ANGEOAnnales GeophysicaeANGEOAnn. Geophys.1432-0576Copernicus PublicationsGöttingen, Germany10.5194/angeo-36-101-2018Evaluation of electromotive force
in interplanetary spaceNaritaYasuhitoyasuhito.narita@oeaw.ac.atVörösZoltánhttps://orcid.org/0000-0001-7597-238XSpace Research Institute, Austrian Academy of Sciences,
Schmiedlstr. 6, 8042 Graz, AustriaInstitute of Physics, University of Graz,
Universitätsplatz 5, 8010 Graz, AustriaDepartment of Geophysics and Space Sciences,
Eötvös University, Budapest, HungaryYasuhito Narita (yasuhito.narita@oeaw.ac.at)24January201836110110629October201719December201719December2017This 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/36/101/2018/angeo-36-101-2018.htmlThe full text article is available as a PDF file from https://angeo.copernicus.org/articles/36/101/2018/angeo-36-101-2018.pdf
Electromotive force plays a central role in the turbulent dynamo mechanism
and carries important information on the nature of the turbulent fields. In
this study, an analysis method is developed for the electromotive force and
the transport coefficients such as those for the α effect (coefficient
α) and the turbulent diffusivity (coefficient β). The method is
applied to a magnetic cloud event observed by the Helios 2 spacecraft in the
inner heliosphere. The electromotive force is enhanced together with the
magnetic cloud event by 1 or 2 orders of magnitude, suggesting that the
magnetic field can locally be amplified in the heliosphere, presumably for a
short time.
Interplanetary physics (interplanetary magnetic fields) – space plasma physics (transport processes)Introduction
The essential part in the dynamo mechanism amplifying the large-scale
magnetic field lies in the existence of electromotive force. In the theory of
mean-field electrodynamics, the electromotive force is defined as a
statistically averaged vectorial quantity, and it is a cross product between the
fluctuating flow velocity δU and the fluctuating magnetic field
δB,
Eem=〈δU×δB〉.
Here, the angular brackets 〈⋯〉 denote the operation of
ensemble averaging over many realizations. The electromotive force has
dimensions of the electric field.
In the mean-field dynamo theory, the electromotive force is assumed to be
linear in the mean magnetic field B0 to the first order, and also in
the spatial gradient of the mean field such as the curl of the mean magnetic
field, ∇×B0 (one may also use the current density
j using Ampère's law), and the curl of the mean flow velocity,
∇×U0 (which is the vorticity). A more comprehensive form
for the electromotive force using up to the first-order derivatives of the
mean fields is expressed as Eem=αB0-β∇×B0+γ∇×U0,
where the first term on the right-hand side in
Eq. () with the coefficient α and the mean
magnetic field serves as the amplification of the mean magnetic field by
small-scale twisting flow motions. The second term with the coefficient
β (which has the same dimension as that of the magnetic diffusivity)
and the curl of the mean magnetic field (which is proportional to the
electric current density for the mean magnetic field) is the turbulent magnetic
diffusion, and the third term with the coefficient γ and the curl of
the large-scale flow velocity the cross helicity dynamo.
A simpler form or a minimal expression of the electromotive force for a
dynamo mechanism is composed of the two terms associated with the mean
magnetic field only :
Eem=αB0-β∇×B0.
Equation () was historically proposed before the
importance of the cross helicity term was recognized in studies on the
sunspot number variation and the accretion disk dynamo
. Here again, the
first term with α is responsible for the growth of the mean magnetic
field, and it is essential to the α effect due to the amplification of
the magnetic field by a turbulent twisted flow motion. The second term with
β is destructive and contributes to an effective diffusion of the
large-scale fields when coupled to the induction equation (turbulent
diffusion). Equation () is obtained from the induction
equation for the fluctuating magnetic field using the first-order smoothing
approximation (see, e.g., , for a concise derivation).
The coefficients α and β are in fact tensors (of rank 2 and 3,
respectively). In the isotropic turbulence treatment, these tensors reduce to
scalars (the α tensor becomes diagonal and the β tensor is
proportional to the Levi-Civita antisymmetric tensor). In our work, we use
the simpler form (Eq. ) for the sake of brevity in
mathematical operations in the spirit of proof of concept.
It is important to note that the electromotive force is a second-order
fluctuation quantity and is in the same class as the energy densities of the
fluctuating fields (for the magnetic field and the flow velocity) and the
helicity densities (magnetic helicity, kinetic helicity, and cross helicity).
The electromotive force appears as off-diagonal elements of the covariance
matrix composed of the fluctuating magnetic field and fluctuating velocity
.
