So far we have not made any assumption about the magnitudes of the rates of change of the zero-order quantities. The linearized equations only require that the perturbation is small. In principle, they describe the time evolution of a perturbation applied to a self-consistent MHD flow that obeys the zero-order equations. In practice, to obtain solutions, we need to make further approximations. One such approximation is to assume that the medium varies only on time- and length scales long compared with the characteristic period τ and wavelength λ of the perturbation. The medium is now assumed to change slowly in space and time. The slow variation can be characterized by a small quantity ϵ such that the parameters of the medium change on a timescale of order τ/ϵ and on a length scale of order λ/ϵ. There are two different approximations to the MHD equations in this problem. One is that the wave perturbation is small, allowing linearization of the equations. The other is the slowly varying approximation which assumes that operators ∂/∂t and ∂/∂xi, operating on zero-order quantities, are of order ϵω and ϵk respectively, while when operating on first-order quantities they are of order ω and k.

The first approximation is found by ignoring terms of order ϵ. Locally, on length scales and timescales of order λ and τ, the wave is approximately harmonic. The first stage in any WKBJ approximation Chapter 9 is to assume a phase variation, insert it into the wave equation, get a non-linear differential equation for the phase, and obtain a first approximate solution in the form of a phase integral. We short-circuit this by assuming a first approximation of phase integral form in both space and time so that the field components vary approximately as exp⁡i∫ωdt-∫kidxi so that ∂/∂t=iω, ∂/∂xi=-iki, and ddt=∂∂t+Vi∂∂xi=iω-kiVi=iω0, where ω0 is the angular frequency in the plasma rest frame. It is Doppler-shifted relative to ω, the angular frequency in the moving frame

The first-order equations to this level of accuracy become 21.13–21.15 ω0ρ0vj=kjp+Bkbkμ0-kkBkμ0bj,ω0pγP=kkvk,ω0bj=-kkBkvj+kkvkBj so that ω0=ω-kiVi and the local dispersion relations are the same as for a uniform stationary medium with the frequency ω replaced by the Doppler-shifted frequency ω0.

The dispersion relations for each type of wave are ω02-kiVA,i2=0 for the transverse Alfvén wave and ω04-k2ω02VA2+VS2-kiVA,i2VS2=0 for the two magnetosonic waves, where VA,i=Biμ0ρ0,VS=γPρ0.

We can write any of these formally as ω0=ϖ0xi,t, where ϖ0 represents the dispersion function in the local rest frame and depends on xi and t through Bi, P, and ρ.

In the special case of a sound wave VA is 0, and this becomes ω0=kVS=kγPρ0.

has discussed ray tracing of MHD waves in a medium that varies both in space and time. The ray defines the path of a wave packet. The inclusion of time variation introduces a new feature. When the motion of the background plasma is steady state with ∂/∂t=0, then the frequency is constant. The ray-tracing equations then consist of a set of three first-order differential equations for the rate of change of position dxi/dt of the wave packet, and a set of three for the rate of change dki/dt of the wave vector. This is no longer true when the properties of the medium depend on time as well as the spatial coordinates. There is an additional Doppler shift associated with the time variation. This leads to the introduction of another first-order equation giving the rate of change of frequency.

The full set of ray-tracing equations is then drxidt=∂ϖ∂ki;drkidt=-∂ϖ∂xi;drωdt=∂ϖ∂t. The subscript r emphasizes that dr/dt represents the rate of change of a quantity with time following the wave packet as it travels along the ray. It may be convenient to express these in terms of the dispersion relation in the local rest frame. Then, from Eq. (), drxidt=∂ϖ0∂ki+Vi=VG,i(0)+Vi,drkidt=-∂ϖ0∂xi-kj∂Vj∂xi,drωdt=∂ϖ0∂t+kj∂Vj∂t.

