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 $\Omega =({\Omega }_1,{\Omega }_2,{\Omega }_3)$. (i) We call the full problem the set of equations governing the temporal behaviour of the flow and temperature for an arbitrary $\Omega$, and by the reduced problem the set of equations for the angular velocity $({\Omega }_1,0,{\Omega }_3)$. 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 $\Omega$ and the rolls axes. We show that $\text{β}=0$ for the least stable mode. When the axis of rotation is horizontal, this is proven analytically, and for ${\Omega }_3\ne 0$, the result is obtained numerically. Taking i into account, we conclude that the critical Rayleigh number for the onset of convection is independent of ${\Omega }_1$ and ${\Omega }_2$ 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 $\Omega =(0,{\Omega }_2,{\Omega }_3)$ 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 ${\Omega }_2$ increases. Namely, on increasing ${\Omega }_2$, the interval of the Rayleigh numbers for which convective rolls are stable increases.

Keywords:

• Boussinesq convection
• critical Rayleigh number
• rotation
• nonlinear convection

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Symmetries of a rotating layer depend on the direction of its rotation axis. In the case of vertical axis of rotation studied in Chandrasekhar (1961) the operator L (6(6) $\begin{array}{r}L\mathbf{w}=\left(\begin{array}{l}P\mathrm{\Delta }\mathbf{v}+\mathrm{PR\theta }{\mathbf{e}}_z-\mathrm{\nabla }{p}^{\prime }+\mathrm{P\tau }\mathbf{v}\times {\mathbf{e}}_{\mathrm{r}}\\ v_z+\mathrm{\Delta \theta }\end{array}\right)\end{array}$(6) ) is invariant under the reflection in the vertical direction, $(x,y,z)\to (x,y,-z)$. Therefore, modes even or odd with respect to this transformation can be assumed. For a layer rotating about a horizontal or inclined axis the operator ceases to be invariant with respect to this reflection, therefore we use the inversion $(x,y,z)\to (-x,-y,-z)$ to distinguish between the even and odd modes. The inversion is a symmetry of L for any inclination of the rotation axis. In the case of vertical rotation axis it is irrelevant, whether we use the transformation $(x,y,z)\to (x,y,-z)$ or $(x,y,z)\to (-x,-y,-z)$ for the splitting into even/odd modes.

Additional information

Funding

The project was financed by the grant No. 22-17-00114 of the Russian Science Foundation, https://rscf.ru/project/22-17-00114/.