Numerical scheme and analytical solutions to the stochastic nonlinear advection diffusion dynamical model

In this study, we give the numerical scheme to the stochastic nonlinear advection diffusion equation. This models is considered with white noise (or random process) having same intensity by changing frequencies. Furthermore, the stability and consistency of proposed scheme are also discussed. Moreover, it is concerned about the analytical solutions, the Riccati equation mapping method is adopted. The different families of single (shock and singular) and mixed (complex solitary-shock, shock-singular, and double-singular) form solutions are obtained with the different choices of free parameters. The graphical behavior of solutions is also depicted in 3D and corresponding contours.

Tropical Cyclone Intensification Simulated in the Ooyama-type Three-layer Model with a Multilevel Boundary Layer

The first successful simulation of tropical cyclone (TC) intensification was achieved with a three-layer model, often named the Ooyama-type three-layer model, which consists of a slab boundary layer and two shallow water layers above. Later studies showed that the use of a slab boundary layer would produce unrealistic boundary layer wind structure and too strong eyewall updraft at the top of TC boundary layer and thus simulate unrealistically rapid intensification compared to the use of a height-parameterized boundary layer. To fully consider the highly height-dependent boundary layer dynamics in the Ooyama-type three-layer model, this study replaced the slab boundary layer with a multilevel boundary layer in the Ooyama-type model and used it to conduct simulations of TC intensification and also compared the simulation with that from the model version with a slab boundary layer. Results show that compared with the simulation with a slab boundary layer, the use of a multilevel boundary layer can greatly improve simulations of the boundary-layer wind structure and the strength and radial location of eyewall updraft, and thus more realistic intensification rate due to better treatments of the surface layer processes and the nonlinear advection terms in the boundary layer. Sensitivity of the simulated TCs to the model configuration and to both horizontal and vertical mixing lengths, sea surface temperature, the Coriolis parameter, and the initial TC vortex structure are also examined. The results demonstrate that this new model can reproduce various sensitivities comparable to those found in previous studies using fully physics models.

Application of SPD-RBF method of lines for solving nonlinear advection–diffusion–reaction equation with variable coefficients

Purpose The purpose of this study is to use the method of lines to solve the two-dimensional nonlinear advection–diffusion–reaction equation with variable coefficients. Design/methodology/approach The strictly positive definite radial basis functions collocation method together with the decomposition of the interpolation matrix is used to turn the problem into a system of nonlinear first-order differential equations. Then a numerical solution of this system is computed by changing in the classical fourth-order Runge–Kutta method as well. Findings Several test problems are provided to confirm the validity and efficiently of the proposed method. Originality/value For the first time, some famous examples are solved by using the proposed high-order technique.

Rotation suppresses giant-scale solar convection

The observational absence of giant convection cells near the Sun’s outer surface is a long-standing conundrum for solar modelers. We herein propose an explanation. Rotation strongly influences the internal dynamics, leading to suppressed convective velocities, enhanced thermal-transport efficiency, and (most significantly) relatively smaller dominant length scales. We specifically predict a characteristic convection length scale of roughly 30-Mm throughout much of the convection zone, implying weak flow amplitudes at 100- to 200-Mm giant cells scales, representative of the total envelope depth. Our reasoning is such that Coriolis forces primarily balance pressure gradients (geostrophy). Background vortex stretching balances baroclinic torques. Both together balance nonlinear advection. Turbulent fluxes convey the excess part of the solar luminosity that radiative diffusion cannot. We show that these four relations determine estimates for the dominant length scales and dynamical amplitudes strictly in terms of known physical quantities. We predict that the dynamical Rossby number for convection is less than unity below the near-surface shear layer, indicating rotational constraint.

Numerical study of 1D and 2D advection-diffusion-reaction equations using Lucas and Fibonacci polynomials

In this work, a numerical scheme based on combined Lucas and Fibonacci polynomials is proposed for one- and two-dimensional nonlinear advection–diffusion–reaction equations. Initially, the given partial differential equation (PDE) reduces to discrete form using finite difference method and $$\theta -$$ θ - weighted scheme. Thereafter, the unknown functions have been approximated by Lucas polynomial while their derivatives by Fibonacci polynomials. With the help of these approximations, the nonlinear PDE transforms into a system of algebraic equations which can be solved easily. Convergence of the method has been investigated theoretically as well as numerically. Performance of the proposed method has been verified with the help of some test problems. Efficiency of the technique is examined in terms of root mean square (RMS), $$L_2$$ L 2 and $$L_\infty $$ L ∞ error norms. The obtained results are then compared with those available in the literature.