Stochastic Climate Models

1
Stochastic Climate Models

• Hasselmann Model of Climate Variability.
• Dynamical timescale modification.
• Ice-Age Model. Stochastic Resonance.

2
SST Observations
3
Hasselmann/Frankignoul Model
Q
Ocean Well Mixed Layer
4
Hasselmann Stochastic Hypothesis

• Fluctuations in Windspeed (W') are white on
  timescales of weeks to a year so the Hasselmann
  equation becomes an approximate Ornstein
  Uhlenbeck Process. The relaxation parameter a can
  be estimated from historical data

In Ito form the SST equation reads
where D is proportional to the windpeed variance.
This equation may be solved using standard
techniques to give
which can be shown to give asymptotically that
the stationary temporal autocorrelation of SST is
5
Hasselmann Stochastic Hypothesis
As is well known in stochastic theory the Fourier
transform of this function gives the stationary
power spectrum which in this case becomes
which has the following graphical form
The observational data fits the theoretical
spectrum and autocorrelation well except at large
horizontal scales where there seems more weight
at the low frequency end of the spectrum. This
result suggests that much low frequency climate
variability is due to stochastic factors and not
intrinsic low frequency dynamics.
6
Elaborations of the Hasselmann Model

• On climate time scales there are a number of
  important physical time scales arising mainly
  from ocean dynamics. Examples include the current
  advection timescale the deep and shallow ocean
  adjustment timescale and the tropical coupled
  ocean-atmosphere relaxation timescale.
• The first two scales are decadal or longer while
  the latter is shorter (around 4 years). Spectral
  peaks in SST can sometimes be seen at these
  frequencies. As an example eastern equatorial SST
  in the Pacific shows a four year peak (El Niño).

In the previous module on El Niño we saw that a
rather simple oscillator explained much of the
observed regular behaviour. This sort of model
can be adapted to explain the spectrum seen above.
7
Two dimensional stochastic models
Consider a linearized version of the dynamical
equations. Let us also assume that only bounded
solutions occur (often the case in the climate
context).
As is well known the nxn real matrix A has n
complex eigenvalues which occur in complex
conjugate pairs. Some of these may be real. The
corresponding eigenvectors are often called
normal modes. In terms of dynamical evolution
the real part of the eigenvalue is the inverse of
the damping time of the mode while the imaginary
part is 2 pi divided by the period of an
oscillation. This oscillation consists of the
following evolution
Where R and I are the real and imaginary parts of
the corresponding (complex) eigenvector. This
pair of patterns are sometimes referred to as
POP pairs in the observational climate
literature. In the case of El Niño it is often
the case that there is a (complex) normal mode
which has by far the longest damping time. The
real and complex parts of the eigenvector
correspond closely with dominant patterns from
the observed El Niño cycle. This mode is
referred to theoretically as the recharge
oscillator. It seems appropriate then to
consider a stochastically forced version of this
simple and important two dimensional subsystem.
8
Two dimensional stochastic models
This two dimensional system can be written as
Where the RHS matrix coefficients are related
algebraically to the damping time and period of
the oscillation (exercise derive these
relations). The stochastic forcing terms on the
RHS can be taken to be white noise and then this
system becomes a two dimensional generalization
of the Hasselmann Ornstein Uhlenbeck process
considered earlier. By an appropriate choice
for the stochastic forcing we can easily
reproduce the observed spectrum for El Niño seen
above. The stochastic model above shall be the
basis for the numerical component of this module.
9
Other simple stochastic models arising from ocean
timescales

• Heat Flux horizontal patterns on climate
  timescales are large scale but white in the time
  domain. Here are the "most common" North Atlantic
  patterns (EOFs)

Note the dipolar nature of the patterns. This
derives from the large scales of the low
frequency atmospheric repsonse. An important
feature of the North Atlantic is the Gulf Stream
which transports a large amount of water (to
large depth) in a northerly (and easterly
direction. Saravanan (1998) suggested a simple
stochastic model to explain the strongly decadal
spectrum of SST in this region.
10
Saravanan Model

• Consider only the meridional direction (latitude)
  and assume that heat flux is large scale like the
  EOFs

The one dimensional ocean temperature equation
can be written as
where is white noise. The second term on
the left is a damping term which depends on
atmospheric feedback and vertical ocean mixing.
Advection is modelled by the third term.
11
Saravanan Model

• Expanding the ocean temperature as a Fourier
  series in y we obtain equations for the first two
  Fourier components. Other components are
  unforced.

