In thermal convection at high Rayleigh numbers, in the hard turbulent regime, a large scale flow is present. When the viscosity of the fluid strongly depends on temperature, the top–bottom symmetry is broken. In addition to the asymmetric temperature profile across the convection cell, the velocity profiles near the plate boundaries show dramatic difference from the symmetric case. We report here that the second derivative of the velocity profiles are of opposite signs in the thermal sublayers, through measurements derived from the power spectrum of temperature time-series. As a result, the stress rate applied at the plates is maintained constant within a factor of 3, while the viscosity changes by a factor of 53, in qualitative agreement with previous theory.

In this paper, we study the numerical methods for solving the time-fractional Schrödinger equation (TFSE) with Caputo or Riemann–Liouville fractional derivative. The numerical schemes are implemented by using the L1 scheme in time direction and Fourier–Galerkin/Legendre-Galerkin spectral methods in spatial variable. We prove that the two schemes are unconditionally stable and numerical solutions converge with the order $\mathcal{O}( \Delta t^{2-\alpha }+N^{-s}+ N^{-m})$O(Δt2−α+N−s+N−m) , where α is the order of the fractional derivative, Δt, N are the step of time and polynomial degree, respectively, m, s are the regularity of u and V. Several numerical results are performed to confirm the theoretical analysis.

We consider the numerical approximation for a time fractional Schrödinger equation whose solution exhibits an initial weak singularity. A fully discrete scheme is constructed using $L1$ scheme on graded mesh for the discretiaztion of temporal Caputo derivative and spectral method for spatial discretization. It is shown that with appropriate choice of the grading parameter, the scheme can attain order $2-\alpha$ convergence in temporal direction, where α $(0<\alpha <1)$ is the order of time Caputo fractional derivative, and spectral accuracy in spatial direction if the solution is sufficiently smooth in its spatial part. Numerical results confirm the sharpness of the error analysis.