Novel Types of Tipping Points.
Sebastian Wieczorek
Mathematics Research Institute
University of Exeter, UK
Prof. Peter Ashwin
Prof. Peter Cox
Dr. Renato Vitolo
March 19-23, 2012, Imperial College, London.
Outline:
1. Tipping types
2. Rate-dependent Tipping:
• Basic examples and existence of critical rates
• Does it occur in real systems?
• Towards a general theory of R-tipping
3. Conclusions
United Nations Framework Convention
on Climate Change (UNFCCC)
“The ultimate objective is. . .
stabilisation of greenhouse gas concentrations in the
atmosphere at a level that would prevent dangerous
anthropogenic interference with the climate system . . .
. . . such a level should be achieved within a time frame
sufficient to allow ecosystems to adapt naturally
to climate change, to ensure food production
is not threatened and to enable economic development
to proceed in a sustainable manner”
dangerous levels and dangerous rates or
climate tipping points
Climate Models and Forced (Nonautonomous) Systems
High−dimensional
ODEs
dλ/dt = g(x, λ)
Full−Scale
Climate Model
(coupled PDEs)
λ(t)
Low−dimensional
λ(t)
x(t)
ODEs with
forcing term
Low−dimensional
ODEs
dx/dt = f (x, λ(t))
dx/dt = f (x, λ)
Isaac M. Held, Bull. Amer. Meteor. Soc. 86, 1609–1614 (2005)
• warns of the widening gap between climate simulations and understanding
• encourages the study of model hierarchies to narrow this gap
Two Tipping Types:
B-tipping
and
|x|
R-tipping
|x|
QSA
QSA
λ
Stability of the unforced system
d
x = f (x, λ)
dt
for different but fixed λ: there is an attractor QSA(λ).
λ
B-tipping
|x|
and
tips
R-tipping
|x|
QSA
QSA
λ
tips for any rate of change
tracks
λ
tracks the QSE
B-tipping
|x|
and
tips
R-tipping
|x|
tips
QSA
QSA
λ
tips for any rate of change
tracks
λ
tracks the QSE below some critical rate
tips above some critical rate!
[C.M. Luke and P. Cox, Eur. J. Soil Sci., 62, 5-12 (2011)]
[S.Wieczorek, P. Ashwin, C.M. Luke, P. Cox, Proc. Roy. Soc. A, May 8 (2011)]
Outline:
1. Tipping types
2. Rate-dependent Tipping:
• Basic examples and existence of critical rates
• Does it occur in real systems?
• Towards a general theory of R-tipping
3. Conclusions
Three Basic Examples of R-tipping
focus & cycle
saddle & node
slow−fast
t=0
t=0
t>0
t>0
Understanding:
• identify the tipping mechanism
• calculate the critical rate (analytically)
Subcritical Hopf Normal Form with Steady Drift
Consider
d
z = (−1 + iω)(z − λ) + |z − λ|2(z − λ),
dt
d
λ=r
dt
z ∈ C,
λ = rt ∈ R
Reduce to a co-moving system for s = z − λ
d
s = (−1 + iω)s + |s|2s − r,
s∈C
dt
Tracking the QSE in the forced syst. ⇒ Stable equilib. in the co-moving syst.
The critical rate problem reduces to a bifurcation problem:
Find r = rc where the stable equilib. of the co-moving system disappers/destabilises.
[P. Ashwin, S.Wieczorek, R. Vitolo, P. Cox, Phil. Trans. Roy. Soc. A, (2012)]
The Tipping Diagram
2
H
r
H
tips
1
tracks
-2
•
BT
S
0
-1
where
|se|2 =
2
1±
3
r
3
1 − (1 + ω 2)
4
1
2
ω
Critical rate:
p
rc(ω) = (1 + 4ω 2)/8
p
rc(ω) = |se|6 − 2|se|4 + (1 + ω 2)|se|2
"
tracks
•BT
#
if
ω 2 ≥ 1/4
if
ω 2 < 1/4
3
Three Basic Examples of R-tipping
focus & cycle
saddle & node
slow−fast
t=0
t=0
t>0
t>0
Critical Rates in Slow-Fast Systems
• (Existence) A forced system with folded slow (critical) manifold:
d
x = f (x, z, λ, ǫ),
dt
ǫ
d
z = g(x, z, λ, ǫ),
dt
d
λ = r,
dt
that preserves a stable equilibrium has a critical rate rc
above which it tips, if the reduced system:
∂g/∂x|S f |S + r ∂g/∂λ|S
d
,
z=−
dt
∂g/∂z|S
d
λ = r,
dt
has a folded singularity for some r > 0.
