Current knowledge on the description of the interplanetary magnetic field is reviewed with an emphasis on the kinematic approach as well as the analytic expression. Starting with the Parker spiral field approach, further effects are incorporated into this fundamental magnetic field model, including the latitudinal dependence, the poleward component, the solar cycle dependence, and the polarity and tilt angle of the solar magnetic axis. Further extensions are discussed in view of the magnetohydrodynamic treatment, the turbulence effect, the pickup ions, and the stellar wind models. The models of the interplanetary magnetic field serve as a useful tool for theoretical studies, in particular on the problems of plasma turbulence evolution, charged dust motions, and cosmic ray modulation in the heliosphere.

The interplanetary magnetic field (IMF)
is a spatially extended magnetic field of the Sun
and forms together with the plasma flow
from the Sun (referred to as the
solar wind) a spatial domain of the heliosphere

IMF is also referred to as the heliospheric magnetic field.

around the Sun surrounded by the local interstellar cloud. Starting with the first direct measurements in the 1960sIn the lowest-order picture, the IMF has an Archimedean spiral
structure, also referred to as the Parker spiral
after

Typical values of the IMF magnitude (in the sense of the mean field)

Model construction of the IMF has immediate applications
in the following plasma physical or astrophysical problems:

Plasma and magnetic field in interplanetary space
develop into turbulence. Early in situ measurements
in the 1960s have already shown that the frequency
spectrum of the fluctuation of the IMF is
a power law over a wide range of frequencies
(typically in the MHz regime)

Dust grains in interplanetary space
have typically a length scale of
nanometers to micrometers and are
electrically charged by various processes,
e.g., sticking of the ambient electrons
onto the dust surface (which makes
the dust charge state negative) or
photo-electrons (which makes the
charge state positive)

A cosmic ray consists mostly (more than 90 %) of protons.
The spectrum of the cosmic ray is well characterized
by a power law as a function of the particle energy
(kinetic energy, strictly speaking)
with a peak at about 1 GeV and a slope of about

Here we review various models of the IMF
with an emphasis on the hydrodynamic approach
and the analytic expression.
This review is intended to complement
a more comprehensive review by

We also limit our review to the analytic expression
as much as possible. Analytic expression of
the magnetic fields is a
useful tool
in space science and has been constructed for various plasma domains
or plasma phenomena in the solar system
other than the solar wind:
solar corona

The advantage of the analytic or semi-analytic expression
is that one can implement the magnetic field models by themselves
for the theoretical studies of the solar system
plasma phenomena. Verification of the magnetic field models
is possible using the existing in situ spacecraft data
from, e.g., the Helios, Voyager, and Ulysses missions as well as
the upcoming measurements in interplanetary space
by Parker Solar Probe

We focus on the kinematic approach such that the flow pattern is given as an external field of a model field. The magnetic field is passive in the sense of the frozen-in field into the plasma. The reaction of the magnetic field onto the plasma motion (such as the Lorentz force acting on the plasma bulk flow) is not considered here.

In this section we review the formulation of the original Parker spiral model of the interplanetary magnetic field.

As suggested by

The Parker model treats the solar wind as a one-dimensional
(in the radial direction), steady-state, isothermal
thermally driven stream. Basic equations are
the continuity equation,

A more detailed analysis of the Parker model with the asymptotic solution
of the flow velocity is presented by

Radial solution of the solar wind

Using the angular velocity of the Sun,

The transformation into the stationary frame (HGI, heliographic inertial)
yields the same expression of the magnetic field as
Eqs. (

Note that, due to a Galilean transformation, the electric field has a
convective contribution in the polar direction

Streamlines in the Parker spiral model of interplanetary magnetic field around the Sun (a filled circle in yellow) in the heliospheric ecliptic plane up to 5 astronomical units (au) under different conditions of the solar wind speed. The orbit of the Earth is marked by a blue curve at a radius of 1 au, that of Mars by a red curve (1.5 au), and that of Jupiter by a green curve (5 au).

We rewrite
Eqs. (

We note that in Eqs. (

The distance to the surface on which an azimuthal angle of 45

Heliocentric distance

Alternatively, the Parker spiral model can be
formulated in terms of the spiral angle

In this formulation the magnitude of the magnetic field is
estimated as

The magnetic vector potential

Another formulation of the vector potential
(again, under the Coulomb gauge) is
to introduce a scalar potential as

Magnetic field lines (black curves) in the Parker spiral
model for different latitude angles

The magnetic field lines for the Parker spiral model are shown in
Fig.

The Parker spiral model well approximates the mean and large-scale structure of the interplanetary magnetic field of our solar system. However, it fails to describe the three-dimensional geometry and evolution in time on various scales.

The Parker model does not recognize the sign reversal
of the dipolar magnetic field over the Northern Hemisphere and
the Southern Hemisphere; the divergence-free nature
of the magnetic field is not well represented.
The hemispheric sign reversal can be incorporated
into the Parker model as follows

A more elaborated analytic model is proposed along with the Ulysses
measurements over the solar polar regions

A model of latitudinal dependence of the magnetic field is constructed by
employing the method of separation of the variable for an axisymmetric
magnetohydrodynamic outflow

The IMF can have a nonzero polar (or latitudinal) component, e.g.,
from the solar dipolar field.
Generalization of the Parker model to the nonzero polar
component case (

The azimuthal angle of the spiral field

Another way of generalization is to use the power-law dependence
using the power-law index

The solar cycle is a periodic change
in the sunspot number over 11 years.
In the plasma physics sense,
the solar cycle is more
associated with the magnetic activity of the Sun
with a period of 22 years (the magnetic polarity
is reversed after one sunspot cycle).
During solar maximum the entire magnetic field of the Sun flips,
thus alternating the polarity of the field every solar cycle.
The solar (magnetic) activity is diverse,
such as solar radiation, ejections of solar material, and
the number and the size of sunspots and
the occurrence rate of solar eruptions.
As a consequence, the periodic change in
the solar magnetic field (or dipolar axis)
affects the polarity of the IMF as well.
To include the time-dependent effect

Two additional effects can further be incorporated
into the IMF model, the polarity

The tilt angle

The wavy, flapping shape of the heliospheric current sheet
is expressed by the equation for the polar angle as follows

The approximation in Eq. (

Shape of the “ballerina skirt” model of the heliocentric current
sheet defined by

A sketch of the topology of the heliospheric current sheet is shown in Fig.

