# Nonlinear Coupling of Alfvén and Slow Magnetoacoustic Waves in Partially Ionized Solar Plasmas: The Effect of Thermal Misbalance

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## Abstract

**:**

## 1. Introduction

## 2. Method and Basic Equations

#### 2.1. Background Model

#### 2.2. First-Order Equations

#### 2.3. Second-Order Equations

## 3. Results

#### 3.1. $\gamma =5/3$, $a=b=0$ (Constant Heating per Unit Volume)

#### 3.2. $\gamma =5/3$, $a=2,b=4$

#### 3.3. $\gamma =5/3$, $a=2,b=6.25$

#### 3.4. $\gamma =5/3$, $a=4$, $b=2$ or $a=6.25,b=2$

#### 3.5. $\gamma =5/3$, ${\tau}_{\mathrm{r}1}={\tau}_{\mathrm{r}2}$

#### 3.6. LTE ${\mathsf{\Gamma}}_{1}$ and NLTE ${\mathsf{\Gamma}}_{1}$

#### 3.7. $\gamma =5/3$, LTE ${\mathsf{\Gamma}}_{1}$ and NLTE ${\mathsf{\Gamma}}_{1}$, $a=2,b=4$ or $a=2,b=6.25$

## 4. CHIANTI7 Radiative Loss Function

## 5. Discussion

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

## References

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**Figure 1.**Dimensionless longitudinal velocity perturbations versus dimensionless time computed (

**left**) at $z=0.5$ for T = 12,000 K (red curve), $T=8000$ K (blue curve), $T=6000$ K (black curve) and $T=4000$ K (brown curve); $a=0$, $b=0$, ${k}_{z}=\frac{\pi}{2}$; and (

**right**) at $z=0.5$ for T = 12,000 K, ${\eta}_{\mathrm{C},0}=0.022$ (red curve), $T=8000$ K, ${\eta}_{\mathrm{C},0}=0.026$ (blue curve), $T=6000$ K, ${\eta}_{\mathrm{C},0}=0.03$ (black curve) and $T=4000$ K, ${\eta}_{\mathrm{C},0}=0.036$ (brown curve). $a=0$, $b=0$, ${k}_{z}=\frac{\pi}{2}$. The characteristic length, ${L}_{0}={10}^{5}$ m, and the Hildner [42] radiative function are used. See text for details.

**Figure 2.**

**Left**: Comparison between the numerical solution for the longitudinal velocity perturbation (red curve) and the analytical solution (blue curve) ($T=4000$ K, $a=b=0$, ${\eta}_{\mathrm{C},0}=0.036$).

**Right**: Temporal behavior of the longitudinal velocity perturbations for different heating mechanisms, with $a=1/2,b=-1/2$ (red curve), $a=1,b=0$ (blue curve), $a=b=1$ (black curve), and $a=b=7/6$ (brown curve); $T=8000$ K, ${\eta}_{\mathrm{C},0}=0.026$. ${L}_{0}={10}^{5}$ m and the Hildner [42] radiative function are used as needed. See text for details.

**Figure 3.**Dimensionless longitudinal velocity perturbations versus dimensionless time computed (

**left**) at $z=0.5$ for T = 12,000 K, ${\eta}_{\mathrm{C},0}=0.022$ (red curve), $T=8000$ K, ${\eta}_{\mathrm{C},0}=0.026$ (blue curve), $T=6000$ K, ${\eta}_{\mathrm{C},0}=0.03$ (black curve) and $T=4000$ K, ${\eta}_{\mathrm{C},0}=0.036$ (brown curve), with ${L}_{0}={10}^{5}$ m; and (

**right**) at $z=0.5$ for T = 12,000 K, ${\eta}_{\mathrm{C},0}=0.00022$ (red curve), $T=8000$ K, ${\eta}_{\mathrm{C},0}=0.00026$ (blue curve), $T=6000$ K, ${\eta}_{\mathrm{C},0}=0.0003$ (black curve) and $T=4000$ K, ${\eta}_{\mathrm{C},0}=0.00036$ (brown curve); $a=2$, $b=4$, ${k}_{z}=\frac{\pi}{2}$, with ${L}_{0}={10}^{7}$ m. The Hildner [42] radiative function is used. See text for details.

