2. Theory and numerical solutions

The kinetic MHD equations with the Bhatnagar–Gross–Krook (BGK) collision operator (Bhatnagar, Gross & Krook Reference Bhatnagar, Gross and Krook1954; Gross & Krook Reference Gross and Krook1956) produce dispersion relations and plasma fluctuations (e.g. magnetic field and pressure) that span between the collisionless and collisional limits (Kulsrud et al. Reference Kulsrud, Sagdeev and Rosenbluth1980; Snyder, Hammett & Dorland Reference Snyder, Hammett and Dorland1997; Sharma, Hammett & Quataert Reference Sharma, Hammett and Quataert2003; Chandran et al. Reference Chandran, Dennis, Quataert and Bale2011). They describe a non-relativistic, magnetised plasma of arbitrary collision frequency (Kulsrud et al. Reference Kulsrud, Sagdeev and Rosenbluth1980; Sharma et al. Reference Sharma, Hammett and Quataert2003). See Appendix B for details on the equations and BGK collision operator. The specific equations, which we refer to as kinetic MHD-BGK (KMHD-BGK), model both the proton and electron responses with the kinetic equation so that Landau and Barnes dampings are retained. The BGK operator is used here to model relaxation processes, not particle collisions, so we use the language of an effective proton collision frequency $\nu _{\mathrm {eff}}$ or mean free path $\lambda _{\mathrm {mfp}}^{\mathrm {eff}} = v_{\mathrm {th}}^{\mathrm {p}} / \nu _{\mathrm {eff}}$, where the proton thermal speed is $v_{\mathrm {th}}^{\mathrm {p}}$.

Assuming plasma motions are slow compared with the gyrofrequency $\varOmega _p$, the second moment of the kinetic equation and the ideal induction equation leads to

(2.1a)\begin{gather} n_{\mathrm{p}} B \frac{\mathrm{d}}{\mathrm{d}t} \left( \frac{p_{{\perp}}^{\mathrm{p}}}{n_{\mathrm{p}} B} \right) ={-} \boldsymbol{\nabla} \boldsymbol{\cdot} ( q^{\mathrm{p}}_{{\perp}} \hat{\boldsymbol{b}}) - q^{\mathrm{p}}_{{\perp}} \boldsymbol{\nabla} \boldsymbol{\cdot} \hat{\boldsymbol{b}} + \frac{\nu_{\mathrm{eff}}}{3} (p_{{\parallel}}^{\mathrm{p}} - p_{{\perp}}^{\mathrm{p}}), \end{gather}

(2.1b)\begin{gather}\frac{n_{\mathrm{p}}^{3}}{2 B} \frac{\mathrm{d}}{\mathrm{d}t} \left(\frac{p_{{\parallel}}^{\mathrm{p}} B^{2}}{n_{\mathrm{p}}^{3}} \right) ={-} \boldsymbol{\nabla} \boldsymbol{\cdot} (q_{{\parallel}}^{\mathrm{p}} \hat{\boldsymbol{b}} ) + q_{{\perp}}^{\mathrm{p}} \boldsymbol{\nabla} \boldsymbol{\cdot} \hat{\boldsymbol{b}} + \frac{2 \nu_{\mathrm{eff}} }{3} ( p_{{\perp}}^{\mathrm{p}} - p_{{\parallel}}^{\mathrm{p}} ), \end{gather}

where $\mathrm {d}/\mathrm {d}t$ is the convective derivative and the quantities are the proton density $n_{\mathrm {p}}$, magnetic-field strength $B$, parallel (perpendicular) proton pressure $p_{\parallel }^{\mathrm {p}} \ (p_{\perp }^{\mathrm {p}})$, field parallel flux of parallel (perpendicular) proton heat $q_{\parallel }^{\mathrm {p}} \ (q_{\perp }^{\mathrm {p}})$ and the unit magnetic-field vector $\hat {\boldsymbol {b}} = \boldsymbol {B} / B$ (Chew, Goldberger & Low Reference Chew, Goldberger and Low1956; Hunana et al. Reference Hunana, Tenerani, Zank, Khomenko, Goldstein, Webb, Cally, Collados, Velli and Adhikari2019). The Alfvén speed is $v_A= B/\sqrt {4 {\rm \pi}n_{\mathrm {p}} m_{\mathrm {p}}}$, the proton gyroradius is $\rho _{\mathrm {p}} = v_{\mathrm {th}}^{\mathrm {p}} / \varOmega _{\mathrm {p}}$ and the ion-acoustic speed is $c_{\mathrm {s}} = \sqrt {(3 k_{\mathrm {B}} T_{\parallel }^{\mathrm {p}} + k_{\mathrm {B}} T_{\parallel }^{\mathrm {e}})/m_{\mathrm {p}}}$, where the parallel proton (electron) temperature is $T_{\parallel }^{\mathrm {p}} (T_{\parallel }^{\mathrm {e}})$.