Here we propose an analysis method to determine the electromotive force using
in situ spacecraft data in space plasma. While a large number of studies on
turbulent space plasmas (e.g., solar wind turbulence) concentrate on
the behavior of the energy and helicity quantities (see, e.g., reviews and
monographs such as ), only few
observational studies have been performed related to the discussion on the
electromotive force in space plasma physics, e.g., an impulsive solar wind
event in the Earth magnetosphere and turbulent solar wind
plasma . In particular,
discovered evidence that the electromotive
force follows a power-law spectrum in the solar wind (indicating turbulent
electromotive force) and is not simply proportional in the mean magnetic
field. The electromotive force is, in contrast to the limited space plasma
studies, one of the primary study targets in the laboratory pinch plasmas
and in the liquid sodium experiment
. This paper is motivated by a wish to fill the gap in
the application of the electromotive force between laboratory and space
plasma studies.
Estimates of transport coefficients
When modeling with B0 and ∇×B0 and neglecting
the other terms or contributions from the cross helicity or the higher-order
derivatives of the mean fields, it is possible to evaluate the transport
coefficients α and β directly from the measurement of the
electromotive force. To this goal, we first build a vector product of the
mean-field model of the electromotive force (Eq. ) with
the mean magnetic field B0 and eliminate the term with the
coefficient α. We obtain an estimator for the coefficient β
after some vector calculus:
βobs=-1H2H⋅B0×Eem,
where
H=B0×∇×B0
and H=|H|. We now build a scalar product of
Eq. () with the mean magnetic field B0. Since the
coefficient β is evaluated in Eq. (), we obtain
an estimator for the coefficient α:
αobs=1B02B0⋅Eem+βB02B0⋅∇×B0.
The inputs to the calculation of the coefficients α and β are
the electromotive force Eem, the mean magnetic
field B0, and the curl of the mean magnetic field ∇×B0. When the measurement is performed by a single-point sensor (or
spacecraft) in a supersonic or super-Alfvénic flow, one may limit the
gradient direction to the flow direction and introduce an assumption that the
spatial derivative is mostly a time derivative with advection by the mean
flow U0,
∇≃-eU1U0∂∂t=-U0U02∂∂t,
where U0=|U0| is the absolute value of the flow speed and
eU=U0/U0 the unit vector in the direction of the
mean flow. Note that Eq. () assumes that the spatial
gradient is parallel to the mean flow, which reduces the problem to one
spatial dimension, and that the spatial derivative is estimated by the time
derivative using Eq. (), which is equivalent to the
stationarity of the mean field, d/dt=∂/∂t+U0⋅∇=0.
In this paper, the time derivative is evaluated using the
first-order-accuracy backward difference due to the irregularly sampled time
series data,
∂B0∂tt≃B0(t)-B0(t-Δt)Δt,
where Δt is the time resolution in the data. Thus, the x component
of the curl of the mean magnetic field is evaluated as
∇×B0x≃1U02-U0y∂B0z∂t+U0z∂B0y∂t≃-U0yU02B0z(t)-B0z(t-Δt)Δt+U0zU02B0y(t)-B0y(t-Δt)Δt.
The y and z components of the curl of the mean magnetic field are
obtained by circulating {x,y,z} into {y,z,x} and {z,x,y} in
Eqs. () or (),
respectively.
Magnetic cloud in interplanetary space
The estimator for the electromotive force and that for the transport
coefficients (α and β) are tested against a magnetic cloud event
in interplanetary space using the magnetic field and plasma (ion) data from
the Helios 2 spacecraft . The magnetic field data are
obtained by the fluxgate magnetometer (also referred to as the saturation
core magnetometer or the Förstersonde magnetometer) and
the plasma data by the electrostatic analyzer
. Merged data between the magnetic field
and the plasma measurements are used. The sampling rate varies from 40 s to
multitudes of 40 s, and there are data gaps as well. Figure 1 displays the
magnetic field magnitude, the proton bulk speed, the proton number density,
and the electromotive force magnitude as time series plots from 17 to
20 April 1978. The dots in black represent the original measurements, and the
solid lines in gray represent the smoothed data. The Helios 2 spacecraft is
in an inbound orbit in the inner heliosphere and moves from a heliocentric
distance of 0.41 AU (astronomical unit) on 17 April 1978 to a distance of
0.37 AU on 20 April 1978. A magnetic cloud passes by the spacecraft around
18:00–20:00 UT on 18 April. The magnetic field magnitude increases from
about 40 nT to about 70 nT, the ion bulk speed from about 500 to
800 km s-1, and the ion number density from about 30 to about
300 cm-3.
