Magnetohydrodynamics

J.W.Haverkort April 2009

Abstract This summary of the basics of magnetohydrodynamics (MHD) as- sumes the reader is acquainted with both ﬂuid mechanics and electrodynamics. The content is largely based on “An introduction to Magneto- hydrodynamics” by P.A. Davidson. The basic equations of electrodynamics are summarized, the assumptions underlying MHD are exposed and a selection of aspects of the theory are discussed.

Contents

1 Electrodynamics 2 1.1 The governing equations ...... 2 1.2 Faraday’s law ...... 2

2 Magnetohydrodynamical Equations 3 2.1 The assumptions ...... 3 2.2 The equations ...... 4 2.3 Some more equations ...... 4

3 Concepts of Magnetohydrodynamics 5 3.1 Analogies between ﬂuid mechanics and MHD ...... 5 3.2 Dimensionless numbers ...... 6 3.3 Lorentz force ...... 7

1 1 Electrodynamics 1.1 The governing equations The Maxwell equations for the electric and magnetic ﬁelds E and B in vacuum are written in terms of the sources ρe (electric charge density) and J (electric current density) ρ ∇ · E = e Gauss’s law (1) ε0 ∂B ∇ × E = − Faraday’s law (2) ∂t ∇ · B = 0 No monopoles (3) ∂E ∇ × B = µ J + µ ε Ampere’s law (4) 0 0 0 ∂t

with ε0 the electric permittivity and µ0 the magnetic permeability of free space. The electric field can be thought to consist of a static curl-free part Es = ∂A −∇V and an induced or in-stationary part Ei = − ∂t which is divergence-free, with V the electric potential and A the magnetic vector potential. The force on a charge q moving with velocity u is given by the Lorentz force f = q(E+u×B). Moving along with a charge, however, the force f = qEr is proportional to the relative electric field mEr so that one finds that non-relativistically the electric field transforms as Er = E + u × B where B = Br. Note that from f = qu × B one can see that the magnetic field is not a real vector, but a pseudo-vector: under a coordinate inversion x → −x both F and u change sign so that B can not. In the continuum limit the Lorentz force per unit volume becomes

F = ρeE + J × B (5) Due to collisions an applied force will only lead to a ﬁnite current density. Assuming a linear relationship yields Ohm’s law:

J = σEr = σ (E + u × B) (6) We conclude this summary of the basic equations of electrodynamics by noting that the divergence of Ampere’s law yields the conservation of charge ∂ρ ∇ · J = − e (7) ∂t and that by taking the divergence of Faraday’s law (∂/∂t)(∇ · B) = 0 which, if true relative to all sets of axes moving uniformly relative to one another, yields ∇ · B = 0.

1.2 Faraday’s law For a closed curve C around a surface S, moving in space with a prescribed velocity u the surface area swept out by a line element dl of the curve is given by δS = dl0 × dl = (u × dl)δt where dl0 = uδt is the inﬁnitesimal displacement of the element dl in a time δt. Therefore for any solenoidal vector ﬁeld G Z Z ∂G I δ G · dS = δt · dS − u × G · δtdl (8) S S ∂t C

2 such that with Stoke’s theorem it follows that

d Z Z ∂G G · dS = − ∇ × (u × G) · dS (9) dt S S ∂t This mathematical identity shows that the flux of G through the surface S can change either due to a change in G itself or due to a change in the surface area of S. Its physical relevance is that whenever a vector field obeys the transport equation ∂G/∂t = ∇ × (u × G) the flux through any material surface, i.e. moving along with the fluid, is conserved. It also allows us to rewrite Faraday’s law to integral form, by adding ∇ × (u × B) to both sides of Eq. 2 to yield H (E + u × B) · dl = −(d/dt) R B · dS or, using the definition H C S e.m.f.≡ C Er · dl: I d Z e.m.f = Er · dl = − B · dS (10) C dt S Somewhat hidden in Faradays law thus lies the information that the e.m.f. in a closed loop can change either due to a time-varying magnetic field, transformer e.m.f. due to Faraday’s law, or through a change in the surface area of the loop, motional e.m.f due to the way the electric field transforms.

