Wave-Particle Interactions in Astrophysical Plasmas

: Dissipation processes derived from the kinetic theory of gases (shear viscosity and heat conduction) are employed to examine the solar wind that interacts with planetary ionospheres. The purpose of this study is to estimate the mean free path of wave-particle interactions that produce a continuum response in the plasma behavior. Wave-particle interactions are necessary to support the fluid dynamic interpretation that accounts for the interpretation of various features measured in a solar wind–planet ionosphere region; namely, (i) the transport of solar wind momentum to an upper ionosphere in the presence of a velocity shear, and (ii) plasma heating produced by momentum transport. From measurements conducted in the solar wind interaction with the Venus ionosphere, it is possible to estimate that in general terms, the mean free path of wave-particle interactions reaches λ H ≥ 1000 km values that are comparable to the gyration radius of the solar wind particles in their Larmor motion within the local solar wind magnetic field. Similar values are also applicable to conditions measured by the Mars ionosphere and in cometary plasma wakes. Considerations are made in regard to the stochastic trajectories of the plasma particles that have been implied from the measurements made in planetary environments. At the same time, it is as possible that the same phenomenon is applicable to the interaction of stellar winds with the ionosphere of exoplanets, and also in regions where streaming ionized gases reach objects that are subject to rotational motion in other astrophysical problems (galactic flow–plasma interactions


Introduction
A common view in studies of the motion of charged planetary particles assimilated by the solar wind that interacts with planetary ionospheres is that they execute gyration trajectories within the magnetic field that the solar wind brings along, and that piles up around the planets' dayside ionosphere.The Larmor radius of the planetary particles can be comparable to a planetary radius (Venus-Mars), and thus they may be subject to laminar motion in large distances along the planetary wakes.Since particle-particle interactions under such conditions are practically non-existent in the solar wind-planet interaction region, it is necessary to consider that other processes should be effective to produce the measured stochastic motion of the plasma particles.Such conditions arise from the frequent variations in the magnitude and direction of the magnetic field that were first reported from the Mariner 5 data as the spacecraft flew by Venus [1], and also from the Venus Express measurements in the inner regions of its flank ionosheath [2].The Mariner 5 spacecraft trajectory is indicated in the lower panel of Figure 1 using cylindrical coordinates to represent its azimuthal and latitudinal position with respect to the sun coordinates.
The magnetic field and the plasma data measured by Venus are presented in the upper panel of Figure 1 to show the frequent and repeated changes of the magnetic field intensity and in its orientation.The large fluctuations in the latter coordinates reveal strong and frequent deviations of the magnetic field direction.At the same time, the same panel of that figure also indicates a persistent change in the value of the density and the bulk speed of the solar wind at and past the feature labeled 2 in their profiles (at −100 min before closest approach).The latter changes are stressed by the shaded areas in that region and are peculiar since they display significantly smaller values of both variables throughout the inner regions of the Venus ionosheath and that lead to a local loss of the solar wind momentum in that region.
Galaxies 2024, 12, x FOR PEER REVIEW 2 of 11 panel of that figure also indicates a persistent change in the value of the density and the bulk speed of the solar wind at and past the feature labeled 2 in their profiles (at −100 min before closest approach).The latter changes are stressed by the shaded areas in that region and are peculiar since they display significantly smaller values of both variables throughout the inner regions of the Venus ionosheath and that lead to a local loss of the solar wind momentum in that region.Calculations have shown that the "missing" momentum flux of the solar wind is comparable to the momentum flux of the ionospheric flow measured by the terminator in the Venus upper ionosphere [3,4].Such a result supports the existence of an efficient transport of momentum between both plasmas with a related increase in the ion thermal speed up to VT ~ 100 km/s values, as noted in the temperature profile in the Mariner 5 data in Figure 1 beginning at label 2 in the inner ionosheath.Evidence of plasma heating in that region was later more extensively reported from measurements conducted with the Venera spacecraft [5][6][7].As shown in Figure 2, enhanced temperatures were measured together with smaller flow speeds.Similar variations were also reported from the VEX Figure 1.(Lower panel) Trajectory of the Mariner 5 spacecraft projected in cylindrical coordinates in its flyby past Venus.Labels 1 to 5 along the trajectory mark important events in the plasma properties (a bow shock is identified at features 1 and 5), and the intermediate plasma transition occurs at features 2 and 4).(Upper panel) Magnetic field intensity and its latitudinal and azimuthal orientation, together with the plasma properties (thermal speed, density, and bulk speed) measured around Venus [1].
Calculations have shown that the "missing" momentum flux of the solar wind is comparable to the momentum flux of the ionospheric flow measured by the terminator in the Venus upper ionosphere [3,4].Such a result supports the existence of an efficient transport of momentum between both plasmas with a related increase in the ion thermal speed up to V T ~100 km/s values, as noted in the temperature profile in the Mariner 5 data in Figure 1 beginning at label 2 in the inner ionosheath.Evidence of plasma heating in that region was later more extensively reported from measurements conducted with the Venera spacecraft [5][6][7].As shown in Figure 2, enhanced temperatures were measured together with smaller flow speeds.Similar variations were also reported from the VEX plasma data   A more complex feature of the solar wind interaction with the Venus ionosphere is related to measurements of distinct ionospheric holes in the nightside hemisphere in the electron density profiles, and that shows narrow deep decreases to low density values [10].Such features have been interpreted as reflecting conditions that can be accounted    A more complex feature of the solar wind interaction with the Venus ionosphere is related to measurements of distinct ionospheric holes in the nightside hemisphere in the electron density profiles, and that shows narrow deep decreases to low density values [10].Such features have been interpreted as reflecting conditions that can be accounted A more complex feature of the solar wind interaction with the Venus ionosphere is related to measurements of distinct ionospheric holes in the nightside hemisphere in the electron density profiles, and that shows narrow deep decreases to low density values [10].Such features have been interpreted as reflecting conditions that can be accounted for in terms of fluid dynamic processes.In particular, those structures reveal the continued erosion of ionospheric particles, and that has been interpreted as resulting from ionospheric channels or ducts produced at the magnetic polar regions of the Venus ionosphere and that extend downstream along the Venus wake [4,11].The dynamics of the plasma configuration within those channels have been examined and reveal features that, as will be illustrated below in Figure 7, have a corkscrew shape formed by vortex structures [12], and of which the width decreases with the downstream distance along the wake [13].This variation is derived from the expansion of the solar wind into the Venus wake from the magnetic polar regions of the ionosphere and produces a gradual decrease in the width of the region where the ionospheric plasma is being dragged by the solar wind.As a result, the width of the vortex structures is also being reduced, leading to the shape indicated in Figure 7, which will be examined in regard to the acceleration of planetary ions along the wake.
The importance of fluid dynamics to validate the information provided by measurements should be justified by a procedure that relies on mathematical aspects, applicable to the discussion.Thus, we examine a mechanism that is applied to derive the mean free path values of wave-particle interactions that are suitable to account for the observations.

Plasma Data Calculations
In the fluid dynamic description of the features measured in the solar wind interaction with the Venus upper ionosphere, it is necessary to identify the physical conditions that allow for such an interpretation.In particular, it is necessary to account for the correlation that exists between fluid dynamic concepts and the physical origin of the processes that produce them.This is the case for the manner in which the solar wind is modified through wave-particle interactions with the Venus upper ionosphere.As a whole, such interactions are derived from the statistical transport of fluid dynamic properties through dissipation processes in a collective medium (shear viscosity and heat conduction).In this sense Liepmann and Roshko [14] (Section 14.9 in p. 372) examined the connection between both phenomena using the kinetic transport of gases and concluded that, independent of the active processes that produce dissipation, their effect is related to the corresponding mean free path λ H in those processes.Accordingly, they first defined the variables: A more complex feature of the solar wind interaction with the Venus ionosphere is related to measurements of distinct ionospheric holes in the nightside hemisphere in the electron density profiles, and that shows narrow deep decreases to low density values [10].Such features have been interpreted as reflecting conditions that can be accounted for in terms of fluid dynamic processes.In particular, those structures reveal the continued erosion of ionospheric particles, and that has been interpreted as resulting from ionospheric channels or ducts produced at the magnetic polar regions of the Venus ionosphere and that extend downstream along the Venus wake [4,11].The dynamics of the plasma configuration within those channels have been examined and reveal features that, as will be illustrated below in Figure 7, have a corkscrew shape formed by vortex structures [12], and of which the width decreases with the downstream distance along the wake [13].This variation is derived from the expansion of the solar wind into the Venus wake from the magnetic polar regions of the ionosphere and produces a gradual decrease in the width of the region where the ionospheric plasma is being dragged by the solar wind.As a result, the width of the vortex structures is also being reduced, leading to the shape indicated in Figure 7, which will be examined in regard to the acceleration of planetary ions along the wake.
The importance of fluid dynamics to validate the information provided by measurements should be justified by a procedure that relies on mathematical aspects, applicable to the discussion.Thus, we examine a mechanism that is applied to derive the mean free path values of wave-particle interactions that are suitable to account for the observations.

