Zero-Prandtl-number convection with slow rotation

Abstract

We present the results of our investigations of the primary instability and the flow patterns near onset in zero-Prandtl-number Rayleigh-B\'enard convection with uniform rotation about a vertical axis. The investigations are carried out using direct numerical simulations of the hydrodynamic equations with stress-free horizontal boundaries in rectangular boxes of size $(2\pi/k_x) \times (2\pi/k_y) \times 1$ for different values of the ratio $\eta = k_x/k_y$. The primary instability is found to depend on $\eta$ and $Ta$. Wavy rolls are observed at the primary instability for smaller values of $\eta$ ($1/\sqrt{3} \le \eta \le 2$ except at $\eta = 1$) and for smaller values of $Ta$. We observed K\"{u}ppers-Lortz (KL) type patterns at the primary instability for $\eta = 1/\sqrt{3}$ and $ Ta \ge 40$. The fluid patterns are found to exhibit the phenomenon of bursting, as observed in experiments [Bajaj et al. Phys. Rev. E {\bf 65}, 056309 (2002)]. Periodic wavy rolls are observed at onset for smaller values of $Ta$, while KL-type patterns are observed for $ Ta \ge 100$ for $\eta =\sqrt{3}$. In case of $\eta = 2$, wavy rolls are observed for smaller values of $Ta$ and KL-type patterns are observed for $25 \le Ta \le 575$. Quasi-periodically varying patterns are observed in the oscillatory regime ($Ta > 575$). The behavior is quite different at $\eta = 1$. A time dependent competition between two sets of mutually perpendicular rolls is observed at onset for all values of $Ta$ in this case. Fluid patterns are found to burst periodically as well as chaotically in time. It involved a homoclinic bifurcation. We have also made a couple of low-dimensional models to investigate bifurcations for $\eta = 1$, which is used to investigate the sequence of bifurcations.Comment: 50 pages, 22 figures and 3 table

Zero Prandtl-number rotating magnetoconvection

“…symmetry of the problem. In absence of magnetic field (Q = 0), stable CR and CR ′ branches would meet at a supercritical pitchfork bifurcation point 28,30,58 . Comparing the magnetic and non-magnetic cases we understand that the dashed cyan curves are the reminiscence of the square saddle which would exist when Q = 0 and we denote it by SQR.…”

“…Therefore, for simplicity here we simultaneously consider Pr → 0 and Pm → 0. In absence of the magnetic field, Pr → 0 limit has already been considered 24,25,28,58 and the reduced equations revealed significant properties of low Prandtl number convection 24,26 . In this limit, the equations (1), (2) and (3) reduces to…”

“…The effects of rotation and external magnetic field on the instabilities and bifurcation structures near the onset of convection when acted separately have recently been investigated in [55][56][57][58][59][60][61][62] and reported very rich dynamics including chaos at the onset of convection. However, instabilities and the associated bifurcation structures near the onset of RBC of electrically conducting fluids in presence of both rotation and uniform external magnetic field have not been studied so far.…”

Effects of a small magnetic field on homoclinic bifurcations in a low-Prandtl-number fluid

“…The flow direction was not found to flip in this experiment. The direct numerical simulations (DNS) of zero-Prandtl-number thermal convection with slow rotation (Taylor number T a < 100) showed the possibility of periodic as well as random bursting 17 of flow patterns without change in flow directions. However, the bursts led to reversal of the flow directions at higher rotation rates (T a ≥ 100).…”

“…A homoclinic bifurcation may lead to the possibility of Shilnikov wiggle 11 , three-frequency quasi-periodic orbits 12,13 , spontaneous gluing of two limit cycles into a large one or spontaneous breaking of a limit cycle into two smaller ones 2,4-6 or homoclinic chaos. This may also lead to the phenomenon of pattern bursting [14][15][16][17] , when a fluid pattern disappears for a finite time and reappears again for a fixed value of the bifurcation parameter.…”