Mathematics of Climate Change - Cornell...
Transcript of Mathematics of Climate Change - Cornell...
![Page 1: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/1.jpg)
The Role of Mathematics in Understanding the Earth’s Climate
Andrew Roberts
![Page 2: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/2.jpg)
Outline• What is climate (change)?
• History of mathematics in climate science
• How do we study the climate?
• Dynamical systems
• Large-scale (Atlantic) ocean circulation
• Ice ages and the mid-Pleistocene transition
• Winter is coming?
![Page 3: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/3.jpg)
Weather vs. Climate• Conditions of the atmosphere
over a short period of time (minutes - months)
• Temp, humidity, precip, cloud coverage (today)
• Snowfall on November 14, 2014
• Heat wave in 2010
• Hurricane
• How the atmosphere “behaves” over a long period of time
• Average of weather over time and space (usually 30-yr avg)
• Historical average November precipitation
• Record high temperature
• Average number and strength of tropical cyclones, annually
![Page 4: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/4.jpg)
Weather vs. Climate
• Climate is what you expect, weather is what you get
• Can view climate as a probability distribution of possible weather
Climate
Weather
-4 -2 2 4
0.1
0.2
0.3
0.4
![Page 5: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/5.jpg)
What is Climate Change?
-4 -2 2 4
0.1
0.2
0.3
0.4
![Page 6: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/6.jpg)
What is Climate Change?
-4 -2 2 4
0.1
0.2
0.3
0.4
![Page 7: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/7.jpg)
What is Climate Change?
-4 -2 2 4
0.1
0.2
0.3
0.4
![Page 8: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/8.jpg)
What is Climate Change?
-4 -2 2 4
0.1
0.2
0.3
0.4
![Page 9: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/9.jpg)
PrecipitationDry areas get dryer, wet areas get wetter
Climate scientists predict more floods and
more droughts!
![Page 10: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/10.jpg)
Mathematics and Climate Change
• Energy balance equation
• : incoming solar radiation
• : proportion absorbed by the Earth
• :heat re-radiated back to space
�T = Q(1� ↵(T ))� �T 4
Q
(1� ↵(T ))
�T 4
![Page 11: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/11.jpg)
Energy Balance
200 220 240 260 280 300T
100
150
200
250
300
350
400
Radiation
�T = 0 ) Q(1� ↵(T )) = �T 4
![Page 12: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/12.jpg)
Greenhouse Effect• Joseph Fourier attempted to
calculate the average temperature of the Earth (c. 1820)
• Hypothesized what has come to be known as the “greenhouse effect” — something is trapping heat in the Earth’s atmosphere
• 50 years before Stefan-Boltzmann energy balance equation
• 75 years before Arrhenius quantified how much colder the Earth “should” be
![Page 13: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/13.jpg)
Greenhouse Effect
220 240 260 280 300T
50
100
150
200
250
300
350
400
Radiation
�T = Q(1� ↵(T ))� "�T 4
![Page 14: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/14.jpg)
Energy Balance Cartoon
![Page 15: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/15.jpg)
• Mid-1700s: speculation that ice ages exists
• 1830s: A few geologists claim ice-ages happend, ideas rejected
• 1842: Joseph Adhémar (mathematician) is first to propose ice-ages caused by variation in solar radiation
Ice Ages
![Page 16: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/16.jpg)
Ice Ages• 1870s: Geologists reach
consensus that ice-ages occurred (James Croll)
• 1912-1924: Milutin Milankovic
• Eccentriciy (100 kyr) —Kepler 1609
• Obliquity/Axial tilt (41 kyr)—Milankovic 1912
• Precession (23 kyr)—Hipparchus 130 B.C.
![Page 17: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/17.jpg)
Milankovic Cycles
![Page 18: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/18.jpg)
Snowball Earth• Budyko and Sellers (1969)
describe spatially dependent energy balance model
• Assume Northern and Southern hemisphere symmetric
• Assume temperature is the same for fixed latitude (y)
• Includes energy transport term
@T (y)
@t= Qs(y)(1� ↵(y, ⌘))� (A+BT (y))� C(T (y)� T (⌘))
![Page 19: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/19.jpg)
Snowball Earth• 2 stable states of ice coverage:
• Warm climate (like now - and even ice ages)
• Snowball climate (entire Earth covered in ice)
• Dismissed as “mathematical artifact” until 1990s
• New consensus: 3 snowball events (all over 600 myr ago)
![Page 20: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/20.jpg)
How do we study the climate?