Plasma Data Calculations
In the fluid dynamic description of the features measured in the solar wind interaction with the Venus upper ionosphere, it is necessary to identify the physical conditions that allow for such an interpretation.In particular, it is necessary to account for the correlation that exists between fluid dynamic concepts and the physical origin of the processes that produce them.This is the case for the manner in which the solar wind is modified through wave-particle interactions with the Venus upper ionosphere.As a whole, such interactions are derived from the statistical transport of fluid dynamic properties through dissipation processes in a collective medium (shear viscosity and heat conduction).In this sense Liepmann and Roshko [14] (Section 14.9 in p. 372) examined the connection between both phenomena using the kinetic transport of gases and concluded that, independent of the active processes that produce dissipation, their effect is related to the corresponding mean free path λ H in those processes.Accordingly, they first defined the variables: (obtained from the momentum and the energy equations of a fluid) where ‫͏ﬠ‬ is the kinematic viscosity coefficient and k/ρc p the corresponding transport parameter for heat diffusion.Both relations can be derived by considering that the media involved in a mixing process (namely the solar wind and the Venus upper ionosphere) only experience small deviations away from equilibrium, and thus we will assume that there are linear relations between stress and the rate of strain for viscous dissipation, and also between the heat flow and the temperature gradient in the case of thermal dissipation.In both cases, the purpose of the approach is to obtain the gas dynamic expressions for the shear viscosity µ = ‫͏ﬠ‬ρ and for the heat conductivity k when the flow is subject to wave-particle interactions.Under such conditions, both quantities in Equations (1a) and (1b) are related to the mean free path λ H through: V T •λ H = α•k/ρc p is for thermal dissipation (1b) (obtained from the momentum and the energy equations of a fluid) where A more complex feature of the solar wind interaction with the Venus ionosphere is related to measurements of distinct ionospheric holes in the nightside hemisphere in the electron density profiles, and that shows narrow deep decreases to low density values [10].Such features have been interpreted as reflecting conditions that can be accounted for in terms of fluid dynamic processes.In particular, those structures reveal the continued erosion of ionospheric particles, and that has been interpreted as resulting from ionospheric channels or ducts produced at the magnetic polar regions of the Venus ionosphere and that extend downstream along the Venus wake [4,11].The dynamics of the plasma configuration within those channels have been examined and reveal features that, as will be illustrated below in Figure 7, have a corkscrew shape formed by vortex structures [12], and of which the width decreases with the downstream distance along the wake [13].This variation is derived from the expansion of the solar wind into the Venus wake from the magnetic polar regions of the ionosphere and produces a gradual decrease in the width of the region where the ionospheric plasma is being dragged by the solar wind.As a result, the width of the vortex structures is also being reduced, leading to the shape indicated in Figure 7, which will be examined in regard to the acceleration of planetary ions along the wake.
The importance of fluid dynamics to validate the information provided by measurements should be justified by a procedure that relies on mathematical aspects, applicable to the discussion.Thus, we examine a mechanism that is applied to derive the mean free path values of wave-particle interactions that are suitable to account for the observations.

Plasma Data Calculations
In the fluid dynamic description of the features measured in the solar wind interaction with the Venus upper ionosphere, it is necessary to identify the physical conditions that allow for such an interpretation.In particular, it is necessary to account for the correlation that exists between fluid dynamic concepts and the physical origin of the processes that produce them.This is the case for the manner in which the solar wind is modified through wave-particle interactions with the Venus upper ionosphere.As a whole, such interactions are derived from the statistical transport of fluid dynamic properties through dissipation processes in a collective medium (shear viscosity and heat conduction).In this sense Liepmann and Roshko [14] (Section 14.9 in p. 372) examined the connection between both phenomena using the kinetic transport of gases and concluded that, independent of the active processes that produce dissipation, their effect is related to the corresponding mean free path λ H in those processes.Accordingly, they first defined the variables: (obtained from the momentum and the energy equations of a fluid) where ‫͏ﬠ‬ is the kinematic viscosity coefficient and k/ρc p the corresponding transport parameter for heat diffusion.Both relations can be derived by considering that the media involved in a mixing process (namely the solar wind and the Venus upper ionosphere) only experience small deviations away from equilibrium, and thus we will assume that there are linear relations between stress and the rate of strain for viscous dissipation, and also between the heat flow and the temperature gradient in the case of thermal dissipation.In both cases, the purpose of the approach is to obtain the gas dynamic expressions for the shear viscosity µ = ‫͏ﬠ‬ρ and for the heat conductivity k when the flow is subject to wave-particle interactions.Under such conditions, both quantities in Equations (1a) and (1b) are related to the mean free path λ H through: is the kinematic viscosity coefficient and k/ρc p the corresponding transport parameter for heat diffusion.Both relations can be derived by considering that the media involved in a mixing process (namely the solar wind and the Venus upper ionosphere) only experience small deviations away from equilibrium, and thus we will assume that there are linear relations between stress and the rate of strain for viscous dissipation, and also between the heat flow and the temperature gradient in the case of thermal dissipation.In both cases, the purpose of the approach is to obtain the gas dynamic expressions for the shear viscosity A more complex feature of the solar wind interaction with the Venus ionosphere is related to measurements of distinct ionospheric holes in the nightside hemisphere in the electron density profiles, and that shows narrow deep decreases to low density values [10].Such features have been interpreted as reflecting conditions that can be accounted for in terms of fluid dynamic processes.In particular, those structures reveal the continued erosion of ionospheric particles, and that has been interpreted as resulting from ionospheric channels or ducts produced at the magnetic polar regions of the Venus ionosphere and that extend downstream along the Venus wake [4,11].The dynamics of the plasma configuration within those channels have been examined and reveal features that, as will be illustrated below in Figure 7, have a corkscrew shape formed by vortex structures [12], and of which the width decreases with the downstream distance along the wake [13].This variation is derived from the expansion of the solar wind into the Venus wake from the magnetic polar regions of the ionosphere and produces a gradual decrease in the width of the region where the ionospheric plasma is being dragged by the solar wind.As a result, the width of the vortex structures is also being reduced, leading to the shape indicated in Figure 7, which will be examined in regard to the acceleration of planetary ions along the wake.
The importance of fluid dynamics to validate the information provided by measurements should be justified by a procedure that relies on mathematical aspects, applicable to the discussion.Thus, we examine a mechanism that is applied to derive the mean free path values of wave-particle interactions that are suitable to account for the observations.

