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Analytical Solutions for a Large-Scale Long-Lived Rotating Layer of Fluid Heated Underneath

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Analytical Solutions for a Large-Scale Long-Lived Rotating Layer of Fluid Heated Underneath Western Michigan University ScholarWorks at WMU Master's Theses Graduate College 12-2013 Analytical Solutions for a Large-Scale Long-Lived Rotating Layer of Fluid Heated Underneath Pouya Jalilian Follow this and additional works at: https://scholarworks.wmich.edu/masters_theses Part of the Mechanical Engineering Commons Recommended Citation Jalilian, Pouya, "Analytical Solutions for a Large-Scale Long-Lived Rotating Layer of Fluid Heated Underneath" (2013). Master's Theses. 437. https://scholarworks.wmich.edu/masters_theses/437 This Masters Thesis-Open Access is brought to you for free and open access by the Graduate College at ScholarWorks at WMU. It has been accepted for inclusion in Master's Theses by an authorized administrator of ScholarWorks at WMU. For more information, please contact [email protected]. ANALYTICAL SOLUTIONS FOR A LARGE-SCALE LONG-LIVED ROTATING LAYER OF FLUID HEATED UNDERNEATH by Pouya Jalilian A thesis submitted to the Graduate College in partial fulfillment ofthe requirements for the degree ofMaster ofScience Mechanical and Aeronautical Engineering Western Michigan University December 2013 Thesis Committee: Tianshu Liu, Ph.D., Chair Parviz Merati, Ph.D. Christopher Cho, Ph.D. Javier Martin Montefort -Sanchez, Ph.D. ANALYTICAL SOLUTIONS FOR A LARGE-SCALE LONG-LIVED ROTATING LAYER OF FLUID HEATED UNDERNEATH Pouya Jalilian, M.S. Western Michigan University, 2013 In this work, the system ofequations for a large- scale long lived rotating layer of fluid with the deformable upper free surface and non-deformable lower free surface heated underneath has been reviewed and derived. The quasi-geostrophic approximation, the beta effect and the method ofmulti-scale expansions have been employed to and as a result, an equation governing the evolution oflarge-scale perturbations, has been derived. The effect of each term present in the upper surface deformation equation has been analyzed and the analytical solutions have been obtained by virtue ofemploying auxiliary Riccati equation method. The soliton solutions obtained contributes to the sustenance of the vortex structure of the long-lived rotating layer of fluid due to the existence of two terms namely the nonlinear term or the so-called beta effect and the diffusion term resulted from the presence ofheating energy from below. The solution obtained, has been also applied to the case of long-lived vortex structure of the Great red spot of Jupiter and the results for the large-scale perturbations and averaged dominant terms of non-dimensional components ofthe velocity fields have been presented. The results show the correlation between the heating of the fluid motion from the lower layers, which is one of the fundamental features of the Great Red spot of Jupiter, and the sustenance ofthe vortex structure. Copyright by Pouya Jalilian 2013 ACKNOWLEDGMENTS First, I would like to deeply appreciate all my family members especially my mother and my father for their full supports, without their help I would not be able to complete this work. Special thanks go to my major advisor, Dr. Tianshu Liu for his valuable comments and perfect guidance. I would like to express my gratitude to my committee members including Dr. Parviz Merati, Dr. Christopher Cho and Dr. Khavier M. Montefort. I am also grateful to the departmental graduate advisor, Dr. Koorosh Naghshineh for useful information he has provided me. Finally, I would like to thank Mr. Sai Kode for his editing help. Pouya Jalilian 11 TABLE OF CONTENTS ACKNOWLEDGMENTS ii LIST OF FIGURES v CHAPTER 1. INTRODUCTION 1 2. MATHEMATICAL MODEL 4 2.1. Basic Assumptions 4 2.2 Derivation ofthe Governing Equations 9 2.2.1 Dimensional Form 9 2.2.2 Non - Dimensional Form 13 2.3 Quasi-Geostrophic Approximation 16 2.4 Method ofMulti-Scale Expansions 19 3. ANALYTICAL SOLUTIONS 28 3.1 Auxiliary Riccati Equation Method 28 3.2 Applying Auxiliary Riccati Equation Method 30 4. RESULTS 40 iii Table ofContents - Continued CHAPTER 5. DISCUSSIONS 49 6. CONCLUSION 53 REFERENCES 54 APPENDICES 56 A. The Static Solutions for Governing Equations 56 B. The Derivation ofthe Equations (2.47) - (2.49) 58 C. The Derivation ofthe Equation (2.55) 61 D. The Derivation ofthe Equations (2.64) - (2.66) 65 E. The Derivation ofthe Equation (3.10) 73 F. Ten Sets ofSolutions for the System ofEquations (3.20) - (3.26) . 