Nonlinear Magnetohydrodynamic Waves

R. V. Polovin

Introduction

The present article is a survey of works devoted to nonlinear one-dimensional magnetohydrodynamic waves. The title of the survey is conditional—what is in fact investigated is a broad class of hyperbolic systems of quasilinear partial differential equations. The article does not claim mathematical rigor.

The survey is intended for mathematicians interested in applications, as well as for mechanicians and physicists interested in mathematical questions of magnetohydrodynamics.

We begin by explaining the meaning of characteristics and their connection with waves of small amplitude (§ 1).

In § 2 the theory of simple waves is presented and the question of Riemann invariants is discussed. In addition, it is shown that, in the absence of discontinuities, only a simple wave can border a region of constant flow.

When a simple wave moves, its profile changes (§ 3), which ultimately leads to the formation of discontinuities. The boundary conditions on surfaces of discontinuity are obtained from the conservation laws written in integral form (§ 4). The conservation laws are satisfied not only by true discontinuities, but also by extraneous solutions. To exclude them, one must add evolutionary conditions to the conservation laws (§ 5).

For waves of small intensity, the difference between a shock wave and an automodel wave (with accuracy up to quantities of second order inclusive) consists only in the sign of the density jump (§ 6). From this, in particular, it follows that the change of a Riemann invariant in a weak shock wave is of third order of smallness. Using these relations, one can solve in general form the problem of the decay of a discontinuity in the initial conditions for waves of small intensity (§ 7).

In § 8 the reasons for the appearance of discontinuities as the dissipative coefficients tend to zero are clarified, and the structure of a shock wave is investigated.

§ 1. Characteristics

1. In the theory of systems of partial differential equations, the concept of a characteristic plays an important role. Characteristics are closely connected with waves of small amplitude, with simple and shock waves.

To find the characteristics, it is convenient to write the system of equations of ordinary or magnetohydrodynamics, describing one-dimensional motion, schematically in the form*

* These equations describe waves in an unbounded medium in nonrelativistic and relativistic ordinary and magnetohydrodynamics, axially symmetric nonlinear waves in a pinch [1], in the theory of plasticity, and so on.

\[ \frac{\partial u_i}{\partial t}+\sum_{k=1}^{n} X_{ik}(u)\,\frac{\partial u_k}{\partial x}=0;\quad i=1,2,\ldots,n, \tag{1.1} \]

where \(u_i\) is the set of magnetohydrodynamic quantities, and \(X_{ik}\) are certain functions of \(u_1,\ldots,u_n\).

Let us first find the characteristics of the system (1.1). We recall the definition of characteristics. If a perturbation is prescribed on some arc \(AB\) in the \((x,t)\) plane, then it affects the solution \(u_i(x,t)\) of the system only in the region bounded by the characteristics passing through the points \(A\) and \(B\).

Since a characteristic separates the perturbed region from the unperturbed one, if all quantities \(u_i\) are prescribed along the characteristics (and consequently also their derivatives \(\partial u_i/\partial s\) along the direction of the characteristic), then the normal derivatives \(\partial u_i/\partial n\) cannot be determined uniquely by means of equations (1.1). Denoting by \(V\) the tangent of the angle of inclination of the characteristics to the \(t\)-axis, we express \(\partial u_i/\partial x\) and \(\partial u_i/\partial t\) in terms of \(\partial u_i/\partial s\) and \(\partial u_i/\partial n\):

\[ \frac{\partial u_i}{\partial t} = \frac{1}{\sqrt{V^2+1}}\frac{\partial u_i}{\partial s} - \frac{V}{\sqrt{V^2+1}}\frac{\partial u_i}{\partial n}, \]

\[ \frac{\partial u_i}{\partial x} = \frac{V}{\sqrt{V^2+1}}\frac{\partial u_i}{\partial s} + \frac{1}{\sqrt{V^2+1}}\frac{\partial u_i}{\partial n}. \]

Substituting these relations into equations (1.1), we obtain

\[ \sum_{k=1}^{n} (X_{ik}-V\delta_{ik})\,\frac{\partial u_k}{\partial n} = -\sum_{k=1}^{n} (VX_{ik}+\delta_{ik})\,\frac{\partial u_k}{\partial s}. \tag{1.2} \]

The right-hand side of equations (1.2) contains known quantities. In order that the normal derivatives \(\partial u_i/\partial n\) could not be determined from these equations, it is necessary that the determinant of the system vanish:

\[ \det (X_{ik}-V\delta_{ik})=0. \tag{1.3} \]

This equation, which is a polynomial of degree \(n\) with respect to \(V\), determines the tangents of inclination of \(n\) characteristics.

Since at different points \((x,t)\) the values of the quantities \(u_i\) (and consequently also \(X_{ik}\)) will be different, a characteristic will, generally speaking, be curvilinear. However, one can indicate an important special case when one of the characteristics is a straight line. This characteristic separates the magnetohydrodynamic wave from the region of steady flow (in which all the quantities \(u_i\) are constant).

1. Characteristics have a simple relation to waves of small amplitude. To find the latter, one must linearize equations (1.1), i.e., regard the quantities \(X_{ik}\) as constants depending on the equilibrium values of \(u_i\). The variable components \(u_i\) must be sought in the form

\[ u_i=r_i e^{i(kx-\omega t)}, \tag{1.4} \]

where \(r_i\) are constants. In this case system (1.1) takes the form

\[ \sum_{k=1}^{n} X_{ik} r_k = \frac{\omega}{k}\, r_i . \tag{1.5} \]

This relation shows that the phase velocities \(\omega/k\) of propagation of small perturbations are the eigenvalues of the matrix \(X_{ik}\), and the amplitudes \(r_i\) of small-amplitude waves are the right eigenvectors of this matrix. We shall assume all eigenvalues to be real and distinct (i.e., we shall assume system (1.1) to be hyperbolic). Comparing (1.3) and (1.4), we find that the tangent of the inclination of the characteristic to the \(t\)-axis is equal to the phase velocity of propagation of a small-amplitude wave.*

1. The system of equations (1.1) assumes a simpler form if it is reduced to characteristic form [2, 3], i.e., transformed so that all derivatives are taken only along characteristic directions.

For this, in addition to the right eigenvector \(r_i^{(j)}\) (the index \(j\) means that the vector belongs to the eigenvalue \(V^{(j)}\)), we shall also need the left eigenvector \(l_i^{(j)}\), defined by the equality

\[ \sum_{i=1}^{n} l_i^{(j)} X_{ik} = V^{(j)} l_k^{(j)} \tag{1.6} \]

(we note that the right and left eigenvectors are defined up to a normalizing factor).

Multiplying equations (1.1) by \(l_i^{(j)}\) and summing over \(i\), we obtain the system of equations in characteristic form

\[ \sum_{i=1}^{n} l_i^{(j)} \frac{du_i}{dj} = 0; \qquad j = 1, 2, \ldots, n, \tag{1.7} \]

where

\[ \frac{d}{dj} = \frac{\partial}{\partial t} + V^{(j)} \frac{\partial}{\partial x} \]

is the derivative along the \(j\)-th characteristic direction.

1. In what follows we shall need the completeness and orthogonality properties [2] of the vectors \(l^{(i)}\) and \(r^{(j)}\).

Let us first show that the vectors \(r^{(j)}\) \((j = 1, 2, \ldots, n)\) are linearly independent (the same is also true for the left eigenvectors). The proof is easily obtained by induction. We shall first show that two vectors \(r^{(1)}\) and \(r^{(2)}\), corresponding to distinct eigenvalues \(V^{(1)}\) and \(V^{(2)}\), are linearly independent.

Suppose the contrary, i.e.,

\[ r^{(2)} = \alpha r^{(1)}. \tag{1.8} \]

Multiplying this equality on the left by the matrix \(X\) and using (1.5), we obtain

\[ V^{(2)} r^{(2)} = \alpha V^{(1)} r^{(1)}, \]

or, by virtue of (1.8), \(V^{(1)} = V^{(2)}\), which contradicts the assumption.

\[ \text{* In magnetohydrodynamics the number of waves is seven } (n=7): \text{ two fast magnetosonic, two slow magnetosonic, two Alfvén, and one entropy wave [47].} \]

We shall now show that the three right eigenvectors \(r^{(1)}\), \(r^{(2)}\), and \(r^{(3)}\) \((V^{(1)} \ne V^{(2)} \ne V^{(3)})\) are linearly independent.

