MHD resonant flow instability in the magnetotail
R. Erdélyi and Y. Taroyan
Space and Atmosphere Research Center (SPARC), Department of Applied Mathematics, University of Sheffield,
The Hicks Building, Hounsfield Road, Sheffield S3 7RH, UK. (e-mail: Robertus@sheffield.ac.uk)
Abstract. Resonant flow instability (RFI) and Kelvin-Helmholtz instability (KHI) are investigated as possible wave generating mechanisms in the mantle-like boundary layer of the Earth’s magnetotail where all equilibrium quantities transition
continuously from magnetosheath values to values more characteristic of the tail lobe. It is shown that as in the case of a sharp
interface the KHI requiring high flow speeds in the magnetosheath is unlikely to be operative under typical conditions. RFI
which is physically distinct from KHI may appear at lower flow speeds due to the inhomogeneity of the mantle-like boundary
layer. It is shown that RFI can be important when the variation length-scale of the flow velocity is smaller than the variation
length-scales of other equilibrium quantities such as density and magnetic field strength. Interpretation in terms of the wave
energy flux is presented and the applicability to the magnetotail is discussed. The obtained results could explain the observed
low power of ULF waves in the tail lobes compared with other parts of the magnetosphere.
INTRODUCTION
teraction means that instead of wave energy dissipation
the wave subtracts energy from the system (where the
source of energy is the reservoir of plasma flow) resulting
in amplified wave amplitudes. Later this phenomenon
was explained in terms of negative energy waves (see,
e.g., [9, 10, 11]. All these works considered discontinuous flow profiles which facilitates the study significantly
but makes them less applicable to the magnetotail where
the flow speed is known to vary continuously.
The aim of the present paper is to find and characterize the main conditions under which RFI could be an
important wave generating mechanism in a tail-like geometry and to check if this mechanism is operative in a
realistic magnetotail model where all equilibrium quantities undergo continuous variation across a mantle-like
boundary layer. Effects introduced by these continuous
variations on the KHI are also examined.
The Kelvin-Helmholtz instability (KHI) on the magnetopause may be important for coupling solar wind momentum into the magnetosphere and is often invoked as a
source mechanism for various magnetospheric phenomena.
Most of the studies mainly concentrate on KHIs at a
tangential discontinuity representing the magnetopause.
However, observations (see, e.g., [1, 2, 3]) have identified
nonuniform boundary layers inside the magnetopause
extending from the subsolar point to the distant tail. Its
thickness tends to increase with distance from the subsolar point. The inclusion of such boundaries leads to
resonances which complicate the analysis. Few works
([4, 5, 6]) have investigated the effects introduced by
nonuniform boundary layers on the KHI at the flanks
of the magnetosphere. However, in all these works resonances were avoided by taking sufficiently high flow
speeds or by imposing additional simplifying constraints
on plasma parameters (e.g., no magnetic field in the magnetosheath and in the boundary layer).
The inclusion of a boundary layer, where one or several equilibrium quantities undergo continuous variation
across the inhomogeneity, not only could modify the
KHI but it could also lead to the appearance of a new type
of instability known as resonant flow instability (RFI)
which is physically distinct from the KHI. In [7, 8] the
authors investigated the effect of velocity shear on the
rate of resonant absorption in a steady plasma and found
negative absorption rate above a certain velocity threshold. Negative absorption rate in a driven wave-plasma in-
MODEL AND EIGENVALUE PROBLEM
The model shown in Fig. 1 represents the tail region
where the plasma flow in the magnetosheath is approximately parallel to the open field lines in the magnetosphere. All equilibrium quantities change only in the direction of the x-axis and divide the plasma into two semiinfinite homogeneous regions separated by a nonuniform layer of thickness L. The equilibrium quantities
in the homogeneous region x < 0 (x > L) are denoted
by subscript 1 (2), respectively. The magnetic field is
directed along the z -axis, i.e., 0 = (0 0B0 (x)). For
the sake of simplicity we assume that the plasma flow
B
CP679, Solar Wind Ten: Proceedings of the Tenth International Solar Wind Conference,
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355
u0 = (0 0u0(x)), the Alfvén speed
cA B0 =p0 0
p
and the cusp speed c T
cs cA = c2s + c2A are linear in
the inhomogeneous layer and constant elsewhere:
8 V
>< h;x x < 0
u0(x) = > V h 0 < x < h
(1)
: 0
x h
8c x 0
>< A1 T 1
cA T (x) = > cA1 T 1 + (cA2 T 2 ;cA1 T 1) Lx 0 < x < L
: cA2 T 2 x L:
(2)
FIGURE 1.
