Riddhi bandyopadhyay biography definition

The scale-dependent measures used to quantify the Alfvénicity of the solar wind are similar to those described by Parashar et al. (2020). In order to include a sensitivity to scale, the following measures will be based on increments and structure functions (or other second-order functions of increments) rather than on ordinary fluctuations that are defined in terms of departures from a mean field (see, e.g., Matthaeus & Goldstein 1982). These measures include the normalized increment cross helicity, the increment Alfvén ratio, the normalized increment residual energy, and the alignment cosine between velocity and magnetic field increments. Following the definitions below, for the remainder of the paper, we will drop the modifier "increment" from these terms, with the understanding that we are employing increment-based measures.

The solar wind velocity and magnetic increments for time lag τ are described as

where ΔB is usually measured in Alfvén speed units with implied division by .

In certain circumstances, it would be useful to instead define the increments in terms of fluctuating fields v ≡ V − 〈V〉 and b ≡ B − 〈B〉. Such increments would differ from those in Equations (3) and (4) when the mean fields are time-dependent, e.g., based on running averages, a difference that becomes more pronounced at larger τ. In the present work, we use Equations (3) and (4) directly and avoid the need for averaging. Indeed, avoiding an averaging or detrending procedure has been considered a substantial advantage of using structure functions rather than Fourier spectra of turbulent fluctuations (Lindborg 1999). In any case, at large τ, the spacecraft locations at times t and t + τ are often in distinct plasma streams with different physical features, so Taylor's frozen-in hypothesis (Taylor 1938) is a poor approximation, and this technique, or indeed any single-spacecraft technique, is no longer providing information about the spatial properties of local turbulence; rather, it informs us about temporal decorrelation as influenced by such stream crossings.

As a first example, we define the increment cross helicity as

It is straightforward to see that, so defined, this quantity behaves qualitatively as the second-order structure functions that are familiar in hydrodynamics (e.g., Frisch 1995; Monin & Yaglom 1999). Note that the bracket 〈...〉 introduced in Equation (5) represents a suitable time average appropriate to local solar wind conditions, and in this work, we average for each value of t used in Equations (3) and (4) with a cadence of 1 NYs over nonoverlapping time intervals, usually of 10 minutes. This quantity measures the correlation between velocity and magnetic increments and is therefore relevant to assessing Alfvénicity (see Equation (1)). As will be explained later, the sign of ΔV · ΔB typically reverses with the sign of B_R, i.e., with magnetic sector crossings. To avoid including both signs within the averaging of Equation (5), in which case the result would depend strongly on the durations of magnetic sectors within the averaging period, we sometimes use a rectified cross helicity defined by

Similar to H_c, one may define a second-order structure function separately for the magnetic field vector increments as

and for the velocity field vector increments as

Note that all of these second-order quantities depend on the time lag τ as an argument, which is suppressed here.

These quantities begin at zero lag at a value of 0 and reach an asymptotic value at large lag of twice the estimated value of the total 〈v · b〉, 〈b²〉, or 〈v²〉 defined in terms of the fluctuating fields v ≡ V − 〈V〉 and b ≡ B − 〈B〉. A relevant physical interpretation is that each function represents the contribution to its respective asymptotic value (energy or cross helicity) due to all fluctuations at timescales less than the lag τ (e.g., Davidson 2004).

This of course corresponds to the behavior of standard second-order structure functions in homogeneous hydrodynamic turbulence. We also note that these functions are related to rugged invariants of incompressible MHD turbulence (see, e.g., Matthaeus & Goldstein 1982).

Some previous work (e.g., Chasapis et al. 2017; Chhiber et al. 2018; Parashar et al. 2018) has considered a structure function, computed from increments, in a form called an "equivalent spectrum" that is directly comparable to but with less scatter than a Fourier spectrum. We can explain the relationship as follows. The Fourier spectrum P(ω) for, say, magnetic fluctuations about a mean value can be normalized to satisfy the relation

In terms of the structure function S_b(τ), as τ → ∞, there is no correlation between b(t + τ) and b(t), so

Now we identify ω with 1/τ and therefore express the integral in terms of 1/τ:

The above derivation also applies if S_b is replaced by S_v or in the following, we use S to refer to any of these quantities. Comparing Equations (9) and (11), (1/4)τ²dS(τ)/dτ as a function of 1/τ can be directly compared with and interpreted in the same way as the Fourier spectrum. As noted above, S(τ) expresses the cumulative contributions of fluctuations at all timescales up to τ, so dS/dτ relates to the specific contribution from the timescale τ. Therefore, we identify

as an equivalent spectrum based on structure functions.