Suppose the contrary, i.e.,
\[ r^{(3)}=\alpha_1 r^{(1)}+\alpha_2 r^{(2)} . \tag{1.9} \]
Multiplying this equality on the left by \(X\), we obtain
\[ V^{(3)} r^{(3)}=\alpha_1 V^{(1)} r^{(1)}+\alpha_2 V^{(2)} r^{(2)} \]
and, using (1.9),
\[ \alpha_1 V^{(3)} r^{(1)}+\alpha_2 V^{(3)} r^{(2)} =\alpha_1 V^{(1)} r^{(1)}+\alpha_2 V^{(2)} r^{(2)} . \]
Since any two vectors \(r^{(1)}, r^{(2)}\) are linearly independent, it follows that \(V^{(3)}=V^{(1)}\), \(V^{(3)}=V^{(2)}\), which contradicts the assumption.

This assertion is proved analogously for a larger number of vectors.

We shall now show that right and left eigenvectors corresponding to distinct eigenvalues are orthogonal.

Let
\[ l^{(i)}X=V^{(i)}l^{(i)}, \qquad Xr^{(j)}=V^{(j)}r^{(j)} \tag{1.10} \]
and \(V^{(i)}\ne V^{(j)}\). Multiplying the first equality (1.10) on the right by \(r^{(j)}\), and the second on the left by \(l^{(i)}\), we obtain
\[ l^{(i)}Xr^{(j)}=V^{(i)}l^{(i)}r^{(j)}, \qquad l^{(i)}Xr^{(j)}=V^{(j)}l^{(i)}r^{(j)} . \]
Subtracting the second relation from the first and dividing by \(V^{(i)}-V^{(j)}\), we obtain
\[ l^{(i)}r^{(j)}=\sum_{k=1}^{n} l_k^{(i)} r_k^{(j)}=0, \quad \text{if } i\ne j . \tag{1.11} \]

We shall now show that
\[ l^{(i)}r^{(i)}\ne 0 . \tag{1.12} \]
Indeed, since the vectors \(r^{(i)}\) are linearly independent and \(l^{(i)}\) is perpendicular to all vectors \(r^{(j)}\) except \(r^{(i)}\), the equality \(l^{(i)}r^{(i)}=0\) would mean that the vector \(l^{(i)}\) is equal to zero.

1. Let us illustrate the theory presented with the example of ordinary hydrodynamics. In this case the system (1.1) takes the form
   \[ \begin{gathered} \rho_t+v\rho_x+\rho v_x=0,\\ v_t+\frac{c^2}{\rho}\rho_x+vv_x+\frac{p_s}{\rho}s_x=0,\\ s_t+v s_x=0, \end{gathered} \tag{1.13} \]
   where \(\rho\) is the density, \(v\) the velocity of the medium, \(c\) the speed of sound, \(p\) the pressure, and \(s\) the entropy per unit mass. Comparing (1.1) with (1.13), we find
   \[ u_1=\rho,\qquad u_2=v,\qquad u_3=s. \]
   The eigenvalues are \(V^{(1)}=v-c\) (a sound wave moving in the negative direction of the \(x\)-axis), \(V^{(2)}=v\) (an entropy wave), and \(V^{(3)}=v+c\) (a sound wave moving in the positive direction of the \(x\)-axis). The right eigenvectors are
   \[ \begin{gathered} r_1^{(1)}=\rho,\qquad r_2^{(1)}=-c,\qquad r_3^{(1)}=0,\\ r_1^{(2)}=p_s,\qquad r_2^{(2)}=0,\qquad r_3^{(2)}=-c^2,\\ r_1^{(3)}=\rho,\qquad r_2^{(3)}=c,\qquad r_3^{(3)}=0 . \end{gathered} \tag{1.14} \]

Right eigenvectors for the equations of magnetohydrodynamics: see [4].

§ 2. SIMPLE WAVES

1. We now turn to waves of large amplitude, described by the nonlinear equations (1.1). An important class of nonlinear waves, for which there exists a general method of finding the solution, is that of simple waves.

Simple waves are those solutions of the system of equations (1.1) which have the form of a traveling wave

\[ u_i=f_i\left(x-V^{(j)}t\right), \qquad i=1,\ 2,\ldots,\ n . \tag{2.1} \]

The quantity \(V^{(j)}\) entering this formula is a function of \(u_1, u_2,\ldots,u_n\). This means that, in the motion of a simple wave, points having different values \(u_1,\ldots,u_n\) move with different velocities. Thus, the profile of a simple wave is, in general, distorted as it propagates. An essential feature of a simple wave, distinguishing it from other nonlinear waves, is that all the quantities \(u_i\) move with the same velocity \(V^{(j)}\). The index \(j\) of the quantity \(V^{(j)}\) denotes the different simple waves. It follows from formula (2.1) that in a simple wave all the quantities \(u_i\) are functions of one of them, for example the density \(\rho\), which in turn depends on the coordinate and time.

Using this dependence, relation (2.1) can be written in the form

\[ x - V^{(j)}(\rho)\cdot t = f(\rho), \tag{2.2} \]

whence it follows that

\[ \frac{\partial \rho}{\partial t} + V^{(j)}(\rho)\frac{\partial \rho}{\partial x} =0. \tag{2.3} \]

1. Simple waves are closely connected with waves of small amplitude. Indeed, substituting into relations (1.1)

\[ \frac{\partial u_i}{\partial t} = \frac{d u_i}{d\rho}\frac{\partial \rho}{\partial t}, \qquad \frac{\partial u_k}{\partial x} = \frac{d u_k}{d\rho}\frac{\partial \rho}{\partial x} \]

and using (2.3), we obtain

\[ \sum_{k=1}^{n} X_{ik}(\rho)\frac{d u_k}{d\rho} = V^{(j)}(\rho)\frac{d u_i}{d\rho}. \tag{2.4} \]

Comparing (2.4) with (1.5), we obtain a system of ordinary differential equations

\[ \frac{d u_1}{r_1^{(j)}}= \frac{d u_2}{r_2^{(j)}}= \ldots = \frac{d u_n}{r_n^{(j)}} . \tag{2.5} \]

We see that to each wave of small amplitude there corresponds a simple wave, and that the velocities \(V^{(j)}\) in formulas (2.1) and (1.3) coincide. It also follows from relation (2.3) that in the \(j\)-th simple wave the quantity \(\rho\) (and with it all the quantities \(u_i\)) is constant along the \(j\)-th characteristic, and that this characteristic is a straight line.

1. The system (2.5) has \((n-1)\) first integrals

\[ I_1^{(j)}(u_i)=\mathrm{const}, \ldots,\qquad I_{n-1}^{(j)}(u_i)=\mathrm{const}. \]

These integrals are called Riemann invariants [5] (an equivalent definition was given earlier by Lax [2]).

Thus, Riemann invariants are functions of \(u_1,\ldots,u_n\) that remain constant in a simple wave. Different simple waves have different Riemann invariants, and each wave has \((n-1)\) invariants.

This definition is also valid for isentropic flows of ordinary hydrodynamics; however, in the latter case one usually uses another definition of a Riemann invariant, which is equivalent to the one given above only for \(n=2\).

In ordinary hydrodynamics, a Riemann invariant is a function \(I\) of the hydrodynamic quantities \(u_1,u_2\) that remains constant along some characteristic. If one were to use this definition, it would follow that in magnetohydrodynamics there are no Riemann invariants [3].

Indeed, under this definition a Riemann invariant \(I(u_1,\ldots,u_n)\) is constant along some characteristic with slope \(V^{(j)}\). This means that

\[ \frac{dI}{dj} = \sum_{i=1}^{n}\frac{\partial I}{\partial u_i}\frac{du_i}{dj} = 0 \left( \frac{d}{dj} = \frac{\partial}{\partial t} + V^{(j)}\frac{\partial}{\partial x} \right). \tag{2.6} \]

Comparing relations (2.6) and (1.7), we find \(l_i^{(j)}=\dfrac{\partial I}{\partial u_i}\), i.e., the components of the left eigenvector must satisfy the equations

\[ \frac{\partial l_i^{(j)}}{\partial u_k} = \frac{\partial l_k^{(j)}}{\partial u_i}. \tag{2.7} \]

The left eigenvector \(l^{(j)}\) is determined by equations (1.6) up to a factor—some function of \(u_1,\ldots,u_n\). This single function must satisfy \(\dfrac{n(n-1)}{2}\) equations (2.7), which for \(n>2\), generally speaking, is impossible. Equations (2.7) could prove compatible because of an accidental degeneracy. It is easy to show that in the case of magnetohydrodynamics there is no such degeneracy and equations (2.7) are incompatible. However, if the velocity of the medium is perpendicular to the magnetic field, then the equations of magnetohydrodynamics reduce to the equations of ordinary hydrodynamics [6], and Riemann invariants in the ordinary sense exist [7–11]. Let us also note that Riemann invariants in the ordinary sense always exist for systems of linear equations [3, 12].

