The solar wind is a magnetized collisionless plasma comprised of ions (mostly protons) and electrons. (Collisionless means that inter-particle collisions that would disrupt the ion or electron orbits in the magnetic and electric fields are rare.) The solar-wind plasma flows at a high velocity radially outward from the Sun, in the range 250 km/s to 1000 km/s. Typical parameters for the solar wind at Earth appear in Table 1. The magnetic field B in the solar-wind plasma originates at the Sun and is carried outward with the supersonic plasma flow. Because of the rotation of the Sun and the radial outflow of the wind, the magnetic field in the solar wind has on average a “Parker-spiral” orientation [1] . The magnetic-field vector has strong localized variations about the Parker-spiral direction: these spatial variations in B are advected outward with the solar-wind flow [2] . Much of the spatial variation of B is owed to the presence of current sheets embedded in the solar-wind plasma [3] [4] , with the value of the magnetic-field vector B jumping across the current sheets.

The motional electric field E of a plasma is a Lorentz transformation of the magnetic field B in the plasma [5] . For a velocity vector v between two reference frames, the Lorentz transformation of the components of the electric and magnetic fields parallel to and perpendicular to v is [6]

where and where we will take the primed frame to be at rest in the solar-wind plasma and the unprimed frame to be the rest frame of the Earth or the Sun. In the rest frame of the solar-wind plasma,. This is because (in the absence of non-electromagnetic forces such as gravity) in a collisionless plasma particle orbit theory says that the ions and the electrons E-cross-B drift with an averaged velocity of (cf. eq. (12.43) of Reference [6] ). In the rest frame of the plasma there can be no E-cross-B drift velocity of the ions and electrons, otherwise it would not be the rest frame of the plasma. Therefore, in the rest frame of the solar-wind plasma. In the reference frame of the Earth or the Sun, the plasma flow is of the order of v/c ~ 10⁻³, so γ @ 1 can be taken. For expression (1d) gives. For γ @ 1, , and, expression (1b) yields the motional electric field E for a plasma moving with a velocity v in a magnetic field B to be

which is the low-velocity Lorentz transformation common for a moving conducting fluid [6] [7] . The solar-wind motional electric field as seen by the Earth (expression (2)) is often taken to be the driver of geomagnetic activity in the Earth’s magnetosphere-ionosphere system [8] (but see Reference [9] for a contrary interpretation).

In this paper, we wish to gain an understanding of the solar-wind motional electric field from first principles. The motional electric field of the solar wind will be examined via the Lorentz transformation of the magnetic-field structure in the solar wind. Because of the spatial variations of the magnetic-field vector B, the motional electric field will have spatial variations. Via Maxwell’s equations, the sources of the motional electric field will be sought: charge densities in the solar wind and time derivatives of current densities in the solar wind. The question will be asked whether the motional electric field of the solar wind is an electrostatic electric field, an

inductive electric field, or both. We wish to gain an understanding of the solar-wind motional electric field from first principles. Identifying the electrostatic versus induction portions of electric fields within the Earth’s magnetosphere has been of interest because the identification yields an understanding of the origin of the electric fields and a capability to add them to numerical models of the magnetosphere [10] [11] .

To avoid confusion, the reader should note that there is also a non-motional electric field in the solar wind associated with the “interplanetary electrostatic potential” or “heliospheric electric potential” [12] . This is a global parallel-to-B electric field associated with the differences in the collisionless transport of the solar-wind electrons and the solar-wind ions along the magnetic field. The sign of this interplanetary electric field is outward along the magnetic field lines, accelerating ions away from the Sun and retarding the outward electron flux. This interplanetary electrostatic potential is responsible for the observed statistical increase of the bulk speed of the solar wind v_sw between 0.3 AU and 1 AU. This interplanetary electrostatic potential is not the subject of the present study.

