Potential Symmetry Studies on a Rotating Fluid System

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 [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]:𝑢𝑢_𝑥+𝑣_𝑦=0,(1.1)_𝑡𝑣+𝑢𝑢_𝑥+𝑣𝑢_𝑦−𝑓𝑣+𝑝_𝑥=0,(1.2)_𝑡+𝑢𝑣_𝑥+𝜌𝑣+𝑣𝑝_𝑦𝑧+𝑓𝑢+𝑝_𝑦=0,(1.3)𝜌=0,(1.4)_𝑡+𝑢𝜌_𝑥+𝑣𝜌_𝑦+𝑤𝜌_𝑧=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), 𝑝_𝑥𝑦=𝑝_𝑦𝑥, 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),𝜎≡⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝𝜎^𝑢𝜎^𝑣𝜎^𝑝𝜎^𝜌𝜎^𝑤⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠,(2.1) is defined as a solution of the linearized equations of (1.1)–(1.5):𝜎^𝑢𝑥+𝜎^𝑣𝑦=0,(2.2)𝜎^𝑢𝑡+𝜎^𝑢𝑢_𝑥+𝑢𝜎^𝑢𝑢_𝑥+𝜎^𝑣𝑢_𝑦+𝑣𝜎^𝑢𝑢_𝑦−𝑓𝜎^𝑣+𝜎^𝑝𝑥=0,(2.3)𝜎^𝑣𝑡+𝜎^𝑢𝑣_𝑥+𝑢𝜎^𝑣𝑥+𝜎^𝑣𝑣_𝑦+𝑣𝜎^𝑣𝑦+𝑓𝜎^𝑢+𝜎^𝑝𝑦=0,(2.4)𝜎^𝜌+𝜎^𝑝𝑧=0,(2.5)𝜎^𝜌𝑡+𝜎^𝑢𝜌_𝑥+𝑢𝜎^𝜌𝑥+𝜎^𝑣𝜌_𝑦+𝑣𝜎^𝜌𝑦+𝜎^𝑤𝜌_𝑧+𝑤𝜎^𝜌𝑧=0,(2.6) which means the model system is form invariant under the transformation⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝𝑤𝑣𝜌𝑝𝑢⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠⟶⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝𝑤𝑣𝜌𝑝𝑢⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠+𝜖⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝𝜎^𝑢𝜎^𝑣𝜎^𝑝𝜎^𝜌𝜎^𝑤⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠(2.7) with