Feedbacks and climate sensitivity

Program on Climate Change,Summer Institute.

Feedbacks are found in many forms:Mechanical feedbacks used in water clocks in ancient Greece.

Float valve, Greek Clepsydra (“water thief”)

Feedbacks are found in many forms:A feedback in every bathroom….

Ballcock assembly

Feedbacks are found in many forms:James Watt’s governor

“Science”Holborn viaduct,

London

Feedbacks are found in many forms:Centrifugal governors in windmills since 16th century.

Centrifugal governor, Dutch Mill

“On governors” Maxwell, 1869

Harold S. Black (1898-1983)

- introduced the concept of negative feedback.

- got the idea on Lackawanna ferry on his way to work.

- took nine years to get granted a patent.

“Our patent application was treated in the same manner one would a perpetual motion machine” Black, H.S. IEEE Spectrum, 1977

Original notes scribbled on NY Times

Feedback analysisFormalized framework for the evaluation of interactions in dynamical systems.

Feedback analysisThe language of feedbacks is ubiquitous in Earth Sciences(Maxwell, 1863; Black, 1927; Cess, 1975; Charney et al., 1979; Hansen et al., 1984; Schlesinger & Mitchell, 1985)

But the language is confused and abused…

U.S. National Research Council report, 2003

- gets definitions of feedbacks wrong…

- Worth standardizing terminology

Feedback analysisDefinition of reference system is intrinsic to feedbacks

reference climate system

forcing, R responseT

Climate sensitivity parameter defined by: T0 = 0 R

Feedback analysisAdding a feedback

reference climate systemR T

c1T

So now T = 0(R + c1T )

Feedback analysisAdding a feedback

reference climate systemR T

c1T

So now T = 0(R + c1T )Additional radn forcing due to system response to R

Feedback analysisAdding a feedback

reference climate systemR T

c1T

So now T = 0(R + c1T )

€

T =0R

1−c10

Rearrange for T

€

Gain=responsewithfeedback

responsewithoutfeedback=

TT0

Feedback factor: f = c10 (f to fraction of output fed back into input)

(Gain is proportion by which system has gained)

€

T =GT0

€

G =1

(1−f)

From before

€

T =0R

1−c10

=T0

1−f

And since :

Feedback analysisTechnobabble

- < f < 0: G < 1 response damped NEGATIVE fdbk.0 < f < 1: G > 1 response amplified POSITIVE fdbk.f > 1: G undef. Planet explodes…

Range of possibilities:

Feedback analysisThe gain curve

Aspects of feedbacks I.The compounding effect of multiple feedbacks

Treference

climate systemR

c1T

c2T

T = 0(R + c1T + c2T)Now have (two nudges)

The effect of one feedback is influenced by the strength of the others..

€

G =TT0

=1

1− fii=1

N

∑

Aspects of feedbacks II.Comparing different feedbacks

Treference

climate system

R

c1T

c2T

The relative importance of two different feedbacks must be evaluated relative to the same reference system.

There is a danger is comparing separate studies where only onepiece of physics has been isolated.

T fo

r 2 x

CO

2 (o C

)

∆T = ∆T0

1 - f

Aspects of feedbacks III.How does uncertainty in feedbacks translate into uncertainty inthe system response?

f

T fo

r 2 x

CO

2 (o C

)

Aspects of feedbacks III.How does uncertainty in feedbacks translate into uncertainty inthe system response?

∆T = ∆T0

1 - f

f

T

T fo

r 2 x

CO

2 (o C

)

Aspects of feedbacks III.How does uncertainty in feedbacks translate into uncertainty inthe system response?

∆T = ∆T0

1 - f

f

T

T fo

r 2 x

CO

2 (o C

)

f

Aspects of feedbacks III.How does uncertainty in feedbacks translate into uncertainty inthe system response?

∆T = ∆T0

1 - f

f

T

T fo

r 2 x

CO

2 (o C

)

T

f

Aspects of feedbacks III.How does uncertainty in feedbacks translate into uncertainty inthe system response?

∆T = ∆T0

1 - f

Systems of strong positive feedbacks inherently less predictable

Aspects of feedbacks IV.The relationship between feedbacks and response time

climate model response (mean & 95% bounds) to step function in forcing

€

τ =Cλ 01− Σ

if i

Positive feedback systems have inherently long response times

Aspects of feedbacks V.Diagnosing feedbacks from models and observations

€

fi ≈0Rαi

⎞

⎠⎟j, j≠i

⋅αi

T

€

ciT =R)j, j≠i

=∂R∂αi

⎞

⎠⎟j, j≠i

dαi

dTTFor ith climate variable:

So feedback factors:

αi - can be a lumped property (like clouds, sea ice, etc.), - or individual model parameter (like entrainment coefficient) - can also calculate spatial variations in fi if desired.

Aspects of feedbacks V.Diagnosing feedbacks from models and observations

Feedback factor of ice albedoon sea-ice thickness (Bitz, 2008)

Springtime snow albedo feedback(Fernandes et al., 2009)

Aspects of feedbacks VI.The variable of interest matters…

The same physical process can be a positive or negative feedbackdepending on the variable of interest.

e.g., dynamic sea-ice is

- a positive feedback on surface air temperatures

- a negative feedback on mixed layer temperature

Strengths of feedback analysis.Good points:

• Feedback analysis powerful representation of system dynamics -system will try to adjust via most negative feedback.

• Can be used to propagate how uncertainty in one process

controls uncertainty in system response.

• Puts different mechanisms in the same non-dimensional language.

e.g., Gaia is just a number….fGaia ~ -0.65 (which is pretty absurd)

Issues with feedback analysis.Not always a useful technique...

