International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 163689 | https://doi.org/10.1155/2012/163689

S. Y. Lou, Man Jia, F. Huang, X. Y. Tang, "Potential Symmetry Studies on a Rotating Fluid System", International Journal of Mathematics and Mathematical Sciences, vol. 2012, Article ID 163689, 16 pages, 2012. https://doi.org/10.1155/2012/163689

Potential Symmetry Studies on a Rotating Fluid System

Academic Editor: Marc de Montigny
Received13 Jul 2011
Revised26 Sep 2011
Accepted26 Oct 2011
Published06 Mar 2012

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,𝑑}. 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 [1–6] 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 [7–9].

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]:𝑒π‘₯+𝑣𝑦𝑒=0,(1.1)𝑑+𝑒𝑒π‘₯+π‘£π‘’π‘¦βˆ’π‘“π‘£+𝑝π‘₯𝑣=0,(1.2)𝑑+𝑒𝑣π‘₯+𝑣𝑣𝑦+𝑓𝑒+𝑝𝑦=0,(1.3)𝜌+π‘π‘§πœŒ=0,(1.4)𝑑+π‘’πœŒπ‘₯+π‘£πœŒπ‘¦+π‘€πœŒπ‘§=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, 𝑒 and 𝑣 are horizontal velocities, and 𝑀 is the vertical velocity.

It should be mentioned that the consistent condition of (1.2) and (1.3), 𝑝π‘₯𝑦=𝑝𝑦π‘₯, 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 [12–14].

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),βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽπœŽπœŽβ‰‘π‘’πœŽπ‘£πœŽπ‘πœŽπœŒπœŽπ‘€βŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ ,(2.1) is defined as a solution of the linearized equations of (1.1)–(1.5):πœŽπ‘’π‘₯+πœŽπ‘£π‘¦=0,(2.2)πœŽπ‘’π‘‘+πœŽπ‘’π‘’π‘₯+π‘’πœŽπ‘’π‘’π‘₯+πœŽπ‘£π‘’π‘¦+π‘£πœŽπ‘’π‘’π‘¦βˆ’π‘“πœŽπ‘£+πœŽπ‘π‘₯=0,(2.3)πœŽπ‘£π‘‘+πœŽπ‘’π‘£π‘₯+π‘’πœŽπ‘£π‘₯+πœŽπ‘£π‘£π‘¦+π‘£πœŽπ‘£π‘¦+π‘“πœŽπ‘’+πœŽπ‘π‘¦=0,(2.4)𝜎𝜌+πœŽπ‘π‘§=0,(2.5)πœŽπœŒπ‘‘+πœŽπ‘’πœŒπ‘₯+π‘’πœŽπœŒπ‘₯+πœŽπ‘£πœŒπ‘¦+π‘£πœŽπœŒπ‘¦+πœŽπ‘€πœŒπ‘§+π‘€πœŽπœŒπ‘§=0,(2.6) which means the model system is form invariant under the transformationβŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽπ‘’π‘£π‘πœŒπ‘€βŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ βŸΆβŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽπ‘’π‘£π‘πœŒπ‘€βŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽπœŽ+πœ–π‘’πœŽπ‘£πœŽπ‘πœŽπœŒπœŽπ‘€βŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ (2.7) with infinitesimal πœ–. The Lie symmetries of (1.1)–(1.5) have the formβŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽπœŽπ‘’πœŽπ‘£πœŽπ‘πœŽπœŒπœŽπ‘€βŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ =βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽπ‘ˆπ‘‰π‘ƒπ‘„π‘ŠβŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽπ‘’βˆ’π‘‹π‘₯𝑣π‘₯𝑝π‘₯𝜌π‘₯𝑀π‘₯βŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽπ‘’βˆ’π‘Œπ‘¦π‘£π‘¦π‘π‘¦πœŒπ‘¦π‘€π‘¦βŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽπ‘’βˆ’π‘π‘§π‘£π‘§π‘π‘§πœŒπ‘§π‘€π‘§βŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽπ‘’βˆ’π‘‡π‘‘π‘£π‘‘π‘π‘‘πœŒπ‘‘π‘€π‘‘βŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ ,(2.8) where 𝑋,π‘Œ,𝑍,𝑇,π‘ˆ,𝑉,𝑃,𝑄, and π‘Š are functions of π‘₯,𝑦,𝑧,𝑑,𝑒,𝑣,𝑝,𝜌, and 𝑀.

