IMPACTS OF SIGMA COORDINATES ON THE EULER AND NAVIER-STOKES EQUATIONS USING CONTINUOUS/DISCONTINUOUS GALERKIN METHODS

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1 Approved for public release; distribution is unlimited IMPACTS OF SIGMA COORDINATES ON THE EULER AND NAVIER-STOKES EQUATIONS USING CONTINUOUS/DISCONTINUOUS GALERKIN METHODS Sean L. Gibbons Captain, United States Air Force B.S. Materials Science, United States Air Force Academy, 2003 Submitted in partial fulfillment of the requirements for the degrees of MASTER OF SCIENCE IN METEOROLOGY MASTER OF SCIENCE IN APPLIED MATHEMATICS from the NAVAL POSTGRADUATE SCHOOL March 2009 Author: Sean L. Gibbons Approved by: Francis Giraldo, Co-Advisor Maj Tony Eckel, Co-Advisor Philip Durkee, Chairman Department of Meteorology Carlos Borges, Chairman Department of Applied Mathematics iii

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3 ABSTRACT In this thesis,... Three test cases are analyzed: A rising thermal bubble, a linear hydrostatic mountain, and a linear nonhydrostatic mountain. The methods will be outlined for the transformation of two sets (set 1 the non-conservative form using Exner pressure, momentum, and potential temperature; set 2 the conservative form using density, momentum, and potential temperature) of the x-z Navier-Stokes equations to x-σ z... The same transformation for sigma-z vertical coordinates used by COAMPS, WRF and other mature mesoscale models will be applied to the two sets of the Navier-Stokes equations of interest. After applying the sigma-z coordinates, the discretization method of choice employs continuous/discontinuous Galerkin techniques. The existing code is in Fortran and all of the necessary modifications will also be made in Fortran. After the modifications have been made to the model, three test cases will be run: rising thermal bubble, linear hydrostatic mountain, and linear non-hydrostatic mountain. The numerical solutions will then be evaluated against either other model solutions (case 1) or the analytic approximations (case 2 and case 3) using root mean squared error and L2 error norms. The resultant data will then be compared to the unmodified solutions. The initial data for the test case is pre-generated by the source code, maintaining uniform initial conditions from which both coordinate systems numerical solutions can be compared. v

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5 TABLE OF CONTENTS I. INTRODUCTION II. BACKGROUND A. GOVERNING EQUATIONS Equation Set 1: Non-conservative Equation Set 2: Non-conservative B. X-Z TO X-σ Z COORDINATE SYSTEM TRANSFORM Gal-Chen and Somerville Basic Transformation Machinery Transformation Functions C. SPATIAL DISCRETIZATION D. TEMPORAL DISCRETIZATION RK III. APPLIED COORDINATE TRANSFORMS A. EQUATION SET Perturbation Method Transform Decomposition Application of the Galerkin Statement B. EQUATION SET Perturbation Method Transform Decomposition Application of the Galerkin Statement IV. TEST CASES A. CASE 1: RISING THERMAL BUBBLE B. CASE 2: LINEAR HYDROSTATIC MOUNTAIN C. CASE 3: LINEAR NON-HYDROSTATIC MOUNTAIN vii

6 V. RESULTS A. OVERVIEW B. CASE 1: RISING THERMAL BUBBLE Accuracy Comparison and Conclusions C. CASE 2: LINEAR HYDROSTATIC MOUNTAIN Accuracy Comparison and Conclusions D. CASE 3: LINEAR NON-HYDROSTATIC MOUNTAIN Accuracy Comparison and Conclusions VI. CONCLUSIONS AND RECOMMENDATIONS LIST OF REFERENCES INITIAL DISTRIBUTION LIST viii

7 LIST OF FIGURES 1. The stability of the explicit leapfrog time-integrator. Figure a) has no time-filter, while figure b) has a time-filter weight of ǫ=.05. The solid lines represent the physical solutions while the dashed lines represent the computational modes ix

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9 LIST OF TABLES xi

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11 ACKNOWLEDGMENTS I would like to thank my advisors, Francis X. Giraldo and Major Tony Eckel, without whom this thesis would not have been possible. I would also like to thank the Naval Postgraduate School, the Meteorology and Mathematics departments at NPS, The Air Force Institute of Technology, The United States Air Force, and The United States Navy. xiii

