tokens: leads to a strong back-reaction of the Lorentz force, which is expressed asThis force triggers latitudinal flows, and in the end, the wind speed is compressed to negative flows because of it, which leads to accretion at the poles. Appendix D Computations for Helicity D.1. Computation of <italic>A</italic><sub><italic>p</italic></sub> with Various Boundary Conditions D.1.1. General Principle The relative magnetic helicity, which is gauge-invariant, is defined aswhere B p = ∇ × A p , but also B p is potential (∇ × B p = 0) and a reference component from B ( B p · n ∣ S = B · n ∣ S ). Thus, we want to find a potential magnetic field B p derived from a scalar magnetic potential φ ( B p = −∇ φ ) that we can project on spherical harmonics: Because it is potential, B p is also poloidal; thus, we can introduce a quantity C p , such as Thus, to obtain B p and A p , we only need to compute C p , which can be linked to φ using the following relation: Under the assumption of axisymmetry, we have, in the end, The computation of and depends on the boundary conditions of the domain. D.1.2. Potential Components with Perfect Conductor and Potential Field For the dynamo, we have a perfect conductor boundary condition at the bottom, which leads to At the top, we have a potential field boundary condition, which implieswith . We combine these two equations to obtain, in the end, D.1.3. Potential Components with Potential Field and Outflow Boundaries For the wind, we have a potential field boundary at the bottom, which leads towith . At the top, we have an outflow boundary condition, which is expressed using a modified outflow boundary condition with ∂ r ( B r r 2 ) = 0 instead of ∂ r B r = 0, which leads to With Equation ( D2 ), we obtain We combine this condition with Equation ( D10 ) to obtain, in the end, D.2. Computation of <bold><italic>A</italic></bold> We have a magnetic field B that derives from a vector potential A such that B = ∇ × A . Knowing B , we want to recover A in 2.5D. To do so, we decompose the magnetic field B in a poloidal and a toroidal term in spherical coordinates: To obtain C , we can use direct integration, given that To obtain A , we use two different methods: direct integration for outside the star and projection on spherical harmonics for inside the star. With direct integration, we have We can project A onto spherical harmonics, which leads to Now, if we take the r component of the projection of the current on spherical harmonics, we obtain We can then link A and C to A using the following relation: D.3. Balance for Helicity Fluxes To make sure that all of the helicity computations were carried out correctly, we have checked the balance described by Equation ( 21 ), which we rewrite here in the formwhere Γ d is the diffusion term due to the electric currentsand F H is the helicity flux The results are presented in Figure 19 . Case DW1 is on the left, and case DW2 is on the right. The top panel is the budget for the dynamo domain,
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