A Mathematical Introduction to Magnetohydrodynamics

Typeset in LATEX 2ε July 26, 2017 A Mathematical Introduction to Magnetohydrodynamics

Omar Maj

Max Planck Institute for Plasma Physics, D-85748 Garching, Germany. e-mail: [email protected]

1 Contents

Preamble 3

1 Basic elements of fluid dynamics 4 1.1 Kinematics of fluids...... 4 1.2 Lagrangian trajectories and flow of a vector field...... 5 1.3 Deformation tensor and vorticity...... 14 1.4 Advective derivative and Reynolds transport theorem...... 17 1.5 Dynamics of fluids...... 20 1.6 Relation to kinetic theory and closure...... 24 1.7 Incompressible flows ...... 32 1.8 Equations of state, isentropic flows and vorticity...... 34 1.9 Effects of Euler-type nonlinearities...... 35

2 Basic elements of classical electrodynamics 39 2.1 Maxwell’s equations...... 39 2.2 Lorentz force and motion of an electrically charged particle...... 52 2.3 Basic mathematical results for electrodynamics...... 57

3 From multi-fluid models to magnetohydrodynamics 69 3.1 A model for multiple electrically charged fluids...... 69 3.2 Quasi-neutral limit...... 74 3.3 From multi-fluid to a single-fluid model...... 82 3.4 The Ohm’s law for an electron-ion plasma...... 86 3.5 The equations of magnetohydrodynamics...... 90

4 Conservation laws in magnetohydrodynamics 95 4.1 Global conservation laws in resistive MHD...... 95 4.2 Global conservation laws in ideal MHD...... 98 4.3 Frozen-in law...... 100 4.4 Flux conservation...... 105 4.5 Topology of the magnetic ﬁeld...... 109 4.6 Analogy with the vorticity of isentropic ﬂows...... 117

5 Basic processes in magnetohydrodynamics 119 5.1 Linear MHD waves...... 119 5.2 Nonlinear shear Alfvénwaves...... 128 5.3 Magnetic field diffusion...... 129

A Energy conservation in extended MHD models 137

B Magnetic vector potential in MHD 141

References 145

2 Preamble

Magnetohydrodynamics is the theory of electrically conducting, neutral fluids in the low-frequency regime. The basic equations of magnetohydrodynamics (MHD) have been proposed by Hannes Alfvén[1, 2], who realized the importance of the electric currents carried by a plasma and the magnetic field they generate. Alfvéncombined the equations of fluid dynamics with Faraday’s and Ampère’s laws of electrodynamics, thus obtaining a novel mathematical theory, which helped understanding space plasmas in Earth and planetary magnetospheres, as well as the physics of the Sun, solar wind, and stellar atmospheres. In fusion research, MHD is crucial to the understanding of plasma equilibria and their stability. Liquid metals and electrolytes, like salt water, can also be modeled by MHD equations. Besides the important physical applications, MHD equations exhibit a re- markably beautiful mathematical structure, with connections to geometry and topology that allows us to understand some of the dynamics of magnetic fields in plasmas in terms of topological ideas [3, 4, 5, 6, 7, 8, 9]. As a dynamical system MHD is an example of infinite-dimensional Hamiltonian system [10, 11]. The scope of this lecture is, in this regard, extremely limited. The goal is to introduce MHD equations in a reasonably self-contained way and to discuss some of their most important features. The style of this lectures is quite similar to the mathematical introduction to fluid dynamics by Chorin and Marsden [12] as the title of this note suggests. Particularly, we shall attempt to introduce the physics modeling in a mathematically precise albeit not always rigorous way. The physics literature on the subject is vast. As a reference for further read- ing, the book by Biskamp [13] provides a clear and comprehensive exposition, while the lectures by Schnack [14] offer a more gradual learning curve. For MHD of the solar atmosphere one can refer to Priest [15] as well as to Aschwanden’s book on the solar corona [16]. For an introduction to magnetohydrodynamics with emphasis on equilibria and stability of fusion plasmas one can refer to the books by Freidberg [17] and Zohm [18], while Goedbloed, Poedts and Keppens address applications to both astrophysical and fusion plasmas [19, 20]. A nice introduction to MHD with a broader perspective which includes applications to metals can be found in Davidson’s book [21]. On the mathematical side, MHD has received considerable attention from applied mathematicians. Its rich mathematical structure has become a paradigm for the application of geometry and topology [9, 22] as well as for structure preserving discretization [23, 24, 25, 26]. As a system of partial differential equations, well-posedness of the Cauchy problem for MHD equations subject to appropriate boundary conditions have been studied first by Duvaut and Lions [27, in French], Sermange and Temam [28], Secchi [29] and more recently, by Chen and coworkers [30] and by Fefferman and coworkers [31, and references therein].

3 1 Basic elements of ﬂuid dynamics

The basic understanding of fluid dynamics is an essential prerequisite to the study of MHD. We shall start by recalling the basic elements thereof, following Chorin and Marsden [12], cf. also Marsden and Hughes [32]. First, we define the physical quantities that describe the dynamical state of a fluid (kinematics) and continue with the equations of motion (dynamics).

1.1 Kinematics of fluids. Fluid dynamics is built on the basis of the continuum hypothesis: A fluid is a distribution of matter occupying a certain region of the continuous three-dimensional space. The considered region of space is a domain (i.e., an open and connected subset) Ω ⊆ R3. We neglect the fact that any fluid is ultimately made of atoms and molecules as we are interested in studying its collective motion on a much larger spatial scale. With the continuum hypothesis, one needs to quantify how matter is dis- tributed in Ω at any time t in a certain time interval I ⊆ R. Thus, the first physical quantity of interest is the mass density, which is a non-negative time- dependent scalar field ρ : I × Ω → R, such that

ρ(t, x) ≥ 0 (1.1) gives the mass per unit of volume at time t ∈ I at the spatial location x ∈ Ω. By definition, the amount of mass contained in an arbitrary volume W ⊆ Ω (to be referred to as a control volume) is given by Z (mass in W at time t) = ρ(t, x)dx, W which implies that the mass density must be at least locally integrable. Physically we think of an infinitesimal volume of fluid centered around a point x ∈ Ω. The volume of this infinitesimal region is mathematically represented by the Lebesgue measure dx in Ω and the mass is represented by the measure ρ(t, x)dx. Such infinitesimal portions of fluid are referred to as fluid elements. In plasma physics the mass density is often replaced by an equivalent non- negative scalar field referred to as the particle number density or simply number density which is defined in terms of the mass density by

n(t, x) = ρ(t, x)/m.

Here the fluid is regarded as a collection of particles that all have the same mass m; thus, the number of particles contained in a control volume W is given by 1 Z Z (number of particles in W at time t) = ρ(t, x)dx = n(t, x)dx. m W W According to this definition, the number of particles does not need to be an integer, due to the continuum hypothesis. Next we need to describe the motion of the fluid. We introduce a velocity field defined as a time-dependent vector field, u : I × Ω → R3 such that,

u(t, x) ∈ R3 (1.2)

4 gives the velocity of the fluid element at the point x ∈ Ω and time t ∈ I. The vector u(t, x) is referred to as the fluid velocity. The mass times the velocity of the fluid element, namely, ρ(t, x)u(t, x)dx, gives the linear momentum of the fluid element, hence Z (momentum in W at time t) = ρ(t, x)u(t, x)dx. W In addition to the quantities ρ(t, x) and u(t, x), that can be regarded as the counterpart in fluid dynamics of mass and velocity of a particle in mechanics, we need to specify another scalar field for the internal energy of the fluid element. Differently from a point-mass particle, a fluid element is a thermodynamical system that can undergo expansions and compressions, thus absorbing and releasing energy. The thermodynamic status of a fluid element is specified by the internal energy density U : I × Ω → R≥0, where R≥0 denotes the set of non-negative real numbers. If each fluid element is regarded as an ideal gas composed by n(t, x)dx particles, we can equivalently express the internal energy density in terms of a new variable. Specifically the laws of thermodynamics for a perfect gas allow us to write the internal energy of a fluid element as 3 U(t, x)dx = n(t, x)k T (t, x)dx, 2 B where kB is the Boltzmann constant and T : I × Ω → R is a strictly positive scalar field, such that T (t, x) > 0 (1.3) represents the local temperature of the fluid element at time t ∈ I and position x ∈ Ω. Therefore, the total energy carried by a fluid element is the sum of the kinetic energy associated to its motion plus the internal energy associated to its thermodynamics, namely, Z 1 2 3 (energy in W at time t) = ρ(t, x)u(t, x) + n(t, x)kBT (t, x) dx. W 2 2 When the law of a perfect gas does not apply we can still define T as above, but now it has the meaning of an effective temperature, which is just a measure of the internal energy and does not imply thermodynamical equilibrium. Summarizing, we shall describe the dynamical state of a fluid by the triple of functions ρ, u, T , where • the mass density ρ is a non-negative scalar field, • the fluid velocity u is a vector field, and • the temperature T is a positive scalar field. The equations of fluid dynamics are a system of partial differential equations governing the time evolution of (ρ, u, T ).

1.2 Lagrangian trajectories and flow of a vector field. Under appropriate hypotheses, we can associate to any velocity field u : I × Ω → R3 a one-parameter family of maps Ft :Ω → Ω, where time t, varying in a possibly smaller interval Iε ⊆ I, is the parameter. Such a family of maps is referred to

5 as the flow of the vector field. It gives an equivalent description of the motion of the fluid, i.e., the vector field u and its flow Ft contain the same information on the fluid motion. In this section we shall define the flow and prove some of its basic properties. Although it is often overlooked in the physics literature, the flow is a key concept in the mathematical theory of fluid dynamics and thus of MHD. Let us start from a given velocity field u : I × Ω → R3. The associated flow Ft is constructed from the solution of the Cauchy problem dx(t) = ut, x(t), x(t ) = x . (1.4) dt 0 0 Physically, the solution t 7→ x(t) represents the trajectory of a fluid element as it moves with the fluid velocity from the initial position x0 ∈ Ω at time t = t0 ∈ I. Such curves are referred to as Lagrangian trajectories. Basic results from the theory of ordinary differential equations (ODE) guarantee the existence and uniqueness of the solution of the Cauchy problem (1.4) at least for a short time. A compact account of results on ODEs can be found, for instance, in Marsden et al. [33] as part of the theory of vector fields. The first chapter of both Hörmander’s[34] and Tao’s [35] lectures on nonlinear partial differential equations gives a very nice and compact overview of the theory. Specifically we have the following standard result that we recall without proof. Let us fix constants τ, ρ ≥ 0 such that the interval Iτ = [t0 − τ, t0 + τ] and 3 the ball Bρ(x0) = {x ∈ R | |x−x0| ≤ ρ} are contained in I and Ω, respectively, and let u be continuous and V = sup |u(t, x)| be the maximum velocity in the restricted domain Iτ × Bρ(x0). Theorem 1.1 (Local existence and uniqueness for ODEs). If u is continuous on Iτ ×Bρ(x0), where we have |u(t, x)| ≤ V , and satisfies the Lipschitz condition

|u(t, x) − u(t, y)| ≤ L|x − y|, with constant L ≥ 0 uniformly in time, then for any positive ε ≤ min{τ, ρ/V } 1 there exists a solution x ∈ C ([t0−ε, t0+ε]) of the Cauchy problem (1.4) and any 1 other solution x˜ ∈ C ([t0 −ε,˜ t0 +ε ˜]) must satisfy x(t) =x ˜(t) on the intersection of the domains. We can see that the upper limit of the domain of definition is determined by the minimum time ρ/V needed to traverse the ball Bρ(x0). The interval [t0 − ε, t0 + ε] is referred to as the lifespan of the solution. This has a physical significance: The lifespan of the solution is determined by how fast the trajectory can travel up to the boundary of the considered ball. In general the maximum lifespan depends on the initial condition x0. For instance, if the initial condition is very close to the boundary of Ω, ρ and thus ε can be rather small. We can at most refine a bit this result and make the lifespan of the solution uniform for all initial conditions in a small neighborhood of x0. This can be established as a corollary of the basic existence result applied to a smaller ball centered on x0: For all initial conditions y0 ∈ Bρ/2(x0), theorem 1.1 with x0 and ρ replaced by y0 and ρ/2, respectively, gives a solution of the Cauchy problem with initial condition x(t0) = y0; then such a solution is contained in Bρ(x0) and the lifespan is ≤ min{τ, ρ/(2V )} for all y0 ∈ Bρ/2(x0). Hence,

6 Corollary 1.2. Let u, t0, x0, ρ, τ, and V be as in theorem 1.1. Then there exists a neighborhood U ⊂ Bρ(x0) and 0 < ε ≤ min{τ, ρ/(2V )} such that for every 1 y0 ∈ U the Cauchy problem (1.4) has a solution x ∈ C ([t0 − ε, t0 + ε]). We shall however work under the assumption that the lifespan of Lagrangian trajectories is uniform on the whole domain Ω, i.e., we assume that there is an ε > 0 depending only on t0, such that for every initial condition x0 ∈ Ω there is a Lagrangian trajectory x : Iε → Ω with Iε = [t0 − ε, t0 + ε]). For a generic ordinary differential equation, this is a very strong assumption. For our problem, however, this is not so strong because, in practice, it just means that the domain Ω and the boundary conditions for the vector field u have been chosen properly, in the sense that “Ω contains the fluid”. We shall fix the initial time to be t0 = 0 and let Iε = (−ε, ε) the interval of existence of the Lagrangian trajectories. At this point we are ready to define the flow of the velocity field.

Deﬁnition 1.1 (Flow). For every t ∈ Iε the map Ft :Ω → Ω is deﬁned by

x0 7→ x(t) = Ft(x0), where x(t) is the Lagrangian trajectory passing through x0 at the time t = 0. In addition this defines a map F : Iε × Ω → Ω given by F (t, x) = Ft(x). The uniqueness of the solution of the Cauchy problem for Lagrangian trajectories is essential in the definition of Ft. In fact, for Ft to be unambiguously defined we need that Ft(x) 6= Ft(y) implies x 6= y; it is not admissible that the same point is mapped into two different points. The reader can check that definition 1.1 is well posed in this sense because of the uniqueness of Lagrangian trajectories. From a physical point of view the flow describes the displacement of the fluid as time advances, i.e., given a control volume W ⊆ Ω, then Ft(W ) ⊆ Ω is the volume occupied by the fluid initially in W after it has evolved for a time t. In summary, we have constructed a one-parameter family of maps Ft which can be used in two different ways, namely,

• t 7→ Ft(x) is the Lagrangian trajectory passing through x at time t = 0;

• x 7→ Ft(x) is the displacement of the point x after a time t. As a consequence of the definition, the flow satisfies, cf. equation (1.4)

 d  F (x ) = ut, F (x ), dt t 0 t 0 (1.5)  F0(x0) = x0, where the initial point x0 ∈ Ω is regarded as a parameter, so that we write a total derivative instead of a partial derivative and consider this an ordinary differential equation rather than a partial differential equation. We shall now establish a few key properties of the flow Ft that essentially descend from equation (1.5) and conclude this section with two examples.

Proposition 1.3 (Semi-group property). For every t, s ∈ Iε such that t+s ∈ Iε we have Ft+s = Ft ◦ Fs = Fs ◦ Ft.

7 Proof. For every x0 ∈ Ω, let us consider the Lagrangian trajectory x(t) corresponding to the initial condition x(0) = x0. By deﬁnition, Ft+s(x0) = x(t + s). 0 0 0 0 0 0 Let us deﬁne the functions t 7→ cs(t ) = x(t + s) and t 7→ ct(t ) = x(t + t). We observe that cs solves the Cauchy problem dc s = u(t0, c ), c (0) = x(s) = F (x ), dt0 s s s 0

0 hence, cs(t ) = Ft0 Fs(x0) = Ft0 ◦ Fs(x0). Then, Ft+s(x0) = x(t + s) = cs(t) = Ft ◦ Fs(x0). The second identity is obtained by applying the same argument to 0 the curve ct(t ). This property is usually referred to as the semi-group property of the ﬂow and it has an immediate consequence.

Corollary 1.4. For every t ∈ Iε, Ft :Ω → Ω is invertible and the inverse is −1 given by Ft = F−t.

Proof. We have F0(x) = x for all x ∈ Ω and proposition 1.3 with s = −t gives x = F0(x) = Ft ◦ F−t(x) = F−t ◦ Ft(x) which shows that F−t is both a left- and a right-inverse of Ft.

Another basic result from ODE theory implies that Ft :Ω → Ω is Lipschitz continuous in Ω for all t ∈ Iε.

Proposition 1.5. If u is Lipschitz continuous uniformly in time and Ft :Ω → Ω is deﬁned on Ω for all t ∈ Iε, then

L|t| Ft(x) − Ft(y) ≤ |x − y|e , for all x, y ∈ Ω and t ∈ Iε. Here L is the Lipschitz constant of u. Proof. Let us ﬁrst consider the half-interval t ≥ 0. For every x, y ∈ Ω ﬁxed, let h(t) = Ft(x) − Ft(y) and, by the Lipschitz condition for u,

dh(t) = u t, Ft(x) − u t, Ft(y) ≤ L|h(t)|. dt We actually need to control the derivative of the norm, rather then the norm of the derivative. With this aim we can estimate

1 d 2 dh(t) dh(t) 2 h(t) = h(t) · ≤ h(t) · ≤ Lh(t) , 2 dt dt dt and for t ≥ 0, d h d i h(t)2e−2Lt = h(t)2 − 2Lh(t)2 e−2Lt ≤ 0, dt dt hence h(t)2e−2Lt ≤ h(0)2 which is equivalent to the claim for t ≥ 0. As for the other half-interval t ≤ 0, let s = −t ≥ 0 and h(s) = F−s(x) − F−s(y) and we notice that dh(s) = −u − s, F (x) + u − s, F (y), ds −s −s and repeat the argument, integrating in the variable s.

8 We now know that Ft is a continuous transformation of Ω into itself for all t ∈ Iε. We also shall need to understand when Ft is diﬀerentiable and in those cases have a convenient way to compute its Jacobian matrix and determinant, namely,

T DFt(x0) = ∇x0 Ft(x0) ,Jt(x0) = det DFt(x0) , where AT denotes the transpose of a tensor A. In this note, the gradient ∇v(t, x) of a generic vector field v(t, x) is defined according to standard dyadic vector calculus, which differs from the definition adopted by Chorin and Marsden [12]. The Jacobian matrix is then denoted by Dv and it is the transpose of the gradient, namely, ∂vj ∇v(t, x) ij = = Dv(t, x) ji. ∂xi We derive an evolution equation for the Jacobian matrix of the flow.

1 Proposition 1.6. If the velocity field u is of class C and Ft :Ω → Ω is defined 1 1 on Ω for all t ∈ Iε, then Ft ∈ C (Ω), the map t 7→ DFt(x) is in C , and satisfies the Cauchy problem,  d  DF (x ) = Dut, F (x )DF (x ), dt t 0 t 0 t 0 (1.6)  DF0(x0) = I, with I being the identity matrix. By induction, if u ∈ Ck for k ≥ 1 then k k+1 Ft ∈ C (Ω) and t 7→ Ft(x) is C .

2 If we can say that the function F (t, x) = Ft(x) is of class C , equation (1.6) is a direct consequence of the chain rule. In fact, the time derivative and the gradient of the ﬂow commute and ∂t∇x0 Ft(x0) = ∇x0 ∂tFt(x0) = ∇x0 u t, Ft(x0) = ∇x0 Ft(x0) · ∇u t, Ft(x0) = ∇Ft(x0) · ∇u t, Ft(x0) .

By transposing this identity and considering the initial condition ∇F0 = I, which follows from F0(x0) = x0 in equation (1.5), we obtain equation (1.6). Without assuming further regularity, the proof require some more work. The full argument can be found in Marsden et al. [33], with the only diﬀerence that here we are assuming that Ft is deﬁned uniformly on the whole domain Ω and not just locally. The evolution equation for the determinant follows from proposition 1.6. Proposition 1.7. Under the same hypotheses of proposition 1.6, we have  d  J (x ) = ∇ · ut, X(t, x )J (x ), dt t 0 0 t 0 (1.7)  J0(x0) = 1.

This result is a special case of the Liouville’s formula, which can be proven from the properties of the determinant.

9 Lemma 1.8 (Liouville’s formula). Let A, ψ be functions from an interval I ⊆ R with values in the space Rn×n of n × n matrices, such that ψ ∈ C1 and dψ/dt = A(t)ψ(t). Then d det ψ(t)/dt = tr A(t) det ψ(t), where tr(A) is the trace of the matrix A. Proof. See, for instance, proposition 1.2.4 in H¨ormander’s lectures [34]. Here however we give a proof which relies on the following basic identity from vector calculus. This establishes a relationships between the volume of a parallelepiped spanned by three vectors, expressed by the scalar triple product of the vectors, and the determinant of the matrix deﬁned by the vectors. Lemma 1.9. Let A be a 3-by-3 matrix and we write it as a column of row T vectors, i.e., A = (A1,A2,A3) with Ai = (aij)j. Then

det(A) = A1 · (A2 × A3) = A2 · (A3 × A1) = A3 · (A1 × A2). Proof. By deﬁnition, the determinant is X det(A) = sign(σ)a1σ(1)a2σ(2)a3σ(3),

σ∈S3 where the sum runs over the set S3 of permutations of three elements (1, 2, 3). We can write this in terms of the completely anti-symmetric symbol (Levi-Civita symbol)  1 (i, j, k) is an even permutation of (1, 2, 3),  ijk = 0 (i, j, k) is not a permutation of (1, 2, 3), −1 (i, j, k) is an odd permutation of (1, 2, 3), namely, X X det(A) = sign(σ)a1σ(1)a2σ(2)a3σ(3) = ijka1ia2ja3k,

σ∈S3 ijk and the right-hand side is just the scalar triple product A1 · (A2 × A3). The other two identities follow on noting that the triple product is invariant under cyclic permutations. It is worth noting that the identity of lemma 1.9 holds in any dimension, but then the cross product has to be replaced by an appropriate anti-symmetric multi-linear operation. The proof of proposition 1.7 now follows by direct calculation.

k k+1 Proof of proposition 1.7. By proposition 1.6, Ft is C in space and C in time. By lemma 1.9, det ∇x0 Ft = ∇x0 X1 · ∇x0 X2 × ∇x0 X3 , where Ft(x0) = X1(t, x0),X2(t, x0),X3(t, x0) and Xi(t, x0) are the Cartesian coordinates of the position vector of the Lagrangian trajectory. We note that ∇Xi are the rows of the Jacobian matrix DFt, hence, from equation (1.6),

∂t∇x0 Xi = ∇x0 X · ∇ui,

10 where u = (u1, u2, u3) is the ﬂuid velocity in Cartesian components, and ∇ui is evaluated at t, Ft(x0) . Then, we compute ∂t det ∇x0 Ft = (∂t∇x0 X1) · ∇x0 X2 × ∇x0 X3 + ∇x0 X1 · (∂t∇x0 X2) × ∇x0 X3 + ∇x0 X1 · ∇x0 X2 × (∂t∇x0 X3) = (∇x0 X · ∇u1) · ∇x0 X2 × ∇x0 X3 + ∇x0 X1 · (∇x0 X · ∇u2) × ∇x0 X3 + ∇x0 X1 · ∇x0 X2 × (∇x0 X · ∇u3) 3 X h i∂u1 = ∇ X · ∇ X × ∇ X x0 i x0 2 x0 3 ∂x i=1 i 3 X h i∂u2 + ∇ X · ∇ X × ∇ X x0 1 x0 i x0 3 ∂x i=1 i 3 X h i∂u3 + ∇ X · ∇ X × ∇ X . x0 1 x0 2 x0 i ∂x i=1 i Since the scalar triple product A · [B × C] vanishes if any pair of its factors are equal, in the ﬁrst sum the only contribution comes from i = 1, in the second sum from i = 2, and in third sum from i = 3. We therefore have

h∂u1 ∂u2 ∂u3 i ∂t det ∇x0 Ft = det ∇x0 Ft + + . ∂x1 ∂x2 ∂x3 In addition we have det ∇x0 Ft(x0) = 1 since ∇x0 F0(x0) = I, so that we can write the initial value problem for the Jacobian determinant.

This concludes our overview of basic properties of the flow Ft. Let us now address a few examples. First we consider the stationary velocity field   x1 u(t, x) = ν −x2 , (1.8) 0 in coordinates x = (x1, x2, x3), ν > 0 being a constant with the dimensions of a frequency. This field is essentially two-dimensional because it is uniform in the third coordinate x3 and the corresponding component u3 is zero. The flow of this field can be computed by solving the ordinary differential equation (1.5), which in this case reads (only the two non-trivial components) ( ∂tX1 = νX1,X1(0, x0) = x0,1,

∂tX2 = −νX2,X2(0, x0) = x0,2, where the initial point is x0 = (x0,1, x0,2, x0,3). Since the system in uniform in x3, the planes x3 = constant are invariant, that is, X3 = x0,3, and the solution for the ﬂow is readily found in the form

 νt  e x0,1 −νt Ft(x0) = e x0,2 . x0,3

11 Figure 1.1: Field lines of the velocity field (1.8) and evolution with the flow Ft of a sample of points arranged in the shape of a square. Blue points are the positions x0 at the time t = 0, while red points are their evolution x = Ft(x0) at time t = 1, with ν = 0.5. One can observe that the square shape is deformed but not rotated by the flow.

We see that x1(t)x2(t) = x0,1x0,2, i.e., the trajectory of the flow (Lagrangian trajectories) are hyperbolas on the plane (x1, x2). Figure 1.1 shows the field lines of the velocity field and the evolution of a sample of points (arranged in the shape of a square) according to the flow map Ft. The Jacobian matrix of the vector field (1.8) is ν 0 0 T Du = (∇u) = 0 −ν 0 , (1.9) 0 0 0 so that the Jacobian matrix of the flow satisfies d (DF ) = ν(DF ) , dt t 1j t 1j d (DF ) = −ν(DF ) , j = 1, 2, 3, dt t 2j t 2j d (DF ) = 0, dt t 3j and, upon accounting for the initial condition DF0 = I, eνt 0 0 −νt DFt =  0 e 0 , 0 0 1 as can be seen directly from the flow. At last, the vector field is divergence free, i.e., ∇ · u(t, x) = 0, so that Jt = J0 = 1, cf. equation (1.7); this can be deduced by inspection of the matrix DFt. Another example, with quite different properties, is given by the flow   x2 u(t, x) = ν −x1 . (1.10) 0

12 Figure 1.2: The same as in figure 1.1, but for the velocity field (1.10). In this case, ν = 1.2π and the final time is t = 1. The initial control volume is rotated, but not stretched.

The ordinary diﬀerential equation for the ﬂow in this case reads ( ∂tX1 = νX2,X1(0, x0) = x0,1,

∂tX2 = −νX1,X2(0, x0) = x0,2, and the x3 = constant planes are again invariant. The solution is now oscillatory, with frequency given by the constant ν > 0,   x0,1 cos(νt) + x0,2 sin(νt) Ft(x0) = −x0,1 sin(νt) + x0,2 cos(νt) . x0,3 Thus Lagrangian trajectories are circles, cf. figure 1.2. The Jacobian matrix of the velocity field is  0 ν 0 T Du = (∇u) = −ν 0 0 , (1.11) 0 0 0 and it is anti-symmetric. Correspondingly, the equation for the Jacobian matrix of the flow amounts to d (DF ) = ν(DF ) , dt t 1j t 2j d (DF ) = −ν(DF ) , j = 1, 2, 3, dt t 2j t 1j d (DF ) = 0, dt t 3j and the solution is the matrix for a clock-wise rotation of an angle νt, namely,  cos(νt) sin(νt) 0 DFt = − sin(νt) cos(νt) 0 , 0 0 1 which could have been computed directly from the flow. As for the previous example, the field is divergence free, ∇ · u = 0, therefore the corresponding flow has unit Jacobian Jt = J0 = 1.

13 It is worth noting that, in both the considered examples, the Lagrangian trajectories coincides with the field lines of the velocity field, since u is independent of time (stationary flow). In addition, they are both examples of two-dimensional potential flows, namely, there exists a scalar field φ(x1, x2), to be referred to as streaming potential, such that

u1 = ∂φ/∂x2, u2 = −∂φ/∂x1.

In the case of equation (1.8), one has φ(x1, x2) = νx1x2, while for the case of 2 2 equation (1.10) the potential is φ(x1, x2) = ν(x1 + x2)/2. One should notice the analogy with Hamilton’s equations on a two-dimensional phase space, φ playing the role of the Hamiltonian function. For generic potential ﬂows, the contours of the potential φ are invariant for the ﬂow, i.e.,

u · ∇φ = ∂x2 φ∂x1 φ − ∂x1 φ∂x2 φ = 0. Geometrically, one can observe that the field u is just the gradient of φ rotated by π/2 clock-wise; this, in particular, implies that u is everywhere tangent to the contours of φ, and thus that the contours of φ coincide with the field lines, which is turn coincide with the Lagrangian trajectories, in the same way as, in two-dimensional Hamiltonian systems, the the contours of the Hamiltonian function coincide with the trajectories. Although the definition of the flow Ft might appear a mere mathematical abstraction, its importance in fluid dynamics cannot be stressed enough. In modern (geometrical) approaches to fluid dynamics, Ft is the main variable specifying the state of a fluid [9, 36]. In the next two sections, we consider two applications of these ideas, that lead to the definition of deformation tensor and vorticity on one hand, and to the Reynolds transport theorem on the other hand.

1.3 Deformation tensor and vorticity. At time t = 0, let us consider 0 two points x0, x0 ∈ Ω and follow them as they evolve with the fluid motion. From the result of the last section, we can write the trajectories of those two point as 0 0 x(t) = Ft(x0), x (t) = Ft(x0). We are interested in studying the evolution of the difference vector, cf. figure 1.3, h(t) = x0(t) − x(t),

0 as the ﬂuid evolves. We shall assume that the initial points x0, x0 are very close to each other so that |h(0)| is small, and consider a time interval so short that |h(t)| can still be considered small; we make no attempt to be mathematically more precise here. In view of equation (1.5) and Taylor expansion we have d d d h(t) = F (x0 ) − F (x ) dt dt t 0 dt t 0 0 = u t, Ft(x0) − u t, Ft(x0) = ut, x0(t) − ut, x(t) = ut, h(t) + x(t) − ut, x(t) = h(t) · ∇ut, x(t) + O(|h(t)|2).

14 Ft(x0)

x0 h(t)

0 x0 0 Ft(x0)

0 Figure 1.3: Evolution of two nearby points x0, x0 with the ﬂuid motion (La- grangian trajectories), and deﬁnition of the vector h(t).

We see that, at least for a short time, the evolution of h(t) is dominated by the gradient of the ﬂuid velocity ∇u. That amounts to a 3-by-3 matrix which we can split into its symmetric and anti-symmetric parts, namely, 1 1 ∇u = ∇u + (∇u)T + ∇u − (∇u)T . 2 2 The symmetric part is referred to as deformation tensor, 1 D = ∇u + (∇u)T . (1.12) 2 The anti-symmetric part is deﬁned by 1 S = − ∇u − (∇u)T , 2 and, as any anti-symmetric tensor, it can be reduced to three degrees of freedom, namely, in matrix form,

 0 −ω ω  1 3 2 S = ω 0 −ω . 2  3 1 −ω2 ω1 0 The scalars ωi can be arranged to form the vector field ω = ω1, ω2, ω3 . In order to understand the relationship between the vector field ω and the fluid velocity u, let v ∈ R3 be an arbitrary vector and consider the identity 1 1 Sv = v · ∇u − ∇u · v = (∇ × u) × v. 2 2

On the other hand, we have v = (v1, v2, v3) and, by matrix-vector multiplication,

 ω v − ω v  1 2 3 3 2 1 Sv = ω v − ω v = ω × v. 2  3 1 1 3  2 −ω2v1 + ω1v2

By comparison of the two expressions for Sv, we deduce that ω must be the curl of the fluid velocity, namely, ω = ∇ × u. (1.13) The vector field ω thus obtained is referred to as vorticity of the fluid.

15 At last the evolution of the vector h(t) reads d 1 h(t) = Dt, x(t)h(t) + ωt, x(t) × h(t) + O(|h(t)|2). dt 2 We can now study separately the effects of the symmetric and anti-symmetric terms. Of course the full dynamics is the result of the combination of the two. Let us start with the deformation tensor D(t, x) for which we have to consider the linear symmetric equation d h(t) = Dt, x(t)h(t). dt For sake of simplicity (we are only interested in qualitative ideas) let us assume that the deformation tensor does not vary too much along the Lagrangian trajectory, i.e., we set it to a constant, D(t, x) = D(0, x0) = D0. Since D0 is by definition symmetric, we can find a set of three orthonormal eigenvector ei, i.e.,

D0ei = λiei, and the eigenvalues λi are real. The set of eigenvectors ei constitutes a bases for vectors in R3, hence we can write

3 X h(t) = ci(t)ei, i=1 and the coefficients ci(t) of the expansion satisfy the scalar ordinary differential equation dci = λ c , c (t) = c (0)eλit, dt i i i i where ci(0) are the coefficient of the expansion of the initial vector h(0). From the full solution 3 X λit h(t) = e ci(0)ei, i=1 we can deduce the effect of D0 on the fluid motion: The fluid element is stretched exponentially along the directions of the eigenvalues of D0, but such directions are invariant (no rotation happens). Here stretching can be either expansion (λi > 0) or compression (λi < 0). The contribution of the vorticity on the other hand can be understood by considering the equation d 1 h(t) = ωt, x(t) × h(t). dt 2 Again we set ω t, x(t) = ω(0, x0) = ω0 and we recognize that this generates a 1 rigid rotation of h about the direction of ω0 with angular frequency 2 |ω0|. In order to see that, we can assume (without loss of generality) that the vorticity is directed along the x1-axis of a Cartesian coordinate system, i.e., ω0 = (|ω0|, 0, 0). Then, the equation of motion for h(t) = h1(t), h2(t), h3(t) becomes

dh1(t)/dt = 0, 1 dh2(t)/dt = − 2 |ω0|h3(t), 1 dh3(t)/dt = 2 |ω0|h2(t),

16 from which we see that the component of h(t) parallel to ω0 does not change, while the perpendicular projection h2(t), h3(t) rotates with angular frequency 1 2 |ω0|, as claimed. Let us consider the two examples of section 1.2. For the velocity ﬁeld (1.8), by inspection of equation (1.9) we see that ∇u = (Du)T is symmetric and thus,

D = ∇u, ω = 0,

The corresponding flow therefore should just deform the fluid element without rotating it, cf. figure 1.1. On the other hand, the velocity field (1.10) is such that ∇u is anti-symmetric, cf. equation (1.11). Hence,

D = 0, ω = (0, 0, −2ν), and we see that the vorticity is pointing along the axis of rotation of the vortex (the third axis in this case) and it is equal to twice the rotation angular frequency, cf. figure 1.2. Summarizing the results of this section, we have shown that the motion of two nearby points in the fluids amounts to the combination of two effects: exponential stretching along prescribed directions (controlled by the deformation tensor) and rigid rotation (controlled by the vorticity).

1.4 Advective derivative and Reynolds transport theorem. The concept of flow of the fluid velocity is central in the proof of the Reynolds transport theorem, by means of which the equations of motion of fluid dynamics are usually formulated. Let us start from a generic scalar function f ∈ C1(I ×Ω, R) defined for t ∈ I and x ∈ Ω; that can represent any scalar physical quantity as the mass density, the temperature or any function thereof. We evaluate the function f along a Lagrangian trajectory, x(t) = Ft(x0). The time derivative of f restricted to the Lagrangian trajectory then reads

d dx(t) ft, x(t) = ∂ ft, x(t) + · ∇ft, x(t). dt t dt Since by deﬁnition dx(t)/dt = u(t, x(t), cf. equation (1.5), we can write

d Df ft, x(t) = t, x(t), dt Dt where we have deﬁned the advective derivative as the operator Df (t, x) = ∂ f(t, x) + u(t, x) · ∇f(t, x). (1.14) Dt t The advective derivative (also referred to as convective derivative, or material derivative) gives the rate of variation of a function along the Lagrangian trajectory passing through the space-time point (t, x). Incidentally, it is worth noting that, if the advective derivative of a function f is identically zero on Ω for all time t ∈ I, then the value of f along any

17 Ft

Figure 1.4: The control volume W0 is dragged along by the ﬂuid motion, thus evolving in time Wt = Ft(W0).

