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

¹ US Naval Research Laboratory Washington DC, USA
² Rice University Houston TX, USA
³ Southwest Research Institute Boulder CO, USA
⁴ Montana State University Bozeman MT, USA

^★ Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 28 April 2025
Accepted: 23 November 2025

Abstract

Context. Steady-state solutions to the Navier-Stokes equations are a valuable tool for constructing quasi-steady models of the solar wind and exploring the various factors that affect the fluxes of mass, energy, and momentum into the heliosphere. These models typically omit the effects of viscosity, which is assumed to be negligible under most coronal and heliospheric conditions; however, the inviscid Navier-Stokes equations are known to admit solutions that are singular at the sonic point, where the solar wind speed becomes equal to the relevant acoustic speed. 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.

Aims. In this paper we revisit the governing hydrodynamic equations for the expanding solar wind, with the inclusion of the viscous stress as defined by the classical (Newtonian) closure, 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 steady-state solar wind profiles from initial conditions in the asymptotic limit (outside of the Sun’s gravitational well); 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, which they deemed to be computationally unfeasible. Our aim, therefore, is to expand this method to include external heating and optically thin radiative losses and show that solutions can be computed outward 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.

Methods. Our approach was to cast the steady-state, field aligned Navier-Stokes equations as a system of five coupled, first order, ordinary differential equations (ODEs) describing the spatial evolution of the mass density, pressure, speed, conductive heat flux, and viscous stress. These equations were then solved using conventional methods, without any special treatment of the governing equations in the vicinity of the sonic point. Physically meaningful solutions were identified by varying the initial conditions at the lower boundary until the solution obtained the correct asymptotic form, which we derived for the particular closures that we employed.

Results. 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. The solutions obtained from this method can be used to initialize time-dependent simulations, to generate large families of steady-state solutions that can then be used to populate the hydrodynamic variables along individual magnetic field lines in global magnetic field models, and to explore how the properties of the quasi-steady solar wind are affected by changes in magnetic geometry and different coronal heating models.