Substituting (2.8) into (2.2)–(2.6), eliminating 𝑒𝑑,𝑣𝑦,𝑝𝑦,𝜌, and 𝑀 via (1.1)–(1.5) and vanishing different powers of the fields 𝑒,𝑣,𝑝,𝜌,𝑀 and their derivatives, one can obtain the determining equations of the functions 𝑋,π‘Œ,𝑍,𝑇,π‘ˆ,𝑉,𝑃,𝑄, and π‘Š. After solving these determining equations, one can find that the general Lie symmetries of (1.1)–(1.5) have the form (2.8) with𝑋=π‘Ž11π‘₯+2ξ€·π‘Ž2𝑓𝑑+2π‘Ž5ξ€Έπ‘¦βˆ’π‘Ž7(𝑑),π‘Œ=π‘Ž11π‘¦βˆ’2ξ€·π‘Ž2𝑓𝑑+2π‘Ž5ξ€Έπ‘₯βˆ’π‘Ž8π‘ξ€·π‘Ž(𝑑),=21βˆ’π‘Ž2𝑧+π‘Ž4,𝑇=π‘Ž2𝑑+π‘Ž3,π‘ˆ=βˆ’π‘Ž7𝑑+ξ€·π‘Ž1+π‘Ž2ξ€Έ1π‘’βˆ’2ξ€·βˆ’π‘Ž2π‘“π‘‘βˆ’2π‘Ž5ξ€Έ1𝑣+2π‘Ž2𝑓𝑦,𝑉=βˆ’π‘Ž8𝑑+12ξ€·βˆ’π‘Ž2π‘“π‘‘βˆ’2π‘Ž5ξ€Έξ€·π‘Žπ‘’+1βˆ’π‘Ž2ξ€Έ1π‘£βˆ’2π‘Ž2ξ€·π‘Žπ‘“π‘₯,𝑃=7π‘‘βˆ’π‘“π‘Ž8ξ€Έπ‘‘ξ€·π‘Žπ‘₯+7𝑓+π‘Ž8π‘‘ξ€Έπ‘‘ξ€·π‘Žπ‘¦+21βˆ’π‘Ž2ξ€Έ1π‘βˆ’4𝑓2π‘Ž2𝑦2+π‘₯2ξ€Έ+𝑐1𝑧+π‘Ž10(𝑑),𝑄=βˆ’π‘1,ξ€·π‘Š=βˆ’3π‘Ž2+2π‘Ž1𝑀,(2.9) which is a linear combination of the following generators:(a)time translation (π‘Ž3-part of (2.8) with (2.9))𝑉1=πœ•π‘‘;(2.10)(b)vertical space translation (π‘Ž4-part of (2.8) with (2.9)),𝑉2=πœ•π‘§;(2.11)(c)generalized π‘₯-translation and Galilean invariance 𝑉3=βˆ’π‘Ž7(𝑑)πœ•π‘₯βˆ’π‘Ž7π‘‘πœ•π‘’+ξ€·π‘“π‘Ž7𝑑𝑦+π‘₯π‘Ž7π‘‘π‘‘ξ€Έπœ•π‘,(2.12) which is π‘₯-translation for π‘Ž7 being constant and Galilean boost in π‘₯ direction for π‘Ž7βˆΌπ‘‘;(d)generalized 𝑦-translation and Galilean invariance 𝑉4=βˆ’π‘Ž8(𝑑)πœ•π‘¦βˆ’π‘Ž8π‘‘πœ•π‘£+ξ€·π‘“π‘Ž8𝑑π‘₯βˆ’π‘¦π‘Ž8π‘‘π‘‘ξ€Έπœ•π‘,(2.13) which is 𝑦-translation for π‘Ž8 being constant and Galilean boost in 𝑦 direction for π‘Ž8βˆΌπ‘‘;(e)time and 𝑧-dependent 𝑝-translation (pressure shift for reference point)𝑉5=𝑐1𝑧+π‘Ž10ξ€»πœ•(𝑑)𝑝,(2.14)(f)space scaling invariance (π‘Ž1-part of (2.8) with (2.9))𝑉6=π‘₯πœ•π‘₯+π‘¦πœ•π‘¦+2π‘§πœ•π‘§+π‘’πœ•π‘’+π‘£πœ•π‘£+2π‘πœ•π‘+2π‘€πœ•π‘€;(2.15)(g)time-independent rotation (π‘Ž5-part of (2.8) with (2.9))𝑉7=βˆ’π‘¦πœ•π‘₯+π‘₯πœ•π‘¦βˆ’π‘£πœ•π‘’+π‘’πœ•π‘£;(2.16)(h)time scaling company with time-dependent rotation (π‘Ž2-part of (2.8) with (2.9)) 𝑉8=12π‘‘π‘“π‘¦πœ•π‘₯βˆ’12𝑑𝑓π‘₯πœ•π‘¦+π‘‘πœ•π‘‘βˆ’2π‘§πœ•π‘§+𝑓2ξ‚Ά(𝑦+𝑣𝑑)βˆ’π‘’π‘£πœ•π‘’βˆ’ξ‚΅π‘“π‘£+2ξ‚Άπœ•(π‘₯+𝑒𝑑)π‘£βˆ’ξ‚΅π‘“2𝑝+24ξ€·π‘₯2+𝑦2ξ€Έξ‚Άπœ•π‘βˆ’3π‘€πœ•π‘€,(2.17) where π‘Žπ‘–,𝑖=1,…,5, and 𝑐1 are arbitrary constants while π‘Ž7,π‘Ž8, and π‘Ž9 are arbitrary functions of 𝑑.

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 𝑒, 𝑣, and 𝑝 constitute a closed subsystem. Whence 𝑒, 𝑣, and 𝑝 are fixed, the density 𝜌 and the vertical velocity 𝑀 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 formβŽ›βŽœβŽœβŽœβŽœβŽπœŽπ‘’πœŽπ‘£πœŽπ‘βŽžβŽŸβŽŸβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽœβŽœβŽπ‘’=𝑋π‘₯𝑣π‘₯𝑝π‘₯βŽžβŽŸβŽŸβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽœβŽœβŽπ‘’+π‘Œπ‘¦π‘£π‘¦π‘π‘¦βŽžβŽŸβŽŸβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽœβŽœβŽπ‘’+π‘π‘§π‘£π‘§π‘π‘§βŽžβŽŸβŽŸβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽœβŽœβŽπ‘’+π‘‡π‘‘π‘£π‘‘π‘π‘‘βŽžβŽŸβŽŸβŽŸβŽŸβŽ βˆ’βŽ›βŽœβŽœβŽœβŽœβŽπ‘ˆπ‘‰π‘ƒβŽžβŽŸβŽŸβŽŸβŽŸβŽ ,(3.1) where 𝑋,π‘Œ,𝑍,𝑇,π‘ˆ,𝑉, and 𝑃 are functions of π‘₯,𝑦,𝑧,𝑑,𝑒,𝑣, and 𝑝. The same procedure as the last section leads to the general solution𝑋=π‘₯π‘Ž1+12π‘Ž2𝑦𝑓𝑑+π‘¦π‘Ž5βˆ’π‘Ž7,1π‘Œ=βˆ’2π‘Ž2π‘₯π‘“π‘‘βˆ’π‘₯π‘Ž5+π‘¦π‘Ž1βˆ’π‘Ž8,𝑍=π‘Ž4(𝑧),𝑇=π‘Ž2𝑑+π‘Ž3,ξ€·π‘Žπ‘ˆ=1βˆ’π‘Ž2ξ€Έ1𝑒+2ξ€·π‘‘π‘“π‘Ž2+2π‘Ž5ξ€Έ1𝑣+2π‘Ž2π‘“π‘¦βˆ’π‘Ž7𝑑,1𝑉=2ξ€·βˆ’2π‘Ž5βˆ’π‘‘π‘“π‘Ž2ξ€Έξ€·π‘Žπ‘’+1βˆ’π‘Ž2ξ€Έ1π‘£βˆ’2π‘Ž2𝑓π‘₯βˆ’π‘Ž8𝑑,ξ€·π‘Žπ‘ƒ=21βˆ’π‘Ž2ξ€Έ1π‘βˆ’4π‘Ž2𝑦2+π‘₯2𝑓2βˆ’ξ€·π‘₯π‘Ž7π‘‘βˆ’π‘¦π‘Ž8𝑑𝑓+π‘₯π‘Ž7π‘‘π‘‘βˆ’π‘¦π‘Ž8𝑑𝑑+π‘Ž9,(3.2) where π‘Žπ‘–, 𝑖=1,…,5, are arbitrary functions of 𝑧 and π‘Ž7, π‘Ž8 and π‘Ž9 are arbitrary functions of {𝑧,𝑑}. It is clear that the Lie symmetries of the last section are just the special case of (3.2) for π‘Ž1,π‘Ž2,π‘Ž3,π‘Ž5 being arbitrary constants, π‘Ž4 and π‘Ž9 being linear functions of 𝑧, and π‘Ž7 and π‘Ž8 being only functions of 𝑑.