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13 I. INTRODUCTION word [1] word [2] blah [3] blah [4] This is a NWP topic that examines coordinate systems and continuous or discontinuous Galerkin methods in relation to the Navier-Stokes equations. The proposed title of the thesis is: Impacts of Sigma Coordinates on the Euler and Navier- Stokes Equations using Continuous or Discontinuous Galerkin Methods. There are multiple sets of governing equations that can be used to describe atmospheric flow. The Navier-Stokes equations, along with their variations, form the most widely used and accepted sets of equations for numerically resolving atmospheric flow. Two specific formulations of the equation sets will be the focus of this study. In order to use discontinuous Galerkin methods to solve the Navier-Stokes equations, the equations have been written in conservation form. Since there is no conservative form of the first set of equations, continuous Galerkin methods have to be used to solve them. Dr. Francis X. Giraldo, implementing continuous/discontinuous Galerkin techniques, developed a 2-D (x-z slice) mesoscale model using Non-Hydrostatic Equations (Euler and Navier-Stokes Equations). The original construct used z for the vertical coordinates. In this study, the current formulation of the Navier-Stokes equations will be transformed using sigma-z vertical coordinates to test their impacts on resolving atmospheric motion in a continuous/discontinuous Galerkin framework. Will implementing sigma-z coordinates significantly improve or diminish the solution of the Navier-Stokes equations over x-z coordinates when using continuous/discontinuous Galerkin methods? 1

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15 II. BACKGROUND A. GOVERNING EQUATIONS 1. Equation Set 1: Non-conservative the Pressure Tendency Equation: π t + u π + R c v π u = 0 (2.1) the Momentum Equations (µ = 0): t + u u + c pθ π = g k (2.2) the Thermodynamic Energy Equation (µ = 0): θ t + u θ = 0 (2.3) 2. Equation Set 2: Non-conservative the Mass Equation: ρ t + (ρ u) = 0 (2.4) the Momentum Equations: t + u u + 1 ρ P = g k (2.5) 3

16 the Thermodynamic Energy Equation: θ t + u θ = 0 (2.6) B. X-Z TO X-σ Z COORDINATE SYSTEM TRANSFORM 1. Gal-Chen and Somerville In 1975, Gal-Chen and Somerville, took the anelasic approximation of the Navier-Stokes Equation (in the cartesian form) and transformed the coordinated system to sigma-z coordinates. The initial equations consisted of: the Continuity Equation: (p 0 u j ), j = 0. the Momentum Equations: ( ) (ρ 0 u i ) + (ρ 0 u i u j ), j = (δ ij p ), j +δ i3 ρ g + τ ij, j. t the Thermodynamic Energy Equation: ( ) δ (ρ 0 θ ) + (ρ 0 θ u j ), j = H j, j. δt the Eddy Viscosity: the Eddy Diffusion: ( ) ] 2δ τ ij = ρ 0 K M [e ij ij (u k, k ). δ ii ( ) θ H i = ρk H δ ij. x j 4

17 the Variable Eddy Viscosity: the Variable Heat Diffusion: [ ( ) ] 1/2 K M = (k ) 2 KH Def 1 (Ri). K M K H /K M = constant = 1/P r. the Modified Richardson Number: Ri : θ 10 3 θ (Ri) 0 = Ri ( p + x θ x p y θ y )/( ρθ (Def) 2 ) : otherwise the Richardson Number: ( ) ( g θ ) Ri = /(Def) 2. θ0 z the (Def) 2 : (Def) 2 = 0.5τ ij e ij /(ρ 0 K M ) = 1 2 eij e ij (2/δ ij )(u k, k ) 2 and e ij : ( ) ( ) u e ij i u j = + x j x i a. Transformation Functions The set of transformations used by Gal-Chen and Somerville were: x = x, ȳ = y, z = H(z z s) (H z s ) z x = z s z H, x H z s z y = z s z H, y H z s z z = H H z s 5

18 ū v w = z s x 1, 0, 0 0, 1, 0 z H z H z s, s y z H H z s, z H H z s u v w with the inverse transformations: x = x, y = ȳ, z = [ z(h z s) ] + z s H u v w = 1, 0, 0 0, 1, 0 zs z H, z H, H z s x H zs y H H ū v w 2. Basic Transformation Machinery This section will outline the basic equation used to set up the Navier-Stokes Equations for transformation. The concept used was the total differential: d x = x x dx + x z dz dσ z = σ z x dx + σ z z dz 6

