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International Journal of Mathematics and Mathematical Sciences
Volume 2012, Article ID 163689, 16 pages
http://dx.doi.org/10.1155/2012/163689
Research Article

Potential Symmetry Studies on a Rotating Fluid System

S. Y. Lou,1,2,3 Man Jia,1 F. Huang,4 and X. Y. Tang1,2

1Department of Physics, Ningbo University, Ningbo 315211, China
2Department of Physics, Shanghai Jiao Tong University, Shanghai 200240, China
3School of Mathematics, Fudan University, Shanghai 200433, China
4Physical Oceanography Laboratory, Ocean University of China, Qingdao 266003, China

Received 13 July 2011; Revised 26 September 2011; Accepted 26 October 2011

Academic Editor: Marc de Montigny

Copyright © 2012 S. Y. Lou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A rotational fluid model which can be used to describe broad vortical flows ranging from large scale to the atmospheric mesoscale and the oceanic submesoscale is studied by the symmetry group theory. After introducing one scalar-, two vector-, and two tensor potentials, we find that the Lie symmetries of the extended system include many arbitrary functions of 𝑧 and {𝑧,𝑡}. The obtained Lie symmetries are used to find some types of exact solutions. One of exact solutions can be used to qualitatively describe the three-dimensional structure of hurricanes.

1. Introduction

Symmetry study is one of the effective methods to study complicated nonlinear problems. Though the Lie symmetry group method have been quite perfectly studied and some excellent books has been published [1], there still exist many important problems to be studied. For instance, to find group invariant solutions related to the generalized symmetries and nonlocal symmetries is still a very difficult topic. We know that for integrable systems there are infinitely many generalized symmetries [16] and nonlocal symmetries. The finite transformation (symmetry group) of the Lie symmetries may be obtained via Lie's first theorem. For many types of the nonlocal symmetries, the finite transformations can also be obtained by using Lie's first theorem; say, the Darboux transformations (DT) are just the finite transformation form of the nonlocal symmetries obtained from DT. This fact implies that some types of generalized symmetries and nonlocal symmetries can be localized to closed Lie symmetry algebra such that the Lie's first principle can be successfully applied. In this paper, we study a special type of nonlocal symmetries, potential symmetries of a rotating fluid model. The potential symmetries have been studied by many authors [79].

It is known that the quasi-geostrophic equations for a Boussinesq fluid in a uniformly rotating and stably stratified environment, which is an idealization for various phenomena in either atmosphere or ocean. In this paper, we consider the following rotating stratified fluid model [10, 11]:𝑢𝑥+𝑣𝑦𝑢=0,(1.1)𝑡+𝑢𝑢𝑥+𝑣𝑢𝑦𝑓𝑣+𝑝𝑥𝑣=0,(1.2)𝑡+𝑢𝑣𝑥+𝑣𝑣𝑦+𝑓𝑢+𝑝𝑦=0,(1.3)𝜌+𝑝𝑧𝜌=0,(1.4)𝑡+𝑢𝜌𝑥+𝑣𝜌𝑦+𝑤𝜌𝑧=0,(1.5) where 𝑓 is the Coriolis parameter, 𝑝 is the pressure perturbation divided by a mean density 𝜌0, 𝜌 is the density perturbation scaled by 𝜌0/𝑔, 𝑢 and 𝑣 are horizontal velocities, and 𝑤 is the vertical velocity.

It should be mentioned that the consistent condition of (1.2) and (1.3), 𝑝𝑥𝑦=𝑝𝑦𝑥, is just the Euler equation for an ideal incompressible two-dimensional fluid. Some authors have studied the symmetry structure of the two-dimensional Euler equations [1214].

In section 2, the Lie symmetries of the original model (1.1)–(1.5) are directly written down because it can be obtained by means of some known methods. Section 3 is devoted to discuss special types of generalized symmetries of the system (1.1)–(1.5) which are only Lie symmetries for the subsystem (1.1)–(1.3). In section 4, to find more symmetries and symmetry groups we discuss the potential symmetries which are Lie symmetries for an enlarged system. In section 5, the full symmetry group related to the Lie symmetries of the enlarged system is directly written down. The section 6 is devoted to find some new exact solutions which display abundant vortex structure and some of them qualitatively display some three-dimensional structure of hurricanes. The final section is a short summary and discussion.

2. The Lie Symmetries of the System (1.1)–(1.5)

A symmetry of (1.1)–(1.5),𝜎𝜎𝑢𝜎𝑣𝜎𝑝𝜎𝜌𝜎𝑤,(2.1) is defined as a solution of the linearized equations of (1.1)–(1.5):𝜎𝑢𝑥+𝜎𝑣𝑦=0,(2.2)𝜎𝑢𝑡+𝜎𝑢𝑢𝑥+𝑢𝜎𝑢𝑢𝑥+𝜎𝑣𝑢𝑦+𝑣𝜎𝑢𝑢𝑦𝑓𝜎𝑣+𝜎𝑝𝑥=0,(2.3)𝜎𝑣𝑡+𝜎𝑢𝑣𝑥+𝑢𝜎𝑣𝑥+𝜎𝑣𝑣𝑦+𝑣𝜎𝑣𝑦+𝑓𝜎𝑢+𝜎𝑝𝑦=0,(2.4)𝜎𝜌+𝜎𝑝𝑧=0,(2.5)𝜎𝜌𝑡+𝜎𝑢𝜌𝑥+𝑢𝜎𝜌𝑥+𝜎𝑣𝜌𝑦+𝑣𝜎𝜌𝑦+𝜎𝑤𝜌𝑧+𝑤𝜎𝜌𝑧=0,(2.6) which means the model system is form invariant under the transformation𝑢𝑣𝑝𝜌𝑤𝑢𝑣𝑝𝜌𝑤𝜎+𝜖𝑢𝜎𝑣𝜎𝑝𝜎𝜌𝜎𝑤(2.7) with infinitesimal 𝜖. The Lie symmetries of (1.1)–(1.5) have the form𝜎𝑢𝜎𝑣𝜎𝑝𝜎𝜌𝜎𝑤=𝑈𝑉𝑃𝑄𝑊𝑢𝑋𝑥𝑣𝑥𝑝𝑥𝜌𝑥𝑤𝑥𝑢𝑌𝑦𝑣𝑦𝑝𝑦𝜌𝑦𝑤𝑦𝑢𝑍𝑧𝑣𝑧𝑝𝑧𝜌𝑧𝑤𝑧𝑢𝑇𝑡𝑣𝑡𝑝𝑡𝜌𝑡𝑤𝑡,(2.8) where 𝑋,𝑌,𝑍,𝑇,𝑈,𝑉,𝑃,𝑄, and 𝑊 are functions of 𝑥,𝑦,𝑧,𝑡,𝑢,𝑣,𝑝,𝜌, and 𝑤.