Plasma Data Calculations
In the fluid dynamic description of the features measured in the solar wind interaction with the Venus upper ionosphere, it is necessary to identify the physical conditions that allow for such an interpretation.In particular, it is necessary to account for the correlation that exists between fluid dynamic concepts and the physical origin of the processes that produce them.This is the case for the manner in which the solar wind is modified through wave-particle interactions with the Venus upper ionosphere.As a whole, such interactions are derived from the statistical transport of fluid dynamic properties through dissipation processes in a collective medium (shear viscosity and heat conduction).In this sense Liepmann and Roshko [14] (Section 14.9 in p. 372) examined the connection between both phenomena using the kinetic transport of gases and concluded that, independent of the active processes that produce dissipation, their effect is related to the corresponding mean free path λ H in those processes.Accordingly, they first defined the variables: (obtained from the momentum and the energy equations of a fluid) where ‫͏ﬠ‬ is the kinematic viscosity coefficient and k/ρc p the corresponding transport parameter for heat diffusion.Both relations can be derived by considering that the media involved in a mixing process (namely the solar wind and the Venus upper ionosphere) only experience small deviations away from equilibrium, and thus we will assume that there are linear relations between stress and the rate of strain for viscous dissipation, and also between the heat flow and the temperature gradient in the case of thermal dissipation.In both cases, the purpose of the approach is to obtain the gas dynamic expressions for the shear viscosity µ = ‫͏ﬠ‬ρ and for the heat conductivity k when the flow is subject to wave-particle interactions.Under such conditions, both quantities in Equations (1a) and (1b) are related to the mean free path λ H through: and for the heat conductivity k when the flow is subject to wave-particle interactions.Under such conditions, both quantities in Equation (1a,b) are related to the mean free path λ H through: where t* 1 and t* 2 are the relaxation times corresponding to the momentum and to the energy of the flow that are modified by wave-particle interactions (the method can be applied to any property of the medium that is altered under such conditions).It should be noted that these relations lead to Equation (1a,b), since the transport coefficients µ and k are connected through the time factors t* 1 and t* 2 with the thermal speed V T (i.e., V T = λ H /t*), providing a relation that can be employed to derive the mean free path λ H value.
Galaxies 2024, 12, 28 5 of 11 Within this framework, we can take V T ~60 km/s for the thermal speed of the solar wind requested in both equations, and that is available in the upper panel of Figure 1.Separately, we can also estimate values for the kinematic viscosity coefficient dissipation processes in a collective medium (shear viscosity and heat conduction).In this sense Liepmann and Roshko [14] (Section 14.9 in p. 372) examined the connection between both phenomena using the kinetic transport of gases and concluded that, independent of the active processes that produce dissipation, their effect is related to the corresponding mean free path λ H in those processes.Accordingly, they first defined the variables: (obtained from the momentum and the energy equations of a fluid) where ‫͏ﬠ‬ is the kinematic viscosity coefficient and k/ρc p the corresponding transport parameter for heat diffusion.Both relations can be derived by considering that the media involved in a mixing process (namely the solar wind and the Venus upper ionosphere) only experience small deviations away from equilibrium, and thus we will assume that there are linear relations between stress and the rate of strain for viscous dissipation, and also between the heat flow and the temperature gradient in the case of thermal dissipation.In both cases, the purpose of the approach is to obtain the gas dynamic expressions for the shear viscosity µ = ‫͏ﬠ‬ρ and for the heat conductivity k when the flow is subject to wave-particle interactions.Under such conditions, both quantities in Equations ( 1a) and (1b) are related to the mean free path λ H through: = µ/ρ that is employed in Equation (1a).A procedure to carry out this effort is to measure the magnitude of the viscous force in the momentum equation of the plasma as the solar wind streams and interacts with the Venus ionosphere.To this effect, we first calculate in the momentum equation the relative value of the viscous force with respect to that of the magnetic field J × B force that applies to the planetary ions that are carried along by the solar wind [15]: The first two terms on the right side indicate the magnetic tension and the magnetic pressure derived from the J × B forces, thus providing their total dimensional value; the third term refers to viscous forces [16].In a dimensionless form, this equation leads to: where B 0 = 10 nT is the reference value of the magnetic field in the freestream solar wind derived from the Mariner 5 measurements shown in the upper panel of Figure 1, L = 6000 km is the Venus radius, and U 0 = 500 km/s is the freestream solar wind speed indicated in the upper panel of the same figure.In turn, ρ 0 = 1.6 × 10 −24 gr cm −3 derives from the free stream density value (n sw ~3 cm −3 ) indicated in the same figure.The first term here indicates the combined dimensional value of the magnetic field pressure component together with that of the magnetic field tension.This equation can in turn be reduced to where V A = B 0 •(µ e •ρ 0 ) −1/2 ≈ 100 km/s is the Alfven speed and R = ρ 0 U 0 L/µ the Reynolds number.At the same time, δ sw ~2000 km is the width of the velocity boundary layer implied by the data in the lower panel of Figure 1 (approximate distance between the Venus ionopause and the Mariner 5 position by the 3 ′ -4 labels at the terminator), and also that δ i = 1000 km is the width of the trans-terminator ionospheric flow revealed by the label "altitude" in the PVO data indicated in Figure 3.A similar value for this latter parameter is available from the H + and O + ion average velocity profiles reported by Lundin et al., (2011) [17] presented in Figure 4.In their profiles, there is a drastic decrease in the flow speed of the H + and O + ions from ~100 km/s by ~10 3 km altitudes to very low ~30 km/s by ~10 2 km altitudes, thus again leading to δ i = 1000 km for the width of the trans-terminator flow.
With these numbers, we can estimate the total value of the factor multiplying ρ 0 U 0 2 within the square parenthesis on the right side of Equation ( 5), since it has to be equal to the number one so that it is equivalent to on the left side.Thus, we can compare the contribution of its terms and require that (L/δ sw ) 2 /R ≈ 1 × since (V A /U 0 ) 2 ~0.04 can be neglected.Using the values for L and δ sw indicated above, we have (L/δ sw ) 2 ~9 so that R ≈ 9, and thus ρ 0 U 0 L/µ = 9, leading to an equivalent kinematic viscosity coefficient e solar wind interaction with the Venus ionosphere is t ionospheric holes in the nightside hemisphere in the hows narrow deep decreases to low density values [10].d as reflecting conditions that can be accounted for in In particular, those structures reveal the continued erohat has been interpreted as resulting from ionospheric magnetic polar regions of the Venus ionosphere and Venus wake [4,11].The dynamics of the plasma conave been examined and reveal features that, as will be e a corkscrew shape formed by vortex structures [12], ith the downstream distance along the wake [13].This nsion of the solar wind into the Venus wake from the phere and produces a gradual decrease in the width of lasma is being dragged by the solar wind.As a result, is also being reduced, leading to the shape indicated d in regard to the acceleration of planetary ions along mics to validate the information provided by measurecedure that relies on mathematical aspects, applicable e a mechanism that is applied to derive the mean free ctions that are suitable to account for the observations.ion of the features measured in the solar wind interacere, it is necessary to identify the physical conditions n.In particular, it is necessary to account for the correamic concepts and the physical origin of the processes e for the manner in which the solar wind is modified with the Venus upper ionosphere.As a whole, such tatistical transport of fluid dynamic properties through medium (shear viscosity and heat conduction).In this ection 14.9 in p. 372) examined the connection between transport of gases and concluded that, independent of dissipation, their effect is related to the corresponding es.Accordingly, they first defined the variables: the energy equations of a fluid) where ‫͏ﬠ‬ is the kinematic corresponding transport parameter for heat diffusion.nsidering that the media involved in a mixing process us upper ionosphere) only experience small deviations we will assume that there are linear relations between cous dissipation, and also between the heat flow and se of thermal dissipation.In both cases, the purpose of ynamic expressions for the shear viscosity µ = ‫͏ﬠ‬ρ and he flow is subject to wave-particle interactions.Under Equations (1a) and (1b) are related to the mean free path = µ/ρ 0 ~3 × 10 5 km 2 /s by using the values for U o and L given above.This value of the kinematic viscosity coefficient is due to the low mass density ρ 0 = 1.6 × 10 −23 gr cm −3 of the solar wind and, at the same time, it is related to the ability of particle motion to eliminate velocity variations that give rise to viscous transport ( [18], see p. 37).Also, the corresponding value of the shear viscosity coefficient Galaxies 2024, 12, 0 ing value of the shear viscosity coefficient µ = ‫•͏ﬠ‬ρ 0 ~5 × 10 −8 gr cm −1 s −1 is a meas internal friction opposing deformation of the flow ( [18], see p. 36).With such va and µ, it is possible to validate Equation ( 5) and, at the same time, employ the kine cosity coefficient ‫͏ﬠ‬ to calculate, together with the thermal speed V T ≈ 10 2 km/s, free path value λ H in Equation (1a).When V T ≈ 10 2 km/s we obtain λ H ~3 × 10 when V T ≈ 60 km/s we have λ H ~5 × 10 3 km.
Galaxies 2024, 12, x FOR PEER REVIEW noted that these relations lead to Equations (1a) and (1b), since the transport c µ and k are connected through the time factors t*1 and t*2 with the thermal spee VT = λH/t*), providing a relation that can be employed to derive the mean fre value.
Within this framework, we can take VT ~ 60 km/s for the thermal speed o wind requested in both equations, and that is available in the upper panel of Separately, we can also estimate values for the kinematic viscosity coefficient ‫ﬠ‬ is employed in Equation (1a).A procedure to carry out this effort is to measure nitude of the viscous force in the momentum equation of the plasma as the s streams and interacts with the Venus ionosphere.To this effect, we first calcu momentum equation the relative value of the viscous force with respect to that o netic field J × B force that applies to the planetary ions that are carried along by wind [15]: The first two terms on the right side indicate the magnetic tension and the pressure derived from the J × B forces, thus providing their total dimensional third term refers to viscous forces [16].In a dimensionless form, this equation le where B0 = 10 nT is the reference value of the magnetic field in the freestream s derived from the Mariner 5 measurements shown in the upper panel of Figure 1 km is the Venus radius, and U0 = 500 km/s is the freestream solar wind speed in the upper panel of the same figure.In turn, ρ0 = 1.6 × 10 -24 gr cm −3 derives from stream density value (nsw ~ 3 cm −3 ) indicated in the same figure.The first term cates the combined dimensional value of the magnetic field pressure componen with that of the magnetic field tension.This equation can in turn be reduced to where VA = B0•(µe•ρ0) −1/2 ≈ 100 km/s is the Alfven speed and R = ρ0U0L/µ the Reyn ber.At the same time, δsw ~ 2000 km is the width of the velocity boundary laye by the data in the lower panel of Figure 1 (approximate distance between the Ve pause and the Mariner 5 position by the 3′-4 labels at the terminator), and als 1000 km is the width of the trans-terminator ionospheric flow revealed by the l tude" in the PVO data indicated in Figure 3.A similar value for this latter pa available from the H + and O + ion average velocity profiles reported by Lundin et [17] presented in Figure 4.In their profiles, there is a drastic decrease in the flow the H + and O + ions from ~100 km/s by ~10 3 km altitudes to very low ~30 km/s b altitudes, thus again leading to δi = 1000 km for the width of the trans-terminato is a measure of the internal friction opposing deformation of the flow ( [18], see p. 36).With such values for 4 of 12 complex feature of the solar wind interaction with the Venus ionosphere is asurements of distinct ionospheric holes in the nightside hemisphere in the ity profiles, and that shows narrow deep decreases to low density values [10].have been interpreted as reflecting conditions that can be accounted for in dynamic processes.In particular, those structures reveal the continued eroheric particles, and that has been interpreted as resulting from ionospheric ucts produced at the magnetic polar regions of the Venus ionosphere and ownstream along the Venus wake [4,11].
The dynamics of the plasma conthin those channels have been examined and reveal features that, as will be low in Figure 7, have a corkscrew shape formed by vortex structures [12], the width decreases with the downstream distance along the wake [13].This erived from the expansion of the solar wind into the Venus wake from the r regions of the ionosphere and produces a gradual decrease in the width of ere the ionospheric plasma is being dragged by the solar wind.As a result, the vortex structures is also being reduced, leading to the shape indicated hich will be examined in regard to the acceleration of planetary ions along ortance of fluid dynamics to validate the information provided by measurebe justified by a procedure that relies on mathematical aspects, applicable ion.Thus, we examine a mechanism that is applied to derive the mean free f wave-particle interactions that are suitable to account for the observations.ta Calculations id dynamic description of the features measured in the solar wind interac-Venus upper ionosphere, it is necessary to identify the physical conditions such an interpretation.In particular, it is necessary to account for the corrests between fluid dynamic concepts and the physical origin of the processes them.This is the case for the manner in which the solar wind is modified -particle interactions with the Venus upper ionosphere.As a whole, such re derived from the statistical transport of fluid dynamic properties through ocesses in a collective medium (shear viscosity and heat conduction).In this nn and Roshko [14] (Section 14.9 in p. 372) examined the connection between ena using the kinetic transport of gases and concluded that, independent of cesses that produce dissipation, their effect is related to the corresponding h λ H in those processes.Accordingly, they first defined the variables: the momentum and the energy equations of a fluid) where ‫͏ﬠ‬ is the kinematic cient and k/ρc p the corresponding transport parameter for heat diffusion.can be derived by considering that the media involved in a mixing process olar wind and the Venus upper ionosphere) only experience small deviations uilibrium, and thus we will assume that there are linear relations between rate of strain for viscous dissipation, and also between the heat flow and re gradient in the case of thermal dissipation.In both cases, the purpose of is to obtain the gas dynamic expressions for the shear viscosity µ = ‫͏ﬠ‬ρ and onductivity k when the flow is subject to wave-particle interactions.Under ns, both quantities in Equations (1a) and (1b) are related to the mean free path and µ, it is possible to validate Equation ( 5) and, at the same time, employ the kinematic viscosity coefficient A more complex feature of the solar wind interaction with the Venus ionosphere is related to measurements of distinct ionospheric holes in the nightside hemisphere in the electron density profiles, and that shows narrow deep decreases to low density values [10].Such features have been interpreted as reflecting conditions that can be accounted for in terms of fluid dynamic processes.In particular, those structures reveal the continued erosion of ionospheric particles, and that has been interpreted as resulting from ionospheric channels or ducts produced at the magnetic polar regions of the Venus ionosphere and that extend downstream along the Venus wake [4,11].
The dynamics of the plasma configuration within those channels have been examined and reveal features that, as will be illustrated below in Figure 7, have a corkscrew shape formed by vortex structures [12], and of which the width decreases with the downstream distance along the wake [13].This variation is derived from the expansion of the solar wind into the Venus wake from the magnetic polar regions of the ionosphere and produces a gradual decrease in the width of the region where the ionospheric plasma is being dragged by the solar wind.As a result, the width of the vortex structures is also being reduced, leading to the shape indicated in Figure 7, which will be examined in regard to the acceleration of planetary ions along the wake.
The importance of fluid dynamics to validate the information provided by measurements should be justified by a procedure that relies on mathematical aspects, applicable to the discussion.Thus, we examine a mechanism that is applied to derive the mean free path values of wave-particle interactions that are suitable to account for the observations.