Equation () shows how the wave packet progresses in space as time advances, while Eq. () shows how the wave vector changes in magnitude and direction as the wave packet advances. They are analogous to Hamilton's equations with frequency corresponding to energy and wave vector to momentum. The equations for ki are conjugate to those for xi. The equation for the frequency (Eq. ) also has a trivial conjugate equation, dτ/dt=1, where τ is the time elapsed since the wave packet left its starting point. Then (xi,ω) and (ki,τ) are the conjugate variables in a four-dimensional space–time. This does not, however, imply that relativity has been taken into account, although it would, in principle, be possible to do so.

The ray-tracing equations do not, of course, provide a solution to the wave equations. This would require expressions for the field components. At this level of approximation the solution is the phase integral, which is to be evaluated along the ray. The rays are the characteristic curves in the problem. A better solution would be one in which the variation of the amplitude of each field component varied along the path. In the WKBJ method for a stationary medium the next stage of the process is to substitute the first approximation (the phase integral) into the non-linear equation for the phase to get a second approximation. This second approximation gives the amplitude of each field component. These all vary in such a manner as to conserve energy as the wave progresses. Another way of getting the same solution is to use energy conservation. In a stationary, slowly varying medium, the wave energy density is an adiabatic invariant. We could find the relationship between each component and the energy flux and use that to determine behaviour of the amplitudes. This is the route that we follow, although wave energy is no longer conserved in this case and it turns out that the appropriate adiabatic invariant is the wave action density.

Sect. 21.5 has carried out this procedure and has derived an expression for the energy balance between a propagated wave and a non-uniform medium. The derivation is for a steady state where the Lagrangian operator following the plasma is ddt≡Vi∂∂xi. The only difference for a time-varying medium is that ddt≡∂∂t+Vi∂∂xi. The derivation does not use the explicit form of d/dt but simply uses Eqs. (), (), (), and () to eliminate the time derivatives of the zero-order quantities in the frame of the plasma. The derivation including time variation of these quantities gives the same result: ∂U∂t=-∂Πi∂xi-Tij∂Vi∂xj, where U=12ρ0v2+12b2/μ0+12p2/γP, Πi=p+Bjbjμ0vi-Biμ0bjvj+UVi, Tij=ρ0vivj+12δijγ-1γp2P-bibj-12δijb2μ0, where U is the energy density in the wave, Πi the energy flux vector, and Tij a stress tensor. There is a factor (γ-1)/γ in the second term of the expression for the stress tensor that does not occur in Walker's expression. This corrects the error noted in the discussion of Eq. ().

The energy flux vector can be shown to be the rate per unit area per unit time at which the internal energy is transported by the wave at the group velocity Eq. 21.26. Πi=UVG,i=UVG,i(0)+Vi

The first term in the stress tensor is the Reynolds stress. It represents the momentum flux associated with the oscillation of the wave. If we consider an element of area dAj, moving with the unperturbed velocity Vi, then ρ0vivjdAj is the rate at which wave momentum ρ0vi is transported across dAj at velocity vj. It thus represents the mechanical force exerted by the wave field on dAj. The second term is simply the normal stress exerted by the isotropic second-order pressure of the wave (see Appendix ). The third and fourth terms represent the Maxwell stress Sect. 1.8 associated with the first-order magnetic perturbation bi. The Maxwell stress, in the same way, can be regarded as the electromagnetic momentum flux tensor.

The last term in the equation for energy balance Eq. () may be written -Tij∂Vi∂xj=-∂(TijVi)∂xj+Vi∂Tij∂xj. The force per unit volume exerted by the wave on the unperturbed medium is -∂Tij/∂xj. Since the medium is moving with velocity Vj, the wave does work on the medium -Vi∂Tij/∂xj and thus the last term in Eq. () represents the gain of energy by the wave as a result of this process. The first term on the right-hand side has the form of the divergence of a flux vector. Explicitly we can write TijVj=ρ0vjVjvi+12(γ-1)p2γPVi+12b2μ0Vi-bjVjμ0bi. Compare this expression with the expression for the flux vector in a uniform medium (Eq. ). The first term is an additional second-order contribution to the transport of internal energy U: the first-order contribution to the energy density 12ρ0(Vj+vj)(Vj+vj) is ρ0Vjvj, and there is a second-order flux term as it is transported by the wave velocity vi. The second term is the acoustic flux associated with the second-order pressure (Eq. ) when the zero-order velocity is non-zero. Compare it with the term pv in Eq. (). The remaining terms represent the flux of magnetic energy and are of the same form as the corresponding terms in Eq. ().

