ANGEOAnnales GeophysicaeANGEOAnn. Geophys.1432-0576Copernicus PublicationsGöttingen, Germany10.5194/angeo-36-1-2018Energy conversion through mass loading of escaping ionospheric ions for different Kp valuesYamauchiMasatoshim.yamauchi@irf.sehttps://orcid.org/0000-0001-7065-9087SlapakRikardhttps://orcid.org/0000-0002-2347-1871Swedish Institute of Space Physics (IRF), P.O. Box 812, 98128 Kiruna, SwedenDivision of Space Technology, Luleå University of Technology, Kiruna, SwedenMasatoshi Yamauchi (m.yamauchi@irf.se)4January20183611127July201722November201723November2017This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/This article is available from https://angeo.copernicus.org/articles/36/1/2018/angeo-36-1-2018.htmlThe full text article is available as a PDF file from https://angeo.copernicus.org/articles/36/1/2018/angeo-36-1-2018.pdf
By conserving momentum during the mixing of fast solar wind flow and slow
planetary ion flow in an inelastic way, mass loading converts kinetic energy
to other forms – e.g. first to electrical energy through charge separation and
then to thermal energy (randomness) through gyromotion of the newly born cold
ions for the comet and Mars cases. Here, we consider the Earth's exterior
cusp and plasma mantle, where the ionospheric origin escaping ions with
finite temperatures are loaded into the decelerated solar wind flow. Due to
direct connectivity to the ionosphere through the geomagnetic field, a large
part of this electrical energy is consumed to maintain field-aligned currents
(FACs) toward the ionosphere, in a similar manner as the solar wind-driven
ionospheric convection in the open geomagnetic field region. We show that the
energy extraction rate by the mass loading of escaping ions (ΔK) is
sufficient to explain the cusp FACs, and that ΔK depends only on the
solar wind velocity accessing the mass-loading region (usw) and the
total mass flux of the escaping ions into this region (mloadFload),
as ΔK∼-mloadFloadusw2/4. The expected distribution
of the separated charges by this process also predicts the observed flowing
directions of the cusp FACs for different interplanetary magnetic field (IMF)
orientations if we include the deflection of the solar wind flow directions
in the exterior cusp. Using empirical relations of u0∝Kp+1.2 and Fload∝exp(0.45Kp) for
Kp=1–7, where u0 is the solar wind velocity upstream of the bow
shock, ΔK becomes a simple function of Kp as
log10(ΔK)=0.2⋅Kp+2⋅log10(Kp+1.2)+constant. The major contribution of
this nearly linear increase is the Fload term, i.e. positive feedback
between the increase of ion escaping rate Fload through the increased
energy consumption in the ionosphere for high Kp, and subsequent
extraction of more kinetic energy ΔK from the solar wind to the
current system by the increased Fload. Since Fload significantly
increases for increased flux of extreme ultraviolet (EUV) radiation, high EUV
flux may significantly enhance this positive feedback. Therefore, the ion
escape rate and the energy extraction by mass loading during ancient Earth,
when the Sun is believed to have emitted much higher EUV flux than at
present, could have been even higher than the currently available highest
values based on Kp=9. This raises a possibility that the ion escape has
substantially contributed to the evolution of the Earth's atmosphere.
The effect of mass loading has long been discussed in relation to solar wind
interaction with comets (Biermann et al., 1967; Kömle and Lichtenegger,
1984; Szegö et al., 2000, and references therein) and unmagnetized bodies,
such as Venus and Mars (Breus et al., 1989; Szegö et al., 2000, and
references therein). This effect was originally considered for newly born
cold ions in cometary environments and planetary exospheres, where
cometary/exospheric atoms are exposed to the solar wind before they are
ionized with negligible initial velocity as compared to the solar wind
velocity in the planetary frame. In these applications, the mass-loading
effect transfers momentum and energy from the solar wind to the newly born
ions by driving these ions to move with finite velocities along cycloid
trajectories in the solar wind electric field and interplanetary magnetic
field (IMF).
The general concept of mass loading can be applied to any mixture of plasmas
with different velocities if the slower component (e.g. planetary ions)
gains momentum by experiencing the motional electric field E=-u×B of the faster component, where u is the velocity
(e.g. the local solar wind velocity) and B is the local magnetic
field (e.g. IMF). The slower component does not have to be cold ions because
the E×B convection velocity will be added even to heated ions.
Also, the momentum transfer does not have to be localized (e.g. can be
remote when it is connected by sufficiently strong magnetic fields) as long
as the motional electric field accelerates the slower component as bulk
plasma (Alfvén and Fälthammar, 1963, Sects. 3.2.1 and 3.11.1). For example,
the ionospheric plasma gaining anti-sunward velocity in the open geomagnetic
field region (Dungey, 1961) through the dynamo-motor mechanisms connected by
the geomagnetic field (Akasofu, 1975) can also be considered as a
mass-loading plasma to the solar wind flow on the same geomagnetic field
lines. As a consequence, the anti-sunward solar wind in the magnetospheric
boundary (e.g. across the polar magnetopause for southward IMF) is expected
to lose its anti-sunward momentum by the J×B force, where
J is the electric current density (in the present case, the
magnetopause Chapman–Ferraro current) flowing against E (Roederer,
1977). The source of the motional electric field is not limited to the open
magnetic field – viscous-like interaction at the magnetospheric boundary
(Axford and Hines, 1961), e.g. through Kelvin–Helmholtz instability
(Hasegawa et al., 2004), can also provide such an electric field in the
magnetosphere.
