# Stability Analysis of Numerical Methods for a 1.5-Layer Shallow-Water Ocean Model

Authors: Guang-an Zou; Bo Wang; Mu Mu
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

A 1.5-layer reduced-gravity shallow-water ocean model in spherical coordinates is described and discretized in a staggered grid (standard Arakawa C-grid) with the forward-time central-space (FTCS) method and the Leap-frog 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 time-space step sizes and model parameters are obtained. It is showed that these stability conditions are more accurate than the Courant-Friedrichs-Lewy (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.


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## Body

## 1. Introduction

The shallow-water model is a set of partial differential equations (PDEs), which derived from the principles of conservation of mass and conservation of momentum (the Navier-Stokes 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 [1]. The vertical integrating allows the vertical velocity to be removed from the equations; this is a classical derivation of the shallow-water system.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 shallow-water 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 [2–10].The shallow-water equations are derived from the Navier-Stokes 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 Navier-Stokes 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 Navier-Stokes existence and smoothness problems; this is one of the seven most important open problems (the Millennium Prize Problems) in mathematics [11]. Therefore, it is also a challenge to make substantial progress toward the exact solution of shallow-water equations.Research on numerical methods for the solution of the shallow-water 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 [12–22]. The numerical models which are based on the shallow-water equations, especially for finite difference numerical models, have been successfully applied to study the ocean circulation; for example, the low-frequency variability and bifurcation structure of wind-driven ocean circulation [5–7], the shallow-water model for the study of the Gulf Stream and its extension region [23–25], the Kuroshio current and its extension system [26–30], and so on. Although few people discuss the stability of numerical models, only several papers give the stability conditions of the simple formulation and linearized shallow-water equations [20, 31–36]. In this paper, we use the discrete Fourier analysis method and the Gerschgorin circle theorem to study the stability of the shallow-water numerical models and give a series of necessary conditions for the selection criteria of time step size.The remainder of the paper is organized as follows. In Section2, the brief description of 1.5-layer reduced-gravity shallow-water ocean model has been introduced. In Section 3, we use the FTCS method and the Leap-frog finite difference scheme to solve the shallow-water equations in a staggered grid. In Section 4, the discrete Fourier method combined with the Gerschgorin circle theorem is used to analyze the stability of these two numerical methods. In Section 5, numerical examples are given to test our results. The conclusions are given in Section 6.


## 2. Matematical Model

The mathematical derivation of the shallow-water equations can be found in many fluid dynamics books [37] and has already been presented by many authors. In our study, 1.5-layer shallow-water equations in spherical coordinates are nondimensionalized using the length scale r0, which is radius of the earth, the mean depth of the upper layer H, a characteristic horizontal velocity scale U, a time scale r0/U, and a wind-stress scale τ0 (see [27, 38]).The nondimensional equations are governed by the following reduced-gravity, nonlinear partial differential equations:(1)ε(∂u∂t+ucosθ∂u∂ϕ+v∂u∂θ-uvtanθ)-vsinθ=-εFcosθ∂h∂ϕ+E(∇2u-ucos2θ-2sinθcos2θ∂v∂ϕ)+Fϕ,(2)ε(∂v∂t+ucosθ∂v∂ϕ+v∂v∂θ+u2tanθ)+usinθ=-εF∂h∂θ+E(∇2v-vcos2θ+2sinθcos2θ∂u∂ϕ)+Fθ,(3)∂h∂t+1cosθ[∂(hu)∂ϕ+∂(hvcosθ)∂θ]=0,in whichu is the zonal velocity, v is the meridional velocity, ϕ is the coordinate in the zonal direction, θ is the coordinate in the meridional direction, and h is the thickness of the upper ocean. The parameters in the equations are ε=U/2Ωr0, E=AH/2Ωr02, and F=g′H/U2, where Ω is the rotation rate of the earth, AH is the lateral friction coefficient, and g′ is the reduced gravity.The termsFϕ and Fθ in (1)-(2) are defined as follows:
(4)Fϕ=ατϕh-μu,Fθ=ατθh-μv,whereα=τ0(2ΩρHU) is wind stress coefficient (τ0 is the amplitude of wind stress) and  τϕ and τθ are the zonal component and meridional component of wind stress, respectively. In addition, μ=γ/(2Ω), where γ is the interfacial friction coefficient.


