Stochastic Climate Models

1、Stochastic Climate ModelsHasselmann Model of Climate Variability.Dynamical timescale modification.Ice-Age Model. Stochastic Resonance.SST ObservationsHasselmann/Frankignoul ModeldTdt=QCpHDT= Sea Surface Temperature SSTQ= Heat Flux out of AtmosphereH= Depth of well mixed layerD= Internal Ocean Dynami

2、csQHOcean Well Mixed LayerQ= kWTRadiationW= WindspeedQ= cT rW thusdT dt= aT bWHasselmann Stochastic HypothesisFluctuations 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 estim

3、ated from historical data:dT = aT dtDdW twhere D is proportional to the windpeed variance. This equation may be solved using standard techniques to givea1= 25 monthsIn Ito form the SST equation readsT t = T 0 eatD0tea ttdW t which can be shown to give asymptotically that the stationary temporal auto

4、correlation of SST isAC= eaHasselmann Stochastic HypothesisAs is well known in stochastic theory the Fourier transform of this function gives the stationary power spectrum which in this case becomesI= I0aa22which has the following graphical form The observational data fits the theoretical spectrum a

5、nd 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 Mode

6、lOn 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 d

7、ecadal 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 Nio). In the previous module on El Nio we saw that a rather simple oscillator explained much

8、 of the observed regular behaviour. This sort of model can be adapted to explain the spectrum seen above.Two dimensional stochastic modelsConsider a linearized version of the dynamical equations. Let us also assume that only bounded solutions occur (often the case in the climate context).ut= AuAs is

9、 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 mod

10、e while the imaginary part is 2 pi divided by the period of an oscillation. This oscillation consists of the following evolution RIRIRWhere 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 obser

11、vational climate literature. In the case of El Nio 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 Nio cycle. This mode is referred to th

12、eoretically 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 modelstu1u2=0 1u1u2F1F2This two dimensional system can be written asWhere the RHS matrix coefficients are

13、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 ear

14、lier.By an appropriate choice for the stochastic forcing we can easily reproduce the observed spectrum for El Nio seen above. The stochastic model above shall be the basis for the numerical component of this module. Other simple stochastic models arising from ocean timescalesHeat Flux horizontal pat

15、terns 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

16、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 assum

17、e that heat flux is large scale like the EOFsThe one dimensional ocean temperature equation can be written as:TtTvTy= Q0sin 2y/Lwhere 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

18、. Q0Saravanan ModelExpanding the ocean temperature as a Fourier series in y we obtain equations for the first two Fourier components. Other components are unforced.tT1GT2= Q0tT2GT1= 0G=2vLThis constitutes another two dimensional Ornstein Uhlenbeck process like that seen for El Nio. General probabili

19、ty 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 dT t = AT t dtBdW twhere A=GGand B=000CSpectrum of Saravanan modelThe spect

20、ral matrix of a stationary multivariate Ornstein Uhlenbeck process is given by (Gardiner equation 4.4.58) S=12Ai11BBtAti11substituting the matrix entries from the Saravanan model we get after some manipulation S=C22122G2 222G iG iG2The spectrum of a linear combination imiTiisi, jmiSijmjso the spectr

21、a of each Fourier component isS1=C222222G2 2S2=C22G222G2 2Spectrum of Saravanan modelThis 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 valuemax= G22which because of the physical nature of

22、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

23、 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

24、(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 10K). In general it is

25、 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

26、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 ModelsdTdt= short wave absorbed long wave lostWe briefly revise some of the content of the second module.Global temperature is ra

27、diatively controlleddTdt= Q0t1TabTQ0is the incoming solar radiation which varies with timeT is the albedo. Larger for ice than land:T =iceTTminT =landTTmaxT =landiceTTminTmaxTminiceTminTTmaxClimate EquilibriaFor equilibrium we have Q0t1T= abTTtTLong WaveShort WaveStable=InterglacialUnstableStable=Ic

28、e AgeDouble Well PotentialBenzi ModelDefine the functionT = Q 1T/ abT 1Then the stable-unstable-stable structure observed above will occur if we chooseT 1TT11TT21TT3This ansatz effectively defines an albedo function and we assume that the equilibria points satisfyT1T2T3Benzi Model (Continued) If we

29、assume that the solar forcing has a small periodic component to represent orbital variations associated with Milankovitch cycles then Q t = Q 1AcostwithA1And finally if we assume the temperature equation has an additive stochastic term representing random changes in factors controlling radiation suc

30、h as cloudiness, volcanos and humidity then we obtain Benzis equation (in Ito form):dT= abT1T1Acost 1 dtdWF T ,t dtdWThis 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. Sto

31、chastic 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 ti

32、me is given byT3=V T2V T3exp2V T2,T3,tV T2,T3,t T2T3V T ,t dtIn 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)

33、 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 cha

34、nges 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 s

35、ince 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 eq

36、uilibria 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 Benzis model which gives an approximately correc

37、t spectral peak frequency. Resonance and spectral peaks can only occur for a particular range of stochastic forcing amplitude. In Benzis model this range is realistic.V T2,T3,tBenzi Model (Continued) Numerical solutions illustrate the behaviour wellLow NoiseHigh NoiseConclusionsSimple linear stochastic models are