19 where the notation indicates the transformed variable. The two total derivatives can then be written as a system of equations: d x x x x z = d z σ z σ z x z dx dz Using vector notation ( over bar) the above system can be simplified to: d x = Jd x where J is the Jacobian. Using the definition for velocity ( u): d x dt = u the transformation becomes: u = J u The inverse transform for velocity can be written as: u = J 1 u The transform was for the gradient operator ( ) is: x = x x x + σ z σ z x z = x x z + σ z σ z z 7

20 The gradient can then be written as a matrix: x z = x x x z σ z x σ z z x σ z Using vector notation the above system can be simplified to: = J T (2.7) the last basic definition used for the transformation is: (AB) T = B T A T (2.8) Proof: (AB) T = a 1 a 2 a 3 a 4 b 1 b 2 b 3 b 4 T T = a 1b 1 + a 2 b 3 a 1 b 2 + a 2 b 4 a 3 b 1 + a 4 b 3 a 3 b 2 + a 4 b 4 = 1b 1 + a 2 b 3 a 3 b 1 + a 4 b 3 a 1 b 2 + a 2 b 4 a 3 b 2 + a 4 b 4 B T A T = 1 b 3 b 2 b 4 a 1 a 3 a 2 a 4 = 1b 1 + a 2 b 3 a 3 b 1 + a 4 b 3 a 1 b 2 + a 2 b 4 a 3 b 2 + a 4 b 4 8

21 3. Transformation Functions The set of transformations are: x = x, ỹ = y, σ z = H(z z s) (H z s ) (2.9) σ z x = z s σ z H, x H z s σ z y = z s σ z H, y H z s σ z z = H H z s (2.10) J = 1, 0 z s σ z H x H z s, H H z s (2.11) J T = 1, z s x σ z H H z s 0, H H z s (2.12) J 1 = zs x 1, 0 σ z H, H z s H H (2.13) (J 1 ) T = σ 1, zs z H x H 0, H z s H (2.14) ũ w = 1, 0 z s σ z H x H z s, H H z s u w (2.15) 9

22 with the inverse transformations: x = x, y = ỹ, z = [ σ z(h z s ) ] + z s (2.16) H u w = zs x 1, 0 σ z H, H z s H H ũ w (2.17) C. SPATIAL DISCRETIZATION To construct the problem we will first consider the we will first consider the generalized 2-D hyperbolic-elliptic PDE: q t + u q = ν 2 q where q = q( x, t), u = u( x), x = (x, z) T, and ν is the viscosity coefficient. Using Galerkin machinery, q N and u where approximated using basis function expansion: q N ( x, t) = MN j=1 ψ j ( x)q j (t) u N ( x, t) = MN j=1 ψ j ( x) u j (t) 10

23 where ψ j is the Lagrange polynomial basis functions. The approximations for q N and u N were then substituted into the PDE, multiplied by a test function, ψ I, and integrated in the global domain, Ω, to get (weak integral form): Ω q N ψ I t Ω dω + ψ I ( u q N )dω = ν ψ I 2 q N dω Ψ H 1 Ω Instead of solving the global problem directly, 2-D local basis functions where constructed and then direct stiffness summation (DSS) was used to construct the global problem. The 2D local basis functions are defined as: ψ i (ξ, η) = h j (ξ) h k (η) where: h j (ξ) = N l = 0 l j ( ) ξ ξl ξ j ξ l Additionally, in order to construct the basis functions and transition between physical and computational space requires known of the metric terms: ξ x = 1 J z η, ξ z = 1 J x η η x = 1 z J ξ, η z = 1 x J ξ 11

24 J = x z ξ η x z η ξ converting the above using integration by parts (IBP): ψ i 2 q N dω e = (ψ i q N )dω e ψ i (q N )dω e Ω e Ω e Ω e then using divergence theorem: ψ i 2 q N dω e = Ω e n (ψ i q N )dγ e Γ e ψ i (q N )dω e Ω e and subbing the result back into the PDE produced: Ω e ψ i q N t dω e + ψ i ( u q N )dω e = ν n (ψ i q N )dγ e ν ψ i (q N )dω e Ω e Γ e Ω e Subbing for the summation approximation for q N and u N yields: MN ψ i Ω e j=1 ψ j q j t dω e + Ω e ψ i ( MN k=1 ) MN ψ k u k ψ j q j dω e j=1 MN MN = ν ψ i n ψ j q j dγ e ν ψ i ψ j q j dω e Γ e Ω e j=1 j=1 12

