Wave energy propagation in drifting plasmas

A. D. M. Walker

School of Pure and Applied Physics, University of Natal, Durban, South Africa

Contents

Abstract

1. Introduction

Recently Walker [1999] has discussed the proper definition of the energy associated with an MHD wave which is propagated in a moving plasma. Such waves are frequently treated as having a negative energy density [Fejer, 1963; McKenzie, 1970, 1972; Mann et al., 1999]. This can lead to difficulties in interpretation when the waves are localized in space. In this paper we present further discussion of the meaning of the wave energy as defined by Walker [1999] and show how it leads to a consistent picture when studying wave energy which is localized in wave packets. This is important when considering MHD Pc3-5 waves at the magnetopause boundary between streaming magnetosheath and approximately stationary magnetosphere, as well as Pc3 waves propagated between the bow shock and magnetopause.

2. The Basic Equations

2.1. The MHD Equations

In a magnetohydrodynamic medium with b 1 the magnetohydrodynamic equations may be written as

(1)

(2)

(3)

(4)

where B is the total magnetic field, r is the mass density, p is the pressure, and v is the plasma velocity.

These are non-linear equations. When dealing with small amplitude waves they are generally linearized by considering the wave perturbation and writing the equations to first order in the small quantities. This is the first stage in a solution by successive approximations. In the discussion which follows we shall need to consider order accuracy. We therefore present below first and second order approximations to the MHD equations.

2.2. First Order Wave Equations

Assume that

(5)

(6)

(7)

(8)

where the constant uniform equilibrium state is represented by subscript 0, and the first and second order corrections by subscripts 1 and 2. We have allowed for the fact that the plasma may have a drift velocity v₀. While it is always possible to consider the problem in the plasma rest frame, our later analysis requires that we allow for this drift velocity.

If these quantities are substituted in equations (1), (2), (3), and (4), and only first order terms are retained, we get the following set of first order equations:

r₀ v₁ t = -r₀ v₀ v₁

(9)

(10)

(11)

(12)

It should be noted that r₁,r₂, v₁, B₁ are exact solutions of these first order differential equations; they are first order approximations to the solutions of (1), (2), (3), and (4).

It can also be seen that the three equations (9), (11), and (12) do not depend on r₁. The first order fields are thus entirely determined by solving only these equations. The continuity equation (10) serves only to determine the momentum density once v₁, p₁, and B₁ have been found.

2.3. Second Order Corrections to the Wave Equations

The second order terms in the equations may be written

r₀ v₂ t+r₀ v₀ v₂+{p₂+ B₀ B₂ m₀}

- B₀ m₀ B₂ = -r₁ v₁ t-r₀ v₁ v₁

(13)

(14)

(15)

B₂ t+ v₀ B₂+ B₀ v₂- B₀ v₂

(16)

If the exact solution of the first order equations, v₁,r₁,p₁, and B₁, are substituted in these, then v₂,r₁,p₂, and B₂ are exact solutions of these second order equations. Then r₀+r₁+r₂, p₀+p₁+p₂, v₀+ v₁+ v₂, and B₀+ B₁+ B₂ are solutions of the MHD equations (1), (2), (3), and (4), correct to second order accuracy.

3. Energy Conservation

3.1. The MHD Energy Equations

Take the scalar product of v with equation (1), and B/m₀ with (4), multiply (2) by (1/2)v² and (3) by 1/(g -1). If the resulting equations are added, after some manipulation we get

(17)

This has the form of an energy equation

(18)

where U is the energy density and S the energy flux.

The three terms of the energy density U represent the kinetic energy density, the thermal energy density, and the magnetic energy density respectively. The terms of the flux have been grouped to give the kinetic energy, thermal and magnetic energy flux, the work done by the pressure force as the plasma moves, and the work done by the Maxwell stresses in the form of magnetic pressure and magnetic field tension. It can easily be verified that the magnetic field terms in the energy flux represent the Poynting vector

(19)

since

(20)

3.2. Wave Energy

When small amplitude waves exist as a perturbation on a uniform background MHD medium then, to a good approximation, the wave fields can be found from the first order equations (9), (10), (11), and (12). We can then construct an energy conservation equation entirely from the first order wave fields [Walker, 1999]. The result is

(21)

where the first order electric field in the rest frame of the drifting plasma is

(22)

and

(23)

If the Poynting vector is expanded using

(24)

the expression for the energy flux becomes

S_wave={p₁+ B₀ B₁ m₀} v₁+U_wave v₀

(25)

Each term in (21) involves the product of two first order small quantities. The question arises of how this energy conservation equation is related to the total energy conservation equation (17).