Equations (2.1) are often discussed when the right-hand sides are zero and are then referred to as the double adiabatic equations or Chew–Goldberger–Low (CGL) invariants (Chew et al. Reference Chew, Goldberger and Low1956). The focus here is on how the CGL invariants are broken, for example, by the heat flux terms in the collisionless limit, and by the effective collisional terms ($\propto \nu _{\mathrm {eff}}$). Therefore, the relative non-conservation of the CGL invariants provides a sensitive test of the equation of state.

Cross-correlations and amplitude ratios, which can be measured, are constructed from the left-hand sides of (2.1),

(2.2a)\begin{gather} C_{{\parallel}} = \frac{ \langle \delta p_{{\parallel}}^{\mathrm{p}}\, \delta (n^{3}_{\mathrm{p}} / B^{2}) \rangle }{ \langle | \delta p_{{\parallel}}^{\mathrm{p}} |^{2} \rangle^{1/2} \langle | \delta (n^{3}_{\mathrm{p}} / B^{2}) |^{2} \rangle^{1/2}} , \end{gather}

(2.2b)\begin{gather}A_{{\parallel}} =\frac{ \langle | \delta (n^{3}_{\mathrm{p}}/ B^{2}) |^{2} \rangle^{1/2} }{ \langle n^{3}_{\mathrm{p}}/ B^{2} \rangle } \frac{ \langle p_{{\parallel}}^{\mathrm{p}} \rangle }{ \langle | \delta p_{{\parallel}}^{\mathrm{p}} |^{2} \rangle^{1/2} } , \end{gather}

(2.2c)\begin{gather}C_{{\perp}} = \frac{ \langle \delta p_{{\perp}}^{\mathrm{p}} \, \delta (n_{\mathrm{p}} B) \rangle }{ \langle | \delta p_{{\perp}}^{\mathrm{p}} |^{2} \rangle^{1/2} \langle | \delta (n_{\mathrm{p}} B) |^{2} \rangle^{1/2}} , \end{gather}

(2.2d)\begin{gather}A_{{\perp}} = \frac{ \langle | \delta (n_{\mathrm{p}} B) |^{2} \rangle^{1/2} }{ \langle n_{\mathrm{p}} B \rangle } \frac{ \langle p_{{\perp}}^{\mathrm{p}} \rangle }{ \langle | \delta p_{{\perp}}^{\mathrm{p}} |^{2} \rangle^{1/2} } , \end{gather}

where $\delta \chi = \chi - \langle \chi \rangle$ is the fluctuation about the average $\langle \chi \rangle$. They describe the relative non-conservation of the CGL invariants. The method compares predictions for (2.2) derived from the slow-mode eigenmodes of the linearised KMHD-BGK system (e.g. $\delta p_{\perp }^{\mathrm {p}}, \, \delta B$ etc.) with solar wind measurements, since the slow mode is the dominant compressive mode at large scales (Howes et al. Reference Howes, Bale, Klein, Chen, Salem and TenBarge2012). A description of the equations and the linear system of equations appears in Appendix B.

The model's free parameters are the propagation angle $\theta _{\hat {\boldsymbol {b}},\hat {\boldsymbol {k}}}$ and proton effective mean free path $\lambda _{\mathrm {mfp}}^{\mathrm {eff}}$; they are determined by fitting to solar wind observations. The wavenumber $k$ and the proton beta $\beta = 8 {\rm \pi}p^{\mathrm {p}} / B^{2}$, where $p^{\mathrm {p}} = 2p^{\mathrm {p}}_{\perp }/3 + p^{\mathrm {p}}_{\parallel }/3$, are set to measured values. The species temperature ratio is set to a typical value for the solar wind $T_{\mathrm {p}} / T_{\mathrm {e}} = 1$ and the effective mean free path species ratio is set to $\lambda _{\mathrm {mfp}}^{\mathrm {eff}} / \lambda _{\mathrm {mfp,electrons}}^{\mathrm {eff}} = 1$ (see Appendix B for details). At small $\beta$, these two ratios $T_{\mathrm {p}} / T_{\mathrm {e}}, \, \lambda _{\mathrm {mfp}}^{\mathrm {eff}} / \lambda _{\mathrm {mfp,electrons}}^{\mathrm {eff}}$ have an insignificant influence on the correlations, (2.2), and nearly no influence at large $\beta$.