## 3. Finite Difference Schemes

If one discretizes the domain to a grid with equally spaced points with a spacing ofΔϕ in the ϕ-direction, Δθ in the θ-direction, and Δt in the t-direction, we define ui,jn=u(θi,ϕj,tn) with θi=θ0+iΔθ, ϕj=ϕ0+jΔϕ, and tn=nΔt for i=0,1,…,M, j=0,1…,N,  and n=1,2,…,Q, where M, N, and Q are positive integers. The variables u, v, and  h are evaluated at a staggered grid (standard Arakawa C-grid) as shown in Figure 1; then the shallow-water equations can be solved by using finite difference method.Figure 1
Staggered grid on space domain.


### 3.1. The FTCS Method

The forward difference approximation is used for the time derivative and the central difference approximation for the spatial derivatives. The difference approximation of (1)–(3) is given by
(5)ui+1/2,jn+1-ui+1/2,jnΔt+u*(ui+1/2,j+1n-ui+1/2,j-1n)2Δϕ+v*(ui+3/2,jn-ui-1/2,jn)2Δθ-f*vi,j+1/2n=-F*(hi,j+1n-hi,j-1n)2Δϕ+E*(ui+1/2,j+1n-2ui+1/2,jn+ui+1/2,j-1nΔϕ2+ui+3/2,jn-2ui+1/2,jn+ui-1/2,jnΔθ2)-E1*(vi,j+3/2n-vi,j-1/2n)2Δϕ+H*τϕ-Φui+1/2,jn,vi,j+1/2n+1-vi,j+1/2nΔt+u*(vi,j+3/2n-vi,j-1/2n)2Δϕ+v*(vi+1,j+1/2n-vi-1,j+1/2n)2Δθ+f*ui+1/2,jn=-F(hi+1,jn-hi-1,jn)2Δθ+E*(vi,j+3/2n-2vi,j+1/2n+vi,j-1/2nΔϕ2+vi+1,j+1/2n-2vi,j+1/2n+vi-1,j+1/2nΔθ2)+E1*(ui+1/2,j+1n-ui+1/2,j-1n)2Δϕ+H*τθ-Φvi,j+1/2n,hi,jn+1-hi,jnΔt+u*(hi,j+1n-hi,j-1n)2Δϕ+v*(hi+1,jn-hi-1,jn)2Δθ+h1*(ui+1/2,j+1n-ui+1/2,j-1n)2Δϕ+h*(vi+1,j+1/2n-vi-1,j+1/2n)2Δθ-h2*vi,j+1/2n=0,in whichu*=ui+1/2,jn/cosθi+1/2,  v*=vi,j+1/2n, f*=ui+1/2,jntanθi+1/2+(1/ε)sinθi, F*=F/cosθi, E*=E/ε, E1*=2Esinθi/εcos2θi, h*=hi,jn, H*=α/εh*, Φ=(μ/ε)-(E/εcos2θi+1/2), h1*=h*/cosθi, and h2*=h*tanθi.


### 3.2. The Leap-Frog Scheme

The Leap-frog 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 [33, 35]; we obtain
(6)ui+1/2,jn+1-ui+1/2,jn-12Δt+u*(ui+1/2,j+1n-ui+1/2,j-1n)2Δϕ+v*(ui+3/2,jn-ui-1/2,jn)2Δθ-f*vi,j+1/2n=-F*(hi,j+1n-hi,j-1n)2Δϕ+E*(ui+1/2,j+1n-1-2ui+1/2,jn-1+ui+1/2,j-1n-1Δϕ2+ui+3/2,jn-1-2ui+1/2,jn-1+ui-1/2,jn-1Δθ2)-E1*(vi,j+3/2n-vi,j-1/2n)2Δϕ+H*τϕ-Φui+1/2,jn,vi,j+1/2n+1-vi,j+1/2n-12Δt+u*(vi,j+3/2n-vi,j-1/2n)2Δϕ+v*(vi+1,j+1/2n-vi-1,j+1/2n)2Δθ+f*ui+1/2,jn=-F(hi+1,jn-hi-1,jn)2Δθ+E*(vi,j+3/2n-1-2vi,j+1/2n-1+vi,j-1/2n-1Δϕ2+vi+1,j+1/2n-1-2vi,j+1/2n-1+vi-1,j+1/2n-1Δθ2)+E1*(ui+1/2,j+1n-ui+1/2,j-1n)2Δϕ+H*τθ-Φvi,j+1/2n,hi,jn+1-hi,jn-12Δt+u*(hi,j+1n-hi,j-1n)2Δϕ+v*(hi+1,jn-hi-1,jn)2Δθ+h1*(ui+1/2,j+1n-ui+1/2,j-1n)2Δϕ+h*(vi+1,j+1/2n-vi-1,j+1/2n)2Δθ-h2*vi,j+1/2n=0.The Leap-frog method allows the direct calculation ofui+1/2,jn+1,  vi,j+1/2n+1,  hi,jn+1 from the known values at time levels n and n-1, which are the explicit difference equations.


