Solution of one class of magnetohydrodynamic equations with magnetic field amplification

Solution of magnetohydrodynamic equations applied to solar magnetic fiel

Limitations of the background field method applied to Rayleigh-Bénard convection

We consider Rayleigh-Bénard convection as modeled by the Boussinesq equations, in case of infinite Prandtl number and with no-slip boundary condition. There is a broad interest in bounds of the upwards heat flux, as given by the Nusselt number $Nu$, in terms of the forcing via the imposed temperature difference, as given by the Rayleigh number in the turbulent regime $Ra \gg 1$. In several works, the background field method applied to the temperature field has been used to provide upper bounds on $Nu$ in terms of $Ra$. In these applications, the background field method comes in form of a variational problem where one optimizes a stratified temperature profile subject to a certain stability condition; the method is believed to capture marginal stability of the boundary layer. The best available upper bound via this method is $Nu \lesssim Ra^{1/3}(\ln Ra)^{1/15}$. it proceeds via the construction of a stable temperature background profile that increases logarithmically in the bulk. In this paper, we show that the background temperature field method cannot provide a tighter upper bound in terms of the power of the logarithm. However, by another method one does obtain the tighter upper bound $Nu \lesssim Ra^{1/3}(\ln \ln Ra)^{1/3}$, so that the result of this paper implies that the background temperature field method is unphysical in the sense that it cannot provide the optimal bound

Do quasi-regular structures really exist in the solar photosphere? I. Observational evidence

Two series of solar-granulation images -- the La Palma series of 5 June 1993 and the SOHO MDI series of 17--18 January 1997 -- are analysed both qualitatively and quantitatively. New evidence is presented for the existence of long-lived, quasi-regular structures (first reported by Getling and Brandt (2002)), which no longer appear unusual in images averaged over 1--2-h time intervals. Such structures appear as families of light and dark concentric rings or families of light and dark parallel strips (``ridges'' and ``trenches'' in the brightness distributions). In some cases, rings are combined with radial ``spokes'' and can thus form ``web'' patterns. The characteristic width of a ridge or trench is somewhat larger than the typical size of granules. Running-average movies constructed from the series of images are used to seek such structures. An algorithm is developed to obtain, for automatically selected centres, the radial distributions of the azimuthally averaged intensity, which highlight the concentric-ring patterns. We also present a time-averaged granulation image processed with a software package intended for the detection of geological structures in aerospace images. A technique of running-average-based correlations between the brightness variations at various points of the granular field is developed and indications are found for a dynamical link between the emergence and sinking of hot and cool parcels of the solar plasma. In particular, such a correlation analysis confirms our suggestion that granules -- overheated blobs -- may repeatedly emerge on the solar surface. Based on our study, the critical remarks by Rast (2002) on the original paper by Getling and Brandt (2002) can be dismissed.Comment: 21 page, 8 figures; accepted by "Solar Physics

Instability of small-amplitude convective flows in a rotating layer with stress-free boundaries

We consider stability of steady convective flows in a horizontal layer with stress-free boundaries, heated below and rotating about the vertical axis, in the Boussinesq approximation (the Rayleigh-Benard convection). The flows under consideration are convective rolls or square cells, the latter being asymptotically equal to the sum of two orthogonal rolls of the same wave number k. We assume, that the Rayleigh number R is close to the critical one, R_c(k), for the onset of convective flows of this wave number: R=R_c(k)+epsilon^2; the amplitude of the flows is of the order of epsilon. We show that the flows are always unstable to perturbations, which are a sum of a large-scale mode not involving small scales, and two large-scale modes, modulated by the original rolls rotated by equal small angles in the opposite directions. The maximal growth rate of the instability is of the order of max(epsilon^{8/5},(k-k_c)^2), where k_c is the critical wave number for the onset of convection.Comment: Latex, 12 pp., 15 refs. An improved version of the manuscript submitted to "Mechanics of fluid and gas", 2006 (in Russian; English translation "Fluid Dynamics"