On the use of combined compact scheme for numerical solution of the two-layer oceanic shallow water equations

Many types of oceanic physical phenomena have a wide range in both space and time. In general, simplified models, such as shallow water model, are used to describe these oceanic motions. The shallow water equations are widely applied in various oceanic and atmospheric extents. By using the two-layer shallow water equations, the stratification effects can be considered too. In this research, the sixth-order combined compact method is investigated and numerically implemented as a high-order method to solve the two-layer shallow water equations. The second-order centered, fourth-order compact and sixth-order super compact finite difference methods are also used to spatial differencing of the equations. The first part of the present work is devoted to accuracy assessment of the sixth-order super compact finite difference method (SCFDM) and the sixth-order combined compact finite difference method (CCFDM) for spatial differencing of the linearized two-layer shallow water equations on the Arakawa's A-E and Randall's Z numerical grids. Two general discrete dispersion relations on different numerical grids, for inertia-gravity and Rossby waves, are derived. These general relations can be used for evaluation of the performance of any desired numerical scheme. For both inertia-gravity and Rossby waves, minimum error generally occurs on Z grid using either the sixth-order SCFDM or CCFDM methods. For the Randall's Z grid, the sixth-order CCFDM exhibits a substantial improvement , for the frequency of the barotropic and baroclinic modes of the linear inertia-gravity waves of the two layer shallow water model, over the sixth-order SCFDM. For the Rossby waves, the sixth-order SCFDM shows improvement, for the barotropic and baroclinic modes, over the sixth-order CCFDM method except on Arakawa's C grid. In the second part of the present work, the sixth-order CCFDM method is used to solve the one-layer and two-layer shallow water equations in their nonlinear form. In one-layer model with periodic boundaries, the performance of the methods for mass conservation is compared. The results show high accuracy of the sixth-order CCFDM method to simulate a complex flow field. Furthermore, to evaluate the performance of the method in a non-periodic domain the sixth-order CCFDM is applied to spatial differencing of vorticity-divergence-mass representation of one-layer shallow water equations to solve a wind-driven current problem with no-slip boundary conditions. The results show good agreement with published works. Finally, the performance of different schemes for spatial differencing of two-layer shallow water equations on Z grid with periodic boundaries is investigated. Results illustrate the high accuracy of combined compact method.

On the sound waves propagation in the Persian Gulf

In this research I focused on the propagation of acoustic rays in shallow water areas then I selected the Persian Gulf and described sound transmission in this region with emphasize on physical properties of water masses and of sediments. Finally I studied on the sound speed variations and sound attention with data collected from this area (NE of Farsi Island & 50 kilometers south of Delware). Sound speed deviation in western part of Strait of Hormuz in winter is between 20-30 m/s and it is between 5-20 m/s in the Oman Sea. Minimum sound speed deviation is at 23-24 degree north & 60-62 degree east. In spring, this deviation varies from 25-35 m/s, which is greater than in winter. In winter, at east of 56 degree east, greater speed are in shallow water coastal areas. In summer, sound speeds are greater than in spring and vary from 35 to 55 m/s at western part of Strait of Hormuz and 20 to 40 m/s in Oman Sea. Finally in autumn, sound speed deviation is 30-45 m/s west of 56 degree east and in Oman Sea is the same. The greatest attenuation rate caused by absorption in Bandar Dayer is between 17 to 27 meters depth, which is from water masses with different densities.