Lagrangian trajectory is constant. A function f(t, x) that has this property satisﬁes the partial diﬀerential equation,

∂tf(t, x) + u(t, x) · ∇f(t, x) = 0, which is referred to as linear advection equation. Vice versa we can use the flow of the velocity field u(t, x) to construct a solution of the initial-value problem for the advection equation. In fact, if f(t, x) is constant on any Lagrangian trajectory, we have f t, Ft(x0) = f0(x0) where f0 is the initial condition at the time −1 t = 0 and x0 ∈ Ω; inverting the flow we have the solution f(t, x) = f0 Ft (x) . This is a particular case of a much more general method to solve initial-value problems for first-order linear and non-linear partial differential equations known as the method of characteristic curves [37]. In this case the Lagrangian trajectories coincide with the characteristic curves of the linear advection equation. Having defined the advective derivative, we can now give a proof of a central result in fluid dynamics due to Reynolds. Let us consider an arbitrary volume Wt which moves along with the fluid. The fact that Wt moves with the fluid can be expressed mathematically in terms of the flow map Ft, namely, Wt = Ft(W0), where W0 is the configuration of the volume at time t = 0, and the application of the map Ft to the set W0 is defined pointwise, i.e., Wt is the set obtained by applying Ft to each point of W0, cf. figure 1.4. For the flows (1.8) and (1.10), figures 1.1 and 1.2 show examples of a control volume W0 (sampled by blue points) which evolves into Wt (sampled by red points). We shall choose W0 compact in Ω and since Ft is continuous, Wt will be compact at any time t for which the flow is defined, namely in the interval Iε, 1 cf. section 1.2. For an arbitrary function f ∈ C (I ×Ω), the restriction to Wt at any given time is bounded hence integrable on Wt and we consider the integral Z Q(t) = f(t, x)dx. Wt We are interested in computing the time derivative of Q. The difficulty here is that both the integrand and the domain of integration depend on time.

18 Theorem 1.10 (Reynolds transport theorem, [32]). Let u ∈ C1(I × Ω) be a velocity field with flow Ft defined on the whole domain Ω for t ∈ Iε ⊆ I. Then 1 1 for all f ∈ C (I × Ω, R), and W0 ⊂ Ω compact we have Q ∈ C (Iε) and Z Z Z d hDf i f(t, x)dx = + f∇ · u dx = ∂tf + ∇ · fu) dx, (1.15) dt Wt Wt Dt Wt where Wt = Ft(W0). 0 Proof. The idea behind this calculation is that we can use the map x = Ft(x ) as a coordinate transformation mapping W0 into Wt; changing coordinates in the integral we have Z Z 0 0 0 f(t, x)dx = f t, Ft(x ) Jt(t, x )dx , Wt W0 0 where Jt is the Jacobian determinant addressed in section 1.2 and dx is the volume element in the primed coordinates. The advantage is that now the domain of integration does not depend on time. In addition all the integrals are finite as both the integrand and its time derivative are the restriction to a bounded domain of continuous functions. We see that we can differentiate under the integral sign and Q ∈ C1. Since the derivative of f restricted to a Lagrangian trajectory gives the advective derivative, we compute Z Z d hDf i 0 f(t, x)dx = Jt + f∂tJt dx dt Wt W0 Dt Z hDf i 0 0 = + f∇ · u Jt(t, x )dx , W0 Dt

0 where the terms in square brackets are evaluated at t, Ft(x ) and we have used equation (1.7) in the second identity. By transforming back to Wt we have d Z Z hDf i f(t, x)dx = (t, x) + f(t, x)∇ · u(t, x) dx. dt Wt Wt Dt At last, we note that Df (t, x) + f(t, x)∇ · u(t, x) = ∂ f(t, x) + ∇ · f(t, x)u(t, x). Dt t This conclude the proof of the Reynolds transport theorem. In this argument we have considered a scalar function f for sake of simplicity; however, the advective derivative and the transport theorem can be applied component-wise to multi-component ﬁelds, such as vectors or tensors. A special case of the Reynolds transport theorem (1.15) is obtained for f ≡ 1, i.e., the function identically equal to one. Then, the integral of f amounts to the volume of W , t Z |Wt| = dx, Wt and from the advective form of (1.15) we have d Z |Wt| = ∇ · u(t, x)dx. dt Wt

19 This equation controls the expansion/compression of a volume of ﬂuid; it can be considered the “macroscopic” form of equation (1.7) for the Jacobian determinant of the ﬂow. We can also notice that, when

∇ · u(t, x) = 0, (1.16) the volume |Wt| is preserved, i.e., the fluid in incompressible and the divergence- free condition (1.16) is referred to as the incompressibility condition. Both examples of figures 1.1 and 1.2 are incompressible flows. By considering the transport of mass, linear momentum, and total energy we shall use the transport theorem to justify the equations of motion of fluid dynamics.

1.5 Dynamics of fluids. In this section we shall consider three basics physics principles, translate them into mathematical statements, and use the Reynolds transport theorem to obtain partial differential equations for the three state variables introduced in section 1.1. The considered physics principles are: • mass conservation, which implies an equation for ρ, • momentum balance (Newton second law of dynamics), which implies an equation for u, • energy balance, which defines the dynamics of the internal energy U.

Mass conservation. Let us consider the mass of fluid in a volume Wt that moves with the fluid. Since the volume Wt is “following” the fluid in its motion, the mass contained therein should be constant, that is, d Z ρ(t, x)dx = 0, dt Wt where ρ(t, x) is the mass density. Then the Reynolds transport theorem of section 1.4 gives Z ∂tρ + ∇ · ρu dx = 0. Wt

This identity must be true for an arbitrary control volume Wt ⊂ Ω, hence the integrand must vanish with the result that ∂tρ + ∇ · ρu = 0. This is the first equation of fluid dynamics expressing the conservation of the fluid mass, and it is referred to as mass continuity equation. Upon integrating over the whole domain Ω and using the Gauss theorem Z Z ∇ · ρudx = ρu · ndS, Ω ∂Ω where n is the outgoing unit normal on the boundary ∂Ω of the domain, we obtain that the total variation of the fluid mass in Ω is d Z Z ρdx = − ρu · ndS, dt Ω ∂Ω

20 that is, ρu is the mass ﬂux through the boundary. If the boundary conditions for the velocity u are appropriately chosen, e.g., either if u = 0 on the boundary (no-slip boundary conditions), or n·u = 0, then the mass of the ﬂuid is conserved.

Momentum balance. The main equation of fluid dynamics follows from the transport of linear momentum, which parallels Newton’s second law, namely, d Z ρudx = (forces acting on Wt). dt Wt The left-hand side of the equation can be treated by means of the transport theorem, but we need to identify the forces acting on the volume of fluid Wt. We have to distinguish between forces acting on the whole body of the fluid volume Wt, and those acting on its boundary ∂Wt. Forces acting on the whole volume Wt can be written as Z (body forces on Wt) = ρ(t, x)f(t, x)dx, Wt where f(t, x) is the force per unit of mass acting of the fluid element ρ(t, x)dx; one might think of such forces as the result of an external force field acting on the region occupied by the fluid, such as gravity or electromagnetic forces for the case of electrically charged or conducting fluid (this will be the case for plasmas). On the other hand, forces acting on the boundary ∂Wt are due to internal interaction among the fluid elements. The force per unit of area acting on the boundary of Wt can be shown to be a linear function of the outgoing unit normal n on ∂Wt, namely, Z (surface forces on ∂Wt) = − P · ndS, ∂Wt where P is a symmetric tensor referred to as Cauchy stress tensor (the linear relationship between surface forces and n is known as Cauchy stress theorem [32]; the proof is not reported here, but this issue will be resolved in the next section.) It is worth noting that P has the dimensions of a pressure. In fact is P is isotropic, i.e., P = pI where p is a scalar, then the expression above reduces to the familiar pressure force Z (pressure force) = − pndS, ∂Wt with pressure p; in general, the surface force might not be exactly normal and this is described by the symmetric tensor P . We can still isolate the isotropic contribution by means of the identity,

P = pI + π, (1.17)

1 where the scalar pressure p = 3 tr P is deﬁned as one third of the trace of P so that the symmetric tensor π is trace-free tr π = 0. The trace-free part of the stress tensor is referred to as viscosity tensor. By the transport theorem, the momentum balance now reads Z Z Z ∂t(ρu) + ∇ · (ρuu) dx = − pn + π · n dS + ρfdx, Wt ∂Wt Wt

21 where uu = u ⊗ u = (uiuj)ij is the tensor product in dyadic notation. The boundary terms can be dealt with by Gauss theorem Z Z pn + π · ndS = ∇p + ∇ · πdx, ∂Wt Wt and since the control volume Wt is arbitrary, we can deduce that

∂t(ρu) + ∇ · (ρuu + π) = −∇p + ρf, which expresses the balance law for the linear momentum density of the ﬂuid, in terms of viscosity, pressure, and external forces. This is referred to as the momentum balance equation or equation of motion in analogy with Newton’s second law.

Energy balance. At last we consider the energy transport, Z d h1 2 3 i ρu + nkBT dx = (rate of energy input on Wt). dt Wt 2 2

The rate at which energy is injected into the fluid contained in Wt is the sum of the work done by the forces on the fluid, plus the energy flux through the boundary as the control volume Wt is embedded in the fluid and can exchange heat with the surroundings, plus the energy produced by heat sources in the volume Wt. If q is the heat flux vector and Q the rate of production per unit of volume, we have Z Z (rate of energy input on Wt) = − u · P · ndS + ρu · fdx ∂Wt Wt Z Z − q · ndS + Qdx, ∂Wt Wt where the first two terms represent the work done by the internal and external forces respectively and the last two terms are the total heat flux through the boundary and the rate of energy produced by heat sources. Again by the Gauss theorem, we can rewrite the boundary terms so that Z (rate of energy input on Wt) = ρu · f + Q − ∇ · (P · u + q) dx. Wt The transport theorem applied to the total energy then yields,

1 2 3 1 2 3 ∂t 2 ρu + 2 nkBT + ∇ · ( 2 ρu + 2 nkBT + p)u + π · u + q = ρu · f + Q, (1.18) which is the total energy balance law. It is worth noting that the energy ﬂux (i.e., the vector within the divergence operator on the left-hand side) amounts to the advection of kinetic energy, internal energy, and pressure, plus the contribution of viscosity and heat ﬂux. We can use the continuity equation and the momentum balance equation to eliminate the kinetic energy terms in the energy transport, thus obtaining an equation for the internal energy only. This requires some calculations. First,

1 2 1 2 ∂t 2 ρu + ∇ · 2 ρu u = ρu · ∂tu + ρu · ∇u · u,

22 where the continuity equation has been accounted for. The momentum balance equation on the other hand can be written as

ρ(∂tu + u · ∇u) + ∇ · π + ∇p = ρf, where again the continuity equation has been accounted for. Upon scalar multiplying by u, the latter gives

ρu · ∂tu + ρu · ∇u · u = ρu · f − (∇ · π) · u − u · ∇p which can be used in equation (1.19) to obtain the identity

1 2 1 2 ∂t 2 ρu + ∇ · 2 ρu u = ρu · f − (∇ · π) · u − u∇p = ρu · f − ∇ · (π · u + pu) + π : ∇u + p∇ · u, where π : ∇u = tr(π · ∇u). Equivalently,

1 2 1 2 ∂t 2 ρu + ∇ · 2 ρu u + pu + π · u = ρu · f + π : ∇u + p∇ · u. (1.19) In view of this identity, the total energy equation implies

3 3 ∂t 2 nkBT + ∇ · 2 nkBT u + q + p∇ · u + π : ∇u = Q, which is the internal energy balance equation. We also notice that the work done by the force f does not contribute to the production of internal energy, but it goes into the kinetic energy only.

Summary of ﬂuid equations. The basic equations of ﬂuid dynamics amount to:

• the mass continuity equation for the mass density ρ(t, x), ∂tρ + ∇ · ρu = 0, (1.20a)

• the momentum balance for the ﬂuid velocity u(t, x),

∂t(ρu) + ∇ · (ρuu + π) = −∇p + ρf, (1.20b)

• and the internal energy balance for the temperature T (t, x),

3 3 ∂t 2 nkBT + ∇ · 2 nkBT u + q + p∇ · u + π : ∇u = Q, (1.20c) where n(t, x) = ρ(t, x)/m. One should note however that equations (1.20) are not closed as the pressure p, the viscosity tensor π, the heat flux q, and the heat source Q, as well as the forces f have not been specified yet. We need to find expressions of those quantities in terms of the basic state variables ρ, u and T : This is known as the closure problem. In order to obtain a physically accurate closure, one needs to account for the microscopic properties of the fluid.

23 1.6 Relation to kinetic theory and closure. In order to find a closure of fluid equations for both gases and plasmas, one usually relies on kinetic theory [38]. Specifically for plasmas, this leads to the transport equations derived by Braginskii [39] that constitute the standard basis for fluid and transport models in plasma physics [40, 41]. In kinetic theory a fluid is viewed as a collection of particles (atoms or molecules for gases, ions and electrons for plasmas) all of the same type, i.e., same mass m > 0 and same electric charge (if any). When particles of different species are present, e.g., ions and electrons, each particle species is treated separately. In the position-velocity phase-space (x, v) ∈ Ω×R3, the fluid is described by 3 the particle distribution function f : I × Ω × R → R≥0, where I ⊆ R is a time interval and R≥0 is the set of non-negative real numbers. Its value f(t, x, v) gives the number of particles per unit of phase-space volume that have position x ∈ Ω and velocity v ∈ R3 at time t ∈ I. The basic equation of the theory is the kinetic equation, which has the form

∂tf + v · ∇xf + a · ∇vf = C(f), where a : I × Ω × R3 → R3 is such that a(t, x, v) represents the acceleration of the particle at the time t, position x, and velocity v, while C(f) is the collision operator, which describes deviations from the motion of the individual particles due to interaction (collisions) with the other particles. The specific expressions for a and C are problem-dependent, but we can work under the following hypotheses: - The acceleration field satisfies

∇v · a = 0.

- The collision operator satisﬁes Z C(f)(t, x, v)dv = 0. R3

The condition on the acceleration field is verified in all physical systems we are interested in, i.e., fluids and plasmas. The condition on the collision operator is also automatically satisfied for the standard collision operators relevant to gases and plasmas. In fact this condition just means that collisions can change the velocity of a particle but cannot change its position and cannot destroy or create particles (differently from chemical reactions and ionization phenomena that are sometimes treated as collisions; we prefer however to distinguish such processes from collisions since they are physically different, and we do not consider them in this note.) In addition the collision operator can have additional properties such as momentum and energy conservation (elastic collisions). However, the results of this section do not rely on such additional properties. The assumption on the acceleration, in particular, implies that

∂tf + ∇x · (vf) + ∇v · (af) = C(f), (1.21) which has the same form as the continuity equation (1.20a), except that it is formulated in phase-space and it has a non-zero right-hand side. In fact the

24 kinetic equation can also be understood on the basis of the Reynolds transport theorem 1.10 in the same way the continuity equation (1.20a) has been constructed. Here the domain Ω is replaced by the phase space Ω × R3, the mass density ρ is replaced by the distribution function f, and the fluid flow is replaced by the laws of particle mechanics, i.e., dx dv = v, = a(t, x, v). dt dt This also means that the characteristic curves of the kinetic equation are the particle orbits. Differently from the continuity equation, however, the right- hand side of the kinetic equation is not zero as the particle can escape a control volume due to collisions. The total number of particle composing the fluid at the time t is given by the integral of the particle distribution on the whole phase-space, namely, Z N(t) = f(t, x, v)dxdv, Ω×R3 hence we need at least f(t, ·) ∈ L1(Ω × R3). The continuity form (1.21) of the kinetic equation, together with the assumption on the collision operator, ensures that the number of particle is constant, i.e., N(t) = N(0). Fluid quantities such as the mass density are related to the partial statistical moments of the phase-space distribution function with respect to the particle velocity, namely, Z vαf(t, x, v)dv, R3 where α = (α1, α2, α3) is a multi-index (a vector of non-negative integers) and α α1 α2 α3 v = v1 v2 v3 . Specifically, the integral of f alone (α = 0) gives the number of particles per unit of volume independently of their velocity, hence the mass density is Z ρ(t, x) = mn(t, x) = m f(t, x, v)dv. R3

Then, if we integrate the kinetic equation (1.21) in velocity and multiply by the mass m we obtain, Z ∂tρ + ∇ · mvfdv = 0. (1.22) R3 which allows us to identify the mass flux by comparison with the mass continuity equation (1.20a), namely, Z ρ(t, x)u(t, x) = m vf(t, x, v)dv. R3 The Cauchy stress tensor and the macroscopic forces acting of the fluid can then be identified by multiplying the kinetic equation (1.21) by mv and integrating, thus obtaining the evolution of the momentum, Z Z ∂t(ρu) + ∇ · mvvfdv = maf + mvC(f) dv, (1.23) R3 R3

25 where we have integrated by part the term involving the acceleration. Equa- tion (1.23) should be compared to the momentum balance equation (1.20b). The second-order moment of f, which appears on the left-hand side, can be rewritten as Z Z mvvfdv = m(v − u + u)(v − u + u)fdv R3 R3 Z = ρuu + m (v − u)(v − u)fdv, R3 from which we see that the Cauchy stress tensor is related to the distribution function by, cf. equation (1.17) Z P (t, x) = p(t, x)I + π(t, x) = m (v − u)(v − u)f(t, x, v)dv. R3 This result can be considered a proof in the framework of kinetic theory of Cauchy stress theorem mentioned in section 1.5. The pressure p is then obtained as the isotropic part of P , namely, 1 Z p(t, x) = m (v − u)2f(t, x, v)dv, 3 R3 which is 2/3 of the kinetic energy of the particles measured in the reference frame moving at the velocity u(t, x). The viscosity is then the trace-free part of P , namely, Z 1 2 π(t, x) = m (v − u)(v − u) − 3 (v − u) I f(t, x, v)dv. R3 At last, the macroscopic forces acting on the fluid split into a component due to the actual particle acceleration a plus a component that accounts for the collision momentum transport, namely, Z Z ρ(t, x)f(t, x) = ma(t, x, v)f(t, x, v)dv + mvC(f)(t, x, v)dv R3 R3 Z Z = ma(t, x, v)f(t, x, v)dv + m(v − u)C(f)(t, x, v)dv, R3 R3 R where we have used the fact that R3 C(f)dv = 0 in the second equality. We examine now the total energy density, which is given by the kinetic 1 2 energy 2 mv carried by each fluid particle times the number of particles per unit of volume in the phase-space, integrated in velocity, Z 1 2 3 1 2 ρu + nkBT = m v fdv. 2 2 2 R3 We recall that here the temperature T is defined as a measure of the internal energy of the fluid and, in general, does not have a thermodynamical meaning. The integral on the right-hand side can be dealt with by writing v = v − u + u and expanding the square with the result 1 Z 1 1 Z m v2fdv = ρu2 + m (v − u)2fdv, 2 R3 2 2 R3

26 where we have accounted for the identity Z m(v − u)fdv = 0. R3 Therefore, Z 3 1 2 U = nkBT = m (v − u) fdv, 2 2 R3 which means that the internal energy of the fluid as defined in section 1.1 is due to the motion of the fluid particles relative to the overall fluid velocity u. On 3 the other hand, the right-hand side equals 2 p, hence we have established a first closure relation, which expresses the pressure in terms of the internal energy, and thus of the temperature, namely,

p = nkBT.

This relationship is somewhat special because it does not depend on the particular form of the distribution function: It follows from the statistical definition of internal energy and temperature. 1 2 Upon multiplying the kinetic equation by 2 mv and integrating, we obtain the transport equation for the total energy in the form Z 1 Z Z 1 ∂ 1 ρu2 + 3 nk T + ∇ · mv2vfdv = mv · afdv + mv2C(f)dv, t 2 2 B 2 2 (1.24) where we have already integrated by part the acceleration term. By using again the identity v = v − u + u and expanding the products, the total energy flux amounts to Z 1 Z 1 mv2vfdv = 1 ρu2 + 3 nk T u + pu + π · u + m(v − u)2(v − u)fdv, 2 2 2 B 2 and by comparison with the total energy flux in equation (1.18) we can deduce the heat flux Z 1 q(t, x) = m(v − u)2(v − u)f(t, x, v)dv. 2 On going back to equation (1.24), the right-hand side amounts to

Z Z 1 mv · afdv + mv2C(f)dv 2 Z 1 Z = ρf · u + m(v − u)2C(f)dv + m(v − u) · afdv, 2 from which we can deduce that the heat sources are Z 1 Z Q(t, x) = m(v − u)2C(f)dv + m(v − u) · afdv. 2 This expression is actually valid when the acceleration a(t, x, v) is a generic function of v satisfying only the divergence-free constraint ∇v · a(t, x, v) = 0. Usually however the forces are such that they do not contribute to heating the ﬂuid, i.e., the second integral vanishes exactly for the physical accelerations we shall consider.

27 As an example of acceleration ﬁeld relevant to plasma physics, let

a(t, x, v) = a0(t, x) + v × b0(t, x),

3 where a0(t, x), b0(t, x) ∈ R are two vector ﬁelds independent of velocity. For those cases, one has (v − u) · a = (v − u) · a0(t, x) + u(t, x) × b0(t, x) , and the right-hand side integrated in velocity against f is zero, i.e., no contribution to the heat source. In general it is however possible that dissipative acceleration ﬁeld contribute to heat sources. For instance, let

a(t, x, v) = −ν(t, x)v, where ν(t, x) > 0 is a scalar coefficient. Then, Z Z (v − u) · afdv = −ν(t, x) (v − u)2fdv, R3 R3 which in general is finite and tends to reduce the internal energy. We can summarize the expressions of fluid quantities in terms of the distribution function f describing the microscopic state of the fluid: • mass density and particle density, Z ρ(t, x) = mn(t, x) = m f(t, x, v)dv; (1.25a)

• linear momentum, Z ρ(t, x)u(t, x) = m vf(t, x, v)dv; (1.25b)

• temperature and pressure, 3 3 1 Z n(t, x)k T (t, x) = p(t, x) = m (v − u)2f(t, x, v)dv; (1.25c) 2 B 2 2

• forces, Z Z ρ(t, x)f(t, x) = ma(t, x, v)f(t, x, v)dv + m(v − u)C(f)(t, x, v)dv; (1.25d) • viscosity tensor, Z 1 2 π(t, x) = m (v − u)(v − u) − 3 (v − u) I f(t, x, v)dv; (1.25e)

• heat ﬂux, 1 Z q(t, x) = m (v − u)2(v − u)f(t, x, v)dv; (1.25f) 2

28 • heat sources Z 1 Z Q(t, x) = m(v − u)2C(f)dv + m(v − u) · afdv. (1.25g) 2

The forgoing derivation proves formally the following Proposition 1.11 (formal). If f is a sufficiently regular solution of the kinetic equation (1.21) with a finite value of the first velocity moments, Z vαf(t, x, v)dv < +∞, |α| ≤ 3, then the quantities defined by (1.25) satisfy identically fluid equations (1.20). We shall now apply this result in order to obtain closure relations in the simplest case, namely, Euler’s equations of fluid dynamics. First we need to choose a specific collision operator. The simplest collision operator is the BGK (Bhatnagar, Gross and Krook [42]) operator. That is the simplest operator for which the distribution function relaxes to a local thermodynamic equilibrium. We say that a gas or a plasma is in local thermodynamical equilibrium, if the distribution is described by a local Maxwell distribution

3 2 m 2 h m v − u(t, x) i fM (t, x, v) = n(t, x) exp − , (1.26) 2πkBT (t, x) 2kBT (t, x) where n(t, x) is the number density (we shall use n and ρ = mn equivalently), u(t, x) the local average velocity, and T (t, x) is the local temperature, measuring the spread of v with respect to its average. The physical meaning of the distribution (1.26) is that in the velocity frame of the fluid velocity u(t, x), the fluid element is in thermodynamical equilibrium with a thermal bath at the local thermodynamical temperature T (t, x), and thus the particle distribution is given by the Boltzmann distribution fM ∝ exp − E/(kBT ) , where E is the kinetic energy of the particle in the moving frame. Let us stress that, if f = fM , then the temperature T has a thermodynamical meaning. For general distribution f we can still define T by means of relation (1.25c), but that has only a statistical meaning measuring the internal energy as the system is not in a local equilibrium. We can now define the map,

M : f 7→ fM = M(f), which associates to an arbitrary distribution function f, with ﬁnite velocity moments, a Maxwell distribution (1.26) with density n, average velocity u, and temperature T computed from f according to equations (1.25a)-(1.25c). The BGK operator is deﬁned in terms of M by

C(f) = −νf − M(f), (1.27) where ν > 0 is the collision frequency. Concerning this definition, let us remark that, in their original work Bhatnagar, Gross and Krook considered C = −ν(f − f0) with a fixed equilibrium f0. Here we adopt the improved definition [38] which is designed to preserve particles, momentum, and energy. The BGK

29 operator (1.27) indeed satisfies the required constraint as the velocity integral of both f and fM is equal to the number density n, hence Z C(f)dv = −νn − n) = 0. R3 More generally, the BGK operator preserves the first three velocity moments since f and M(f) have exactly the same number density, average velocity and temperature. In fact, formally at least, we have the following property. Let f be a sufficiently regular solution of the Cauchy problem

∂tf = C(f), f(0) = f0.

2 3 2 If, in addition, f(t), ∂tf(t) ∈ L (Ω × R ), then f relaxes in L -norm to the Maxwell distribution with n, u, and T given by f0. A formal proof of this statement follows on noticing that, if a solution exists, it necessarily has ﬁnite n, u, and T . Then, Z Z ∂tn = ∂tfdv = −ν f − M(f) dv = 0, R3 R3 and analogously,

Z Z ∂t(nu) = v∂tfdv = −ν v f − M(f) dv = 0, R3 R3 Z Z 2 2 (2/m)∂tU = (v − u) ∂tfdv = −ν (v − u) f − M(f) dv = 0, R3 R3 which proves that n, u and T are constant. In addition we have Z 1 d 2 kf − M(f)kL2 = f − M(f) ∂t f − M(f) dxdv. 2 dt Ω×R3

Since the moments are not changing, we have M(f) = M(f0), ∂tM(f) = 0 and Z 1 d 2 kf − M(f)kL2 = f − M(f) ∂tfdxdv 2 dt Ω×R3 Z 2 2 = −ν f − M(f) dxdv = −νkf − M(f)kL2 . Ω×R3 We conclude 2 2 −2νt kf − M(f0)kL2 = kf0 − M(f0)kL2 e , 2 hence f approaches the Maxwell distribution M(f0) in the L -norm. If the full kinetic equation (1.21) is accounted for, the relaxation process is much more complicated. We might expect however that for a collision- dominated gas or plasma, the advection term might have a small eﬀect and the distribution will become nearly Maxwellian in the long time. This suggests the possibility of an asymptotic solution of the kinetic equation. In order to represent mathematically “strong collisions” let us scale the collision frequency according to ν = ν0/ with ν0 > 0 ﬁxed and let ∈ (0, 1] tend to zero.

30 We consider the Hilbert expansion of the distribution function [38, and references therein], X n f = f ∼ fn, n≥0 where the symbol “∼” means asymptotic convergence, i.e., for all integers N > 0 there are constants Cα,N such that

N α X n N ∂ f − fn ≤ Cα,N , n=0 for all multi-indices α with |α| ≤ k, where k is the required regularity, e.g., n k = 1. In plain words this means that for every N the partial series of fn is a good approximation of f for suﬃciently small, even if the full series might not converge for any ﬁxed value of . We notice that X n M(f ) ∼ Mn, n≥0 where Mn depends on f0, . . . fn and

M0 = M(f0).

Then the kinetic equation (1.21) becomes

ν0 X f − M(f ) + n−1∂ f + v · ∇ f + a · ∇ f 0 0 t n−1 x n−1 v n−1 n≥1 + ν0 fn − Mn(f0, . . . , fn) ∼ 0, and separating diﬀerent powers of the small parameter, we have ν0 M(f0) − f0 = 0, for n = 0, ν0 Mn(f0, . . . , fn) − fn = ∂tfn−1 + v · ∇xfn−1 + a · ∇vfn−1, for n ≥ 1.

The lowest order equation (n = 0) tells us that f0 is a Maxwell distribution, with arbitrary moments ρ0, u0, and T0. Given such moments, we can solve recursively for fn, n ≥ 1 provided that we can invert the operators Mn(f0, . . . , fn−1, ·)−Id, and we have corrections ρn, un, and Tn. The simplest possible closure is obtained considering only the lowest order so that f = fM , ρ = ρ0, u = u0,T = T0, and equations (1.25e)-(1.25g) give the closure relations for the remaining quantities. Considering the case an acceleration ﬁeld which does not contribute to heat, such relations read

π = 0, q = 0, and Q = 0, (1.28a) and there is no contribution of collisions to the body force (1.25d), Z ρ(t, x)f(t, x) = ma(t, x, v)f(t, x, v)dv. (1.28b)

31 This is the simplest possible closure which leaves us with the equations, cf. equations (1.20), ∂tρ + ∇ · ρu = 0,

∂t(ρu) + ∇ · (ρuu) = −∇p + ρf, (1.29) 3 3 ∂t 2 nkBT + ∇ · 2 nkBT u + p∇ · u = 0, or the equivalent advective form (with equation (1.25c), Dρ = −ρ∇ · u, Dt Du ρ = −∇p + ρf, (1.30) Dt Dp = −γp∇ · u, Dt where we have defined the adiabatic index 2 5 γ = + 1 = . (1.31) 3 3 Equations (1.30), or equivalently their conservative form (1.30), are referred to as Euler equations for inviscid fluids (viscosity is zero). Neglecting viscosity is usually a tremendous simplification both for ordinary fluids and plasmas. However, in many applications the lowest order closure will not suffice and corrections should be added thus obtaining non-trivial transport coefficients, namely, viscosity and heat fluxes. For plasmas in presence of a magnetic field, such transport coefficients are much more complicated, even with strong collisions, cf. the seminal paper by Braginskii [39]. In its basic form, magnetohydrodynamics is build upon Euler’s equations and thus we shall not need the intricacies of Braginskii’s closure. As a last remark, let us note that the Hilbert expansion method does not produce a uniform approximation of the solution and it is expected to fail when steep gradients build up [38]. In those cases one can obtain stronger results with the more sophisticated Chapman-Enskog method [38, and references therein]. For the specific case of plasmas, apart from some notable exceptions, collisions are weak since the collision frequency satisfies [43]

ν ∝ nT −3/2, and we have relatively low density and high temperature. Then transport is dominated by turbulence rather then by collisions. In such situations, the current trend is to give up with the search for the appropriate closure and focus on the numerical modeling of turbulent transport leading to multi-scale codes in which the equation of ﬂuid dynamics are coupled to kinetic codes.

1.7 Incompressible ﬂows The closure relations obtained on the basis of the kinetic equation (1.21), as well as the corresponding thermodynamic relations such as equation (1.25c) between pressure and temperature, are valid as long as the kinetic model describes correctly the microscopic dynamics of the considered ﬂuid. This is the case for important physical systems that are made by atoms or molecules such as gases or by ions and electrons such as plasmas.

32 Even stellar systems such as globular clusters and galaxies can be described in terms of a kinetic equation and thus, of the corresponding fluid quantities [44]. There are situations however, in which this does not apply and closure relations have to be inferred in other ways. One such case is that of incompressible flows. For an incompressible flows, cf. equation (1.16), the closure obtained above is not appropriate. In fact, accounting for the incompressibility condition in Euler’s equations (1.30) yields an over-determined system. In order to see this, let us consider an initial condition at the time t = 0 with uniform density and pressure. According to Euler’s equations (1.30), for an incompressible flow both density and pressure are constant along Lagrangian trajectories, hence they remain uniform for all t ≥ 0. On the other hand, the divergence of the momentum equation together with the incompressibility condition gives

∇u : ∇u = ∇ · f, (pressure is constant), which cannot be satisfied in general. The problem is that the closure discussed above does not apply to incompressible flows: Thermodynamic relations that hold for compressible fluids and gases do not hold for incompressible fluids. If we still maintain that viscosity vanishes, but accept that pressure and internal energy are not just proportional to each other (as they are in kinetic theory), we see that equation (1.20c) is decoupled from the system. The remaining equations Dρ = 0, Dt Du (1.32) ρ = −∇p + ρf, Dt ∇ · u = 0, describe an incompressible flow and are referred to as incompressible Euler’s equations. One should notice that the divergence-free condition implicitly determine the pressure p, as follows by taking the divergence of the momentum equation. For non-ideal fluids viscosity cannot be neglected. In order to write a closure for π, one usually makes the assumption that viscosity is a linear function of ∇u. Since π must be symmetric and trace free, we write 1 π = −2µD − tr(D)I, 3 where D is the deformation tensor, i.e., the symmetric part of ∇u and µ is the viscosity coefficient. For incompressible flows,

∇ · π = −µ∆u,

P 2 2 where ∆ = i ∂ /∂xi is the Laplace operator, so that the equation of motion becomes, Dρ = 0, Dt Du (1.33) ρ = −∇p + ρf + µ∆u, Dt ∇ · u = 0.

33 This is the celebrated Navier-Stokes equation, in which the hyperbolic char- acter of Euler’s equation is terribly complicated by the apparently innocuous Laplace operator. The eﬀect of viscosity consists in dissipating kinetic energy which is converted into internal energy and thus temperature according to equation (1.20c). The internal energy is however unrelated to pressure as the closure relation (1.25c) does not apply in this case. Hence the energy balance equation (1.20c) decouples from the systems and one needs to solve it only if one is interested in heat transport in the ﬂuid.

1.8 Equations of state, isentropic ﬂows and vorticity. Concerning compressible Euler’s equations, in many situations the pressure equation does not need to be explicitly solved. In fact, if the mass density is uniformly bounded away from zero, i.e., there is a constant ρ0 > 0 such that ρ ≥ ρ0, we can express ∇ · u from the continuity equation, 1 Dρ ∇ · u = − , ρ Dt and we can substitute it into the pressure equation with the result that Dp γp Dρ − = 0. Dt ρ Dt

This implies that pρ−γ is constant along the Lagrangian trajectories of the solution of Euler’s equations. In order to see this, we compute D Dp γp Dρ ργ pρ−γ = − = 0, Dt Dt ρ Dt

−γ and since ρ ≥ ρ0 > 0, we have D(pρ )/Dt = 0. If the initial conditions for equations (1.30) are chosen so that pρ−γ = C = constant at the initial time, then pρ−γ = C for all time. Under those conditions we can replace the pressure equation with the algebraic relation p = Cργ . Other closure relations may be considered in which the pressure is assigned as a function of the other fluid variables, and particularly as a function of the density, P : R≥0 → R≥0, p = P(ρ). The relation defining the pressure in terms of the density is referred to as equation of state. For the specific case of Euler’s equations discussed above, we have P(ρ) = Cργ , but one can also consider other equations of state. The following is a particularly interesting class thereof. Definition 1.2 (Isentropic flows). A flow is called isentropic if there exists a function h : R≥0 → R, to be referred to as enthalpy, such that ∇h(ρ) = ∇p/ρ. With reference to the momentum balance in Euler’s equation (1.30) one can notice that ∇p/ρ plays an important role, as it provides a drive for the fluid velocity. For isentropic flows, this driving force is potential and the momentum balance equation reduces to

∂tu + u · ∇u = −∇h + f. (1.34)

As a consequence the vorticity ω = ∇ × u of an isentropic ﬂow satisﬁes

∂tω − ∇ × (u × ω) = ∇ × f. (1.35)

34 This is readily derived by making use of the vector calculus identity

u · ∇u − ∇u · u = (∇ × u) × u.

The second term on the left-hand side is an exact gradient, hence, the momentum equation (1.34) can be rewritten as

2 ∂tu − u × (∇ × u) = −∇(h + u /2) + f.