• (Computing) General condition for critical rate rc with
dependence on system parameters and initial conditions.
[S.Wieczorek, P. Ashwin, C.M. Luke, P. Cox, Proc. Roy. Soc. A, May 8 (2011)]
The Phase Space of the Slow-Fast Problem
Slow (critical) manifold S = Sa ∪ L ∪ Sr
Fold L is given by ∂g/∂z|S = 0
No equilibrium points because r > 0
Sr
QSE
L
The reduced system:
∂g/∂x|S f |S + r ∂g/∂λ|S
d
z=−
,
dt
∂g/∂z|S
d
λ= r
dt
Sa
x
λ
z
Outline:
1. Tipping types
2. Rate-dependent Tipping:
• Basic examples and existence of critical rates
• Does it occur in real systems?
• Towards a general theory of R-tipping
3. Conclusions
Example: Climate-Carbon Cycle Models
Carbon Facts:
Peatland soils contain 400–1000 billion tones of carbon.
Question:
How will peatlands respond to global warming?
Photo: Peatland fires in Russia in Summer 2010
Simple Soil-Carbon & Temperature Model
d
C = Π − C r0 eαT ,
dt
ǫ
d
λ
T = − (T − Ta) + C r0 eαT ,
dt
A
where ǫ =
µ
= 0.064
A
d
Ta = r.
dt
C–soil carbon content, T –soil temperature, Ta–atmospheric temperature
From the general condition get the critical rate of global warming
rc =
r0(1 − αAΠ/λ)
exp αTa0 + 1 ≈ 8 ◦C/100 years
α
[C.M. Luke and P. Cox, Eur. J. Soil Sci., 62, 5-12 (2011)]
[S.Wieczorek, P. Ashwin, C.M. Luke, P. Cox, Proc. Roy. Soc. A, May 8 (2011)]
The Compost-Bomb Instability
r = 7.5 ◦ C/100 years < rc
L
10 3
QSE
F
T [◦ C]
10 2
kg
C[ m
2]
Sa
10 1
103
QSE
Ta [ C]
Sr
◦
102
0
101
40
80
time [years]
r = 9.0 C/100 years > rc
◦
L
10 3
QSE
T [◦ C]
F
10 2
kg
C[ m
2]
QSE
Sa
Ta [◦ C]
Sr
103
10
1
102
101
T [◦ C]
0
40
time [years]
80
Outline:
1. Tipping types
2. Rate-dependent Tipping:
• Basic examples and existence of critical rates
• Does it occur in real systems?
• Towards a general theory of R-tipping
3. Conclusions
Towards a General Theory of R-tipping
The unforced system has a QSE(λ(t)) with a tipping radius R > 0.
Consider a linear forced system
d
x = A (x − QSE(t)) for |x − QSE(t)| < R.
dt
R
QSE(t)
x(t)
The system tracks the QSE up to time t if
kA−1k · |vmax(t)| < R,
where
The system R-tips at time t if
dQSE dQSE dλ = sup vmax(t) = sup dt
dλ dt u≤t
u≤t
kAk−1 · |vmax(t)| = R.
[P. Ashwin, S.Wieczorek, R. Vitolo, P. Cox, Phil. Trans. Roy. Soc. A, (2012)]
Towards a General Theory of R-tipping:
The question of timescales.
The slowest timescale of the unforced system (leading eigenvalue)
kA−1k−1 (units of s−1)
should be compared with timescale for the motion of the QSE:
R−1
dQSE dλ (units of s−1)
dλ dt If R ≈ 1 and
d
dλ QSE
≈ 1, tipping may occur when |dλ/dt| ≈ kA−1k−1
If R ≈ 1 and
d
dλ QSE
≈ 1/ǫ, tipping may occur when |dλ/dt| ≈ ǫkA−1k−1
Summary of Different Tipping Mechanisms
“B-tipping” where the state of the system changes abruptly or qualitatively
due to a bifurcation of a quasi-static attractor
(Thompson & Sieber, Kuehn).
“N-tipping” where noisy fluctuations result in the system departing from a
neighbourhood of a quasi-static attractor.
(work on noise-induced escape from attractors)
“R-tipping” where the forced system fails to track a continuously changing
quasi-static attractor.
Outlook
• General Theory of R-tipping
(non-autonomous bifurcations, singular perturbation theory)
• A unifying description of the interplay between
different tipping mechanisms