Effects of Nonzero-frequency Fluctuations on Turbulence Spectral Observations

In situ observations of turbulence spectra in space plasmas are usually interpreted as wavenumber spectra, assuming that the fluctuation frequency is negligible in the plasma flow frame. We explore the effects of nonzero frequency in the plasma flow frame on turbulence spectral observations. The finite frequency can be caused by either propagating waves or nonlinear broadening of nonpropagating structures. We show that the observed frequency spectrum can be modified by the nonzero frequency of turbulent fluctuations in several ways. Specifically, (i) frequency broadening results in a minor modification to the observed spectrum, primarily acting as a smoothing kernel of the spectrum near the spectral break, while the asymptotic spectral index remains unchanged; (ii) wave propagation can affect the observed spectral index for anisotropic turbulence. The effect is significant at low frequencies and weaker at high frequencies, leading to a “concave” shape of the observed perpendicular spectrum; (iii) the Doppler shift for forward- and backward-propagating Elsasser modes can result in a nonzero cross helicity for critical-balanced turbulence since the effect of the Doppler shift favors outward-propagating waves systematically, resulting in an observed imbalance. These results may have important implications for the interpretation of solar wind flows observed by Parker Solar Probe.


Introduction
Spectral observations are routinely performed for in situ turbulence measured by spacecraft (SC).The standard spectral analysis yields the power spectrum, which is a decomposition of fluctuation energy in frequency space.The frequency spectrum can be interpreted as a wavenumber spectrum based on Taylor's hypothesis, which translates the measured frequency into wavenumber through k = Ω/V sw , where k is the wavenumber, Ω the measured frequency in the SC frame, and V sw the speed of the plasma flow relative to the SC (Taylor 1938).While being useful, Taylor's hypothesis assumes that the characteristic propagation speed of the fluctuations is much smaller than the plasma flow speed.The assumption is typically well satisfied in the highly supersonic and super-Alfvénic solar wind, such as near the Earth, but not necessarily true near the Sun, where the solar wind speed tends to be lower.Thus, the launch of Parker Solar Probe (PSP) has stimulated investigations of the validity of Taylor's hypothesis for turbulence data analysis (e.g., Klein et al. 2015;Bourouaine & Perez 2018, 2020;Perez et al. 2021;Zhao et al. 2022c;Zank et al. 2022).
It is conventional that turbulence is described in terms of the power spectrum in wavenumber space (e.g., Frisch 1995).However, recent progress in magnetized plasma turbulence has recognized the importance of finite-frequency effects (e.g., Dmitruk & Matthaeus 2009;Andrés et al. 2017;Markovskii & Vasquez 2020;Perez & Bourouaine 2020;Brodiano et al. 2021;Papini et al. 2021;Gan et al. 2022;Yuen et al. 2023).Compared to the wavenumber spectrum, the frequency (ω)wavevector (k) spectrum (or 4D spectrum) is a more complete description of turbulence as it includes both the spatial (kspace) and temporal (ω-space) information of the turbulent fluctuation.The 4D spectrum represents the distribution of fluctuation energy in frequency-wavenumber space.A typical 4D spectrum of turbulence can be understood as a superposition of propagating waves with distinct wavevectors and frequencies as well as zero-frequency modes.Some of the waves can be explained by theories such as linearized MHD, which contains three types of waves: fast and slow magnetosonic waves, and Alfvén waves.Each linearized mode follows a distinct dispersion relation between its wavevector and frequency.However, as noted by many previous literatures, turbulent fluctuations do not necessarily follow linear wave dispersion relations.This is hinted at, e.g., in simulations by Dmitruk et al. (2004), Andrés et al. (2017), Perez & Bourouaine (2020), and Yuen et al. (2023).More recently, a more computationally demanding 4D spectral analysis has become available for 3D simulations.For example, Fu et al. (2022) and Gan et al. (2022) show that the 4D spectra of magnetic and density fluctuations in compressible MHD simulations are dominated by "nonwave" fluctuations that do not follow any linear dispersion relations.Similar results have been presented by Markovskii & Vasquez (2020), Brodiano et al. (2021), andPapini et al. (2021) for various types of simulations.An important component of the nonwave-like turbulent fluctuations are nonpropagating structures with zero frequency.In magnetized plasma turbulence, these structures usually have wavevectors perpendicular to the mean magnetic field, and are thus commonly referred to as "quasi-2D" turbulence (Zank & Matthaeus 1992, 1993).Observationally, the presence of quasi-2D turbulence is demonstrated by the "Maltese-cross"-like structure of the correlation function (Matthaeus et al. 1990;Dasso et al. 2005), by the anisotropic turbulence measurements (Bandyopadhyay & McComas 2021;Zhao et al. 2022a), and its consistency with cosmic-ray mean Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.free paths (Bieber et al. 1994;Zhao et al. 2017).Multi-SC observations also provide evidence of low-frequency fluctuations (Sahraoui et al. 2010) and nonpropagating modes (Zank et al. 2023;Zhao et al. 2023) in the solar wind.