75 IV LIST OF FIGURES 2.1 The sketch ofa single layer model heated underneath 5 2.2 The thin shell offluid and the local coordinate frame at latitude 9 6 4.1 The time evolution ofthe perturbation h = (x,y) 42 4.2 The time evolution and decaying ofthe perturbation h = (x,y) 43 4.3 The decaying ofthe perturbation h = (x,y) 43 4.4 The time evolution ofthe non-dimensional mean u^ at t=0 44 4.5 The time evolution ofthe non-dimensional mean ifi*) at t=l 45 4.6 The time evolution ofthe non-dimensional mean u^ at t=2 45 4.7 The time evolution ofthe non-dimensional mean ifi^ at t=3 45 4.8 The decaying ofthe non-dimensional mean u^ att=8 46 4.9 The decaying of the non-dimensional mean u^ att=10 46 4.10 The time evolution ofthe non-dimensional mean v^ att=0 47 4.11 The time evolution ofthe non-dimensional mean v^ at t=l 47 4.12 The time evolution ofthe non-dimensional mean v^ att=2 47 4.13 The time evolution ofthe non-dimensional mean v^ at t=3 48 4.14 The decaying ofthe non-dimensional mean v^ att=8 48 4.15 The decaying ofthe non-dimensional mean i;(0) att=10 48 CHAPTER 1 INTRODUCTION Analytical solutions for a rotating layer of fluid heated underneath is of great interest. In the past six decades, there have been many studies devoted to the numerical solutions for a rotating layer of fluid heated underneath. However, very few analytical solutions for the above fundamental phenomenon can be found in the literature and very important features ofthe above phenomenon such as the deformation ofthe upper surface due to the onset of convection was not taken into account in the previous analytical solutions. The first analytical solutions for a rotating layer of fluid heated from below were obtained by Chandrasekhar in 1953 [1]. He obtained the analytical solutions for three cases ofboundary conditions: both boundaries free, one boundary free and the other one rigid, both boundaries rigid. In obtaining his solutions, nonlinear terms in the governing equations were neglected (linear theory of stability). Also, the deformation of the upper surface and the beta effect were neglected in his analytical solutions. He showed that the Coriolis force or the angular velocity H has an inhibitive effect on the onset of convection. In other words, an increase in the angular velocity results in the inhibition of the onset ofconvection [1]. 1 Zhang and Roberts [2] developed the solution obtained by Chandrasekhar. They employed asymptotic analysis and obtained asymptotic solutions for a rotating layer of fluid heated from below. Also, the exact solutions constructed works well in the case that the Prandtl number is sufficiently small. However, nonlinear terms were neglected in the governing equations. Petviashvili in 1980 [3] also obtained analytical solution for an inviscid layer of fluid subject to the Coriolis force. His solution is the first soliton solution accounts for the sustenance of the vortex structure of a rotating layer of fluid. In his governing equations the effect ofconvection and viscosity has been disregarded [3-5] Busse [6,7] has also made significant contributions to the solution of a rotating layer of fluid heated from below, with application to planetary cases, both analytically and numerically. The model that was used by him is known as the deep model characterized by the Taylor-Proudman theorem. Linear analysis of the governing equation associated with weakly nonlinear analysis was carried out by him. Analytical solutions give us better understanding and help us to explain the physics of the problem in a clear way. Therefore, we have been motivated to obtain analytical solutions for a large-scale long-lived rotating layer of fluid heated underneath with the deformable upper and non-deformable lower surfaces. In deriving the system of governing equations, the Coriolis effect is considered with the assumption that the rotation period of flow is long compared to the rotation period of the overall system. Therefore, the quasi-geostrophic approximation has been employed together with the Boussinesq approximation and the beta plane approximation or the so-called beta effect. Heating of fluid from below, viscosity of fluid and the deformation of the upper surface have been taken into account in deriving the system of governing equations. However, the effect ofthe surface tension has been neglected at the deformable upper boundary condition. We have employed the method of multi-scale expansions in order to obtain the equation for the upper surface deformation. In order to solve the equation for the upper surface deformation analytically, the auxiliary equation method [8,9] along with the Riccati expansion method [10,11] have been used. The soliton solutions of the equation for the upper surface deformation, account for the sustenance ofRossby waves or vortices. One ofthe applications ofthe mathematical model in our study is to model large- scale and long-lived vortex structures in planetary atmospheres such as the mysterious problem oflong-lived vortex structures ofthe Great Red Spot ofJupiter which have been observed for more than 300 years. It is proposed that the Great Red Spot of Jupiter is a Rossby solitary vortex and retains its shape for a long time [12,13]. The other important feature of the Great Red Spot of Jupiter is the heating of the atmosphere from the lower layers [14,15], which has been taken into consideration in the mathematical model.
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