This constitutes another two dimensional Ornstein
Uhlenbeck process like that seen for El Niño.
General probability solutions as well as
covariance and spectral matrices are well known
(see Gardiner p109-111). Physically these
equations represent a stochastically forced
damped oscillator. In standard matrix Ito form
they can be written
12
Spectrum of Saravanan model
The spectral matrix of a stationary multivariate
Ornstein Uhlenbeck process is given by (Gardiner
equation 4.4.58)?
substituting the matrix entries from the
Saravanan model we get after some manipulation
The spectrum of a linear combination
is
so the spectra of each Fourier component is
13
Spectrum of Saravanan model
This shows that the spectrum varies with y and a
simple calculation shows that at one quarter and
three quarters through the channel the spectrum
may peak at the value
which because of the physical nature of these
parameters will be in the decadal range. Note
that for the points at the ends and center of the
channel the spectrum is not peaked (it is
actually more strongly peaked at lower
frequencies than the Hasselmann spectrum). Thus
the domain gains spectral weight in the decadal
range. The above model is obviously too simple
(there are more than one heat flux patterns for
example) however the enhanced decadal spectral
intensity due to the slow advection time scale of
the Gulf Stream is plausible. Whether an actual
peak in the low frequency spectrum results for
more realistic models is not clear (good research
problem).
14
Stochastic Paleoclimate Models

• Climate records of global temperature over very
  long time periods can show very strong spectral
  peaks which correspond to ice age and
  interglacial periods. The changes in average
  global temperature can be very large (order
  10K).
• In general it is thought these changes are due
  to to orbital changes (Milankovitch forcing) such
  as rotation axis angle changes.
• Such changes in external forcing are quite
  small so something in the climate system must
  strongly magnify the forcing changes.
• One theory is changes in biosphere CO2 as this
  tends to lag forcing but act via the greenhouse
  effect as a magnifier.
• Another theory is stochastic resonance (due to
  Benzi and co-workers) which we review here.

15
Global Energy Balance Models
We briefly revise some of the content of the
second module.... Global temperature is
radiatively controlled
16
Climate Equilibria
For equilibrium we have
Long Wave
Short Wave
StableInterglacial
Unstable
StableIce Age
Double Well "Potential"
17
Benzi Model
Define the function
Then the stable-unstable-stable structure
observed above will occur if we choose
This ansatz effectively defines an albedo
function and we assume that the equilibria points
satisfy
18
Benzi Model (Continued)?
If we assume that the solar forcing has a small
periodic component to represent orbital
variations associated with Milankovitch cycles
then
And finally if we assume the temperature equation
has an additive stochastic term representing
random changes in factors controlling radiation
such as cloudiness, volcanos and humidity then we
obtain Benzi's equation (in Ito form)
This is a stochastically forced time dependent
double well potential. If the stochastic forcing
is not present then temperature fluctuates close
to the initial equilibrium chosen. Stochastic
forcing is required to transition between
equilibria.
19
Benzi Model (Continued)?
Stochastically forced double well potential
equations are a highly studied area and the
distributions for first exit times from one
equilibrium to the other are known (see Gardiner
Chapter 9). The mean first exit time is given by
In addition it is known that the first exit time
is distributed according to an exponential
distribution which implies for a time independent
V that approximately the decorrelation time of
the independent variable (global temperature
here) is also exponential with a decay time given
by the (constant) mean exit time. Since the
spectrum is the Fourier transform of this
decorrelation it follows that the spectrum will
not have a peak unlike the observations.
20
Benzi Model (Continued)?
If we consider instead the case where V varies
due to changes in orbital shortwave radiation
forcing then Benzi shows that even if this
variation is small as a percentage of total solar
radiation that in his model
shows significant variations with
time. This implies that the exit time from a
particular equilibria varies strongly with the
orbital forcing since the exit time is the
exponential of this function. In addition it can
be shown that variance of this first exit time
decreases markedly as the exit time falls. This
effect is called stochastic resonance.
Physically this means that the system spends
considerable time in the vicinity of either
equilibria but when the astronomical forcing is
favourable then the stochastic forcing can knock
the system into the other stable point. This
behaviour results in a strong spectral peak. The
first exit time is typically of the order of
100,000 years for Benzi's model which gives an
approximately correct spectral peak frequency.
Resonance and spectral peaks can only occur for a
particular range of stochastic forcing amplitude.
In Benzi's model this range is realistic.
21
Benzi Model (Continued)?
Numerical solutions illustrate the behaviour well
High Noise
Low Noise
22
Conclusions

• Simple linear stochastic models are able to
  explain much of the observed climate variability
  on timescales varying from annual through to
  centennial.
• Observed broad spectral peaks in climate records
  can be explained by the incorporation of
  dynamical time scales into linear stochastic
  models.
• The basic paleoclimate dynamics are non-linear
  due to the ice albedo positive feedback. A
  stochastic model with this feedback is able to
  explain the way in which small amplitude
  astronomical variations in solar radiation may
  induce the large observed spectral peak
  associated with ice ages.

23
References

• Hasselmann K., Stochastic climate models, Part I,
  Theory, Tellus, 28, p.473, 1976.
• Frankignoul, C. Hasselmann, K., Stochastic
  climate models, part II. Application to
  sea-surface temperature anomalies and thermocline
  variability, Tellus, 29, p.289, 1977.
• R. Kleeman. Stochastic theories for the
  irregularity of ENSO. Phil. Trans. Roy. Soc. A.,
  166, p2511, 2008.
• Saravanan, R. and McWilliams, J. C., Advective
  oceanatmosphere interaction An analytical
  stochastic model with Implications for decadal
  variability, Journal of Climate, 11, p165, 1997.
• Benzi, R., Parisi, G., Sutera, A. and Vulpiani,
  A., A theory of stochastic resonance in climatic
  change, Siam J. Appl. Math., 43, p565, 1983.