Substituting (2.8) into (2.2)–(2.6), eliminating 𝑢𝑡,𝑣𝑦,𝑝𝑦,𝜌, and 𝑤 via (1.1)–(1.5) and vanishing different powers of the fields 𝑢,𝑣,𝑝,𝜌,𝑤 and their derivatives, one can obtain the determining equations of the functions 𝑋,𝑌,𝑍,𝑇,𝑈,𝑉,𝑃,𝑄, and 𝑊. After solving these determining equations, one can find that the general Lie symmetries of (1.1)–(1.5) have the form (2.8) with𝑋=𝑎11𝑥+2𝑎2𝑓𝑡+2𝑎5𝑦𝑎7(𝑡),𝑌=𝑎11𝑦2𝑎2𝑓𝑡+2𝑎5𝑥𝑎8𝑍𝑎(𝑡),=21𝑎2𝑧+𝑎4,𝑇=𝑎2𝑡+𝑎3,𝑈=𝑎7𝑡+𝑎1+𝑎21𝑢2𝑎2𝑓𝑡2𝑎51𝑣+2𝑎2𝑓𝑦,𝑉=𝑎8𝑡+12𝑎2𝑓𝑡2𝑎5𝑎𝑢+1𝑎21𝑣2𝑎2𝑎𝑓𝑥,𝑃=7𝑡𝑓𝑎8𝑡𝑎𝑥+7𝑓+𝑎8𝑡𝑡𝑎𝑦+21𝑎21𝑝4𝑓2𝑎2𝑦2+𝑥2+𝑐1𝑧+𝑎10(𝑡),𝑄=𝑐1,𝑊=3𝑎2+2𝑎1𝑤,(2.9) which is a linear combination of the following generators:(a)time translation (𝑎3-part of (2.8) with (2.9))𝑉1=𝜕𝑡;(2.10)(b)vertical space translation (𝑎4-part of (2.8) with (2.9)),𝑉2=𝜕𝑧;(2.11)(c)generalized 𝑥-translation and Galilean invariance 𝑉3=𝑎7(𝑡)𝜕𝑥𝑎7𝑡𝜕𝑢+𝑓𝑎7𝑡𝑦+𝑥𝑎7𝑡𝑡𝜕𝑝,(2.12) which is 𝑥-translation for 𝑎7 being constant and Galilean boost in 𝑥 direction for 𝑎7𝑡;(d)generalized 𝑦-translation and Galilean invariance 𝑉4=𝑎8(𝑡)𝜕𝑦𝑎8𝑡𝜕𝑣+𝑓𝑎8𝑡𝑥𝑦𝑎8𝑡𝑡𝜕𝑝,(2.13) which is 𝑦-translation for 𝑎8 being constant and Galilean boost in 𝑦 direction for 𝑎8𝑡;(e)time and 𝑧-dependent 𝑝-translation (pressure shift for reference point)𝑉5=𝑐1𝑧+𝑎10𝜕(𝑡)𝑝,(2.14)(f)space scaling invariance (𝑎1-part of (2.8) with (2.9))𝑉6=𝑥𝜕𝑥+𝑦𝜕𝑦+2𝑧𝜕𝑧+𝑢𝜕𝑢+𝑣𝜕𝑣+2𝑝𝜕𝑝+2𝑤𝜕𝑤;(2.15)(g)time-independent rotation (𝑎5-part of (2.8) with (2.9))𝑉7=𝑦𝜕𝑥+𝑥𝜕𝑦𝑣𝜕𝑢+𝑢𝜕𝑣;(2.16)(h)time scaling company with time-dependent rotation (𝑎2-part of (2.8) with (2.9)) 𝑉8=12𝑡𝑓𝑦𝜕𝑥12𝑡𝑓𝑥𝜕𝑦+𝑡𝜕𝑡2𝑧𝜕𝑧+𝑓2(𝑦+𝑣𝑡)𝑢𝑣𝜕𝑢𝑓𝑣+2𝜕(𝑥+𝑢𝑡)𝑣𝑓2𝑝+24𝑥2+𝑦2𝜕𝑝3𝑤𝜕𝑤,(2.17) where 𝑎𝑖,𝑖=1,,5, and 𝑐1 are arbitrary constants while 𝑎7,𝑎8, and 𝑎9 are arbitrary functions of 𝑡.

It should be pointed out that the time scaling (2.17) is linked with the time-dependent rotation. This property may hint us that we should uncover more symmetries of the model.

3. The Lie Symmetries of the Subsystem (1.1)–(1.3)

From (1.1)–(1.3), we know that the fields 𝑢, 𝑣, and 𝑝 constitute a closed subsystem. Whence 𝑢, 𝑣, and 𝑝 are fixed, the density 𝜌 and the vertical velocity 𝑤 can be simply obtained from (1.4) and (1.5), respectively, via differentiations.