Figure 1 demonstrates the ability of the KMHD-BGK equations to resolve the dynamics of the compressive slow mode from collisional (lighter blue) to collisionless (black). Numerical predictions for (2.2a), (2.2c) (bottom panels of figure 1) show distinct differences at $\beta > 1$ for different $k_{\parallel } \lambda _{\mathrm {mfp}}^{\mathrm {eff}}$, which can be compared with observations. The MHD ($k_{\parallel } \lambda _{\mathrm {mfp}}^{\mathrm {eff}} \ll 1$) and collisionless ($k_{\parallel } \lambda _{\mathrm {mfp}}^{\mathrm {eff}} \gg 1$) limits are illustrated in magenta, for $C_{\perp }$ (bottom right panel) these two limits produce similar trends, therefore it is necessary to make comparisons at multiple $k_{\parallel }$ to measure $\lambda _{\mathrm {mfp}}^{\mathrm {eff}}$.

Figure 1.

Numerical solutions of the KMHD-BGK equations for a range of $k_{\parallel } \lambda _{\mathrm {mfp}}^{\mathrm {eff}}$ (see colour bar). The imaginary (real) part of the complex frequency is denoted $\gamma \ (\omega _r)$. The dotted (dashed) magenta lines are the long (short) limit of $\lambda _{\mathrm {mfp}}^{\mathrm {eff}}$ corresponding to the collisionless (collisional) slow-mode/ion-acoustic branch for $\theta _{\hat {\boldsymbol {b}},\hat {\boldsymbol {k}}} = 88^{\circ }$ (Howes et al. Reference Howes, Cowley, Dorland, Hammett, Quataert and Schekochihin2006; Verscharen et al. Reference Verscharen, Chen and Wicks2017).

3. Measurements

The dataset consists of Wind spacecraft measurements of the pristine solar wind during years 2005–2010. The electrostatic analyser, 3DP, records onboard moments of the proton density, velocity and pressure tensor, and the magnetometer MFI records the magnetic field, at a nominal ${\sim }3s$ cadence (Lepping et al. Reference Lepping, Acũna, Burlaga, Farrell, Slavin, Schatten, Mariani, Ness, Neubauer and Whang1995; Lin et al. Reference Lin, Anderson, Ashford, Carlson, Curtis, Ergun, Larson, McFadden, McCarthy and Parks1995).

The dataset is restricted to time intervals satisfying three criteria: (i) 95 % of the data are available (the remaining are then linearly interpolated); (ii) the median density must be greater than 1 particle per cm$^{3}$; and (iii) the average norm of the non-gyrotropic tensor ($\mathit{\boldsymbol{\Pi}} ^{\mathrm {p}} =\boldsymbol{ \mathsf{ p}}^{\mathrm {p}} - \hat {\boldsymbol {b}} \hat {\boldsymbol {b}} \, p^{\mathrm {p}}_{\parallel } - (1 - \hat {\boldsymbol {b}} \hat {\boldsymbol {b}} ) p_{\perp }^{\mathrm {p}}$), must be less than 30 % of the average norm of the pressure tensor $\boldsymbol{ \mathsf{ p}}^{\mathrm {p}}$. The final point here is satisfied $\sim$94 % of the time at the ${\sim }3s$ time interval and more so at longer time intervals.