Then one can compute (formally)

∂tω = ∇ × (∂tu) = ∇ × (u × ω) + ∇ × f, as the curl of a gradient vanishes identically, and this is equation (1.35). One should notice that the curl of the forces f is the only drive of vorticity, i.e., of rotation of the fluid. A potential force cannot create vortices. The equation of state P(ρ) = Cργ obtained for Euler’s flows is isentropic. In fact we compute ∇p/ρ = γCργ−2∇ρ. On the other hand, d γ γCργ−2 = Cργ−1 , dρ γ − 1 hence we have ∇p/ρ = ∇h(ρ) where γ h(ρ) = Cργ−1, γ − 1 is the enthalpy associated to Euler’s flows. Therefore for any solution of Euler’s equations satisfying the equation of state at the initial time, one has that the vorticity solves equation (1.35).

1.9 Effects of Euler-type nonlinearities. In virtue of the Reynolds transport theorem, the advective derivative D/Dt along the flow plays a central role in the equations of fluid dynamics. In the momentum balance equation, the velocity u is advected by its own flow, producing the typical Euler nonlinearity Du/Dt = ∂tu + u · ∇u. In order to understand the behavior of this operator let us briefly study a prototypical case of Euler nonlinearity in one spatial dimension. Specifically, we study the Cauchy problem

∂tu + u∂xu = 0, u(0, x) = u0(x), (1.36)

1 for u : R≥0 × R → R, with smooth initial data u0 ∈ C (R). This equation is referred to as Hopf equation or inviscid Burgers equation and it is discussed in details in Hörmander’s lectures [34], which we follow closely here. It is a nonlinear first-order partial differential equation which can be dealt with by means of the characteristics method sketched in section 1.4. 1 R 0 Proposition 1.12. For any initial condition u0 ∈ C ( ) with −u0(x) ≤ b, b ≥ 0, there exists a unique classical solution u ∈ C1[0,T ), 1/T = b, of the initial value problem (1.36).

35 Proof. If u ∈ C1 is a solution, then along Lagrangian trajectories, deﬁned by

dx(t, y) = ut, x(t, y), x(t, y) = y, dt we have d ut, x(t, y) = (∂ + u∂ u)(t, x(t, y)) = 0, dt t x and u(0, x(0, y)) = u0(y). We can solve the two coupled ordinary diﬀerential equations analytically, with the result that u t, x(t, y) = u0(y), x(t, y) = y + u0(y)t. (1.37)

The ﬂow map y 7→ x(t, y) is invertible as long as ∂x/∂y 6= 0, which is the case if

0 1 + u0(y)t ≥ 1 − bt > 0, that is for t < T and 1/T = b. We construct a ﬁeld u : [0,T ) × R → R by u(t, x) = u0 y(t, x) , where y(t, x) is the solution of x = y + u0(y)t for ﬁxed (t, x). By the inverse function theorem [33] the function y is in C1 and so is u. Since u is constant along the Lagrangian trajectories, it is a solution of the Cauchy problem. Uniqueness follows from the uniqueness of the solution of the associated characteristics equations.

If the initial condition is bounded, then sup |u(t, ·)| = sup |u0|, due to the solution of the characteristics (1.37). If T is ﬁnite, however, for t → T the spatial derivatives blows up since ∂x/∂y approaches zero and

h∂x(t, y)i−1 ∂ ut, x(t, y) = u0 (y), x ∂y 0 by the inverse function theorem. The classical solution breaks with a singularity in the gradient. By inspection of the solution for the characteristics, one can see the effect of the Euler nonlinearity: Each point y on the t = 0 line moves with a constant speed given by the value of the initial condition u0(y) at the initial point. Hence, if the initial condition is decreasing (negative derivative) the trajectory starting from y1 < y2 have velocity u0(y1) > u0(y2) and it must overtake the trajectory issued from y2 at some time in the future. At this time both points y1, y2 are mapped into the same point x by the flow, which means that the flow is no longer invertible and the classical solution breaks. It is still possible to define weak solutions that capture the shock due to the crossing of trajectories [34]. A regularized version of the Hopf equation (1.36) is the Burgers equation

2 ∂tu + u∂xu = µ∂xu, u(0, x) = u0(x), (1.38) in which a ﬁnite viscosity µ > 0 is added. For µ suﬃciently small, the behavior of the Burgers equation is dominated by the nonlinear advection up to a time near the breakdown time T . There characteristics are getting closer to each other thus amplifying derivatives of u

36 Figure 1.5: Solution of the Cauchy problem (1.38) for the Burgers equation 2 2 corresponding to the initial condition u0(x) = exp(−(x − π) /a ) with a = 0.2 and µ = 0.002. The profiles of the solution u(t, ·) at various instants of time are shown on the top panel. One can see the effect of the nonlinear advection steepening the profiles where the spatial derivative is negative. The full solution is shown on the bottom-left panel as a function of (t, x). A detail of the contours of the solution is shown on the bottom-right panel: The contours are almost straight lines, thus follwiong the characteristics of the inviscid equation, up to the dissipative shoch where viscosity starts to play a role.

37 2 so that the terms µ∂xu becomes large (even for a small viscosity). The eﬀect of this term is to balance the formation of a singularity in the gradient. Figure 1.5 shows the numerical solution of the Cauchy problem for the Burg- ers equation (1.38) with a Gaussian initial conditions. One can see that the 0 initial condition is distorted so that the part of the proﬁle where u0(x) < 0 is steepened by the Euler nonlinearity increasing the slope up to the point where the viscosity balances the nonlinear advection and dissipative shock is formed.

38 2 Basic elements of classical electrodynamics

This section gives a short summary of classical electrodynamics. We shall split this in two parts. The first part is a quick introduction on Maxwell’s equations and the calculation of the electromagnetic fields given their sources, namely electric charges and currents. In the second part, we consider the Lorentz force and the motion of an electrically charged particle under the influence of a given electromagnetic field. A standard reference on the foundations of classical electrodynamics is the book by Jackson [45]. In a final addendum, two basic mathematical results are briefly presented, namely, Poisson equation (repeatedly used in the main discussion) and on the Cauchy problem for Maxwell’s equations on the whole space.

2.1 Maxwell’s equations. Maxwell’s equations are diﬀerential equations describing the electromagnetic ﬁeld (E,B) where

E : I × Ω → R3, and B : I × Ω → R3, are the electric field and the magnetic field, respectively, defined for time t ∈ I and position x in a domain Ω ⊆ R3. As we shall see, it is appropriate (both physically and mathematically) to treat E and B as elements of the same object, the electromagnetic field (E,B). For this reason, throughout this note, Gauss (c.g.s.) units are used, so that both E and B are measured with the same dimensions, a property which is broken in the international system (SI). Conversion formulas between Gauss and SI units and related definitions can be found, e.g., in Jackson’s book [45] or in the NRL plasma formulary [43]. The unit system in electrodynamics have to be precisely specified as the constants in Maxwell’s equations (unfortunately) depend on it. However, when dealing with static, i.e., time-independent, phenomena the electric field and the magnetic field were initially studied separately as two different physical objects. Electrostatics and magnetostatics, respectively, developed as two separate physics theories. Then Faraday discovered experimentally that a time-dependent magnetic field induces an electric field, thus establishing a connection between the two separate worlds. It was the mathematical physicist James Clerk Maxwell who formulated a consistent system of equations, namely, Maxwell’s equations, that contain electrostatics, magnetostatics and Faraday’s induction laws as special cases. In addition Maxwell’s theory predicted electromagnetic waves that were later confirmed experimentally. For our purposes, it is important to develop a certain familiarity with all the four main aspects of Maxwell’s equations: electrostatics, magnetostatics, Faraday’s induction (which is a key element of MHD), and the full system of Maxwell’s equations. Let us begin with the definition of the sources of electromagnetic fields, namely, electric charge densities and electric current densities.

Sources. In addition to mass, elementary particles can carry an electric charge which can be either positive or negative (diﬀerently from mass which has one sign only). Analogously to the continuum hypothesis of ﬂuid dynamics, cf.

39 section 1.1, at a macroscopic level we can model charges as a continuum with an associated electric charge density. By deﬁnition, the electric charge density is a function ρc : I × Ω → R such that, for any control volume W ⊆ Ω, Z (electric charge in W at time t) = ρc(t, x)dx. W This is the analogous of the mass density introduced in section 1.1, but with the important diﬀerence that ρc takes values in the whole real line R as electric charges can be either positive or negative. It is a basic principle of physics that electric charge can neither be created nor destroyed. Analogously to the mass continuity equation (1.20a), we have

∂tρc + ∇ · J = 0, (2.1) which is referred to as charge continuity equation. The flux of electric charge J : I × Ω → R3 is by definition the electric current density. Physically an electric current I(t) through a certain surface Σ, for instance the cross-section of a conductor, is defined as the rate at which charge is flowing through it. If W is a control volume and Σ = ∂W is its boundary, we have Z dQ(t) I(t) = J · ndS = − , ∂W dt where Q(t) is the change in W at time t. For a standard conductor, like an electric wire in a circuit, dQ/dt = 0 and, by Gauss theorem, the current density J is divergence-free.

Electrostatics. Let us consider a time-independent charge density ρc :Ω → R deﬁned on a bounded domain Ω ⊂ R3. 3 The static charge density ρc generates a static electric ﬁeld E :Ω → R , according to the Gauss law,

∇ · E = 4πρc, (2.2a) subject to the constraint that E must be irrotational, namely, ∇ × E = 0. (2.2b) Equipped with proper boundary conditions, equations (2.2) deﬁne electrostatics. If the domain Ω is simply connected, then condition (2.2b) is equivalent to E being a potential ﬁeld, i.e., there exists a scalar potential φ :Ω → R such that E = −∇φ, and Gauss law takes the form of a Poisson equation for φ, namely,

− ∆φ = 4πρc, (2.3) where ∆ denotes the Laplace operator, defined by ∆φ = ∇ · ∇φ. The potential φ is referred to as the electrostatic potential. It is worth noting that the electrostatic potential is defined apart from constants. This means that the value of φ in a point does not have a physical meaning; only differences of potential between two points is physical. More generally physical quantities should never depend on the arbitrary offset of the potential. We therefore have some degree of freedom in choosing the boundary conditions for equation (2.3).

40 Magnetostatics. Let us now consider a static current density J :Ω → R3. That generates a static magnetic ﬁeld B :Ω → R3 according to Amp`ere law 4π ∇ × B = J, (2.4a) c where c is a constant determined by the unit system and in Gauss (c.g.s.) units has the dimensions of a velocity, while B is subject to the constraint

∇ · B = 0. (2.4b)

Ampèrelaw implies, as a necessary condition for the existence of a solution, ∇ · J = 0, since ∇ · ∇ × B = 0, and by charge conservation (2.1) one must have no change in time of the electric charge distribution, ∂tρc = 0. Under proper conditions on the domain Ω and with boundary conditions, equations (2.4) determine uniquely the magnetic field B and define magnetostatics. Again the task of computing the solution B can be reduced to solving a Poisson equation. With this aim let us recall that if the domain Ω is simply connected, equation (2.4b) is equivalent to the existence of a vector potential A :Ω → R3 such that B = ∇ × A. (2.5) This is called magnetic vector potential. It should not be confused with the magnetic scalar potential which is used for the homogeneous magnetostatic problem, i.e., when J = 0, so that ∇ · B = ∇ × B = 0 and we can set B = −∇Φ with the magnetic scalar potential Φ satisfying the Laplace equation ∆Φ = 0. This technique is used for potential field extrapolation in the solar corona [16]. As for both scalar potentials, the vector potential is not uniquely determined. In fact, one can add to A the gradient of a function f without changing the value of B, since ∇×∇f = 0. It is important therefore to make sure that any physical quantity defined in terms of A does not depend on the choice of the arbitrary function f. A specific choice or constraint that remove this arbitrariness in the definition of the potential is referred to as a gauge. A particular choice of the gauge, to be referred to as the Coulomb gauge, corresponds to the requirement that the vector potential is divergence-free,

∇ · A = 0. (2.6)

If the domain is bounded and regular enough, it is always possible to guarantee the existence of this gauge. In fact, if we find a vector potential A˜ that satisfies (2.5) but not (2.6), then we can redefine A = A˜ + ∇f with a gauge function f chosen so that ∇ · A = ∇ · A˜ + ∆f = 0, which is again a Poisson equation for f with ∇ · A˜ as a source term. Equipped with suitable boundary conditions, this equation as a unique solution and the resulting potential A satisfies the Coulomb gauge condition. Ampère’slaw (2.4a) in terms of the vector potential reads 4π ∇ × (∇ × A) = J, c

41 and the vector-calculus identity

∇ × (∇ × A) = ∇(∇ · A) − ∆A, together with the Coulomb gauge (2.6) implies that Ampère’slaw is equivalent to a “vector Poisson equation”, namely, 4π −∆A = J, c where the Laplace operator ∆ is applied to A component-wise. This is a system of three decoupled Poisson equations for the three components of the vector potential. The boundary conditions for such decoupled Poisson equations must be chosen so that the Coulomb gauge (2.6) is satisfied. We notice that, if A is a sufficiently regular solution, 4π −∆∇ · A = ∇ · J = 0, c i.e., ∇ · A satisfies the Laplace equation in Ω. If we choose the boundary conditions such that, e.g., ∇ · A = 0 on the boundary, then the Laplace equation has a unique solution and that is ∇ · A = 0, i.e., the solution will automatically satisfy the Coulomb gauge condition. A particularly simple case is that of periodic boundary conditions, i.e., Ω is a square identified with the 3-torus T3. Then all possible solutions of the Laplace equation are the constants and we have to choose such constant equal to zero.

Faraday’s induction. Let us now turn our attention to time-dependent electric and magnetic fields, E,B : I × Ω → R3. Essentially Faraday’s induction law is expressed mathematically as a relation between the time-derivative of the magnetic field and the curl of the electric field, namely, 1 ∂B ∇ × E = − . (2.7) c ∂t This can be seen as a generalization of constraint (2.2b) and replaces it when time-dependent fields are considered. One remarkable consequence is that for time dependent fields the electric field E is no longer a potential field, since Faraday’s law has replaced (2.2b). However, the magnetic field still satisfies the divergence-free condition (2.4b) and thus, in simply connected domains we can find A : I × Ω → R3 such that (2.5) holds point-wise in time. From Faraday’s law, it follows that 1 ∂A ∇ × E + = 0, c ∂t −1 which means that the vector E + c ∂tA = 0 is irrotational and thus, if the domain Ω is simply connected, there exists a time-dependent scalar potential φ : I × Ω → R such that 1 ∂A E = −∇φ − , c ∂t (2.8) B = ∇ × A.

42 In view of the relation between A and the electric ﬁeld, the gauge transformation A 7→ A0 = A + ∇f introduced above has to be dealt with more carefully. If we change the gauge for A we need to change the potential φ as well in order for the electric ﬁeld E to be invariant. This leads us to the full gauge transformation 1 ∂f φ 7→ φ0 = φ − , c ∂t (2.9) A 7→ A0 = A + ∇f.

Then both E and B are invariant with respect to this transformation. As before we ﬁx the gauge to be the Coulomb gauge (2.6), for which we have 1 ∂ ∇ · E = −∆φ − ∇ · A = −∆φ. c ∂t Hence, in the Coulomb gauge, Poisson equation (2.3) holds unchanged for the full time-dependent potential φ(t, x).

Maxwell’s equations. The equations from electrostatics, magnetostatics, and the generalization of condition (2.2b) due to Faraday’s induction amount to the system ∇ · E = 4πρc, 1 ∂B ∇ × E = − , c ∂t (not consistent!) 4π ∇ × B = J, c ∇ · B = 0, but this is not consistent with the principle of charge conservation. The problem is due to Amp`ere’slaw which implies

∇ · J = 0, thus violating the charge continuity equation (2.1) when time dependent fields, and thus sources are considered. Ampère’slaw was in fact justified for the case of static fields. Maxwell saw that a simple way to fix the problem is adding a term proportional to the time derivative of the electric field to the Ampère’sequation in a symmetric way with respect to the other “curl equation” namely Faraday’s law. This leads us to Ampère-Maxwell law, 4π 1 ∂E ∇ × B = J + , c c ∂t and the added term was interpreted as an effective current density 1 ∂E J = , disp 4π ∂t referred to as the displacement current. Now the divergence of Ampère-Maxwell law gives 4π 1 ∂ 0 = ∇ · J + ∇ · E, c c ∂t

43 which, upon accounting for Gauss law, becomes 4π 4π ∂ρ 0 = ∇ · J + c , c c ∂t consistently with the general charge continuity equation (2.1). With the modiﬁed Amp`ere-Maxwell law, the full system reads

∇ · E = 4πρc, 1 ∂B ∇ × E = − , c ∂t (2.10) 4π 1 ∂E ∇ × B = J + , c c ∂t ∇ · B = 0, and those are Maxwell’s equations of classical electrodynamics. It is important to observe that: 1. The charge continuity equation is a necessary condition for the existence of solutions, since it follows from Gauss and Ampère-Maxwell law. 2. With time-independent fields and sources, Maxwell’s equations for E and for B decouples, recovering electrostatics and magnetostatics, respectively. 3. Faraday’s induction is included as one of Maxwell’s equations. 4. The two equations for the divergence of the fields are constraints on the initial conditions, while Faraday and Ampère-Maxwell equations define the dynamics.

The last point explicitly means that, if E0 and B0 are initial data at the time t = t0 satisfying the two divergence equations, i.e.,

∇ · E0(x) = 4πρc(0, x), ∇ · B0(x) = 0, then the solution of the Cauchy problem  ∂tE − c∇ × B = −4πJ,  ∂tB + c∇ × E = 0,   E(0, x) = E0(x),B(0, x) = B0(x), automatically satisﬁes the divergence equations at later time t ≥ t0. Indeed, if (E,B) is the solution of the above problem, we have from Faraday’s equation

∂t∇ · B = 0, ∇ · B(0, x) = ∇ · B0(x) = 0, which implies ∇ · B(t, x) = 0 for all t ≥ t0. Analogously, the divergence of the Amp`ere-Maxwell equation gives

∂t∇ · E = −4π∇ · J = ∂t(4πρc),

∇ · E(0, x) − 4πρc(0, x) = ∇ · E0(x) − 4πρc(0, x) = 0, where the charge continuity equation has been accounted for, and thus Gauss law ∇ · E(t, x) − 4πρc(t, x) = 0 is satisﬁed at all later time t ≥ t0.

44 A physical proof of the correctness of Maxwell’s equations came with the discovery of electromagnetic waves. In fact a direct implication of Maxwell’s equations is the existence of solutions for the electromagnetic ﬁeld in which a perturbation of E and B propagates like a wave. The standard way to see that is via another gauge known as the Lorenz 1gauge, which is deﬁned by the condition 1 ∂φ ∇ · A + = 0. (2.11) c ∂t Under non-restrictive hypotheses, it is possible to satisfy this condition by choosing an appropriate gauge transformation. Let us assume that we have a pair of potentials (φ,˜ A˜) which do not satisfy the Lorenz condition. Then we apply a ˜ −1 gauge transformation (2.9) and require that the new potentials φ = φ − c ∂tf and A = A˜ + ∇f satisfy (2.11), that is,

1 ∂φ 1 ∂φ˜ 1 ∂2f 0 = ∇ · A + = ∇ · A˜ + + ∆f − , c ∂t c ∂t c2 ∂t2 which is equivalent to 1 ∂φ˜ f = ∇ · A˜ + c ∂t where 1 ∂2 = − ∆, c2 ∂t2 denotes the D’Alembert operator (which is the Laplacian with a Lorentz metric). We recognize the D’Alembert wave equation with sources which we can solve for the function f, thus forcing the Lorenz gauge. In terms of potentials, Gauss law becomes 1 ∂A ∇ · − ∇φ − = 4πρ c ∂t c and substituting ∇ · A from the Lorenz gauge condition (2.11) we have

1 ∂2φ − ∆φ = 4πρ . c2 ∂t2 c On the other hand, Amp`ere-Maxwell law in terms of potentials reads 4π 1 ∂ 1 ∂A ∇ × (∇ × A) = J + − ∇φ − , c c ∂t c ∂t and using again the identity ∇ × (∇ × A) = ∇(∇ · A) − ∆A, we have

1 ∂2A h 1 ∂φi 4π − ∆A + ∇ ∇ · A + = J, c2 ∂t2 c ∂t c and the term in square brackets is zero in the Lorenz gauge. At last we have obtained two decoupled wave equations for the potentials,

φ = 4πρc, 4π (in the Lorenz gauge). (2.12) A = J. c 1Notice that this is not a spelling error: The gauge is named after Ludvig Lorenz not Heinrich Lorentz [46].

45 This proves (formally at least) the existence of propagating wave solutions to Maxwell’s equations. In addition we can give a physical meaning to the constant c introduced after equation (2.4a). We see from the definition of the D’Alembert operator that c is the propagation speed of the electromagnetic waves, i.e., c is the speed of light. It is worth noting that using potentials is not essential (although that is the standard way). The same conclusion could be deduced directly from Maxwell’s equations for the electric and magnetic field. In plasma physics it is common to work with the electric field. We differentiate Ampère-Maxwell law in time and divide by c, 1 ∂B 4π ∂J 1 ∂2E ∇ × = + . c ∂t c2 ∂t c2 ∂t2 On substituting the derivative of the magnetic field from Faraday’s law, one finds a decoupled equation for E, namely,

1 ∂2E 4π ∂J + ∇ × (∇ × E) = − . (2.13) c2 ∂t2 c2 ∂t It is less obvious that equation (2.13) supports propagating wave solutions, and yet the left-hand side can be written as

1 ∂2E − ∆E + ∇(∇ · E) = E + 4π∇ρ , c2 ∂t2 c where we have accounted for Gauss law in the last equality. Hence the equation for E is equivalent to 4π ∂J E = −4π∇ρ − , c c2 ∂t which is a wave-equation with sources. The same equation for E could also be obtained by applying the D’Alembert operator to equation (2.8) and by using the wave equations for the potentials. From this argument however, we can understand that the term 1 ∂2E , c2 ∂t2 in the D’Alembert operator E stems from the displacement current! Maxwell’s intuition of adding the displacement current is critical for the existence of wave- like solutions.

Plane electromagnetic waves. It is instructive to look for plane-wave solutions of Maxwell’s equations without sources. This relatively straightforward technique applies to any constant-coeﬃcient wave equation. We search for solutions of  ∂tE − c∇ × B = 0,  ∂tB + c∇ × E = 0,  ∇ · E = ∇ · B = 0, with oscillatory complex-valued initial conditions of the form

ik·x ik·x E(0, x) = E0e ,B(0, x) = B0e ,

46 3 3 where k ∈ R is a given real vector, k 6= 0, and E0,B0 ∈ C are complex vectors. The argument of the exponential, namely,

ψ0(x) = k · x, is the phase of the initial oscillation. The initial condition for both E and B is constant on the surfaces ψ0(x) = C = constant, and since ∇ψ0 = k such surfaces are planes orthogonal to k. The divergence equations give us constraints on initial data, that is,

∇ · E = 0, ⇒ k · E0 = 0,

∇ · B = 0, ⇒ k · B0 = 0,

We look for a solution of such a Cauchy problem in the form of a plane wave

E(t, x) = Ee−i(zt−k·x),B(t, x) = Be−i(zt−k·x), (2.14) where z ∈ C is a possibly complex number and E, B ∈ C3 are complex vectors all depending on the ﬁxed vector k ∈ R3. We must have

k ·E = k ·B = 0.

A physical electromagnetic ﬁeld E,B should however be real-valued. We can obtain a real-valued solution by summing to the complex solution its complex conjugate. This is possible since Maxwell’s equations are linear and have real coeﬃcients, so that if a plane wave is a solution, then its complex conjugate is again a solution, and so is the sum of the two. The exponential in equation (2.14) is oscillatory with time-dependent phase

ψt(x) = k · x − ωt, ω = Re(z), where ω is the frequency. We see that, if a point x0 belongs to a phase front ˆ at the time t = 0, i.e., ψ0(x0) = C for some constant C, then xt = x0 + vphkt, ˆ with k = k/|k| and vph = ω/|k|, belongs to the same phase front at the later time t ≥ 0, i.e., ψt(xt) = C with the same constant C. In fact, ˆ ψt(xt) = k · x0 + k · (vphkt) − ωt = k · x0 = C.

We can say that phase fronts move in the direction of vector k, which is referred to as the wave vector, at the speed vph, which is called phase velocity. It should be observed that in many spatial dimensions the definition of phase velocity is merely a convention: If an infinite plane is moving parallel to itself, then the definition of the velocity of its points is ambiguous. In fact, for any w ∈ R3 ˆ such that k · w = 0 but otherwise arbitrary, the velocity v = vphk + w will move points from a phase front to the next, cf. figure 2.1. The choice w = 0 is a convention. In general, if z ∈ C with γ = Im(z) 6= 0, we may also have and exponential growth (γ > 0) or damping (γ < 0) of the wave. We shall see however that, for the problem at hand, z must be real. Substitution of the plane wave into equation (2.13) without sources yields

(z2/c2 − k2)E = 0.

47 ψt(x) = C

xt w

ˆ vphkt ˆ (vphk + w)t

ψ0(x) = C

Figure 2.1: Definition of the phase velocity of a wave in three dimensions. The conventional definition takes the direction of the wave vector kˆ which is orthogonal to the phase fronts. However, for an infinite plane, one can add an orthogonal velocity w, k · w = 0, obtaining the same translation of the plane.

Non-trivial solutions require E 6= 0, therefore, the complex number z must be solution of the algebraic equation

z2 − c2k2 = 0, which is referred to as dispersion equation. For electromagnetic waves the dispersion equation is particularly simple and it has a pair of real-valued solutions z = ω (no exponential growth or damping) with

ω = ±c|k|.

The relation between the real frequency ω and the wave vector k is referred to as the dispersion relation. Geometrically this dispersion relation defines a double-sided cone in the space (ω, k) ∈ R4, cf. figure 2.2, which is referred to as light cone. One should also observe that the dispersion relation is invariant with respect to any rotation of the wave vector, i.e., electromagnetic waves are isotropic waves. Then Faraday’s equation gives the magnetic field in the form ck B = × E = sign(ω)N × E, ω where N = ck/|ω| is the refractive-index vector. We have |N| = c|k|/|ω| = 1 because of the dispersion relation. In virtue of this relation B is orthogonal to both k and E and, if E is real, then B is also a real vector. A representation of the orthogonal oscillations of E and B is given in figure 2.3. 3 3 In conclusion, given k ∈ R , k 6= 0, and initial vectors E0,B0 ∈ C both orthogonal to k, we have unique solution of the Cauchy problem for homogeneous

48 Figure 2.2: Light cone represented for two spatial dimensions with c = 1.

Figure 2.3: Electric (red curve) and magnetic (blue curve) oscillations in space at fixed time for a plane wave with k = (1, 0, 0) and with E0 = (0, 0, 1), all quantities are normalized. Then, for the positive root of the dispersion relation, ω > 0, B is oscillating in the plane orthogonal to both the propagation direction (black straight line) and the electric field and with opposite phase with, i.e., B < 0 where E > 0. For the negative root, ω < 0, the fields are in phase. The figure refers to the positive root.

49 Maxwell’s equations in form of a plane-wave. In fact the general solution is the linear combination of the two branches ω = ω± = ±c|k|, namely,

ˆ ˆ E = E+eik·(x−ckt) + E−eik·(x+ckt), ˆ ˆ B = N × E+eik·(x−ckt) − N × E−eik·(x+ckt), where E± are free and have to be determined by the initial condition. One wave is characterized by phase fronts moving toward the positive k direction and it is referred to as progressive wave, while the other wave has phase fronts moving toward the negative k direction and it is referred to as regressive wave, although this nomenclature is not universally used. In order to match the general solution to the initial ﬁelds we must have

+ − E0 = E + E , + − N × B0 = −E + E , where we have used the fact that E± are orthogonal to N and N 2 = 1. This is a system of two algebraic equations that have a unique solution, namely, 1 1 E+ = E − N × B , E− = E + N × B , 2 0 0 2 0 0 and that completely deﬁnes the plane-wave solution corresponding to the given initial conditions. The initial conditions for E0,B0 determine the fraction of amplitude carried by the two branches of the dispersion relation.

Poynting theorem and energy conservation (informal version). An important aspect of Maxwell’s equations is the energy balance which is usually referred to as Poynting’s theorem [45]. We shall first derive it formally, assuming that we have a sufficiently regular solution of Maxwell’s equations. Mathematically this is a conservation law for the L2-norm of the electromagnetic field (E,B), again treating both fields as components of the same object. The result follows from the two Maxwell’s equations for the curl of the fields, namely Ampère-Maxwell and Faraday laws. When we scalar-multiply it on the left by the electric field, Ampère-Maxwell equation becomes ∂E E · − cE · ∇ × B = −4πJ · E. ∂t Analogously from Faraday law we have ∂B B · + cB · ∇ × E = 0, ∂t and thus,

1 2 2 2 ∂t |E| + |B| + c B · ∇ × E − E · ∇ × B = −4πJ · E. The second term on the left-hand side is an exact divergence, namely,

B · ∇ × E − E · ∇ × B = ∇ · (E × B).

50 This identity is valid for any two vector ﬁelds E and B and it can be proven by means of the Levi-Civita symbol ijk introduced in section 1.2; with the short-hand notation ∂i = ∂xi for derivatives, one computes

∇ · (E × B) = ikj∂i(EjBk)

= ijkBk∂iEj + ijkEj∂iBk

= kjiBk∂iEj − jikEj∂iBk = B · ∇ × E − E · ∇ × B. We have obtained a balance law for the norm of the vector (E,B), namely,

1 2 2 2 ∂t |E| + |B| + ∇ · (cE × B) = −4πJ · E. Dividing by the constant 4π we obtain Poynting’s theorem in form of a continuity equation, namely, ∂twem + ∇ · P = −J · E, (2.15a) where 1 w = |E|2 + |B|2, (2.15b) em 8π is energy density associated to the electromagnetic field, with |E|2/(8π) and |B|2/(8π) being the electric and magnetic energy densities respectively, while c P = E × B, (2.15c) 4π is the electromagnetic energy flux, also referred to as the Poynting vector. The factor 4π in the definition of the energy can only be understood if we consider electromagnetic fields together with particle dynamics which is addressed in section 2.2. For complex-valued solutions, we have to modify slightly the derivation, as well as the definition of the Poynting vector, but the result holds nonetheless. Instead of multiplying by E and B the evolution equations, we can scalar- multiply by the complex conjugate E, B and we find

E · ∂tE + B · ∂tB + cB · (∇ × E) − E · (∇ × B) = −4πE · J. In this case, we ﬁnd neither the total derivative of the norm of the ﬁeld, which is, e.g., |E|2 = E∗E = E · E where E∗ = tE is the Hermitian conjugate, nor a divergence term. However, we can consider the complex conjugate of the evolution equations and multiply those on the right by E and B respectively, thus obtaining

∂tE · E + ∂tB · B − c(∇ × B) · E + c(∇ × E) · B = −4πJ · E. The sum of the two foregoing equations amounts to

2 2 ∂t |E| + |B| + c B · (∇ × E) − (∇ × B) · E + c(∇ × E) · B − E · (∇ × B) = −4π(E∗J + J ∗E), where J ∗ = tJ and analogously E∗ = tE are the Hermitian conjugate (i.e., the transpose of the complex conjugate) of J and E, respectively. Now the terms in square brackets are divergences and we ﬁnd

∗ ∂twem + ∇ · P = − Re(J E),

51 where the Poynting vector for complex-valued ﬁelds amounts to c P = Re(E × B). 4π As an example, let us compute the Poynting vector for a plane wave. By using Faraday’s equation in the form B = ±N × E we ﬁnd c P = ± Re[E × (N × E], 4π but the refractive-index vector N is orthogonal to E and thus to E, so that

E × (N × E) = |E|2N − (E · N)E = |E|2N.

For a plane wave the energy ﬂux amounts to |E|2 P = ±c kˆ = ±ckwˆ , 4π em where we have used the fact that, for an electromagnetic wave the electric and 2 the magnetic energy are equal, hence wem = |E| /(4π). The energy ﬂux of a plane wave corresponds to the advection of the wave energy density at the speed of light c, toward the positive k-direction for the root ω > 0 (progressive wave) and opposite to it for the negative root (regressive wave).

2.2 Lorentz force and motion of an electrically charged particle. In this section we consider the motion of an electrically charged test particle in a given electromagnetic field. The word “test particle” indicates that we neglect the effect that the charged particle has on the electromagnetic field. Let ep and mp be the electric charge and mass of the considered test particle. The charge is usually a multiple of the elementary charge e > 0, which is defined so that −e is the charge of an electron; hence, ep = Zpe where Zp is an integer. The Lorentz-force law states that the force acting on a charged particle in presence of an electromagnetic field (E,B) is given by FL(t, x, v) = ep E(t, x) + v × B(t, x)/c , (2.16) where x ∈ Ω is the position of the particle and v ∈ R3 is its velocity. If no other force acts on the particle but the Lorentz force, the equation of motion for a non-relativistic test particle (Newton second law) takes the form dx = v, dt (2.17) dv e = p E(t, x) + v × B(t, x)/c. dt mp This is a system of first order equations for which we pose a Cauchy problem with generic initial conditions

x(0) = x0, v(0) = v0. In virtue of theorem 1.1, if the electromagnetic field is locally Lipschitz uniformly in time, we have a solution (x, v) defined in a possibly small interval (−ε, +ε) and such that v ∈ C1, but x ∈ C2 due to the first equation.

52 We can check how the Lorentz force affect the kinetic energy of a particle, which is defined by 1 K(v) = m v2. 2 p Along a solution we have d dv K(v) = m v · = e v · E(t, x). dt p dt p We can immediately see that, due to the cross-product structure, the magnetic part of the Lorentz force does not do work on the particle and thus it does not change its energy. The electric field on the other hand can either accelerate the particle, when it is directed toward its velocity, i.e., v · E > 0, or decelerate it when it is directed opposite to its velocity. Instead of a single particle, let us consider a gas of many identical particles with phase-space distribution function f(t, x, v), cf. section 1.6 for the definition. In view of the results of kinetic theory, cf. section 1.6, we already know that the contribution to heat sources of the Lorentz force, as computed by means of equation (1.25g), is zero. Therefore, the only effect of the Lorentz force on the energy balance is the work done by the average force, namely, Z epv · E(t, x)f(t, x, v)dv = J · E, R3 where the last identity follows on noting that Z J(t, x) = ep vf(t, x, v)dv = epn(t, x)u(t, x) (2.18) R3 defines the flux of electric charge, and thus the current density, in terms of the distribution function. This result justify the choice of the constants in the Poynting theorem (2.15), in which the term J · E appears with a minus sign: When J ·E is positive, the energy is taken by the particles and therefore removed from the electromagnetic field. For the simple case of uniform, i.e., constant in both time and space, electromagnetic fields, there is a rather simple analytical solution for the motion of a charged particle. First, we note that, if E and B in equations (2.17) are independent of x, the equation for v decouples from that for x; once a solution for v is known, we can integrate the equation for x by quadrature. With uniform fields E,B and with B 6= 0, let us consider

dv ep ep 3 = E + v × B v(0) = v0 ∈ R . dt mp mpc In the case E = 0, the equation reduces to dv = ±ω v × ˆb, (2.19) dt c where the sign is determined by the sign of the electric charge of the particle,

|epB| ωc = > 0, (2.20) mpc

53 is referred to as the cyclotron frequency, and ˆb = B/|B| is the unit vector along the direction of the magnetic ﬁeld. We recognize the equation of rigid rotation around the direction of the magnetic ﬁeld. Without loss of generality, we can choose ˆb = t(0, 0, 1) and thus the equation for v becomes

v   v  d 1 2 v = ±ω −v dt  2 c  1 v3 0 from which the have v3(t) = v0,3 = constant, and

d2v 1 = −ω2v , dt2 c 1 which has general integral,

v1(t) = C1 cos(ωct) + C2 sin(ωct).