In a first-order approximation of momentum transfer from the solar wind to
the ionosphere, the mass (or inertia) of the magnetospheric plasma is
normally ignored compared to that of the ionospheric plasma above the
E region. However, ion data from Cluster and Polar have revealed that a large
amount of ionospheric ions are kept supplied to the exterior cusp, plasma
mantle and magnetosheath, with a total flux of about
1024-26s-1
(Peterson et al., 2008; Nilsson, 2011; Slapak et al., 2013, 2015, 2017) and
with a local number density ratio to the solar wind of about 1%. This
amount is sufficient to substantially decelerate the solar wind by about
10 % because of 16 times heavier mass of O+ than H+, and might
significantly influence the electrodynamics of the polar ionosphere, e.g. by
the formation of a localized independent current system in the cusp region
(Yamauchi and Slapak, 2017).
Since the spatial distribution of the escaping ions above the ionosphere is
very localized near the cusp and auroral zone (e.g. Peterson et al., 2008),
the mass-loading effect should also be localized and completely independent
of the mass-loading effect of the ionospheric E region mentioned above.
Therefore, this effect must be considered differently than the traditional
solar wind–magnetosphere–ionosphere coupling. Although localized to the cusp
region at the ionospheric altitude, this region is mapped to a large area at
high altitudes with a scale size much larger than the ion gyroradius (in the
plasma mantle, the geomagnetic field strength is about 25–30 nT where the ion
flux of escaping ions is maximized (Slapak et al., 2017), and the gyroradius
is <180km for 1 keVH+ and <1300km for 3 keVO+). Therefore, we can
assume in the model that the mass loading will be completed within the plasma
mantle.
Due to a strong dependence on Kp and solar EUV of the ion outflow
rate in the cusp and its vicinity (Norqvist et al., 1996; Cully et al., 2003;
Peterson et al., 2006; Slapak et al., 2017), the expected current system
should have a stronger dependence on Kp and EUV than what the
current system relevant to the global electrodynamics has (e.g. Fujii and
Iijima, 1987; Thomsen, 2004, and references therein). One obvious candidate
for this current system is the cusp current system (Potemra, 1994; Yamauchi
and Slapak, 2017, and references therein), which is composed of one or two
pairs of cusp field-aligned currents (FACs) flowing mainly inside the cusp
(equatorward and poleward) and the ionospheric currents closing the current
circuit with these FACs.
Many different terminologies have been used for the FAC part of the cusp
current system, as summarized in Yamauchi and Slapak (2017), and we here use
the terminology of cusp Region 1 FAC and cusp Region 0 FAC for the
equatorward part and the poleward part of the cusp FACs, respectively. Cusp Region 1 FAC and cusp Region 0 FAC are located approximately at the
extended location of the dayside (non-cusp) Region 1 FAC and the Region 0 FAC
toward midday, respectively (Iijima and Potemra, 1976; Heikkila, 1984;
Yamauchi et al., 1993b; Potemra, 1994), while multi-spacecraft satellite
observations at low altitudes demonstrated that they are independent of each
other in terms of locations (e.g. Ohtani et al., 1995; for review, Yamauchi
and Slapak, 2017). Here, the dayside (non-cusp) Region 1 FAC is the strongest
persistent FAC flowing into the ionosphere in the dawn sector and flowing out
from the ionosphere in the afternoon sector in both hemispheres (Iijima and
Potemra, 1976; Potemra, 1994).
To develop the scenario of explaining the cusp FACs by energy conversion
(dynamo) through mass loading, it is important to examine the distribution
and strength of the expected current system by such a “mass-loading
dynamo” for different IMF directions, different Kp, and
different solar wind parameters. Among these parameters, only the Kp
dependence can be estimated from the statistics of ion outflow, because the
solar wind and IMF dependences on the outflow in the exterior cusp and
plasma mantle are not well understood (only the Kp dependence has
been obtained). Considering these limitations, this paper makes a general
formulation for this mass-loading dynamo, and estimates its dependence on
Kp and the relevant current system distribution for different IMF
orientations.
Energy conversion by mass loading
The deceleration of fast plasma inflow (flowing in the -x direction, e.g.
the solar wind) with speed u and mass density ρ, after full mass
loading with newly born planetary/cometary heavy ions that have zero bulk
velocity (i.e. the bulk velocities of both species become the same), is
described by the conservation of momentum flux:
(ρ+Δρ)⋅(u+Δu)2=ρu2,
where Δρ is the mass density of the mass-loading ions, and u+Δu is the final speed after full mass loading. This relation is valid
even when mass-loading ions are introduced into the system from a different
direction than x (we call this direction as z), e.g. along the
geomagnetic field for the Earth's case as long as the bulk motion in the z
direction is not affected. Here, small letters x, y and z represent
general coordinates used for ideal mathematical formulation. Large
letters XY, and Z that appear later (e.g. in Fig. 1) represent the
global coordinates in the actual Sun–Earth geometry, and the Greek letters ξ
and η (e.g. Fig. 3) represent local coordinates of the actual
flow in the exterior cusp, i.e. ξ∼x and η∼y. Note that the
mass flux, which has the dimension of momentum per unit volume, is not
conserved, due to addition of Δρ into the system as either
newly born ions (planetary/cometary cases) or through the z direction
(Earth's case).
