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The Astrophysical Journal, 525:L65–L68, 1999 November 1 q 1999. The American Astronomical Society. All rights reserved. Printed in U.S.A.

VISCOUS FORCES IN VELOCITY BOUNDARY LAYERS AROUND PLANETARY IONOSPHERES H. Pe´rez-de-Tejada Institute of Geophysics, National University of Me´xico, Ensenada, Baja California, Me´xico Received 1999 January 19; accepted 1999 August 31; published 1999 October 4

ABSTRACT A discussion is presented to examine the role of viscous forces in the transport of solar wind momentum to the ionospheric plasma of weakly magnetized planets (Venus and Mars). Observational data are used to make a comparison of the Reynolds and Maxwell stresses that are operative in the interaction of the solar wind with local plasma (planetary ionospheres). Measurements show the presence of a velocity boundary layer formed around the flanks of the ionosphere where the shocked solar wind has reached super-Alfve´nic speeds. It is found that the Reynolds stresses in the solar wind at that region can be larger than the Maxwell stresses and thus are necessary in the local acceleration of the ionospheric plasma. From an order-of-magnitude calculation of the Reynolds stresses, it is possible to derive values of the kinematic viscosity and the Reynolds number that are suitable to the gyrotropic motion of the solar wind particles across the boundary layer. The value of the kinematic viscosity is comparable to those inferred from studies of the transport of solar wind momentum to the earth’s magnetosphere and thus suggest a common property of the solar wind around planetary obstacles. Similar conditions could also be applicable to velocity boundary layers formed in other plasma interaction problems in astrophysics. Subject headings: acceleration of particles — MHD — planets and satellites: individual (Mars, Venus) — plasmas

behavior of the solar wind around planetary ionospheres is the presence of a plasma transition separate from the bow shock and the ionopause (upper boundary of the ionosphere) along the flanks of the ionosheath. That transition has been identified in the plasma data of all the spacecraft that have probed the Venus plasma environment (Bridge et al. 1967; Shefer et al. 1979; Romanov et al. 1979; Pe´rez-de-Tejada et al. 1995) and marks the outer extent of a velocity boundary layer. Currently there is ample evidence of this plasma transition in the Venus ionosheath located at about 1000 km above the ionopause by the terminator after emerging from the dayside mantle (Pe´rezde-Tejada 1995, Fig. 5). In this latter region the velocity boundary layer implied by that plasma transition should be narrow and its thickness should increase with the downstream distance from the planet. Similar conditions have also been detected in the Mars plasma environment, where the Phobos data also indicate a velocity boundary layer bounded by a plasma transition external to the ionosphere (Lundin et al. 1990; Sauer et al. 1992). The effects of the acceleration produced by the solar wind on the ionosphere are illustrated by plasma clouds detected in the ionosheath flow (Brace et al. 1982; Dubinin et al. 1996). The reports on these features indicate that they can be dragged off by the solar wind from both planets and represent a loss of plasma to the upper ionosphere. While the accumulation of the interplanetary magnetic field around the dayside ionosphere can produce large magnetic pressure forces and hence drive away the ionospheric plasma clouds, it is necessary to quantify to what extent these forces can be operative along the flanks of the ionosphere and in the wake. An issue here is to substantiate with experimental data the importance of viscous forces to explain the acceleration of the plasma clouds downstream from the planet and to discuss whether their motion can instead be solely produced by J 3 B forces. These matters will be addressed below through the analysis of plasma data obtained in the Venus near wake.