Then, 1 dv1 v2(t) = ± = ∓C1 sin(ωct) ± C2 cos(ωct). ωc dt

From the initial conditions we have the constants C1 = v0,1 and C2 = v0,2. We note the the projection of v onto the plane orthogonal to the magnetic ﬁeld has constant modulus. Indeed, we have

2 2 2 2 2 2 2 v1(t) + v2(t) = C1 + C2 = v0,1 + v0,2 = v⊥, (2.21)

p 2 2 where we have defined the constant value v⊥ = v1 + v2. The projection onto the direction of the magnetic field is also constant, namely, v3(t) = v0,3 = vk. Hence the velocity vector is moving on a cone with circular section and constant angle with the magnetic field. Upon re-orienting the x1-axis of the Cartesian reference system by a rotation around its x3 axis (the magnetic field direction), we can assume v0,1 = v⊥ and v0,2 = 0, so that  v1(t) = v⊥ cos(ωct),  v2(t) = ∓v⊥ sin(ωct), (2.22a)   v3(t) = vk, and integrating in time, we obtain that the particle position describes a spiral around the magnetic field direction  x1(t) = x0,1 + ρL(v⊥) sin(ωct),  x2(t) = x0,2 ± ρL(v⊥)(cos(ωct) − 1), (2.22b)   x3(t) = x0,3 + tvk, where ρL(v⊥) = v⊥/ωc > 0, (2.23) is referred to as the Larmor radius of the particle. Figure 2.4 show an example of the helical orbit of a particle.

54 Figure 2.4: Trajectories of two particles with positive (blue trajectory) and negative (green trajectory) electric charge in a uniform magnetic ﬁeld directed vertically. Both particle are in x = 0 at the initial time. The projection onto the x1-x2 (panel on the left-hand side) shows the circular motion, clockwise and counter-clockwise for the positive and negative charge respectively. The parallel velocity is such that vk/v⊥ = 0.1.

In presence of a uniform electric ﬁeld this solution is still mostly valid but the electric ﬁeld introduces a drift of the axis of the helix. In order to see that, let us introduce the change of variable

v = vE + w, (2.24) where the constant velocity E × B v = c , (2.25) E B2 is called E × B-drift velocity. We say that the E × B drift is ambipolar, i.e., it does not depend on the sign of the electric charge of the particle. Substituting the change of variables into the Lorentz force gives v × B (E × B) × B w × B E + = E + + , c B2 c and the second term on the right-hand side can be evaluated by means of the vector calculus identity (E × B) × B = B × (B × E) = −B2[E − ˆb(ˆb · E)]. Hence, v × B w × B E + = E ˆb + , c k c ˆ where Ek = b · E is the component of the electric field parallel to the magnetic field. The contribution from E × B-drift velocity cancels out exactly the component of the electric field in the subspace normal to the direction ˆb of the magnetic field. Then the equation for the new variable w reads (vE is constant), dw = e E ˆb ± ω w × ˆb, dt p k c

55 Figure 2.5: The same as in figure 2.4 but with a uniform electric field. The parallel electric field accelerates the particles in different direction, while the E × B-drift makes the center of the gyration move in the same direction for both charges (ambipolarity). The initial condition is such that vk = 0 at the initial time. The normalized electric field is cE/(B0w⊥) = (−0.05, 0.1, −2.0). which is the same as equation (2.19), apart for a parallel acceleration due to the parallel electric field. Again we can choose the third axis of the reference ˆ frame directed like b, but this time we orient the first axis toward v0 − vE = w0 so that w0,1 = w⊥ and w0,2 = 0. The conservation of perpendicular energy, p 2 2 equation (2.21), holds for the w-variables and w⊥ = w1 + w2 = constant. Then the solution for w in this coordinate system reads  w1(t) = w⊥ cos(ωct),  w2(t) = ∓w⊥ sin(ωct), (2.26a)   w3(t) = vk + epEkt, which differs from (2.22a) for the parallel acceleration only. If Ek 6= 0 the angle between w and the magnetic field direction is changing and the velocity is no longer moving on a cone. The actual particle velocity v is obtained by adding the E × B drift which is perpendicular to the magnetic field. At last, we obtain that the particle position describes a spiral motion around a drifting axis, namely, x(t) = x0 + vEt + y(t), (2.26b) and  y1(t) = ρL(w⊥) sin(ωct),  y2(t) = ±ρL(w⊥)(cos(ωct) − 1), (2.26c)  2  y3(t) = tvk + epEkt /2. The E×B-drift plays a crucial role in plasma physics particularly for strongly magnetized plasmas as those used in fusion experiments. In MHD the fluid velocity is closely related to the E × B drift. Figure 2.5 shows an example of drifting orbits.

56 2.3 Basic mathematical results for electrodynamics. We conclude this section with basic mathematical considerations that are central in electrodynamics: the boundary value problem for the Poisson equation, which has been used in both electrostatics and magnetostatics, and the Cauchy problem for Maxwell’s equations on the whole space R3.

Dirichlet problem for Poisson’s equation. In the physics discussion of both electrostatics and magnetostatics we have relied on the solution of the Poisson equation, − ∆u = f, (2.27) for a scalar ﬁeld u where f is a known source function. The unknown can be either the electrostatic potential φ, or a component of the magnetic vector potential A. We start with the following technical observation about the Laplace operator. Proposition 2.1. The function v(x) = 1/|x|a, for x ∈ Rd, a = d − 2, and 1 Rd d ≥ 3, is in Lloc( ) and

−∆v = (d − 2)Adδ, in sense of distributions, δ being the Dirac mass in x = 0 and Ad the area of the unit sphere Sd−1 in the d-dimensional space. Proof. The function v is smooth everywhere except in the origin x = 0, where 1 Rd there is an integrable singularity. Hence, v ∈ Lloc( ) and thus v is a distribution, i.e., an element of the space D 0(Rd) which is deﬁned as the space of ∞ Rd continuous linear functionals on C0 ( ). In sense of distributions, the Laplace operator of v is the linear functional ∞ Rd acting on ϕ ∈ C0 ( ) according to Z h−∆v, ϕi = −hv, ∆ϕi = − |x|−a∆ϕ(x)dx, Rd where hu, ϕi = u(ϕ) denotes the action of a distribution u ∈ D 0 on a test- ∞ Rd 1 Rd function ϕ ∈ C0 ( ). The integral is well-deﬁned since v ∈ Lloc( ) and ϕ has compact support. We compute the integral in polar coordinates (r, ϑ) where r = |x| is the radial coordinate and ϑ ∈ Sd−1 are coordinate on the unit sphere. The Laplacian of ϕ is spherical coordinates amounts to 1 ∂ h ∂ϕi 1 ∆ϕ = rd−1 + ∆ ϕ, rd−1 ∂r ∂r rd−1 ϑ where ∆ϑ is the part of the Laplace operator on the unit sphere. The volume element is dx = rd−1drdω, where dω is the surface element on the unit sphere Sd−1 so that Z Ad = dω, Sd−1

57 is the area of the unit sphere in the d-dimensional space. We have, Z ∆ϑϕ(r, ϑ)dω = 0, Sd−1 hence Z +∞ ∂ h ∂ψ i hv, ∆ϕi = r−a rd−1 dr, 0 ∂r ∂r where we have deﬁne Z ψ(r) = ϕdω. Sd−1 The integral can be evaluated by integration by parts

Z +∞ hv, ∆ϕi = a r−a−1rd−1ψ0(r)dr 0 Z +∞ = (d − 2) ψ0(r)dr = −(d − 2)ψ(0). 0 where the fact that a = d − 2 has been accounted for and ψ0 = dψ/dr. At last, we use Taylor formula ϕ(x) = ϕ(0)+rϕ1(x) to show that ψ(r) = Adϕ(0)+O(r), hence, h−∆v, ϕi = (d − 2)Adϕ(0), and ϕ(0) = hδ, ϕi. Proposition 2.1 is usually summarized in the statement that 1 E(x) = |x|2−d, (d − 2)Ad is the fundamental solution of the Laplace operator. In general, one has the following

Definition 2.1 (Fundamental solutions). A distribution E ∈ D 0(Rd) is the fundamental solution of the constant-coefficient partial differential operator P if the identity P E = δ holds in D 0(Rd). In three dimensions we have dω = sin θdθdφ where (θ, φ) are the standard latitude and longitude angles of spherical coordinates, the Laplace operator amounts to 1 h ∂ϕi 1 h 1 ∂ ∂ϕ 1 ∂2ϕi ∆ϕ = r2 + sin θ + , r2 ∂r r2 sin θ ∂θ ∂θ sin2 θ ∂φ2 from which we can deduce ∆ωϕ. The area is Z π Z 2π A3 = sin θdθdφ = 4π, 0 0 and −∆v = 4πδ. Physically by comparison with (2.3), one could regard the function v as the potential generated by a unitary point charge located at x = 0. If we have a

58 particle of charge ep in x = 0, the charge density is given by ρc = epδ and we have and the electrostatic potential is e V (x) = p , (2.28) |x| which is referred to as Coulomb potential. This is the electrostatic equivalent of Newton’s gravitational potential for a point mass. The associated electric field is e E(x) = −∇V (x) = p x,ˆ (2.29) |x|2 wherex ˆ = x/|x| is the unit vector in the radial direction. If a test particle of 0 charge ep is placed in this electric field, it experiences a force given by equation (2.16) with B = 0, namely, 0 epe F = p x,ˆ (2.30) C |x|2 which is referred to as Coulomb force: The force between two point charges is inversely proportional to the square of their distance and it is attractive if the 0 charges have opposite sign (epep < 0) and repulsive is they have the same same 0 sign (epep > 0). We can also make use of the fundamental solution of the Laplacian to construct solutions of equation (2.27) on Rd when the right-hand side is a distribution with compact support, that is, f ∈ E 0(Rd). The the convolution v ∗ f is well defined in D 0(Rd) and we have

−∆(v ∗ f) = (−∆v) ∗ f = (d − 2)Adδ ∗ f = (d − 2)Adf, where we have used the properties of the convolution and the fact that δ∗f = f. In addition, since a convolution with a smooth compactly supported function is regularizing, when f is smooth we have u ∈ C∞. We can summarize this result as follows. Theorem 2.2. For every f ∈ E 0(Rd), d ≥ 3 the convolution 1 u = v ∗ f, (d − 2)Ad ∞ Rd satisﬁes equation (2.27) in the sense of distributions. If f ∈ C0 ( ), then u ∈ C∞(Rd). We now turn to the Dirichlet problem in a bounded domain Ω ⊂ Rd with a smooth boundary ∂Ω, ( −∆u = f, in Ω, u = 0, on ∂Ω.

Lemma 2.3 (Green’s identities). For any two functions u, v ∈ C2(Ω) we have Z Z Z ∇u · ∇vdx = − u∆vdx + u∂nvdS, (2.31a) Ω Ω ∂Ω Z Z (v∆u − u∆v)dx = (v∂nu − u∂nv)dS, (2.31b) Ω ∂Ω where ∂n = n · ∇ and n is the unit outward normal on the boundary ∂Ω.

59 Proof. Equation (2.31a) follows from the identity

∇ · u∇v = u∆v + ∇u · ∇v, and the Gauss theorem. Reversing the role of u and v and subtracting the one gets equation (2.31b). The ﬁrst Green identity, equation (2.31a), implies uniqueness of a classical solutions [47, 48]. Proposition 2.4. A solution u ∈ C2(Ω) of the Dirichlet problem is unique.

2 Proof. Let ui ∈ C (Ω), i = 1, 2, be two solutions of the Dirichlet problem with the same right-hand side f ∈ C(Ω). Then, identity (2.31a) with u = v = u1 −u2 amounts to Z 2 |∇(u1 − u2)| dx = 0, Ω 2 and thus ∇(u1 − u2) = 0, pointwise since ui are in C . In addition, both functions must vanish on the boundary, hence u1 − u2 = 0. The second Green identity, equation (2.31b), allows us to write an integral representation of the solution in terms of Green’s functions. For the speciﬁc case of the Dirichlet problem on a bounded domain Ω with smooth boundary, the Green’s function can be constructed by correcting the fundamental solution according to

G(x, y) = E(x − y) + K(x, y), x ∈ Ω, y ∈ Ω

2 where for every x ∈ Ω, Kx = K(x, ·) ∈ C (Ω) is the solution of

∆Kx = 0, in Ω, with Kx(y) = −E(x − y), for y ∈ ∂Ω. If the correction K exists, we have the possibility to write an explicit solution in the form of a convolution with the source function plus a boundary integral. Proposition 2.5 (Green’s functions). If G is a Green function constructed as described above, then (i) G is C2 away from the diagonal {(x, y) | x = y},

1 (ii) Gx = G(x, ·) ∈ L (Ω),

0 (iii) −∆Gx = δx in D (Ω), and Gx|∂Ω = 0 for all interior points x ∈ Ω.

2 (iv) If u ∈ C (Ω) is a solution of −∆u = f, with u|∂Ω = g, then Z Z u(x) = G(x, y)f(y)dy − g∂nGxdS. Ω ∂Ω

Proof. Claim (i) and (ii) follow from the deﬁnition, since K ∈ C2(Ω) and E is ∞ Rd 1 Rd C on \{0} and Lloc on . As for (iii), for every interior point x ∈ Ω, Gx is continuous near the boundary and its restriction to ∂Ω is zero by construction; ∞ ∞ Rd in addition, for every ϕ ∈ C0 (Ω) ,→ C0 ( ) we have Z h−∆Gx, ϕi = h−∆E(x − ·), ϕi − ∆Kxϕdy. Ω

60 The last integral is zero since ∆Kx = 0 and change of variable in the distribution gives h−∆E(x − ·), ϕi = h−∆E, ϕ(x − ·)i = ϕ(x) = hδx, ϕi, since −∆E = δ. The last point comes from the second Green identity applied to the domain Ω \ B(x) where B(x) is a ball centered on an interior point 2 x ∈ Ω with radius so small that B(x) ⊂ Ω. If u is a C solution and G 2 is the Green function constructed above, we have u, Gx ∈ C Ω \ B(x) and equation (2.31b) with v = Gx gives Z Z Z − Gxfdy = − (Gx∂nu − u∂nGx)dS − u∂nGxdS Ω\B(x) ∂B(x) ∂Ω where we have used the fact that ∆Gx = 0 away from y = x, −∆u = f, and the boundary condition Gx = 0 on ∂Ω. Since f is uniformly continuous on Ω and Gx is integrable, Z Z Gxfdy → G(x, y)f(y)dy, Ω\B(x) Ω while Z Z (Gx∂nu−u∂nGx)dS = (E(x−y)∂nu(y)−u(y)∂nE(x−y))dS(y) ∂B(x) ∂B(x) Z + (Kx∂nu − u∂nKx)dS. ∂B(x) The last integral involves only continuous functions and thus it is bounded by d−1 CAd where C is the maximum of the integrand. In the remaining term we can change variable so that it reduces to 1 Z h 1 −(d − 2)i d−2 ∂ru − u d−1 dS, (d − 2)Ad ∂B(0) r r where we have used polar coordinates around the point x, and dS = rd−1dω, dω being the area element on the unit sphere. The normal derivative is the same as the derivative with respect to the radial coordinate r. The ﬁrst term in the integral goes like rdω and thus is O() for → 0+. The second term on the other hand gives a ﬁnite contribution, namely, 1 Z 1 Z udω = u(x) + (u − u(x))dω → u(x), Ad ∂B(0) Ad ∂B(0) since u is continuous. We have obtained that the limit for → 0+ of the second Green identity yields the claimed result. Although the representation of the solution in terms of Green’s functions can be quite useful, Green’s functions for complicated domains are not easily available. Moreover proposition 2.5 is not an existence result: Existence of a regular solution has been assumed. Therefore both for practical and theoretical reasons one usually consider the 1 weak formulation of the Dirichlet problem in the Sobolev space H0 (Ω), that is, 1 one looks for u ∈ H0 (Ω) such that 1 (∇u, ∇v)L2(Ω) = F (v), for all v ∈ H0 (Ω),

61 where the right-hand side can be as general as any continuous linear functional 1 R F : H0 (Ω) → , but usually has the form Z F (v) = fvdx, Ω for f ∈ L2(Ω). If u ∈ C2(Ω) is a solution of the Dirichlet problem with f ∈ C(Ω) and u|∂Ω = 0, then it solves the weak formulation as well. However weak solutions might not in general be smooth. 1 Existence and uniqueness of weak solutions in H0 (Ω) follows from Riesz representation theorem together with Poincaréinequality which is stated here without proof, cf. chapter 10 of Tartar [49], as well as Hunter’s lecture notes [47] and Evans textbook [48]. Lemma 2.6 (Poincaréinequality). If Ω ⊂ Rd is a bounded connected domain then there exists a constant C(Ω) depending on the domain Ω, such that 1 kukL2(Ω) ≤ C(Ω)k∇ukL2(Ω) for u ∈ H0 (Ω). This fact allows us to show that Theorem 2.7. Let Ω ⊂ Rd be a bounded connected domain. For every lin- 1 R 1 ear continuous functional F : H0 (Ω) → there exists a unique u ∈ H0 (Ω) satisfying 1 (∇u, ∇v)L2(Ω) = F (v), for all v ∈ H0 (Ω). 1 1 R Proof. Let us consider the bi-linear form a : H0 (Ω) × H0 (Ω) → defined by Z a(u, v) = ∇u · ∇vdx. Ω This is symmetric, non-negative and if a(u, u) = 0, then ∇u = 0 in L2 and by the Poincaréinequality u = 0 in L2. Hence a(·, ·) satisfies the conditions for a 1 scalar product. The space H0 (Ω) is a Hilbert space with the product

(u, v)H1(Ω) = (u, v)L2(Ω) + (∇u, ∇v)L2(Ω), which means that any Cauchy sequence in the topology induced by (·, ·)H1(Ω) 1 has a limit in H0 (Ω). In view of the Poincar´einequality we have 2 2 2 2 2 k∇ukL2(Ω) ≤ kukH1(Ω) = kukL2(Ω) + k∇ukL2(Ω) ≤ Kk∇ukL2(Ω), that is, the topology induced by the standard product is equivalent to the topology induced by the product a(·, ·). In particular, any Cauchy sequence for the product a(·, ·) is a Cauchy sequence for the standard product, and thus it has 1 1 a limit in H0 (Ω). This means that H0 (Ω) with the product a(·, ·) is a Hilbert space. In addition, a functional F , which by hypothesis is continuous with respect to the topology of the standard product, is continuous also in the topology of a(·, ·). We can invoke Riesz representation theorem for which any continuous linear functional F can be represented as a scalar product with a unique element 1 u ∈ H0 (Ω), that is, F (v) = a(u, v), 1 for all v ∈ H0 (Ω). Then u is the solution of the Poisson problem. 1 −1 Theorem 2.7 also show that −∆ is a bijective map : H0 (Ω) → H (Ω), −1 1 where H (Ω) is the space of continuous linear functionals on H0 (Ω).

62 Cauchy problem for symmetric hyperbolic systems in Hs(Rd). Let us ﬁrst consider the two Maxwell’s equations for the curl of the ﬁelds in the form

∂tE − c∇ × B = −4πJ, (2.32) ∂tB + c∇ × E = 0. This system belongs to the following class of first-order equations: Definition 2.2 (Linear symmetric hyperbolic systems [50]). Let d, n be positive d n×n integers, Ai : R × R → R matrix-valued functions, bounded with bounded derivatives, and f : R × Rd → Rn. The first-order partial differential equation

d X ∂tu + Ai(t, x)∂xi u = f, i=1 for u(t, x) ∈ Rn is a linear symmetric hyperbolic system if the symbol of the spatial operator deﬁned, for ξ ∈ Rd, by

d X A(t, x, ξ) = Ai(t, x)ξi, i=1 is a symmetric matrix. For the case of Maxwell’s equations (2.32), let d = 3 n = 6 and u = (E,B). By direct calculation we ﬁnd from (2.32),

3 −c∇ × B X 0 A˜i ∂ E = t , +c∇ × E A˜i 0 ∂x B i=1 i where A˜i are 3 × 3 blocks given by 0 0 0 0 0 −c  0 c 0 ˜ ˜ ˜ A1 = 0 0 c , A2 = 0 0 0  , A3 = −c 0 0 . 0 −c 0 c 0 0 0 0 0

The block structure of the matrices Ai immediately show symmetry, hence equations (2.32) constitute a symmetric hyperbolic system with constant coeﬃcients. In this case the symbol depends only on ξ and we write,

3 3 X R6×6 X A(ξ) = Aiξi ∈ ,A(∂x) = Ai∂xi , i=1 i=1 the latter being a formal short-hand notation only. Another important example of constant-coeﬃcient symmetric hyperbolic system comes from D’Alembert wave equation,

2 2 ∂t v − c ∆v = f, for a scalar ﬁeld v : R × Rd → R. We can in fact write it as the ﬁrst-order system ( ∂tv + c∇ · w = f,

∂tw + c∇v = 0,

63 where we have introduced a vector field w : R×Rd → Rd. By applying the time derivative to the first equation and substituting from the second, one recovers d D’Alembert equation for v. If ei ∈ R are the unit vectors in the direction of the axes, the evolution equation for the combined variable u = t(v, w) can be written as d t X 0 ei ∂u f ∂tu + c = , ei 0 ∂x 0 i=1 i which is a symmetric hyperbolic system, as claimed. Therefore if we address the Cauchy problem for constant-coefficient symmetric hyperbolic system, we address at once both Maxwell’s equations and the D’Alembert wave equation. For the general theory of linear symmetric hyperbolic system including varying coefficients, we refer to the book by Rauch [50]. Here, we want to study the Cauchy problem for the constant-coefficient case,

( d ∂tu + A(∂x)u = f, t ≥ 0, x ∈ R , (2.33) d u(0) = u0, t = 0, x ∈ R , on the whole space Rd. With this aim we shall employ the Fourier transform [51, Chapter VII]. We recall that the Fourier transform of a function u in the Schwartz space S (Rd) is deﬁned by Z uˆ(ξ) = Fu(ξ) = e−iξ·xu(x)dx, Rd the integral being absolutely convergent, andu ˆ(ξ) belongs to S (Rd), that is, Fourier transform is an endomorphism of the Schwartz space. One can show that it is actually an isomorphism, i.e., it is bijective. In the theory of distributions this allows us to extend the Fourier transform to an isomorphism on the space of tempered distributions S 0(Rd), which is the dual of the Schwartz space. For s ∈ R, let us look for solutions in the Sobolev spaces

Hs(Rd) = {u ∈ S 0(Rd) | (1 + |ξ|2)s/2uˆ(ξ) ∈ L2(Rd)}.

The natural norm on those spaces is given by Z 2 −d 2 s 2 kukHs(Rd) = (2π) (1 + |ξ| ) |uˆ(ξ)| dξ. Rd and with this norm, Hs(Rd) is complete and thus a Banach space. Since the Schwartz space S (Rd) is dense in L2(Rd), for every function u ∈ s d d 2 s/2 H (R ) we can ﬁnd φj ∈ S (R ) such that φj(ξ) approaches (1 + |ξ| ) uˆ(ξ) 2 d 2 −s/2 in L (R ) for j → +∞. Then ϕj(ξ) = (1 + |ξ| ) φj(ξ) is again a sequence of functions in S (Rd) and Z Z 2 s 2 2 s/2 2 (1 + |ξ| ) |ϕj(ξ) − uˆ(ξ)| dξ = φj(ξ) − (1 + |ξ| ) uˆ(ξ) dξ → 0. Rd Rd

−1 d Then the inverse Fourier transform F ϕj yields a sequence in S (R ) that ap- proximates u in Hs(Rd), that is, the Schwartz space S (Rd) is dense in Hs(Rd).

64 For constant-coeﬃcient equations, Fourier transform is a powerful tool since it turns diﬀerential operators into algebraic operators. In fact, we have

F(∂xi u) = iξiu.ˆ and thus d d X X F A(∂x)u = AiF(∂xi u) = i Aiξiuˆ = iA(ξ)û, (2.34) i=1 i=1 which justifies the definition of the symbol A(ξ) of the operator.

s d s d Theorem 2.8. If s ∈ R, u0 ∈ H (R ), and f ∈ C R,H (R ) , the Cauchy problem (2.33) has a unique solution u ∈ CR,Hs(Rd) ∩ C1R,Hs−1(Rd) which depends continuously on the data, namely, Z t 0 0 ku(t)kHs(Rd) ≤ ku0kHs(Rd) + kf(t )kHs(Rd)dt , t ≥ 0, (2.35) 0 with equality if f = 0. Proof. 1. First we prove existence for highly regular data, speciﬁcally for ∞ Rd ∞ R1+d u0 ∈ C0 ( ), and f ∈ C0 ( ).

If u ∈ C∞(R1+d) is a solution such that u(t, ·) ∈ S (Rd) then we can Fourier- transform the equation and, accounting for (2.34), ˆ ∂tuˆ(t, ξ) + iA(ξ)ˆu(t, ξ) = f(t, ξ), withu ˆ(0, ξ) =u ˆ0(ξ) at the initial time t = 0. Since this is an ordinary diﬀerential equation with a unique solution, there is one and only one such solution, and that is Z t −itA(ξ) −i(t−t0)A(ξ) ˆ 0 0 uˆ(t, ξ) = e uˆ0(ξ) + e f(t , ξ)dt , 0 with exp − itA(ξ) being a unitary matrix since A(ξ) is symmetric. We see thatu ˆ ∈ C∞(R1+d). The fact thatu ˆ(t, ·) ∈ S (Rd) follows from the identity

β α −iσA(ξ) X −iσA(ξ) β α ξ ∂ξ e g(ξ) = Cα,γ e ξ ∂ξ g(ξ), γ≤α

∞ d for all multi-indices α, β and g ∈ C (R ), Cα,β being constants depending only on the matrices Ai and σ ≥ 0. This can be shown by induction over α. The inverse Fourier transform then gives Z t 0 0 0 u(t, x) = U(t)u0(x) + U(t − t )f(t , x)dt , 0 where the propagator U(t): S (Rd) → S (Rd) is deﬁned by 1 Z U(t)v(x) = eiξ·x−itA(ξ)vˆ(ξ)dξ, (2π)d and we have exchanged the time integral with the inverse Fourier transform since the integrand is L1.

65 2. The regular solution constructed so far is such that 1 Z v(t, x) = 1 − ∆)s/2u(t, x) = eiξ·x(1 + |ξ|2)s/2uˆ(t, ξ)dξ, (2π)d belongs to C∞(R1+d) and, for every ﬁxed t, v(t, ·) ∈ S (Rd). We compute 1 Z ∂ v(t, x) = eiξ·x(1 + |ξ|2)s/2∂ uˆ(t, ξ)dξ t (2π)d t 1 Z = eiξ·x(1 + |ξ|2)s/2 − iA(ξ)ˆu(t, ξ) + fˆ(t, ξ)dξ (2π)d s/2 = −A(∂x)v(t, x) + (1 − ∆) f(t, x).

Then 1 ∂ |v(t, x)|2 = v(t, x) · ∂ v(t, x) 2 t t 1 X = − ∂ v(t, x) · A v(t, x) + v(t, x) · (1 − ∆)s/2f(t, x), 2 xi i i and integrating over space we have Z 1 d 2 s/2 kv(t)kL2(Rd) = v(t, x) · (1 − ∆) f(t, x)dx 2 dt Rd s/2 ≤ kv(t)kL2(Rd)k(1 − ∆) f(t)kL2(Rd), where Cauchy-Schwarz inequality in L2 has been used. This implies

d s/2 kv(t)k 2 Rd ≤ k(1 − ∆) f(t)k 2 Rd , dt L ( ) L ( ) hence Z t d h s/2 0 0i kv(t)kL2(Rd) − k(1 − ∆) f(t )kL2(Rd)dt ≤ 0, dt 0 or Z t s/2 0 0 kv(t)kL2(Rd) − k(1 − ∆) f(t )kL2(Rd)dt ≤ kv(0)kL(Rd). 0

Since kv(t)kL2(Rd) = ku(t)kHs(Rd), this proves estimate (2.35) for the case of ∞ data in C0 . From the derivation one should notice that the inequality becomes an equality if f = 0. In addition, for every T > 0 we have

Z +T 0 0 ku(t)kHs(Rd) ≤ ku0kHs(Rd) + kf(t )kHs(Rd)dt (2.36) −T uniformly for t ∈ [−T, +T ]. s d s d 3. Let us now consider generic data u0 ∈ H (R ) and f ∈ C R,H (R ) . Following the standard approximation argument [50], let us choose sequences

∞ Rd s Rd u0,j ∈ C0 ( ), u0,j → u0 in H ( ), ∞ Rd 1 R s Rd fj ∈ C0 ( ), fj → f in Lloc ,H ( ) .

66 For every j we can construct the solution corresponding to the regular data u0,j and fj as before, and for every j, k we have

∂t(uj − uk) + A(∂x)(uj − uk) = fj − fk, (uj − uk)|t=0 = u0,j − u0,k. Estimate (2.36) gives

sup kuj(t) − uk(t)kHs(Rd) t∈[−T +T ] Z +T ˆ 0 ˆ 0 0 ≤ ku0,j − u0,jkHs(Rd) + kfj(t ) − fk(t )kHs(Rd)dt , −T

s d and thus uj is a Cauchy sequence in C [−T, +T ],H (R ) . Completeness implies that there exists a limit u ∈ C[−T, +T ],Hs(Rd). As for the spatial operator, one has

s−1 d A(∂x)uj → A(∂x)u in C [−T, +T ],H (R ) .

Passing to the limit the equation one has that the weak time derivative ∂tu of the limit u is in C[−T, +T ],Hs−1(Rd) and the equation is satisﬁed in C[−T, +T ],Hs−1(Rd). Since T ≥ 0 is arbitrary we can extend the solution to CR,Hs(Rd) ∩ C1(R,Hs−1(Rd). 4. If u is the solution constructed above, then using

ku(t)kHd(Rd) ≤ kuj(t)kHs(Rd) + ku(t) − uj(t)kHs(Rd), and analogous inequalities for u0,j and fj, we can establish estimate (2.35) in general. 5. As for uniqueness, it is sufficient to show that zero is the only solution corresponding to zero initial condition and zero sources. If u is one such solution then estimate (2.35) implies u(t) = 0. Having established the well-posedness of equation (2.32), we can now consider the full system of Maxwell’s equations. We have already discussed formally that the two equations for the divergence of the field are actually identically satisfied provided that they are satisfied for the initial data. This argument apply to solutions in CR,Hs(Rd) as well.

s d 1 s−1 d Theorem 2.9. With initial data E0,B0 ∈ H (R ) and ρ ∈ C R,H (R ) , J ∈ CR,Hs(Rd) satisfying

s−1 d ∇ · E0 = 4πρ(0), in H (R ), s−1 d ∇ · B0 = 0, in H (R ), s−1 d ∂tρc + ∇ · J = 0, in C R,H (R ) ,

Maxwell’s equations (2.10) have a unique solution E,B ∈ CR,Hs(Rd) ∩ C1(R,Hs−1(Rd) and we have

Z t 0 0 Es(t) ≤ Es(0) + k4πJ(t )kHs(Rd)dt , t ≥ 0, 0

2 2 2 with equality if J = 0. Here, Es (t) = kE(t)kHs(Rd) + kB(t)kHs(Rd).

67 2 The inequality reduces to an exact equality if J = 0, and for s = 0, E0 (t) is proportional to the energy carried by the electromagnetic ﬁeld. Hence we have a mathematically precise version of the Poynting theorem (2.15) obtained formally in section 2.1. Proof. Theorem 2.8 applied to system (2.32), yields the unique solution E,B satisfying the two Maxwell’s equation for the curl of the ﬁelds and the estimate for Es(t) follows directly from the estimate (2.35) since with u = (E,B),

2 2 2 ku(t)kHs(Rd) = kE(t)kHs(Rd) + kB(t)kHs(Rd). We need to show that the equations for the divergence are identically satisﬁed under the hypotheses. In Fourier space equations (2.32) take the form

∂tEˆ(t, ξ) − icξ × Bˆ(t, ξ) = −4πJˆ(t, ξ),

∂tBˆ(t, ξ) + icξ × Eˆ(t, ξ) = 0.

Multiplying by the Fourier variable iξ we have ∂t iξ · Eˆ(t, ξ) − 4πρˆc = 0, ∂t iξ · Bˆ(t, ξ) = 0. and in the ﬁrst identity we have used the charge continuity equation in the form

∂tρˆc(t, ξ) + iξ · Jˆ(t, ξ) = 0,

s−1 d which holds in C R,H (R ) by hypothesis. We deduce that ∇ · E − 4πρc and ∇ · B ∈ CR,Hs−1(Rd) are both constant in time and thus equal to zero in view of the initial conditions.

68 3 From multi-ﬂuid models to magnetohydrodynamics

In this section we introduce MHD equations as a single fluid theory for plasmas obtained from a more fundamental multi-fluid model under appropriate conditions that will be clarified. One could as well write MHD equations directly on the basis of a bit of physics modeling without the detailed derivation. A simple physical argument to obtain MHD equations reads as follows. Let us consider an inviscid fluid which is electrically neutral but which carries an electric current J. We can describe it by Euler’s equations (1.30) with an appropriately chosen force term. In presence of an electromagnetic fields (E,B) the current density carried by the fluid produces a net Lorentz force ρf = J × B/c, where c is the speed of light in free space and Gaussian units are used throughout this note for electromagnetic quantities. Euler’s equations then read Dρ = −ρ∇ · u, Dt Du J × B ρ = −∇p + , Dt c Dp = −γp∇ · u. Dt We still need equations for the current density J and for the electromagnetic fields (E,B) which can be affected by the current carried by the conducting fluid. For low-frequency phenomena, the electromagnetic fields can be described by Maxwell’s equations without the displacement current term, namely, 4π ∇ × B = J, c ∂tB + c∇ × E = 0, ∇ · B = 0.

In view of the motion of the fluid, the standard Ohm’s law for an electrically conducting body, i.e., E0 = ηJ with resistivity η, takes the form u × B E + = ηJ, c where E0 = E + u × B/c is the electric field in the reference frame of the fluid motion. The foregoing equations form a closed system which constitutes in fact the resistive MHD model. This derivation is appealing due to its clear physical argument and its simplicity. It has, however, the usual disadvantage of qualitative physics modeling, that is, we cannot establish precisely (that is, quantitatively) the limits of validity of the proposed model and its relationship to other models in use to describe the same physical system. Therefore, one should prefer, when possible, a derivation which is obtained from other more fundamental models. In the case of MHD equations, let us start from a multi-fluid model of plasmas.