infinitesimal 𝜖. The Lie symmetries of (1.1)–(1.5) have the form⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝𝜎^𝑢𝜎^𝑣𝜎^𝑝𝜎^𝜌𝜎^𝑤⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝𝑊𝑈𝑉𝑄𝑃⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠−𝑋⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝𝑢_𝑥𝑣_𝑥𝑝_𝑥𝜌_𝑥𝑤_𝑥⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠−𝑌⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝𝑢_𝑦𝑣_𝑦𝑝_𝑦𝜌_𝑦𝑤_𝑦⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠−𝑍⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝𝑢_𝑧𝑣_𝑧𝑝_𝑧𝜌_𝑧𝑤_𝑧⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠−𝑇⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝𝑢_𝑡𝑣_𝑡𝑝_𝑡𝜌_𝑡𝑤_𝑡⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠,(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 𝑢_𝑡,𝑣_𝑦,𝑝_𝑦,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=𝑎₁𝑥+12𝑎²𝑌=𝑎₁𝑓𝑡+2𝑎₅𝑦−𝑎₇(𝑡),𝑦−12𝑎²𝑍=2𝑎₁𝑓𝑡+2𝑎₅𝑥−𝑎₈(𝑡),𝑇=𝑎−₂𝑎₂𝑧+𝑎₄,𝑈=−𝑎_7𝑡𝑡+𝑎₃,+𝑎₁+𝑎₂𝑢−12−𝑎²𝑓𝑡−2𝑎₅𝑣+12𝑎²𝑉=−𝑎