3.3. Relation Between Total MHD Energy and Wave Energy

The energy equation (21) is constructed from the first order wave equations. As has been noted, the perturbation quantities v₁,r₁,p₁, and B₁ are exact solutions of these equations. The energy conservation equation thus represents the conservation of a quantity constructed entirely from the first order wave solutions. It is clearly not the total energy and cannot straightforwardly be obtained by expanding the energy density obtained from (17) to second order. If one does so one obtains a number of first order energy terms as well as second order terms which are not included in U_wave. To see why this is so let us consider the derivation of (17) in more detail.

The first order equations can be written in matrix form

(26)

where F_ij⁽⁰⁾ is the 8 8 matrix of zero order operators in (9), (10), (11), and (12), g_j⁽¹⁾={v_x⁽¹⁾,v_y⁽¹⁾,v_z⁽¹⁾,r₁,p₁, B⁽¹⁾_x,B⁽¹⁾_y,B_z⁽¹⁾} is a column of the first order field variables and summation over the repeated suffix is assumed.

Similarly, the second order equations may be written

(27)

The sum of (26) and (27)

(28)

is an approximation to the field equations correct to second order, and each of the quantities in braces is separately zero.

Let the row matrix f_i be given by

(29)

with, zero, first and second order parts f_i⁽⁰⁾, f_i⁽¹⁾, f_i⁽²⁾. Then the process of finding the general energy equation (17), described above, when carried out to second order accuracy, gives

(30)

To second order accuracy, this is equivalent to (17). Each quantity in braces is equal to zero. The equation thus shows clearly that, when an equilibrium plasma suffers a small perturbation, the total energy conservation law is the sum of three separate equations, each representing the conservation of a different quantity. The first grouping represents the first order terms in the energy conservation. The last grouping represents equations involving the second order field quantities. The middle grouping may be written

r₀ v₁ v₁ t = -r₀ v₁ ( v₀ v₁)

(31)

(32)

(33)

B₁ m₀ B₁ t

(34)

This is not exactly the same as the operation carried out by Walker [1999]. To show how it is related to that process consider (33). Because p is a function of r through the adiabatic relation

(35)

we can express the relationship between p and r through a Taylor expansion which, to second order is

(36)

where drr -r₀.

If we differentiate this with respect to t and use (2) we get

(37)

so that

(38)

If this equation is added to (31) and (34), after some manipulation we get the wave energy equation (21). The fourth equation is separately satisfied and does not need to be included, it may be lumped with the other second order terms in (28).

4. Harmonic Waves

4.1. Dispersion Relation

Consider an infinite plane wave varying in time and space as exp{-iwt+i k r} so that

(39)

Substitution of these values in the first order wave equations show that nontrivial solutions only occur when the plane waves obey dispersion relations

(40)   (41)

where

(42)

and k²=k²_x+k²_y+k²_z. The frequency w₀ is that observed in the rest frame of the plasma and w is the Doppler shifted value observed in the frame in which the plasma is moving. The relation D=0 represents the coupled slow and fast waves. The relation F=0 represents the transverse Alfvén wave which, in a uniform medium, is decoupled from the other characteristic waves.

4.2. Refractive Index Surfaces - Group and Phase Velocity

In an anisoptropic medium a wave packet can be constructed as a Fourier synthesis of plane harmonic waves [e.g. Walker, 1993, Chapter 3]. Plane waves move through such a wave packet with the phase velocity

(43)

The wave packet moves with the group velocity

(44)

where _k is the gradient operator in k -space. In general the group and phase velocities have different directions. The ray velocity V_R is defined as the component of the phase velocity in the direction of the group velocity. We can also define a velocity

(45)

Note that this is not the group velocity but its component in the direction of the wave normal.

It is useful to normalize the wave vector. The usual way is to define a refractive index vector c k/w where c is the speed of light in free space. For MHD waves this leads to inconveniently large refractive indices. We shall use a "refractive index'' c₀ k/w where c₀ is some convenient characteristic speed such as the Alfvén speed. An appropriate c₀ can be chosen for each problem.

It is convenient to define a number of surfaces which help to define the properties of waves. A more detailed treatment, together with a history of this topic, is given by Walker [1993]. We summarize some of the features here:

- The wave normal surface is a surface in k space traced out by the tip of the vector for possible directions of propagation. It is usually plotted in normalized form as a refractive index surface.