3.1 A model for multiple electrically charged ﬂuids. Plasmas are ionized gasses composed by charged particles of various species: electrons and ions of various elements. Each species of charged particles, labeled by the index

69 α, is characterized by the mass mα and the electric charge eα = Zαe where −e is electron change and Zα the charge state of the particle. For ions, the latter does not always coincide with the atomic number of the corresponding element, as the ion does not need to be completely ionized (this is the case for impurities in the edge of fusion plasmas, low temperature plasmas in general, and the chromosphere of the Sun). Due to different mass and electric charge, electromagnetic forces as well as gravity act on each different particle species in a different way. In general, it is therefore necessary to treat each species individually. In the context of fluid models, each species is regarded as a fluid with number density nα(t, x), fluid velocity uα(t, x) and temperature Tα(t, x), all occupying the same spatial domain Ω. In order to write the fluid equations governing the motion of each particle species, we need to compute the forces per unit of mass acting on the plasma fluid. Generally, we have electromagnetic forces due to an electric field E(t, x) and a magnetic field B(t, x), and in astrophysical plasmas, possibly, a gravitational acceleration field g(t, x). Each particle of the α-th species experience an acceleration,

eα h v × B(t, x)i aα(t, x, v) = E(t, x) + + g(t, x). mα c According to the arguments of section 1.6, the force per unit of mass on the α-th species is then, cf. equation (1.25d), h u × B i m n f = e n E + α + m n g + R , α α α α α c α α α where Rα is total contribution of collisions. The gravitational field is a potential field, namely, g = −∇Φg, where Φg is the gravitational potential. In most physics problems, the gravity field is externally imposed by the presence of a massive object such as the Sun or more generally a star, i.e., the gravitational field generated by the mass of plasma particles can be neglected. On the other hand, electromagnetic fields must be computed self-consistently from Maxwell’s equations including as sources the charge and current density generated by plasma particles, namely, X X ρc = eαnα,J = eαnαuα. (3.1) α α Therefore we write a system of fluid equations for each particle species, coupled to Maxwell’s equations, namely, ∂tnα + ∇ · nαuα = 0,

∂t(mαnαuα) + ∇ · (mαnαuαuα + πα) = −∇pα

uα×B + eαnα E + c + mαnαg + Rα, 3 3 ∂t 2 pα + ∇ · 2 pαuα + qα + pα∇ · uα + πα : ∇uα = Qα, (3.2)

∂tE − c∇ × B = −4πJ,

∂tB + c∇ × E = 0,

∇ · E = 4πρc, ∇ · B = 0,

70 where the index α runs over the set of all particle species and relation (1.25c) between internal energy and pressure has been accounted for. Viscosity tensors πα, collision forces Rα, heat fluxes qα, and heat sources Qα are assumed to be given by appropriate closure relations. At this level, we only assume that collisions do not change the total momentum or the total energy, i.e., they are such that X X Rα = 0, uα · Rα + Qα = 0. (3.3) α α If all the three components of vector fields are counted individually, model (3.2) comprises 5×number of species+8 equations. In low temperature plasmas such as those in the scrape-off layer of fusion devices or in the solar chromosphere, the number of species can be as high as one hundred and more. In most cases, however, one has a dominant ion species plus electrons, and possibly impurities, which have a very low concentration and can be treated as perturbations. Model (3.2) has an extremely rich dynamics, that includes phenomena with various time scales, ranging from very fast plasma oscillations and electromagnetic wave modes down to relatively slow plasma waves. Moreover, model (3.2) allows for the possibility of electrically charged plasmas (spatial integral of ρc here can be non-zero), which is a situation usually not encountered in both fusion and astrophysical plasmas (though one should mention that pure electron plasmas and particle beam dynamics constitute interesting and active fields of research). From a computational point of view as much as for physics understanding, it is convenient to optimize the considered model to the physical phenomena of interest, rather then solve an unnecessarily general model. E.g., electromagnetic waves in model (3.2) introduce stiffness in the problem, but do not play any role in the low-frequency dynamics of the plasma. Analytical theory and, in particular, asymptotic methods, allows us to obtain approximations of a given model (in this case model (3.2)) that are tailored to the problem at hand. Magnetohydrodynamics is one such model that can approximate the behavior of the solution of the multi-fluid model (3.2) under special conditions which we shall examine in the next sections. Before continuing however, it seems important to add a methodological remark. In building model (3.2) we have taken different physical systems, namely, multiple fluids together with electromagnetic fields, and coupled them “by hand” in the most physically reasonable way, namely, the total charge and current density carried by the fluids generate the electromagnetic fields which in turn act on the fluids via the Lorentz force. This coupling mechanism is physically sound, but that’s in general not enough: Individually, fluid equations as well as Maxwell’s equations, without external forces or sources, enjoy various conservation laws, energy in particular; for fluids, that is the very basis from which equation (1.20c) has been derived; for Maxwell’s equations energy conservation is guaranteed by Poynting theorem [45], cf. also section 2.1. It is natural to ask whether the coupled system also have similar conservation laws. If we are interested in the description of an isolated physical system, at least energy and momentum conservation should be satisfied. For open systems that interact with an environment, one has to check the energy and momentum balance. It is, in general, not guaranteed a priory that the coupling mechanism of choice

71 has all the physically relevant conservation laws. For the speciﬁc case of the multi-ﬂuid model (3.2) we can prove that this is the case.

Mass and electric charge conservation. The multi-ﬂuid model (3.2) implies the conservation of the total mass: If we multiply each of the continuity equations by the mass mα of the corresponding particles ad sum over all species we obtain X ∂tρ + ∇ · mαnαuα = 0, α which is a conservation law for the mass density X ρ = mαnα, α

The electric charge is also conserved; speciﬁcally, equations (3.2) implies the charge continuity equation ∂tρc + ∇ · J = 0, which follows by multiplying the particle continuity equations by eα and summing over the species. As discussed in section 2.1, this is a necessary condition for Maxwell’s equations.

Energy conservation. If the friction forces and heat sources satisfy the second condition in equation (3.3), we have conservation of the total energy, Z (total energy) = w(t, x)dx, Ω the total energy density w being given by X w = wα + wem, (3.4) α where 1 p 5 w = m n u2 + α + m n Φ , γ = , α 2 α α α γ − 1 α α g 3 is the energy density (kinetic plus internal plus gravitational) carried by the ﬂuid of species α, and |E|2 |B|2 w = + , em 8π 8π is the energy density carried by the electromagnetic ﬁeld, cf. equation (2.15b). Here, the internal energy is written in terms of pressure pα = nαkBTα, in virtue of (1.25c), and the adiabatic index γ. In order to prove energy conservation, we derive a continuity equation for w. For every species, we have, cf. equation (1.18),

1 2 pα ∂t( 2 mαnαuα + γ−1 ) 1 2 pα + ∇ · ( 2 mαnαuα + γ−1 )uα + pαuα + πα · uα + qα

= eαnαuα · E − mαnαuα · ∇Φg + uα · Rα + Qα.

72 Then,

1 2 pα ∂twα = ∂t( 2 mαnαuα + γ−1 ) + mαΦg∂tnα

1 2 pα = ∂t( 2 mαnαuα + γ−1 ) − mαΦg∇ · (nαuα), where the continuity equation has been accounted for. Combining the two foregoing equations, one can notice that the terms involving the gravitational potentials combine into an exact divergence, namely,

−mαnαuα · ∇Φg − mαΦg∇ · (nαuα) = −∇ · (mαnαΦguα), with the result that ∂twα + ∇ · uαwα + pαuα + πα · uα + qα = eαnαuα · E + uα · Rα + Qα, where only the electric field contributes to the right-hand side as the magnetic field cannot do work on charged particles. By summing over plasma species and using the second of conditions (3.3), one finds an energy balance equation for the fluid part of the model, namely, X X ∂t wα + ∇ · Γwα = J · E, (3.5) α α

where the fluxes are Γwα = uαwα + pαuα + πα · uα + qα. The only source of energy for the fluids comes from the work done by the electric field on the plasma current. On the other hand, we have Poynting theorem for Maxwell’s equations, cf. equation (2.15),

∂twem + ∇ · P = −J · E, where c P = E × B, 4π is the Poynting flux. The sum of fluid energy balance and Poynting theorem gives ∂tw + ∇ · Γw = 0, (3.6) P where the total energy flux is Γw = α Γwα + P. This is a continuity equation for the total energy density of the system and, upon integrating it over the domain Ω with boundary conditions such that Γw · n = 0, one obtains that the conservation of the total energy as claimed. The crucial point is that the energy density per unit of time transferred to the fluid by the electric field, is equal to the energy density per unit of time lost by the electromagnetic fields, so that those energy-exchange terms cancel each other in the total energy balance.

Momentum conservation. The total momentum density is deﬁned by (not to be confused with the current density J)

X E × B j = j + , (3.7) α 4πc α

73 where jα = mαnαuα and the momentum density carried by the electromagnetic field is P/c2. The sum over the species of each fluid momentum balance equation gives, X X J × B ∂ j + ∇ · Γ = ρ E + + ρg, (3.8) t α jα c c α α where the momentum fluxes are Γjα = mαnαuαuα + πα + pαI, I being the identity tensor, and the first of conditions (3.3) have been accounted for. From Maxwell’s equations on the other hand, we have

∂tE × B = c(∇ × B) × B − 4πJ × B,

E × ∂tB = −cE × (∇ × E), so that ∂ E × B 1 J × B = (∇ × B) × B − E × (∇ × E) − . ∂t 4πc 4π c We can make use of the vector calculus identity

(∇ × B) × B = −∇(B2/2) + B · ∇B = −∇(B2/2) + ∇ · (BB), where we have used ∇ · B = 0, and analogously

E × (∇ × E) = ∇(E2/2) − E · ∇E = ∇(E2/2) − ∇ · (EE) + (∇ · E)E 2 = ∇(E /2) − ∇ · (EE) + 4πρcE, where we have used the Gauss law ∇ · E = 4πρc. Hence the electromagnetic momentum density satisﬁes the balance equation

∂ E × B |E|2 + |B|2 EE + BB J × B + ∇ · I − = −ρ E − . (3.9) ∂t 4πc 8π 4π c c The sum of the ﬂuid momentum balance (3.8) and the electromagnetic momentum balance (3.9) gives ∂tj + ∇ · Γj = ρg, (3.10) where the momentum density ﬂux,

X |E|2 + |B|2 EE + BB Γ = Γ + I − , j jα 8π 4π α is the sum of the ﬂuid stress tensors plus the Maxwell’s stress tensor. The total momentum of the system is conserved provided that g = 0. Of course, gravity breaks momentum conservation since it is an external force.

3.2 Quasi-neutral limit. Model (3.2) hides a very small parameter that can be exposed by scaling the equations. That means writing the equations in terms of normalized variables, with normalization constants chosen to represent

74 the typical magnitude of the corresponding variable for the speciﬁc physics process under consideration. The following scaling argument has been applied by Degond, Deluzet and Savelief [52] in the context of asymptotic preserving schemes for the Euler- Maxwell’s system. With normalized variables denoted by over-bars, we write

t = τt, x = Lx, nα = Nenα, uα = V uα,Tα = T T α, −1 pα = NekBT pα, πα = NekBT πα,Qα = τ NekBT Qα, qα = NekBTV qα, k T c k T E = B E,B = B B,J = eN V J, ρ = eN ρ , eL V eL e c e c V N MV g = g, R = e R , m = Mm , e = ee = eZ , τ α τ α α α α α α where τ, L, and V are the time, space, and velocity scales, respectively, while we have used a reference value for the electron density Ne as a normalization scale for all nα. The typical energy scale in a plasma is given by the thermal energy, namely, kBT , T being the temperature scale; therefore, pressure, viscosity, heat sources, and heat fluxes have all been normalized using this energy scale. Both electromagnetic fields (c.g.s units are used here) have the dimensions of energy per unit of electric charge per unit of length, but we have scaled the magnetic field by c/V , i.e., we normalize B/c rather then B. At last, the gravitational acceleration g is normalized to the natural acceleration scale, friction Rα to the natural scale for momentum variation, with the mass scale M being, e.g., the mass of the ion species with the highest concentration. The substitution of normalized variables into system (3.2) gives

Ne NeV τ ∂tnα + L ∇x · nαuα = 0, 2 MNeV MNeV NekB T τ ∂t(mαnαuα) + L ∇x · (mαnαuαuα) + L ∇x · πα + ∇xpα = NekB T MNeV + L eαnα E + uα × B + τ mαnαg + Rα , NekB T 3 τ ∂t 2 pα + NekB TV 3 NekB T L ∇x · 2 pαuα + qα + pα∇x · uα + πα : ∇xuα = τ Qα,

kB T 2 kB T eLτ ∂tE − c eL2V ∇x × B = −4πeNeV J, kB T kB T c eLV τ ∂tB + c eL2 ∇x × E = 0, kB T eL2 ∇x · E = 4πeNeρc,

∇x · B = 0.

At this point, we introduce crucial physical assumptions relating the various scales. First, we recall the deﬁnition of a basic plasma parameter, the Debye length (c.g.s. units) r kBT λD = 2 , (3.11) 4πe Ne which controls how electrons shield an ion charge. Then, we can state the basic assumptions: L V λ τ = ,MV 2 = k T, = D . (3.12) V B c L

75 The first assumption relates the time scale τ to the advection time scale L/V . Although natural for fluid equations, this scaling has an important consequence on Maxwell’s equations. The second assumption sets the value of the velocity scale to the typical thermal speed, p V = vth = kBT/M, and thus simplifies the scaling of the momentum and heat transport equations. The third assumption is less obvious since the two dimensionless parameters V/c and λD/L are physically independent. Setting them equal to each other is justified a posteriori: One obtains that in the Ampère-Maxwell law, ∇x × B and J have comparable magnitude. In fact, with assumptions (3.12), the scaled system becomes ∂tnα + ∇x · nαuα = 0, ∂t(mαnαuα) + ∇x · mαnαuαuα + πα = − ∇xpα + eαnα E + uα × B + mαnαg + Rα, 3 3 ∂t 2 pα + ∇x · 2 pαuα + qα + pα∇x · uα + πα : ∇xuα = Qα, (3.13)

∂tE − ∇x × B = −J,

∂tB + ∇x × E = 0,

∇x · E = ρc,

∇x · B = 0, which depends on a single dimensionless parameter

λ2 V 2 = D = 1. (3.14) L2 c2

For most plasmas, is extremely small since the Debye length λD is usually very short as compared to the typical spacial scale L and the relevant velocity scale is negligible as compared to the speed of light. The parameter multiplies the displacement current term in the Ampère-Maxwell law and the divergence of the electric field in the Gauss law. The fact that the displacement current is scaled by follows from the first of assumptions (3.12) which rules out the possibility of high-frequency oscillations of the electric field. The scaling of the Gauss law on the other hand, implies that the charge density ρc is negligible. The remaining equations in the system are unchanged by the scaling. We are interested in the limit → 0+, which is referred to as the quasi- neutral limit, since formally ρc = O(), i.e., the plasma is neutral at the leading order in → 0+. The idea is that since the physical value of , although finite, is very small, the limit solution for → 0 will be a good approximation of the full solution. By such an approximation one hopes to obtain a system of equations which is on one hand, computationally simpler and on the other hand, closer to the physics of interest (exposing only essential terms) than the original system. In this case, the formal limit of the scaled system could be obtained by setting = 0 directly into the equations. However it is instructive to follow step-by-step the general procedure of formal asymptotic expansions.

76 As a general rule, one should not approximate directly the equations. Rather one should try to construct an approximation of the solution. In this case, we try an asymptotic expansion in powers of the parameter > 0, namely,

+∞ X χ(t, x) ∼ jχj(t, x), j=0 where χ(t, x) stands for any of the normalized unknown of the scaled system, viewed as a function of > 0. The series is not required to converge absolutely, thus we write “∼” instead of “=”. The precise deﬁnition of “∼” is given in in section 1.6 where the expansion of the phase-space distribution function is considered. (The interested reader can refer to Boyd’s paper [53] for an instructive discussions on asymptotic series). The asymptotic series for each variable is substituted into the system and terms with the same power of are collected. The leading order terms, corresponding to j with j = 0, give a closed system for the leading term χ0 in the asymptotic series of each normalized quantity, namely,

0 0 0 ∂tnα + ∇x · nαuα = 0, 0 0 0 0 0 0 ∂t(mαnαuα) + ∇x · mαnαuαuα + πα = 0 0 0 0 0 0 0 − ∇xpα + eαnα E + uα × B + mαnαg + Rα, 3 0 3 0 0 0 0 0 0 0 0 ∂t 2 pα + ∇x · 2 pαuα + qα + pα∇x · uα + πα : ∇xuα = Qα,

0 0 X 0 0 ∇x × B = J = eαnαuα, α 0 0 ∂tB + ∇x × E = 0, 0 X 0 0 = ρc = eαnα, α 0 ∇x · B = 0, (3.15) which is formally identical to the full scaled system except for two equations: Ampère-Maxwell law (which is now without displacement current) and Gauss law (which is reduced to the quasi-neutrality condition). As anticipated this could have been obtained by setting = 0 in the scaled system. We see that, in this regime, the plasma must be quasi-neutral, i.e., the total charge density is zero to the lowest order in → 0. This, however, does not imply that the lowest order electric field E0 must be divergence-free, as one might expect from Gauss law. This apparent paradox is easily understood computing the equations for the first order correctors, i.e., the terms χ1 in the asymptotic series. We can consider only the correctors for the electromagnetic fields, namely,

0 1 1 ∂tE − ∇x × B = −J , 0 0 ∂tB + ∇x × E = 0, (3.16) 0 1 ∇x · E = ρc . We see that the time-variation of the zero-order electric ﬁeld produces a displacement current that adds to the ﬁrst-order current as a source for the corrector

77 B1 of the magnetic field. In the same way, the divergence of the zero-order 1 electric field defines a charge density ρc which corresponds to a small charge separation (of electrons and ions) in the plasma. Thus, Ampère-Maxwell and Gauss laws are satisfied as they should be, even though the lowest order field E0 has a non-zero time derivative and divergence. The quasi-neutral limit has a very important consequence on the conserved energy which is critical to understand energetics in MHD. Let us consider the electromagnetic energy in terms of normalized quantities. The natural scale for 2 energy density is MNeV = NekBT , hence,

2 2 2 |E| |B| h kBT 1 2 c kBT 1 2i + = NekBT 2 2 |E| + 2 2 2 |B| 8π 8π 4πe NeL 2 V 4πe NeL 2 h 1 i = N k T |E|2 + |B|2 , e B 2 2 and formally passing to the limit → 0 yields

|E|2 |B|2 h 1 i 1 + = N k T |E|2 + |B|2 → N k T |B0|2. (3.17) 8π 8π e B 2 2 2 e B Surprising as it might seems, the electric ﬁeld does not contribute to the lowest- order electromagnetic energy in the quasi-neutral limit. Similarly, the electromagnetic momentum density is a ﬁrst order quantity in :

E × B N k T λ2 P/c2 = = e B D E × B = N MV E × B → 0. (3.18) 4πc V L2 e Upon restoring physical units in system (3.15), we obtain ∂tnα + ∇ · nαuα = 0, ∂t(mαnαuα) + ∇ · mαnαuαuα + πα =

uα×B − ∇pα + eαnα E + c + mαnαg + Rα, 3 3 ∂t 2 pα + ∇ · 2 pαuα + qα + pα∇ · uα + πα : ∇uα = Qα,

4π X (3.19) ∇ × B = J = e n u , c α α α α

∂tB + c∇ × E = 0, X 0 = ρc = eαnα, α ∇ · B = 0, where for simplicity of notation we have dropped the superscript “0” and it is implied from now on that all quantities are identified with their quasi-neutral limit. According to equation (3.17), the electric field energy density does not contribute to the lowest order in the quasi-neutral limit, hence, the energy w defined in equation (3.4) becomes (in physical quantities)

2 X |B| 1 pα w = w + , w = m n u2 + + m n Φ . (3.20) α 8π α 2 α α α γ − 1 α α g α

78 System (3.19) preserves this energy, that is, the asymptotic expansion carried out above is energetically consistent. In order to prove this claim, let us notice that equation (3.5) still holds true, as the fluid equations in system (3.19) are unchanged, but Poynting theorem must be replaced by an equation for the magnetic energy density. Ampère’slaw scalar-multiplied by the electric field E gives c E · ∇ × B = E · J, 4π while from Faraday’s law one gets

∂ B2 c + B · ∇ × E = 0. ∂t 8π 4π The diﬀerence of the two equations yields

∂ B2 + ∇ · P = −E · J, ∂t 8π which, together with the energy balance (3.5) for the fluid energy, yields a continuity equation for energy (3.20). As for the momentum balance, in the quasi-neutral limit the electromagnetic momentum density is a first order quantity, cf. equation (3.18), so that the lowest order total momentum density is (in physical units), X X j = jα = mαnαuα. (3.21) α α The limit system (3.19) preserves the limit (3.21) of the total momentum. In fact, equation (3.8) still holds with the difference that quasi-neutrality cancels the electric force, namely

X J × B ∂ j + ∇ · Γ = + ρg. t jα c α For the J × B force, we have

J × B 1 B2 BB = (∇ × B) × B = −∇ + ∇ · , c 4π 8π 4π which is in divergence form, so that the total momentum balance (3.10) still holds in the quasi-neutral limit with j given by (3.21) and ﬂux

X |B|2 BB Γ = P + I − , j α 8π 4π α where only the magnetic part of Maxwell’s stress tensor enters. The charge continuity equation (2.1) now becomes

∇ · J = 0, (3.22) which is implied by Amp`ere’slaw as well as by particle continuity equations together with charge quasi-neutrality ρc = 0. System (3.19) requires some more care. First, it does not feature an explicit equation for the electric ﬁeld E. Second, equations for particle number densities

79 nα might not be consistent with the charge quasi-neutrality condition. At last, the constraint (3.22) might not be consistent with the momentum equations for the fluid velocities uα that define the current density. We therefore have to wonder if the limit problem (3.19) is well posed, or at least we have to check that there are sufficient equations to determine all the variables. With this aim, let us consider an equivalent reformulation of system (3.19). One can notice that Ampère’slaw can be used to express the current J in terms of the magnetic field, and J thus defined satisfies automatically condition (3.22). Then particle continuity equations yield ∂tρc = 0 which means that the quasi- neutrality constraint ρc = 0 is satisfied at all time, provided that it is satisfied by the initial conditions. The critical point is therefore obtaining an equation for the electric field such that Ampère’slaw is consistent with the sum over α of the momentum balance equations, so that c X J = ∇ × B = e n u , 4π α α α α and all constraints are satisfied. If one multiples the momentum balance equation of the species α by the charge-to-mass ratio eα/mα and sum over the species index α, the result is

2 X eα X e nα uα × B X eα ∂ J + ∇ · Γ = α E + + R , t m jα m c m α α α α α α α

where we have introduced the total stress tensor Γjα = mαnαuαuα + πα + pαI for simplicity, and the gravitational force term vanishes on account of quasi- neutrality. On the other hand, from Amp`ereand Faraday laws we have

c c2 ∂ J = ∇ × (∂ B) = − ∇ × (∇ × E), t 4π t 4π and thus

X 1 h∇ · Γj uα × B Rα i ∇ × (∇ × E) + d−2E = α − − , (3.23) p d2 e n c e n α α α α α α where we have deﬁned the characteristic lengths,

−2 X −2 dα = c/ωp,α, dp = dα , (3.24) α that are referred to as the skin depth of the species α and the plasma skin depth, respectively. Here, the quantity s 2 4πeαnα ωp,α = , mα is the plasma frequency of the species α (c.g.s. units). Equation (3.23) has the form of a vectorial Helmholtz equation for the electric ﬁeld E and, together with appropriate boundary conditions, fully determines the electric ﬁeld given the skin depths and the right-hand side.

80 This suggests the following reformulation of system (3.19), where equation (3.23) is added to the system: ∂tnα + ∇ · nαuα = 0, ∂t(mαnαuα) + ∇ · mαnαuαuα + πα =

uα×B − ∇pα + eαnα E + c + mαnαg + Rα, 3 3 ∂t 2 pα + ∇ · 2 pαuα + qα + pα∇ · uα + πα : ∇uα = Qα,

4π ∇ × B = J, c (3.25) ∂tB + c∇ × E = 0, ∇ · B = 0,

−2 X 1 ∇·Γjα uα×B Rα ∇ × (∇ × E) + dp E = 2 − − , dα eαnα c eαnα α X 0 = ρc = eαnα, α X J = eαnαuα. α Since we have shown that the new equation for E is implied by the quasi-neutral multi-fluid model (3.19), we have that any solution of the original quasi-neutral multi-fluid model is also a solution of the reformulated model (3.25). The advantage of the reformulation is that now we can formally prove that the the last two equation in system (3.25) are just constraints on the initial conditions. Specifically we claim that, if at the initial time t = 0 we pose initial conditions such that X X c e n = ρ = 0, and e n u = J = ∇ × B, α α c α α α 4π α α then the same conditions holds for later time t ≥ 0. Hence, three equations in the system are actually automatically satisfied and we do not need to solve them explicitly. We have already observed that, if the claim holds for the current, then the particle continuity equations imply that ρc is constant and thus zero for t ≥ 0. It is therefore sufficient to prove the claim for the current. With this aim, we consider again the sum over particle species of the momentum-balance equations weighted by the charge-to-mass ratio eα/mα. With the same steps as before, we can write the result in the form 2 X c h 1 X 1 ∇ · Γj − Rα uα × B i ∂ e n u = E − α − . t α α α 4π d2 d2 e n c α p α α α α P It is worth noting that we cannot identify α eαnαuα with J yet, as that is the equation we want to prove. However, now we know the equation for the electric field which implies that the right-hand side is equal to c2 c − ∇ × (∇ × E) = ∇ × (∂ B), 4π 4π t

81 where Faraday’s law has been accounted for. Therefore we have X c ∂ e n u − ∇ × B = 0. t α α α 4π α Using the Ampèrelaw as a definition of J, we have X ∂t eαnαuα − J = 0, α c which means that J = 4π ∇ × B is consistent with the sum of eαnαuα, provided that the initial conditions satisfy Ampèrelaw, namely, 4π X ∇ × B = e n u , c α α α α which is a physically reasonable request in the quasi-neutral limit. Therefore the last two equations in system (3.25) are identically satisfied provided that the initial conditions are properly chosen. In addition, this shows that a solution of the reformulated system (3.25) is also a solution of the original quasi-neutral model (3.19), which is then equivalent to the reformulated one. Although this argument is purely formal, it shows that system (3.25) has hope to be well posed, or at least we have enough equations to determine all the variables compatibly with the constraints. From now on we always implicitly assume that the initial conditions are well prepared in the sense explained above. Therefore, system (3.19) and system (3.25) are regarded as two equivalent formulations of the quasi-neutral multi-fluid model for plasmas.

3.3 From multi-fluid to a single-fluid model. Let us consider the sub- system of equations for fluid variables nα, uα, and Tα in the fluid models of section 3.1 and 3.2. It is remarkable that the multi-fluid equations imply a single-fluid model obtained by averaging over the species. Let us begin from the definition of the variables that describe such a single fluid. The mass density ρ of the single fluid is given by the total mass density of the plasma, namely, X ρ = mαnα, (3.26) α and this is intuitively natural. On the other hand, the single-fluid velocity u is identified with the velocity of the center of mass of fluid elements, that is X ρu = mαnαuα. (3.27) α The single-fluid total energy density is given by the sum of the energies of the fluid element of each species, namely

X X 1 pα w = m n u2 + + m n Φ . α 2 α α α γ − 1 α α g α α The kinetic energy term is X 1 X 1 X 1 m n u2 = m n (u − u + u)2 = m n (u − u)2 + 1 ρu2, 2 α α α 2 α α α 2 α α α 2 α α α

82 from which we have

X 1 X pα 1 w = ρu2 + + m n (u − u)2 + ρΦ α 2 γ − 1 2 α α α g α α 1 p = ρu2 + + ρΦ , 2 γ − 1 g with γ − 1 = 2/3 and the single-ﬂuid pressure is given by

X 1 p = p + m n (u − u)2. (3.28) α 3 α α α α We can now proceed with the derivation of equations for the single-fluid variables ρ, u, and p from the multi-fluid system (3.25). The mass continuity equation for the single-fluid is obtained from the particle continuity equations in (3.25). Upon multiplying by mα and summing over species, one gets X X ∂t mαnα + ∇ · mαnαuα = 0, α α which reads ∂tρ + ∇ · (ρu) = 0, (3.29) where we have used definition (3.26) of the total mass density and definition (3.27) of the center of mass velocity. In fact, this result justifies the choice of the center-of-mass velocity as the fluid velocity u. The sum over α of the momentum balance equations in (3.25) gives

X X ∂t mαnαuα + ∇ · (mαnαuαuα + πα) α α X J × B + ∇ p = + ρg, α c α P where the quasi-neutrality condition ρc = eαnα = 0 and the identity J = P α α eαnαuα (that are both implied by system (3.25)) have been accounted for, as well as the first of conditions (3.3). In addition, we compute X X mαnαuαuα = mαnα(uα − u + u)(uα − u + u) α α X = ρuu + mαnα(uα − u)(uα − u), α X 1 = ρuu + m n (u − u)(u − u) − m n (u − u)2I α α α α 3 α α α α 1 + m n (u − u)2I . 3 α α α The tensor in square bracket is symmetric and trace-free. We can therefore define the single-fluid viscosity X 1 π = π + m n (u − u)(u − u) − m n (u − u)2I , (3.30) α α α α α 3 α α α α

83 so that X X ∇ · (mαnαuαuα + πα) + ∇ pα = ∇ · (ρuu + π) + ∇p, α α where the single-fluid pressure p has been defined in (3.28). At last, the sum of momentum balance equations can be written in the form J × B ∂ (ρu) + ∇ · (ρuu + π) = −∇p + + ρg. (3.31) t c As a consequence of quasi-neutrality and conservation properties of the collision operator, cf. equation (3.3), the only forces acting on the plasma in the single- fluid picture are the classical J × B force due to a current flowing in presence of a magnetic field and (if present) gravity. At last, we need to address the transport of internal energy of the single- fluid. With this aim it is convenient to start from the balance equation for the total fluid energy, namely, equation (3.5), in which the energy flux is

X X 1 2 5 Γwα = [ 2 mαnαuαuα + mαnαΦguα + 2 pαuα + πα · uα + qα]. α α We need to recast the energy ﬂux in a better form. With this aim, we can make use of the identity

X 1 2 X 1 2 2 mαnαuαuα = 2 mαnα(uα − u + u) (uα − u + u) α α X 1 2 2 = 2 mαnα((uα − u) + 2u · (uα − u) + u )(uα − u + u) α 1 2 X = 2 ρu u + u · mαnα(uα − u)(uα − u) α 1 X 2 1 X 2 + 2 mαnα(uα − u) u + 2 mαnα(uα − u) (uα − u), α α together with X X X pαuα = pα u + pα(uα − u), α α α X X X πα · uα = πα · u + πα · (uα − u). α α α The combination of the foregoing identities and the deﬁnitions of single-ﬂuid pressure and viscosity, cf. equations (3.28) and (3.30), give

X 1 2 p Γwα = 2 ρu + γ−1 + ρΦg)u + π · u + up + q, α where the single-fluid heat flux is defined by

X 5 1 2 q = qα + 2 pα(uα − u) + πα · (uα − u) + 2 mαnα(uα − u) (uα − u) . (3.32) α

84 Correspondingly, the energy balance equation (3.5) takes the form of the usual total energy balance of ﬂuid dynamics, cf. equation (1.18),

1 2 p 1 2 p ∂t 2 ρu + γ−1 + ρΦg + ∇ · ( 2 ρu + γ−1 + ρΦg)u + π · u + up + q = J · E, (3.33) the only energy source being the J · E Ohmic heating term. We have therefore derived a single-fluid equations, namely equations (3.29), (3.31), and (3.33), from the multi-fluid model. One should note however that there is a difference with respect to ordinary fluids: In equation (1.18) the work done by the forces is just ρu · f where ρf is the acting force on the right-hand side of the momentum balance. In the case of plasmas, cf. equation (3.31), that would give u · (J × B)/c, which differs from the Ohmic heating term on the right-hand side of the energy balance (3.33). Therefore, a literal transcription of fluid equations to the case of plasmas (just by substitution of ρf with the J × B force plus gravity) would break energy conservation, cf.e Poynting theorem in section 3.1, leading to physically incorrect results. As a consequence the transport equation for the internal energy differs from that of ordinary fluids. We proceed as for equation (1.20c),

1 p h 1 p i ∂ ρu2 + + ρΦ + ∇ · ( ρu2 + + ρΦ )u + π · u + up + q t 2 γ − 1 g 2 γ − 1 g p h p i = ∂ + ∇ · u + q + p∇ · u + π : ∇u + ρu · ∇Φ t γ − 1 γ − 1 g

+ u · (ρ∂tu + ρu · ∇u + ∇ · π + ∇p), while the momentum balance equation (3.31) yields J × B ρ∂ u + ρu · ∇u + ∇ · π + ∇p = − ρ∇Φ . t c g By means of the two foregoing identities, equation (3.33) can be rewritten in the form of a transport equation for the single-ﬂuid internal energy, namely,

p p ∂t γ−1 + ∇ · γ−1 u + q + p∇ · u + π : ∇u = J · E − u · (J × B)/c. The rather complicated heat-exchange term on the right-hand side can be physically understood by means of the vector calculus identity

u · (J × B) = J · (B × u) = −J · (u × B), for which the heat transport equation can be written as

p p u×B ∂t γ−1 + ∇ · γ−1 u + q + p∇ · u + π : ∇u = J · (E + c ). (3.34) The right-hand side now can be understood as the Ohmic heating term J · E0 with the electric field E0 = E + u × B/c being the electric field transformed into the reference frame of the local fluid velocity u in the non-relativistic limit (cf. Jackson’s book [45] for Lorentz transformations of electromagnetic fields). In summary, we have shown for the multi-fluid model that the center-of- mass fluid, with variables defined by equations (3.26), (3.27) and (3.28), satisfies the equations of ordinary fluid dynamics given by equations (3.29), (3.31),

85 and (3.34). Let us note that this is an exact result: No approximations have been introduced. Such single-fluid equations, however, are not closed since (i) viscosity (3.30) and heat flux (3.32) depend on all the multi-fluid variables, and (ii) the electrodynamic quantity, particularly the electric field E, need to be computed from the electrodynamics part of model (3.25), which again depends on the multi-fluid variables. The simplest answer to the first point is to consider a situation where the Euler’s closure π = 0, q = 0 applies, cf. section 1.6, and that yields Euler’s equations for a compressible inviscid plasma, namely, Dρ + ρ∇ · u = 0, Dt Du J × B ρ = −∇p + + ρg, (3.35) Dt c D p γ u × B + p∇ · u = J · (E + ), Dt γ − 1 γ − 1 c where D/Dt = ∂t +u·∇ is the advective derivative. The validity of this system, which is just the Euler’s system (1.30) with the appropriate force and heating terms, is of course limited by the closure conditions. It is physically meaningful to consider a case where each species is close to a local Maxwellian, hence, πα ≈ 0 and qα ≈ 0 for all α, but this assumption alone does not suffice. By inspection of equations (3.30) and (3.32) we see that we need

uα ≈ u, (3.36) which means that all plasma species move with approximately the same velocity. Here, the symbol “≈” must be interpreted as “approximately equal as functions”, that is, not just the values of the functions but also the values of their derivatives is approximately the same. This, in particular, implies that the pressure is approximated by the Dalton’s law of partial pressures, cf. equation (3.28), X p ≈ pα, α thus neglecting the kinetic energy associated to the relative velocity uα−u. Con- dition (3.36) is of course very strong. We shall discuss this point in section 3.4 together with the electrodynamics part of the model. As a last remark, let us note that even if we can obtain a closed system for single-ﬂuid quantities, in general we cannot infer from a solution thereof a solution for the individual particle species. This is however possible for the speciﬁc case of an electron-ion plasma addressed in section 3.4.

3.4 The Ohm’s law for an electron-ion plasma. In a multi-fluid model, different species can have a significantly different dynamics. The calculation of section 3.3 shows in full generality that we can obtain a single-fluid model for the center-of-mass fluid provided that we take into account the stress and heat-flux due to the relative motion uα − u of the different species. In most cases, however, a single ion species (majority ions) accounts for most of the mass and positive electric charge in the plasma; other ion species

86 are impurities with small concentrations. Electrons provide the neutralizing negative charge. It is therefore natural to consider a plasma with a single ion species plus electrons, with the idea that additional impurities have to be treated as perturbations. In that case, the single ﬂuid quantities are, cf equations (3.26), (3.27), 0 = Zini − ne,

ρ = mini + mene

ρu = miniui + meneue,

J = ene(ui − ue), where Zi = ei/e is the charge state of the majority ion species; here, α = i refers to ions and α = e refers to electrons. It follows that, for a quasi-neutral electron- ion plasma, there is a one-to-one relation between single-fluid quantities (on the left-hand side) and multi-fluid quantities (on the right-hand side). Therefore, the single-fluid model and the multi-fluid model (which is referred to as two-fluid for an electron-ion plasma) are totally equivalent provided that the stress and heat flux due to the relative velocity are accounted for. Nonetheless, such an equivalent single-fluid formulation would be rather complicated so that the two- fluid model with electrons and ions treated independently should be preferred. At the price of some approximations, we can however derive a convenient single-fluid model which will eventually lead to magnetohydrodynamics. In fact with one single ion species, it is easier to satisfy condition (3.36), due to the small electron mass me/mi ≈ 1/1836. We have the exact relations, m h Z i J u = u + e i , i me mi 1 + Zi ene mi (3.37) J m h Z i J u = u − + e i . e me ene mi 1 + Zi ene mi

For the ion velocity, we see that ui ≈ u apart from a correction of order me/mi, so that ions well satisfy condition (3.36) as far as J/(ene) stays bounded. For the electron velocity, in addition to the same term of order me/mi, we find the J/(ene) velocity which accounts for the drift of electrons with respect to ions, necessary to sustain a current density. Hence, condition (3.36) for electrons requires |J| |u|, (3.38) ene which is much more difficult to verify. Under assumption (3.38), we can make use of Euler’s equations (3.35), the solution of which allows us to reconstruct a good approximation of the two-fluid variables by relations (3.37). As anticipated at the end of section 3.3, in order to close the single-fluid system, we need to write the electrodynamics part of model (3.25) in terms of single fluid variables only. The issue has to do with the equation for the electric field, which depends on multi-fluid variables. That is equivalent to, cf.

