The Greenhouse Effect

The Greenhouse Effect

Gareth E. Roberts

Department of Mathematics and Computer Science College of the Holy Cross Worcester, MA, USA

Seminar in Mathematics and Climate MATH 392-01 Spring 2018 February 1, 2018

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Lecture Outline

Observations from Computer Project #1: Bifurcations and Hysteresis The Greenhouse Effect Ice Core Data and Human-Induced Climate Change

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Computer Project #1: Observations

Modeling is an iterative process. Sometimes the results are poor or counter-intuitive. Adjust your model.

Units are important (e.g., working in kelvin or Celsius, metric system or not).

There are typically lots of parameters (, Q, , ) and the outcome of a model can vary greatly even for a small change in the value of a parameter--bifurcations.

Tuning: Sometimes we adjust the parameters to make our model agree with known data. This looks good, but can also be misleading to those evaluating the model. Example: Climate Model #3 introduced to model the greenhouse effect and obtain the current average temperature of the Earth. No physics used at all: Q(1 - ) = T 4

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Climate Model #5

C dT dt

= Ein - Eout = (1 - (T ))Q - T 4

where

T = global average surface temperature, in K e(T -265)/5

(T ) = 0.7 - 0.4 1 + e(T -265)/5 (albedo) Q = 1/4 of the solar constant S, 342 W/m2 = 5.67 ? 10-8 W/(m2 ? K4) = greenhouse effect parameter

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dT/dt

Climate Model #5: = 0.6163 400

Ein Eout

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T

dT/dt

Climate Model #5: = 0.54 350

Ein Eout

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T

Climate Model #5: = 0.488 300

Ein Eout

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Climate Model #5: = 0.45 300

Ein Eout

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dT/dt

dT/dt

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Figure: The bifurcation that arises when decreasing in Climate Model #5.

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