University of Applied Sciences Eberswalde &
Potsdam Institute for Climate Impact Research
www.fh-eberswalde.de/gcm
International Master Study Programme Global Change Management
Module lectured by Manfred Stock:
Physical Fundamentals of Global Change Processes
Lecture 4 - 2008-12-04
1. Classical Mechanics:
From Kepler’s laws of planetary motion
to Newton’s generalized laws of motion
2. Radiation and some elements of climate variability
Physical Fundamentals of GC
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04-1
Kepler’s three laws of planetary motion (1630)
1. The orbit of a planet about a star is an
ellipse with the star at one focus.
X
2. A line joining a planet and its star sweeps out
equal areas during equal intervals of time.
3. The squares of the orbital periods of planets
are directly proportional to the cubes of the
semi-major axis of the orbits.
tor2 ∝ ror 3
tor = orbital period of planet
axis of
orbit
ror = semimajor
T = orbital period
of planet
C
a = semimajor axis of orbit
Physical Fundamentals of GC
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04-2
Conservation Laws
a particular measurable property of an
isolated physical system does not
change as the system evolves:
• Conservation of linear momentum
• Conservation of angular momentum
• Conservation of energy
• Conservation of mass
(in classical mechanics)
Physical Fundamentals of GC
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04-3
1. From Kepler’s laws of planetary motion
to Newton’s generalized laws of motion
• Newton formulated his three laws of motion 1687 together with a general
theory of gravitation on the basis of
Kepler’s three laws of planetary motion from 1630.
r
• Kepler’s 1st law of planetary motion:
X
– The orbit of a planet about a star is an
ellipse with the star at one focus.
• Newton’s 1st law of motion:
– An object of mass m will stay at rest or move at a
constant velocity in a straight line unless acted upon
by an unbalanced force.
– Conservation of Momentum: I = m v
• What kind of Force can force a planet to spin around the sun?
• Newton’s 2nd law of motion:
– The rate of change of the momentum I of a body is directly proportional
to the net force acting on it, and the direction of the change in
momentum takes place in the direction of the net force: F = dI/dt
Physical Fundamentals of GC
www.pik-potsdam.de/~stock
04-4
Gravitational Force
• Newton’s 2nd law of motion:
•
– The rate of change of the momentum I of a body is directly
proportional to the net force acting on it, and the direction of
the change in momentum takes place in the direction of the
net force: F = dI/dt
space r
Angular Motion:
– Angular position:
θ
– Angular velocity:
ω = dθ/dt
– Angular accelleration: α = dω/dt
v=ωr
a
• Calculation of the acting force:
– Orbital period tor in the space of position:
tor v = 2 π r
– Orbital period tor in the space of velocity:
tor a = 2 π v
•⇒
a = - v²/r
Physical Fundamentals of GC
α
space v
a=αr
v
⇒ F = m a = - m v²/r
www.pik-potsdam.de/~stock
04-5
Centripetal Motion in the Phase Space R(r,v)
V
m
ay
r
vx
-vy
-vx
ry
-ax
vy
rx
-rx
ax
-ry
-vy
vx
-ay
Physical Fundamentals of GC
tor = 2π r / v
-vx
tor = 2π v / a
⇒ a = + v² / r
centrifugal acceleration
a = - v² / r
vy
centripetal acceleration
⇒ Fc = - m v² / r
centripetal force
Using the phase space is a rather elegant and
simple method to calculate the centripetal or
centrifugal force, compared to the standard
method via vector and differential calculus.
The phase space is the coordinate system of
all variables describing the state of a system.
www.pik-potsdam.de/~stock
04-6
2. From Kepler’s laws of planetary motion
to Newton’s generalized laws of motion
• Kepler’s 2nd law of planetary motion:
– A line joining a planet and its star
sweeps out equal areas during
equal intervals of time.
