What’s the point of climate dynamics?Gerard Roe & David Battisti, Seattle, WA

The ultimate source of it all

(NASA, TRACE)

Its quite big…

and shiny…

Firstly, the fix on the annual mean, and look at the radiation budget

Incoming solar flux

W m

-2 (daily average top

of atmosphere)

80 N60 N40 N20 N0 20 S40 S60 S80 S

latitude

[Peixoto and Oort, 1992]

Percent reflected

(top of atmosphere albedo) %

~30%

Annual mean radiation budget

80 N60 N40 N20 N0 20 S40 S60 S80 S

latitude

[Peixoto and Oort, 1992]

Amount solar absorbed Q0(1-

Amount of longwaveemitted to space, F:

The difference:W

m-2

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

[n.b. diff. tops out at 100 W m-2 high latitudes; more typically ~30-50Wm-2

divergence

conv. conv.

Seasonal cycle:Daily mean insolation at top of atmosphere

Hartmann, 1994

-peaks at poles in summer solstice.

-is zero at poles at winter solstice.

- global average ~ 342 Wm-2 la

titud

e

month

Northern summer radiation budget (Jun. Jul. Aug.)

80 N60 N40 N20 N0 20 S40 S60 S80 S

latitude

[Peixoto and Oort, 1992]

Amount solar absorbed Q0(1-

Amount of longwaveemitted to space, F:

The difference:W

m-2

[n.b. on seasonal time scales also have to worry about storage]

80 N60 N40 N20 N0 20 S40 S60 S80 S

latitude

[Peixoto and Oort, 1992]

Amount solar absorbed Q0(1-

Amount of longwaveemitted to space, F:

The difference:W

m-2

Northern winter radiation budget (Dec. Jan. Feb.)

Back to annual mean picture

Hartmann, 1994

- except at high latitudes, convergence/divergence small (~20%),compared to radiation terms

- “Climate is 80% radiation and 20% dynamics”

dynamicalheat flux

- worth seeing how far radiation picture of climate can get you.

Three lectures: outlineTry 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…

U.S. National Research Council report, 2003

Defines climate feedbacks incorrectly!

Three lectures: outline2nd lecture (David)

What is our basic, default understanding of role of dynamics fora) 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?

Three lectures: outline3rd 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 defaultpicture?

Or,

- does the proxy record compel us to change (i.e., advance)our understanding

Candidate examples (us, plus)

€

F−S =0

€

F=A+BT

€

S =Q0(1−

€

T =Q0(1−−A

BSolve for T:

F = outgoing infra-red radiation:

S = absorbed solar radiationLet:

and

[n.b. if T F]

Obtain climate from energy balance:

Q0 = incident radn, = albedo

Typical values: Q0=340 Wm-2, =0.3, A=200 Wm-2, B=2 Wm-2oC-1

1. Climate at a point: the simplest model

T = 19oC (c.f. 15oC not too shabby)

[T in oC]

1. Climate at a point: the response to a radiative forcing, N

Old climate:

New climate:

Define climate sensitivity, , to create an objective measureof the system response to forcing:

€

F−S =0

€

′ F − ′ S =N

€

F−S =N

€

T =N

[this is a difference eqn for adjustment of F,S to radn forcing]

Question: How does the climate change?

Take difference:

is change in T per unit change in N]

1. Climate at a point: the response to a radiative forcing, N

€

N=F−S

€

N=dFdT

−dSdT

⎛ ⎝ ⎜

⎞ ⎠ ⎟T

€

=1

dFdT

−dSdT

⎛ ⎝ ⎜

⎞ ⎠ ⎟

€

dFdT

=B,dSdT

=0

€

0 =1B

Perturbation balance:

Taylor series in temperatureto 1st order:

And so, climate sensitivity is:

For our world, [()0 denotes reference climate model. ]

, so

Typical numbers: B=2 W m-2 oC-1 0=0.5 oC/Wm-2

(2 x CO2 4 Wm-2 at tropopause T = 0N = 2oC)

2. Climate at a point: what if there is a feedback?

Question: what is a feedback?

System(e.g., climate!)

input output

Answer: an interaction where the input to a system is a partial function of the output from the system.

