A robust empirical relationship between speed and turbulence energy in the near-Earth solar wind

TL;DR

This paper addresses how to incorporate turbulence in heliospheric modeling by deriving an empirical law that links bulk solar-wind speed to turbulence energy, $Z^2 = v^2 + b^2$, using 25 years of ACE observations. It demonstrates a robust positive correlation between speed and turbulence and tests multiple functional forms, with a quadratic fit providing the best representation of the data. The model reproduces the observed log-normal distribution of turbulence energy and achieves a mean predictive accuracy with a Pearson correlation around $0.61$ over long timescales, notably performing better during solar maximum. The approach offers a practical tool to estimate turbulence amplitudes from low-resolution speed data, with applications in space-weather forecasting, SEP diffusion modeling, and enabling turbulence-aware simulations where high-resolution turbulence data are unavailable.

Abstract

The connection between turbulence and solar-wind acceleration, long known in space physics, is further developed in this Letter by establishing a robust empirical law that relates the bulk-flow speed to the magnetohydrodynamic-scale fluctuation energy in the plasma. The model is based on analysis of twenty-five years of near-Earth observations by NASA's Advanced Composition Explorer. It provides a simple way to estimate turbulence energy from low-resolution speed data -- a practical approach that may be of utility when high-resolution measurements or advanced turbulence models are unavailable. Potential heliospheric applications include space-weather forecasting operations, remote imaging datasets, and energetic-particle transport models that require turbulence amplitudes to specify diffusion parameters.

Figures (4)

• Figure 1: Probability density functions (PDFs) of log of turbulence energy ($\log Z^2$), with horizontal axes showing corresponding $Z^2$ values on a logarithmic scale. (a): Thick blue histogram is for ambient solar wind (sw) intervals and thin orange histogram is for intervals that include ICMEs. Best-fit Gaussian to $\log Z^2$ (i.e., a log-normal fit to $Z^2$) for the ambient sw is shown as a magenta curve. Sample means are marked with arrows of corresponding color; with $1\sigma$ spread, these are $2.7\pm2.1 \times10^3 ~(\text{km/s})^2$ and $4.1\pm4.0\times10^3~(\text{km/s})^2$, respectively. (b) The observed PDF for ambient sw is compared with the PDF obtained from the modeled $Z^2$ (thin tomato-colored histogram) and its best-fit log-normal (maroon curve). The model shown here is the quadratic fit (see Table \ref{['tab:fit_paramas']}). Sample means for the two cases are marked with arrows as in (a); these almost overlap. The model $1\sigma$ is $1.2\times10^3 ~(\text{km/s})^2$. Gaussian fits are performed using the IDL function gaussfit.pro.
• Figure 2: Joint histograms of mean speed $V$ and (a) total turbulence energy $Z^2$, (b) magnetic turbulence energy $b^2_A\equiv b^2$, and (c) Alfvén ratio $r_A$. Colorbars show interval abundance. Panels (a) and (b) show three empirical fits each of $Z^2$ and $b^2_A$ to $V$, respectively. Fit parameters are stated in Table \ref{['tab:fit_paramas']}. Panel (c) shows the mean and most probable values of $r_A$ in bins of $V$.
• Figure 3: Evaluation of model performance over a 25-year period. Top panel: For context, mean solar-wind speed is shown at 30-day cadence (see text) as brown connected circles that map to the left vertical axis. Monthly averages of solar sunspot number are shown as dark blue curve that maps to the right vertical axis. Middle panel: Dashed dark-blue curve with circles shows observed $Z^2$ at 30-day cadence; light-blue solid curve shows a 360-day running average. Red curve shows the modeled $Z^2$ at 30-day cadence. Here, model values are obtained from a quadratic fit - $Z^2=A_0+A_1 V + A_2V^2$ - based on observations preceding 2015, i.e., from the annotated "Training period"; $A_0=-3533,~A_1=15,~A_2=-0.002$ (cf. Table \ref{['tab:fit_paramas']}). Model values after the training period constitute a type of prediction, shown on a beige-shaded background. PCC between the 30-day cadence observations and model values is 0.61. Bottom panel: Circles show percentage error between observed and modeled values of $Z^2$, at 30-day cadence. Color of circles maps to colorbar on the right, indicating number of 12-hour intervals within each 30-day period represented by a circle.
• Figure 4: (a) Scatterplot of percent error between modeled and observed $Z^2$, and monthly sunspot number (both from Fig. \ref{['fig:time1']}). The PCC is $-0.37$. Colors of circles map to colorbar in Fig. \ref{['fig:time1']}. (b) A "zoomed in" comparison of observed and modeled $Z^2$, spanning a period of around 4 months. Dashed dark-blue curve with circles shows observations at a 1-day cadence, with the corresponding (quadratic) model curve in red. Pale blue and red curves show the respective 10-day running averages. The PCC between the daily observations and model values is 0.61.