According to the Helmholtz decomposition [13] [14] , any vector field can be decomposed into the sum of an irrotational (curl-free) field plus a solenoidal (divergence-free) field. Hence, any vector electric field E can be written as the superposition

where E_es is an electrostatic electric field that is everywhere curl-free and E_ind is an induction electric field that is everywhere divergence-free. Further, the field E can be determined everywhere in space from knowledge of the values of the divergence of the field E and the curl of the field E everywhere in space [14] . This determination of E from and is simplified by the decomposition as follows. When one takes one finds the regions where and when one takes one finds the regions, where. From all of the regions of one can obtain E_es everywhere in space and from all of the regions where one can obtain E_ind everywhere in space. Knowing E_es everywhere in space and E_ind everywhere in space, one can obtain E everywhere in space.

For this paper the following equivalence of adjectives for each of the two types of electric fields is declared. First type: inductive = divergence-free = solenoidal = non-potential = transverse. Second type: electrostatic = curl-free = irrotational = potential = longitudinal. For an electrostatic field E_es, the electric-field lines end; for an induction field E_ind, the electric-field lines do not end.

Here we will define a purely electrostatic electric field E_es as an irrotational field having its origin in regions where the divergence is nonzero. Likewise, we will define a purely induction electric field E_ind as a solenoidal field having its origin in regions where the curl is nonzero. From this and from expression (3) it follows that and that. From Coulomb’s law

it is seen that the electrostatic field originates from regions of nonzero charge density ρ_q. From Faraday’s law of induction

it is seen that the induction field originates from regions of nonzero ¶B/¶t. It is important to explore expression (5) a little further. Taking the curl of Faraday’s law, using Ampere’s law with the displacement current neglected (negligible in the limit [15] ) to replace, and taking in the expansion yields

(with) for the induction field, which indicates that the induction electric field E_ind has its origin in regions where the time derivative of the current density ¶j/¶t is nonzero.

The motional electric field of the solar wind is sometimes called the induced or inductive electric field of the solar wind because it is “induced” by the motion of the plasma. In this paper more care will be taken with the “induction” nomenclature.

In Figure 1, the solar-wind speed and two components transverse to radial (from the Sun) of solar-wind motional electric field near Earth are plotted as functions of time for 100 days in the year 2004. The GSE (Geocentric Solar Ecliptic) right-handed coordinate system is used where X points from the Earth toward the Sun, Y points transverse to the Sun-Earth line in the ecliptic plane opposite to the direction of the Earth’s motion about the Sun, and Z is normal to the ecliptic plane pointing in the northward direction. In the top panel the solar-wind radial velocity v_sw is plotted, in the middle panel and bottom panel the north-south component E_z and the dawn-dusk component E_y of the solar-wind motional electric field are plotted. The black points are 64-s time resolution values and the red points are 24-hr running averages. The electric field is calculated from measurements of the bulk-flow vector v_sw by the SWEPAM instrument [16] on the ACE spacecraft in the solar wind and from measurements of the magnetic-field vector B from the MAG instrument [17] on ACE. The value of the observed motional electric field depends on the reference frame of the observer, in this case the ACE spacecraft in approximately the rest frame of the Earth. Note the reversals in the sign of the north-south component E_z (e.g. Days 111, 122, 139, 149, 165, 177, and 191): these reversals occur at sector reversals of the solar-wind magnetic field wherein the into-the-Sun versus out-of-the-Sun polarity of the solar-wind magnetic field changes. These magnetic sector reversals typically occur prior to intervals of high-speed wind.

A higher-time-resolution look at the motional electric field of the solar wind appears in Figure 2. Here, plasma [18] and magnetic-field [19] measurements from the WIND spacecraft with 3-s time resolution are used to calculate. In the top panel E_y and E_z are plotted for two hours of fast solar wind and in the bottom panel E_y and E_z are plotted for two hours of slow solar wind. Note in Figure 2 that the temporal behavior of the motional electric field is dominated by sudden changes of the values of E_y and E_z; these sudden changes occur when a current sheet in the solar wind passes the spacecraft. This is the “mesoscale structure” of the solar wind, dominated by magnetic flux tubes bounded by current sheets.