In the case of ordinary hydrodynamics, the Riemann invariants, according to (2.5), (1.14), will be

\[ I_1^{(1)}=v+\int \frac{c(\rho)}{\rho}\,d\rho, \qquad I_2^{(1)}=s, \]

\[ I_1^{(2)}=v, \qquad I_2^{(2)}=p, \tag{2.8} \]

\[ I_1^{(3)}=v-\int \frac{c(\rho)}{\rho}\,d\rho, \qquad I_2^{(3)}=s \]

(\(v\) is the velocity of the medium, \(c\) is the speed of sound, \(s\) is the entropy).

1. Using the Riemann invariants, one can show that, in the absence of discontinuities, a region of constant flow can border only

with a simple wave. We shall give the proof due to Friedrichs (see [2]; another proof is given in [13, 14]).

Let a region of constant flow border on some flow along the \(k\)-th characteristic. From relations (2.5) it follows that \((n-1)\) \(k\)-invariants of Riemann \(I_1^{(k)}, \ldots, I_{n-1}^{(k)}\) satisfy the equation

\[ \sum_{j=1}^{n} r_j^{(k)} \frac{\partial I_i^{(k)}}{\partial u_j} = 0, \qquad i = 1, 2, \ldots, n-1. \tag{2.9} \]

This means that the \((n-1)\) vectors \(\operatorname{grad} I_i^{(k)} = \left( \dfrac{\partial I_i^{(k)}}{\partial u_1}, \ldots, \dfrac{\partial I_i^{(k)}}{\partial u_n} \right)\) \((i = 1, 2, \ldots, n-1)\) are orthogonal to the vector \(r^{(k)} = (r_1^{(k)}, \ldots, r_n^{(k)})\). Since, according to (1.11), any vector \(l^{(j)}\) for \(j \ne k\) is also orthogonal to \(r^{(k)}\), the vector \(l^{(j)}\) lies in the subspace spanned by the \((n-1)\) vectors \(\operatorname{grad} I_1^{(k)}, \ldots, \operatorname{grad} I_{n-1}^{(k)}\),

\[ l_i^{(j)} = \sum_{l=1}^{n-1} \alpha_{jl}\,\frac{\partial I_l^{(k)}}{\partial u_i}. \tag{2.10} \]

Substitute relations (2.10) into (1.7):

\[ \sum_{i=1}^{n} \sum_{l=1}^{n-1} \alpha_{jl}\,\frac{\partial I_l^{(k)}}{\partial u_i}\,\frac{d u_i}{d_j} = 0 \]

or

\[ \sum_{l=1}^{n-1} \alpha_{jl}\,\frac{d I_l^{(k)}}{d_j} = 0, \qquad j \ne k. \tag{2.11} \]

System (2.11) is a system of \((n-1)\) equations in the characteristic form (1.7), and the characteristics of system (2.11) are all the characteristics of system (1.1) except the \(k\)-th. It follows that the Riemann invariants \(I_1^{(k)}, \ldots, I_{n-1}^{(k)}\) are constant on both sides of the \(k\)-th characteristic. Therefore, the flow bordering on a constant flow is a \(k\)-th simple wave.

§ 3. CHANGE OF THE WAVE PROFILE

1. The form of a simple wave, i.e. its profile, does not remain constant during its motion. We shall investigate the nature of the change in the profile and show that in a number of cases this change leads to the appearance of a discontinuity, i.e. a shock wave.

Assume, for definiteness, that the simple wave moves in the laboratory frame of reference in the positive direction of the \(x\)-axis, i.e. that \(V > 0\).

If, moreover, the condition

\[ \frac{dV(\rho)}{d\rho} > 0 \tag{3.1} \]

is satisfied (the value of the derivative \(\dfrac{dV}{d\rho}\) depends on the equation of state of the medium,

in ordinary and magnetic hydrodynamics condition (3.1) is satisfied for almost all media), then points having greater density move faster than points with smaller density. Therefore, during propagation of a simple wave its profile changes. Rarefaction regions, on which \(\dfrac{\partial \rho}{\partial x}>0\), will be stretched out in the course of time, while compression regions, on which \(\dfrac{\partial \rho}{\partial x}<0\), will narrow. In other words, in rarefaction regions the gradients will decrease, and in compression regions they will increase (in absolute value).

The exception is the case when \(\dfrac{dV}{d\rho}=0\), or, according to (2.5),

\[ \sum_{i=1}^{n}\frac{\partial V}{\partial u_i}r_i \equiv 0 . \tag{3.2} \]

A simple wave for which this identity holds propagates without change of profile (in magnetic hydrodynamics this is an Alfvén wave).

We shall henceforth assume that

\[ V'(\rho)\equiv \sum_{i=1}^{n}\frac{\partial V}{\partial u_i}r_i \ne 0 . \tag{3.3} \]

1. To determine the wave profile, we differentiate equality (2.2) with respect to \(x\) at constant \(t\); substituting \(\dfrac{\partial \rho}{\partial x}=\rho_x(t)\), \(\dfrac{1}{f'}=\rho_x(0)\), we obtain

\[ \rho_x(t)=\frac{\rho_x(0)}{1+V'(\rho)t\cdot \rho_x(0)} . \tag{3.4} \]

In regions which at the initial instant were regions of rarefaction, according to (2.3), the condition \(\rho_x(0)>0\) is fulfilled; in compression regions \(\rho_x(0)<0\). It follows from formula (3.4) that for \(V'(\rho)>0\), in compression regions at the instant

\[ t=-\frac{1}{V'(\rho)\rho_x(0)} \]

the quantity \(\rho_x(t)\) becomes infinite, i.e., a discontinuity is formed. If \(V'(\rho)<0\), then the discontinuity is formed in a rarefaction region.

The indicated change of the wave profile leads to the fact that as \(t\to\infty\) the profile takes on a sawtooth form.

1. An important special case of simple waves is self-similar waves, in which all magnetohydrodynamic quantities depend on the coordinates and time only through the combination \(x/t\).

Since the equations of magnetic hydrodynamics without allowance for dissipation (1.1) are invariant with respect to the group of transformations \(x\to Cx,\ t\to Ct\), self-similar waves arise when the initial and boundary conditions are invariant with respect to this group of transformations. Such waves arise, for example, in the problem of the decay of a discontinuity in the initial conditions or in the problem of the motion of a piston with constant velocity.

Automodel waves correspond to \(f \equiv 0\) in formula (2.2):

\[ V^{(j)}(\rho)=-\frac{x}{t}. \tag{3.5} \]

Putting \(\rho_x(0)\to\infty\) in equality (3.4), we obtain for automodel waves

\[ \rho_x(t)=\frac{1}{V'(\rho)t}. \tag{3.6} \]

It is evident from this formula that if \(V'(\rho)>0\), then the automodel wave is a rarefaction wave, while if \(V'<0\), it is a compression wave. Moreover, the gradients of all quantities in an automodel wave decrease with time, and “overturning” of the wave profile is impossible.

Since the right-hand side of (3.5) decreases as \(t\) increases and \(x\) is constant, in the \(j\)-th automodel wave the phase velocity \(V_{(j)}\) always decreases as the wave (moving to the right) passes.

1. Let us now consider the degenerate case, when \(V'(\rho)=0\). For such waves relation (3.2) is satisfied. In magnetic hydrodynamics this relation is satisfied for Alfvén waves. It follows from what has been set out that the profile of an Alfvén simple wave does not change with time. Hence, in particular, it follows that an Alfvén wave cannot be automodel.

§ 4. DISCONTINUITIES

If the magnetohydrodynamic quantities are discontinuous functions of \((x,t)\), then, for the solution to be single-valued, \(n\) boundary conditions on each discontinuity line must be appended to equations (1.1).