To probe a set of wavenumbers, the four quantities in (2.2) are measured, along with the average radial solar wind velocity $\langle V_{\mathrm {SW}} \rangle$ and the average proton beta for a set of time intervals $\tau = [30\ \mathrm {s}, 1\ \mathrm {min.}, 2\ \mathrm {mins.}, \ldots, 128\ \mathrm {mins.}]$. The time scales are converted to wavenumber $k_{\mathrm {SW}} = 1/ \tau \langle V_{\mathrm {SW}} \rangle$ via Taylor's frozen-in-flow (TFF) assumption (Taylor Reference Taylor1938). Outliers in the distribution of $k_{\mathrm {SW}}$ are removed and then three bins of equal probability density are obtained where the median of each bin is $k_{\mathrm {SW}} = [0.288, 1.41, 6.34] \times 10^{-5} \ \mathrm {km}^{-1}$, which lies within the inertial range of the magnetic-field power spectrum at 1 AU (Kiyani, Osman & Chapman Reference Kiyani, Osman and Chapman2015). The choice of bins provides enough separation in wavenumber to resolve differences in the measured quantities (e.g. figure 1) and sufficient sampling; the wavenumber bins contain $[2.98, 16.6, 70.0] \times 10^{5}$ samples.

For bin $k_{\mathrm {SW}} = 0.288 \times 10^{-5} \ \mathrm {km}^{-1}$ the $\beta$-conditioned probability functions of (2.2), mapped to a common colour bar, are displayed in figure 2. The $\beta$-trend lines in magenta (see caption) capture statistically significant differences between $\beta \lessgtr 1$. From the correlations $C_{\parallel }, C_{\perp }$ it is clear that the CGL invariants are rarely conserved $(C_{\parallel }, C_{\perp } = 1)$, but display similar trends to the theoretical expectations seen in figure 1. The amplitude ratios $A_{\parallel }, A_{\perp }$ demonstrate a relative decrease in fluctuation amplitude of the pressure components at $\beta > 1$.

Figure 2.

The $\beta$-conditioned probability functions of the quantities in (2.2) for the wavenumber bin $k_{\mathrm {SW}} = 0.288 \times 10^{-5} \ \mathrm {km}^{-1}$. The thin black line is a contour of probability equal to 0.01. The magenta lines are mean (dashed), median (solid) and maximum (dotted) conditioned on $\beta$.

4. Comparison of measurements and numerical solutions

The theoretical predictions for (2.2) from the numerical model in § 2 are compared with the observations in figure 2 to determine the most probable effective mean free path $\lambda _{\mathrm {mfp}}^{\mathrm {eff}}$ and propagation angles $\theta _{\hat {\boldsymbol {b}},\hat {\boldsymbol {k}}}$. The numerical predictions can be fitted to the observations by altering the parameters (e.g. the effective mean free path), but a degeneracy in parametrisation must be dealt with. The numerical solutions primarily depend on $k_{\parallel } \lambda _{\mathrm {mfp}}^{\mathrm {eff}} = k \cos (\theta _{\hat {\boldsymbol {b}},\hat {\boldsymbol {k}}}) \lambda _{\mathrm {mfp}}^{\mathrm {eff}}$ (Sharma et al. Reference Sharma, Hammett and Quataert2003), which implies that $\lambda _{\mathrm {mfp}}^{\mathrm {eff}}$ and $\theta _{\hat {\boldsymbol {b}},\hat {\boldsymbol {k}}}$ are degenerate. To address this, a scale-dependent anisotropy model $(k_{\parallel } \sim k_{\perp }^{\alpha })$ is introduced, which relates $k$ and $\theta _{\hat {\boldsymbol {b}},\hat {\boldsymbol {k}}}$,

(4.1)\begin{equation} k = \frac{k_{\mathrm{iso}}}{\sqrt{2}} \,\big[ \mathrm{sin} \big( \theta_{\hat{\boldsymbol{b}},\hat{\boldsymbol{k}}} \big) \big]^{\alpha/(1-\alpha)} \,\big[\cos\big(\theta_{\hat{\boldsymbol{b}},\hat{\boldsymbol{k}}} \big) \big]^{1/(\alpha - 1)}, \end{equation}

where $k_{\mathrm {iso}}$ is the isotropic wavenumber ($k = k_{\mathrm {iso}}$ when $k_{\perp } = k_{\parallel }$) and $\alpha$ is the anisotropy exponent, generalised from turbulence models (Goldreich & Sridhar Reference Goldreich and Sridhar1995). The wavenumber model provides $\theta _{\hat {\boldsymbol {b}},\hat {\boldsymbol {k}}}$, given $k$, parameterised by $\alpha, k_{\mathrm {iso}}$, so that $\lambda _{\mathrm {mfp}}^{\mathrm {eff}}$ can be determined at the given wavenumber. Then comparing solutions parameterised by $\lambda _{\mathrm {mfp}}^{\mathrm {eff}}$, $\alpha$, $k_{\mathrm {iso}}$ at multiple wavenumbers $k = k_{\mathrm {SW}}$ clears the degeneracy and allows all three parameters to be measured. Additionally, the model permits quantification of the observed increase of obliqueness with wavenumber of the compressive fluctuations (Chen et al. Reference Chen, Mallet, Schekochihin, Horbury, Wicks and Bale2012; Chen Reference Chen2016).