In the upper two panels of Figure 3 the vector tips of the solar-wind motional electric field are plotted in the E_y-versus-E_z plane. The left panel is for 24-hr averages of the electric field (from the multispacecraft OMNI2 data set [20] ) and the right panel is for 64-s values of the electric field (from the ACE spacecraft). The 24-hr averages are much more dominated by a north-south orientation to E, whereas the 64-s values in the right panel are more isotropic. Basically, the deviation of from purely north-south is owed to the deviation of the solar-wind magnetic field B from the Parker-spiral direction. In the lower two panels of Figure 3 the vector tips of the solar-wind magnetic field B are plotted in the B_y-versus-B_z plane, left panel being 24-hr averages of the magnetic field (from OMNI2) and the right panel being 64-s values (from ACE). The 24-hr averages of the electric and magnetic field (left panels) are dominated by a Parker-spiral orientation with B_y > 0 and E_z > 0 for out-of-the-Sun (away) magnetic fields and B_y < 0 and E_z < 0 for into-the-Sun (toward) magnetic fields.

Owing to the 27-day rotation of the Sun, the magnetic field in the solar-wind plasma has a global Archimides- spiral pattern. In (r, θ, ϕ) spherical coordinates the Parker-spiral magnetic field B at the radius r from the Sun and colatitude θ in the heliosphere is given by [1]

where B_o is a reference magnetic-field strength near the Sun (with B_o > 0 for an out-of-the-Sun (away) field and B_o < 0 for an into-the-Sun (toward) field); r_o is the reference radius where B_o is measured; ω » 2.9 ´ 10⁻⁶ radians/s is the angular rotation rate of the solar surface at the solar equator; and v_sw is the solar-wind radial velocity. The angle θ = 0˚ at the north pole of the Sun and θ = 180˚ at the south pole. In general, the Parker-spiral magnetic field of the heliosphere is of one sector (away or toward) in the northern hemisphere of the heliosphere and is of the opposite sector (toward or away) in the southern hemisphere, with a heliospheric current sheet separating the two morphologies [21] . In Figure 4 the Parker-spiral motional electric field of the solar wind will be sketched with the heliospheric current sheet lying in the equatorial plane bounding an away sector (B_o > 0) in the northern hemisphere and a toward sector (B_o < 0) in the southern hemisphere.

For the Parker-spiral magnetic field given by expression (7), the motional electric field of expression (2) for a radial solar-wind velocity v_sw = (v_sw, 0, 0) is

this electric field is sketched as the red field lines in Figure 4. The motional electric field E for the Parker spiral is purely poloidal: E is northward for an away-from-the-Sun magnetic-field orientation and E is southward for a toward-the-Sun magnetic-field orientation. Note in expression (8) that E_θ does not depend on v_sw.

Examining expression (8) in spherical coordinates one finds everywhere for the Parker-spiral electric field. The distributed charge density in the solar-wind plasma that supports this global electrostatic electric field is given by Coulomb’s law: taking the divergence of expression (8) in spherical coordinates yields

a distributed charge density throughout the heliosphere. This charge density is sketched as the blue “+” and “−“ symbols in Figure 4. According to expression (9) the charge density is strong near the polar regions of the heliosphere (θ ~ 0˚ and θ ~ 180˚) and weak near the equatorial regions (θ ~ 90˚). For an away-from-the-Sun magnetic-field orientation (B_o > 0) the distributed charge density ρ_q is negative in the northern hemisphere and positive in the southern hemisphere and for a toward orientation (B_o < 0) the distributed charge density ρ_q is positive in the northern hemisphere and negative in the southern hemisphere. Note in expression (9) that ρ_q does not depend on the solar-wind velocity v_sw.

With everywhere, the global Parker-spiral electric field is purely electrostatic with its origin in regions where. Note in Figure 4 that the electric-field lines of the global Parker-spiral motional electric field end.

The distributed current density j in the solar-wind plasma that supports the global Parker-spiral magnetic-field geometry is calculated by applying Ampere’s law to the Parker-spiral magnetic field given by expression (7): taking the curl of expression (7) in spherical coordinates yields

This distributed current density is purely radial and it is strong in the polar regions of the heliosphere and weak in the equatorial regions, with j_r negative (toward the Sun) when the magnetic field is away from the Sun (B_o > 0) and j_r is positive (away from the Sun) when the magnetic field is toward the Sun (B_o < 0).