Therefore, in this section we study the Lie symmetries of the subsystem (1.1)–(1.3) which is a solution of (2.2)–(2.4) with the solution form𝜎𝑢𝜎𝑣𝜎𝑝𝑢=𝑋𝑥𝑣𝑥𝑝𝑥𝑢+𝑌𝑦𝑣𝑦𝑝𝑦𝑢+𝑍𝑧𝑣𝑧𝑝𝑧𝑢+𝑇𝑡𝑣𝑡𝑝𝑡𝑈𝑉𝑃,(3.1) where 𝑋,𝑌,𝑍,𝑇,𝑈,𝑉, and 𝑃 are functions of 𝑥,𝑦,𝑧,𝑡,𝑢,𝑣, and 𝑝. The same procedure as the last section leads to the general solution𝑋=𝑥𝑎1+12𝑎2𝑦𝑓𝑡+𝑦𝑎5𝑎7,1𝑌=2𝑎2𝑥𝑓𝑡𝑥𝑎5+𝑦𝑎1𝑎8,𝑍=𝑎4(𝑧),𝑇=𝑎2𝑡+𝑎3,𝑎𝑈=1𝑎21𝑢+2𝑡𝑓𝑎2+2𝑎51𝑣+2𝑎2𝑓𝑦𝑎7𝑡,1𝑉=22𝑎5𝑡𝑓𝑎2𝑎𝑢+1𝑎21𝑣2𝑎2𝑓𝑥𝑎8𝑡,𝑎𝑃=21𝑎21𝑝4𝑎2𝑦2+𝑥2𝑓2𝑥𝑎7𝑡𝑦𝑎8𝑡𝑓+𝑥𝑎7𝑡𝑡𝑦𝑎8𝑡𝑡+𝑎9,(3.2) where 𝑎𝑖, 𝑖=1,,5, are arbitrary functions of 𝑧 and 𝑎7, 𝑎8 and 𝑎9 are arbitrary functions of {𝑧,𝑡}. It is clear that the Lie symmetries of the last section are just the special case of (3.2) for 𝑎1,𝑎2,𝑎3,𝑎5 being arbitrary constants, 𝑎4 and 𝑎9 being linear functions of 𝑧, and 𝑎7 and 𝑎8 being only functions of 𝑡.

Because of the entrance of arbitrary functions, the Lie symmetries (3.1) of the subsystem (1.1)–(1.3) become the generalized symmetries for the full system (1.1)–(1.5). The generalized symmetries of (1.1)–(1.5) are expressed by (2.8) with (3.2) and 𝑎𝑄=2𝑡+𝑎3𝑧𝑝𝑡+𝑥𝑎1+12𝑎2𝑦𝑓𝑡+𝑦𝑎5𝑎7𝑧𝑝𝑥+𝑦𝑎112𝑎2𝑥𝑓𝑡𝑥𝑎5𝑎8𝑧𝑝𝑦+𝑥𝑎7𝑡𝑡𝑦𝑎8𝑡𝑡𝑦𝑓𝑎7𝑡+𝑥𝑓𝑎8𝑡𝑎10+14𝑎2𝑦2+𝑥2𝑓2𝑧𝑎4𝑧𝑎𝜌21𝑎2𝑧𝑎𝑝+21𝑎2𝜌,𝑊=2𝑎4𝑧2𝑎1+2𝑎2𝑝1𝑧𝑧×2𝑇𝑧𝑝𝑧𝑡2𝑋𝑧𝑝𝑥𝑧2𝑌𝑧𝑝𝑦𝑧𝑋𝑧𝑧𝑝𝑥𝑌𝑧𝑧𝑝𝑦𝑇𝑧𝑧𝑝𝑡+4𝑎2𝑧+𝑎4𝑧𝑧4𝑎1𝑧𝑎𝜌+21𝑧𝑧𝑎2𝑧𝑧1𝑝4𝑦2+𝑥2𝑓2𝑎2𝑧𝑧+𝑦𝑓𝑎7𝑧𝑧𝑡𝑥𝑓𝑎8𝑧𝑧𝑡+𝑥𝑎7𝑧𝑧𝑡𝑡+𝑦𝑎8𝑧𝑧𝑡𝑡+𝑝0𝑧𝑧𝑤𝑣𝑓𝑎7𝑧𝑡𝑢𝑓𝑎8𝑧𝑡𝑝𝑦𝑦𝑣𝑦𝑎1𝑧𝑣𝑥𝑎5𝑧𝑣𝑎8𝑧+𝑥𝑎7𝑧𝑡𝑡𝑡+𝑦𝑎8𝑧𝑡𝑡𝑡+𝑣2𝑎13𝑎2𝑣𝑎4𝑧𝑝𝑦𝑧+2𝑎1𝑎4𝑧3𝑎2𝑝𝑧𝑡+(𝑦𝑓+𝑢)𝑎7𝑧𝑡𝑡(𝑥𝑓𝑣)𝑎8𝑧𝑡𝑡+2𝑎13𝑎2𝑧𝑝𝑡𝑎2𝑡+𝑎3𝑧𝑝𝑡𝑡+𝑝0𝑧𝑡12𝑢𝑥𝑓+𝑣𝑦𝑓𝑝𝑦𝑦𝑣𝑥𝑡𝑓𝑎2𝑧+𝑦𝑎5𝑧+𝑎7𝑧𝑢𝑎4𝑧𝑥𝑎1𝑧12𝑦𝑓𝑎2𝑧𝑡𝑢𝑎3𝑧𝑢𝑎2𝑧𝑡+𝑢2𝑎13𝑎2𝑝𝑥𝑧(𝑦𝑣𝑢𝑥)𝑎5𝑧+𝑣𝑎7𝑧+𝑢𝑎8𝑧(𝑣𝑥+𝑢𝑦)𝑎1𝑧+12𝑡(𝑢𝑥𝑦𝑣)𝑓𝑎2𝑧𝑝𝑥𝑦+𝑢𝑎1𝑧12(𝑣𝑡+𝑦)𝑓𝑎2𝑧𝑣𝑎5𝑧+𝑎7𝑧𝑡2𝑢𝑎2𝑧𝑝𝑥+𝑎1𝑧1𝑣+2(𝑢𝑡+𝑥)𝑓𝑎2𝑧+𝑢𝑎5𝑧2𝑣𝑎2𝑧+𝑎8𝑧𝑡𝑝𝑦+𝑢𝑦𝑎5𝑧𝑥𝑎1𝑧12𝑦𝑓𝑎2𝑧𝑡+𝑎7𝑧𝑝𝑥𝑥+12𝑥𝑓𝑎2𝑧𝑡+𝑎8𝑧+𝑥𝑎5𝑧𝑦𝑎1𝑧𝑣𝑎3𝑧𝑣𝑎2𝑧𝑡𝑝𝑦𝑡𝑝1𝑧𝑧.(3.3)

