The quality of Taylor's frozen-in flow hypothesis can be measured by estimating the amount of the fluctuation energy mapped from the streamwise wavenumbers onto the Doppler-shifted frequencies in the spectral domain. For a random sweeping case with a Gaussian variation of the large-scale flow, the mapping quality is expressed by the error function which depends on the mean flow speed, the sweeping velocity, the frequency bin, and the frequency of interest. Both hydrodynamic and magnetohydrodynamic treatments are presented on the error estimate of Taylor's hypothesis with examples from the solar wind measurements.

Taylor's frozen-in flow hypothesis

One may expect that Taylor's hypothesis is valid in a supersonic or
super-Alfvénic flow because the intrinsic time variation such as sound
waves or Alfvén waves is negligible. The presence of a large-scale flow
variation, however, causes a Doppler broadening of the frequencies, and the
quality of the mapping from the frequencies onto the wavenumbers becomes
degraded. A good example and a bad example of Taylor's hypothesis are
sketched in Fig.

Here we develop a method to estimate the error of Taylor's hypothesis in the
spectral domain using the random sweeping model

The problem comes from the property of the wavenumber–frequency spectrum.
According to the random sweeping model, the power-law index of the energy
spectrum is the same between the frequency domain and the wavenumber domain
even though there is significant frequency broadening by the sweeping
velocity and Taylor's hypothesis is violated

It is true that the energy spectrum is measurable and is model-independent. On the other hand, the knowledge on the frequency broadening plays an important role in validating Taylor's hypothesis. The strength of the presented method is in its applicability to estimate the error of Taylor's hypothesis using only a small set of parameters without looking at the wavenumber–frequency spectrum.

Sketches of wavenumber–frequency spectra for a good case of Taylor's hypothesis with a small sweeping velocity (upper panel) and a bad case with a large sweeping velocity (lower panel).

The random sweeping model is one representation of the turbulent fields and
is based on the picture that the sweeping velocity is statistically of
Gaussian nature, and is assumed to be not a function of the frequencies or
wavenumbers. For a scale-dependent formulation of the large-scale fluctuating
fields, the two-scale direct interaction approximation

The essence of the random sweeping model lies in introducing the frequency
broadening function

In the random sweeping model, a Gaussian distribution is used as the
broadening function. The first-order moment of the broadening function is the
Doppler shift

The measure of the validity of Taylor's hypothesis is defined as how much the
fluctuation energy is transferred on the frequency bin between the
Doppler-shifted frequency

The mean flow speed

The sweeping velocity

The frequency bin

The applied frequency

For example, the mean flow velocity and the sweeping velocity are of
comparable order for oceanic turbulence,

The hydrodynamic formulation using the sweeping velocity is applicable to magnetohydrodynamic (MHD) turbulence under several assumptions. Complications arise in MHD turbulence in two aspects: (1) the Doppler shift becomes split into dispersion relations; (2) the concept of the sweeping velocity needs to be separated from the root mean square of the flow velocity variation.

The wavenumber–frequency spectrum for MHD turbulence is

As is seen in Eq. (

Wavenumber–frequency spectra with strong splitting (upper panel) and weak splitting (lower panel) for magnetohydrodynamic turbulence.

The direction of

The number of the free parameters in modeling the wavenumber–frequency
spectrum is larger in MHD turbulence (Eq.

The MHD wavenumber–frequency spectra show two peak lines as a result of
splitting of the Doppler shift into the forward-shifted Alfvén speed and
the backward-shifted Alfvén speed (Fig.

In the case of zero cross helicity (

A further approximation is possible by expanding
the spectrum into a Taylor series and
looking at the coefficient on the second-order term.
We introduce temporary variables

A practical approach regarding Eq. (

Parameters for the error estimate of Taylor's hypothesis for two
events of Cluster spacecraft measurements of solar wind turbulence: event 1
from

Let us apply the error estimate to measurements of turbulent fluctuations in
the solar wind. Under a typical condition, one may use a mean flow speed of
about 400 km s

The profile of the measure

Measure of Taylor's hypothesis accuracy

It is instructive to note that the random sweeping model predicts that the
one-dimensional frequency spectrum (integrated over the wavenumbers) appears
as a power law irrespective of the sweeping velocity if the wavenumber
spectrum (integrated over the frequencies) is a power law. Furthermore, the
spectral index is the same both in the frequency domain and in the wavenumber
domain

In the hydrodynamic case, the random sweeping velocity can be regarded as the root-mean-square large-scale flow velocity variation. In the MHD case, the notion of the sweeping velocity is generalized into a renormalized quantity. Even though many parameters are involved in the formulation of the MHD wavenumber–frequency spectrum, the formulation with the single sweeping velocity provides a useful method to estimate the error of Taylor's hypothesis. A more theoretical formulation of the MHD sweeping velocity and the observational test is left for a future work.

The assumption of Gaussian frequency distribution breaks down if large-scale
field variations show intermittency and phase coherence. Previous studies

In the ion kinetic domain (at scales of the order of 100 to 400 km in the solar wind), the electromagnetic waves become dispersive such as ion cyclotron waves, kinetic Alfvén waves and whistler waves. The dispersion relation is no longer linear between the wavenumbers and the frequencies. The kinetic extension of the error estimate for Taylor's hypothesis needs to include the dispersion effect.

It is true that one needs access to the wavenumber–frequency spectrum to fully validate Taylor's hypothesis and therefore multi-spacecraft missions from an observational point of view. But the applicability of the error estimate is not limited to multi-spacecraft missions. In particular, one can perform direct numerical simulations and study the error of Taylor's hypothesis systematically. The method in the paper can be used as a test for Taylor's hypothesis under different conditions of turbulent flow.

The random sweeping model can be extended to a magnetohydrodynamic treatment.
We begin with a set of linearized magnetohydrodynamic equations:

We construct a state vector using the small-scale flow

The solution can be formulated as a linear combination of Alfvén waves with
the Doppler shift and broadening in two different directions forward and
backward with respect to the flow. There are two contributions to the
frequency broadening. One is the random sweeping by the large-scale flow
variation (such as eddies) and the other is the large-scale Alfvén waves.
The first term on the right-hand side in Eq. (

The energy spectrum is determined from the covariance matrix

The author declares that he has no conflict of interest.

This work is financially supported by the Austrian Space Applications Programme
at the Austrian Research Promotion Agency, FFG ASAP-12 SOPHIE,