- Is the system linear enough that is makes sense to isolate theindividual feedbacks? (The Humpty Dumpty test)

- Is the reference system and variable of interest clear when comparing different feedbacks?

- Feedback analysis can get blurry when physics has differenttimescales (what’s a forcing, and what’s a feedback?)

Climate sensitivity.A benchmark of climate change.An envelope of uncertainty.

Eqm. response of global, annual mean sfc. T to 2 x CO2.

6,000 model runs, perturbed physics.

Slab ocean, Q-flux 12 model params. varied

What governs the shape of this distribution:

a) in observations?

b) in models?

1,200,000+ integrations, 75,000,000+ yrs model time(!);

Climate sensitivity.Estimates from observations.

€

Rf = F + −1T

€

T2xCO2= Rf 2xCO2

Global energy budget:

forcing storage(ocean)

atmospheric response

In principle, get Rf, F, T from observations, solve for , then:

= +

Climate sensitivity.Estimates from modern observations.

€

=T

Rf −F

€

÷

IPCC 2007 (mainly, plus a bit from Kyle)

€

T =0.76 ±0.1oC (1σ)

Temperature change Forcing change

Climate sensitivity.Estimates from modern observations.

€

=T

Rf −F

€

÷

IPCC 2007 (mainly, plus a bit from Kyle)

€

T =0.76 ±0.1oC (1σ)

€

Rf −F =0.9 ±0.56Wm−2 (1σ)

Temperature change Forcing change

Climate sensitivity.Estimates from modern observations.

€

=T

Rf −F

€

÷

€

T =0.76 ±0.1oC (1σ)

€

=

€

Rf −F =0.9 ±0.56Wm−2 (1σ)

Climate sensitivity.Estimates from last glacial maximum observations.

€

=T

Rf −F

€

÷

€

T =5.0 ±1.0oC (1σ)

€

=

€

Rf =6.5 ±1.5Wm−2 (1σ) Hansen et al.1984

n.b. F is assumed zero (not necessarily true)

Climate sensitivity.Estimates from models.

€

T =0Rf

1 − f i∑

Soden & Held (2006):

€

f =0.62; σ f =0.13

€

f =0.70; σ f =0.14Colman (2003):

• How does this uncertainty in physics translate to uncertainty in climate sensitivity?

Individual feedbacksuncorrelated among models, so can be simply combined:

Climate sensitivity.Estimates from models.

€

T =0Rf

1 − f i∑

€

f =0.65σ f =0.14

for:

Climate sensitivity.Estimates from models.

€

T =0Rf

1 − f i∑

€

f =0.65σ f =0.14

for:

Climate sensitivity.Estimates from models.

€

T =0Rf

1 − f i∑

• GCMs produce climate sensitivity consistent with the compounding effect of essentially-linear feedbacks.

Climate sensitivity.An aside: nonlinearity of feedbacks

Taking ~12 different studies:

€

G =1

1−f−T2

dfdT

€

R =dRdT

T +12

d2R

dT2T2 +O(T3 )

€

R =dRdT

T +O(T2 )From basic analysis:

But can take quadratic terms…

giving…

€

−0.04K−1 ≤dfdT

≤+0.04K−1

Climate sensitivity.An aside: nonlinearity of feedbacks

So not a big deal…..

Climate sensitivity.Models and observations.

• All look pretty similar.

• How to do better?

Climate sensitivity.How to do better?

1. Combine different estimates?Very hard to establish the degree of independence of individualestimates.

2. Use other observations?(e.g., NH vs. SH; pole-to-eq. T; seasonality, trop. water vapor)Structural errors among models highly uncertain (see Knutti et al, 2010).

3. Transient climate response?Clim. Sens. is an equilibrium property, short observations onlyhave limited resolving power.

Fortunately...

Feedbacks don’t exist, and

climate sensitivity doesn’t matter!

Feedbacks don’t exist.

They are just a Taylor series in disguise

€

Rf =T0

−c1T −c2T −c3T −c4T −K

(Thanks Kyle and Aaron)

reference system

Treference

climate system

R

c1T

c2T

Feedbacks don’t exist.

They are just a Taylor series in disguise

€

Rf =T0

−c1T −c2T −c3T −c4T −K

€

Rf =10

−c1

⎛

⎝⎜

⎞

⎠⎟T −c2T −c3T −c4T −K

why not this?

Treference

climate system

R

c1T

c2T

(Thanks Kyle and Aaron)

Feedbacks don’t exist.

They are just a Taylor series in disguise.

€

Rf =T0

−c1T −c2T −c3T −c4T −K

€

Rf =10

−c1

⎛

⎝⎜

⎞

⎠⎟T −c2T −c3T −c4T −K

€

Rf =10

−c1 −c2

⎛

⎝⎜

⎞

⎠⎟T −c3T −c4T −K

or this??

• Feedbacks are entirely in the eye of the beholder!

Treference

climate system

R

c1T

c2T

(Thanks Kyle and Aaron)

Climate sensitivity doesn’t matter.Constraining climate sensitivity is not terribly relevant for projecting climate change…

(Allen and Frame, 2007)

Stabilization target of 450 ppm at 2100

High end sensitivities take a long, long time to be realized…

(Allen and Frame, 2007)

Concentrationtarget adjustedat 2050.

Geoengineering = the human feedback.

Climate sensitivity doesn’t matter.Constraining climate sensitivity is not terribly relevant for projecting climate change…

Miscellany

Time dependent climate change:The role of the ocean

• The ocean heat uptake acts as a (transient) negative feedback.