It has been noted that the full anisotropic turbulence spectra cannot be unambiguously determined by the observed frequency spectra (e.g., Fredricks & Coroniti 1976;Forman et al. 2011).Thus, spectral modeling is an effective way of guiding the observational analysis of turbulence.Bieber et al. (1996) considered a 2D+slab decomposition of magnetic fluctuations, where the 2D and slab components have wavevectors perpendicular and parallel to the mean magnetic field, respectively.Their analysis suggested that the 2D component is the dominant component in solar wind turbulence.Forman et al. (2011) compare the 2D+slab model with the critical balance model of turbulence (Goldreich & Sridhar 1995).They conclude that observational data cannot unambiguously distinguish the two models.However, none of these comparisons included finite frequency information, but only considered the theoretical spectrum in 3D wavenumber space.Spectral modeling is important for interpreting the radial evolution of turbulence, especially in the PSP era.Because of the Parker spiral magnetic field (Parker 1958), the changing angle between the flow velocity and the mean magnetic field with radial distance can lead to changes in the observed turbulence properties, which may be misinterpreted as radial evolution of turbulence intrinsic properties.A key to disentangling this effect is modeling the turbulence power spectrum.
In this Letter, we consider the general 4D frequencywavenumber description of plasma turbulence.The SCobserved frequency spectrum can be derived from hypothetical turbulence spectra models.Here we consider 2D, slab, and critical-balanced turbulence models.The advantage of our approach is that it incorporates the effects of wave propagation (we consider Alfvén waves in this Letter).Another effect considered is the possibility of nonlinear spectral broadening in frequency, particularly for nonpropagating 2D turbulence.The Letter is organized as follows.In Section 2, we will discuss the functional form of 4D frequency-wavenumber spectra of different turbulence models.Section 3 will focus on deriving the respective observed 1D frequency spectrum, which can be constructed directly from a time series measured by a single SC.A summary is provided in Section 4.

Frequency-wavenumber Spectrum of Turbulence
We denote the 4D frequency-wavenumber spectrum by P(ω, k), where ω is the fluctuation frequency in the plasma-flow frame and k is the wavevector.Although there have been various suggestions for the power spectrum in 3D wavevector k space (e.g., Goldreich & Sridhar 1995;Narita & Marsch 2015;Zank et al. 2020), the exact functional form of the power spectrum in 4D frequency-wavevector space is not typically specified from a theoretical standpoint.Observationally, it is difficult to determine even a 3D spatial spectrum because of the aliased frequency in SC measurements (Fredricks & Coroniti 1976).Nevertheless, particular spectral models have been demonstrated to be consistent with solar wind data in general (Forman et al. 2011).These models can be used to investigate the statistical properties of anisotropic turbulence quantitatively.We start with the traditional 2D turbulence, which can be described by a power spectrum of the form P(ω, k) = P(k ⊥ )δ(k ∥ )δ(ω).Obviously, this functional expression ignores finite-frequency effects.Here, we introduce the nonlinear frequency broadening to 2D turbulence and consider a 4D spectrum of 2D turbulence model in the following form, where F(ω, k) is the frequency response function constructed such that at any given k, the 4D power spectrum is broadened around ω = 0. We require that ò for normalization.We also caution that the broadening function F(ω, k) used in this work depends only on the magnitude of the perpendicular wavevector k ⊥ , i.e., F(ω, k ⊥ ).The possibility of frequency broadening can be included in the frequency response, as suggested phenomenologically by Bourouaine & Perez (2019) and verified in numerical simulations by Perez & Bourouaine (2020) based on the "sweeping" hypothesis of Kraichnan (1964), with the broadening effect being modeled as a Gaussian function, i.e., The random sweeping by a large-scale flow fluctuation δv leads to the frequency broadening factor Δω ∝ k ⊥ δv.In the sweeping hypothesis of Kraichnan (1964), large-scale motions convect small-scale structures without distorting them, resulting in a time correlation related to the statistics of the large-scale motions, which are assumed to be a Gaussian function.However, recent MHD simulations by Gan et al. (2022) suggest that the broadening may be more accurately described as a Lorentzian function (Yuen et al. 2023), They suggest that nonlinear interactions may introduce an imaginary part ω i to the real frequency ω r due to linear wave propagation, so that the autocorrelation may be expressed as e i .Physically, the exponential decay of the temporal correlation function suggests that the system behaves like a harmonic oscillator damped by nonlinear interactions (Howes & Nielson 2013;Howes et al. 2014;TenBarge et al. 2014).The exact form of broadening is outside the scope of the present work, and here we simply demonstrate the effects of broadening using these two models.For both cases, frequency broadening leads to nonzero fluctuation power at nonzero frequency, i.e., P(ω, k) ≠ 0 for ω ≠ 0. These two frequency-broadening functions are illustrated in Figure 1.Compared with the Gaussian function at the same level of broadening, i.e., Δω = 2, the Lorentzian function has a heavier tail so that the power is spread out to a wider range of frequencies.
It is evident that 2D turbulence is a simplification of reality.One aspect that is not captured in the 2D model is wave propagation.In particular, Alfvén waves are often recognized as an important constituent of plasma turbulence in the solar wind (e.g., Belcher & Davis 1971).The critical balance theory proposed by Goldreich & Sridhar (1995; aka GS95/GS model) requires counterpropagating Alfvén waves with equal energy and is often used to explain the observed solar wind turbulence anisotropy (e.g., Horbury et al. 2008;Sioulas et al. 2023).
