The General Circulation of the
Atmosphere: A Numerical Experiment
NormanA.Phillips(1956)
PresentationbyLukasStrebel andFabianThüring
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-planeapproximation
•
Geostrophicwinddominates
5
Governing Equations
Quasi-geostrophic
theory
•
Beta-planeapproximation
•
Geostrophicwinddominates
Non-adiabaticheat
changesand friction
•
Non-adiabatic
•
Friction
Du
Dt
fv =
@
@⌧x
+ Av r 2 u + g
@x
@p
6
Governing Equations
Quasi-geostrophic
theory
•
Beta-planeapproximation
•
Geostrophicwinddominates
Non-adiabaticheat
changesand friction
•
Non-adiabatic
•
Friction
Du
Dt
fv =
2-levelgeostrophic
model
@
@⌧x
+ Av r 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
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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)
@✓
⇠ const
@p
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Further Assumptions
§ No variation of vertical stability (2 layer model)
@✓
⇠ const
@p
§
dQ
dt
will be interpreted as the average non-adiabatic heating
24
Further Assumptions
§ No variation of vertical stability (2 layer model)
@✓
⇠ const
@p
§
dQ
dt
will be interpreted as the average non-adiabatic heating
§ Release of latent heat
25
Further Assumptions
§ No variation of vertical stability (2 layer model)
@✓
⇠ const
@p
§
dQ
dt
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)
@✓
⇠ const
@p
§
dQ
dt
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
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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)
Prognosticstep
37
Numerical Scheme (QGPV)
Prognosticstep
Diagnosticstep
38
Numerical Scheme (QGPV)
Prognosticstep
Diagnosticstep
39
Numerical Scheme (QGPV)
§ Finite Differences
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Numerical Scheme (QGPV)
§ Finite Differences
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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
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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
Non-adiabatic heating
Lateral eddy-viscosity
Direct meridional
circulation
Poleward sensible
heat transport
Loss by friction
Convergence of mean
eddy momentum
transport
Vertical circulation
Lateral eddy-viscosity
Loss by friction
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