1. INTRODUCTION

A major issue related to particle acceleration in planetary plasma environments has been the role of forces that lead to flow motion. In addition to the magnetic J 3 B forces which accelerate the contaminant pickup particles that are mass loaded to the solar wind, there are indications that additional forces are also operative in the solar wind interaction with the ionospheric plasma of weakly magnetized planets. For example, pressure gradient forces have been suggested as producing the nightside-driven flow that is observed in the Venus ionosphere (Knudsen et al. 1980). The later flow has supersonic speeds and thus requires external constraints (nozzle effects) on the pressure forces in order to produce the required acceleration (Pe´rez-de-Tejada et al. 1993; Whitten et al. 1993). Viscous forces have also been considered as necessary to drive the Venus ionospheric plasma through momentum transport processes exerted by the solar wind adjacent to the ionosphere (Pe´rez-de-Tejada 1986). The later forces have been assumed to produce the sharp velocity boundary layer that extends along the sides of a planetary ionosheath and that implies a large deficiency of the oncoming solar wind momentum (Pe´rez-deTejada 1995). In this view there is an important question regarding the origin of the viscous forces given the collisionless character of the solar wind. In the absence of particle-particle collisions, it has been necessary to assume that wave-particle interactions are involved in the momentum transport, and it is possible that small-phase speed waves derived from lower hybrid electrostatic modes could be involved in producing that effect through Landau damping processes (Shapiro et al. 1995). Independent support for the existence of viscous forces has been the observation of enhanced plasma temperatures across the velocity boundary layer (Shefer et al. 1979; Romanov et al. 1979). A notable temperature increase is measured in that region with respect to the values in the outer ionosheath, and it is possible that local dissipative processes may produce it (Pe´rez-de-Tejada 1982). An important feature related to the L65

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2. PLASMA DATA

Despite the fact that much of what has been learned on the Venus plasma environment derives from observations conducted with the Pioneer Venus Orbiter (PVO), there are important data that were obtained in previous experiments. Most notable is the information obtained during the flyby pass of the Mariner 5 spacecraft through the Venus wake (Bridge et al. 1967; Shefer et al. 1979) and the following measurements carried out with the Venera 9–10 spacecraft (Romanov et al. 1979). Useful evidence of the behavior of the solar wind around Venus is provided by the profiles of plasma parameters reported from the Mariner 5 measurements. These are reproduced in Figure 1 along the trajectory of the spacecraft as it approached Venus from the wake and later moved upstream from the planet. The data show the identification of an inbound and outbound crossing of a bow shock (events 1 and 5) as well as an additional plasma boundary present within the shocked solar wind region (events 2 and 4). This latter (intermediate) transition is external to the ionosphere (whose upper boundary was not crossed by that spacecraft) and contains plasma variations different from those across the bow shock. In particular, the plasma density and the magnetic field intensity have smaller values in the downstream region. Equally relevant is the fact that at and downstream from the intermediate transition (labeled expansion wave in the Mariner 5 data) there is a strong change in the flow speed with much smaller values in the inner ionosheath (this variation is seen inbound between events 2 and 3 and outbound at events 39 and 4). As a result there is a velocity boundary layer within which there is an important deficiency in the kinetic momentum of the oncoming solar wind. A similar velocity boundary layer was also reported from the Venera observations (Romanov et al. 1979) with a strong temperature increase across a region whose outer extent is also bounded by a sharp transition in the ionosheath. Further and more abundant information on a plasma transition within the shocked solar wind is available from the PVO plasma data, which consistently show a decrease of the magnetic field intensity to low values in the inner ionosheath and that is comparable to the change seen at event 2 in the upper panel of Figure 1 in the Mariner 5 data (Pe´rez-de-Tejada et al. 1995). The lower panel of Figure 1 shows the approximate position of the bow shock and the intermediate transition that marks the outer boundary of the velocity boundary layer and that also flares away from the tail with the downstream distance from the planet. An important issue derived from the data of Figure 1 is the calculation of the kinetic energy density rU 2 of the flow along the spacecraft trajectory. With the (≥300 km s21) flow speeds and (≥1 cm23) plasma densities measured within the velocity boundary layer, it turns out that rU 2 1 1029 ergs cm23, which is larger than the (!1029 ergs cm23) values of the magnetic energy density implied by the (≤10 g) magnetic field intensities measured in that region (the estimated rU 2 values correspond to a solar wind proton population since heavier planetary ions streaming with those speeds would lead to even larger values). As a result the flow remains superAlfve´nic with values of the Alfve´n Mach number MA that are larger in the outer parts of the velocity boundary layer. Under these conditions, the kinetic energy density of the flow can be substantially larger than the local magnetic energy density. 3. EVALUATION OF THE VISCOUS FORCES

