# 3D Variable Coefficient KdV Equation and Atmospheric Dipole Blocking.

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 Schrodinger-type equations and variable coefficient KdV- and Schrodinger-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 Section 2; 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:

[mathematical expression not reproducible], (1)

which is a highly nonlinear equation. It is very difficult to solve. In (1), f is the stream function; [[beta].sub.0] = ([[omega].sub.0]/[R.sub.0])/cos [[phi].sub.0], in which [R.sub.0] is the earth's radius, [[omega].sub.0] is the angular frequency of the earth's rotation, and 00 is the latitude; [[lambda].sub.0] = [f.sub.0]/[square root of gH], in which [f.sub.0] 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 variable x in the atmospheric system; thus, the stream function is rewritten as

[psi] = [[PSI].sub.0] (y, t) + [[epsilon].sup.2] [psi]', (2)

where [psi]' is the perturbation stream function, [epsilon] < 1 is a small parameter, and base flow field [[PSI].sub.0] (y, 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

[mathematical expression not reproducible]. (3)

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:

[mathematical expression not reproducible], (4)

where [c.sub.0] is an arbitrary constant. From (4), we have

[mathematical expression not reproducible]. (5)

Consequently,

[mathematical expression not reproducible], (6)

[mathematical expression not reproducible]. (7)

Substituting (5)-(7) into (3) yields

[mathematical expression not reproducible]. (8)

Expand the perturbation stream function [psi] in terms of [epsilon] in the form

[psi] = [[psi].sub.0] + [epsilon] [[psi].sub.1] + [[epsilon].sup.2] [[psi].sub.2] + .... (9)

Substituting (9) into (8), and then requiring all the coefficients of different powers of [epsilon] to be zero, we obtain the following first-order equation of [epsilon]:

[mathematical expression not reproducible]. (10)

We assume that [[psi].sub.0] has the following variable separation solution:

[[psi].sub.0] = A(X, Y, T) [G.sub.0] (y, T). (11)

Substituting (11) into (10), we obtain

[[PSI].sub.0yy] = [[lambda].sup.2.sub.0] [[PSI].sub.0] + [F.sub.1], (12)

[mathematical expression not reproducible], (13)

where [F.sub.1] = [F.sub.1](y) is an arbitrary integral function of variable y.

In order to obtain solitary wave amplitude equation, we continue solving second-order equation about e:

[mathematical expression not reproducible]. (14)

Then, we assume that [[psi].sub.1] has the following variable separation solution:

[[psi].sub.1] = [A.sub.Y] (X, Y, T)[G.sub.1] (y,T). (15)

Substituting (15) into (14), we have

[mathematical expression not reproducible]. (16)

The governing equation of solitary wave amplitude still cannot be obtained from (14), and we continue solving the following high-order problem:

[mathematical expression not reproducible]. (17)

Substituting (11), (13), and (15) into (17), the following can be obtained:

[mathematical expression not reproducible]. (18)

In the derivation of many nonlinear equations, the y-average method is the traditional method and commonly utilized, but we remove this treatment and introduce higher order [[psi].sub.2] as

[mathematical expression not reproducible], (19)

where [B.sub.i] (y, T) (i = 1, 2, ..., 5) are arbitrary functions of variables y and T.

Therefore, we have

[mathematical expression not reproducible], (20)

[mathematical expression not reproducible]. (21)

Substituting (20) and (21) into (18), the following can be obtained:

[mathematical expression not reproducible], (22)

where [B.sub.i] = [B.sub.i] (y, T) (i = i, 2, ..., 5).

When [B.sub.i] (i = 1, 2, ..., 5) satisfy the relationships,

[mathematical expression not reproducible], (23)

a (2 + i)-dimensional variable coefficient KdV (3D VCKdV) equation is derived as follows:

[mathematical expression not reproducible], (24)

where [e.sub.i] = [e.sub.i] (T) (i = i, 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:

[P.sub.[tau]] + 2P[P.sub.[xi]] + [P.sub.[xi][xi][xi]] + [P.sub.[xi][eta][eta]] = 0, (25)

which is taken as a two-dimensional form of constant coefficient KdV equation, and it has the following solitary wave solution [18]:

P([xi], [eta], [tau]) = 6[K.sup.2] [sech.sup.2] (K[xi] + K[eta] - 8[K.sup.3] [tau]), (26)

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:

A = [alpha](X, Y, T) + [beta](X, Y, T) P ([xi] (X, Y, T), [eta](X, Y, T), [tau](X, Y, T)) [equivalent to] [alpha] + [beta]P([xi], [eta], [tau]), (27)

where P([xi], [eta], [tau]) satisfy (25).

