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# Ecm3730: Mathematics Of Climate Change

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ECM3730: MATHEMATICS OF CLIMATE CHANGE

ECM3730: Mathematics of Climate Change

ECM3730: Mathematics of Climate Change

ECM3730: Mathematics of Climate Change

ASSESSED PROBLEM SHEET 3

Deadline: 12pm Thursday 28thMarch (hand-in to room 158)

Water Vapor Feedback

Lapse-

Rate

Feedback

Surface

Albedo

Feedback

Cloud

Feedback

Total

Feedback

Factor

?T2xCO2 (K)

Mean ??(W m-2 K-1)

-1.8

0.8

-0.3

-0.7

0.265

0.53

Min ? ?(W m-2 K-1)

-2.0

0.4

-0.4

-1.2

0.265

8.48

Max ? ?(W m-2 K-1)

-1.5

1.1

-0.2

-0.1

0.265

0.1855

Uncertainty in ?T2xCO2 (K)

0.84 - 0.71

0.63 - 0.45

0.556 - 0.503

0.662 - 0.371

NOTE: Please give numbers to a maximum of 3 significant figures!!

The table at the top of this problem sheet lists mean, maximum and minimum feedback factors from across an ensemble of GCMs.

Estimate the blackbody feedback, ?bb, assuming an effective radiative temperature of Te=255K, and using the Stefan-Boltzmann constant s = 5.67x10-8 W m-2 K-4.?Hence complete the “Total Feedback Factor” column.

Solution

The reference-system climate sensitivity parameter, ?0, can be found from

?0 =

Calculate the climate sensitivity to doubling CO2, and thereby complete the “?T2xCO2 (K)” column of this table.Assume a radiative forcing due to doubling CO2 of 3.71 W m-2.

The climate sensitivity parameter, ?, is de?ned as the constant of proportionality that relates the two:

? =

Estimate the uncertainty in the climate sensitivity to doubling CO2 arising from each feedback by varying the feedback in question from its minimum to its maximum value whilst keeping all other feedbacks at their mean values. Thereby complete the “Uncertainty in ?T2xCO2 (K)” row of the table. 

Uncertainty in ?T2xCO2 (K): ? =

What is the mean overall gain factor? By what factor is the global warming amplified (with respect to the blackbody) by feedbacks?

G = = = = 0.36

Which of the listed individual feedback factors (i) has the largest mean positive feedback, (ii) has the largest mean negative feedback, (iii) leads to the greatest uncertainty in the climate sensitivity to doubling CO2?

Largest mean positive feedback = Water Vapor Feedback

Largest mean negative feedback = Lapse-Rate Feedback

Greatest uncertainty in the climate sensitivity to doubling CO2 = Cloud feedback

How would climate-carbon cycle feedbacks affect the climate sensitivity to doubling CO2?

It will increase the humidity and hence affect the overall sensitivity of the system.

Consider the simplest linear model of the increment ?T to the Earth's mean surface temperature arising from a forcing ?Q and a feedback factor, :

C. dT/dt + ???. ?T = ?Q

Where C is the effective heat capacity of the climate system.

Show that a Total Feedback Factor ?< 0 is inconsistent with a stable climate. What is the associated stability condition for the overall gain factor?

For stable climate conditions variations will be zero.

d?T/dt = 0

Solving the equation for total feedback factor we get,

?. ?T = ?Q - C. dT/dt

? =

Putting, d?T/dt = 0

We get,

? =

Since,

Therefore,

? < 0

Hence it is proved that for the given at stable environmental condition the total feedback factor will remain inconsistent.

Draw a feedback diagram for cloud feedbacks, showing the impacts of both changes in cloud cover and changes in cloud height. If the direction of a causal link is unclear use the symbol “?” rather than “+” or “-“. Hence explain why positive cloud feedbacks can arise from ...