A&A 388, 667–675 (2002)DOI: 10.1051/0004-6361:20020128c© ESO 2002

Astronomy&

Astrophysics

Dynamics of pick-up ions in collisionless velocity shears

E. Perez-Tijerina1, M. Reyes-Ruiz2, H. Perez-de-Tejada3, and H. Durand-Manterola4

1 Posgrado en Fisica Espacial, Instituto de Geofisica, UNAM Km. 103 Carr. Tijuana-Ensenada, 22860 Ensenada,B.C. Mexico

2 Instituto de Astronomıa, UNAM, Km. 103 Carr. Tijuana-Ensenada, 22860 Ensenada, B.C. Mexico3 Instituto de Geofısica, UNAM, Ensenada, B.C. Mexico4 Instituto de Geofısica, UNAM, Mexico D.F., Mexico

Received 20 March 2001 / Accepted 15 January 2002

Abstract. We study the motion of charged particles in large-scale velocity shears that are produced in the in-teraction of magnetized plasma winds and plasma obstacles. The purpose of the analysis is to account for theobservation of strongly energetic contaminant ions in the region of interaction of the solar wind with plane-tary/cometary non-magnetic ionospheric obstacles (Venus, Mars, comets). The convective electric field set upby the streaming plasma is incorporated to the equation of motion of ions born in a velocity shear. Neglectingcollisions and the back reaction of the contaminant particles on the wind, the trajectories of the particles arecomputed as a function of the shear properties as well as of the mass of the ions and the magnetic field con-figuration. For a linear dependence of the wind velocity across the shear, the problem is solved analytically andwe find that the particle velocity can have either a purely oscillatory behavior or grow exponentially with timedepending on the value of a dimensionless parameter proportional to the product of the velocity gradient and thecyclotron frequency of the ion trajectories. In the latter case a strong acceleration of the contaminant ions can beachieved. Adopting magnetic field and flow properties appropriate for cometary and planetary environments, weexplore the potential importance of the mechanism discussed to explain the presence of superthermal ions andfilamentary structures in such regions.

Key words. acceleration of particles – plasmas – comets – planets and satelites

1. Introduction

It has been suggested that the tenuous solar wind thatflows around non-magnetic ionospheric obstacles givesrise to boundary layers over their magnetic polar re-gions (these are defined by the orientation of the in-terplanetary magnetic field), similar to those resultingfrom viscous flows that stream around spherical objects(Perez-de-Tejada 1986). Evidence for this view has beenprovided by multiple satellite observations in Venus, Marsand several comets. However, the collisionless nature of thesolar wind has led to issues related to the physical sourcesof the anomalous viscosity that are responsible for mo-mentum transport across the boundary layer. Today, theobservational evidence that is available has led to sug-gestions on microscopic processes (e.g. wave-particle in-teractions) that could account for momentum transfer inthe collisionless solar wind (Shapiro et al. 1995). As a re-sult, it is important to carry out a thorough investigationof the likely consequences of the existence of boundary

Send offprint requests to: M. Reyes-Ruiz,e-mail: maurey@astrosen.unam.mx

layers and verify the consistency of momentum transferwith regard to other observational data.

In the present study we will analyze the motion ofindividual ions born in the vicinity of ionospheric ob-stacles (mass loading) subject to the convective electricfield set up by the solar wind and in the presence of avelocity shear. This problem was initially examined byParker (1958) for the case in which the wind velocity, themagnetic field, and the velocity gradient are transverseto each other. The pick-up ions execute cycloidal mo-tion at the local velocity of the flow within the boundarylayer. A later study was conducted by Perez-de-Tejada &Durand-Manterola (1996, hereafter referred to as PD96)who revised this problem and calculated the ion densityexpected from the particle trajectories. From mass fluxconservation these authors derived a gradient for the pick-up ion density that is opposed to the velocity gradientand also derived values that are consistent with densitymeasurements made in the vicinity of comet Giacobinni-Zinner (Bame et al. 1986). Our main interest will nowbe to extend the analysis of ion trajectories to arbitraryorientations between the magnetic field and the velocity

Article published by EDP Sciences and available at http://www.aanda.org or http://dx.doi.org/10.1051/0004-6361:20020128