87 section 3.2,

X X eα ∂ J + ∇ · (e n u u ) + (∇ · π + ∇p ) t α α α α m α α α α α 2 X e nα uα × B X eα = α E + + R . m c m α α α α α For a quasi-neutral electron-ion plasma, we have X eαnαuαuα = ene(uiui − ueue) α∈{i,e} J J = ene uiui − (ui − )(ui − ) ene ene JJ = uiJ + Jui − ene JJ m 2 JJ = uJ + Ju − + e , me ene mi 1 + Zi eni mi where the ﬁrst of equations (3.37) has been accounted for in the last equality. Thereby, one gets

me h JJ i 1 2 ∂tJ + ∇ · uJ + Ju − + ∇ · (πe + peI) e ne ene ene u × B J × B Re me h ui × B − E + − − = Zini E + c enec ene mi c R 1 2m n JJ i + i − ∇ · π + p I + e e , i i me 2 ene eni 1 + Zi (ene) mi where all terms of order me/mi have been isolated on the right-hand side. The collision forces Rα for an electron-ion plasma have been computed by Braginskii [39, 43]. Here, we take into account only the friction force which tends to equalize electron and ion velocities, namely,

mene me −Ri = Re = (ui − ue) = J, τe eτe where [43] √ 3/2 3 me(kBTe) τe = √ , (3.39) 4 4 2πe neΛ is the electron collision time, and Λ is the Coulomb logarithm. With respect to the full Braginskii expression, thermal forces and plasma anisotropy have been neglected. With this expression for the collision force, we obtain

me h JJ i 1 2 ∂tJ + ∇ · uJ + Ju − + ∇ · (πe + peI) e ne ene ene u × B J × B me h ui × B − E + − − ηJ = Zini E + c enec mi c 1 2m n JJ i − ηJ − ∇ · π + p I + e e , (3.40) i i me 2 eni 1 + Zi (ene) mi

88 where the plasma resistivity is given by

me η = 2 . (3.41) e neτe Equation (3.40) is equivalent to a vectorial Helmholtz equation for the electric field, as discussed in section 3.2, and it is known as the generalized Ohm’s law for an electron-ion plasma. This form of the generalized Ohm’s law is entirely general: Apart from a specific choice of the collision forces, the only condition that has been used is quasi-neutrality. In magnetohydrodynamics, the generalized Ohm’s law is further simplified in order to capture the essential physics. The most straightforward approximation is neglecting the terms of order me/mi on the right-hand side of the Ohm’s law. With the assumption of electron (nearly) in local thermodynamical equilibrium, the electron viscosity πe is also neglected. The magnitude of the remaining terms is less obvious to establish, and different choices lead to different flavors of MHD, cf. appendix A. In standard MHD theory the Ohm’s law is approximated by u × B E + = ηJ, (3.42) c which is just the classical Ohm’s law E0 = ηJ written in the reference frame of the local fluid velocity, E0 = E + u × B/c being the Lorentz-transformed electric field in the weakly relativistic limit. Here, the electron inertia term, 2 proportional to me/(e ne) in the generalized Ohm’s law (3.40), and the Hall effect, given by the J × B term, have been neglected by condition (3.38). Last, the electron pressure gradient ∇pe is neglected compared to the electric force eneE. In correspondence to this choice of the Ohm’s law, the heating term in the pressure equation of the Euler’s system (3.35) becomes

J · E0 = ηJ 2, (3.43) which is the classic form of resistive heating. It is worth noting that the perpendicular (with respect to B) part of the single-ﬂuid velocity is entirely determined by the Ohm’s law (3.42) in terms of the electromagnetic ﬁelds and the current, namely,

B × (B × u) E × B B × J u = − = c + η . ⊥ B2 B2 B2 We see that in absence of resistivity (η = 0), the component of the fluid velocity perpendicular to the magnetic field is given by the celebrated E × B-drift velocity, namely, E × B v = c . (3.44) E B2 This velocity field plays an important role in plasma physics and can be understood by considering the motion of a particle of electric charge eα and mass mα under the influence of the Lorentz force, cf section 2.2.

89 3.5 The equations of magnetohydrodynamics. We have now built the necessary basis to state magnetohydrodynamics equations. Let us summarize the results obtained in the previous sections, in order to have a complete overview of the derivation. We have shown in section 3.3, that the quasi-neutral model (3.19), or equivalently its reformulated version (3.25), implies single-fluid equations (3.29), (3.31), and (3.34) exactly, provided that stresses and heat fluxes due to the relative velocity of different species are accounted for. For the specific case of an electron-ion plasma, such stresses and heat fluxes can be neglected provided that condition (3.38) is fulfilled. In addition, if all species are close to local thermodynamical equilibrium, Euler’s equations (3.35) apply in the sense that given a solution thereof one can construct electron and ion fluid variables by means of identities (3.37), and those are expected to be good approximations of a solution of the quasi-neutral two-fluid model. At last, we have the MHD form (3.42) of Ohm’s law which holds when election inertia, electron pressure, and Hall effect can be neglected. Under the conditions summarized above and for an electron-ion plasma, we can replace the multi-fluid equations in (3.25) by single-fluid Euler’s equations (3.35) coupled to Ohm’s law (3.42), with the result that Dρ + ρ∇ · u = 0, Dt Du J × B ρ = −∇p + + ρg, Dt c D p γ + p∇ · u = ηJ 2, Dt γ − 1 γ − 1 u × B (3.45) E + = ηJ, c ∂tB + c∇ × E = 0, 4π ∇ × B = J, c ∇ · B = 0.

This is a closed system of equations describing the dynamics of an ideal compressible electrically conducting quasi-neutral fluid with finite resistivity, that is referred to as magnetohydrodynamic equations or more precisely resistive magnetohydrodynamic equations. The derivation of MHD equations in the framework of multi-fluid theory allows us to state validity conditions, that are repeated and commented here for sake of completeness: • Quasi-neutrality and low-frequency dynamics, cf. equations (3.12), L V λ τ = ,MV 2 = k T, = D 1. (3.46a) V B c L The condition of the time-scale τ limits the range of frequencies where MHD applies. E.g., high frequency phenomena such as ion cyclotron, whistler, and electron cyclotron waves are well outside the applicability of MHD equations. The last two conditions on the other hand are rather weak, i.e., easy to satisfy in most fusion and astrophysical plasmas.

90 • Electron-ion plasma with small electron drift velocity, cf. equation (3.38),

|J| |u|. (3.46b) ene This is the most difficult condition. Within standard MHD large localized current densities (current sheets) can occur. Near a current sheet, J can be very large, signaling that electron motion is very much different from ion motion. Then, a two-fluid model is to be preferred to plain MHD. • In the MHD form (3.42) of Ohm’s law,

|∇pe| ene|E|, (3.46c)

while inertia and Hall effects have been dropped in virtue of (3.46b). By inspection of equations (3.45), one can see that condition ∇ · B = 0, in virtue of Faraday’s induction law, amounts to a constraint on the initial conditions only; we shall always assume that initial data are compatible with this constraint. In addition, the current density J and the electric field E are algebraically given in terms of the magnetic field and velocity. They can therefore be easily removed from the system. With reference to the current density, let us consider the J × B force in the momentum balance equation. By vector calculus,

J × B 1 B2 1 = (∇ × B) × B = −∇ + B · ∇B, (3.47) c 4π 8π 4π that is, the force generated by the magnetic field on the fluid amounts to two contributions. The first is the gradient of the magnetic-field energy density and has a similar effect as the fluid pressure: It produces a force directed from regions with high magnetic field intensity to regions with low magnetic field intensity; for this reason the magnetic-field energy density is also referred to as magnetic pressure. The dimensionless number comparing the fluid and the magnetic pressure is the plasma β parameter, defined by, 8πp β = . B2 A low value of the plasma β, e.g., β ≈ 10−3, 10−2, usually implies a moderate plasma temperature and density with a high magnetic field. Those are rather dilute plasmas that can be well confined by the magnetic field. Examples com- prise tokamak plasma and the low density regions of the solar corona (coronal holes [54]). On the other hand, high β, e.g., β ≈ 1, suggests that the fluid pressure can be comparable or even overcome the magnetic field. The solar wind, coronal loops and helmet streamers [16] are example of high-β plasmas. The second component of the J × B force, namely, B · ∇B is referred to as field-line bending force: One can notice that B · ∇B vanishes when B is constant along magnetic field lines, i.e., when the field lines are straight. Any curvature of the field lines activates this force. We can compare the order of magnitude of the field-line bending force with the advection terms u · ∇u in the momentum equation, by means of the scaling argument which is typical in fluid dynamics, cf.

91 also section 3.2; if ρ, B, and V are the typical magnitudes of density, magnetic ﬁeld, and velocity respectively, and L the scale of the gradient, we estimate

2 2 4πρ|u · ∇u| V 2 −2 2 B ≈ 2 = Mm = Al ,VA = , |B · ∇B| VA 4πρ where VA is the Alfvénspeed and the dimensionless number Mm = V/VA is referred to as the magnetic Mach number; equivalently, its inverse Al, the Alfvén number, is also used. For large magnetic Mach numbers (or small Alfvénnum- bers), the flow is dominated by its inertia and the effect of the magnetic field is less important, while the converse holds for small magnetic Mach numbers (or large Alfvénnumbers). As for the magnetic field, Faraday’s induction equation together with the Ohm’s law and Ampère’slaw amounts to

ηc2 ∂ B − ∇ × (u × B) = −∇ × ∇ × B , (3.48) t 4π which is refereed to as the (resistive) MHD induction equation. The term on the right-hand side can be written as

∇ × (u × B) = −u · ∇B + B · ∇u − (∇ · u)B, where we have used the fat that ∇ · B = 0. This is a hyperbolic operator describing the transport of the magnetic ﬁeld B with the ﬂuid velocity u. The left-hand side, on the other hand, is a parabolic operator,

T −∇ × (κη∇ × B) = ∇ · κη ∇B − (∇B) c2η = ∇ · κ ∇B − ∇B · ∇κ , κ = , η η η 4π which describes diffusion with diffusion coefficient κη. This identity can be proven by making use of the Levi-Civita tensor ijk for which we have X ijkkmn = δimδjn − δinδjm, k together with the condition ∇ · B = 0 in the last equality. An equivalent form of the induction equation then reads DB − B · ∇u − (∇ · u)B = ∇ · κ ∇B − (∇B)T . (3.49) Dt η The strength of the diffusion operator relatively to the hyperbolic terms can be estimated by a dimensionless parameter. Proceeding as above, we estimate

|u · ∇B| V B/L VL ≈ = , T 2 ∇ · κη ∇B − (∇B) κηB/L κη and deﬁne the dimensionless number VL 4πV L Rm = = 2 , κη c η

92 which is referred to as the magnetic Reynolds number in analogy with the classical Reynolds number of fluid dynamics, namely, VL Re = , µ which measures inertia as compared to viscosity in the Navier-Stokes equation (1.33). The regime of large magnetic Reynolds numbers (Rm 1) is particularly interesting since the resistivity (3.41) is inversely proportional to the collision time (3.39), which in turn scales like T 3/2: the hotter the plasma the larger the magnetic Reynolds number. In this regimes the resistivity can be neglected and the dynamics of the magnetic field is purely hyperbolic. This is called ideal magnetohydrodynamics: The plasma is ideal as a fluid (satisfies Euler’s equations) and as a conductor (zero electrical resistance). In the opposite regime (Rm 1), the magnetic field diffusion is dominant: Neglecting the hyperbolic terms in (3.48) removes the fluid velocity u from the induction equations, and the only coupling mechanism between the magnetic field and the fluid part of the system is provided by resistivity. Particularly, in the case of constant resistivity, the induction equation decouples from the fluid-dynamic part of the system. At last, let us summarize the equations for both resistive and ideal MHD. • Resistive MHD equations: Dρ + ρ∇ · u = 0, Dt Du 1 ρ = −∇p + (∇ × B) × B + ρg, Dt 4π D p γ κ (3.50a) + p∇ · u = η |∇ × B|2, Dt γ − 1 γ − 1 4π c2η ∂ B − ∇ × (u × B) = −∇ × (κ ∇ × B), κ = . t η η 4π

• Ideal MHD equations: Dρ + ρ∇ · u = 0, Dt Du 1 ρ = −∇p + (∇ × B) × B + ρg, Dt 4π (3.50b) Dp + γp∇ · u = 0, Dt ∂tB − ∇ × (u × B) = 0.

In both cases ﬂuid equations are coupled to the magnetic ﬁeld B, which is the only electrodynamic variable left in the problem. This should explain the name magnetohydrodynamics. In case of ideal MHD, the pressure equation can actually be removed from the system by accounting for the results of section 1.8. In fact we can take p = Cργ where C is a constant, as a solution of the pressure equations. Then,

93 • Ideal compressible MHD: Dρ + ρ∇ · u = 0, Dt Du 1 (3.50c) ρ = −∇p + (∇ × B) × B + ρg, p = Cργ , Dt 4π ∂tB − ∇ × (u × B) = 0.

For an incompressible ﬂow, ∇ · u = 0, the density ρ is constant along the Lagrangian trajectories, hence ρ = constant solves the corresponding equation; the equation for the pressure can be dropped as discussed for ordinary ﬂuids, cf. section 1.6. It is also convenient to make use of the form (3.47) of the J × B force together with the advection form (3.49) of the induction equation. • Ideal incompressible MHD: 1 ∂ u + u · ∇u = −∇P + B · ∇B + g, t 4πρ (3.50d) ∂tB + u · ∇B − B · ∇u = 0, ∇ · u = 0, where P = ρ−1(p + B2/8π) combines kinematic and magnetic pressures.

94 4 Conservation laws in magnetohydrodynamics

The qualitative behavior of solutions of the equations of magnetohydrodynamics is constrained by a set of conservation laws that are implied by the equations. Such conservation laws are crucial for the understanding of the plasma dynamics from a physical point of view, but they are also important for the mathematical analysis of the equations as well as for the design of numerical scheme that respect the basic qualitative properties of the solution.

4.1 Global conservation laws in resistive MHD. We start examining the full system of resistive MHD equations (3.50a) and show that it preserves mass, momentum, and energy even in presence of arbitrary resistivity (provided that the appropriate Ohmic heating term is accounted for on the right-hand side of the pressure transport equation). We consider a solution (ρ, u, p, B) in a spatial domain Ω and satisfying the boundary conditions n · u|∂Ω = 0, n · B|∂Ω = 0, (4.1) where n is the outgoing unit normal to the boundary ∂Ω of the domain. Such boundary conditions impose that the velocity ﬁeld and the magnetic ﬁeld are tangential to the boundary and are natural conditions for both resistive and ideal MHD, as it will become apparent in the following. Proposition 4.1. Resistive MHD equations (3.50a) equipped with boundary conditions (4.1) imply following conservation laws: • Mass conservation, d Z ρdx = 0. (4.2a) dt Ω • Momentum conservation,

d Z Z B2 ρudx = − p + ndS, (g = 0), (4.2b) dt Ω ∂Ω 8π provided that gravitational ﬁeld is g = 0. In presence of an externally imposed gravitational ﬁeld the momentum conservation is broken. • Energy conservation, d Z c Z wdx = − B · n × (ηJ)dS, (g = −∇Φg), (4.2c) dt Ω 4π ∂Ω where the energy density is

1 p |B|2 w = ρu2 + + ρΦ + , 2 γ − 1 g 8π and we have assumed that the gravitational ﬁeld, if present, is potential, i.e., g = −∇Φg. The energy density comprises the kinetic energy per unit of volume of the ﬂuid element, the internal energy density expressed in terms of pressure, the gravitational energy, and the magnetic energy density, while the electric energy

95 density does not appear as it vanishes to first-order in the quasi-neutral limit. Analogously, the momentum density corresponds to the momentum associated to the fluid motion, as the electromagnetic momentum density vanishes to first- order in the quasi-neutral limit. The non-vanishing right-hand sides in equation (4.2b) and (4.2c) mean that without additional boundary conditions there can be a non-zero momentum and/or energy flux through the boundary of the considered domain. Nonethe- less, those equations still express a conservation law: Any variation in the total momentum or energy can only come from fluxes through the boundary. The rest of this section is dedicated to the derivation of the above-claimed conservation laws. To this aim we could use partial results from section 3, but it is more instructive to develop the argument in a self-contained way from equations (3.50a). One should note that mass and momentum conservation do not depend of the specific form of the MHD Ohm’s law, as they follow directly from mass continuity and Euler’s equation, respectively. As for energy conservation, a more general argument, which is valid even for non-standard choices of the Ohm’s law, is briefly reported in appendix A.

Mass conservation. Mass conservation follows directly from the equation for the mass density in conservation form by integrating over the whole spatial domain Ω and using the Gauss theorem for the divergence term, d Z Z Z ρdx = − ∇ · (ρu)dx = − ρ(n · u)dS = 0, dt Ω Ω ∂Ω where the boundary conditions (4.1) have been taken into account.

Momentum conservation. In order to obtain (4.2b), we make use of the mass continuity equation to re-write the momentum balance equation in conservative form, namely,

1 ∂t(ρu) + ∇ · (ρuu + pI) = − 4π B × (∇ × B) + ρg. The J ×B-term on the right-hand side can be dealt with by means of the vector identity

B × ∇ × B = ∇B · B − B · ∇B 1 2 = ∇ 2 B − ∇ · (BB), where ∇ · B = 0 has been accounted for. Therefore,

BB B2 ∂t(ρu) + ∇ · ρuu − 4π + p + 8π I = ρg, which expresses the momentum balance in MHD. One can notice that gravity breaks momentum conservation. Under the assumption that g = 0, the integration over the whole domain Ω gives,

d Z Z h BB i Z B2 ρudx = − n · ρuu − dS − p + ndS. dt Ω ∂Ω 4π ∂Ω 8π The result follows from boundary conditions (4.1).

96 1 2 Energy conservation. First we multiply the mass continuity equations by 2 u , dot-multiply by u the momentum balance equation with g = −∇Φg, and sum the two resulting equations thus obtaining

1 1 u2∂ ρ + ρu · ∂ u + 1 u2u · ∇ρ + ρu2∇ · u + ρu · ∇u · u 2 t t 2 2 = −u · ∇p + u · J × B/c − ρu · ∇Φg.

The ﬁrst two terms give the time derivative of the kinetic energy density. To- gether with the vector identity

1 2 1 2 1 2 ∇ · 2 ρu u = 2 ρu ∇ · u + ρu · ∇u · u + 2 u u · ∇ρ, that gives

1 2 1 2 ∂t 2 ρu + ∇ · 2 ρu u = −u · ∇p + u · J × B/c − ρu · ∇Φg. As for the internal energy, the equation for the pressure in (3.50a) divided by γ − 1 amounts to

p p 2 ∂t γ−1 + ∇ · γ−1 u + p∇ · u = ηJ . Upon using again the mass continuity equation, one gets

∂t(ρΦg) = Φg∂tρ = −Φgu · ∇ρ − ρΦg∇ · u.

The sum of the foregoing three equations at last yields

1 2 p 1 2 p ∂t 2 ρu + γ−1 + ρΦg + ∇ · 2 ρu + γ−1 + ρΦg u + pu = ηJ 2 + u · J × B/c. (4.3)

When we dot-multiply the induction equation by B we have

1 2 ∂t( 2 B ) − B · ∇ × (u × B) = −cB · ∇ × (ηJ). This can be expressed in the form of a transport equation for the magnetic energy density in virtue of the vector identity

v1 · ∇ × v2 = v2 · ∇ × v1 + ∇ · (v2 × v1), which is valid for any two vector ﬁelds v1, v2. Particularly, one has

B · ∇ × (u × B) = (u × B) · (∇ × B) + ∇ · (u × B) × B, B · ∇ × (ηJ) = ηJ × (∇ × B) + ∇ · (ηJ × B), and thus,

B2 c u×B 2 ∂t 8π + 4π ∇ · ηJ − c × B = −ηJ + J · u × B/c, (4.4)

c where we have divided by 4π and used J = 4π ∇×B. The sum of equations (4.3) and (4.4) together with the identity J · (u × B) = u · (B × J) = −u · (J × B) yields ∂tw + ∇ · wu + pu + P = 0, (4.5)

97 where c u × B c P = ηJ − × B = E × B, 4π c 4π is the Poynting flux with the electric field being expressed from the MHD Ohm’s law, cf. equations (2.15) and (3.42). Equation (4.5) is the differential form of the energy conservation law in MHD. Upon integrating equation (4.5) on the whole domain Ω and making use of the Gauss theorem, one gets d Z Z Z wdx = − (w + p)(u · n)dσ − P · ndS. dt Ω ∂Ω ∂Ω Explicitly the Poynting vector on the boundary is proportional to

E × B = ηJ × B − B × (B × u)/c = ηJ × B − (B · u)B + B2u, thus its normal component is proportional to

n · (E × B) = n · (ηJ × B) − (B · u)(n · B) + B2(n · u), = B · n × (ηJ) − (B · u)(n · B) + B2(n · u).

One can see that all ﬂuxes through the boundary vanish if the solution satisﬁes the boundary conditions (4.1) except the term B · [n × (ηJ)], and this proves energy conservation. This result is true independently on the value of the resistivity η: The internal energy produced by the resistive term (right-hand side of equation (4.3)) is taken at the expenses of the magnetic energy (right-hand side of equation (4.4).

4.2 Global conservation laws in ideal MHD. The conservation laws derived in section 4.1 hold for the given boundary conditions (4.1) independently of the resistivity. Hence, the mass and momentum conservation laws (4.2a) and (4.2b) hold unchanged in the ideal case, while the boundary term in the energy balance (4.2c) is identically zero when η = 0. In addition, there are two more invariants due to the very special form of the ideal induction equation, namely, the magnetic helicity and the cross-helicity, deﬁned by Z Z Hm = A · Bdx, Hc = u · Bdx, (4.6) Ω Ω where A is the vector potential associated to the magnetic ﬁeld B, i.e., ∇ × A = B, cf. appendix B. One might notice that with boundary conditions (4.1) the magnetic helicity is a gauge-independent quantity, that is a gauge transformation A0 = A + ∇ϕ changes the magnetic helicity according to Z Z 0 Hm = Hm + ∇ϕ · Bdx = Hm + ϕB · ndS, Ω ∂Ω and the boundary integral vanishes if either ϕ|∂Ω = 0 or B · n = 0. In summary, we can state Proposition 4.2. Ideal MHD equations (3.50b) with boundary conditions (4.1) imply the following conservation laws:

98 • Mass conservation, d Z ρdx = 0. (4.7a) dt Ω • Momentum conservation, d Z Z B2 ρudx = − p + ndS, (g = 0). (4.7b) dt Ω ∂Ω 8π • Energy conservation, d Z wdx = 0, (g = −∇Φg), (4.7c) dt Ω where the energy density w is the same as in the resistive case. • Magnetic helicity, d Z A · Bdx = 0. (4.7d) dt Ω • Cross-helicity, d Z u · Bdx = 0, (4.7e) dt Ω with g = −∇Φg and provided that the plasma is isentropic [12], i.e., ρ−1∇p = ∇h for some function h is referred to as the enthalpy per unit of mass; this is always the case if we take p = Cργ as a solution of the pressure equation, cf. comments after equation (3.50b). Mass momentum and energy conservation have already been proven in section 4.1. It remains to show the conservation of magnetic and cross-helicity.

Conservation law for magnetic helicity. Magnetic helicity involves the magnetic ﬁeld only and the corresponding vector potential and the conservation of magnetic helicity is a direct consequence of the induction equation

∂tB − ∇ × (u × B) = 0. The equation for the vector potential A is derived from the induction equation in appendix B and for η = 0 it reads

∂tA − u × ∇ × A = ∇χ, where χ is an arbitrary scalar ﬁeld which accounts for the gauge freedom. Then, Z Z d A · Bdx = A · ∂tB + B · ∂tA dx dt Ω Ω Z = A · ∇ × (u × B) + B · (u × ∇ × A) + B · ∇χdx Ω Z = (∇ × A) · (u × B) − ∇ · (A × (u × B)) Ω + B · (u × ∇ × A) + B · ∇χdx Z = − ∇ · (A × (u × B) + Bχ)dx Ω Z = − (A × (u × B) + Bχ) · ndS, ∂Ω

99 where we have used ∇ × A = B, B · (u × B) = 0, and in the last equation, the Gauss theorem. The conservation of magnetic helicity follows from the identity A × (u × B) = (A · B)u − (A · u)B and boundary conditions (4.1).

Conservation law for the cross-helicity. The calculation of the time derivative of the cross-helicity requires the momentum balance equation which we write in the form

−1 2 ∂tu − u × ∇ × u = −ρ ∇p − ∇(u /2 + Φg) + J × B/(ρc), in view of the identity u · ∇u = ∇(u2/2) − u × ∇ × u. As discussed after equation (3.50b), we shell assume p = Cργ so that the plasma is isentropic

ρ−1∇ρ = Cγργ−2∇ρ = ∇h(ρ), where the enthalpy function h is given by [12],

Z ρ Cγ h(ρ) = Cγrγ−2dr = ργ−1. γ − 1 The equation for the velocity ﬁeld takes the form

∂tu − u × ∇ × u = −∇χ + J × B/(ρc),

2 where we set χ = h(ρ) + u /2 + Φg for sake of brevity. Then we compute Z Z d u · Bdx = B · ∂tu + u · ∂tB dx dt Ω Ω Z = B · (u × ∇ × u) − B · ∇χ + u · ∇ × (u × B)dx. Ω As before, the integral of B · χ = ∇ · (Bχ) amounts to zero in view of the boundary conditions (4.1). As for the last term

u · ∇ × (u × B) = (∇ × u) · (u × B) − ∇ · u × (u × B), hence, d Z Z u · Bdx = − u × (u × B) · ndS, dt Ω ∂Ω and the boundary integral vanishes in view of the identity u × (u × B) = (u · B)u − u2B and boundary conditions (4.1). This proves the conservation law for the cross-helicity. The physical consequences of the two additional invariants in ideal MHD will be outlined below, cf. section 4.5, after discussing the most celebrated properties of ideal MHD, namely the “freezing” of magnetic field lines in the plasma flow and the conservation of the magnetic flux.

4.3 Frozen-in law. The conservation laws analyzed so far can be considered global, since they deal with integral quantities over the whole domain of interest. In the ideal regime, however, MHD has additional conservation laws that constrain the dynamics locally as well. The most celebrated one is probably the frozen-in law which concerns magnetic ﬁeld lines.

100 Let us recall that, at each fixed time t, the magnetic field line passing through a point y ∈ Ω is the curve determined by  dx(σ)  = Bt, x(σ), dσ (t fixed), (4.8)  x(0) = y, where σ is a parameter along the curve. There a similarity between equation (4.8) defining field lines and equation (1.5) defining the flow of the vector field, but one important difference is that here time is fixed. Another difference is that the “velocity” at which the field line is traced has no physical meaning. More precisely we have the freedom to re-parametrize the curve, i.e., re-scale the parameter σ. If f(t, x) > 0 is a smooth strictly positive function, we can introduce the new parameter Z σ λ(σ) = ft, x(σ0)dσ0, 0 so that dλ/dσ = ft, x(σ) and we can re-write the equation for the field line in the form dx(λ) dσ dx(σ) Bt, x(λ) = = , dλ dλ dσ ft, x(λ) which is the same as re-scaling the vector field by 1/f(t, x). With some abuse of notation we have denoted x(λ) = xσ(λ) for simplicity. As an example, let us consider the arc-length Z σ dx(σ0) Z σ 0 0 0 s(σ) = dσ , = B t, x(σ ) dσ , 0 dσ 0 in a region where |B(t, x| > 0. This corresponds to f(t, x) = |B(t, x)| for which we have dx(s) = bt, x(s), ds where b(t, x) = B(t, x)/|B(t, x)| is the unit vector in the direction of the magnetic field, and the parameter s has the meaning of length of the curve. In general, the parameter σ does not even have the dimensions of a length, but it is just a label of the points on the line. The configuration of a set of magnetic field lines with carefully chosen initial points is often sufficient to give a good representation of the magnetic field at any given time t, cf. figure 4.1. The frozen-in law constrains the way magnetic field lines evolve: We say that they are frozen into the plasma flow since they move with the plasma fluid velocity u(t, x). Precisely, let us consider a generic time-dependent vector field w(t, x) and a given velocity field u(t, x) with flow Ft. Definition 4.1. We say that the vector field w ∈ C1(I × Ω) is frozen in the flow Ft :Ω → Ω if, for any field line x0(σ) of w at the time t = 0, the curve xt(σ) = Ft x0(σ) , is a field line of w at time t, for all t ≥ 0.

101 Figure 4.1: A single magnetic field line of the reconstructed equilibrium magnetic field in ASDEX upgrade (shot #27764). The field line alone provides already an idea of the topology of the field near the magnetic axis (red circle). The color code represents the strength of the magnetic field, from high field inside the torus (light blue) to low field outside the torus (dark blue).

The condition that xt is a field line of w at time t writes dx (σ) t = ft, x (σ)wt, x (σ), (4.9) dσ t t allowing for a possible re-scaling factor f = f(t, x) > 0. Let us remark that the curve xt = xt(σ) is the evolution of x0 = x0(σ) along the Lagrangian trajectories of u and in general that does not coincide with any field line of w at time t. The frozen-in law poses a strong constraint on the evolution of the magnetic field in ideal MHD. Proposition 4.3. For a solution of ideal MHD equations (3.50b) such that 1 2 ρ, u, B ∈ C , ρ > 0, and with flow F (t, x) = Ft(x) in C , the magnetic field B is frozen into the plasma flow. In addition, for incompressible flows the condition ρ > 0 can be dropped. The remaining part of this section is dedicated to the proof of this claim. Definition 4.1 is not easy to check directly. We shall first derive a sufficient condition which can be expressed in terms of a partial differential equation for the field w and then check that, in ideal MHD, the magnetic field (properly re-scaled) satisfies such condition.

1 Lemma 4.4. Let u, w ∈ C (I × Ω) be two vector ﬁelds and let Ft :Ω → Ω be 2 the ﬂow of u with the hypothesis that F (t, x) = Ft(x) is C on I × Ω. Then the following statements are equivalent:

(i) For any ﬁeld line x0 = x0(σ) of w at t = 0,

dF x (σ) t 0 = wt, F (x (σ)). (4.10a) dσ t 0

(ii) For all x ∈ Ω, w t, Ft(x) = w(0, x) · ∇Ft(x). (4.10b)

102 (iii) The field w satisfies the partial differential equation

∂tw + u · ∇w − w · ∇u = 0. (4.10c)

Let us remark that lemma 4.4 establishes stronger conditions then that required in deﬁnition 4.1, as it does not allow for a re-scaling factor, cf. equation (4.9) as compared to (4.10a).

Proof. 1. (i) ⇒ (ii). If (i) is true, then for any field line x0 of w at time t = 0 we have dF x (σ) dx (σ) wt, F (x (σ)) = t 0 = 0 · ∇F x (σ), t 0 dσ dσ t o where we have used the chain rule for differentiation in the second identity. Since x0 is a field line of w(0, ·), we obtain w t, Ft(x0(σ)) = w 0, x0(σ) · ∇Ft xo(σ) , identically in σ and for all field lines x0. Evaluating at σ = 0, we obtain equation (4.10b) with x = x0(0). 2. (ii) ⇒ (iii). If (ii) holds, we can differentiate equation (4.10b) in time. The derivative of the left-hand side reads dF (x) ∂ wt, F (x) + t · ∇wt, F (x) t t dt t = ∂tw t, Ft(x) + u t, Ft(x) · ∇w t, Ft(x) , where equation (1.5) has been accounted for. As for the time derivative of the right-hand side, we use the hypothesis that F (t, x) is in C2(I ×Ω), which allows us to exchange the gradient and the time derivative, with the result that dF (x) w(0, x) · ∇ t = w(0, x) · ∇ut, F (x) dt t = w(0, x) · ∇Ft(x) · ∇u t, Ft(x) = w t, Ft(x) · ∇u t, Ft(x) , and we have used again (ii) in the last equality. At last, we have ∂tw t, Ft(x) + u t, Ft(x) · ∇w t, Ft(x) = w t, Ft(x) · ∇u t, Ft(x) , which evaluated at x = F−t(y) gives equation (4.10c) at the arbitrary point y. 3. (iii) ⇒ (i). Let us consider the quantity ϕσ(t) = dFt x0(σ) /dσ − w t, Ft(x0(σ)) , defined for a given field line x0 of the field w at t = 0. We assume that (iii) is true and want to prove (i) which is equivalent to ϕσ = 0 identically for all σ and all field lines x0. At t = 0 we have ϕσ(0) = 0 for all σ, since x0 is a field line of w(0, ·) by hypothesis. We compute

dϕ (t) d hdF x (σ)i σ = t 0 − ∂ wt, F (x (σ)) dt dt dσ t t 0 − u t, Ft(x0(σ)) · ∇w t, Ft(x0(σ)) .

103 In the first term on the right-hand side we can exchange the order of the deriva- 2 2 tives as, by hypothesis, the flow is C in both time and space, while x0 ∈ C for a C1-vector field w. Hence,

d hdF x (σ)i d hdF x (σ)i d h i t 0 = t 0 = ut, F (x (σ)) dt dσ dσ dt dσ t 0 dF x (σ) = t 0 · ∇ut, F (x (σ)). dσ t 0 By using (iii) we have dϕ (t) σ = ϕ (t) · ∇ut, F (x (σ)), dt σ t 0 which is a linear ordinary differential equation depending on the parameter σ. Since ϕσ(0) = 0, the unique solution is ϕσ(t) = 0 for all σ and all initial field lines x0. We can now go ahead and check whether the differential equation (4.10c) holds true for the magnetic field in ideal MHD. Proof of proposition 4.3. Let us first consider the case of incompressible ideal MHD, cf. system (3.50d). The induction equation for the magnetic field B takes the form ∂tB + u · ∇B − B · ∇u = 0, which is just condition (4.10c) with w = B. We can conclude directly that in ideal incompressible MHD the magnetic field is frozen in the plasma flow and the field lines evolve with the flow according to equation (4.9). The general case of compressible ideal MHD, cf. system (3.50c) is a bit less obvious and we need to use the freedom of re-scaling the magnetic field. In the compressible case, ∇ · u 6= 0 and the induction equation for the magnetic field reads ∂tB + u · ∇B − B · ∇u + (∇ · u)B = 0, which is not in the form (4.10c). However, from the continuity equation for the mass density under the (physically reasonable) assumption ρ > 0, one has 1 ∇ · u = − [∂ + u · ∇]ρ, ρ t and thus

∂tB + u · ∇B − B · ∇u + (∇ · u)B B = ∂ B + u · ∇B − B · ∇u − ∂ + u · ∇ρ = 0 t ρ t and dividing by ρ, we obtain

∂t(B/ρ) + u · ∇(B/ρ) − (B/ρ) · ∇u = 0, which is in the form of condition (4.10c) with w = B/ρ. We conclude that, in the compressible case, the magnetic field re-scaled by the mass density B/ρ is frozen in the plasma flow and magnetic field lines evolve with the plasma flow according to equation (4.9) with f = 1/ρ.

104 St

Ct = ∂St

Figure 4.2: Sketch of magnetic ﬁeld lines intersecting a compact surface St bounded by a simple curve Ct.

The key point of this remarkable result is the frozen-in condition, either in the integral form (4.10b) or in the diﬀerential form (4.10c). This has a deep geometrical meaning which can be fully appreciated in the framework of modern tensor calculus [33].

