The General Circulation of the Atmosphere: A Numerical Experiment Norman A. Phillips (1956) Presentation by Lukas Strebel and Fabian Thüring
Goal of the Model Numerically predict the mean state of the atmosphere 2
Goal of the Model Numerically predict the mean state of the atmosphere Explore the validity of the geostrophic theory in explaining the general circulation 3
Goal of the Model Numerically predict the mean state of the atmosphere Explore the validity of the geostrophic theory in explaining the general circulation Investigate the energetics of the atmosphere 4
Governing Equations Quasi-geostrophic theory Beta-plane approximation Geostrophic wind dominates 5
Governing Equations Quasi-geostrophic theory Non-adiabatic heat changes and friction Beta-plane approximation Non-adiabatic Geostrophic wind dominates Friction Du Dt fv = @ @x + A vr 2 u + g @ x @p 6
Governing Equations Quasi-geostrophic theory Non-adiabatic heat changes and friction 2-level geostrophic model Beta-plane approximation Non-adiabatic Geostrophic wind dominates Friction Du Dt fv = @ @x + A vr 2 u + g @ x @p 7
2-level geostrophic model Equation of momentum and continuity (pressure coordinates) 8
2-level geostrophic model Thermodynamic energy equation 9
2-level geostrophic model Thermodynamic energy equation 10
2-level geostrophic model Vertical levels 11
2-level geostrophic model Vertical levels Vertical boundary conditions Vertical velocities 12
2-level geostrophic model Vertical levels Vertical boundary conditions Vertical velocities Frictional stresses 13
2-level geostrophic model Vertical levels Vertical boundary conditions Vertical velocities Frictional stresses 14
2-level geostrophic model Geometry 15
2-level geostrophic model Geometry Boundary conditions in x are periodic 16
2-level geostrophic model Geometry Boundary conditions in y are defined by walls 17
2-level geostrophic model Lateral boundary conditions in y are defined by walls 18
2-level geostrophic model Lateral boundary conditions in y are defined by walls Normal geostrophic velocity vanishes at the walls 19
2-level geostrophic model Lateral boundary conditions in y are defined by walls Normal geostrophic velocity vanishes at the walls Disturbed vorticity vanishes at the walls (arbitrary) 20
2-level geostrophic model Lateral boundary conditions in y are defined by walls Normal geostrophic velocity vanishes at the walls Disturbed vorticity vanishes at the walls (arbitrary) Integrating momentum equation w.r.t to BC 21
2-level geostrophic model Lateral boundary conditions in y are defined by walls Normal geostrophic velocity vanishes at the walls Disturbed vorticity vanishes at the walls (arbitrary) Integrating momentum equation w.r.t to BC 22
Further Assumptions No variation of vertical stability (2 layer model) @ @p const 23
Further Assumptions No variation of vertical stability (2 layer model) @ @p const will be interpreted as the average non-adiabatic heating dq dt 24
Further Assumptions No variation of vertical stability (2 layer model) dq dt @ @p const will be interpreted as the average non-adiabatic heating Release of latent heat 25
Further Assumptions No variation of vertical stability (2 layer model) dq dt @ @p const will be interpreted as the average non-adiabatic heating Release of latent heat Radiation 26
Further Assumptions No variation of vertical stability (2 layer model) dq dt @ @p const will be interpreted as the average non-adiabatic heating Release of latent heat Radiation Small scale lateral eddy diffusion 27
Quasi-geostrophic equations Quasi-geostrophic vorticity equation 28
Quasi-geostrophic equations Quasi-geostrophic vorticity equation 29
Quasi-geostrophic equations Quasi-geostrophic vorticity equation 30
Quasi-geostrophic equations Thermodynamic energy equation (at interface) 22
Quasi-geostrophic equations Thermodynamic energy equation (at interface) 22
Quasi-geostrophic equations Thermodynamic energy equation (at interface) 22
Quasi-geostrophic equations Thermodynamic energy equation (at interface) Geostrophic stream function Modified Rossby deformation radius 22
Quasi-geostrophic potential vorticity Define Quasi-geostrophic potential vorticity 35
Quasi-geostrophic potential vorticity Define Quasi-geostrophic potential vorticity Define prognostic equations for qi 36
Numerical Scheme (QGPV) Prognostic step 37
Numerical Scheme (QGPV) Prognostic step Diagnostic step 38
Numerical Scheme (QGPV) Prognostic step Diagnostic step 39
Numerical Scheme (QGPV) Finite Differences 40
Numerical Scheme (QGPV) Finite Differences 41
Numerical Scheme (QGPV) Finite Differences 42
Energy Transformation Kinetic energy of the mean zonal flow Kinetic energy of the disturbed flow Potential energy of the mean zonal flow Potential energy of the disturbed flow 43
Total change in energy 1. A loss of energy due to lateral eddy viscosity A 44
Total change in energy 1. A loss of energy due to lateral eddy viscosity A 2. A loss due to effect of surface friction 45
Total change in energy 1. A loss of energy due to lateral eddy viscosity A 2. A loss due to effect of surface friction 3. A change due to the non-adiabatic heating 46
Energy Flow Diagram Lateral eddy-viscosity Lateral eddy-viscosity Non-adiabatic heating Direct meridional circulation Loss by friction Poleward sensible heat transport Convergence of mean eddy momentum transport Vertical circulation Loss by friction Lateral eddy-viscosity Lateral eddy-viscosity 47
Development of the flow Experimental Setup Meridional extent 10 000 km Zonal extent 6000 km Initial atmosphere at rest 130 day forecast without eddies 48
Development of the flow Experimental Setup Meridional extent 10 000 km Zonal extent 6000 km Initial atmosphere at rest 130 day forecast without eddies Interpretation The wave moves eastward 1800 km /day The waves begins as a warm low Tilted troughs and ridges 49
Development of the flow Interpretation Indication of cold and warm fronts in 1000 mb contours 50
Development of the flow Interpretation Occlusion of cyclones Numerical instability after 26 days 51
Development of the flow Variation of u 1 at 250 mbs with latitude (j) and time. Unit are m sec -1. Regions of easterly winds are shaded 52
Development of the flow Variation of u 4 at 1000 mbs with latitude (j) and time. Unit are m sec -1. 53
Energy transformation 54
Conclusion of the experiment + Easterly and westerly distribution of the surface zonal wind +Existence of a jet +Model achieves net poleward transport of energy +Qualitative agreement of the energy transformation processes 55
Conclusion of the experiment + Easterly and westerly distribution of the surface zonal wind +Existence of a jet +Model achieves net poleward transport of energy +Qualitative agreement of the energy transformation processes Same order of magnitude of trade winds and polar easterly Uncertainty of input parameters ( ) Instability of the numerics 56
Questions? 57