In addition to the symmetries (2.8) with (3.3) are not point symmetries, the time scaling invariance related to 𝑎2=𝑎2(𝑧)𝑉=𝑎2𝑡𝜕𝑡+12𝑎2𝑦𝑓𝑡𝜕𝑥12𝑎2𝑥𝑓𝑡𝜕𝑦+12𝑎2(𝑡𝑓𝑣2𝑢+𝑓𝑦)𝜕𝑢12𝑎2(𝑡𝑓𝑢+2𝑣+𝑓𝑥)𝜕𝑣14𝑎2𝑦8𝑝+2+𝑥2𝑓2𝜕𝑝+14𝑎2𝑧4𝑡𝑝𝑡+2𝑦𝑓𝑡𝑝𝑥2𝑥𝑓𝑡𝑝𝑦+𝑦2+𝑥2𝑓2𝜕𝜌+2𝑎2𝑝1𝑧𝑧𝑎2𝑧𝑥𝑓𝑡𝑝𝑦𝑧2𝑡𝑝𝑧𝑡𝑦𝑓𝑡𝑝𝑥𝑧+1+4𝜌2𝑎2𝑧𝑧𝑥𝑓𝑡𝑝𝑦𝑡𝑦𝑓𝑝𝑥2𝑡𝑝𝑡𝑎2𝑧𝑧12𝑝+4𝑦2+𝑥2𝑓2𝑤12𝑝𝑦𝑦𝑣𝑥𝑡𝑢𝑥𝑓𝑣𝑦𝑓𝑓𝑎2𝑧3𝑎2𝑝𝑦𝑧+𝑝𝑧𝑡𝑎2𝑧3𝑝𝑡+𝑡𝑝𝑡𝑡12(𝑦𝑓+2𝑢)𝑎2𝑧𝑡+6𝑎2𝑢𝑝𝑥𝑧+12𝑡(𝑢𝑥𝑦𝑣)𝑓𝑎2𝑧𝑝𝑥𝑦12[]𝑎(𝑣𝑡+𝑦)𝑓+4𝑢2𝑧𝑝𝑥+12[(]𝑎𝑢𝑡+𝑥)𝑓4𝑣2𝑧𝑝𝑦12𝑢𝑦𝑓𝑎2𝑧𝑡𝑝𝑥𝑥+12𝑎2𝑧(𝑥𝑓𝑡2𝑣𝑡)𝑝𝑦𝑡𝑝1𝑧𝑧𝜕𝑤(3.4) is still companied by the time-dependent rotation.