## 3.1. The FTCS Method

The forward difference approximation is used for the time derivative and the central difference approximation for the spatial derivatives. The difference approximation of (1)–(3) is given by
(5)ui+1/2,jn+1-ui+1/2,jnΔt+u*(ui+1/2,j+1n-ui+1/2,j-1n)2Δϕ+v*(ui+3/2,jn-ui-1/2,jn)2Δθ-f*vi,j+1/2n=-F*(hi,j+1n-hi,j-1n)2Δϕ+E*(ui+1/2,j+1n-2ui+1/2,jn+ui+1/2,j-1nΔϕ2+ui+3/2,jn-2ui+1/2,jn+ui-1/2,jnΔθ2)-E1*(vi,j+3/2n-vi,j-1/2n)2Δϕ+H*τϕ-Φui+1/2,jn,vi,j+1/2n+1-vi,j+1/2nΔt+u*(vi,j+3/2n-vi,j-1/2n)2Δϕ+v*(vi+1,j+1/2n-vi-1,j+1/2n)2Δθ+f*ui+1/2,jn=-F(hi+1,jn-hi-1,jn)2Δθ+E*(vi,j+3/2n-2vi,j+1/2n+vi,j-1/2nΔϕ2+vi+1,j+1/2n-2vi,j+1/2n+vi-1,j+1/2nΔθ2)+E1*(ui+1/2,j+1n-ui+1/2,j-1n)2Δϕ+H*τθ-Φvi,j+1/2n,hi,jn+1-hi,jnΔt+u*(hi,j+1n-hi,j-1n)2Δϕ+v*(hi+1,jn-hi-1,jn)2Δθ+h1*(ui+1/2,j+1n-ui+1/2,j-1n)2Δϕ+h*(vi+1,j+1/2n-vi-1,j+1/2n)2Δθ-h2*vi,j+1/2n=0,in whichu*=ui+1/2,jn/cosθi+1/2,  v*=vi,j+1/2n, f*=ui+1/2,jntanθi+1/2+(1/ε)sinθi, F*=F/cosθi, E*=E/ε, E1*=2Esinθi/εcos2θi, h*=hi,jn, H*=α/εh*, Φ=(μ/ε)-(E/εcos2θi+1/2), h1*=h*/cosθi, and h2*=h*tanθi.


## 3.2. The Leap-Frog Scheme

The Leap-frog 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 [33, 35]; we obtain
(6)ui+1/2,jn+1-ui+1/2,jn-12Δt+u*(ui+1/2,j+1n-ui+1/2,j-1n)2Δϕ+v*(ui+3/2,jn-ui-1/2,jn)2Δθ-f*vi,j+1/2n=-F*(hi,j+1n-hi,j-1n)2Δϕ+E*(ui+1/2,j+1n-1-2ui+1/2,jn-1+ui+1/2,j-1n-1Δϕ2+ui+3/2,jn-1-2ui+1/2,jn-1+ui-1/2,jn-1Δθ2)-E1*(vi,j+3/2n-vi,j-1/2n)2Δϕ+H*τϕ-Φui+1/2,jn,vi,j+1/2n+1-vi,j+1/2n-12Δt+u*(vi,j+3/2n-vi,j-1/2n)2Δϕ+v*(vi+1,j+1/2n-vi-1,j+1/2n)2Δθ+f*ui+1/2,jn=-F(hi+1,jn-hi-1,jn)2Δθ+E*(vi,j+3/2n-1-2vi,j+1/2n-1+vi,j-1/2n-1Δϕ2+vi+1,j+1/2n-1-2vi,j+1/2n-1+vi-1,j+1/2n-1Δθ2)+E1*(ui+1/2,j+1n-ui+1/2,j-1n)2Δϕ+H*τθ-Φvi,j+1/2n,hi,jn+1-hi,jn-12Δt+u*(hi,j+1n-hi,j-1n)2Δϕ+v*(hi+1,jn-hi-1,jn)2Δθ+h1*(ui+1/2,j+1n-ui+1/2,j-1n)2Δϕ+h*(vi+1,j+1/2n-vi-1,j+1/2n)2Δθ-h2*vi,j+1/2n=0.The Leap-frog method allows the direct calculation ofui+1/2,jn+1,  vi,j+1/2n+1,  hi,jn+1 from the known values at time levels n and n-1, which are the explicit difference equations.


