What

1
Whats the point of climate dynamics?
Gerard Roe David Battisti, Seattle, WA
The ultimate source of it all
2
Its quite big
(NASA, TRACE)
3
and shiny Firstly, focus on the annual mean,
and look at the radiation budget
Incoming solar flux
(daily average top of atmosphere)
W m-2 ?
Percent reflected
(top of atmosphere albedo)

30
latitude
Peixoto and Oort, 1992
4
Annual mean radiation budget
Amount solar absorbed Q0(1-a)
Amount of longwave emitted to space, F
W m-2 ?
conv.
conv.
The difference
divergence
n.b. diff. tops out at 100 W m-2 high
latitudes more typically 30-50Wm-2
latitude

• surplus of energy in low latitudes, deficit in
  high latitudes.
• this radiation imbalance drive climate dynamics.

Peixoto and Oort, 1992
5
Seasonal cycle
Daily mean insolation at top of atmosphere

• peaks at poles in
• summer solstice.
• is zero at poles at
• winter solstice.
• - global average
• 342 Wm-2

latitude
month
Hartmann, 1994
6
Northern summer radiation budget (Jun. Jul. Aug.)
Amount solar absorbed Q0(1-a)
Amount of longwave emitted to space, F
W m-2 ?
The difference
latitude
n.b. on seasonal time scales also have to worry
about storage
Peixoto and Oort, 1992
7
Northern winter radiation budget (Dec. Jan. Feb.)
Amount solar absorbed Q0(1-a)
Amount of longwave emitted to space, F
W m-2 ?
The difference
latitude
Peixoto and Oort, 1992
8
Back to annual mean picture
dynamical heat flux

• except at high latitudes, convergence/divergence
  small (20),
• compared to radiation terms
• - Climate is 80 radiation and 20 dynamics

- worth seeing how far radiation picture of
climate can get you.
Hartmann, 1994
9
Three lectures outline
Try to build up a default picture of our
understanding of climate dynamics, to put climate
records, and climate challenges in context. This
lecture, think about radiation balance at a point
(i.e., climate without dynamics). Can get
demonstrate some very basic properties of
climate - climate sensitivity - climate
feedbacks - timescale of climate
variability These are fundamental to climate,
dynamic systems in general, and hold in much more
complicated (and realistic) systems) 95 of
climate can be explained by these
processes (Axel
Timmermann) Secondly, an area in which there is
huge amounts of confusion
10
U.S. National Research Council report, 2003
Defines climate feedbacks incorrectly!
11
Three lectures outline
2nd lecture (David)
What is our basic, default understanding of role
of dynamics for a) climatology? b)
seasonality? c) variability?
- Paleoclimate motivation Past is prologue -
Dynamics perspective The present is
precedent - Perhaps the best starting point for
how things might have worked in the past is how
things work today?
12
Three lectures outline
3rd lecture (joint and everyone)
Where is this default picture of climate dynamics
challenged?

• When particular climate proxy records are
  considered
• is it likely the proxy record can be explained
  using this default
• picture?
• Or,
• does the proxy record compel us to change (i.e.,
  advance)
• our understanding

