# 3D Variable Coefficient KdV Equation and Atmospheric Dipole Blocking

Authors: Juanjuan Ji; Yecai Guo; Lanfang Zhang; Lihua Zhang
Journal: Advances in Meteorology
 (2018)

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## Abstract

A (2 + 1)-dimensional variable coefficient Korteweg-de Vries (3D VCKdV) equation is first derived in this paper by means of introducing 2-dimensional space and time slow-varying variables and the multiple-level approximation method from the well-known barotropic and quasi-geostrophic potential vorticity equation without dissipation. The exact analytical solution of the 3D VCKdV equation is obtained successfully by making use of CK’s direct method and the standard Zakharov–Kuznetsov equation. By some arbitrary functions and the analytical solution, a dipole blocking evolution process with twelve days’ lifetime is described, and the result illustrates that the central axis of the dipole is no longer perpendicular to the vertical direction but has a certain angle to vertical direction. The comparisons with the previous researches and Urals dipole blocking event demonstrate that 3D VCKdV equation is more suitable for describing the complex atmospheric blocking phenomenon.


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## Body

## 1. Introduction

Atmospheric blocking is a nonlinear phenomenon with a long lifetime occurring in mid-high latitude regions. Its dynamical study has been an important research topic in the atmospheric science field because of its significant influence on disaster weathers and extreme cold events. In the past decades, many investigators have proposed various nonlinear theories such as Rossby soliton [1–4], envelope Rossby soliton that is described by the Schrodinger equation [5, 6], and others to explain the formation of atmospheric blocking and its life process. Although the eddy-forced envelope soliton model can describe a life cycle of atmospheric dipole blocking [7, 8], the KdV-type soliton cannot represent the time variation or life of atmospheric dipole blocking [2, 3]. However, by considering the time slow- varying basic flow, the KdV-type Rossby solitary wave can represent the life cycle of dipole blocking [9]. Previously derived constant coefficient KdV- and Schrödinger-type equations and variable coefficient KdV- and Schrödinger-type equations are all one-dimensional models on the space [10–15]. However, in the nature, propagation of a solitary wave is usually two-dimensional on the space, and only thinking about one-dimensional model may not be enough.The purpose of this paper is to extend the solitary Rossby wave model to the 3D case, namely, two-dimensional space and time. We aim to derive a (2 + 1)-dimensional variable coefficient KdV (3D VCKdV) equation from the barotropic and quasi-geostrophic potential vorticity equation without dissipation on a beta-plane.The structure of this paper is as follows: the 3D VCKdV equation is derived in Section2; in Section 3, the exact analytical solution of the equation is obtained; Section 4 is devoted to study atmospheric dipole blocking by some arbitrary functions and parameters, and a comparison with the previous model and Urals dipole blocking is made. In the last section, some conclusions are given.