Taking the heliospheric current sheet to lie in the equatorial plane of the heliosphere there is a radial component of the current in the sheet associated with the jump in the azimuthal component of the Parker-spiral magnetic field across the current sheet and there is an azimuthal current in the sheet associated with the reversal of the radial component of the magnetic field across the current sheet. Taking the thickness of the heliospheric current sheet to be d, Ampere’s law gives

for the magnitudes of the radial and azimuthal current densities in the heliospheric current sheet, note that the thickness d of the heliospheric current sheet could be a function of r, so the current density in the heliospheric current sheet could vary with r differently than the right-hand sides of expressions (11a) and (11b).

Note that the poloidal motional electric field of the solar wind reverses across the current sheet owing to the reversal of the direction of B across the sheet as v_sw remains radially outward. This electric-field reversal across the current sheet is sketched in Figure 4. There is a nonzero charge density in the equatorial current sheet associated with the reversal of the poloidal electric field: Coulomb’s law gives this charge density ρ_q as

where again d is the thickness of the current sheet, and where again d could be a function of r. The charge density of the heliospheric current sheet is positive if the magnetic field is outward (from the Sun) in the northward hemisphere and inward in the southern hemisphere (as drawn in Figure 4): the charge density of the heliospheric current sheet is negative if the magnetic field is inward in the northern hemisphere and outward in the southern hemisphere. The source of this charge density is the relativistic transformation of current density into charge density [22] , which will be discussed in Section 5.1.

Owing to the orbital motion of the Earth around the Sun, the radial component B_r of the Parker-spiral magnetic field gives rise to an additional north-south poloidal electric field as seen by the Earth. The orbital velocity of the Earth around the Sun is v_orbit » 29.8 km/s in the dawnward direction, which is small compared with v_sw and so the orbital motional electric field is small compared with the Parker-spiral motional electric field. The motional electric field in the reference frame of the Earth is southward when the Earth is in an away magnetic sector and northward when the Earth is in a toward magnetic sector, always opposite to the sign of the Parker-spiral motional electric field.

In the real solar wind there are several complications to this ideal-Parker-spiral picture. First, the global solar wind contains solar ejecta, which are impulsively ejected large-scale plasmas with non-Parker-spiral-oriented magnetic fields. Second, the solar-wind velocity is not uniform everywhere: rather there are regions on the rotating Sun that emit fast wind and regions that emit slower wind, disturbing the uniform-Parker-spiral picture with compressions and rarefactions. Third, there is a tilt to the solar magnetic dipole, plus magnetic multipoles that invalidate the uniform-northern-hemisphere uniform-southern-hemisphere picture of Figure 4. Fifth, the heliospheric current sheet was drawn to be planar in Figure 4 but in reality it is tilted and warped. Finally, there are large-amplitude fluctuations in the magnetic-field orientation about the Parker-spiral direction representing a mesoscale spatial structure of the solar-wind magnetic field. This mesoscale structure is considered in the following section.

In this section, the mesoscale spatial structure of the solar-wind motional electric field in the reference frame of the Earth is examined. That electric-field spatial structure is associated with the mesoscale magnetic-field spatial structure of the solar-wind plasma, which is largely flux tubes separated by current sheets [3] [4] . The advection of these magnetic spatial structures past the Earth results in temporal variations of E and B at the Earth manifested as sudden large changes in the vectors E and B associated with each current sheet passing the Earth (cf. Figure 2).

A Helmholtz decomposition of the motional electric field in the solar wind was made and the electrostatic versus induction nature of the field was analyzed as was the charge-density origin versus the time-varying-current- density origin of the field. To calculate the motional electric fields, Lorentz transformations were used; when Lorentz transforming the electric and magnetic fields, the currents and charges also had to be Lorentz transformed. Just as the transformation of the motional electric field E from B is important at non-relativistic solar-wind velocities, the transformation of the charge density ρ_q from the current density j is important at non-re- lativistic velocities.