25 The resulting matrix problem is: M (e) q j ij t + A(e) ij ( u)q j = B (e) ij q j L (e) ij q j D. TEMPORAL DISCRETIZATION RK4 13

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27 III. APPLIED COORDINATE TRANSFORMS A. EQUATION SET 1 1. Perturbation Method For this section both π and θ with be broken into two components mean ( π and θ) and their perturbations (π and θ ). the Pressure Tendency Equation 2.1 becomes: ( π + π ) t + u ( π + π ) + R c v ( π + π ) u = 0 π t + u π + w π z + R c v ( π + π ) u = 0 (3.1) the Momentum Equations 2.2 becomes: t + u u + c p( θ + θ ) ( π + π ) = g k [( π t + u u + c p( θ + θ ) x + π ) ( )] π + z x + π = g z k d π dz = g c p θ [ t + u u + c p( θ + θ ) g ( )] π k + c p θ x + π = g z k ( ) π t + u u + c p( θ + θ ) x + π g z k g θ θ k = g k t + u u + c p( θ + θ ) π = g θ θ k (3.2) 15

28 the Thermodynamic Energy Equation 2.3 becomes: ( θ + θ ) t + u ( θ + θ ) = 0 θ t + u θ + w θ z = 0 (3.3) 2. Transform Using the basic machinery prescribed in equations the set of the nonconservative Navier-Stokes (equations ) was prepared for transformation: Applying the machinery to the Pressure Tendency Equation yields: π t + ut π + w π z + R c v ( π + π ) u = 0 π t + (J 1 u) T (J T )π + w π σ z + R ( π + π )(J T ) T (J 1 u) = 0 z σ z c v π ( t + ( u) T (J 1 ) T (J T )( )π + ũ z s σ z H x H + wh z ) s π σ z H σ z z + R c v ( π + π )( ) T (J T ) T (J 1 )( u) = 0 π ( t + ( u) T ( )π + ũ z s σ z H x H + wh z ) ( ) s π H + R ( π + π )( H σ z H z s c ) T ( u) = 0 v 16

29 π t + u π ũ ( σz H H z s ) zs x π + w π + R ( π + π ) σ z σ z c u = 0 v π t + u π ũ ( σz H H z s ) zs π π + w + R ( π + π ) σ z x σ z c u = 0 v π t + u π + w π σ z + R c v ( π + π ) u = 0 (3.4) Applying the machinery to equation 3.2 yields: t + ut u + c p ( θ + θ ) π = g θ θ k t + (J 1 u) T (J T ) u + cp ( θ + θ )(J T )π = g θ θ k t + ( u) T (J 1 ) T (J T )( ) u + c p ( θ + θ )(J T )π = g θ θ k t + ( u) T ( ) u + c p ( θ + θ )(J T )π = g θ θ k 17

30 t + u u + c p ( θ + θ )(J T )π = g θ θ k (3.5) Applying the machinery to equation 3.3 yields: θ t + ut θ + w θ z = 0 θ t + (J 1 u) T (J T )θ + w θ σ z = 0 z σ z θ ( t + ( u) T (J 1 ) T (J T )( )θ + ũ z s σ z H x H + wh z ) s θ σ z H σ z z = 0 θ ( t + ( u) T ( )θ + ũ z s σ z H x H + wh z ) ( ) s θ H = 0 H σ z H z s θ t + u θ ũ σ z H z s θ + w θ = 0 H z s x σ z σ z θ t + u θ ũ σ z H z s θ θ + w = 0 H z s σ z x σ z 18

31 θ t + u θ + w θ σ z = 0 (3.6) 3. Decomposition The Pressure Tendency Equation 3.4: π t + u π + w π σ z + R c v ( π + π ) u = 0 decomposed becomes: π [ ] t + ũ π π + w + w π + R [ ũ ( π + π ) x σ z σ z c v x + w ] = 0 (3.7) σ z The Momentum Equation 3.5: t + u u + c p ( θ + θ )(J T )π = g θ θ k decomposed becomes: [ u t + ũ u ] [ ( ) ] u π + w + c p ( θ + θ ) x σ z x + zs σ z H π = 0 (3.8) x H z s σ z and 19

