A 1.5layer reducedgravity shallowwater ocean model in spherical coordinates is described and discretized in a staggered grid (standard Arakawa Cgrid) with the forwardtime centralspace (FTCS) method and the Leapfrog finite difference scheme. The discrete Fourier analysis method combined with the Gershgorin circle theorem is used to study the stability of these two finite difference numerical models. A series of necessary conditions of selection criteria for the timespace step sizes and model parameters are obtained. It is showed that these stability conditions are more accurate than the CourantFriedrichsLewy (CFL) condition and other two criterions (Blumberg and Mellor, 1987; Casulli, 1990, 1992). Numerical experiments are proposed to test our stability results, and numerical model that is designed is also used to simulate the ocean current.
The shallowwater model is a set of partial differential equations (PDEs), which derived from the principles of conservation of mass and conservation of momentum (the NavierStokes equations). Because the horizontal length scale is much greater than the vertical length scale, under this condition, the conservation of mass implies that the vertical velocity of the fluid is very small. It can be shown from the momentum equations that horizontal pressure gradients are due to the displacement of the pressure surface (or free surface) in a fluid, and that vertical pressure gradients are nearly hydrostatic [
The situations in fluid dynamics where the horizontal length scale is much greater than the vertical length scale are very common; that is to say, the vertical acceleration of the fluid can be negligible. The flow of water over a free surface is a ubiquitous physical phenomenon that has aroused many scientists and engineers’ interest. For instance, if we consider the Coriolis forces in shallowwater model (the Coriolis term exists because we describe flows in a reference frame fixed on earth), this set of equations is particularly well suited for the study and numerical simulations of a large class of geophysical phenomena, such as atmospheric flows, ocean circulation, coastal flows, tides, tsunamis, and river and lake flows [
The shallowwater equations are derived from the NavierStokes equations that are nonlinear partial differential equations, which describe the motion of fluids. The nonlinearity makes most problems difficult or impossible to solve and it is the main contributor to the turbulence. Mathematicians and physicists believe that the turbulence can be found through an understanding of solutions to the NavierStokes equations. However, in the mathematical field, mathematicians have not yet proven that in three dimensions solutions always exist (existence), or that if they do exist, then they do not contain any singularity (that is smoothness). These are called the NavierStokes existence and smoothness problems; this is one of the seven most important open problems (the Millennium Prize Problems) in mathematics [
Research on numerical methods for the solution of the shallowwater system has attracted much attention; numerical simulation is an effective tool in solving them and a great variety of numerical methods have been developed to solve this system [
The remainder of the paper is organized as follows. In Section
The mathematical derivation of the shallowwater equations can be found in many fluid dynamics books [
The nondimensional equations are governed by the following reducedgravity, nonlinear partial differential equations:
in which
The terms
where
If one discretizes the domain to a grid with equally spaced points with a spacing of
Staggered grid on space domain.
The forward difference approximation is used for the time derivative and the central difference approximation for the spatial derivatives. The difference approximation of (
in which
The Leapfrog differences are used for time derivatives and centered differences for space derivatives; the diffusion terms are lagged by one time step following the previous studies [
The Leapfrog method allows the direct calculation of
In this section we present the stability conditions of the finite difference numerical models by using the discrete Fourier analysis method and the Gerschgorin circle theorem. The specific analysis of procedure is given in the following parts.
First of all, we set
in which
In a similar fashion, (
where
The twodimensional discrete Fourier transform of
for
We begin by taking the discrete Fourier transform of both sides of (
By making the change of variables
Similarly, we have
Thus using the expressions (
where
in which
The discrete Fourier transform is used to deal with the vector
The threedimensional discrete Fourier transform of
for
Taking the discrete Fourier transform (
where the growth matrix
in which
The difference schemes (
for
Suppose
If
Then the timespace steps size and model parameters yield the following conditions (one takes
Therefore, one has
in which
Similarly, one obtains the stable condition of the explicit Leapfrog finitedifference scheme (
where
The derivations of stability conclusions in this study are still valid for both Agrid and Bgrid; the results depend mainly on the choice of vector
As a matter of fact, when the rotation, eddy viscosity, wind stress, and interfacial friction are neglected, the second expression in (
This is the CourantFriedrichsLewy condition (CFL condition; see [
In fact, this is the same as the conditions identified by Blumberg and Mellor in 1987 [
This condition is the same as [
In this section, numerical examples are given to test our results. In the present study, we take the FTCS scheme, for example (because the stability criterions of the Leapfrog finitedifference scheme are similar to the FTCS scheme). The domain of integration is set as a part of the North Pacific basin (25°–35°N, 132°–140°E). We use a realistic coastline and the 200 m depth contour as the continental boundary [
The standard values of parameters in the model.
Parameter  Value 


6.37 × 10^{6} m 

500 m 

0.1 Pa 

0.1 m s^{−1} 

4.3752 × 10^{−8} s^{−1} 

9.8 m s^{−2} 

1023.5 kg m^{−3} 

450 m^{2} s^{−1} 

7.292 × 10^{−5} s^{−1} 

0.044 m s^{−2} 
We take
In the light of the CFL condition (
With the expression (
From the stability criterion (
Setting a timestep size
The figures show the values of the zonal velocity
The figures show the values of the zonal velocity
When we choose the step size
The figures show the values of the zonal velocity
The model is run with a time step size of
The figures show the values of the zonal velocity
The figures show the values of the zonal velocity
It is obvious that the stability condition (
In this example, the 1.5layer shallowwater numerical model that is designed by us is used to simulate the ocean current. Based on Example 1, the ocean basin is also adopted with the part of the North Pacific basin (25°–35°N, 132°–140°E). The timespace step sizes and the standard values of parameters in the model are the same as Case
The figures show the meridional velocity of the ocean current throughout a period of the timedependent solution that evolves in one day.
The FTCS and the Leapfrog finite difference scheme for solving 1.5layer shallowwater equations in spherical coordinates have been presented. The stability conditions of these two types of difference schemes are given, which include the CFL condition and other two criterions [
This study was provided by the National Basic Research Program of China (Grant no. 2012CB417404), the National Nature Scientific Foundation of China (41230420), and the Basic Research Program of Qingdao Science and Technology Plan (111495jch).