Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 1981 A finite-element sea-breeze model Lang-Ping Chang Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Atmospheric Sciences Commons Recommended Citation Chang, Lang-Ping, "A finite-element sea-breeze model " (1981). Retrospective Theses and Dissertations. 6873. https://lib.dr.iastate.edu/rtd/6873 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. CHANG, LANG-PING A FINITE-ELEMENT SEA-BREEZE MODEL Iowa State University University Microfilms I n 16 r n 3t i 0 n â I 300 X. Zeeb Road, Ann Arbor, MI 48106 A finite-element sea-breeze model by Lang-Ping Chang A Dissertation Submitted to the Graduate Faculty in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY Department; Earth Sciences Major: Meteorology Approved: Signature was redacted for privacy. In Cha: nor Signature was redacted for privacy. For the Mag/^r Depart ent Signature was redacted for privacy. For the Graduate Coilege Iowa State University Ames, Iowa 1981 11 TABLE OF CONTENTS Page I. INTRODUCTION 1 II. THE FINITE-ELEMENT METHOD 5 A. A Brief History 5 B. Fundamentals 7 1. General description 7 2. Local representation of the shape functions 11 3. Convergence requirements 15 4. Methods of weighted residuals 21 5. Assemblage and restraining of the element matrices 25 6. Computer solution of the global matrix equation 28 C. Application of the Finite-Element Method to Meteorolog­ ical Problems 30 III. THE SEA BREEZE 35 A. General Review 35 B. Literature Review 38 1. Theoretical studies 39 2, Observational studies 44 C. Usefulness of Vazious Numerical Sea-Breeze Models 48 IV. THE MODEL 53 A. Governing Equations and Turbulence Parameterization 53 1. Shallow-convection Boussinesq approximation 55 2. Equations for the mean quantities 61 3. Closure problem 68 4. O'Brien scheme of exchange-coefficient profile 71 B. Finite-Element Description 82 1. Weighted-residual approximation of the governing equations 82 2. Numerical integration for determining the element matrices 91 ill Page a. Coordinate transformation and Isoparametric elements 91 b. Numerical integration by Gaussian quadrature 94 3. Modified Crank-Nicholson time-differing scheme 97 4. Profile Gauss-elimination solver 98 C. Basic Structure of the Code 103 V. PRELIMINARY STUDIES 105 A. Classical Ekman Spiral 105 B. Steady-State Wind Profile for a Neutral Horizontally- Homogeneous Planetary Boundary Layer 113 C. Vertical Motion Due to Pure Differential Roughness 124 VI. SEA-BREEZE SIMULATION AND DISCUSSION 129 A. Domain Discretization and the Initial and Boundary Conditions 130 B. Determination of the Horizontal Diffusion Coefficient 136 C. Results of Simulation 139 1. Sea-breeze with a calm synoptic wind 140 a. Characteristics of the flow velocity field 140 b. Upward warping pattern of the Isentropes near the front 154 c. General pressure pattern and evolution of the surface pressure 160 d. Coriolis effect in the sea breeze 164 e. Development and inland penetration of the sea- breeze front 168 f. Oversmoothlng effect of two large a horizon­ tal diffusion coefficient 172 2. Sea breeze with a light offshore synoptic wind 173 a. Velocity field 177 b. Characteristics of the frontal movement 190 3. The PBL height and the profile of the vertical exchange coefficient 193 iv Page VII. CONCLUDING REMARKS 200 VIII. REFERENCES 203 IX. ACKNOWLEDGMENTS 209 V LIST OF TABLES Page Table 1. Weights and locations of Gauss quadrature 96 Table 2, Comparison of the computed and the analytic solutions for the simulation of the Ekman spiral. Units used are: (m) for the height, (m/s) for the velocity, (°K) for the potential temperature, and 10"^ (m^/s/^K ) for the Exner pressure function. Nine-node biquadratic shape functions are used for the velocity and the potential temperature; and four-node bilinear shape functions are for the pressure. See text for explanation tion of the computed pressure and 111 vi LIST OF FIGURES Page Figure 1. A schematic domain discretization using nine-node Lagrangian elements 8 Figure 2a. A four-node rectangular element centered at (0,0) 12 Figure 2b. A general four-node bilinear element. (x,y) and (Ç,r|) represent global and local coordinates, respectively. Note that an element is always square-shaped in (Ç,ri) 12 Figure 3a, Contour plot of the shape function of a four-node bilinear element 16 Figure 3b. Stereographical plot of 16 Figure 4a. Contour plot of