4. Potential Symmetries of (1.1)–(1.5)

From (1.1), we know that it is natural to introduce a special scalar potential, the stream function 𝜓, such that𝑢=𝜓𝑦,𝑣=𝜓𝑥.(4.1) We call a symmetry potential symmetry if it is explicitly dependent on a potential, say, 𝜓, (or its derivatives) for the fields 𝑢,𝑣,𝑝,𝜌, and 𝑤 described by (1.1)–(1.5). Obviously, a potential symmetry for the field system {𝑢,𝑣,𝑝,𝜌,𝑤} is a nonlocal one though it may be a Lie symmetry for the enlarged system {𝑢,𝑣,𝑝,𝜌,𝑤,𝜓}. From the general symmetry expression (3.3), we know that it is useful to introduce the following potential vector fields𝜓𝑣=𝑥,𝜓𝑦,𝜓𝑧,𝜓𝑡=𝑣1=𝑣,𝑣2=𝑢,𝑣3,𝑣4,𝑝𝑓=𝑥,𝑝𝑦,𝑝𝑧,𝑝𝑡=𝑓1,𝑓2,𝑓3=𝜌,𝑓4,(4.2) and symmetric potential tensor fields𝜓𝑖𝑗=𝜓𝑗𝑖=𝜓𝑥𝑖𝑥𝑗,𝑖,𝑗=1,2,3,4,𝑥1=𝑥,𝑥2=𝑦,𝑥3=𝑧,𝑥4𝑝=𝑡,𝑖𝑗=𝑝𝑗𝑖=𝑝𝑥𝑖𝑥𝑗,𝑖,𝑗=1,2,3,4.(4.3) Now using the standard method as the last two sections, we can obtain the Lie symmetries of the enlarged system (1.1)–(1.5) and (4.1)–(39) for thirty-one fields 𝑢=𝑣2, 𝑣=𝑣1,𝑝,𝑤,𝜓,𝑣3,𝑣4,𝑓1,𝑓2, 𝜌=𝑓3,𝑓4,𝜓𝑖𝑗 and 𝑝𝑖𝑗, 𝑖𝑗=1,2,3,4. The result has the following generators {𝑉1,𝑉2,𝑉3,𝑉4,𝑉5,𝑉6,𝑉7,𝑉8,𝑉9,𝑉10}𝒱 for the original fields 𝑢,𝑣,𝑤,𝑝,𝜌 and the potential 𝜓:𝑉1=𝑎1𝑥𝜕𝑥+𝑦𝜕𝑦+2𝜓𝜕𝜓+𝑢𝜕𝑢+𝑣𝜕𝑣+2𝑝𝜕𝑝+2𝜌𝜕𝜌+𝑎1𝑧×𝑥𝑓1+𝑦𝑓2𝜕2𝑝𝜌+𝑝133×2𝑤2𝜌+𝑥𝑝13+𝑦𝑝23+𝑢𝑥𝑝11+𝑦𝑝12𝑓1+𝑣𝑥𝑝12+𝑦𝑝22𝑓22𝑓4+𝑥𝑃13+𝑦𝑝24𝜕𝑤+𝑎1𝑧𝑧𝑝133𝑤𝑥𝑓1+𝑦𝑓2𝜕2𝑝𝑤,𝑉(4.4)2=𝑎2𝑡𝜕𝑡𝜓𝜕𝜓𝑢𝜕𝑢𝑣𝜕𝑣+(𝑓𝜓2𝑝)𝜕𝑝𝑓𝑣3𝜕+2𝜌𝜌+𝑤+𝑓𝑝133𝑤𝜓33+𝜓34+𝑢𝜓13+𝑣𝜓23𝜕𝑤+𝑎2𝑧𝑡𝑓4𝜕𝑓𝜓+2𝑝𝜌+𝑎2𝑧2𝑡𝑤2+𝑝1332𝑤2𝜌+𝑓𝑣3+𝑡𝑢𝑝13+𝑡𝑣𝑝23𝑢2𝑓1+𝑡𝑝14𝑣2𝑓2+𝑡𝑝24+𝑣4𝑓3𝑓4𝑡𝑝44𝜕𝑤+𝑎2𝑧𝑧𝑝133𝑤𝑓𝜓2𝑝𝑡𝑓4𝜕𝑤,(4.5)𝑉3=𝑎3𝜕𝑡+𝑎3𝑧𝑓4𝜕𝜌+2𝑤2+𝑝1332𝑤𝑢𝑝13+𝑣𝑝23𝑢𝑝14𝑣𝑝24𝑝44𝜕𝑤𝑎3𝑧𝑧𝑝133𝑤𝑓4𝜕𝑤,𝑉(4.6)4=𝑎4𝜕𝑧𝑎4𝑧𝜌𝜕𝜌+𝑤𝜕𝑤+𝑎4𝑧𝑧𝑝133𝑤𝜌𝜕𝑤,𝑉(4.7)5=𝑎5𝑥𝜕𝑦𝑦𝜕𝑥𝑣𝜕𝑢+𝑢𝜕𝑣+𝑎5𝑧𝑥𝑓2𝑦𝑓1𝜕𝜌+𝑝1332𝑦𝑝13𝑥𝑝23𝑤+𝑦𝑝11𝑥𝑝12𝑓2𝑓𝑢+1+𝑦𝑝12𝑥𝑝22𝑣+𝑦𝑝14𝑥𝑝24𝜕𝑤+𝑎5𝑧𝑧𝑝133𝑦𝑓1𝑥𝑓2𝑤𝜕𝑤,𝑉(4.8)6=𝑎6𝑥𝑡𝜕𝑦𝑦𝑡𝜕𝑥+12𝑥2+𝑦2𝜕𝜓(𝑣𝑡+𝑦)𝜕𝑢+(𝑢𝑡+𝑥)𝜕𝑣+124𝜓+𝑓𝑥2+𝑓𝑦2𝜕𝑝2𝑣3𝜕𝜌+2𝑝33𝑢𝜓13+𝑣𝜓23+𝑤𝜓33+𝜓34𝜕𝑤𝑎6𝑧122𝑡𝑥𝑓22𝑡𝑦𝑥14𝜓𝑓𝑥2𝑓𝑦2𝜕𝜌+𝑝133𝑢𝑓𝑥𝑓2𝑡𝑥𝑡𝑝12+𝑦𝑡𝑝11+𝑣𝑦𝑓+𝑡𝑓1𝑥𝑡𝑝22+𝑦𝑡𝑝12+2𝑤2𝑣3+𝑦𝑡𝑝13𝑥𝑡𝑝23+𝑦𝑓1+2𝑣4𝑥𝑡𝑝24+𝑦𝑡𝑝14𝑥𝑓2𝜕𝑤12𝑎6𝑧𝑧𝑝1334𝜓+2𝑦𝑡𝑓12𝑥𝑡𝑓2+𝑓𝑥2+𝑓𝑦2𝑤𝜕𝑤,𝑉(4.9)7=𝑎7𝜕𝑥𝑎7𝑡𝑦𝜕𝜓+𝜕𝑢𝑓𝑦𝜕𝑝𝑎7𝑡𝑡𝑥𝜕𝑝𝑎7𝑧𝑓1𝜕𝜌𝑝1332𝑤𝑝13+𝑢𝑝11+𝑝14+𝑣𝑝12𝜕𝑤+𝑎7𝑧𝑡𝑓𝑦𝜕𝜌𝑝133𝑓1𝜕+𝑓𝑣𝑤+𝑎7𝑧𝑡𝑡𝑝133(𝑢+𝑓𝑦)𝜕𝑤𝑎7𝑧𝑧𝑡𝑥𝜕𝜌𝑝133𝑤𝑦𝑓𝜕𝑤+𝑎7𝑧𝑡𝑡𝑡𝑥+𝑎7𝑧𝑧𝑡𝑡𝑥𝑤+𝑎7𝑧𝑧𝑓1𝑤𝑝133𝜕𝑤,𝑉(4.10)8=𝑎8𝜕𝑦𝑎8𝑡𝑥𝜕𝜓+𝜕𝑣+𝑥𝑓𝜕𝑝+𝑎8𝑧𝑧𝑤𝑓2+𝑎8𝑧𝑡𝑡𝑡𝑦𝑎8𝑧𝑧𝑡𝑝𝑤𝑥𝑓133𝜕𝑤+𝑎8𝑡𝑡𝑦𝜕𝑝𝑎8𝑧𝑓2𝜕𝜌𝑝1332𝑤𝑝23+𝑣𝑝22+𝑝24+𝑢𝑝12𝜕𝑤+𝑎8𝑧𝑧𝑡𝑡𝑝133𝑦𝑤𝜕𝑤+𝑎8𝑧𝑡𝑓𝑥𝜕𝜌+𝑝133𝑓2𝜕𝑓𝑢𝑤𝑎8𝑧𝑡𝑡𝑦𝜕𝜌𝑝133(𝑣𝑥𝑓)𝜕𝑤,𝑉(4.11)9=𝑎9𝜕𝑝𝑎9𝑧𝜕𝜌𝑎9𝑡𝑡𝑤+𝑎9𝑧𝑡𝑝133𝜕𝑤,𝑉(4.12)10=𝑎10𝜕𝜓,(4.13) where 𝑎𝑖, 𝑖=1,,6, are arbitrary functions of 𝑧 and 𝑎7,𝑎8,𝑎9,𝑎10 are arbitrary functions of {𝑧,𝑡}.