_8𝑡𝑓𝑦,+12−𝑎²𝑓𝑡−2𝑎₅𝑢+𝑎₁−𝑎₂𝑣−12𝑎²𝑃=𝑎_7𝑡𝑓𝑥,−𝑓𝑎₈_𝑡𝑥+𝑎₇𝑓+𝑎_8𝑡_𝑡𝑦+2𝑎₁−𝑎₂𝑝−14𝑓²𝑄=−𝑐₁𝑎₂𝑦²+𝑥²+𝑐₁𝑧+𝑎₁₀(𝑡),𝑊=−3𝑎₂,+2𝑎₁𝑤,(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)𝜕_𝑥−𝑎_7𝑡𝜕_𝑢+𝑓𝑎_7𝑡𝑦+𝑥𝑎_7𝑡𝑡𝜕_𝑝,(2.12) which is 𝑥-translation for 𝑎₇ being constant and Galilean boost in 𝑥 direction for 𝑎₇∼𝑡;(d)generalized 𝑦-translation and Galilean invariance 𝑉₄=−𝑎₈(𝑡)𝜕_𝑦−𝑎_8𝑡𝜕_𝑣+𝑓𝑎_8𝑡𝑥−𝑦𝑎_8𝑡𝑡𝜕_𝑝,(2.13) which is 𝑦-translation for 𝑎₈ being constant and Galilean boost in 𝑦 direction for 𝑎₈∼𝑡;(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𝜕_𝑦−𝑣𝜕_𝑢+𝑢𝜕_𝑣;(2.16)(h)time scaling company with time-dependent rotation (𝑎₂-part of (2.8) with (2.9)) 𝑉₈=12𝑡𝑓𝑦𝜕^𝑥−12𝑡𝑓𝑥𝜕^𝑦+𝑡𝜕_𝑡−2𝑧𝜕_𝑧+𝑓2(𝑦+𝑣𝑡)−𝑢𝑣𝜕^𝑢−𝑣+𝑓2(𝑥+𝑢𝑡)𝜕^𝑣−2𝑝+𝑓²4𝑥²+𝑦²𝜕_𝑝−3𝑤𝜕_𝑤,(2.17) where 𝑎_𝑖,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 form⎛⎜⎜⎜⎜⎝𝜎^𝑢𝜎^𝑣𝜎^𝑝⎞⎟⎟⎟⎟⎠=𝑋⎛⎜⎜⎜⎜⎝𝑢_𝑥𝑣_𝑥𝑝_𝑥⎞⎟⎟⎟⎟⎠+𝑌⎛⎜⎜⎜⎜⎝𝑢_𝑦𝑣_𝑦𝑝_𝑦⎞⎟⎟⎟⎟⎠+𝑍⎛⎜⎜⎜⎜⎝𝑢_𝑧𝑣_𝑧𝑝_𝑧⎞⎟⎟⎟⎟⎠+𝑇⎛⎜⎜⎜⎜⎝𝑢_𝑡𝑣_𝑡𝑝_𝑡⎞⎟⎟⎟⎟⎠−⎛⎜⎜⎜⎜⎝𝑈𝑉𝑃⎞⎟⎟⎟⎟⎠,(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+12𝑎²𝑌=−1𝑦𝑓𝑡+𝑦𝑎₅−𝑎₇,2𝑎²𝑥𝑍𝑓=𝑡−𝑎𝑥₄𝑎₅+𝑦𝑎₁−𝑎₈,𝑇=𝑎₂(𝑧),𝑈=𝑎₁𝑡+𝑎₃,−𝑎₂𝑢+12𝑡𝑓𝑎²+2𝑎₅𝑣+12𝑎²𝑉=1𝑓𝑦−𝑎_7𝑡,2−2𝑎⁵−𝑡𝑓𝑎₂𝑢+𝑎₁−𝑎₂𝑣−12𝑎²𝑃=2𝑎₁𝑓𝑥−𝑎_8𝑡,−𝑎₂𝑝−14𝑎²𝑦²+𝑥²𝑓²−𝑥𝑎_7𝑡−𝑦𝑎_8𝑡𝑓+𝑥𝑎_7𝑡𝑡−𝑦𝑎_8𝑡𝑡+𝑎₉,(3.2) where 