Alternatively, we can define a function , giving us another expression of the equivalent spectrum:

According to Figure 1, our second-order functions often have a power-law form S ∝ τ^α, and we find that they can be described by the Kolmogorov index α = 2/3, at least over certain ranges of τ. Then we can approximate the equivalent spectrum as (1/6)τS(τ).

Figure 1 compares the approximate equivalent spectra (1/6)τS(τ) based on second-order functions of rectified cross helicity, magnetic field, and velocity (black, red, and blue lines, respectively) with the corresponding Fourier transforms (light blue). Each equivalent spectrum is similar to but much smoother than the corresponding Fourier transform.

At this point, we introduce normalized quantities that are formed as ratios involving the above elementary second-order functions of increments and that provide physical interpretations related to solar wind properties. See Bruno & Carbone (2013) for other applications of increments to solar wind studies.

The normalized cross helicity is defined as

Note that 〈∣ΔV∣²〉 + 〈∣ΔB∣²〉 ± 2〈ΔV · ΔB〉 ≥ 0, so −1 ≤ σ_c ≤ 1. This measure allows us to assess the cumulative degree of correlation between the velocity and magnetic field fluctuations, a characteristic of Alfvénic fluctuations, as a function of scale. To be precise, recalling that a structure function or related quantity represents the cumulative contribution of fluctuations at scales below and up to the lag τ, σ_c is the ratio of the cumulative cross helicity to the cumulative energy up to the scale τ. A value of ∣σ_c∣ close to 1 is indicative of highly Alfvénic fluctuations.

The Alfvén ratio is the ratio of the velocity field structure function to the magnetic field structure function:

It represents the ratio of cumulative kinetic energy density to cumulative magnetic fluctuation energy density up to the scale τ. In a spectrum of normal unidirectionally propagating Alfvén waves, the value of r_A is equal to 1; other values indicate deviation from ideal Alfvénicity.

Another measure that we use to characterize Alfvénicity in solar wind plasma is the normalized residual energy, which combines information on the Alfvén ratio and the fluctuation energy. It is normally defined as the difference in kinetic and magnetic energies normalized by their sum. In keeping with the current approach based on structure functions and related quantities, it is defined here as

For Alfvén waves and ideal Alfvénicity, there is equal energy in the velocity and magnetic fluctuations, and the value of σ_r is equal to 0. However, solar wind observations usually indicate a negative value (Matthaeus & Goldstein 1982; Bruno et al. 1985).

The final measurement is the global alignment cosine angle between the magnetic and velocity fluctuations. This measure is defined as

This indicates the degree of directional alignment between the velocity and magnetic field fluctuations. A value of 1 indicates that the two quantities are perfectly aligned, while a value of −1 indicates that they are perfectly antialigned. Either of these values is consistent with ideal Alfvénicity.

These measures of Alfvénicity are based on second-order functions of increments, and their contributions come from fluctuations at scales smaller than τ. Power spectra in the solar wind are observed to rise with increasing τ (see Figure 1), so the main contribution to such a second-order function comes from fluctuations at scales close to τ, and our measures of Alfvénicity are generally dominated by fluctuations of scale ∼τ.

We also note that these quantities are not all independent. For example,

and

which implies that

Equivalently (Matthaeus & Goldstein 1982),

The Elsässer increments are defined as

where Z⁻ is the Elsässer field that propagates parallel to the mean magnetic field and Z⁺ has antiparallel propagation. In further analysis, the scalar notation ΔZ^± refers to the rms value of this vector increment.

Figure 2 shows scatter plots of the normalized cross helicity and residual energy during PSP solar encounters 8, 9, and 10 using the time lag τ = 87.4 s, which is smaller than the correlation time measured during E1 of 300–600 s (Parashar et al. 2020), in order to study Alfvénicity in the inertial range. Each 10 minute data point is colored according to the distance of PSP from the Sun. According to Equation (20), σ_c and σ_r are constrained to lie within the unit circle.

At a small radial distance from the Sun, ∣σ_c∣ tends to approach 1, indicating the predominance of waves that propagate in one direction (either parallel or antiparallel to the mean field), with σ_r tending toward 0 according to the constraint of Equation (20). These properties are indicative of high Alfvénicity in the solar wind.

However, as the heliocentric distance increases to r ≳ 0.35 au, ∣σ_c∣ decreases. σ_r almost always exhibits a negative value. Indeed, σ_r is mostly distributed near the minimum value allowed by the constraint, and in some time intervals with σ_c near 0, σ_r is as low as −0.8 to −0.9 (though this is far from typical). Our results are consistent with previous results (Chen et al. 2020; Shi et al. 2021), which indicate a decrease in ∣σ_c∣ and σ_r with increasing heliocentric distance r.