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 ξ€·π‘Žπ‘„=2𝑑+π‘Ž3𝑧𝑝𝑑+ξ‚€π‘₯π‘Ž1+12π‘Ž2𝑦𝑓𝑑+π‘¦π‘Ž5βˆ’π‘Ž7𝑧𝑝π‘₯+ξ‚€π‘¦π‘Ž1βˆ’12π‘Ž2π‘₯π‘“π‘‘βˆ’π‘₯π‘Ž5βˆ’π‘Ž8𝑧𝑝𝑦+ξ‚€βˆ’π‘₯π‘Ž7π‘‘π‘‘βˆ’π‘¦π‘Ž8π‘‘π‘‘βˆ’π‘¦π‘“π‘Ž7𝑑+π‘₯π‘“π‘Ž8π‘‘βˆ’π‘Ž10+14π‘Ž2𝑦2+π‘₯2𝑓2ξ‚π‘§βˆ’π‘Ž4π‘§ξ€·π‘ŽπœŒβˆ’21βˆ’π‘Ž2ξ€Έπ‘§ξ€·π‘Žπ‘+21βˆ’π‘Ž2ξ€Έξ€½πœŒ,π‘Š=2π‘Ž4π‘§βˆ’2π‘Ž1+2π‘Ž2βˆ’π‘βˆ’1π‘§π‘§Γ—ξ€Ίβˆ’2π‘‡π‘§π‘π‘§π‘‘βˆ’2𝑋𝑧𝑝π‘₯π‘§βˆ’2π‘Œπ‘§π‘π‘¦π‘§βˆ’π‘‹π‘§π‘§π‘π‘₯βˆ’π‘Œπ‘§π‘§π‘π‘¦βˆ’π‘‡π‘§π‘§π‘π‘‘+ξ€·4π‘Ž2𝑧+π‘Ž4π‘§π‘§βˆ’4π‘Ž1π‘§ξ€Έξ€·π‘ŽπœŒ+21π‘§π‘§βˆ’π‘Ž2𝑧𝑧1π‘βˆ’4𝑦2+π‘₯2𝑓2π‘Ž2𝑧𝑧+π‘¦π‘“π‘Ž7π‘§π‘§π‘‘βˆ’π‘₯π‘“π‘Ž8𝑧𝑧𝑑+π‘₯π‘Ž7𝑧𝑧𝑑𝑑+π‘¦π‘Ž8𝑧𝑧𝑑𝑑+𝑝0π‘§π‘§π‘€βˆ’ξ‚†ξ€»ξ€Ύπ‘£π‘“π‘Ž7π‘§π‘‘βˆ’π‘’π‘“π‘Ž8π‘§π‘‘βˆ’π‘π‘¦π‘¦ξ€·π‘£π‘¦π‘Ž1π‘§βˆ’π‘£π‘₯π‘Ž5π‘§βˆ’π‘£π‘Ž8𝑧+π‘₯π‘Ž7𝑧𝑑𝑑𝑑+π‘¦π‘Ž8𝑧𝑑𝑑𝑑+𝑣2π‘Ž1βˆ’3π‘Ž2ξ€Έβˆ’π‘£π‘Ž4𝑧𝑝𝑦𝑧+ξ€·2π‘Ž1βˆ’π‘Ž4π‘§βˆ’3π‘Ž2𝑝𝑧𝑑+(𝑦𝑓+𝑒)π‘Ž7π‘§π‘‘π‘‘βˆ’(π‘₯π‘“βˆ’π‘£)π‘Ž8𝑧𝑑𝑑+ξ€·2π‘Ž1βˆ’3π‘Ž2ξ€Έπ‘§π‘π‘‘βˆ’ξ€·π‘Ž2𝑑+π‘Ž3𝑧𝑝𝑑𝑑+𝑝0π‘§π‘‘βˆ’12𝑒π‘₯𝑓+π‘£π‘¦π‘“βˆ’π‘π‘¦π‘¦ξ€Έπ‘£π‘₯π‘‘π‘“π‘Ž2𝑧+ξ‚ƒβˆ’π‘¦π‘Ž5𝑧+π‘Ž7π‘§βˆ’π‘’π‘Ž4π‘§βˆ’π‘₯π‘Ž1π‘§βˆ’12π‘¦π‘“π‘Ž2π‘§π‘‘βˆ’π‘’π‘Ž3π‘§βˆ’π‘’π‘Ž2𝑧𝑑+𝑒2π‘Ž1βˆ’3π‘Ž2𝑝π‘₯π‘§βˆ’ξ‚ƒ(π‘¦π‘£βˆ’π‘’π‘₯)π‘Ž5𝑧+π‘£π‘Ž7𝑧+π‘’π‘Ž8π‘§βˆ’(𝑣π‘₯+𝑒𝑦)π‘Ž1𝑧+12𝑑(𝑒π‘₯βˆ’π‘¦π‘£)π‘“π‘Ž2𝑧𝑝π‘₯𝑦+ξ‚ƒπ‘’π‘Ž1π‘§βˆ’12(𝑣𝑑+𝑦)π‘“π‘Ž2π‘§βˆ’π‘£π‘Ž5𝑧+π‘Ž7π‘§π‘‘βˆ’2π‘’π‘Ž2𝑧𝑝π‘₯+ξ‚ƒπ‘Ž1𝑧1𝑣+2(𝑒𝑑+π‘₯)π‘“π‘Ž2𝑧+π‘’π‘Ž5π‘§βˆ’2π‘£π‘Ž2𝑧+π‘Ž8𝑧𝑑𝑝𝑦+π‘’βˆ’π‘¦π‘Ž5π‘§βˆ’π‘₯π‘Ž1π‘§βˆ’12π‘¦π‘“π‘Ž2𝑧𝑑+π‘Ž7𝑧𝑝π‘₯π‘₯+ξ‚€12π‘₯π‘“π‘Ž2𝑧𝑑+π‘Ž8𝑧+π‘₯π‘Ž5π‘§βˆ’π‘¦π‘Ž1π‘§βˆ’π‘£π‘Ž3π‘§βˆ’π‘£π‘Ž2π‘§π‘‘ξ‚π‘π‘¦π‘‘ξ‚‡π‘βˆ’1𝑧𝑧.(3.3)