𝑎_𝑖, 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 𝑄=𝑎₂𝑡+𝑎₃_𝑧𝑝_𝑡+𝑥𝑎₁+12𝑎²𝑦𝑓𝑡+𝑦𝑎₅−𝑎₇_𝑧𝑝_𝑥+𝑦𝑎₁−12𝑎²𝑥𝑓𝑡−𝑥𝑎₅−𝑎₈_𝑧𝑝_𝑦+−𝑥𝑎_7𝑡𝑡−𝑦𝑎_8𝑡𝑡−𝑦𝑓𝑎_7𝑡+𝑥𝑓𝑎_8𝑡−𝑎₁₀+14𝑎²𝑦²+𝑥²𝑓²_𝑧𝑊=−2𝑎𝑎_4𝑧𝜌_4𝑧−2𝑎₁−𝑎₂_𝑧𝑝+2𝑎₁−𝑎₂𝜌,−2𝑎₁+2𝑎₂−𝑝^−1𝑧𝑧×−2𝑇_𝑧𝑝_𝑧𝑡−2𝑋_𝑧𝑝_𝑥𝑧−2𝑌_𝑧𝑝_𝑦𝑧−𝑋_𝑧𝑧𝑝_𝑥−𝑌_𝑧𝑧𝑝_𝑦−𝑇_𝑧𝑧𝑝_𝑡+4𝑎_2𝑧+𝑎_4𝑧𝑧−4𝑎_1𝑧𝜌+2𝑎_1𝑧𝑧−𝑎_2𝑧𝑧𝑝−14𝑦²+𝑥²𝑓²𝑎_2𝑧𝑧+𝑦𝑓𝑎_{7𝑧𝑧𝑡}−𝑥𝑓𝑎_{8𝑧𝑧𝑡}−𝑣+𝑓𝑥𝑎𝑎_{7𝑧𝑧𝑡𝑡7𝑧𝑡}+𝑦𝑎_{8𝑧𝑧𝑡𝑡}+𝑝_0𝑧𝑧𝑤+𝑥𝑎_{7𝑧𝑡𝑡𝑡}−𝑢𝑓𝑎_8𝑧𝑡−𝑝_𝑦𝑦𝑣𝑦𝑎_1𝑧−𝑣𝑥𝑎_5𝑧−𝑣𝑎_8𝑧+𝑦𝑎_{8𝑧𝑡𝑡𝑡}+𝑣2𝑎₁−3𝑎₂−𝑣𝑎_4𝑧𝑝_𝑦𝑧+2𝑎₁−𝑎_4𝑧−3𝑎₂𝑝_𝑧𝑡+(𝑦𝑓+𝑢)𝑎_{7𝑧𝑡𝑡}−(𝑥𝑓−𝑣)𝑎_{8𝑧𝑡𝑡}+2𝑎₁−3𝑎₂_𝑧𝑝_𝑡−𝑎₂𝑡+𝑎₃_𝑧𝑝_𝑡𝑡+𝑝_0𝑧𝑡−12𝑢𝑥𝑓+𝑣𝑦𝑓−𝑝^𝑦𝑦𝑣𝑥𝑡𝑓𝑎_2𝑧+−𝑦𝑎_5𝑧+𝑎_7𝑧−𝑢𝑎_4𝑧−𝑥𝑎_1𝑧−12𝑦𝑓𝑎^2𝑧𝑡−𝑢𝑎_3𝑧−𝑢𝑎_2𝑧𝑡+𝑢2𝑎₁−3𝑎₂𝑝_𝑥𝑧−(𝑦𝑣−𝑢𝑥)𝑎_5𝑧+𝑣𝑎_7𝑧+𝑢𝑎_8𝑧−(𝑣𝑥+𝑢𝑦)𝑎_1𝑧+12𝑡(𝑢𝑥−𝑦𝑣)𝑓𝑎^2𝑧𝑝^𝑥𝑦+𝑢𝑎_1𝑧−12(𝑣𝑡+𝑦)𝑓𝑎^2𝑧−𝑣𝑎_5𝑧+𝑎_7𝑧𝑡−2𝑢𝑎_2𝑧𝑝^𝑥+𝑎_1𝑧𝑣+12(𝑢𝑡+𝑥)𝑓𝑎^2𝑧+𝑢𝑎_5𝑧−2𝑣𝑎_2𝑧+𝑎_8𝑧𝑡𝑝^𝑦+𝑢−𝑦𝑎_5𝑧−𝑥𝑎_1𝑧−12𝑦𝑓𝑎^2𝑧𝑡+𝑎_7𝑧𝑝^𝑥𝑥+12𝑥𝑓𝑎^2𝑧𝑡+𝑎_8𝑧+𝑥𝑎_5𝑧−𝑦𝑎_1𝑧−𝑣𝑎_3𝑧−𝑣𝑎_2𝑧𝑡𝑝_𝑦𝑡𝑝^−1𝑧𝑧.(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𝜕_𝑥−12𝑎²𝑥𝑓𝑡𝜕_𝑦+12𝑎²(𝑡𝑓𝑣−2𝑢+𝑓𝑦)𝜕_𝑢−12𝑎²(𝑡𝑓𝑢+2𝑣+𝑓𝑥)𝜕_𝑣−14𝑎²8𝑝+𝑦²+𝑥²𝑓²𝜕_𝑝+14𝑎^2𝑧4𝑡𝑝_𝑡+2𝑦𝑓𝑡𝑝_𝑥−2𝑥𝑓𝑡𝑝_𝑦+𝑦²+𝑥²𝑓²𝜕_𝜌+2𝑎₂−𝑝^−1𝑧𝑧𝑎^2𝑧𝑥𝑓𝑡𝑝_𝑦𝑧−2𝑡𝑝_𝑧𝑡−𝑦𝑓𝑡𝑝_𝑥𝑧+4𝜌+12𝑎^2𝑧