2 Magnetohydrodynamical Equations 2.1 The assumptions Taking the divergence of Ohm’s law, Eq. 6, using conservation of Charge, Eq. 7, and inserting Gauss’s law, Eq. 1 yields

∂ρe ρe + + σ∇ · (u × B) = 0 with τe = ε0/σ (11) ∂t τe In a stationary conductor the charge density will therefore exponentially decay with a characteristic time-scale τe = ε0/σ, which for a typical conductor is −18 τe ∼ 10 s. If one is not interested in these very small time-scales, ∂ρe/∂t can thus be neglected. Note that in steady state a moving conductor can maintain a small electric charge by magnetic forces:

ρe = −ε0∇ · (u × B) = ε0(u · ∇ × B − (∇ × u) · B) (12) Where use has been made of a vector relation. Using Ampere’s law and Ohm’s law yields

ρe u · E = u · µ0J − ω · B = − ω · B (13) ε0 λ

with λ = 1/µ0σ the ‘magnetic diffusivity’ and ω ≡ ∇ × u the fluid vorticity. From Eq. 12 ρe ∼ ε0uB/l, while Ohm’s law implies that usually E ∼ J/σ. We thus find for electric forces relative to magnetic forces ρeE/JB ∼ uτe/l, which again involves the extremely small number τe. Assuming the characteristic length scale l corresponding to ∇ · (u × B) is not too small, the electric force can therfore safely be neglected in the Lorentz force. This however does not mean that the presence of electric volume charges can be neglected altogether.

3 In the dynamics of the electric current density as stated by Ohm’s law it plays a significant role, i.e. through the the electric field it produces. Through the −12 smallness of ε0 = 8.85 · 10 F/m a small charge can already create a significant divergence in the electric field that can act to prevent magnetic forces from creating a divergence in the current density, i.e. a change in the charge density. Because the current density directly enters the Lorentz force, volume charges can indirectly have an impact on the flow. Their direct influence via the Lorentz force, however, can be safely neglected in magnetohydrodynamics.

In a similar way we can do without the displacement current term in Ampere’s law (the µ0ε0∂E/∂t term) because by Ohm’s law this term is ∼ µ0τe∂J/∂t. Again, when we are not interested in the very small time-scales associated with charge equilibration: µ0τe∂J/∂t ∼ µ0τe∂(u × B)/∂t µ0J we can neglect the displacement current. To summarize the assumptions behind magnetohydrodynamical theory:

1. One is not interested in the very small time-scales τe = ε0/σ of charge equilibration such that ∂ρe/∂t is neglected from the conservation of charge 2. The characteristic length-scale l corresponding to ∇ · (u × B) is assumed to be large enough such that ρeE/JB ∼ uτe/l 1 and the electric force can be neglected in the Lorentz force 3. The current density J = σ(E+u×B) varies on the equilibration time-scale τe due to the fast dynamics of the electric ﬁeld E and on much longer time- scales due to the ‘macroscopic’ time-scales associated with the magnetic part u × B. One neglects the fast dynamics µ0ε0∂E/∂t ∼ µ0τe∂J/∂t µ0J such that the displacement current can be neglected in Ampere’s law

2.2 The equations The equations of magnetohydrodynamics thus contain the reduced Maxwell equations ∂B ∇ × B = µ J and ∇ × E = − with ∇ · B = 0 (14) 0 ∂t and the Lorentz force density F = J × B which is added as a body force to the Navier-Stokes equations for a Newtonian ﬂuid Du = −∇(p/ρ) + ν∇2u + J × B/ρ (15) Dt 2.3 Some more equations Inserting Ohm’s law and the reduced Ampere’s equation into Faraday’s equation:

∂B = −∇ × E = −∇ × (J/σ − u × B) = ∇ × (u × B − ∇ × B/µ σ) (16) ∂t 0 2 With ∇ · B = 0 we have ∇ × ∇ × B = −∇ B yielding, with λ = 1/σµ0 the magnetic diﬀusivity, the ‘induction equation’

4 ∂B = ∇ × (u × B) + λ∇2B (17) ∂t Note the striking similarity with the vorticity equation: ∂ω = ∇ × (u × ω) + ν∇2ω + ∇ × (J × B)/ρ (18) ∂t We can therefore draw an analogy between the dynamics of the vorticity ω = ∇ × u and the magnetic field B = ∇ × A or similarly between the fluid velocity u and the magnetic vector potential A. If we ‘uncurl’ Eq. 16 or Eq. 17 we get an equation equation for A very similar to the Navier-Stokes equations: ∂A = −∇φ + λ∇2A + u × (∇ × A) (19) ∂t 3 Concepts of Magnetohydrodynamics 3.1 Analogies between fluid mechanics and MHD Note that the analogy between ω and B is not perfect, both by the occurrence of a source term in the vorticity equation which is absent in the induction equation, and by the fact that the vorticity is functionally related to u in a way that B is not. Furthermore through the fluid velocity u the dynamics of ω and B is usually coupled. Alternatively through the Lorentz force in the Navier-Stokes equations and the magnetic part in Ohm’s law the dynamics of u and B is coupled.