**Figure 4.**Dimensionless longitudinal velocity perturbations versus dimensionless time computed (

**left**) at $z=0.5$ for T = 12,000 K, ${\eta}_{\mathrm{C},0}=0.022$ (red curve), $T=8000$ K, ${\eta}_{\mathrm{C},0}=0.026$ (blue curve), $T=6000$ K, ${\eta}_{\mathrm{C},0}=0.03$ (black curve) and $T=4000$ K, ${\eta}_{\mathrm{C},0}=0.036$ (brown curve), with ${L}_{0}={10}^{5}$ m; and (

**right**) at $z=0.5$ for T = 12,000 K, ${\eta}_{\mathrm{C},0}=0.00022$ (red curve), $T=8000$ K, ${\eta}_{\mathrm{C},0}=0.00026$ (blue curve), $T=6000$ K, ${\eta}_{\mathrm{C},0}=0.0003$ (black curve) and $T=4000$ K, ${\eta}_{\mathrm{C},0}=0.00036$ (brown curve) with $a=2$, $b=6.25$, ${k}_{z}=\frac{\pi}{2}$, and ${L}_{0}={10}^{7}$ m. The Hildner [42] radiative function is used. See text for details.

**Figure 5.**Dimensionless longitudinal velocity perturbations versus dimensionless time computed at $z=0.5$ for ${L}_{0}={10}^{7}$ m (red curve), ${\eta}_{\mathrm{C},0}=0.00026$, ${L}_{0}={10}^{6}$ m (blue curve), ${\eta}_{\mathrm{C},0}=0.0026$, ${L}_{0}={10}^{5}$ m (black curve), ${\eta}_{\mathrm{C},0}=0.026$. $T=8000$ K; $a=2$, $b=6.25$. The Hildner [42] radiative function is used.

**Figure 6.**Dimensionless longitudinal velocity perturbations versus dimensionless time computed (

**left**) at $z=0.5$ for T = 12,000 K, ${\eta}_{\mathrm{C},0}=0.022$, $a=2,b=4$ (red curve) and $a=4,b=2$ (blue curve), and (

**right**) at $z=0.5$ for T = 12,000 K, ${\eta}_{\mathrm{C},0}=0.022$, $a=2,b=6.25$ (red curve) and $a=6.25,b=2$ (blue curve); ${k}_{z}=\frac{\pi}{2}$. ${L}_{0}={10}^{5}$ m, and the Hildner [42] radiative function are used.

**Figure 7.**

**Left**: thermal misbalance times versus b exponent: ${\tau}_{\mathrm{r}1}$ (red curve), ${\tau}_{\mathrm{r}2}$ (blue curve).

**Right**: $\left(\frac{1}{{\tau}_{\mathrm{r}2}}\right.$ − $\left.\frac{1}{{\tau}_{\mathrm{r}1}}\right)$ versus b exponent, T = 12,000 K, $a=2$, ${k}_{z}=\frac{\pi}{2}$. ${L}_{0}={10}^{5}$ m and the Hildner [42] radiative function are used. See text for details.

**Figure 8.**The threshold values for (

**left**) the b exponent versus the a exponent and for (

**right**) the a exponent versus the b exponent. $\gamma =5/3$ and the Hildner [42] radiative function are used. See text for details.

**Figure 9.**(

**Left**): $\left(\frac{1}{{\tau}_{\mathrm{r}2}}\right.$ − $\left.\frac{1}{{\tau}_{\mathrm{r}1}}\right)$ versus a and b exponents. (

**Right**): ${\omega}_{\mathrm{i}}$ versus a and b exponents. T = 12,000 K. The Hildner [42] radiative function is used. See text for details.

**Figure 10.**Thermal time, ${\tau}_{\mathit{T}}$ (

**left**), and (

**right**) ${\tau}_{\mathit{T}}$ (red) and ambipolar diffusion time, ${\tau}_{\mathrm{AD}}$ (blue), versus a and b exponents. T = 12,000 K. The Hildner [42] radiative function is used. See text for details.

