Wind Braking Torque: Solar Cycle Dynamics

• Wind braking torque is the measure of angular momentum loss due to magnetized stellar winds, defined by the mass loss rate and the effective lever arm from the Alfvén surface.
• Modeling techniques, including the Weber & Davis formalism and MHD simulations, reveal how cyclic changes in magnetic field geometry modulate wind dynamics.
• Solar cycle variations shift the location of open flux, altering the lever arm and thereby affecting both mass and angular momentum loss rates.

Wind braking torque denotes the angular momentum loss rate that a star experiences due to its magnetized stellar wind. In solar and stellar contexts, the outflowing plasma, interacting with large-scale magnetic fields, extracts angular momentum through a process governed by both the wind’s mass loss rate and the geometry and strength of the open magnetic flux. This mechanism enforces an effective lever arm, set by the Alfvén surface, and is dynamically modulated by magnetic cycles—controlling the rotational evolution of the Sun and other solar-type stars. The coupling of solar dynamo evolution to the coronal magnetic field, and to the consequent wind morphology and dynamics, produces substantial cyclic variability in wind braking torque and thus in the rate of solar angular momentum loss.

1. Mathematical Formalism of Wind Braking Torque

The quantitative foundation for wind braking torque is rooted in the Weber & Davis (1967) formalism, in which the angular momentum loss from a magnetized star is set by the transport of mass embedded in a rotating magnetic field. For the Sun,

$\tau = -\dot{M} \, \Omega_0 \langle r_A^2 \rangle$

where:

• $\tau$ is the wind braking torque,
• $\dot{M}$ is the total mass loss rate,
• $\Omega_0$ is the stellar (solar) surface rotation rate,
• $\langle r_A^2 \rangle$ is the mass-flux weighted mean squared cylindrical Alfvén radius.

The Alfvén surface is defined by the locus where the wind speed equals the local Alfvén speed, $c_A = B/\sqrt{4\pi \rho}$, with $B$ the magnetic field and $\rho$ the plasma density. The torque depends critically on both the mass flux and the geometry of the magnetic field as encoded in $\langle r_A \rangle$ (the average lever arm). The mass-flux weighted mean Alfvén radius is specified as

$$\langle r_A \rangle = \frac{\int r^2 \sin\theta \, \rho v \cdot r_A(\theta) \, d\theta}{\int r^2 \sin\theta \, ||\rho v|| \, d\theta}$$

with the integrations performed over the outflow.

Owing to the quadratic dependence of torque on lever arm length, modest changes in the effective Alfvén radius, as occur across the solar cycle, yield order-of-magnitude variations in torque.

2. Solar Wind Structure and Latitudinal Variability

Global MHD simulations, employing a kinematic dynamo code (STELEM) coupled to a 2.5D isothermal MHD coronal code (DIP), demonstrate that wind properties—including velocity and angular momentum loss—exhibit marked latitudinal and temporal structure over the 11-year solar cycle.

During solar minimum, open magnetic flux is confined largely to polar coronal holes. Wind emerging from these high-latitude regions is fast due to strong, unimpeded expansion along open field lines. In contrast, during activity maxima and polarity reversal, closed magnetic regions expand, producing smaller, lower-latitude coronal holes. The resulting surface wind channels at mid and low latitudes make a disproportionately large contribution to the net angular momentum loss, as the effective emitting area scales with $\sin\theta$. Despite the more localized surface extent, these regions dominate the integrated torque due to geometrical projection effects.

At activity maximum, the wind becomes generally slower; this is attributed to increased flux-tube expansion factors and altered topology at the surface–corona interface.

3. Cycle-Dependent Mass and Momentum Losses

The simulation results reveal that the overall mass loss rate, given by

$$\dot{M} = 2\pi R_0^2 \int_0^{\pi} \rho V_r \sin\theta d\theta$$

varies by a factor $\sim 1.6$ during the activity cycle—specifically from $4.2 \times 10^{-14}\, M_\odot$/yr at minimum to $6.9 \times 10^{-14}\, M_\odot$/yr at maximum. This variation is driven principally by the geometry and distribution of open flux channels: while wind speeds may not differ drastically, the emergence of low-latitude, high-expansion-factor wind channels at maximum, with larger emitting areas, significantly raises the net mass loss.