Similarities between the vorticity and the magnetic field include that both are solenoidal, the vorticity by definition and the magnetic field by one of the Maxwell equations. Another similarity is that both the magnetic field and the vorticity are pseudo-vectors, due to the fact that they can be expressed as the curl of a ‘real’ vector (the velocity and the magnetic vector potential respectively).

We expand the curl-term in Eqs. 17 and 18 with G = B or ω solenoidal ∇ × (u × G) = −(u · ∇)G + (G · ∇)u − G(∇ · u). The first term then denotes advection of G. The second term gives ‘vortex’ stretching, i.e. the effect that when there is a positive velocity gradient in the fluid (i.e. streamlines converge) in the direction of G this quantity intensifies. Physically for the vorticity this is due to conservation of angular momentum. For the magnetic field this is due to the conservation of magnetic flux.

In highly conducting ﬂuids, the magnetic diﬀusivity λ = 1/σµ0 is negligible, such that in this case we can borrow some theorems from inviscid vortex dynamics.

• Kelvin’s theorem, which states that since ∇·ω = 0 Gauss’s law over a flux tube gives H ω·dS = 0. The flux of vorticity R ω·dS = H u·dl ≡ Γ, with Cm Γ the circulation or the ‘strength’ of the vortex tube, is constant along the length of a vortex tube. This is because no flux crosses the side of the tube. This theorem directly follows from Eq. 9 and the inviscid vorticity

5 equation. For the magnetic field the analogous theorem implies that the magnetic flux R B·dS = H A·dl linking any ‘material’ loop co-moving Sm Cm with the fluid, is constant. Note that this is essentially information already contained in Faraday’s law in integral form, Eq. 10, with Er = J/σ → 0 such that (d/dt) R B · dS Sm • Helmholtz’s first law, which states that fluid elements continue to lie on the same vortex line, i.e. the vortex lines are frozen into the fluid. This result follows from the rate of change of an infinitesimal line element dl which moves with the fluid: Ddl/Dt = u(x + dl) − u(x) = (dl · ∇)u. This is the same equation of motion as for the vorticity in case the fluid is incompressible, proving the theorem. For the magnetic field the analogous theorem implies that for an incompressible fluid the fluid elements continue to lie on the same magnetic field lines, i.e. the field lines are frozen into the fluid.

The analogues of Kelvin’s theorem and Helmholtz’s first theorem for the magnetic field are collectively called Alfvén’stheorem. Furthermore the concept of helicity from fluid mechanics can be extended to MHD

• Using the inviscid and incompressible Navier-Stokes and vorticity equation, the material derivative of the quantity u · ω can be written as D(u · ω)/Dt = Du/Dt · ω + Dω/Dt · u = −∇(p/ρ) · ω + (ω · ∇u) · u = ∇ · ((u2/2 − p/ρ)ω) such that the quantity h ≡ R u · ωdV = H (u2/2 − Vω Sω p/ρ)ω · dS is conserved in time for any material volume Vω for which ω · dS = 0. Note that indeed in flows with non-zero helicity fluid particles move in a spiralling helical motion. Evaluating for two vortex tubes the he- R H R 2 H licity in all space h = (u·ω)dV = C +C u·( ω·dSdl) = Σi=1Φi C u·dl. H 1 2 H i If the two flux tubes are linked u·dl = Φ2 and u·dl = Φ1 such that C1 C2 the helicity h = 2Φ1Φ2 but if they are not linked h = 0 such that the helicity is a direct measure of the topology of the vortex tubes. By an analogous derivation it follows from the ideal induction equation (λ = 0) and ideal R Eq. 19 that the ‘magnetic helicity’ hm ≡ A · BdV is conserved, show- VB ing the conservation of both the flux and topology of the flux tubes. When the fluid under consideration is both inviscid (ν = 0) and ideal (λ = 0) a similar derivation, using the inviscid Navier-Stokes equation and the ideal induction equation, shows the conservation of ‘cross-helicity’ R B · udV , VB representing the degree of linkage of the vortex and magnetic field lines.