Candidate examples (us, plus)
13
1. Climate at a point the simplest model
Let
Q0 incident radn, a albedo
S absorbed solar radiation
and
T in oC
F outgoing infra-red radiation
n.b. if T? F?
Obtain climate from energy balance
Solve for T
Typical values Q0340 Wm-2, a0.3, A200 Wm-2,
B2 Wm-2oC-1
?T 19oC (c.f. 15oC ?not too shabby)
14
1. Climate at a point the response to a
radiative forcing, DN
Question How does the climate change?
Define climate sensitivity, l, to create an
objective measure of the system response to
forcing
l is change in T per unit change in N
Old climate
New climate
this is a difference eqn for adjustment of F,S
to radn forcing
Take difference
15
1. Climate at a point the response to a
radiative forcing, DN
Perturbation balance
Taylor series in temperature to 1st order
And so, climate sensitivity is
For our world,
, so
()0 denotes reference climate model.
Typical numbers B2 W m-2 oC-1 ? l00.5
oC/Wm-2
(2 x CO2 ? 4 Wm-2 at tropopause ? DT l0DN
2oC)
16
2. Climate at a point what if there is a
feedback?
Question what is a feedback?
Answer an interaction where the input to a
system is a partial function of the output
from the system.
( hearing loss)
n.b. a feedback is only meaningfully defined in
terms of a reference system without that
feedback (hence the switch)
17
2. Climate at a point what if there is a
feedback?
Propose albedo feedback
e.g., T? ??
i.e., let a be a function of T
Input, S, now a fn of output, T
? is a feedback
Revisit radn balance eqn now with extra term
Rearrange for DT
Gives new climate sensitivity
18
3. Climate at a point some terminology
Defining a system Gain, G, provides an objective
measure of the climate response to feedbacks
Defn of Gain
In our world
i.e. a 1 change in ? per 1oC change in T
Example suppose ? 0.3 x (1-0.01T)
For our typical numbers gives G ? 2.
Allowing an albedo feedback has doubled the
sensitivity of our climate to a change in
forcing!
19
3. Climate at a point some terminology
Define a feedback factor
So, in our world, albedo feedback factor is
n.b. G?2?f?0.5
n.b. It can be shown from above that the
feedback factor is equal to the fraction of the
output which is fed back into the input (hence
the name).
Some possibilities -? ? f lt 0 ? G lt 1 ? response
damped. 0 lt f lt 1 ? G gt 1 ? response amplified. f
gt 1 ? G undefined ? planet explodes/melts.
f
n.b. the maths has gone weird b/c we assumed a
new equilibrium existed with the change in
forcing, which is not true in a catastrophic
runaway ve feedback (AND you have forgotten
some physics).
20
4. Climate at a point what if there is more than
one feedback?
Suppose longwave radiation is also affected by
water vapor, q, and cloud fraction, fc That
is F F(T,q,fc)
- What is the total response of the system to
combined feedbacks?
Previous radn balance eqn
Now, we have partial derivatives to deal with
Rearranging
Hence
n.b. The crux is that the partial derivatives
appear in the denominator
21
4. Climate at a point what if there is more than
one feedback?
For general variables, xi
Or, can write
where
n.b. fi is a function of l0, and hence depends
on the reference climate state
Crucial point 1
Individual feedback factors combine linearly for
total response,whereas individual gains
definitely do not combine linearly!
22
4. Climate at a point what if there is more than
one feedback?

• Example 1
• suppose water vapor enhances response by 50 ?
  G1.5, f1/3
• suppose alb. fdbck. enhances response by 100 ?
  G2.0, f1/2

Linear combination of G would be 1 0.5 1.0
2.5
Actual
6.0
i.e., more than double, because feedbacks
reenforce
Example 2 -two cases where the two gains would
cancel Case A G11.2, G20.8 Case B
G11.8, G20.2 f11/6, f2 -1/4
f14/9, f2-4 GT12/13 ?
0.92 GT9/41 ? 0.22
Case B much more strongly damped than case A
23
5. Climate at a point time dependence
Allow for storage of heat ( warming/cooling)
C heat capacity (or thermal
intertia)
mean temp.
Linearize
pert. temp.
So eqn becomes
But from before
So get simple form
24
5. Climate at a point time dependence
Let T ' T0 at t 0
Governing eqn
has solution
where

• is the response timescale of the system to a
  perturbation
• i.e., the characteristic timescale of climate
  variability.
• or, equivalently ? is memory of the system -
  the duration over
• which it retains information about previous
  states.