## 4. Stability Analysis

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 setXi,jn=(ui+1/2,jn,vi,j+1/2n,hi,jn)T, and after some rearrangement, (5) can be written as
(7)Xi,jn+1=A1Xi+1,jn+C1Xi-1,jn+B1Xi,jn+D1Xi,j+1n+E1Xi,j-1n+en,in whichen=(ΔtH*τϕ,ΔtH*τθ,0)T, (8)A1=(-Δtv*2Δθ+ΔtE*Δθ2000-Δtv*2Δθ+ΔtE*Δθ2-ΔtF2Δθ0-Δth*2Δθ-Δtv*2Δθ),C1=(Δtv*2Δθ+ΔtE*Δθ2000Δtv*2Δθ+ΔtE*Δθ2ΔtF2Δθ0Δth*2ΔθΔtv*2Δθ),B1=(1-ΔtΦ-2ΔtE*(1Δϕ2+1Δθ2)Δtf*0-Δtf*1-ΔtΦ-2ΔtE*(1Δϕ2+1Δθ2)00Δth2*1),D1=(-Δtu*2Δϕ+ΔtE*Δϕ2-ΔtE1*2Δϕ-ΔtF*2ΔϕΔtE1*2Δϕ-Δtu*2Δϕ+ΔtE*Δϕ20-Δth1*2Δϕ0-Δtu*2Δϕ),E1=(Δtu*2Δϕ+ΔtE*Δϕ2ΔtE1*2ΔϕΔtF*2Δϕ-ΔtE1*2ΔϕΔtu*2Δϕ+ΔtE*Δϕ20Δth1*2Δϕ0Δtu*2Δϕ).In a similar fashion, (6) can be written as the following matrix equations:
(9)Xi,jn+1=A2Xi-1,jn+B2Xi+1,jn+C2Xi,j-1nXi,jn+1=+D2Xi,j+1n+E2Xi,jn+F2Xi+1,jn-1+H2Xi-1,jn-1Xi,jn+1=+M2Xi,j+1n-1+N2Xi,j-1n-1+P2Xi,jn-1+en,whereen=(2ΔtH*τϕ,2ΔtH*τθ,0)T,
(10)A2=-B2=(Δtv*Δθ000Δtv*ΔθΔtFΔθ0Δth*ΔθΔtv*Δθ),C2=-D2=(Δtu*ΔϕΔtE1*ΔϕΔtF*Δϕ-ΔtE1*ΔϕΔtu*Δϕ0Δth1*Δϕ0Δtu*Δϕ),E2=(-2ΔtΦ2Δtf*0-2Δtf*-2ΔtΦ002Δth2*0),F2=H2=(2ΔtE*Δθ20002ΔtE*Δθ20000),M2=N2=(2ΔtE*Δϕ20002ΔtE*Δϕ20000),P2=(1-4ΔtE*(1Δϕ2+1Δθ2)0001-4ΔtE*(1Δϕ2+1Δθ2)0001).Definition 1.
The two-dimensional discrete Fourier transform ofu∈ℓ2 is the function u†∈L2([-π,π]×[-π,π]) defined by (see [39])
(11)u†(ξ,β)=12π∑m,k=-∞∞‍exp(-i(mξ+kβ))um,k
forξ,β∈[-π,π].We begin by taking the discrete Fourier transform of both sides of (7), and we can obtain
(12)X†n+1(ξ,β)=12π∑m,k=-∞∞‍exp(-i(mξ+kβ))Xm,kn+1=12π∑m,k=-∞∞‍exp(-i(mξ+kβ))×(A1Xm+1,kn+C1Xm-1,kn+B1Xm,kn+D1Xm,k+1n+E1Xm,k-1n+en)=A12π∑m,k=-∞∞‍exp(-i(mξ+kβ))Xm+1,kn+C12π∑m,k=-∞∞‍exp(-i(mξ+kβ))Xm-1,kn+B12π∑m,k=-∞∞‍exp(-i(mξ+kβ))Xm,kn+D12π∑m,k=-∞∞‍exp(-i(mξ+kβ))Xm,k+1n+E12π∑m,k=-∞∞‍exp(-i(mξ+kβ))Xm,k-1n+12π∑m,k=-∞∞‍exp(-i(mξ+kβ))en.By making the change of variablesl=m±1 we get
(13)12π∑m,k=-∞∞‍exp(-i(mξ+kβ))Xm±1,kn=12π∑l,k=-∞∞‍exp[-i((l∓1)ξ+kβ)]Xl,kn=exp(±iξ)12π∑l,k=-∞∞‍exp(-i(lξ+kβ))Xl,kn=exp(±iξ)X†n(ξ,β).Similarly, we have(14)12π∑m,k=-∞∞‍exp(-i(mξ+kβ))Xm,k±1n=exp(±iβ)X†n(ξ,β).Thus using the expressions (13)-(14) in (12) leads to
(15)X†n+1(ξ,β)=G1(ξ,β)X†n(ξ,β)+en′,whereen′=(1/2π)∑m,k=-∞∞exp(-i(mξ+kβ))en, and the growth matrix
(16)G1(ξ,β)=A1exp(iξ)+C1exp(-iξ)+B1+D1exp(iβ)+E1exp(-iβ)=(a11a12a13a21a22a23a31a32a33),in which(17)a11=a22=1-ΔtΦ-2ΔtE*(1Δϕ2+1Δθ2)a11=+2ΔtE*(cosβΔϕ2+cosξΔθ2)-iΔt(u*sinβΔϕ+v*sinξΔθ),a12=Δtf*-iΔtE1*Δϕsinβ,a13=-iΔtF*Δϕsinβ,a21=-Δtf*+iΔtE1*Δϕsinβ,a23=-iΔtFΔθsinξ,a31=-iΔth1*Δϕsinβ,a32=Δth2*-iΔth*Δθsinξ,a33=1-iΔt(u*Δϕsinβ+v*Δθsinξ).Remark 2.