32 [ w t + ũ w ] [ ( ) ] w H π + w + c p ( θ + θ ) = g θ x σ z H z s σ z θ (3.9) The Thermodynamic Energy Equation 3.6: θ t + u θ + w θ σ z = 0 decomposed becomes: θ [ t + ũ θ x ] θ + w + w θ = 0 (3.10) σ z σ z 4. Application of the Galerkin Statement π [ ] t + ũ π π + w + w π + R [ ũ ( π + π ) x σ z σ z c v x + w ] = 0 σ z [ u t + ũ u ] [ ( ) ] u π + w + c p ( θ + θ ) x σ z x + zs σ z H π = 0 x H z s σ z [ w t + ũ w ] [ ( ) ] w H π + w + c p ( θ + θ ) = g θ x σ z H z s σ z θ 20

33 θ [ t + ũ θ x ] θ + w + w θ = 0 σ z σ z B. EQUATION SET 2 1. Perturbation Method ρ t + (ρ u) = 0 ρ t + u ρ + ρ u = 0 ( ρ + ρ ) t + u ( ρ + ρ ) + ( ρ + ρ ) u = 0 ρ t + u ρ + w ρ z + ( ρ + ρ ) u = 0 (3.11) the Momentum Equations: t + u u + 1 ρ P = g k t + u u + 1 ( ρ + ρ ) ( P + P ) = g k t + u u + 1 ( ρ + ρ ) P 1 + P ( ρ + ρ ) z k = g k 21

34 t + u u + 1 ( ρ + ρ ) P ρg ( ρ + ρ ) k = g k t + u u + 1 ( ρ + ρ ) P + ρ g ( ρ + ρ ) k = 0 t + u u + 1 ( ρ + ρ ) P = ρ g ( ρ + ρ ) k (3.12) the Thermodynamic Energy Equation: θ t + u θ = 0 ( θ + θ ) t + u ( θ + θ ) = 0 θ t + u θ + w θ z = 0 (3.13) 2. Transform Using the basic machinery prescribed in equations the set of the non-conservative Navier-Stokes (equations ) was prepared for transformation where the Mass Equation 3.11 becomes: ρ t + u ρ + w ρ z + ( ρ + ρ ) u = 0 ρ t + ut ρ + w ρ z + ( ρ + ρ ) T u = 0 22

35 ρ t + (J 1 u) T (J T )ρ + w ρ σ z + ( ρ + ρ )(J T ) T J 1 u = 0 z σ z ρ ( t + ( u) T (J 1 ) T (J T )ρ + ũ z s σ z H x H + wh z ) s ρ σ z H σ z z + ( ρ + ρ )( ) T (J T ) T J 1 u = 0 ρ ( t + ( u) T ρ + ũ z s σ z H x H + wh z ) ( ) s ρ H + ( ρ + ρ )( H σ z H z ) T u = 0 s ρ t + u ρ + w ρ σ z + ( ρ + ρ ) u = 0 (3.14) the Momentum Equations 3.12 becomes: t + u u + 1 ( ρ + ρ ) P = ρ g ( ρ + ρ ) k t + ut u + 1 ( ρ + ρ ) P = ρ g ( ρ + ρ ) k t + (J 1 u) T J T 1 u + ( ρ + ρ ) JT P = ( ρ + ρ ) k ρ g 23

36 t + u T (J 1 ) T J T 1 u + ( ρ + ρ ) JT P = ρ g ( ρ + ρ ) k t + u T 1 u + ( ρ + ρ ) JT P = ρ g ( ρ + ρ ) k t + u u + 1 ( ρ + ρ ) JT P = ρ g ( ρ + ρ ) k (3.15) the Thermodynamic Energy Equation 3.13 becomes: θ t + u θ + w θ z = 0 θ t + ut θ + w θ z = 0 θ t + (J 1 u) T J T θ + w θ σ z = 0 z σ z θ ( t + ( u) T (J 1 ) T J T θ + ũ z s σ z H x H + wh z ) s θ σ z H σ z z = 0 24

37 θ ( t + ( u) T θ + ũ z s σ z H x H + wh z ) ( ) s θ H = 0 H σ z H z s θ t + u θ + w θ σ z = 0 (3.16) 3. Decomposition The Mass Equation 3.14: ρ t + u ρ + w ρ σ z + ( ρ + ρ ) u = 0 decomposed becomes: ρ [ ] t + ũ ρ + w ρ + w ρ [ ũ + ( ρ + ρ ) σ z σ z σ z x + w ] = 0 (3.17) σ z The Momentum Equation 3.15: t + u u + 1 ( ρ + ρ ) JT P = ρ g ( ρ + ρ ) k decomposed becomes: [ u t + ũ u ] u + w + x σ z [ ( 1 P ( ρ + ρ ) x + zs x σ z H H z s ) ] P = 0 (3.18) σ z 25