It would seem that these boundary conditions can simply be obtained from equations (1.1). First of all, let us transform these equations so that they take the form of conservation laws

\[ \frac{\partial u_i}{\partial t}+\frac{\partial}{\partial x}F_i(u_1,\ldots,u_n)=0,\qquad i=1,\ldots,n. \tag{4.1} \]

From the differential conservation laws (4.1) we pass to the integral ones

\[ \iint_Q \frac{\partial u_i}{\partial t}\,dxdt+\iint_Q \frac{\partial F_i}{\partial x}\,dxdt=0, \tag{4.2} \]

where \(Q\) is an arbitrary domain. The double integrals (2.2) can be transformed into integrals over the contour \(\Gamma\) enclosing the domain \(Q\):

\[ \int_\Gamma [u_i\,dx-F_i(u)\,dt]=0. \tag{4.3} \]

Integrating expression (4.3) over a narrow loop-shaped contour \(\Gamma\), enclosing a discontinuity line with tangent slope

\[ \frac{dx}{dt}=S, \]

we obtain the boundary conditions on the discontinuity line

\[ S\,\Delta u_i=\Delta F_i(u). \tag{4.4} \]

(The symbol \(\Delta\) denotes the jump of the corresponding quantity. Another method for determining discontinuous solutions was proposed in works [15, 16]).

However, if one starts from equations (1.1), then a number of difficulties arise. First of all, is it always possible to reduce equations (1.1) to the conservation laws (4.1)?

To answer this question, let us write equations (4.1) in the form

\[ \frac{\partial u_i}{\partial t}+\sum_{k=1}^{n}\frac{\partial F_i}{\partial u_k}\,\frac{\partial u_k}{\partial x}=0, \]

whence

\[ \frac{\partial F_i}{\partial u_k}=X_{ik}. \tag{4.5} \]

Thus, the \(n\) unknown functions \(F_1,\ldots,F_n\) must satisfy \(n^2\) equations (4.5), which, generally speaking, is impossible. However, in the case of the equations of ordinary and magnetic hydrodynamics, owing to degeneracy, it is not only possible to reduce equations (1.1) to conservation laws, but the number of conservation laws turns out to be greater than \(n\) [17].

Thus, for example, from the three equations of ordinary hydrodynamics (1.13) there follow four conservation laws [18]:

\[ \frac{\partial \rho}{\partial t}+\frac{\partial}{\partial x}\rho v=0, \tag{4.6} \]

\[ \frac{\partial}{\partial t}\rho v+\frac{\partial}{\partial x}(\rho v^2+p)=0, \tag{4.7} \]

\[ \frac{\partial}{\partial t}\left(\frac{\rho v^2}{2}+\rho \varepsilon\right) +\frac{\partial}{\partial x}\rho v\left(\frac{v^2}{2}+w\right)=0, \tag{4.8} \]

\[ \frac{\partial}{\partial t}\rho s+\frac{\partial}{\partial x}\rho vs=0, \tag{4.9} \]

where \(\varepsilon\) and \(w\) are the internal energy and the heat function per unit mass, equal to

\[ w=\varepsilon+\frac{p}{\rho}. \]

This apparent contradiction between the number of unknowns and the number of conservation laws is resolved by the fact that, in reality, the initial equations are not equations (1.1), but the conservation laws (4.1). In this case, in ordinary hydrodynamics only the conservation laws of mass, momentum, and energy (4.6)—(4.8) are always satisfied. As for the entropy conservation law, it is satisfied approximately only for small values of the coefficients of viscosity and thermal conductivity and for small gradients of the hydrodynamic quantities. Within the profile of a shock wave, where these gradients are large, the entropy conservation law (4.9) is not satisfied. The use of the conservation law (4.9) in a shock wave has been the source of a number of errors (see [19, 20]).

If one takes the conservation laws in the differential form (4.1), then one can obtain several different boundary conditions (4.4). For example, from a single conservation law

\[ \frac{\partial u}{\partial t}+\frac{\partial}{\partial x}u^2=0 \tag{4.10} \]

with the aid of transformations (4.2), (4.3) we obtain the boundary condition

\[ S\Delta u=\Delta u^2. \]

If, however, equation (4.10) is multiplied by \(u\) and then again put in the form of a conservation law,

\[ \frac{\partial}{\partial t}u^2+\frac{\partial}{\partial x}\frac{2}{3}u^3=0, \]

then we obtain another boundary condition

\[ S\Delta u^2=\Delta \frac{2}{3}u^3 . \]

This apparent contradiction is explained by the fact that the fundamental laws are the conservation laws not in differential form (4.1), but in integral form (4.2). From the conservation laws in integral form the boundary conditions follow uniquely [21].

§ 5. EVOLUTIONARITY CONDITIONS

1. The differential equations (1.1), together with the boundary conditions (4.4) on the discontinuity line, do not always determine a unique solution [3, 21—23]. For example, when a piston is pushed out of a tube, two solutions are formally mathematically possible: 1) a simple rarefaction wave, 2) a rarefaction shock wave.

Usually the second solution is rejected on thermodynamic grounds: in a rarefaction shock wave the entropy decreases, which is impossible. In magnetohydrodynamics, however, cases are possible in which several discontinuous solutions correspond to given initial and boundary conditions, and the entropy increases on each of the shock waves.

For example, when an ideally conducting piston is pushed into a magnetohydrodynamic medium along the direction of the magnetic field [24], in a certain interval of velocities two solutions are formally possible: 1) the same shock wave as in the absence of a magnetic field, 2) two magnetohydrodynamic shock waves.

In reality, however, not all shock waves for which the boundary conditions are satisfied and the entropy increases can occur. It is also necessary that the solution depend continuously on the initial and boundary conditions, i.e., that an infinitely small perturbation of the magnetohydrodynamic quantities produce an infinitely small change in the solution.

1. To determine the change in the solution under a change of the initial data, let us prescribe an arbitrary perturbation of the quantities \(u_i\) on both sides of the surface of discontinuity. This perturbation will lead to the appearance of various waves of small amplitude. If the perturbation is different from zero only in a neighborhood of the discontinuity, then only those waves that emanate from the discontinuity line should be taken into account. After linearizing the boundary conditions (4.4), one obtains \(n\) linear homogeneous algebraic equations for the amplitudes of the various waves on both sides of the discontinuity line. In doing so, it should be taken into account that the shock-wave velocity \(S\) will also acquire an infinitely small increment \(\delta S\). After eliminating the quantity \(\delta S\), there remain \((n-1)\) independent equations (boundary conditions). Therefore the total number of outgoing waves must be equal to \((n-1)\). Discontinuities satisfying this condition are called evolutionary [25, 46]. In non-evolutionary discontinuities an infinitely small perturbation causes a finite change in the solution, namely, the splitting of the original discontinuity into several discontinuities (or simple waves) of finite magnitude

2. Differential equations.

[26]. Nonevolutionary discontinuities thus turn out to be unstable with respect to splitting and therefore cannot exist.

The necessity of the evolutionarity conditions can be explained visually as follows. If the number of outgoing waves is greater than the number of independent boundary conditions on the discontinuity line, then the initial perturbation gives rise to an infinite set of solutions, including solutions in which the frequency has an arbitrarily large imaginary part, i.e., arbitrarily rapidly growing waves [18]. If, on the other hand, the number of outgoing waves is less than the number of independent boundary conditions, then independent perturbations to the right and to the left of the discontinuity line will lead to a violation of the boundary conditions (4.4), which in turn will lead to the splitting of the initial shock wave.

1. Let us now determine the domains of evolutionarity of discontinuities. To this end, arrange the phase velocities of waves of infinitesimal amplitude in increasing order:

\[ V^{(1)} < V^{(2)} < \ldots < V^{(n)} . \]

Suppose that to the left of the discontinuity surface there emerge \((k-1)\) waves, i.e.,

\[ V^{(k-1)}(u_l) < S^{(k)} < V^{(k)}(u_l), \tag{5.1} \]

where \(u_l\) is the value of the vector \((u_1,\ldots,u_n)\) to the left of the discontinuity line. Since the total number of outgoing waves must be equal to \((n-1)\), \((n-k)\) waves must emerge to the right, i.e.,

\[ V^{(k)}(u_r) < S^{(k)} < V^{(k+1)}(u_r), \tag{5.2} \]

where \(u_r\) is the value of the vector \((u_1,\ldots,u_n)\) to the right of the discontinuity line. The evolutionarity conditions (5.1) define \(n\) different types of evolutionary discontinuities [2]. (For a single equation the evolutionarity conditions were formulated in [27—29].)

Let us note that as the intensity of the shock wave tends to zero, \(u_l \to u_r\), i.e., \(V^{(k)}(u_l) \to V^{(k)}(u_r)\), and the shock-wave velocity \(S^{(k)}\) tends to the phase velocity of propagation of an infinitesimal perturbation \(V^{(k)}(u)\).