From (27), we have

[A.sub.t] = [[alpha].sub.T] + [[beta].sub.T]P + [beta][P.sub.T], (28)

[a.sub.x] = [[alpha].sub.X] + [[beta].sub.X]P + [beta][P.sub.X], (29)

[A.sub.XXX] = [[alpha].sub.XXX] + [[beta].sub.XXX] P + 3[[beta].sub.XX] [P.sub.X] + 3[[beta].sub.X] [P.sub.XX] + [beta][P.sub.XXX], (30)

[mathematical expression not reproducible], (31)

[P.sub.T] = [P.sub.[xi]] [[xi].sub.T] + [P.sub.[eta] [[eta].sub.T] + [P.sub.[tau]] [[tau].sub.T], (32)

[P.sub.X] = [P.sub.[xi]] [[xi].sub.X] + [P.sub.[eta] [[eta].sub.T] + [P.sub.[tau]] [[tau].sub.X], (33)

[P.sub.Y] = [P.sub.[xi]] [[xi].sub.Y] + [P.sub.[eta] [[eta].sub.Y] + [P.sub.[tau]] [[tau].sub.Y], (34)

[mathematical expression not reproducible], (35)

[mathematical expression not reproducible], (36)

[mathematical expression not reproducible], (37)

[mathematical expression not reproducible], (38)

[mathematical expression not reproducible]. (39)

Substituting (27)-(39) into (24), the following can be obtained:

[mathematical expression not reproducible], (40)

where

[mathematical expression not reproducible]. (41)

Comparing the coefficients of (25) and (40), we obtaii

[mathematical expression not reproducible], (42)

where [[beta].sub.1] [not equal to] 0 and C are constants, and these variable co efficients must satisfy the following equation:

[e.sub.1] [e.sub.2] [e.sub.4] - [e.sub.1T] [e.sub.2] + [e.sub.1] [e.sub.2T] = 0. (43)

Thus, the exact analytical solution of 3D VCKdV equation is as follows:

[mathematical expression not reproducible]. (44)

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 assume C = [e.sub.4] = [e.sub.5] = 0, [e.sub.1] = [e.sub.2] = [e.sub.3] = P, P [not equal to] 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:

[mathematical expression not reproducible]. (45)

When taking the basic flow [[PSI].sub.0] (y, t) as

[[PSI].sub.0] = [b.sub.0] y + [b.sub.1], (46)

where [b.sub.0] and [b.sub.1] are arbitrary constants, by (12) and (13), we obtain

[mathematical expression not reproducible], (47)

where R([[epsilon].sub.3]t) is an arbitrary function of variable t; here, we suppose

R([[epsilon].sub.3]t) = sech [30[[epsilon].sub.3] (t - 9)] (48)

and other parameters as follows:

[mathematical expression not reproducible]. (49)

A dipole blocking evolution with a life cycle of twelve days is displayed in Figure 1.

Figure 1 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]:

[A.sub.[tau]] + [e.sub.1] [A.sub.[xi][xi][xi]] + [e.sub.2] [A.sub.[xi]] + [e.sub.3] [A.sub.[xi]] + [e.sub.4]A + [e.sub.5] = 0. (50)

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

[mathematical expression not reproducible]. (51)

Taking [e.sub.3] = [e.sub.5] = [[xi].sub.0] = 0, [[lambda].sub.0] = 0.25, [a.sub.0] = -1, [a.sub.1] = 3, [[beta].sub.0] = 1, K = 8, and F = sech [20[[epsilon].sup.3/2] (t - 4)], as in [13], a dipole-type blocking evolution is shown in Figure 2.

Obviously, the lifetime of dipole-type blocking in Figure 2 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 Figure 3.

It can be found from Figure 3 that there is a strong dipole blocking around Urals (67[degrees]N, 66[degrees]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 variable y, 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.

https://doi.org/10.1155/2018/4329475

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper and regarding the funding that the authors have received.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (61673222), the Major Project of Nature Science Foundation of Higher Education Institution of Jiangsu Province of China (13KJA51QQQ1), the Research Innovation Program for College Graduates of Jiangsu Province of China (KYLX15_0873), and the Natural Science Research Project of Education Department of Anhui Province under Grant no. KJ2017A368.

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Juaonjuan Ji (iD), (1,2) Yecai Guo (iD), (1) Lanfang Zhang, (2) and Lihua Zhang (2)

(1) College of Atmospheric Science, School of Electronic and Information Engineering, Nanjing University of Information Science and Technology, Nanjing, Jiangsu 210044, China

(2) School of Physics and Electrical Engineering, Anqing Normal University, Anqing, Anhui 246133, China

Correspondence should be addressed to Juanjuan Ji; jjj0721@126.com; and Yecai Guo; guo-yecai@163.com

Received 4 January 2018; Revised 5 March 2018; Accepted 14 March 2018; Published 17 May 2018

Caption: 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 Q, (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.

Caption: 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.

Caption: Figure 3: Geopotential height at the 500 hPa pressure level of the Urals dipole blocking case from 25 January 1986 to 5 February 1986. The x 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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