the corner shape function of the nine-node biquadratic element 17 Figure 4b. Stereographical plot of Y^(C,n) 17 Figure 4c. Contour plot of the midside shape function Vg(C,n) of a nine-node biquadratic element 18 Figure 4d. Stereographical plot of Wg(g,n) 18 Figure 4e. Contour plot of the central shape function Vg(C,n) of a nine-node biquadratic element 19 Figure 4f. Stereographic plot of Yg(5,n) 19 Figure 5. Schematic representation of the banded and profile elimination solvers for a symmetrical global stiffness matrix. Notice the different sizes of matrices given in parentheses 31 Figure 6. Schematic representation of the O'Brien profile for the vertical exchange coefficients for momentum and heat 74 Figure 7. A plot of the inverse Prandtl number K^/Kg as a function of the non-dimensional height Ç. Ç > 0 is for the stable PEL, and Ç < 0 the unstable PBL. The neutral value of the ratio at Ç = 0 is 1.35 78 Figure 8a. The global nodal numbers for the example on p. 100. Roman numerals in parentheses represent element numbers 101 vli Page Figure 8b. The sequential unknown numbers of the variable x. The other six nodes take on prescribed boundary values 101 Figure 9. An illustration of the basic structure of the code 104 Figure 10. The domain discretization for the simulation of the constant-K Ekman spiral. Notice that the z-scale is not linearly plotted. For each of the six nine- node elements indicated by Roman numerals, velocity and potential temperature are defined at all nine nodes, but pressure is only defined at the four corner nodes 110 Figure 11. Comparison of the computed and the analytical Ekman spirals. The solid curve represents the analytical solution, and the +*s give the computed result 112 Figure 12. Hodograph of the computed wind profiles for rough­ ness lengths 0.05 m and 0.01 m. Notice the larger surface cross-isobaric angle and the slower increase in wind with height for the rougher surface. Dots represent heights in real domain 116 Figure 13. Vertical profiles of horizontal velocities normalized by u /u^ at (zf)/u* = 1 for the x-component and v^/u* at (zf)/u* = 1 for the y-component 118 Figure 14. Nondimenslonal profiles of the stresses K^CSu^^/Sz) and K™(3v /3z) normalized by their surface values, respectively 120 Figure 15. Nondlmensional profile of the vertical exchange coefficient k"" f normalized by the surface value of K^Ou^/az)^ 121 Figure 16. Inertial circles at two different levels (10.68 m and 1043 m) in the circulation of a 1-D steady-state wind profile. The geostrophic windspeed is 10 m/s at the upper boundary, and the roughness length is 0.01 m. u and v represent characteristic values of u and v™at a gïven height. They are, respectively, 6.490 m/s and 1.022 m/s for z = 10.68 m, and 10.096 m/s and 0.500 m/s for z = 1043 m. The time interval between two successive points is approximately 2.5 hours 123 viil Page Figure 17. The steady-state vertical velocity field in cm/s for the case of pure differential roughness in a neutrally stratified atmosphere. Because of the neutral atmospheric stability, the vertical velocity has little change with height above 450 m. 128 Figure 18a. The computational domain and its finite-element discretization for the calm synoptic wind case. Solid lines outline the four boundaries of each biquadratic element, and dashed lines help define positions of the five non-corner nodes of an element 131 Figure 18b. Same as 18a except for the 2.5 m/s offshore synoptic wind case 132 Figure 19a. Computed u (solid line and in m/s) and w (dashed line and in cm/s) at t = 1 hour for the calm synoptic wind case 141 Figure 19b. Computed u (solid line and In m/s) and w (dashed line and In cm/s) at t = 2 hours for the calm synoptic wind case 142 Figure 19c, Computed u (solid line and in m/s) and w (dashed line and in cm/s) at t = 3 hours for the calm synoptic wind case 143 Figure 19d. Computed u (solid line and In m/s) and w (dashed line and in cm/s) at t = 4 hours for the calm svnoptic wind case» 144 Figure 19e. Computed u (solid line and in m/s) and w (dashed line and In cm/s) at t = 5 hours for the calm synoptic wind case 145 Figure 19f. Computed u (solid line and in m/s) and w (dashed line and in cm/s) at t = 6 hours for the calm synoptic wind case 146 Figure 19g. Computed u (solid line and In m/s) and w (dashed line and in cm/s) at t = 7 hours for the calm synoptic wind case 147 Figure 19h.