Magnetohydrodynamic turbulent flows for viscous incompressible fluids through the lenses of harmonic analysis (original) (raw)
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Recent numerical and observational studies contain conflicting reports on the spectrum of magnetohydrodynamic turbulence. In an attempt to clarify the issue we investigate anisotropic incompressible magnetohydrodynamic turbulence with a strong guide field B 0. We perform numerical simulations of the reduced MHD equations in a special setting that allows us to elucidate the transition between weak and strong turbulent regimes. Denote k , k ⊥ characteristic field-parallel and field-perpendicular wavenumbers of the fluctuations, and b λ the fluctuating field at the scale λ ∼ 1/k ⊥. We find that when the critical balance condition, k B 0 ∼ k ⊥ b λ , is satisfied, the turbulence is strong, and the energy spectrum is E(k ⊥) ∝ k −3/2 ⊥. As the k width of the spectrum increases, the turbulence rapidly becomes weaker, and in the limit k B 0 ≫ k ⊥ b λ , the spectrum approaches E(k ⊥) ∝ k −2 ⊥. The observed sensitivity of the spectrum to the balance of linear and nonlinear interactions may explain the conflicting numerical and observational findings where this balance condition is not well controlled.
Journal of Evolution Equations, 2017
We investigate the well posedness of the stochastic Navier-Stokes equations for viscous, compressible, non-isentropic fluids. The global existence of finite-energy weak martingale solutions for large initial data within a bounded domain of R d is established under the condition that the adiabatic exponent γ > d/2. The flow is driven by a stochastic forcing of multiplicative type, white in time and colored in space. This work extends recent results on the isentropic case, the main contribution being to address the issues which arise from coupling with the temperature equation. The notion of solution and corresponding compactness analysis can be viewed as a stochastic counterpart to the work of Feireisl (Dynamics of viscous compressible fluids, vol 26. Oxford University Press, Oxford, 2004). By including the noise, system (1.1) becomes a model for a turbulent compressible flow, with variable density and temperature. The mathematical study of the compressible stochastic Navier-Stokes equations has seen several recent developments. In the barotropic case, the first result is due to [8], which used a deterministic approach in the case where the coefficients σ k are independent of the fluid variables. In this setting, one can make a convenient change of variables which turns the SPDE into a random PDE. In a more general setting, a stochastic approach is required in order to give a meaning to the noise. Three sets of authors [3,14,17] studied independently the case of more general noise coefficients, establishing the existence of global martingale solutions to (1.1) in the barotropic regime. It should be noted that the article [3] of Breit/Hofmanova was the first to appear on ArXiv, while the article [17] of D. Wang/H. Wang was the first to appear in print. The main focus of the present article is to extend the results of [3,14,17] to the more general setting of (1.1), which allows for changes in the temperature of the fluid. We use a combination of the arguments in [7,14] to treat the stochastic and deterministic sides of the problem, respectively. In particular, we follow the work of E. Feireisl [7] for the notion of solution to the temperature equation. We also adapt various compactness arguments from [7], modified appropriately to the frame of arguments in [14] for building weak martingale solutions. We now turn to a more precise statement of our results.
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Acta Applicandae Mathematicae, 1995
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