After full mass loading, the total kinetic energy flux (we use the terminology
“kinetic power” hereafter to avoid confusion) flowing through a
surface S perpendicular to the x direction changes from K=ρu3S/2
to K+ΔK=(ρ+Δρ)(u+Δu)3S/2, with a relation
of
K+ΔKK=u+Δuu⋅(ρ+Δρ)⋅(u+Δu)2ρu2=ρρ+Δρ1/2,
where we have used Eq. (), or
ΔKK=1+Δρρ-1/2-1.
For very small Δρ→dρ, this becomes
dKK≈-dρ2ρ.
If the planetary ions are kept supplied by new ionization of the
exospheric neutrals (Mars and Venus) or outgassing neutrals (comet), we
simply replace ρ and Δρ by mswnsw and mloadploadτ in Eq. (), where msw and mload are
the average masses of the fast inflow and newly produced mass-loading ions,
respectively, nsw is the number density of the fast inflow, and τ
is the duration that the inflow spends travelling through the ion production
region with mass-loading ions in which the volume ion production rate is
pload. By assuming O+ for the mass-loading ions and H+ for
incident flow, we have
ΔKK=1+16⋅ploadτnsw-1/2-1.
The converted energy is used first to azimuthally accelerate O+ along the motional electric field, which is simultaneously subject
to the Lorentz force proportional to the speed, to make cycloid motion. If
pload is significantly high, the azimuthal acceleration causes
detectable azimuthal bulk deflection of the incident H+ flow as observed
in the comet environment of 67P/Churyumov–Gerasimenko by the Rosetta Plasma
Consortium Ion Composition Analyser (RPC-ICA), which showed that such a
deflection is even more drastic than expected, whatever the heliocentric
distance (Behar et al., 2016, 2017).
Amount of converted kinetic energy by the escaping O+
The mass-loading region that we consider (red). Solar wind H+
ions (blue arrows) access this region through the blue area (cross section
S), with a width of about 4–10 RE in the Y direction and a
thickness of about 5–6 RE in the Z direction (although there is
no solid observational statistics for the plasma mantle, these are good
estimates, and these numbers disappear from the final formula anyway). The
extent of the region in the anti-flow direction (-X direction) is L.
Bulk velocity profiles of H+ (red) and O+ (blue) along the
average O+ flow direction over plasma mantle as defined by Fig. 6 of
Slapak et al. (2017) and obtained from the same statistics (Cluster,
2001–2005). Error bars defined by the standard deviation are given.
Instead of ionization, we here consider escaping ionospheric ions flowing
into the terrestrial exterior cusp and plasma mantle as shown in Fig. 1. The
largest differences from the new ionization (cometary) case are as follows: (a) the
mass-loading regions are geomagnetically connected to the ionosphere through
which separated charges can flow, and (b) the mass-loading ions are already
heated with non-zero gyrating velocity when they are supplied from one
boundary along the magnetic field perpendicular (in the z direction) to the
inflow. In Cluster observations at the Earth, these mass-loading ions coming
from the ionosphere have thermal velocities of >100kms-1 in Nilsson et
al. (2006) and Waara et al. (2011). This value is comparable to the decelerated
solar wind speed (about 100 kms-1 in the plasma mantle) as shown in Fig. 2, in
which the bulk velocities of O+ and H+ along the average O+ stream
line in the plasma mantle (as defined in Slapak et al., 2017) are plotted.
Fig. 2 further showed that H+ is decelerated whereas O+ is accelerated
along a region which is nearly parallel to the O+ outflow. Such velocity
profiles of O+ and H+ in the plasma mantle are consistent with mass
loading, i.e. momentum transfer from the solar wind H+ to the escaping
O+. Liao et al. (2015) also showed extra O+ acceleration between the
cusp and nightside magnetosphere other than the centrifugal acceleration
(Cladis, 1986; Nilsson et al., 2008).
We first estimate the energy conversion rate using the density ratio in a
similar way as Yamauchi and Slapak (2017). For simplicity, we ignore the -x
component of the bulk velocity of the mass-loading O+ at the boundary.
This means that the mass loading effect mainly adds anti-sunward bulk
velocity (E×B convection velocity) but not very much
gyro-velocity to the mass-loading ions. We also assume values of parameters
as follows: the local solar wind density and velocity just before accessing
the mass-loading region are nsw=5cm-3, usw=200kms-1,
respectively, and the cross section of the solar wind access to the
mass-loading region (see Fig. 1) is S=20–60 RE2=1–3 ×109km2, where RE is the Earth's radius (S depends strongly
on the solar wind and IMF condition, but disappears from the final formula
later). The total (number) flux and power of the solar wind inflow accessing
this mass-loading region are estimated as nswuswS=1–3 ×1027s-1 and
Kin=mpnswusw3S2=3-10×1010W.
The estimated values of the number flux indicate that the escaping flux of
O+ into this mass-loading region is 1–2 ×1025s-1 on
average (Nilsson, 2011; Slapak et al., 2017), which is around 1 % of the
incident solar wind flux (dρ/ρ about 10–20 % after multiplying
with the mass ratio of 16). Therefore, we expect 5–10 % conversion rate
from the kinetic power of the solar wind according to Eq. (),
i.e. ΔK=0.1–1×1010W, into other energy forms.