The large-scale time-averaged electric field of the solar wind associated with the outward advection of the Parker-spiral magnetic field in the solar-wind plasma is an electrostatic electric field having its source in the distributed charge density in the solar-wind plasma and the more-concentrated charge density of the heliospheric current sheet. This large-scale electric field is in the poloidal direction, northward at Earth in away magnetic sectors and southward at Earth in toward magnetic sectors. The origin of the large-scale electric field is a relativistic transformation of the large-scale magnetic field in the solar-wind plasma and the origin of the distributed charge density is the same relativistic transformation of the distributed current density in the solar-wind plasma that supports the Parker-spiral magnetic field.

The mesoscale structure of the solar-wind electric field spatially varies from being dominantly electrostatic to being dominantly inductive. At the Earth the time-varying electric field will be dominantly inductive near current sheets that have their normal vectors nearly aligned with the Sun-Earth line and the time-varying electric field will be dominantly electrostatic near current sheets that have their normal vectors nearly perpendicular to the Sun-Earth line. In general the mesoscale electric field will be a combination of electrostatic and induction electric fields, varying with time as differently shaped magnetic structures pass over the Earth. In compression regions of the solar wind time-varying induction fields will tend to dominate, and in rarefaction regions of the solar wind time-varying electrostatic fields will tend to dominate.

Note that in the actual solar wind the solar-wind velocity vector v_sw also has spatial structure, on both large scales and mesoscales. On the mesoscale, there is usually a jump in the plasma velocity vector v as well as the magnetic-field vector B across the current sheets in the solar wind [27] . The jump in v across the current sheet amounts to a vorticity in the sheets; in a plasma there is a charge density ρ_q associated with vorticity [28] with the charge density in a vorticity layer given by eq. (11) of [9]

(Note that for charge density associated with vorticity there is no reference frame in which the charge density is zero.) In general the change in velocity ∆v across a solar-wind current sheet is much smaller in magnitude than the bulk flow velocity v_sw of the solar-wind plasma, so the inhomogeneity of the flow-velocity field is typically a less-than-10% effect on magnitude of the motional electric fields of the solar wind.

To conclude, a first-principles understanding of the motional electric field of the solar wind associated with the advection of magnetic structure in the solar-wind plasma has been developed. This understanding is based on the Lorentz transformations of electric and magnetic fields, the Lorentz transformations of current densities and charge densities, and the application of Maxwell’s equations.

As seen in Sections 4 and 5, there is a nonzero charge density ρ_q in the solar-wind plasma associated with the motional electric field of the solar-wind: a distributed charge density associated with the advection of the global Parker-spiral magnetic field and more-concentrated charge densities in the current sheets of the solar wind. The physical origin of that charge density is the difference in the Lorentz contractions of volumes containing the positive ions of the solar wind and the negative electrons of the solar wind when the ions and electrons move at (slightly) different velocities owing to the presence of an electrical current. The differences in ion versus electron velocity are on the order of 1 km/s in the solar-wind current sheets.

The charge density in the solar-wind plasma associated with the motional electric field is frame dependent, being zero in the reference frame of the plasma and nonzero in other reference frames such as the frame of the Earth or the Sun.

In the directions perpendicular to B, the solar-wind plasma behaves on large scales as an ideal-MHD fluid: in the parallel-to-B direction the MHD description of the collisionless solar wind can fail (cf. Sect. 1 of [29] for MHD tests that are failed by the solar wind). Sometimes in space physics textbooks it is declared that MHD has (e.g. eq. (5.22) of Parks [30] , eq. (4.82) of Gombosi [31] , eq. (6.1.31) of Gurnett and Bhattacharjee [32] , or Table 6-6 of Knipp [33] ) or equivalently textbooks declare that ρ_q = 0 (e.g. eq. (5.23) of Parks [30] or Sect. 6.1 of Gurnett and Bhattacharjee [32] ), which cannot be true relativistically (cf. expressions (13) and (17)) and which is not true in MHD computer simulations of the solar-wind flow (cf. Sect. 2.3 of reference [9] ).

In deriving ideal MHD, a quasineutrality condition n_p » n_e (ion number density approximately equals the electron number density) is often assumed. The assumption of quasineutrality does not mean that Coulomb’s law is irrelevant or inconsistent with the application of ideal MHD.