- The direction of the group velocity for a particular direction of k is normal to the refractive index surface at the point where k intersects the surface.

- The ray surface is the surface swept out by the ray velocity. It represents the shape of the wave front emitted by an isotropic source. The phase velocity surface is sometimes erroneously stated to represent the shape of a wave front (see Walker [1993] for a full discussion.)

In Figure 1 we illustrate the refractive index surfaces, normalized in terms of the Alfvén speed, in a frame in which the plasma is at rest. The upper diagram corresponds to V_SA and the lower to V_S>V_A. The magnetic field is in the z direction and the surfaces are surfaces of revolution about the z axis. Several different wave vectors are shown and the corresponding normals to the surfaces represent the direction of the group velocity. Clearly, for the transverse Alfvén wave, energy propagation is exactly along the magnetic field. This is approximately true for the slow wave, while the fast wave is more nearly isotropic.

When the plasma is moving these surfaces are greatly modified. Figures 2 and 3 show two cases, the first where the plasma drift speed is less than and the second where it is greater than the characteristic wave speeds. In each case the magnetic field is in the z direction and the drift velocity is in the y direction. Consider Figure 2. On the left hand side three-dimensional representations of the refractive index surfaces for the fast, slow and transverse Alfvén waves are shown. On the right hand side three cuts through these surfaces are shown. These are in the x-z plane, the y-z plane, and the x-y plane through the origin. The fast wave surface is not affected much. It is still roughly spheroidal in shape but, because the drift velocity is added to the wave velocity, it is shifted relative to the origin. The slow and Alfvén waves are modified more drastically. Although the characteristic speeds V_A,V_S, and (V²_A+V²_S)^1/2 are all greater than the plasma drift speed, the phase speeds of these waves can be very small for some directions of the wave vector. This is because the surfaces are open so that the refractive index approaches infinity for some wave vector directions. The result is that there is a locus in k space on which the component of the phase speed in the direction of the plasma drift is equal to the drift speed. This locus is a straight line where the surfaces are singular. The refractive index surfaces for the slow and Alfvén waves cross on this line. The ray picture breaks down on this locus.

Figure 3 shows the situation when the plasma drift speed is greater than the characteristic speeds of the waves. The format is the same as that for Figure 2. Here it is the fast wave which is most dramatically modified. The plasma drift speed can now exceed the component of the wave speed for some directions of the wave vector. The consequence is that the fast wave surface is no longer closed. It is now roughly the shape of hyperboloids.

Careful study of these and similar surfaces gives insight into the nature of wave propagation. They are used to determine the direction of the group velocity, and hence the boundary conditions at infinity in the negative energy wave picture [McKenzie, 1970; Mann et al., 1998] described below.

4.3. Energy Conservation for Harmonic Waves

For harmonic waves second order quantities in the energy conservation equation can be replaced by their values averaged over one period. First order quantities average to zero. The time average of the product of two first order quantities, for example p and v, is given by

(46)

where the tilde ~ represents the complex conjugate. The energy density and energy flux are defined in this sense. In the rest frame of the drifting plasma, in which v₀=0, the energy conservation equation (21) becomes

(47)

where w₀ is the frequency observed in the rest frame. In a frame in which the plasma drifts with a velocity v₀, the quantity w₀ is the Doppler shifted value of the frequency w. It can be shown [Walker, 1999] that

(48)

where V_G,0=_kw₀ is the group velocity.

In the frame of reference in which the plasma is moving, the energy conservation equation can be written in terms of w rather than w₀ :

(49)

This may be written

(50)

Thus, in this frame, the energy density and the energy flux are

(51)   (52)

5. Negative Energy Waves

An alternative picture of energy flux in moving media involves the concept of negative energy waves [Fejer, 1963; McKenzie, 1970, 1972; Mann et al., 1998; Sturrock, 1960]. This approach has been compared with the approach described above by Walker [1999]. In this section we compare these two approaches, amplifying the discussion in that paper.