In addition to the symmetries (2.8) with (3.3) are not point symmetries, the time scaling invariance related to π‘Ž2=π‘Ž2(𝑧)𝑉=π‘Ž2π‘‘πœ•π‘‘+12π‘Ž2π‘¦π‘“π‘‘πœ•π‘₯βˆ’12π‘Ž2π‘₯π‘“π‘‘πœ•π‘¦+12π‘Ž2(π‘‘π‘“π‘£βˆ’2𝑒+𝑓𝑦)πœ•π‘’βˆ’12π‘Ž2(𝑑𝑓𝑒+2𝑣+𝑓π‘₯)πœ•π‘£βˆ’14π‘Ž2𝑦8𝑝+2+π‘₯2𝑓2ξ€»πœ•π‘+14π‘Ž2𝑧4𝑑𝑝𝑑+2𝑦𝑓𝑑𝑝π‘₯βˆ’2π‘₯𝑓𝑑𝑝𝑦+𝑦2+π‘₯2𝑓2ξ€»πœ•πœŒ+2π‘Ž2βˆ’π‘βˆ’1π‘§π‘§ξ‚ƒπ‘Ž2𝑧π‘₯π‘“π‘‘π‘π‘¦π‘§βˆ’2π‘‘π‘π‘§π‘‘βˆ’π‘¦π‘“π‘‘π‘π‘₯𝑧+1+4𝜌2π‘Ž2𝑧𝑧π‘₯π‘“π‘‘π‘π‘¦βˆ’π‘‘π‘¦π‘“π‘π‘₯βˆ’2π‘‘π‘π‘‘ξ€Έβˆ’π‘Ž2𝑧𝑧12𝑝+4𝑦2+π‘₯2𝑓2π‘€βˆ’ξ‚ƒ12𝑝𝑦𝑦𝑣π‘₯π‘‘βˆ’π‘’π‘₯π‘“βˆ’π‘£π‘¦π‘“π‘“π‘Ž2π‘§βˆ’3π‘Ž2𝑝𝑦𝑧+π‘π‘§π‘‘ξ€Έβˆ’π‘Ž2𝑧3𝑝𝑑+π‘‘π‘π‘‘π‘‘ξ€Έβˆ’12ξ€Ί(𝑦𝑓+2𝑒)π‘Ž2𝑧𝑑+6π‘Ž2𝑒𝑝π‘₯𝑧+12𝑑(𝑒π‘₯βˆ’π‘¦π‘£)π‘“π‘Ž2𝑧𝑝π‘₯π‘¦βˆ’12[]π‘Ž(𝑣𝑑+𝑦)𝑓+4𝑒2𝑧𝑝π‘₯+12[(]π‘Žπ‘’π‘‘+π‘₯)π‘“βˆ’4𝑣2π‘§π‘π‘¦βˆ’12π‘’π‘¦π‘“π‘Ž2𝑧𝑑𝑝π‘₯π‘₯+12π‘Ž2𝑧(π‘₯π‘“π‘‘βˆ’2𝑣𝑑)π‘π‘¦π‘‘ξ‚„π‘βˆ’1π‘§π‘§ξ‚‡πœ•π‘€(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 πœ“, such that𝑒=βˆ’πœ“π‘¦,𝑣=πœ“π‘₯.(4.1) We call a symmetry potential symmetry if it is explicitly dependent on a potential, say, πœ“, (or its derivatives) for the fields 𝑒,𝑣,𝑝,𝜌, and 𝑀 described by (1.1)–(1.5). Obviously, a potential symmetry for the field system {𝑒,𝑣,𝑝,𝜌,𝑀} is a nonlocal one though it may be a Lie symmetry for the enlarged system {𝑒,𝑣,𝑝,𝜌,𝑀,πœ“}. From the general symmetry expression (3.3), we know that it is useful to introduce the following potential vector fieldsβƒ—ξ€·πœ“π‘£=π‘₯,πœ“π‘¦,πœ“π‘§,πœ“π‘‘ξ€Έ=𝑣1=𝑣,𝑣2=βˆ’π‘’,𝑣3,𝑣4ξ€Έ,⃗𝑝𝑓=π‘₯,𝑝𝑦,𝑝𝑧,𝑝𝑑=𝑓1,𝑓2,𝑓3=βˆ’πœŒ,𝑓4ξ€Έ,(4.2) and symmetric potential tensor fieldsπœ“π‘–π‘—=πœ“π‘—π‘–=πœ“π‘₯𝑖π‘₯𝑗,𝑖,𝑗=1,2,3,4,π‘₯1=π‘₯,π‘₯2=𝑦,π‘₯3=𝑧,π‘₯4𝑝=𝑑,𝑖𝑗=𝑝𝑗𝑖=𝑝π‘₯𝑖π‘₯𝑗,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 𝑒=βˆ’π‘£2, 𝑣=𝑣1,𝑝,𝑀,πœ“,𝑣3,𝑣4,𝑓1,𝑓2, 𝜌=βˆ’π‘“3,𝑓4,πœ“π‘–π‘— and 𝑝𝑖𝑗, 𝑖≀𝑗=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 𝑒,𝑣,𝑀,𝑝,𝜌 and the potential πœ“:𝑉1=π‘Ž1ξ€·π‘₯πœ•π‘₯+π‘¦πœ•π‘¦+2πœ“πœ•πœ“+π‘’πœ•π‘’+π‘£πœ•π‘£+2π‘πœ•π‘+2πœŒπœ•πœŒξ€Έ+π‘Ž1𝑧×π‘₯𝑓1+𝑦𝑓2ξ€Έπœ•βˆ’2π‘πœŒ+π‘βˆ’133Γ—ξ€Ίξ€·2𝑀2𝜌+π‘₯𝑝13+𝑦𝑝23ξ€Έξ€·+𝑒π‘₯𝑝11+𝑦𝑝12βˆ’π‘“1ξ€Έξ€·+𝑣π‘₯𝑝12+𝑦𝑝22βˆ’π‘“2ξ€Έβˆ’2𝑓4+π‘₯𝑃13+𝑦𝑝24ξ€»πœ•π‘€ξ€Ύ+π‘Ž1π‘§π‘§π‘βˆ’133𝑀π‘₯𝑓1+𝑦𝑓2ξ€Έπœ•βˆ’2𝑝𝑀,𝑉(4.4)2=π‘Ž2ξ€Ίπ‘‘πœ•π‘‘βˆ’πœ“πœ•πœ“βˆ’π‘’πœ•π‘’βˆ’π‘£πœ•π‘£+(π‘“πœ“βˆ’2𝑝)πœ•π‘βˆ’ξ€·π‘“π‘£3ξ€Έπœ•+2𝜌𝜌+𝑀+π‘“π‘βˆ’133ξ€·π‘€πœ“33+πœ“34+π‘’πœ“13+π‘£πœ“23πœ•ξ€Έξ€Έπ‘€ξ€»+π‘Ž2𝑧𝑑𝑓4ξ€Έπœ•βˆ’π‘“πœ“+2π‘πœŒ+π‘Ž2𝑧2𝑑𝑀2+π‘βˆ’133ξ€Ίξ€·2𝑀2𝜌+𝑓𝑣3+𝑑𝑒𝑝13+𝑑𝑣𝑝23ξ€Έξ€·βˆ’π‘’2𝑓1+𝑑𝑝14ξ€Έξ€·βˆ’π‘£2𝑓2+𝑑𝑝24ξ€Έ+𝑣4π‘“βˆ’3𝑓4βˆ’π‘‘π‘44πœ•ξ€»ξ€Ύπ‘€+π‘Ž2π‘§π‘§π‘βˆ’133π‘€ξ€·π‘“πœ“βˆ’2π‘βˆ’π‘‘π‘“4ξ€Έπœ•π‘€,(4.5)𝑉3=π‘Ž3πœ•π‘‘+π‘Ž3𝑧𝑓4πœ•πœŒ+ξ€Ί2𝑀2+π‘βˆ’133ξ€·ξ€·2𝑀𝑒𝑝13+𝑣𝑝23ξ€Έβˆ’π‘’π‘14βˆ’π‘£π‘24βˆ’π‘44πœ•ξ€Έξ€»π‘€ξ€Ύβˆ’π‘Ž3π‘§π‘§π‘βˆ’133𝑀𝑓4πœ•π‘€,𝑉(4.6)4=π‘Ž4πœ•π‘§βˆ’π‘Ž4π‘§ξ€·πœŒπœ•πœŒ+π‘€πœ•π‘€ξ€Έ+π‘Ž4π‘§π‘§π‘βˆ’133π‘€πœŒπœ•π‘€,𝑉(4.7)5=βˆ’π‘Ž5ξ€Ίπ‘₯πœ•π‘¦βˆ’π‘¦πœ•π‘₯βˆ’π‘£πœ•π‘’+π‘’πœ•π‘£ξ€»+π‘Ž5𝑧π‘₯𝑓2βˆ’π‘¦π‘“1ξ€Έπœ•πœŒ+π‘βˆ’133ξ€Ί2𝑦𝑝13βˆ’π‘₯𝑝23𝑀+𝑦𝑝11βˆ’π‘₯𝑝12βˆ’π‘“2𝑓𝑒+1+𝑦𝑝12βˆ’π‘₯𝑝22𝑣+𝑦𝑝14βˆ’π‘₯𝑝24ξ€»πœ•π‘€ξ€Ύ+π‘Ž5π‘§π‘§π‘βˆ’133𝑦𝑓1βˆ’π‘₯𝑓2ξ€Έπ‘€πœ•π‘€,𝑉(4.8)6=βˆ’π‘Ž6π‘₯π‘‘πœ•π‘¦βˆ’π‘¦π‘‘πœ•π‘₯+12ξ€·π‘₯2+𝑦2ξ€Έπœ•πœ“βˆ’(𝑣𝑑+𝑦)πœ•π‘’+(𝑒𝑑+π‘₯)πœ•π‘£+12ξ€·4πœ“+𝑓π‘₯2+𝑓𝑦2ξ€Έπœ•π‘βˆ’2𝑣3πœ•πœŒ+2𝑝33ξ€·π‘’πœ“13+π‘£πœ“23+π‘€πœ“33+πœ“34ξ€Έπœ•π‘€ξ‚Ήβˆ’π‘Ž6𝑧12ξ€·2𝑑π‘₯𝑓2βˆ’2𝑑𝑦π‘₯1βˆ’4πœ“βˆ’π‘“π‘₯2βˆ’π‘“π‘¦2ξ€Έπœ•πœŒ+π‘βˆ’133𝑒𝑓π‘₯βˆ’π‘“2π‘‘βˆ’π‘₯𝑑𝑝12+𝑦𝑑𝑝11ξ€Έξ€·+𝑣𝑦𝑓+𝑑𝑓1βˆ’π‘₯𝑑𝑝22+𝑦𝑑𝑝12ξ€Έξ€·+2𝑀2𝑣3+𝑦𝑑𝑝13βˆ’π‘₯𝑑𝑝23ξ€Έ+𝑦𝑓1+2𝑣4βˆ’π‘₯𝑑𝑝24+𝑦𝑑𝑝14βˆ’π‘₯𝑓2ξ‚„πœ•π‘€ξ‚‡βˆ’12π‘Ž6π‘§π‘§π‘βˆ’133ξ€·4πœ“+2𝑦𝑑𝑓1βˆ’2π‘₯𝑑𝑓2+𝑓π‘₯2+𝑓𝑦2ξ€Έπ‘€πœ•π‘€,𝑉(4.9)7=βˆ’π‘Ž7πœ•π‘₯βˆ’π‘Ž7π‘‘ξ€·βˆ’π‘¦πœ•πœ“+πœ•π‘’βˆ’π‘“π‘¦πœ•π‘ξ€Έβˆ’π‘Ž7𝑑𝑑π‘₯πœ•π‘βˆ’π‘Ž7𝑧𝑓1πœ•πœŒβˆ’π‘βˆ’133ξ€·2𝑀𝑝13+𝑒𝑝11+𝑝14+𝑣𝑝12ξ€Έπœ•π‘€ξ€Έ+π‘Ž7π‘§π‘‘ξ€Ίπ‘“π‘¦πœ•πœŒβˆ’π‘βˆ’133𝑓1ξ€Έπœ•+𝑓𝑣𝑀+π‘Ž7π‘§π‘‘π‘‘π‘βˆ’133(𝑒+𝑓𝑦)πœ•π‘€βˆ’π‘Ž7𝑧𝑧𝑑π‘₯πœ•πœŒβˆ’π‘βˆ’133π‘€π‘¦π‘“πœ•π‘€ξ€Έ+ξ€·π‘Ž7𝑧𝑑𝑑𝑑π‘₯+π‘Ž7𝑧𝑧𝑑𝑑π‘₯𝑀+π‘Ž7𝑧𝑧𝑓1π‘€ξ€Έπ‘βˆ’133πœ•π‘€,𝑉(4.10)8=βˆ’π‘Ž8πœ•π‘¦βˆ’π‘Ž8𝑑π‘₯πœ•πœ“+πœ•π‘£+π‘₯π‘“πœ•π‘ξ€Έ+ξ€·π‘Ž8𝑧𝑧𝑀𝑓2+π‘Ž8π‘§π‘‘π‘‘π‘‘π‘¦βˆ’π‘Ž8𝑧𝑧𝑑𝑝𝑀π‘₯π‘“βˆ’133πœ•π‘€+π‘Ž8π‘‘π‘‘π‘¦πœ•π‘βˆ’π‘Ž8𝑧𝑓2πœ•πœŒβˆ’π‘βˆ’133ξ€·2𝑀𝑝23+𝑣𝑝22+𝑝24+𝑒𝑝12ξ€Έπœ•π‘€ξ€»+π‘Ž8π‘§π‘§π‘‘π‘‘π‘βˆ’133π‘¦π‘€πœ•π‘€+π‘Ž8𝑧𝑑𝑓π‘₯πœ•πœŒ+π‘βˆ’133𝑓2ξ€Έπœ•βˆ’π‘“π‘’π‘€ξ€»βˆ’π‘Ž8π‘§π‘‘π‘‘ξ€Ίπ‘¦πœ•πœŒβˆ’π‘βˆ’133(π‘£βˆ’π‘₯𝑓)πœ•π‘€ξ€»,𝑉(4.11)9=π‘Ž9πœ•π‘βˆ’π‘Ž9π‘§πœ•πœŒβˆ’ξ€·π‘Ž9𝑑𝑑𝑀+π‘Ž9π‘§π‘‘ξ€Έπ‘βˆ’133πœ•π‘€,𝑉(4.12)10=π‘Ž10πœ•πœ“,(4.13) where π‘Žπ‘–, 𝑖=1,…,6, are arbitrary functions of 𝑧 and π‘Ž7,π‘Ž8,π‘Ž9,π‘Ž10 are arbitrary functions of {𝑧,𝑑}.

