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University of Wisconsin Physics Department Department of Energy National Science Foundation

Background

The role of magnetic fields in astrophysical processes has gained greater attention in recent times. The plasma physics community is making efforts to create experiments that explore processes which are fundamental to the evolution of planets, stars, and galaxies. One such process is the generation of the magnetic fields we observe in each of these astrophysical bodies.

The dynamic formation of these magnetic fields is explained by the dynamo. Flowing conduncting fluids can bend and stretch magnetic field lines so that they amplify the magnetic field. If the geometry of the flow also provides a positive feedback, the magnetic field continues to grow until it becomes strong enough to affect the flow. The result is a system whereby kinetic energy in the flow is spontaneously converted into magnetic energy.

Our experiment is designed to create a flow in a liquid metal that is predicted to produce a dynamo that can be studied in the laboratory. We are interested in answering questions about what types of flow are required for a dynamo, what are the effects of turbulence on a laminar dynamo, and what causes the saturation of the magnetic field growth in a dynamo.

The Kinematic Dynamo:

The evolution of the velocity and the magnetic field is governed by magnetodyrodynamics. The equations which describes the evolution of the magnetic field is the magnetic induction equation which is non-linearly coupled to the Navier-Stokes equation which describes the evolution of the velocity. Solving this system of non-linear equations is a formidable task. The kinematic dynamo addresses this problem by assuming that the magnetic field is sufficiently weak so that the Lorentz force, the force the magntic field applies to the fluid, is weak. The magnetic induction equation becomes decoupled from the fluid equation and the problem reduces to solving a linear partial differential equation for which we assume that the velocity field is fixed.

The growth rate of the field increases as the magnetic Reynolds number is increased.

In solving the magnetic induction equation, we find that there are two important criteria for the production of a dynamo. First, the fluid must be able to move the magnetic field faster than it can diffuse away. A parameter which describes this comparison of advection to diffusion is the magnetic Reynolds number Rm. The magnetic Reynolds number depends on the fluid conductivity, the size of the system, and the characteristic speed of the fluid. A dynamo typically has a critical magnetic Reynolds number. When the fluid is sufficiently conductive, large, and fast the magnetic Reynolds number is large enough to result in a dynamo. For fluid flow that is below the critical Reynolds number, the magnetic field decays away with some characteristic decay rate that approaches zero as the magnetic Reynolds number reaches criticality. To the left is shown a plot of the growth rate of the magnetic field as magnetic Reynolds number is increased. The critical magnetic Reynolds number is at about 82 for this flow. Notice as well that the gain, the ratio of the magnetic response to the applied field, starts to grow rapidly just before criticality.

The stretch-twist-fold mechanism of the double-vortex dynamo.

The second criterion is that the flow have the correct geometry for amplifying the magnetic field and creating a feedback loop that leads to magnetic field growth. This requirement is illustrated in the figure to the right. The magnetic field that is excited by the flow we are studying is created by a process similar to the stretch-twist-fold rope dynamo of Vainshtein and Zeldovich. The figure shows how the magnetic field is carried by the fluid flow in the limit of infinite conductivity. A magnetic field line lies tangent to the axis of rotation of the fluid as indicated by the arrows in the figure. Calculating the new position of the fluid element in which the field rests allows us to follow the motion of the field in the flow. Progressing from (a) to (d), the magnetic field is stretched (or amplified) and twisted by the flow. In (d) we can see that there is a new magnetic field line lying along the original straight field line indicating the generation of new magnetic flux. The complicated structure in the core would tend to be smoothed out by resistive diffusion resulting in a stronger field. We can also see that the degree that the field is twisted plays an important role in generating the dynamo. If the field line were twisted too far, or not far enough, then the resulting field would not align itself with the original field line. This angle of twist is governed by the geometry of the flow.

Here is a movie depicting the evolution of a pair of magnetic field lines.

Turbulence:

The geodynamo relies on the large size of the Earth to achieve the critical magnetic Reynolds number. In a laboratory experiment, however, we must rely on larger flow speeds. Fluids at these speeds (in excess of 20 m/s) are turbulent resulting in a number of interesting changes to the dynamics of the system.

Mean-field Electrodynamics addresses the presence of fluctuations in the velocity and magnetic field by assuming the fluctuations are small compared to some mean value and performing statistical averages over the fluctuating quantities to determine the appropriate modifications to the equations.

Given the assumptions of homogeneous, isotropic turbulence, the theory predicts a turbulent EMF described by two terms: the alpha-effect and the beta-effect.

The alpha-effect arises due to the small-scale helical motions of the turbulent flow. These small helicies can produce the same stretch-twist-fold mechanism described above, but on a smaller scale. The alpha-effect produces current anti-parallel to the direction of the magnetic field line. If the helical motions of the fluctuations are correlated, the net current generated can produce a large-scale magnetic field.

The beta-effect describes the increased transport of magnetic flux due to turbulent stirring. In the laminar dynamo, the dissipation of magnetic flux is governed by the rate of diffusion. Turbulent stirring, however, can increase the transport of magnetic flux from one region to another. The result is an enhanced diffusivity of the fluid. This can also be thought of as a reduction in the effective conductivity in the fluid. Since the critical magnetic Reynolds number depends on the conductivity, the beta-effect raises the critical magnetic Reynolds number for the flow, thus making it harder to produce a dynamo in the laboratory.

Saturation:

The magnetic Reynolds number decreases and the critical magnetic Reynolds number increases leading to saturation of the field.

Once the magnetic field begins to grow, the Lorentz force will become strong enough to modify the flow. The kinematic dynamo model breaks down and in order to capture the dynamics of the system, we have to solve the full non-linear set of MHD equations. One anticipates that the flow will be modified in such a way to limit growth of the magnetic field. Simulations that solve the fully non-linear system show that this saturation is accomplished in two ways. First, the magnetic field acts as a breaking force which slows down the fluid flow. The result is that the magnetic Reynolds number (which depends on the characteristic speed of the fluid) is reduced. Second, the magnetic field also changes the geometry of the flow, thereby increasing the critical magnetic Reynolds number for the flow. Saturation occures when the magnetic Reynolds number and the critical magnetic Reynolds number are equal as depicted in the figure to the right.

An example of planetary dynamos:

The magnetic field of Uranus is somewhat similar to the field which we are attempting to generate. Click here to see a movie of thetime-history of the magnetic field of Uranus. (Super big 6M file...don't dare do this with a modem!)

The magnetic dipole moment of Uranus is not aligned with the planet's axis of spin. Our experiment will produce a similar field.