668 E. Perez-Tijerina et al.: Pickup-ions in velocity shears

Fig. 1. Basic geometry of the problem being solved. The mag-netic field, B, has an arbitrary orientation with the respect tothe flow U which varies only in the z-direction.

gradient and derive an analytical solution for the constantvelocity gradient case as well as numerical solutions suit-able for classical boundary layers. While a linear profile forthe flow velocity is supported by observations in comets(PD96) it is important to study different magnetic fieldconfigurations as well as to examine the sensitivity of theresults to other forms of the velocity gradient using the-oretically justified profiles. In light of our results we willdiscuss novel potential applications of these ideas to ex-plain the acceleration of charged particles in space andastrophysical phenomena.

The paper is organized as follows: Sect. 2 we presentthe formulation of the problem to be solved. Section 3deals with the analytic solution of the equation of motionfor charged particles in linear velocity shears. In Sect. 4we discuss the results obtained focusing on the processof particle acceleration. In Sect. 5 we show that our re-sults may have potential applications in the study of theinteraction of the solar wind with unmagnetized bodies,leaving a detailed, quantitative analysis for future contri-butions. And finally, in Sect. 6 we summarize our resultsand conclusions.

2. Formulation of the problem

We consider the case in which a uniform magnetic field,frozen to a streaming plasma, is oriented arbitrarily withrespect to a one-dimensional velocity gradient along aheight coordinate over an immobile obstacle. The basicconfiguration, with a linear velocity profile, is illustratedin Fig. 1.

Using Cartesian coordinates the flow streams along they axis and the velocity gradient is directed in the z axis.The streaming plasma flows with velocity U = [0, U(z), 0]carrying a “frozen-in” magnetic field B = [Bx, By, Bz].Our reference frame is considered at rest in the obstacleso that a convective electric field E = −(U ×B)/c, is feltthere. The equation of motion for an individual particleof mass m, charge q and position x, is:

md2x

dt2= qE +

q

c

dxdt×B (1)

whose components (with x = [x, y, z]) are:

d2x

dt2= −ΩzU(z) + Ωz

dydt− Ωy

dzdt

(2)

d2y

dt2= Ωx

dzdt− Ωz

dxdt

(3)

d2z

dt2= ΩxU(z) + Ωy

dxdt− Ωx

dydt

(4)

where,

Ωx =qBxmc

, Ωy =qBymc

and Ωz =qBzmc

(5)

and

Ω =√

Ωx2 + Ωy2 + Ωz2. (6)

While Ω is always positive Ωx, Ωy and Ωz can be negativedepending on the sign of the charge and the respectivefield component.

We present results for the case in which the velocitydepends linearly with distance above the obstacle and de-rive particle trajectories using an analytical solution. Inthe discussion section we will later analyze the problemwith a velocity gradient as that resulting from a standardboundary layer in which the velocity increases as z awayfrom the obstacle.

The set of equations can be easily solved for a linear ve-locity gradient U(z) = Kz by taking their Laplace trans-form for a given set of initial conditions. In the following,we will assume that the particles start their motion fromrest, at some point [0, 0, z0] on the z-axis, i.e.

x(t = 0) = y(t = 0) = 0

x(t = 0) = y(t = 0) = z(t = 0) = 0.

These initial conditions are based on the physical prob-lems we intend to study since the pick-up ions arise froma relatively small region, of the order of the size of theobstacle, in comparison with the scale of the flow. Thiscontext also justifies our assumption that ions are bornat rest with respect to the obstacle since typical thermalvelocities in planetary/cometary atmospheres are muchsmaller than the solar wind velocity.

