Spatially Highly Resolved Solar-wind-induced Magnetic Field on Venus

A popular IMF estimation approach compares the IMF conditions immediately before the inbound bow-shock crossing and after the outbound crossing (e.g., Zhang et al. 2015). However, the current study focuses on the magnetosphere over the northern polar cap, which is typically much closer to one bow-shock crossing, either inbound or outbound, than the other in any given orbit. This situation is similar to the situation illustrated by He et al. (2017), where the authors estimated the IMF for MESSENGER's Mercurian magnetospheric transit using the actual IMF observation from the nearest SW encounter. Following the IMF estimation approach in He et al. (2017), we estimate the IMF conditions as the average of actual IMF observations in a remote 20-minute-wide interval of VEX's nearest SW encounter at the closest bow-shock crossing, either inbound or outbound. The bow-shock crossings are identified manually and detailed in Rong et al. (2014). In each orbit, we first identify the VEX maximum latitude arriving as a reference point of the magnetospheric transit, and calculate the temporal separation between the reference point and the corresponding closest bow-shock crossing. The temporal separations of all orbits are distributed in Figure 2, which peaks at about 17 minutes with an average of 20.5 minutes. On average, the reference point is separated away from the center of the SW interval by Δt = δ t/2 + 20.5 minutes = 30.5 minutes, where δ t = 20 minutes is the sampling window width. Approximately, IMF is estimated as the actual IMF observations 30 minutes preceding or succeeding. Below, we evaluate the estimation using actual IMF observations through a correlation analysis.

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For the correlation analysis, we first select the magnetic fields measured in the SW and average them within discrete 20-minute-wide intervals. We denote B ^IMF(t) as the average in the interval from t − 10 min to t + 10 minutes. Then, B ^IMF(t ± 30 min) are "estimations" of B ^IMF(t). In total, the VEX SW observations enable about N_pair = 6.5 × 10⁵ independent pairs of [ B ^IMF(t), B ^IMF(t ± 30 min)], the three components of which are scatter-plotted in the three panels in Figure 3. The points are color-coded according to the IMF variability defined as ${\sigma }_{20\min }^{\mathrm{total}}$ $=\sqrt{{\sigma }_{{Bx}}^{2}+{\sigma }_{{By}}^{2}+{\sigma }_{{Bz}}^{2}}$. Here σ_Bx , σ_By , and σ_Bz are the standard deviation of the three magnetic components within the 20-minute sampling window centered at t ± 30 min. According to the ascending ${\sigma }_{20\min }^{\mathrm{total}}$ and following He et al. (2017), we sort the N_pair = 6.5 × 10⁵ pairs into 10 groups evenly, and calculate the correlation coefficient r between B ^IMF(t) and B ^IMF(t ± 30 min) in each group, illustrating that r decreases with increasing ${\sigma }_{20\min }^{\mathrm{total}}$ (not shown). For a robust analysis (following Section 2.5, He et al. 2017), we exclude the VEX orbits associated with disturbed IMF states, characterized by ${\sigma }_{20\min }^{\mathrm{total}}\gt$ 3.7 nT, from the following analyses. In the end, we obtain N_o = 2945 orbits for which bow-shock crossings are identified when IMF states are not disturbed.

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After removing the 15% most disturbed IMF pairs from the N_pair = 6.5 × 10⁵ pairs, we carry out a bootstrapping analysis as follows. First, we randomly select N_o = 2945 samples from the reduced set of 6.5 × 10⁵ pairs with replacements to calculate the correlation coefficient r and p-value. The sampling is repeated for N_s = 5000 times, yielding N_s values of r and p. Here, N_s = 5000 > N_o is selected subjectively. The average and standard deviation of r are present in each panel of Figure 3, indicating that the correlations are significant for all IMF components and demonstrating that the IMF estimation is reasonable. The correlation coefficients of the three components are close to each other, suggesting that their predictability is comparable. This is different from the IMF predictability at Mercury (He et al. 2017), where the predictability of the different components is significantly different.

Venus does not have a significant intrinsic magnetic field, neither a global dipole field nor a regional crustal field, that is magnetically spherically symmetrical. The interaction between the SW and the Venusian ionosphere induces a global magnetic field B ^V and a magnetosphere. The interaction is largely magnetically rotationally symmetrical with respect to the axis along v^SW. Below, we describe the symmetry mathematically and use it to discuss the VSE system and introduce a new coordinate system.

2.2.1. The Rotational Symmetry in VSO

The large-scale topology of B ^V is characterized by a draped configuration, determined mainly by the upstream SW conditions, mainly its magnetic field B ^IMF (e.g., He et al. 2016). Mathematically,

$$\begin{eqnarray}&&{{\boldsymbol{B}}}^{V}={{\boldsymbol{B}}}^{V}\left({\boldsymbol{r}}| {{\boldsymbol{B}}}^{\mathrm{IMF}}\right).\end{eqnarray} \tag{ 1 }$$

Here, B ^IMF is assumed to be homogeneous on the planetary scale at Venus. Therefore it is not a function of r , but a function of time, ${{\boldsymbol{B}}}^{\mathrm{IMF}}\left(t\right)$.