While the critical balance model may be applicable to an imbalanced turbulence (e.g., Lithwick et al. 2007;Chandran 2008;Beresnyak & Lazarian 2009;Perez & Boldyrev 2009), some modifications to the formalization of the spatial spectrum are needed.The effect of the extended GS95 theory applicable to the imbalanced turbulence on the observed 1D frequency spectrum deserves further investigation.In this Letter, we focus on critically balanced turbulence to which the classical GS95 theory applies.Based on the assumption of an equal Alfvén time τ A and nonlinear eddy turnover time τ nl , the GS95 theory suggests that the 3D spatial spectrum for critically balanced turbulence follows a functional form of where C 0 determines the spectral power and L 0 represents the energy injection scale.The function g(...) is an unspecified function that decays at large argument.Under certain normalization conditions for the function g, Equation (4) ensures a −5/3 perpendicular spectral index and a −2 parallel one.The exact functional form of g(...) is unclear theoretically, but the idea is that spectral power is attenuated where τ A = τ nl , which corresponds to a large argument in the g function (Oughton & Matthaeus 2020).Suggestions of the specific g function include exponential, Dirac delta, and Heaviside step functions, etc. Numerical simulation data seem to be in best agreement with an exponential function, i.e., Goldreich 2001;Forman et al. 2011).
Figure 2 shows the distribution of the spatial spectral power P(k) in the k ∥ -k ⊥ plane calculated by Equation (4), assuming g is an exponential function, L 0 = 5 × 10 4 km, and C 0 = 1.The scaling =  k k 2 3 holds for critical balance turbulence due to the assumption τ A = τ nl and is plotted as the white line in Figure 2. The contour lines are color-coded by the spectral power in the logarithmic scale.It can be seen that the region τ nl < τ A has greater power compared to the region τ nl > τ A .Since the GS turbulence model is based on the interaction of Alfvén waves, it is natural to set the finite frequency associated with the GS model based on the dispersion relation of Alfvén waves, ω = ± k ∥ V A where V A is the Alfvén speed.The ± sign indicates two possible propagation directions for the waves (along or opposite to the mean magnetic field).In our description, ω is taken to be always positive (as well as V A ).The functional form of the 4D spectrum of the GS model can be expressed as the following, Here, the frequency broadening occurs around the Alfvén wave frequency, and P(ω, k) = 0 for ω < 0. In the limit of k ∥ = 0, the Alfvén wave frequency tends to 0, which is consistent with the 2D turbulence description.Equation (5) applies to the total energy spectrum, assuming equal partitioning or balance between counterpropagating Alfvén waves.The 4D spectra of Elsasser variables can be considered by including the propagation direction in the frequency response factor, i.e., where the ± subscript represents the z ± Elsasser modes, respectively.Distinct from the critical balance model, another possible component of solar wind turbulence is slab turbulence, which contains fluctuations with parallel wavevectors k ∥ .We consider an ensemble of forward-and backward-propagating Alfvén waves for (possibly imbalanced) slab turbulence, and the 4D (ω, k) spectrum for the Elsasser slab spectrum can be expressed as A For imbalanced slab turbulence, the normalized cross helicity σ c will be nonzero and bounded by σ c = ±1.The slab component may be superimposed on the 2D component as the 2D + slab turbulence model (see the Appendix for details).

Observations of Turbulence Power Spectra
In general, the observed turbulence power spectrum can be related to the 4D spectrum via where Ω is the observed frequency (in the instrument frame), and V is the SC velocity relative to the plasma flow (Fredricks & Coroniti 1976).More specifically, V = v sc − U, where v sc is the SC velocity with respect to an inertial frame fixed relative to the Sun and U is the solar wind velocity.Far away from the Sun, v sc is normally much smaller than U, in which case V is approximately −U.However, at the closer distances to the Sun sampled by PSP, v sc may not be negligible.In such cases, the solar wind speed measured in the SC frame can be considered as −V.Hereafter, we refer to V as the SC velocity for simplicity.The wavevector k is measured in the plasma flow frame.For simplicity, we assume that the mean magnetic field B 0 is along the z-direction, and the SC velocity V is in the x-z plane (V y = 0).We also assume that z points toward the Sun so that V z > 0 (since the solar wind flows outward).We consider gyrotropic spectra with a rotational symmetry with respect to the mean magnetic field.
