Given as an invited talk in Tromso Norway 29th September 2014

1. Generalised and fractional
Langevin equations-implications
for energy balance models
Nick Watkins
(nww@pks.mpg.de)
Norklima meeting, Tromso, 29th September, 2014
Paper in prep, with Sandra, David, Rainer Klages and Aleksei Chechkin

2. Description of Slide Deck
Given as an invited talk on 29th September 2014 in the kickoff workshop of NORKLIMA project
“Long-range memory in Earth’s climate response and its implications for future global
warming” at Quality Hotel Saga in Tromso, Norway, September 29th-October 1st
Papers currently in preparation with co-authors listed on cover page: Sandra Chapman,
David Stainforth, Rainer Klages and Aleksei Chechkin.
Uploaded to Slideshare in pdf on 8th January 2015 by Nick Watkins
(NickWatkins@mykolab.com).

3. Summary
• Energy balance models (EBMs) longstanding field [e.g. Sellers 1969, Budyko 1969, Ghil 1984]
• Stochastic EBMs can be used to study time dependent problems of climate sensitivity
[e.g. Padilla et al, 2011; Rypdal, 2012; Rypdal and Rypdal, 2014]
• 1st question: Can we make use of what we already know about the Langevin equation, and
the full Generalised Langevin Equation (GLE) that it approximates [Kubo, 1966; Haenggi, 1978],
in order to study EBMs with memory i.e. a Generalised Stochastic EBM ?
• 2nd Question: Can we make use of what we know about fluctuation-dissipation theorem ?
• 3rd question: If the memory kernel in the GLE takes a power law form, we get the Fractional
Langevin equation or FLE. If we do the same with the Generalised Stochastic EBM we
will get a fractional stochastic EBM, is it useful ?

4. Climate Sensitivity via Energy Balance Models
1
   eq
d Q
S T F
dt
  Q C T 0
0
  
  
Q Q Q
T T T
e.g. Rypdal, JGR, 2012

5. Solving Linear Energy Balance Models
1
1



   

   
 
   
 
eq
d T
C T F
dt S
d T
C T F
C
dt
d
C T F
dt
Green function
( )
1
( )
solves homogene
e
ous d.
xp( / )
1
( ) ( ) exp( / )F(s
e
1
)




     
 
 
     
d
C G t t
dt
G t t
C
T G t s F s s
C
e.g. Rypdal and Rypdal, J Climate, 2014

6. Stochastic version … looks familiar
Stochasti
1
c EBM


 
    
 
d
C T F
dt
'
Langevin equation


 
    
 
d
M v U
dt M
1st Question for this workshop is: Can we make
use of what we already know about the
Langevin equation, and the full Generalised
Langevin Equation (GLE) that it
is an approximation to, in order to study
EBMs with memory ?
One advantage of doing this might be the ability
to handle more types of memory than purely
short or long-ranged.

7. EBM – Langevin correspondence
1


 
    
 
d
C T F
dt
'
LE: when ( / ) ( )


  
 
    
 
 
d
M v U
dt M
M t t
1st Question for this workshop is: Can we use what we know about Langevin equation,
& Generalised Langevin eqn. [Kubo, 1966; Haenggi,1978] on stochastic EBMs with memory ?
0
(t) (t t') v(t')dt' '
GLE: notation Lutz,as .2001
 
 
     
 


td
M v U
dt
v x
What goes here ?

8. What would generalised stochastic EBM be ?
1


 
    
 
d
C T F
dt
Obvious guess is just to make same substitutions as were made for EBM
0
(t) (t t') (t')dt' 
 
      
 

td
C T T F
dt
'
LE: when ( / ) ( )


  
 
    
 
 
d
M v U
dt M
M t t
0
(t) (t t') v(t')dt' '
GLE: notation Lutz,as .2001
 
 
     
 


td
M v U
dt
v x

9. Why fluctuation dissipation theorems ?
2nd Question for this workshop is: Can we make use of what we know about the Fluctuation-dissipation theorem ?
To see why an FDT must exist, set acceleration and deterministic force both equal to zero. FDT then stops particle
speeding up or slowing down purely due to noise effects (which we don’t want). In EBM this would be T increase/decrease
(t) (t t') v(t')dt' ' 
 
     
 

d
M v U
dt
“We know that the complementary force … is indifferently positive and
negative and that its magnitude is such as to maintain the agitation of
the particle, which, given the viscous resistance, would stop without it ”-
Langevin, 1908, quoted in Lemons, 2002.

10. FDT for Generalised Langevin equation
1
( ') ( ) (t')
which in ordinary Langevin case becomes
( ') ( / ) ( ')
Note in EBM case, will not a be factor
  
  
   
  
B
M t t t
k
t t
T
M t t
kT

11. Fractional Langevin equation (FLE)
3rd Question for workshop is:
If the kernel in the GLE takes
a power law form, we get
the Fractional Langevin
equation or FLE. If we do the
same with the Generalised
Stochastic EBM we will
get an equation,
is it useful ?
0
0
1
1
(t) (t t') v(t')dt' '
( )
Now ~
( )
can be replaced by fractional derivative
(t
Lutz
, so integral
) v(t) '
Details in no, 2001. t white noisNote



 
 

 




 
     
 


 
    
 


t
t
d
M v U
dt
v t
t
t t
d
M v U
dt t
e

12. Fractional Langevin equation (FLE)
• FLE has 3 key properties from point of view of this workshop.
• Can be given a detailed and physical derivation in terms of a coordinate
coupled to a heat bath [see e.g. NWW Dresden talk on Slideshare,
and Kupferman, 2004]
• Its noise term is proportional to fractional Gaussian noise [Kupferman, op. cit.]
• Known to obey an Fluctuation-Dissipation Theorem [e.g. Metzler et al, 2014]

13. Fluctuation DissipationTheorem for FLE
[Metzler et al, in press, 2014]

14. So is this a useful fractional stochastic EBM ?
1
1
1
1
T(t) (t)
c.f. (t) v(t) '

 

 
 
 




 
     
 
 
    
 
d
T F
dt t
d
M v U
dt t
C

15. Can we solve the fractional stochastic EBM ?
1
1
T(t) (t)
Can we solve by fractionally integrating both sides ?

 
 


 
     
 
d
T F
dt
C
t

16. Summary
• Energy balance models (EBMs) longstanding field [e.g. Sellers 1969, Budyko 1969, Ghil 1984]
• EBMs can be used to study time dependent problems of climate sensitivity
[e.g. Padilla et al, 2011; Rypdal, 2012; Rypdal and Rypdal, 2014]
• 1st question: Can we make use of what we already know about the Langevin equation, and
the full Generalised Langevin Equation (GLE) that it approximates [Kubo, 1966; Haenggi, 1978],
in order to study EBMs with memory i.e. a Generalised Stochastic EBM ?
• 2nd Question: Can we make use of what we know about fluctuation-dissipation theorem ?
• 3rd question: If the memory kernel in the GLE takes a power law form, we get the Fractional
Langevin equation or FLE. If we do the same with the Generalised Stochastic EBM we
will get a fractional stochastic EBM, is it useful ?