The drift motion depends on the sign of

A more refined magnetic field model is constructed by

The Fisk angle

The tilt angles

The models of the solar wind and the interplanetary magnetic field
can be extended from kinematic or hydrodynamic treatments
to magnetohydrodynamic (MHD) treatments.
An overview of the MHD wind models is given by

An MHD model is proposed for an axisymmetric, one-dimensional,
centrifugal-force-driven wind on the solar equatorial plane

The momentum balance equation by

In the two-dimensional picture,
the energy conservation (the generalized Bernoulli equation)
and the conservation law perpendicular to the
magnetic field (the generalized Grad–Shafranov equation)
are derived using the force balance equation
among the advection of the flow itself
(flow nonlinearity such as steepening and eddies),
the pressure gradient, the Lorentz force, and the gravitational
attraction by the Sun, the mass flux conservation,
the induction equation, and the adiabatic condition along the flow

It is useful to introduce
the poloidal–toroidal expression of the magnetic field
in the two-dimensional MHD treatment:

Solar wind models can further be improved by considering turbulent diffusion and pickup ions.

Turbulence on smaller spatial scales serves as an energy sink to
large-scale mean fields, which leads to the notion of turbulent diffusion
(mean-field electrodynamics). To see this more clearly, one may decompose the
magnetic field into a large-scale mean field

Pickup ions from interstellar neutral hydrogen atoms are one of the
ingredients to the solar wind and contribute to additional mass of the
plasma, which results in deceleration of the solar wind expansion and in
increase in the plasma temperature. Pickup ions originate in (1) charge
exchange with the solar wind protons and (2) photoionization by the solar
radiation. Steady-state MHD equations for the wind including pickup ions are
introduced by

The continuity equation in the one-fluid sense (mixture of electrons, solar
wind protons, and pickup ions of interstellar origin) has a contribution from
the photoionization as a source term and is written for the steady state as

Various outflow models have been proposed for the stellar wind. For
example, a wind model is constructed and numerically studied for the
thermally driven hydrodynamic outflow from low-mass stars

Stellar winds can be detected by the spectroscopic investigation.
A line spectrum becomes distorted to
blue-shifted absorption and red-shifted emission
by the retarding stellar wind (away from the observer),
known as the P Cygni profile.
One type of the stellar wind models is
the Lucy model

There is an increasing amount of models for the interplanetary magnetic field. Starting with the Parker model, the magnetic field model can be extended to include the latitudinal dependence, the poleward component, the time dependence, and the polarity and tilt effect even in the analytic or semi-analytic treatment. Which model to choose would depend on the application, e.g., if the solar cycle is to be included or not, or if the latitudinal dependence is to be or not. In the temporal sense, cosmic ray diffusion has the shortest timescale, about 13 h for relativistic particles nearly at the speed of light to travel over 100 au distance in the heliosphere. In contrast, plasma turbulence evolves together with the solar wind, and the timescale is intermediate, being of the order or days; i.e., the solar wind travel time from the Sun to the Earth orbit, 1 au, is about 100 h or roughly 4 d. Charged dust motions and modulation of the cosmic ray flux in the heliosphere evolve on the longest timescale among the three applications, of the order of of years (secular variation of the orbital parameters).

The accuracy or the uncertainty of the reviewed models needs to be verified using in situ magnetic field measurements from the previous, current, and upcoming spacecraft missions. Above all, the magnetic field in the inner heliosphere will be extensively studied with Parker Solar Probe, BepiColombo (in particular, the cruise-phase measurements), and Solar Orbiter.

It is interesting to note that the analytic expression is also available for the coronal magnetic field (during the solar minimum) and the local interstellar magnetic field surrounding the heliosphere. Hence, naively speaking, one may expect to construct a more complete model of the magnetic field from the Sun to the local interstellar medium. Such a model, once smoothly and rationally connected from one region to another, enables one to improve the accuracy of theoretical studies on plasma turbulence evolution, charged dust motions, and diffusion of cosmic ray and energetic particles.

It is also worth noting the limits of the models. First, the magnetic fields are highly irregular in structure in the solar corona and at the solar surface. At some distance sufficiently close to the Sun, the interplanetary magnetic field should smoothly be connected to the coronal magnetic field. Second, the outer heliosphere has the termination shock and the heliopause, which are not included in the models in this review. Third, the solar variability includes not only the 11-year sunspot number variation or the 22-year magnetic structure variation, but also modulations of the solar cycle on long timescales such as 100 or even 1000 years.

No data sets were used in this article.

This article has been prepared by both authors in equal parts.

The authors declare that they have no conflict of interest.

This work is financially supported by the Austrian Space Applications Programme FFG ASAP-12 SOPHIE at the Austrian Research Promotion Agency under contract 853994 and the Austrian Science Funds (FWF) under contract P30542-N27.

This paper was edited by Manuela Temmer and reviewed by two anonymous referees.