Assuming that the broadening function F(ω, k) depends only on the magnitude of perpendicular wavevector k ⊥ (Bourouaine & Perez 2018), the spectral model of 2D turbulence shown in Equation (1) can be rewritten as w and the observed frequency spectrum for 2D turbulence can be further simplified as Several examples of the 2D frequency spectra calculated from Equation (11) are shown in Figure 3.The effects of different levels of frequency broadening are demonstrated in different panels.For the 2D turbulence, we assume the SC velocity V⊥B 0 .At large heliocentric distances, the magnetic field is on average more tangential, thus perpendicular sampling is easily satisfied.At small distances, although the magnetic field is more radial, perpendicular sampling can also occur in some cases where there are variations in the direction of the field, such as near the current sheets during the disturbed periods (e.g., Perrone et al. 2020;Zhao et al. 2021;Réville et al. 2022).In panels (a)-(d), we set the fluctuating velocity amplitude δv = 10, 50, 200, and 400 km s −1 , respectively, while keeping V x = 400 km s −1 , so that the broadening width Δω = k ⊥ δv is different in each panel.We note that 400 km s −1 is not a typical value for solar wind velocity fluctuations.However, what is more important is the fluctuation relative to the plasma flow.Our results demonstrate the effects of broadening for a range of relative velocity fluctuations δv/V between ∼0 and 1. Depending on the value of δv/V, the effect of broadening may tend to be stronger closer to the Sun.For the broadening function F(ω, k) in Equation (11), we consider three cases, (i) without broadening (F being the Dirac delta function), (ii) a Gaussian description (Equation ( 2)), and a Lorentzian description (Equation ( 3)), shown as a solid black line, an orange dashed line, and a blue dotted-dashed line in each panel, respectively.For all cases, we set the wavenumber spectrum to be a broken power law: 5 3 for k ⊥ k 0 , and P(k ⊥ ) = P 0 for k ⊥ < k 0 .The constants are P 0 = 3.4 × 10 12 km 3 s −2 and k 0 = 2 × 10 −5 km −1 .With a low level of broadening (i.e., small δv and hence small Δω), the three cases are nearly the same, as shown in Panel (a), where δv = 10 km s −1 .The Lorentzian broadening has a slightly more noticeable deviation from the others due to its fatter tail (Figure 1).Panels (b)-(d) show that stronger broadening makes the spectral break smoother while the asymptotic power-law shape is unchanged.These results are as expected since the broadening function effectively acts as a smoothing kernel on the frequency spectrum, as also reported in previous work (e.g., Narita 2017; Bourouaine & Perez 2019).
Similarly, in the case of slab turbulence with outward propagating Alfvén waves, the 4D spectrum should have the form of , , and the corresponding observed frequency spectrum from Equation (8) reduces to Since the perpendicular wavenumber k ⊥ of slab turbulence is zero and the frequency broadening we discussed earlier is proportional to the perpendicular wavenumber, frequency broadening effects will not be considered for slab turbulence, simplifying the Equation (12) to which means that the slab turbulence spectrum keeps the exact shape of the corresponding wavenumber spectrum.However, see the discussion in the Appendix discussing a more refined form of frequency broadening of slab turbulence that can be developed in the future.
For the GS turbulence model, assuming that the broadening function depends only on the magnitude of the perpendicular wavenumber, the observed frequency spectrum derived from Equations (5) and (8) corresponds to Figure 4 shows the observed frequency spectra for the GS model calculated from Equation (14).Here, we have included a transition between the inertial range and the energy-containing range by requiring that the maximum value of the 3D spatial spectrum is achieved at k ⊥ = 1/L 0 , k ∥ = 0 (brown contour line in Figure 2) and assuming a flat spectrum where the value based on Equation (4) exceeds the maximum (bottom left corner bounded by the brown contour line in Figure 2).The SC speed relative to the plasma flow V 0 = |V| is considered to be 400 km s −1 , L 0 = 5 × 10 4 km, and power normalization factor C 0 = 1 in all cases.We consider four cases here, (i) without broadening and the Alfvén wave frequency (F is the Dirac delta function at ω = 0, solid black lines); (ii) without broadening but with Alfvén wave propagation (F is the Dirac delta function at ω = |k z |V A , solid cyan lines); (iii) with Gaussian broadening and Alfvén frequency (Equation (2) but using ω − |k z |V A instead of ω, orange dashed lines); and (iv) with Lorentzian broadening and Alfvén frequency (Equation (3) but using ω − |k z |V A instead of ω, blue dotted-dashed lines).For the Figure 3.The "observed" frequency (Ω) spectra of the 2D turbulence obtained by integrating its 4D frequency-wavenumber spectral model P(ω, k) (Equation ( 11)) with different broadening functions F(ω, k).The tangential SC velocity relative to plasma flow V x = 400 km s −1 in all cases.The wavenumber spectrum is set to a broken power law.The solid black lines represent the spectra without broadening effects, where the Dirac delta description of the broadening function F is used.The orange dashed lines denote the spectra calculated using a Gaussian form of the broadening function F, and the blue dotted-dashed lines represent the spectra calculated using a Lorentzian broadening function.Panels (a)-(d) show the cases of different broadening widths Δω = k ⊥ δv where the