To properly treat the dependence on the parameters at both boundaries (for
the solar wind and ion escape), we next consider an integrated effect of the
entire mass-loading region with volume L⋅S (red area in Fig. 1), where
L is the length of the mass-loading region in the -x direction (the
location slides anti-sunward along the escaping ion trajectory), as
illustrated in Fig. 1. In this case we should replace dρ/ρ in
Eq. () by the ratio of the total mass fluxes into a box of
volume S(x)dx, i.e. mloadfload′(x)dx for mass loading ions and
ρ(x)u(x)S(x) for the solar wind ions, where fload′(x)dx is the
O+ outflow rate flowing into S(x)dx at x from the lower boundary. By
integrating this over dx, we have
ΔKKin=-12∫Lmloadfload′(x)dxρ(x)u(x)S(x),=-mload2mpnswusw2S(x=0)∫Lfload′(x)u(x)dx,
where we have used momentum conservation of ρ(x)u2(x)S(x) (total force of
the solar wind over S(x)=const).
Here, we assume low mixing ratio of mass-loading ions in the solar wind flow
such that changes of u(x) and fload′(x) along x are not large over
distance L. This assumption is reasonable for the observed very low mixing
ratio (∼1% level). In such a case, the integral part in
Eq. () (which, strictly speaking, depends on the
deceleration profile, and hence on nsw and S(x=0)) does not depend
much on these parameters. Therefore, by inserting Kin from
Eq. (), we finally have
ΔK=-14mloadusw∫Lfload′(x)u(x)dx∼-14mloadFloadusw2,
where Fload=∫Lfload′(x)dx is the total escaping flux into the
mass-loading region. Both the solar wind density nsw and the
cross section of the solar wind access (S) disappear from the expression of
ΔK in Eq. (), because a larger S or a higher
nsw means more incident kinetic power but at the same time a smaller
conversion rate due to a lower mass mixing rate.
According to Eq. (), a usw=200kms-1 solar wind
inflow just before the mass-loading region and a O+ escape rate of
Fload=1–2×1025s-1 into the mass-loading region predict
ΔK=0.3–0.5×1010W, narrowing the above estimate. This
value is comparable to the power that the cusp current carries (up to a total
of 106A over a 104V potential drop for relatively strong IMF; Yamauchi
and Araki, 1989), and hence the kinetic energy extracted by the mass loading
is indeed a good candidate for its supplier.
In this estimate, the kinetic energy of the bulk flow in the height (z)
direction is ignored. In reality, the effect of the anti-sunward bending of
the geomagnetic field converts the kinetic energy of the escaping O+ from
the z direction to the -x direction by the centrifugal acceleration, and
the mirror effect of the dipole-like spread of the magnetic field converts
the O+ gyromotion to the -x direction (e.g. Alfvén and Fälthammar,
1963, Sect. 2.3.6). These additional contributions to momentum to the escaping ions in the
-x direction reduce the amount of the kinetic energy conversion from the
solar wind to the escaping ions. However, the strongest escape is mainly
found in the dayside rather than in the nightside polar cap (e.g. Peterson
et al., 2008; Nilsson et al., 2012; Slapak et al., 2017), and this effect
should not significantly reduce the energy extraction rate ΔK.
Destination of the converted energy
The main destination of the converted energy cannot be heating (cycloid-like
gyromotion) of the escaping O+, because they are already heated with
comparable gyro-velocity as the decelerated solar wind before entering the
mass-loading region, as mentioned above. On the other hand, deceleration of
the solar wind means charge separation in the azimuthal (η) and
longitudinal (ξ) directions with respect to the local flow direction (see
Fig. 3 for directions) due to finite inertia (finite gyroradius) effects
(Willis, 1975; Yamauchi et al., 1993a) because the direction of the azimuthal
charge separation is opposite to the motional electric field, as illustrated
in Fig. 3 (Willis, 1975; Yamauchi and Slapak, 2017). Thus, electrostatic
energy accumulates during this process. One example of charge separation by
the solar wind deceleration is the bow shock, where the electric current is
expected to flow in the dynamo direction, i.e. against the motional electric
field of the solar wind (e.g. Tanaka, 1995).
Illustration of azimuthal charge separation in a constant magnetic
field B, when the solar wind inflow is decelerated by the start of mass
loading with the outflowing ionospheric ions (the boundary of the start of
the outflowing region is shown by an orange plane). The coordinates ξ and
η are defined by the flow direction at this boundary, and ξ is not
necessarily aligned with the solar wind direction (-X) outside the
magnetosphere when IMF points dawnward or duskward. The positive (negative)
charges are always deflected opposite to (along) the motional electric field
E of the flow, resulting in a dynamo current that causes a magnetic
deviation (ΔB). Because of the ionosphere, where charges can
move along the electric field across the magnetic field, the accumulated
charges form a current circuit through the field-aligned currents (J//
with blue and red arrows giving directions along B) closing in the
ionosphere.
