A Conservative Scheme for the Multi-layer Shallow-water System based on Nambu Representation and the ICON Grid Philipp Griewank November 24, 2009 Diplomarbeit Gutachter: PD Dr. Peter N´evir Prof. Dr. Ulrich Cubasch Institut f¨urMeteorologie To Mom & Dad Abstract In this thesis a conservative multi-layer shallow-water spacial dis- cretization scheme is constructed for the ICON grid. The term \con- servative scheme" implies that the spatial discretization preserves en- ergy and potential enstrophy numerically exact. This scheme is based on the equations in Nambu form, applying the method presented in Salmon (2005), which enables deriving conservative schemes by dis- cretizing Nambu brackets so they maintain their antisymmetry. The multi-layer scheme introduced and tested in this thesis is an extension of the single-layer scheme developed by Sommer and N´evir(2009) for the ICON grid. An additional scheme based on the shallow-water equa- tions in momentum form is developed to serve as a reference and both schemes are tested to ensure proper implementation. It is shown that the Nambu based scheme is superior in respect to energy and potential enstrophy preservation as well as stability, but has a higher computa- tional cost. Zusammenfassung In dieser Arbeit wird ein konservatives numerisches Schema des mehr-schichtigen Flachwasser-Modells auf dem ICON Gitter entwickelt, welches auf den Flachwassergleichungen in Nambu-Darstellung basiert. Der Begriff \konservatives Schema" setzt eine numerisch exakte Er- haltung der Energie und potentiellen Enstrophie durch die r¨aumliche Diskretisierung voraus. Diese Erhaltung wird erreicht, indem die Nambu- Klammern der Gleichungen in Nambu-Darstellung so diskretisiert wer- den, dass ihre Antisymmetrie erhalten bleibt. Diese Methode wurde in Salmon (2005) vorgeschlagen und erl¨autert. Das mehr-schichtige Schema dieser Arbeit ist eine Erweiterung des ein schichtigen, konserva- tiven Schemas welches in Sommer and N´evir(2009) vorgestellt worde. Aus den Flachwassergleichungen in der Impulsdarstellung wird ein weit- eres numerisches Schema konstruiert, welches als Vergleichsmass dient. Die korrekte Implementierung beider Schemata wurde sichergestellt. Nachgewiesen wurde, dass Energie und potentielle Enstrophie von dem deutlich stabileren, aber rechen intensiveren, konservativen Schema wesentlich genauer erhalten werden. 3 Contents 1 Introduction 8 1.1 Motivation . 8 1.2 Thesis Structure . 10 1.3 Summary: Current State of the Art . 10 1.3.1 Energy-Vorticity Theory . 10 1.3.2 Numerical Applications . 11 1.3.3 ICON Project . 11 1.3.4 Current Trends in Model Development . 12 1.3.5 Model Evaluation . 12 2 Energy-Vorticity Theory 12 2.1 Introduction . 13 2.2 Hamilton Mechanics . 13 2.3 Nambu Mechanics . 14 2.4 Energy-Vorticity Theory as Nambu Field Theory . 17 3 Shallow-water System 19 3.1 Introduction . 19 3.2 Single-layer Shallow-water Equation . 20 3.3 Multi-layer Shallow-water Equations . 20 3.4 Conserved Quantities . 21 4 Shallow-water Equations in Nambu Form 22 4.1 Single-layer Shallow-water Equations . 22 4.2 Multi-layer Shallow-water Equations . 23 5 Model 25 5.1 Basic Description . 25 5.2 Temporal Discretization . 26 5.3 ICON Grid . 26 5.4 Discretization Schemes . 30 5.4.1 Momentum Discretization - ICON-Scheme . 30 5.4.2 Conservative Scheme Based on Nambu Representation - Nambu-scheme . 32 6 Numerical Results 35 6.1 Model Tests . 36 6.1.1 Stationarity . 36 6.1.2 Gravity Waves . 37 6.1.3 Potential Enstrophy Conservation . 39 6.2 Time Resolution Dependencies . 41 6.2.1 Initial Conditions - Rossby-Haurwitz Wave . 42 6.2.2 Results . 44 4 6.3 Spatial Resolution Dependencies . 46 6.3.1 Initial Conditions . 47 6.3.2 Results . 47 6.4 Vertical Resolution Dependencies . 49 6.4.1 Initial Conditions . 50 6.4.2 Conservation . 50 6.4.3 Stability . 52 6.5 Baroclinic Instability . 55 7 Summary & Conclusion 59 7.1 Summary . 59 7.2 Conclusion . 61 Appendices 62 A Notation 62 B Additional Figures 64 B.1 Stationarity Test . 64 B.2 Gravity Wave Test . 65 B.3 Rossby-Haurwitz Wave . 66 C Functionals and Their Derivatives 71 5 List of Figures 1 ICON grid level 2 . 27 2 ICON grid level 3 and level 4 . 28 3 Local stencil: mass points i, vorticity points ν and wind edges l 28 4 Initial height fields of stationarity test: n = 0, ∆t = 75 s, level 6 grid resolution, α = 0:01, no diffusion, 4 layers: ρ1 = 1:0; ρ2 = 0:9; ρ3 = 0:5; ρ4 = 0:2 . 37 5 Initial height fields of gravity wave test: n = 0, ∆t = 100 s, level 5 grid resolution, α = 0:01, no diffusion, 2 layers: ρ1 = 1:0; ρ2 = 0:9 ........................... 