Angular momentum loss, and hence wind braking torque, is impacted by both mass loss rate and the effective lever arm, which is controlled by the coronal field’s topology. During maximum, while the mass loss increases, the effective Alfvén radius (and thus the lever arm) decreases—meaning cycle-averaged angular momentum loss is sensitive to the co-evolution of field geometry and mass flux.

4. Coronal Magnetic Field Dynamics and the Alfvén Surface

The coupled kinematic dynamo/coronal MHD model captures critical aspects of field topology evolution. As the solar activity cycle progresses,

• the cyclic dynamo generates time-varying poloidal and toroidal fields,
• the coronal response (computed with the isothermal MHD DIP code) rapidly reorganizes large-scale open and closed field regions,
• the open–flux topology, especially near polarity reversal, is directly linked to remapping of wind source zones, thus modulating both wind velocity distribution and torque.

This coupling accelerates the reversal of the open flux at high coronal altitudes compared to the photosphere. Consequently, the geometry (and thus lever arm) over which angular momentum is extracted evolves rapidly—resulting in significant changes in $\langle r_A \rangle$ (from $\sim 9\,R_\odot$ at minimum to $\sim 2.2\,R_\odot$ at maximum).

5. Cyclic Modulation and Spatial Migration of Braking Zones

Cyclic changes in field topology and open-flux source latitude modulate the efficiency and spatial application of wind braking torque. As the sources of open flux migrate from polar to low-latitude regions over the cycle, the zones of effective torque application traverse the surface. The reduction in Alfvén radius at maximum, despite increased mass loss, reduces the overall angular momentum-extraction efficiency. Conversely, at minimum, the longer lever arm leads to higher torque even though the mass-loss rate is lower.

Thus, over the solar cycle,

• the highest braking torque is obtained at minimum, driven by the extended lever arm,
• at maximum, the torque drops as the lever arm shortens (with open field lines coupled closer to the rotation axis),
• the spatial application zones of the torque evolve with the migration of open flux.

6. Implications for Solar and Stellar Rotational Evolution

The MHD modeling framework elucidates why wind braking torque is not temporally invariant: instead, it is a dynamically modulated quantity determined by the deep interaction between dynamo-driven field evolution and coronal-wind structure. Even moderate time-dependent rearrangements of coronal topology, as driven by the dynamo, produce large-amplitude modulations in the mean torque—hence, the rotational evolution of the Sun (and by analogy, Sun-like stars) is intricately tied to the star’s magnetic cycle, and not simply to mean values of mass loss and surface rotation.

In the context of solar and stellar spin-down, this complexity necessitates dynamic models that incorporate not just steady wind solutions and mean-field properties, but also self-consistent coronal field evolution, cyclic modulation, and feedback between the surface dynamo and the corona.

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Table: Cycle-Dependent Quantities in Wind Braking Torque

Solar Cycle Phase	Mean Alfvén Radius ($\langle r_A \rangle$)	Mass Loss Rate ($\dot{M}$) ($10^{-14}\, M_\odot$/yr)	Braking Torque ($\tau$)
Activity Minimum	$\sim$9 $R_\odot$	4.2	Maximal (long lever arm)
Activity Maximum	$\sim$2.2 $R_\odot$	6.9	Reduced (short lever arm)

The joint temporal variation of mass-loss rate and lever arm length sets the temporal profile of angular momentum loss.

7. Summary

Wind braking torque in the solar context is a dynamically modulated loss mechanism that depends crucially on the mass–flux weighted lever arm set by the Alfvén surface and the spatiotemporal evolution of open magnetic flux. Cyclic dynamo activity produces major changes in both mass loss and topology, resulting in strong variations in braking torque over the 11-year activity cycle. The feedbacks among the dynamo, the coronal field, and wind dynamics establish a variable angular momentum loss rate, fundamentally governing the Sun’s rotational evolution on both short and long timescales. The physical insights, mathematical formalism, and simulation methodologies presented in this framework provide a benchmark for understanding solar and stellar wind torque variability (Pinto et al., 2011).