Table of Contents

• Abstract
  
  
• Introduction
  
  
• Convection in Rotating Frames
  
  
• The Experiment
  
  
• Results
  
  
• Conclusions
  
  
• Acknowledgements
  
  
• References
  
  

There has been a major literature on convection in rotating frames since planetary as well as stellar convection depends upon an understanding of the interaction of the Coriolis force with buoyancy. However, here we are interested in just one circumstance of this problem, namely the case of an isolated buoyant or driven element in a rotating frame (See Pedlosky¹ for the generalization of circulation in these flows). This is because the α - Ω dynamo requires, for optimum gain, an episodic non-axisymmetric production of an average helicity (Cowling's anti-dynamo theorem² applied to the axisymmetric Ω flow). Although convection in a rotating frame has been extensively studied, the peculiar requirement of episodic, off-axis production of helicity is more obscure. It remains to be demonstrated elsewhere that such flows should occur in astrophysical circumstances. For our investigations we are analyzing the characteristics of pulsed jets analogous to thermals or plumes whose buoyancy is driven by thermal, pressure or density gradients.

The most extensive experiments and analysis of thermal plumes in a rotating frame have been performed by Helfrich³ where negatively buoyant elements marked by dye were released both in a rotating non-stratified and stratified environments. The entrainment of the thermals as a function of depth, with and without rotation, were observed and analyzed along with the depth of penetration with and without stratification. In our case we are most concerned with the magnetic flux already entrained in the fluid before displacement by the pulsed jet and then subsequently displaced by it. Thus in order to observe motions corresponding to rotation of magnetic flux, we release our markers ahead of the pulsed jet. It is comforting to note that Helfrich's observation of the anti-cyclonic rotation of the entrained fluid will add to the rotation of the displaced fluid ahead of the jet (presumably with entrained flux) and so will add to the α-effect. Because of the enhanced flux transported by the induced turbulence of entrainment and the calculations of rotation on turbulence, we expect the near-laminar displacement of the fluid ahead of the pulsed jet will be most effective for dynamo gain (Chan⁴). In agreement with Helfrich's observations, we observe and quantify the rotation angle in both the near-laminar flow ahead of the pulsed jet and the turbulent entrainment within it. Recently Fernando, Chen and Ayotte⁵ have studied such buoyancy-driven plumes in a rotating frame, but where the axis of the plume was coincident with that of rotation. Under these circumstances no off-axis helicity is generated, but the change in rotation and the modification of convection was observed. Similarly, Lavelle⁶ studied plumes with rotation and shear from the standpoint of modifying the entrainment, and Sánchez⁷ et al. analyzed and demonstrated plume entrainment with small initial perturbations. Golytsyn⁸ studied the effects of rotation on convection, but not the rotation angle in both the laminar flow ahead of the plume and the turbulently entrained volume. Similarly Jones and Marshal⁹ studied the effects of rotation on convection in the ocean and Narimousa¹⁰ in a two-layer fluid. These studies indeed confirm the effects of rotation on convection, but the degree of effectiveness of the α-deformation or helicity of a pulsed jet must be evaluated separately.

A. The Dynamo

The α - Ω dynamo was an attempt by Parker¹¹ to produce a dynamo from predictable, coherent motions. The problem was the difficulty in finding a natural source of a coherent fluid motion to produce an α-effect. The observation of a relatively coherent behavior of pulsed jets as compared to the purely random motions of turbulence has led to the possibility of this new source of helicity. This flow property requires both an axial translation and a finite rotation of the flow relative to a rotating frame. In dynamos, helicity serves the function of changing the field of one direction, toroidal, to an orthogonal one, poloidal. The kinetic helicity density of a fluid flow is defined as:

α = v • (∇ × v).

(1)

The integral of this kinetic helicity density over the volume of the dynamo in a conducting fluid flow becomes the α-deformation of magnetic flux as the α-effect of the α - Ω dynamo. We recognize that in dynamo theory the α-effect may be distributed and far more subtle as in the α² dynamo and is actually a relationship between the mean field and the mean current and is not proven here. In this paper we do not attempt to evaluate the effectiveness of the pulsed jet rotation in producing gain of magnetic flux in an α - Ω dynamo. We only note that the larger the volume of fluid that rotates and translates as a unit, the larger the contribution to the magnetic flux conversion from toroidal to poloidal field orientations. The Ω-component of the conducting fluid flow is differential rotation, dΩ/dR < 0, typical of Keplerian flow in a gravitational field or of annular rotational Couette flow in the laboratory. Subsequent simulations must prove that the combination of these two flows will produce a dynamo in the astrophysical and laboratory circumstances.

There is a significant difference between divergent and convergent flow. In divergent flow the vorticity decreases and so the differential rotation is bounded by that of the frame. In convergent flow, the vorticity can become nearly infinite. This is not useful for the dynamo because then the total rotation of the jet becomes very large and the mean rotation angle averages to zero. Therefore this work is limited to the case of the diverging pulsed jet only. It was further recognized that in laboratory experiments this translation and rotation of a buoyant element could be approximated by a driven pulse of fluid, a jet. The pulse of fluid is driven through an orifice by a transient pressure pulse. The flow velocity may be large as compared to the relatively lower velocities produced by convective forces. This then made possible a laboratory experiment demonstrating the α - Ω dynamo using liquid sodium (Colgate¹²). The two flows, the differential rotation of the Ω-flow and the helicity of the α-flow, can be driven by high power and high pressure resulting in high velocities v, and consequently high magnetic Reynolds number, Re_mag = Lv/η in a conducting fluid of resistivity η and dimension L. A high magnetic Reynolds number Re_mag > 100, α-helicity and Ω-differential rotation of a conducting fluid are the conditions for creating a positive gain α - Ω dynamo.

We conclude this section with a simple analogy of the problem in question. Imagine viewing a rising thunderstorm from a geostationary satellite in orbit in space. Then ask for the differential rotation angle of the lenticular clouds formed ahead of the rising plume and of the turbulently entraining plume itself during the lifetime of the cloud. If the fluid were conducting, thunderstorms would be the major source of helicity for a geo-atmospheric dynamo.