E. Perez-Tijerina et al.: Pickup-ions in velocity shears 669

3. Results

The complete calculation of the solution of Eqs. (2)–(4) isstraightforward but tedious. An extensive discussion of themethod employed can be found in Perez-Tijerina (1999)and leads to:

x(t) = X1(er1t − 1) +X2(er2t − 1) +X3(er3t − 1) (7)

y(t) = Y1(er1t − 1) + Y2(er2t − 1) + Y3(er3t − 1) (8)

z(t) = Z1er1t + Z2er2t + Z3er3t (9)

where r1, r2, r3 are the roots of equation

r3 + (Ω2 −KΩx)r +KΩyΩz = 0 (10)

and the coefficients Xi, Yi and Zi for i = 1, 2, 3 aregiven by

Xi =[− K(ΩxΩy + riΩz)z0

ri(3ri2 + Ω2 −KΩx)

](11)

Yi =[

K(Ω2x + Ω2

z)z0

ri(3ri2 + Ω2 −KΩx)

](12)

Zi =[

(r2i + Ω2)z0

3ri2 + Ω2 −KΩx

]· (13)

The roots of the characteristic Eq. (10) are:

r1 = −12

(s1 + s2) + i

√3

2(s1 − s2), (14)

r2 = −12

(s1 + s2)− i√

32

(s1 − s2), (15)

and

r3 = s1 + s2, (16)

where:

s1 =

[−KΩyΩz

2+

√ω6

27+

(KΩyΩz)2

4

] 13

, (17)

s2 =

[−KΩyΩz

2−√ω6

27+

(KΩyΩz)2

4

] 13

, (18)

and,

ω2 = Ω2

(1− KΩx

Ω2

)· (19)

According to the theory of cubic equations r3 will be real,and r1 will be the conjugate of r2, if:

D =[

Ω2

27

(1− KΩx

Ω2

)]3

+(

12KΩyΩz

)2

> 0. (20)

Under different conditions the three roots can be realor imaginary with equal or unequal values depending onwhether the value of the discriminant is smaller, greateror equal to zero.

Hence, allthough the solutions (7)–(9) looks rathercomplicated, the possible values of r1, r2 and r3 clearly in-dicate the existence of three qualitatively different types ofsolutions; purely oscillatory, purely exponential (growingor decaying) or a mixture of both. As discussed in Perez-Tijerina et al. (2000) a constant velocity drift can exist inoscillatory solutions which will be considered below.

The emergence of either type of solution depends onthe velocity gradient K, the magnetic field componentsthrough Ωx, Ωy and Ωz and the particle properties, namelyits mass and electrical charge. These characteristics of theflow and particle determine which of the forces presentin Eqs. (2)–(4) dominates. For example, the particle willexecute mainly gyromotion if the terms involving first or-der time derivatives of position, on the right hand sideof Eqs. (2)–(4), greatly dominate over the terms leadingto exponential acceleration (from the height dependentelectric force) i.e. those involving U(z). As a critical ra-tio between these two is overcome, or if the electric forcehas no component along the magnetic field, qualitativelydifferent solutions are obtained.

We thus divide the presentation of our results into twomain cases, D ≥ 0 and D < 0.

3.1. Results for D ≥ 0

A positive discriminant (D > 0) implies that s1 and s2

are real. In this case the solution can be written as:

x(t) = (2Ax cos bt−2Bx sin bt+Cxe−3at)eat−2Ax−Cx(21)

y(t) = (2Ay cos bt−2By sin bt+Cye−3at)eat−2Ay−Cy(22)

z(t) = (2Az cos bt− 2Bz sin bt+ Cze−3at)eat (23)

where,

Ax =X1 +X2

2, (24)

Bx =X1 −X2

2i, (25)

and,

Cx = X3, (26)

with similar expressions for Ay, By, Cy and Az, Bz, Cz.The “frequencies” a and b result from writting the rootsof Eq. (10) as:

r1 = a+ ib, (27)

r2 = a− ib (28)

and

r3 = −2a. (29)

Different solutions are obtained for the following cases:when a = 0 and b 6= 0 we have purely oscillatory solu-tions. When b = 0 and a 6= 0 they are purely exponential.And solutions consisting of growing or damped oscillationscorrespond to non-zero values for a and b. We will analyzeparticular solutions for each of these cases separately.

670 E. Perez-Tijerina et al.: Pickup-ions in velocity shears

Fig. 2. Trajectories of oscillatory type for particles originatingat 3 different initial positions for the case in which D > 0 withΩx = Ωy and Ωz = 0.

