Bifurcation analysis of a two-dimensional magnetic Rayleigh-Bénard problem

TL;DR

The paper develops and applies deflated continuation to a 2D magnetic Rayleigh–Bénard system to map how a vertical magnetic field, quantified by the Chandrasekhar number $Q$, reshapes the bifurcation landscape as the Rayleigh number $ ext{Ra}$ grows. By combining deflation with continuation in $Q$ and carefully constructing initial guesses, the authors compute a rich network of steady states at $Q=10^3$ and $ ext{Ra} eq 0$, uncovering numerous pitchfork, Hopf, and saddle-node bifurcations and tracing how magnetic coupling delays instabilities and biases patterns aligned with the background field. The study shows that some branches stabilize at high $Q$, and that the magnetic field can discriminate among solution profiles, with implications for pattern control in MHD convection. The methods, including a structure-preserving finite-element discretization and an augmented Lagrangian formulation, provide a robust framework for multiparameter bifurcation analysis in magnetohydrodynamic systems and suggest avenues for extension to more realistic parameter regimes and three-dimensional settings.

Abstract

We perform a bifurcation analysis of a two-dimensional magnetic Rayleigh--Bénard problem using a numerical technique called deflated continuation. Our aim is to study the influence of the magnetic field on the bifurcation diagram as the Chandrasekhar number $Q$ increases and compare it to the standard (non-magnetic) Rayleigh--Bénard problem. We compute steady states at a high Chandrasekhar number of $Q=10^3$ over a range of the Rayleigh number $0\leq \Ra\leq 10^5$. These solutions are obtained by combining deflation with a continuation of steady states at low Chandrasekhar number, which allows us to explore the influence of the strength of the magnetic field as $Q$ increases from low coupling, where the magnetic effect is almost negligible, to strong coupling at $Q=10^3$. We discover a large profusion of states with rich dynamics and observe a complex bifurcation structure with several pitchfork, Hopf, and saddle-node bifurcations. Our numerical simulations show that the onset of bifurcations in the problem is delayed when $Q$ increases, while solutions with fluid velocity patterns aligning with the background vertical magnetic field are privileged. Additionally, we report a branch of states that stabilizes at high magnetic coupling, suggesting that one may take advantage of the magnetic field to discriminate solutions.

Figures (8)

• Figure 1: Bifurcation diagrams of the magnetic Rayleigh--Bénard problem using the kinetic energy $\|\mathbf{u}\|_2^2$ (top row), potential energy $\|T\|_2^2$ (middle row), and magnetic energy $\|\mathbf{B}\|_2^2$ (bottom row) as diagnostic. The diagrams display the evolution of the branches studied in this paper, i.e., the first four primary instabilities at $Q=10^3$ and their secondary bifurcations, in the range $0\leq \mathrm{Ra}\leq 10^5$ at $Q=1$ (a), $1\leq Q\leq 10^3$ at $\mathrm{Ra}=10^5$ (b), and $0\leq \mathrm{Ra}\leq 10^5$ at $Q=10^3$ (c). The first four primary branches for the bifurcation diagrams in (c) are displayed with solid lines while the secondary branches are indicated with dashed lines. The dark line in the diagrams using the potential energy corresponds to the conducting state satisfying $\|T_0\|_2^2=1/3$. The $x$-axes of the diagrams in panel (c) are reversed to analyze the evolution of the branches as the strength of the magnetic field increases.
• Figure 2: Growth rates of the conducting state in different parameter regimes as a function of the Chandrasekhar number $Q$. The growth rates are computed using the leading eigenvalues of the linearized system around the conducting state (see \ref{['app:unstableEigenmodes']}). In panel (b), the red lines indicate that the imaginary part of the eigenvalues is non-zero and grows as $Q$ increases, eventually leading to Hopf bifurcations when the corresponding growth rates vanish.
• Figure 3: (a-c) Evolution of the positive growth rates of the conducting state \ref{['eq:trivialsol']} as the Rayleigh and Chandrasekhar numbers vary in the intervals $0\leq \mathrm{Ra} \leq 10^5$ and $1\leq Q\leq 10^3$. (d) First ten eigenmodes of the primary bifurcations that emanate from the conducting state in the range $0\leq \mathrm{Ra} \leq 10^5$ at $Q=1$. The plots display the magnitude of the velocity, where blue colors indicate the zero-velocity magnitude and red colors correspond to a high magnitude. (e) Eigenmodes associated with the first five primary bifurcations that emanate from the conducting state in the range $0\leq \mathrm{Ra} \leq 10^5$ at $Q=10^3$. Note that, as before, the x-axis is reversed in (c).
• Figure 4: Evolution of the steady states in the first primary branch (1) with respect to the Rayleigh number, where the Chandrasekhar is fixed to $Q=10^3$, illustrated via the kinetic energy (a), potential energy (b), and magnetic energy (c). The plots in these panels display the magnitude of the velocity, temperature, and magnitude of the magnetic field. The color schemes are defined as follows. For the velocity and magnetic fields, blue corresponds to zero while red stands for high magnitude (the color maps are rescaled to the data range). For the temperature, blue corresponds to $T=0$ and red to $T=1$. The corresponding largest growth rates are reported in (g). (d-f) Same as (a-c) but the evolution is for the steady states in the secondary branch (6), with the largest growth rates plotted in (h). The green dots and red triangles indicate the presence of pitchfork bifurcations (eigenvalue $\lambda=0$) and Hopf bifurcations (pair of complex conjugate purely imaginary eigenvalues).
• Figure 5: Evolution of the steady states in the second primary branch (2) as $Q$ increases at $\mathrm{Ra}=10^5$, illustrated via the kinetic energy (a), potential energy (b), and magnetic energy (c). (d-f) Same as (a-c) but the evolution is displayed over $\mathrm{Ra}$ at $Q=10^3$. Panel (g) shows the largest growth rates (real parts of the eigenvalues) corresponding to the states in (a-c), while (h) reports the largest growth rates over $\mathrm{Ra}$ at $Q=10^3$, i.e., states in (d-f).