[n.b. a feedback is only meaningfully defined in terms of a reference system without that feedback (hence the switch)]

e.g., microphone amplifier speaker (= hearing loss)

2. Climate at a point: what if there is a feedback?

Propose albedo feedback:

€

=(T

€

S =Q0(1−(T

€

N=dFdT

−dSdT

⎛ ⎝ ⎜

⎞ ⎠ ⎟T

€

T =1

B+Q0ddT

⎛ ⎝ ⎜

⎞ ⎠ ⎟N

€

=0

1+ 0Q0ddT

Input, S, now a fn of output, T

Revisit radn balance eqn

now with extra term:

Rarrange for T:

Gives new climate sensitivity:

i.e., let be a function of T:

is a feedback

e.g., T

3. Climate at a point: some terminology

Defining a system Gain, G, provides an objective measure of the climate response to feedbacks

€

G =TT0

=responsewithfeedbck

responsewithoutfeedbckDefn of Gain:

In our world:

€

G =TT0

=1

1−0Q0ddT

Example: suppose = 0.3 x (1-0.01T) [i.e. a 1% change in per 1oC change in T]

For our typical numbers gives G 2.

Allowing an albedo feedback has doubles the sensitivity of our climate to a change in forcing!

3. Climate at a point: some terminology

Define a feedback factor

€

G =1

1−fSo, in our world, albedo feedback factor is

€

f=−0Q0ddT

[n.b. G2f0.5]

[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 < 0 G < 1 response damped.0 < f < 1 G > 1 response amplified.f > 1 G undefined planet explodes/melts.

[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)].

f

G

1

1

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:

€

N=dFdT

−dSdT

⎛ ⎝ ⎜

⎞ ⎠ ⎟T

Now, we have partial derivatives to deal with:

€

N=∂F∂T

+∂F∂q

∂q∂T

+∂F∂fc

∂fc∂T

−∂S∂

∂∂T

⎛ ⎝ ⎜

⎞ ⎠ ⎟T

€

T =0N

1+ 0∂F∂q

∂q∂T

+ ∂F∂fc

∂fc∂T

−∂S∂

∂∂T

⎛ ⎝ ⎜

⎞ ⎠ ⎟

Rearranging…

Hence

€

=0N

1+ 0∂F∂q

∂q∂T

+ ∂F∂fc

∂fc∂T

−∂S∂

∂∂T

⎛ ⎝ ⎜

⎞ ⎠ ⎟

[n.b. The crux is that the partial derivatives appear in the denominator]

4. Climate at a point: what if there is more than one feedback?

For general variables, xi:

€

G =TT0

=0

=1

1+ 0∂F∂xi

∂xi

∂Ti

∑ −∂S∂xi

∂xi

∂Ti

∑ ⎛ ⎝ ⎜

⎞ ⎠ ⎟

Or, can write

€

G =1

1− fii

∑; =

0

1− fii

∑

where

€

fi =−0∂F∂xi

∂xi

∂T, 0

∂S∂xi

∂xi

∂T

[n.b. fi is a function of 0, 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!

4. Climate at a point: what if there is more than one feedback?

Example 1-suppose water vapor enhances response by 50% G=1.5, f=1/3-suppose alb. fdbck. enhances response by 100% G=2.0, f=1/2

Linear combination of G would be 1 + 0.5 + 1.0 = 2.5

€

G =1

1− fii

∑ = 6.0 [i.e., more than double, because feedbacks reenforce]

Example 2: -two cases where the two gains would cancel:

Case A: G1=1.2, G2=0.8 Case B: G1=1.8, G2=0.2 f1=1/6, f2 = -1/4 f1=4/9, f2=-4 GT=12/13 0.92 GT=9/41 0.22

Case B much more strongly damped than case A:

Actual

5. Climate at a point: time dependence

Allow for storage of heat:(& warming/cooling)

€

CdTdt

=S −F

€

T =T + ′ T

€

Cd ′ T dt

=−dFdT

−dSdT

⎛ ⎝ ⎜

⎞ ⎠ ⎟′ T

€

=1

dFdT

−dSdT

⎛ ⎝ ⎜

⎞ ⎠ ⎟=

0

1− fii

∑

€

d ′ T dt

=−′ T

C

C = heat capacity (or thermal intertia)

Linearize:

So eqn becomes

But from before

So get simple form:

€

T

€

′ T = mean temp.

