Data-driven discovery of dynamo cycle equations

Abstract

Many low-mass stars like the Sun host periodic, oscillatory magnetic fields that lead to variable levels of stellar activity, driving space weather that affects the habitability and detection of exoplanets. Owing to the intrinsic difficulty in modeling stellar magnetohydrodynamics across scales, realistic numerical simulations of this process are very challenging, and developing reduced-order models is of interest. In this work, we develop a framework to recover such models directly from numerical data by using a combination of Dynamic Mode Decomposition (DMD) to identify coherent magnetic structures, and the Sparse Identification of Nonlinear Dynamics (SINDy) framework to model their dynamics. We compare these models to those obtained using the classic mathematical method of weakly nonlinear (WNL) analysis. This approach is implemented on a one-dimensional mean-field dynamo model that parameterizes the main components of a convective dynamo in a low-mass star -- helical convection and differential rotation. We recover oscillatory dynamo models as a function of the dynamo strength parameter $D\sim αΩ'$, magnetic dissipation parameter $κ$, and a comprehensive dynamo model that predicts the magnetic state for a combination of these two parameters. Our results suggest that equations discovered with SINDy are more robust than equations from WNL analysis, and can predict the saturation amplitude of magnetic fields in parameter regimes far from the onset of dynamo action characterized by stiff nonlinearities. This includes unstable, and typically unknown, subcritical branches. Further to this, SINDy is able to find equations in parameter regimes where the nonlinearity is not analytic and WNL analysis cannot be applied. These properties of data-driven SINDy models suggest them as a viable alternative for modeling of stellar dynamo cycles directly from the data.

Figures (5)

• Figure 1: Dynamo topology at the onset of instability, as a function of latitude $x$ and time $t$. (a) Magnetic potential $A$; (b) Toroidal field $B$. Subcritical dynamo with $\kappa_1=0.4$, $\kappa_2 = 0.005$, $D=262.725$ ($\epsilon = 0.1$). (c) Real and imaginary part of the eigenvector (solid grey); real and imaginary part of the first DMD mode in the steady state. Red dashed line, supercritical case; blue dotted line, subcritical case. During the linear regime, DMD modes are identical to the eigenmodes. (d) DMD spectrum of the dynamo for supercritical case (in blue) and subcritical case (in red). Full symbols, DMD of the linear dynamo phase; empty symbols, DMD of saturated state.
• Figure 2: Temporal coefficient of the leading DMD mode calculated with adjoint DMD. (a) DMD on the linear evolution phase, $t<50$, $r=5$; (b) DMD on steady-state data, $t>270$, $r=9$. Blue, subcritical dynamo bifurcation; red, supercritical dynamo bifurcation. $\epsilon = 0.1$; $D = 262.725$.
• Figure 3: (a) Comparison of eigenvector amplitude $H$, DMD modal coefficient $c_{DMD}^G$, third-order and fifth-order WNL models for subcritical case: $\kappa_1 =0.4$, $\kappa_2 = 0.005$. (b) Relative mean squared error of the solution as a function of SINDy cut-off threshold $\Theta_{cr}$ along the entire trajectory in figure \ref{['fig:adj_coef']} for supercritical case (in blue) and subcritical case (in red). (c) Temporal integration of the SINDy models (dashed), compared to the data (black): (c) supercritical equations \ref{['eq:sup_wnl_sindy']}, $\kappa=0$; (d) subcritical equations \ref{['eq:sub_wnl5_sindy']}, $\kappa_1 =0.4$, $\kappa_2 = 0.005$. $D=262.7$ ($\epsilon=0.1$) for all panels.
• Figure 4: (a) Temporal evolution supercritical dynamo ($\kappa=0$) given by models \ref{['eq:sup_varD']} and \ref{['eq:sup_cart_varD']} with $D=265$. Solid lines correspond to the data, dotted lines to polar model \ref{['eq:sup_varD']}, dashed to Cartesian model \ref{['eq:sup_cart_varD']}. Radius is denoted in red and phase in blue. (b) Magnitude of the dynamo cycle, normalized with the magnitude at $D=265$. Circles, DMD coefficients from the data; dash-dotted line, third order WNL \ref{['eq:H_normal_form']}; dashed, fifth order WNL \ref{['eq:wnl_H_fifth']}; solid line, third order SINDy model \ref{['eq:sup_cart_varD']}. (c) Same as (a) but for subcritical bifurcation, $\kappa_1 = 0.4$, $\kappa_2 = 0.005$. (d) Same as (b) but for subcritical bifurcation. Circles, DMD coefficients from the data; dashed, fifth order WNL model \ref{['eq:wnl_H_fifth']}; solid line, fifth order SINDy model \ref{['eq:sub_cart_varD']}.
• Figure 5: (a) Contributions to $\chi$, as a function of $\kappa$ at third order. Vertical dotted line denotes $\kappa=0.0017$, where the dynamo becomes subcritical. The data taken at $D=270$ ($\epsilon=0.2$, the highest value in the training range). (b) Bifurcation diagram of the large full dynamical model of the system as a function of $D$ and $\kappa$. Black solid lines, WNL model \ref{['eq:wnl_H_fifth']}, in colour, SINDy model \ref{['eq:sindy_varkappa_varD']}. Filled circles, amplitudes of the DMD coefficients; empty circles, solution branches obtained with Newton-Krylov method. In blue, supercritical bifurcation, in red, subcritical bifurcation. Increasing color intencity: $\kappa_1 = [0,0.2,0.34,0.4, 0.44]$, $\kappa_2 = 0.005$. Dashed lines, unstable dynamo branches.