Plasma Data Calculations
In the fluid dynamic description of the features measured in the solar wind interaction with the Venus upper ionosphere, it is necessary to identify the physical conditions that allow for such an interpretation.In particular, it is necessary to account for the correlation that exists between fluid dynamic concepts and the physical origin of the processes that produce them.This is the case for the manner in which the solar wind is modified through wave-particle interactions with the Venus upper ionosphere.As a whole, such interactions are derived from the statistical transport of fluid dynamic properties through dissipation processes in a collective medium (shear viscosity and heat conduction).In this sense Liepmann and Roshko [14] (Section 14.9 in p. 372) examined the connection between both phenomena using the kinetic transport of gases and concluded that, independent of the active processes that produce dissipation, their effect is related to the corresponding mean free path λ H in those processes.Accordingly, they first defined the variables: (obtained from the momentum and the energy equations of a fluid) where ‫͏ﬠ‬ is the kinematic viscosity coefficient and k/ρc p the corresponding transport parameter for heat diffusion.Both relations can be derived by considering that the media involved in a mixing process (namely the solar wind and the Venus upper ionosphere) only experience small deviations away from equilibrium, and thus we will assume that there are linear relations between stress and the rate of strain for viscous dissipation, and also between the heat flow and the temperature gradient in the case of thermal dissipation.In both cases, the purpose of the approach is to obtain the gas dynamic expressions for the shear viscosity µ = ‫͏ﬠ‬ρ and for the heat conductivity k when the flow is subject to wave-particle interactions.Under such conditions, both quantities in Equations (1a) and (1b) are related to the mean free path to calculate, together with the thermal speed V T ≈ 10 2 km/s, the mean free path value λ H in Equation (1a).When V T ≈ 10 2 km/s we obtain λ H ~3 × 10 3 km and when V T ≈ 60 km/s we have λ H ~5 × 10 3 km.1000 km is the width of the trans-terminator ionospheric flow revealed by the label "altitude" in the PVO data indicated in Figure 3.A similar value for this latter parameter is available from the H + and O + ion average velocity profiles reported by Lundin et al., (2011) [17] presented in Figure 4.In their profiles, there is a drastic decrease in the flow speed of the H + and O + ions from ~100 km/s by ~10 3 km altitudes to very low ~30 km/s by ~10 2 km altitudes, thus again leading to δi = 1000 km for the width of the trans-terminator flow.A separate manner to derive λ H is available by using the relation: αk = 0.75 n sw B VT •λ H where B = 1.4 × 10 −16 ergs • K −1 is the Boltzmann number, and that derives from studies of magnetic field fluctuations applied to particle motion [19].At the same time, the value αk ~6 ergs cm −1 K −1 s −1 with the thermal conductivity is obtained by combining Equation (1a,b) which lead to αk = ρc p V T λ H ; so that by using nsw = 3 cm −3 and V T = 60 km/s for freestream conditions in Figure 1 we have: thus implying λ H ~3 × 10 4 km (see Appendix A) and that varies with the V T value (α ~0.6).As a whole, the mean free path derived here is in a value range nearly one order of magnitude larger than that implied above from considerations on the viscous transport of momentum as discussed by Liepmann and Roshko [14].In summary, two methods leading to similar results have been proposed to derive mean free path values of the solar wind that interacts with the Venus upper ionosphere, either through wave-particle interactions in terms of their relationship between the viscous transport of momentum and thermal conductivity, and also by applying magnetic field fluctuations to particle motion.That difference may be characteristic of wave-particle interactions.
A peculiar property in both procedures that lead to the mean free path λ H value is that as shown in Equation (1a) they provide a relationship between this parameter and the thermal speed V T of the flow particles.In both procedures higher V T speeds imply lower λ H values.This variation indicates that the thermal speed profile V T in Figure 1 leads to two different speed values, namely ~60 km/s in the solar wind before the inbound bow shock crossing marked as (1) and after the outbound crossing marked as (5), and differently, they reach ~100 km/s values through the inbound ionosheath between the marks labeled 2 and 3.As shown in Equation (1a) an increase in the thermal speed implies a decrease in the magnitude of the mean free path.This is the case in the solar wind where V T = 60 km/s λ H = 5 × 10 3 km, and also in the inner ionosheath where we have V T = 100 km/s thus implying λ H = 3 × 10 3 km.A similar variation is also applicable from the relation αk = 0.75 n sw BV T λ H that derives from studies of magnetic field fluctuations applied to particle motion.In particular, with With these numbers, we can estimate the total value of the factor multiplying ρ0U0 2 within the square parenthesis on the right side of Equation ( 5), since it has to be equal to the number one so that it is equivalent to on the left side.Thus, we can compare the contribution of its terms and require that (L/δsw) 2 /R ≈ 1 × since (VA/U0) 2 ~ 0.04 can be neglected.Using the values for L and δsw indicated above, we have (L/δsw) 2 ~ 9 so that R ≈ 9, and thus ρ0U0L/µ = 9, leading to an equivalent kinematic viscosity coefficient ‫ﬠ‬ = µ/ρ0 ~ 3 × 10 5 km 2 /s by using the values for Uo and L given above.This value of the kinematic viscosity coefficient is due to the low mass density ρ0 = 1.6 × 10 −23 gr cm −3 of the solar wind and, at the same time, it is related to the ability of particle motion to eliminate velocity variations that give rise to viscous transport ( [18], see p. 37).Also, the corresponding value of the shear viscosity coefficient µ = ‫•ﬠ‬ρ0 ~ 5 × 10 −8 gr cm −1 s −1 is a measure of the internal friction opposing deformation of the flow ( [18], see p. 36).With such values for ‫ﬠ‬ and µ, it is possible to validate Equation ( 5) and, at the same time, employ the kinematic viscosity coefficient ‫ﬠ‬ to calculate, together with the thermal speed VT ≈ 10 2 km/s, the mean free path value λH in Equation (1a).When VT ≈ 10 2 km/s we obtain λH ~ 3 × 10 3 km and when VT ≈ 60 km/s we have λH ~ 5 × 10 3 km.
A separate manner to derive λH is available by using the relation: αk = 0.75 nswBVT•λH where B = 1.4 × 10 −16 ergs °K−1 is the Boltzmann number, and that derives from studies of magnetic field fluctuations applied to particle motion [19].At the same time, the value αk ~ 6 ergs cm −1 K −1 s −1 with the thermal conductivity is obtained by combining equations 1a and 1b which lead to αk = ρcp VT λH; so that by using nsw = 3 cm −3 and VT = 60 km/s for freestream conditions in Figure 1 we have: λH = αk × [0.75 nswBVT] −1 which leads to λH = (6 ergs cm −1 s −1 °K−1 ) × [0.75 × (3 cm −3 ) × (1.4 × 10 −16 ergs °K−1 ) × (6 × 10 6 cm s −1 )] −1 =3 × 10 9 cm thus implying λH ~ 3 × 10 4 km (see Appendix A) and that varies with the VT value (α ~ 0.6).As a whole, the mean free path derived here is in a value range nearly one order of magnitude larger than that implied above from considerations on the viscous transport of momentum as discussed by Liepmann and Roshko [14].In summary, two methods leading to similar results have been proposed to derive mean free path values of the solar wind that interacts with the Venus upper ionosphere, either through wave-particle interactions in terms of their relationship between the viscous transport of momentum and thermal conductivity, and also by applying magnetic field fluctuations to particle motion.That difference may be characteristic of wave-particle interactions.
A peculiar property in both procedures that lead to the mean free path λH value is that as shown in Equation (1a) they provide a relationship between this parameter and the thermal speed VT of the flow particles.In both procedures higher VT speeds imply lower λH values.This variation indicates that the thermal speed profile VT in Figure 1 leads to two different speed values, namely ~60 km/s in the solar wind before the inbound bow shock crossing marked as (1) and after the outbound crossing marked as ( 5), and differently, they reach ~100 km/s values through the inbound ionosheath between the marks labeled 2 and 3.As shown in Equation (1a) an increase in the thermal speed implies a decrease in the magnitude of the mean free path.This is the case in the solar wind where VT = 60 km/s implies λH = 5 × 10 3 km, and also in the inner ionosheath where we have VT = 100 km/s thus implying λH = 3 × 10 3 km.A similar variation is also applicable from the relation αk = 0.75 nswBVT λH that derives from studies of magnetic field fluctuations applied to particle motion.In particular, with ‫ﬠ‬ = 3 × 10 5 cm 2 /s for the viscosity coefficient we obtain λH = 3 × 10 4 km when VT = 6 × 10 6 cm s −1 in the solar wind, and λH = 2 × 10 4 km when VT = 10 7 cm s −1 in the inner ionosheath (see Appendix A).These latter values exceed by an order of magnitude those derived from Equation (1a) and thus suggest conditions that may not be applicable to wave particle interaction.In fact, no magnetic fluctuations can = 3 × 10 5 cm 2 /s for the viscosity coefficient we obtain λ H = 3 × 10 4 km when V T = 6 × 10 6 cm s −1 in the solar wind, and λ H = 2 × 10 4 km when V T = 10 7 cm s −1 in the inner ionosheath (see Appendix A).These latter values exceed by an order of magnitude those derived from Equation (1a) and thus suggest conditions that may not be applicable to wave particle interaction.In fact, no magnetic fluctuations can be identified in the solar wind or in the outer ionosheath.Different conditions are encountered in the inner ionosheath where V T = 100 km/s.
A notable aspect of the mean free path values obtained so far is the up to one order of magnitude difference between those obtained with the magnetic field fluctuation cal-culation and those derived from the wave-particle interaction procedure.This is better illustrated in Figure 5 where the linear variation in both traces reflects the relation between λ H and V T that states as to how they are connected.It should be noticed, however, that the criteria used in both cases is different and thus it should not be expected that they lead to identical values.Despite their distinct peculiarity it is of interest to note that since V T is larger in the inner ionosheath smaller λ H values should occur in that region.The wo different connecting traces in Figure 5 have been included to state the similar variation for each different procedure.As noted above, the different width value of the mean free path in the inner and in the outer ionosheath may be related to the more enhanced accumulation of the plasma particles in the inner ionosheath where the thermal speed is larger.Much research with detail calculations should be conducted to account for this behavior.
each different procedure.As noted above, the different width value of the mean free path in the inner and in the outer ionosheath may be related to the more enhanced accumulation of the plasma particles in the inner ionosheath where the thermal speed is larger.Much research with detail calculations should be conducted to account for this behavior.
The values derived in these calculations provide an approximate estimate of the variables involved since only a few of them can be measured.For example, the kinematic viscosity coefficient ‫ﬠ‬ was inferred from the general position of the Mariner 5 spacecraft as it moved across the terminator plane over the Venus ionosphere.Thus, it is possible that a more general ‫ﬠ‬ ~ 10 5 km 2 /s value for the kinematic viscosity coefficient could also be employed thus leading to a smaller λH ~ 10 4 km value for the mean free path being closer to those inferred from the wave-particle interactions.
The validity of Equation (1a) can also be supported noting, through an unrelated example that is applicable to air flows where there is also a relationship between the viscosity coefficient and the mean free path value.In fact, with VT = 4.5 × 10 4 cm s −1 for the thermal speed of air particles at room temperature, together with λ = 4 × 10 −5 cm for their mean free path value [14], Equation (1a) leads to ‫ﬠ‬ = 1.8 cm 2 /s for the kinematic viscosity coefficient of air flows at atmospheric pressures.Such value is comparative to those reported by Hughes and Brighton (1967) [20] under such conditions.The implication is that the viscosity coefficient and the thermal temperature of the solar wind derived from measurements may also lead to approximate mean free path values that satisfy Equation (1a).Mean free path values λ H of the solar wind obtained in wave-particle interactions and also in magnetic field fluctuations using the solar wind thermal speed V T and its kinematic viscosity coefficient ρ0U0L/µ = 9, leading to an equivalent kinematic viscosity coefficient ‫ﬠ‬ = µ/ρ0 ~ 3 × 10 5 km 2 /s by using the values for Uo and L given above.This value of the kinematic viscosity coefficient is due to the low mass density ρ0 = 1.6 × 10 −23 gr cm −3 of the solar wind and, at the same time, it is related to the ability of particle motion to eliminate velocity variations that give rise to viscous transport ( [18], see p. 37).Also, the corresponding value of the shear viscosity coefficient µ = ‫•ﬠ‬ρ0 ~ 5 × 10 −8 gr cm −1 s −1 is a measure of the internal friction opposing deformation of the flow ( [18], see p. 36).With such values for ‫ﬠ‬ and µ, it is possible to validate Equation ( 5) and, at the same time, employ the kinematic viscosity coefficient ‫ﬠ‬ to calculate, together with the thermal speed VT ≈ 10 2 km/s, the mean free path value λH in Equation (1a).When VT ≈ 10 2 km/s we obtain λH ~ 3 × 10 3 km and when VT ≈ 60 km/s we have λH ~ 5 × 10 3 km.
