Environmental Physics

1
Environmental Physics

• Dr Bryan Gallagher Room C139
• bryan.gallagher_at_nottingham.ac.uk
• Course web site. www.nottingham.ac.uk/ppzblg/e
  nvironment.htm
• Course notes available at website (No suitable
  textbook).
• PowerPoint slides placed on website.
• Pre-requisites A level Physics and Maths or
  equivalent.
• Coursework Students doing physics courses
  First set out 6th Oct. hand in 13th Oct. Other
  students collect from and return to your first
  physics lecture of the week. 15 of assessment.

Content Basic physics required to understand a
range of global environmental issues inc. Global
Warming, Ozone Depletion, potential for renewable
nuclear energy etc.
2
Stability of Environments
1.1 Static Equilibrium
Consider an object, such as a ball, moving in a
one dimensional potential U(x).
State at x3 ?
stable equilibrium since
State at x2 ?
unstable equilibrium since
3
Metastability
State at x1? stable equilibrium ?
Yes, but it is metastable equilibrium
because x1 is not the position of lowest
potentials energy. A perturbation could cause
the ball to move into the potential minimum at x3
.
Bi-stability
potential energy at adjacent minima almost the
same. Perturbations can cause transitions
between two states.
4
Response to perturbations

• A system which is stable against small
  perturbations may not be stable against large
  perturbations.
• Past stability of environments is no guide to
  the behaviour under new types of perturbations.

Small variations in the Solar energy arriving at
the earth result in small changes in temperature.

Larger changes can trigger transitions into and
out of ice ages (world climate system is
bi-stable).
The previously stable ozone layer has been
destabilised by very small amounts of CFCs.
5
1.2 Dynamical Equilibrium
Stability often results from a balance between
opposing dynamical processes. For the last 104
years biological chemical processes have put
2 x 1014 kg of carbon into the atmosphere each
year. Other processes have taken out 2 x 1014
of kg carbon each year.
There was a dynamical equilibrium with a stable
CO2 concentration of 280ppm (part per million).
CO2 is currently increasing by about 4 x 1012 kg
yr-1
Co2 level is about 30 up on pre-industrial
levels .
Small perturbations of a dynamical balance can,
over time, lead to very large changes.
6
1.3 Feedback Mechanisms

• POSITIVE FEEDBACK

If a change in a system occurs then processes
that enhance or amplify the change are said to
provide positive feedback.
A Positive Feedback Loop
NEGATIVE FEEDBACK
Processes that act against and lessen the effects
of a change.
7
Will Global warming lead to less snow and ice?
Positive Feedback
Negative Feedback
8
1.4 Predictability and Chaos
Many natural systems, for example the weather,
show chaotic behaviour

• Classical determinism Laplace If we knew the
  positions and velocities of all particles in the
  universe we could uniquely determine all future
  events.

• This is untrue in macroscopic classical systems
  showing chaotic behaviour.

• CHAOS Stochastic behaviour in a deterministic
  system

• STOCHASTIC Predictable only in the statistical
  sense i.e. we can only state the probability of
  an event

9
Chaotic Rabbit Breeding
http//www.kimvdlinde.com/professional/biology/pop
dyn/popdyn_en.html Let the number of rabbits be
described by xt1 ?xt   xt is the number at
time t, xt1 the number in the following
generation and ? is the average number of
offspring per rabbit. If we start off with 2
rabbits and l 10 we would have
Generations (t) 1 2 3 4 5
No. of rabbits (x) 2 20 200 2000 20000
What really happens is that some negative
feedback mechanism, such as shortage of food,
comes into play. This can be modelled by
xt1 ?xt(1 xt)   Where xt is now the
fraction of some absolute maximum population for
which extinction would occur in one generation.
10
Steady Population
Population Dynamics (Try this out on your
calculator or computer)
Expect that the negative feedback factor will
limit the population growth and that the
population will reach some steady level. For
small values of ? this is correct.
xt1 ?xt ( 1 xt )
11
Oscillatory Population
As ? is increased the competition between the
positive and negative feedback leads to
oscillatory behaviour.
This is the sign of the population becoming
unstable.
12
Chaotic Population
As ? is further increased a point is reached at
which an apparently chaotic behaviour sets in.
13
Chaotic Population Sensitivity
Chaotic behaviour is unpredictable.
If we start with initial conditions differing by
only a very small amount then, after some short
time, the behaviour should be quite different.
Solid line initial value of x0 0.300 Dashed line
initial value of x0 0.301
Chaotic systems are extremely sensitive to
initial conditions.
Very small perturbations lead to very large
effects (Butterfly effect). One needs infinite
precision to look into the future.
14
Non-Linearity

• The underlying mathematical reason for chaos is
  non-linearity.
• Using the equation for the population
• we can write x2 in term of the initial value xo
  as
•  x2 ?2xo - (l ?) ?2 xo2 2?3 x3o - ?3 xo4
• After two generations the number of rabbits
  depends upon xo4.
• After four generations the number of rabbits
  depends upon xo16.

xt1 ?xt (1 - xt)
Non-linearity leads to rapid amplification of
perturbations.
Chaos can be regarded as persistent instability.
15
Predictions of climate models, IPCC 2001
Can still make predictions about average
properties (Climate) but not about the detailed
behaviour (weather)
16
1.5 Co-evolution of Life and Climate

• Living organisms often modify their environment
  in ways that improve their chances of surviving.
  Living organism also modify the global
  environment.
• The energy arriving at the Earth from the Sun has
  increased by 30 over the last 4 billion years.
  The average surface temperature of the Earth has
  remained at an almost constant value, compatible
  with life, during this period.
• 4 billion years ago the greenhouse effect was
  very strong with high levels of CO2 in the
  atmosphere. The modification of the CO2 level
  over time by living organisms has kept the
  temperature almost constant.
• It has been argued that the interaction between
  living organisms and the atmosphere can be seen
  as a feedback loop, living systems regulating the
  environment so as to maintain the conditions
  necessary for their survival.