## 2. Derivation of 3D VCKdV Equation

The barotropic and quasi-geostrophic potential vorticity equation without dissipation on a beta-plane in the atmospheric dynamical system is as follows:(1)∂∂t+∂ψ∂x∂∂y−∂ψ∂y∂∂x∇2ψ−λ02ψ+β0∂ψ∂x=0,which is a highly nonlinear equation. It is very difficult to solve. In (1), ψ is the stream function; β0=ω0/R0/cosϕ0, in which R0 is the earth’s radius, ω0 is the angular frequency of the earth’s rotation, and ϕ0 is the latitude; λ0=f0/gH, in which f0 is the Coriolis parameter, g is the gravitational acceleration, and H is the atmospheric average height.Let us assume that there is a base flow independent of variablex in the atmospheric system; thus, the stream function is rewritten as(2)ψ=Ψ0y,t+ε2ψ′,where ψ′ is the perturbation stream function, ε<1 is a small parameter, and base flow field Ψ0y,t is a function of variables y and t; in the previous studies, it is often taken only as a function of y. For simplicity of notation, the prime is dropped out in the remaining of this paper.Substituting (2) into (1), we obtain(3)∂∂t−Ψ0y∂∂x∇2ψ+Jψ,∇2ψ−λ02∂ψ∂t+β0+Ψ0yyy∂ψ∂x−λ02−∂2∂y2Ψ0t=0.Because of the multiple time-space scale features of the solitary wave in the fluid, we introduce 2-dimensional space and time slow-varying variables:(4)X=εx−c0t,Y=εy,y=y,T=ε3t,where c0 is an arbitrary constant. From (4), we have(5)∂∂x=ε∂∂X,∂2∂x2=ε2∂2∂X2,∂∂t=−c0ε∂∂X+ε3∂∂T,∂∂y=ε∂∂Y+∂∂y,∂2∂y2=ε2∂2∂Y2+2ε∂2∂y∂Y+∂2∂y2.Consequently,(6)∇2ψ=ε2∂2ψ∂X2+ε2∂2ψ∂Y2+2ε∂2ψ∂y∂Y+∂2ψ∂y2,(7)Jψ,∇2ψ=∂ψ∂x∂∂y∇2ψ−∂ψ∂y∂∂x∇2ψ=ε∂ψ∂Xε∂∂Yε2∂2ψ∂X2+ε2∂2ψ∂Y2+2ε∂2ψ∂y∂Y+∂2ψ∂y2+∂∂yε2∂2ψ∂X2+ε2∂2ψ∂Y2+2ε∂2ψ∂y∂Y+∂2ψ∂y2−εε∂ψ∂Y+∂ψ∂y∂∂Xε2∂2ψ∂X2+ε2∂2ψ∂Y2+2ε∂2ψ∂y∂Y+∂2ψ∂y2=ε∂ψ∂Xε3∂3ψ∂Y∂X2+ε3∂3ψ∂Y3+3ε2∂3ψ∂y∂Y2+3ε∂3ψ∂y2∂Y−εε∂ψ∂Y+ε2∂3ψ∂y∂X2+∂3ψ∂y3+∂ψ∂yε2∂3ψ∂X3+ε2∂3ψ∂Y2∂X+2ε∂3ψ∂y∂Y∂X+∂3ψ∂y2∂X.Substituting (5)–(7) into (3) yields(8)ε3∂∂T−εc0∂∂X−εΨ0y∂∂Xε2∂2ψ∂X2+ε2∂2ψ∂Y2+2ε∂2ψ∂y∂Y+∂2ψ∂y2+ε3∂ψ∂Xε3∂3ψ∂Y∂X2+ε3∂3ψ∂Y3+3ε2∂3ψ∂y∂Y2+3ε∂3ψ∂y2∂Y+ε2∂3ψ∂y∂X2+∂3ψ∂y3−ε3ε∂ψ∂Y+∂ψ∂yε2∂3ψ∂X3+ε2∂3ψ∂Y2∂X+2ε∂3ψ∂y∂Y∂X+∂3ψ∂y2∂X+ελ02c0∂ψ∂X−ε3λ02∂ψ∂T+εβ0+Ψ0yyy∂ψ∂X−ελ02−∂2∂y2Ψ0T=0.Expand the perturbation stream functionψ in terms of ε in the form(9)ψ=ψ0+εψ1+ε2ψ2+⋯.Substituting (9) into (8), and then requiring all the coefficients of different powers of ε to be zero, we obtain the following first-order equation of ε:(10)ε1:c0+Ψ0y∂∂X∂2ψ0∂y2−λ02c0+β0+Ψ0yyy∂ψ0∂X=Ψ0yyT−λ02Ψ0T.We assume thatψ0 has the following variable separation solution:(11)ψ0=AX,Y,TG0y,T.Substituting (11) into (10), we obtain(12)Ψ0yy=λ02Ψ0+F1,(13)Ψ0y+c0G0yy−G0λ02c0+β0+F1y+λ02Ψ0y=0,where F1=F1y is an arbitrary integral function of variable y.In order to obtain solitary wave amplitude equation, we continue solving second-order equation aboutε:(14)ε2:c0+Ψ0y∂∂X∂2ψ1∂y2−λ02c0+β0+Ψ0yyy∂ψ1∂X=−2c0+Ψ0y∂3ψ0∂X∂y∂Y.Then, we assume thatψ1 has the following variable separation solution:(15)ψ1=AYX,Y,TG1y,T.Substituting (15) into (14), we have(16)Ψ0y+c0G1yy−G1λ02c0+β0+F1y+λ02Ψ0y=−2Ψ0y+c0G0y.The governing equation of solitary wave amplitude still cannot be obtained from (14), and we continue solving the following high-order problem:(17)ε3:c0+Ψ0y∂∂X∂2ψ2∂y2−λ02c0+β0+Ψ0yyy∂ψ2∂X=−2c0+Ψ0y∂3ψ1∂X∂y∂Y+∂∂T∂2ψ0∂y2−c0+Ψ0y∂3ψ0∂X3−c0+Ψ0y∂3ψ0∂X∂Y2+∂ψ0∂X∂3ψ0∂y3−∂ψ0∂y∂3ψ0∂y2∂X−λ02∂ψ0∂T.Substituting (11), (13), and (15) into (17), the following can be