# Toroidal and poloidal energy in rotating Rayleigh–Bénard convection

###### Abstract

We consider rotating Rayleigh–Bénard convection of a fluid with a Prandtl number of in a cylindrical cell with an aspect ratio . Direct numerical simulations were performed for the Rayleigh number range and the inverse Rossby number range . We propose a method to capture regime transitions based on the decomposition of the velocity field into toroidal and poloidal parts. We identify four different regimes. First, a buoyancy dominated regime occurring as long as the toroidal energy is not affected by rotation and remains equal to that in the non-rotating case, . Second, a rotation influenced regime, starting at rotation rates where and ending at a critical inverse Rossby number that is determined by the balance of the toroidal and poloidal energy, . Third, a rotation dominated regime, where the toroidal energy is larger than both, and . Fourth, a geostrophic turbulence regime for high rotation rates where the toroidal energy drops below the value of non-rotating convection.

S. Horn and O. Shishkina]Susanne Horn^{†}^{†}thanks: Email address for correspondence: susanne.horn@ds.mpg.deand Olga Shishkina

Key words:

## 1 Introduction

Turbulent flows, driven by thermal convection and affected by rotation, are ubiquitous phenomena in geo- and astrophysics. Examples are the convection in stars, in the interior of gaseous planets and in the Earth’s atmosphere and oceans, to mention only a few. And even though, these phenomena are also shaped by other processes, such as magnetic fields, stratification or liquid-gas phase transition, convection under the influence of the Coriolis force is crucial to their description. Thus to increase our fundamental understanding of the physics behind it, we study rotating Rayleigh–Bénard convection, i.e. a fluid rotated about its vertical axis which is heated from below and cooled from above.

The commonly used control parameters of rotating Rayleigh–Bénard convection are the Rayleigh number Ra, the Prandtl number Pr and the convective Rossby number Ro, defined by

(1.0) |

where is the isobaric expansion coefficient, the acceleration due to gravity, the vertical distance between the top and bottom plate, the imposed adverse temperature difference, the thermal diffusivity, the viscosity and the angular speed. Instead of Ro occasionally also the Taylor number Ta and the Ekman number Ek are used to characterise the importance of rotation, which are given by

Ta | (1.0) | ||||

Ek | (1.0) |

Apart from that, the geometry of the container, in particular its aspect ratio, also plays an important role. However, the preferred aspect ratio has changed over the years. Starting from investigating convection in cylindrical containers with large diameter-to-height aspect ratios, , recent developments in numerical and experimental studies, rather go to smaller and smaller of He2012; Ahlers2012 or even Stevens2011. Most of the earlier studies were about the onset of convection and pattern formation Chandrasekhar1961, thus, the aim was to mimic an infinite lateral extent, where analytical relations are available. On the contrary, most of the current investigations focus on turbulent thermal convection, including the transition to the so-called “ultimate state” (Grossmann2011), thus, the aim is to achieve high Ra and, hence, practical considerations demand a small . The development to smaller is not only true for “ordinary”, but also for rotating convection Oresta2007; Stevens2012; Ecke2013. Yet, the finite size has serious implications for rotating Rayleigh–Bénard convection. Not only, does the destabilising effect of the lateral wall yield a lower critical Ra for the onset of convection at fast rotation rates (Buell1983) because of drifting wall modes (Zhong1991; Ecke1992; Kuo1993; Herrmann1993; Goldstein1993; Goldstein1994), but also determines the bifurcation point , at which, for and higher Ra, heat transfer enhancement sets in Weiss2010; Weiss2011a.

The increased heat transport, expressed in terms of the Nusselt number Nu, is usually used as an indicator for the different turbulent states occurring in rotating turbulent thermal convection, suggesting a division into three regimes Kunnen2011. In the weak rotation regime, Nu remains nearly constant, but as soon as is increased to values above , after a sharp onset, a continuous increase of Nu is observed for moderate rotation rotates, which coincides with the generation of columnar vortex structures Weiss2010; Stevens2011. After it has reached a peak, which marks the transition to the regime of strong rotation, it drops rapidly with the rotation rate due to the suppression of vertical velocity fluctuations (cf. also the recent review by Stevens2013a). However, this classification of regimes is only valid for fluids with ; for no heat transfer enhancement is expected Stevens2010b.

As the change of Nu is closely connected to the columnar vortices, the number of vortices serves as another criterion to determine the point where rotation dominates over buoyancy. However, extracting these vortices is relatively cumbersome, and involves a certain arbitrariness in choosing what constitutes a vortex. Furthermore, for the larger diffusivity results in only short vortices that dissipate quickly when they reach the bulk, which complicates matters. Conversely, one can also look at the large-scale circulation (LSC) or more specifically at the rotation rate when it breaks-down (Kunnen2008a; Weiss2011b; Stevens2012). In experiments, this is frequently obtained by analysing the temperature signal at the sidewall. However, one has to be careful with two-vortex states or multiple-roll states occurring in Rayleigh–Bénard cells with small aspect ratios. Evidently, also the crossover of the boundary layer thicknesses (Rossby1969; King2009; King2012) cannot be applied to fluids with , where the thermal boundary layer is thicker than the viscous one even without rotation.

Here, we offer an alternative method for the characterisation of the different regimes in rotating Rayleigh–Bénard convection. Motivated by the work by Breuer2004, who have shown, that the toroidal and poloidal energy are characteristic for the distinctive types of dynamics in low and high Prandtl number flows in non-rotating convection, namely, that the toroidal energy is highest for fluids with Breuer2004, and vanishes for Busse1967b, we analyse the contribution of the toroidal and poloidal energy in rotating convection. This is a very natural approach. The poloidal energy is the energy contained in cellular or roll motion, such as the LSC or double-roll states, i.e. the dominant motion without rotation. The toroidal energy, on the other hand, is contained in swirling motion in the horizontal plane, i.e. with a vertical vorticity (Olson1991), which is the dominant motion in rotating convection. This means, we are able to distinguish different regimes of rotating convection based on global quantities, namely the time and volume averaged toroidal and poloidal energy without a restriction to certain Prandtl numbers or aspect ratios.

## 2 Numerical method

We study rotating Rayleigh–Bénard convection by means of direct numerical simulations (DNS) using a fourth order finite volume code for cylindrical domains. Details about the code can be found in Shishkina2005 and Horn2013a. Additionally, we implemented a term describing the Coriolis force, whereas the centrifugal potential can be incorporated in the reduced pressure, and, hence, does not need to be considered explicitly.

We neglect any effects due to centrifugal buoyancy whose importance can be estimated by calculating the Froude number

(2.0) |

Since numerically dimensionless equations are solved, we use the parameters of the High-Pressure Convection Facility (HPCF) at the Max Planck Institute for Dynamics and Self-Organization in Göttingen, Germany, to evaluate Fr. It is a cylindrical cell, with a height of