Investigating the details of Equation (1) is sort of an essential task of SW and Venus interactions. The independent variables of Equation (1) are two vectors that can be denoted with six scale variables: the position vector r := (r_x , r_y , r_z ) and ${{\boldsymbol{B}}}^{\mathrm{IMF}}:=\,({B}_{x}^{\mathrm{IMF}},{B}_{y}^{\mathrm{IMF}},{B}_{z}^{\mathrm{IMF}})$. In VSO, the six scale variables could be reduced to five, or a three- and a two-dimensional vector, r and ${{\boldsymbol{B}}}_{\perp }^{\mathrm{IMF}}:=\,(0,{B}_{y}^{\mathrm{IMF}},{B}_{z}^{\mathrm{IMF}})$, if we neglect B^IMF _x for simplicity. Although B^IMF _x also plays an important role in shaping the magnetosphere (Zhang et al. 2009), it is less capable to explain the induced magnetic variance near Venus (as quantified in Figure 10 in He et al. 2016), and therefore is often neglected for a first-order approximation. Accordingly, Equation (1) is often simplified as

$$\begin{eqnarray}&&{{\boldsymbol{B}}}^{V}={{\boldsymbol{B}}}^{V}\left({\boldsymbol{r}}| {{\boldsymbol{B}}}_{\perp }^{\mathrm{IMF}}\right).\end{eqnarray} \tag{ 2 }$$

Theresponse of B ^V to the IMF is characterized by rotational symmetry. Mathematically, for any angle ω,

$$\begin{eqnarray}&&{\boldsymbol{R}}\left(\omega \right){{\boldsymbol{B}}}^{V}\left({\boldsymbol{r}}| {{\boldsymbol{B}}}_{\perp }^{\mathrm{IMF}}\right)={{\boldsymbol{B}}}^{V}\left({\boldsymbol{R}}\left(\omega \right){\boldsymbol{r}}| {\boldsymbol{R}}\left(\omega \right){{\boldsymbol{B}}}_{\perp }^{\mathrm{IMF}}\right).\end{eqnarray} \tag{ 3 }$$

Here, a product of a vector with the rotation matrix ${\boldsymbol{R}}\left(\omega \right)\,:=\,\left[\begin{array}{ccc}1 & 0 & 0\\ 0 & \cos \omega & -\sin \omega \\ 0 & \sin \omega & \cos \omega \end{array}\right]$ represents a rotation of the vector with respect to the x-axis (i.e., − v^SW) by ω.

2.2.2. In VSE

Specifically, we define the IMF clock angle ${\theta }^{\mathrm{IMF}}:=\arctan ({B}_{y}^{\mathrm{IMF}},{B}_{z}^{\mathrm{IMF}})$. Substituting − θ^IMF for ω in Equation (3) yields

$$\begin{eqnarray}&&\begin{array}{l}{\boldsymbol{R}}\left(-{\theta }^{\mathrm{IMF}}\right){{\boldsymbol{B}}}^{V}\left({\boldsymbol{r}}| {{\boldsymbol{B}}}_{\perp }^{\mathrm{IMF}}\right)\\ \ \ =\,{{\boldsymbol{B}}}^{V}\left({\boldsymbol{R}}\left(-{\theta }^{\mathrm{IMF}}\right){\boldsymbol{r}}| {\boldsymbol{R}}\left(-{\theta }^{\mathrm{IMF}}\right){{\boldsymbol{B}}}_{\perp }^{\mathrm{IMF}}\right).\end{array}\end{eqnarray} \tag{ 4 }$$

For an arbitrary vector a , its product with ${\boldsymbol{R}}\left(-{\theta }^{\mathrm{IMF}}\right)$ defines a coordinate transformation from VSO to VSE: ${{\boldsymbol{a}}}^{\mathrm{VSE}}\,={\boldsymbol{R}}\left(-{\theta }^{\mathrm{IMF}}\right){{\boldsymbol{a}}}^{\mathrm{VSO}}$ (see Figure 4(a)). Therefore Equation (4) writes ${\left({{\boldsymbol{B}}}^{V}\right)}^{\mathrm{VSE}}={{\boldsymbol{B}}}^{V}\left({{\boldsymbol{r}}}^{\mathrm{VSE}}| {\left({{\boldsymbol{B}}}_{\perp }^{\mathrm{IMF}}\right)}^{\mathrm{VSE}}\right)$, which could be simplified as

$$\begin{eqnarray}&&{\left({{\boldsymbol{B}}}^{V}\right)}^{\mathrm{VSE}}={{\boldsymbol{B}}}^{V}\left({{\boldsymbol{r}}}^{\mathrm{VSE}}| \parallel {{\boldsymbol{B}}}_{\perp }^{\mathrm{IMF}}\parallel \right)\end{eqnarray} \tag{ 5 }$$

since ${\left({{\boldsymbol{B}}}_{\perp }^{\mathrm{IMF}}\right)}^{\mathrm{VSE}}\equiv (0,\parallel {{\boldsymbol{B}}}_{\perp }^{\mathrm{IMF}}\parallel ,0)$. Equation (5) has four independent scale variables, one less than in the VSO representation in Equation (2), which can therefore facilitate data processing, analysis, and interpretation. However, the VSE position vector r ^VSE is determined using the IMF estimation. Therefore the r ^VSE determination is subject to the error in the IMF estimation and r ^VSE cannot be determined at a singular point $\parallel {{\boldsymbol{B}}}_{\perp }^{\mathrm{IMF}}\parallel \approx 0$. To conquer these difficulties, we develop a cylindrical coordinate system in the following subsection.

