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Stationary MHD waves modified by Hall current coupling—II. Incompressible flow
Planet. Space Sci., Vol. 42, NO. 6, pp. 481489, 1994 Copyright 0 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0032-0633/94 $7.00 + 0.00
Pergamon 0032_0633(94)E0073-Y
Stationary MHD waves modified by Hall current coupling-II. Incomprekible flow T. I. Woodward Is2and J. F. McKenzie’x3 ’ Physics Department, University of Natal, Durban, South Africa ’ Space and Atmospheric Physics, The Blackett Laboratory, Imperial College, London, U.K. 3Max-Planck-Institut fiir Aeronomie, D-37189 Katlenburg-Lindau, Germany Received 22 August 1993 ; revised 14 March 1994 ; accepted 21 March 1994
Abstract. In this paper we consider the incompressible limit of the general development presented in Woodward and McKenzie [Planet. Space Sci. 42, 463-479, 19941, who analyze the system of stationary waves generated by a point source current immersed in a flowing magnetoplasma including Hall current effects. There are two wave modes present in this limit, namely the fast and slow modes, both of which are semi-fieldaligned, and the slow mode experiences a resonance at the proton gyrofrequency. In contrast to the compressible situation, the magnetic pressure perturbation displays a Laplacian decay in addition to the characteristics of the fast and slow modes. Furthermore in the very sub-AlfvCnic flow limit, the parallel current and magnetic pressure disturbances together with the associated electric potential perturbation are predominantly of fast mode character.
1. Introduction In this paper we consider the incompressible limit (i.e. the p -+ 00 limit) of the general stationary wave system generated by a point source current in a flowing magnetized plasma including Hall current effects, which is developed in Woodward and McKenzie (1994a) (hereafter referred to as paper I). Such a limit would be appropriate to the case of a hot, low density plasma. The two wave modes, fast and slow, which propagate in this limit arise from the Hall current splitting of the incompressible limit of the corresponding MHD system: i.e. from the constant density limit of the shear Alfvtn and slow magnetoacoustic modes. Both modes are semifield-aligned and dispersive, with the characteristic length scale being determined by the proton gyrofrequency and Correspondence to : T. I. Woodward
the AlfvCn speed (Lighthill, 1960; Woodward, 1993 ; Woodward and McKenzie, 1994b). Furthermore the slow mode experiences a resonance at the proton gyrofrequency above which it is evanescent, while the fast mode continues to propagate. The stationary wavecrest surfaces generated in a flowing magnetoplasma reflect the propagation characteristics of the mode to which they correspond. It is found that the perturbations of the parallel current density and magnetic pressure exhibit a hybrid nature of both fast and slow mode characteristics, with the magnetic pressure disturbance displaying in addition a Laplacian decay. Further the electric potential disturbance displays the characteristics of both modes along with a contamination by an Alfven-like potential operator, together with an intrinsic potential (paper I) arising from the inclusion of Hall current effects. The organization of the paper follows that for the cold case discussed in paper I. Section 2.1 deals with a derivation of the wave equations in the frame & of the source from the general wave equations developed in section 2.2 of paper I, the topological structure of the two stationary wavecrest surfaces generated in this limit, and solutions for the magnetic field variables. Following this, in section 2.2, we discuss the electric potential perturbation appropriate to this incompressible limit, and finally summarize our findings in section 3.
2.1. Wave equations, wavecrest surfaces and$eld
variables
in C,
The formulation follows from the general development given in section 2 of paper I by letting the sound speed a,, + CO; i.e. taking the density to be constant and the dilatation (A = divu) to be zero. Thus in C, the general wave equations ((32) and (33)) in section 2.2 of paper I yield the following expressions in the incompressible limit :
482
T. I. Woodward
Y;V’B,, = pu,J, M2R2V2~
az 2
-9;
(1)
-roJoM3R~V’a(x)~‘(y)6(z);
S$curl,,B = -p,JoM3R-$
-poJo
M2R2V2$
-9;
99’Z+M2&_572 A
f
(3)
ax2 a22 ’
in which _Y’i is the Doppler shifted AlfvCn wave operator defined in paper I, equation (30). The Fourier transforms of equations (1) and (2) yield, B,,k =
-AJO
x [i{k:-M2(1+R2k2)k~}k~+M3Rk,,kz’k2]k,,
(4) >
k2 _5?;,
x [iM3Rk: + {kz - M2(l +R’k’)k:}k,]kx~ Zk
(5) 3
where _Y;,‘= Aik,4 - B,k; + Ci ,
(6)
in which Ai = M2(M2-R’k;) .
Bi = M2k;(2+R2(k;+k;)) Ci E k;
(7)
1
B,, and curl,,B may be obtained by inverting equations (4) and (5) with a three-fold Fourier integral of the type given in equation (54) of paper I. The singularities of the integrands correspond to the natural modes of the system and lie on the wave number surfaces defined by Zik = 0. The wave number surfaces. The dispersion surfaces may be written in the explicit form : k2
i
=
k2
:* = _
Bit_JB,Z_4AiCi 2A,
’
(8)
with Ai, Bi and Cj as defined in equations (7). The surface k= = k=+(_, refers to the Doppler shifted fast (slow) mode. It is noted that neither the fast nor the slow modes’ real k-surfaces cease to exist at a critical Mach number as was the case with the fast Alfvkn mode in the cold compressible limit discussed in paper I. The Doppler shifted k-surface for the fast mode, shown in Fig. 1, essentially consists of a three-sided channel which is reflected in kz < 0, having k,y = + M/R as blocking planes which give rise to the two vertical sides of the channel, while the bottom is formed by the flattening out of the k,k,-section curves as k, decreases to zero. It is noted that the remnant of the
: Stationary
MHD waves
“Alfvtn wedge” (Woodward and McKenzie, 1993) obtains in the limit of k, and k: small. In fact (see below) at these small values of k, and k; there is a “double wedge”, the second one arising from the slow mode. The k-surface for the slow mode (Fig. 2) reflects the proton gyroresonance at the blocking planes kz = + l/MR where lkxl -+ co, a feature also displayed by the corresponding surface for the cold Alfven ion-cyclotron mode (paper I). For k,_ and k, small,
(2)
where 3:
and J. F. McKenzie
k,_ N i;,
representing the remnant of the AlfvCn wedge, giving the second wedge of the double Alfvtn wedge mentioned above. Furthermore, letting R-0 (i.e. neglecting the Hall current effect), retrieves the dispersionless double Alfvtn wedges indicative of the fact that the two modes in this limit propagate unidirectionally at the same speed (Cowling, 1957 ; Dungey, 1968 ; Lighthill, 1960). The method of stationary phase provides us with the topologies of the stationary wavecrest surfaces associated with the two wave modes present (Woodward, 1993). The stationary wavecrest surface associated with the fast mode (Fig. 3) is found in line with and downstream of the source. It is shed from two points on either side of the source which are along the magnetic field direction and remains upstream of the AlfvCn lines (McKenzie et al., 1993 ; Woodward and McKenzie, 1993) in the near-field region, only tending towards these lines at large distances from the source. Furthermore, its spatial extent perpendicular to the plane containing the flow, uO, and the background magnetic field, Bo, is small in comparison to the distance from the source. On the other hand, the wavecrest surface generated by the slow mode (Fig. 4) emanates from a single point directly downstream of the source and tends to the Alfvtn lines in the far-field region, otherwise being downstream of these lines. In addition, the ‘body’ of the surface is contained in the v-shaped region formed by the Alfvtn lines downstream of the source, in contrast to the fast mode wavecrest (Fig. 3) which is upstream of these lines. As with the fast mode, there is only a small ‘y-deviation’ of the surface, where y is in the direction perpendicular to the plane containing u0 and Bo. Expressions for B,, and curl,,B are developed in an approximate fashion in Appendix A, where it is shown that both perturbations exhibit a hybrid nature of fast and slow mode characteristics with, in the case of B,,, the additional effect of the Laplacian operator which appears on the left-hand side of its wave equation (equation (1)). In very sub-AlfvCnic flow regimes (M <
in which 27r5’2 ZBliFLX -
3112M3R [
M,,6
z 0 2x
7’3
1
JRX
c-y.“~&-iq7
T. I. Woodward and J. F. McKenzie : Stationary MHD waves
483
20.0
-40.a
Fig. 1. Doppler shifted fast incompressible mode wave number surface plotted for M = 1.1. Because of the view angle chosen in this diagram, the labelling on the k,-axis is not clear : k, ranges from - 1.2 to 1.2. kx,y,r+ are measured in units of R-’
POJO
(curl,,B),
N - 167~’
(curl,,B),
N
p,, Jox3” n312R1Pz2M1/2
x
with z ‘SC
M(2+W) J
2~3
N-
ZM”2
-2x~3/2'
(13)
In these expressions BIIC,,,, refer to the contributions to the B,, perturbation from the fast and slow modes, respeccorrespond to the “pure mode” contributions, tively. zBII~Fs~ while Zv2CF,sjrepresent the effects of the Laplacian operator mentioned above on the fast and slow mode contributions, respectively. Furthermore with (curl,,B)(,,s, representing the contributions from the fast and slow modes respectively to the curl,,B perturbation, it is shown in Appendix A that
(&J2wcos [& -;I.
