PERCEPTION OF EDUCATIONAL INCLUSION IN THE UNIVERSITY CONTEXT: AN APPROXIMATE STUDY IN THE FCEE OF GRANADA

Magnetohydrodynamics (MHD) describes the bulk motion of charged particles subject to the presence of internal and external magnetic fields. This is an alternative method for describing the motion of charged particles, instead of examining the motion of single particles, MHD treats the plasma as a single conducting fluid.

The one-fluid theory assumes that plasma is made up of two particle species, ions and electrons (subscripted as i and e respectively) and for simplicity, it assumes ions are singly-charged, and neglects the difference between the two constituent particles. It also considers the plasma as a conducting fluid carrying magnetic and electric fields and currents. When considering this quasi-neutral plasma, the key variables are given for

n= mene+mini me+mi (2.21) m=me+mi=mi 1 +me mi (2.22) v= minivi+meneve mene+mini (2.23) ρq=e(ni−ne) (2.24) j=e(nivi−neve) (2.25)

where n is the fluid number density, m is the fluid mass,vis the fluid velocity, ρq is the charge density, and jis the current density (Baumjohann and Treumann,1997).

The continuity equation is given by

∂ρ

∂t +∇ ·ρv= 0 (2.26)

where ρ = nm is the fluid mass density. This equation shows that the one-fluid theory satisfies the conservation of mass in that the amount of mass entering the system is equal to the amount of mass leaving the system plus some accumulation of mass within it (Pedlosky,1992).

The equation of motion for electrons is given by

neme

dve

and the equation of motion for ions is given by

nimi

dvi

dt =nimig− ∇ ·Pi+nieE+nievi×B (2.28)

whereρ is the mass density, ρq is the charge density,P is pressure,jis current density and g is the acceleration due to gravity.

In order to calculate the overall motion of a plasma, equations 2.27 and 2.28 must be added together. It must also be assumed that the plasma is charge neutral, meaning ne'ni. Finally, by applying equations2.24and 2.25, themomentum equation for a quasi-neutral plasma is formed

ρdv

dt =−∇ ·P+ρg+j×B (2.29)

This equation describes the change of the centre of mass velocity of a quasi-neutral plasma element (Schunk and Nagy,2009).

2.4.1 Ohm’s Law

The generalised Ohm’s law describes the variation of current density with electromag- netic fields and is found by multiplying equations2.27and2.28bymeandmirespectively and then subtracting them from each other:

E+v×B=ηj+ 1 nej×B− 1 ne∇ ·Pe+ me ne2 ∂j ∂t (2.30)

Whereηis the resistivity,Pe is electron pressure tensor. The equation actually consists of several terms on the right-hand side; the first term is the resistive term, the second is the Hall term, the third term is the anisotropic electron pressure and the final term is the contribution of electron inertia to the current flow. The algebra is also simplified by neglecting terms with small mass ratios, me/mi and assuming quasi-neutrality, ne'ni ' n.

When considering the case of ideal MHD, the plasma resistivity reduces to zero (η = 0), and as such the conductivity (σ) must tend to infinity , there is no electron pressure gradient and current density is assumed to vary slowly in time. In addition, the electric field cannot have a component parallel to the magnetic field and there is no electric field in the plasma’s rest frame. Furthermore, the magnetic flux through a surface S (lying perpendicular toB) must remain constant, even if the surface changes shape or position.

If all of the above assumptions are used, then the last three terms in equation2.30drop out and Ohm’s law reduces to

j=σ(E+v×B) (2.31)

where σ, plasma conductivity is the inverse of η, plasma resistivity (Baumjohann and Treumann,1997).

2.4.2 Magnetic Pressure and Tension

The Hall term in equation 2.30, j × B introduces an effect specific to MHD called

magnetic tension. Using equation 2.5, the Hall term can be rewritten as

j×B= 1 µ0 (∇ ×B)×B=−∇B 2 2µ0 + 1 µ0 (B· ∇)B (2.32)

The first term in equation2.32 gives the gradient in magnetic pressure, PB, where:

PB =

B2

2µ0

(2.33)

The second term is themagnetic tension, TB

TB= B2 µ0 ˆ n Rc (2.34)

where Rc is the radius of curvature and ˆn is the outward normal. Magnetic tension acts towards the centre of curvature, meaning that the force acts to straighten curved magnetic field lines.