4.4 Flux conservation. Let us consider a time-dependent compact surface St ⊂ Ω which moves with the plasma flow, that is, St = Ft(S0) where S0 is the configuration of the surface at time t = 0 and Ft is the flow of the plasma fluid. In virtue of the frozen-in law (section 4.3) the magnetic field lines that intersect St are the same at all time as both surface and field lines are frozen in the plasma. It is therefore natural to ask if the magnetic flux through St is constant and we shall obtain that under suitable hypotheses, this is the case. This situation is schematically represented in figure 4.2. First let us recall briefly a few basic concepts about surfaces and curves. A curve γ given parametrically by X :[a, b] 3 σ 7→ X(σ) ∈ Rd is of class Ck if the parametrization X is of class Ck on the interval [a, b] ⊂ R. We say that the curve is regular if X ∈ C1([a, b]) and X0(σ) 6= 0, while the curve is called simple if X is injective. Geometrically, the latter condition means that the curve does not cross itself. A closed curve is by definition a curve such that X(a) = X(b), i.e., it closes on itself. One should notice that the curve σ 7→ (σ, 0, 0) ∈ R3 for σ ∈ [0, 1] is not closed in this sense, but it is a closed as a set in R3: The two concept should not be confused. A simple closed curve (or Jordan curve) is a closed curve with X injective in the open interval (a, b). In a similar manner, surfaces without boundary are locally the image of a map X : U 3 y 7→ X(y) ∈ Rd where U is an open set of R2 with coordinates R2 y = (y1, y2). Surfaces with boundary are locally modeled on open sets in + = R2 R2 {(y1, y2) ∈ | y2 ≥ 0}, that is U ⊂ + can include a boundary. A surface is Ck-regular if the parametrization X is of class Ck. If X ∈ C1 and d = 3, the

105 n

C = ∂S t

Figure 4.3: Positive orientation of the boundary of an oriented surface. unit normal is defined by the local parametrization, ∂X(y) ∂X(y) n ∝ × , ∂y1 ∂y2 and we always take n normalized, i.e., |n| = 1. We say that a surface is orientable if we can choose a normal vector globally and smoothly on the surface. The Möbiusstrip and the Klein bottle are examples of non-orientable surfaces. A surface is connected if it is path-wise connected as a set, i.e., any two points of the surface can be joined by a continuous path on the surface. We are interested in compact surfaces that is, surfaces that are compact as topological spaces. As a special case, closed surfaces are compact surfaces without a boundary. The classification theorem for compact surfaces restrict significantly the possibilities: Closed orientable surfaces are topologically equivalent to one of the following model surfaces: (1) a sphere, (2) a torus, or (3) a connected sum or tori. (Pre- cisely, two surfaces are “topologically equivalent” if they are mapped into each other by a continuous map with continuous inverse, i.e., a homeomorphism.) In particular, closed orientable surfaces enclose a bounded region of space. We also need to recall Stokes theorem and related definitions. Let S be any compact connected orientable surface with unit normal n and boundary C = ∂S, which is a simple closed regular curve with tangent t. We always choose the positive orientation of the boundary C with respect to the orientation of S. Here “positive orientation” means that the unit tangent vector of the curve C = ∂S is oriented counter-clockwise with respect to the unit normal n of the surface S, cf. figure 4.3. Under those conditions, Stokes theorem for a generic C1-vector-field w states that the flux of the curl of w equals its circulation (line integral) on the boundary, namely, Z Z (∇ × w) · ndS = w · tds. (4.11) S ∂S From Stokes theorem (4.11) it follows that the circulation of a gradient is zero, Z Z ∇f · tds = (∇ × ∇f) · ndS = 0, (4.12) ∂S S in view of the identity ∇ × ∇f = 0.

106 If σ 7→ X(σ) is a given parametrization of the curve C, the unit tangent amounts to −1 dX(σ) dX(σ) t = , dσ dσ while the arc-length is given by, cf. section 4.3,

dX(σ) ds = dσ, dσ hence the oriented line element takes the form dX(σ) tds = dσ. (4.13) dσ We can now state the flux-conservation theorem of ideal MHD. At the time t = 0, we consider a compact connected orientable surface S0 possibly with boundary. In case S0 has a non-empty boundary, we assume further that the boundary C0 = ∂S0 is a regular closed curve with a parametrization [0, 1] 3 σ 7→ X0(σ) ∈ Ω, (4.14) 2 such that X0 ∈ C ([0, 1]). At a generic point t in time, the surface is St = Ft(S0), where Ft :Ω → Ω is the flow of the velocity field u. As for the frozen-in law discussed in section 4.3, we shall consider the case of sufficiently regular flows, namely, the map F (t, x) = 2 Ft(x) is assumed to be in C (I × Ω). Then St = Ft(S0) is a compact connected 2 orientable surface of class C . Particularly, since Ft is continuous and invertible with continuous inverse (homeomorphism), we have Ct = ∂St = Ft(C0), i.e., points on the boundary stay on the boundary at all time, and Ct is a regular curve of class C2.

Proposition 4.5. Let St be a compact connected orientable surface constructed 2 as described above with the ﬂow Ft such that F (t, x) = Ft(x) is in C (I × Ω). Then, d Z B · ndS = 0, (4.15) dt St for any solution B = ∇ × A, A ∈ C2, of the ideal MHD induction equation

∂tB − ∇ × (u × B) = 0, ∇ · B = 0, where u is the velocity field associated to the flow Ft. One should notice that proposition 4.5 does not refer to a full solution of ideal MHD equations, but consider only the induction equation with a velocity field u. Regular solutions of ideal MHD equations follow as special cases. When ∂St is non-empty, Stokes theorem allows us to rewrite the magnetic flux through St in terms of the circulation of the vector potential A, namely, Z Z Z B · ndS = (∇ × A) · ndS = A · tds, (4.16) St St Ct and we check that the time derivative of the circulation of A on Ct is zero.

107 Proof of proposition 4.5. If St is closed, i.e., without boundary, the flux is identically zero. This follows from the fact that closed orientable surfaces enclose a finite volume Wt. Since B is divergence-free, Z Z 0 = ∇ · Bdx = B · ndS, Wt St by Gauss theorem, with normal n oriented in the outward direction. Since this holds at any time t, the statement is trivially true. For the case of a surface with non-empty boundary, the composition of Ft with the parametrization (4.14) of the initial boundary C0 gives a parametrization of the boundary Ct, namely, [0, 1] 3 σ 7→ Xt(σ) = Ft(X0 σ) ∈ Ω, and, in virtue of the assumptions on Ft and X0, we have that (t, σ) 7→ Xt(σ) 2 belongs to C . The circulation of the vector potential A on the curve Ct reads Z Z 1 ∂Xt(σ) A · tds = A t, Xt(σ) · dσ, Ct 0 ∂σ where equation (4.13) has been accounted for. The integral on the right-hand side is absolutely convergent and the integrand is differentiable in time. Com- puting the derivative, we have

Z Z 1 d dXt(σ) ∂Xt(σ) A · tds = ∂tA t, Xt(σ) + · ∇A t, Xt(σ) · dσ dt Ct 0 dt ∂σ Z 1 d hdXt(σ)i + A t, Xt(σ) · ds. 0 dt dσ

In the last term we can exchange the order of derivatives since Xt is the composition of two C2 maps and thus it is C2 in the combined variables (t, σ). Since,

d hdX (σ)i d hdX (σ)i d h i dX (σ) t = t = ut, X (σ) = t · ∇ut, X (σ), dt dσ dσ dt dσ t dσ t we obtain Z Z 1 d h A · tds = ∂tA t, Xt(σ) + u t, Xt(σ) · ∇A t, Xt(σ) dt Ct 0 i dX (σ) + ∇ut, X (σ) · At, X (σ) · t ds. t t dσ If B is a solution of MHD induction equation with zero resistivity (ideal MHD) the magnetic vector potential solves, cf. appendix B,

∂tA − u × (∇ × A) = ∇χ, where the scalar function χ accounts for the gauge freedom. Equivalently we can write ∂tA + u · ∇A = ∇χ + ∇A · u,

108 so that Z Z 1 d dXt A · tds = ∇ χ(t, Xt) + A(t, Xt) · u(t, Xt) · ds dt Ct 0 dσ Z = ∇(χ + A · u) · tds = 0, Ct since the circulation of a gradient is zero, cf. equation (4.12). At last one can deduce the conservation of the magnetic ﬂux from Stokes theorem (4.16). A few consequences of ﬂux-conservation are outlined in section 4.5.

4.5 Topology of the magnetic field. In this section we review a few key concepts that are used to characterize the magnetic field topology in plasma physics. The definition of such topological objects relies on the flux-conservation result obtained in section 4.4 for ideal MHD.

Flux surfaces. A surface spanned by magnetic field lines in plasma physics is referred to as a flux surface. Let us formulate a precise definition. Definition 4.2. Given a magnetic field B = B(t, x) at least continuous, an orientable surface S with unit normal n is a flux surface at time t if B(t, x)·n = 0 for all points x ∈ S.

In ideal MHD, flux conservation, proposition 4.5, guarantees that flux surfaces evolve into flux surfaces. Precisely, Proposition 4.6. If the magnetic field B and the velocity field u satisfy the hypotheses of proposition 4.5 and S0 is a smooth flux surface at time t = 0, then St = Ft(S0) is a flux surface at all time.

Proof. We have to show that B · n = 0 on St = Ft(S0), given the fact that S0 is a flux surface at time t = 0. For every area patch At ⊂ St such that At is a compact surface with boundary Ct satisfying the hypotheses of proposition 4.5 we have Z Z B(t, x) · ndS = B(0, x) · ndS = 0, At A0 since A0 = F−t(At) is a patch on a flux surface. If there is a point x0 on St where B(t, x0)·n(t, x0) 6= 0 then by continuity we can find a patch At sufficiently small that the flux is non-zero contradicting flux conservation. It follows that B · n = 0 on St at all time. Flux surfaces are particularly important in describing static MHD equilibria, i.e., time-independent solutions of ideal MHD with u = 0. In this limit, MHD equations (3.50c) reduce to a set of conditions, namely, 4π ∇p = J × B/c, ∇ × B = J, ∇ · B = 0. c The first condition expresses the exact balance of the forces on the right-hand side of the momentum equation in ideal MHD, cf. system (3.50b). This system of equations is not closed even when boundary conditions are added: There are

109 many possible MHD equilibria compatible with physically reasonable boundary conditions. Let us consider the case of one such equilibrium for which at the point x0 in the given domain Ω the pressure attains a regular value p0 = p(x0) in the sense on the implicit function theorem, i.e., ∇p(x0) 6= 0. Then locally near x0 −1 the set p (p0) is a regular surface Σ with normal n ∝ ∇p. Such surfaces are called pressure surfaces. We have that for MHD equilibria pressure surfaces are ﬂux surfaces. From the force balance we have in fact that

B · n ∝ B · ∇p = B · (J × B)/c = 0.

Geometrically pressure surfaces contains magnetic ﬁeld lines. Indeed we also have the same for the current density,

J · n ∝ J · ∇p = J · (J × B)/c = 0.

Physically in static MHD equilibria, the Lorentz force due to current and magnetic field tangent to pressure surfaces balances the pressure gradient of the plasma. In fusion plasma physics the knowledge of equilibrium flux surfaces is critical and define much of the geometry of the fusion experiment. The shape of such surfaces is constrained by two basic results of topology.

1 Theorem 4.7. Let B, j, and p in C (Ω) satisfy the force balance, and let p0 ∈ R −1 be a regular value of p such that Σ = p (p0) is a closed connected orientable surface of constant pressure. If B 6= 0 then Σ is a flux surface homeomorphic to a connected sum of tori of genus g ≥ 1. This result is often reported informally in basic plasma physics text books [40, Chapter 3]. The proof relies on the classification theorem recalled at the beginning of section 4.4 and on the so called “hairy ball theorem”. Theorem 4.8 (Hairy ball theorem [33].). The sphere Sk has a continuous field of non-vanishing tangent vectors if and only if the dimension k is odd. Proof of theorem 4.7. We have already shown that Σ is a flux surface which contains both magnetic field lines and current lines. If it is closed and orientable, by the classification theorem in must be homeomorphic to either a sphere S2, a torus, or a connect sum of tori. We known however that B is nowhere vanishing and tangent to the surface hence, by the hairy ball theorem we can rule out the possibility of a sphere. An example of realistic flux surface (from the ASDEX upgrade tokamak) is shown in figure 4.4. Existence of flux surfaces in a cylindrically symmetric equilibrium such as that of a tokamak is guaranteed and they form a family of nested tori, cf. figure 4.4. In a full three-dimensional configuration, on the other hand, existence of flux surfaces is in general not possible: Three dimensional perturbations resonate with special surfaces and destroy them [55]. For this reason, the com- putation of three-dimensional MHD equilibria is still a challenging problem [56].

110 Figure 4.4: Reconstructed magnetic equilibrium for a plasma in the ASDEX upgrade tokamak (shot number 27764). Toroidal flux surfaces are cut by a plane in order to show interior nested surfaces. A magnetic field line is also shown, cf. figure 4.1, and one can observe that it stays tangent to a magnetic surface.

S2 S3

S 1 B

Figure 4.5: Sketch of a ﬂux tube.

111 Flux tubes. A flux tube is a connected region of space W ⊂ Ω defined by the congruence of a bundle of magnetic field lines and thus bounded by a flux surface as shown in figure 4.5. For definiteness we can choose a central field line which represent the overall trajectory of the flux tube in space. To any flux tube we can associate a constant which is defined as the magnetic flux through any cross-section of the tube. This is possible in view of the following result. Proposition 4.9. Under the conditions of proposition 4.5, let T be a flux tube, and S a cross-section of T oriented along the direction of the field lines. Then the magnetic flux Z κ = B · ndS. (4.17) S is independent of the choice of S and constant in time. Proof. A cross-section of the tube is a compact connected orientable surface and thus the flux through it is constant in time in virtue of proposition 4.5. The fact that the flux is independent on the chosen cross-section follows from ∇ · B = 0. Consider two non-intersecting surfaces S1 and S2 delimiting a finite volume W of the flux tube as in figure 4.5. At any given time, Z Z Z Z 0 = ∇ · Bdx = B · nW dS + B · nW dS + B · nW dS, W S1 S2 S3 where S3 is part of the outer boundary of the flux tube and by definition is a flux surface, hence the last integral vanishes. Here nW is the outer normal of W . In view of the chosen orientation, the normal on S1 is n1 = −nW while the normal on S2 is n2 = nW and thus, Z Z − B · n1dS + B · n2dS = 0, S1 S2 that is, S1 and S2 intersect the same flux. In the pathological case in which S1 and S2 intersect each other, let us choose a third section S˜ that does not intersect either of them. We can use the same argument to show that both S1 and S2 intersect the same flux as S˜.

Magnetic helicity and topology of flux tubes. Magnetic helicity is related to the topology of flux tubes in a given configuration. This was first understood by Moffatt [6] and Arnold [57]. We report here the qualitative (discrete) argument proposed by Moffatt and just state the precise mathematical equivalent by Arnold. Let us consider a set of regular simple closed magnetic-field lines Ci, i = 1, 2,..., in the domain Ω of interest at a given time t fixed. Let us also consider the extremely idealized situation in which the magnetic field is non-zero only in narrow flux tubes around the lines Ci and such flux tubes do not intersect each other. Figure 4.6 gives a schematic representation of this idealized field configuration. To each flux tube we can associate a constant κi which is the flux through a cross section. In virtue of proposition 4.9, κi is well-defined (i.e., does not depend on the choice of the cross section) and constant in time.

112 C2

Figure 4.6: Linked flux tubes projected on a plane. Each oriented simple curve Cj should be thought of as a narrow flux tube carrying a constant flux κj. The intersection of a the curve with a surface Si with boundary Ci = ∂Si contributes to the circulation of the vector potential A along Ci.

In this configuration we can compute the flux Φi of the magnetic field through any surface Si bounded by Ci = ∂Si and positively oriented with respect to the boundary. Since the magnetic field vanishes everywhere away from the considered field lines, the contribution of Cj to the flux Φi is +κj if the curve Cj intersects Si in the direction of the normal, −κj if it intersects Si in the opposite direction and 0 if it does not intersect. Hence, we have

Z X Φi = B · ndS = αijκj, Si j where the constants αij depend only on the topology of the field lines. Since the flow Ft of the plasma motion is continuous and the magnetic field is frozen-in due to proposition 4.3, the coefficients αij are independent of time. Essentially αij counts the number of links between Ci and Cj and it is therefore referred to as linking number. Figure 4.7 gives a few examples of linking numbers. We now go ahead with a formal calculation of the magnetic helicity,

Z X Z Hm(t) = A(t, x) · B(t, x)dx = A(t, x) · B(t, x)dx, Ω i Ωi where Ωi is the volume of the narrow flux tube around the curve Ci. Within each flux tube we switch to adapted coordinates (s, y) where s is a coordinate that reduces to the arc-length on Ci and y = (y1, y2) are two coordinates spanning the cross-section normal to B in the flux tube. The volume element in Ωi is therefore dx = dsdS(y) where dS(y) is the area element on the cross-section spanned by y. In addition in Ωi we have

B = |B|n, |B| = B · n,

113 C1 C2 C1 C1

C2 C2

α12 = α21 = 0 α12 = α21 = +1 α12 = α21 = −1

C2 C1

C2 C1 α12 = α21 = +2 α12 = α21 = −2

Figure 4.7: A few examples of linked curves following Moﬀatt’s paper [6]. where n is the normal to the surface spanned by y. Then we have,

X Z Hm(t) = A(t, x) · B(t, x)dx i Ωi X Z Z = A · BdS(y) ds i Ci X Z Z ≈ A|Ci · t |B|dS(y) ds i Ci X Z Z = A|Ci · t B · ndS(y) ds i Ci X = κiΦi, i where we have converted the circulation of A into the flux Φi across a surface bounded by Ci. The approximations used in the second step are: (1) n ≈ t where t is the unit tangent on Ci and (2) A ≈ A|Ci both due the the narrow cross-section of the flux tube. In the limit of zero-width flux tubes we obtain the formal (but elegant) result X Hm(t) = αijκiκj. (4.18) i,j

The magnetic helicity is a quadratic form of the flux strengths κi of each flux tube with coefficients αij depending only on the topology of the field lines.

114 C C1

α12 = +1

Figure 4.8: Unknotting the trefoil knot [6]. The curve C is equivalent to the trefoil knot and can be unknotted by adding two properly oriented segments (dashed lines). The result is two linked unknotted curves, C1 and C2, with linking number α = 1.

We can slightly generalize the result by allowing the curves Ci to be knotted. It is possible to decompose a knotted curve into several links by adding two segments carrying opposite fluxes, thereby not changing the total value of the magnetic helicity. A knotted field line (trefoil knot) is shown in figure 4.8 together with its equivalent decomposition in two linked curves following Moffatt’s paper [6]. In this case, equation (4.18) holds with the constants αij accounting for the combination of the number of links and knots in the system. Equation (4.18) shows at once that the conservation of Hm(t) is a direct consequence of the conservation of the topology of the magnetic field lines, i.e., the constants αij, as well as of the flux through each flux tube. We also see that the magnetic helicity measures how linked are the magnetic field lines, since Hm = 0 if flux tubes are linked with each other (αij = 0). A variation in Hm, due to non-ideal effects such as a finite resistivity or finite electron inertia, signals a change in the topology of the field lines. In 1973 Arnold proposed a theorem which generalizes equation (4.18) to the case of a realistic field [57]. Such generalization reads [58], Z Z Hm(t) = λ(x1, x2)dx1dx2, (4.19) Ω Ω where λ ∈ L1(Ω × Ω) is the asymptotic linking number of two field lines issued from the points x1 and x2, respectively. This is operatively defined by the following procedure: Given two point x1 and x2, one computes the two field lines issued from such points following them for a length L1 and L2. In general, the two field lines will not close on themselves, but one can close them by adding a straight line connecting the end-points to the corresponding starting point. Such segments will not intersect each other apart for a set of measure zero of points x1 and x2; then one has two closed simple curves and we can compute their linking number which will depend on L1, L2; we divide the linking number by L1 · L2 and let L1,L2 → +∞; the limit is the asymptotic linking number λ(x1, x2). The rigorous proof of equation (4.19) has been completed by Volgel [59] later in 2003.

115 Topological bounds on magnetic energy. Conservation of magnetic helicity implies a lower bound on the magnetic energy. This observation has important consequences for the dynamics of plasmas as observed by Woltjer [5] and later by Taylor [7] in his approach to the MHD equilibrium problem which is now called Taylor relaxation. An overview of the role of magnetic helicity in plasma confinement is given by Yoshida [60]. Let us consider a bounded simply connected domain Ω with smooth boundary ∂Ω and a magnetic field configuration B ∈ L2(Ω) at a certain time, satisfying ∇ · B = 0 in Ω and B · n = 0 on ∂Ω. The corresponding potential A such that ∇ × A = B with the gauge fixed by the condition ∇ · A = 0 is given by the following result. Theorem 4.10 (Theorem 3.6 in Girault and Raviart [61]). Let Ω ⊂ R3 be a bounded simply connected domain with smooth connected boundary ∂Ω and v ∈ L2(Ω) satisfying

∇ · v = 0 in Ω and v · n = 0 weakly on ∂Ω.

Then there is one potential w ∈ H1(Ω) such that v = ∇ × w with tangential boundary condition w × n = 0 on ∂Ω. The potential is characterized as a unique solution of  − ∆w = ∇ × v in H−1(Ω),  ∇ · w = 0 in Ω,  w × n = 0 on ∂Ω. We shall also need the following Poincar´e-type inequality. Theorem 4.11 (Lemma 3.4 in Girault and Raviart [61]). With Ω as in theorem 4.10, there is a constant CP such that kwkL2(Ω) ≤ CP k∇ × wkL2(Ω) + k∇ · wkL2(Ω) , for all w :Ω → R3 such that w ∈ L2(Ω) with weak curl and divergence ∇ × w, ∇ · w ∈ L2(Ω) and satisfying the boundary conditions w × n = 0 on ∂Ω. As a direct consequence of theorem 4.10 and 4.11 one has the claimed lower bound on the energy. Theorem 4.12 (Arnold [57]). With Ω as in theorem 4.10, let B ∈ L2(Ω) be a given magnetic ﬁeld and A ∈ H1(Ω) the corresponding vector potential constructed as in theorem 4.10. Then

2 |Hm| ≤ CP kBkL2(Ω), where Hm = (A, B)L2(Ω) is the magnetic helicity. Proof. Schwartz inequality gives

Hm = (A, B)L2(Ω) ≤ kAkL2(Ω) · kBkL2(Ω), and A ∈ H1(Ω) obtained from theorem 4.10 satisﬁes in particular ∇ · A = 0 in Ω and A × n = 0 on ∂Ω. Hence theorem 4.11 implies

kAkL2(Ω) ≤ CP k∇ × AkL2(Ω),

116 and thus kAkL2(Ω) ≤ CP kBkL2(Ω). When this is inserted in the estimate for the helicity, we have

2 Hm ≤ kAkL2(Ω) · kBkL2(Ω) ≤ CP kBkL2(Ω), which is the claimed estimate. Conservation of magnetic helicity implies that the magnetic energy, which 2 is proportional to kBkL2(Ω) cannot decrease arbitrarily. Since magnetic helicity is essentially related to the topological properties of the magnetic field, we have that the topology imposes a constraint on the relaxation of the magnetic field: not all magnetic energy can be transferred to kinetic or internal energy of the plasma (total energy is conserved). It was proposed by Woltjer and Taylor that even in presence of non-ideal effects (viscosity and resistivity) magnetic helicity decays slowly on the time scale of interest and thus is approximately preserved and poses a strong constraint on the dissipation and relaxation to equilibrium.

4.6 Analogy with the vorticity of isentropic flows. Many of the results stated for the magnetic field where first discovered for isentropic flows. There is in fact a strong formal analogy between the equations of ideal MHD and those for isentropic flows. If one recalls the results of section 1.8, the momentum balance equation and the vorticity equation for an inviscid isentropic fluid without external forces take the form ∂tρ + ∇ · (ρu) = 0,

∂tu − u × (∇ × u) = ∇χ(ρ, u), (4.20)

∂tω − ∇ × (u × ω) = 0, with ω = ∇ × u, and we recognize the same structure as in ideal MHD where u plays the role of the the magnetic vector potential A, and the vorticity ω that of the magnetic field. The only difference is the in the case of equations (4.20) the velocity field is both the potential for ω and the advecting field, but this difference is inessential here. We can therefore conclude at once the following results for the vorticity. First we have an exact invariant the helicity of the flow which is the analogous of the magnetic helicity in MHD. Proposition 4.13. For solutions of (4.20) the helicity is conserved, d Z u · ωdx = 0. (4.21) dt Ω The proof follows in the same way as for magnetic helicity (proposition 4.2). In this case it is possible to draw a physical interpretation of this quantity. The vorticity ω describes the rotation of a fluid element around a reference Lagrangian trajectory, cf. section 1.3, and the quantity u · ω is positive if the rotation is “right-handed” with respect to u, i.e., anti-clockwise as in figure 4.3. For particle with a spin this is called helicity and the name has been adopted in fluid mechanics [6]. We also have an equivalent of the frozen-in law which one can prove exactly in the same way.

117 Proposition 4.14. For a solution of equations (4.20) such that ρ, u, ω ∈ C1, 2 ρ > 0 and the flow F (t, x) = Ft(x) is in C , the magnetic field B is frozen into the plasma flow. In addition, for incompressible flows the condition ρ > 0 can be dropped. Analogously we have flux conservation. For the case of isentropic flows, however, this result is usually formulated as a conservation of the circulation of the velocity and is referred to as Kelvin’s theorem.

Proposition 4.15 (Kelvin’s circulation theorem). Let ω = ∇×u and u ∈ C2 be solution of equations (4.20) and let St be a compact connected orientable surface with boundary evolving with the flow Ft of the velocity field u. If F (t, x) = Ft(x) is in C2(I × Ω), then d Z u · tds = 0. (4.22) dt ∂St The surface satisfying the condition ω · n = 0 are referred to as vortex sheets [12] and they are preserved by the dynamics in the same way as flux surfaces are preserved in MHD (the precise statement can be deduced from proposition 4.6). A tube of vorticity field lines is referred to as a vortex tube. The analogous of proposition 4.9 is the Helmholtz theorem which is formulated in terms circulations rather then fluxes.

Proposition 4.16 (Helmholtz theorem). Let u, ω and F as in proposition 4.15 and let T be a vortex tube. Then the circulation Z κ = u · tds. (4.23) C of the velocity on a simple closed curve, which bounds a cross-section of the tube, is independent of the chosen cross-section and constant in time. The constant κ can be associate to the vortex tube in the same way the magnetic flux can be associated to a flux tube. The same consideration on linked and knotted flux tubes of section 4.5 apply here to vortex tubes. Particularly, the conservation of fluid helicity implies a lower bound for the L2-norm of the vorticity.

118 5 Basic processes in magnetohydrodynamics

The solution of MHD equations (except for very few simple cases) necessarily requires computer codes even for very idealized problems. A good understanding of qualitative features of the solutions is, however, an essential prerequisite to the development and application of such MHD codes as well as for the physics interpretation of the results. In this section, some basic physics processes oc- curring in magnetohydrodynamics are illustrated in simple situations.

5.1 Linear MHD waves. Given a uniform plasma (where “uniform” means constant in time and homogeneous in space), we shall study the evolution of a small disturbance in terms of normal modes (plane waves). This allows us to identify the natural wave modes of the system and their propagation speeds which often set the characteristic time scales for physical processes and have to be accounted for in the design of either stable explicit numerical schemes or preconditioners for implicit schemes. The method is relatively simple and can be carried out for most plasma physics models. The technique employed here is at least as important as the results themselves. Let us consider the full resistive MHD system (3.50a). We are interested in solutions describing a uniform plasma without flow, i.e., with constant density ρ0, no fluid velocity u0 = 0, constant pressure p0, and constant magnetic field B0. One can check that this is a solution only in absence of gravity, which typically produces stratification in a fluid, hence we assume g = 0 in this section. We study the evolution of a small amplitude perturbation

ρ = ρ0 + ρ1,

u = u1, (u0 = 0), (5.1) p = p0 + p1,

B0 = B0 + B1. where “small amplitude” means that, on substituting the perturbation (5.1) into MHD equation (3.50a), nonlinear terms are neglected. This procedure is known as linearization. For the speciﬁc case under consideration, the linearized system reads

∂tρ1 + ρ0∇ · u1 = 0, 1 ρ ∂ u = −∇p + (B · ∇B − ∇B · B ), 0 t 1 1 4π 0 1 1 0 (5.2) ∂tp1 + γp0∇ · u1 = 0,

∂tB1 − B0 · ∇u1 + (∇ · u1)B0 = κη∆B1 where we have assumed constant resistivity and all nonlinear terms have been dropped. We see in particular that the Ohmic heating term at the right-hand c side of the pressure equation is zero, since J0 = 4π ∇ × B0 = 0 for a uniform equilibrium. Therefore we expect that, in general, the linearized system does not conserve energy: The ﬁxed background provides an inexhaustible energy sink/source.

119 We look for solutions of the linearized system in the form of plane waves,     ρ1 ρ˜ u1  u˜ −izt+ik·x   =   e , (5.3) p1   p˜ B1 B˜ where z = ω + iν ∈ C is a complex frequency and k ∈ Rd the wave vector. Let us note that this is not a Fourier transform, but rather a particular family of smooth functions depending parametrically on (z, k). The substitution of the plane wave (5.3) into the linearized system (5.2) gives

−izρ˜ + iρ0k · u˜ = 0, i −izρ0u˜ = −ikp˜ + (B0 · kB˜ − kB0 · B˜), 4π (5.4) −izp˜ + iγp0k · u˜ = 0, 2 −izB˜ − iB0 · ku˜ + iB0(k · u˜) = −κηk B,˜ which is a linear algebraic system for the complex amplitudesρ, ˜ u,˜ p˜, and B˜. It is worth noting that for Euler’s equations (1.30), the linearization procedure would give the system (5.4) with B0 = 0 and B˜ = 0. Let us examine this case ﬁrst. The linearized momentum balance equation without the magnetic ﬁeld and multiplied by iz reads

2 z ρ0u˜ = ik(−izp˜) = iz(−iγp0k · u˜), which can be rewritten in the form

2 z I − (γp0/ρ0)kk · u˜ = 0, (5.5) where I is the identity tensor. The quick way to solve this equation is a scalar multiplication by k, which gives

2 2 ˆ ˆ (z − (γp0/ρ0)k )(k · u˜) = 0, k = k/|k|, on one hand, and by k×, which gives

z2k × u˜ = 0, on the other hand. The latter implies that for z 6= 0,u ˜ is parallel to k, and the former reduces to the algebraic condition

2 2 z = (γp0/ρ0)k , which has two real valued solutions, z = ω with p ω = ±cS|k|, cS = γp0/ρ0. (5.6)

This describes a wave which moves in the direction of k with constant speed cS and is characterized by a velocity perturbation in the same direction k of the propagation, which means that the wave is purely compressional and therefore it

120 is accompanied by density and pressure oscillations. This is commonly referred to as the sound wave, and its propagation speed cS is the sound speed in the ﬂuid. The fact that cS is constant means, in particular, that the propagation speed does not depend of the direction of k, i.e., the wave is isotropic. Equation (5.6) speciﬁes the frequency of the wave, given the wave vector and it is referred to as the dispersion relation. On going back to (5.4), we note that the equation forρ ˜ andp ˜ can be explicitly solved in terms of the velocity perturbationu ˜; the induction equation requires some more care because of the resistivity term,

˜ iz −izB = 2 B0 · ku˜ − B0k · u˜ , z + iκηk for ω = Re(z) 6= 0 so that the denominator does not vanish for real k. The equation foru ˜ multiplied by z amounts to 1 −iz2ρ u˜ = k(−izp˜) − B · k(−izB˜) − kB · (−izB˜), 0 4π 0 0 1 iz h = −iγp0k(k · u˜) − 2 B0 · k B0 · ku˜ − B0k · u˜ 4π z + iκηk i − kB0 · B0 · ku˜ − B0k · u˜

= −iγp0k(k · u˜) 2 |B0| iz h 2 ˆ ˆ i − 2 kku˜ − kk(bk + kb) · u˜ + k(k · u˜) , 4π z + iκηk ˆ where b = B0/|B0| is the unit vector along the direction of the background magnetic field B0. Upon dividing by −iρ0 we can recognized the sound speed in the first term on the right-hand side, cf. equation (5.6), and the Alfvénspeed s 2 |B0| cA = , (5.7) 4πρ0 cf. section 3.5. Therefore, we have

2 2 2 2 ˆ ˆ z I − cskk − cAζη kkI − kk(bk + kb) + kk · u˜ = 0, (5.8)

2 where ζη = z/(z + iκηk ) accounts for the eﬀect of resistivity; for ideal MHD we have ηη = 1. Equation (5.8) is the MHD analogous of equation (5.5) and in fact it reduces to (5.5) when cA = 0. The presence of the magnetic ﬁeld induces a much richer wave motion. Equation (5.8) has the form

D(z, k)˜u = 0, (5.9) where the complex-matrix-valued function D(z, k) is referred to as the dispersion tensor. A non-trivial solutionu ˜ 6= 0 of (5.9) exists if and only if the matrix D is not invertible, that is, det D(z, k) = 0, (5.10) which should be solved for the complex frequency z, given a real wave vector k [62]. To every solution z(k) ∈ C it corresponds a plane wave solution of the

121 x3 B0 kk k

k⊥

Figure 5.1: Reference frame used to study plasma waves [63]. The magnetic field direction defines the direction of the x3 axis, while the x1 and x2 axes are rotated so that k2 = 0. This, in particular, yields the matrix form (5.12) of the dispersion tensor in equation (5.8). linearized MHD system (5.2) oscillating at the frequency ω(k) = Re z(k) and with amplitude depending on time exponentially ∼ eν(k)t where ν(k) = Im z(k), cf. equation (5.3). The wave can be stable, i.e., uniformly bounded for t ≥ 0, if ν(k) ≤ 0, or unstable, if ν(k) > 0. In the latter case, the wave amplitude grows exponentially in time, up to the point where the linearization (i.e., neglecting nonlinear terms in the perturbation) is no longer valid and a fully nonlinear treatment has to be considered. For the specific case of equation (5.8) the problem can be simplified by a suitable choice of the coordinate system. Without loss of generality we can choose the coordinate system so that the third axis is directed along the background ˆ magnetic field, i.e., b =e ˆ3 = (0, 0, 1), and so that the wave vector k belongs to ˆ the plane spanned bye ˆ1, eˆ3, i.e., k = (k⊥, 0, kk) = k⊥eˆ1 + kkeˆ3, where kk = b · k 2 2 2 and k⊥ = k − kk are the components of k parallel and perpendicular to the background magnetic field. The angle ϑ between the direction of the wave vector k and the magnetic field is the propagation angle,

kk = |k| cos ϑ, k⊥ = |k| sin ϑ. (5.11)

This coordinate system is referred to as the Stix frame in the theory of plasma waves [63] and it is sketched in ﬁgure 5.1. In the Stix frame, the dyadic products in (5.8) amount to the matrices

 2  k⊥ 0 kkk⊥ kk =  0 0 0  , 2 k⊥kk 0 kk and     0 0 k⊥ 0 0 0 ˆbk = 0 0 0  , kˆb =  0 0 0  . 0 0 kk k⊥ 0 kk

122 Correspondingly, the dispersion tensor in (5.8) for linear MHD waves reads

 2 2 2 2 2 2  z − csk⊥ − cAζηk 0 −cskkk⊥ 2 2 2 D(z, k) =  0 z − cAζηkk 0  . (5.12) 2 2 2 2 −cskkk⊥ 0 z − cskk

Let us ﬁrst study the ideal case, i.e., κη = 0 and ζη = 1. By inspection of the matrix (5.12) with ζη = 1, one can notice the solution

2 2 2 z − cAkk = 0, u˜1 =u ˜3 = 0, u˜2 6= 0, which means that there exists a normal mode with real-valued frequency

2 2 2 ω = cAkk, (5.13a) and polarized so that k · u˜ = 0,B0 · u˜ = 0. (5.13b) This normal mode is referred to as Alfvénwave or shear Alfvénwave and it is the most characteristic wave mode in MHD [1]. The fact that the frequency of shear Alfvénwaves is real means that the wave is stable and undamped for ideal MHD on a uniform plasma background. The first polarization condition in (5.13b) is equivalent to incompressibility ∇ · u1 = 0 and implies that density and pressure perturbations are zero, cf. equation (5.4). As for the magnetic perturbation, from the induction equation in (5.4) we have B˜ = ∓ |B0|/cA u,˜ (5.13c) that is, the magnetic field perturbation B˜ is either anti-parallel or parallel to the velocity perturbationu ˜ depending on the root of the dispersion relation (5.13a), namely, anti-parallel for ω = +cAkk or parallel for ω = −cAkk. The divergence- free condition k·B˜ = 0 is automatically satisfied, while the second of polarization conditions (5.13b) implies B0 · B˜ = 0, which means that the perturbation is orthogonal to the background magnetic field. Figure 5.2 shows a simple example of Alfvénwave propagating parallel to the background magnetic field. One can see that, the perturbation makes the field lines oscillate like strings without compressing the plasma. The fluid flow bends the magnetic field lines, thus creating an electric current J ∝ ∇ × B, which in presence of a magnetic field reacts back on the fluid by the field-line bending force, cf. section 3.5, opposing the bending of field lines and thus generating the wave oscillation. At last let us note that the shear Alfvénwave is the only wave mode in the incompressible limit. This can be checked by imposing the incompressibility condition k·u˜ = 0 in equation (5.4) and noticing that in this case the momentum balance equation implies the condition B0 ·u˜ = 0, which in turn gives B0 ·B˜ = 0 from the induction equation. On going back to the full problem, we can proceed systematically and apply the general condition (5.10) to the dispersion tensor (5.12) again in the ideal case, i.e., with ζη = 1. That gives

2 2 2 2 2 2 2 2 2 2 2 4 2 2 z − cAkk (z − cSk⊥ − cAk )(z − cSkk) − cSkkk⊥ = 0. (5.14)

123 Figure 5.2: Velocity and magnetic field for the simplest case of an Alfvénwave in a uniform plasma, shown at a given point in time. The background magnetic field is B0 = |B0|eˆ3 and the wave vector is k = |k|e3, i.e., the propagation is parallel to the background magnetic field. The amplitude of the perturbations is |B1|/|B0| = |u1|/cA = 0.05. Parallel black curves correspond to the field lines of the background magnetic field, while the thick blue lines are those of the total field, including the perturbation. The little arrows represent the velocity field (in red) and the magnetic field perturbation (in light blue).