**Figure 11.**

**Left**: dimensionless longitudinal velocity perturbations versus dimensionless time, computed at $z=0.5$ for ${L}_{0}={10}^{5}$ m, $T=4000$ K, ${\eta}_{\mathrm{C},0}=0.036$, $a=2$, and $b=5.9$. The Hildner [42] radiative function is used.

**Right**: LTE first adiabatic exponent, ${\mathsf{\Gamma}}_{1}$ (blue curve) and NLTE ${\mathsf{\Gamma}}_{1}$ (red curve) versus temperature. See text for details.

**Figure 12.**Dimensionless longitudinal velocity perturbations versus dimensionless time computed (

**left**) at $z=0.5$ for $T=8000$ K, ${\eta}_{\mathrm{C},0}=0.026$, $\gamma =5/3$ (red curve), ${\eta}_{\mathrm{C},0}=0.028$, LTE ${\mathsf{\Gamma}}_{1}=1.38$ (blue curve), ${\eta}_{\mathrm{C},0}=0.026$, NLTE ${\mathsf{\Gamma}}_{1}=1.15$ (black curve), $a=2$, $b=4$, and (

**right**) at $z=0.5$ for $T=8000$ K, ${\eta}_{\mathrm{C},0}=0.026$, $\gamma =5/3$ (red curve), ${\eta}_{\mathrm{C},0}=0.026$, LTE ${\mathsf{\Gamma}}_{1}=1.38$ (blue curve), ${\eta}_{\mathrm{C},0}=0.026$, NLTE ${\mathsf{\Gamma}}_{1}=1.15$ (black curve), $a=2$, $b=6.25$. ${k}_{z}=\frac{\pi}{2}$. ${L}_{0}={10}^{5}$ m and the Hildner [42] radiative function are used.

**Figure 13.**Dimensionless longitudinal velocity perturbations versus dimensionless time computed (

**left**) at $z=0.5$ for T = 12,000 K, ${\eta}_{\mathrm{C},0}=0.022$ (red curve), $T=8000$ K, ${\eta}_{\mathrm{C},0}=0.026$ (blue curve), $T=6000$ K, ${\eta}_{\mathrm{C},0}=0.03$ (black curve) and $T=4000$ K, ${\eta}_{\mathrm{C},0}=0.036$ (brown curve), with $a=2$ and $b=4$, and (

**right**) at $z=0.5$ for T = 12,000 K, ${\eta}_{\mathrm{C},0}=0.022$ (red curve), $T=8000$ K, ${\eta}_{\mathrm{C},0}=0.026$ (blue curve), $T=6000$ K, ${\eta}_{\mathrm{C},0}=0.03$ (black curve) and $T=4000$ K, ${\eta}_{\mathrm{C},0}=0.036$ (brown cirve), with $a=2$ and $b=6.25$. The CHIANTI7 radiative function [43,44,45] is used.

**Figure 14.**The threshold values of (

**left**) the b exponent versus ${\mathsf{\Gamma}}_{1}$ for $\alpha =7.4$ Hildner [42] (red curve) and $\alpha =8.06$ CHIANT7 (blue curve) radiative loss functions with $a=2$, and (

**right**) of the b exponent versus the exponent a for ${\mathsf{\Gamma}}_{1}=5/3$ (red curve), $1.38$ (blue curve), $1.15$ (black curve) using the CHIANTI7 radiative loss function.

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**MDPI and ACS Style**

Ballester, J.L.
Nonlinear Coupling of Alfvén and Slow Magnetoacoustic Waves in Partially Ionized Solar Plasmas: The Effect of Thermal Misbalance. *Physics* **2023**, *5*, 331-351.
https://doi.org/10.3390/physics5020025

**AMA Style**

Ballester JL.
Nonlinear Coupling of Alfvén and Slow Magnetoacoustic Waves in Partially Ionized Solar Plasmas: The Effect of Thermal Misbalance. *Physics*. 2023; 5(2):331-351.
https://doi.org/10.3390/physics5020025

**Chicago/Turabian Style**

Ballester, José Luis.
2023. "Nonlinear Coupling of Alfvén and Slow Magnetoacoustic Waves in Partially Ionized Solar Plasmas: The Effect of Thermal Misbalance" *Physics* 5, no. 2: 331-351.
https://doi.org/10.3390/physics5020025