(15)
In accordance with the M << 1 approximation, clearly the “pure” fast mode contribution (ZBIIF)makes the dominant contribution to the B,, perturbation. The effects of the Laplacian terms Zv2CF,sjare negligible in this (M << 1) approximation in comparison to the pure mode contributions ZBIICFs). Similarly curl,,B is predominantly determined by the fast mode contribution (curl,,B),. The solution strategy adopted in Appendix A for B,, and curl,,B initially assumes different approximations for the fast and slow mode Doppler shifted wave number surfaces (see equations (A.6) and (A.7) in Appendix A) in order to continue the development in an analytic fashion, and to illustrate the use of the method of station-
T. I. Woodward and J. F. McKenzie:
484
-5.0
I
I
-3.0
I
I 1.0
-1.0
kx
I
Stationary MHD waves
I 3.0
B -0 -
5.0
Fig. 2. Doppler shifted slow incompressible mode wave number surface for M = 0.9. The labelling on the k,,-axis is unclear due to the choice of view angle. The actual k,-range is - 10 to 10. kX,,,,2_ are measured in units of R-’
Fig. 3. Sketch of the stationary wavecrest surface generated by the fast incompressible mode
Fig. 4. Sketch of the stationary wavecrest surface associated with the slow incompressible mode. Sections in planes perpendicular to the flow u0 are included to emphasize the three-dimensional nature of the (hollow) surface
T. I. Woodward
and J. F. McKenzie
: Stationary
MHD waves
The approximation for the slow mode wave number surface (equation (A.6)) assumes k,_ to be independent of k,. and proportional to JlkxliR for M << I. However, the same type of approximation for the fast mode would push the blocking planes (referred to above) to k, = 0 and give rise to an imaginary wave number surface and thus evanescence. Nevertheless by approximating k,, for values of k, between the blocking planes (equation (A.7)) the analysis is rendered tractable, if only through a stationary phase development, and allows the nonevanescent contribution from the fast mode to be taken into account. Finally the fast mode contribution may be ordered according to an M CC1 approximation so as to ascertain which mode, fast or slow, is dominant. ary phase.
485
where Qi,t is the intrinsic potential given by equation (68) correspond to the terms arising from the Of paper I. 'B,,,,,, “pure” fast and slow mode B,, perturbations, while ZV2CF,s) represent the influences of the Laplacian decays of the fast and slow mode B,, perturbations. ZP,,,,,, are associated with the effects of the potential operator YQ (equation (44) of paper I) on the fast and slow mode B,, driving terms. Finally ZPpzand Z,,, arise from a source term contributed by the particle pressure p, and deriving from Laplacian and Tip, residues, respectively. Furthermore an approximate solution for cDi,valid in very sub-Alfvenic flows (i.e. M <
ZB,,,N --~L~‘~M”~R”‘x
2.2. Electric potential
associated
with the incompressible
limit
In this limit the equation for the electric potential, denoted by cDi,is given by the general potential equation (43) of paper I in which B, is now determined by its wave equation in this limit (i.e. equation (l)), pe is yet to be specified, and an equation prescribing particle pressurep is required. An equation for p is easily derived from the linearized equation of motion (equation (12) of paper I) by taking the divergence of that equation and noting the incompressibility condition, div u G 0, to yield
V2 (p+$Bx)=
divM, = B,J06(x)6(y)6’(z),
IBllF=
-
xe-,i(“,‘-1)cos[3(&r3Gx+:];
(20)
log (y)};
ZvzCF,sjN n2MR -_$ c-
(16) I@2 N -2r?${$log~)};
in which M, E -J, x B. is the momentum source term, and J, is taken to be J,,S(x)6(y)6(z)(O,l,O) (paper I). The integrated version of this equation shows that the total pressure P = p + BOBJp,, decays away from the source in a two-dimensional dipole-like fashion : i.e. p+$Bx=
(21)
(22)
N_p-y 4z=R a & py*J2M ay y2
Z
x%[&exp[--&(2--?)]
- --
Thus particle pressure p is given by expression (17) with B, prescribed by equations (9). In accordance with the M CC1 approximation and since we are concerned with far-field estimates, p = - (B,/po) B,. Hence in the case of a plasma with a cold ion population and a warm electron fluid, for which p = pe, the particle pressure is almost exactly balanced by the magnetic pressure and the effect of the total pressure driving term on the electric potential is far less significant than those of the remaining driving terms, namely the two-dimensional (x, z)-Laplacian derivative of B,, and the “intrinsic” terms arising from the purely source current terms (see paper I, equation (43)). In Appendix B it is shown that mi exhibits a hybrid character which may be written in the form POUOJo =----
(2n)3
xexp[X-/&I +Gexp
-& [
x
exp
2+-
MxY z /-I
(
pc+Jzy]:
2
(23)
where p and r are given by p s Jy’+z’,
and i+ (= x+z/M) are the AlfvCn characteristics. Further, it is shown that ZP,,,,, cancel each other to this order of the approximation. Thus the major contribution to the modified potential @,_ POVoR I
B.
”
arises from the “nure” (or far-field) fast mode term Z,,,,,
486
with negligible
T. I. Woodward and J. F. McKenzie : Stationary MHD waves
effects from the slow mode (IB,,), Laplacian
(&,s) and +) and ym (&. and L,,,,,) terms, and the intrinsic potential Oint. Moreover considering a plasma with a warm electron fluid and a cold ion population so that Pefp”
_B,B,, PO
and noting that the B, perturbation is predominantly of fast mode character (see equations (9)), clearly Oj adopts largely a fast mode character.