124 The Alfv´enmode appears as a factor, leading to the shear Alfv´enwave addressed above. The factor in square brackets amounts to a quadratic equation for the square of the complex frequency z2, namely,

4 2 2 2 2 2 2 2 2 z − (cS + cA)k z + cScAkkk = 0, which is readily solved by the real valued frequencies 1h q i ω2 = (c2 + c2 ) ± (c2 + c2 )2 − 4c2 c2 cos2 ϑ k2. (5.15) 2 S A S A S A Those two wave modes are referred to as magnetosonic waves as they combine the sound speed with the Alfv´enspeed. It is customary to characterize magnetosonic waves in terms of their phase velocity vph deﬁned in general by

ω(k) v = . (5.16) ph |k|

For instance, the phase velocity of the shear Alfv´enwave is

vA ph = cos ϑ, (5.17) cA and it is natural to normalize the phase velocity to the Alfv´enspeed. For the case of the two magnetosonic waves, one has

± 1 v 1h 1 i 2 ph = (2 + γβ) ± (2 + γβ)2 − 8γβ cos2 ϑ 2 , (5.18) cA 2 where we have used the identity

2 cS 1 2 = γβ, cA 2 relating the ratio of sound and Alfvénspeeds to the plasma β defined in sec- + − tion 3.5. One can check that vph ≥ vph with equality for ϑ = 0, i.e., for parallel + propagation. Therefore, the wave corresponding to vph is referred to as the fast magnetosonic wave, while the other one is the slow magnetosonic wave. Figure 5.3 shows a polar plot of phase velocities of ideal MHD wave modes as function of the propagation angle: The curves are defined as vph(ϑ)(cos ϑ, sin ϑ) for ϑ ∈ [0, 2π), i.e., a straight line drawn from the center of the plot in the direction ϑ intersects one of the phase-velocity curves exactly at the distance vph(ϑ); when no intersection occurs, the wave mode is not propagating in that direction. From such a representation it is possible to appreciate the highly anisotropic behavior of the slow and the shear Alfvénwaves (in this plot an isotropic wave would be represented by an exact circle of radius given by its phase velocity). The most isotropic mode is the fast wave which is just the sound wave modified by the presence of the magnetic field. We conclude this section by examining the effect of a small but finite resistivity on ideal MHD waves. Equation (5.10) with dispersion tensor given in equation (5.12), gives 2 2 2 z − cAζηkk = 0, (5.19)

125 Figure 5.3: Polar plot of phase velocities of ideal MHD waves normalized to cA according to equations (5.17) and (5.18), as functions of the propagation angle ϑ, cf. equation (5.11) for the case γβ = 1. for the shear Alfv´enwave and

4 2 2 2 2 2 2 2 2 z − (cS + cAζη)k z + cScAζηkkk = 0 (5.20) for the two magnetosonic waves, where ζη depends, in particular, on the complex frequency z with the result that the solution of the dispersion equations is less simple than in the ideal case. The case of the Alfv´enwave can be dealt with dividing the corresponding complex dispersion relation by ζη with the result that 2 2 2 (z + iκηk )z − cAkk = 0, and this is a quadratic equation for the complex frequency z ∈ C. The solutions are q 2 2 1 2 4 i 2 z = ± cAkk − 4 κηk − 2 κηk . (5.21)

For κη = 0 we recover the frequency of the ideal Alfvénwave, while for a finite resistivity κη one can notice two effects. On one hand, the frequency has a real part that depends non-linearly on |k| (wave dispersion). On the other hand, it has acquired an imaginary part which means that the wave can either decrease (damping) or increase (instability) exponentially in time depending on whether Im(z) < 0 or Im(z) > 0, respectively. By inspection of the solutions, one notices that the largest imaginary part is obtained when kk = 0, which gives 1 Im(z) = (±1 − 1) κ k2 ≤ 0, (k = 0) 2 η k hence, resistivity introduces damping of the Alfvénwave. In the limit of small resistivity (and assuming cos θ 6= 0), one has

2 3 κη|k| i 2 2 4 2 2 z ≈ ±cA|kk| ∓ 2 − κηk , (for κηk /(4cAkk) 1), (5.22) 8cA cos θ 2

126 2 which shows that wave dispersion is O(κη) while resistive damping is O(κη). As for magnetosonic waves, in presence of resistivity the dispersion relation (5.20) amounts to an algebraic equation of forth degree: Again dividing by ζη we have 4 2 3 2 2 2 2 2 2 2 2 2 2 2 z + iκηk z − (cS + cA)k z − iκηk cSk z + cScAkkk = 0. However, if one is interested in the effects of a small resistivity on the ideal magnetosonic waves, the exact solution of this equation is not needed. It is possible to obtain a result analogous to equation (5.22) by a perturbation argument. For 2 simplicity, let us denote by Q(z, k) = Q0(z, k) + iκηk Q1(z, k) the polynomial above, where Q0(z, k) = 0 is the dispersion relation of the ideal magnetosonic mode and Q1(z, k) accounts for the effects of resistivity. We look for deforma- tions of an ideal root z = ω(k), with ω(k) given in (5.15), for small κη. With this 2 aim let us introduce the new complex variable w defined by z = ω(k) + iκηk w so that, by Taylor formula, 2 2 2 0 = Q0 ω(k) + iκηk w, k + iκηk Q1 ω(k) + iκηk w, k 2 0 = iκηk Q0 ω(k), k w + Q1 ω(k), k + ··· , 0 where Q0 denotes the complex derivative of the polynomial Q0 with respect to z and the dots stand for terms of higher order in the resistivity; the ideal dispersion relation Q0 ω(k), k = 0 has been accounted for. We can now solve for w, obtaining the lowest order correction

2 2 2 Q1 ω(k), k ω(k) − cSk w = − 0 = − 2 2 2 2 , Q0 ω(k), k 4ω(k) − 2(cS + cA)k and correspondingly the ﬁrst-order correction to the frequency of resistive magnetosonic waves amounts to 2 2 2 i 2 ω(k) − cSk z ≈ ω(k) − κηk 2 2 2 2 . 2 2ω(k) − (cS + cA)k The ﬁrst order correction could describe either a damping or an instability depending on its sign; physically we expect damping as resistivity dissipates the currents associated to the wave into heat. Upon accounting for the ideal dispersion relation, one has 1 1√ ω(k)2 − c2 k2 = (c2 − c2 )k2 ± ∆k2 S 2 A S 2 2 2 2 2 2 2 where ∆ = (cS + cA) − 4cScA cos θ, and ω(k)2 − c2 k2 1h c2 − c2 i S A√ S 2 2 2 2 = 1 ± , 2ω(k) − (cS + cA)k 2 ∆ and thus 2 2 i 2h cA − cS i z ≈ ω(k) − κηk 1 ± √ . (5.23) 4 ∆ On noting that √ q 2 2 2 2 2 2 ∆ = (cS + cA) − 4cScA cos θ q q 4 4 2 2 2 4 4 2 2 2 2 = cS + cA + 2cScA(1 − 2 cos θ) ≥ cS + cA − 2cScA = cA − cS ,

127 we can conclude that c2 − c2 |c2 − c2 | A√ s ≤ A S = 1, 2 2 ∆ |cA − cS| which implies that the ﬁrst-order eﬀect of resistivity on magnetosonic wave is damping as expected.

5.2 Nonlinear shear Alfvénwaves. The linear MHD waves discussed in section 5.1 are approximations of a solution of the equations of magnetohydrodynamics valid when the amplitude of the wave is sufficiently small to justify the linearization. However, a remarkable property of the shear Alfvénwave is that it actually corresponds to an exact solution of the fully nonlinear MHD equations, even with arbitrarily large amplitude. This is a direct consequence of the specific polarization of the Alfvénwave for which the magnetic field is either parallel or anti-parallel to the velocity field, cf. equation (5.13b). The Alfvénwave corresponds to an incompressible disturbance of an ideal plasma (η = 0), therefore let us consider the incompressible ideal MHD equations (3.50d). We consider incompressible perturbations of arbitrary amplitude of a uniform plasma with constant mass density ρ0 and magnetic field B0 in a steady state, i.e., u0 = 0. In view of incompressibility of the perturbation, the density is constant and remains equal to ρ0. Upon denoting by u the perturbation in the velocity field and by B the perturbation in the magnetic field, so that the total magnetic field is B0 + B, equations (3.50d) amount to  1  ∂tu + u · ∇u − (B0 + B) · ∇B = −∇P, 4πρ0

 ∂tB + u · ∇B − (B0 + B) · ∇u = 0, where we have accounted for the fact that B0 is a constant background field. It is convenient to introduce new variables B z± = u ± √ , (5.24) 4πρ0 which are referred to as Elsässervariables. For incompressible perturbations, ∇ · u = 0, and thus ∇ · z± = 0. In terms of Elsässervariables, incompressible ideal MHD equations with a uniform guide field B0 read

∂tz± ∓ vA · ∇z± + z∓ · ∇z± = −∇P, (5.25) √ where vA = B0/ 4πρ is the vectorial Alfvénvelocity (|vA| = cA where cA is the Alfvénspeed defined in section 5.1). It is worth noting that in the above system the pressure is determined implicitly by the constraints ∇ · z± = 0; it is sufficient to have a single scalar function P to enforce both constraints since the divergence of the equation for both z+ and z− yields the same equation for the pressure, namely, ∆P = ∇z+ : ∇z−. Equations (5.25) are fully non-linear and describe the exact dynamics of a perturbation (u, B) of the considered uniform plasma. One can note however that the only nonlinear terms describe advection of one of the Elsässervariable by the other one. It follows that if z− = 0, then the equation for z+ is linear and the constraint ∇ · z+ = 0 is automatically satisfied so that P is constant. Viceversa if z+ = 0, then the equation for z− is linear and again ∇ · z− = 0

128 with P = constant. Summarizing, we have found two classes of exact solutions, namely,  z− = 0,  ∂tz+ − vA · ∇z+ = 0, (5.26a)   ∇ · z+ = 0, and  z+ = 0,  ∂tz− + vA · ∇z− = 0, (5.26b)   ∇ · z− = 0. In both systems (5.26) the divergence-free constraint is identically satisfied for all time t ≥ t0 if it is at the initial time t0. According to this systems, an initial disturbance is transported rigidly in a direction parallel to the background field B0, either backward for the case of (5.26a), regressive wave, or forward for the case of (5.26b), progressive wave. Fourier transform shows that each harmonic composing the solution of system (5.26a) has the same dispersion and the same polarization as a regressive Alfvénwave, cf. equations (5.13a) and (5.13c), and analogously for the progressive wave. Therefore, a solution of either one of systems (5.26) represents a wave packet of shear Alfvénwaves. Large-amplitude Alfvénwaves are frequently encountered in space plasmas. Such waves, in view of their large amplitude, can trigger nonlinear processes such as parametric decays which have been used to explain various energy transfer mechanisms in the solar corona and solar wind [64].

5.3 Magnetic field diffusion. Let us consider the regime of low magnetic Reynolds numbers, Rm 1 with a constant plasma resistivity. This leads to pure magnetic field diffusion, which is a highly idealized situation hardly to be found in nature; nonetheless, this study allows us to understand how a finite plasma resistivity affects the magnetic field. For Rm 1, the hyperbolic terms in the induction equation (3.49) can be neglected and the latter decouples from the rest of the MHD system; with constant resistivity, we have

d ∂tB(t, x) = κη∆B(t, x), t > 0, x ∈ R . (5.27) Here, ∆ is the Laplace operator in Cartesian coordinates and d = 3, but the following results are valid in a generic number d of dimensions. We set up an initial value problem with the initial condition

B(0, x) = B0(x), ∇ · B0(x) = 0, given at time t = 0. Equation (5.27) is known as the heat equation as it is the same equation describing the diffusion of temperature in a thermally conducting body. This is also one of the the simplest examples of linear constant-coefficient partial differential equations, and the prototype of parabolic partial differential equations. Let us start by noting two important properties of this equation, that alone give a good intuition on the behavior of the solutions. The first property is that

129 the average field is conserved. If B(t, x) is a regular solution of (5.27) such that for every t, B(t, x) is integrable in space, we can define the average field Z B(t) = B(t, x)dx, Rd and, since κη∆B = ∇ · (κη∇B), Gauss theorem yields Z Z d B(t) = lim ∇ · κη∇B(t, x) dx = lim κηn · ∇Bdσ = 0, dt r→+∞ |x|≤r r→+∞ |x|=r since ∇B restricted to the sphere |x| = r approaches zero as r → +∞. Therefore the average field B(t) is a constant of motion

B(t) = constant. (5.28)

The second property is the dissipation of the L2-norm. If B(t, x) is a classical solution with a ﬁnite L2-norm, Z 2 2 kB(t)kL2(Rd) = |B(t, x)| dx, Rd then, Gauss theorem again gives Z Z d 2 kB(t)kL2(Rd) = 2 B(t, x) · ∂tB(t, x)dx = 2κη B(t, x) · ∆B(t, x)dx dt d d R R Z d Z 2 X 2 = κη ∇ · ∇B(t, x) dx − 2κη |∇Bi(t, x)| dx Rd d i,j=1 R d Z X 2 = −2κη |∇Bi(t, x)| dx ≤ 0 d i=1 R where the integral of the divergence vanishes as before. It follows that

2 2 kB(t)kL2(Rd) ≤ kB(0)kL2(Rd). (5.29) Those two properties alone allow us to have a fairly accurate qualitative picture 2 of the dynamics: If the L -norm of initial magnetic field B0 must be dissi- pated while conserving the average field, the maximum value of the field should decrease and the solution should spread over a larger volume. In addition, the dissipation of the L2-norm implies that B = 0 is the unique regular squared-integrable solution of the heat equation corresponding to the zero initial condition, B0 = 0. In fact, if kB0kL2(Rd) = 0, inequality (5.29) implies kB(t)kL2(Rd) = 0 for all t ≥ 0 and thus B = 0. For linear equations this is equivalent to uniqueness of the solution in the class of squared-integrable functions. Indeed if B1(t, x) and B2(t, x) are two regular squared-integrable solutions corresponding to the same initial condition B0, by linearity of the equation, their difference B1 −B2 is again a solution corresponding to the initial condition B1(0) − B2(0) = B0 − B0 = 0, thus, B1(t) − B2(t) = 0, which means that B1 = B2. We can construct solutions of the heat equation by means of a partial Fourier transform in space. This technique is important in itself as it can be applied

130 to any linear constant-coeﬃcient partial diﬀerential equation describing a time evolution on the full space Rd. Let us look for solutions that can be represented as Z 1 ik·x ˆ B(t, x) = d e B(t, k)dk, (2π) Rd with Fourier amplitude Bˆ(t, k). This is possible only if the initial condition B0 admits a similar representation with Fourier amplitude Bˆ0(k). We take initial conditions in the Schwartz space S (Rd) of rapidly decreasing functions, which, in particular, is invariant under Fourier transform. Then, Z −ik·x Bˆ0(k) = e B0(x)dx, Rd belongs to S (Rd). After Fourier transform equation (5.27) becomes

2 ∂tBˆ(t, k) + κηk Bˆ(t, k) = 0.

If the Fourier amplitude Bˆ(t, k) satisfies the ordinary differential equation d Bˆ(t, k) = −κ k2Bˆ(t, k), Bˆ(0, k) = Bˆ (k), (5.30) dt η 0 where the Fourier dual variable k is treated as a parameter, then B(t, x) is a solution of the heat equation (5.27) with initial condition B0(x). The obtained ordinary differential equation has the unique solution

2 −κη tk Bˆ(t, k) = e Bˆ0(k), and we have

1 Z 2 ik·x−κη tk ˆ B(t, x) = d e B0(k)dk (2π) Rd

1 Z Z 0 2 ik·(x−x )−κη tk 0 0 = d e B0(x )dx dk. (2π) Rd Rd For t > 0, the integral is absolutely convergent and we can exchange the integration order. With this aim we compute the Gaussian integral

1 Z 2 ik·x−κη tk K(t, x) = d e dk (2π) Rd −x2/(4κ t) √ e η 1 Z 2 = e−(ξ−ix/ 4κη t) dξ, (t > 0) d/2 d/2 (4πκηt) π Rd and the remaining integration gives a factor πd/2. Then,

κη t∆ B(t, x) = e B0(x) 2 Z e−x /(4κη t) (5.31) = K(t, x − x0)B (x0)dx0,K(t, x) = . 0 d/2 Rd (4πκηt)

The operator eκη t∆ : S (Rd) → S (Rd) is the heat ﬂow operator and its kernel K(t, x) is referred to as the heat kernel. As t → 0+, the kernel approaches the

131 Dirac’s δ-function K(t, x − x0) → δ(x − x0) in the space S 0(Rd) of tempered distributions, and thus the heat flow approaches the identity operator, eκη t∆ → I, thus recovering the initial condition. Since the heat kernel acts on the initial condition as a convolution with a rapidly decreasing function (for t > 0), we can extend this solution to initial conditions in the space of tempered distributions S 0(Rd), i.e., to a rather large class of (generalized) functions. For instance, it is sufficient that B0 is bounded by a polynomial in x for the integral in (5.31) to be absolutely convergent for all t > 0 and x ∈ Rd; the result solves the heat equation (5.27) for t > 0 and tends to the initial condition for t → 0+. We note that B(t, x) given by (5.31) is smooth in x for every t > 0 (strict inequality!) even when B0 is not, as a consequence of the regularity of the heat kernel: The heat flow is the prototype of regularizing (or smoothing) operator. The solution obtained in terms of the heat flow operator satisfies the constraint ∇ · B = 0 if the initial condition B0 does,

d Z X ∂K 0 0 0 ∇ · B(t, x) = (t, x − x )B0,i(x )dx Rd ∂x i=1 i d Z X ∂K 0 0 0 = − 0 (t, x − x )B0,i(x )dx Rd ∂x i=1 i d Z X ∂ 0 0 0 = − 0 K(t, x − x )B0,i(x )]dx Rd ∂x i=1 i Z 0 0 0 + K(t, x − x )∇ · B0(x )dx = 0, Rd where the first term vanishes because it is a divergence and the second because of the initial condition satisfies the constraint ∇ · B0 = 0. Since the heat kernel is normalized, i.e., Z K(t, x)dx = 1, Rd one can check that (5.28) is automatically satisfied too. The reader is encouraged to check the property (5.29) directly from (5.31). We can now consider two specific solutions in order to illustrate the diffusion of magnetic field. First, we consider a regular initial condition, namely a Gaussian tube of magnetic field lines. The initial condition is

−(x2+x2)/(2a2) B0(x) = Bceˆ3e 1 2 , a > 0, wheree ˆ3 is the unit vector in the direction of x3 and Bc is a constant magnetic ﬁeld amplitude (which is added for dimensional reasons). The solution amounts to the Gaussian integral

Z (y−y0)2 (y0)2 Bceˆ3 − − B(t, x) = e 4κη t 2a2 dy, 4πκηt R2

0 where the integration in x3 have been performed and y = (x1, x2). It remains

132 to compute a two-dimensional Gaussian integral; we note that 0 2 0 2 2 (y − y ) (y ) y 1 1 0 2 + 2 = 2 + + 2 y − η(y) 4κηt 2a 2a + 4κηt 4κηt 2a 2 y 2 = 2 + z , 2a + 4κηt 2 0 where η(y) = y/(1 + 2κηt/a ), and by the change of variable y 7→ z one gets

2 x2+x2 2a − 1 2 2a2+4κη t B(t, x) = Bceˆ3 2 e . (5.32) 2a + 4κηt Therefore, the magnetic field profile preserves its Gaussian shape, but the width p 2 √ increases by a factor 1 + 2κηt/a ∼ κηt as the solution spreads over the 2 −1 whole space; correspondingly the amplitude decreases ∼ (2κηt/a ) , so that the average field is constant. From the form of the heat kernel, we could have −d/2 expected a scaling of the amplitude of the form ∼ (κηt) where d is the dimension of the space; here, the physical dimension is d = 3, but the solution is essentially two-dimensional, hence we get the scaling with d = 2. As mentioned above we can consider much more general initial conditions. One particularly interesting case is given by the initial condition   − 1, x1 < 0,  B0(x) = Bceˆ3 H(x1) − H(−x1) = Bceˆ3 0, x1 = 0,   + 1, x1 > 0, which is a one-dimensional configuration with a jump at the plane x1 = 0, with the Heaviside step function being H(z) = 0 for z < 0, H(0) = 0, and H(z) = 1 for z > 0. The corresponding current density is given by c cB J (x) = ∇ × B (x) = c ∇H(x ) − H(−x ) × eˆ 0 4π 0 4π 1 1 3 cB cB = −2 c H0(x )ê × eˆ = c δ(x )ê × eˆ 4π 1 1 3 2π 1 1 3 cB = − c δ(x )ê , 2π 1 2 0 where δ(x1) is the Dirac distribution in x1 and we have used the identity H = δ. This represents a singular current density directed along the direction orthogonal to both the magnetic field and its gradient and localized exactly at the jump point. Such a field/current configuration is referred to as current sheet or current layer. Formation and persistence (stability) of current layers is particularly discussed in various aspects of MHD applications such as magnetic reconnection and equilibria. In the low magnetic Reynolds number regime, the current sheet diffuses. The solution can be computed analytically. Since the field is essentially one dimensional, the integration over x2 and x3 in (5.31) can be carried out first, with the result that −(x −x0 )2/(4κ t) Z e 1 1 η B(t, x) = B eˆ H(x0 ) − H(−x0 )dx0 c 3 1/2 1 1 1 R (4πκηt) Z +∞ Z 0 Bceˆ3 h i −(x −x0 )2/(4κ t) 0 = − e 1 1 η dx . 1/2 1 (4πκηt) 0 −∞

133 Figure 5.4: Normalized magnetic ﬁeld component B3/Bc at diﬀerent time as a function of x1 at x2 = x3 = 0, for the cases of solutions (5.32), left-hand-side panel, and (5.33), right-hand-side panel. The spatial coordinate x as well as all √ 1 lengths including κηt are normalized. We choose the normalization length so that a = 1 in the case of solution (5.32), while the current sheet solution (5.33) p depends only the dimensionless quantity x1/ 4κηt so that the result is scale invariant.

The remaining Gaussian integrals can be expressed in terms of the error function which is deﬁned by

z 2 Z 2 erf(z) = √ e−u du, z ∈ R. π 0

0 0 p In fact, the change of variable x1 7→ u = (x1 − x1)/ 4κηt yields √ Z x1/ 4κη t Z +∞ Bceˆ3 h i −u2 B(t, x) = √ − √ e du π −∞ x1/ 4κη t √ √ 0 x1/ 4κη t ∞ x1/ 4κη t B eˆ h Z Z Z Z i 2 = √c 3 + − + e−u du ( π −∞ √0 0 0 Z x1/ 4κη t 2 −u2 = Bceˆ3 √ e du, π 0 p from which one recognizes the error function of argument x1/ 4κηt, and p B(t, x) = Bceˆ3 erf x1/ 4κηt . (5.33) p The error function is essentially a smoothed jump at x1 = 0, and 4κηt is the spatial scale of the field variation near the jump: Again we find that such a p scale increases in time like 4κηt. Both solutions (5.32) and (5.33) are essentially one-dimensional diffusion processes as the magnetic field has only one non-vanishing component and that depends on one variable. Specifically, in the case of solution (5.32) one has diffusion in the radial direction (due to symmetry), while in the case of solution (5.33) one has diffusion in the x1 coordinate. Figure 5.4 shows the time-evolution of B3 for the two considered examples. Diffusion yields a broadening of the Gaussian profile in the case of (5.32), and a smoothing of the current sheet of the case

134 Figure 5.5: Magnetic field diffusion in a “Gaussian flux tube” (left-hand-side panels) and in a current sheet (right-hand-side panels). The uppermost panels show the respective initial condition and time is increasing from top to bottom. The same cases as in figure 5.4 are considered, with the same normalizations. The magnetic field is represented by arrows of different color and length, while the contour plot at the bottom of each panel shows the profile of B3/Bc in the (x1, x2)-plane. The color scale is constant in time, while the size of the arrows is normalized to the maximum norm of B at the given time.

135 of (5.33). Figure 5.5 gives a three-dimensional representation of the magnetic field according to solutions (5.32) and (5.33). For the case of a Gaussian flux tube, the field has the same direction everywhere and just spreads over a larger area, thus reducing its intensity. In the case of the current sheet, the magnetic field has different directions depending on the value of x1; diffusion tends to average out the field in space so that in regions where both opposite directions of the magnetic field are present, the convolution with the heat kernel averages to a zero (or small) value of the field with the results that the current layer broadens as time goes by. A plasma with a magnetic Reynolds number so low is however not really common; in fact resistivity is usually small as it scales with temperature like −3/2 1/τe ∼ T , cf. equations (3.39) and (3.41). Nonetheless, magnetic field diffusion plays a role near current sheets even if resistivity is small because there the magnetic field can develop gradients so large that locally the magnetic Reynolds number is reduced and magnetic field diffusion quantitatively and qualitatively modifies the ideal dynamics.

136 A Energy conservation in extended MHD models

In section 4.1, the conservation of mass, momentum, and energy has been proven for the case of resistive MHD equations (3.50a), making explicit use of the specific Ohm’s law (3.42) of standard MHD. Such conservation laws however are valid even for more general forms of the Ohm’s law. In section 4.1 we have already noted that mass and momentum conservation are direct consequences of the mass continuity equation and the Euler’s equation, respectively, and therefore holds independently of the chosen form of the Ohm’s law. Energy conservation on the other hand requires some more comments. Let us assume that we want to consider a generalized form of the Ohm’s law which we write as u × B E + − ηJ = F (ρ, u, p, B), (A.1) c where F represents a possibly nonlinear operator acting on MHD state variables (ρ, u, p, B). For instance F can represent all the remaining terms in equation (3.40). Alternatively only the most important terms can be retained as appropriate for the considered physics problem, thus leading to a family of models, usually referred to as extended MHD [65, 66]. Such models are useful when condition (3.38) breaks down, e.g., due to the build-up of strong current layers. By retaining the effects of a large current density and electron pressure gradient, extended MHD models stand between standard MHD and a complete two-fluid description. However, when such large currents are present, assumption (3.36) becomes questionable stresses and heat flux due to differences in the ion and electron fluid velocity must be taken into account. Stresses are particularly important as the single-fluid momentum balance equation (3.31) and the generalized Ohm’s law (3.40) must be derived consistently (a fact that is sometimes overlooked in the literature where only the Ohm’s law is extended without consistently extend- ing the momentum equation). The importance of stresses due to the relative motion of the ion and electron fluids becomes apparent if one recalls that, equations (3.31) and (3.40) are equivalent to the system of the momentum balance laws for the inividual ion and electron fluids, the relation being established by means of (3.37). On the other hand, the single-fluid heat flux (3.32) enters the equation for the pressure (3.28) only, which, in addition to the sum of partial pressures of the plasma species, accounts for their kinetic energy relative to the center-of-mass fluid. For a plasma with one ion species and electrons, equations (3.37) allow us to write the differences ui − u and ue − u in terms of J/(ene); in addition both the electron ne and ion ni densities are related to the mass density ρ by the quasi-neutrality condition ne = Zini. It follows that we can obtain a closure for single-fluid equations of section 3.3 in the form

π = π(ρ, u, J), q = q(ρ, u, J). (A.2)

We have the choice of computing the functions F , π, and q exactly from the two- fluid model by means of (3.37) or to introduce various levels of approximations, each leading to a different flavour of MHD. Some popular extended models are breifly recalled at the end of this section.

137 Upon accounting for both the closure relations (A.2) and the extended Ohm’s law (A.1) in their general form, the single-ﬂuid system (3.35) must be replaced by the full system of equations (3.29), (3.31), and (3.34), coupled to the quasi- neutral limit of Maxwell’s equations. We obtain an extended version of MHD equations (3.45) which reads

Dρ + ρ∇ · u = 0, Dt Du J × B ρ = −∇p − ∇ · π + + ρg, Dt c D p γ h i + p∇ · u = J · (ηJ + F ) − ∇ · q − π : ∇u , Dt γ − 1 γ − 1 u × B (A.3) E + = ηJ + F , c 4π ∇ × B = J, c ∂tB + c∇ × E = 0, ∇ · B = 0.

In spite of the additional terms, namely, the functions F , π, and q, this system preserves much of the structure of the standard MHD equations (3.45). Particularly, the system is formulated in terms of single fluid quantities, and it interpolate from standard MHD, which corresponds to F = 0, π = 0, q = 0, to the full two-fluid model, which corresponds to the case of the full generalized Ohm’s law (3.40) together with the exact closures for π and q. Independently of the form of F , π, and q, one has mass, momentum, and energy conservation exactly. Therefore, all the physics models obtained by the various approximations of F , π, and q, enjoy those three basic conservation laws. Specifically, we have: • Mass conservation, d Z ρdx = 0. (A.4a) dt Ω • Momentum conservation,

d Z Z h B2 i ρudx = − π · n + p + n dS, (g = 0). (A.4b) dt Ω ∂Ω 8π

• Energy conservation, d Z c Z wdx = − B · n × (ηJ + F )dS, (g = −∇Φg), (A.4c) dt Ω 4π ∂Ω where the energy density is

1 p |B|2 w = ρu2 + + ρΦ + 2 γ − 1 g 8π 1 p + p |B|2 = m n u2 + m n u2 + i e + ρΦ + . 2 i i i e e e γ − 1 g 8π

138 In the second expression of the energy w, we have explictly accounted for deﬁ- nition (3.28), thus separation the sum of partial pressures pi + pe according to Dalton’s law, from the kinetic energy of electrons and ions relative to u. Mass and momentum conservation laws, equations (A.4a) and (A.4b), respectively, follow directly from the mass continuity and momentum balance equations. Particularly, the stress tensor π enters the momentum equation as a divergence and thus only contributes to the boundary term. Energy conservation on the other hand needs to be checked carefully. If gravitational forces are potential, i.e., g = −∇Φg, then the continuity equation, the momentum balance equation, and the pressure equation imply, cf. equation (3.33),

1 2 p ∂t 2 ρu + γ−1 + ρΦg 1 2 p + ∇ · ( 2 ρu + γ−1 + ρΦg)u + π · u + up + q = J · E. With the Ohm’s law (A.1), the right-hand side is u × B J · E = J · (ηJ + F ) − J · , c while the induction equation for the magnetic ﬁeld reads

∂tB − ∇ × (u × B) + c∇ × (ηJ + F ) = 0, which gives

|B|2 1 c ∂t 8π − 4π B · ∇ × (u × B) + 4π B · ∇ × (ηJ + F ) = 0. The combination of the foregoing balance laws gives

1 2 p |B|2 1 2 p ∂t 2 ρu + γ−1 + ρΦg + 8π + ∇ · ( 2 ρu + γ−1 + ρΦg)u + pu 1 c − 4π B · ∇ × (u × B) + 4π B · ∇ × (ηJ + F ) = J · (ηJ + F ) + u · (J × B)/c. By means of the same vector calculus identities used in section 4.1 we obtain Z Z d c wdx = B × (ηJ + F ) · ndS. dt Ω 4π ∂Ω This expresses the conservation of the total energy apart from the boundary terms, which is usually zero in practice in view of the chioce of the domain Ω. Hence the conservation laws of section 4.1 are so fundamental that apply to a fairly general family of extended MHD models. Let us conclude this section with a bird-eye view of two popular extended MHD models, both discussed in details by Kimura and Morrison [65, and references therein].

Hall MHD. As in standard electrodynamics, the (classical) Hall effect consists in an electric field established when a current density J is subject to an external magnetic field B. Specifically, the Lorentz force acting on the current (carried by electron) amounts to J × B −en E + , e c

139 and this must vanish at the equilibrium. Hence the electric ﬁeld balancing the magnetic force is, J × B EH = , enec and this is referred to as the Hall ﬁeld (usually expressed in terms of a potential for ordinary cunductors). We can reconize the term EH in equation (3.40). Hall MHD is obtained by retaining the Hall term in the Ohm’s law (some authors include the electron pressure gradient as well, [65]) and setting stressed to zero. This corresponds to the choice, J × B π = 0, F = EH = . enec The latter in particular, gives the Ohm law, cf. equation (A.1),

u × B E + e = ηJ, c where the flow of the electron fluid is defined by ue = u − J/(ene). We see that, when η = 0, the magnetic field is frozen into the electron fluid, cf. section 4.3.

Inertial MHD. The electron inertia eﬀects are described by the ﬁrst term on the left-hand side of the generalized Ohm’s law (3.40). Retaining that term corresponds to the choice

me h JJ i F = 2 ∂tJ + ∇ · uJ + Ju − . e ne ene Consistently one should retain the eﬀect of a ﬁnite current J while neglecting terms of O(me/mi). Then, equatuons (3.37) amount to ui = u, and ue = u − J/(ene. Therefore, the pressure (3.28) and the viscosity tensor (3.30) with πi = πe = 0 become

1 J 2 p = pi + pe + mene , 3 ene me 1 J 2 π = − 2 JJ + mene . e ne 3 ene The foregoing closure relations yield an extended MHD model known as inertial MHD [65, 66, 67]. It is worth noting that the pressure and viscosity forces in the momentum equation amount to

me J ∇ · π + pI = ∇(pe + pi) + J · ∇ , e ene and it includes an “extra-force” (the last term on the right-hand side) as discussed by Kimura and Morrison [65].

140 B Magnetic vector potential in MHD

The magnetic vector potential is a vector ﬁeld A such that [45]

B = ∇ × A.

This is not uniquely determined by the ﬁeld B, as the gauge transformation A 7→ A0 = A + ∇ϕ leaves the magnetic ﬁeld B invariant. In terms of the magnetic vector potential A and including resistivity, the MHD induction equation, cf. system (3.50a), takes the form ∇ × ∂tA − u × ∇ × A + κη∇ × ∇ × A = 0.

This implies that, at least locally, there exists a scalar function χ such that

∂tA − u × ∇ × A + κη∇ × ∇ × A = ∇χ, (B.1) and this the relevant equation for the magnetic vector potential. The scalar function χ accounts for the gauge transformation in the sense that a gauge- transformed ﬁeld A0 = A + ∇ϕ satisﬁes the same equation with χ replaced by 0 χ = χ + ∂tϕ.

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