A separate manner to derive λH is available by using the relation: αk = 0.75 nswBVT•λH where B = 1.4 × 10 −16 ergs °K−1 is the Boltzmann number, and that derives from studies of magnetic field fluctuations applied to particle motion [19].At the same time, the value αk ~ 6 ergs cm −1 K −1 s −1 with the thermal conductivity is obtained by combining equations 1a and 1b which lead to αk = ρcp VT λH; so that by using nsw = 3 cm −3 and VT = 60 km/s for freestream conditions in Figure 1 we have: λH = αk × [0.75 nswBVT] −1 which leads to λH = (6 ergs cm −1 s −1 °K−1 ) × [0.75 × (3 cm −3 ) × (1.4 × 10 −16 ergs °K−1 ) × (6 × 10 6 cm s −1 )] −1 =3 × 10 9 cm thus implying λH ~ 3 × 10 4 km (see Appendix A) and that varies with the VT value (α ~ 0.6).As a whole, the mean free path derived here is in a value range nearly one order of magnitude larger than that implied above from considerations on the viscous transport of momentum as discussed by Liepmann and Roshko [14].In summary, two methods leading to similar results have been proposed to derive mean free path values of the solar wind that interacts with the Venus upper ionosphere, either through wave-particle interactions in terms of their relationship between the viscous transport of momentum and thermal conductivity, and also by applying magnetic field fluctuations to particle motion.That difference may be characteristic of wave-particle interactions.
A peculiar property in both procedures that lead to the mean free path λH value is that as shown in Equation (1a) they provide a relationship between this parameter and the thermal speed VT of the flow particles.In both procedures higher VT speeds imply lower λH values.This variation indicates that the thermal speed profile VT in Figure 1 leads to two different speed values, namely ~60 km/s in the solar wind before the inbound bow shock crossing marked as (1) and after the outbound crossing marked as (5), and differently, they reach ~100 km/s values through the inbound ionosheath between the marks labeled 2 and 3.As shown in Equation (1a) an increase in the thermal speed implies a decrease in the magnitude of the mean free path.This is the case in the solar wind where VT = 60 km/s implies λH = 5 × 10 3 km, and also in the inner ionosheath where we have VT = 100 km/s thus implying λH = 3 × 10 3 km.A similar variation is also applicable from the relation αk = 0.75 nswBVT λH that derives from studies of magnetic field fluctuations applied to particle motion.In particular, with ‫ﬠ‬ = 3 × 10 5 cm 2 /s for the viscosity coefficient we obtain λH = 3 × 10 4 km when VT = 6 × 10 6 cm s −1 in the solar wind, and λH = 2 × 10 4 km when VT = 10 7 cm s −1 in the inner ionosheath (see Appendix A).These latter values exceed by an order of magnitude those derived from Equation (1a) and thus suggest conditions that may not be applicable to wave particle interaction.In fact, no magnetic fluctuations can during the Mariner 5 trajectory in Figure 1.The connecting line labeled "W" refers to a value in the Venus inner and in the outer ionosheath that is implied by wave-particle interactions.The connecting line labeled "F" is implied by the magnetic field fluctuations.
The values derived in these calculations provide an approximate estimate of the variables involved since only a few of them can be measured.For example, the kinematic viscosity coefficient With these numbers, we can estimate the total value of the factor multiplying ρ0U0 2 within the square parenthesis on the right side of Equation ( 5), since it has to be equal to the number one so that it is equivalent to on the left side.Thus, we can compare the contribution of its terms and require that (L/δsw) 2 /R ≈ 1 × since (VA/U0) 2 ~ 0.04 can be neglected.Using the values for L and δsw indicated above, we have (L/δsw) 2 ~ 9 so that R ≈ 9, and thus ρ0U0L/µ = 9, leading to an equivalent kinematic viscosity coefficient ‫ﬠ‬ = µ/ρ0 ~ 3 × 10 5 km 2 /s by using the values for Uo and L given above.This value of the kinematic viscosity coefficient is due to the low mass density ρ0 = 1.6 × 10 −23 gr cm −3 of the solar wind and, at the same time, it is related to the ability of particle motion to eliminate velocity variations that give rise to viscous transport ( [18], see p. 37).Also, the corresponding value of the shear ergs °K−1 is the Boltzmann number, and that derives from studies of magnetic field fluctuations applied to particle motion [19].At the same time, the value αk with the thermal conductivity is obtained by combining equations 1a and 1b which lead to αk = ρcp VT λH; so that by using nsw = 3 cm −3 and VT = 60 km/s for freestream conditions in Figure 1 Appendix A) and that varies with the VT value (α ~ 0.6).As a whole, the mean free path derived here is in a value range nearly one order of magnitude larger than that implied above from considerations on the viscous transport of momentum as discussed by Liepmann and Roshko [14].In summary, two methods leading to similar results have been proposed to derive mean free path values of the solar wind that interacts with the Venus upper ionosphere, either through wave-particle interactions in terms of their relationship between the viscous transport of momentum and thermal conductivity, and also by applying magnetic field fluctuations to particle motion.That difference may be characteristic of wave-particle interactions.
A peculiar property in both procedures that lead to the mean free path λH value is that as shown in Equation (1a) they provide a relationship between this parameter and the thermal speed VT of the flow particles.In both procedures higher VT speeds imply lower λH values.This variation indicates that the thermal speed profile VT in Figure 1 leads to two different speed values, namely ~60 km/s in the solar wind before the inbound bow shock crossing marked as (1) and after the outbound crossing marked as (5), and differently, they reach ~100 km/s values through the inbound ionosheath between the marks labeled 2 and 3.As shown in Equation (1a) an increase in the thermal speed implies a decrease in the magnitude of the mean free path.This is the case in the solar wind where VT = 60 km/s implies λH = 5 × 10 3 km, and also in the inner ionosheath where we have VT = 100 km/s thus implying λH = 3 × 10 3 km.A similar variation is also applicable from the relation αk = 0.75 nswBVT λH that derives from studies of magnetic field fluctuations applied to particle motion.In particular, with ‫ﬠ‬ = 3 × 10 5 cm 2 /s for the viscosity coefficient we obtain λH = 3 × 10 4 km when VT = 6 × 10 6 cm s −1 in the solar wind, and λH = 2 × 10 4 km when VT = 10 7 cm s −1 in the inner ionosheath (see Appendix A).These latter values exceed by an order of magnitude those derived from Equation (1a) and thus suggest conditions that may not be applicable to wave particle interaction.In fact, no magnetic fluctuations can was inferred from the general position of the Mariner 5 spacecraft as it moved across the terminator plane over the Venus ionosphere.Thus, it is possible that a more general aries are marked on the right-hand side as the I-sphere (the ionopause (IP), and the ionosheath (IMB) (from Lundin et al., (2011) [17]).
With these numbers, we can estimate the total value of the factor multiplying ρ0U0 2 within the square parenthesis on the right side of Equation ( 5), since it has to be equal to the number one so that it is equivalent to on the left side.Thus, we can compare the contribution of its terms and require that (L/δsw) 2 /R ≈ 1 × since (VA/U0) 2 ~ 0.04 can be neglected.Using the values for L and δsw indicated above, we have (L/δsw) 2 ~ 9 so that R ≈ 9, and thus ρ0U0L/µ = 9, leading to an equivalent kinematic viscosity coefficient ‫ﬠ‬ = µ/ρ0 ~ 3 × 10 5 km 2 /s by using the values for Uo and L given above.This value of the kinematic viscosity coefficient is due to the low mass density ρ0 = 1.6 × 10 −23 gr cm −3 of the solar wind and, at the same time, it is related to the ability of particle motion to eliminate velocity variations that give rise to viscous transport ( [18], see p. ergs °K−1 is the Boltzmann number, and that derives from studies of magnetic field fluctuations applied to particle motion [19].At the same time, the value αk with the thermal conductivity is obtained by combining equations 1a and 1b which lead to αk = ρcp VT λH; so that by using nsw = 3 cm −3 and VT = 60 km/s for freestream conditions in Figure 1 Appendix A) and that varies with the VT value (α ~ 0.6).As a whole, the mean free path derived here is in a value range nearly one order of magnitude larger than that implied above from considerations on the viscous transport of momentum as discussed by Liepmann and Roshko [14].In summary, two methods leading to similar results have been proposed to derive mean free path values of the solar wind that interacts with the Venus upper ionosphere, either through wave-particle interactions in terms of their relationship between the viscous transport of momentum and thermal conductivity, and also by applying magnetic field fluctuations to particle motion.That difference may be characteristic of wave-particle interactions.
A peculiar property in both procedures that lead to the mean free path λH value is that as shown in Equation (1a) they provide a relationship between this parameter and the thermal speed VT of the flow particles.In both procedures higher VT speeds imply lower λH values.This variation indicates that the thermal speed profile VT in Figure 1 leads to two different speed values, namely ~60 km/s in the solar wind before the inbound bow shock crossing marked as (1) and after the outbound crossing marked as (5), and differently, they reach ~100 km/s values through the inbound ionosheath between the marks labeled 2 and 3.As shown in Equation (1a) an increase in the thermal speed implies a decrease in the magnitude of the mean free path.This is the case in the solar wind where VT = 60 km/s implies λH = 5 × 10 3 km, and also in the inner ionosheath where we have VT = 100 km/s thus implying λH = 3 × 10 3 km.A similar variation is also applicable from the relation αk = 0.75 nswBVT λH that derives from studies of magnetic field fluctuations applied to particle motion.In particular, with ‫ﬠ‬ = 3 × 10 5 cm 2 /s for the viscosity coefficient we obtain λH = 3 × 10 4 km when VT = 6 × 10 6 cm s −1 in the solar wind, and λH = 2 × 10 4 km when VT = 10 7 cm s −1 in the inner ionosheath (see Appendix A).These latter values exceed by an order of magnitude those derived from Equation (1a) and thus suggest conditions that may not be applicable to wave particle interaction.In fact, no magnetic fluctuations can ~10 5 km 2 /s value for the kinematic viscosity coefficient could also be employed thus leading to a smaller λ H ~10 4 km value for the mean free path being closer to those inferred from the wave-particle interactions.
The validity of Equation (1a) can also be supported by noting, through an unrelated example that is applicable to air flows where there is also a relationship between the viscosity coefficient and the mean free path value.In fact, with V T = 4.5 × 10 4 cm s −1 for the thermal speed of air particles at room temperature, together with λ = 4 × 10 −5 cm for their mean free path value [14], Equation (1a) leads to Galaxies 2024, 12, x FOR PEER REVIEW 6 dusk Meridian (left panel) and of the noon-midnight Meridian (right panel).Regions and bo aries are marked on the right-hand side as the I-sphere (the ionopause (IP), and the ionosheath (from Lundin et al., (2011) [17]).
With these numbers, we can estimate the total value of the factor multiplying within the square parenthesis on the right side of Equation ( 5), since it has to be equ the number one so that it is equivalent to on the left side.Thus, we can compare the tribution of its terms and require that (L/δsw) 2 /R ≈ 1 × since (VA/U0) 2 ~ 0.04 can be negle Using the values for L and δsw indicated above, we have (L/δsw) 2 ~ 9 so that R ≈ 9, and ρ0U0L/µ ergs °K−1 is the Boltzmann number, and that derives from stud magnetic field fluctuations applied to particle motion [19].At the same time, the valu ~ 6 ergs cm −1 K −1 s −1 with the thermal conductivity is obtained by combining equatio and 1b which lead to αk = ρcp VT λH; so that by using nsw = 3 cm −3 and VT = 60 km freestream conditions in Figure 1  km (see Appendix A) and that varies with the VT value 0.6).As a whole, the mean free path derived here is in a value range nearly one ord magnitude larger than that implied above from considerations on the viscous transp momentum as discussed by Liepmann and Roshko [14].In summary, two methods ing to similar results have been proposed to derive mean free path values of the solar that interacts with the Venus upper ionosphere, either through wave-particle interac in terms of their relationship between the viscous transport of momentum and the conductivity, and also by applying magnetic field fluctuations to particle motion.difference may be characteristic of wave-particle interactions.
A peculiar property in both procedures that lead to the mean free path λH val that as shown in Equation (1a) they provide a relationship between this parameter the thermal speed VT of the flow particles.In both procedures higher VT speeds i lower λH values.This variation indicates that the thermal speed profile VT in Figure 1 to two different speed values, namely ~60 km/s in the solar wind before the inbound shock crossing marked as (1) and after the outbound crossing marked as (5), and d ently, they reach ~100 km/s values through the inbound ionosheath between the m labeled 2 and 3.As shown in Equation (1a) an increase in the thermal speed imp decrease in the magnitude of the mean free path.This is the case in the solar wind w VT = 60 km/s implies λH = 5 × 10 3 km, and also in the inner ionosheath where we have 100 km/s thus implying λH = 3 × 10 3 km.A similar variation is also applicable from relation αk = 0.75 nswBVT λH that derives from studies of magnetic field fluctuation plied to particle motion.In particular, with ‫ﬠ‬ = 3 × 10 5 cm 2 /s for the viscosity coefficien obtain λH = 3 × 10 4 km when VT = 6 × 10 6 cm s −1 in the solar wind, and λH = 2 × 10 4 km w VT = 10 7 cm s −1 in the inner ionosheath (see Appendix A).These latter values exceed b order of magnitude those derived from Equation (1a) and thus suggest conditions may not be applicable to wave particle interaction.In fact, no magnetic fluctuation = 1.8 cm 2 /s for the kinematic viscosity coefficient of air flows at atmospheric pressures.Such value is comparative to those reported by Hughes and Brighton (1967) [20] under such conditions.The implication is that the viscosity coefficient and the thermal temperature of the solar wind derived from measurements may also lead to approximate mean free path values that satisfy Equation (1a).

