Climate Change:some basic physicalconcepts and simplemodels
David Andrews

1Some of you have used mytextbook ‘An Introduction to
Atmospheric Physics’ (IAP)

I am now preparing a 2nd edition.
The main difference will be a new chapter on
Climate Physics.
This lecture will cover some of the new material.
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Outline• The physics of climate change is avery complex subject!
• However, many of the mostimportant ideas can be describedusing very simple models. (‘Toymodels’.)• An example is the Energy Balance
Model (EBM).
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I shall focus on:• Response of EBM to radiative forcing– Several examples– CO2 stabilisation– Clarification of Andrews and Allen figure fromprevious lecture• Introduction to climate feedbacks[if there is time]4Energy Balance Model• Simplest case: the ‘climate system’
includes the atmosphere, land
surface and ‘mixed layer’ (= top100m of ocean), but not deep ocean:Atmosphere
Land

Global-meantemperature T

Mixed layer
5Radiative power input and output(per unit area)
Solar input F↓

Earth’s output F↑

Climate system• Solar input F↓ : ‘short-wave’ radiation (visible,ultra-violet), wavelength ≤ 4 µm
• Earth’s output F↑ : ‘long-wave’ radiation (infrared), also called ‘thermal’ radiation, wavelength≥ 4 µm.

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In equilibrium:
Solar input = Earth’s output

Out of equilibrium:
Solar input ≠ Earth’soutput
Climate change

7Equilibrium climate: simple
models~ 1370 W m-2 shortwave radiation from
Sun, of which ~ 30%
is reflected[See IAP, 1.3.1]
8• Earth intercepts solar beam over its crosssectional area, πa 2
• So incident power = 1370 x 0.7 x
πa 2 Watts (short wave).
• But it emits (long-wave) power from all its
surface, 4πa 2.
• Assume Earth acts like a black body at(absolute) temperature T, so that it emits
power σT4 per unit area, whereσ = 5.67 x 10-8 W m-2 K-4
is the Stefan-Boltzmann constant.
910• This temperature (255 K) is much
lower than typical observed globalmean T ~ 290 K!
• So this model misses some
important physics, especially theeffects of greenhouse gases.

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Greenhouse gases• Are gases that absorb and emit infra-red
radiation but allow solar radiation to pass throughwithout significant absorption. They affect F↑ butnot F↓.
• Examples: Water vapour (H2O) and carbon
dioxide (CO2) are the two most important indetermining the current climate.
• Increasing CO2 is a major contributor to climatechange.
• Effect of H2O in climate change is more complex.
12• Very simple model of effect ofgreenhouse gases in producing
current climate: see IAP, 1.3.2.

• In this lecture I shall mostly focus onclimate change.

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Consider unit area of climatesystem14

Steady-state (equilibrium)
climate15

Perturbed climate

Taylor expansion
16Radiative forcing
Define radiative forcing by

Example: U’ might be an increase
in CO2 concentration

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Climate feedback
Define by
Examples:• ‘Black body feedback’: warmer Earthemits more long-wave radiation,
λ > 0, negative feedback•Water-vapour feedback: warmer
atmosphere contains more water vapour,traps more long-wave radiation,
λ < 0, positive feedback.

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Equation for EBM
Radiative forcing

Heat
capacity

Feedback
Rate of change of heat
content of climate system

19Solution of EBM for somespecified time-dependent
radiative forcings F(t)F(t) could represent, say, radiative
forcing due to an increase ofgreenhouse gases such as CO2.
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General solution

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Some examples of F(t)t22

Equilibrium climate
sensitivity

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F(t)

Solution to EBM equation (∗)
for step function forcing

T’ approaches
S as t → ∞Temperature
response / S

Readas T’Time / feedback response time
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t25

F(t)

Temperature
response linear
at large t

Temperature response
initially quadratic
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CO2 stabilisationtime

EquilibriumtemperatureF(t) initially grows
linearly and thenlevels off

Linear phase29

What can this model tell us about how theclimate might respond to CO2 stabilisation?

• Can we forecast the equilibrium
temperature, given information atsome earlier time?
e.g. predict thistemperature…
… given this?
(This temperature is called
the transient climate
response, TCR.)
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• This is easy if– we have a perfect model– we know all the parameters.

• But it is difficult to do in practice!
• Even if we believe our EBM
represents the physics quite well(???), we still have to know thevalues of parameters C and λ.
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A possible approach• Forecast (say) 100 years ahead withvery complex general circulation
model.
• Fit results for global-meantemperature to EBM over this 100years.
• Then use EBM for forecast beyond
100 years.
32• But there will be uncertainties in thefitted values of C and λ.
(Note: the value of the heat capacity C can’t
be determined “from first principles”,
because of likely importance of the deepocean.)• This may make it difficult to get anaccurate value for equilibrium
temperature.
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Example
Take ramp time = 70 years.
Fix λ = 1.2 Wm-2 K-1
Equilibrium

Vary τ = C/ λ: 10 years,
40 years, 70 years.

Forcing
Temperature
response34

τ = 10 years: 70-year T’ is
88% of equilibrium valueτ = 40 years: 70-year T’ is
54% of equilibrium valueτ = 70 years: 70-year T’ is
38% of equilibrium value35• So given the temperature at 70years, the equilibrium value depends
strongly on the (poorly-known) valueof C.
• But actually, the value of thefeedback parameter λ is even morepoorly known!

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Use of EBM to provide
diagnostics for IPCC climate
forecasting models(Andrews & Allen, Atmos. Sci. Lett. 9,7-12, 2008)

Each GCM run (numbered dot)
gives an estimate ofparameters such as:
TCR,
ECS = S,
Heat capacity = C,Feedback response time = τ
These are not all independentof each other.
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Contours: likelihood calculated for an
observationally-constrained ensembleof 28,800 simple climate models

Projection of likelihood onto axis

Clearest interpretation of GCMresults is in terms offeedback response time andtransient climate response.
The GCMs systematicallyunderestimate the TCR
compared with observations

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Climate Feedbacks

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Simplest case: black body feedback40

Water vapour feedback

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The End43