The electromotive force Eem is evaluated by
constructing the mean field and the fluctuation field from the time series
data. The detailed procedure is as follows.
Mean field determination. The mean field is determined by assuming
the ergodic hypothesis and regarding the mean values in the time domain as
statistically representative of ensemble average. We use a fixed time window
of 3 h (90 min before the window center and after the center). The
averaging period is determined as a compromise such that the fluctuation
fields have zero mean values under the conditions of the shortest periods and
a sufficient number of data points within the time window. In this work, the
smoothing is computed typically over 130 to 270 data points, depending on
data availability (due to changes in the sampling rate) within each fixed
time window of 3 h. The mean fields for the magnetic field, the flow
velocity (for the protons), and the density (again for the protons) at the
mth time record tm are determined by the box-car averaging method using
the number of data points N within the time window, e.g., B0(tm)=1N∑n=0N-1B(tm+n-N/2), for the magnetic
field. Other possibilities of the mean field determination are discussed in
the last section of the paper.
Fluctuation field determination. The fluctuation fields are obtained
by subtracting the mean field from the measured field, e.g., δB(tm)=B(tm)-B0(tm), for the magnetic field.
Electromotive force. The electromotive force is determined in the
time domain by building a cross product between the fluctuation flow velocity
and the fluctuation magnetic field and then averaging over the same time
window as that for the mean fields (again, using the box-car averaging),
Eem(tm)=1N∑n=0N-1δU(tm+n-N/2)×δB(tm+n-N/2).
The electromotive force increases from about 102 mV km-1 before
the magnetic cloud event to above 103 mV km-1 during the magnetic
cloud event. For reference, the order of the electromotive force is
1 mV km-1 for a flow velocity fluctuation of 1 km s-1 and a
magnetic field fluctuation of 1 nT.
The transport coefficients α and β are evaluated using
Eqs. () and (), respectively,
and their magnitudes are displayed as a function of time in the bottom two
panels in Fig. . The order of the coefficient α is
1 km s-1 for an electromotive force of 1 mV km-1 and a mean
magnetic field of 1 nT, and that of β is 1 km2 s-1 for an
electromotive force of 1 mV km-1 and a curl of the magnetic field of
1 nT km-1.
The coefficient α fluctuates around 1 km s-1 overall with an
excursion to about 101 to 102 km s-1 during the magnetic cloud
event. If we use an estimate of 50 nT for the mean magnetic field and
103 mV km-1 for the electromotive force, we obtain the coefficient
α at about 20 km s-1, which roughly agrees with the peak value
of α.
In contrast, the coefficient β has by far larger values throughout the
observed time interval and varies by about 4 orders of magnitude between
1010 km2 s-1 at the beginning of 17 April 1978 and a peak of
nearly 1014 km2 s-1 around 19:00 UT on 19 April 1978 at the
time of magnetic cloud passing. It is interesting to observe that the
enhancement of the coefficient β is found not only during the magnetic
cloud event but also during other periods, for example, around
02:00–03:00 UT on 18 April 1978 when the magnetic field magnitude is at a
local minimum. Another interesting feature is that the variation sense of the
coefficient β changes from an anticorrelation sense to that of the
coefficient α (e.g., 00:00–06:00 UT on 17 April 1978) into a
positive correlation sense (entire period on 18 April 1978) and then back
into the anticorrelation sense (06:00–18:00 UT on 19 April 1978). Since
the coefficient β is an index of turbulent magnetic diffusion, the
variation profile of β indicates that turbulence occurs in an
inhomogeneous way in the solar wind; some variations are accompanied by the
magnetic cloud, others not. A naive estimate for the reason of the large
values of the coefficient β is as follows. We use values of
102 mV km-1 for the electromotive force and
10-5 nT km-1 (which is 10-17 V s m-3) for the curl
of the mean magnetic field and obtain a value of the coefficient β
about 107 km2 s-1, which is the lower limit of the measured
profile for the coefficient β. Here we used a flow speed of
500 km s-1 and a varying timescale of 1000 s for the advection and
a magnetic field of 10 nT.