Discussion
The main argument examined in this study has been to employ a procedure used in gas dynamic theories to support a fluid view that describes the behavior of the solar wind in its interaction with the Venus upper ionosphere.The procedure is useful in the sense that despite the absence of collisions among the particles of both populations, it is possible to transfer their statistical properties through dissipation processes that are necessary to validate a continuum flow approach.Under such circumstances, the continuum flow contact between both plasma populations is provided via wave-particle interactions that are produced through the viscous transport of momentum and thermal conductivity.
As a whole, the procedure is different from that used in standard techniques where the combined motion of individual particles is followed as they lead to plasma instabilities and electric current systems that in the end produce wave-particle interactions as well [21][22][23].In all those studies, it is desirable to validate weather Maxwell-Boltzmann velocity distributions are adequate to represent the solar wind motion.Suitable examples of its velocity profiles are derived from measurements conducted with the PVO spacecraft in the Venus ionosheath that are presented in Figure 6 [24].Those profiles show shapes that resemble Maxwell-Boltzmann distribution functions in the outer ionosheath (labeled I and IV in the upper panel), while an energy profile with lower values is derived from measurements in the inner ionosheath (labeled II).A more complicated distribution (labeled III) shows two peak values that describe different conditions in the outer and in the inner ionosheath.Between those peak values, there is a plasma transition, as identified in Figure 2, that is related to the effects of dissipative viscous processes responsible for the plasma heating revealed by the temperature profile shown in that figure .Galaxies 2024, 12, x FOR PEER REVIEW 8 of 11 coefficient ‫ﬠ‬ during the Mariner 5 trajectory in Figure 1.The connecting line labeled "W" refers to a value in the Venus inner and the outer ionosheath that is implied by wave-particle interactions.The connecting line labeled "F" is implied by the magnetic field fluctuations.

