ANGEOAnnales GeophysicaeANGEOAnn. Geophys.1432-0576Copernicus GmbHGöttingen, Germany10.5194/angeo-33-1011-2015Profile of a low-Mach-number shock in two-fluid plasma
theoryGedalinM.gedalin@bgu.ac.ilKushinskyY.BalikhinM.Department of Physics, Ben-Gurion University, Beer-Sheva, IsraelDepartment of Automatic Control and Systems
Engineering, University of Sheffield,
Sheffield, UKM. Gedalin (gedalin@bgu.ac.il)18August2015338101110179March201513July201510August2015This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://angeo.copernicus.org/articles/33/1011/2015/angeo-33-1011-2015.htmlThe full text article is available as a PDF file from https://angeo.copernicus.org/articles/33/1011/2015/angeo-33-1011-2015.pdf
Magnetic profiles of low-Mach-number collisionless shocks in space plasmas
are studied within the two-fluid plasma theory. Particular attention is given
to the upstream magnetic oscillations generated at the ramp. By including
weak resistive dissipation in the equations of motion for electrons and
protons, the dependence of the upstream wave train features on the ratio of
the dispersion length to the dissipative length is established
quantitatively. The dependence of the oscillation amplitude and spatial
damping scale on the shock normal angle θ is found.
Collisionless shocks represent one of the most ubiquitous strong nonlinear
phenomena in space plasmas. The main role of the collisionless shocks is to
convert the energy of the directed flow into heating for the bulk of the
plasma and into the acceleration of a fraction of the initial particle
distribution to high energies. While the acceleration mechanism works mainly
on large scales, the heating and the beginning of the acceleration occurs at
the shock transition itself. Since the shock width is on the order of or
smaller than the convective ion
gyroradius ,
the macroscopic electric and magnetic fields inside the transition layer
govern the charged-particle motion there. Therefore, knowledge of the shock
front structure is crucial for understanding the processes related to shocks.
Observations show that, with the increase in the Mach number, the shock
transition becomes progressively more
complicated .
In low-Mach-number shocks the transition region is known to be almost
monotonic or accompanied by magnetic oscillations decaying upstream and
downstream .
Low-Mach-number shocks are subcritical or marginally critical collisionless
shocks in magnetized
plasmas , where
the number of reflected ions is negligible. Although there is no
theoretically established upper limit for the Mach number for such shocks,
usually shocks with Alfvénic Mach numbers lower than 3 fall within this
category .
Most of observed interplanetary shocks in the heliosphere are low-Mach-number
shocks . Many
cosmological shocks, responsible for heating and acceleration, may be
low-Mach-number
shocks .
So far, there has been no satisfactory theoretical description of the shock
front even for low-Mach-number shocks. The magnetohydrodynamic (MHD)
approach, which treats the plasma as a single conductive fluid, does not
possess any characteristic spatial lengths and does not allow us to resolve
the transition layer. If no dissipation is included, MHD provides only the
relation between the upstream and downstream plasma parameters in the form of
Rankine–Hugoniot relations (RHs) .
Derivation of the latter requires the introduction of additional assumptions,
such as specifying the state equations for plasma species. RHs refer only to
the asymptotic values of the plasma and magnetic field parameters, while the
transition from one asymptotic value to the other remains unknown. Invoking
dissipation in MHD, it becomes possible to describe a
nonzero width transition. The typical way of including dissipation is to add
resistivity to Ohm's law and/or add ad hoc viscosity terms to the equation of
motion for the plasma. Yet, the width is determined by the dissipation alone
and should be significantly larger than the ion inertial length for MHD to be
valid . During the
last decades supercritical quasi-perpendicular
shocks and
quasi-parallel shocks have attracted more
attention because of their complicated pattern and the role of kinetic
effects. Yet a theoretical description of the magnetic profile of even a
low-Mach-number shock remains a challenge. Observations show that the ramp
width and the upstream oscillation wavelength is on the order of or less than
the ion inertial length even in low-Mach-number
shocks .
In this regime MHD breaks down and the two-fluid plasma theory (TFPT) has to
be used . TFPT describes each species
(electrons and protons) with the use of the continuity equation and Euler
equation with the pressure and Lorenz force included. The plasma equations
are coupled to the Maxwell equations via the charge and current densities
produced by both species together. The pressure pij is only the second
moment of the distribution function f(v), as follows: pij=m∫(vi-Vi)(vj-Vj)f(v)dv, where
Vi is the bulk velocity .