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2.2.3. A Cylindrical Coordinate System

The previous subsection uses the IMF clock angle θ^IMF to define the rotation matrix ${\boldsymbol{R}}\left(-{\theta }^{\mathrm{IMF}}\right)$. We can also define ${\boldsymbol{R}}\left(-{\theta }^{\mathrm{VEX}}\right)$ using the clock angle ${\theta }^{\mathrm{VEX}}:=\arctan ({r}_{y}^{\mathrm{VEX}},{r}_{z}^{\mathrm{VEX}})$, where ${{\boldsymbol{r}}}^{\mathrm{VEX}}:=\,[{r}_{x}^{\mathrm{VEX}},{r}_{y}^{\mathrm{VEX}},{r}_{z}^{\mathrm{VEX}}]$ is the VEX position vector. Then, the product ${\boldsymbol{R}}\left(-{\theta }^{\mathrm{VEX}}\right){{\boldsymbol{a}}}^{\mathrm{VSO}}$ defines a coordinate transformation of the vector a ^VSO from VSO to a cylindrical coordinate system along the VSO x-axis. The three components of ${\boldsymbol{R}}\left(-{\theta }^{\mathrm{VEX}}\right){{\boldsymbol{a}}}^{\mathrm{VSO}}$ correspond, in order, to the axial, azimuth, and radial directions, which could be arranged into the conventional order (ρ, ϕ, and ζ) of cylindrical coordinate systems by multiplying with a permutation matrix ${\boldsymbol{U}}=: \left[\begin{array}{ccc}0 & 1 & 0\\ 0 & 0 & -1\\ -1 & 0 & 0\end{array}\right]$. As sketched in Figures 4(b) and 4(c), the ζ-axis is the cylindrical axis pointing from the Sun to Venus, the ρ-axis is the radial direction parallel to the position vector ${{\boldsymbol{r}}}_{\perp }^{\mathrm{VEX}}:=\,(0,{r}_{y}^{\mathrm{VEX}},{r}_{z}^{\mathrm{VEX}})$, and the ϕ-axis is the azimuth coordinate normal to the ρ-ζ plane. This coordinate system is called hereafter the solar-Venus-cylindrical (SVC) coordinate system. The product a ^SVC:= ${\boldsymbol{U}}{\boldsymbol{R}}\left(-{\theta }^{\mathrm{VEX}}\right){{\boldsymbol{a}}}^{\mathrm{VSO}}$ transforms the vector a ^VSO from VSO to SVC. Specifically, the VEX position in SVC reads $[{r}_{\rho }^{\mathrm{VEX}},{r}_{\phi }^{\mathrm{VEX}},{r}_{\zeta }^{\mathrm{VEX}}$] := ${\boldsymbol{U}}{\boldsymbol{R}}\left(-{\theta }^{\mathrm{VEX}}\right){{\boldsymbol{r}}}^{\mathrm{VEX}}$. Substituting the definitions of R and U results in ${r}_{\phi }^{\mathrm{VEX}}\equiv 0$. (Geometrically, ${r}_{\phi }^{\mathrm{VEX}}\equiv 0$ denotes that VEX is on the plane determined by the Sun, Venus, and VEX). Therefore all VEX observations are distributed two-dimensionally on the ρ-ζ plane in the SVC system.

We substitute U R ( − θ^VEX) for R(ω) in Equation (3), yielding ${\boldsymbol{U}}{\boldsymbol{R}}(-{\theta }^{\mathrm{VEX}}){{\boldsymbol{B}}}^{V}({\boldsymbol{r}}| {{\boldsymbol{B}}}_{\perp }^{\mathrm{IMF}})={{\boldsymbol{B}}}^{V}({\boldsymbol{U}}{\boldsymbol{R}}\left(-{\theta }^{\mathrm{VEX}}\right)$ r ∣ U R (−θ^VEX) ${{\boldsymbol{B}}}_{\perp }^{\mathrm{IMF}})$, which can be denoted as

$$\begin{eqnarray}&&\begin{array}{l}{\left({{\boldsymbol{B}}}^{V}\right)}^{\mathrm{SVC}}={{\boldsymbol{B}}}^{V}\left({{\boldsymbol{r}}}^{\mathrm{SVC}}| {\left({{\boldsymbol{B}}}_{\perp }^{\mathrm{IMF}}\right)}^{\mathrm{SVC}}\right)\\ \ \,=\,{{\boldsymbol{B}}}^{V}\left({{\boldsymbol{r}}}^{\mathrm{SVC}}| {B}_{\rho }^{\mathrm{IMF}},{B}_{\phi }^{\mathrm{IMF}}\right).\end{array}\end{eqnarray} \tag{ 6 }$$

Distributing all on the ρ-ζ plane, the VEX observations allow only investigating the induced field B ^V in the ρ-ζ plane. Accordingly, Equation (6) writes

$$\begin{eqnarray}&&{\left({{\boldsymbol{B}}}^{V}\right)}^{\mathrm{SVC}}={{\boldsymbol{B}}}^{V}\left({r}_{\rho },{r}_{\zeta }| {B}_{\rho }^{\mathrm{IMF}},{B}_{\phi }^{\mathrm{IMF}}\right).\end{eqnarray} \tag{ 7 }$$

Similar to Equation (5) in VSE, Equation (7) also comprises four independent variables and describes the response of B ^V to ${B}_{\rho }^{\mathrm{IMF}}$ and ${B}_{\phi }^{\mathrm{IMF}}$ in the ρ-ζ plane. The SVC position vector is not subject to the IMF estimation and therefore is capable of handling the VSE singular point at $\parallel {{\boldsymbol{B}}}_{\perp }^{\mathrm{IMF}}\parallel \approx 0$. In the following subsections, we construct B ^V (r_ρ , r_ζ ) under five different ${\left({{\boldsymbol{B}}}_{\perp }^{\mathrm{IMF}}\right)}^{\mathrm{SVC}}$ conditions.