Meanwhile, the azimuthal deflection also adds extra gyromotion to the H+
flow, i.e. additional non-laminar velocity to the H+ motion, increasing
the ion temperature. Distribution of the extracted kinetic energy to
different destinations, i.e. to the dynamo current system (charge
separation; see Fig. 3) and to thermal energy (extra gyration), depends on
the degree of the solar wind deceleration (degree of the compression over a
limited region in space), with sharper compression (e.g. under high Mach
number) leading to a larger fraction converting into thermal energy. This
applies to the exterior cusp: the pressure increase (solar wind deceleration)
by mass-loading effect should be enhanced in the narrowed cusp geometry, and
might even become a shock-like structure (Yamauchi and Lundin, 1997).
However, the location of the mass loading is not limited to the exterior
cusp, but is extended in both the azimuthal and flow directions. The plasma
mantle is in fact the mass loading area, as illustrated in Fig. 1. Therefore,
we ignore such kind of mixing effect in the following formulation. Without a
shock formation that converts a large part of the kinetic energy into thermal
energy, a substantial amount of the kinetic energy will be converted to
electrostatic potential energy.
Unlike the bow shock, the region of compression by mass loading is
electromagnetically connected to the ionosphere through relatively straight
geomagnetic fields (curvature might cause non-aligned currents, and we ignore
it here), where Pedersen currents in the electric field direction flow
perpendicular to the geomagnetic field. Then, the electric current (separated
charges) relevant to the mass loading will be consumed by a current circuit
that closes in the ionosphere. In such a case, a large fraction of the
original kinetic energy is expected to be converted to electrostatic energy
and to the current system.
Kp dependence and IMF dependenceKp dependence for Kp≤7
Next we consider the Kp dependence. Slapak et al. (2017) recently
showed a clear Kp dependence of O+ outflow rate in the plasma
mantle as ∼3.9×1024exp(0.45Kp) s-1 for
Kp=1–7, from 0.5×1025s-1 for Kp=1 to 10×1025s-1 for Kp=7, with a 1–2 orders of magnitude range for
a given geomagnetic activity condition (note that O+ in the
magnetosheath is
about one-third of this value). Applying this relation to
Eq. (), we have a proportionality relation
ΔK∝usw2exp(0.45Kp).
To obtain the Kp dependence of the energy conversion, we need to
know the Kp dependence of the solar wind velocity. Newell et al. (2007, 2008) derived a best empirical prediction of Kp from solar
wind parameters only, and found that any Kp dependence primarily
represents the solar wind velocity (u0) dependence with some contribution
from IMF and a very small contribution from the solar wind density n0
(u0 and n0 are defined upstream of the bow shock, whereas usw and
nsw are defined just before entering exterior cusp, because we are
interested in ΔK only in the mass-loading region).
(a) Average relation between Kp and solar wind velocity
(usw) obtained from hourly data (from NASA/OMNI dataset) averaged over
50 years (1965–2014). (b) Predicted increase of the extraction rate of the
solar wind kinetic energy ΔK as a function of Kp compared to Kp=0
given in Eq. (). Since the Kp–escape relation and the solar wind–Kp
relation is reliable only up to Kp=5-6, the estimation for Kp>6 is given in the dashed line, where we have ignored the effect of the
nonlinearly enhanced ion flux for extreme high Kp (≥8+) as reported
by Schillings et al. (2017). The red dashed line illustrates possible
enhancement after adding such an effect for extreme high Kp.
Therefore, a simple empirical relation between u0 (∝usw) and Kp is enough for the present purpose.
Figure 4a shows the average relation using 50 years of OMNI hourly values since 1965, which is reduced to u0=135⋅(Kp+1.2).
This relation also traces the peak percentage in the occurrence frequency plot of Kp for given u0 by Elliott et al. (2013, Fig. 3b).
Using this linear relation, Eq. () becomes
ΔK∝(Kp+1.2)2exp(0.45Kp)orlog10(ΔK)=0.2⋅Kp+2⋅log10(Kp+1.2)+constant.
Equation () predicts a nearly linear relationship between
log10(ΔK) and Kp as shown in Fig. 4b. This linearity
comes from the Fload term in Eq. (), i.e. positive
feedback between the increase of ion escaping rate Fload through the
increased energy consumption in the ionosphere for high Kp, and
extraction of more kinetic energy ΔK from the solar wind to the
current system by the increased Fload.
7$}?>Kp dependence for Kp>7
For the relationship given above, we expect additional nonlinear effects,
mainly for very high Kp, for two reasons: one is the extremely high escape
rate observed by the Cluster satellites during Kp≥8+, which
was much higher than the expected values from the empirical Fload–Kp
relation (Schillings et al., 2017). The resultant ΔK for these
“extreme events” should therefore be higher than the predicted
value from Eq. (), as hand-drawn in Fig. 4b with the red
dashed line. Schillings et al. (2017) found such an effect for Kp≥8+, while future studies will be needed to establish the geomagnetic activity
level (including Kp) when such a nonlinearity appears. The second possible
factor to add a nonlinear effect in Eq. () is the significant
enhancement of ion outflow during periods when the solar extreme ultraviolet
(EUV) flux is enhanced (Cully et al., 2003; Peterson et al., 2006). This
additional effect activates the nonlinear feedback between the ion escaping
rate Fload and energy extraction rate ΔK. This would cause much
higher geomagnetic activities, causing extremely high Fload.