𝑧𝑥𝑓𝑡𝑝_𝑦−𝑎_2𝑧𝑧−𝑡𝑦𝑓𝑝_𝑥−2𝑡𝑝_𝑡−12𝑝+41𝑦²+𝑥²𝑓²𝑤2𝑝^𝑦𝑦−1𝑣𝑥𝑡−𝑢𝑥𝑓−𝑣𝑦𝑓𝑓𝑎_2𝑧−3𝑎₂𝑝_𝑦𝑧+𝑝_𝑧𝑡−𝑎_2𝑧3𝑝_𝑡+𝑡𝑝_𝑡𝑡2(𝑦𝑓+2𝑢)𝑎^2𝑧𝑡+6𝑎₂𝑢𝑝_𝑥𝑧+12𝑡(𝑢𝑥−𝑦𝑣)𝑓𝑎^2𝑧𝑝_𝑥𝑦−12[(𝑣𝑡+𝑦)𝑓+4𝑢]𝑎^2𝑧𝑝_𝑥+12[(𝑢𝑡+𝑥)𝑓−4𝑣]𝑎^2𝑧𝑝_𝑦−12𝑢𝑦𝑓𝑎^2𝑧𝑡𝑝_𝑥𝑥+12𝑎^2𝑧(𝑥𝑓𝑡−2𝑣𝑡)𝑝_𝑦𝑡𝑝^−1𝑧𝑧𝜕^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=−𝜓_𝑦,𝑣=𝜓_𝑥.(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 fields𝑣⃗𝑓=⃗=𝜓_𝑥𝑝,_𝑥𝜓_𝑦,𝜓_𝑧,𝜓_𝑡=𝑣₁=𝑣,𝑣₂=−𝑢,𝑣₃,𝑣₄,,𝑝_𝑦,𝑝_𝑧,𝑝_𝑡=𝑓₁,𝑓₂,𝑓₃=−𝜌,𝑓₄,(4.2) and symmetric potential tensor fields𝜓_𝑖𝑗=𝜓_𝑗𝑖=𝜓_{𝑥𝑖𝑥𝑗},𝑝𝑖,𝑗=1,2,3,4,𝑥₁=𝑥,𝑥₂=𝑦,𝑥₃=𝑧,𝑥₄=𝑡,_𝑖𝑗=𝑝_𝑗𝑖=𝑝_{𝑥𝑖𝑥𝑗},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,𝜓_𝑖𝑗 and 𝑝_𝑖𝑗, 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 𝑢,𝑣,𝑤,𝑝,𝜌 and the potential 𝜓:𝑉₁=𝑎₁𝑥𝜕_𝑥+𝑦𝜕_𝑦+2𝜓𝜕_𝜓+𝑢𝜕_𝑢+𝑣𝜕_𝑣+2𝑝𝜕_𝑝+2𝜌𝜕_𝜌+𝑎_1𝑧×𝑥𝑓₁+𝑦𝑓₂−2𝑝𝜕_𝜌+𝑝⁻¹³³×2𝑤+2𝜌𝑣+𝑥𝑥𝑝𝑝₁₃₁₂+𝑦𝑝₂₃+𝑢𝑥𝑝₁₁+𝑦𝑝₁₂−𝑓₁𝑉+𝑦𝑝₂₂−𝑓₂−2𝑓₄+𝑥𝑃₁₃+𝑦𝑝₂₄𝜕_𝑤+𝑎_1𝑧𝑧𝑝⁻¹³³𝑤𝑥𝑓₁+𝑦𝑓₂−2𝑝𝜕_𝑤,(4.4)₂=𝑎₂𝑡𝜕_𝑡−𝜓𝜕_𝜓−𝑢𝜕_𝑢−𝑣𝜕_𝑣+(𝑓𝜓−2𝑝)𝜕_𝑝−𝑓𝑣₃+2𝜌𝜕_𝜌+𝑤+𝑓𝑝⁻¹³³𝑤𝜓₃₃+𝜓₃₄+𝑢𝜓₁₃+𝑣𝜓₂₃𝜕_𝑤+𝑎_2𝑧𝑡𝑓₄−𝑓𝜓+2𝑝𝜕_𝜌+𝑎_2𝑧−𝑣2𝑡𝑤2𝑓²₂+𝑝⁻¹³³2𝑤2𝜌+𝑓𝑣₃+𝑡𝑢𝑝₁₃+𝑡𝑣𝑝₂₃−𝑢2𝑓₁+𝑡𝑝₁₄+𝑡𝑝₂₄+𝑣₄𝑓−3𝑓₄−𝑡𝑝₄₄𝜕_𝑤+𝑎_2𝑧𝑧𝑝⁻¹³³𝑤𝑓𝜓−2𝑝−𝑡𝑓₄𝜕_𝑤,(4.5)𝑉₃=−𝑎₃𝑎𝜕_𝑡+_3𝑧𝑧𝑎_3𝑧𝑓₄𝜕_𝜌+2𝑤²+𝑝⁻¹³³2𝑤𝑢𝑝₁₃+𝑣𝑝₂₃−𝑢𝑝₁₄−𝑣𝑝₂₄−𝑝₄₄𝜕_𝑤𝑉𝑝⁻