obtained:(18)c0+Ψ0y∂∂X∂2ψ2∂y2−λ02c0+β0+Ψ0yyy∂ψ2∂X=ATG0yy−λ02G0+G0G0yyy−G0yG0yyAAX−c0+Ψ0yG0AXXX−c0+Ψ0y2G1y+G0AXYY+G0yyT−λ02G0TA=F1y+β0c0+Ψ0yG0AT+G02λ02c0+λ02Ψ0y+F1y+β0c0+Ψ0yyAAX−c0+Ψ0yG0AXXX−c0+Ψ0y2G1y+G0AXYY+F1y+β0c0+Ψ0yG0TA=G02λ02Ψ0yy+F1yyc0+Ψ0y−Ψ0yyλ02c0+λ02Ψ0y+F1y+β0c0+Ψ0y2AAX+F1y+β0c0+Ψ0yG0AT−c0+Ψ0yG0AXXX−c0+Ψ0y2G1y+G0AXYY+G0TF1y+β0c0+Ψ0y−Ψ0yTF1y+βc0+Ψ0y2G0A.In the derivation of many nonlinear equations, they-average method is the traditional method and commonly utilized, but we remove this treatment and introduce higher order ψ2 as(19)ψ2=B1y,TA2+B2y,TAXX+B3y,TAYY+B4y,TX+B5y,T∫AdX,where Biy,T (i=1,2,…,5) are arbitrary functions of variables y and T.Therefore, we have(20)∂ψ2∂X=2B1y,TAAX+B2y,TAXXX+B3y,TAXYY+B4y,T+B5y,TA,(21)∂∂X∂2ψ2∂y2=2B1yyy,TAAX+B2yyy,TAXXX+B3yyy,TAXYY+B4yyy,T+B5yyy,TA.Substituting (20) and (21) into (18), the following can be obtained:(22)AT+G02λ02Ψ0yy+F1yyc0+Ψ0y−Ψ0yyλ02c0+λ02Ψ0y+F1y+β0c0+Ψ0y2−2c0+Ψ0yB1yy+2λ02c0+λ02Ψ0y+F1y+β0B1c0+Ψ0yF1y+βG0AAX−c0+Ψ0yG0+c0+Ψ0yB2yy−c0λ02+λ02Ψ0y+F1y+βB2c0+Ψ0yF1y+βG0AXXX−c0+Ψ0y2G1y+G0+c0+Ψ0yB3yy−c0λ02+λ02Ψ0y+F1y+βB3c0+Ψ0yF1y+βG0AXYY−Ψ0yTF1y+βc0+Ψ0y2G0+c0+Ψ0yB5yy−G0TF1y+β0c0+Ψ0y−c0λ02+λ02Ψ0y+F1y+βB5c0+Ψ0yF1y+βG0A+c0λ02+λ02Ψ0y+F1y+βB4−c0+Ψ0yB4yyc0+Ψ0yF1y+βG0=0,where Bi=Biy,T (i=1,2,…,5).WhenBi (i=1,2,…,5) satisfy the relationships,(23)G02λ02Ψ0yy+F1yyc0+Ψ0y−Ψ0yyλ02c0+λ02Ψ0y+F1y+β0−2c0+Ψ0y3B1yy+2c0+Ψ0y2λ02c0+λ02Ψ0y+F1y+β0B1−G0e1c0+Ψ0yF1y+β=0,c0+Ψ0yc0λ02+λ02Ψ0y+F1y+βB2−c0+Ψ0y2G0−c0+Ψ0y2B2yy−G0e2F1y+β=0,c0+Ψ0yc0λ02+λ02Ψ0y+F1y+βB3−c0+Ψ0y2B3yy−c0+Ψ0y22G1y+G0−G0e3F1y+β=0,c0+Ψ0y2c0λ02+λ02Ψ0y+F1y+βB5−c0+Ψ0y3B5yy−Ψ0yTF1y+βG0+G0Tc0+Ψ0yF1y+β0−G0e4c0+Ψ0yF1y+β=0,c0+Ψ0yc0λ02+λ02Ψ0y+F1y+βB4−c0+Ψ0y2B4yy−G0e5F1y+β=0,a (2 + 1)-dimensional variable coefficient KdV (3D VCKdV) equation is derived as follows:(24)AT+e1AXXX+e2AAX+e3AXYY+e4A+e5=0,where ei=eiT (i=1,2,…,5) are arbitrary functions of variable T.