3.1.1. Oscillatory solutions

The growth rate a in Eqs. (21)–(23) will vanish when s1 =−s2. This can only be achieved if

KΩyΩz = 0. (30)

For a wind flow with velocity gradient this condition cor-responds to a magnetic configuration in which the field iscontained in the xz or xy planes. In either case, as wellas when the field is along the x axis, the solution will bepurely oscillatory.

From Eqs. (14)–(16) and (27), (29) the frequency ofthe oscillatory motions is: b = ω.

The coefficients in the solution (21)–(23) are easily ob-tained once a field configuration is specified. The calcu-lation of the terms corresponding to the third root areobtained as a limit when a→ 0.

Particle trajectories have been derived and analyzedfor magnetic configurations with B = (Bx, 0, 0) andB = (Bx, By, 0) by Perez-de-Tejada & Durand-Manterola(1996) and Perez-Tijerina et al. (2000) respectively. Ouranalysis for a general magnetic field orientation allows usto easily derive those results. Solutions in which the par-ticles move on cycloidal trajectories drift in a directionparallel to B × ∇U are exemplified in Fig. 2. We referthe reader to the work of Perez-Tijerina et al. (2000), andreferences therein for a careful analysis of these solutions.Here we focus, instead, on the novel results of our analysis.

It is worth mentioning that in the work of Perez-Tijerina et al. (1999) a parameter ε = KΩx/Ω2 is usedto characterize the solutions in a manner analogous to thediscriminant D. When KΩyΩz = 0 as is the case for theconfigurations studied by Perez-Tijerina et al. (2000), thecondition ε < 1 is equal to our inequality (20). The con-dition for ε, equivalent to D > 0, in the case of generalorientation of the magnetic field is

ε < 1 + 27[KΩyΩz

2Ω3

]2/3

(31)

Fig. 3. Trajectories of oscillatory type for particles originatingat 3 different initial positions for the case in which D > 0 withΩx = Ωz and Ωy = 0.

which reduces to ε < 1 in the configurations previouslystudied.

Similar cycloidal trajectories are found in the case ofmagnetic field contained in the xz plane. Solutions (21)–(23) in this case simplify to:

x(t) =KΩzzo

ω2(cosωt− 1), (32)

y(t) =KΩ2zo

ω2t − KΩ2zo

ω3sinωt, (33)

and

z(t) =Ω2zo

ω2− KΩxzo

ω2cosωt. (34)

Figure 3 shows a typical trajectory for a magnetic fieldB = (Bx, 0, Bz) and the additional restriction D > 0 orε < 1. The exact value of ε is 0.05.

3.1.2. Exponential solutions

Solutions purely exponential arise only when D = 0. Inthis case s1 = s2, b = 0 and a is real. The parameter εmust be:

ε =KΩxΩ2

= 1 + 27[KΩyΩz

2Ω3

]2/3

, (35)

which is equal to one only when KΩyΩz = 0. In suchcase however, both a and b are zero and one has a trivialsolution.

When all field components are non-zero, satisfyingEq. (35), solutions as those shown in Fig. 4 are obtained.The form of such solutions is in general:

x(t) = 2Ax(eat − 1) + Cx(e−2at − 1), (36)

y(t) = 2Ay(eat − 1) + Cy(e−2at − 1), (37)

E. Perez-Tijerina et al.: Pickup-ions in velocity shears 671

0.00.1

0.20.3

0.4-0.4

-0.3-0.2

-0.10.00.0

0.2

0.4

0.6

0.8

1.0

Z

X

Y

Fig. 4. Exponential trajectories of particles originating at 3different initial positions for the case in which D = 0 andΩx = Ωy.

and

z(t) = 2Azeat + Cze−2at. (38)

Regardless of the value of a, solutions for this case willgrow exponentially.

3.1.3. Mixed solutions

Particles will move on exponentially growing or decayingoscillatory trajectories when both a and b are real andnon-zero. This will be the case with D > 0, if the magneticfield has at least components in the y and z directions.

Under these conditions, and for all values of K andfield configurations the s1 > 0 and s2 < 0. This impliesthat a and b are positive. Thus, in solutions (21)–(23)the purely exponential term of the form Cwe−2at for w =x, y, or z only contributes initially and rapidly becomesnegligible in comparison to the oscillatory terms.