Remark 4.1. 𝑧-dependent time scaling symmetry 𝑉2 given by (4.5) is independent of the time-dependent rotation symmetry (4.9) thanks to the inclusion of the potential symmetry.

Remark 4.2. The symmetries 𝑉2 and 𝑉6 are nonlocal symmetries of the original system {𝑢,𝑣,𝑝,𝜌,𝑤} while they are local Lie symmetries for the enlarged system.

Remark 4.3. All the symmetries except 𝑉10 are generalized symmetries for the original system and they all become the Lie symmetries for the enlarged system.

5. Finite Transformation Group of the System (1.1)–(1.5) Related to Lie Symmetries of the Enlarged System

To find the finite transformation group of the system (1.1)–(1.5) related to the Lie symmetry algebra 𝒱, we have to solve the initial valued problemd𝑥=𝑋d𝑦=𝑌d𝑧=𝑍d𝑡=𝑇d𝜓=Ψd𝑢+𝑈d𝑣=𝑉d𝑤=𝑊d𝑝𝑥𝑃==𝑑𝜖,,𝑦,𝑧,𝑡,𝜓,𝑢,𝑣,𝑤,𝑝,𝜖=0={𝑥,𝑦,𝑧,𝑡,𝜓,𝑢,𝑣,𝑤,𝑝,},(5.1) where 𝑥,𝑦,𝑧,𝑡,𝑢,𝑣,𝑤,𝑝,, are functions of 𝜖 and 𝑋,𝑌,𝑍,𝑇,𝑈,𝑉,𝑃, can be read off from the linear combination of the generators of the algebra 𝒱. For instance,𝑋=𝑎7+𝑎1𝑥𝑡2𝑦𝑔𝑦𝑡,𝑌=𝑎8+𝑎1𝑦+𝑡2𝑥+𝑔𝑥𝑡,𝑍=𝑎4𝑧,𝑇=𝑡0+𝑡𝑡1,𝑈=𝑎7𝑡+𝑐2𝑡1𝑢𝑡2𝑣𝑔𝑣𝑡+𝑦,𝑉=𝑎8𝑡+𝑐2𝑡1𝑣+𝑡2𝑢+𝑔𝑢𝑡+𝑥,𝑃=𝑓𝑎7𝑡𝑦𝑥𝑎7𝑡𝑡𝑓𝑎8𝑡𝑥+𝑦𝑎8𝑡𝑡,,(5.2) with {𝑎1𝑎1(𝑧), 𝑎3𝑎3(𝑧), 𝑎2𝑎2(𝑧), 𝑎5𝑎5(𝑧), 𝑔𝑔(𝑧), 𝑎4𝑎4(𝑧),} being arbitrary functions of 𝑧 and {𝑎7𝑎7(𝑧,𝑡), 𝑎8𝑎8(𝑧,𝑡),} being arbitrary functions of the indicated variables.

Because the density 𝜌 and the vertical velocity 𝑤 can be simply obtained from (1.4) and (1.5) when the fields {𝑢,𝑣,𝑝,𝜓} are known, we write down only the transformation group theorem on the fields {𝑢,𝑣,𝑝,𝜓} here.

Theorem 5.1. If {𝑈(𝑥,𝑦,𝑧,𝑡), 𝑉(𝑥,𝑦,𝑧,𝑡), 𝑃(𝑥,𝑦,𝑧,𝑡), Ψ(𝑥,𝑦,𝑧,𝑡)} is a solution of the system (1.1)–(1.3) and (4.1), so is {𝑢,𝑣,𝑝,𝜓} with 𝑢=𝛼6𝑦𝛼8𝑡sin𝜃+𝛼7𝑡cos𝜃+𝛼2𝛼11[],𝑈(𝜉,𝜂,𝜁,𝜏)cos𝜃𝑉(𝜉,𝜂,𝜁,𝜏)sin𝜃𝑣=𝛼6𝑥+𝛼7𝑡sin𝜃+𝛼8𝑡cos𝜃+𝛼2𝛼11[],𝛼𝑉(𝜉,𝜂,𝜁,𝜏)cos𝜃+𝑈(𝜉,𝜂,𝜁,𝜏)sin𝜃𝑝=2𝛼21𝛼2𝑃(𝜉,𝜂,𝜁,𝜏)+𝑓+2𝛼6𝑓𝛼2Ψ(𝜉,𝜂,𝜁,𝜏)+𝛼9+𝛼62𝑓+𝛼6𝑥2+𝑦2+(𝑥cos𝜃+𝑦sin𝜃)𝑓+2𝛼6𝛼8𝑡𝛼7𝑡𝑡+(𝑥sin𝜃𝑦cos𝜃)𝑓+2𝛼6𝛼7𝑡+𝛼8𝑡𝑡,𝛼𝜓=2𝛼21Ψ(𝜉,𝜂,𝜁,𝜏)+𝛼10+𝛼62𝑦cos(𝛼6𝑡)𝑥sin(𝛼6𝑡)2+𝛼𝑦sin6𝑡𝛼+𝑥cos6𝑡2𝛼+𝑥8𝑡cos𝜃+𝛼7𝑡𝛼sin𝜃+𝑦8𝑡sin𝜃𝛼7𝑡,cos𝜃(5.3) where 𝜉=𝛼1(𝑦sin𝜃+𝑥cos𝜃𝛼7), 𝜂=𝛼1(𝑦cos𝜃𝑥sin𝜃𝛼8), 𝜁=𝛼4, 𝜏=𝛼2𝑡+𝛼3, 𝜃𝛼6𝑡+𝛼5, {𝛼1,𝛼2,𝛼3,𝛼4,𝛼5} and 𝛼6 are arbitrary functions of 𝑧 while {𝛼7,𝛼8,𝛼9} and 𝛼10 are arbitrary functions of {𝑧,𝑡}.

To prove the theorem, one can solve the initial problem (5.1) or directly substitute (5.3) to (1.1)–(1.3) and (4.1). Actually, it is more straightforward to verify that the infinitesimal form of (5.3) is just the linear combination of the generators given in the last section by taking𝛼11+𝜖𝑎1,𝛼21+𝜖𝑎2,𝛼4𝑧+𝜖𝑎4,𝛼𝑖𝜖𝑎𝑖,𝑖1,2,4,(5.4)

with the infinitesimal parameter 𝜖.