When Vi is constant, the two terms on the right-hand side of Eq. () cancel and energy is conserved. The term involving the stress tensor in Eq. () is the difference between the work done per unit volume and the energy transported and represents the net effect of the interaction between wave and unperturbed medium.

The rest frame frequency ω0 is given by the dispersion relation () and is a function of xi, ki, and t. Its rate of change following a wave packet is, therefore, drω0dt=∂ω0∂t+∂ω0∂xidrxidt+∂ω0∂kidrkidt. We can substitute for drxi/dt from Eq. () and for drki/dt from Eq. () to write this as drω0dt=∂ω0∂t+Vi∂ω0∂xi-kjVG,i(0)∂Vj∂xi=dω0dtk-kjVG,i(0)∂Vj∂xi, where ddtk≡∂∂t+Vi∂∂xi is an operator representing the rate of change with respect to time of a quantity following the motion of the unperturbed plasma. The subscript k emphasizes that this is done keeping k constant.

We can also see how U varies following the wave packet by noting that the equation for transport of energy can be written in terms of the rate of change along a ray. If we use Eq. (), the divergence of the flux vector can be expanded as ∂∂xiUVG,i(0)+Vi=VG,i(0)+Vi∂U∂xi+U∂∂xiVG,i(0)+Vi. This can be substituted in () to yield drUdt≡∂U∂t+VG,i∂U∂xi=-U∂VG,i∂xi-Tij∂Vi∂xj.

In order to establish the principle of conservation of wave action in any particular case, we compare Eqs. () and (). The right-hand side of Eq. () is evaluated from the appropriate dispersion relation, while in Eq. () we express Tij in terms of U. We proceed to do this for a number of special cases.

Sect. 4.6 has discussed the conservation of wave action for sound waves where the background motion is a steady-state flow. His book draws attention to the importance of the concept of conservation of wave action, but he nowhere considers the case of non-stationary flow. As we have seen for steady-state flow, the frequency remains constant and this simplifies the analysis. Sound waves are a special case of MHD waves when B→0, and they provide a good introductory example.

For a sound wave, where the magnetic field is 0, the expressions for energy density (Eq. ), flux vector (Eq. ), and stress tensor (Eq. ) become U=12ρ0v2+12p2/γP,Πi=pvi,Tij=ρ0vivj+12δijγ-1γp2P.

If we use these expressions for the flux vector and stress tensor in Eq. (), we get the rate of change of U following a wave packet: drUdt≡∂U∂t+VG,i∂U∂xi=-U∂VG,i∂xi-ρ0vivj∂Vj∂xi-12(γ-1)p2γP∂Vj∂xj.

If we multiply Eq. () by vi and use Eq. (), we see that ρ0vivj=kiVG,jUω0. Also if we use the equipartition relation () in Eq. (), we get 12(γ-1)p2γP=12(γ-1)U. If we substitute these in Eq. (), we get 1UdrUdt=-∂VG,i(0)∂xi-∂Vj∂xj-kiVG,jω0∂Vi∂xj-12(γ-1)∂Vj∂xj.

Now compare this with Eq. (), the rate of change of ω0 following the ray. On the right-hand side (dω0/dt)k represents the rate of change of the rest frame frequency following the zero-order motion of the plasma and keeping k constant. It is to be found from the sound dispersion relation (). We use Eqs. () and () for the derivatives of P and ρ0 following the plasma. Then dω0dtk≡d(kVS)dtk≡kddtγPρ0=12kVs1PdPdt-12kVs1ρ0dρ0dt=-12kVsγ∂Vj∂xj+12kVs∂Vj∂xj so that (noting that kVs/ω0=1) 1ω0drω0dt=-12(γ-1)∂Vj∂xj-kiVG,j(0)ω0∂Vi∂xj.