It also follows from formulas (5.1), (5.2) that

\[ V^{(k)}(u_l) > S^{(k)} > V^{(k)}(u_r). \tag{5.3} \]

This means that the jump in the quantity \(V^{(k)}\) in the shock wave is always positive (the wave moves to the right): \(\Delta V^{(k)} > 0\). Therefore, when \(dV^{(k)}/d\rho > 0\) the shock wave is a compression wave, and in the opposite case it is a rarefaction wave. Comparing with the results of § 3, we find that a shock wave formed when the profile of a simple wave “overturns” is always evolutionary (for sufficiently small intensities).

1. In the case of ordinary hydrodynamics, for shock waves moving to the right, we obtain from (5.1), (5.2)

\[ v_l < S < v_l + c_l,\qquad v_r + c_r < S. \tag{5.4} \]

Noting that the index \(r\) refers to the medium ahead of the shock wave, and the index \(l\) to the medium behind the wave, we see from (5.4) that the shock wave moves relative to the medium ahead of it with supersonic speed, and relative to the medium behind it with subsonic speed.

The existence and uniqueness of the solution of the Cauchy problem in the class of continuous and discontinuous evolutionary solutions in the case of a single equa-

tion was proved in [21, 25, 27, 30—33] (see [3]). In the case of a system of equations this was proved only in certain special cases [33—37] (see [3]).

1. The evolutionary conditions can also be given another meaning [38]. Evolutionary shock waves are discontinuities that arise from simple waves under a continuous increase of the steepness of the profile. This definition is less general and, using it, one can arrive at the erroneous conclusion that Alfvén discontinuities [4] are non-evolutionary, since they cannot arise through the “overturning” of the profile of a simple wave (see § 3). The necessity of the existence of Alfvén discontinuities follows from the fact that, generally speaking, they are formed in the decay of a discontinuity in the initial conditions (see § 7).

2. Let us now turn to the question of whether a shock wave located on the boundary of the region of evolutionary character is evolutionary, i.e., when the conditions

\[ V^{(k-1)}(u_l)=S^{(k)} \tag{5.5} \]

or

\[ S^{(k)}=V^{(k+1)}(u_r) \tag{5.6} \]

are satisfied (see inequalities (5.1), (5.2)). This question may seem scholastic, since it is enough to change the intensity of the shock wave by an infinitesimal amount for it to become either evolutionary or non-evolutionary. However, there exist such values of the quantities \((u_1,\ldots,u_n)\) ahead of or behind the shock wave that the conditions (5.5), (5.6) are satisfied for any intensity. Thus, if the magnetic field on one side of the shock wave is directed along the normal to the discontinuity surface, then the velocity of the medium on the other side of the shock wave relative to the discontinuity surface is equal to the Alfvén velocity, i.e. condition (5.5) or (5.6) is satisfied (the so-called special shock wave [39, 40]). When conditions (5.5) or (5.6) are fulfilled, the phase velocity of one of the waves into which the initial disturbance decomposes is equal to zero (in the reference frame in which the discontinuity is at rest). It can be shown [26] that such waves should be classified as incoming. Using this rule, we arrive at the conclusion that special shock waves are non-evolutionary.

On the other hand, special shock waves are formed in the decay of a discontinuity in the initial conditions if the magnetic field on one side of the discontinuity surface is directed along the normal (see § 7). This indicates that the conclusion drawn above about the non-evolutionary character of special shock waves is incorrect. The error consisted in our distinguishing outgoing disturbances from incoming ones according to the phase velocity of the linearized disturbances, whereas a more accurate definition is to determine whether the characteristics describing nonlinear disturbances will be outgoing or incoming [25]. If an infinitesimal disturbance is imposed on a special wave, then the magnetic field will no longer be directed along the normal, and therefore in nonlinear theory special waves are evolutionary.

The foregoing is illustrated in the figure, where \(AB\) is the discontinuity line, the straight lines \(CA\) and \(AD\) are the incoming and outgoing characteristics of the linear theory (the positive direction on a characteristic is deter-

Evolutionary character of a special shock wave:

\(AB\) — discontinuity line; \(CA\) and \(AD\) — incoming and outgoing characteristics of the linear theory; \(AE\) — characteristic that is incoming in the linear theory and outgoing in the nonlinear theory.

is determined by the direction of increasing time \(t\)). The curve \(AE\) is the initial characteristic in the nonlinear theory. Since this characteristic concerns the line of discontinuity at the point \(A\), in the linear theory it coincides with the line \(BA\), i.e., it is regarded as incoming, which is incorrect.

§ 6. WAVES OF SMALL INTENSITY

1. Let us first show, following Lax [2], that the relations between the jumps of various magnetohydrodynamic quantities in a shock wave of small intensity are the same as in a simple wave, with accuracy up to terms of second order of smallness.

Let \(\varepsilon\) be a certain parameter characterizing the intensity of the shock wave, with \(\varepsilon = 0\) ahead of the wave. Thus, \(u_l = u(\varepsilon)\), \(u_r = u(0)\) (the wave moves to the right).

First consider a simple wave. It follows from relations (2.5) that the parameter \(\varepsilon\) can be chosen so that the relations

\[ \dot{u}_i(0) = r_i(0), \tag{6.1} \]

\[ \ddot{u}_i(0) = \dot{r}_i(0) \tag{6.2} \]

hold (the dot denotes differentiation with respect to \(\varepsilon\)).

To prove the theorem it is enough to show that, also in a shock wave, \(\varepsilon\) can be chosen in such a way that relations (6.1), (6.2) hold.

To prove this, we differentiate relation (4.4) with respect to \(\varepsilon\). Using formula (4.5) and denoting the derivative with respect to \(\varepsilon\) by a dot, we obtain

\[ S\dot{u}_i+\dot{S}\Delta u_i=\sum_k X_{ik}\dot{u}_k, \tag{6.3} \]

Differentiate once more with respect to \(\varepsilon\):

\[ S\ddot{u}_i+2\dot{S}\dot{u}_i+\ddot{S}\Delta u_i = \sum_k X_{ik}\ddot{u}_k+\sum_k \dot{X}_{ik}\dot{u}_k. \tag{6.4} \]

Putting \(\varepsilon=0\) in (6.3), (6.4), and at the same time \(\Delta u_i=0\), and observing that \(S(0)=V(0)\), we obtain

\[ V(0)\dot{u}_i(0)=\sum_k X_{ik}(0)\dot{u}_k(0), \tag{6.5} \]

\[ V(0)\ddot{u}_i(0)+2\dot{S}(0)\dot{u}_i(0) = \sum_k X_{ik}(0)\ddot{u}_k(0)+\sum_k \dot{X}_{ik}(0)\dot{u}_k(0). \tag{6.6} \]

It follows from (6.5) that the vector \(\dot{u}_i(0)\) is parallel to the right eigenvector \(r_i(0)\):

\[ \dot{u}_i(0)=\alpha r_i(0). \]

Changing the parameter \(\varepsilon\), i.e., introducing instead of \(\varepsilon\) a new parameter \(\eta=\eta(\varepsilon)\), one can make the coefficient \(\alpha\) equal to unity, i.e., make relation (6.1) hold. For this it is enough to take such a function \(\eta(\varepsilon)\) that \(\eta(0)=0\), \(\eta'(0)=\alpha\).

Let us proceed to the proof of relation (6.2) for shock waves. Differentiate (1.5) with respect to \(\varepsilon\), regarding \(\omega/k \equiv V\) as a function of \(\varepsilon\), and then put \(\varepsilon=0\):

\[ V(0)\dot{r}_i(0)+\dot{V}(0)r_i(0) = \sum_k X_{ik}(0)\dot{r}_k(0)+\sum_k \dot{X}_{ik}(0)r_k(0). \tag{6.7} \]

Multiply (6.6) and (6.7) by the components of the left eigenvector \(l_i(0)\) (corresponding to the same eigenvalue) and sum over \(i\). Taking (6.1) into account, we obtain

\[ V\sum_i l_i \ddot u_i+2\dot S\sum_i l_i r_i =\sum_{i,k} l_i X_{ik}\ddot u_k+\sum_{i,k} l_i \dot X_{ik} r_k, \]

\[ V\sum_i l_i \dot r_i+\dot V\sum_i l_i r_i =\sum_{i,k} l_i X_{ik}r_k+\sum_{i,k} l_i \dot X_{ik}r_k . \tag{6.8} \]

By the definition of the left eigenvector (1.6), the first terms on the left- and right-hand sides of equations (6.8) cancel each other, and we obtain

\[ 2\dot S\sum_i l_i r_i=\sum_{i,k}l_i\dot X_{ik}r_k, \]

\[ \dot V\sum_i l_i r_i=\sum_{i,k}l_i\dot X_{ik}r_k . \tag{6.9} \]

Subtracting the equalities (6.9) one from the other and using the orthogonality relation (1.12), we obtain

\[ \dot S(0)=\frac{1}{2}\dot V(0). \tag{6.10} \]

Substituting (6.10) and (6.1) into (6.6), (6.7) and subtracting one of the resulting equations from the other, we find

\[ V(\ddot u_i-\dot r_i)=\sum_k X_{ik}(\ddot u_k-\dot r_k). \]

This relation means that the vector \(\ddot u_i-r_i\) is parallel to \(r_i\):

\[ \ddot u_i(0)-\dot r_i(0)=\beta r_i(0). \tag{6.11} \]

By changing the parameter \(\varepsilon\), one can make the coefficient \(\beta\) in this formula become zero. For this it is sufficient to choose such a function \(\eta(\varepsilon)\) that, when the condition \(du_i/d\eta=r_i\) is satisfied, the equalities \(\eta(0)=0\), \(\eta'(0)=1\), \(\eta''(0)=\beta\) hold. Then relation (6.11) will turn into (6.2), which proves the assertion.