The model cartoon .
or
RESULTS
The magnetosheath is usually taken to be a high plasma, whereas the plasma in the tail lobes has a very
small (of the order of 10 ;4 ) and, therefore, it can be
considered as cold. For representative numerical evaluations we set = 5=3, 1 = 2:6 in the magnetosheath
and (a) 2 = 0:6 or (b) 2 = 0 (i.e., cold plasma approximation) in the tail lobes. Case (a) gives a more general
picture of existing wave modes and their interactions,
whereas case (b) is more realistic representing the cold
plasma in the tail lobes. The equation of magnetohydrostatic balance yields B02 =B01 = 1:5 or B02 =B01 = 1:89,
respectively. The ratio of the Alfvén speeds on both sides
is taken to be vA2 =vA1 = 4:5 and, according to Eq. (2),
it is an increasing function in the boundary layer. One
can show that in case (a) the cusp speed is an increasing function in the inhomogeneous layer, whereas in case
(b) it is a decreasing function. A propagation angle of
= arctan ky =kz = 50 with respect to the magnetic
field is chosen. Here ky = k sin , kz = k cos and k is
the wavenumber. The length, speed and magnetic field
strength are normalized with respect to 1=k , v A1 and
B01 , respectively.
Since the equilibrium quantities depend on the xcoordinate only, the perturbed quantities can be Fourier
analyzed with respect to y , z and t. We thus seek solutions of the form f (xk y kz ! ) exp i(ky y + kz z ; !t):
The linearized ideal MHD equations are then reduced to
a set of two coupled ordinary differential equations for
the normal component of the Lagrangian displacement
(x ) and the Eulerian perturbation of total pressure (P ).
In the homogeneous regions 1 and 2 the solutions can be
represented analytically. Inside the nonuniform boundary layer the set of equations has regular singularities at
the positions x = xA and x = xT where
! = kz u0 (x) cA(x)]
we numerically integrate the ideal MHD equations. If a
resonance is encountered during the calculations, then
connection formulae are applied. After having passed
through the resonant layer the integration of the ideal
equations is resumed. The same procedure is repeated
whenever any other resonance is encountered until the
final point x = L is reached. The application of the
continuity conditions for x and P at x = L yields the
dispersion relation.
Narrow shear flow layer (h=L = 0:1)
Let us now consider an inhomogeneous boundary
layer in the tail region across which all equilibrium quantities vary continuously from their characteristic values
in the magnetosheath to their counterparts in the tail
lobes. In general the inhomogeneity length-scales can
be different for the various physical quantities (e.g., velocity, density, etc.). First we consider the case when
the variation length-scale of the boundary layer plasma
flow (h) is small compared to the variation length-scales
of other equilibrium quantities (L), i.e., h=L 1. Further we assume that kL = 0:1 which corresponds to
the long wavelength approximation. When the magnetosheath flow, V , is high enough a given mode at a given
frequency can be in resonance with backward and forward continua simultaneously. The real and imaginary
! = kz u0 (x) cT (x)]
(3)
respectively. Connection formulae crossing the regular
singularities were originally derived in [12, 13, 14, 15]
for plasmas where the only dissipative process is resistivity.
The procedure for solving the eigenvalue problem for
MHD surface modes is a shooting method from x = 0 to
x = L. Starting with the analytical solutions for x < 0,
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its frequency consecutively enters the forward Doppler
shifted cusp and Alfvén continua (e.g., at V = 2:5 and
V = 2:9, correspondingly, in Fig. 2b). Note that for the
case shown in Figs. 2c, d the cusp RFI has very narrow
velocity ranges. Unlike the RFI the KHI is present when
the inhomogeneous boundary layer is reduced to a true
tangential discontinuity (L=0). When 2 = 0 the KHI1
disappears because of vanishing modes on side 2. Note
also that in Figs. 2b and d the magnetosheath flow thresholds for RFI overlap the KHI thresholds.
DISCUSSION
The existence of an extended geomagnetic tail has been
known for many decades. It was suggested relatively
early that the large size of this cavity might allow the
existence of MHD eigenmodes with frequencies of the
order of mHz and below, as an explanation for observations of such frequencies in magnetometer data. The
KHI at the magnetopause is a possible external mechanism for the generation of waves. However, as shown
by several authors, the flow speeds required for the KHI
are too high compared with the flow speeds usually observed in the magnetosheath. The introduction of the
mantle-like boundary layer, where equilibrium quantities undergo continuous variation, leads to the appearance of RFI. The assumption of a discontinuous flow
profile adopted in previous studies [10, 11] of RFI is
not valid for realistic magnetotail. According to [2] near
the tail boundary beyond about 100 R E GEOTAIL often encounters a plasma mantle-like boundary layer in
which the plasma flowing tailward transitions smoothly
from magnetosheath values of speed and density to much
smaller values, more characteristic of the tail lobe. We
have studied the possible resonant amplification of the
surface waves at this boundary layer for different representative variation profiles of the equilibrium quantities.