• Newton’s 2nd law of motion:
– The rate of change of the momentum I of a body is directly
proportional to the net force acting on it, and the direction of the
change in momentum takes place in the direction of the net force:
F = dI/dt
• Conservation of Angular Momentum
– L=rxI=rxmv=rxmrω
= m r² dθ/dt ≅ m r² Δθ / Δt
Area A = (Δθ/2π) π r² (circle)
⇒ A / Δt = ½ L/m = constant
Physical Fundamentals of GC
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I
04-7
The Three Laws of Motion (Newton 1687)
Newton's Laws of Motion describe the motion of a body as a
whole and are valid for motions relative to a reference frame:
1. An object of mass m will stay at rest or move at a constant
velocity in a straight line unless acted upon by an unbalanced
force.
2. The rate of change of the momentum I of a body is directly
proportional to the net force acting on it, and the direction of
the change in momentum takes place in the direction of the
net force: F = dI/dt
3. To every action (force applied) there is an equal but opposite
reaction (equal force applied in the opposite direction).
⇒ gravitational force = centripetal force
= Fg(r) = - m v²/r = - m (2πr/tor)²/r
Physical Fundamentals of GC
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04-8
3. From Kepler’s laws of planetary motion
to Newton’s generalized laws of motion
• Kepler’s 3rd law of planetary motion:
The squares of the orbital periods
of planets are directly proportional
to the cubes of the semi-major axis
of the orbits
tor2 ∝ ror 3
tor = orbital period of planet
ror = semimajor axis of orbit
• gravitational force
= Fg(r) = - m v²/r = - m (2πr/tor)²/r = 4π² m/tor² r
⇒
Fg(r) = - G m r-2
Physical Fundamentals of GC
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04-9
Laws of Gravitation and Planetary Motion
gravitational force between two bodies of mass m1 and m2
separated by the distance r12
FG = - G m1 m2 / r122
G = 6.673 10-11 N m² kg-2
centfugal force acting against gravitation, accelerating a body of
mass m1 and speed v1 on an orbit with radius r12
Fc = m1 v1² / r12
We have shown that Kepler’s 3rd law is a consequence of Newton’s
3rd law expressed in the the relation: FG + Fc = 0
Task: calculate the Earth’s orbital speed, using this relation and the
parameters: mE = 5.98 1024 kg, mS =1.988 1030 kg, rES = 1.49 1011 m
Solution:
Epot = - FG * rES = G mE mS / rES = 5.3*1033 J
⇒ vE = 29.8*103 m/s
= Fc * rES = mE vE²
Physical Fundamentals of GC
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04-10
Titius-Bode “rule” for planetary distances
Planet
n
DTB
real av.
distance
DTB = 0,4 + 0,3 x 2(n-1) x sgn(n)
Distances in astronomical units, AU.
One AU is the average distance
between Earth and Sun, roughly
Mercury
0
0.4
0.39
Venus
1
0.7
0.72
Earth
2
1.0
1.00
AU = 149 598 000 km
Mars
3
1.6
1.52
Ceres1
4
2.8
2.77
Jupiter
5
5.2
5.20
The rule was proposed in 1766 by Johann
Daniel Titius and "published" without
attribution in 1772 by the director of the
Berlin Observatory, Johann Elert Bode.
Saturn
6
10.0
9.54
Uranus
7
19.6
19.2
Neptune
Pluto1
8
38.8
30.06
39,50
Eris1
9
77,2
68,00
Physical Fundamentals of GC
There is no solid theoretical explanation of
the rule, but it is likely a combination of
orbital resonance and shortage of degrees
of freedom.
1
dwarf planet
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04-11
Conservation Laws
a particular measurable property of an
isolated physical system does not
change as the system evolves:
• Conservation of linear momentum
• Conservation of angular momentum
• Conservation of energy
• Conservation of mass
(in classical mechanics)
Physical Fundamentals of GC
www.pik-potsdam.de/~stock
04-12
Energy = "the potential for causing changes."
