# The Interplay between Coronal Holes and Solar Active Regions from Magnetohydrostatic Models

^{1}

^{2}

^{3}), Universitat de les Illes Balears (UIB), E-07122 Palma de Mallorca, Spain

## Abstract

**:**

## 1. Introduction

^{−3}in the core to ${10}^{15}$ m

^{−3}in the halo.

## 2. The Problem of Magnetohydrostatic Equilibrium in 2D

## 3. Magnetic Configuration

#### 3.1. The Unipolar Configuration: Coronal Hole

#### 3.2. The Bipolar Configuration: Active Region

#### 3.3. Coronal Hole plus an Active Region

## 4. Coupling the Magnetic Field to the Plasma

#### 4.1. Coronal Hole Thermal Structure

#### 4.2. Active Region Thermal Structure

#### 4.3. Coronal Hole Plus Active Region: Basic Model

#### 4.3.1. Symmetric Case

#### 4.3.2. Non-Symmetric Case

#### 4.4. Extended Model

#### 4.4.1. Symmetric Case

#### 4.4.2. Non-Symmetric Case

## 5. Numerical Results

## 6. Conclusions and Discussion

- When the closest foot of the bipolar configuration to the CH and the CH itself have the same magnetic polarity, a solution resembling a typical AR with closed field lines embedding a hot and dense core is obtained.
- When the magnetic field of the bipolar foot is below $-{B}_{0}{w}_{0}/{w}_{1}$ and therefore the CH and the closest AR foot have an opposite polarity, one again obtains a case of an AR with a hot and dense core. However, the magnetic field lines near the footpoints of the AR are essentially open and the plasma has low values of the pressure, temperature and density there. The AR is surrounded by cold and light plasma, instead. Hence, the configuration may resemble that of multiple CHs located near the AR or even a dark halo.
- When the polarity of the CH and the closest foot of the bipolar region are opposite, but the magnetic field of the foot and ${B}_{1}$ is in the range from 0 to $-{B}_{0}{w}_{0}/{w}_{1}$, one finds closed magnetic field lines filled with plasma with the pressure, temperature and density below the coronal reference values and reaching a minimum at CH values. The plasma conditions are therefore that of typical CHs but with closed magnetic field lines in nearby locations.

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Flux function as a function of the horizontal position at $z=0$ for the combination of a coronal hole (CH) and a bipolar region. Here, ${B}_{1}={B}_{0}=-{B}_{2}$, ${x}_{1}/h=4$, ${x}_{2}/h=8$, ${w}_{0}/h=1$, ${w}_{1}/h={w}_{2}/h=1/4$, and $C=0$. See text for details.

**Figure 2.**The maximum temperature (32) at the core of the active region (AR) as a function of the magnetic field in the bipolar region. The dotted line corresponds to the value (33), while the dashed-dotted lines correspond to the values (34) and ${B}_{1}=0$. The dashed-dotted lines separate the domain in three intervals that lead to different types of solutions. Here, ${w}_{0}={w}_{1}$.

**Figure 3.**The maximum density (35) at the core of the AR as a function of the magnetic field in the bipolar region. The lower dashed line corresponds to the coronal density, ${\rho}_{\mathrm{C}}/{\rho}_{\mathrm{CH}}=3.3$, while the upper dashed line represents the asymptotic value (36). Here, ${w}_{0}={w}_{1}$.

**Figure 4.**The maximum temperature (37) at the core of the AR as a function of the magnetic field in the bipolar region for an asymmetric case. The dotted line corresponds to the value (38). The thin dashed line represents the coronal temperature, ${T}_{\mathrm{C}}/{T}_{\mathrm{CH}}=1.2$, while the thick dashed line is the asymptotic temperature (40). The vertical dash-double-dotted line represents the value (39). Here, ${w}_{2}={w}_{1}/2$ ($\alpha =1/2$) and ${w}_{1}={w}_{0}$.

**Figure 5.**Asymptotic temperature (40) at the core of the AR in the limit of large enough $|{B}_{1}/{B}_{0}|$ as a function of asymmetry parameter $\alpha $. The thin horizontal dashed line represents the coronal temperature, ${\left({T}_{\mathrm{max}}\right)}_{a}/{T}_{\mathrm{CH}}=1.2$.

