Erratum: “Nonlinear instability of nonhomogeneous thermal structures” [Phys. Plasmas 4, 618 (1997)]

Nonlinear time evolution of thermal structures

“…These asymptotic cases were considered in previous analytical works. [23][24][25][26] According to inequality ͑10͒, the above change of conductive regime occurs for high values of the angle and for low values of the temperature. Furthermore, in the range 0рр/2, () is an increasing or decreasing function of according Ͻ⑀ * 1/5 or Ͼ⑀ * 1/5 , respectively.…”

“…Additionally, the scope of the linear approximation and the pattern that would develop just at the onset of the nonlinear regime were also studied. In particular, it was found [23][24][25][26] that ͑1͒ the response of a thermal structure in a steady state to an arbitrary disturbance depends on the sign of the perturbation ͑i.e., if the disturbance tends to increase or decrease the initial temperature͒ and on the amplitude of the disturbance; and ͑2͒ there are threshold values for the amplitude of the disturbance, beyond which a first-order stable thermal equilibrium solution of the heat transport equation destabilizes. The above results have been obtained for states close enough to the marginal state of any thermal structure initially in isothermal equilibrium ͑the trivial solution of the steady state equation͒.…”

“…[1][2][3][4][5][6][7][8][9][10][11][12][13][14] In contrast, the nonlinear regime is not well understood ͑even when only the bare thermal effects are considered͒ and much of the progress in this area has been done from numerical simulations of particular structures. [15][16][17][18][19][20][21][22] In previous works [23][24][25][26] the stability of thermal structures initially in thermal equilibrium was carried out up to the second order, and explicit conditions for asymptotic and supercritical stability, as well as for superexponential and subcritical instability, were obtained. The method proposed by to study turbulence was used, and the main interest was focused on the onset of the nonlinear regime rather than to follow the evolution of the structure in a welladvanced nonlinear stage.…”

Propagation of sound and thermal waves in an ionizing-recombining hydrogen plasma: Revision of results

“…An approximate method for studying such problems involves the use of linear analysis, in which a perturbed equilibrium state is assumed and the equations of gas dynamics are solved by neglecting second-and higher-order terms. [12][13][14][15][16] The propagation of sound and thermal waves in a conducting, ionizing-recombining medium has been studied in the linear approximation by Ibáñez and Mendoza. In the past four decades, this method has been applied widely to study the stability of astrophysical flows under varied assumptions and physical effects, including thermal conduction 1 along with prescribed heat-loss functions, 2,3 chemical reactions, 4 -7 radiative transfer, 8,9 and heating by an external radiation field.…”

Nonlinear thermal instability in two dimensions

“…In previous works, [20][21][22][23] the thermal stability analysis up to the second order was carried out for the different steady state solutions of the heat balance equation, taking into account heat diffusion, heating and cooling. Explicit conditions for asymptotic and supercritical stability, as well as for superexponential and subcritical instability, were obtained by applying the successive approximation method proposed by Landau 24,25 to study the onset of the turbulence.…”