Examples of this behavior are seen in Figs. 5 and 6.The first case corresponds to the simplest geometry forthis type of solution, that is a magnetic field with Ωx = 0and Ωy = Ωz. The second case, Fig. 6, corresponds toparticle trajectories in a magnetic field with all three com-ponents being equal. The velocity shear is assumed to beK = 0.1Ω in both cases but the precise choice will bediscussed later.

Let us analyze solutions for the simplest configura-tion although more general cases are qualitatively similar.When Ωx = 0, calculating s1 and s2 from (17) and (18) weobtain a ≈ 0.03Ω and b ≈ 0.6Ω The particle trajectorieswill be described by Eqs. (21)–(23) with all coefficientsreal. Close analysis of the solutions indicates that thesecan be written as

x(t) = (2Ax cos bt− 2Bx sin bt)eat + xc(t), (39)

y(t) = (2Ay cos bt− 2By sin bt)eat + yc(t), (40)

Fig. 5. Trajectory of mixed type, exponential and oscillatory,for a particle originating at the top of the boundary layer forthe case in which D > 0 and Ωx = 0,Ωy = Ωz.

and

z(t) = (2Az cos bt− 2Bz sin bt)eat + zc(t), (41)

where the position of the center of oscilation is given by

xc(t) = Cx(e−2at − 1)− 2Ax, (42)

yc(t) = Cy(e−2at − 1)− 2Ay, (43)

and

zc(t) = Cze−2at. (44)

The particle will execute oscillatory motion with an ampli-tude that grows exponentially at a rate a while the centerof the oscillation moves according to Eqs. (42)–(43). Notethat since a > 0, as time progresses, the center of the os-cillation will lead to a finite, non-zero value in x and ydepending on the initial position and the field configura-tion. Also, within the restrictions of the case analyzed inthis subsection particles will move toward z = 0 regardlessof the magnetic field configuration or the velocity shear.

We will return to the discussion of the implications ofthese redistribution and acceleration of the particles in thenext section.

3.2. Results for D < 0

When the discriminant D is negative then, according toEqs. (17) and (18), s1 and s2 will be the conjugate of eachother. This implies that a is real and b is imaginary andthus all roots of Eq. (10) are real and the solutions arepurely exponential.

Once again we will analyze in detail a simple case forthis regime when Ωy = Ωz = 0. For such configuration,the condition ε = K/Ω > 1 is equivalent to a negative

672 E. Perez-Tijerina et al.: Pickup-ions in velocity shears

Fig. 6. Trajectory of mixed type for a particle originating atthe top of the boundary layer for the case in which D > 0 andΩx = Ωy = Ωz.

Fig. 7. Trajectories of particles originating at 4 different initialpositions for the case in which ε > 1 and Ωy = Ωz = 0.

discriminant. Solutions, shown in Fig. 7 can be writtenas:

x(t) = 0, (45)

y(t) =εzo

2(ε− 1)3/2(e$t − e−$t)− Kzot

2(ε− 1), (46)

and

z(t) =εzo

2(ε− 1)(e$t + e−$t)− zo

ε− 1, (47)

where $ = Ω(ε− 1)1/2. After a few timescales ∼1/$ thesolution is dominated by the exponential growth term withthe particle escaping rapidly from the source region.

More general magnetic field orientations, with D < 0lead to similar solutions. An example corresponding toΩx = Ωy = Ωz and K = 1.6Ωx is shown in Fig. 8.

0.000.13

0.250.38

0.50-0.6-0.5

-0.4-0.2

-0.10.00.0

0.2

0.4

0.6

0.8

1.0

Z

X

Y/100

Fig. 8. Trajectories of particles originating at 3 different initialpositions for the case in which ε > εcrit and Ωx = Ωy = Ωz.

4. Discussion

The most notable results that have been described are re-lated to the exponential acceleration and the migration ofthe contaminant ions that move within the velocity shear.Acceleration on mixed trajectories, with a constant veloc-ity drift, will only occur whenever the magnetic field hasnon-zero By and Bz components. If either of these com-ponents is zero, exponential particle acceleration can takeplace if ε > 1.

