^{1}

^{2}

In this paper, we propose a numerical method based on semi-Lagrangian approach for solving quasi-geostrophic (QG) equations on a sphere. Using potential vorticity and stream-function as prognostic variables, two-order centered difference is suggested on the latitude-longitude grid. In our proposed numerical scheme, advection terms are expressed in a Lagrangian frame of reference to circumvent the CFL restriction. The pole singularity associated with the latitude-longitude grid is eliminated by a smoothing technique for the initial flow. Error analysis is provided for the numerical scheme.

The quasi-geostrophic (QG) equations on a sphere are the major system of interest in weather forecasting and climate prediction [

Analytic solutions are rarely available due to the relatively complex nature of QG equations on a sphere; numerical simulations play an important role in the exploration of the QG equations, for example, [

Some efforts have made to alleviate the difficulties of the pole singularity. Kurihara [

A framework of semi-Lagrangian semi-implicit methods was developed by Robert [

In this paper, we consider the following non-dimensional 2D QG equations on a sphere:

where

The rest of paper is structured as follows. In Section 2, we present the semi-Lagrangian discrete scheme of QG equations on a sphere. In Section 3, we carry out the detailed accuracy analysis of the method and explain some details. The conclusions are summarized in Section 4.

In this section, we introduce the semi-Lagrangian scheme for QG equations. We will focus on the time discretization and ignore the space variable for the moment. The first equation of QG model is expressed as

where

Then the semi-Lagrangian scheme for above equation is

Here subscripts a and d refer to evaluation at an arrival and departure point, respectively. We firstly iteratively calculate departure point using some first guess and an interpolation formula, the order of the interpolation is much less important. So we use linear interpolation here. Secondly, an cubic interpolation formula is adopted to evaluate

Using the two-stage trajectory calculation, Equation (3) can be computed as follows,

where

the departure point. Comparing to the classical Runge-Kutta midpoint method, a slight difference is that the first stage uses

Let us introduce, for example, cubic Lagrange interpolation. Set

and similarly for q. When

With the propositions above, we have

alternatively, it can be expanded as follows,

In the case

When

and similarly for the case

For spatial approximation, we adopt unstaggered grid that is uniform in longitude and latitude. Stream-function

More specifically, for fixed

For the second term of right hand side of Equation (6), we use the following approximation

With the definition of the discrete operators above, the spatial discretization for (6) is defined as follows. When

When

Similarly when

In this section, we will carry out the error estimate of our new algorithm. We let

Employ Taylor expansion for the terms in the above equation at

Substituting (12)-(17) into (11), notice that

Similarly, we consider the truncation error of the semi-Lagrangian discrete scheme. The semi-Lagrangian scheme of the first equation in (1) reads

Here subscript d refers to evaluation at an departure point. By expanding

where

We first calculate the error of the Runge-Kutta discretization

Similarly

Since the right hand side of Equation (20) matches the Taylor series expansion of

Now consider the error that generated by Runge-Kutta scheme used to estimate the back-trajectory in semi-Lagrangian approach. The data are available only at discrete

points on a space-time grid in many practical dynamics,

evaluated by interpolation or extrapolation. Before examining the errors introduced by such interpolation and extrapolation, consider the cases where

Noticing

Similarly we can get

Employ Taylor expansions of

Substituting (22) into (23) gives

Which implies that the Runge-Kutta scheme (4) generates an

Now suppose that the velocity data

lation between times

Suppose that the extrapolated velocity field

Substituting the preceding equation into (20) shows that the use of

In this paper, we present a numerical method which combines semi-Lagrangian method with two-order centered difference scheme for solving two-dimensional quasi- geostrophic equations on a sphere. In our approach, potential vorticity and stream- function are used as prognostic variables. Advection terms are expressed in a Lagrangian frame of reference to avoid the necessity of stable constraint. The pole singularity is eliminated by means of employ a smoothing technique. An error analysis is presented.

We thank the Editor and the referee for their comments. This work is supported by the National Natural Science Foundation of China (Nos. 11447017, 11471166 and 11401294).

Zhu, Q.Y. and Yang, Y. (2016) A Semi-Lagrangian Type Solver for Two-Dimensional Quasi-Geostro- phic Model on a Sphere. Applied Mathematics, 7, 2296-2306. http://dx.doi.org/10.4236/am.2016.718181