In the absence of collisions, the distribution function does not have to be
isotropic so that the pressure is, in general, a tensor. Yet in many cases
adopting the approximation of a scalar pressure pij=pδij,
δij being the Kronecker symbol, is widely accepted. TFPT equations
are completed with the state equations p(n) for each species (here n is
the number density). The polytropic law p∝nG is widely used as a
state equation. In addition to the pressure and Lorenz force, dissipative
terms which describe phenomenologically the momentum exchange between the
species phenomenologically can be added to the equations of
motion . In a collisionless plasma this
momentum exchange is usually related to anomalous
resistivity .
Standard dissipation-free polytropic TFPT allows solitons and periodic
nonlinear waves but does not allow solutions decaying to different
asymptotical
states .
In fact, TFPT with scalar ion pressure is valid only in the upstream region,
since ions begin to gyrate just behind the ramp, which results in
non-gyrotropic distributions and in a non-scalar
pressure .
Before entering the ramp, however, ions are expected to remain gyrotropic.
Numerical simulations of low-Mach-number, low-β (β≲1)
quasi-parallel shocks have shown that, apart
from the effects due to backstreaming ions, their structure is not very
different from those of quasi-perpendicular ones. The main additional feature
is the appearance of the upstream whistler wave train. It should be
mentioned, however, that the whistler wavelength rapidly decreases with the
increase in the shock angle and will become smaller than the simulation cell
size, unless the latter is on the order of the electron inertial length.
Ignoring the kinetic effects at the first stage, it is reasonable to apply
TFPT with a simple polytropic pressure law in the upstream region, up to the
plasma entry to the ramp, for both quasi-perpendicular and quasi-parallel
geometries. In the present study we restrict ourselves to the upstream region
of the shock. A shock-like profile cannot form without some
dissipation . It is expected that such
dissipation can be produced by
microinstabilities . In the upstream
region one can expect to have weak anomalous resistivity.
Including resistive dissipation in a scalar polytropic TFPT, we describe the
profiles of low-Mach-number shocks for a wide range of angles between the
shock normal and the upstream magnetic field. Our approach is similar to that
of , who treated a shock ramp as a
discontinuity constantly generating large-amplitude waves propagating
upstream. We assume that a finite width ramp is established which acts as a
large-amplitude perturbation for the upstream region. Time stationary
magnetic oscillations would damp with the distance from the ramp due to the
dissipation.
Basic equations
We consider a one-dimensional stationary plasma within TFPT, where all
variables depend only on the coordinate x along the shock normal. With the
increase in the Mach number the collisionless shock front is known to develop
deviations from planarity and become nonstationary see,
e.g.,. Yet observations show that, in
low-Mach-number shocks, these deviations are
weak .
Electrons are treated as a massless fluid. The ion and electron kinetic
pressures are assumed to be scalar and polytropic state equations are used.
The resistive dissipation is included as a friction term between the two
fluids. Quasi-neutrality, ni=ne=n, is assumed, which is natural for the
spatial and temporal scales under consideration. With these assumptions the
equations take the following form :
miv∂xvi=eE+ecvi×B-x^∂xpi-ν(vi-ve),0=-eE-ecve×B-x^∂xpe-ν(ve-vi),x^×∂xB=4πcne(vi-ve),nv=J=const,E⟂=const,Bx=const,
where x^ is the unity vector along the x axis, v=v⋅x^, and ⟂ denotes components perpendicular to
x^. Here E and B are the total electric
and magnetic field, respectively, vs is the bulk velocity
of the species s=e,i, ns is the number density of the species
s, and ps is the pressure of the species s. The momentum
exchange (friction between the electrons and protons) is described by the
term with ν. We have also taken into account that ni=ne means
vi=ve. Summing the x components of Eqs. () and
() and taking into account Eq. (), one gets see,
e.g.,nmiv2+B28π+p=const,p=pi+pe.
For the perpendicular components one has miv∂xvi⟂=e[E⟂+vcx^×B⟂]+eBxcvi⟂×x^-ν(vi⟂-ve⟂),0=-e[E⟂+vcx^×B⟂]-eBxcve⟂×x^+ν(vi⟂-ve⟂),x^×∂xB⟂=4πcne(vi⟂-ve⟂).
Here subscript ⟂ denotes vectors perpendicular to x^.