fluctuating velocity amplitude δv = 10, 50, 200, and 400 km s −1 , respectively.The insets in panels (a), (b), and (c) show a zoom of the spectral break to illustrate the smoothing effect of two different broadening functions on spectral breaks.spectra calculated with the Alfvén wave frequency, the Alfvén speed V A is taken to be 50 km s −1 .Since GS turbulence includes both k ∥ and k ⊥ spectra, the plasma flow velocity can be at any angle to the magnetic field.We consider two cases, θ VB = 90°(panels (a) and (b)) and θ VB = 5°(panels (c) and (d)) to represent k ⊥ and k ∥ spectra.In each case, two different broadening levels are presented for spectra with broadening effects, i.e., δv = 50 (panels (a) and (c)) and 400 km s −1 (panels (b) and (d)), respectively.The GS theory predicted perpendicular −5/3 and parallel −2 spectra are shown as reference (solid yellow lines).As shown in the figure, for the spectrum without broadening effects and Alfvén wave propagation it follows strictly - k 5 3 and -  k 2 scaling within the inertial range.However, the inclusion of Alfvén wave propagation effects results in a perpendicular spectrum with θ VB = 90°deviating from the −5/3 scaling and becoming steeper, while the parallel spectrum experiences little change.This can be seen by comparing the black and cyan lines in each panel and is illustrated in more detail below with different Alfvén wave speeds (Figure 5).For the additional frequency broadening effects introduced (by comparison of the cyan line with the orange and blue lines), the results are consistent with expectations as in the case of 2D turbulence shown in Figure 3.The transition of the spectra from the energycontaining range to the inertial range is smoothed due to the convolution of the spatial spectrum with a broadening function.We also note that the spectral transition for the parallel spectrum occurs at a slightly large frequency or wavenumber compared to the perpendicular cases.The transition wavenumber is defined by the brown contour line in Figure 2, and it is obvious that the transition k ∥ is larger than k ⊥ .We caution that this is an artificial shift determined by our method of  3, but for GS turbulence calculated using Equation (14).Four cases are considered, (i) without broadening and the Alfvén wave frequency (solid black lines); (ii) without broadening but with the Alfvén frequency (cyan solid lines); (iii) with Gaussian frequency broadening and the Alfvén frequency (orange dashed lines); and (iv) with Lorentzian broadening and the Alfvén frequency (blue dashed-dotted lines).Panels (a) and (b) are for perpendicular scaling with θ VB = 90°, and panels (c) and (d) are for parallel scaling with θ VB = 5°.Two different levels of broadening (e.g., δv = 50 and 400 km s −1 ) are presented for the parallel and perpendicular spectra, respectively.The Alfvén velocity used in calculating the Alfvén wave frequency is set to 50 km s −1 , and the SC speed relative to the plasma flow V 0 = |V| in all cases is 400 km s −1 .introducing a broken power law and that no important conclusions should be drawn based on it.Since the frequency broadening does not change the asymptotic behavior of the frequency spectrum in the instrument frame, we may neglect the broadening effects when the fluctuating velocity δv is small compared to the plasma flow speed in the SC frame and the focus is not in the transition region of the power spectrum.In such cases, we can consider the wave frequency only as indicated by the cyan lines in Figure 4, and the integral in Equation ( 14) can be simplified as Figure 5 shows the frequency spectrum (instrument frame) of the GS turbulence model calculated from Equation ( 14) for several angles between the SC velocity relative to the plasma flow and the mean magnetic field θ VB .In this plot, we fix the fluctuating velocity δv = 50 km s −1 , SC speed relative to the plasma flow V 0 = 400 km s −1 , and use the Gaussian broadening function (Equation ( 2)).We focus on the spectrum in the inertial range beyond the transition region, in which cases the effect of frequency broadening is actually negligible, as shown in Figure 4.The spectra are compensated by Ω 5/3 .The top left panel neglects the Alfvén velocity (V A = 0), illustrating the case in the absence of wave frequency effects, and the remaining three panels assume a nonzero Alfvén velocity, i.e., V A = 50, 250, and 400 km s −1 , respectively.We consider the sampling angle θ VB = 5°, 45°, 60°, 75°, and 90°for each case.
When the Alfvén wave frequency is neglected, as shown in panel (a), the perpendicular spectrum scales exactly as Ω −5/3 and the parallel spectrum as Ω −2 .However, when wave frequency is included (V A > 0), the spectral shape is slightly modified.The perpendicular spectrum is no longer a strict power law but shows a concave shape with the low-frequency part being steeper than a Ω −5/3 power law and the highfrequency part being slightly flatter.This can be understood as the parallel propagation of Alfvén waves inducing the observer to identify parallel variation, thereby affecting the perpendicular variation even though the sampling angle θ VB = 90°(see also the discussion in the Appendix).The effect is particularly significant at low frequencies.Since the observed spectrum at high frequency is mostly contributed by large wavenumber fluctuations, which are dominated by perpendicular fluctuations, the effect of parallel propagating waves is weak (see also, Perez et al. 2021).These results may also apply to more general turbulence models with spectral index anisotropy.
