The key conclusions from my simulations on $\alpha^2-\omega$ dynamos are:

Results:

• The parameter space of $R_\omega\in[-20, 0]$ and $R_\alpha \in [0, 14.5]$ is roughly divided into three regions as $R_\alpha$ increases, with the oscillatory solutions lying in the regions corresponding to small $|R_{\omega}|$ and large $|R_\omega|$. (Ref figure (1))

  The results indicate that, for large values of $R_\alpha$, the range of values of $R_\omega$ for which the galactic mean fields are oscillatory increases. This implies that as turbulence increases, the fields will oscillate for a broader range of sheer values. Oscillatory mean fields could introduce variations in the gas density and star formation rates of the galaxy, and hence through observational constraints on $R_\omega$ and $R_\alpha$, we can predict these features using galactic dynamo theory.

• The oscillation frequencies are high for high values of $R_\alpha$ and $R_\omega$, and are also high near the parameter coordinate $(R_\omega, R_\alpha) = (-0.5, 8.0)$ (Ref figure (7))

  High frequency oscillations in mean fields would affect galaxy evolution at short timescales, hence, to study the long term effects of mean fields on galaxies, we would be interested in low frequency solutions to the mean field dynamo equation. This would narrow our parameter space of $R_\alpha$ and $R_\omega$, and help us constrain the properties of galaxies, especially the ones that are old.

Limitations:

• One limitation of my analysis is the inability of the FFT algorithm to capture really low oscillations. Since we would be interest in these slowly varying solutions to understand long term evolution of galaxies, my code is best useful for studying galaxy evolution for a few diffusion timescales.

Possible Improvements:

• One possible improvement to the results would be to incorporate the unresolvable region of the parameter space in figure (1), by improving the code to allow the evaluation of very large values of $B_{tot}$. This region could change the conclusions if we find a significant portion of it to be non-oscillatory.
• Another improvement would be to simulate the system for longer times. My current code takes 8 hours to do the computations for $40 \times 40 = 1600$ systems, but we can run it for much longer. But, this would not be helpful unless we first work on making sure that the code can resolve large values.

In conclusion, studying the $R_\alpha$ and $R_\omega$ parameter space is useful to understand the large scale properties of a galaxy. Oscillatory mean fields can affect the long term evolution of galaxies, and modelling the galacitic dynamo as an $\alpha^2 - \omega$ allows us to understand the effect of turbulence and sheer forces on galactic mean field evolution.

I am very grateful to my instructor Dr. Luke Chamandy for assigning me this project, I learned a lot!