Figure 5 displays the IMF estimations for all VEX magnetospheric transits in the SVC ρ-ζ plane. Each point corresponds to one vector of 4-second-resolved B ^V observed below the altitude h < 4400 km and within the range of solar zenith angle (SZA) between 65° and 150°, which is the targeting region of the current work. The arc-shaped traces reflect cylindrical rotations of VEX with respect to ${{\boldsymbol{B}}}_{\perp }^{\mathrm{IMF}}$. One arc represents one targeting region transit. The portion colored in cyan corresponds to the 15% lowest magnitude $\parallel {{\boldsymbol{B}}}_{\perp }^{\mathrm{IMF}}\parallel$, while the remaining portions, in magenta, blue, green, and red, denote four 60° wide ranges of the clock angle θ^IMF. The five colors approximately represent the five SW/IMF conditions, namely, (${B}_{\rho }^{\mathrm{IMF}}\,\approx \,$0, ${B}_{\phi }^{\mathrm{IMF}}\,\approx$ 0), (${B}_{\rho }^{\mathrm{IMF}}\,\gt \,$0, ${B}_{\phi }^{\mathrm{IMF}}\approx$ 0), (${B}_{\rho }^{\mathrm{IMF}}\,\approx \,$0, ${B}_{\phi }^{\mathrm{IMF}}\lt$ 0), (${B}_{\rho }^{\mathrm{IMF}}\,\lt \,$0, ${B}_{\phi }^{\mathrm{IMF}}\,\approx$ 0), and (${B}_{\rho }^{\mathrm{IMF}}\,\approx \,$0, ${B}_{\phi }^{\mathrm{IMF}}\,\gt$ 0). The last four conditions correspond to the transit of the polar caps of $+{B}_{\perp }^{\mathrm{IMF}}$, −E^SW, $-{B}_{\perp }^{\mathrm{IMF}}$, and +E^SW hemispheres in the VSE coordinate system. Accordingly, these SW/IMF conditions are denoted hereafter as $0{B}_{\perp }^{\mathrm{IMF}}$, $+{B}_{\perp }^{\mathrm{IMF}}$, −E^SW, $-{B}_{\perp }^{\mathrm{IMF}}$, and +E^SW.

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Under each of these conditions, we combine the VEX observations to construct B ^V in ρ-ζ depiction, using the method detailed in the following subsection.

We distribute the B ^V observations under the $0{B}_{\perp }^{\mathrm{IMF}}$ condition, colored in cyan in Figure 5, equally into 14 altitude levels, then into 14 SZA bins at each altitude level, resulting in 14 × 14 spatial bins with approximately identical sampling count, e.g., N_sample = 478–479 in all bins. In each bin, the median of the position vector (r_ρ , r_ζ ) of the 478–479 samples is shown as a point in the ρ-ζ plane in Figure 6(a) and is color-coded with the median of the component B_ρ . On each point, the black cross is the error bar that represents the upper and lower quartiles of (r_ρ , r_ζ ) of the 478–479 samples. Similarly, the ${B}_{\phi }^{V}$ and ${B}_{\zeta }^{V}$ components and the magnitude ∥ B ^V ∥ are also constructed in Figures 6(b), (c), and (d), respectively. For the other four IMF conditions, ${{\boldsymbol{B}}}^{V}\left({r}_{\rho },{r}_{\zeta }\right)$ are also constructed and displayed in Figures 6(e)–(h), (i)–(l), (m)–(p), and (q)–(t). In addition, to check the details, we zoom and project ${B}_{\phi }^{V}$ and ∥ B^V ∥, namely, Figures 6(b), (d), (f), (h), (j), (l), (n), (p), (r), and (t), into the SZA-altitude depiction in Figure 8.

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The IMF dependence of the Venusian magnetospheric B ^V distribution has been observationally investigated in both cases (e.g., Zhang et al. 2012) and statistical studies (e.g., Dubinin et al. 2014a; Zhang et al. 2015), which are mostly based on manually selected observations and therefore are subject to potential prior expectations. Some studies have also constructed unbiased statistical B ^V under different SW/IMF conditions (Chai et al. 2016; Zhang et al. 2010), using data without any guiding selection. These unbiased studies focused mainly on large structures at planetary scales because they used a popular data-averaging approach. The approach divides the VEX samplings spatially uniformly into cubes and averages data in each cube separately (e.g., Du et al. 2013; Zhang et al. 2010). The resultant maps have uniform spatial resolutions but a nonuniform standard error of the mean (SEM) that is proportional to $1/\sqrt{{N}_{\mathrm{sample}}}$. Here, N_sample counts observations in a bin that is highly spatially nonuniform (Figure 7(a); also see Du et al. 2013; Chai et al. 2016). In the current study, we map VEX magnetic observations with a uniform SEM by averaging the same number of samplings N_sample in each bin, which yields a uniform significance with a nonuniform spatial resolution, different from the uniformly spatially resolved works (e.g., Du et al. 2013).

Our resolution is about 1°–1.5° in SZA and up to about 50 km in altitude over 90° SZA at 200 km altitude, at least one order of magnitude higher than the typical resolution of results on the uniform spacing grid (e.g., about 600 × 600 km in Du et al. 2013; Chai et al. 2016).