In this section we discuss the conditions and limits ofthe mechanism of particle acceleration.

4.1. Conditions for particle acceleration

As we have shown above, particle acceleration can takeplace whenever Ωy, Ωz and K are different from zero. Fora given flow and magnetic field configuration the rate atwhich particles are accelerated depends on their mass. Forexample, for the particular case shown in Fig. 5 the am-plitude of the oscillations, both in position and velocity,grows at a rate a ≈ 0.03qB/mc and hence, for this con-figuration, the more massive ions will be accelerated at alesser rate.

For cases in which there are purely exponential trajec-tories, such as those shown in Figs. 4 and 7, there is acritical value of ε,

εcrit = 1 + 27[KΩyΩz

2Ω3

]2/3

, (48)

at and above which, such solutions are obtained.For a given magnetic field and flow properties, ions

with mass such that ε is greater than the critical value, willmove on exponential trajectories instead of the oscillatorymotion derived from previous studies.

For the simple magnetic configuration when the fieldis in the xy plane the condition for particle acceleration is

E. Perez-Tijerina et al.: Pickup-ions in velocity shears 673

simply ε > 1. In terms of the ion mass the condition forhaving exponential trajectories is:

mpcKBxq(B2

x +B2y)> 1. (49)

So the critical mass for this particular magnetic configu-ration is

mcrit =q(B2

x +B2y)

cKBx· (50)

Ions with mass less than this critical value will executecycloidal motion drifting with the flow, while ions withmass greater than the critical value will describe orbits ofexponentially increasing size and velocity.

4.2. Limits of particle acceleration

Accelerated ions also migrate in the z direction, upwardsfor purely exponential trajectories, or downwards for os-cillatory exponential trajectories as those described inSect. 3.1.3. The limits for particle acceleration are thenset by the fact that ions will eventually leave the regionwhere the velocity shear is present.

The width of the velocity shear region in our modelcan be prescribed or determined by the adopted velocitygradient, K and an assumed velocity for the free flowingwind, i.e. for the wind outside the velocity shear, Uw. Weadopt the latter approach so that the width of the shearregion is given by H = Uw/K.

Our analysis, and the predicted acceleration, will nolonger be applicable when an ion reaches the bottom orthe top of the shear layer. Ions moving on exponential tra-jectories will be accelerated when they move within thewidth H. Ions on mixed trajectories will undergo expo-nentially growing oscillations moving downwards eventu-ally reaching the bottom of the velocity shear, i.e. z = 0.

We analyze the maximum acceleration possible forboth cases separately by looking at the behavior of theparticle kinetic energy as the particles migrates in the z-direction. Exponential trajectories are exemplified by thecase in which the magnetic field is: B = (B, 0, 0) andε > 1. Acceleration for particles on mixed type trajecto-ries is exemplified by the special case in which Bx = 0 andBy = Bz.

4.2.1. B = (B , 0 , 0 )

The trajectory of particles subject to this configuration isobtained from Eqs. (32)–(34) by taking Ωz = 0. Rewritingin terms of ε the particle velocity as a function of time is:

vy(t) =Kzo

1− ε(1− cosωt) (51)

and,

vz(t) =Kzo√1− ε

sinωt (52)

0 2000 4000 6000 8000 100000

5

10

15

20

25

30

35

40

45

50

mp=32

mp=28

mp=18

mp=16

mp=12

mp=2

mp=28

mp=32

mp=2

mp=12

mp=16

mp=18

Tm

ax

[ke

V]

zO [km]

Fig. 9. Maximum kinetic energy attained by the ions as func-tion of the initial position in the gradient. Each curve corre-sponds to different ion mass. The horizontal lines are the maxi-mum energy attained by the ions in a flux without shear. In thiscase we considered B = 10γ, H = 104 km, Uw = 500 km s−1.

with no component in the x direction for this particularcase. The kinetic energy for exponential solutions (ε > 1)as a function of time is given by

T =mK2z2

o

2(1− ε)[2− 2 cosh$t− ε sinh2$t

]. (53)

The maximum kinetic energy attained by ions on expo-nential trajectories is obtained after finding the time atwhich they reach the top of shear layer. The calculationof the time to reach H, and the resulting kinetic energy,are easily carried out but it does not lead to a simpleanalytical expression. The maximum kinetic energy is afunction of the mass of the accelerated ions as well as themagnetic field strength (through $), the velocity gradi-ent K, the streaming wind velocity, which determines H,and the position z0 where the ion is born. The maximumkinetic energy is shown in Fig. 9 as a function of the ionmass and the initial position for the case relevant to thesolar wind flow past a non-magnetic planet.