For the charge density in the solar wind, one can assess just how quasineutral the solar-wind plasma is by calculating the fractional charge imbalance n_q/n_o, where n_q = ρ_q/e is the number density of excess charges and where n_o » n_p » n_e is the number density of the plasma. A current sheet of the mesoscale structure of the solar wind plasma is considered. If the magnetic-field vector jumps by ∆B across the current sheet of thickness w, then the current density in the sheet is. The charge density ρ_q in the current sheet from the transformation is then and. If the magnetic-field vector across the current sheet jumps by ∆B = 5 ´ 10⁻⁵ Gauss, if the width of the current sheet is w = 2500 km, and if the speed of the solar wind is v_sw = 438 km/s (cf. Table 1), then the estimate for the number density of charges is n_q = 4.8 ´ 10⁻⁸ cm⁻³. A typical number density of the solar-wind plasma is n_o = 6.8 cm⁻³ (cf. Table 1). Thus n_q/n_o ~ 10⁻⁸ for the charge density of the solar wind.

This amount of charge imbalance is small, but it is not zero. Even with n_q/n_o ~ 10⁻⁸ in the solar wind, the limit ρ_q = 0 cannot be taken and: this is demonstrated with the following example. One can compare the solar-wind degree of quasineutrality to the degree of quasineutrality in a lightning bolt. The linear charge density Q/L in a conducting lightning channel is about Q/L = 10⁻⁴ C/m [34] . Dividing by e, this is a number of excess charges per unit length N/L = 6.2 ´ 10⁶ cm⁻¹. A conducting lightning channel has a radius r of about r = 1 cm [35] , so the net number density of charges n_q in the channel is n_q = 2 ´ 10⁶ cm⁻³_. The number density n_o of free electrons in the highly ionized lightning channel is about n_o = 10¹⁸ cm⁻³ [36] . Hence, the fractional charge density in the channel is n_q/n_o ~ 2 ´ 10⁻¹². (In actuality, the charge density n_q inside the channel is lower than 2 ´ 10⁶ cm⁻³ since the strong radial electrostatic electric field of the charged lightning channel pushes excess charge out of the channel where it migrates radially outward into the unionized air, into a “corona sheath” of radius of a meter or so [37] ). Note that the lightning channel, with an electrostatic potential of ~10⁸ V [34] , is several orders of magnitude more charge neutral than the n_q/n_o ~ 10⁻⁸ of the solar-wind plasma. Lightning is an electrodynamic problem in which Coulomb’s law (Gauss’ law, Poisson’s equation, capacitance calculations) cannot be ignored (e.g. [38] [39] ).

Further, the condition of quasineutrality does not mean that the electric field is negligibly small. Certainly in calculating test-particle orbits in a plasma (or MHD fluid) both E and B are needed [40] . In Gaussian-cgs units the electric field E and the magnetic field B have the same units (1 gauss = 1 Stat Volt/cm = 1 gm^1/2cm^−1/2s⁻¹) so their magnitudes are directly comparable: even though the magnitude of the motional electric field is v/c times smaller than the magnitude of the magnetic field (cf. expression (2)), the electric force on a particle qE is c/v times larger than the magnetic force q(1/c)v ´ B on a particle.

The author wishes to thank Joachim Birn, Mick Denton, Raluca Ilie, and Dan Welling for useful conversations. All spacecraft data utilized are freely available from NASA data servers http://cdaweb.gsfc.nasa.gov and http://omniweb.gsfc.nasa.gov. This work was supported at the Space Science Institute by the NSF Solar-Terre- strial Program via grant AGS-1261659, by the NASA Heliophysics LWS TRT program via grants NNX14- AN90G and NNX16AB75G, and by the NASA Heliophysics Guest Investigator Program via grant NNX14- AC15G and at the University of Michigan by the NASA Geospace SR&T Program via grant NNX12AD29G.

Joseph E. Borovsky, (2016) Relativity and the Solar Wind: The Maxwell-Equation Origins of the Solar-Wind Motional Electric Field. Journal of Electromagnetic Analysis and Applications,08,133-151. doi: 10.4236/jemaa.2016.88014