6. Exact Solutions and Applications

To find some types of new exact solutions via the finite symmetry group theorem of the last section, we should find some simple exact seed solutions at first. Fortunately, it is not difficult to verify that the model (1.1)–(1.5) possesses the following known trivial solution:𝑢=2𝐹(𝑟,𝑧)𝑦,𝑟𝑥2+𝑦2,𝐹𝑣=2𝑥𝐹(𝑟,𝑧),𝑝=𝐹(𝑟,𝑧)(2𝐹(𝑟,𝑧)+𝑓)𝑑𝑟,𝜌=𝑧1(𝑟,𝑧)(4𝐹(𝑟,𝑧)+𝑓)𝑑𝑟,𝑤=𝑝𝑧𝑧𝑝𝑧𝑡+𝑢𝑝𝑥𝑧+𝑣𝑝𝑦𝑧,(6.1) with arbitrary function 𝐹𝐹(𝑟,𝑧) of {𝑟,𝑧}.

Applying the group theorem of the last section to solution (6.1), we have𝑢=𝛼6𝑦𝛼8𝑡sin𝜃+𝛼7𝑡cos𝜃2𝛼2𝛼11𝜂𝐹𝑅,𝛼4cos𝜃+𝜉𝐹𝑅,𝛼4,sin𝜃𝑣=𝛼6𝑥+𝛼7𝑡sin𝜃+𝛼8𝑡cos𝜃+2𝛼2𝛼11𝜉𝐹𝑅,𝛼4cos𝜃𝜂𝐹𝑅,𝛼4,𝛼sin𝜃𝑝=2𝛼21𝛼2𝐹𝑅,𝛼42𝐹𝑅,𝛼4+𝑓𝑑𝑅+𝑓+2𝛼6𝑓𝛼2𝐹𝑅,𝛼4𝑑𝑅+𝛼10+𝛼62𝑓+𝛼6𝑥2+𝑦2+(𝑥cos𝜃+𝑦sin𝜃)𝑓+2𝛼6𝛼8𝑡𝛼7𝑡𝑡+(𝑥sin𝜃𝑦cos𝜃)𝑓+2𝛼6𝛼7𝑡+𝛼8𝑡𝑡,𝜌=𝑝𝑧,1𝑤=𝑝𝑧𝑧𝑝𝑧𝑡+𝑢𝑝𝑥𝑧+𝑣𝑝𝑦𝑧,(6.2) with𝑅𝛼21𝑦sin𝜃+𝑥cos𝜃𝛼72+𝑦cos𝜃𝑥sin𝜃𝛼82.(6.3)

The different selections of the arbitrary functions appeared in the exact solution (6.2) may lead to various vortex and circumfluence structures. It was demonstrated that the vortex and circumfluence solutions could be applied to tropical cyclones. A concrete application of a two-dimensional vortex solution to Hurricane Katrina 2005 was discussed in [16]. Similar to the two dimensional case, solution (6.2) displays some interesting phenomena. The first part, 𝑢𝛼6𝑦,𝑣𝛼6𝑥 and the 𝐹(𝑅, 𝛼4) parts of the solution exhibit abundant vortex structure. {𝑥=𝛼7, 𝑦=𝛼8} is expressed as the center of the vortices while the second part ({𝑢𝛼8𝑡sin𝜃+𝛼7𝑡cos𝜃, 𝑣𝛼7𝑡sin𝜃+𝛼8𝑡cos𝜃}) of the solution shows us that the induced flow (the background wind) is related to the moving of the vortex center. This fact can be used to predict regular cyclone's track [16]. In order to find possible three-dimensional structure of the cyclones, one has to fix the arbitrary functions. Here we just take a special form of (6.2),𝑅𝑢=𝐹1,𝑧𝑦,𝑅1𝑥2+𝑦2𝑅,𝑣=𝐹1𝛼,𝑧𝑥,𝑤=10𝑧𝑡(𝑧,𝑡)𝑅4𝐹1𝐹,𝑧+𝑓𝑧𝑧𝑅1,𝑧+4𝐹2𝑧𝑅1,𝑧2d𝑅1+𝛼10𝑧𝑧(,𝑧,𝑡)(6.4) with only two arbitrary functions 𝐹(𝑅1,𝑧) and 𝛼10(𝑧,𝑡) to qualitatively display the three-dimensional vortex structure.

For the sake of capturing the known features of cyclone's structure, we here assume the arbitrary functions to be a more special form such that𝑢=𝑐0𝑐𝑦sech1𝑧+𝑐5𝑐tanh1𝑧+𝑐5𝑅sech1𝑐2,(6.5)𝑣=𝑐0𝑐𝑥sech1𝑧+𝑐5𝑐tanh1𝑧+𝑐5𝑅sech1𝑐2,(6.6)𝑤=𝑐42𝑐3+𝑐20𝑐21𝑐2sech2𝑐1𝑧+𝑐510sech4𝑐1𝑧+𝑐511sech2𝑐1𝑧+𝑐5𝑅+2tanh1𝑐2+𝑐0𝑐21𝑐2𝑐𝑓tanh1𝑧+𝑐5𝑐sech1𝑧+𝑐516sech2𝑐1𝑧+𝑐5𝑅arctanexp1𝑐21,(6.7) with arbitrary constants 𝑐𝑖, 𝑖=0,1,2,,5.

For simplicity, in the special selection, we have set the cyclone's center to be located at (0,0,𝑧). The parameters 𝑐2 and 𝑐1 determine typhoon's horizontal and vertical scales, respectively. The sign of 𝑐0 decides the rotating direction of the vortex. The value of 𝑐5/𝑐1 gives the turning plane where typhoon changes its rotating direction, namely, from cyclone to anticyclone.