Finally, the predictions of (2.2) from the numerical solutions are normalised to the measured $\beta$-conditioned mean value (dashed magenta lines in figure 2) of $C_{\parallel }, C_{\perp }, A_{\parallel }, A_{\perp }$ at $\beta \simeq 10^{-1}$. This is to account for the fact that linear wave properties are only approximately observed in strong turbulence (Chen Reference Chen2016; Grošelj et al. Reference Grošelj, Chen, Mallet, Samtaney, Schneider and Jenko2019).

The ranges $\alpha = [0.05,1.0]$, $k_{\mathrm {iso}} = [5 \times 10^{-9}, 5 \times 10^{-7}]\ \mathrm {km}^{-1}$ and $\lambda _{\mathrm {mfp}}^{\mathrm {eff}} = [3.5 \times 10^{4}, 2.1 \times 10^{6}] \ \mathrm {km}$ are chosen for computing numerical solutions. The ranges of $\alpha, k_{\mathrm {iso}}$ are consistent with previous observations (Chen et al. Reference Chen, Mallet, Schekochihin, Horbury, Wicks and Bale2012; Chen Reference Chen2016). The range of $\lambda _{\mathrm {mfp}}^{\mathrm {eff}}$ returns numerical solutions of (2.2) that compare qualitatively well with the observations (seen in figure 2). The Spitzer–Härm mean free path returns the (collisionless) ion-acoustic dispersion relation which is inconsistent with the measurements.

To make a quantitative comparison, we compute the ‘goodness of fit’

(4.2)\begin{equation} R = \sqrt{N^{{-}1} \sum_i^{N} \, (\bar{y}_i - \hat{y}_i)^{2}}, \end{equation}

where $\hat {y}_i$ ($\bar {y}_i$) is the local numerical solution (local measured mean), summed over $i$, denoting the $i$th $\beta$-bin. Here, $R(k_{\mathrm {SW}}; \alpha, k_{\mathrm {iso}}, \lambda _{\mathrm {mfp}}^{\mathrm {eff}})$ is calculated for each wavenumber $k_{\mathrm {SW}}$, where the mean $\bar {y}_i$ is respective to the wavenumber bin. The $R$-values are inverted for unnormalised weights ($w = R^{-1}$), divided by the maximum weight ($w_{\mathrm {max}}$), then summed over wavenumber $\mathcal {W}(\alpha, k_{\mathrm {iso}}, \lambda _{\mathrm {mfp}}^{\mathrm {eff}}) = \sum _k w(k; \alpha, k_{\mathrm {iso}}, \lambda _{\mathrm {mfp}}^{\mathrm {eff}})/ w_{\mathrm {max}}$ to break the aforementioned degeneracy. From this volume, weighted geometric means $\mu _x$, covariances $\sigma ^{2}_{x,y}$, and two sigma confidence intervals $\mathrm {CI}_{x}$ are calculated (see Appendix C for the statistics; Norris Reference Norris1940; Kendall & Stuart Reference Kendall and Stuart1977). This is the method employed to measure the quantities $\lambda _{\mathrm {mfp}}^{\mathrm {eff}}, \alpha, k_{\mathrm {iso}}$, providing the main results of the Letter.

To visualise the weighted parameter space for $C_{\perp }$, figure 3 illustrates the weight volume $\mathcal {W}(\alpha, k_{\mathrm {iso}}, \lambda _{\mathrm {mfp}}^{\mathrm {eff}})$ numerically integrated over each parameter axis $\chi$,

(4.3)\begin{equation} \mathcal{W}_{\chi} = \int_{\chi_0}^{\chi_n} \, {\rm d} \chi \, \frac{\mathcal{W}(\alpha, k_{\mathrm{iso}}, \lambda_{\mathrm{mfp}}^{\mathrm{eff}})}{\chi_n - \chi_0}, \end{equation}

where $\chi _n, \chi _0$ are limits of the range. The weighted means in figure 3 lie in the maximum regions of $\mathcal {W}_{\chi }$, within the confidence intervals, indicating the weighted geometric statistics are a good representation of the observations.