The discrete Fourier transform is used to deal with the vectorXi,jn=(ui+1/2,jn,vi,j+1/2n,hi,jn)T. Therefore, the individual variables u*, v*, and h* in the coefficient matrix of expression (7) can be treated as constants. Consequently, the coefficient matrixes can be extracted from the Fourier transform in the expression (12). These are similar to the frozen coefficients approach for discussing the stability of numerical solution of variable coefficient partial differential equation (see [40–42]).Definition 3.
The three-dimensional discrete Fourier transform ofu∈ℓ2 is the function u†∈L2([-π,π]×[-π,π]×[-π,π]) defined by
(18)u†(ξ,β,φ)=12π∑m,k,n=-∞∞‍exp(-i(mξ+kβ+nφ))um,kn
forξ,β,φ∈[-π,π].Taking the discrete Fourier transform (18) to the both sides of (9), similar to the derivation of (15), we have
(19)X†n+1(ξ,β,φ)=G2(ξ,β,φ)X†n(ξ,β,φ)+en′,where the growth matrix(20)G2(ξ,θ,β)=A2exp(-iξ)+B2exp(iξ)+C2exp(-iβ)G2(ξ,θ,β)=+D2exp(iβ)+E2+F2exp[i(ξ-φ)]G2(ξ,θ,β)=+H2exp[-i(ξ+φ)]+M2exp[i(β-φ)]G2(ξ,θ,β)=+N2exp[-i(β+φ)]+P2exp(-iφ)G2(ξ,θ,β)=(b11b12b13b21b22b23b31b32b33),in which(21)b11=b22=-2ΔtΦ-2iΔt(u*Δϕsinβ+v*Δθsinξ)b11=+[1-4ΔtE*(1Δϕ2+1Δθ2)]exp(-iφ)b11=+2ΔtE*Δϕ2[exp(i(β-φ))+exp(-i(β+φ))]b11=+2ΔtE*Δθ2[exp(i(ξ-φ))+exp(-i(ξ+φ))]b12=2Δtf*-2iΔtE1*Δϕsinβ,b13=-2iΔtF*Δϕsinβ,b21=-2Δtf*+2iΔtE1*Δϕsinβ,b23=-2iΔtFΔθsinξ,b31=-2iΔth1*Δϕsinβ,b32=2Δth2*-2iΔth*Δθsinξ,b33=exp(-iφ)-2iΔt(u*Δϕsinβ+v*Δθsinξ).Theorem 4.
The difference schemes (7) and (9) are stable in the ℓ2 norm; then there exist constants Δx0, Δy0, Δt0, and C, independent of Δx, Δy, and Δt, so that
(22)|ρ(G)|≤1+CΔt
for0<Δt≤Δt0, and 0<Δx≤Δx0, 0<Δy≤Δy0, and for all ξ,θ,φ∈[-π,π]. ρ(G) is the spectral radius of the matrix G (G1 and G2) [39].Theorem 5 (Gerschgorin circle theorem, see [39]).
SupposeA=(aij) is a general n×n matrix, and ρs=∑j=1,j≠sn|asj| is the sum of the absolute values of the elements in the sth row except for the diagonal element. For each eigenvalue λ of A, there exists an s such that
(23)|λ-ass|≤ρs,s=1,2,…n.
Ifλ is an eigenvalue of G1, by using Gerschgorin circle theorem and the triangular inequality, one has
(24)|λ|-|a11|≤|a12|+|a13|,|λ|-|a22|≤|a21|+|a23|,|λ|-|a33|≤|a31|+|a32|.