Summing up Eqs. () and () and using
Eq. (), one gets
vi⟂=BxB⟂4πminv+V,ve⟂=vi⟂-c4πnex^×∂xB⟂,
where V is a constant vector which is determined by the conditions at
a particular reference point. There is significant freedom in choosing the
reference point and the conditions there. This freedom will be used later,
after deriving the equations in the general form. Respectively, V is
specified below in Eq. (). Substituting all this into
Eq. (), one arrives at the following equation for the magnetic
field cf.:
Bx4πn∂xBy-cν4πne∂xBz=eFy-evcBz1-Bx24πminv2,Bx4πn∂xBz+cν4πne∂xBy=eFz+evcBy1-Bx24πminv2,F=E⟂+BxcV×x^.
These equations should be completed with nv=J=const and
nmiv2+By2+Bz28π+p(n)=Q=const.
The stationary points of Eqs. (–) are the points where
∂xBy=∂xBz=0, that is,
0=eFy-ev0cBz01-Bx24πmin0v02,0=eFz+ev0cBy01-Bx24πmin0v02,
where subscript 0 denotes values at (one of) the stationary point(s). Using
the freedom in choosing the coordinate axes, we put By0=0 and introduce
the angle θ between the magnetic field at the stationary point and the
normal, so that Bz0=B0sinθ and Bx=B0cosθ. Respectively,
one has Fz=0 and
Fy=v0cB0sinθ1-B02cos2θ4πmin0v02.
For the ion velocities at the stationary point we find
viy0=Vy,vz0=B0sinθcosθ4πmin0v0+Vz.
Choosing the reference frame in which vi⟂0=0 (this is the
well-known normal incidence frame) sets
Vy=0,Vz=-B0sinθcosθ4πmin0v0.
Using Eq. () one obtains Ez=0 and
Ey=v0cB0sinθ.
It is natural to define the Alfvén velocity and the Alfvénic Mach
number at the stationary point as
vA2=B024πmin0,M=v0vA.
In order to achieve a better physical understanding, we introduce the
following normalized variables:
b=B⟂B0sinθ,V=vv0,N=nn0=1V.
Then the derived equations take the following shape:
lw∂xby-ld∂xbz=N(1-s)-(1-Ns)bz,lw∂xbz+ld∂xby=(1-Ns)by,1N+yb2+xf(N)=1+y+x.
Here
s=cos2θM2,y=sin2θ2M2,x=β2M2,β=8πp0B02,f(N)=p(N)p0,
and the dispersion and dissipative length are
lw=cB0cosθ4πn0ev0=ccosθMωpi,ld=c2ν4πn0e2v0=c2η4πv0,
where ωpi2=4πn0e2/mi and η is the resistivity. The
length lw is easily recognizable as the inverse wave number of a
low-frequency whistler wave standing in the frame which is moving with the velocity v0: v0=ω/k=kccosθB0/4πn0e. Hereafter, lw is referred
to as “whistler wavelength”. It is worth mentioning that ld may
depend on θ and M only implicitly, via a possible dependence on
ν. At the same time, lw
rapidly drops with the increase in M and/or θ.
The nature of the stationary point is determined by the linearized equations
lw∂xδby-ld∂xδbz=δN-(1-s)δbz,lw∂xδbz+ld∂xδby=(1-s)δby,-δN+2yδbz+GxδN=0,
where G=(1/p0)(dp/dN)N=1 so that vs2=GβvA2/2 is the local sound speed at the reference point. Assuming δby,δbz,δN,δV∝exp(kx), one has
k2(lw2+ld2)-C1kld+C2=0,k1,2=C1ld±C12ld2-4C2(lw2+ld2)2(lw2+ld2),C1=(v2-vs2)(v2-vI2)+(v2-vF2)(v2-vSL2)v2(v2-vs2),C2=(v2-vSL2)(v2-vI2)(v2-vF2)v4(v2-vs2),
where vI, vSL, and vF are the usual
velocities of intermediate, slow, and fast MHD waves,
respectively , defined locally at the
stationary point:
vI2=vA2cos2θ,vF2=12vA2+vs2+(vA2+vs2)2-4vA2vs2cos2θ,vSL2=12vA2+vs2-(vA2+vs2)2-4vA2vs2cos2θ.
