# Detailed mathematical and numerical analysis of a dynamo model

Abstract: We investigate the role of nonlinear feedback by α-quenching, flux losses, and feedback by differential rotations in dynamo. Specifically, by studying the nonlinear dynamo model analytically and numerically, we unfold how frequency p of magnetic field, magnetic field strength |B|, and phase φ are influenced by different types of nonlinear feedback in the limit of a very weak mean and/or fluctuating differential rotation. We find that p and φ are controlled by both flux losses with no influence by α-quenching when there is no back reaction because of fluctuating differential rotation. We find a similar effect of poloidal flux loss and toroidal flux loss on p and |B| in the absence of a back reaction of shear. Their effect becomes totally different with the inclusion of this back reaction. Detailed investigations suggest that toroidal flux loss tends to have more influence than poloidal flux loss (with or without α-quenching) in the presence of fluctuating shear. Furthermore, the effect of α-quenching is boosted when combined with toroidal flux loss, indicating that the dynamic balance of dynamo is optimized in the presence of both α-quenching and flux loss. These results highlight the importance of nonlinear transport coefficients and differential rotation in the regulation of a dynamo.

...read more

##### Citations

4 citations

3 citations

### Cites background from "Detailed mathematical and numerical..."

..., in parameterisations of transport coefficients in MHD simulations)” (Sood and Kim, 2014)....

[...]

3 citations

1 citations

### Cites background or methods or result from "Detailed mathematical and numerical..."

...We propose a dynamical model for the evolution of rotation rate and magnetic field in spindown by extending a previous nonlinear dynamo model (Sood & Kim, 2013, 2014; Weiss et al. 1984)....

[...]

...In particular, Sood & Kim (2013, 2014) incorporated various nonlinear transport coefficients such as α-quenching and flux losses and took the control parameter D known as the dynamo number to scale with rotation rate as D ∝ Ω2....

[...]

...The parameters ν, ν0, κ and λ1,2 are much the same as in our previous work (Sood & Kim 2013, 2014)....

[...]

...We note that Sood & Kim (2013, 2014) have demonstrated that nonlinear feedback plays a vital role in the generation and destruction of magnetic fields as well as self-regulation of the dynamo....

[...]

...To this end, we evolve the stellar rotation and magnetic field simultaneously over the stellar evolution time by extending our previous work (Sood & Kim 2013, 2014) which incorporates the nonlinear feedback mechanisms on rotation and magnetic fields via α-quenching and magnetic flux losses as well…...

[...]

##### References

1,764 citations

### "Detailed mathematical and numerical..." refers background in this paper

...Based upon the traditional α − Ω dynamo (Parker 1955; Moffat 1978; Parker 1979), the dynamo number in this work, which controls the efficiency of dynamo action, is proportional to the product of α and Ω effects, which generate a toroidal field from a poloidal field by cyclonic convection (α effect)…...

[...]

1,518 citations

1,451 citations

### "Detailed mathematical and numerical..." refers methods in this paper

...It will also be of great interest to utilize our model to understand the spin down of solar type stars (Leprovost & Kim 2010; Kepens et al. 1995) by using a magnetic field that is consistently obtained from a dynamo model (work in progress)....

[...]

937 citations

### "Detailed mathematical and numerical..." refers background in this paper

...…+ κ1|B|2, F2 = 1 + λ1|B|2, F3 = 1 + λ2|B|2, where κ1, λ1, and λ2 are constant parameters; F1 represents the α source term (i.e., helicity by the magnetic field) due to the back reaction by magnetic field that was studied in previous works including Pouquet et al. (1976) and Kleeorin et al. (1982)....

[...]

##### Related Papers (5)

[...]