Magnetohydrodynamics

The magnetohydrodynamics ( MHD) is a branch of physics. Describes the behavior of electrically conductive fluids, which are penetrated by the magnetic ( and electrical ) fields. The magnetohydrodynamics ( MHD) in the strict sense treated liquids, the magneto gas dynamics ( MGD ) treated gases, in particular deals with the MHD plasma ( magnetosphere plasma dynamics, MPD). In the context of MHD plasmas are described as fluids (fluids). The mathematical treatment is a combination of the Navier -Stokes equations of hydrodynamics and Maxwell's equations of electrodynamics.

Typical areas of application are the MHD flow control and flow measurement in metallurgy and semiconductor single crystal growth as well as the description of plasmas in stellar atmospheres and fusion reactors.

• 2.1 Geo - and solar dynamo
• 2.2 Stellar atmospheres
• 2.3 Electromagnetic Flow Control
• 2.4 Electromagnetic flow measurement
• 2.5 MHD generator
• 2.6 MHD propulsion
• 2.7 MHD sensor

Theory

To describe the dynamics of plasma in detail, it would be necessary to solve the equation of motion for each particle. Of course, this approach does not make sense, since the number of particles is generally very large. In order to solve the problem anyway, choosing a statistical approach. This means in practice that one, rather than looking at the trajectories of the individual particles, macroscopic quantities (mean values ​​) were used.

An example of such a variable is the flow rate of a flow. Overall, the river water moves toward the sea in one direction. Every single water molecule may be, but quite moving " criss-cross " because it constantly interacts with other molecules. Crucial to the macroscopic behavior is that the mean velocity of flow rate corresponds. With such macroscopic quantities such as velocity and pressure can be then all the phenomena of Marine adequately describe without ever comes a single molecule into play.

Something like it is with the MHD. Also, it is basically a hydrodynamic theory, in the sense that plasmas are treated similarly to liquids. Again, no individual particles but are " averaged quantities " treated. The MHD combines elements of hydro-and electrodynamics is a theory " of electrically conducting plasmas " to mold.

Usually some other approximations are made ​​yet. These are, first, electrical neutrality, secondly, the validity of Ohm's law and thirdly quasi-equilibrium. The last approximation means that the plasma is developed but slowly in the balance adjusts itself relative to the time. This assumption represents a major simplification of the equations, as they, for example, ensures that the pressure can be treated as a scalar quantity, while one has to do it in general with a ( direction-dependent ) tensor. Electrical neutrality eliminates the charge density of the equations. Under these conditions, the equations of MHD can now be formulated.

The MHD equations

The basic equations of MHD are ( in CGS units) as follows:

The occurring values ​​are the mass density of the plasma velocity, the pressure, the electric current, the magnetic field, the electric field, the gravitational acceleration and the speed of light. In equation (2) gravity was considered explicitly; the term implies that in general more forces must be taken into account such as, for example, frictional forces, which were not explicitly specified here. Equation (1 ) describes the conservation of mass, equation (2) is the " equation of motion " or " momentum equation ", the equations ( 3), ( 4) and (5 ) come from electrodynamics, where neglected in equation (3 ), the displacement current been. Equation (6) is the Ohm's law to the electrical conductivity. This is the only equation to be considered in the relativistic effects.

To close the system of equations, one further equation is needed. This can for example be the equation of state of matter or the equation of local energy conservation.

The equations of the MHD may be derived in different ways. One way is to start from the Boltzmann equation, to get to the above result.

The induction equation

By combining the laws of Ampere, Faraday and Ohm ( equations 3, 4 and 6), one obtains the so-called induction equation.

In the equation, the magnetic diffusivity. This equation plays an important role in understanding magnetic plasmas.

Ideal MHD

The solution of the MHD equations can be very complicated. In many cases, however, the equations can be simplified by further assumptions to facilitate solution. Assuming that the electrical conductivity of the plasma is infinitely large (), so it therefore has no electrical resistance, the resulting theory is called "Ideal MHD " as opposed to " resistive MHD " with finite conductivity. In the ideal case, equation (6).

Examples of the applicability of the MHD ideals can be found for example in the calculation of plasma flows in the context of nuclear fusion or stellar coronas. In addition, it is often assumed that the plasma is non-compressible (ie ) is and there is no internal friction ( viscosity).

Frozen magnetic field

The MHD combines the hydrodynamic and electrodynamic properties of plasmas. Thus, material movements and behavior of the magnetic field are not independent. Plasma motions lead to further magnetic and electric fields, which in turn lead inside the plasma to electric currents.