¹³³𝑤𝑓₄𝜕_𝑤,(4.6)₄𝑉=𝑎₄𝜕_𝑧−𝑎_4𝑧𝜌𝜕_𝜌+𝑤𝜕_𝑤+𝑎_4𝑧𝑧𝑝⁻¹³³𝑤𝜌𝜕_𝑤,(4.7)₅=+−𝑎𝑎_55𝑧𝑥𝜕_𝑦−𝑦𝜕_𝑥−𝑣𝜕_𝑢+𝑢𝜕_𝑣𝑥𝑓₂−𝑦𝑓₁𝜕_𝜌+𝑎+_5𝑧𝑧𝑝⁻¹³³2𝑦𝑝₁₃−𝑥𝑝₂₃𝑤+𝑦𝑝₁₁−𝑥𝑝₁₂−𝑓₂𝑢+𝑓₁+𝑦𝑝₁₂−𝑥𝑝₂₂𝑣+𝑦𝑝₁₄−𝑥𝑝₂₄𝜕_𝑤𝑉𝑝⁻¹³³𝑦𝑓₁−𝑥𝑓₂𝑤𝜕_𝑤,(4.8)₆=−𝑎₆𝑥𝑡𝜕^𝑦−𝑦𝑡𝜕_𝑥+12𝑥²+𝑦²𝜕_𝜓−(𝑣𝑡+𝑦)𝜕_𝑢+(𝑢𝑡+𝑥)𝜕_𝑣+124𝜓+𝑓𝑥²+𝑓𝑦²𝜕_𝑝−2𝑣₃𝜕_𝜌+2𝑝₃₃𝑢𝜓₁₃+𝑣𝜓₂₃+𝑤𝜓₃₃+𝜓₃₄𝜕_𝑤−𝑎_6𝑧122𝑡𝑥𝑓²−2𝑡𝑦𝑥₁−4𝜓−𝑓𝑥²−𝑓𝑦²𝜕_𝜌+𝑝⁻¹³³𝑢+𝑓2𝑥𝑤−𝑓2𝑣₂𝑡₃−𝑥𝑡𝑝₁₂+𝑦𝑡𝑝₁₁+𝑣𝑦𝑓+𝑡𝑓₁−𝑥𝑡𝑝₂₂+𝑦𝑡𝑝₁₂−1+𝑦𝑡𝑝₁₃−𝑥𝑡𝑝₂₃+𝑦𝑓₁+2𝑣₄−𝑥𝑡𝑝₂₄+𝑦𝑡𝑝₁₄−𝑥𝑓₂𝜕^𝑤2𝑎^6𝑧𝑧𝑉𝑝⁻¹³³4𝜓+2𝑦𝑡𝑓₁−2𝑥𝑡𝑓₂+𝑓𝑥²+𝑓𝑦²𝑤𝜕_𝑤,(4.9)₇=−𝑎₇𝜕_𝑥−𝑎_7𝑡−𝑦𝜕_𝜓+𝜕_𝑢−𝑓𝑦𝜕_𝑝−𝑎_7𝑡𝑡𝑥𝜕_𝑝+−𝑎𝑎_{7𝑧7𝑧𝑡}𝑓₁𝜕_𝜌−𝑝⁻¹³³2𝑤𝑝₁₃+𝑢𝑝₁₁+𝑝₁₄+𝑣𝑝₁₂𝜕_𝑤𝑓𝑦𝜕_𝜌−𝑝⁻¹³³𝑓₁+𝑓𝑣𝜕_𝑤+𝑎_{7𝑧𝑡𝑡}𝑝⁻¹³³(𝑢+𝑓𝑦)𝜕_𝑤𝑉−𝑎_{7𝑧𝑧𝑡}𝑥𝜕_𝜌−𝑝⁻¹³³𝑤𝑦𝑓𝜕_𝑤+𝑎_{7𝑧𝑡𝑡𝑡}𝑥+𝑎_{7𝑧𝑧𝑡𝑡}𝑥𝑤+𝑎_7𝑧𝑧𝑓₁𝑤𝑝⁻¹³³𝜕_𝑤,(4.10)₈=−𝑎₈𝜕_𝑦−𝑎_8𝑡𝑥𝜕_𝜓+𝜕_𝑣+𝑥𝑓𝜕_𝑝+𝑎_8𝑧𝑧𝑤𝑓₂+𝑎_{8𝑧𝑡𝑡𝑡}𝑦−𝑎_{8𝑧𝑧𝑡}𝑤𝑥𝑓𝑝⁻¹³³𝜕_𝑤+𝑎_8𝑡𝑡𝑦𝜕_𝑝−𝑎_8𝑧𝑓₂𝜕_𝜌−𝑝⁻¹³³2𝑤𝑝₂₃+𝑣𝑝₂₂+𝑝₂₄+𝑢𝑝₁₂𝜕_𝑤+𝑎_{8𝑧𝑧𝑡𝑡}𝑝⁻¹³³𝑦𝑤𝜕_𝑤𝑉+𝑎_8𝑧𝑡𝑓𝑥𝜕_𝜌+𝑝⁻¹³³𝑓₂−𝑓𝑢𝜕_𝑤−𝑎_{8𝑧𝑡𝑡}𝑦𝜕_𝜌−𝑝⁻¹³³(𝑣−𝑥𝑓)𝜕_𝑤,(4.11)₉𝑉=𝑎₉𝜕_𝑝−𝑎_9𝑧𝜕_𝜌−𝑎_9𝑡𝑡𝑤+𝑎_9𝑧𝑡𝑝⁻¹³³𝜕_𝑤,(4.12)₁₀=𝑎₁₀𝜕_𝜓,(4.13) where 𝑎_𝑖, 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^𝑋𝑥′=d𝑌𝑦′^=d𝑍𝑧′^=d𝑇𝑡′^=dΨ𝜓′^=d𝑈𝑢′^+d𝑉𝑣′^=d𝑊𝑤′^=d𝑃𝑝′^=⋯=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,𝑋𝑌′′==−−𝑎𝑎^7_8+𝑎^1𝑥′−𝑡^