Sometimes in nearly ideal fluids the flux tubes get so twisted that the gra- dients become large enough to have some diffusion in spite of the smallness of λ. Magnetic field lines can then cross and reconnect, a process called magnetic reconnection. Although an interesting and sometimes important phenomena, in nearly ideal fluids the conservation of field line topology is a very useful and accurate representation.

3.2 Dimensionless numbers By adding a body force to the Navier-Stokes equation, one adds an extra dimensionless number to the familiar Reynolds number. The characteristic ratio

6 between the Lorentz force term and inertia is given by the ‘interaction param- eter’, or Stuart number:

|J × B| σB2l Ha2 N = = = (20) |ρu · ∇u| ρu Re Here the dimensionless Hartmann number Ha alternatively gives the characteristic ratio between Lorentz forces and viscous forces s |J × B| r σ √ Ha = = Bl = NRe (21) |ν∇2u| ρν In the induction equation on the other hand a ‘magnetic Reynolds number’ can be deﬁned as the characteristic ratio between the advection and diﬀusion term

|∇ × (u × B)| ul Re = = = µ σul (22) m |λ∇2B| λ 0

Using as a characteristic velocity scale the Alfv´enspeed vA this quantity is called the Lundqvist number Lu such that the ratio of the two gives the Alfv´en −7 3 Mach number Rem/Lu = v/vA = MA. Using the values ρ ∼ 4 · 10 kg/m (n ∼ 1020m−3), σ ∼ 2.4 · 109 S/m (∼40 times that of copper), l ∼ 4 m, B ∼ 4 T, u ∼ 105 m/s (M∼0.15), and ν ∼ 1m2/s (neoclassical), typical values for an ITER-like tokamak, yield

12 5 9 9 N ∼ 4 · 10 , Re ∼ 4 · 10 , Ha ∼ 10 and Rem ∼ 10 (23) (λ ∼ 3 · 10−4m2/s). We thus see that Lorentz forces there are much larger than inertial forces (N 1) and that inertial forces are in turn much larger than viscous forces (Re 1). In the induction equation non-ideal eﬀects can safely be neglected (Rem 1). Note that the used numerical values, especially the characteristic velocity and viscosity, are highly speculative and vary widely depending on the situation. E.g. there are orders of magnitude between the classical, neoclassical and turbulent viscosity. The velocity depends heavily on the heating techniques used. Other values than macroscopic lengths and velocities might also be more suitable depending on the situation.

3.3 Lorentz force

The Lorentz force J×B can, with J = ∇×B/µ0 and the use of a mathematical identity, be rewritten to

B2 J × B = (B · ∇)B − ∇ (24) 2µ0 which can be written in terms of a Maxwell stress tensor T

1 B2 J × B = −∇ · T with T = BB − I (25) µ0 2µ0 where the second term is the magnetic pressure and the ﬁrst are called Faraday or Maxwell tensions in the ﬁeld lines. The latter terminology arises

7 from the fact that the resulting force dF = 1 BB·dS is zero on the sides of flux µ0 tubes. At the ‘ends’ of field lines these tensile stresses cause a non-zero tension force on the flux tubes. By Gauss’s law the body force on a fluid element can be thought of as the surface integrated effect of these stresses and pressure. Going back to the concept of a body force on fluid elements one can decompose the anisotropic part of J × B in tangential and normal curvilinear coordinates attached to a field line

∂B B2 (B · ∇)B = B eˆ − eˆ (26) ∂s t R n with R the local radius of curvature of the field line. We note that this decomposition is equal to that which can be made of the non-linear advection term (u·∇)u in the Navier-Stokes equations, apart from an important difference in sign. The first term is in this case a force opposing acceleration associated with inertia, i.e. acceleration of a fluid element. The second term is then the centripetal force directed outwards. The terms in Eq. 26 represent respectively an accelerating force whenever the magnetic field increases along a field line and an inwards force when the field lines are curved. These forces have important implications for the stability of a plasma. Whereas the inertial force opposes acceleration, the first component in Eq. 26 causes acceleration in the direction of increasing magnetic field strength. The second component acts to straighten the magnetic field lines, in accordance with the idea of tensile stresses acting on the ends of field lines. In terms of these stresses the force in the direction of larger magnetic field arises due the difference in magnitude between the opposing stresses on the ends of field lines.

In summary, the Lorentz force on the fluid can be thought of as resulting from the magnetic field lines being under tension. As these field lines are to some extent ‘frozen into the fluid’ these tensions thereby act as a body force on the fluid. As as result the field lines may act similar to a string under tension and cause wave motion called ‘Alfvénwaves’, where the necessary inertia is provided by the fluid.