In Figure 6, the most notable features in the three magnetic components are three antisymmetries, between ${B}_{\rho }\{+{B}_{\perp }^{\mathrm{IMF}}\}$ and ${B}_{\rho }\{-{B}_{\perp }^{\mathrm{IMF}}\}$, between B_ϕ {+E^SW} and B_ϕ {−E^SW}, and between ${B}_{\zeta }\{+{B}_{\perp }^{\mathrm{IMF}}\}$ and ${B}_{\zeta }\{-{B}_{\perp }^{\mathrm{IMF}}\}$, illustrated in Panels (m, q), (f, j), and (o, s), respectively. The antisymmetries in B_ρ and B_ζ reveal the draped IMF: the magnetic field is bending from the ±ρ direction in the upstream SW to the ±ζ direction in the magnetosphere (for a three-dimensional schematic draping pattern, we refer to He et al. 2016).

Although above, the three magnetic components exhibit a dependence on SW/IMF conditions, ∥ B ∥ exhibits similarities under different SW/IMF conditions. The first similarity is the distinctive day-night difference: on the dayside, e.g., at SZA = 70°, ∥ B ∥ is stronger than that on the nightside at SZA > 90°, and it maximizes vertically at about 400–800 km altitude (see Figures 8(b), (d), (f), (h), and (j), which are zoomed versions of Figure 6(d), (h), (l), (p), and (t)), which corresponds to the magnetic barrier. For example, under +E^SW in Figures 6(h) and 8(d), ∥ B ∥ in the magnetic barrier is a few times stronger than the IMF magnitude: ∥ B ^barrier∥ > 40 nT versus 〈∥ B ^IMF∥〉 = 7.3 nT. Because ∥ B ∥ in Figures 6 and 8 shares similarities between different IMF conditions, in Figure 9(a) we depict ∥ B ∥ as a function of h using all VEX observations at SZA = 70°–75° without considering IMF conditions.

In Figure 9, ∥ B ∥ maximizes at 30–35 nT between 400 and 900 km altitude. In this altitude range, the probability density of ∥ B ∥ exhibits two maxima, at 25–40 and 0–6 nT, as illustrated in Figure 9(d). The 25–40-nT maximum corresponds to the magnetic barrier, whereas the 0–6-nT maximum reveals the unmagnetized ionosphere that exists below the ionopause. We display the probability p of ∥ B ∥ < 6 nT as a function of altitude in Figure 9(c) to use the weak ∥ B ∥ as an indicator of the unmagnetized ionosphere. At 400–900 km, for about p < 10%, the ionosphere is unmagnetized. (Note that this indicator is imperfect. Otherwise, (1) p should maximize at 100% at a low altitude and decrease monotonously with h, and (2) its vertical gradient should be proportional to the probability density of in situ occurrences of the ionopause). In Figure 9(a), below and above the 400–900 km region, ∥ B ∥ decreases with decreasing and increasing h, respectively. The vertical gradients are signatures of meridional currents flowing oppositely on the ionopause and IMB, as sketched in Figure 19 in Baumjohann et al. (2010), which is quantified in Figure 9(b). In Figure 9(b), the ∥ B ∥ gradient maximizes at h = 320 km, with a full width at half maximum (FWHM) of about 90 km, extending from 280 to 370 km. Note that the gradient peak at h = 280–370 km describes the altitude of the maximum probability density of ionopause occurrences, rather than the complete occurring range or thickness of the ionopause. The ionopause at every instant is a thin layer without a vertical extension. The altitude h = 280–370 km is above the exobase (150–200 km; see Table 2 in Hodges 2000), suggesting that during VEX operation, the dayside (SZA < 70°–75°) bottom ionosphere is unmagnetized for most of the time. The low-ionosphere magnetization reported in VEX observations (e.g., Dubinin et al. 2014a; Zhang et al. 2012) occurs mostly around the terminator (as detailed below in Section 4.2) because most of the VEX low-ionosphere magnetic observations are distributed not far from the terminator due to the constraints of the VEX obit, as displayed in Figure 7 and also pointed out by Dubinin et al. (2014a). At h < 200 km, 75.7% of the observations are distributed at 85° ≤ SZA ≤ 95° (Figure 7(b)).

The ∥ B ∥ similarities in Figures 6(d), (h), (l), (p), and (t) are associated with different magnetic components under different SW/IMF conditions. Under ±E^SW (in Figures 6(e)–(g) and (i)–(k)), the gradient is sharper in B_ϕ than that in the other two components, whereas under $\pm {B}_{\perp }^{\mathrm{IMF}}$ (in Figures 6(m)–(o) and 6(q)–(s)), the gradient in B_ζ is sharpest. The gradients of these components suggest that the dominant components of the ionopause current in the ±E^SW and $\pm {B}_{\perp }^{\mathrm{IMF}}$ hemispheres are in the directions of ±ζ-and ±ϕ-axis, respectively, as displayed by the arrows, cross, and point in Figures 8(d), (f), (h), and (j). Figures 10(a) and 10(b) sketch the topology of the ionopause current in VSE coordinates in the plane r_x = Rv/4 and r_y = 0, respectively. In the current work, Rv = 6052 km denotes the radius of Venus. The currents exhibit symmetry between $\pm {B}_{\perp }^{\mathrm{IMF}}$ hemispheres and antisymmetry between the ±E^SW hemispheres. On the dayside, ± j_ϕ in the $\pm {B}_{\perp }^{\mathrm{IMF}}$ hemispheres closes + j_ζ in the +E^SWSW hemisphere with − j_ζ in the −E^SW hemisphere. Over the poles of the ±E^SW hemispheres, ± j_ζ flows across the terminator and then closes through the SW or through currents on the IMB or in the magnetic barrier. Furthermore, the vertical B_ϕ gradient in, e.g., Figures 6(f) and 8(c) allows estimating the ionopause current density: the drop by more than 20 nT from the magnetic barrier maximum to low altitude is associated with a current with an intensity higher than 15 A km⁻¹ under the assumption of a sheet-like current distribution.

