Direct calculation of steady-state hydrodynamic solar wind solutions with newtonian viscosity

TL;DR

The problem addressed is the sonic-point singularity in steady-state inviscid solar wind solutions. The authors incorporate field-aligned Newtonian viscosity to remove these singularities and recast the governing equations as five coupled first-order ODEs in the spatial coordinate $s$, including external heating and radiative losses. They analyze inviscid polytropic and inviscid heating limits and the viscous isothermal and viscous nonideal-heating limits using a matrix representation with a determinant criterion, showing that the viscous closures render $\det(\uuline{M}\uuline{N})$ nonzero and well-posed. The method enables direct, fast extrapolation of solar wind profiles from near-solar initial conditions through the transition region to the outer heliosphere, suitable for initialization of time-dependent simulations and for populating global field-line models.

Abstract

Steady-state solutions to the Navier-Stokes equations are known to admit solutions that are singular at the sonic point. Consequently, inviscid solar wind models require special treatment of the solution near the sonic points, and this has proven to be a significant impediment to efficient modeling of the solar wind. In this paper we revisit the governing hydrodynamic equations for the expanding solar wind, with the inclusion of the classical (Newtonian) viscous stress , and we show how this inclusion eliminates the singularities that emerge from the inviscid equations. This result has been previously reported and used to generate solar wind profiles from initial conditions in the asymptotic limit; however, those studies did not include realistic treatments of the inner corona, and generally rejected the prospect of extrapolating solutions outward from the Sun into the heliosphere. Here, we expand this method to include external heating and optically thin radiative losses and show that solutions can be computed from initial conditions near the solar surface, thereby capturing the entire range of scales from below the transition region to the outer heliosphere in a single solution. Our approach is to cast the steady-state Navier-Stokes equations as a system of five coupled, ordinary differential equations (ODEs), which we solve using conventional methods, without any special treatment of the governing equations in the vicinity of the sonic point. The representative solutions that we present here demonstrate the utility and efficiency of this extrapolation method, which is considerably more realistic than commonly used analytical or empirical models. This method provides a direct approach to generating accurate solar wind profiles subject to observationally motivated initial conditions near the solar surface, at a fraction of the computational cost of comparable relaxation-based models.

Figures (3)

• Figure 1: Comparison of the viscous isothermal solutions to the analytical (inviscid) Parker solution. The black curves represent contours of De Leval's equation \ref{['Eqn::Parker']}. Of these, only the black dashed-dotted curve(s) exhibits a sonic point at the allowed critical point, consistent with Parker's description of a transonic solar wind. The variously colored blue, yellow, and red curves are numerical solutions to the viscous isothermal equations. The dashed blue curve is a subsonic breeze solution for which to $u\rightarrow0$ as $s\rightarrow\infty$. The dashed yellow (divergent wind) and red (shocked breeze) curves are transonic solutions that exhibit either $u\rightarrow\infty$ or $u\rightarrow0$, respectively, as $s\rightarrow\infty$. The constraint matching solution (solid teal curve) satisfies the boundary condition $\sigma(s=1\rm \,AU) = 0$.
• Figure 2: Solution profiles of the complete set of viscous nonideal solar wind equations. The black dashed curves are generated from a guess at the initial conditions, which is reasonable given the assumed mass flux of the solar wind, but which does not result in a constraint satisfying solution. The solid teal curves are the constraint satisfying solution, for which the initial conditions were systematically tuned (beginning with the initial guess) to obtain a solution that matches the asymptotic boundary condition.
• Figure 3: Asymptotic profiles of the constraint satisfying solution plotted with respect to heliocentric radius ($s$). Top panel: Profiles of $T/T_*$, $(u_\infty - u ) / (u_\infty - u_*)$, and an $s^{-2/7}$ power law for reference. Middle panel: Profiles of $f/f_*$, and $\sigma/\sigma_*$, with an $s^{-2}$ power law for reference. In both cases, the computed solutions are in good agreement with the representative power laws in the asymptotic region ($r \gtrsim 10^4 R_\odot$). Bottom panel: Total energy flux associated with heat conduction ($f$), viscous stress ($\sigma u$), bulk kinetic energy ($\rho u^3 / 2$), as well as saturation limits associated with free-streaming of electrons ($p_e v_e$) and ion thermal energy density ($p_i u$). The total energy flux, which is proportional to the sum of ($\rho u^3 / 2 + f + \sigma u$), is shown in the black dashed-dotted curve.