Both extremely high Kp and high EUV flux are particularly important from
the viewpoint of ion escape on a geological timescale. According to the
result from the Sun-in-Time project (Dorren and Guinan, 1994), in which
G-type stars at different stages are compared in order to model the past Sun,
the early Sun is believed to have emitted much higher EUV flux, produced
faster solar wind, and caused stronger IMF and coronal mass ejections due to
a more active dynamo under much faster rotation compared to present day
(Ribas et al., 2005; Wood, 2006; Airapetian and Usmanov, 2016). The faster
solar wind and stronger IMF made the past nominal conditions equivalent to
the major storm conditions in the present time (Krauss et al., 2012), i.e.
extremely high Kp. With these effects only, Slapak et al. (2017)
assumed Kp=10 to estimate the ion escape rate for the ancient Earth from
their empirical result of the ion escape rate as a function of Kp.
It is possible to define Kp=10 for such a purpose because Kp is defined
as the logarithmic of the magnetic disturbance.
In addition, high EUV flux in the ancient time should mean a much higher
extraction rate from the kinetic energy for the same solar wind and
magnetospheric conditions. Therefore, we expect a higher O+ escape rate in
the ancient time than the empirical estimation and higher total amount of
O+ escape over the geological history (e.g. 4 billion years) than the
estimate by Slapak et al. (2017) or by Krauss et al. (2012), which already
reach 5×1017kg, i.e. 40 % of the total oxygen mass in the
atmosphere using the empirical estimation (Slapak et al., 2017) or 130 % of
the total oxygen mass in the atmosphere using a proxy method (Krauss et al.,
2012). Although the reservoir of oxygen molecules is continuously supplied by
the evaporation of the oceans, and although the ancient atmosphere contained
a large amount of CO2, we should not ignore the ion escape when considering
atmospheric evolution.
Kp dependence of velocities of H+ and O+ in the plasma mantle using the same dataset as Fig. 2.
IMF dependence
The remaining major task on this mass-loading dynamo model is the examination
of the dependence on different IMF clock angles defined as θc= atan(BY/BZ), where BY is the duskward component and BZ is the
northward component. The prediction of the distribution of separated charges
(i.e. distribution of the FACs) must qualitatively agree with observations.
Here, in addition to the azimuthal charge separation of the solar wind by the
mass-loading deceleration as depicted in Fig. 3, we also include the
modification of the solar wind inflow in the exterior cusp by θc.
According to the observations (for review, Potemra, 1994; Yamauchi and
Slapak, 2017), the northern cusp (injection region of the solar wind) shifts
toward postnoon for the IMF BY>0 and prenoon for the IMF BY<0. Also,
the solar wind flow in the northern cusp is deflected dawnward for the IMF
BY>0 and duskward for BY<0, respectively, flowing toward the
noon–midnight meridian. The directions of the shift and deflection are
reversed in the Southern Hemisphere for both the cusp location and the solar
wind flow. The shift of the cusp location has been successfully modelled by
the anti-parallel merging model that predicts the region of minimum magnetic
field strength (Crooker, 1979). Also, both the pressure consideration
(Yamauchi and Lundin, 1997) and magnetic tension force consideration (e.g.
Cowley et al., 1991) predict the deflection direction of the solar wind in
the exterior cusp.
We now include the effect of mass loading as illustrated in Fig. 3.
When we apply charge separation as illustrated in Fig. 3, we must align the flow direction and the -ξ direction.
Therefore, the -ξ direction is no longer aligned with the solar wind direction (-X) outside the magnetosphere.
With this modification, the predicted directions of the charge separation in the Northern Hemisphere are illustrated in Fig. 6.
Expected distribution of the positive/negative charges (FAC into/out
from the ionosphere) in the Northern Hemisphere in the exterior cusp–plasma
mantle region for three different IMF orientations: (a) purely southward
IMF, (b) duskward IMF, and (c) strongly northward IMF. Here, both the
mass-loading effect and the IMF effect on the solar wind flow in the exterior
cusp are considered. Morphologically, the downstream (plasma mantle for
southward IMF and LLBL northward IMF) corresponds to the lobe region; see
text for further explanations. Longitudinal charge separation (Yamauchi et
al., 1993a) is not included here. Blue colour represents flow and charge
separation of positive charges (H+) of the solar wind, and resultant FACs
(flowing into the ionosphere). Red colour represents flow and charge
separation of negative charges (e-) of the solar wind, and resultant FACs
(flowing out from the ionosphere). Coordinate systems are shown in two ways:
X-Y-Z based on the solar wind direction outside the magnetosphere
(similar to GSM coordinate), and ξ-η-Z based on the local solar
wind direction inside the exterior cusp (in the same manner as Fig. 3). The
abbreviations “R1-sense” and “R0-sense” indicate
polarities of the charge accumulation consistent with the flowing directions
of cusp Region 1 FAC and cusp Region 0 FAC, respectively. Note that the
different gyroradii between the solar wind protons and electrons are
overemphasized in the figure (a 100 kms-1 proton in a 30 nT magnetic field has
a gyroradius of only 35 km).
For purely southward IMF (θc=180∘, BY=0), the mass-loading
dynamo will separate solar wind protons (P+) and electrons (e-) in the
Y direction is illustrated in Fig. 6a in both hemispheres, which is
consistent with the flowing direction of the cusp Region 1 currents in both
hemispheres (Ohtani et al., 1995; Yamauchi and Slapak, 2017). To connect the
polarities of the separated charges and the FAC direction, we marked these
charges as “R1-sense” in the figure (“R0-sense” is
defined in the same way).