Figure 9 shows the maximum energy attained by aparticle as function of initial position, z0, and ion mass.In this figure we can observe two types of behaviour forthe maximum kinetic energy depending on the value ofepsilon or mp, since ε = Kmpc/qB in this configuration.

674 E. Perez-Tijerina et al.: Pickup-ions in velocity shears

For the case of exponential acceleration, i.e. if mp issuch that ε > 1, the maximum kinetic energy of the par-ticles is always less than the kinetic energy of ions in thefree streaming wind. Alltough the acceleration increaseswith z0, ions born above a certain height escape from theboundary layer so quickly that no significant accelerationis achieved. Hence, particles on purelly exponential trajec-tories are not even accelerated to the velocity of the freelystreaming wind.

It is to be noted that a greater maximum kinetic energycan be attained by particles with mass such that ε < 1,which move on cycloidal trajectories. This is also shown inFig. 9 where it is seen that Tmax increases as z0 increasesuntil, again, particles exit the boundary layer before theyattain the maximum velocity possible in their gyromotion.

The behavior of the kinetic energy for diferent mag-netic fields and velocity gradients leading to exponentialtrajectories, is qualitatively similar.

4.2.2. B = (0 ,By ,Bz)

As discussed in Sect. 3.1.3 particle acceleration will alsooccur when both By and Bz are different from zero. Ionswill then move on exponentially growing oscillations givenby mixed type solutions. The particles will migrate down-wards until they reach the bottom of the boundary layerwhich will set the limit of the acceleration process.

Once again we illustrate our results looking at the sim-plest configuration, namely, that in which B is entirely inthe yz plane and the components have equal values. Theparticle trajectories in this case are given by Eqs. (39)–(44). The calculation of the kinetic energy is a direct butrather tedious exercise leading to a complicated expres-sion. The evolution of the particle kinetic energy is exem-plified by the case of ion mass mp = 28mH, magnetic con-figuration, Bz = By, and velocity gradient, K = 0.001 Ω,shown in Fig. 10. The particles gain kinetic energy as theymigrate downwards until they reach the bottom of the ac-celeration region.

The maximum kinetic energy attained by particles inthis configuration can be much greater than that of par-ticles in the cases discussed in the previous section. Thisis due to the fact that particles, specially those startingnear the upper bound of the boundary layer, spend a greatamount of time in the region, gaining energy in each os-cillation.

The maximum kinetic energy depends strongly on theion mass, the initial position, and the field configuration.Figure 11 shows the maximum kinetic energy attained bya particle starting from z = H as a function of its massand the magnetic field direction is in the yz plane, char-acterized by the angle φ = atan(By/Bz). In Fig. 11 wecan see that the maximun kinetic energy attained by theion increases with φ angle; as φ increases, the drift in thez direction decreases and the ion stays a longer time inthe aceleration zone, e.g., if φ = 70o and the ion mass ismp = 28mH, the ion arives to the bottom of the region

0 1x105 2x105 3x105 4x105 5x105

0

500

1000

1500

2000

2500

|V|

[km

/s]

Z [ km ]

Fig. 10. An example of the evolution of the total speed of aparticle that “born” in the highest layers of a velocity sheardescending to layers of low velocity. In this case B = 10γ,H = 5×105 km. Uw = 500 km s−1. The magnetic configurationis Bz = By, Bx = 0. The horizontal line is the speed of ionsin the free streaming wind. The inclined solid line is the localbackground flow velocity.

with an energy 25 times greater the energy of similar ionsin the free streaming wind. Furthermore, the energy of theion is approximately 3000 times the energy of particles inthe background flow at a height of 50 000 km.