1. As was shown in §§ 2, 5, in a self-similar wave \(\Delta V<0\), while in a shock wave \(\Delta V>0\) (the waves move to the right); therefore the jumps of all magnetohydrodynamic quantities in the shock and self-similar waves have opposite signs.

The theorem proved above means, in particular, that in the second approximation shock waves are reversible, and that the entropy jump is of third order of smallness. Moreover, in this approximation, when the boundary conditions on the shock wave are obtained, all the paradoxes mentioned at the beginning of § 4 disappear.

We shall now determine the sign of the quantity \(\varepsilon\). Noting that \(S(0)=V(u_r)\), \(S(\varepsilon)=S\), we write equality (6.10) in the form

\[ S=V(u_r)+\frac{1}{2}\dot V(0)\varepsilon. \tag{6.12} \]

Comparing it with the obvious relation

\[ V(u_l)=V(u_r)+\dot V(0)\varepsilon, \tag{6.13} \]

we find that a shock wave of small intensity satisfies the evolutionarity conditions (5.1), (5.2) if

\[ \dot V(0)\varepsilon>0. \tag{6.14} \]

Let us note that from formulas (6.12), (6.13) it also follows that the velocity of the shock wave is the arithmetic mean of the phase velocities ahead of and behind the wave [2]:

\[ S=\frac{1}{2}[V(u_r)+V(u_l)]. \tag{6.15} \]

1. We shall now show that the change of the Riemann invariants in the shock wave is of order \(\varepsilon^3\) [2].

For this it is sufficient to show that

\[ \dot I(0)=0,\quad \ddot I(0)=0. \tag{6.16} \]

Since \(I\) depends on \(u_i\), which in turn depend on \(\varepsilon\), taking (6.1), (6.2) into account, we obtain

\[ \dot I(0)=\sum_i \frac{\partial I}{\partial u_i}\, r_i(0), \tag{6.17} \]

\[ \ddot I(0)=\sum_i \frac{d}{d\varepsilon}\left(\frac{\partial I}{\partial u_i}\right) r_i(0) +\sum_i \frac{\partial I}{\partial u_i}\, \dot r_i(0). \tag{6.18} \]

It is not difficult to verify that the expressions (6.17), (6.18) are equal to zero by differentiating the identity (2.9). This proves the assertion.

From the fact that the change of the Riemann invariants in the shock wave is of third order of smallness, it follows that entropy is one of the Riemann invariants.

Let us note that, according to (6.12), the difference between the velocity of the shock wave \(S\) and the phase velocity of propagation of the linear wave \(V\) is a linear function of the amplitude [4]. Therefore, in the interaction of a shock wave and a simple wave of the same type* in the quadratic approximation, the shock wave does not affect the simple wave, whereas the simple wave, according to (4.4), causes a perturbation of the shock wave.

§ 7. DECAY OF A DISCONTINUITY

1. We shall investigate the decay of a discontinuity in the initial conditions [2], for which the relations (4.4) are not satisfied. This problem is the first step on the way to proving the theorem of existence and uniqueness of the solution of the Cauchy problem in the class of evolutionary discontinuities [25] (see also [3]).

If the values of the magnetohydrodynamic quantities on both sides of the discontinuity at the initial instant are constant, then the waves arising in the decay of the discontinuity can be only discontinuous or self-similar.

Let us consider the decay of a discontinuity of small intensity.

In the first approximation with respect to the amplitudes of the waves formed, the relations between the jumps of the magnetohydrodynamic quantities at the shock—

* That is, corresponding to one and the same eigenvalue.

linear and self-similar waves are identical. Using relations (2.5), we obtain for the \(j\)-th wave

\[ \Delta_j u_i=\varepsilon_j r_i^{(j)}, \tag{7.1} \]

where \(\varepsilon_j\) is a quantity characterizing the amplitude of the wave. Denoting by \(\Delta u_i\) the initial jumps of the quantities, we obtain

\[ \sum_{j=1}^{n}\varepsilon_j r_i^{(j)}=\Delta u_i,\qquad i=1,\ldots,n. \tag{7.2} \]

Since the vectors \(r^{(j)}\) are linearly independent, all the quantities \(\varepsilon_j\) can be found uniquely from the system (7.2). The sign of \(\varepsilon_j\) determines the character of the \(j\)-th wave (shock or self-similar).

Let us note that, for any values of \(u_1,\ldots,u_n\) on one side of the discontinuity line, the jumps \(\Delta u_1,\ldots,\Delta u_n\) can be chosen so that the amplitudes of all waves \(\varepsilon_j\) are different from zero. From this, in particular, follows the necessity of the occurrence of special shock waves (if the magnetic field on one side of the discontinuity line is directed along the normal) and Alfvén discontinuities.

1. The decay of a discontinuity of small intensity in magnetohydrodynamics was investigated in [41]. The decay of non-evolutionary magnetohydrodynamic shock waves in a number of particular cases was considered in [42—44]. A qualitative investigation of the general problem of the decay of an arbitrary magnetohydrodynamic discontinuity and of a non-evolutionary shock wave of arbitrary intensity was carried out in [45, 48, 49].

A special case of the problem of the decay of a discontinuity in the initial conditions is the piston problem. If the piston begins at the instant \(t=0\) to move with constant velocity, then a discontinuity arises between the velocity of the piston and the velocity of the medium.

The piston problem in various cases was solved in [50—55].

The problems of collision of shock waves and of reflection of shock waves from a wall and from a region of strong magnetic field also reduce to the problem of the decay of a discontinuity [56—58].

1. We now turn to the question of the existence and uniqueness of the solution of the problem of the decay of an arbitrary discontinuity.

For an ideal gas, the existence of a solution was proved by V. V. Gogosov [45]. In order to prove uniqueness of the solution, it is sufficient to show that the system of equations of magnetohydrodynamics is “convex” [3]. Apparently, the proof given in the review [3] can also be carried over to magnetohydrodynamics.

If the theorem on the existence and uniqueness of the solution of the problem of the decay of an arbitrary discontinuity is true, then it follows directly from it that an evolutionary shock wave cannot split, whereas a non-evolutionary shock wave must split into a number of self-similar and shock (evolutionary) waves.

§ 8. STRUCTURE OF SHOCK WAVES

1. The equations of an ideal medium are obtained by discarding the terms describing the effects of viscosity, thermal conductivity, and Joule heating (due to electrical resistance). Since these terms contain higher derivatives, discarding them lowers the order of the differential equations. This leads to a decrease in the num-

to the constants of integration, i.e., to the impossibility of satisfying the boundary conditions. In order to satisfy the boundary conditions, it is necessary to consider discontinuous solutions [59]. Let us clarify this by an example. Suppose it is required to find a solution of the differential equation

\[ \varepsilon \frac{d^2 y}{d x^2} + y \frac{dy}{dx} = 0,\quad \varepsilon > 0, \tag{8.1} \]

satisfying the boundary conditions

\[ y(+\infty)=1,\quad y(-\infty)=-1. \tag{8.2} \]

The solution of this problem has the form

\[ y=\operatorname{th}\frac{x}{2\varepsilon}. \tag{8.3} \]

On the other hand, for small \(\varepsilon\) in equation (8.1) one may discard the first term,

\[ y\frac{dy}{dx}=0. \tag{8.4} \]

The solution of the simplified equation (8.4) has the form

\[ y=C=\mathrm{const} \tag{8.5} \]

and cannot satisfy the two boundary conditions (8.2). However, as we shall now see, the solution (8.5) satisfies equation (8.1) for \(|x|\gg \varepsilon\). Indeed, passing in expression (8.3) to the limit \(\varepsilon \to 0\), we obtain

\[ y=1,\quad \text{if } x>0, \tag{8.6} \]

\[ y=-1,\quad \text{if } x<0. \tag{8.7} \]

Solution (8.6) corresponds to formula (8.5) with \(C=1\), and solution (8.7)—with \(C=-1\). Thus, as \(\varepsilon \to 0\), the continuous solution (8.3) tends to the discontinuous solution (8.6), (8.7).