Although Section 3.1 illustrates the similarity of ∥ B ∥ between the ±E^SW hemispheres, the ∥ B ∥ intensity is stronger in the +E^SW hemisphere than in the other hemisphere. Referring to ∥ B ∥ = 25 nT as a boundary, the magnetic barrier extends to about 85° SZA under the +E^SW condition (Figure 8(d)), but about 75° SZA under the −E^SW (Figure 8(f)). To quantify the ±E^SW difference, in Figure 11(a) we display ∥ B {+E^SW}∥ − ∥ B {−E^SW}∥, namely, the difference between Figures 6(h) and 6(l). The main contributing component to the asymmetric ∥ B ∥ is B_ϕ , which exhibits an ±E^SW antisymmetric polarity (Figures 6(h) and (l)), as discussed in Section 3.1. To quantify the break of the antisymmetry between Figure 6(h) and 6(l), we display B_ϕ {+E^SW} + B_ϕ {−E^SW} in Figure 11(b). The ±E^SW difference reflects the asymmetry of the IMF wrapping between the ±E^SW poles, as reported in Zhang et al. (2010).

In Figure 11(a), the ∥ B ∥ difference exhibits three peaks, on the dayside in the magnetic barrier, on the nightside in the very near magnetotail, and exactly over the terminator at low altitude (in the ionosphere; see the contour line of 13 nT). The barrier and terminator ∥ B ∥ peaks are associated with peaks in the B_ϕ difference in Figure 11(b) (above 13 nT; see the contour lines). The barrier and magnetotail peaks extend into global scales, whereas the terminator peak is restricted quite regionally. The global-scale asymmetries are known as the magnetic ±E^SW asymmetry (investigated in both observations and simulations; Saunders & Russell 1986; Zhang et al. 1991; Jarvinen et al. 2013), which also exists at Mars (Vennerstrom et al. 2003). The underlying mechanism is still under debate. Finite Larmor radii effects of pickup ions were suggested to account for the asymmetry (Phillips et al. 1987), but hybrid simulations (Brecht 1990) could reproduce the phenomenon without planetary ions, and indicated that the Hall effect contributes significantly to the asymmetry. In addition, global magnetohydrodynamics (MHD) simulations with multifluid treatment at Mars (Najib et al. 2011) reproduced the ±E^SW asymmetry, suggesting that the decoupling of separate ion fluids could serve as an alternative explanation.

Compared with the global-scale asymmetries, the regional asymmetry over the terminator is relatively less well understood and is discussed below in Section 4.2.

A cylindrical symmetrical preference of positive B_ϕ was reported on a cylinder in the magnetotail, which was called the induced global magnetic field looping (Chai et al. 2016). The looping structure is denoted by the pink region in Figures 1 and 12(a). For a comparison with Chai et al. (2016), in Figure 11(c) we average B_ϕ under the SW/IMF conditions $\pm {B}_{\perp }^{\mathrm{IMF}}$ and ±E^SW (Figures 6(f), (j), (n), and (r)). As a reference, the black box in Figure 11(c) sketches the distribution of the B_ϕ symmetry according to Chai et al. (2016). In Figure 11(c), the tail-like B_ϕ maximum on the nightside is largely consistent with the black box and wrapping slightly toward low latitude.

The tail-like B_ϕ maximum exhibits a quantitative difference between the different IMF conditions (in Figures 6(f), (j), (n), and (r)). The B_ϕ maximum is strongest under +E^SW (Figure 6(f)) and weaker under $\pm {B}_{\perp }^{\mathrm{IMF}}$ (Figures 6(n) and 6(r)). Under −E^SW (in the black rectangle in Figure 6(j)), B_ϕ even partially reverses to negative. This IMF dependence suggests a cylindrical asymmetry, as sketched in Figure 12(b). Chai et al. (2016) explained the symmetry of +B_ϕ preference in terms of currents on two cylinders with the identical longitudinal axis pointing toward ζ and − ζ (as sketched in Figure 4(b) in Chai et al. 2016, and here in Figure 12(a) by the red symbols). The breaking of the symmetry of the +B_ϕ preference under the −E^SW condition reported here suggests that the current system is not cylindrically complete in the near tail. A correction of the looping structure and two-cylinder currents is sketched in Figure 12(b).

This corrected looping structure could be explained as the superposition of the classical magnetic draping structure with an additional draping configuration. This superposition theory was proposed to explain a similar structure on Mars (Dubinin et al. 2019). Similar to Venus, Mars does not have an internal dipole magnetic field either, and its interaction with SWs produces a similar looping structure (Chai et al. 2019). Dubinin et al. (2019) attributed this looping structure to an asymmetrical pileup of IMF and an associated additional draping configuration in which magnetic fields bend toward the −E^SW direction in the planes normal to the SW velocity. The authors also reproduced this structure in hybrid simulations. The additional draping can produce closed loops over the −E^SW hemisphere and weaken the magnetic field and potentially also magnetic reconnection (also see Rong et al. 2014; Ramstad et al. 2020).

Another morphological character of this looping in Figure 11(c) is that it does not overlap with but is isolated from the low-ionospheric preference of B_ϕ over the terminator, as detailed in the following subsection.