**Figure 6.**The maximum density (35) at the core of the AR as a function of the magnetic field in the bipolar region for an asymmetric case. The vertical dotted line corresponds to the value (38). The horizontal dotted line corresponds to the value (41). The thin dashed line represents the coronal density, ${\rho}_{\mathrm{max}}/{\rho}_{\mathrm{CH}}=3.3$, while the thick dashed line is the asymptotic density, given by Equation (36). The vertical dash-double-dotted line represents the value (39). Here, ${w}_{2}={w}_{1}/2$ ($\alpha =1/2$) and ${w}_{1}={w}_{0}$.

**Figure 8.**Temperature (

**upper**) and density (

**lower**) distributions for the case of a CH plus an AR with $\beta =0.01$. The left foot of the AR and the CH have the same magnetic polarity. In this magnetohydrostatics (MHS) equilibrium, ${p}_{\mathrm{CH}}/{p}_{\mathrm{C}}=1/4$, ${T}_{\mathrm{CH}}/{T}_{\mathrm{C}}=0.8$, ${w}_{0}/h=1$, ${w}_{1}/h={w}_{2}/h=1/2$, ${B}_{1}/{B}_{0}=1/2$, ${B}_{2}/{B}_{0}=-1/2$, ${x}_{1}/h=8$ and ${x}_{2}/h=12$. The blue lines represent the magnetic field and the arrows indicate its direction. See text for more details.

**Figure 9.**Temperature (

**upper**) and density (

**lower**) distributions for the case of a CH plus an AR with $\beta =0.01$. The left foot of the AR and the CH have opposite magnetic polarities. In this MHS equilibrium, ${p}_{\mathrm{CH}}/{p}_{\mathrm{C}}=1/4$, ${T}_{\mathrm{CH}}/{T}_{\mathrm{C}}=0.8$, ${w}_{0}/h=1$, ${w}_{1}/h={w}_{2}/h=1/2$, ${B}_{1}/{B}_{0}=3$, ${B}_{2}/{B}_{0}=-3$. The blue lines represent the magnetic field and the arrows show its direction. See text for more details.

**Figure 10.**Temperature (

**upper**) and density (

**lower**) distributions for the case of a CH plus an AR with $\beta =0.01$. The left foot of the AR and the CH have opposite magnetic polarities. In this MHS equilibrium, ${p}_{\mathrm{CH}}/{p}_{\mathrm{C}}=1/4$, ${T}_{\mathrm{CH}}/{T}_{\mathrm{C}}=0.8$, ${w}_{0}/h=1$, ${w}_{1}/h={w}_{2}/h=1/2$, ${B}_{1}/{B}_{0}=-1$, ${B}_{2}/{B}_{0}=1$. The blue lines represent the magnetic field and the arrows show its direction. See text for more details.

**Table 1.**Conditions that lead to realistic or unrealistic models according to the values of the magnetic field ${B}_{0}$ at the coronal hole and ${B}_{1}$ at the active region. The three regimes correspond to the three intervals that correspond to the intersection of the continuous line with the horizontal dashed line in Figure 2. ${w}_{0}$ and ${w}_{1}$ are the corresponding characteristic spatial widths.

Case 1: ${\mathit{B}}_{1}{\mathit{B}}_{0}>0$ | Case 2: ${\mathit{B}}_{1}<-{\mathit{B}}_{0}{\mathit{w}}_{0}/{\mathit{w}}_{1}$ | Case 3: $-{\mathit{B}}_{0}{\mathit{w}}_{0}/{\mathit{w}}_{1}<{\mathit{B}}_{1}<0$ |
---|---|---|

Realistic | Realistic | Unrealistic |

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**MDPI and ACS Style**

Terradas, J.
The Interplay between Coronal Holes and Solar Active Regions from Magnetohydrostatic Models. *Physics* **2023**, *5*, 276-297.
https://doi.org/10.3390/physics5010021

**AMA Style**

Terradas J.
The Interplay between Coronal Holes and Solar Active Regions from Magnetohydrostatic Models. *Physics*. 2023; 5(1):276-297.
https://doi.org/10.3390/physics5010021

**Chicago/Turabian Style**

Terradas, Jaume.
2023. "The Interplay between Coronal Holes and Solar Active Regions from Magnetohydrostatic Models" *Physics* 5, no. 1: 276-297.
https://doi.org/10.3390/physics5010021