38 and [ w t + ũ w ] [ ( ) ] w 1 H P + w + x σ z ( ρ + ρ ) H z s σ z = ρ g ( ρ + ρ ) (3.19) The Thermodynamic Energy Equation 3.16: θ t + u θ + w θ σ z = 0 decomposed becomes: θ [ t + ũ θ x ] θ + w + w θ = 0 (3.20) σ z σ z 4. Application of the Galerkin Statement ρ ( [ ( ) ] [ ρ t + ũ x + zs σ z H ρ ( ) ]) H ρ + w x H z s σ z H z s σ z + w ρ [ ũ + ( ρ + ρ ) σ z x + w ] = 0 σ z [ u t + ũ u ] u + w + x σ z [ ( 1 P ( ρ + ρ ) x + zs x σ z H H z s ) ] P = 0 σ z 26

39 [ w t + ũ w ] [ ( ) ] w 1 H P + w + x σ z ( ρ + ρ ) H z s σ z = ρ g ( ρ + ρ ) θ [ t + ũ θ x ] θ + w + w θ = 0 σ z σ z 27

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41 IV. TEST CASES A. CASE 1: RISING THERMAL BUBBLE For this test case there is no terrain: z surf = 0, x which lead to: z surf x = 0, x making the transformation of the coordinate system (x-z σ ) reduced to x-z. B. CASE 2: LINEAR HYDROSTATIC MOUNTAIN For this test case the terrain is represented by: z surf = hc ( 1 + ( ) 2 ), x x xc ac which lead to: z surf = hc ( ( ) ) x xc 2 1 z surf = hc 1 + ac [ (ac) 2 + (x 2 2(xc)(x) + (xc) 2 ) (ac) 2 ] 1 29

42 1000 A 1000 B z z x C x z x a) Figure 1. The stability of the explicit leapfrog time-integrator. Figure a) has no time-filter, while figure b) has a time-filter weight of ǫ=.05. The solid lines represent the physical solutions while the dashed lines represent the computational modes. z surf = (hc)(ac) 2 [ (ac) 2 + x 2 2(xc)(x) + (xc) 2] 1 z surf x = ( 1)(hc)(ac)2 (2x 2(xc)) [(ac) 2 + x 2 2(xc)(x) + (xc) 2 ] 2 z surf x = ( 2)(hc)(ac)2 (x xc) [(ac) 2 + (x xc) 2 ] 2, x 30

43 C. CASE 3: LINEAR NON-HYDROSTATIC MOUNTAIN 31

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45 V. RESULTS A. OVERVIEW B. CASE 1: RISING THERMAL BUBBLE 1. Accuracy 2. Comparison and Conclusions 33

46 C. CASE 2: LINEAR HYDROSTATIC MOUNTAIN 1. Accuracy 2. Comparison and Conclusions D. CASE 3: LINEAR NON-HYDROSTATIC MOUNTAIN 1. Accuracy 2. Comparison and Conclusions 34

47 VI. CONCLUSIONS AND RECOMMENDATIONS This study will aid in determining the usefulness of applying a specific coordinate system in the future, when developing meteorological and oceanographic models for the US Naval Research Laboratory (NRL) by constituents at the Naval Postgraduate School. In addition, the successful conversion of the non-hydrostatic x-z models to x-sigma-z will allow for the straightforward extension of these models to global non-hydrostatic models. 35

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49 LIST OF REFERENCES [1] M. Restelli F.X. Giraldo. A study of continuous and discontinuous galerkin methods for the navierstokes equations in nonhydrostatic mesoscale atmospheric modeling: Equation sets and test cases. Journal of Computational Physics, 227: , [2] C.J. Somerville T. Gal-Chen. On the use of a coordinate transformation for the solution of the navier-stokes equations. Journal of Computational Physics, 17: , [3] R.M. Hodur. The naval research laboratorys coupled ocean/atmosphere mesoscale prediction system (coamps). Monthly Weather Review, 125: , [4] W.C. Skamarock J.B. Klemp J. Dudhia D.O. Gill D.M. Baker W. Wang J.G. Powers. A description of the advanced research wrf version 2. NCAR Technical Note NCART/TN-468+STR,

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51 INITIAL DISTRIBUTION LIST 1. Defense Technical Information Center Ft. Belvoir, Virginia 2. Library, Code 52 Naval Postgraduate School Monterey, California 39