We address the question of whether there is a possibility of an interplanetary magnetic field reaching Venus' surface by magnetic diffusion across the ionosphere. We present a model calculation, estimate the magnetic diffusion time at Venus, and find out that the typical diffusion timescale is in a range between 12 and 54 h, depending on the solar activity and the ionospheric magnetic field condition. The magnetic field can thus permeate Venus' surface and even its interior when the solar wind is stationary (i.e., no magnetic field reversal) on the timescale of half a day to several days.

Venus, being the nearest neighbor to Earth,
differs from Earth in that the intrinsic magnetic
field is absent. Nevertheless, a magnetospheric cavity
is formed around Venus with a standing shock wave (bow shock)
and a magnetotail as the solar wind becomes
deflected by Venus' ionosphere and the
interplanetary field drapes
around
the planet.
In situ measurements by the Pioneer Venus Orbiter
studied Venus' magnetic environment
in detail, such as a tail structure

A low-altitude profile of Venus' magnetic field
was first obtained during the Pioneer Venus
Orbiter
entry in the nightside ionosphere

We address the question of whether the magnetic field of interplanetary origin can
ever reach Venus' surface. Hybrid code simulations, for example in

To answer the question on the magnetic field at Venus' surface, we estimate the magnetic diffusion time in Venus' atmosphere. Two competing scenarios are possible. In scenario 1, the magnetic field can reach the planetary surface and even penetrate the planetary body, which is achieved when Venus' atmosphere is sufficiently diffusive and the interplanetary magnetic field surrounding Venus is stationary for a longer period of time. In scenario 2, on the other hand, the magnetic diffusion process at low altitudes becomes reset when the external field (in the induced magnetic field) reverses its orientation. Here, we mean by the “reset” a change in the sunward or antisunward direction of the interplanetary magnetic field. Since the diffusive transport process is local and linear in the magnetic field, the diffusive transport problem is not affected by the amount of magnetic energy supplied to the ionosphere.

The problem of the surface magnetic field at Venus is formulated as a competition between the diffusion time (such that the field reaches the surface on a detectable level) and the reset time (such that the field diffusion process is reset by the change in the interplanetary magnetic field). The interplanetary magnetic field has a four-sector structure in the solar ecliptic plane in the solar minimum phase. Therefore, the longest time length for a stable interplanetary magnetic field (without the field reversion due to the sector boundary crossing) is about 6 to 7 days. We take the four-sector structure of the interplanetary magnetic field (IMF) to infer the longest time interval (as the upper time limit) of the stable IMF. There is no large-scale pattern known to Venus' induced magnetosphere, unlike the Earth substorm case. Solar minimum is more relevant to our theoretical model because the four-sector structure holds well and the coronal mass ejection (CME) occurrence rate (which shortens the time length for the stable IMF) is minimum.

Here we find that the magnetic diffusion time in Venus' atmosphere is of the order of 44 000 to 194 000 s, that is, in the range between 12 and 54 h. It is thus likely that the interplanetary magnetic field reaches Venus' surface and further into Venus' interior for a long time period of stationary solar wind. Our conclusion will be tested against the upcoming magnetic field measurements of the low-altitude region (down to 1000 km) during the BepiColombo flyby at Venus.

It is worth mentioning that the convective transport of the magnetic field is also an important transport mechanism, and the magnetic Reynolds number gives an estimate of the ratio of the convective transport to the diffusion. However, our study works on a more simplified situation to give an estimate by reducing the convective–diffusive problem to a diffusive problem. The reason for this is that the convective transport does not enter the problem of the vertical diffusion (in the sense of radial direction from the planet) and the plasma flow is in the horizontal direction (tangential to the planet surface). The convective transport makes the penetration time longer, and not shorter. Therefore, our study gives an estimate of the lower limit (i.e., the shortest time) of the magnetic field penetration through the ionosphere.

We first estimate the diffusion time in an order-of-magnitude fashion.
Magnetic diffusion time

In general, conductivity in a magnetized plasma is a tensor, whose components
are (1) Pedersen conductivity, (2) Hall conductivity, and (3) field-aligned
or parallel conductivity. Above all, the Pedersen conductivity is relevant to
the problem of diffusion time estimate. The reason for this is that magnetic
diffusion takes time because energy is dissipated along the way (magnetic
energy is converted to heat). It is the Pedersen current by which the energy
dissipation is achieved. The Hall current, in contrast, has no energetic
effect. From a geometrical point of view, the Pedersen conductivity (or the
current, to be more precise) can transmit the magnetic field (say, in the

We take the length scale (or thickness in altitude)

In reality, the conductivity depends on the altitude, the ionospheric
condition, and the solar activity. We estimate the diffusion time more
quantitatively by numerically integrating the differential diffusion time

Electron number density

The task is to evaluate the electric conductivity as a function of the
altitude. Since we work on the Pedersen conductivity for the magnetic
diffusion problem, the electron density, the collision frequency, and the
magnetic field profiles are needed to calculate the conductivity before
performing the height integration. The procedure of the diffusion time
estimate is summarized as follows.

The ion term in the Pedersen conductivity (Eq.

Results from the diffusion time estimate.
The symbol

Diffusion time varies in the range from about 44 000 s (about 12 h) to
about 194 000 s (54 h). See Table 1. Solar activity and the local magnetic
field in the ionosphere influence the diffusion time. A minimum of 12 h
(half a day) for the diffusion time is needed for the magnetic field to
penetrate Venus' ionosphere and atmosphere. If the electron density is higher
or the local magnetic field weaker, the diffusion time can scale up to 54 h
(more than 2 days). Therefore, Venus' surface may exhibit a nonzero magnetic
field when the solar wind is stationary (in the sense that the interplanetary
magnetic field does not show a reversal) on the timescale of half a day to
several days. For reference, we repeat the calculation of the diffusion time
using only the electron term in the conductivity. The diffusion time from the
electron term is in the range from about 40 000 s (11 h) to about
146 000 s (40 h). The ion contribution makes a difference in the diffusion
time by about 10 %–20 %. Note that the peak Pedersen conductivity is
about

Comparison of Pedersen conductivity estimate at the peak altitudes
of the conductivity;

It is interesting to observe the difference between the simulation and
analytic estimates of the diffusion time by about 2 orders of magnitude.
The major reason behind this difference most likely lies in the neglect of
the electron-neutral collisions and the numerical diffusion in the hybrid
simulations.

The magnetic diffusion time is

It is also interesting to observe that the Pedersen conductivity on Venus
is much higher than on Earth by about 4 orders of magnitude. This
difference can be explained as follows. We write formulas separately for the
electron Pedersen conductivity

The Pedersen conductivity on Venus

Using the facts that (1) the electron density is almost the same
(

We conclude the diffusion time estimate with the following notes. First, a
stationary solar wind condition on a timescale of half a day to several days
is likely occurring in Venus' environment. The interplanetary magnetic
field can theoretically reach (under the condition of stationary solar wind)
Venus' surface and justifies the nonzero field measurements by Venus
Express. Second, further improvement is possible by including the ion-neutral
collisions and the solar activity influence on the collision frequency.
Third, the upcoming missions such as Parker Solar Probe

Input data used in this work are graphically available in the articles by Kilore and Luhmann (1991) for the electron number density, Dubinin et al. (2014) for the collision frequency, and Villarreal et al. (2015) and Zhang et al. (2016) for the magnetic field.

YN carried out the calculation, writing, and revision work. UM came up with the original idea and handled the discussion and finalization.

The authors declare that they have no conflict of interest.

The authors thank those at the Europlanet workshop