There are several types of energy:
Example
associated with
• kinetic energy
motion
(formula)
Ekin = ½ m v²
∫
R2
• potential energy
position, work
Epot =
• internal energy
thermal energy
atomic speed
temperature T
heat Q, entropy S
dU = δQ + δW
= TdS + pdV
• radiant energy
radiation,
luminosity = E/t
• chemical energy
chemical bonds,
type of potential energy
Physical Fundamentals of GC
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R1
F(r) dr
L = 4πR² σ T4 [ W ]
04-13
Linear motion on Earth with constant acceleration g
Fg
=mg
; weight
h
h
⇒ Epot = 0∫ F(r) dr = Fg h = m g h
g = - 9.81 m s-2 ; acceleration
v(t) = - g t
; velocity
y(t) = ∫ v(t) dt
= h - ½ g t²
; position
h = ½ g tf²
= ½ vf²/g
; tf time of impact
⇒ Ekin = Epot = m g h
= ½ m vf ²
Physical Fundamentals of GC
; kinetic Energy
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04-14
Physical Quantities and Units
quantity
symbol
unit
equations
distance, position
r
m
area
volume
A
V
m²
m³
time
t
s
velocity
v
m/s
acceleration
a
m/s²
mass
m
kg
momentum
I
kg m/s
I =m*v
angular
momentum
L
kg m²/s
L =rxI
force
F
N, kg m/s²
F =m*a
work
W
J, Nm
energy
E
J, Nm
power
P
W, J/s
temperature
T
K
pressure
p
Pa, N/m²
Physical Fundamentals of GC
Asphere = 4 π r²
Vsphere = 4/3 π r³
W = R1∫
R2
F(r) dr
W/t, E/t
0 K = –273.15 °C
F/A
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04-15
Milankovitch Cycles
- circular orbits:
e=0
- elliptic orbits:
0<e<1
- parabolic trajectories: e = 1
- hyperbolic trajectories: e > 1
Physical Fundamentals of GC
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04-16
Milankovitch Cycles
- circular orbits:
e=0
- elliptic orbits:
0<e<1
- parabolic trajectories: e = 1
- hyperbolic trajectories: e > 1
Physical Fundamentals of GC
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04-17
Braun, H., M. et al., 2005: Solar forcing of abrupt glacial climate change in a coupled climate
system model. Nature, 438, 208-211.
Physical Fundamentals of GC
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04-18
Global Change Processes & Physical Fundamentals
• Planetary motion in solar systems:
1. Habitable planetary climates
• Natural climate change
–
–
–
–
– Kinematics: parameters of motion,
position, velocity, angular velocity
– Dynamics: Newton’s laws of motion,
mass, force, acceleration, momentum,
angular momentum
Milankovitch cycles
Orbital forcing
albedo
Greenhouse gas effect
• Radiative forcing – radiation laws
• Climate zones and seasons
• Atmospheric layering and
compositon
2. Climate variability
– Optics: reflection and refraction
– Electrodynamics: absorption and
emission
• Meteorology, atmospheric forces:
– Oceanic conveyer belt
– El Niño, ENSO, NAO
– Monsoon
3. Weather phenomena
– Low & High pressure, winds, cloud
formation, precipitation
– Sea level rise
• Extreme weather events
– Tropical and winter storms
– Tornados, thunderstorms
Physical Fundamentals of GC
– Pressure Gradient Force
– Gravity
– Coriolis Force
– Friction
– Centrifugal Force
• Meteorology, thermodynamics
– Radiation
– Evaporation
– Condensation
– Convection
www.pik-potsdam.de/~stock
04-19
Radiation
Energy is transferred by
electromagnetic waves
Types of radiant energy:
ƒX-rays
ƒRadio waves
ƒLight (sunlight)
ƒMicrowaves
visible
light ultraviolet
infrared
microwaves
Low
Energy
1000
100
λ (μm)
Physical Fundamentals of GC
10
1
0.1
0.7 to 0.4 μm
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x-rays
High
Energy
0.01
04-20
Black Body Radiation and Temperature
A black body is an object that
absorbs all electromagnetic
radiation that falls onto it.