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 {𝑒,𝑣,𝑝,𝜌,𝑀} 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.

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 problemdπ‘₯ξ…ž=𝑋′dπ‘¦ξ…ž=π‘Œβ€²dπ‘§ξ…ž=𝑍′dπ‘‘ξ…ž=𝑇′dπœ“ξ…ž=Ξ¨β€²dπ‘’ξ…ž+π‘ˆβ€²dπ‘£ξ…ž=𝑉′dπ‘€ξ…ž=π‘Šβ€²dπ‘ξ…žξ€½π‘₯𝑃′=β‹―=π‘‘πœ–,ξ…ž,π‘¦ξ…ž,π‘§ξ…ž,π‘‘ξ…ž,πœ“ξ…ž,π‘’ξ…ž,π‘£ξ…ž,π‘€ξ…ž,π‘ξ…žξ€Ύβˆ£,β€¦πœ–=0={π‘₯,𝑦,𝑧,𝑑,πœ“,𝑒,𝑣,𝑀,𝑝,…},(5.1) where π‘₯ξ…ž,π‘¦ξ…ž,π‘§ξ…ž,π‘‘ξ…ž,π‘’ξ…ž,π‘£ξ…ž,π‘€ξ…ž,π‘ξ…ž,…, are functions of πœ– and π‘‹ξ…ž,π‘Œξ…ž,π‘ξ…ž,π‘‡ξ…ž,π‘ˆξ…ž,π‘‰ξ…ž,π‘ƒξ…ž,… can be read off from the linear combination of the generators of the algebra 𝒱. For instance,𝑋′=βˆ’π‘Žξ…ž7+π‘Žξ…ž1π‘₯β€²βˆ’π‘‘ξ…ž2π‘¦β€²βˆ’π‘”ξ…žπ‘¦ξ…žπ‘‘ξ…ž,π‘Œβ€²=βˆ’π‘Žξ…ž8+π‘Žξ…ž1𝑦′+π‘‘ξ…ž2π‘₯β€²+π‘”ξ…žπ‘₯ξ…žπ‘‘ξ…ž,π‘ξ…ž=π‘Žξ…ž4ξ€·π‘§ξ…žξ€Έ,π‘‡ξ…ž=π‘‘ξ…ž0+π‘‘ξ…žπ‘‘ξ…ž1,π‘ˆξ…ž=βˆ’π‘Žξ…ž7𝑑′+ξ€·π‘ξ…ž2βˆ’π‘‘ξ…ž1ξ€Έπ‘’ξ…žβˆ’π‘‘ξ…ž2π‘£ξ…žβˆ’π‘”ξ…žξ€·π‘£ξ…žπ‘‘ξ…ž+π‘¦ξ…žξ€Έ,π‘‰ξ…ž=βˆ’π‘Žξ…ž8𝑑′+ξ€·π‘ξ…ž2βˆ’π‘‘ξ…ž1ξ€Έπ‘£ξ…ž+π‘‘ξ…ž2π‘’ξ…ž+π‘”ξ…žξ€·π‘’ξ…žπ‘‘ξ…ž+π‘₯ξ…žξ€Έ,𝑃′=π‘“π‘Žξ…ž7π‘‘β€²π‘¦ξ…žβˆ’π‘₯ξ…žπ‘Žξ…ž7π‘‘β€²π‘‘β€²βˆ’π‘“π‘Žξ…ž8𝑑′π‘₯ξ…ž+π‘¦ξ…žπ‘Žξ…ž8𝑑′𝑑′,…,(5.2) with {π‘Žξ…ž1β‰‘π‘Ž1(𝑧′), π‘Žξ…ž3β‰‘π‘Ž3(𝑧′), π‘Žξ…ž2β‰‘π‘Ž2(𝑧′), π‘Žξ…ž5β‰‘π‘Ž5(𝑧′), 𝑔′≑𝑔(𝑧′), π‘Žξ…ž4β‰‘π‘Ž4(𝑧′),…} being arbitrary functions of 𝑧′ and {π‘Žξ…ž7β‰‘π‘Ž7(𝑧′,𝑑′), π‘Žξ…ž8β‰‘π‘Ž8(π‘§ξ…ž,𝑑′),…} being arbitrary functions of the indicated variables.