3. Discussion
and conclusion
In the limit of an incompressible plasma, two wave modes, the fast and slow, appear. These modes result from the Hall current splitting of the corresponding single MHD mode which possesses the propagation characteristics of the MHD shear AlfvCn mode and propagates both the parallel current and the Laplacian derivative of the magnetic pressure term along the background magnetic field. The presence of source currents prevents the purely onedimensional propagation of the plasma flow and magnetic field perturbations. The slow mode experiences a resonance at the proton gyrofrequency, while the fast mode continues to propagate. Both modes are reasonably well field-guided, especially for very low frequencies (CC Q,), a fact which is consistent with them both deriving from the incompressible limit of the MHD modes. In a perpendicular flow, neither of the stationary wavecrest surfaces associated with the two modes ceases to exist at a critical flow Mach number (i.e. real flow characteristics exist for both modes for all Mach numbers, which in turn means that neither of the modes is evanescent), and the topology of these surfaces reflects the dispersive nature of the wave system. Both wavecrest surfaces are disconnected from the source and tend towards the Alfven lines in the far-field region. The fast mode surface is found in line with and downstream of the source, being shed from two points on either side of the source which are along the background magnetic field. It remains upstream of the Alfvtn lines in the near-field and approaches them at large distances from the source, at all times being well confined perpendicular to the plane containing the flow and the background magnetic field. On the other hand, the slow mode surface emanates from a single point directly downstream of the source, therefore being downstream of the Alfven lines close to the source tends towards them in the far-field. Furthermore, it is similarly well confined perpendicular to the plane containing the flow and the background magnetic field. Both the magnetic pressure and parallel current perturbations exhibit a hybrid character of the slow and fast modes, and, in the case of the magnetic pressure, a Laplacian decay. In very sub-Alfvenic flows both perturbations are predominantly of fast mode character, and are therefore found just upstream of the Alfvtn lines on the stationary fast mode wavecrest surface described above. Finally, the electric potential displays a mixed nature of fast and slow mode characters combined with the
characteristics of the Alfven-like potential operator and Laplacian decay, in addition to the properties of the intrinsic potential. In very sub-Alfvenic flows the major contribution to the electric potential comes from the fast mode with less dominant potential operator contributions (including the intrinsic potential), and negligible slow mode and Laplacian influences. This is in contrast to the cold case in which the fast mode is evanescent and gives way to the influence of the potential operator characteristics. Acknowledgements. J.F.McK. wishes to thank the FRD (South Africa) for their partial support. T. I. W. acknowledges financial support from the FRD (South Africa) in the form of a Special Merit Scholarship, and thanks the Max-Planck Institut fiir Aeronomie (Lindau) for support during the tenure of a stipendium, during which part of the work discussed in this article was completed. Further we wish to thank Dr G. M. Webb (Tucson) for invaluable discussions on the subject of this paper. Finally we gratefully acknowledge the effort and perseverance of Neal Powell for producing Figs 3 and 4.
References Cowling, T. G., Magnetohydrodynamics. New York, Interscience, 1957. Dungey, J. W., in Physics of Geomagnetic Phenomena (edited by S. Matsushita and W. H. Campbell), pp. 913-934. Academic Press, New York, 1968. Lighthill, M. J., Studies on magneto-hydrodynamic waves and other anisotropic wave motions. Phil. Trans. Roy. Sot. London, Ser A 252,391430, 1960. McKenzie, J. F., Woodward, T. I. and Inhester B., Magnetoacoustic and Alfvtn potentials for Stationary waves in a moving plasma. J. geophys. Res. 98, 9201-9213, 1993. Woodward, T. I., Linear and non-linear waves in space plasmas,
PhD Thesis. University 1993.
of Natal, Durban,
South Africa,
Woodward, T. I. and McKenzie, J. F., Stationary MHD structures. Planet. Space Sci. 41,217-228, 1993. Woodward, T. I. and McKenzie, J. F., Stationary MHD waves modified by Hall current coupling-I. Cold compressible flow. Planet. Space Sci. 42,463479, 1994a. Woodward, T. I. and McKenzie, J. F., Stationary MHD waves modified by Hall current coupling : I Cold compressible flow. M. P. Ae. I. Rep. No. MPAE-W-100-93-15, 1994b.
Appendices The methods of solution for the magnetic variables (BN and curl,,B) and for the electric potential ($) are sketched in these appendices. More details of the mathematical developments may be found in Woodward (1993), and Woodward and McKenzie (1994b).
A. The field variables In this appendix approximate expressions for B,, and curl,,B appropriate to the incompressible limit (equations (1) and (2), respectively) are developed. The k,-integrals are done using Cauchy’s residue theorem, noting that there are always four hyperbolic poles regardless of the Mach number M. This shows that the B,, and curl,,B perturbations are hybrid in nature. In particular curl,,B consists
487
T. I. Woodward and J. F. McKenzie : Stationary MHD waves of fast mode ((curlFB)F) and slow mode ((cur$B)s) contributions, while the B,, perturbation exhibits a similar mixed character with an additional influence arising from the Laplacian operator on the left hand side of the differential equation for B,, (equation (l)), giving rise to Laplacian terms associated with each of the two wave modes. The fast and slow mode contributions to B,, are denoted BliFand Blis, respectively. Thus we derive
N(2 +W)
k;,
where dk, dk,
x
(eik,(+,_,r
+e-ik,,+.-,z)ei(k,x+k~Y~
iv $3; F.S
(A3)
M(2 +M*)
lTm
(2+M2)
J
=
-
-
s
m e-~~+iW+k,y)
dk,
dk,,
2 f --a0 --m
--k%,+,-,
(A51 in which we define
In order grations in For very expression yields
N5
and hence the effect of the Laplacian operator on the two wave mode contributions is the same in this approximation. This integral has been done in Appendix A of paper I, leading to the solution for &z~~,~~ (equation (12)) in the main text.
~~~~‘~;,.-,~Ik:-M2(1+R2(k.~+k:+k:+,_,))k~,,_,}k,,]k~; 2&t%+
(Al)
3
In this approximation we have left out the k,,-dependence in order to simplify the subsequent algebra. This i&reasonable for values of k, sufficiently close to the blocking planes. Using the idea of stationary phase for this mode, we expect the solution to display singtdarities on the fast mode stationary wavecrest (Fig. 3). However, approximation (A7) has a different topology near lkxl cc M/R to that of the exact expression (equation (8) and Fig. l), and we re-emphasize the illustrative nature of this approach. Integrals IBxf and burl,,B)~ both arise from the fast mode and may be treated in a similar fashion. Using approximations (A7) and the fact that M << 1, the method of stationary phase may be used to obtain the approximate solutions (10) and (14) for ZBIF and (curl,@),, respectively. lSRrand &rlSj, are similar in structure since they correspond to the slow mode contributions and may be dealt with together. Using approximations (A6), noting MC 1, and again appealing to stationary phase arguments we estimate IBysand (curl,B), as in the text equations (11) and (15), respectively. Finally, an approximate solution to Iv~~r,~~(equation (A4)) may be obtained as follows. Using approximations (A@ &‘(F,S)
X
0 < k, < ;
N
2A,(k,2, - kz”_)= Ji3‘j27_4A,c
and
,
to proceed with the subsequent k,- and k,-intean analytic fashion we approximate k,, as follows. low AlfvCn Mach numbers M, expanding the for k,2_ (equation (8)) using the Binomial theorem
On the other hand, such an expansion fork:, yields an imaginary k,+ since the k,-range in which the real k,, surface exists (i.e. -M/R G k, < M/R) is not ‘seen’ by the process. However, for illustrative purposes it is worth approximating ki, near the ‘blocking’ planes (see the discussion on the Doppler shifted wave number surfaces below equation (8) of the text), which is equivalent to considering Ai -3 0, i.e.