Finally, subtract Eq. () from Eq. () to get ω0UdrdtUω0=-∂VG,j(0)∂xj-∂Vj∂xj. Then, noting that drdt=∂∂t+VG,j(0)+Vj∂∂xj, we obtain, using Eq. (), the result ∂∂tUω0=-∂∂xjUω0VG,j(0)+Vj=-∂∂xjΠjω0, which shows that the wave action, with density U/ω0, is conserved.

For the transverse Alfvén wave we follow the same procedure as for a sound wave with the appropriate dispersion relation () and relations between the field components. It is an incompressible shear wave with p=0, Bibi=0, and kivi=0. The relations between the field components () and () are then ω0ρ0vj=-kkBkμ0bj,ω0bi=-kkBkvi. Multiply Eq. () by bi/ω0 and Eq. () by bi/ω0μ0, and we see that ρ0vivj=bibjμ0.

From the energy equipartition relation () the energy density is U=12ρ0v2+12b2/μ0=b2/μ0. As we have noted above, bi is normal to Bi so that the flux vector from Eq. () is Πi=-Bibjvjμ0 and the stress tensor from Eqs. (), (), and () is Tij=12δijb2μ0=12δijU.

If these are substituted in Eq. (), we get 1UdrUdt=-∂∂xiVG,i(0)+Vi-12∂Vi∂xi.

We now use the dispersion relation () to find (dω0/dt)k in Eq. (). The derivatives of ρ and Bi are found from Eqs. () and () so that dω0dtk=kiVA,j(0)∂Vi∂xj-12kiVA,i(0)∂Vj∂xj. Thus 1ω0drω0dt=-12kiVA,i(0)∂Vj∂xj. Finally subtract this from Eq. (), and we get drdtUω0=Uω0∂VG,i(0)∂xi or, equivalently, ∂∂tUω0=∂∂xjUω0VG,j(0), which shows that, once again, the wave action is conserved along the ray.

The magnetosonic waves are algebraically more complicated, but we can follow the same process. The dispersion relation () is in the form of a biquadratic as it describes both waves. It would be possible to solve the quadratic equation for ω02, but the solution contains a square root and we must choose which sign applies to which wave, and this is not as convenient as finding a solution that applies to both magnetosonic waves.

In a uniform medium we can use Eqs. () and () to eliminate p and bi from Eq. (). The result is a set of three equations relating the three components of vi. We use a set of local magnetic-field-aligned coordinates x‖, xn, and x⟂, where x‖ is parallel to B, xn is normal to the plane containing B and k, and x⟂ is normal to B in this plane. The three equations then separate into two sets Sect. 7.5.1, one involving vn representing the Alfvén wave, and the other two involving only v‖ and v⟂ representing the magnetosonic waves. The latter two equations may be written ω02-k2VA2-k⟂2Vs2-k⟂k‖VS2-k⟂k‖VS2ω02-k‖2VS2v⟂v‖=0.

These are two homogeneous linear equations which are only consistent if the determinant of the matrix is 0. This determines the dispersion relation () which, in these coordinates, can be written ω04-ω02k2VA2+VS2+k2k‖2VA2VS2=0

In Eq. (), representing the rate of change of U following a wave packet, we note that Tij∂Vi∂xj=ρ0vivj-bibjμ0∂Vi∂xj+12(γ-1)p22γP+b22μ0∂Vi∂xi.

If we multiply Eq. () by vi, Eq. () by bi, subtract the second from the first, and use Eq. (), we get ρ0vivj-bibjμ0=p+Bkbkμ0kivjω0-kkvkω0Bjbiμ0=kiω0Πj(0)+Bjvkω0μ0kibk-kkbi=kiω0VG,j(0)U+Bjvkω0μ0kibk-kkbi.

