A numerical investigation of the barotropic instability on the equatorial beta-plane

作者：Kacimi Abderrahim^*; Khouider Boualem

来源：Theoretical and Computational Fluid Dynamics, 2013, 27(3-4): 491-512.

DOI：10.1007/s00162-012-0260-3

• Arakawa Jacobian
• Essentially non-oscillatory schemes
• Equatorial Rossby waves
• Atmospheric circulation
• Barotropic instability
• Vorticity
• Shear flow
• Stream function

The barotropic instability of horizontal shear flows is investigated by using two numerical algorithms to solve the equatorial beta-plane barotropic equations. The first is the Arakawa Jacobian method (Arakawa, in J Comput Phys 1:119-143, 1966), which is a second-order-centered finite differences scheme that conserves energy and enstrophy, and the second is the fourth-order essentially non-oscillatory scheme for non-linear PDE%26apos;s of Osher and Shu (SIAM J Numer Anal 28:907-922, 1991), which is designed to track sharp fronts. We are interested in the performance of these two methods in tracking the long-time behavior of the instability, under the influence of the non-linearity, in the simple case of a Helmholtz shear layer. The associated linear problem is solved analytically, and the linear solution is used as an initial condition for the numerical simulations. We run a series of numerical simulations using both methods with various grid refinements and with two different amplitudes of the initial perturbation. A small viscosity term is added to the vorticity equation to damp the grid-scale waves for Arakawa%26apos;s method. This is not necessary for the high-order ENO-4 scheme, which has its own grid-scale dissipation. At high resolution, the two methods are in good agreement; they yield qualitatively and quantitatively the same solution in the long run: for small disturbances, the total flow stabilizes into a steady-state meridional shear with a smooth profile near the equator, while strong disturbances merge together to form a single large-scale vortex that propagates westward, along the equator, consistent with the African easterly waves and the monsoons trough circulation. At coarse resolution, however, Arakawa%26apos;s method seems to be much superior to the fourth-order ENO-4 scheme as it provides solutions that are more consistent with the fine resolution one.

• 出版日期2013-6