B. The incompressible electric potential [Di In this appendix an approximate solution for the electric potential CD,associated with the incompressible modes is derived. “ftiis determined from equation (43) of paper I in which B,,is described by equation (1) and p by equation (16) of this paper. We begin by Fourier transforming these equations, substituting into the resulting algebraic equation describing the Fourier form of Qti (i.e. at,) from the corresponding equations for B, and p, solving for dD{k in terms of source terms, and finally inverting the consequent expression with a three-dimensional Fourier integral (exactly as for the cold case studied in paper I, except with Qj @ik)
replacing
%
(%k)).
The Fourier transforms of equations (16) and (1) for p and B, appropriately substituted into that of CD? and solved for tD& yields
kk ----_+%, Ymkk2
(Bl)
488
T. I. Woodward and J. F. McKenzie : Stationary MHD waves
where Yak is the Fourier transform of uOVQand is given by -[l-iMRk,,]k~)(k:+k_~+k:,+,_,);
Look E M2 ki - (I - iMRk,) k: ,
032)
the properties of which are discussed in section 3.2 of paper I. kf, are the zeros of _%‘zk , the Fourier form of the Doppler shifted incompressible wave operator (equation (3)) and are given by equation (8), in which Ai, B, and C, are defined in equations (7). Finally
NV2 = {(1+M2)k~+M2k;}k~~k,?; Dv~(F,s) =
(M2-R2k.:)(kS+-kt_)(k:+k_:+k;+,_,) x((l+M’)k:-iMRkyk:+M2kj);
N~~=k~:(l+~-ixk,,)[i{l-(l+R2[k;+k;
represents the Fourier form of the terms in parentheses. Note that pe is unspecified as yet and we shall invert equation (Bl) to obtain a solution for the difference Di--
>I;
POVoR
B,,
“’
and comment later on an expression for pc (see section 2.2 of the text). We begin by doing the Fourier inversion integrals with the k,integration, making use of Cauchy’s residue theorem to obtain PO&l JCI
Pe = -
+$(l-iMRk,,)])(l-iMRk,,)}k;
D ~p,,,-(~2-R2k:)(kf+-k,2_)
x $
>
(I -iMRk,)-ki+,_,
.
k,,i are defined in paper I (equations
(63) and (64)) and correspond to the real and imaginary parts of the pole contributed by yak. @,,t corresponds to the intrinsic potential referred to in the general development of paper I, a solution for which has been derived in the analysis of the cold compressible electric potential in Appendix B of that article. Thus it is seen that the incompressible electric potential Qi exhibits a hybrid nature, consisting of contributions from the fast and slow modes, the influence of the potential operator Yo, a limit- and mode-independent intrinsic potential together with a term arising from the electron pressure, namely
(2RS
in which dk, dk, x e-ze+i(k,x+kyy)
NV2
(J34)
&(F.S)
x
zPV2Girc
e-k,z
dk, dk, e-zm J-m
ei(krz+kxx+k~d;
ki
>
0 ;
&(k,x+k,y)
J--m
k,.
037)
x (1+M2)k:-iMRk,k:+M2k:’ Z
pp*
R E
-
-
M2
(~56)
ss m m
_a
dk
__m x
*+-_lkxl MR’
k2 &
e-k,z
ei(k,r+k,x+k,.y)
’ ik,
X
;
k,>O;
(B8)
with = NB ,,Fs)
D BI,Fsl=
In equations (B4)-(B8) ZBlllFS) correspond to the terms arising from the “pure” (or far-field) fast and slow mode B,, perturbations, while &~(~,s)represent the effects of the Laplacian decays associated with the fast and slow mode B,, perturbations. The terms ZY,,,SIare associated with the influence of the potential operator 4p, on the fast and slow mode B,, driving terms. Finally Zpvzand ZpPpg arise from a source term contributed by the particle pressure p. In order to proceed further in an analytic fashion we will approximate the integrands for small Mach numbers M. In such a limit k,.; may be approximated as in Appendix B of paper I (equations (B6) and (B7)), and k,2_ may be taken as in Appendix A (equation (A6)). For kt+ we shall use
except in integrals Ia , where k,, appears in the exponential and approximation (g9) would lead to strong attenuation. In ZBRF we shall take the expression for kj, used in Appendix A (equation (A7)), i.e. approximate kj, near the ‘blocking planes’ (see discussion on the Doppler shifted wave number surfaces for the incompressible modes which appears below equation (8) of the text). This approach should be considered illustrative of the nature of the method, rather than a complete solution.
(k~+k~+,_,)[i{k:-M2(1+R2(k:+kz
+k~+,-,))k~+,-,}k~+M3~,~k~+.-,~k~+k~+k~+,_,)l;
Integrals I,,‘
(MZ-R2k~)(k~+,-,-k~-,+~)(M2k~+,-~
Using the relevant approximations
we have
T. I. Woodward
x
{
and J. F. McKenzie
(e”e+‘k“+e-
: Stationary
MHD waves
(elllL-(kr)+e-ix~-_(k,))},
4+(U)+
have been written xrj+ (k,) in which +k (k,) = k,f.z&/ (x,/%). Noting that M << I, the method of stationary phase may be used to confirm the estimate for IBllsgiven in the text equation (19).
N
use of approximation
iR3
M3(2+M2)
(A7) for kj,
14
ss
m1 0X -e-:~COS 0 o k:
(k,x) cos (k,,y) dk, dk,
(B13)
we can write
ss * dk, -03
Integrals IPm(F,sjand IPJ,
m dk, 0
x {(eird+(k,)+e-‘xO+(k,t)+(elx~-(k,)+e-”~-(k~))} x(k:+ki+)
where
Thus, to this order of approximation, the Laplacian terms arising from the B,, driving terms, namely Iv+ and lvzs, are equal. Furthermore integrals I in expression (B13) have already been tackled in Appendix B of paper I. Hence the text equations (21) and (22) for ZV~(r,sjand ZPv2,respectively.
Integrals IB,,F
I+
@W
(~10)
where the k,-integral has been done and the k,-integration range is now from 0 to co. In (BlO) the phases of the integrands
Making
489
e’k,’
M*( - iR* k,: + MRk,.)kj+ (M*kt+-(I-iMRk,.)k:) i(ki--M*kI+)kf
+(k~+k~+kf+)(M*kf+-(I-iMRky)k:) in which k,, = JM(2+M2)/(2R’(M/R-kJ), phases of the integrands have been denoted
These terms all arise from the elliptic-hyperbolic pole contributed by the Fourier form of the electric potential operator YO,, and may therefore be treated together. Using approximations (A6) for k:_ , (B9) for kf, , and (B6) and (B7) of Appendix B of bapkr I for k,,i we obtain
’ P11) 11 and where the x4+ (k,) with
#& (k,) = k,k (z/x) JM(2+ M2)/(2R3(M/R/-k,)). The k,integral is estimated using a stationary phase development, leaving the k,.-integration which may be done using Cauchy’s residue theorem. Finally noting that M << 1 we confirm expression (20) of the text for I,,,,.
Integrals Iv~cF,Sj and Ipv2 These may be dealt with together since they all derive from a Laplacian pole in the k,-integration. Using approximations (A6) for kt_, and (B9) for k:, , it is seen that
i71 m )zf
m
ss_m -m
R* i zkf
-Rk, lk.4
e-k,z+@,z+k,x++) dk,dk,;k,
> 0;
(B14)
Thus the fast and slow mode Y@ contributions, namely ZP,, On the other and &,, cancel to this order of approximation. hand, IPy, is of order M-’ and the remaining double integral may be estimated using the method of stationary phase for both the remaining k,- and k,-integrals, thereby confirming equation (23) for ZPYpm.