Figure 3 shows the equivalent spectra of rectified cross helicity, , and the second-order structure functions, (1/6)τS_b and (1/6)τS_v, for PSP data from different ranges of radial distance from the Sun. These spectra represent the average values of the second-order functions during encounters 8–10 with the heliospheric current sheet (HCS) crossing events removed, plotted as functions of 1/τ, which can be interpreted as the fluctuation frequency f. Because previous work has reported an inertial range magnetic frequency spectrum proportional to f^−5/3 at larger r and f^−3/2 at smaller r in PSP data (Chen et al. 2020), we indicate these spectral dependences with dotted lines, as well as the f⁻¹ spectral dependence that has been found at lower frequencies (Russell 1972; Goldstein et al. 1984).

All of the equivalent spectra exhibit qualitatively similar trends, gradually but clearly bending from a steeper frequency dependence at high frequency to a flatter dependence at low frequency. The frequency of transition from the inertial range to the energy-containing f⁻¹ range decreases with increasing heliocentric distance, which is consistent with the classic results from the Helios spacecraft (e.g., Bavassano et al. 1982; Bruno & Carbone 2013; Wu et al. 2021). In the upper frequency range, corresponding to the inertial range of turbulence, our analysis is truncated near 1 NYs, the native cadence of the solar wind data from PSP/SWEAP. Thus, we analyze only about 1 order of magnitude of the inertial range and cannot precisely determine the power-law index of the frequency dependence, which seems consistent with a power-law index of either −3/2 or −5/3. At lower frequency, all equivalent spectra for r between 15 and 55 R_s are consistent with a gradual transition at decreasing frequency toward an f⁻¹ spectrum.

Our results confirm that magnetic field energy dominates over kinetic energy in all heliospheric distance ranges and at all scales examined. Closer to the Sun, the difference between S_b and S_v magnitude is small; however, this difference increases greatly as the heliocentric distance increases. Interestingly, is similar to S_v at all distances at scales, but because S_b becomes much greater than both of these with increasing r, the normalized cross-helicity magnitude decreases greatly.

Figure 4 shows that ∣σ_c∣, σ_r, and , for which a higher value indicates stronger Alfvénicity, usually exhibit a dependence on the time lag, especially at distances close to the Sun. The blue shaded region represents the range of τ corresponding to the correlation length, where we employ the value obtained from Cuesta et al. (2022). Our analysis of Alfvénicity does not reach ion kinetic scales.

We find that these measures of Alfvénicity have their highest values for lags on the order of the correlation scale. For τ below the correlation scale, ∣σ_c∣, σ_r, and always decrease with decreasing τ, indicating decreasing Alfvénicity. An exception is σ_r in the highest distance range of 45–55 R_s, which fluctuates around a constant value and does not exhibit systematic scale dependence. This trend of decreasing Alfvénicity as the lag decreases in the inertial range was previously reported by Parashar et al. (2018), who observed this phenomenon in both the magnetosheath and the solar wind using data from the Magnetospheric Multiscale spacecraft.

As the lag increases beyond the correlation length, the values of ∣σ_c∣, σ_r, and again decrease, with the same exception as noted above. With further increases in the lag, there is evidently weaker correlation between increments of the velocity and magnetic fields.

We also note that while all measures of Alfvénicity systematically decrease with increasing r, there is a sudden change in the scale-dependent cross-helicity magnitude ∣σ_c∣ and the alignment measure from the distance range of [35, 45) R_s to [45, 55) R_s. Furthermore, all measures of Alfvénicity shown in Figure 4 become nearly independent of scale at r > 45 R_s.

Figure 5 illustrates the time dependence of Alfvénicity near perihelion during PSP solar encounters E8–E10. In the context of the dependence on the radial distance described in Section 4.1, note that all these time periods have r < 25 R_s, corresponding to the innermost range of r considered in that section. Here every parameter was calculated over nonoverlapping 10 minute time windows.

Panel (a) displays the second-order structure functions from the velocity and magnetic increments. Throughout most of the time, we observed that 〈ΔB²〉 was slightly higher than 〈ΔV²〉, resulting in σ_r < 0 and r_A < 1.

Panels (b) and (c) show the normalized cross helicity and cosine of the global alignment angle, respectively, which exhibit very similar time series. It is generally observed that Alfvénic fluctuations propagate predominantly outward from the Sun along magnetic field lines. Because a wave in Z⁺ = V + B propagates antiparallel to the mean field, when B_R < 0 (see panel (g)), the strongest increments are in ΔZ⁺, with ΔV · ΔB > 0 and therefore σ_c > 0 and . The predominance of ΔZ⁺ can be seen from the generally high value of the ratio of rms values, ΔZ⁺/ΔZ⁻ ≫ 1, when B_R < 0 (see panel (f)). On the other hand, when B_R > 0, we generally observe stronger increments in ΔZ⁻ with σ_c < 0, , and ΔZ⁺/ΔZ⁻ ≪ 1. A notable exception to this rule is that during a magnetic switchback, which can be defined as a temporary reversal in the B_R, the cross helicity is observed to be unchanged, as the predominant propagation direction of Alfvén waves continues to follow the field lines and reverses together with B_R (McManus et al. 2020). However, here we plot 10 minute averaged quantities, and switchbacks are too small to be seen on this scale (Raouafi et al. 2023).