1. In order to investigate the structure of a shock wave, one must add dissipative terms to the equations of an ideal medium (1.1):

\[ \frac{\partial u_i}{\partial t} + \sum_k X_{ik}(u)\frac{\partial u_k}{\partial x} = \mu \sum_k \frac{\partial}{\partial x} B_{ik}(u)\frac{\partial u_k}{\partial x}. \tag{8.8} \]

We shall regard the quantity \(\mu\), which characterizes the magnitude of the dissipative coefficients, as small. The matrix \(B_{ik}(u)\) appears under the derivative sign, since equations (8.8) must have the form of conservation laws (4.1); moreover, when dissipation is taken into account, the quantities \(F_i\) entering formula (4.1) depend not only on \(u_k\), but also on their derivatives \(\dfrac{\partial u_k}{\partial x}\).

The matrix \(B_{ik}\) must satisfy the “dissipativity condition,” which consists in the fact that all plane waves \(u_i=a_i e^{ikx-i\omega t}\) that are solutions of system (8.8) must decay as \(t\to+\infty\).

1. Let us determine the magnitude of the dissipation in the \(k\)-th wave. Using the smallness of \(\mu\), we shall apply perturbation theory. Normalizing the \(k\)-th eigenvectors by

\[ \sum_{i=1}^{n} l_i^{(k)} r_i^{(k)} = 1, \tag{8.9} \]

we obtain for the imaginary part of the frequency \(\gamma=-\operatorname{Im}\omega\)

\[ \gamma=\mu k^{2}\sum_{i,j} l_i^{(k)} B_{ij} r_j^{(k)}, \tag{8.10} \]

where \(k\) is the wave vector. The condition of dissipativity consists in the fact that the quantity \(\gamma\) must be positive, i.e.,

\[ \mu \sum_{i,j} l_i B_{ij} r_j>0 . \tag{8.11} \]

1. Let us now determine the structure of the established (i.e., time-independent) shock wave of small intensity. To this end we pass to a reference frame moving together with the shock wave, by means of the change of variables \(\xi=St-x\) (the wave moves to the right; in front of the wave \(\xi=-\infty\), behind it \(\xi=+\infty\); all quantities depend on \(x\) and \(t\) only through \(\xi\)). Then equation (8.8) takes the form

\[ S\frac{du_i}{d\xi}-\sum_j X_{ij}(u)\frac{du_j}{d\xi} = \mu \sum_j \frac{d}{d\xi} B_{ij}\frac{du_j}{d\xi}. \tag{8.12} \]

Since all the quantities \(u_1,\ldots,u_n\) depend only on \(\xi\), they are all functions of one of them, for example \(\eta\). Normalizing the right eigenvector so that, for any value of \(\eta\), the equalities

\[ \frac{du_i}{d\eta}=r_i(\eta), \tag{8.13} \]

hold, we transform (8.12):

\[ S r_i(\eta)\frac{d\eta}{d\xi} - \sum_j X_{ij}r_j\frac{d\eta}{d\xi} = \mu \sum_j \frac{d}{d\xi} B_{ij}r_j\frac{d\eta}{d\xi}. \tag{8.14} \]

(Strictly speaking, the relations (8.13) hold for \(\mu=0\), but for small \(\mu\) the vectors \(r_i\) acquire a small perturbation, which may be neglected. This is not correct if \(\eta\) is a Riemann invariant, since then there is no zeroth approximation for \(r_i\).) We multiply the \(i\)-th equality (8.14) by \(l_i\) and sum over \(i\). Using relations (1.5), (8.9), we obtain

\[ [S-V(\eta)]\frac{d\eta}{d\xi} = \mu \sum_{i,j} l_i \frac{d}{d\xi} B_{ij} r_j \frac{d\eta}{d\xi}. \tag{8.15} \]

Since \(\mu\) is a small parameter, the differentiation operator \(\dfrac{d}{d\xi}\) must also be considered small. Therefore, in the right-hand side of equation (8.15), the quantities \(l_i\), \(B_{ij}\), and \(r_j\) may be regarded as constant. Multiplying (8.15) by \(d\xi\) and integrating, we obtain, putting \(\eta=0\), \(d\eta/d\xi=0\) for \(\xi=-\infty\) (i.e., in front of the shock wave),

\[ S\eta-\int_0^\eta V(\eta)\,d\eta = \mu \sum_{i,j} l_i B_{ij} r_j \frac{d\eta}{d\xi}. \tag{8.16} \]

Putting \(V(\eta)=V(0)+\eta\,\dfrac{dV}{d\eta}\) and using (6.15), we find

\[ \frac{\Delta V}{2}\eta-\frac{1}{2}\frac{dV}{d\eta}\eta^2 = \mu \sum_{i,j} l_i B_{ij} r_j \frac{d\eta}{d\xi}. \tag{8.17} \]

Using the relation \(dV/d\eta=\Delta V/\Delta\eta\), where \(\Delta\eta\) is the value of the parameter \(\eta\) behind the shock wave \((\xi=+\infty)\), we integrate equation (8.17):

\[ \frac{2\eta}{\Delta\eta}-1=\operatorname{th}\frac{\xi}{L}, \tag{8.18} \]

where

\[ L=\frac{4\mu \displaystyle\sum_{i,j} l_i B_{ij} r_j}{\Delta V}. \tag{8.19} \]

1. Let us now determine the course of variation, in the shock wave, of the Riemann invariant. To this end we pass from the variables \(u_1,\ldots,u_n\) to the \((n-1)\)-th Riemann invariant \(I_1^{(k)},\ldots,I_{n-1}^{(k)}\) (corresponding to the \(k\)-th wave) and to some quantity \(\eta\). In this case equations (8.8) take the form

\[ \frac{\partial I_i}{\partial t} +\sum_{j=1}^{n-1}Y_{ij}\frac{\partial I_j}{\partial x} +Y_{in}\frac{\partial\eta}{\partial x} = \mu\sum_{j=1}^{n-1}\frac{\partial}{\partial x}C_{ij}\frac{\partial I_j}{\partial x} +\mu\frac{\partial}{\partial x}C_{in}\frac{\partial\eta}{\partial x} \tag{8.20} \]

\[ (i=1,\ldots,n-1); \]

\[ \frac{\partial\eta}{\partial t} +\sum_{j=1}^{n-1}Y_{nj}\frac{\partial I_j}{\partial x} +Y_{nn}\frac{\partial\eta}{\partial x} = \mu\sum_{j=1}^{n-1}\frac{\partial}{\partial x}C_{nj}\frac{\partial I_j}{\partial x} +\mu\frac{\partial}{\partial x}C_{nn}\frac{\partial\eta}{\partial x}. \tag{8.21} \]

If we put \(\mu=0\), then the values \(I_i=\mathrm{const}\) are solutions of this system (simple waves). Therefore \(Y_{in}=0\). Neglecting the terms

\[ \frac{\partial}{\partial x}C_{ij}\frac{\partial I_j}{\partial x} \]

in comparison with \(Y_{ij}\dfrac{\partial I_j}{\partial x}\), we obtain from (8.20)

\[ \frac{\partial I_i}{\partial t} +\sum_{j=1}^{n-1}Y_{ij}\frac{\partial I_j}{\partial x} = \mu\frac{\partial}{\partial x}C_{in}\frac{\partial\eta}{\partial x}. \tag{8.22} \]

Passing in equations (8.22) to the variable \(\xi=St-x\), solving the resulting equations with respect to \(dI_i/d\xi\), and integrating with respect to \(\xi\), using relation (8.18), we obtain

\[ I_i=\mu\sum_{j=1}^{n-1}D_{ij}C_{jn}\, \frac{\Delta\eta}{2L}\, \frac{1}{\operatorname{ch}^{2}\dfrac{\xi}{L}} . \tag{8.23} \]

(We assume that the value of the Riemann invariant ahead of the shock wave is equal to zero; \(D_{ij}\) is the matrix inverse to \((S\delta_{ij}-Y_{ij})\), and the determinant of the matrix \((S\delta_{ij}-Y_{ij})\) is nonzero; this is clear from the fact that the system of equations (8.22) for \(\mu=0\) is equivalent to system (2.11), and therefore the eigenvalues of the matrix \(Y_{ij}\) are all the values \(V^{(j)}\), except \(V^{(k)}\).)