Pergamon 0032_0633(94)E0073-Y
Stationary MHD waves modified by Hall current coupling-II. Incomprekible flow T. I. Woodward Is2and J. F. McKenzie’x3 ’ Physics Department, University of Natal, Durban, South Africa ’ Space and Atmospheric Physics, The Blackett Laboratory, Imperial College, London, U.K. 3Max-Planck-Institut fiir Aeronomie, D-37189 Katlenburg-Lindau, Germany Received 22 August 1993 ; revised 14 March 1994 ; accepted 21 March 1994
Abstract. In this paper we consider the incompressible limit of the general development presented in Woodward and McKenzie [Planet. Space Sci. 42, 463-479, 19941, who analyze the system of stationary waves generated by a point source current immersed in a flowing magnetoplasma including Hall current effects. There are two wave modes present in this limit, namely the fast and slow modes, both of which are semi-fieldaligned, and the slow mode experiences a resonance at the proton gyrofrequency. In contrast to the compressible situation, the magnetic pressure perturbation displays a Laplacian decay in addition to the characteristics of the fast and slow modes. Furthermore in the very sub-AlfvCnic flow limit, the parallel current and magnetic pressure disturbances together with the associated electric potential perturbation are predominantly of fast mode character.
1. Introduction In this paper we consider the incompressible limit (i.e. the p -+ 00 limit) of the general stationary wave system generated by a point source current in a flowing magnetized plasma including Hall current effects, which is developed in Woodward and McKenzie (1994a) (hereafter referred to as paper I). Such a limit would be appropriate to the case of a hot, low density plasma. The two wave modes, fast and slow, which propagate in this limit arise from the Hall current splitting of the incompressible limit of the corresponding MHD system: i.e. from the constant density limit of the shear Alfvtn and slow magnetoacoustic modes. Both modes are semifield-aligned and dispersive, with the characteristic length scale being determined by the proton gyrofrequency and Correspondence to : T. I. Woodward
the AlfvCn speed (Lighthill, 1960; Woodward, 1993 ; Woodward and McKenzie, 1994b). Furthermore the slow mode experiences a resonance at the proton gyrofrequency above which it is evanescent, while the fast mode continues to propagate. The stationary wavecrest surfaces generated in a flowing magnetoplasma reflect the propagation characteristics of the mode to which they correspond. It is found that the perturbations of the parallel current density and magnetic pressure exhibit a hybrid nature of both fast and slow mode characteristics, with the magnetic pressure disturbance displaying in addition a Laplacian decay. Further the electric potential disturbance displays the characteristics of both modes along with a contamination by an Alfven-like potential operator, together with an intrinsic potential (paper I) arising from the inclusion of Hall current effects. The organization of the paper follows that for the cold case discussed in paper I. Section 2.1 deals with a derivation of the wave equations in the frame & of the source from the general wave equations developed in section 2.2 of paper I, the topological structure of the two stationary wavecrest surfaces generated in this limit, and solutions for the magnetic field variables. Following this, in section 2.2, we discuss the electric potential perturbation appropriate to this incompressible limit, and finally summarize our findings in section 3.
2.1. Wave equations, wavecrest surfaces and$eld
variables
in C,
The formulation follows from the general development given in section 2 of paper I by letting the sound speed a,, + CO; i.e. taking the density to be constant and the dilatation (A = divu) to be zero. Thus in C, the general wave equations ((32) and (33)) in section 2.2 of paper I yield the following expressions in the incompressible limit :
482
T. I. Woodward
Y;V’B,, = pu,J, M2R2V2~
az 2
-9;
(1)
-roJoM3R~V’a(x)~‘(y)6(z);
S$curl,,B = -p,JoM3R-$
-poJo
M2R2V2$
-9;
99’Z+M2&_572 A
f
(3)
ax2 a22 ’
in which _Y’i is the Doppler shifted AlfvCn wave operator defined in paper I, equation (30). The Fourier transforms of equations (1) and (2) yield, B,,k =
-AJO
x [i{k:-M2(1+R2k2)k~}k~+M3Rk,,kz’k2]k,,
(4) >
k2 _5?;,
x [iM3Rk: + {kz - M2(l +R’k’)k:}k,]kx~ Zk
(5) 3
where _Y;,‘= Aik,4 - B,k; + Ci ,
(6)
in which Ai = M2(M2-R’k;) .
Bi = M2k;(2+R2(k;+k;)) Ci E k;
(7)
1
B,, and curl,,B may be obtained by inverting equations (4) and (5) with a three-fold Fourier integral of the type given in equation (54) of paper I. The singularities of the integrands correspond to the natural modes of the system and lie on the wave number surfaces defined by Zik = 0. The wave number surfaces. The dispersion surfaces may be written in the explicit form : k2
i
=
k2
:* = _
Bit_JB,Z_4AiCi 2A,
’
(8)
with Ai, Bi and Cj as defined in equations (7). The surface k= = k=+(_, refers to the Doppler shifted fast (slow) mode. It is noted that neither the fast nor the slow modes’ real k-surfaces cease to exist at a critical Mach number as was the case with the fast Alfvkn mode in the cold compressible limit discussed in paper I. The Doppler shifted k-surface for the fast mode, shown in Fig. 1, essentially consists of a three-sided channel which is reflected in kz < 0, having k,y = + M/R as blocking planes which give rise to the two vertical sides of the channel, while the bottom is formed by the flattening out of the k,k,-section curves as k, decreases to zero. It is noted that the remnant of the
: Stationary
MHD waves
“Alfvtn wedge” (Woodward and McKenzie, 1993) obtains in the limit of k, and k: small. In fact (see below) at these small values of k, and k; there is a “double wedge”, the second one arising from the slow mode. The k-surface for the slow mode (Fig. 2) reflects the proton gyroresonance at the blocking planes kz = + l/MR where lkxl -+ co, a feature also displayed by the corresponding surface for the cold Alfven ion-cyclotron mode (paper I). For k,_ and k, small,
(2)
where 3:
and J. F. McKenzie
k,_ N i;,
representing the remnant of the AlfvCn wedge, giving the second wedge of the double Alfvtn wedge mentioned above. Furthermore, letting R-0 (i.e. neglecting the Hall current effect), retrieves the dispersionless double Alfvtn wedges indicative of the fact that the two modes in this limit propagate unidirectionally at the same speed (Cowling, 1957 ; Dungey, 1968 ; Lighthill, 1960). The method of stationary phase provides us with the topologies of the stationary wavecrest surfaces associated with the two wave modes present (Woodward, 1993). The stationary wavecrest surface associated with the fast mode (Fig. 3) is found in line with and downstream of the source. It is shed from two points on either side of the source which are along the magnetic field direction and remains upstream of the AlfvCn lines (McKenzie et al., 1993 ; Woodward and McKenzie, 1993) in the near-field region, only tending towards these lines at large distances from the source. Furthermore, its spatial extent perpendicular to the plane containing the flow, uO, and the background magnetic field, Bo, is small in comparison to the distance from the source. On the other hand, the wavecrest surface generated by the slow mode (Fig. 4) emanates from a single point directly downstream of the source and tends to the Alfvtn lines in the far-field region, otherwise being downstream of these lines. In addition, the ‘body’ of the surface is contained in the v-shaped region formed by the Alfvtn lines downstream of the source, in contrast to the fast mode wavecrest (Fig. 3) which is upstream of these lines. As with the fast mode, there is only a small ‘y-deviation’ of the surface, where y is in the direction perpendicular to the plane containing u0 and Bo. Expressions for B,, and curl,,B are developed in an approximate fashion in Appendix A, where it is shown that both perturbations exhibit a hybrid nature of fast and slow mode characteristics with, in the case of B,,, the additional effect of the Laplacian operator which appears on the left-hand side of its wave equation (equation (1)). In very sub-AlfvCnic flow regimes (M <
in which 27r5’2 ZBliFLX -
3112M3R [
M,,6
z 0 2x
7’3
1
JRX
c-y.“~&-iq7
T. I. Woodward and J. F. McKenzie : Stationary MHD waves
483
20.0
-40.a
Fig. 1. Doppler shifted fast incompressible mode wave number surface plotted for M = 1.1. Because of the view angle chosen in this diagram, the labelling on the k,-axis is not clear : k, ranges from - 1.2 to 1.2. kx,y,r+ are measured in units of R-’
POJO
(curl,,B),
N - 167~’
(curl,,B),
N
p,, Jox3” n312R1Pz2M1/2
x
with z ‘SC
M(2+W) J
2~3
N-
ZM”2
-2x~3/2'
(13)
In these expressions BIIC,,,, refer to the contributions to the B,, perturbation from the fast and slow modes, respeccorrespond to the “pure mode” contributions, tively. zBII~Fs~ while Zv2CF,sjrepresent the effects of the Laplacian operator mentioned above on the fast and slow mode contributions, respectively. Furthermore with (curl,,B)(,,s, representing the contributions from the fast and slow modes respectively to the curl,,B perturbation, it is shown in Appendix A that
(&J2wcos [& -;I.