Discussion
The main argument examined in this study has been to employ a procedure used in gas dynamic theories to support a fluid view that describes the behavior of the solar wind in its interaction with the Venus upper ionosphere.The procedure is useful in the sense that despite the absence of collisions among the particles of both populations, it is possible to transfer their statistical properties through dissipation processes that are necessary to validate a continuum flow approach.Under such circumstances, the continuum flow contact between both plasma populations is provided via wave-particle interactions that are produced through the viscous transport of momentum and thermal conductivity.
As a whole, the procedure is different from that used in standard techniques where the combined motion of individual particles is followed as they lead to plasma instabilities and electric current systems that in the end produce wave-particle interactions as well [21][22][23].In all those studies, it is desirable to validate weather Maxwell-Boltzmann velocity distributions are adequate to represent the solar wind motion.Suitable examples of its velocity profiles are derived from measurements conducted with the PVO spacecraft in the Venus ionosheath that are presented in Figure 6 [24].Those profiles show shapes that resemble Maxwell-Boltzmann distribution functions in the outer ionosheath (labeled I and IV in the upper panel), while an energy profile with lower values is derived from measurements in the inner ionosheath (labeled II).A more complicated distribution (labeled III) shows two peak values that describe different conditions in the outer and in the inner ionosheath.Between those peak values, there is a plasma transition, as identified in Figure 2, that is related to the effects of dissipative viscous processes responsible for the plasma heating revealed by the temperature profile shown in that figure.As a result of momentum transfer between the solar wind and the Venus upper ionosphere, there is evidence in the solar wind speed profile of the H + ions depicted in Figure 4 that there is a significant speed decrease from the freestream 300 km/s values Galaxies 2024, 12, 28 9 of 11 measured above the interaction region in the dawn-dusk Meridian (left panel) to ~10 km/s values measured across a layer that extends from ~10 4 km altitudes to ~10 2 km altitudes.A similar contour is also encountered in the noon-midnight speed profile (right panel) but with variations.In particular, in the latter case, there are smaller speed values measured within that layer suggesting the spacecraft transit through a plasma channel that mostly develops by the vicinity of the midnight plane (Pérez-de-Tejada, 2023 [25], see Figure 4.11).In addition, the sudden increase in the speed values of the planetary H + ions with an altitude from ~10 km/s to ~50 km/s above ~5 × 10 3 km in the upper part of the speed profiles in both panels of Figure 4 also be accounted for.Such a change is consistent with the entry of a spacecraft through the narrow section of a corkscrew flow shape as that indicated in Figure 7 (see [13], Figure 5.15).In fact, momentum flux conservation requires higher flow speed values within the corkscrew when its cross-section decreases.This would be the case when the spacecraft moves through the thinner region of the corkscrew as it moves along the wake.Wave-particle interactions are necessary to allow for momentum flux conservation to increase the flow speed depending on the width of the corkscrew configuration.Such a change in the shape of the corkscrew flow, shown in Figure 7, implies a necessary increase in the local flow speed at high altitudes across its thin regions, as reported in both panels of Figure 4.
time (their position is noted in the upper panel along the PVO trajectory).Positions A, B, and C in spectrum III mark the time when the ion fluxes were obtained ( [24]).
As a result of momentum transfer between the solar wind and the Venus upper ionosphere, there is evidence in the solar wind speed profile of the H + ions depicted in Figure 4 that there is a significant speed decrease from the freestream 300 km/s values measured above the interaction region in the dawn-dusk Meridian (left panel) to ~10 km/s values measured across a layer that extends from ~10 4 km altitudes to ~10 2 km altitudes.A similar contour is also encountered in the noon-midnight speed profile (right panel) but with variations.In particular, in the latter case, there are smaller speed values measured within that layer suggesting the spacecraft transit through a plasma channel that mostly develops by the vicinity of the midnight plane (Pérez-de-Tejada, 2023 [25], see Figure 4.11).In addition, the sudden increase in the speed values of the planetary H + ions with an altitude from ~10 km/s to ~50 km/s above ~5 × 10 3 km in the upper part of the speed profiles in both panels of Figure 4 should also be accounted for.Such a change is consistent with the entry of a spacecraft through the narrow section of a corkscrew flow shape as that indicated in Figure 7 (see [13], Figure 5.15).In fact, momentum flux conservation requires higher flow speed values within the corkscrew when its cross-section decreases.This would be the case when the spacecraft moves through the thinner region of the corkscrew as it moves along the wake.Wave-particle interactions are necessary to allow for momentum flux conservation to increase the flow speed depending on the width of the corkscrew configuration.Such a change in the shape of the corkscrew flow, shown in Figure 7, implies a necessary increase in the local flow speed at high altitudes across its thin regions, as reported in both panels of Figure 4. Its geometry is equivalent to that of a vortex flow in the Venus wake, with its width and position varying during the solar cycle.Near the solar cycle minimum, the vortex is located closer to Venus (located by the right side) and there are also indications that its width becomes smaller with increasing distance downstream from the planet [13].
A distinct characteristic of the procedure described here and that is based on the effects of dissipation processes is that the calculations of values for the mean free path of the solar wind particles subject to such conditions are comparable to those of the Larmor radius in gyrotropic trajectories.The latter are due to their motion within the ~10 nT intensity of the magnetic field measured in the solar wind.The difference between both situations is that in both cases, the particle trajectory is entirely different with stochastic variations being dominant in the region where a fluid description is applicable.Gyrotropic trajectories occur, on the other hand, far from the interaction region.
It is of interest to note that that there is also observational evidence of vortex flow structures measured in the solar wind-Mars ionosphere boundary, as inferred from plasma features in the vicinity of the Mars ionosphere detected with the plasma data of the Maven spacecraft.Ruhunusiri et al. (2016) [26] identified vortex plasma waves with Its geometry is equivalent to that of a vortex flow in the Venus wake, with its width and position varying during the solar cycle.Near the solar cycle minimum, the vortex is located closer to Venus (located by the right side) and there are also indications that its width becomes smaller with increasing distance downstream from the planet [13].
A distinct characteristic of the procedure described here and that is based on the effects of dissipation processes is that the calculations of values for the mean free path of the solar wind particles subject to such conditions are comparable to those of the Larmor radius in gyrotropic trajectories.The latter are due to their motion within the ~10 nT intensity of the magnetic field measured in the solar wind.The difference between both situations is that in both cases, the particle trajectory is entirely different with stochastic variations being dominant in the region where a fluid description is applicable.Gyrotropic trajectories occur, on the other hand, far from the interaction region.
It is of interest to note that that there is also observational evidence of vortex flow structures measured in the solar wind-Mars ionosphere boundary, as inferred from plasma features in the vicinity of the Mars ionosphere detected with the plasma data of the Maven spacecraft.Ruhunusiri et al. (2016) [26] identified vortex plasma waves with average periods of nearly 3 min by the boundary of the Mars ionosphere that are comparable to those derived from the Venus ionosheath measurements [13].
A useful outcome in the onset of wave-particle interactions for the solar wind that mixes up with planetary/cometary plasmas is that similar conditions should also be applicable in the interaction of stellar winds with exoplanets.In such cases, there should also be evidence of a fluid dynamic response in the behavior of the interacting plasmas.Similarly, it would be of interest to examine whether a fluid dynamic approach is also applicable to larger (galactic)-scale plasma flow interactions involving plasma-directed flows mixing with rotating plasmas.
A separate manner to derive λH is available by using the relation: αk = 0.75 nswBVT•λH where B = 1.4 × 10 −16 ergs °K−1 is the Boltzmann number, and that derives from studies of magnetic field fluctuations applied to particle motion [19].At the same time, the value αk ~ 6 ergs cm −1 K −1 s −1 with the thermal conductivity is obtained by combining equations 1a and 1b which lead to αk = ρcp VT λH; so that by using nsw = 3 cm −3 and VT = 60 km/s for freestream conditions in Figure 1 we have: λH = αk × [0.75 nswBVT] −1 which leads to λH = (6 ergs cm −1 s −1 °K−1 ) × [0.75 × (3 cm −3 ) × (1.4 × 10 −16 ergs °K−1 ) × (6 × 10 6 cm s −1 )] −1 =3 × 10 9 cm thus implying λH ~ 3 × 10 4 km (see Appendix A) and that varies with the VT value (α ~ 0.6).As a whole, the mean free path derived here is in a value range nearly one order of magnitude larger than that implied above from considerations on the viscous transport of momentum as discussed by Liepmann and Roshko [14].In summary, two methods leading to similar results have been proposed to derive mean free path values of the solar wind that interacts with the Venus upper ionosphere, either through wave-particle interactions in terms of their relationship between the viscous transport of momentum and thermal conductivity, and also by applying magnetic field fluctuations to particle motion.That difference may be characteristic of wave-particle interactions.
A peculiar property in both procedures that lead to the mean free path λH value is that as shown in Equation (1a) they provide a relationship between this parameter and the thermal speed VT of the flow particles.In both procedures higher VT speeds imply lower λH values.This variation indicates that the thermal speed profile VT in Figure 1 leads to two different speed values, namely ~60 km/s in the solar wind before the inbound bow shock crossing marked as (1) and after the outbound crossing marked as (5), and differently, they reach ~100 km/s values through the inbound ionosheath between the marks labeled 2 and 3.As shown in Equation (1a) an increase in the thermal speed implies a decrease in the magnitude of the mean free path.This is the case in the solar wind where VT = 60 km/s implies λH = 5 × 10 3 km, and also in the inner ionosheath where we have VT = 100 km/s thus implying λH = 3 × 10 3 km.A similar variation is also applicable from the relation αk = 0.75 nswBVT λH that derives from studies of magnetic field fluctuations applied to particle motion.In particular, with ‫ﬠ‬ = 3 × 10 5 cm 2 /s for the viscosity coefficient we obtain λH = 3 × 10 4 km when VT = 6 × 10 6 cm s −1 in the solar wind, and λH = 2 × 10 4 km when VT = 10 7 cm s −1 in the inner ionosheath (see Appendix A).These latter values exceed by an order of magnitude those derived from Equation (1a) and thus suggest conditions that may not be applicable to wave particle interaction.In fact, no magnetic fluctuations can = 3 × 10 15 cm 2 •s −1 and using n sw = 3 cm −3 from Figure 1