A comparison of the magnitude of J # B forces with respect to that of viscous forces present in the velocity shear around

Fig. 1.—Top: Magnetic field intensity with the latitudinal and azimuthal angles of the magnetic field direction, thermal speed, plasma density, and flow speed, measured during the Mariner 5 encounter with Venus. Bottom: Trajectory of the Mariner 5 on a plane in which the vertical ordinate is the distance to the Sun-Venus axis (after Bridge et al. 1967 and Shefer et al. 1979).

planetary ionospheres can be conducted by examining the momentum equation r(U · =)U = (B · =)B/me 2 =(B 2/2me ) 1 m∇ 2U,

(1)

where r is the plasma density, U and B are the velocity and magnetic field vectors, m is a suitable viscosity coefficient, and me is the magnetic permeability. Other forces (due to electric fields, thermal pressure, gravity, and collisions with other particle populations) are not considered, and steady state conditions have been assumed. In an order-of-magnitude calculation it is possible to replace this equation by r 0U02/L ∼ B 02/(2me L) 1 mU0 /d 2,

(2)

where L is the length of the obstacle (which can be assumed to be the Venus radius) and d is the width of the velocity boundary layer where the viscous forces apply. The approximation made to derive equation (2) with the free-stream values r0, U0, and B0 is based on various assumptions; namely, that a strong change in the value of the magnetic field intensity under the magnetic pressure forces occurs across the width L of the obstacle and that a comparable change in the value of

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the velocity field through viscous forces takes place across the width d of the velocity boundary layer. The later simplification requires that the main change in the flow speed is on a direction transverse to that of the main flow and that the boundary layer has a geometry consistent with this condition (boundary layer approximation; Roberts 1967, p. 161). At the same time, the constraints assumed on the magnetic forces are that the magnetic field intensity varies mostly across distances comparable to the scale distance L but not across the width d. From the data shown in Figure 1 it is possible to make this assumption since there is no appreciable change in the magnetic field intensity across the region where the flow speed decreases significantly (between events 2 and 3). The implication here is that across the velocity boundary layer the magnetic field pressure remains nearly uniform. It is to he noted, however, that the enhanced magnetic field intensity seen between events 4 and 5 does not occur within the velocity boundary layer and, most likely, represents the outer part of the magnetic barrier formed around the dayside ionosphere. Also important is the fact that the large change in the azimuthal angle a seen across the wake (between events 2 and 4) implies strong magnetic field rotations and hence nonuniform magnetic tension forces. However, if the curvature of the magnetic field lines along the Mariner 5 trajectory is assumed to be comparable to the scale length of the obstacle, then the pressure and the tension forces should have the same order of magnitude. By introducing the Alfve´n speed VA = B 0 /[me r 0 ]1/2 and the Reynolds number R = (r 0 /m)U0 L, equation (2) can be reduced to r 0U02 ∼ r 0U02[(VA/U0 ) 2/2 1 (L/d) 2/R],

(3)

where it is possible to compare the relative importance of both terms in the right-hand side. It should be first noted that the magnitude of the magnetic force term will be small in superAlfve´nic flows (MA = U0 /VA 1 1) and hence that the viscous force term will have a large participation on the value of the inertial force under such conditions. In particular, since the value of the square bracket in equation (3) should be equal to the one factor that multiplies r 0U02 in the left-hand side, it is necessary that the contribution of the viscous force term is close to that value if the magnetic force term becomes very small. From the numbers derived above with the Mariner 5 data, it is permissible to assume that the contribution of the magnetic force term in equation (3) becomes very small in the outer parts of the velocity boundary layer where the Alfve´n Mach number is large, MA ≈ 5 (the flow remained superAlfve´nic throughout the region probed by the Mariner 5). In this case it is possible to consider the implications of assuming that the viscous force term is dominant and thus take R ∼ (L/d) 2. In this limit we have R ≈ 36 given the L ∼ 6 # 10 3 km and d ∼ 10 3 km values that were indicated above for the obstacle size and the width of the velocity boundary layer (a comparable value of the Reynolds number was derived earlier from an estimate of the shape of the boundary layer; Pe´rezde-Tejada 1982). An independent calculation of the Reynolds number can be made by estimating the kinematic viscosity suitable to gyrotropic motion. Following Parker (1958) we will consider the effects of momentum transport produced by the thermal motion of the solar wind particles across a velocity shear. In this case the momentum that can be exchanged by solar wind protons through their Larmor radius rL is m p rL­UT /­y (mp is the proton mass and UT is their thermal speed). For convenience we will assume that the momentum transport occurs through undefined