The mass-loading dynamo makes only one pair of FACs and therefore, does not
predict the relatively weak cusp Region 0 FAC in the plasma mantle for purely
southward IMF (Iijima et al., 1978; Bythrow et al., 1988). Since the cusp
Region 0 FAC for purely southward IMF is relatively weak (e.g. Yamauchi et
al., 1993b), this can be explained in different ways – such as the longitudinal
charge separation (Yamauchi et al., 1993a) or widening the flow after narrow
channel (e.g. Saunders, 1992).
We next consider strongly dawnward or duskward IMF cases (|BY|>|BZ|). As illustrated in Fig. 6b for
duskward IMF (BY>0) case, the strong deflection of the solar wind inflow
makes the azimuthal (η direction) separation of the charges rotate
counter-clockwise, i.e. positive charges toward equatorward of the cusp and
negative charges toward poleward of the cusp, respectively, in the Northern
Hemisphere, instead of prenoon–postnoon pairs. In the Southern Hemisphere,
the pattern rotates clockwise, i.e. negative charges toward the equatorward
and positive charges toward the poleward, respectively. As a result, we
expect a pair of cusp Region 1 FAC and cusp Region 0 FAC, which is
consistent with observations in both hemispheres. The resultant FAC
distribution becomes similar to those predicted as a part of the global
current system in the simple magnetohydrodynamics (MHD) concept (e.g. Cowley
et al., 1991). But as summarized in the Introduction, this single MHD model
does not explain the observed independence of the FACs in the cusp and
outside the cusp.
Finally, we consider strongly northward IMF cases (BZ≫|BY|). In such a case, the plasma flow inside the cusp is very
slow (Woch and Lundin, 1992; Lavraud et al., 2004) without the plasma mantle
(Sckopke et al., 1976; Yamauchi and Lundin, 1993), and therefore we cannot
simply apply our model to the cusp flow. However, the solar wind flow may
still exist in the magnetosphere, through the prenoon boundary and postnoon
of the cusp, flowing toward the low-latitude boundary layer: LLBL (Mitchell
et al., 1987; Raeder et al., 1997). In such a situation, we can still apply
the mass loading as a combination of strong dawnward IMF and duskward IMF, as
illustrated in Fig. 6c, with the entire plasma flow deflected to both sides
of the boundary layer. Since we have two different flows towards +Y and
-Y directions, we expect two pairs of FACs. The predicted senses of the
separated charges are consistent with the observed flowing directions of FACs
in both hemispheres (Potemra et al., 1994; Yamauchi and Slapak, 2017, and
references therein). However, it is difficult to evaluate whether the observed
deceleration is mainly caused by the mass-loading effect, by the magnetic
tension force, or by Kelvin–Helmholtz instability. Therefore, the degree of
contribution from mass loading during northward IMF is not clear.
Discussion
As mentioned in the Introduction, mass-loading deceleration of the solar wind
in the open geomagnetic field region in the magnetosphere with the “load” originating in the ionosphere is not a new concept. The new
addition in the present study is dividing the mass-loading ions into two
independent components. One is the ionospheric (mainly E region) ions, where
the majority of the electrodynamic energy – that is originally coming from
the deceleration of the solar wind – is consumed on global scales. The other
component corresponds to the escaping ionospheric ions that interact directly
with the incoming solar wind in a localized region in the exterior cusp, its
vicinity, and the plasma mantle.
These two mass-loading mechanisms are independent of each other in spatial
distribution, timescale, and dependence on the solar wind and IMF. The
resultant current systems are also independent of each other. Yet, the
flowing directions of the current systems by these mechanisms are similar to
each other in the cusp region. This is because the present model simply
separated the deceleration (by mass loading) and deflection of the solar wind
flow (by magnetic tension force) instead of explaining both with the magnetic
tension force only.
The estimation of the energy extraction rate ΔK by the local mass
loading associated with the escaping ion (0.3–0.5×1010W for
200kms-1 solar wind inflow and an O+ escape rate of 1–2×1025s-1 according to Eq. ) is sufficient to
explain the cusp current system, a localized yet large-scale current
system that closes with the cusp Region 1 and cusp Region 0 FACs and the
ionospheric currents closing these FACs, in the cusp and its vicinity. The
predicted location and IMF dependence of this dynamo is also consistent with
the cusp Region 1 and cusp Region 0 FACs, which are found to be independent
of the dayside Region 1 FACs in the observations (e.g. Ohtani et al., 1995;
for review, Yamauchi and Slapak, 2017).
Independent of the mass loading, a compressional structure is predicted
near the exterior cusp due to the geometry of the exterior cusp (Yamauchi and
Lundin, 1994; Tanaka, 1995). However, the current system due to such
compression without the mass-loading effect simply becomes a part of the
global current system in the solar wind–magnetosphere interaction. On the
other hand, there is a strong seasonal variation (and hence the EUV
dependence) of the cusp current system (Fujii and Iijima, 1987; Thomsen,
2004), and this can be a result of combination of the strong EUV dependence
of the ion outflow (Cully et al., 2003; Peterson et al., 2008) in the
mass-loading model and a geometry effect in the compression model (the
exterior cusp is facing the magnetosheath flow in the summer hemisphere).