An important special case arises for the case of ideal MHD. Here, the electric resistance of the plasma disappears and the equation takes the form of induction

Of. One can show that in this case are coupled to the plasma and the magnetic field; the magnetic field follows the plasma flows. One speaks in this case of a "frozen" magnetic field or in connection with the ideal induction equation, even from a " flow -preserving " equation. The morphology of the magnetic field can also be subjected to considerable changes in this case, it is important that any topological changes (hence, no change in the context of the magnetic field ) may occur.

Whether or not the magnetic field of the plasma or the plasma followed by the magnetic field, depends on which component is dominant, therefore, exerts a larger pressure. In the sun in both cases can be observed. While the plasma in the photo- sphere is very tight and the plasma flows from the convection dominate the behavior of the magnetic field in the plasma of the corona is very thin, so that the structure is dominated by the magnetic field of the corona.

MHD waves

Inside the plasma, different wave modes can form. These are the fast and slow magneto acoustic waves, and the so -called Alfvén waves. These modes represent special cases for waves propagating either parallel or perpendicular to the magnetic field. They provide insight into the basic physical phenomena. in general case, we will, however, have to deal with a superimposition of all modes.

Applications

The MHD finds various applications in both engineering as well as in the natural sciences.

Geo - and solar dynamo

Earth's magnetic field is generated by the so-called Geodynamo, which is described by equations of the MGD. The Earth's magnetic field is created in the outer core, which consists mainly of liquid iron (see formation and maintenance of the geomagnetic field ( geodynamo ) ).

Appearing within the context of a geodynamo partial differential equations can be solved only in very simplified cases analytically. Numerical methods provide since the mid- 1990s, the first steps to understanding the dynamics of the Earth's magnetic field.

The situation is similar with the magnetic field of the sun and other stars similar spectral type. The origin of their magnetic fields resulting from the interaction of several factors in the outer layers of the star ( dynamo theory ) and is also described based on the MHD models.

Stellar atmospheres

The corona (outermost atmosphere ) of the sun and other stars is a highly structured region of very low plasma densities where the magnetic field is the dominating, structuring component. MHD effects here are very important to describe and understand the distribution and dynamics of the outer stellar atmosphere. Also in the underlying atmospheric layers of the chromosphere and the photosphere the MHD plays an important role.

Electromagnetic Flow Control

Magnetic fields can be used in metallurgy, in order to influence the flow of liquid metal, such as steel or aluminum. When the application is to distinguish between static and time-dependent magnetic fields. Static ( time-independent ) magnetic fields lead to a damping of turbulence and are therefore in the form of magnetic braking (English: electromagnetic brakes ) used in continuous casting of steel. Time-dependent magnetic fields are applied to the electromagnetic supporting the casting of aluminum (electromagnetic casting ).

Electromagnetic flow measurement

Moving an electrically conductive fluid under the influence of a magnetic field, as in the fluid, a voltage is generated which is the strength of the magnetic field and the flow rate is proportional. This principle was discovered in 1832 by Michael Faraday, who (unsuccessfully) tried in this way to determine the flow rate of the Thames. The principle is used today in the form of electromagnetic flowmeters (also called inductive flow meter ) wide application in food industry, the chemical industry and in water treatment plants. During the movement of an electrically conductive fluid in a magnetic field created in the fluid on the basis of the induced voltage and eddy currents. These generate a Lorentz force, which counteracts the movement of the fluid. This physical effect is applied in the designated as Lorentz force velocimetry non-contact flow measurement method.

MHD generator

A technical application of magnetohydrodynamics is the magnetohydrodynamic generator ( MHD generator ). Here, a plasma between two conductive electrodes through shot. Perpendicular to the electrodes, a magnetic field is applied, then the negatively charged electrons and positively charged ions separate due to the opposite charge. This produces a voltage difference between the plates. In this way, kinetic energy is converted directly into electrical energy without the use of mechanical components (turbines and generators ). Despite intensive efforts, particularly in the 1960s, has not been able to develop long -term stable electrodes. For this reason, the MHD generator currently has no practical significance.

MHD propulsion

An electrically conductive substance, such as a plasma, can be accelerated in an electromagnetic field ( magnetoplasma dynamic drive ). Since sea water and ionized air as ion mixtures are conductive, they can also be accelerated in an electromagnetic field ( Magneto Hydrodynamic drive ). These properties can in principle be used for the propulsion of ships, submarines and aircraft. However, the efficiency of such systems is low. For this reason, the MHD propulsion has no practical significance today.

MHD sensor

Magneto Hydro Dynamic sensors are used to measure angular velocities. The accuracy increases with the size. One application is the aerospace. The principle of MHD sensor that helps a whole understand the basic idea of magnetohydrodynamics ( MHD), is shown in this sketch.