This approach relates to harmonic plane waves. For the negative energy wave interpretation we follow the treatment of McKenzie [1970, Appendix I]. He notes that the Galilean transformation of the energy density U and momentum density M are [Sturrock, 1960]

(53)   (54)

where the subscript 0 denotes the frame in which the plasma is at rest. He then takes the momentum density to be the component of the energy flux in the wave normal direction k S divided by the square of the velocity V_L=dw /dk, which in the case of MHD waves is equal to the phase velocity w /k. The value of k S is U₀V_L. Thus

(55)

so that

(56)

Consider the second term in (49). Previously we considered it to be part of the energy flux and transposed it to the right hand side where its interpretation was the divergence of that part of the flux associated with the transport of wave energy density associated with the plasma drift. In the negative energy interpretation it is interpreted as part of the energy density and (56) is used so that

(57)

We have seen that the component of energy flux parallel to k in the rest frame is w₀U₀/k. The energy flux component in the wave normal direction in the frame in which the plasma moves is the energy density multiplied by the parallel component of the group velocity in that frame. If we define S_Gal=wU_Gal/k then we see that

(58)

and U_Gal also obeys the wave energy conservation equation:

(59)

A consequence of this definition of energy density and flux is that, if the component of the plasma streaming velocity in the wave normal direction is greater than the phase speed of the wave then w₀ is negative. In these circumstances the energy density and energy flux are negative, leading to the concept of a negative energy wave.

For harmonic waves either approach is possible. For waves which are not harmonic with a well-defined frequency w a more detailed discussion shows that w₀U/w is more properly the wave action in the Hamiltonian sense.

6. Active Boundaries

Mann et al. [1999] have shown how energy can be extracted from streaming magnetosheath plasma in order to excite waveguide modes in the magnetosphere. The approach they use is to study the reflection of waves at the boundary between two uniform media one of which has a uniform streaming velocity parallel to the boundary. They follow the approach of McKenzie [1970] using the negative energy wave concept. Walker [1999] has discussed the same problem using the wave energy defined in (21).

Walker [1999] considers an MHD wave incident on the boundary between two uniform media. The boundary is in the x - z plane, the magnetic field is in the z direction, and the plasma in the second medium flows with a uniform velocity V in the y direction. He defines energy flux according to (21). He then calculates the energy flux flowing into and out of the boundary, getting the result

(60)

The angle brackets denote a time average, x is normal to the boundary in the direction of the incident wave, the first and second media are denoted by subscripts 1 and 2, and propagation in the positive (negative) direction is denoted by a superscript +(-). The term on the left hand side is the normal component of the incident energy flux. The first term on the right is the transmitted energy flux and the second term on the right is minus the reflected energy flux so that these two terms together represent the total flux transmitted away from the boundary. The last term is zero if the Doppler shifted frequency w₀ is equal to w, i.e. if V=0. In this case energy is conserved at the boundary. Otherwise it represents the difference between the incident energy and the energy propagated away from the boundary. Depending on conditions, it may be positive or negative, so that wave energy may be either lost or gained in the reflection and transmission process.

McKenzie [1970], on the other hand, gives expressions for wave reflection and transmission coefficients which are equivalent to those of Walker [1999]. Using (58) as a definition of energy flux he finds that the ratio of the energy propagated away from the interface to the incident energy is always unity, showing that energy is always conserved at the interface. In this interpretation the reflected energy flux can, in appropriate circumstances, be larger than the incident flux. Conservation of energy is achieved because the transmitted wave carries negative energy.

Each of these interpretations gives correct answers in the context in which it is applied. The negative energy wave interpretation is applied in the context of infinite harmonic plane waves in which the frequency is well defined. Since the energy density is negative in the second medium, the energy flux vector, which is the product of the energy density and the group velocity, is in the opposite direction to the to the group velocity. The boundary conditions for the second medium require that the group velocity has a normal component which points away from the boundary. Thus, energy is flowing into the boundary from the second medium as well as from the first, in which the source is located. This need not trouble us in the case of waves which are uniform in space and time. It is well known that any definition of an energy flux vector is not unique; the curl of an arbitrary vector field of appropriate dimensions can be added to it without affecting energy conservation (see, for example, Panofsky and Phillips [1962]). Unique results are only obtained when integration over a closed surface is performed. Integration over such a surface which lies entirely in the medium on either side of the boundary shows that that there is no net flux into or out of the surface. Only if the surface includes part of the boundary is there a net flux into or out of it. This implies an energy source or sink in the boundary and nowhere else. It can be argued Panofsky and Phillips [1962] that there is no way to use the energy flux vector to locate energy flow.

Despite this fact, physicists often feel a sense of unease with a definition of energy flux which is inconsistent with the location of known sources of energy. If it is possible to find a definition of energy flux which accords with simple minded ideas of localized energy flow they prefer it. It is worth while consulting a well-known "elementary'' but sophisticated source [Feynman et al., 1964, Chapter 27] for an enjoyable discussion of this point.

