A model of non-elliptic wavevector anisotropy is developed for the inertial-range spectrum of magnetohydrodynamic turbulence and is presented in the two-dimensional wavevector domain spanning the directions parallel and perpendicular to the mean magnetic field.

The non-elliptic model is a variation of the elliptic model with different scalings along the parallel and the perpendicular components of the wavevectors to the mean magnetic field.

The non-elliptic anisotropy model reproduces
the smooth transition of the power-law spectra
from an index of

The spatial structure of space and astrophysical plasma turbulence
is fundamentally different from that of ordinary fluid turbulence
because the large-scale magnetic field imposes a special direction,
causing anisotropy in the energy spectrum in the wavevector domain.
On spatial scales larger than the ion gyro-radius and inertial length,
the behavior of plasmas can be treated as magnetohydrodynamics (MHD).
Evidence for anisotropy in MHD turbulence can be found
in numerical simulations

Various models have been proposed to explain or reproduce
the anisotropic energy spectra in the wavevector domain:
the density fluctuation model

Single-spacecraft measurements show that the magnetic energy spectra
in the frequency domain (in the spacecraft frame) have different
values of the spectral index in the range from

The critical balance model
is so far known to explain the dependence of the spectral index
on the angles between the flow direction and the mean magnetic field direction
in solar wind turbulence.

Here we develop a non-elliptic model of wavevector anisotropy
for the inertial-range spectrum of MHD turbulence.
The non-elliptic anisotropy is obtained as a generalization of
the elliptic anisotropy by including
different scalings along the parallel and the perpendicular components
of the wavevectors to the mean magnetic field.
It is worth noting that the multi-spacecraft
measurements in the solar wind
do not provide strong evidence for the critical balance so far,
but the wavevector spectrum is of elliptic type or nearly
elliptic

The non-elliptic spectrum model is important because it expands the list of possible models to explain the spectral anisotropy in the solar wind.

The currently known spectral models cannot account for the observation of
a nearly elliptic formation of the energy spectra in the wavevector domain
in solar wind turbulence showing a transition of the spectral extension
from the parallel direction to the mean magnetic field into the perpendicular
direction

In particular, although the critical balance model was successfully
applied to explaining solar wind turbulence essentially using one free parameter
(except for the degree of freedom
in the positive symmetric function),
the model relies on both assumptions that
obliquely propagating Alfvén waves co-exist with eddies
and that an axially symmetry exists in the directions around
the mean magnetic field. The direct measurements of the dispersion relations
using the Cluster spacecraft data indicate that obliquely propagating
Alfvén waves unlikely exist (or at least do not occur on the
analyzed time intervals) both in the MHD and the kinetic
ranges

The advantage of the non-elliptic model is in its mathematical construction. The spectral anisotropy in the wavevector domain is described. The model does not require assumption of any particular wave modes or fluctuation types such as Alfvén waves or eddies.

The non-elliptic model is particularly suited for testing in the solar wind, since the angle dependence of the spectral slopes has been studied in detail using in situ spacecraft measurements. The model is constructed as an inertial-range spectrum for MHD turbulence. Applications to laboratory plasmas and astrophysical plasmas are in principle possible, too.

Elliptic shape of the energy spectrum in the two-dimensional
wavevector domain is constructed as a natural extension of
the isotropic spectrum by introducing the shape coefficients,

Note that
polarizations in the elliptic model by

To simplify the argument, the other coefficients in the inertial-range spectrum
such as the Kolmogorov or Iroshikov–Kraichnan constant

The elliptic spectrum assumes a symmetry
with respect to changing the sign of the wavevector components,
i.e., the cross helicity is implicitly zero.
The scale invariance holds under the transformation

The coefficient

The coefficient

The power-law index of the frequency spectra in the solar wind
has been found to depend on the angle of the flow (streamwise direction)
from the large-scale magnetic field

We construct a non-elliptic wavevector spectrum by generalizing the
quadratic dependence of one of the wavevector components.
We seek a generalization of the dependence on the parallel wavevector component
and replace

The coefficients

Figure

The values are motivated by the direct measurements of the
wavevector spectra

The wave-number range is representative of
the MHD range in the solar wind,
well below the ion inertial wave number for protons
at about

The Alfvén speed is typically in the range
between

Anisotropy is moderate in case (a) and becomes
clearer at smaller ratios of the coefficients

Two-dimensional energy spectrum for non-elliptic anisotropy
(Eq.

Two-dimensional spectra are rotated around the axis at

The transformation of the wavevector components
(or the coordinate system) is given by the following relations:

The model is constructed as an inertial-range spectrum for MHD turbulence. It is assumed that the inertial range is infinitely long in the integration over the wavevector components.

The limit of the integration range is set to

One-dimensional streamwise energy spectra obtained from
the non-elliptic two-dimensional spectra
for the four coefficient sets shown in Fig.

Variation in the spectral index is studied quantitatively as
a function of the projection angle

The wave-number range from

The transition from a slope of

The angle dependence of the spectral index is then compared with
that obtained from the single-spacecraft measurements in the solar wind

The differences in the observed spectral slopes represent different solar wind realizations. The spectra are measured at different radial distances from the Sun, different heliospheric latitudes, and for different components of the fluctuating fields.

Power-law index

The concept of the non-elliptic anisotropy is a likely candidate to explain the power-law spectra at arbitrary projection angles from the mean magnetic field and the smooth change in the spectral index from the parallel to the perpendicular projection to the mean magnetic field.

The major lesson from the non-elliptic anisotropy is that the angle dependence of the spectral index can be explained without incorporating the critical balance.

Non-elliptic anisotropy implies that the scale invariance is broken in the wavevector domain such that the sense of anisotropy turns from a parallel extension of the spectrum at lower wave numbers into a perpendicular extension at higher wave numbers.

The broken scale invariance means that the spectrum
(or spectral shape) is not self-similar under the
simultaneous transformation of
the wavevector components

More detailed verification of the non-elliptic anisotropy is possible
using both single-spacecraft and multi-spacecraft methods.
In the former case, the angle dependence of the spectral index
(as shown in Fig.