Then the time-space steps size and model parameters yield the following conditions (one takesC≤ϵ/Δt, where 0<ϵ≤1):
(25)Δt≤ϵ|f*|+|Φ|,Δt≤ϵ|h2*|(ξ=β=0),(26)Δt(|f*|+|Φ|+4|E*|Δϕ2+4|E*|Δθ2)≤ϵ(ξ=β=π),(27)ΔtΔϕ(|u*|+|E1*|+|F*|)+Δt|v*|Δθ+Δt(|Φ|+|f*|)+2Δt|E*|Δϕ2+2Δt|E*|Δθ2≤ϵ,ΔtΔϕ(|u*|+|E1*|)+ΔtΔθ(|v*|+|F|)+Δt(|Φ|+|f*|)+2Δt|E*|Δϕ2+2Δt|E*|Δθ2≤ϵ,ΔtΔϕ(|u*|+|h1*|)+ΔtΔθ(|v*|+|h*|)+Δt|h2*|≤ϵΔtΔϕ(|u*|+|h1*|)+ΔtΔθ(|v*|+|h*|)(ξ=β=π2).
Therefore, one has|λ|≤1+CΔt, and the FTCS scheme is conditionally stable. The condition is as follows:
(28)Δt≤min{ϵ|f*|+|Φ|,ϵ|h2*|,ϵδ1,ϵδ2,εδ3,ϵδ4},
in whichδ1=|f*|+|Φ|+(4|E*|/Δϕ2)+(4|E*|/Δθ2), δ2=(1/Δϕ)(|u*|+|E1*|+|F*|)+(|v*|/Δθ)+|Φ|+|f*|+(2|E*|/Δϕ2)+(2|E*|/Δθ2),δ3=(1/Δϕ)(|u*|+|E1*|)+(1/Δθ)(|v*|+|F|)+|Φ|+|f*|+(2|E*|/Δϕ2)+(2|E*|/Δθ2), and δ4=(1/Δϕ)(|u*|+|h1*|)+(1/Δθ)(|v*|+|h*|)+|h2*|.
Similarly, one obtains the stable condition of the explicit Leap-frog finite-difference scheme (9):
(29)Δt≤min{ϵ2(|f*|+|Φ|),ϵ2|h2*|,ϵ2δ1,ϵ2δ2,ϵ2δ3,ϵ2δ4},
whereδ1, δ2, δ3, and δ4 are the same as in expression (28).Remark 6.
The derivations of stability conclusions in this study are still valid for both A-grid and B-grid; the results depend mainly on the choice of vectorXi,jn; for example, in the A-grid, we take Xi,jn=(ui,jn,vi,jn,hi,jn)T. In addition, it is easy to prove that the stability conditions derived from C-grid are the same for both A-grid and B-grid.
As a matter of fact, when the rotation, eddy viscosity, wind stress, and interfacial friction are neglected, the second expression in (27) can be written as (ϵ=1)
(30)Δt|u*|Δϕ+Δt|v*|Δθ+Δt|F|Δθ≤1.
This is the Courant-Friedrichs-Lewy condition (CFL condition; see [13, 34, 42]) in two-dimensional case. Assuming the terms |f*|=0, |Φ|=0 and ϵ=1 in the expression (26), then we have
(31)4Δt|E*|Δϕ2+4Δt|E*|Δθ2≤1.
In fact, this is the same as the conditions identified by Blumberg and Mellor in 1987 [43]. When the rotation, wind stress and interfacial friction terms are neglected and set ϵ=1, the first and second expressions in (27) are given as
(32)Δt|u*|Δϕ+Δt|v*|Δθ+12(Δt|E*|Δϕ2+Δt|E*|Δθ2)≤1.
This condition is the same as [15, 20] that given by Casulli and Cheng. The stability criteria (30)–(32) have been widely applied to the numerical model for the selection of the time-step size. However, these three conditions are only special cases in our results.