The ratio of plasma pressure to magnetic pressure is called theplasma beta,β, where

β = p

B2_/2µ 0

(2.35)

p = k(niTi + neTe) and k is the Boltzmann constant. A plasma is said to be cold if

Figure 2.11: Diffusion of magnetic field lines (Baumjohann and Treumann, 1997).

2.4.3 Diffusion and Frozen-in Flux

The transportation of magnetic field lines and plasma can be investigated through the combination of Ohm’s law, Amp`ere-Maxwell’s law and Faraday’s law, equations 2.30, 2.5and 2.4 respectively to create theinduction equation.

∂B

∂t =∇ ×(v×B) +

1

µ0σ

∇2_{B (2.36)}

The right-hand side of this equation contains terms for convection and diffusion of mag- netic fields.

Magnetic fields tend to diffuse across plasma when met with finite resistance and act to smooth out local inhomogeneities, as in Figure2.11. This change in topological structure occurs over a period of time called the magnetic diffusion time

τd=µ0σL2B (2.37)

In collisionless geophysical plasma regions, conductivities can often be very high and length scales very large, so from equation 2.37it is clear that large τd values can arise, meaning magnetic fields may no longer diffuse across the plasma. When this occurs, the diffusion term in the induction equation (equation 2.36) tends to zero and only the convection term is left

∂B

∂t =∇ ×(v×B) (2.38)

This is the condition for frozen-in flux (Alfv´en,1942) and it implies that when convection is dominant, plasma and magnetic field lines are intrinsically connected together meaning that they experience frozen-in flow, where plasma must move with the magnetic fields and therefore the magnetic fields must also move with the plasma. This theorem is often referred to as the hydromagnetic theorem, the Frozen-in flux theorem and also

Alfv´en’s theorem.

The frozen-in theorem gives rise to the notion of flux tubes, which are essentially a generalised cylinder containing a constant amount of magnetic flux. It implies that all particles and magnetic flux within the volume of a flux tube must remain in that tube independent of any motion the tube experiences or any distortion to its overall shape. Faraday’s law, equation 2.4, can be used to replace the partial differential in equation 2.38 to produce an equivalent form of the frozen-in flux condition:

E+v×B= 0 (2.39)

Equation 2.39 indicated that in an infinitely conducting plasma, electric fields are zero in a frame moving with the plasma. The condition also demonstrates through its cross product that there are no electric fields parallel to magnetic fields when in an infinitely conducting plasma.

The induction equation, equation 2.36indicated that the motion of plasma is governed by both diffusion and convection. By considering the ratio of these properties, a more precise definition of frozen-in can be established, providing the magnetic Reynolds number:

This value enables the determination of whether diffusion or convection is dominant within a plasma. If Rm 1, convection dominates and diffusion can be entirely ne- glected, meaning that magnetic field lines flow with the plasma. When Rm ≈ 1 then diffusion cannot be ignored and may become dominant, meaning that the frozen-in the- orem is not satisfied and magnetic field lines may slip across the plasma. However, because space plasmas generally involved large scale lengths, in this regime it can read- ily be assumed that convective flow dominates and the contribution from diffusion is negligible (Baumjohann and Treumann,1997).

2.4.4 Particle Motion Summary

In general, the MHD theory offers an accurate description of plasma behaviour, holding true for plasmas as long as magnetic fields vary slowly in space and time in comparison to particle gyroradii and gyroperiods. The solar wind is a perfect example of this, with few collisions and relatively consistent magnetic field over large spatial scales, it satisfies the frozen-in theory. However, the breakdown of the frozen-in theory is an important mechanism to facilitate the mixing of plasma and the initiation of magnetic reconnection. An example of this is seen when the solar wind meets the Earth’s magnetopause, it experiences sharp magnetic field gradients and when the IMF is directed southward, there is also a shear in the magnetic field system. The magnetic shear configuration is a precursor to magnetic reconnection, where magnetic diffusion of plasma from the field line gives rise to plasma mixing (when the diffusion term in Eqn 2.38 becomes significant). Both phenomena signify the breakdown of the frozen-in theory, which is key to solar wind - magnetosphere coupling. This is MHD.