The next objective is to express the products of the field components on the right-hand side in terms of the wave energy density U. Then we will be able to divide Eq. () by U to get an expression for (1/U)drU/dt that only depends on the zero-order quantities and their derivatives. This is best done by using the local field-aligned coordinates and making use of the properties of the magnetosonic waves described above. The details are shown in Appendix . From Eqs. () and () we get (γ-1)p22γP+b22μ0=12(γ-1)ω02-k2VA2+ω02-k2VS22ω02-k2VA2+VS2U. Also from Eqs. () and () ρ0vivj-bibjμ0∂Vi∂xj=kiVG,j(0)ω0∂Vi∂xjU+k‖k⟂k2VA2VS2ω0ω02-k‖2VS2∂V‖∂x⟂-ω0ω02-k2VS22ω02-k2VA2+VS2∂V‖∂x‖U. These two results can be substituted in Eq. () with the result that 1UdrUdt=-∂VG,i(0)∂xi-∂Vi∂xi-kiVG,j(0)ω0∂Vi∂xj-12(γ-1)ω02-k2VA2+ω02-k2VS22ω02-k2VA2+VS2∂Vi∂xi-k‖k⟂k2VA2VS2ω02-k‖2VS2∂V‖∂x⟂+ω02-k2VS22ω02-k2VA2+VS2∂V‖∂x‖.

The next step is to evaluate dω0/dtk in Eq. () by differentiating the dispersion relation. For the magnetosonic waves this is algebraically tedious and is carried out in Appendix . The result is 1ω0drω0dt=-kiVG,j(0)ω0∂Vi∂xj-12(γ-1)ω02-k2VA2+ω02-k2VS22ω02-k2VA2+VS2∂Vi∂xi-k‖k⟂k2VA2VS2ω02-k‖2VS2∂V‖∂x⟂-ω02-k2VS22ω02-k2VA2+VS2∂V‖∂x‖. If the second of these is subtracted from the first, we get drdtUω0≡∂∂tUω0+VG∂∂xiUω0=-Uω0∂VG,i(0)∂xi-Uω0∂Vi∂xi=-Uω0∂VG,i∂xi or ∂∂tUω0=-∂∂xiUω0VG,i, which expresses the conservation of the wave action for the magnetosonic waves.

We start with the simplest possible example. Consider a plane sound wave propagated in the z direction in a gas that undergoes a slow adiabatic compression. The gas is assumed to be in a long cylinder with axis in the z direction and a diameter very large compared with the wavelength. The compression is assumed to take place by reducing the diameter of the cylinder. The zero-order velocity Vi of the gas resulting from the compression is then at right angles to the direction of propagation so that there is no Doppler shift and ω=ω0. This velocity is a function of xi and t, but the uniform compression means that its spatial derivatives are independent of position being functions of time only. The sound speed VS (Eq. ) is also a function of t only. The ray-tracing (Eq. ) with the group velocity equal to the sound speed become drxdt=Vx(t),drydt=Vy(t),drzdt=VS(t)),

drkxdt=-kx∂Vx∂x,drkydt=-ky∂Vy∂y,drkzdt=0,drωdt=∂ϖ∂t=∂{kzVS(t)}∂t.

The first three equations (Eqs. , , ) give the path of a wave packet while the compression takes place. They show that the velocity of a wave packet during the compression has a component along the z axis equal to the sound speed. In addition the wave packet is carried towards the axis of the cylinder with the velocity of the medium. This is illustrated in Fig. . Consider portions of rays AB and CD on either side of the axis OP of the container, at time t0 just before the compression starts. During the compression a wave packet such as that at A moves along a ray path AA′ and reaches A′ at time t when the compression ends. Other wave packets such as B move along parallel paths such as BB′. The behaviour is analogous to the behaviour of streak lines in time-varying fluid dynamics. The net effect is that the ray pencil AB moves inwards as it progresses until it reaches A′B′, and CD does the same to reach C′D′. Clearly, because the ray pencils move with the gas, the cross-sectional area of any ray tube changes proportionally to the cross-sectional area of the container.