The total energy radiated per
unit area per unit time Lb by a
black body is related to its
temperature T in K and the
Stefan-Boltzmann constant σ as
follows:
Lb = σ T4
with
σ = 5.67e-8 Wm-2K-4
spectral intensity according to
Planck's law of black body radiation
Physical Fundamentals of GC
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04-21
Heat radiation from a human body
The net power (energy/time) emitted is
the difference between what someone
absorbs from their surroundings and
what they radiate themselves:
Pnet = Pemit – Pabsorb
= A σ (Th4 – T04) ≅ 100 W
Calculate with:
A = 2 m², Th = 28°C T0 = 20°C
Physical Fundamentals of GC
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04-22
Solar Radiation and Greenhouse Energy Balance
4π R² IROUT
Refl.: π R² αS
1.0
0.9
Albedo α
0.8
IN: π R² S
T0(S,α)
TG(S,α,GF)
0.7
0.6
0.5
0.4
0.3
0.2
EGAIN = π R² S (1- α) + GF
EIR-OUT = 4 π R² σ T4
EGAIN = EIR-OUT
0.1
0
⇒ T0 = 255 K (-18°C) ⇒ TG = 288 K (+15°C)
Physical Fundamentals of GC
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04-23
Physical Fundamentals of GC
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04-24
Calculate Surface Temperatures in the Solar System
1. Mean surface temperature T(α)
of the Earth for different albedo α
without greenhouse effect.
α surface
T(α) K
°C
0 black
0.1 water, forest
0.3 actual mean
0.8 ice cover
DSE = 1.496e11 m, dist. Sun-Earth
RS = 0.696e9 m
RE = 6.373e6 m
DMe = 5.834e10 m
DVe = 1.077e11 m
DMa = 2.274e11 m
DPl = 5.909e12 m
Physical Fundamentals of GC
2. Surface temperature of the Sun
TS =
3. Surface Temperature of planets
(α = 0, no greenhouse effect):
a) Mercury (trot=torb)
b) Venus
c) Mars
d) Pluto
www.pik-potsdam.de/~stock
04-25
Calculate Surface Temperatures in the Solar System
1. Mean surface temperature T(α)
of the Earth for different albedo α
without greenhouse effect.
α surface
T(α) K
°C
278,
5
0 black
271,
-2
0.1 water, forest
255, -18
0.3 actual mean
186, -87
0.8 ice cover
2. Surface temperature of the Sun
DSE = 1.496e11 m , dist. Sun-Earth
5773, 5500
TS =
RS = 0.696e9 m
3. Surface Temperature of planets
RE = 6.373e6 m
(α = 0, no greenhouse effect):
DMe = 5.834e10 m
a) Mercury (trot=torb) 630, 357
DVe = 1.077e11 m
328, 55
b) Venus
DMa = 2.274e11 m
226, -47
c) Mars
DPl = 5.909e12 m
44, -229
d) Pluto
Physical Fundamentals of GC
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04-26
Solar Radiation: spatial variation and energy balance
Top of atmosphere (global
mean) S0 = 1360/4 W/m²
= 340 W/m²
IR = 227 W/m²
2/3 S0 = 227 W/m²
(67%)
45%
30% 15%
45%
global mean net solar energy
Sne ≅ 100 W/m²
for evapotranspiration, photosynthesis and other processes
Physical Fundamentals of GC
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04-27
Atmospheric electromagnetic transmittance or opacity
Physical Fundamentals of GC
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04-28
Insolation and atmospheric absorption
Physical Fundamentals of GC
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04-29
Climate Variability
- Elements and Mechanisms
• Incoming Solar Radiation Forcing
– Geographical Distribution
– Seasonal Distribution
• Forcing, Response and Feedback
– Energy Exchange by
Oceanic and Atmospheric Flow
– Seasonal and Orographical Variability
– Vegetation and Precipitation
Physical Fundamentals of GC
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04-30
Elements of Earth‘s Climate System
Physical Fundamentals of GC
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04-31
Long Term Climate Variability
Physical Fundamentals of GC
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04-32
Layers of the Atmosphere
hPa
Physical Fundamentals of GC
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04-33
Metric: 1 hectopascal (hPa) is:
Physical Fundamentals of GC
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04-34
Global atmospheric flux processes
Physical Fundamentals of GC
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04-35
What is the Mass of the Atmosphere?
