Dispersive Properties of MHD Waves in the Expanding Solar Wind for a Parker Spiral Geometry

TL;DR

The paper develops an analytical framework to study how radial solar wind expansion in a Parker spiral background reshapes MHD wave dispersion. By formulating linear MHD in the Expanding Box Model and deriving the dispersion tensor, it yields the eigenfrequencies $ω(k,R)$, magnetic compressibility $C_B$, and the ratio $|δE_∥|/|δB_⊥|$ for Alfvén, fast, and slow magnetosonic modes, explicitly showing the impact of the expansion factor $a(t)$ and the spiral parameter $ξ$. The results show that expansion slows the magnetosonic phase speeds relative to simple radial rescalings, induces a minimum in $C_B$ at small radii, and increases $C_B$ with distance, in good agreement with observations, while expansion also suppresses the growth of the electric-to-magnetic amplitude ratio, especially for the slow mode. Overall, expansion redistributes energy between magnetically compressive and transverse fluctuations, providing a robust mechanism to explain the radial evolution of solar wind turbulence signatures in the inner heliosphere. The framework sets the stage for future extensions to nonlinear, multi-fluid, and oblique-propagation regimes relevant to inner-heliospheric turbulence.

Abstract

In this work, we quantify the effects of solar wind expansion on the dispersive properties of the three normal modes of ideal MHD using the Expanding Box Model, under a background magnetic field that follows the Parker spiral geometry. From the linearized MHD-EBM equations, we construct the dispersion tensor and derive analytical expressions for the eigenfrequencies $ω(k,R)$, magnetic compressibility $C_B$, and the ratio of the parallel electric field to the perpendicular magnetic field $|δE_\parallel|/|δB_\perp|$ of the magnetosonic modes to quantify how radial solar wind expansion reshapes the character of compressive fluctuations in the solar wind. Magnetic compressibility increases with heliocentric distance, and this trend shows a better alignment with in-situ observations when expansion is included from the MHD-EBM framework. $C_B$ shows a well-defined minimum at small radii and then increases linearly with distance, which naturally reproduces the observed transition from Alfvénic to compressive fluctuations between $\sim$0.3-1 AU. The ratio $|δE_\parallel|/|δB_\perp|$ reveals opposite behaviors for the fast and slow modes: while the fast mode becomes more electrostatic with increasing distance, the slow mode evolves to a more magnetically dominated character. Expansion reduces the growth of their electromagnetic/compressive balance at large radii. Our results demonstrate that solar wind expansion actively redistributes energy between magnetically compressive modes and purely transverse fluctuations with respect to the background magnetic field, playing a major role in shaping the radial evolution of wave dynamics throughout the inner heliosphere.

Figures (6)

• Figure 1: Coordinate transformation in the EBM. The radial expansion of the plasma is described through a Cartesian approximation. The plasma box expands away from a fixed reference system $S$ at a radial distance $R(t)$. The EBM introduces a non-inertial reference system $S'$, co-moving with the plasma box at a constant velocity $V_0$. The perpendicular coordinates are renormalized in the co-moving frame by the expansion parameter $a(t)$, maintaining a constant volume of the box.
• Figure 2: Background magnetic field lines represented in the x-y plane for each given value of the $\xi$ parameter, which represents fast solar wind (left) and slow solar wind (right). A larger magnitude of $\xi$ implies a more tightly spiraled magnetic field profile.
• Figure 3: Normalized phase speeds of fast magnetosonic, slow magnetosonic waves as a function of heliocentric distance $R$, computed for a Parker-spiral background magnetic field and a wave vector aligned with the x-axis. The parameter $\xi$ distinguishes the slow solar wind case ($\xi = -0.1$, red line) from the fast solar wind case ($\xi = -0.05$, black line). Solid lines show the phase speeds obtained within the MHD-EBM framework, including plasma expansion from first principles as given by Equations \ref{['fast']}-\ref{['alfven']}, while dashed lines show the corresponding values derived from a parametric rescaling of the non-expanding case, as given by Equations \ref{['disp1']}-\ref{['disp3']}. Here $R_0 = 0.1$ AU.
• Figure 4: Relative percentage difference between expanding and non-expanding phase speeds for fast and slow modes, shown vs heliocentric distance $(\beta = 0.25)$ and plasma beta ($R = 0.5$ AU), for both $\xi$ values. Horizontal lines indicate noise floor percentage assuming $B_0 \sim 150$nT at $R_0 = 0.1$ AU.
• Figure 5: Magnetic compressibility $C_B$ as a function of heliocentric distance $R$, computed for a Parker-spiral background magnetic field where a wave vector $\mathbf k$ is oriented at an angle $\theta$ with respect to the background magnetic field, and $R_0 = 0.1$AU. The value of $\xi$ distinguishes between fast solar wind $(\xi = - 0.05$) and slow solar wind $(\xi = - 0.1$). Solid lines represent the expanding case as obtained from the MHD-EBM framework given by Equation \ref{['cb']}, while dashed lines represent the non-expanding case given by Equation \ref{['cbnoexp']}.