For the energy inertial range, the time required for error to reach wavenumber k from very high wavenumbers is t=Aε−1/3k−2/3, where ε is the rate of energy transfer per unit mass and A≈10 in three dimensions or A≈2.5 in two dimensions. For the enstrophy inertial range in two dimensions the time for error to propagate from k′ down to k≪k′ (k and k′ both in the inertial range) is t≈4η−1/2{[ln(k′/ k1)]2/3−[ln(k/ k1)]2/3}, where η is the rate of enstrophy transfer and k1 marks the bottom of the enstrophy inertial range. Error growth is also computed for a two-dimensional spectrum that fits the energy spectrum of planetary waves in the atmosphere. An initial state determined with a horizontal resolution feasible with a satellite-based observing system results in significant predictability of large-scale motions for more than a week. It is argued that the test-field model probably underestimates rather than overestimates predictability times.