38 6 Total height of gravity wave test, internal and external gravity wave labelled \int" and \ext": n = 0; 500; 1000; 1900, ∆t = 100 s, level 5 grid resolution, α = 0:01, no diffusion, 2 layers: ρ1 = 1:0; ρ2 = 0:9 ........................ 40 7 Relative potential enstrophy error of potential enstrophy con- servation test: n = 1 ::: 10000, ∆t = 50 s, level 5 grid resolu- tion, α = 0:01, no diffusion, 2 layers: ρ1 = 1:0; ρ2 = 0:9 . 41 8 Initial height fields of Rossby-Haurwitz wave: n = 0, level 4 grid resolution, α = 0:01, no diffusion, 2 layers: ρ1 = 1:0; ρ2 = 0:5................................. 43 9 Relative energy and potential enstrophy error H and E over M M time, sample Rossby-Haurwitz wave run, Nambu-scheme and ICON-scheme: n = 1 ::: 1666, ∆t = 90 s, level 4 grid resolu- tion, α = 0:01, no diffusion, 2 layers: ρ1 = 1:0; ρ2 = 0:5 . 45 10 Time-averaged absolute relative energy and potential enstro- phy error of multiple runs after 1.7 days arranged by Courant number: n = 1000;:::; 75000, ∆t = 2;:::; 150 s, level 4 grid resolution, α = 0:01, no diffusion, 2 layers: ρ1 = 1:0; ρ2 = 0:5 46 11 Time-averaged absolute relative energy error and potential en- strophy error of multiple runs after 0.5 days arranged by aver- age edge length: n = 2000, ∆t = 20 s, level 2-7 grid resolution, α = 0:01, no diffusion, 2 layers: ρ1 = 1:0; ρ2 = 0:9 . 48 12 Relative conservation error of single-layer, four-layer and ten- layer run over days: ∆t = 50 s, level 6 grid resolution, α = 0:1, no diffusion . 51 13 Days passed until potential enstrophy is twice the initial value for 1 to 10 layers, initial conditions detailed in Subsection 6.4.1: Subfigure (a): ∆t = 100 s, level 5 grid resolution, α = 0:1, no diffusion. Subfigure (b): ∆t = 50, level 6 grid resolution, α = 0:1, no diffusion . 53 14 Second layer height field of four-layer model: ∆t = 100 s, level 5 grid resolution, α = 0:1, no diffusion, 4 layers: ρ1 = 1:00; ρ2 = 0:75; ρ3 = 0:50; ρ4 = 0:25 . 54 6 15 Zonal wind distribution of baroclinic initial condition . 55 16 Height fields of baroclinic instability test: ∆t = 50, level 6 grid resolution, α = 0:01, no diffusion, 2 layers: ρ1 = 1:0; ρ2 = 0:9 57 17 Vorticity field of lower and upper layer, northern hemisphere: ∆t = 50, level 6 grid resolution, α = 0:01, no diffusion, 2 layers: ρ1 = 1:0; ρ2 = 0:9 .................... 58 18 Total height fields of ICON and Nambu-scheme after ≈ 4.5 days: n = 5000, ∆t = 75 s, level 6 grid resolution, α = 0:01, no diffusion, 4 layers: ρ1 = 1:0; ρ2 = 0:9; ρ3 = 0:5; ρ4 = 0:2 64 19 Height fields at end of gravity wave test: n = 1900, ∆t = 100 s, level 5 grid resolution, α = 0:01, no diffusion, 2 layers: ρ1 = 1:0; ρ2 = 0:9 ........................... 65 20 Height fields of Rossby-Haurwitz wave: n = 1666, ∆t = 90 s, level 4 grid resolution, α = 0:01, no diffusion, 2 layers: ρ1 = 1:0; ρ2 = 0:5 ........................... 66 21 Initial layer height of single-layer (upper plot) and ten-layer (lower plot) Rossby-Haurwitz test. ∆t = 100 s, level 5 grid res- olution, α = 0:1, no diffusion, 1-10 layers: ρ = 1:0; 0:9; 0:8; :::; 0:1 67 22 Bottom layer height field of two-layer Rossby-Haurwitz wave: ∆t = 100 s, level 5 grid resolution, α = 0:1, no diffusion, 2 layers: ρ1 = 1:0; ρ2 = 0:5 .................... 68 23 Top layer height field of two-layer Rossby-Haurwitz wave: ∆t = 100 s, level 5 grid resolution, α = 0:1, no diffusion, 2 layers: ρ1 = 1:0; ρ2 = 0:5 ........................ 69 24 Total height field of two-layer Rossby-Haurwitz wave: ∆t = 100 s, level 5 grid resolution, α = 0:1, no diffusion, 2 layers: ρ1 = 1:0; ρ2 = 0:5 ........................ 70 List of Tables 1 Overview of Hamilton and Nambu mechanics. 16 2 Overview of Hamiltonian fluid dynamics and energy-vorticity theory. 17 3 Grid characteristics of the ICON grid for resolution levels 1 to 7 29 4 Notation used for the ICON grid . 29 5 Definition of field variables on the ICON grid . 30 6 Flowchart of ICON-scheme . 31 7 Flowchart of Nambu-scheme . 34 7 1 Introduction The key idea behind this work is to create an energy and potential enstrophy conserving numerical multi-layer shallow-water model using the equations written in Nambu form. Although a multitude of shallow-water models exist this newly constructed multi-layer model is unique because it is based on the equations in Nambu form and the ICON grid, a geodesic icosahedral grid with C-Type staggering. The crucial step is to devise a spatial discretization that substitutes the par- tial differential equations of continuous fields with approximating ordinary differential equations of finite differences while preserving the conservation properties.