## 5. Numerical Experiments

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 Leap-frog finite-difference 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 [27]. The horizontal resolution is 0.2° × 0.2°; that is, the space-step size Δθ=Δϕ=0.2. Standard parameter values in the shallow-water model are shown in Table 1.Table 1
The standard values of parameters in the model.
Parameter
Value
r
0
6.37 × 106 m
H
500 m
τ
0
0.1 Pa
U
0.1 m s−1
γ
4.3752 × 10−8 s−1
g
9.8 m s−2
ρ
1023.5 kg m−3
A
H
450 m2 s−1
ω
7.292 × 10−5 s−1
g
′
0.044 m s−2


### 5.1. Example  1

We takeϵ=0.3, the zonal velocity u=0.3, and the meridional velocity v=0.3. According to the expression (28), it is easy to obtain Δt<3/233020; multiplying by the time scale, we have Δt<821 s.In the light of the CFL condition (30), we have Δt<5790 s.With the expression (31), we obtain Δt<4.6×108 s.From the stability criterion (32), we get Δt<2.2×107 s.Case 1.
Setting a time-step sizeΔt=825 s, after running the model with 30 steps (55/8 hours), Figure 2 gives the values (the dimensionless quantity, as well as the following results) of the zonal velocity u and the meridional velocity v at the latitude 30°N; Figure 3 gives the results at the longitude 135°E. It is not difficult to find that the current velocities u and v are not in accord with the actual condition of ocean.The figures show the values of the zonal velocityu and the meridional velocity v at the latitude 30°N after running the model with 30 steps.
(a)
(b)The figures show the values of the zonal velocityu and the meridional velocity v at the longitude 135°E after running the model with 30 steps.
(a)
(b)Case 2.
When we choose the step sizeΔt=1200 s, the results in Figure 4 give the values of the zonal velocity u and the meridional velocity v at the latitude 29°N after a time integration of 5 hours. Obviously, the results are also unreasonable. Moreover, after continuing the calculation of model, we find that the results start to overflow after running the model with 17 steps (17/3 hours).The figures show the values of the zonal velocityu and the meridional velocity v at the latitude 29°N after a time integration of 5 hours.
(a)
(b)Case 3.
The model is run with a time step size ofΔt=300 s, Figures 5 and 6 show the values of the zonal velocity u, the meridional velocity v, and the layer thickness h after a time integration of 8 hours and 8 years, respectively. These results illustrate that the model is integrated for long periods of time, and the results are still reasonable.The figures show the values of the zonal velocityu, the meridional velocity v, and the layer thickness h after a time integration of 8 hours.
(a)
(b)
(c)The figures show the values of the zonal velocityu, the meridional velocity v, and the layer thickness h after a time integration of 8 years.
(a)
(b)
(c)It is obvious that the stability condition (28) is reasonable, because the numerical model is unstable when we take the time step size Δt>821 s (as shown in Cases 1 and 2). On the other hand, it is easy to see that our results are more strict and accurate than the CFL condition (30) and other two criterions (31) and (32).


