Oscillatory instability and fluid patterns in low-Prandtl-number Rayleigh-B\'{e}nard convection with uniform rotation

We present the results of direct numerical simulations of flow patterns in a low-Prandtl-number ($Pr = 0.1$) fluid above the onset of oscillatory convection in a Rayleigh-B\'{e}nard system rotating uniformly about a vertical axis. Simulations were carried out in a periodic box with thermally conducting and stress-free top and bottom surfaces. We considered a rectangular box ($L_x \times L_y \times 1$) and a wide range of Taylor numbers ($750 \le Ta \le 5000$) for the purpose. The horizontal aspect ratio $\eta = L_y/L_x$ of the box was varied from $0.5$ to $10$. The primary instability appeared in the form of two-dimensional standing waves for shorter boxes ($0.5 \le \eta < 1$ and $1 < \eta < 2$). The flow patterns observed in boxes with $\eta = 1$ and $\eta = 2$ were different from those with $\eta < 1$ and $1 < \eta < 2$. We observed a competition between two sets of mutually perpendicular rolls at the primary instability in a square cell ($\eta = 1$) for $Ta < 2700$, but observed a set of parallel rolls in the form of standing waves for $Ta \geq 2700$. The three-dimensional convection was quasiperiodic or chaotic for $750 \le Ta < 2700$, and then bifurcated into a two-dimensional periodic flow for $Ta \ge 2700$. The convective structures consisted of the appearance and disappearance of straight rolls, rhombic patterns, and wavy rolls inclined at an angle $\phi = \frac{\pi}{2} - \arctan{(\eta^{-1})}$ with the straight rolls.Comment: 32 pages, 14 figures, 1 tabl

Dynamics of market states and risk assessment

Based on previous developments of the concept of market states using correlation matrices, in the present paper we address the dynamical evolution of correlation matrices in time. This will imply minor modifications to the market states themselves, due to increased attention to the transition matrix between the states. We will introduce trajectories of the correlation matrices by considering one day shifts for the epoch used to calculate the correlation matrices and will visualize both the states and the trajectories after dimensional scaling. This approach using dynamics improves the options of risk assessment and opens the door to dynamical treatments of markets and shows noise suppression in a new light.Comment: 22 pages and 27 figures. arXiv admin note: text overlap with arXiv:2003.0705