# Month of Equations: What Does The Magnetic Reynold's Number Really Mean?

### October 25: Magnetic Reynold's Number

Meaning of the Equation:

Reynold's number gives the ratio of the relative effects of magnetic field induction by the motion of a conducting fluid to the magnetic diffusion in the medium.

First predicted by Osborne Reynold's in 1883, Reynold's number in fluids indicates that the transition of laminar flow of a fluid to a turbulent one can be indicated by a ratio of the product of flow velocity and length of the tube to the kinematic viscosity of the fluid. We use two approaches to mathematically model a plasma- either we consider the differently charged species as fluids or we take single particle effects into consideration. The Reynold's number is relevant to the fluid description of plasmas.

In plasmas, Reynold's number, also known as the magnetic Reynold's number, helps to identify the dependence of the plasma generation process on the fluid velocity.  To understand its significance, let us begin with understanding the basic term associated with it :

Diffusion of the magnetic field:

In most of the universe, the magnetic field behaves as if it is frozen in the plasma. This means that the magnetic field lines may be said to move around as the particles move in the plasma. However, in some regions, the magnetic field can slip through the plasma and reconnect, with far-reaching consequences such as magnetic reconnection that leads to the conversion of magnetic energy into bulk kinetic energy, heat and fast particle energy. This is called the diffusion of the magnetic field.

Induction of the magnetic field:

The relationship between the magnitude of the magnetic field and the velocity of a conducting fluid is presented by the induction equation. In other words, the magnitude of the magnetic field generated by the flow of a conducting fluid, determined by Maxwell's law and the Ohm's law is called induction.

Reynold's number in fluids helps to predict the critical velocity above which the turbulence occurs in a fluid. In plasmas, the different ranges of the magnitude of the magnetic Reynold's number give different pieces of information about the fluid. In general, the magnetic Reynold's number helps to find out the critical flow velocity of the fluid above which the magnetic field induction dominates the diffusion and hence, the plasma gets magnetized.

• For Rm>>1, the magnetic field diffusion dominates and the field lines tend to completely relax. In this case, the magnetic field is not dominated by the flow velocity rather by the boundary conditions.
• For Rm <<1, the diffusion is not dominant on the characteristic length scale L and the magnetic field lines tend to be frozen into the plasma.

The magnetic Reynold's number also gives the critical conditions below which the flow is laminar and above which the flow becomes turbulent. One may about what meant by the turbulent flow?

Turbulent flow is defined as an irregular motion of the fluid resulting from the formation of eddies, and vortices within the fluid.

In plasmas, At Rm<2000, the flow is laminar or in other words streamlined, whereas turbulence is set when Reynold's number becomes greater than 4000. A snake-like propagation of plasma plumes has been reported in experiments that are found to be related to the turbulence caused by the gas flow being greater than the critical Reynold's number. So, identifying the magnetic Reynold's number in a particular plasma delivers a lot of insight about the plasma system under consideration.

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