(15)
In accordance with the M << 1 approximation, clearly the “pure” fast mode contribution (ZBIIF)makes the dominant contribution to the B,, perturbation. The effects of the Laplacian terms Zv2CF,sjare negligible in this (M << 1) approximation in comparison to the pure mode contributions ZBIICFs). Similarly curl,,B is predominantly determined by the fast mode contribution (curl,,B),. The solution strategy adopted in Appendix A for B,, and curl,,B initially assumes different approximations for the fast and slow mode Doppler shifted wave number surfaces (see equations (A.6) and (A.7) in Appendix A) in order to continue the development in an analytic fashion, and to illustrate the use of the method of station-
T. I. Woodward and J. F. McKenzie:
484
-5.0
I
I
-3.0
I
I 1.0
-1.0
kx
I
Stationary MHD waves
I 3.0
B -0 -
5.0
Fig. 2. Doppler shifted slow incompressible mode wave number surface for M = 0.9. The labelling on the k,,-axis is unclear due to the choice of view angle. The actual k,-range is - 10 to 10. kX,,,,2_ are measured in units of R-’
Fig. 3. Sketch of the stationary wavecrest surface generated by the fast incompressible mode
Fig. 4. Sketch of the stationary wavecrest surface associated with the slow incompressible mode. Sections in planes perpendicular to the flow u0 are included to emphasize the three-dimensional nature of the (hollow) surface
T. I. Woodward
and J. F. McKenzie
: Stationary
MHD waves
The approximation for the slow mode wave number surface (equation (A.6)) assumes k,_ to be independent of k,. and proportional to JlkxliR for M << I. However, the same type of approximation for the fast mode would push the blocking planes (referred to above) to k, = 0 and give rise to an imaginary wave number surface and thus evanescence. Nevertheless by approximating k,, for values of k, between the blocking planes (equation (A.7)) the analysis is rendered tractable, if only through a stationary phase development, and allows the nonevanescent contribution from the fast mode to be taken into account. Finally the fast mode contribution may be ordered according to an M CC1 approximation so as to ascertain which mode, fast or slow, is dominant. ary phase.
485
where Qi,t is the intrinsic potential given by equation (68) correspond to the terms arising from the Of paper I. 'B,,,,,, “pure” fast and slow mode B,, perturbations, while ZV2CF,s) represent the influences of the Laplacian decays of the fast and slow mode B,, perturbations. ZP,,,,,, are associated with the effects of the potential operator YQ (equation (44) of paper I) on the fast and slow mode B,, driving terms. Finally ZPpzand Z,,, arise from a source term contributed by the particle pressure p, and deriving from Laplacian and Tip, residues, respectively. Furthermore an approximate solution for cDi,valid in very sub-Alfvenic flows (i.e. M <
ZB,,,N --~L~‘~M”~R”‘x
2.2. Electric potential
associated
with the incompressible
limit
In this limit the equation for the electric potential, denoted by cDi,is given by the general potential equation (43) of paper I in which B, is now determined by its wave equation in this limit (i.e. equation (l)), pe is yet to be specified, and an equation prescribing particle pressurep is required. An equation for p is easily derived from the linearized equation of motion (equation (12) of paper I) by taking the divergence of that equation and noting the incompressibility condition, div u G 0, to yield
V2 (p+$Bx)=
divM, = B,J06(x)6(y)6’(z),
IBllF=
-
xe-,i(“,‘-1)cos[3(&r3Gx+:];
(20)
log (y)};
ZvzCF,sjN n2MR -_$ c-
(16) I@2 N -2r?${$log~)};
in which M, E -J, x B. is the momentum source term, and J, is taken to be J,,S(x)6(y)6(z)(O,l,O) (paper I). The integrated version of this equation shows that the total pressure P = p + BOBJp,, decays away from the source in a two-dimensional dipole-like fashion : i.e. p+$Bx=
(21)
(22)
N_p-y 4z=R a & py*J2M ay y2
Z
x%[&exp[--&(2--?)]
- --
Thus particle pressure p is given by expression (17) with B, prescribed by equations (9). In accordance with the M CC1 approximation and since we are concerned with far-field estimates, p = - (B,/po) B,. Hence in the case of a plasma with a cold ion population and a warm electron fluid, for which p = pe, the particle pressure is almost exactly balanced by the magnetic pressure and the effect of the total pressure driving term on the electric potential is far less significant than those of the remaining driving terms, namely the two-dimensional (x, z)-Laplacian derivative of B,, and the “intrinsic” terms arising from the purely source current terms (see paper I, equation (43)). In Appendix B it is shown that mi exhibits a hybrid character which may be written in the form POUOJo =----
(2n)3
xexp[X-/&I +Gexp
-& [
x
exp
2+-
MxY z /-I
(
pc+Jzy]:
2
(23)
where p and r are given by p s Jy’+z’,
and i+ (= x+z/M) are the AlfvCn characteristics. Further, it is shown that ZP,,,,, cancel each other to this order of the approximation. Thus the major contribution to the modified potential @,_ POVoR I
B.
”
arises from the “nure” (or far-field) fast mode term Z,,,,,
486
with negligible
T. I. Woodward and J. F. McKenzie : Stationary MHD waves
effects from the slow mode (IB,,), Laplacian
(&,s) and +) and ym (&. and L,,,,,) terms, and the intrinsic potential Oint. Moreover considering a plasma with a warm electron fluid and a cold ion population so that Pefp”
_B,B,, PO
and noting that the B, perturbation is predominantly of fast mode character (see equations (9)), clearly Oj adopts largely a fast mode character.
