This repository complete and extend the ParkerSolarWind respository that can be found at https://github.com/STBadman/ParkerSolarWind. The IsopolySolarWind respository contains Python codes which solve two-fluids hydrodynamic solar wind equations, for a 1D radial trans-sonic flow in spherical expansion, including super-radial expansion, based on Eugene Parker's theory of the solar wind (Parker 1958, Parker 1960) and expansion factor modeling (Kopp & Holzer 1976).
The codes follows the recent "isopoly" Parker's-like model by Dakeyo et al. (2022), which model an isothermal evolution close to the Sun, followed by a polytropic evolution somewhere above the critical radius. It also include the extension of the isopoly equations by Dakeyo et al. (2024b) accounting for the expansion factor modeling in the near Sun region, and the resulting "f-subsonic" and "f-supersonic" solutions, respectively subsonic and supersonic in the super-expansion region. This split of solution is due to the deLaval nozzle effect applied on the wind expansion (Kopp & Holzer 1976), that induces two possible critical radius values for a single coronal temperature value.
There are three main files : function_to_run_iso_poly_dakeyo2024b, isopoly_solar_wind_solve_and_plot and main_iso_poly_dakeyo2024b. All the files must be stored in the same respository for the execution. The former contains the functions to solve the equations themselves, the second file runs the solver and plots the solution, and the latter is the main code for the user, which takes the inputs of the model, runs the first two files to display the solutions, and provides the output isopoly parameters. An example of the use of the main code main_iso_poly_dakeyo2024b is provided in the Jupyter Notebook ExampleNotebook.ipynb.
The equations are solved by a finite difference scheme (explicit method). The combination of input parameters results in three different possible thermal regimes :
- A fully isothermal solar wind (
$\gamma_p = \gamma_e = 1$ )- This follows Parker 1958, in which the solar wind fluid is held at a fixed temperature. Mass flux conservation results in a negative density gradient and in turn an outwards directed pressure gradient force. For sufficiently hot
$T_0$ , this outwards force outcompetes gravitation, resulting in a trans-sonic solar wind flow out to infinity. While such a constant temperature is non-physical in the heliosphere, it is a reasonable first approximation to behavior in the solar corona where coronal heating operates (equivalent version also available at https://github.com/STBadman/ParkerSolarWind). - A two fluid isopoly solar wind with single transition (
$\gamma_p$ and/or$\gamma_e \neq 1$ , with$r_{iso|p} = r_{iso|e}$ )- Here, the solar wind temperature is allowed to cool with heliocentric distance, as is observed to actually occur in the solar wind (e.g. Dakeyo et al. (2022). This consists of an initial isothermal evolution (isothermal layer) up to a boundary distance called the "isothermal radius"
$r_{iso}$ , which can be interpreted in a first approximation as the region to which the coronal heating extends (as an abstract physical process, i.e. not related to the actual size of the corona). In the present solution case, both protons and electrons share the same transition, i.e.$r_{iso} = r_{iso|p} = r_{iso|e}$ . For$r \gt r_{iso}$ , the solar wind is constrained to follow a polytropic evolution which is initialized by the outer boundary conditions of the isothermal region. Protons and electrons can follow differentiate polytropic evolution ($\gamma_p \neq \gamma_e$ is possible). For most combinations of physical conditions, the trans-sonic critical point is located within the isothermal region. As long as the isothermal boundary is at sufficiently high altitude that the solar wind stays super-sonic at the transition to polytropic behavior, the solution remains on the asymptotically accelerating solution branch and a reasonable solar wind solution is obtained. The unphysical discontinuity at the regime transition can be smoothed by considering slowly varying polytropic indexes at the transition between the two regions, but this feature is not addressed here and may require in-depth work. - A two fluid isopoly solar wind with double transition (
$\gamma_p$ and/or$\gamma_e \neq 1$ , with$r_{iso|p} \neq r_{iso|e}$ )- This case is closely similar to the single transition solution, at the difference that protons and electrons do not share the same isothermal radius.
All the thermal regimes can be declined in both "f-subsonic" and "f-suersonic" type of solution, depending the influence of the expansion factor profile.
- f-subsonic solutions
- The f-subsonic solutions are the more commonly used in solar wind modeling and space weather. They embed relatively slowly accelerating wind, and a critical radius location between 3 and 8
$r_\odot$ for coronal temperature of the order of 0.5 - 3 MK. For this type of solution, the influence of the expansion factor is to create a deceleration region within the super-expansion region. - f-supersonic solutions
- The f-supersonic solutions are known, but less used in the space weather community. They induce a rapidly accelerating solar wind solution with a critical radius very close to the Sun inside the super-expansion region, ranging between 1 and
$\sim$ 3$r_\odot$ , for coronal temperatures of the order of 0.5 - 3 MK. They also induce a deceleration region, but approximately from the end of the super-expansion region to$\sim$ 8$r_\odot$ .
The numerical and physical justification for the choice between the two solutions depends on the determination of the critical radius, which must respects the requirements of the transonic solutions. More details are available in Dakeyo et al. (2024b).
Recommendations for appropriate isopoly input values are provided at the end of the ExampleNotebook.ipynb file in the repository.
Since all solutions are computed with the same set of codes, each of the above solutions can be obtained by modifying the inputs parameters. The main_iso_poly_dakeyo2024b code returns an array of heliocentric distances (
All the outputs are numpy array.
In the following example we solve and plot an f-subsonic isopoly solution with double transition. All thermal regimes related to the evolution of the two species (proton and electron) are indicated by a different color: the fully isothermal region is in red, the region after the first thermal transition (one species isothermal and the other polytropic) is in light blue, and the region after the second thermal transition (fully polytropic) is in blue.
# Importation required to run this code
import isopoly_solar_wind_solve_and_plot as ipsw
#########################################
# Inputs of the model
#########################################
# Length of the output model
N = 7e4
L = 1.496e11 # set to 1au by default
# Polytropic ind
gamma_p_max = 1.45
gamma_e_max = 1.25
# Coronal temperature
Tp0 = 2e6
Te0 = 1.5e6
# Isothermal radius (in solar radii)
r_iso_p = 4
r_iso_e = 7
# Expansion factor parameters
fm = 25
r_exp = 1.9 # in solar radii
sig_exp = 0.15 # in solar radii
#########################################
# Plotting option
plot_f = True
plot_gamma = False
plot_unT = True
plot_energy = False
#########################################
###############################################################
# Running of the main function
(r, n, u, Tp, Te, gamma_p, gamma_e, f, bol_super) = ipsw.solve_isopoly(
N, L, gamma_p_max, gamma_e_max,
Tp0, Te0, r_iso_p, r_iso_e,
fm, r_exp, sig_exp, plot_f,
plot_gamma, plot_speed,
plot_density, plot_temp, plot_energy)
###############################################################
Other examples for all the thermal regimes and types of solution (f-subsonic and f-supersonic) can be seen in ExampleNotebook.ipynb.
At the end of the same file, another function stream_calc_dakeyo2024a allows to trace the Parker's like spiral (streamline) associated with the computed isopoly solution. The code follows the backmapping method including acceleration and corotational effect presented in Dakeyo et al. (2024a), that is also used in Dakeyo et al. (2024b).