2𝑦′−𝑔^𝑦^𝑡^,𝑍+_𝑎^1𝑦′+𝑡^2𝑥′+𝑔^𝑥^𝑡^,𝑇_=𝑎^4𝑧^,𝑈_=𝑡^0+𝑡^𝑡^1,𝑉_=−𝑎^7𝑡_′+𝑐^2−𝑡^1𝑢^−𝑡^2𝑣^−𝑔^𝑣^𝑡^+𝑦^,=𝑃′−=𝑎𝑓^8𝑡_′𝑎+_7𝑡′𝑐^2−𝑡^1𝑣^+𝑡^2𝑢^+𝑔^𝑢^𝑡^+𝑥^,𝑦^−𝑥^𝑎…^7𝑡_′𝑡′−,𝑓𝑎^8𝑡_′𝑥^+𝑦^𝑎^8𝑡_′𝑡′,(5.2) with {𝑎^1≡𝑎₁(𝑧′), 𝑎^3≡𝑎₃(𝑧′), 𝑎^2≡𝑎₂(𝑧′), 𝑎^5≡𝑎₅(𝑧′), 𝑔′≡𝑔(𝑧′), 𝑎^4≡𝑎₄(𝑧′),…} being arbitrary functions of 𝑧′ and {𝑎^7≡𝑎₇(𝑧′,𝑡′), 𝑎^8≡𝑎₈(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 𝑣𝑢==−𝛼₆𝛼₆𝑦−𝛼_8𝑡sin𝜃+𝛼_7𝑡cos𝜃+𝛼₂𝛼¹⁻¹[𝑈(𝜉,𝜂,𝜁,𝜏)cos𝜃−𝑉(𝜉,𝜂,𝜁,𝜏)sin𝜃],𝑝=𝛼₂𝑥+𝛼_7𝑡sin𝜃+𝛼_8𝑡cos𝜃+𝛼₂𝛼¹⁻¹[𝑉(𝜉,𝜂,𝜁,𝜏)cos𝜃+𝑈(𝜉,𝜂,𝜁,𝜏)sin𝜃],𝛼²¹+(𝑥𝛼co₂𝑃s𝜃(𝜉+,𝜂𝑦,s𝜁i,n𝜏𝜃))+𝑓𝑓++22𝛼𝛼₆₆−𝑓𝛼₂Ψ(𝜉,𝜂,𝜁,𝜏)+𝛼₉+𝛼2⁶𝑓+𝛼⁶𝑥²+𝑦²𝜓=𝛼₂𝛼_8𝑡−𝛼_7𝑡𝑡+(𝑥sin𝜃−𝑦cos𝜃)𝑓+2𝛼₆𝛼_7𝑡+𝛼_8𝑡𝑡,𝛼²¹Ψ(𝜉,𝜂,𝜁,𝜏)+𝛼₁₀+𝛼⁶2𝑦cos(𝛼⁶+𝑥𝛼_8𝑡𝑡)−𝑥sin(𝛼₆𝑡)²+𝑦sin𝛼₆𝑡+𝑥cos𝛼₆𝑡²cos𝜃+𝛼_7𝑡sin𝜃+𝑦𝛼_8𝑡sin𝜃−𝛼_7𝑡cos𝜃,(5.3) where 𝜉=𝛼₁(𝑦sin𝜃+𝑥cos𝜃−𝛼₇), 𝜂=𝛼₁(𝑦cos𝜃−𝑥sin𝜃−𝛼₈), 𝜁=𝛼₄, 𝜏=𝛼₂𝑡+𝛼₃, 𝜃≡𝛼₆𝑡+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 taking𝛼₁⟶1+𝜖𝑎₁,𝛼₂⟶1+𝜖𝑎₂,𝛼₄⟶𝑧+𝜖𝑎₄,𝛼_𝑖⟶𝜖𝑎_𝑖,𝑖≠1,2,4,(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:𝜌𝑢𝑝===−−2𝐹𝐹𝐹𝑣_𝑧(𝑟(=𝑟,,𝑧𝑧)2)(𝑥2𝑦𝐹𝐹,((𝑟𝑟,𝑟,𝑧𝑧≡)),+𝑥²𝑓+)𝑑𝑦𝑟²,,𝑤=−1(𝑟,𝑧)(4𝐹(𝑟,𝑧)+𝑓)𝑑𝑟,𝑝_𝑧𝑧𝑝_𝑧𝑡+𝑢𝑝_𝑥𝑧+𝑣𝑝_𝑦𝑧,(6.1) with arbitrary function 𝐹≡𝐹(𝑟,𝑧) of {𝑟,𝑧}.

