Fluctuations of finite-time Lyapunov exponents in an intermediate-complexity atmospheric model: a multivariate and large-deviation perspective

The stability properties as characterized by the fluctuations of finite-time Lyapunov exponents around their mean values are investigated in a three-level quasi-geostrophic atmospheric model with realistic mean state and variability. Firstly, the covariance structure of the fluctuation field is examined. In order to identify dominant patterns of collective excitation, an empirical orthogonal function (EOF) analysis of the fluctuation field of all of the finite-time Lyapunov exponents is performed. The three leading modes are patterns where the most unstable Lyapunov exponents fluctuate in phase. These modes are virtually independent of the integration time of the finite-time Lyapunov exponents. Secondly, large-deviation rate functions are estimated from time series of finite-time Lyapunov exponents based on the probability density functions and using the Legendre transform method. Serial correlation in the time series is properly accounted for. A large-deviation principle can be established for all of the Lyapunov exponents. Convergence is rather slow for the most unstable exponent, becomes faster when going further down in the Lyapunov spectrum, is very fast for the near-neutral and weakly dissipative modes, and becomes slow again for the strongly dissipative modes at the end of the Lyapunov spectrum. The curvature of the rate functions at the minimum is linked to the corresponding elements of the diffusion matrix. Also, the joint large-deviation rate function for the first and the second Lyapunov exponent is estimated.

On the origin of Sérsic profiles of galaxies and Einasto profiles of dark-matter halos

The surface-brightness profiles of galaxies I(R) and the density profiles of dark-matter halos ρ(r) are well represented by the same analytic function, named after either Sérsic, I∝e−(R/R*)1/m, or Einasto, ρ∝e−(r/r*)α, where R* and r* are characteristic radii. Systems with high Sérsic index m (or low Einasto index α) have steep central profiles and shallow outer profiles, while systems with low m (or high α) have shallow central profiles and steep profiles in the outskirts. We present the results of idealized numerical experiments which suggest that the origin of these profiles can be traced back to the initial density fluctuation field: high-α (low-m) systems form in smooth regions via few mergers, while low-α (high-m) systems form in clumpy regions via several mergers.

Pressure fluctuations beneath instability wavepackets and turbulent spots in a hypersonic boundary layer

To investigate the pressure-fluctuation field beneath turbulent spots in a hypersonic boundary layer, a study was conducted on the nozzle wall of the Boeing/AFOSR Mach-6 Quiet Tunnel. Controlled disturbances were created by pulsed-glow perturbations based on the electrical breakdown of air. Under quiet-flow conditions, the nozzle-wall boundary layer remains laminar and grows very thick over the long nozzle length. This allows the development of large disturbances that can be well-resolved with high-frequency pressure transducers. A disturbance first grows into a second-mode instability wavepacket that is concentrated near its own centreline. Weaker disturbances are seen spreading from the centre. The waves grow and become nonlinear before breaking down to turbulence. The breakdown begins in the core of the packets where the wave amplitudes are largest. Second-mode waves are still evident in front of and behind the breakdown point and can be seen propagating in the spanwise direction. The turbulent core grows downstream, resulting in a spot with a classical arrowhead shape. Behind the spot, a low-pressure calmed region develops. However, the spot is not merely a localized patch of turbulence; instability waves remain an integral part. Limited measurements of naturally occurring disturbances show many similar characteristics. From the controlled disturbance measurements, the convection velocity, spanwise spreading angle, and typical pressure-fluctuation field were obtained.