25
5. Climate at a point time dependence
More on characteristic timescale
Crucial point 2
Four fundamental properties of the climate system
- the characteristic timescale of climate
variability, the thermal inertia of the system,
the climate sensitivity, and climate feedbacks -
are all intrinsically and tightly linked to each
other c.f. ei? -1.
Crucial point 3
It is a very strong expectation that this
relationship applies to most aspects of the
climate system - (fundamental to all dynamic
systems).
26
6. Climate at a point the effect of random noise
Can always expect random noise in the climate
system
What is the effect of this noise on the behavior
of the system?
For time dep. eqn with random noise, dN
rearranging
Random forcing drives T' away from eqm.
restoring force - a relaxation back to
equilibrium (T'0)
27
6. Climate at a point the effect of random noise
Discretize eqn into time steps, ?t
Equation becomes
where ?t is white noise
current state
random noise
memory of last state
- How does the systems response to noise vary as
a function of the memory, ? ?(l,C, fi)?
28
6. Climate at a point the effect of random noise
The effect of varying t on the response of T to
forcing by noise.
note multi-decadal excursions here
note century-scale excursions here
The greater the memory, the longer the timescale
of the variability
29
6. Climate at a point the effect of random noise
Climate is defined the statistics of weather.
(i.e. the mean and standard deviation of
atmospheric variables) Therefore a constant
climate has a constant standard deviation (i.e.,
even in a constant climate there is
variability) Crucial point 4 In
paleoclimate, if the proxy variable (i.e.,
glacier, lake, tree, elephant, etc.) has long
memory, say, t 25 yrs, that proxy will have
long timescale (i.e., centennial) variability
even in a constant climate.
30
7. Climate at a point power spectrum of response
to noise
Quick intro to power spectra they are an
alternative way of describing a time series
Time series
Power spectrum
Gives power (energy) at each sine wave frequency
that makes up the time series (analogous to
spectrum of light)
31
7. Climate at a point power spectrum of response
to noise
Power spectra of response to noise as a function
of t
As ?? there is more damping at high
frequencies

• The long t is, the redder the power spectrum,
• i.e., the greater the relative amount of long
  period variability.

32
7. Climate at a point power spectrum of response
to noise
Eqn for this power spectra
t

• Can show 50 of power
• in the spectrum occurs at
• periods which are greater
• than 2? x ?.

t
t
Crucial point 5
Long period variability is driven by short
timescale physics.
33
7. Climate at a point power spectrum of response
to noise
Why the factor of 2??
i) Hand-wavy analogy with pendulum
Physical timescale
Oscillation period
ii) Real reason
Real time-behavior
Projected onto sines and cosines
34
8. Example the Pacific Decadal Oscillation
Dominant pattern of sea surface temperatures in
North Pacific
35
8. Example the Pacific Decadal Oscillation
Power spectrum of PDO index
Lots of variability at multi-decadal time scale
But.
Best fit is a red noise process with a ? of
1.2?0.3 years i.e., indistinguishable from an
annual timescale.
36
8. Example the Pacific Decadal Oscillation
PDO index
Random noise 1.2yr memory
Random noise 1.2yr memory
n.b. note the apparent long timescale
variability even with short memory
37

• Summary
• get definitions of feedbacks right!
• ?, l, C,and fi are all inextricably intertwined.
• robust property of natural, nonlinear dynamical
  systems.
• if a climate proxy has memory, expect long
  timescale variations
• even if climate is not changing.
• changes the question from what was the climate
  change
• to was there a climate change?
• Long period variability is driven by short
  timescale physics,
• default expectation is that multi-decadal
  variability comes from
• sub-decadal physics!

38
Small print
In general, feedback strengths and sensitivity
are functions of the mean state
For a black-body
Stefan-Boltzmann law
We linearized into ABT
Our basic climate sensitivity l01/B, is a
sensitive function of T.
Feedback analyses are completely linear, can be
trouble when strongly nonlinear behavior is
being studied. Feedback analyses rely on
defining an appropriate reference state, against
which to test models/observations. You have to be
careful that the same reference state is being
use when comparing feedbacks.