While σ_c and usually remain quite close to ±1, the deviation from ideal Alfvénicity is more apparent when examining the normalized residual energy σ_r and Alfvén ratio r_A (see panels (d) and (e), respectively). For ideal Alfvénicity, we would have σ_r = 0 and r_A= 1, but in these observations, we almost always find σ_r < 0 and r_A < 1.

Sub-Alfvénic regions (highlighted in blue) are identified by M_A < 1 (see panel (h)), where the Alfvénic Mach number is defined by M_A ≡ V/V_A. In the sub-Alfvénic solar wind, we typically observed high Alfvénicity and a low ratio of the mean squared parallel to perpendicular velocity increments (see panel (i)), which is consistent with the variance anisotropy in the sub-Alfvénic solar wind reported in Bandyopadhyay et al. (2022). We observe no noticeable changes in Alfvénicity between sub-Alfvénic and super-Alfvénic solar wind over this range of heliospheric distance.

We observed upward spikes in σ_r and r_A at the edges of sub-Alfvénic regions at some but not all of the times when σ_c and changed sign, usually in association with PSP crossing boundaries between magnetic structures with different polarities or at full or partial HCS crossings. Some of these events involved sharply higher electron density, resulting in lower V_A and higher M_A, and if the wind outside the high-density structure was sub-Alfvénic, the sharply higher M_A inside the structure caused the wind there to be super-Alfvenic. Hence, some of these decreases in σ_c and , in some cases with upward spikes in σ_r and r_A, occurred at the boundaries of sub-Alfvénic time periods.

We consider the time intervals where ∣σ_c∣ < 0.5 for a time increment of τ = 87.4 s to have low Alfvénicity. This definition allows us to identify and study the brief periods of time when the normalized cross helicity of the solar wind plasma is relatively low, indicating that the velocity and magnetic field fluctuations are not strongly correlated. By analyzing the characteristics of the solar wind during these periods, we can gain insights into the processes that are driving the anomalously low Alfvénicity, i.e., the factors that are contributing to the weaker correlation between the velocity and magnetic field fluctuations.

In contrast to the usual behavior that ΔZ⁺/ΔZ⁻ ≫ 1 when B_R < 0 and ΔZ⁺/ΔZ⁻ ≪ 1 when B_R > 0, during these periods of low Alfvénicity, we observed a mixture of inward- and outward-propagating waves, as indicated by ΔZ⁺/ΔZ⁻ ≈ 1. Low Alfvénicity is commonly observed near the HCS. The times when PSP crosses the HCS are marked by a change in the polarity of the radial magnetic field B_r from one polarity to another, corresponding with the change in degree of the suprathermal electron pitch angle distribution from 0° to 180°. Within the plasma sheet that surrounds the HCS, the properties of solar wind turbulence are different from the normal solar wind. There is an increase in ion density, temperature, and flow speed, accompanied by a weakening of the magnetic field strength (Smith 2001).

A reduction in Alfvénicity is also observed during incomplete or partial crossings. In these partial crossings, PSP approaches the HCS without crossing to the other side, as indicated by B_R approaching zero but not fully reversing to the opposite direction, and there is a smaller magnetic field rotation compared with a complete crossing (Phan et al. 2021). The plot from encounter 10 illustrates one such partial crossing occurring between two sub-Alfvénic regions as reported by Zhao et al. (2022). When PSP encounters such structures, we also observe that the velocity field is dominated by the parallel velocity component, unlike the highly Alfvénic solar wind.

Away from HCS crossings, we observed two additional time intervals near perihelia that exhibited low Alfvénicity, as indicated by the red shading in Figure 5. In Figure 6, we show expanded plots for those brief non-Alfvénic time periods, showing σ_c, σ_r, and various solar wind parameters. Note that vector components are expressed in radial–tangential–normal coordinates. Both intervals exhibited σ_c approaching zero, which indicates counterpropagating waves during this interval, and negative σ_r.

During the first interval of interest (19:30–21:00 UTC on 2021 April 28), the magnitude of the magnetic field was not constant and was accompanied by fluctuations in n_e, T, and β. The variability of these parameters suggests incompatibility with the Alfvén wave equation. Additionally, we observed a bidirectional electron pitch angle distribution. In the second interval of low Alfvénicity (22:20–23:10 UTC on 2021 August 8), we observed sudden changes in all magnetic field components, V_R, n_e, T, and β.