Stochastic Climate Models

Hasselmann Model of Climate Variability.

Dynamical timescale modification.

Ice-Age Model. Stochastic Resonance.

SST Observations

Hasselmann/Frankignoul Model

dTdt

= QCpH

D

T=Sea Surface Temperature SST Q=Heat Flux out of AtmosphereH=Depth of well mixed layerD=Internal Ocean Dynamics

Q

HOcean Well Mixed Layer

Q=−kWTRadiation

W=Windspeed

Q'=−cT '−rW ' thusdT 'dt

=−aT '−bW'

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:

dT '=−aT 'dtDdW t

where D is proportional to the windpeed variance. This equation may be solved using standard techniques to give

a−1=2−5 months

In Ito form the SST equation reads

T ' t =T '0e−atD∫0

te−a t−t 'dW t '

which can be shown to give asymptotically that the stationary temporal autocorrelation of SST is

AC =e−a∣∣

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

I=I0a

a22

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.

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.

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).

∂ u∂ t

=A u

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

R I−R−I R

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.

Two dimensional stochastic models

∂∂ t u 1

u 2=0

1 u1

u2F 1

F 2

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.

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 transportsa 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.

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:

T tTvT y=Q0sin2 y /L

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.

Q0

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.

∂tT1−GT 2=Q0

∂tT2GT1=0

G=2vL

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

d T t =A T t dtBd W t

where A=−−G

G− and B=000C

Spectrum of Saravanan model

The spectral matrix of a stationary multivariate Ornstein Uhlenbeck process is given by (Gardiner equation 4.4.58)

S = 12

Ai1 −1 B Bt At−i1−1

substituting the matrix entries from the Saravanan model we get after some manipulation

S = C2

21

22G22 22

G −i−G i−

G2 The spectrum of a linear combination ∑

i

miT iis ∑

i , j

mi Sij mj

so the spectra of each Fourier component is

S1= C2

222

22G22S2= C2

2G2

22G22

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

max=G2−2

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).

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 10°K). 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.

Global Energy Balance Models

dTdt

=short wave absorbed− long wave lost

We briefly revise some of the content of the second module....Global temperature is radiatively controlled

dTdt

=Q0 t 1−T −a−bT

Q0 is the incoming solar radiation which varies with time

T is the albedo. Larger for ice than land:

T =ice TTmin

T =land TTmax

T =land−ice T−Tmin

Tmax−Tminice TminTTmax

Climate Equilibria

For equilibrium we have

Q0t 1−T =abT

Tt

T

Long Wave

Short Wave

Stable=Interglacial

Unstable

Stable=Ice Age Double Well "Potential"

Benzi Model

Define the function

T = Q 1−T /abT −1

Then the stable-unstable-stable structure observed above will occur if we choose

T ≡ 1− TT 1 1− T

T 2 1− TT 3

This ansatz effectively defines an albedo function and we assume that the equilibria points satisfy

T 1T 2T 3

Benzi Model (Continued)

If we assume that the solar forcing has a small periodic component to represent orbital variations associated with Milankovitch cycles then

Q t = Q 1Acos t

with A≪1

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):

dT=abT 1 T 1Acos t −1 dtdW≡F T ,t dtdW

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.

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

⟨T 3 ⟩=

V ''T2V '' T3exp−2V T 2 ,T 3 ,t

V T 2,T3 ,t ≡∫T 2

T 3

V T ,t dt

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.

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.

V T 2 ,T 3 ,t

Benzi Model (Continued)

Numerical solutions illustrate the behaviour well

Low Noise High Noise

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.

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., 166166, , p2511, 2008.

Saravanan, R. and McWilliams, J. C., Advective ocean–atmosphere 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.