Figure 1 .
Figure 1.(Lower panel) Trajectory of the Mariner 5 spacecraft projected in cylindrical coordinates in its flyby past Venus.Labels 1 to 5 along the trajectory mark important events in the plasma properties (a bow shock is identified at features 1 and 5), and the intermediate plasma transition occurs at features 2 and 4).(Upper panel) Magnetic field intensity and its latitudinal and azimuthal orientation, together with the plasma properties (thermal speed, density, and bulk speed) measured around Venus [1].
for the planetary O + fluxes and the electron component measured by the flanks of the Venus ionosheath ([8] see Figures 2 and 3 by 01:50 UT).Galaxies 2024, 12, x FOR PEER REVIEW 3 of 11 plasma data with higher temperatures for the planetary O + fluxes and the electron component measured by the flanks of the Venus ionosheath ([8] see Figures 2 and 3 by 01:50 UT).

Figure 2 .
Figure 2. Ion speed and temperature measured along the orbit of Venera 10 on 19 April 1976.The Venera orbit in cylindrical coordinates is shown at the top.The temperature burst at position 1 was recorded during a flank crossing of a bow shock.A boundary layer is apparent by the increase in temperature and decrease in speed, and is initiated by the intermediate transition at the position labeled 2. A latter discontinuity in the boundary layer temperature profile corresponds to the boundary of the magneto-tail (from [6]).

Figure 3 .
Figure 3. Vector velocity speeds of the trans-terminator flow in the Venus upper ionosphere measured with instruments onboard the Pioneer Venus Orbiter spacecraft [9].

Figure 2 .
Figure 2. Ion speed and temperature measured along the orbit of Venera 10 on 19 April 1976.The Venera orbit in cylindrical coordinates is shown at the top.The temperature burst at position 1 was recorded during a flank crossing of a bow shock.A boundary layer is apparent by the increase in temperature and decrease in speed, and is initiated by the intermediate transition at the position labeled 2. A latter discontinuity in the boundary layer temperature profile corresponds to the boundary of the magneto-tail (from [6]).

Galaxies 2024 ,
12, x FOR PEER REVIEW 3 of 11 plasma data with higher temperatures for the planetary O + fluxes and the electron component measured by the flanks of the Venus ionosheath ([8] see Figures 2 and 3 by 01:50 UT).

Figure 2 .
Figure 2. Ion speed and temperature measured along the orbit of Venera 10 on 19 April 1976.The Venera orbit in cylindrical coordinates is shown at the top.The temperature burst at position 1 was recorded during a flank crossing of a bow shock.A boundary layer is apparent by the increase in temperature and decrease in speed, and is initiated by the intermediate transition at the position labeled 2. A latter discontinuity in the boundary layer temperature profile corresponds to the boundary of the magneto-tail (from [6]).

Figure 3 .
Figure 3. Vector velocity speeds of the trans-terminator flow in the Venus upper ionosphere measured with instruments onboard the Pioneer Venus Orbiter spacecraft [9].

Figure 3 .
Figure 3. Vector velocity speeds of the trans-terminator flow in the Venus upper ionosphere measured with instruments onboard the Pioneer Venus Orbiter spacecraft [9].

Figure 4 .
Figure 4. Measured flow velocities versus VEX altitude for solar wind H + ions, and ionospheric H + and O + ions.The curve marked vesc illustrates escape velocity versus altitude above Venus.The data points represent average values in 50 km altitude intervals sampled within Y = +0.5 of the dawn-

Figure 4 .
Figure 4. Measured flow velocities versus VEX altitude for solar wind H + ions, and ionospheric H + and O + ions.The curve marked v esc illustrates escape velocity versus altitude above Venus.The data points represent average values in 50 km altitude intervals sampled within Y = +0.5 of the dawn-dusk Meridian (left panel) and of the noon-midnight Meridian (right panel).Regions and boundaries are marked on the right-hand side as the I-sphere (the ionopause (IP), and the ionosheath (IMB) (from Lundin et al., (2011) [17]).

alaxies 2024 ,
12, x FOR PEER REVIEW 6 of 11 dusk Meridian (left panel) and of the noon-midnight Meridian (right panel).Regions and boundaries are marked on the right-hand side as the I-sphere (the ionopause (IP), and the ionosheath (IMB) (from Lundin et al., (2011) [17]).

Figure 5 .
Figure 5. Mean free path values λH of the solar wind obtained in wave-particle interactions and also in magnetic field fluctuations using the solar wind thermal speed VT and its kinematic viscosity

= 9 ,
leading to an equivalent kinematic viscosity coefficient ‫ﬠ‬ = µ/ρ0 ~ 3 × 10 5 k by using the values for Uo and L given above.This value of the kinematic viscosity c cient is due to the low mass density ρ0 = 1.6 × 10 −23 gr cm −3 of the solar wind and, a same time, it is related to the ability of particle motion to eliminate velocity variations give rise to viscous transport ([18], see p. 37).Also, the corresponding value of the viscosity coefficient µ = ‫•ﬠ‬ρ0 ~ 5 × 10 −8 gr cm −1 s −1 is a measure of the internal frictio posing deformation of the flow ([18], see p. 36).With such values for ‫ﬠ‬ and µ, it is pos to validate Equation (5) and, at the same time, employ the kinematic viscosity coeffi ‫ﬠ‬ to calculate, together with the thermal speed VT ≈ 10 2 km/s, the mean free path valu in Equation (1a).When VT ≈ 10 2 km/s we obtain λH ~ 3 × 10 3 km and when VT ≈ 60 km have λH ~ 5 × 10 3 km.A separate manner to derive λH is available by using the relation: αk = 0.75 nswB where B = 1.4 × 10 −16

Figure 6 .
Figure 6.(Upper panel) Trajectory of the PVO in orbit 87 projected on one quadrant in cylindrical coordinates.The bow shock, the intermediate transition, and the ionopause are indicated.(Lower panel) Ion flux values measured as a function of energy in cycles I, II, III, and IV state their start

Figure 6 .
Figure 6.(Upper panel) Trajectory of the PVO in orbit 87 projected on one quadrant in cylindrical coordinates.The bow shock, the intermediate transition, and the ionopause are indicated.(Lower panel) Ion flux values measured as a function of energy in cycles I, II, III, and IV state their start time (their position is noted in the upper panel along the PVO trajectory).Positions A, B, and C in spectrum III mark the time when the ion fluxes were obtained ([24]).

Figure 7 .
Figure 7.View of a corkscrew vortex flow in fluid dynamics.Its geometry is equivalent to that of a vortex flow in the Venus wake, with its width and position varying during the solar cycle.Near the solar cycle minimum, the vortex is located closer to Venus (located by the right side) and there are also indications that its width becomes smaller with increasing distance downstream from the planet[13].

Figure 7 .
Figure 7.View of a corkscrew vortex flow in fluid dynamics.Its geometry is equivalent to that of a vortex flow in the Venus wake, with its width and position varying during the solar cycle.Near the solar cycle minimum, the vortex is located closer to Venus (located by the right side) and there are also indications that its width becomes smaller with increasing distance downstream from the planet[13].