Section 3.2 mentioned the ±E^SW asymmetry over the terminator (highlighted in the black circle in Figure 11(b)). In this small region, the polarity of the induced magnetic component B_ϕ is unresponsive to the upstream ${B}_{\phi }^{\mathrm{IMF}}$ polarity and prefers to be positive (see the circle in Figure 6(j) and the green isoline in Figure 8(e)). Otherwise, in all other regions, the B_ϕ polarity is consistent with the IMF polarity of ${B}_{\phi }^{\mathrm{IMF}}$. The asymmetry has been investigated using selected cases (e.g., Dubinin et al. 2014a; Zhang et al. 2015) and was called the ±E^SW asymmetrical response of low-ionospheric magnetization to the ${B}_{\perp }^{\mathrm{IMF}}$ polarity (e.g., Dubinin et al. 2014a). Here, Figures 11(b) and 8(c) and (e) present the unbiased statistic at high spatial resolution.

Not surprisingly, our unbiased statistical results are significantly different from those based on selected cases. For example, the maximum B_ϕ is about 13 nT at +E^SW in Figure 8(c) and about 4 nT at −E^SW in Figure 8(e), which are significantly lower than the averaged magnetic intensity, 45 nT, of the 77 selected events in Zhang et al. (2015). This significant difference between +E^SW and −E^SW is also inconsistent with the conclusion in Zhang et al. (2015) that the asymmetry does not show a preference for any particular IMF orientation. Furthermore, our results illustrate that the asymmetry occurs in a very narrow region, at about solar zenith angle 85° < SZA < 95° and altitude h < 230 km, referring to B_ϕ > 4 nT in Figure 8(e). To specify further the extensions in h and SZA, we construct the one-dimensional distribution of B_ϕ as a function of h and SZA in Figure 13 under ±E^SW conditions. The vertical dashed line in each panel displays the half-maximum of B_ϕ , referring to which, B_ϕ peaks at about h = 185–210 km, h = 185–250 km, SZA = 88°–95°, and SZA = 88°–93° in Figures 13(a), (b), (c), and (d), respectively. (Note that B_ϕ maxima will be smeared out if using broader h or SZA windows. Therefore the estimation of the B_ϕ peak width is subject to the sampling window widths.) Of the identifications of these four peaks, the peak identification in Figure 13(b) is most vulnerable to disturbances of the half-maximum of B_ϕ . For example, if the half-maximum of B_ϕ in Figure 13(b) increases by 2 nT, the peak width will shrink from h = 185–250 km to 200–230 km. We therefore describe below the altitude variations, referring mainly to Figure 13(a). The regional distribution implicates that simulations to reproduce the structure entail a high spatial resolution, at about 1° in SZA and 10 km in altitude. Note that this SZA range was not included in the simulation in Figure 7(b) by Villarreal et al. (2015), where the simulation presents the magnetic distribution in the VSO x–y plane at z = 1 Rv, as sketched by the yellow line in Figure 1 and by the cyan line in Figure 6(h) and Figure 11(a).

Several efforts were made to interpret the asymmetry detailed in Section 4.2, e.g., in terms of the giant flux rope, magnetic belt, intrinsic field, or induced field, and antisunward transports of low-altitude magnetic belts (Zhang et al. 2012; Dubinin et al. 2014a; Villarreal et al. 2015). The potential interpretations were discussed qualitatively in Dubinin et al. (2014a) and Dubinin et al. (2014b), which suggested that the most promising explanation is a Cowling current. As sketched in Figure A1 and detailed in Appendix, a Cowling channel entails a particular electromagnetic configuration, including a narrow band of high Hall conductivity σ_H , elongated perpendicular to an ambient magnetic field B ₀, and a primary electric field E ₀, perpendicular both to the gradient of the σ_H band and to B ₀. A Cowling channel might occur over the poles of ± E^SW hemispheres with an orientation as sketched in Figure 14. In the daytime Venusian ionosphere, the Hall conductivity σ_H maximizes vertically nearby the maximum vertical peak of the ionospheric electron density (V2 layer; see Dubinin et al. 2014a; Pätzold et al. 2007). The peak is thin vertically and extends broadly in horizontal directions, which is typically under the ionopause and therefore is unmagnetized. However, as suggested by Figures 6 and 8, the ionopause does not cross the terminator, but is below about 85° SZA at 200–400 km altitude. Therefore the SW magnetic field ${{\boldsymbol{B}}}_{\perp }^{\mathrm{IMF}}$ might penetrate the ionosphere over the ± E^SW poles (Dubinin et al. 2014a). Over the poles, the V2 layer is normal to the ρ-axis and σ_H gradient, while the ${{\boldsymbol{B}}}_{\perp }^{\mathrm{IMF}}$ is parallel to the ϕ-axis and might serve as B ₀. Under this configuration, an antisunward primary electric field E _0ζ might induce a Cowling current along the ζ-axis, j _{C ζ} . The magnetic field associated with j _{C ζ} might account for the low-ionosphere ±E^SW asymmetry. However, a few issues about the Cowling current are still open. Here we discuss the specific distribution of the current and the primary field E _0ζ .