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wave-particle interactions and that the solar wind protons feed the background of waves in their gyration. If such an interaction takes place with a frequency Qp and np is the particle density, then the total rate of momentum transferred per unit time and per unit area (viscous stress) is n p rLQ p[m p rL­UT /­y] and the kinematic viscosity is n = m/(n p m p ) = rL2 Q p , which is the form obtained by Thomson (1961) and Newcomb (1966) in the analysis of gyroviscosity in the Vlasov equation. With UT = 80 km s21 and B = 10g for the flow outside the velocity boundary layer in Figure 1, we have rL = 80 km so that if we assume that the interactions occur each time the protons complete a Larmor gyration, i.e., Q p = UT /rL, we obtain n = 6 # 10 3 km2 s21 which in turn leads to R ∼ 480 (using U0 = 500 km s21 and L = 6 # 10 3 km). Larger values of the kinematic viscosity leading to smaller values of the Reynolds number like that derived from equation (3) can also be obtained with a larger interaction frequency, thus implying conditions in which the protons are more strongly connected to the wave field along their Larmor gyration. Despite imprecise information on the wave-particle interaction frequency, the above results are indications that the Reynolds number is not very large, thus implying a thick velocity boundary layer around the obstacle. The unrelated evaluation of the Reynolds number based on momentum transport in gyrotropic motion and on the calculation of the viscous term in equation (3) lead to approximate estimates of the effects of viscous forces in the solar wind interaction with planetary ionospheres. Values of the kinematic viscosity of the solar wind have also been derived from calculations of the transport of momentum to the earth’s magnetosphere (Eviatar & Wolf 1968; Miura 1987). In the analysis of that component it is estimated that n is in the 103–104 km2 s21 range with values that may be sufficient to drive the plasma circulation pattern in the Earth’s magnetosphere suggested by Axford (1964). Despite the different origin of the perturbations that produce the transport of momentum across the earth magnetopause (Kelvin-Helmholtz instability, two-stream instability), it is significant that they lead to n values that are comparable to those estimated for the solar wind interaction with planetary ionospheres. It is possible that such values may be representative of the kinematic viscosity of the solar wind in velocity boundary layers around planetary obstacles.

4. DISCUSSION

There are important implications that can be inferred if viscous forces are present in the solar wind interaction with planetary ionospheres. The main aspect is related to the transport of solar wind momentum to the bulk of the ionospheric plasma through a drag force that produces an overall displacement in the solar wind direction. The transterminator flow in the Venus upper ionosphere is a useful example with an integrated momentum flux that is comparable to the deficiency of solar wind momentum across the boundary layer adjacent to the ionosphere (Pe´rez-de-Tejada 1986). Independent of complications related to the acceleration of pickup ions through the convective electric field of the solar wind (Kasprzak et al. 1991, Fig. 9), viscous forces could be responsible for removing a significant fraction of the solar wind momentum through processes that are mostly operative in a well-defined velocity boundary layer. Even though there is evidence of this feature near and downstream from the terminator of the Venus ionosphere, the conditions near the subsolar region should be different. The large

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(∼100g) magnetic field intensities and the low flow speeds expected in that region lead to a sub-Alfve´nic regime in which magnetic stresses should be more important than the kinetic

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forces. Differences in the value of these parameters with respect to those at the terminator should also be important in other astrophysical boundary layer problems.

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