## 3. Exact Solution of 3D VCKdV Equation

It is not easy to obtain the exact analytical solution of 3D VCKdV equation (24) in the case that 5 variable coefficients are kept arbitrary. As we know, quite a few methods for obtaining solitary wave solution of nonlinear systems have been proposed, for instance, the hyperbolic function method [16], generalized Darboux transformation [17], and CK’s direct method [15]. CK’s direct method is a very simple and effective method. In this section, we are going to construct the exact analytical solution of 3D VCKdV equation by CK’s direct method.The Zakharov–Kuznetsov equation is as follows:(25)Pτ+2PPξ+Pξξξ+Pξηη=0,which is taken as a two-dimensional form of constant coefficient KdV equation, and it has the following solitary wave solution [18]:(26)Pξ,η,τ=6K2sech2Kξ+Kη−8K3τ,where K is an arbitrary constant.According to CK’s direct method, we suppose the exact analytical solution of 3D VCKdV equation in the following form:(27)A=αX,Y,T+βX,Y,TPξX,Y,T,ηX,Y,T,τX,Y,T≡α+βPξ,η,τ,where Pξ,η,τ satisfy (25).From (27), we have(28)AT=αT+βTP+βPT,(29)AX=αX+βXP+βPX,(30)AXXX=αXXX+βXXXP+3βXXPX+3βXPXX+βPXXX,(31)AXYY=αXYY+βXYYP+2βXYPY+βXPYY+βYYPX+2βYPXY+βPXYY,(32)PT=PξξT+PηηT+PττT,(33)PX=PξξX+PηηX+PττX,(34)PY=PξξY+PηηY+PττY,(35)PXX=PξξξX2+2PξηξXηX+2PξτξXτX+2PητηXτX+PηηηX2+PτττX2,(36)PYY=PξξξY2+2PξηξYηY+2PξτξYτY+2PητηYτY+PηηηY2+PτττY2,(37)PXY=PξξξXξY+PξηξXηY+ξYηX+PξτξXτY+ξYτX+PητηXτY+ηYτX+PηηηXηY+PτττXτY,(38)PXXX=PξξξξX3+3PξξηξX2ηX+3PξξτξX2τX+3PξηηξXηX2+6PξητξXηXτX+3PξττξXτX2+3PηττηXτX2+3PηηττXηX2+PηηηηX3+PττττX3,(39)PXYY=PξξξξXξY2+Pξξη2ξXξYηY+ξY2ηX+Pξξτ2ξXξXτY+ξY2τX+Pξητ2ξYηXτY+ξYηYτX+ξXηYτY+PηηηηXηY2+Pξηη2ξYηXηY+ξXηY2+Pξττ2ξYτXτY+ξXτY2+Pηττ2ηYτXτY+ηXτY2+PττττXτY2.Substituting (27)–(39) into (24), the following can be obtained:(40)γ0+γ1P+γ2Pξ+γ3Pη+γ4Pτ+γ5PPξ+γ6PPη+γ7PPτ+γ8Pξξ+γ9Pξη+γ10Pξτ+γ11Pητ+γ12Pηη+γ13Pττ+γ14Pξξξ+γ15Pξξη+γ16Pξξτ+γ17Pξηη+γ18Pξητ+γ19Pξττ+γ20Pηττ+γ21Pηητ+γ22Pηηη+γ23Pτττ=0,where(41)γ0=αT+e2ααX+e1αXXX+e3αXYY+e4α+e5,γ1=βT+e2αβX+e2αXβ+e2ββX+e1βXXX+e3βXYY+e4β,γ2=βξT+e2αβξX+3e1βXXξX+2e3βXYξY+e3βYYξX,γ3=βηT+e2αβηX+3e1βXXηX+2e3βXYηY+e3βYYηX,γ4=βτT+e1αβτX+3e2βXXτX+2e3βXYτY+e3βYYτX,γ5=e2β2ξX,γ6=e2β2ηX,γ7=e2β2τX,γ8=3e1βXξX2+e3βXξY2+2e3βYξXξY,γ9=3e1βX2ξXηX+e3βX2ξYηY+2e3βYξXηY+2e3βYξYηX,γ10=3e1βX2ξXτX+e3βX2ξYτY+2e3βYξXτY+2e3βYξYτX,γ11=3e1βX2ηXτX+e3βX2ηYτY+2e3βYηXτY+2e3βYηYτX,γ12=3e1βXηX2+e3βXηY2+2e3βYηXηY,γ13=3e1βXτX2+e3βXτY2+2e3βYτXτY,γ14=e1βξX3+e3βξXξY2,γ15=3e1βξX2ηX+2e3βξXξYηY+e3βξY2ηX,γ16=3e1βξX2τX+2e3βξXξXτY+e3βξY2τX,γ17=3e1βξXηX2+2e3βξYηXηY+e3βξXηY2,γ18=6e1βξXηXτX+2e3βξYηXτY+2e3βξYηYτX+2e3βξXηYτY,γ19=3e1βξXτX2+2e3βξYτXτY+e3βξXτY2,γ20=3e1βηXτX2+2e3βηYτXτY+e3βηXτY2,γ21=e1β3τXηX2+2e3βηYηXτY+e3βηY2τX,γ22=e1βηX3+e3βηXηY2,γ23=e1βτX3+e3βτXτY2.Comparing the coefficients of (25) and (40), we obtain(42)α=Ce−∫0Te4dφ−e−∫0Te4dφ∫0Te5e∫0Te4dφdφ,β=β1e1e2,ξ=β12X−β12∫0Tαe2dφ,η=β1e12e3Y,τ=∫0Te2β1e1dφ,where β1≠0 and C are constants, and these variable coefficients must satisfy the following equation:(43)e1e2e4−e1Te2+e1e2T=0.Thus, the exact analytical solution of 3D VCKdV equation is as follows:(44)AX,Y,T=Ce−∫0Te4dφ−e−∫0Te4dφ∫0Te5e∫0Te4dφdφ+6K2β1e1e2sech2Kβ12X−Kβ12∫0TCe−∫0Te4dφ−e−∫0Te4dφ∫0Te5e∫0Te4dφdφdφ+Kβ1e12e3Y−8K3∫0Te2β1e1dφ.


