Alfven wave propagation from the photosphere to the corona: temporal evolution against stationary results

TL;DR

The paper addresses how time-dependent propagation of Alfvén waves from the photosphere to the corona compares with stationary predictions in the presence of chromospheric reflection and dissipation. It uses a 1D, gravity-stratified chromospheric model based on the FALC background with a vertical magnetic field $B_0$, solving the linearized MHD equations with Cowling diffusion and applying boundary conditions via Elsässer variables to obtain time-dependent reflectivity, transmissivity, and absorption, which are then compared to stationary results. The main finding is that the time-dependent coefficients converge quickly to stationary values across most of the frequency range, with transmissivity matching after a few Alfvén crossing times and reflectivity and absorption converging conditionally depending on frequency and resolution; the study also highlights the critical role of spatial resolution in accurate convergence. This work validates the stationary approach for broad parameter regimes while clarifying limitations due to the simplified 1D geometry and linear assumptions, and it points toward future multi-dimensional and nonlinear extensions to capture phase mixing and coupling effects.

Abstract

Recent observations have confirmed that a significant fraction of the coronal Alfvenic wave spectrum originates in the photosphere. These waves travel from the photosphere to the corona, overcoming the barriers of reflection and dissipation posed by the chromosphere. Previous studies have theoretically calculated the chromospheric reflection, transmission, and absorption coefficients for pure Alfven waves under the assumption of stationary propagation. Here, we relax that assumption and investigate the time-dependent propagation of Alfven waves driven at the photosphere. Using an idealized chromospheric background model, we compare the coefficients obtained from time-dependent simulations with those derived under the stationary approximation. Additionally, we examine the impact of the spatial resolution in the numerical simulations. Considering a spatial resolution of 250 m, we find that the time-dependent transmission coefficient converges to the stationary value across the entire frequency range after only a few chromospheric Alfven crossing times, while the reflectivity displays a good convergence for frequencies lower than 30 mHz. The absorption coefficient also converges for wave frequencies above 1 mHz, for which chromospheric dissipation is significant. In contrast, at lower frequencies, wave energy dissipation is weak and the time-dependent simulations typically overestimate the absorption. Inadequate spatial resolution artificially enhances the chromospheric reflectivity, reduces wave transmission to the corona, and poorly describes the wave energy absorption. Overall, the differences between the stationary and time-dependent approaches are only minor and gradually decrease as spatial resolution and simulation time increase, which reinforces the validity of the stationary approximation.

Figures (7)

• Figure 1: Background atmospheric model. Dependence with height over the photosphere of density (top), temperature (mid), and Cowling's coefficient (bottom). Three different values of $B_0$ are considered in the bottom panel.
• Figure 2: Stationary results. Reflectivity (top), transmissivity (mid), and absorption (bottom) as functions of the Alfvén wave frequency. Three different values of $B_0$ are considered.
• Figure 3: Time-distance diagrams of the Elsässer variables $Z^\uparrow$ (left) and $Z^\downarrow$ (right) for an Alfvén wave with a frequency of 5 mHz. The amplitudes of the Elsässer variables are expressed in arbitrary units. Time and distance are normalized with respect to the Alfvén crossing time and domain length, respectively.
• Figure 4: Time-dependent results for an Alfvén wave with $f=$ 5 mHz. Top: Reflectivity, transmissivity, and absorption coefficients as functions of time. The horizontal dashed lines denote the corresponding results in the stationary regime. Bottom: Percentage differences (in absolute value) between the time-dependent and the stationary coefficients as functions of time. Time is normalized with respect to the Alfvén crossing time.
• Figure 5: Time-dependent results for an Alfvén wave with $f=$ 5 mHz. Percentage differences (in absolute value) between the time-dependent and the stationary reflectivity (top), transmissivity (mid), and absorption (bottom) coefficients as functions of time. The various color lines correspond to different spatial resolutions. Time is normalized with respect to the Alfvén crossing time.