![Page 21: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/21.jpg)
How do we study the climate?
![Page 22: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/22.jpg)
How do we study the climate?
![Page 23: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/23.jpg)
Models!
How do we study the climate?
![Page 24: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/24.jpg)
Model Hierarchies
![Page 25: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/25.jpg)
Conceptual Models• Examples: Energy balance
models
• Typically model 1 or 2 processes/phenomena
• Large-scale average behavior
• Help explain climate to non-experts
• Motivate large experiments
• Pros:
• Simple enough to be analyzed by a person
• Can explore all possibilities
• Intuition
• Cons:
• Too simple to prove scientific results definitively
• Adding more processes could destroy phenomenon
![Page 26: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/26.jpg)
Intermediate Complexity and Process Models
• Some spatial resolution
• More processes (but not too many)
• Simple enough for some interpretation
• Too complex to analyze “by hand”
![Page 27: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/27.jpg)
GCMs and ESMs• Too complicated to interpret
causality
• Too complicated to explore all possibilities (where do we look?)
• Millions of lines of code (bugs?)
• Expensive (financially and computationally)
• Treated as “experimental Earths”
• Useful for prediction*
![Page 28: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/28.jpg)
Weather Prediction
Model
Observation of Current State
![Page 29: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/29.jpg)
Weather Prediction
Model
Observation of Current State
Prediction (1 hour)
![Page 30: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/30.jpg)
Weather prediction
Model
Observation of Current State
Prediction (1 hour)
Model
Prediction (2 hour)
![Page 31: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/31.jpg)
Observations have errorObserved
state
Actual (initial) state
![Page 32: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/32.jpg)
Error grows
Model
Observed state
Actual (initial) state
1 hour prediction
State after 1 hr
![Page 33: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/33.jpg)
and grows…
Model
Observed state
Actual (initial) state
1 hour prediction
State after 1 hr
Model
2 hour prediction
State after 2 hr
![Page 34: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/34.jpg)
Lorenz Butterly
![Page 35: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/35.jpg)
Climate Prediction
![Page 36: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/36.jpg)
Climate Prediction
![Page 37: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/37.jpg)
Climate Prediction
![Page 38: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/38.jpg)
Climate Prediction
Where do observations come in?
![Page 39: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/39.jpg)
Confronting Models with Data
![Page 40: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/40.jpg)
Confronting Models with Data
1st IC
Model
Data
![Page 41: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/41.jpg)
Confronting Models with Data
1st IC
Model
Data
![Page 42: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/42.jpg)
Data Assimilation
1st IC
Model
Data2nd IC
![Page 43: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/43.jpg)
Data Assimilation
1st IC
Model
Data2nd IC
Model (w/o DA)
Data
Model (w/ DA)
![Page 44: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/44.jpg)
Data Assimilation
1st IC
Model
Data2nd IC
Data
![Page 45: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/45.jpg)
The Role of Mathematics in Climate Science
Observation Data
LabFieldSimulations
as ExperimentsESMs GCMsTheory
Conceptual Models
Math
![Page 46: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/46.jpg)
Dynamical Systems
dx
dt
= f(t)
Derivative (from Calculus)
Example
dx
dt
= t
3 � t+ k
x(t) =t
4
4� t
2
2+ kt+ C
![Page 47: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/47.jpg)
What if dx
dt
= f(x)?
Dynamical Systems
dx
dt
= f(t)
Derivative (from Calculus)
Example
dx
dt
= t
3 � t+ k
![Page 48: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/48.jpg)
More than one variable?