## 4. Atmospheric Dipole Blocking Phenomenon

The analytical solution of 3D VCKdV equation (44) including the three variables X, Y, and T and five arbitrary functions can be used to describe some complex atmospheric phenomena such as atmospheric blocking.If we assumeC=e4=e5=0, e1=e2=e3=P, P≠0, which is an arbitrary constant, returning the variables x, y, and t, the first-order approximation solution of basic system equation (1) can be obtained:(45)ψ≈Ψ0+G0A=Ψ0y,t+G0y,t6K2β1sech2Kβ12εx+y−c0t−8K3β1ε3t.When taking the basic flowΨ0y,t as(46)Ψ0=b0y+b1,where b0 and b1 are arbitrary constants, by (12) and (13), we obtain(47)G0y,t=Rε3tsin−c0λ02+β0b0+c0y,where Rε3t is an arbitrary function of variable t; here, we suppose(48)Rε3t=sech30ε3t−9and other parameters as follows:(49)λ0=0.25,β0=1,β1=2,b0=−5,b1=8,K=5.A dipole blocking evolution with a life cycle of twelve days is displayed in Figure1.Figure 1
An atmospheric dipole blocking evolution with time from the theoretical solution of (45); other functions and parameters are given by (46)–(49). (a) Day 1, (b) day 2, (c) day 3, (d) day 4, (e) day 5, (f) day 6, (g) day 7, (h) day 8, (i) day 9, (j) day 10, (k) day 11, and (l) day 12.
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(l)Figure1 clearly displays the onset, development, maintenance, and decay of a dipole blocking with a life cycle of twelve days. Generally speaking, the dipole blocking has a timescale of 10–20 days. Clearly, our model is able to describe such a timescale. From day 1 to day 4, the dipole blocking is at the growth status (Figures 1(a)–1(d)); at day 7, the dipole blocking is at the strongest status (Figure 1(g)); then, it becomes weaker and finally vanishes after the eleventh day (Figures 1(h)–1(l)), and the dipole always slowly moves westward in the whole life cycle. More importantly, we can find that the central axis of the dipole is not perpendicular to the vertical direction, but it has a certain angle to the vertical direction.Next, we make a comparison with the theoretical model VCKdV equation in [13]:(50)Aτ+e1Aξξξ+e2AAξ+e3Aξ+e4A+e5=0.Obviously, (50) is one-dimensional on the space model, but our derived 3D VCKdV equation (24) is two-dimensional on the space.Furthermore, [13] gives a corresponding analytical solution of (50) as(51)A=−e3e2−6β0e1ξ0e2−e1e2∫e1e2e5−e1e3τ+e1τe3e12dτ+2K2β0e1e2sech2Kβ06εx−c0t+K∫e1ξ0dτ+β06e1∫e1e2e5−e1e3τ+e1τe3e12dτ−69K2β03/2∫e1dτ.Takinge3=e5=ξ0=0, λ0=0.25, a0=−1, a1=3, β0=1, K=8, and F=sech20ε3/2t−4, as in [13], a dipole-type blocking evolution is shown in Figure 2.Figure 2
A dipole blocking life cycle from the theoretical solution of Equation (114) in [13]. (a) Day 1, (b) day 2, (c) day 3, (d) day 4, (e) day 5, (f) day 6, (g) day 7, and (h) day 8.
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(h)Obviously, the lifetime of dipole-type blocking in Figure2 is eight days, and the central axis of the dipole is perpendicular to the vertical direction.On the other hand, in order to enhance practical significance of our work, we take the Urals dipole blocking case happened from 25 January 1986 to 5 February 1986 as a simple example. Making use of the National Centers for Environmental Prediction/National Center for Atmospheric Research (NCEP/NCAR) reanalysis data, the dipole blocking event is depicted in Figure3.Figure 3
Geopotential height at the 500 hPa pressure level of the Urals dipole blocking case from 25 January 1986 to 5 February 1986. Thex axis is the longitude, and the y axis is the latitude. (a) 25 January, (b) 26 January, (c) 27 January, (d) 28 January, (e) 29 January, (f) 30 January, (g) 31 January, (h) 1 February, (i) 2 February, (j) 3 February, (k) 4 February, and (l) 5 February.
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(l)It can be found from Figure3 that there is a strong dipole blocking around Urals (67°N, 66°E) with the lifetime of twelve days, and the dipole moves slowly westward. More importantly, the central axis of the dipole is not perpendicular to the vertical direction, but it has a certain angle to the vertical direction. In previous researches [5, 10, 12, 13], the central axis of the dipole is always perpendicular to the vertical direction. Making a comparison to Figures 1–3, Figure 1 is more similar to Figure 3. Consequently, our derived model is more suitable for describing the complex atmospheric blocking phenomenon.


## 5. Conclusions

In this paper, by making use of perturbation expansions and stretching transformation of a two-dimensional time and space method, a 3D VCKdV equation is derived. This equation includes three variables and 5 arbitrary functions, and its exact analytical solution still keeps 5 arbitrary variable coefficients. Under the influence of the zonal variabley, the dipole blocking depicted by exact analytical solution has a certain angle to the vertical direction. As a result, our derived model is more suitable for describing the complex atmospheric blocking phenomenon compared to the previous result and real observation.



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