While a statistical study of the EUV (illumination at the ionospheric height)
dependence of the O+ flow in the exterior cusp and the plasma mantle is
needed in the future to diagnose the relative importance of these two
mechanisms, if these two mechanisms are coupled, we expect a significant
modification of the compressional structure in its location, enhancement of
the compression, and enhancement of the Kp dependence (Yamauchi and
Lundin, 1997). At the moment it is not possible to say whether such a coupled structure
generates the current system independently from that generated by the global
solar wind–magnetosphere–ionosphere coupling.
The expected compression (deceleration) of the solar wind also causes a
longitudinal charge separation (e.g. Yamauchi et al., 1993a) as well as in
the azimuthal direction that is illustrated in Fig. 3. Such a longitudinal
charge separation can actually be important according to comet 67P/C-G
observations by Rosetta/ICA: the observation showed that the momentum
transfer is not limited to the solar wind flow direction, but occurs also in
the azimuthal direction (Nilsson et al., 2015; Behar et al., 2016). This
indicates that the direction of the charge separation is not necessarily in
the azimuthal direction but could largely be in the solar wind direction even
without extra factors such as FACs, i.e. as the result of mass loading of
newly born ions. The resultant deflection of the solar wind observed by
Rosetta/ICA is very often more than 90∘ although the speed did not
decrease very much when we see the deflected solar wind (Behar et al., 2017;
Nilsson et al., 2017). The longitudinal charge separation was also observed
at the Earth where FACs flow. Cluster observation found a travelling
compressional structure in the inner magnetosphere, which actually caused
evening sector aurora, i.e. strong FACs (Yamauchi et al., 2009).
Yamauchi (1994) made a MHD simulation of FACs by such a longitudinal charge
separation in a compressional structure, and showed that its intensity is
sufficient in explaining the cusp current system with flowing directions
consistent with the observations. Since what we showed in the above
formulation is that the loss of the kinetic power (ΔK) is sufficient
in powering the cusp current system, the distribution of the energy
destination between this type (longitudinal) of current system and the Fig. 3
type (azimuthal) current system is not clear. This problem cannot be solved
without numerical simulations that include mass-loading effects.
Such a numerical formulation of the mass-loading effect on the proton flow
should be possible even within the MHD regime (and of course with hybrid
simulations and the other high-accuracy models) if we violate the
conservation of the mass-flux and electric charges, allowing the accumulated
charges to flow toward the ionosphere (outside the simulation box). In such a
formulation, energy conservation should be case by case because the energy
conversion to the thermal energy should be different between a comet case
(additional O+ is cold) and the Earth's case (additional O+ is already
heated).
Conclusions
By considering the entire region of the mass loading by the escaping ions
(exterior cusp and plasma mantle), we estimated the total kinetic energy
extraction rate (in power unit) ΔK from the solar wind inflow in this
region. Since the escaping ions are already heated with large gyro-velocity
before entering the mass-loading region, ΔK is converted mainly to
other energy forms than thermal energy (through gyromotion), unlike the mass
loading of newly ionized atoms of comets and unmagnetized planets. The
estimated value of the extracted energy depends mainly on the solar wind
velocity accessing to the mass-loading region (usw) and the total mass
flux of the escaping ions into the mass-loading region (mloadFload),
i.e. ΔK∼-mloadFloadusw2/4, which is about
109-10W, depending on the solar wind and magnetospheric conditions.
This agrees well with the energy of the cusp current system (up to
106A
over a 104V potential drop for a strong-IMF case). Since the IMF BY dependence can also be explained with this dynamo by considering the
BY-dependent deflection of the solar wind flow in the exterior cusp,
the mass-loading dynamo is a good candidate as the source mechanism of the
cusp current system.
Using the observed Kp dependence of the ionospheric ion outflow flux
in the mass-loading region, we also estimated the Kp dependence of
the mass-loading dynamo as ΔK∝(Kp+1.2)2exp(0.45Kp), or log10(ΔK)=0.2⋅Kp+2⋅log10(Kp+1.2)+constant. Since Fload
significantly increases for increased EUV flux, high EUV flux may
significantly activate this positive feedback, causing much higher kinetic
energy extraction rate by mass loading than the above prediction and hence
much higher ion escape rate than the empirical relation. Therefore, the ion
escape during the ancient times, when the Sun is believed to have emitted
much higher EUV flux than present days, could have been even higher than the
currently available highest values based on Kp=9. This raises a possibility
that the total O+ escape over 4 billion years could have been much larger
than the current atmospheric mass, and thus it could have substantially
contributed to the evolution of the Earth's atmosphere.
The Cluster CODIF and FGM data
are provided by Institut de Recherche en Astrophysique et Planetologie in
Toulouse, France, and Imperial College London, UK, through the Cluster
Science Archive (www.cosmos.esa.int/web/csa). The solar wind OMNI data are provided by NASA and Kp is
provided by GFZ, Adolf-Schmidt-Observatory Niemegk, Germany. Both datasets are
taken from NASA/GSFC OMNI site (http://omniweb.gsfc.nasa.gov/ow.html).
The authors declare that they have no conflict of interest.
Acknowledgements
This work is partly supported by the Swedish National Space Board. The Cluster
project is performed by the European Space Agency.
The topical editor, Vincent Maget, thanks two anonymous referees for help in evaluating this paper.
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