Figure 3.

The three panels display the integrated weight volume, (4.3), for each parameter $\chi$, for $C_{\perp }$. The magenta crosses indicate the weighted geometric means and two sigma confidence intervals.

To check the scale dependence, figure 4 displays the observed $\beta$-conditioned means of (2.2) and the numerical solutions corresponding to the maximum $\mathcal {W}$. The numerical solutions and observations trend similarly with wavenumber, indicating the scale dependence of the effective collisionality has been well modelled. The parameters of the maximum (recorded in figure 4) do not correspond exactly to the weighted geometric means of $\mathcal {W}$ (seen in figure 3), reflecting the statistical nature of the measured quantities.

Figure 4.

The panels (a–d) show the $\beta$-conditioned mean of the four quantities in (2.2) for the three median wavenumber bins $k_{\mathrm {SW}} = [0.288, 1.41, 6.34] \times 10^{-5} \ \mathrm {km}^{-1}$ as solid (black, blue, magenta) lines, respectively. Statistical uncertainties on the mean trends can be seen in figure 2. The dashed lines are the numerical solutions corresponding to the maximum $\mathcal {W}$; the parameters of the maximum are reported in the panels.

The method of calculating statistics for $\lambda _{\mathrm {mfp}}^{\mathrm {eff}}$, $\alpha$, $k_{\mathrm {iso}}$ displayed in figure 3 for $C_{\perp }$ produces similar statistics for $C_{\parallel }, A_{\parallel }, A_{\perp }$ (see Appendix C). Therefore, in table 1, combined statistics are reported. The measured effective mean-free-path and mean proton thermal speed (measured with this dataset) give an effective collision frequency of $\nu _{\mathrm {eff}} = v_{\mathrm {th}}^{\mathrm {p}} / \lambda _{\mathrm {mfp}}^{\mathrm {eff}}$ = 1.11 $\times 10^{-4}$ s$^{-1}$.

Table 1.

Combined weighted geometric mean $\mu _x$, standard deviation $\sigma _{x,x}$ and the two sigma confidence interval $\mathrm {CI}_{x}$.

The transition frequency, where $\nu _{\mathrm {eff}} \simeq \omega$, can be estimated with $\nu _{\mathrm {eff}}$ and the ion-acoustic dispersion relation $\omega _{\mathrm {IA}} = k_{\parallel } \, c_{\mathrm {s}}$, giving the parallel transition wavenumber $k^{\mathrm {trans}}_{\parallel } = v_{\mathrm {th}}^{\mathrm {p}}/c_{\mathrm {s}} \, \lambda _{\mathrm {mfp}}^{\mathrm {eff}}$. Using the wavenumber model, (4.1), the transition wavenumber is

(4.4)\begin{equation} k^{\mathrm{trans}} = \frac{v_{\mathrm{th}}^{\mathrm{p}}}{c_{\mathrm{s}} \, \lambda_{\mathrm{mfp}}^{\mathrm{eff}} } \, \sqrt{1 + \left[\frac{2 \big( v_{\mathrm{th}}^{\mathrm{p}} \big)^{2} }{ \big(\lambda_{\mathrm{mfp}}^{\mathrm{eff}} \, k_{\mathrm{iso}} \, c_{\mathrm{s}}\big)^{2} } \right]^{(1 - \alpha)/\alpha} } \end{equation}

(more details are provided in Appendix D). Inserting the combined statistics from table 1, using a typical value of $v_{\mathrm {th}}^{\mathrm {p}} / c_{\mathrm {s}} = \sqrt {1/2}$ for the solar wind, and using the TFF assumption, the transition wavenumber in spacecraft-frame frequency at 1 AU is $\langle V_{\mathrm {SW}} \rangle \, k^{\mathrm {trans}} = f^{\mathrm {trans}} =0.19$ Hz and $\mathrm {CI}_{f^{\mathrm {trans}}} = [0.046, 0.33 ]$ Hz. The uncertainties are propagated from $V_{\mathrm {SW}}$ and the four estimates of $k^{\mathrm {trans}}$ from $C_{\parallel }, \, A_{\parallel }, \, C_{\perp }, \, A_{\perp }$.