The velocity field of a cyclone is plotted in Figure 1 with (6.5)–(6.8) for the fixed parameters𝑐0=𝑐5=1,𝑐1=10,𝑐3=1000,𝑐4=𝑓=1,𝑐2=5.(6.8)

fig1
Figure 1: The profiles of the velocity field given by the horizontal velocities 𝑢, 𝑣 and the vertical velocity 𝑤 expressed by (6.5), (6.6), and (6.7), respectively, with the parameters (6.8) in the horizontal 𝑥𝑦 plane (a) at the bottom 𝑧=0, (b) at the top 𝑧=0.14, and (c) in the vertical 𝑦𝑧 plane with 𝑥=0.

Figure 1(a) shows the vortex structure rotating anticlockwise in lower level. While the rotation becomes clockwise at higher altitude, as observed in Figure 1(b). It is clear from Figure 1(c) that around the high 𝑧=0.1, the atmospheric flow turns from cyclone to anticyclone, which is one of the typical characters of a cyclone. It means that the energy moves inside at the lower layer and then radiates at the higher. Besides, the cyclone center and radius can be easily distinguished in the profiles of the velocity field.

The corresponding three-dimensional structure is depicted in Figure 2. In order to make it more clear how the atmosphere flows to form a cyclone, one can solve a simple system of equations coming from the relationsd𝑥d𝑡=𝑢,d𝑦d𝑡=𝑣,d𝑧d𝑡=𝑤.(6.9) Since it is difficult to obtain an analytical solution of system (6.9), we solve it numerically with the initial conditions 𝑥(0)=𝑧(0)=0 and 𝑦(0)=2. It is obviously revealed from the line depicted in Figure 2 that typhoon has a spiral pattern. The air close to the ground spirals upward and anticlockwise and then changes its rotation direction when approaching a higher altitude, which is consistent with the real observations. One can see a much closer comparison of our analytical results with a recent high-resolution hurricane simulation picture (Figure 3) using the WRF-ARW [15].

163689.fig.002
Figure 2: The-three dimensional spiral pattern of a typhoon vortex structure with the velocities 𝑢,𝑣, and 𝑤 expressed by (6.5)–(6.8), respectively. The spiral line is a numerical solution of (6.9) with (6.5)–(6.8) and the initial conditions 𝑥(0)=𝑧(0)=0 and 𝑦(0)=2.
163689.fig.003
Figure 3: Picture of a recent high-resolution hurricane simulation using the WRF-ARW, cited from [15].

7. Summary and Discussions

The symmetries of the rotational fluid model (1.1)–(1.5) are studied in three ways. Firstly, we directly study the Lie symmetries of the full system (1.1)–(1.5) for the fields {𝑢,𝑣,𝑝,𝜌,𝑤}. The result shows us that the Lie symmetries do not contain arbitrary functions of 𝑧 and the time scaling is companied by a time-dependent rotation. Secondly, the Lie symmetries of the subsystem (1.1)–(1.3) for the fields {𝑢,𝑣,𝑝} exhibit eight arbitrary functions with five of them being arbitrary functions of z and others being arbitrary functions of {𝑧,𝑡} while the time scaling invariance is still companied by a time-dependent rotation. Though these symmetries are the Lie symmetries for the subsystem {𝑢,𝑣,𝑝}, they are generalized symmetries for the full system {𝑢,𝑣,𝑝,𝜌,𝑤}. Finally, we enlarged the original system to an enlarged one which contains the stream function, four-dimensional vector forms 𝑣 and 𝑓 shown by (4.2) and the 4×4 symmetric tensors 𝜓𝑖𝑗 and 𝑝𝑖𝑗 shown by (4.3). For the enlarged system, the Lie symmetries display much abundant structure which possesses six arbitrary functions of 𝑧 and four arbitrary functions of {𝑧,𝑡}. It is shown that because of the entrance of the stream function, the time scaling invariance and time-dependent rotation invariance become independent potential symmetries which are nonlocal symmetries for the original system {𝑢,𝑣,𝑝,𝜌,𝑤} though they are local Lie symmetries for the enlarged system.

The corresponding finite symmetry transformation group related to both the generalized local symmetries and nonlocal symmetries for the subsystem {𝑢,𝑣,𝑝} are obtained because the symmetries are only Lie symmetries for the enlarged system. The finite symmetry transformation group can be used to find quite general solutions from simple ones. Especially a known trivial solution is transformed to produce a quite general vortex solution. A special form of the obtained vortex solution is used to qualitatively display the three-dimensional structure of cyclones.

All results of the paper in fact are based on the fact that system (1.1)–(1.5) is degenerate in certain sense. Namely, subsystem (1.1)–(1.3) involves the independent variable 𝑧 as a parameter. It is obvious that any local (i.e., generalized [1]) symmetry of subsystem (1.1)–(1.3) can be extended to a local symmetry of the entire system (1.1)–(1.5) using (1.4) and (1.5) and the standard prolongation procedure. The same assertion is true for finite symmetry transformations and potential symmetry associated with the stream function as a potential. Similarly, any solution of subsystem (1.1)–(1.3) can be extended to a solution of the entire system (1.1)–(1.5) using (1.4) and (1.5) for defining 𝜌 and 𝑤. Moreover, the equation for the stream function coincides with the classical vorticity equation for an ideal incompressible fluid. Subsystem (1.1)–(1.3) is reduced to the two-dimensional Euler equations via simple transformation of the pressure involving the stream function. This is why the simplest algorithm for finding exact solutions system (1.1)–(1.5) is the following. Take any solution of the vorticity equation assuming that all constant and functional parameters additionally depend on the variable 𝑧. Then find the corresponding values of 𝑢,𝑣,𝑝,𝜌, and 𝑤 using (4.1) and (1.2)–(1.5) respectively.

Acknowledgments

The authors are grateful for the referee's agreement to add the comment paragraph (the last paragraph of the paper) to the paper. This work was supported by the National Natural Science Foundation of China (nos. 10735030, 11175092, 10905038, and 40305009), the National Basic Research Program of China (nos. 2007CB814800 and 2005CB422301), the Specialized Research Fund for the Doctoral Program of Higher Education (no. 20070248120), SRF for ROCS, SEM and K. C. Wong Magna Fund in the Ningbo University.

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