= pert. temp.

€

′ T =T0 exp(−t/ t

€

t =C0

1− fii

∑

€

d ′ T dt

=−′ T

C

5. Climate at a point: time dependence

Let T ' = T0 at t = 0

has solution

Governing eqn

where

t is the response timescale of the system to a perturbation i.e., the characteristic timescale of climate variability. or, equivalently t is ‘memory’ of the system - the duration over which it retains information about previous states.

5. Climate at a point: time dependence

€

t =C0

1− fii

∑

More on characteristic timescale:

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. eip = -1].

Crucial point #2

It is a very strong expectation that this relationship applies to most aspects of the climate system - (fundamental to alldynamic systems).

Crucial point #3

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’:

€

Cd ′ T dt

=−dFdT

−dSdT

⎛ ⎝ ⎜

⎞ ⎠ ⎟′ T + d ′ N

€

d ′ T dt

=−′ T t+d ′ N C

rearranging

restoring ‘force’ - a relaxation back to equilibrium (T'=0)

Random forcingdrives T' away from eqm.

€

d ′ T dt

=Tt −Tt−t

t

€

′ T t =1 ′ T t−t +0ut

€

0 =|d ′ N |t

C, 1 =1−

tt

1 =exp(−tt ift ≈t

⎛ ⎝ ⎜

⎞ ⎠ ⎟

6. Climate at a point: the effect of random noise

Discretize eqn into time steps, t:

Equation becomes

€

currentstate memory of

last state

random noise

- How does the system’s response to noise vary as a function of the memory, t = t(,C, fi)?

where nt is white noise

The greater the memory, the longer the timescale of the variability

6. Climate at a point: the effect of random noise

The effect of varying ton the response of T’to forcing by noise.

note century-scale excursions here

t = 1 yr

t = 25 yrs

t = 5 yrs

Time (yrs)

note multi-decadalexcursions here

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 yrs, that proxy will have long timescale (i.e., centennial) variability even in a constant climate.

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)

ampl

itude

log(

pow

er)

log(period=1/frequency)

time

7. Climate at a point: power spectrum of response to noise

Power spectra of response to noise as a function of t

-The long t is, the ‘redder’ the power spectrum, i.e., the greater the relative amount of long period variability.

log(period=1/frequency)

log(

pow

er)

t =25 yrs

t =5 yrs

t =1 yr

As t there is more damping at high frequencies

7. Climate at a point: power spectrum of response to noise

log(period=1/frequency)

log(

pow

er)

t =25 yrs

t =5 yrs

t =1 yr

€

P(f =0

1+ pft(

Eqn for this power spectra

- Can show 50% of power in the spectrum occurs at periods which are greater than 2p x t.

Long period variability is driven by short timescale physics.Crucial point #5

t

t

t

€

t =g

€

exp(−t/ t

€

exp(iwt, w =p

period

€

pg

Why the factor of 2p?

7. Climate at a point: power spectrum of response to noise

i) Hand-wavy analogy with pendulum:

Physical timescale

Oscillation period

ii) Real reason:

Real time-behavior ~

Projected onto sines and cosines ~

8. Example: the Pacific Decadal Oscillation

Dominant pattern of sea surface temperatures in North Pacific

8. Example: the Pacific Decadal Oscillation

Power spectrum of PDO index

Best fit is a red noise process with a t of 1.20.3 yearsi.e., indistinguishable from an annual timescale.

Lots of variability at multi-decadal time scale

But….

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]

Summary:

- get definitions of feedbacks right!

- t, , C,and fi are all inextricably intertwined.

- robust property of natural, nonlinear dynamical systems.

- if a climate proxy has memory, expect long timescale variationseven 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!

€

F=sT4

€

A=sT 4, B =4sT 3

Small print:

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.

In general, feedback strengths and sensitivity are functions ofthe mean state:

For a black-body

We linearized into A+BT:

[Stefan-Boltzmann law]

Our basic climate sensitivity: 0=1/B, is a sensitive function of T.