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Weakly nonlinear stability analysis of MHD channel flow using an efficient numerical approach

We analyze weakly nonlinear stability of a flow of viscous conducting liquid driven by pressure gradient in the channel between two parallel walls subject to a transverse magnetic field. Using a non-standard numerical approach, we compute the linear growth rate correction and the first Landau coefficient, which in a sufficiently strong magnetic field vary with the Hartmann number as $μ_{1}\sim(0.814-\mathrm{i}19.8)\times10^{-3}\textit{Ha}$ and $μ_{2}\sim(2.73-\mathrm{i}1.50)\times10^{-5}\textit{Ha}^{-4}$. These coefficients describe a subcritical transverse velocity perturbation with the equilibrium amplitude $|A|^{2}=\Re[μ_{1}]/\Re[μ_{2}](\textit{Re}_{c}-\textit{Re})\sim29.8\textit{Ha}^{5}(\textit{Re}_{c}-\textit{Re})$ which exists at Reynolds numbers below the linear stability threshold $\textit{Re}_{c}\sim 4.83\times10^{4}\textit{Ha}.$ We find that the flow remains subcritically unstable regardless of the magnetic field strength. Our method for computing Landau coefficients differs from the standard one by the application of the solvability condition to the discretized rather than continuous problem. This allows us to bypass both the solution of the adjoint problem and the subsequent evaluation of the integrals defining the inner products, which results in a significant simplification of the method.