Applying the group theorem of the last section to solution (6.1), we have𝑣𝑢==−𝛼₆𝛼₆𝑦−𝛼_8𝑡sin𝜃+𝛼_7𝑡cos𝜃−2𝛼₂𝛼¹⁻¹𝜂𝐹𝑅,𝛼₄cos𝜃+𝜉𝐹𝑅,𝛼₄sin𝜃,𝑝=𝛼₂𝑥+𝛼_7𝑡sin𝜃+𝛼_8𝑡cos𝜃+2𝛼₂𝛼¹⁻¹𝜉𝐹𝑅,𝛼₄cos𝜃−𝜂𝐹𝑅,𝛼₄sin𝜃,𝛼²¹𝛼²𝐹𝑅,𝛼⁴2𝐹𝑅,𝛼₄+𝑓𝑑𝑅+𝑓+2𝛼₆−𝑓𝛼₂𝐹𝑅,𝛼₄𝑑𝑅+𝛼₁₀+𝛼⁶2𝑓+𝛼⁶𝑥²+(𝑥sin𝜃−𝑦cos𝜃+)𝑦²𝑓++2(𝛼𝑥₆cos𝜃+𝑦sin𝜃)𝑓+2𝛼₆𝛼_8𝑡−𝛼_7𝑡𝑡𝜌=−𝑝_𝑧𝛼_7𝑡+𝛼_8𝑡𝑡,𝑤=−1,𝑝_𝑧𝑧𝑝_𝑧𝑡+𝑢𝑝_𝑥𝑧+𝑣𝑝_𝑦𝑧,(6.2) with𝑅≡𝛼²¹𝑦sin𝜃+𝑥cos𝜃−𝛼⁷²+𝑦cos𝜃−𝑥sin𝜃−𝛼₈².(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, 𝑢∼−𝛼₆𝑦,𝑣∼𝛼₆𝑥 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 ({𝑢∼−𝛼_8𝑡sin𝜃+𝛼_7𝑡cos𝜃, 𝑣∼𝛼_7𝑡sin𝜃+𝛼_8𝑡cos𝜃}) 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),𝑤=−𝑢=−𝐹𝑅₁,𝑧𝑦,𝑅₁≡𝛼𝑥²_10𝑧𝑡+𝑦²,𝑣=𝐹𝑅₁,𝑧𝑥,(𝑧,𝑡)∫4𝐹𝑅₁,𝑧+𝑓𝐹_𝑧𝑧𝑅₁,𝑧+4𝐹^2𝑧𝑅₁,𝑧²d𝑅¹+𝛼_10𝑧𝑧(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=−𝑐₀𝑦sech𝑐₁𝑧+𝑐₅tanh𝑐₁𝑧+𝑐₅sech𝑅¹𝑣=𝑐₀𝑐₂,(6.5)𝑥sech𝑐₁𝑧+𝑐₅tanh𝑐₁𝑧+𝑐₅sech𝑅¹𝑤=𝑐₄𝑐₂,(6.6)2𝑐³+𝑐²⁰𝑐²¹𝑐₂sech²𝑐₁𝑧+𝑐₅10sech⁴𝑐₁𝑧+𝑐₅−11sech²𝑐₁𝑧+𝑐₅+2tanh𝑅¹+𝑐₀𝑐₂𝑐²¹𝑐₂𝑓tanh𝑐₁𝑧+𝑐₅sech𝑐₁𝑧+𝑐₅1−6sech²𝑐₁𝑧+𝑐₅arctanexp𝑅¹𝑐₂⁻¹,(6.7) with arbitrary constants 𝑐_𝑖, 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)

(a)

(b)

(c)

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 𝑥−𝑦 plane (a) at the bottom 𝑧=0, (b) at the top 𝑧=0.14, and (c) in the vertical 𝑦−𝑧 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,dd𝑦𝑡=𝑣,dd𝑧𝑡=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].

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.

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 𝑣⃗ and 𝑓⃗ shown by (4.2) and the 4×4 symmetric tensors 𝜓_𝑖𝑗 and 𝑝_𝑖𝑗 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.