3. Discussion
and conclusion
In the limit of an incompressible plasma, two wave modes, the fast and slow, appear. These modes result from the Hall current splitting of the corresponding single MHD mode which possesses the propagation characteristics of the MHD shear AlfvCn mode and propagates both the parallel current and the Laplacian derivative of the magnetic pressure term along the background magnetic field. The presence of source currents prevents the purely onedimensional propagation of the plasma flow and magnetic field perturbations. The slow mode experiences a resonance at the proton gyrofrequency, while the fast mode continues to propagate. Both modes are reasonably well field-guided, especially for very low frequencies (CC Q,), a fact which is consistent with them both deriving from the incompressible limit of the MHD modes. In a perpendicular flow, neither of the stationary wavecrest surfaces associated with the two modes ceases to exist at a critical flow Mach number (i.e. real flow characteristics exist for both modes for all Mach numbers, which in turn means that neither of the modes is evanescent), and the topology of these surfaces reflects the dispersive nature of the wave system. Both wavecrest surfaces are disconnected from the source and tend towards the Alfven lines in the far-field region. The fast mode surface is found in line with and downstream of the source, being shed from two points on either side of the source which are along the background magnetic field. It remains upstream of the Alfvtn lines in the near-field and approaches them at large distances from the source, at all times being well confined perpendicular to the plane containing the flow and the background magnetic field. On the other hand, the slow mode surface emanates from a single point directly downstream of the source, therefore being downstream of the Alfven lines close to the source tends towards them in the far-field. Furthermore, it is similarly well confined perpendicular to the plane containing the flow and the background magnetic field. Both the magnetic pressure and parallel current perturbations exhibit a hybrid character of the slow and fast modes, and, in the case of the magnetic pressure, a Laplacian decay. In very sub-Alfvenic flows both perturbations are predominantly of fast mode character, and are therefore found just upstream of the Alfvtn lines on the stationary fast mode wavecrest surface described above. Finally, the electric potential displays a mixed nature of fast and slow mode characters combined with the
characteristics of the Alfven-like potential operator and Laplacian decay, in addition to the properties of the intrinsic potential. In very sub-Alfvenic flows the major contribution to the electric potential comes from the fast mode with less dominant potential operator contributions (including the intrinsic potential), and negligible slow mode and Laplacian influences. This is in contrast to the cold case in which the fast mode is evanescent and gives way to the influence of the potential operator characteristics. Acknowledgements. J.F.McK. wishes to thank the FRD (South Africa) for their partial support. T. I. W. acknowledges financial support from the FRD (South Africa) in the form of a Special Merit Scholarship, and thanks the Max-Planck Institut fiir Aeronomie (Lindau) for support during the tenure of a stipendium, during which part of the work discussed in this article was completed. Further we wish to thank Dr G. M. Webb (Tucson) for invaluable discussions on the subject of this paper. Finally we gratefully acknowledge the effort and perseverance of Neal Powell for producing Figs 3 and 4.
References Cowling, T. G., Magnetohydrodynamics. New York, Interscience, 1957. Dungey, J. W., in Physics of Geomagnetic Phenomena (edited by S. Matsushita and W. H. Campbell), pp. 913-934. Academic Press, New York, 1968. Lighthill, M. J., Studies on magneto-hydrodynamic waves and other anisotropic wave motions. Phil. Trans. Roy. Sot. London, Ser A 252,391430, 1960. McKenzie, J. F., Woodward, T. I. and Inhester B., Magnetoacoustic and Alfvtn potentials for Stationary waves in a moving plasma. J. geophys. Res. 98, 9201-9213, 1993. Woodward, T. I., Linear and non-linear waves in space plasmas,
PhD Thesis. University 1993.
of Natal, Durban,
South Africa,
Woodward, T. I. and McKenzie, J. F., Stationary MHD structures. Planet. Space Sci. 41,217-228, 1993. Woodward, T. I. and McKenzie, J. F., Stationary MHD waves modified by Hall current coupling-I. Cold compressible flow. Planet. Space Sci. 42,463479, 1994a. Woodward, T. I. and McKenzie, J. F., Stationary MHD waves modified by Hall current coupling : I Cold compressible flow. M. P. Ae. I. Rep. No. MPAE-W-100-93-15, 1994b.
Appendices The methods of solution for the magnetic variables (BN and curl,,B) and for the electric potential ($) are sketched in these appendices. More details of the mathematical developments may be found in Woodward (1993), and Woodward and McKenzie (1994b).
A. The field variables In this appendix approximate expressions for B,, and curl,,B appropriate to the incompressible limit (equations (1) and (2), respectively) are developed. The k,-integrals are done using Cauchy’s residue theorem, noting that there are always four hyperbolic poles regardless of the Mach number M. This shows that the B,, and curl,,B perturbations are hybrid in nature. In particular curl,,B consists
487
T. I. Woodward and J. F. McKenzie : Stationary MHD waves of fast mode ((curlFB)F) and slow mode ((cur$B)s) contributions, while the B,, perturbation exhibits a similar mixed character with an additional influence arising from the Laplacian operator on the left hand side of the differential equation for B,, (equation (l)), giving rise to Laplacian terms associated with each of the two wave modes. The fast and slow mode contributions to B,, are denoted BliFand Blis, respectively. Thus we derive
N(2 +W)
k;,
where dk, dk,
x
(eik,(+,_,r
+e-ik,,+.-,z)ei(k,x+k~Y~
iv $3; F.S
(A3)
M(2 +M*)
lTm
(2+M2)
J
=
-
-
s
m e-~~+iW+k,y)
dk,
dk,,
2 f --a0 --m
--k%,+,-,
(A51 in which we define
In order grations in For very expression yields
N5
and hence the effect of the Laplacian operator on the two wave mode contributions is the same in this approximation. This integral has been done in Appendix A of paper I, leading to the solution for &z~~,~~ (equation (12)) in the main text.
~~~~‘~;,.-,~Ik:-M2(1+R2(k.~+k:+k:+,_,))k~,,_,}k,,]k~; 2&t%+
(Al)
3
In this approximation we have left out the k,,-dependence in order to simplify the subsequent algebra. This i&reasonable for values of k, sufficiently close to the blocking planes. Using the idea of stationary phase for this mode, we expect the solution to display singtdarities on the fast mode stationary wavecrest (Fig. 3). However, approximation (A7) has a different topology near lkxl cc M/R to that of the exact expression (equation (8) and Fig. l), and we re-emphasize the illustrative nature of this approach. Integrals IBxf and burl,,B)~ both arise from the fast mode and may be treated in a similar fashion. Using approximations (A7) and the fact that M << 1, the method of stationary phase may be used to obtain the approximate solutions (10) and (14) for ZBIF and (curl,@),, respectively. lSRrand &rlSj, are similar in structure since they correspond to the slow mode contributions and may be dealt with together. Using approximations (A6), noting MC 1, and again appealing to stationary phase arguments we estimate IBysand (curl,B), as in the text equations (11) and (15), respectively. Finally, an approximate solution to Iv~~r,~~(equation (A4)) may be obtained as follows. Using approximations (A@ &‘(F,S)
X
0 < k, < ;
N
2A,(k,2, - kz”_)= Ji3‘j27_4A,c
and
,
to proceed with the subsequent k,- and k,-intean analytic fashion we approximate k,, as follows. low AlfvCn Mach numbers M, expanding the for k,2_ (equation (8)) using the Binomial theorem
On the other hand, such an expansion fork:, yields an imaginary k,+ since the k,-range in which the real k,, surface exists (i.e. -M/R G k, < M/R) is not ‘seen’ by the process. However, for illustrative purposes it is worth approximating ki, near the ‘blocking’ planes (see the discussion on the Doppler shifted wave number surfaces below equation (8) of the text), which is equivalent to considering Ai -3 0, i.e.
