Stability of convective rolls in a horizontal layer rotating about an inclined axis

Abstract

We present three results on stability of rolls in Boussinesq convection in a plane horizontal layer with rigid boundaries that is rotating about an inclined axis with the angular velocity $Ω=(Ω_1,Ω_2,Ω_3)$. i. We call the full problem the set of equations governing the temporal behaviour of the flow and temperature for an arbitrary $Ω$, and by the reduced problem the set of equations for the angular velocity $(Ω_1,0,Ω_3)$. Here $x,y$ are horizontal Cartesian coordinates in the layer and $z$ is the vertical one. We prove that a $y$-independent solution to one of the two problems is also a solution to the second one. ii. We calculate the critical Rayleigh number for the monotonic onset of convection. The instability mode in the form of rolls (a flow independent of a horizontal direction) is assumed. Let $β$ be the angle between the horizontal projection of $Ω$ and the rolls axes. We show that $β=0$ for the least stable mode. Taking i into account, we conclude that the critical Rayleigh number for the onset of convection is independent of $Ω_1$ and $Ω_2$ and the emerging flow are rolls with axis aligned with the horizontal component of the rotation vector. iii. We study the behaviour of convective flows by integrating numerically the three-dimensional equations of convection for $Ω=(0,Ω_2,Ω_3)$ and a range of the Rayleigh numbers, other parameters of the problem being fixed. We assume square horizontal periodicity cells, whose sides are equal to the period of the most unstable mode. The computations indicate that, in general, in the nonlinear regime convective rolls become more stable as $Ω_2$ increases. Namely, on increasing $Ω_2$, the interval of the Rayleigh numbers for which convective rolls are stable increases.