The Budyko Energy Balance Models of the Earth’s...

The Budyko Energy Balance Models of the Earth’s Climate

Esther Widiasih

Summer SeminarJuly 22 2013

Tuesday, July 23, 2013

Where we were left off

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Global EBMT = global annual average

from Samantha’s talk



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Global EBMT = global annual average

Next: zonal EBM with

ice albedo feedback

from Samantha’s talk


Recall: Earth’s Energy Budget

energy received = energy re-emitted

temperature change = (heat capacity) (heat imbalance)


Recall: Earth’s Energy Budget

energy received = energy re-emitted

temperature change = (heat capacity) (heat imbalance)

to use in a course: explain the numbers?Tuesday, July 23, 2013

Global Energy Balance Models

heat imbalance = insolation - reflection - reemission

The Earth’s climate is represented as a temperature over one “point”

RdT

dt= Q(1− α)− (A+BT )

all variables and parameters are global averages.

incoming solar radiation short wave long wave


Zonal Energy Balance Models

The climate on Earth is represented by a function of temperature over an interval

(temperature profile).

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0 1

T(y)

equator pole

y



Why might we want to consider this type of model?

What is the advantage of the spatial dimension?



heat imbalance = insolation - reflection - reemission - transport

transport processes eg. wind, storm, the gulf streammodeling transport eg. diffusion, relaxation to global average

0 1

equator pole

yy




reemission processes include the effects of green house gassesmodeling reemission is often based on the Stefan-Boltzman’s law of black body radiation

0 1

equator pole

yy




GRAVES ET AL..' NEW PARAMETERIZATIONS AND SENSITIVITIES 5027

300

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Scatter plot for OLR vs T (10 years)

•F..•

::::::::::. :: .

---- :-:-:b-•. ..:.:

........... i!ig ...... ' • ....... .: ........... . ,• • ,• ......... . .......... •[•i ';':[ :-:•:•. .

// :•=:'::•':'•"•:: =:si::.• •;'...• ß E:>;•.•:::::-::: •::;•

I ' I ' I ' I ' I ' I ' I '

-40 -30 -20 -10 0 10 20

Surface Temperature (in C)

I

0.0%

1.0%

Fig. 2. Scatter plot of OLR versus surface temperature from 30øN to 90øN from the 10-year data set. The scale on the right indicates the percent of total. Note that there is a cosine of latitude weighting to account for the differing grid point areas.

only in the response of the IR to the surface temperature at the annual frequency but at lower and higher frequencies as well, since there is always the possibility of bias entering that is peculiar to the systematic poleward and equatorward motions of cloud patterns.

2.1. OLR Data and Method of Analysis Most of our analysis was conducted with 10 years of OLR

data (July 1975 through June 1985) which were deconvoluted from the wide field-of-view radiometers aboard the Nimbus 6 and 7 satellites. The details on the construction of this data set have been discussed by Bess and Smith [ 1987a, b; Bess et al., 1992]. There were 4 missing months in the data set (July 1978 through October 1978) which were replaced by the 9-year mean for those calendar months. The data were obtained in spherical harmonic format truncated triangularly at degree 12, although OLR gridded fields were computed by expansion on to a 15 ø x 15 ø grid.

We also analyzed an additional 9 months of OLR data from the ERBE mission (February, March, May, June, July, September, October, November, of 1985 and January 1986), which were provided by NESDIS.

Monthly mean surface temperature climatology as well as anomalies analyzed in this study were obtained from the National Center for Atmospheric Research. The temperature anomalies are from those used by Jones et al. [ 1986] and Jones [1988]. The missing temperature anomalies were filled using spatial linear interpolation. The interpolation was more extensive in the southern hemisphere so we concentrate our discussion on the northern hemisphere results. The temperature data were obtained on a 5 ø x 5 ø but were smoothed to 15 ø x 15 ø to match spatial resolution of the OLR data. Additionally, only those months for which the OLR data were available were analyzed.

The analysis is based on the linear regression given by (2),

where the regression coefficients are computed for a variety of filtered I and T s time series. The parameter B is of most concern in this study since it is inversely proportional to the sensitivity of temperature anomalies to radiative cooling. That is, the larger the value of B the faster temperature anomalies are radiatively damped. In the absence of feed- backs such as snow-albedo B -] is also a measure of a climate model's sensitivity to changes in external control parameters. When the regression was done at a particular grid point, no area weighting was applied, however when large regional estimates of B were obtained, the analysis included area weighting (i.e., a cosine of latitude weighting at each grid point).

2.2. OLR Analysis 2.2.1. The 10-year mean. The regression analysis was

first applied to each grid point for the entire 10 years (120 months). A map of the locally computed regression coefficient B and the corresponding correlation coefficient are shown in Figure 1. As with all plots in this study, some nearest neighbor averaging was applied to smooth out the contours. A pronounced difference between the tropics and middle latitudes can be observed in both the values of B and the correlation coefficient in Figure 1. In much of the middle and high latitudes the value of B is near 2.0 W m-2 øC-•, and the transition region tends to follow the position of the climatological storm tracks along the east coasts of the two northern hemisphere land masses. In middle and high latitudes the correlation coefficient is more zonally symmetric with values greater than 0.6.

In the tropics, the value of B is highly longitude dependent reflecting the strong regional cloud-surface temperature de- pendence in the tropics. Of course, the dynamic range of T is also very small in the tropics. The three locations with large negative values of B are in the vicinity of regions of

Outgoing Longwave Radiation = A+BT



heat imbalance = insolation - reflection - reemission - transportBackground Albedo

Albedo Observations

The albedo, or reflectivity, of the Earth’s surface is much higher for ice than for ocean or land.