Because the density 𝜌 and the vertical velocity 𝑀 can be simply obtained from (1.4) and (1.5) when the fields {𝑒,𝑣,𝑝,πœ“} are known, we write down only the transformation group theorem on the fields {𝑒,𝑣,𝑝,πœ“} here.

Theorem 5.1. If {π‘ˆ(π‘₯,𝑦,𝑧,𝑑), 𝑉(π‘₯,𝑦,𝑧,𝑑), 𝑃(π‘₯,𝑦,𝑧,𝑑), Ξ¨(π‘₯,𝑦,𝑧,𝑑)} is a solution of the system (1.1)–(1.3) and (4.1), so is {𝑒,𝑣,𝑝,πœ“} with 𝑒=βˆ’π›Ό6π‘¦βˆ’π›Ό8𝑑sinπœƒ+𝛼7𝑑cosπœƒ+𝛼2𝛼1βˆ’1[],π‘ˆ(πœ‰,πœ‚,𝜁,𝜏)cosπœƒβˆ’π‘‰(πœ‰,πœ‚,𝜁,𝜏)sinπœƒπ‘£=𝛼6π‘₯+𝛼7𝑑sinπœƒ+𝛼8𝑑cosπœƒ+𝛼2𝛼1βˆ’1[],𝛼𝑉(πœ‰,πœ‚,𝜁,𝜏)cosπœƒ+π‘ˆ(πœ‰,πœ‚,𝜁,𝜏)sinπœƒπ‘=2𝛼21𝛼2𝑃(πœ‰,πœ‚,𝜁,𝜏)+𝑓+2𝛼6βˆ’π‘“π›Ό2ξ€ΈΞ¨ξ€»(πœ‰,πœ‚,𝜁,𝜏)+𝛼9+𝛼62𝑓+𝛼6π‘₯ξ€Έξ€·2+𝑦2ξ€Έ+(π‘₯cosπœƒ+𝑦sinπœƒ)𝑓+2𝛼6𝛼8π‘‘βˆ’π›Ό7𝑑𝑑+(π‘₯sinπœƒβˆ’π‘¦cosπœƒ)𝑓+2𝛼6𝛼7𝑑+𝛼8𝑑𝑑,π›Όπœ“=2𝛼21Ξ¨(πœ‰,πœ‚,𝜁,𝜏)+𝛼10+𝛼62𝑦cos(𝛼6𝑑)βˆ’π‘₯sin(𝛼6𝑑)2+𝛼𝑦sin6𝑑𝛼+π‘₯cos6𝑑2𝛼+π‘₯8𝑑cosπœƒ+𝛼7𝑑𝛼sinπœƒ+𝑦8𝑑sinπœƒβˆ’π›Ό7𝑑,cosπœƒ(5.3) where πœ‰=𝛼1(𝑦sinπœƒ+π‘₯cosπœƒβˆ’π›Ό7), πœ‚=𝛼1(𝑦cosπœƒβˆ’π‘₯sinπœƒβˆ’π›Ό8), 𝜁=𝛼4, 𝜏=𝛼2𝑑+𝛼3, πœƒβ‰‘π›Ό6𝑑+𝛼5, {𝛼1,𝛼2,𝛼3,𝛼4,𝛼5} and 𝛼6 are arbitrary functions of 𝑧 while {𝛼7,𝛼8,𝛼9} and 𝛼10 are arbitrary functions of {𝑧,𝑑}.

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 taking𝛼1⟢1+πœ–π‘Ž1,𝛼2⟢1+πœ–π‘Ž2,𝛼4βŸΆπ‘§+πœ–π‘Ž4,π›Όπ‘–βŸΆπœ–π‘Žπ‘–,𝑖≠1,2,4,(5.4)

with the infinitesimal parameter πœ–.

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:𝑒=βˆ’2𝐹(π‘Ÿ,𝑧)𝑦,π‘Ÿβ‰‘π‘₯2+𝑦2,ξ€œξ€œπΉπ‘£=2π‘₯𝐹(π‘Ÿ,𝑧),𝑝=𝐹(π‘Ÿ,𝑧)(2𝐹(π‘Ÿ,𝑧)+𝑓)π‘‘π‘Ÿ,𝜌=βˆ’π‘§1(π‘Ÿ,𝑧)(4𝐹(π‘Ÿ,𝑧)+𝑓)π‘‘π‘Ÿ,𝑀=βˆ’π‘π‘§π‘§ξ€·π‘π‘§π‘‘+𝑒𝑝π‘₯𝑧+𝑣𝑝𝑦𝑧,(6.1) with arbitrary function 𝐹≑𝐹(π‘Ÿ,𝑧) of {π‘Ÿ,𝑧}.