A close inspection shows that the curl of the mean magnetic field is even
much smaller than the estimate above and is about 10-10 nT km-1,
which gives the coefficient β of the order of
1012 km2 s-1. There are various contributions that suppress
the values of the curl of the mean field. The assumption of the
one-dimensional advected structure is motivated by the use of single
spacecraft data. On the one hand, the alignment of the mean magnetic field with
its curl implies the use of a force-free field or a field-aligned current
configuration, realized in various space plasma environments. The alignment
may have a more fundamental nature. On the other hand, the constraint to the
one-dimensional advected structure in the data analysis can be tested against
multi-point measurements, from Cluster or Magnetospheric
Multiscale (MMS) . For example, it would be interesting to
compare two different scenarios upon the direction of the spatial gradient
using the multi-point data: first, if the gradient of the large-scale fields
is mostly aligned with the mean flow direction and, second, if the gradient is
perpendicular to the mean magnetic field direction as is the case for solar
wind turbulence on ion kinetic scales (at about 400 km)
. Perhaps the alignment may be scale-dependent from the
fluid picture of plasma (on the length scales of the order of 10 000 km) down
to ion-kinetic scale (of the order of 400 km), and the small-scale field-aligned
currents may be an important component to the turbulent solar wind. Also,
continuous measurements at a higher sampling rate are available with the
Cluster and MMS missions in comparison to the plasma measurements by the
Helios spacecraft limited to 40 s.
Time series plots of magnetic field magnitude B, ion bulk speed
U, ion density n, estimated electromotive force Eem
(in the same units as those of the electric field), coefficients α and
β for a magnetic cloud event observed by Helios 2 spacecraft in the
inner heliosphere from 17 to 20 April 1978.
Outlook
The electromotive force is a second-order quantity such as the energy
densities (magnetic energy and kinetic energy) and the helicity densities
(magnetic helicity, kinetic helicity, and cross helicity) of the fluctuating
fields, but its analysis using the in situ spacecraft data in space plasma
has largely been overlooked in earlier studies. Although assumptions have
to be incorporated, such as the use of one-dimensional advected structure in
the time series data, it is possible to observationally evaluate the
electromotive force and determine the transport coefficients using the mean
field model for the dynamo theory. Studies on the transport coefficients in
the turbulence and dynamo theories can be performed not only by the analytic
or numerical methods but also by the observational method. The advantage of
the presented method is that even data with different sampling rates can be
used to the studies of the electromagnetic force and the transport
coefficients at the cost of first-order accuracy approximation in the
gradient computation.
Although the proposed method is rather a simple or a naive one, the analysis
shows an enhancement of the electromotive force at the magnetic cloud event
by 1 or 2 orders of magnitude. This result indicates a scenario that the
enhanced or strong magnetic fields in the heliosphere are not merely
generated in the Sun or in the solar atmosphere and stream into the
heliosphere, but they can be actively amplified in the heliosphere by the flow
shear or twist. We believe, however, that the dynamo action is unlikely to
occur in the heliosphere. The enhanced electromotive force or the α
effect was associated with the magnetic cloud event for a short time. The
electromotive force is largest during the magnetic cloud, but the transport
coefficients appear to be large also at other times, notably around the
minimum magnetic field. For the magnetic field to grow by a dynamo mechanism,
the transport coefficients must stay at larger values for a much longer time
period and must also operate on different components (toroidal and poloidal
components) of the magnetic field.
There are various ways to improve the method presented here. First, from an
accuracy point of view, the regularly sampled data are preferred because the
second-order central difference method can be applied to the calculation of
the derivatives. Second, for a further evaluation of the mean-field dynamo
theory, one may test the relations on the transport coefficients α=-13τδU⋅(∇×δU) (proportional to the kinetic helicity density) and
β=13τδU⋅δU (proportional to the kinetic energy density), where τ
denotes the characteristic time of turbulence. To achieve this test, the
quantity τ needs to be determined from the time series data. Third, one
may include the cross helicity term in the test for the mean-field dynamo
theory. Fourth, the use of multi-point data is helpful to relax the
assumptions of time stationarity and the alignment of the spatial gradient
with the mean flow direction.
Helios plasma and magnetic field data are available at CDAWeb
https://cdaweb.sci.gsfc.nasa.gov.
The authors declare that they have no conflict of
interest.
Acknowledgements
This work is financially supported by the Austrian Space Applications Programme
(ASAP) at Austrian Research Promotion Agency, FFG ASAP12 SOPHIE, under
contract 853994 and Austrian Science Fund (FWF) under contract P28764-N27.
Discussion and collaboration with Philippe Bourdin and Bernhard Hofer in the
preparation of the manuscript are acknowledged in the framework of the ASAP project.
Yasuhito Narita is grateful to Masahiro Hoshino and his group at the
University of Tokyo for their hospitality during his research stay, which was supported by
the Japan Society for the Promotion of Science, Invitational Fellowship for
Research in Japan (short-term) under grant FY2017 S17123. The topical editor, Georgios Balasis, thanks Octav
Marghitu for help in evaluating this paper.
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