Satellite data from NASA

Anna M. Barry Climate Transitions in a Conceptual Model

NASA Observation surface dependent albedo

snow/ iceocean




0 1

equator pole

yy

McGehee & Lehman Ice-Albedo Feedback

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McGehee and Lehman, 2012.Tuesday, July 23, 2013

Zonal Energy Balance Models:

What affects insolation



The Budyko Zonal Energy Balance Models

R∂T

∂t=

Qs(y)� �� insolation

−Qs(y)(α(η, y)� �� reflection

− (A+BT (y))� �� reemission

−C�T (y)− T

��

transport

R = planetary heat capacity

T = T (y) = T (y, t)

s(y) = a distribution function

α(η, y) = the albedo at y given that the ice line is at η

T =

� 1

0T (ξ)dξ

A, B, C are parameters

Q and s(y) contain Earth orbital elements


Budyko, 1969. KK Tung, 2007.


The Budyko Zonal EBM

T(y, t) forward in timeWater covered

Ice coveredIce cap boundary


The Budyko Zonal EBM: Steady State

if the ice cap boundary is fixed at η, the steady state is

R∂T

∂t= Qs(y)(1− α(η, y))− (A+BT (y))− C(T (y)− T )

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

�40

�30

�20

�10

0

10

20

30

40

η

T ∗(η, y) =Q · s(y) · (1− a(η)(y)) + C

� 10 T ∗(η, ξ)dξ −A

B + C


The Budyko Zonal EBM with Ice Albedo Feedback

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Could the Earth’s climate reach extreme states? What makes the current small ice cap possible?



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What do we need to do to the Budyko EBM?



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Need to evolve the ice (cap) boundary.


What do we need to do to the Budyko EBM?


The Budyko Zonal EBM with Evolving Ice Boundary

R∂T


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

�40

�30

�20

�10

0

10

20

30

40

η



R∂T


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

�40

�30

�20

�10

0

10

20

30

40

η

Assumptions to evolve 1. Ice forms or melts at a much slower rate2. At the ice line, ice forms above and melts below a critical temperature

η



R∂T


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

�40

�30

�20

�10

0

10

20

30

40

ηdη

dt=�(T (η)− Tc)

Assumptions to evolve 1. Ice forms or melts at a much slower rate2. At the ice line, ice forms above and melts below a critical temperature

η


The Budyko Zonal EBM and a Slow Moving Ice Line

T(y, t) forward in time

Water covered

Ice covered

Ice cap boundary


The Budyko Zonal EBM and a Slow Moving Ice Line

T(y, t) forward in time

Water covered

Ice covered

Ice cap boundary

Where would the ice line (the ice cap

boundary) end up?Tuesday, July 23, 2013

The Budyko EBM with a Slow Moving Ice Line:

Some Results



Some Results

R∂T


dη

dt= �(T (η)− Tc)

T ∗(y, η) =Q · s(y) · (1− a(η, y)) + C

� 10 T ∗(η)(ξ)dξ −A

B + C

Budyko EBM

Slow moving ice line

Temperature profile steady state



Some Results

1. The temperature profile T(y, t) quickly approaches T*(y, η), while the ice line η moves slowly.

2. When the temperature profile is near T*(y, η), the dynamics of the ice line η is approximately the following

R∂T


dη

dt= �(T (η)− Tc)

T ∗(y, η) =Q · s(y) · (1− a(η, y)) + C

� 10 T ∗(η)(ξ)dξ −A

B + C

Budyko EBM





Some Results

1. The temperature profile T(y, t) quickly approaches T*(y, η), while the ice line η moves slowly.

2. When the temperature profile is near T*(y, η), the dynamics of the ice line η is approximately the following

dη

dt= �(T ∗(η, η)− Tc)

R∂T


dη

dt= �(T (η)− Tc)

T ∗(y, η) =Q · s(y) · (1− a(η, y)) + C

� 10 T ∗(η)(ξ)dξ −A

B + C

Budyko EBM





Some Results

dη

dt= �(T ∗(η, η)− Tc)

R∂T


dη

dt= �(T (η)− Tc)

dimension: ∞

dimension: one Tc

T*(η, η )



Some Results

dη

dt= �(T ∗(η, η)− Tc)

R∂T


dη

dt= �(T (η)− Tc)

dimension: ∞

dimension: one Tc

T*(η, η )

Quiz: What is (are) the stable ice line state(s)?The unstable one(s)?



Preliminary Bifurcation Analysis

Tc

T*(η, η )T ∗(y, η) =Q · s(y) · (1− a(η, y)) + C

� 10 T ∗(η)(ξ)dξ −A

B + C

dη

dt= �(T ∗(η, η)− Tc)



Preliminary Bifurcation Analysis

Tc

T*(η, η )T ∗(y, η) =Q · s(y) · (1− a(η, y)) + C

� 10 T ∗(η)(ξ)dξ −A

B + C

dη

dt= �(T ∗(η, η)− Tc)

Some Interesting questions:1. What are the effects of increasing green house gas? --see Jim’s session for more details

2. What are the effects of solar insolation variability ie. the Milankovitch cycle? -- see Dick’s session.

3. What climate states can this simple model predict? -- see Anna’s session


Suggested Exercises

• Software oriented-page 10, Section 1.2 : modeling, stability, green house effect(walking through the details of this lecture)

• Minimum software-page 10, Section 1.2

-page 21, Section 2.1: stability, bifurcation-pageSection 2.2: dimension reduction of the Budyko EBM-page 30, Section 2.5: Budyko EBM with contraction-page 33, Section 2.6: Snowball Earth, Jormungand state


Thank you for your attention!


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