x = f(x, y)
y = g(x, y)
System of Differential Equations
Defines a Vector Field
![Page 49: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/49.jpg)
Vector FieldsSystem of
Differential Equations
Defines a Vector Field
x = y � x
3 + x
y = x� 2y + k
-3 -2 -1 0 1 2 3
-3
-2
-1
0
1
2
3
![Page 50: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/50.jpg)
Equilibrium Pointsx = y � x
3 + x
y = x� 2y + k
-3 -2 -1 0 1 2 3
-3
-2
-1
0
1
2
3
Equilibrium points occur when
x = 0
y = 0
![Page 51: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/51.jpg)
Solutionsx = y � x
3 + x
y = x� 2y + k
Even if equations can’t be solved,
we can understand
Qualitative Behavior-3 -2 -1 0 1 2 3
-3
-2
-1
0
1
2
3
![Page 52: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/52.jpg)
Varying k
-3 -2 -1 0 1 2 3
-3
-2
-1
0
1
2
3
x = y � x
3 + x
y = x� 2y + 0
x = y � x
3 + x
y = x� 2y � 2
-3 -2 -1 0 1 2 3
-3
-2
-1
0
1
2
3
![Page 53: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/53.jpg)
Bifurcation in Algebra I
Quadratic equation
ax
2 + bx+ c = 0
![Page 54: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/54.jpg)
Bifurcation in Algebra I
Bifurcation parameter: Discriminant
b2 � 4ac > 0
2 Real Roots
Quadratic equation
ax
2 + bx+ c = 0
-4 -2 2 4
5
10
![Page 55: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/55.jpg)
Bifurcation in Algebra I
Bifurcation parameter: Discriminant
b2 � 4ac > 0
2 Real Roots
Quadratic equation
ax
2 + bx+ c = 0
No qualitative change for small change
in equation
-4 -2 2 4
-5
5
10
15
![Page 56: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/56.jpg)
Bifurcation in Algebra I
Bifurcation parameter: Discriminant
0 Real Roots
Quadratic equation
ax
2 + bx+ c = 0
Big enough change in system leads to
qualitatively different solutions
b2 � 4ac < 0
-4 -2 2 4
-5
5
10
15
![Page 57: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/57.jpg)
Bifurcation in Algebra I
Bifurcation parameter: Discriminant
1 Real Root
Quadratic equation
ax
2 + bx+ c = 0
Bifurcation occurs when solutions
collide
b2 � 4ac = 0
-4 -2 2 4
5
10
15
20
![Page 58: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/58.jpg)
Bifurcations as Tipping Points
![Page 59: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/59.jpg)
Bifurcations as Tipping Points
![Page 60: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/60.jpg)
Bifurcations as Tipping Points
![Page 61: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/61.jpg)
Bifurcations as Tipping Points
![Page 62: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/62.jpg)
Hysteresis
![Page 63: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/63.jpg)
Bifurcation vs. Intrinsic Dynamics
• Idea of bifurcations assumes modeler has control over how parameters change — i.e., do NOT depend on state of system
• Snowball Earth: bifurcation “parameter” depends on GHGs (which in turn depend on temperature and ice)
• How does behavior change?
![Page 64: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/64.jpg)
Fast/Slow Dynamics• Fast variable is like state of
system as before
• Slow variable is acts partly like parameter, partly like state variable
• Example of parameter: Milankovic cycles depend only on time (influence climate, but not influenced by climate)
• Examples of slow variable: GHGs, Ice coverage
x = f(x, y)
y = "g(x, y)
" ⌧ 1
x = f(x;�)
�(t) = g(t)
![Page 65: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/65.jpg)
Picturing the difference
x = f(x;�)
�(t) = g(t)
-4 -2 2 4 x
-2
-1
1
2λ
x = f(x, y)
y = "g(x, y)
" ⌧ 1
-4 -2 2 4 x
-2
-1
1
2y
![Page 66: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/66.jpg)
Example in Ocean Circulation
![Page 67: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/67.jpg)
Stommel’s Circulation Model
Circulation variable:
![Page 68: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/68.jpg)
Stommel’s Circulation Model
y = µ� y �A|1� y|y
y ⇠ Se � Sp
x ⇠ Te � Tp ! 1Model Reduces:
Get one state variable:
Bifurcation parameter:
µ ⇠ �SA
�TA
![Page 69: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/69.jpg)
µ ! slow variable
y = µ� y �A|1� y|yµ = �0(�� y)
![Page 70: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/70.jpg)
Mixed-mode Oscillations• 2D dynamical systems can have
up to 3 end states:
• Fixed equilibrium
• Periodic equilibrium (with fixed amplitude and period)
• Run-away behavior
• MMOs have big and small oscillations—need 3D system!