B. The incompressible electric potential [Di In this appendix an approximate solution for the electric potential CD,associated with the incompressible modes is derived. “ftiis determined from equation (43) of paper I in which B,,is described by equation (1) and p by equation (16) of this paper. We begin by Fourier transforming these equations, substituting into the resulting algebraic equation describing the Fourier form of Qti (i.e. at,) from the corresponding equations for B, and p, solving for dD{k in terms of source terms, and finally inverting the consequent expression with a three-dimensional Fourier integral (exactly as for the cold case studied in paper I, except with Qj @ik)
replacing
%
(%k)).
The Fourier transforms of equations (16) and (1) for p and B, appropriately substituted into that of CD? and solved for tD& yields
kk ----_+%, Ymkk2
(Bl)
488
T. I. Woodward and J. F. McKenzie : Stationary MHD waves
where Yak is the Fourier transform of uOVQand is given by -[l-iMRk,,]k~)(k:+k_~+k:,+,_,);
Look E M2 ki - (I - iMRk,) k: ,
032)
the properties of which are discussed in section 3.2 of paper I. kf, are the zeros of _%‘zk , the Fourier form of the Doppler shifted incompressible wave operator (equation (3)) and are given by equation (8), in which Ai, B, and C, are defined in equations (7). Finally
NV2 = {(1+M2)k~+M2k;}k~~k,?; Dv~(F,s) =
(M2-R2k.:)(kS+-kt_)(k:+k_:+k;+,_,) x((l+M’)k:-iMRkyk:+M2kj);
N~~=k~:(l+~-ixk,,)[i{l-(l+R2[k;+k;
represents the Fourier form of the terms in parentheses. Note that pe is unspecified as yet and we shall invert equation (Bl) to obtain a solution for the difference Di--
>I;
POVoR
B,,
“’
and comment later on an expression for pc (see section 2.2 of the text). We begin by doing the Fourier inversion integrals with the k,integration, making use of Cauchy’s residue theorem to obtain PO&l JCI
Pe = -
+$(l-iMRk,,)])(l-iMRk,,)}k;
D ~p,,,-(~2-R2k:)(kf+-k,2_)
x $
>
(I -iMRk,)-ki+,_,
.
k,,i are defined in paper I (equations
(63) and (64)) and correspond to the real and imaginary parts of the pole contributed by yak. @,,t corresponds to the intrinsic potential referred to in the general development of paper I, a solution for which has been derived in the analysis of the cold compressible electric potential in Appendix B of that article. Thus it is seen that the incompressible electric potential Qi exhibits a hybrid nature, consisting of contributions from the fast and slow modes, the influence of the potential operator Yo, a limit- and mode-independent intrinsic potential together with a term arising from the electron pressure, namely
(2RS
in which dk, dk, x e-ze+i(k,x+kyy)
NV2
(J34)
&(F.S)
x
zPV2Girc
e-k,z
dk, dk, e-zm J-m
ei(krz+kxx+k~d;
ki
>
0 ;
&(k,x+k,y)
J--m
k,.
037)
x (1+M2)k:-iMRk,k:+M2k:’ Z
pp*
R E
-
-
M2
(~56)
ss m m
_a
dk
__m x
*+-_lkxl MR’
k2 &
e-k,z
ei(k,r+k,x+k,.y)
’ ik,
X
;
k,>O;
(B8)
with = NB ,,Fs)
D BI,Fsl=
In equations (B4)-(B8) ZBlllFS) correspond to the terms arising from the “pure” (or far-field) fast and slow mode B,, perturbations, while &~(~,s)represent the effects of the Laplacian decays associated with the fast and slow mode B,, perturbations. The terms ZY,,,SIare associated with the influence of the potential operator 4p, on the fast and slow mode B,, driving terms. Finally Zpvzand ZpPpg arise from a source term contributed by the particle pressure p. In order to proceed further in an analytic fashion we will approximate the integrands for small Mach numbers M. In such a limit k,.; may be approximated as in Appendix B of paper I (equations (B6) and (B7)), and k,2_ may be taken as in Appendix A (equation (A6)). For kt+ we shall use
except in integrals Ia , where k,, appears in the exponential and approximation (g9) would lead to strong attenuation. In ZBRF we shall take the expression for kj, used in Appendix A (equation (A7)), i.e. approximate kj, near the ‘blocking planes’ (see discussion on the Doppler shifted wave number surfaces for the incompressible modes which appears below equation (8) of the text). This approach should be considered illustrative of the nature of the method, rather than a complete solution.
(k~+k~+,_,)[i{k:-M2(1+R2(k:+kz
+k~+,-,))k~+,-,}k~+M3~,~k~+.-,~k~+k~+k~+,_,)l;
Integrals I,,‘
(MZ-R2k~)(k~+,-,-k~-,+~)(M2k~+,-~
Using the relevant approximations
we have
T. I. Woodward
x
{
and J. F. McKenzie
(e”e+‘k“+e-
: Stationary
MHD waves
(elllL-(kr)+e-ix~-_(k,))},
4+(U)+
have been written xrj+ (k,) in which +k (k,) = k,f.z&/ (x,/%). Noting that M << I, the method of stationary phase may be used to confirm the estimate for IBllsgiven in the text equation (19).
N
use of approximation
iR3
M3(2+M2)
(A7) for kj,
14
ss
m1 0X -e-:~COS 0 o k:
(k,x) cos (k,,y) dk, dk,
(B13)
we can write
ss * dk, -03
Integrals IPm(F,sjand IPJ,
m dk, 0
x {(eird+(k,)+e-‘xO+(k,t)+(elx~-(k,)+e-”~-(k~))} x(k:+ki+)
where
Thus, to this order of approximation, the Laplacian terms arising from the B,, driving terms, namely Iv+ and lvzs, are equal. Furthermore integrals I in expression (B13) have already been tackled in Appendix B of paper I. Hence the text equations (21) and (22) for ZV~(r,sjand ZPv2,respectively.
Integrals IB,,F
I+
@W
(~10)
where the k,-integral has been done and the k,-integration range is now from 0 to co. In (BlO) the phases of the integrands
Making
489
e’k,’
M*( - iR* k,: + MRk,.)kj+ (M*kt+-(I-iMRk,.)k:) i(ki--M*kI+)kf
+(k~+k~+kf+)(M*kf+-(I-iMRky)k:) in which k,, = JM(2+M2)/(2R’(M/R-kJ), phases of the integrands have been denoted
These terms all arise from the elliptic-hyperbolic pole contributed by the Fourier form of the electric potential operator YO,, and may therefore be treated together. Using approximations (A6) for k:_ , (B9) for kf, , and (B6) and (B7) of Appendix B of bapkr I for k,,i we obtain
’ P11) 11 and where the x4+ (k,) with
#& (k,) = k,k (z/x) JM(2+ M2)/(2R3(M/R/-k,)). The k,integral is estimated using a stationary phase development, leaving the k,.-integration which may be done using Cauchy’s residue theorem. Finally noting that M << 1 we confirm expression (20) of the text for I,,,,.
Integrals Iv~cF,Sj and Ipv2 These may be dealt with together since they all derive from a Laplacian pole in the k,-integration. Using approximations (A6) for kt_, and (B9) for k:, , it is seen that
i71 m )zf
m
ss_m -m
R* i zkf
-Rk, lk.4
e-k,z+@,z+k,x++) dk,dk,;k,
> 0;
(B14)
Thus the fast and slow mode Y@ contributions, namely ZP,, On the other and &,, cancel to this order of approximation. hand, IPy, is of order M-’ and the remaining double integral may be estimated using the method of stationary phase for both the remaining k,- and k,-integrals, thereby confirming equation (23) for ZPYpm.