Stationary points of the autonomous system of two first-order ordinary
differential equations are classified according to the exponents k1 and
k2see, e.g.,. If C2<0, then both k1 and k2
are real and k1k2<0, that is, the stationary point is a saddle point,
which means that one special solution ends at the stationary point and
another special solution starts at this point, while all other solutions do
not arrive at this point at all. If C2>0 and
D=C12ld2-4C2(lw2+ld2)>0, then the
stationary point is a node, which means that either all solutions end there
or start there and the magnetic field vector rotates by a finite angle only.
If C2>0 and D<0, the stationary point is a focus, which is similar to
the node, with the only difference being that the angle of the magnetic field
rotation is infinite.
In order to have a shock solution with different upstream (x→-∞) and downstream (x→∞) asymptotic states, we need
the magnetic perturbation, caused by the ramp, to decay toward ±∞.
Thus, Rek1>0 and Rek2>0 are required at the
upstream stationary point, and Rek1<0 and Rek2<0 are required at the downstream stationary point. For a fast
magnetosonic shock in the upstream asymptotic state at x→-∞, one has v>vF, which means that both C1 and C2 are
positive. This point is a node for strong dissipation, D>0, and a focus for
weak dissipation, D<0. In both cases Rek1>0 and Rek2>0 are ensured by the presence of the dissipation. In the downstream
asymptotic state (x→∞), the
evolutionarity conditions require that the velocity should be in the range
vI<v<vF. If vs<vI, the asymptotic point is
a saddle and Rek1Rek2<0. If vs>vI and
v<vs, the asymptotic point is a node. Since in this case C1>0, the
exponents Rek1>0 and Rek2>0 and magnetic
perturbations do not damp toward x→∞. Thus, a resistive
dissipation does not allow a fast shock solution for any Mach number.
Upstream region
As mentioned above, one cannot really expect that the simple scalar pressure
TFPT be applicable behind the ramp. Indeed, the gyration of the ion
distribution as a whole breaks down these approximations. The objectives of
the present paper do not include the analysis of a non-gyrotropic pressure.
However, the region upstream of the ramp should be well approximated by the
approach adopted for sufficiently low Mach numbers when ion reflection in
negligible. Moreover, we have shown that in an asymptotically super-fast
magnetosonic flow, magnetic perturbation damps toward x→-∞.
From
dNdb2=yN2GxNG+1-1,
the maximum achievable density is Nc=(Gx)-1/(G+1), which corresponds to the maximum possible
bc2=1y1+y+x-G+1GNc,
with the maximum magnetic compression of Bt/B0=cos2θ+bc2sin2θ.
We could not solve Eqs. (–) analytically. For
numerical visualization below we have chosen the fast magnetosonic Mach
number to be fixed at MF=2, while θ and β are
varied. A polytropic pressure is chosen, p(N)=NG, with G=5/3 so that
vs2/vA2=Gβ/2.
The Alfvénic Mach number is M=MFvF/vA and also varies with θ and β. The ratio of the dissipation and dispersion lengths in terms of resistivity is
ϵ=ldlw=ωpiηcosθωpiΩi=ωpiηcosθcvA=Mωpiηcosθcv0.
For the solar wind conditions, v0/c∼10-3, one would have
ηωpi∼10-3ϵ(cosθ/M). For the visualization
below we have chosen ϵ=0.05/cosθ, which gives an approximate idea of the
dependence of ϵ on the
shock angle for constant resistivity. For the chosen parameters the latter is
in the range η∼(10-5-10-4)ωpi-1.
Figure compares the profiles obtained for the following four
cases: 1) θ=70∘, β=0.2 (top left); 2)
θ=30∘, β=0.2 (top right); 3) θ=70∘,
β=1.2 (bottom left); and 4) θ=30∘, β=1.2 (bottom
right). Coordinate x is measured in the whistler wavelengths
lw. The calculated Alfvénic Mach numbers are given in the
figure caption. The profiles are rather similar. Longer wave trains for
smaller θ are due to lower ϵ.
The total magnetic field (solid line) and the two perpendicular
components, Bz/B0 (dotted) and By/B0 (dash-dotted) for the four
cases: 1) θ=70∘, β=0.2, M=2.14 (top left); 2)
θ=30∘, β=0.2, M=2.05 (top right);
3) θ=70∘, β=1.2, M=2.79 (bottom left); and 4) θ=30∘, β=1.2, M=2.14
(bottom right). In all cases the fast Mach number MF=2.