The surface waves can have several resonances with local
cusp or Alfvén waves throughout the entire inhomogeneous boundary layer. When due to an increased Doppler
shift the backward propagating primary mode p b changes
its direction of propagation, i.e., starts to propagate tailward in the rest frame of the magnetosphere, and its frequency enters the forward cusp and/or Alfvén continua
it can become resonantly unstable. However, this is not
necessarily true and numerical investigation is required,
since at the same time the pb mode is subject to backward
resonances.
Let us finally investigate how the threshold of RFI and
KHI depends on the thickness of the shear flow layer. The
dependence of critical velocities of KH and RF instabilities on the parameter h=L is plotted in Fig. 3. When
2 = 0 (in the limit of a cold plasma in the magneto-
FIGURE 2. The real and imaginary parts of oscillation frequencies as function of the magnetosheath flow, V , for a narrow shear flow layer in the long wavelength approximation
(h=L
: , kL
: ), (a,b) for 2
: and (c,d) for 2
.
=0 1
=0 1
=0 6
=0
parts of the mode frequencies as function of the flow
strength are plotted in Fig. 2. The first two plots (a, b)
show the eigenfrequencies when there is a non-zero kinetic pressure in the magnetosphere ( 2 = 0:6) and the
remaining two plots (c, d) correspond to 2 = 0 (cold
plasma in the magnetosphere). Dashed lines (long dashed
lines) indicate the lower and upper boundaries of forward (backward) Doppler shifted cusp and Alfvén continua. Forward propagating modes are denoted by dotted lines and backward propagating modes are denoted
by solid lines. The modes discussed in the previous subsection (primary and modes) are still present with the
only difference that the modes on side 2 are absent in
Figs. 2c, d, since 2 = 0. However, now both the forward
and backward propagating primary modes (p f and pb )
are damped in the absence of flow due to the mechanism
of resonant absorption (e.g., Fig. 2b). The damping rate
changes due to the Doppler shift. At a certain value of
magnetosheath flow (V = 1:3) the backward propagating primary mode p b reverses its direction of propagation. It starts to propagate in the Sun-Earth direction in
the rest frame of the magnetosphere remaining a backward propagating mode in the rest frame of the magnetosheath. The pb mode becomes resonantly unstable as
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unaffected by the change in continuous variation profiles
of equilibrium quantities within the boundary layer. If
MHD waves are generated by the mechanism of RFI,
then this should indicate a flow velocity variation lengthscale being smaller than the variation length-scales of
other equilibrium quantities, such as the density and
the magnetic field strength. Conversely, if the variation
length-scale of the flow velocity is small then the outer
boundary of the magnetotail should be subject to RFIs.
ACKNOWLEDGMENTS
RE acknowledges M. Kéray for patient encouragement.
RE also acknowledges NSF, Hungary (OTKA, Ref.
No. TO32462) and The Nuffield Foundation (Ref. No.
NAL/99-00) for financial support. YT is grateful to the
White Rose Consortium for financial support.
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FIGURE 3. Critical velocities for the RFI and KHI as a
function of h=L, where L
: .
=01
sphere) the KHI1 is absent since there are no modes
on side 2. The critical threshold velocities for both KHI1
and KHI2 are not affected very much by the introduction of the inhomogeneous boundary layer and the variation of the dimensionless parameter h=L, where L = 0:1.
The critical velocity for the KHI2 increases slightly with
increasing h. The plasma in the tail lobes is usually
of the order of 10 ;4 . In that case, as shown in Fig. 3b,
KHI1 is absent and the critical velocity required for the
KHI2 is approximately equal to c A1 + cA2 which is too
high for realistic flow seeds. The critical velocity for the
RFI is lower, however, it becomes higher when the parameter h=L is increased. In our study we have taken
h=L 1. Based on this study, it is not difficult to conclude that the critical velocities for RFIs become less important and may become even higher than those for the
KHIs when h=L 1. It is also important to note the following: in the present study we have taken a flow parallel
with the magnetic field. The antiparallel case could be included if negative velocities were considered. However,
taking into account symmetry considerations we can easily show that in both cases the resonantly unstable waves
will propagate tailward away from the Earth and that the
same critical velocities will be present with positive or
negative signs.
In conclusion the KHI threshold velocities are almost
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