Numerical solution of the geostrophic mesoscale eddy in the shallow water model

Geostrophic currents profoundly influence various oceanographic events and different interactive processes between the atmosphere and the ocean. According to satellite images, eddies of Persian Gulf with an average diameter of 40-90 km and a speed of 3-6 cm/s, have been observed mostly near the coasts of Bushehr and the central part of the Persian Gulf. Hence, by the purpose of numerical solution, a geostrophic eddy current is dissolved in one layer fluid model called the shallow water model.

The present study uses numerical solutions for geostrophic equations provided by the Woods Hole Oceanographic Institution, entitled "Coriolis Guide" by James F. Price and is available in four separate sections for free research studies. It can cover basic equations based on the Coriolis force (such as the geostrophic equations). Using dynamic realities such as ocean shallowness, hydrostatic approximation, and the relatively constant nature of the vertical structure of the ocean, a geostrophic eddy is solved in MATLAB software based on the geostrophic equations. The origin of the coordinate system is on the center of the eddy and the results are plotted along the diameter of the eddy. The Coriolis parameter, f, is constant (f-plane coordinate system) and the friction is ignored. In this numerical solution, the finite difference numerical method has been used and the model development has been done on a structured grid and regular rectangular shape. The studies have been performed at latitude 28˚N (similar to the Persian Gulf) for a geostrophic eddy with a width of 50 km and a peak height of 3 m relative to its sides. In this numerical solution, the eddy properties of the Persian Gulf are tested as much as possible. The simulation is performed as a dense layer of fluid with a rectangular peak for 10 days from the initial time of zero and from a standstill (initial velocity of zero in the center of the peak). The energy source for the eddy activity is provided by the potential energy stored in the initial stationary state of the fluid package. Therefore, the total energy is stored in this shallow water model.

The Rossby deformation radius was 28.5 km and eddy velocity was 0.029 m/s, which was in good agreement with the satellite images of previous studies. The results show that due to the approximation of the geostrophic current, the resulting eddy shape is symmetrical and Gaussian. The eddy peak begins to fall by the release of the initial rectangular peak, at zero time for a few seconds under gravity at the beginning of the simulation, and it releases potential energy and produces a current. The current moves at a constant speed in the eddy. Moreover, at the beginning of the simulation, many instabilities are observed on the water surface and around the main geostrophic ring, which as the end of the simulation approaches, these pulses almost disappear and only the main geostrophic ring of eddy remains relatively stable. In addition, the initial rectangular eddy becomes a soft curved eddy. The results show well the two opposite directions of (horizontal components of) velocities at the right and left of the center of the geostrophic eddy. By shifting along the current due to the right and left of the geostrophic eddy ring relative to its center, a vortex is created that creates rotation for a balanced geostrophic eddy. Kinetic energy and potential energy will be equal on the last day of the simulation when the model reaches a steady state and perturbations on the basin surface disappear.

From the results, it can be clearly seen that the Coriolis force and gravity lead to an eddy with a geostrophic equilibrium. By changing the vertical intensity of the fluid column strength, the rotation speed of the eddy also changes, and the larger eddy diameters have more energy.