The equations for the transverse components of the wave vector (Eqs. , ) show that, if kx and ky are initially 0, they will remain 0. Equation () shows that kz remains constant. The wave thus remains a plane wave propagated in the z direction with a constant wave length. Finally Eq. () can be integrated directly to show that ω varies with time as ω(t)=kzVS(t). We thus see that, as VS changes as a result of the compression, the frequency also changes. This is a kind of Doppler shift that does not result from the motion of the medium, the source, or the observer.

The cross section of the ray tubes changes in proportion to the cross-sectional area A of the container. The density is inversely proportional to A through the adiabatic relation P∝A-γ. Thus VS∝A-(γ-1)2. During this change the wave action is conserved. If the wave energy density is U, the quantity UA/ω remains constant. The energy density thus varies as U∝VS/A∝A-(γ+1)2. In the sound wave the magnitude of the energy flux Π is UVS so that Π∝VS2/A∝A-γ. The total energy per unit length W of the wave train changes as W∝UA∝A-(1)3. For example in an ideal gas γ=5/3 and W∝A-(γ-1)2.

Since γ>1 the total wave energy increases when the gas is compressed. This means that the work done in the adiabatic compression is greater than that needed to increase the internal energy of the gas. For a simple solar wind model with large plasma β so that the fast wave approximates a sound wave, if there is a spectrum of waves in existence, we can infer that temporal compression will increase not only the temperature but also the wave energy. Explicit calculation for a more realistic fast wave would show the same behaviour.

First we consider a steady-state situation in which the unperturbed medium does not change with time. Consider an isotropic Alfvén wave in a shear layer. The simplest possible case is that for which the density is constant, the pressure negligible, and the magnetic field is constant and normal to the z axis. The velocity is in the x direction and is a function of z. The fast wave is then an isotropic Alfvén wave. We consider the case where the wave vector lies in the x–z plane and has the dispersion relation ω0=ω-kxV(z)=kVA.

The ray-tracing equations (Eqs. , , ) for this case become dxdt=∂ω∂kx=kxkVA+V(z),dzdt=∂ω∂kz=kzkVA,dkxdt=0,dkzdt=-∂ω∂z=-kxdVdz,dωdt=0. This immediately shows that ω is constant as expected in the steady state, and that kx is constant, which is an expression of Snell's law. The right-hand sides of Eqs. () and () are the x and z components of the group velocity.

Figure  shows the results of a ray tracing in a shear layer by integrating these equations using a Runge–Kutta process. The dependence of the velocity on z is given by V=12V01+tanh⁡(z/w). The dependence of V on z is shown in Fig. a, and the parameters for the shear layer used are shown in the figure. The maximum shear occurs at z=0. Each ray is started at the point O below the shear layer where the medium has a very small velocity and is essentially at rest. The initial direction of the wave normal is spaced at equal angles so that a fan of rays spreads out from O. In Fig. b the interruptions in the rays occur at equal time intervals for a point on the ray moving at the ray velocity (which for these waves is the same as the group velocity) Sect. 13.19. This is the velocity of the point of intersection of a plane wavefront travelling with the phase velocity and the ray. In a stationary MHD medium it is the same as the phase velocity Sect. 9.2.5, but this is no longer true in a moving medium. These interruptions therefore represent successive positions of a wave crest.

There are several noteworthy features in this ray pattern. Were it not for the shear, the medium would be uniform and the rays would travel radially from O, producing spherical wave fronts. This is the situation near O, where V is very small. As the velocity increases, since the group velocity is the sum of the group velocity in the rest frame of the medium and the velocity of the medium itself, the rays are swept to the right. Those rays with negative kx initially make a positive angle with the z axis and are rotated clockwise relative to it. No matter how large the initial angle, when kx is negative, the right-hand side of Eq. () eventually becomes 0 and the motion in the x direction is reversed. When kx is positive and small, the initial x component of the ray velocity is positive, so there is no reversal of direction. The magnitude of the x component of the ray velocity, however, increases substantially and the ray is still swept to the right. For positive values of kx the value of kz decreases along the ray, as can be seen from Eq. (). It eventually becomes zero and changes sign, and hence from Eq. () the ray is reflected. Another point of interest is the focusing of the reflected rays seen at the bottom right-hand corner of the ray pattern.