The Elsasser spectra for the GS model P + (Ω) and P − (Ω) can be obtained by replacing |k z |V A with mk z V A in Equations ( 14) and (15).We note that even though we have assumed balanced turbulence between counterpropagating Alfvénic fluctuations, the observed frequency spectrum is not necessarily balanced, depending on the relative magnitude between plasma flow velocity and wave propagation speed.Specifically, the observed cross-helicity spectrum can be inferred to be An example is shown in Figure 6 where we consider GS turbulence in a sub-Alfvénic flow, i.e., V 0 = 134 km s −1 , V A = 200 km s −1 , δv = 20 km s −1 , and the angle θ VB = 9°.The effect of finite wave frequency is significant when V A is comparable to V 0 , and the sub-Alfvénic solar wind typically satisfies this condition.The left panel of Figure 6 shows the frequency spectra P + (Ω) (black line) and P − (Ω) (blue line) of the Elsasser variables z + and z − for the GS-balanced turbulence, calculated using Equation (14) with Gaussian broadening function.Note that the spatial spectra of z + and z − are balanced for GS turbulence, i.e., the cross-helicity spectrum in k-space should be zero, which should correspond to the red line in Figure 6(b).Since the flow is basically aligned with the mean magnetic field in the interval (θ VB = 9°), the spectra of both z + and z − exhibit GS parallel scaling with a spectral index close to −2.However, it should be noted that the relative power between z + and z − is no longer balanced when the wave propagation speed cannot be neglected and the wave frequency needs to be included in z ± modes.As shown in the left bottom panel, when the Alfvén wave frequency is considered for both z + and z − modes, the observed cross helicity σ c (Ω) for balanced turbulence, calculated from Equation (16), can be as much as ∼0.6.On the right panel of Figure 6, we show the dependence of the observed cross helicity on the sampling angle θ VB .The other parameters remain the same as in the left panel.It can be seen from the figure that the observed cross helicity of GS-balanced turbulence is zero only when θ VB = 90°.When there is a nonzero parallel plasma flow speed in the SC frame, nonzero cross helicity can be generated with finite wave frequency included, and the value of the observed cross helicity increases with decreasing angle θ VB .In fact, the nonzero cross helicity is naturally understood by considering a Doppler shift.Since the solar wind travels radially outward, the outward-propagating modes are essentially "blue-shifted" to higher apparent frequency while the inward-propagating modes are "red-shifted" to lower apparent frequency (in the sense that blue/red shift corresponds to waves propagating in the same/opposite direction of the velocity of the wave source relative to the observer).Given an observed frequency, the observed fluctuations are a combination of outward-propagating modes that have longer wavelengths and inward-propagating modes that have shorter wavelengths.The turbulence power spectrum indicates that longer-wavelength fluctuations contain more power, which leads to the observed nonzero cross helicity favoring outwardpropagating modes.Goldstein et al. (1986) have noted that the effect of the Doppler shift favors outward-propagating waves systematically.Although perhaps being underappreciated due to the dominance of solar wind speed over Alfvén speed near 1 au and beyond, the concept of imbalance due to the Doppler Figure 6.Left panel: the "observed" Elsasser frequency spectra P + (Ω) (for z + mode, black line) and P − (Ω) (for z − mode, blue line) and the corresponding cross helicity for the critical-balanced GS turbulence model in a sub-Alfvénic flow when finite Alfvén wave frequency is included.Right panel: the "observed" cross helicity spectrum for the GS model at different sampling angles θ VB from 1°to 90°.shift has become of greater interest as PSP moves ever closer to the Sun.Our results suggest that the high cross helicity observed by PSP may not represent the true extent of the imbalance between sunward and outward fluctuations, since the cross helicity is not measured in the plasma rest frame.
To demonstrate an application of the analysis, we consider another sub-Alfvénic solar wind interval observed by PSP on 2021 November 22 from 02:40 UT to 09:55 UT during its Encounter 10 (Zhao et al. 2022b).The crossing of this sub-Alfvénic flow, characterized by an Alfvén Mach number less than 1, lasts about 7 hr.Specifically, the plasma flow speed in the SC frame during the interval is about 115 km s −1 , the average Alfvén speed V A = 253 km s −1 , the angle between the mean magnetic field and SC frame mean velocity field q =  46 VB sc , and the amplitude of the velocity fluctuation in the SC frame |δV| = 36 km s −1 .
In Figure 7, we consider both the 2D+slab spectral model and the GS spectral model in an effort to fit the observed Elsasser spectra and cross helicity.The 2D + slab model used here consists of a superposition of 2D turbulence with Gaussian frequency broadening and slab turbulence with the Alfvén wave frequency included.The 2D component is imposed on both z + and z − modes and the slab component consists of the z + mode only, representing an excess of outward-propagating Alfvénic fluctuations.The GS spectral model used here includes both the Alfvén wave frequency and Gaussian frequency broadening.The top panels show the z ± frequency spectra during this sub-Alfvénic flow and the bottom panels show the cross-helicity frequency spectrum.The left panels show the 2D+slab modeling results, and the right panels show the GS model comparison.However, we find that the GS turbulence model alone is insufficient to fit the observed Elsasser spectra.There are two main reasons for this: (a) the observed spectra are flatter than the GS model predicted and (b) the PSP-observed cross helicity is too high to be explained by the balanced GS model even with the inclusion of Alfvén frequency.This is because θ VB is relatively large, and the power difference between z + and z − decreases, resulting in a nonzero but small cross helicity (panel (b) in Figure 6).However, it remains an open question whether the extended GS model applicable to an imbalanced turbulence (e.g., Lithwick et al. 2007;Chandran 2008;Beresnyak & Lazarian 2009;Perez & Boldyrev 2009) is consistent with the observation and requires further investigation.To fit the data, we rather arbitrarily add an imbalanced slab turbulence component (despite this violating the explicit assumption of balanced turbulence for the GS model) to the z + mode for the GS model fitting.This ad hoc assumption allows us to better fit the observed cross helicity s c obs .It is difficult to justify this model physically within the framework of GS theory.The black and green lines in the top panels are the best-fitting model results for the z + and z − mode spectra, respectively.The black line shown in the bottom panels represents the cross helicity calculated from the corresponding Elsasser z ± spectra.During the fitting procedure, we consider a frequency range of 10 −3 − 2.5 × 10 −2 Hz and assume that the 2D component follows a power law with a spectral index of −5/3, and the slab component has a power-law exponent of −3/2 at the considered frequency range (Zank et al. 2020).The spectral index of the observed z − spectrum in the considered frequency range is about −1.64 (close to 2D turbulence), which is much steeper compared to the observed z + spectrum with a spectral exponent of ∼ − 1.46.In panel (a), we use the 2D turbulence frequency spectrum model (Equation ( 11)) with Gaussian broadening function (Equation ( 2)) plus the slab turbulence frequency spectrum model (Equation ( 13)) to fit the z + spectrum, while the z − spectrum is modeled by 2D turbulence only.We do not rule out the presence of slab turbulence in the z − mode.The model combination here is chosen solely for the purpose of best fitting the observations.The other parameters in our model equations are all taken from the measurements, i.e., the SC speed with respect to the plasma flow V, Alfvén speed V A , velocity fluctuation amplitude δv, energy injection scale L 0 , and the associated wavenumber k 0 .We use the same proportion of 2D turbulence to fit the z + and z − spectra.The ratio between the 2D and slab components in fitting the z + mode spectrum is approximately 0.07, which is determined by the standard minimization of chi-square.The ratio is relatively small, indicating that the z + mode is dominated by slab turbulence, i.e., outward-propagating Alfvén waves.As shown in the figure, the model results are in good agreement with the observed Elsasser spectra.The corresponding cross helicity from the modeled spectra of z ± also agrees well with the observed cross-helicity spectrum in the considered frequency range.For the right figure of the GS model comparison, we follow a similar procedure as the 2D+slab model fitting.However, the spectral index anisotropy ( -  k 2 and - k 5 3 ) predicted by the balanced GS turbulence limits the agreement with the observed Elsasser spectra.For the GS model alone, the spectral indices for both z + and z − modes are close to −2 when the Alfvén wave frequency is considered, and the calculated cross helicity, as shown by the black dashed line in the bottom panel, is about 0.3 and differs significantly from the observed cross helicity s c obs .The nonzero but small cross helicity here is due to the nonnegligible wave propagation speed and the relatively small parallel flow velocity during the interval, as discussed in Figure 6.To achieve a better comparison with the observed Elsasser spectra, we add imbalanced slab turbulence with a specified spectral form of -  k 1.5 to the z + mode.The z − -component is included in the GS turbulence only.Such a model is inherently imbalanced but is fundamentally inconsistent with GS theory.Nonetheless, the combination of critical-balanced GS turbulence and the slab model has been used to account for the observed solar wind features, such as wavevector anisotropy and spectral anisotropy (e.g., Forman et al. 2011;He et al. 2013).As shown in the figure, the spectral index of the z + mode fitted by the critical-balanced GS turbulence plus slab model is −1.54, while the z − spectrum modeled only with balanced GS turbulence still has a spectral index close to −2, although θ VB = 46°.This is due to the inclusion of wave propagation effects, as shown in Figure 5.However, the cross helicity (black solid line in the bottom panel) calculated from this combination is significantly enhanced and is in good agreement with the observed cross helicity s c obs .

Conclusions
Single-SC measurements have provided much knowledge about the spectral properties of solar wind turbulence.Standard Fourier spectral analysis constructs frequency spectra from observed time series data of quantities such as velocity and magnetic field.However, it is impossible to infer a complete picture of anisotropic turbulence with single-SC measurements due to the aliasing effects that Doppler shifts multiple fluctuation modes to a single observed frequency.Spectral modeling is a way to extract valuable information about turbulence from single-SC measurements.In this Letter, we summarize the 4D spectral models f (ω, k) of the 2D+slab and balanced GS turbulence.Their respective 1D frequency spectra in the instrument frame are discussed.Finally, an application to the sub-Aflvénic solar wind observed by PSP is presented.Our main results are as follows.
1. Frequency broadening leads to a (slight) modification of the observed frequency spectrum.For a broken powerlaw spectrum, the broadening mostly affects the spectra near the spectral break, while the asymptotic power-law index remains unchanged.This is due to the convolution of the spatial spectrum and the broadening function, such that the broadening function acts as a smoothing kernel in the functional form of the observed frequency spectrum (Narita 2017;Bourouaine & Perez 2019).2. For anisotropic turbulence such as critical-balanced GS turbulence, including finite wave frequency can affect the observed spectral index.When considering Alfvén waves, the parallel propagation of Alfvén waves enables the sampling of parallel fluctuations even when the flow parallel velocity is zero (i.e., θ VB = 90°).The effect is particularly significant at low frequencies, while the observed spectrum at high frequencies is mostly contributed by large-wavenumber fluctuations, which are dominated by perpendicular fluctuations.Therefore, the wave propagation effect is weaker at high frequencies, which results in a "concave" shape of the perpendicularly sampled spectrum.3. The Doppler shift for forward-and backward-propagating Elsasser modes can produce a nonzero cross helicity for balanced turbulence.Assuming a critical-balanced GS turbulence, the cross helicity can reach ∼0.6 for conditions comparable to the sub-Alfvénic solar wind observed near the Sun (e.g., wave speed exceeds flow velocity; parallel sampling).A nonzero cross helicity can be expected considering the Doppler shift since the effect of the Doppler shift favors outwardly propagating waves systematically (Goldstein et al. 1986).The importance of imbalance due to the Doppler shift will be increasingly pronounced as the solar wind speed and the wave propagation speed become increasingly similar.