1. Formulas (8.18), (8.19), (8.23) determine the structure of a shock wave of small intensity. A similar investigation in the case of ordinary hydrodynamics was carried out in [18], in the case of nonrelativistic magnetohydrodynamics in [60], and in the case of relativistic magnetohydrodynamics in [61].

It is clear from formula (8.23) that the Riemann invariant (and, in particular, the entropy) changes nonmonotonically, and has a maximum inside the shock wave. For \(\xi=\pm\infty\), formula (8.23) gives identical values \(I_i=0\); this is connected with the fact that the total change of the Riemann invariant \(\Delta I_i\) is a quantity of third order in \(\Delta\eta\), whereas the change of \(I_i\) inside the shock wave is of second order.

1. Expression (8.19) for the width of the shock wave can be obtained more simply by equating the rate of contraction of the wave profile \(\Delta V\), caused by the convergence of the characteristics, to the rate of spreading of the wave profile, equal to \(L\gamma\), or, by virtue of relations (8.10) and \(k\sim \dfrac{1}{L}\), equal to

\[ \frac{1}{L}\,\mu \sum_{i,j} l_i B_{ij} r_j . \]

Since for waves of small intensity the evolutionary condition has the form \(\Delta V>0\), formula (8.18) shows that the evolutionary conditions mean the presence of a dissipative structure, i.e., an increase of entropy. For waves of large intensity this is not the case. In magnetohydrodynamics there exist shock waves in which the entropy increases, but which are not evolutionary. On the other hand, it can be shown that if a shock wave is evolutionary, then the entropy always increases on it [62].

For Alfvén discontinuities [47] \(\Delta V=0\). Therefore an Alfvén discontinuity has no stationary structure [60, 63], as a result of which the existence of Alfvén discontinuities was called into question. On the other hand, as we saw in § 7, an Alfvén discontinuity necessarily arises in the decay of a discontinuity in the initial conditions. Such a discontinuity has a nonstationary structure, and its thickness increases with time the more slowly, the smaller the values of the dissipative coefficients.

1. It would seem that, in order to determine the “admissible” discontinuities, one could, instead of the evolutionary conditions, require that the discontinuous solution of system (1.1) be the limit of a continuous solution of system (8.8) with “fictitious viscosity” as \(\mu\to 0\). In this case the matrix \(B_{ij}(u)\) must satisfy the dissipativity condition (8.11). However, such a method is suitable only in the case of a single equation [21, 28, 29, 64]. In the case of a system of equations, if the matrix \(B_{ij}\) is a function of \(u_k\), the limiting discontinuous solution depends on the specific form of \(B_{ij}\). Examples showing that different matrices \(B\) lead to different discontinuous solutions have been constructed in [65—70]. Incidentally, for ordinary and magnetic hydrodynamics the latter assertion is a consequence of the fact that the conservation laws (4.1) cannot be obtained uniquely from equations (1.1).

In connection with the failure of the method of “fictitious viscosity,” the question arises whether admissible shock waves cannot be regarded as limits of solutions of system (8.8) with real dissipative matrices (viscosity, thermal conductivity, Joule dissipation) as \(\mu\to 0\). In the case of magnetohydrodynamics such an investigation was carried out by Germain [71] (Germain’s work contains an error, corrected by A. G. Kulikovskii and G. A. Lyubimov [72]). Evolutionary shock waves have a unique continuous structure for arbitrary values of the dissipative coefficients. Nonevolutionary shock waves either have no continuous structure, or have an infinite number of structures corresponding to the prescribed state in front of and behind the shock wave (the realization of the first or the second case depends on the relation between the coefficients of viscosity, thermal conductivity, and electrical conductivity).

In this connection one should make a remark about the “through-counting method.” In this method the system of equations (1.1) is replaced by the “Navier–Stokes equations” (8.8). The latter differential equations, in turn, are replaced by difference equations. The shock wave is regarded as the limit of a solution of the difference equations. From what has been set forth above it follows that the problem of solving the difference equations is ill-posed in the case of nonevolutionary shock waves.

1. In finding the structure of strong shock waves with the aid of the Navier–Stokes equations (8.8), the various dissipative coefficients play unequal roles. If the viscosity is nonzero, then a continuous structure exists for an arbitrarily large intensity of the shock wave; on the contrary, for zero viscosity, discontinuities appear in the structure of a shock wave of sufficiently large intensity. This is connected with the fact that, for zero viscosity, the system of Navier–Stokes equations is parabolic, whereas for nonzero viscosity it becomes hyperbolic [7]. For example, in ordinary hydrodynamics, after linearization of the system (8.8), we obtain

\[ \frac{4}{3}\gamma \chi \nu^{2} u_{xxxxt} +\chi c^{2}u_{xxxx} -\left(\frac{4}{3}\nu+\gamma\chi\right)u_{xxt} +u_{ttt} -c^{2}u_{xxt}=0, \tag{8.24} \]

where \(u\) is the perturbation of any of the hydrodynamic quantities; \(\gamma\) is the ratio of specific heats \(c_p/c_v\); \(\nu\) is the coefficient of viscosity; \(\chi\) is the coefficient of thermal conductivity; \(c\) is the speed of sound.

Let us note that this assertion about the special role of the viscosity coefficient has no physical meaning. Indeed, in deriving the Navier–Stokes equation one usually assumes that the momentum-flux density tensor \(\Pi_{ik}\) (which determines the force acting on a unit surface [18]) is equal to

\[ \Pi_{ik} = p\hat{\delta}_{ik} +\rho v_i v_k -\eta\left( \frac{\partial v_i}{\partial x_k} +\frac{\partial v_k}{\partial x_i} -\frac{2}{3}\hat{\delta}_{ik}\frac{\partial v_l}{\partial x_l} \right) -\zeta \hat{\delta}_{ik}\frac{\partial v_l}{\partial x_l}. \tag{8.25} \]

This expression does not take into account the fact that the momentum-flux density \(\Pi_{ik}\) becomes equal to the quasi-equilibrium value (8.25) only after a relaxation time \(\tau\), which is of the order of the mean free time. If one assumes that the velocity distribution at each point is Maxwellian, then, integrating the kinetic equation over velocities, one obtains [73] on the left-hand side of (8.25), instead of \(\Pi_{ik}\), the expression \(\Pi_{ik}+\frac{1}{\tau}\frac{\partial \Pi_{ik}}{\partial t}\). This leads to the Navier–Stokes equation becoming not parabolic but hyperbolic (see also [74]). Let us note that for a non-Maxwellian distribution formula (8.25) loses its meaning altogether.

Strictly speaking, the Navier–Stokes equations can be used only to find the structure of a shock wave of small intensity. For waves of large intensity one should solve the kinetic equation.

1. In connection with the difficulty of solving the kinetic equation, various approximate methods are of interest. In the theory of small perturbations the collision integral is usually replaced by the expression \(\frac{f_0-f}{\tau}\), where \(f_0\) is the equilibrium distribution function and \(\tau\) is the relaxation time. When finding the structure of a strong shock wave it is unclear what should be understood by \(f_0\), since on the two sides of the shock wave the equilibrium distribution functions are different. It is natural to choose

as \(f_0\) the Maxwellian function having the same density, mean velocity, and mean square velocity as the instantaneous distribution function \(f\) [75]. However, such an investigation can be carried through to the end only for shock waves of small intensity [76–78].

To find the structure of strong shock waves one may use another method (Mott-Smith [79]). Since particles practically do not undergo collisions while traversing distances of the order of the thickness of the shock wave, in the transition zone there is a mixture of cold particles, corresponding to the state ahead of the shock wave, and hot particles, corresponding to the state behind the shock wave. Therefore the distribution function may be regarded as equal to the sum of two Maxwellian distributions with variable densities. If the particles are neutral, then each distribution corresponds to a constant mean velocity. For charged particles the mean velocity depends on the coordinate. The structure of the shock wave is found with the aid of the first few moments of the kinetic equation [80–86].

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Received by the editors
December 28, 1964

Physico-Technical Institute, Academy of Sciences of the Ukrainian SSR,
Kharkov