Total magnetic field for the four cases: (1) θ=70∘,
β=0.2 (solid line); (2) θ=30∘, β=0.2 (dash-dotted
line);
(3) θ=70∘, β=1.2 (dotted line); and (4) θ=30∘, β=1.2 (dashed line).
In the quasi-perpendicular case, θ=70∘, one of the
perpendicular components of the magnetic field, By, always remains
substantially smaller than the other, Bz, so that the polarization of the
wave train is close to linear. In the quasi-parallel case,
θ=30∘, the two components are comparable, and the polarization
is elliptical, approaching a circular polarization. High-β cases should
be treated with caution since ion reflection may be noticeable even at low
Mach numbers. Figure shows the magnetic field profiles (total
magnetic field) for all four cases. The wavelengths for θ=70∘
are smaller than the wavelengths for θ=30∘. For
θ=30∘ the wavelength is smaller for lower β.
Comparison of the profiles for θ=70∘ (solid line) and θ=30∘ (dash-dotted line). Here the coordinate is measured in ion inertial lengths.
In the above figures the coordinate is measured in whistler wavelengths
lw=ccosθ/Mωpi. When keeping MF and
β constant, the whistler wavelength rapidly decreases with the increase
in θ. Our analysis is done using the massless electron approximation
and can be valid only if lw≫c/ωpe, that is, for
cosθ/M≫(me/mi)1/2. Figure compares the
profiles for θ=70∘ and θ=30∘ when the
coordinate is measured in ion inertial lengths c/ωpi.
The representative values of θ and β were chosen to illustrate
the differences. Figures and show the profiles with
additional sets of parameters.
The total magnetic field (solid line) and the two perpendicular
components, Bz/B0 (dotted) and By/B0 (dash-dotted). Top left:
θ=15∘; M=2.01. Top right: θ=45∘; M=2.09.
Bottom left: θ=60∘; M=2.13. Bottom right:
θ=85∘; M=2.16. Other parameters: β=0.2;
MF=2.
Figure shows the magnetic profiles for MF=2 and
β=0.2 and four values of the angle between the shock normal and the
upstream magnetic field (θ=15∘,45∘,60∘,85∘). The number of oscillations drops rapidly with the increase in
the angle. For the nearly perpendicular shock, θ=85∘, the
upstream wave train reduces to a magnetic dip just ahead of the ramp. In
Fig. the upstream plasma is hotter, β=0.5, while
MF and θ are the same as in Fig. . There is
little difference between the behavior of the magnetic profiles for the same
MF and θ and different β. However, with the increase
in β for given MF and θ, the Alfvénic Mach
number is higher. The whistler wavelength is smaller. Therefore, when
measured in the ion inertial lengths, the profiles with higher β will
look “tighter” (compare with Fig. ).
The total magnetic field (solid line) and the two perpendicular
components, Bz/B0 (dotted) and By/B0 (dash-dotted). Top left:
θ=15∘; M=2.04. Top right: θ=45∘; M=2.24.
Bottom left: θ=60∘; M=2.31. Bottom right:
θ=85∘; M=2.38. Other parameters: β=0.5;
MF=2.
Conclusions
Simple TFPT with polytropic pressure and resistive dissipation is able to
reproduce quantitatively the basic features of the upstream side of a
low-Mach-number collisionless shock in a wide range of θ and β.
In the dimensionless variables, the profiles are rather similar for constant
MF, with only a weak dependence on θ and β. The
length of the whistler wave train depends on the ratio between the
dissipation and whistler wavelengths and rapidly decreases with the increase
in this ratio. This similarity should not be surprising since the kinetic
effects are negligible for low Mach numbers in the upstream region. Unless
θ is too small, the species motion is governed by the magnetic field.
The approximation of massless electrons is valid if cosθ/M≫(me/mi)1/2. Thus, in a wide range of angles the upstream parts of
low-Mach-number shocks should be similar. Observations would not show this
similarity as long as the measurements are normalized not with the whistler
wavelength but with the ion inertial length, as is widely accepted. The
whistler wavelength rapidly decreases with the increase in the angle, while
the ratio of the dissipation-to-whistler wavelength increases. In the
quasi-perpendicular case the shock front should be much narrower and exhibit
fewer oscillations.
Acknowledgements
M. Gedalin was partly supported by the Israel Science Foundation (grant No.
368/14). The topical editor G. Balasis
thanks two anonymous referees for help in evaluating this paper.
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