• 3D dynamics much more complicated (chaos)
![Page 71: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/71.jpg)
Ice Ages over the last 400 kyr
x = y � x
3 + 3x� k
y = "[p(x� a)2 � b�my � (�+ y � z)]
z = "r(�+ y � z)
![Page 72: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/72.jpg)
Ice Ages over the last 400 kyr
x = y � x
3 + 3x� k
y = "[p(x� a)2 � b�my � (�+ y � z)]
z = "r(�+ y � z)
ice volume
atmospheric carbon
oceanic carbonx ⇠
y ⇠
z ⇠
![Page 73: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/73.jpg)
Ice Ages over the last 400 kyr
x = y � x
3 + 3x� k
y = "[p(x� a)2 � b�my � (�+ y � z)]
z = "r(�+ y � z)
Fast Slow
![Page 74: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/74.jpg)
Ice Ages over the last 400 kyr
x = y � x
3 + 3x� k
y = "[p(x� a)2 � b�my � (�+ y � z)]
z = "r(�+ y � z)
Change in ice volume depends on temperature, but temperature depends on the amount of ice
and how much GHGs are in the atmosphere
![Page 75: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/75.jpg)
Ice Ages over the last 400 kyr
x = y � x
3 + 3x� k
y = "[p(x� a)2 � b�my � (�+ y � z)]
z = "r(�+ y � z)
Land-atmosphere carbon flux
![Page 76: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/76.jpg)
Ice Ages over the last 400 kyr
Ocean-atmosphere carbon flux
x = y � x
3 + 3x� k
y = "[p(x� a)2 � b�my � (�+ y � z)]
z = "r(�+ y � z)
![Page 77: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/77.jpg)
Ice Ages over the last 400 kyr
![Page 78: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/78.jpg)
Ice Ages over the last 400 kyr
![Page 79: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/79.jpg)
Ice Ages over the last 400 kyr
![Page 80: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/80.jpg)
Ice Ages over the last 400 kyr
![Page 81: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/81.jpg)
Ice Ages over the last 400 kyr
![Page 82: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/82.jpg)
El Niño-Southern Oscillation
![Page 83: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/83.jpg)
How does ENSO work?
![Page 84: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/84.jpg)
The Data
![Page 85: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/85.jpg)
Predicting ENSOAugust predictions
![Page 86: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/86.jpg)
Predicting ENSO
• August Prediction
• ?
• September prediction
• Probability of ENSO: low
• October Prediction
• Probability of ENSO: 0.68
• November Prediction
• 58% chance of ENSO
• Normal to weak ENSO
![Page 87: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/87.jpg)
ENSOx = "(x2 � ax) + x
x+ y � nz + d� c
✓x� x
3
3
◆�
y = �"(ay + x
2)
z = m
⇣k � z � x
2
⌘
![Page 88: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/88.jpg)
ENSO
x
z
temperature gradient
temp of W Pacific
thermocline dept in W Pacific
x = "(x2 � ax) + x
x+ y � nz + d� c
✓x� x
3
3
◆�
y = �"(ay + x
2)
z = m
⇣k � z � x
2
⌘
y
![Page 89: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/89.jpg)
ENSOx = "(x2 � ax) + x
x+ y � nz + d� c
✓x� x
3
3
◆�
y = �"(ay + x
2)
z = m
⇣k � z � x
2
⌘
Upwelling feedback
![Page 90: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/90.jpg)
ENSO
Thermocline adjustment
x = "(x2 � ax) + x
x+ y � nz + d� c
✓x� x
3
3
◆�
y = �"(ay + x
2)
z = m
⇣k � z � x
2
⌘
![Page 91: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/91.jpg)
ENSO
Advection
x = "(x2 � ax) + x
x+ y � nz + d� c
✓x� x
3
3
◆�
y = �"(ay + x
2)
z = m
⇣k � z � x
2
⌘
![Page 92: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/92.jpg)
ENSO
![Page 93: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/93.jpg)
Simulation 1
![Page 94: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/94.jpg)
Simulation 1
![Page 95: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/95.jpg)
Simulation 2
![Page 96: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/96.jpg)
Simulation 2
![Page 97: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/97.jpg)
Model Output
![Page 98: Mathematics of Climate Change - Cornell Universitypi.math.cornell.edu/.../5080/2014nov/roberts-climate.pdfOutline • What is climate (change)? • History of mathematics in climate](https://reader031.fdocuments.us/reader031/viewer/2022011907/5f4f1547aae59d08f13f1808/html5/thumbnails/98.jpg)
Winter?