In order to study the conservation of action in this geometry, it is necessary to know how the cross-sectional area of a ray tube changes along its length. This is not a trivial problem and we shall postpone it. Instead we shall consider a plane wave situation. Consider one of the rays with negative kx and suppose that it originates from a very distant source. Then the wave fronts are approximately plane and the separation of adjacent rays in the x direction remains constant.

Consider Fig. a. It shows a tube of rays passing through the boundary of a rectangular area A where the group velocity is VG. The tube penetrates the shear layer and reaches a level where the group velocity is VG′. The flux of wave action into the tube is equal to the flux out. Thus UVG,zAω0=U′VG,z′Aω0′. If the starting level is well below the shear layer, then ω0=ω and ω0′=ω-kxV0. The ratio of the energy flowing out of the tube to that flowing in is therefore U′VG,z′UVG,z=ω0′ω0=ω-kxV0ω. We see then that for negative kx there is a gain in energy. However those rays that start with positive kx lose energy. It might be thought that if V0 were sufficiently large they would acquire negative energy, but the reflection condition ensures that this cannot occur. The situation is illustrated in Fig. b. Thus the reflected waves do, in fact, lose energy as they are propagated upwards, but this is regained after they are reflected so that there is no net change.

So far this calculation has applied to the total energy in the ray tube and would apply equally well to the diverging ray tubes of Fig. a. If we want to calculate the flux vector, and hence the energy density and the amplitudes of the field components, we need to know the cross-sectional area of the ray tube. For diverging ray tubes such as those of Fig. a the calculation is not trivial and will not be pursued in this paper. For non-diverging ray tubes such as those of Fig. b matters are simpler. The cross-sectional area of the ray tube is simply A0=Acos⁡θ, where θ is the angle between the ray tube and the z axis. Equation () can just as well be written UVGA0ω0=U′VG′A0′ω0′ so that Π′Π=A0ω0′A0′ω0=ω0′cos⁡θω0cos⁡θ′. For this example cos⁡θ can be found as a function of z. The only dependence on z is through V(z). Both ω0 and kz depend on V through ω0=ω-kxV(z),kz=ω0(z)Va, which leads to cos⁡θ=kx2VA2+ω0(ω-ω0)ω0kx2VA22ωω0-1+(ω-ω0)2, which allows the cross-sectional area and hence the flux vector and energy density along the ray to be found. The direct calculation is only possible because the plane geometry allows it. In general some means of calculating the cross-sectional area of the flux tube as the integration progresses must be used. has proposed a method of doing this, but, surprisingly, it does not appear to have been applied in practice. Work is in progress in applying it to MHD waves.

Next we consider another problem in which the plasma conditions vary as a function of time. traced rays using Eqs. (), (), and (). He considered a uniform region of plasma containing a sheet of current flowing in the y direction, and situated on the plane z=0. A uniform magnetic field was associated with the current sheet. For z>0 it flowed in the x direction and for z<0 in the -x direction. This can be regarded as a crude model of the magnetotail near its axis. A transverse Alfvén wave was assumed to be propagated along the x axis while the system changed slowly in time. The surface current density in the current sheet was increased slowly and quasistatically, by which was meant that any acceleration was so small that the plasma remained in quasi-equilibrium as conditions changed. The magnetic field increased uniformly, and the electromotive force arising from the changing magnetic flux provided an electric field associated with a plasma drift inwards to the current sheet, at a velocity consistent with the frozen-in field line theorem. The surface current density of the sheet changes with time in such a manner that, over a period of time T, the magnetic field magnitude changes smoothly from an initial value B0 to a final value BF according to B=B01+12f1+tanh⁡t/T, where f is the fractional change in B. As this change takes place, the frozen-in field lines move towards the current sheet and the plasma is compressed.