4.3.1. The Distribution of the Cowling Current

4.3.2. The Primary Electric Field

In magnetized plasma, the Lorenz force drives charged particles drifting gyratorily and periodically perpendicular to the ambient magnetic field B ₀. As a result, the particles are bound to magnetic field lines. The gyratory drift might be interrupted by collisions with other particles, either charged or neutral. In an extreme collisional case, the gyratory is relatively negligible, and particles are freed from the magnetic field and move freely at the bulk velocity of the ambient particles. A measure of the relative importance between the gyratory drift and collision is the ratio of the gyrofrequency to collision frequency $\kappa =\tfrac{{\rm{\Omega }}}{\nu }$. The ratio κ quantifies the expectation of the number of gyratory drifting circles that one particle could achieve in between two collisions with other particles. The collision frequency is proportional to the particle densities. In planetary atmospheres, the collision frequency therefore decreases exponentially with increasing altitude as the neutral density decreases, whereas the gyrofrequency is relatively less dependent on altitude. Accordingly, κ overall increases with altitude. At low altitude, collisions are superior to gyratory drifts (κ ≪ 1), whereas at high altitude, the collisions are rare and gyratory drifts are more important (κ ≫ 1). In between the low and high altitudes, there is a transitional altitude h^κ≈1, where κ ≈ 1. However, the vertical increasing rate $\tfrac{d\kappa }{{dh}}$ is different for different species, steeper for electrons than for ions $\tfrac{d{\kappa }_{e}}{{dh}}\gt \tfrac{d{\kappa }_{i}}{{dh}}$, yielding higher transitional altitudes for ions than for electrons ${h}_{i}^{\kappa \approx 1}\gt {h}_{e}^{\kappa \approx 1}$. Here, i and e in the subscripts denote ions and electrons, respectively.

As a result, the atmospheric electrical conductivity and its isotropy are altitude dependent. Above ${h}_{i}^{\kappa \approx 1}$ when plasma is collisionless (κ_i ≫ 1, κ_e ≫ 1), in response to an applied electric field E ₀, both ions and electrons drift toward the direction of E ₀ × B ₀ on the macroscopic scale, and no net current is generated. Under ${h}_{e}^{\kappa \approx 1}$ in the collisional case (κ_i ≪ 1, κ_e ≪ 1), the applied field E ₀ drives ions and electrons flowing along E ₀ oppositely, generating a current j _P = σ_P E ₀ parallel to E ₀, independent of B ₀. The current j _P is known as Pedersen current, and the coefficient σ_P is known as Pedersen conductivity. Between ${h}_{i}^{\kappa \approx 1}$ and ${h}_{e}^{\kappa \approx 1}$ in the intermediate case (κ_i ≪ 1, κ_e ≫ 1), electrons are already bound to B ₀, whereas ions could still essentially move with ambient winds. The applied field E ₀ decouples electrons from ions: electrons drift in the direction E ₀ × B ₀, whereas the ions prefer following toward E ₀. The caused current comprises a component j _P following along E ₀ and a component ∥ j _H ∥ = σ_H ∥ E ₀∥ parallel to E ₀ × B ₀. The current j _H is known as the Hall current and the coefficient σ_H is known as Hall conductivity.

For references, the range (${h}_{i}^{\kappa \approx 1}$, ${h}_{e}^{\kappa \approx 1}$) approximately corresponds to (90 km, 150 km) in the terrestrial equatorial ionosphere (see Figure 2.5 in Kelley 2009) and (130 km, 270 km) at Venus (see Dubinin et al. 2014a).

In a particular configuration of conductivity and magnetic field, the electric current in the direction of E ₀ might be much intenser than j _P due to a superposition of j _H . The geometry of the configuration is characterized by a band of σ_P that is elongated infinitely in one dimension, but is restricted in the perpendicular direction. Figure A1(a) sketches the configuration and the causative relations of the superposition through three steps. In the first step, a primary electric field E ₀, parallel to the elongated direction and within the ambient magnetic field B ₀ that is normal to the plane spanned by ∇σ_P and E ₀ generates the Hall and Pedersen current j _H and j _P . Second, on the boundary where ∥∇σ_P ∥ > 0, polarization charges and electric field ${\boldsymbol{E}}^{\prime}$ are created by preventing a divergence of j _H . In the last step, the secondary field ${\boldsymbol{E}}^{\prime}$ further generates secondary Hall and Pedersen currents ${\boldsymbol{j}}{{\prime} }_{H}$ and ${\boldsymbol{j}}{{\prime} }_{P}$. ${\boldsymbol{j}}{{\prime} }_{H}$ flows along j _P , and therefore the total current parallel to E ₀ equals j _C = j _P +${\boldsymbol{j}}{{\prime} }_{H}$, which is more intense than j _P . The reinforcement of the current in the E ₀ direction is called Cowling effect (e.g., Yoshikawa et al. 2013), the reinforced current j _C is known as Cowling current (e.g., Tang et al. 2011), σ_C :=∥ j _C / E ₀∥ is the Cowling conductivity, and the whole current system is called the Cowling channel. Note that the horizontal separations of elements shown in Figure A1(a) represent only the causative relations in the explanation, but not the spatial distribution. In space, the elements coexist under an electrodynamic equilibrium, as sketched in Figure A1(b). The three steps illustrated in Figure A1(a) are not a unique explanation for the equilibrium. In the end, only the equilibrium matters.

For detailed discussions, we refer to the literature in the context of two instances of Cowling channels in Earth's ionospheric E-layer, over the equator and auroral zone, respectively (e.g., Kelley 2009; Fujii et al. 2011, and references therein). The corresponding Cowling currents in these two instances are well known as the equatorial electrojet (EEJ; Forbes 1981, and references thereafter) and auroral electrojet (AEJ; Boström 1964, and references thereafter). The orientation of these two Cowling channels is sketched in Figures A1(c) and (d). In both cases, the Cowling channels are zonally elongated and the geomagnetic field serves as B ₀. In the case of EEJ, B ₀ is northward, the gradient ∇σ_P is vertical, and the eastward component of the electric field induced by the tidal wind serves as the primary electric field E ₀. For the boreal westward AEJ, B ₀ is downward, ∇σ_P is in the magnetic meridional direction, and the zonal electric field mapped from the magnetosphere serves as E ₀.

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