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 electro- dynamics. The content is largely based on \An introduction to Magneto- hydrodynamics" by P.A. Davidson. The basic equations of electrodynam- ics 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 r · E = e Gauss's law (1) "0 @B r × E = − Faraday's law (2) @t r · B = 0 No monopoles (3) @E r × B = µ J + µ " Ampere's law (4) 0 0 0 @t with "0 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 r · J = − e (7) @t and that by taking the divergence of Faraday's law (@=@t)(r · B) = 0 which, if true relative to all sets of axes moving uniformly relative to one another, yields r · 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 = − r × (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 = r × (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 r × (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 + + σr · (u × B) = 0 with τe = "0/σ (11) @t τe In a stationary conductor the charge density will therefore exponentially decay with a characteristic time-scale τe = "0/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 = −"0r · (u × B) = "0(u · r × B − (r × 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 − ! · B = − ! · B (13) "0 λ with λ = 1/µ0σ the `magnetic diffusivity' and ! ≡ r × u the fluid vorticity. From Eq. 12 ρe ∼ "0uB=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 characteris- tic length scale l corresponding to r · (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 "0 = 8:85 · 10 F/m a small charge can already create a signifi- cant 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 µ0"0@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 magnetohydrody- namical theory: 1. One is not interested in the very small time-scales τe = "0/σ of charge equilibration such that @ρe=@t is neglected from the conservation of charge 2. The characteristic length-scale l corresponding to r · (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 µ0"0@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 r × B = µ J and r × E = − with r · 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) + νr2u + J × B/ρ (15) Dt 2.3 Some more equations Inserting Ohm's law and the reduced Ampere's equation into Faraday's equa- tion: @B = −∇ × E = −∇ × (J/σ − u × B) = r × (u × B − r × B/µ σ) (16) @t 0 2 With r · B = 0 we have r × r × B = −∇ B yielding, with λ = 1/σµ0 the magnetic diffusivity, the `induction equation' 4 @B = r × (u × B) + λr2B (17) @t Note the striking similarity with the vorticity equation: @! = r × (u × !) + νr2! + r × (J × B)/ρ (18) @t We can therefore draw an analogy between the dynamics of the vorticity ! = r × u and the magnetic field B = r × 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 + λr2A + u × (r × A) (19) @t 3 Concepts of Magnetohydrodynamics 3.1 Analogies between fluid mechanics and MHD Note that the analogy between ! and B is not perfect, both by the occurrence of a source term in the vorticity equation which is absent in the induction equa- tion, 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 ! 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.