Difficulties with the negative energy wave interpretation appear when we consider waves limited in space and time. The definitions (56) and (58) of energy density and flux only apply when there is a well-defined frequency. As has already been stated, the energy density so defined is actually the wave action. The most general definition of wave action involves integration over the generalized coordinates describing the wave and is beyond the scope of this paper. Any wave variable can be described by a Fourier synthesis of plane waves. If the Fourier amplitude is narrowly peaked in frequency and wave number, the wave is propagated as a well-defined wave packet with a well-determined frequency and wavelength as illustrated in Figure 4. This figure shows three epochs in the history of a wave packet. At t=t₁ the wave packet is travelling towards the boundary between two media. The lower medium is at rest and the upper medium is streaming parallel to the boundary with a uniform velocity V. The wave packet moves with the group velocity while wave fronts move through it with the phase velocity which, in a dispersive anisotropic medium, does not have the same direction and magnitude as the phase velocity. These velocities are schematically shown and are not intended to show the actual relative directions and magnitudes of an MHD wave. At time t=t₂ the wave packet has been partially reflected and partially transmitted at the boundary. At time t=t₃ the transmitted wave has progressed further and the reflected wave is no longer within the illustrated region. We emphasize that this is an observable phenomenon. The wave packet is a physical disturbance with its own time history.

Now consider what the negative energy wave picture requires us to accept. Suppose that the velocity V is large enough so that the reflected wave has a larger amplitude than the incident wave. It therefore carries more energy than the incident wave. The negative energy wave picture requires that energy be conserved in the reflection and transmission process. The transmitted wave must therefore carry negative energy away from the boundary. Consider the volume ABCD. If we integrate the energy density over this volume at times t=t₁ and t=t₂ the result is zero. Evaluation of the integral at t=t₃ leads to a negative result. This is hard to reconcile with the fact that this is the time at which the wave packet has arrived within the volume. This is an observable fact. If there were a detector within the volume it would be excited, presumably extracting energy from the already negative wave energy reservoir. Another aspect causing difficulty is the non-locality. At time t=t₂ the total energy at the location of the reflected wave packet is augmented at the expense of the energy of the transmitted wave packet which is reduced below zero.

The wave energy picture raises none of these difficulties. Both transmitted and reflected wave packets carry positive energy. As described by Walker [1999], in these circumstances, both acquire additional energy at the boundary. The mechanism can be understood by looking at the details of propagation within the thin boundary layer. The velocity gradient in the boundary layer leads to additional Reynolds and Maxwell stresses, which result in work being done on the wave by the streaming background plasma.

Why does the negative energy picture produce correct results for harmonic waves but provide puzzling problems when applied to wave packets which are limited in space and time? First, we should note that the Galilean transformation should be applied to the total energy; it is, however, applied to the wave energy in the rest frame. The expression used is the wave energy in that frame and omits zero and first order terms. When the Galilean transformation is applied to find the energy in the frame in which the plasma is moving, first order terms are introduced. In addition the period over which time averages are taken is different in the rest and moving frames. The implication is that some of the energy associated with the wave in the moving frame is associated with the background in the wave energy picture. It is possible to resolve the inconsistencies but only at the cost of elaborate argument. For example, if the wave is absorbed as described above, it is necessary to consider not only the negative energy associated with the flux but also what work is done by the streaming background plasma at the interface with the absorber. None of these difficulties arise with the wave energy picture.

7. Discussion and Conclusions

It is likely that interaction of waves with a streaming background plasma is of considerable importance in understanding the generation and propagation of MHD waves. It already appears likely Mann et al. [1999] that waveguide modes of Pc5 and longer period waves can be excited by the counterstreaming magnetosheath and magnetopause plasmas at the magnetopause. It is also probable that upstream Pc3 waves are affected by the magnetosheath flows between the bow shock and the magnetopause and this may be an important effect in their production. Such a problem would be best treated by ray tracing in the moving medium. In such a case wave action is conserved along the ray tubes but wave energy is not; energy is continually exchanged between background plasma and wave through the action of the stresses associated with the velocity gradients. This problem requires further study. In all such problems it is necessary to have a proper understanding of the wave energy. There are different methods of defining energy density which can give correct results if consistently applied. It is our opinion that the wave energy as defined here and by Walker [1999] is the most satisfying definition, with an intuitively obvious interpretation.

Acknowledgments

References

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