Figure a shows the result of a ray tracing as the background parameters are changed. Initially, at t=0 there is a pencil of transverse Alfvén waves propagated to the left along the line ABCD. As the current increases, the points on the ray pencil ABCD move successively to A1B1C1D1, A2B2C2D2, A3B3C3D3, and A4B4C4D4 respectively along the separate ray paths shown. The net effect is that the pencil of rays remains on the same field line as it is convected towards the current sheet. At all times the wave normal remains aligned with the magnetic field, but the ray follows a path that keeps the wave packet on the same convecting field line. The wavelength in the x direction remains unchanged so that kx is constant. In the time-varying case, just as in the simple case of a sound wave in a medium undergoing compression, successive points along a pencil of rays do not follow the same ray path.

Figure b shows a ray tube of rectangular cross-sectional area of width A=wh. As the wave packet follows the ray path, the plasma is compressed in the z direction so that the cross-sectional area becomes A1=wh1. At t=0 the degree of compression is imperceptible and the ray is directed parallel to the field; at t=t4 the compression is essentially complete and the same is true. The medium is at rest in each case, and the group velocity for the transverse Alfvén wave is VA=B/μ0ρ0 parallel to the magnetic field. The magnetic flux through A is constant so that B′B=AA′=hh′. The compression is in the z direction, so the density is inversely proportional to h. Thus VA′VA=hh′.

The action flux vector in Eq. () is UVGω0=UVAkxVA=Ukx, where we have used the dispersion relation (). The conservation of wave action implies that the flux of wave action into the ray tube through A is equal to the flux out through A′: UVGω0A=U′VG′ω0′A′ or Uh=U′h′ since A∝h. The energy density thus increases by a factor h/h′ during the compression. The net gain of energy can be found from Eq. (). The power delivered to the wave in the ray tube at A is UVGA and that extracted at A′ is U′VG′A′. The ratio is G=U′VG′A′UVGA=ω0′ω0=VA′VA=hh′.

We see, therefore, that the energy of the wave is increased in the compression. External work has been done on the unperturbed plasma by the compression, and some of this energy has been used to do work on the wave through the stress term in Eq. ().

This situation is more complicated than the steady-state problem of the previous section. We have taken care to look at the energy flux into the volume represented by the ray tube in Fig. b at an initial time before the compression has started and a final time when it is complete. At each of these times the motion is steady state and we can calculate the flux of energy into or out of the volume from the steady-state flow. While the compression is in progress, the energy flux at an intermediate time through a cross section of the ray tube will not be constant. Before the compression takes place, it will be 0. As the pencil sweeps through the cross section, it will rise to a maximum and then fall again, as will the action density. We need to calculate the flux as we follow a wave packet in time.

The results of our computations show that, when the velocity of the medium is a function of both space and time, the wave exchanges energy with the background. This means that the zero-order flow that has been assumed is modified by the wave. This happens on temporal and spatial timescales that are large compared with the period and wavelength and does not invalidate our assumptions. As outlined by Sect. 21.5.1 there are other second-order terms in the total MHD energy conservation equation that represent the energy change in the background. The neglect of these second-order terms is equivalent to a quasi-linear approximation analogous to that used in the study of unstable plasma distribution functions close to quasi-equilibrium.

We expect, therefore, that a spectrum of MHD waves propagated in a moving non-uniform plasma such as the solar wind will be modified as they flow, consequently modifying the flow of the wind. As described by Chapter 4 for internal atmospheric waves, the energy exchange is most efficient at a critical layer where ω0≡ω-kiVi=0. For the MHD waves that we are considering, such critical layers are inaccessible in a slowly varying medium; they only occur where the wave is evanescent. For the atmospheric sound waves described by they only occur when the Earth's gravitational field is significant when the waves are described as internal waves. Similarly MHD waves would only show accessible critical layers sufficiently near the surface of the Sun, where the gravitational field is large. The theory of this paper would have to be extended to include this.