5. Applications

Evidence of velocity shears in the region of interac-tion of the solar wind with plasma obstacles is availablefrom measurements made at Venus, Mars and cometsGiacobinni-Zinner, Halley and Gregg-Skjellerup. As awhole the velocity shear is seen along the flanks and down-stream from the plasma obstacle with a thickness thatincreases with distance along the tail direction. In thenear wake of Venus and Mars the velocity shear has a1000–2000 km width (Bridge et al. 1967; Romanov et al.1979; Lundin et al. 1991) and thus is much smaller thanthe (∼100 000 km values that have been inferred frommeasurements made in cometary plasma environments(Johnstone et al. 1986; Balsiger et al. 1986; Gloecker et al.1986). An important feature of the velocity shear seen inall these cases is that it is bounded by a sharp plasma tran-sition across which there is a strong change in the velocitygradient in the direction transverse to the solar wind flow

E. Perez-Tijerina et al.: Pickup-ions in velocity shears 675

-10 0 10 20 30 40 50 60 70 80 90 100

5x105

1x106

2x106

2x106

mp=2

mp=16

mp=32

mp=32, v=sw

Tm

ax

[ke

V]

φ [deg]

Fig. 11. Maximum kinetic energy attained by the ions as func-tion of the angle φ (φ = atan (By/Bz)). Each curve cor-responds to different ion mass. In this case we consideredB = 10γ, H = 5× 105 km and Uw = 500 km s−1.

(Perez-de-Tejada et al. 1995). Measurements show thatthe flow speed begins a strong decrease from conventional300–400 km s−1 solar wind values at that boundary to asmall fraction seen closer to the plasma obstacle. Furtherinformation is that across the velocity shear the plasma ishotter and becomes more turbulent than what it is seenabove it. At the same time there are indications that theplasma flux intensity within the velocity shear becomesmuch smaller than the values measured further outside.

While these variations in the plasma properties of thesolar wind together with the transterminator flow seen inthe upper ionosphere (Knudsen et al. 1980) are consis-tent with the overall behavior expected from the effectsof momentum transport through viscous-like processes(Perez-de-Tejada 1986), there are unrelated observationsthat reveal further important conditions within the shearlayer. In particular, measurements conducted in the Marsmagnetosheath with a particle detector system of thePhobos spacecraft indicate the presence of energetic par-ticle fluxes at the lowest (∼900 km) altitudes from Marsthat were approached along trajectories near the flank re-gions. These include enhanced values in the 30–200 keVenergy range far above those expected from the standardpick-up process for O+ ions in the freestream solar wind(Afonin et al. 1989). While it has been suggested that suchaccelerated particles may be related to a pronounced com-pression of the interplanetary magnetic field lines close tothe planet it is important to point out that the location

where they are observed agrees very closely with the re-gion where the flow speed decreases to very small values.Measurements conducted with the ASPERA experimentin the Phobos spacecraft show, for example, that alongthat section of the Phobos trajectory the speed of theshocked solar wind decreases to very low (∼50 km s−1)values (Lundin et al. 1991). The presence of a sharp ve-locity shear in that region of space may thus provide con-ditions suitable to the local acceleration of plasma parti-cles as was discussed above. A similar agreement can alsobe inferred from measurements conducted in cometaryplasma environments. Energetic particle fluxes have beenobserved within the wide velocity shear across the plasmatail of comets Giacobinni Zinner and Halley (Gloeckeret al. 1986). Thus it is conceivable that particle accelera-tion within a velocity shear may be an important contrib-utor to the population of energetic particles seen in thoseregions of space.

6. Summary

We have analyzed the trajectories of pick-up ions in linearvelocity shears as those expected as the solar wind flowsaround nonmagnetic bodies (Venus, Mars, comets). Themost interesting result we found, exclusively present in ve-locity shears, is the possibility of accelerating pick-up ionsto velocities much greater than those in the free stream-ing wind. We have discussed the conditions and limits ofthe acceleration mechanism and speculated on possibleapplications to explain the presence of high energy ions invarious scenarios.

Acknowledgements. We thank CONACYT-Mexico for supportfor E.P. through a scholarship for graduate studies. M.R. ac-knowledges support from CONACYT grant J32990E.

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