• The complex way:
Mat = ∫ ∫ ∫ ρ(λ,θ,h) dλ dθ dh
= FE * ∫ ρ(h) dh
• The simple way:
Mat = FE * p(0) /g
= 5.27e18 kg
with
FE = 4 π RE²
RE = 6.373e6 m
p(0) = 101,325 Pa
g = 9.81 m/s²
Physical Fundamentals of GC
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= 5.105e14 m²
(Pa = kg m-1 s-2)
04-36
Water Vapor Pressure
hPa
Clausius-Clapeyron Equation:
pV= nRT
R = 8.314472 J K-1 mol-1
dp/dT = (Sg –Sf)/(Vg – Vf)
dp
ΔQ
ΔQ
── = ──────── ≅ ───
Vg T
dT
(Vg – Vf) T
heat of evaporation:
ΔQ = ΔU + p·ΔV
= 2088 kJ/kg + 169 kJ/kg
= 2.26 MJ/kg
Physical Fundamentals of GC
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04-37
What is the order of the mass of water in the atmosphere
and its evaporation energy ?
Mat = FE * p(0) /g
= 5.27e18 kg
⇒
Mw = Mat pw/p(0) mw/mat
= 1.25e16 kg
Ew = Mw * ΔQ
= 2.82e22 J
= 7,84e15 kWh
with
mw/mat = 18/30 (molecular mass ratio)
pw(-5°C) = 400 Pa
(Pa = kg m-1 s-2)
ΔQ = 2.26 MJ/kg
World energy demand:
1.07e14 kWh/a
Terrestrial solar energy:
1.49e22 kWh/a
• The simple way:
Physical Fundamentals of GC
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04-38
Global Hydrological Cycle: Reservoirs (10³ km³) and Flux (10³ km³/a)
www.wbgu.de,
D 1.3-1 (1997)
Physical Fundamentals of GC
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04-39
Tropical cyclone = Carnot cycle
The thermodynamic structure of the hurricane
can be modelled as a heat engine running
between sea temperature of about 300K and
the tropopause which has temperature of
about 200K.
Parcels of air traveling close to the surface
take up moisture and warm, ascending air
expands and cools releasing moisture (rain)
during the condensation. This release of
latent heat energy during the condensation
provides mechanical energy for the hurricane.
Both decreasing temperature of upper
troposphere or increasing temperature of
atmosphere close to the surface will increase
on maximum winds observed in hurricanes.
When applied to hurricane dynamics it
defines Carnot heat engine cycle and predicts
maximum hurricane intensity.
http://www.uncw.edu/phy/documents/Woods_06.pdf
Physical Fundamentals of GC
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04-40
Gefährdung durch Tropenstürme
Zugbahnen und Intensitäten der letzten 150 Jahre
Physical Fundamentals of GC
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04-41
Hurricane Energy and Sea Surface Temperature
Hurricane Power (PDI)
Sea Surface Temperature (August-October)
Global Mean Temperature
Atlantic
Observed data:
Hurricane energy closely linked to SST, and increasing
(Emanuel, Nature 2005)
Physical Fundamentals of GC
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04-42
Next Lecture (planned)
Processes behind Climate Variability
Physical Fundamentals of GC
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04-43