Rapidly rotating Rayleigh-Bénard convection with a tilted axis

We numerically explore the dynamics of an incompressible fluid heated from below, bounded by free-slip horizontal plates and periodic lateral boundary conditions, subject to rapid rotation about a distant axis that is tilted with respect to the gravity vector. The angle $\varphi$ between the rotation axis and the horizontal plane measures the tilting of the rotation axis; it can be taken as a proxy for latitude if we think of a local Cartesian representation of the convective dynamics in a rotating fluid shell. The results of the simulations indicate the existence of three different convective regimes, depending on the value of $\varphi$: (1) sheared, intermittent large-scale winds in the direction perpendicular to the plane defined by the gravity and rotation vectors, when rotation is “horizontal” ($\varphi =0^{\circ }$); (2) a large-scale cyclonic vortex tilted along the rotation axis, when the angle between the rotation axis and the gravity vector is relatively small ($\varphi$ between about ${45}^{\circ }$ and ${90}^{\circ }$); and (3) a new intermediate regime characterized by vertically sheared large-scale winds perpendicular to both gravity and rotation. In this regime, the winds are organized in bands that are tilted along the rotation axis, with unit horizontal wave number in the plane defined by gravity and rotation at values of $\varphi$ less than about ${60}^{\circ }$. This intermediate solution, studied for the first time in this work, is characterized by weaker vertical heat transport than the cases with large-scale vortices. For intermediate values of $\varphi$ (between about ${45}^{\circ }$ and ${60}^{\circ }$), the banded, sheared solution coexists with the large-scale vortex solution, with different initial conditions leading to one or the other dynamical behavior. A discussion of the possible implications of these results for the dynamics of rapidly rotating planetary atmospheres is provided.

Figure 1

Time series of the Nusselt number for all simulations. Each line identifies a value of $\varphi$, indicated in the legend (in degrees).

Figure 2

Horizontal and vertical sections of the temperature field computed at $t=0.75$ for different values of $\varphi$, increasing upward. Horizontal sections (leftmost column) are taken at mid-depth, vertical sections in the $x\text{−}z$ and $y\text{−}z$ planes are computed at $y=\pi$ and at $x=\pi$, respectively. The rotation axis lies in the $y\text{−}z$ plane (rightmost column) and its orientation is shown by the solid line. Rising hot plumes are shown in yellow (light gray), and falling cold plumes are shown in dark purple (dark gray).

Figure 3

Horizontal and vertical sections of the $u$ velocity component, averaged over 0.025 thermal times centered on $t=0.75$. Horizontal sections (leftmost column) are taken at mid-depth, and vertical sections in the $x\text{−}z$ and $y\text{−}z$ planes are computed at $y=\pi$ and at $x=\pi$, respectively. The rotation axis lies in the $y\text{−}z$ plane (rightmost column) and is shown as a solid line.

Figure 4

Horizontal and vertical sections of the $v$ velocity component, averaged over 0.025 thermal times centered on $t=0.75$. Horizontal sections (leftmost column) are taken at mid-depth, and vertical sections in the $x\text{−}z$ and $y\text{−}z$ planes are computed at $y=\pi$ and at $x=\pi$, respectively. The rotation axis lies in the $y\text{−}z$ plane (rightmost column) and is shown as a solid line.

Figure 5

Horizontal and vertical sections of the vorticity component along the rotation axis, averaged over 0.025 thermal times centered on $t=0.75$. Horizontal sections (leftmost column) are taken at mid-depth, vertical sections in the $x\text{−}z$ and $y\text{−}z$ planes are computed at $y=\pi$ and at $x=\pi$, respectively, for $\varphi \le {30}^{\circ }$. For $\varphi \ge {45}^{\circ }$ the sections are taken through the large-scale cyclonic vortex that has formed. The solid lines in the horizontal sections show the positions of the two vertical sections. The rotation axis lies in the $y\text{−}z$ plane (rightmost column) and is shown as a solid line.

Figure 6

Horizontal and vertical sections of the temperature and velocity components along the $x$ direction at $\varphi ={15}^{\circ }$ and $\varphi ={30}^{\circ }$ at $t=0.3$. From top to bottom: temperature sections for $\varphi ={30}^{\circ }$ and $\varphi ={15}^{\circ }$; $u$ velocity sections at $\varphi ={30}^{\circ }$ and $\varphi ={15}^{\circ }$; $v$ velocity sections at $\varphi ={30}^{\circ }$ and $\varphi ={15}^{\circ }$. Velocity sections are time averaged over 0.025 thermal times and centered on $t=0.3$. Horizontal sections are taken at mid-depth, and vertical sections in the $x\text{−}z$ and $y\text{−}z$ planes are computed at $y=\pi$ and at $x=\pi$, respectively. The rotation axis lies in the $y\text{−}z$ plane (rightmost column) and is shown as a solid line.

Figure 7

Vertical profiles of the $u$ velocity component averaged along the $x$ direction and time averaged over 0.025 thermal times centered on $t=0.3$ [(a) and (b)] and $t=0.75$ [(c) and (d)] for $\varphi ={15}^{\circ }$ [left column, panels (a) and (c)] and $\varphi ={30}^{\circ }$ [right column, panels (b) and (d)]. Profiles are computed at four $y$ locations: $y=\pi /2$, $y=\pi$, $y=3\pi /2$, $y=2\pi$. Profiles at $\varphi ={15}^{\circ }$ exhibit the same shape at $t=0.3$ (a) and $t=0.75$ (c) since no intermediate transition occurs at this angle, unlike for the $\varphi ={30}^{\circ }$ case, in which the lower value of the Nusselt number at $t=0.75$ is associated with the change in the profiles (d).

Figure 8

Horizontal and vertical sections for the simulations at $\varphi ={45}^{\circ }$ and $\varphi ={60}^{\circ }$ restarted from the solution at $\varphi ={30}^{\circ }$ at $t=0.9$, taken after $\mathrm{\Delta }t=0.25$. The panels show temperature (upper two rows, denoted as $T$), velocity component along the $x$ direction (middle two rows, denoted as $u$) and vorticity component along the rotation axis (bottom two rows, denoted as ${\xi }_r$). The velocity component along the $y$ direction is very weak and their sections are thus omitted. The velocity and vorticity fields are time averaged over 0.025 thermal times. The solid line in the rightmost column indicates the direction of the rotation axis. Horizontal sections (leftmost column) are taken at mid-depth, and vertical sections in the $x\text{−}z$ and $y\text{−}z$ planes are computed at $y=\pi$ and at $x=\pi$, respectively