Hopf bifurcations in plasma layers between rigid planes in thermal magnetohydrodynamics, via a simple formula
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CLASSICAL AND QUANTUM PLASMAS
Hopf bifurcations in plasma layers between rigid planes in thermal magnetohydrodynamics, via a simple formula Salvatore Rionero1,2 Received: 18 March 2020 / Accepted: 10 September 2020 / Published online: 16 October 2020 © The Author(s) 2020
Abstract The phenomenon produced by the Hopf bifurcations is of notable importance. In fact, a Hopf bifurcation—guaranteeing the existence of an unsteady periodic solution of the linearized problem at stake—is also an optimum limit cycle candidate of the nonlinear associated problem and, if non linearly globally attractive, is an absorbing set and an effective limit cycle. The present paper deals with the onset of Hopf bifurcations in thermal magnetohydrodynamics (MHD). Precisely, it is devoted to characterization—via a simple formula—of the Hopf bifurcations threshold in horizontal plasma layers between rigid planes, heated from below and embedded in a constant transverse magnetic field. This problem, remarked clearly and notably by the Nobel Laureate Chandrasekhar (Nature 175:417–419, 1955), constitutes a difficulty met by him and—for plasma layers between rigid planes electricity perfectly conducting—is, as far as we know, still not removed. Let m0 be the thermal conduction rest state and let Pr , Pm , R, Q , be the Prandtl, the Prandtl magnetic, the Rayleigh and the Chandrasekhar number, respectively. Recognized (according to Chandrasekhar) that the instability of m0 via Hopf bifurcation can occur only in a plasma with Pm > Pr , in this paper it is shown that the Hopf bifurcations occur if and only if
Q > Qc =
4𝜋 2 [1 + Pr (𝜇∕2𝜋)4 ] , Pm − Pr
with 𝜇 = 7.8532 . Moreover, the critical value of R at which the Hopf bifurcation occurs is characterized via the smallest zero of the second invariant of the spectrum equation governing the most destabilizing perturbation. The critical value of Q, 1 in the free-rigid and rigid-free cases is shown to be of the previous value. 4 Keywords Thermal MHD · Instability · Hopf bifurcations · Plasma layers between rigid planes
1 Introduction
* Salvatore Rionero [email protected] 1
Department of Mathematics and Applications ‘R. Caccioppoli’, Complesso Universitario Monte S. Angelo, University of Naples Federico II, Via Cinzia, 80126 Naples, Italy
Accademia Nazionale dei Lincei, via della Lungara 10, Rome, Italy
2
Let L be a horizontal layer heated from below, filled by a plasma and embedded in a transverse uniform magnetic field and let m0 denote the thermal conduction in L: the temperature field arising when the fluid is in rest and L is heated from below by a constant transverse gradient of temperature. The linear stability of m0—in the scheme of the nonrelativistic thermal MHD—has been analyzed by Chandrasekhar in the early of 1950 (Chandrasekhar 1952, 1954) and it appeared in 1961 in the celebrated monograph (Chandrasekhar 1981). The Chandrasekhar results—for any type of boundaries (rigid–rigid, rigid–free, free–rigid, free–free)—have been since then a basic paradigm for all the subsequent resear
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