A consequence of our results is that wave propagation effects cannot be neglected when the wave propagation speed is close to or larger than the solar wind flow speed in the SC frame.We demonstrate an example of using (i) a combination of 2D and slab models, and (ii) a critical-balanced GS turbulence plus slab model to fit the Elsasser spectra observed by PSP in a sub-Alfvénic interval during the Encounter 10.It seems that the 2D +slab model agrees better with the observations in terms of spectral index.However, we note that the 2D+slab model is not necessarily the best model for the case.In fact, the assumed spectral index of −5/3 in the 2D turbulence model tends to be slightly steeper than the observed spectral index of z − mode for some sub-Alfvénic intervals observed by PSP.Nevertheless, the combination of 2D and slab turbulence provides a better comparison with the observed Elsasser spectra.Another consequence of the third conclusion is that we may infer the intrinsic turbulence cross helicity from the observed cross helicity.For example, separate coefficients C + and C − in the turbulence spectral model may be used as fitting parameters, so that the level of imbalance can be obtained from observation.Finally, the only propagating waves we consider here are Alfvén waves.We expect that other waves (such as the compressible fast and slow modes) may also affect the observed frequency spectrum and can be incorporated into the functional form of the 4D frequency-wavenumber spectra based on their specific dispersion relations.
In summary, the description of the 4D frequency-wavenumber spectrum presented in this Letter, which combines both the "sweeping" effect and the Alfvén propagator, represents a general modification of Taylor's hypothesis and can be directly applied to the PSP's measurements close to the Sun.

Figure 1 .
Figure 1.Broadening function F(ω, k) associated with the Gaussian (Equation (2); black curve) and Lorentzian (Equation (3); orange curve) assumptions with the same broadening width Δω.Figure 2. Power spectral density P(k ⊥ , k ∥ ) of GS95 theory assuming the function g in Equation (4) is an exponential function.The white line indicates the critical balance scaling =  k k 2 3 based on the assumption of τ nl = τ A .The brown contour curve represents the power P at k ⊥ = 1/L 0 and k ∥ = 0 and is used below to construct a broken power law.

Figure 2 .
Figure 1.Broadening function F(ω, k) associated with the Gaussian (Equation (2); black curve) and Lorentzian (Equation (3); orange curve) assumptions with the same broadening width Δω.Figure 2. Power spectral density P(k ⊥ , k ∥ ) of GS95 theory assuming the function g in Equation (4) is an exponential function.The white line indicates the critical balance scaling =  k k 2 3 based on the assumption of τ nl = τ A .The brown contour curve represents the power P at k ⊥ = 1/L 0 and k ∥ = 0 and is used below to construct a broken power law.
and k y are the components parallel and perpendicular to B 0 , respectively, i.e., k ∥ = |k z | and =

Figure 4 .
Figure 4. Similar to Figure3, but for GS turbulence calculated using Equation (14).Four cases are considered, (i) without broadening and the Alfvén wave frequency (solid black lines); (ii) without broadening but with the Alfvén frequency (cyan solid lines); (iii) with Gaussian frequency broadening and the Alfvén frequency (orange dashed lines); and (iv) with Lorentzian broadening and the Alfvén frequency (blue dashed-dotted lines).Panels (a) and (b) are for perpendicular scaling with θ VB = 90°, and panels (c) and (d) are for parallel scaling with θ VB = 5°.Two different levels of broadening (e.g., δv = 50 and 400 km s −1 ) are presented for the parallel and perpendicular spectra, respectively.The Alfvén velocity used in calculating the Alfvén wave frequency is set to 50 km s −1 , and the SC speed relative to the plasma flow V 0 = |V| in all cases is 400 km s −1 .

Figure 5 .
Figure5.Compensated frequency spectra in the instrument frame for the GS turbulence model from Equation (14).The panels from top left to bottom right represent the spectra at different Alfvén wave propagation speeds, i.e., assuming V A = 0, 50, 250, and 400 km s −1 , respectively.The parallel and perpendicular scaling are indicated by the values of θ VB , including 5°, 45°, 60°, 75°, and 90°, respectively.The SC speed relative to the plasma flow V 0 = 400 km s −1 and the fluctuating velocity δv = 50 km s −1 for all cases.

Figure 7 .
Figure 7.Comparison of different spectral models with PSP-observed sub-Alfvénic solar wind flow during Encounter 10.The bottom panels show the comparison between the theoretical cross helicity (s c M and s c GS ) and PSP-observed cross helicity s c obs .The theoretical cross helicity is computed from the Elsasser spectra, which include both Alfvén frequency and broadening effects.Panel (a) shows the 2D+slab model comparison, and panel (b)shows primarily GS model comparison, but with additional imbalanced slab turbulence imposed on the z + spectrum to better fit the observed cross helicity.The solid black and green, and black dashed lines represent the modeling results, and the blue, yellow, and pink curves represent the observed z ± frequency spectra and cross-helicity s c obs spectra.In the bottom of panel (b), the black dashed line represents the cross helicity calculated from the balanced GS model alone, and the black solid line represents the cross helicity calculated with an imbalanced slab component included in the z + mode.