### 5.2. Example  2

In this example, the 1.5-layer shallow-water 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 time-space step sizes and the standard values of parameters in the model are the same as Case3 in Example  1. Figure 7 gives the meridional velocity of the ocean current throughout a period of the time-dependent solution that evolves in one day. We will make an attempt to use this explicit shallow-water numerical model to simulate the Kuroshio current and its extension system in further studies.Figure 7
The figures show the meridional velocity of the ocean current throughout a period of the time-dependent solution that evolves in one day.


## 5.1. Example  1

We takeϵ=0.3, the zonal velocity u=0.3, and the meridional velocity v=0.3. According to the expression (28), it is easy to obtain Δt<3/233020; multiplying by the time scale, we have Δt<821 s.In the light of the CFL condition (30), we have Δt<5790 s.With the expression (31), we obtain Δt<4.6×108 s.From the stability criterion (32), we get Δt<2.2×107 s.Case 1.
Setting a time-step sizeΔt=825 s, after running the model with 30 steps (55/8 hours), Figure 2 gives the values (the dimensionless quantity, as well as the following results) of the zonal velocity u and the meridional velocity v at the latitude 30°N; Figure 3 gives the results at the longitude 135°E. It is not difficult to find that the current velocities u and v are not in accord with the actual condition of ocean.The figures show the values of the zonal velocityu and the meridional velocity v at the latitude 30°N after running the model with 30 steps.
(a)
(b)The figures show the values of the zonal velocityu and the meridional velocity v at the longitude 135°E after running the model with 30 steps.
(a)
(b)Case 2.
When we choose the step sizeΔt=1200 s, the results in Figure 4 give the values of the zonal velocity u and the meridional velocity v at the latitude 29°N after a time integration of 5 hours. Obviously, the results are also unreasonable. Moreover, after continuing the calculation of model, we find that the results start to overflow after running the model with 17 steps (17/3 hours).The figures show the values of the zonal velocityu and the meridional velocity v at the latitude 29°N after a time integration of 5 hours.
(a)
(b)Case 3.
The model is run with a time step size ofΔt=300 s, Figures 5 and 6 show the values of the zonal velocity u, the meridional velocity v, and the layer thickness h after a time integration of 8 hours and 8 years, respectively. These results illustrate that the model is integrated for long periods of time, and the results are still reasonable.The figures show the values of the zonal velocityu, the meridional velocity v, and the layer thickness h after a time integration of 8 hours.
(a)
(b)
(c)The figures show the values of the zonal velocityu, the meridional velocity v, and the layer thickness h after a time integration of 8 years.
(a)
(b)
(c)It is obvious that the stability condition (28) is reasonable, because the numerical model is unstable when we take the time step size Δt>821 s (as shown in Cases 1 and 2). On the other hand, it is easy to see that our results are more strict and accurate than the CFL condition (30) and other two criterions (31) and (32).


## 5.2. Example  2

In this example, the 1.5-layer shallow-water 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 time-space step sizes and the standard values of parameters in the model are the same as Case3 in Example  1. Figure 7 gives the meridional velocity of the ocean current throughout a period of the time-dependent solution that evolves in one day. We will make an attempt to use this explicit shallow-water numerical model to simulate the Kuroshio current and its extension system in further studies.Figure 7
The figures show the meridional velocity of the ocean current throughout a period of the time-dependent solution that evolves in one day.


## 6. Conclusions

The FTCS and the Leap-frog finite difference scheme for solving 1.5-layer shallow-water 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 [15, 20, 43]. The numerical experiments are proposed for testing the stability of the FTCS scheme; the numerical results illustrate that our stability conditions are effective and reasonable. Moreover, the present stability criterion is shown to be more accurate than other criterions that this research mentioned. The theory of stability analysis in this paper can also be used to study the complex coupled atmosphere-ocean models.



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*Source: 478054-2013-11-06.xml*