Applying the group theorem of the last section to solution (6.1), we have𝑒=βˆ’π›Ό6π‘¦βˆ’π›Ό8𝑑sinπœƒ+𝛼7𝑑cosπœƒβˆ’2𝛼2𝛼1βˆ’1ξ€Ίξ€·πœ‚πΉπ‘…,𝛼4ξ€Έξ€·cosπœƒ+πœ‰πΉπ‘…,𝛼4ξ€Έξ€»,sinπœƒπ‘£=𝛼6π‘₯+𝛼7𝑑sinπœƒ+𝛼8𝑑cosπœƒ+2𝛼2𝛼1βˆ’1ξ€Ίξ€·πœ‰πΉπ‘…,𝛼4ξ€Έξ€·cosπœƒβˆ’πœ‚πΉπ‘…,𝛼4ξ€Έξ€»,𝛼sinπœƒπ‘=2𝛼21𝛼2ξ€œπΉξ€·π‘…,𝛼4ξ€·ξ€Έξ€·2𝐹𝑅,𝛼4ξ€Έξ€Έξ€·+𝑓𝑑𝑅+𝑓+2𝛼6βˆ’π‘“π›Ό2ξ€Έξ€œπΉξ€·π‘…,𝛼4𝑑𝑅+𝛼10+𝛼62𝑓+𝛼6π‘₯ξ€Έξ€·2+𝑦2ξ€Έ+(π‘₯cosπœƒ+𝑦sinπœƒ)𝑓+2𝛼6𝛼8π‘‘βˆ’π›Ό7𝑑𝑑+(π‘₯sinπœƒβˆ’π‘¦cosπœƒ)𝑓+2𝛼6𝛼7𝑑+𝛼8𝑑𝑑,𝜌=βˆ’π‘π‘§,1𝑀=βˆ’π‘π‘§π‘§ξ€·π‘π‘§π‘‘+𝑒𝑝π‘₯𝑧+𝑣𝑝𝑦𝑧,(6.2) with𝑅≑𝛼21𝑦sinπœƒ+π‘₯cosπœƒβˆ’π›Ό7ξ€Έ2+𝑦cosπœƒβˆ’π‘₯sinπœƒβˆ’π›Ό8ξ€Έ2ξ‚„.(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, π‘’βˆΌβˆ’π›Ό6𝑦,π‘£βˆΌπ›Ό6π‘₯ and the 𝐹(𝑅, 𝛼4) parts of the solution exhibit abundant vortex structure. {π‘₯=𝛼7, 𝑦=𝛼8} is expressed as the center of the vortices while the second part ({π‘’βˆΌβˆ’π›Ό8𝑑sinπœƒ+𝛼7𝑑cosπœƒ, π‘£βˆΌπ›Ό7𝑑sinπœƒ+𝛼8𝑑cosπœƒ}) 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),𝑅𝑒=βˆ’πΉ1ξ€Έ,𝑧𝑦,𝑅1≑π‘₯2+𝑦2𝑅,𝑣=𝐹1𝛼,𝑧π‘₯,𝑀=βˆ’10𝑧𝑑(𝑧,𝑑)βˆ«ξ‚ƒξ€·ξ€·π‘…4𝐹1𝐹,𝑧+𝑓𝑧𝑧𝑅1ξ€Έ,𝑧+4𝐹2𝑧𝑅1ξ€Έ,𝑧2ξ‚„d𝑅1+𝛼10𝑧𝑧(,𝑧,𝑑)(6.4) with only two arbitrary functions 𝐹(𝑅1,𝑧) and 𝛼10(𝑧,𝑑) 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 that𝑒=βˆ’π‘0𝑐𝑦sech1𝑧+𝑐5𝑐tanh1𝑧+𝑐5𝑅sech1𝑐2ξ‚Ά,(6.5)𝑣=𝑐0𝑐π‘₯sech1𝑧+𝑐5𝑐tanh1𝑧+𝑐5𝑅sech1𝑐2ξ‚Ά,(6.6)𝑀=𝑐4ξ‚Έ2𝑐3+𝑐20𝑐21𝑐2sech2𝑐1𝑧+𝑐5ξ€Έξ€·10sech4𝑐1𝑧+𝑐5ξ€Έβˆ’11sech2𝑐1𝑧+𝑐5𝑅+2tanh1𝑐2ξ‚Ά+𝑐0𝑐21𝑐2𝑐𝑓tanh1𝑧+𝑐5𝑐sech1𝑧+𝑐5ξ€Έξ€·1βˆ’6sech2𝑐1𝑧+𝑐5𝑅arctanexp1𝑐2ξ‚Άξ‚Άξ‚Ήβˆ’1,(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 βˆ’π‘5/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=βˆ’1,𝑐1=10,𝑐3=1000,𝑐4=𝑓=1,𝑐2=5.(6.8)

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 relationsdπ‘₯d𝑑=𝑒,d𝑦d𝑑=𝑣,d𝑧d𝑑=𝑀.(6.9) Since it is difficult to obtain an analytical solution of system (6.9), we solve it numerically with the initial conditions π‘₯(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].

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 {𝑒,𝑣,𝑝,𝜌,𝑀}. 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 {𝑒,𝑣,𝑝} exhibit eight arbitrary functions with five of them being arbitrary functions of z and others being arbitrary functions of {𝑧,𝑑} while the time scaling invariance is still companied by a time-dependent rotation. Though these symmetries are the Lie symmetries for the subsystem {𝑒,𝑣,𝑝}, they are generalized symmetries for the full system {𝑒,𝑣,𝑝,𝜌,𝑀}. 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 4Γ—4 symmetric tensors πœ“π‘–π‘— 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,𝑑}. 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 {𝑒,𝑣,𝑝,𝜌,𝑀} 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 {𝑒,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 𝑀. 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 𝑒,𝑣,𝑝,𝜌, and 𝑀 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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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.


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