Model Reduction

To model low frequency dynamics one aims to retain the large scale features while approximating the small fast features, often by some stochastic process. The model

reduction procedure consists of two steps. First an optimal basis to represent the dynamics is chosen and then the system is truncated. Secondly a closure scheme is used to account for the affects of the unresolved variables on the retained modes. This split is often designed to separate the large scale, slow modes from the small, fast modes. The closure procedure could be based on fitting linear stochastic damp- ing terms empirically [Selten, 1995] or predicting stochastic corrections using theory valid in the limit of complete time scale separation [Majda et al., 1999].

Crommelin and Majda [2004] investigated the importance of the choice of basis for the reduced system. Empirical Orthogonal Functions (EOFs) are often used as a basis. EOFs are constructed by finding the mode that accounts for the most variability in the system. Then subsequent modes account for the most vari- ability subject to being orthogonal to the first and so on. EOFs are calculated by computing the eigenvectors of the covariance matrix. They can drastically reduce the number of dimensions in a system while retaining nearly all of the variance (see e.g Preisendorfer [1988]). However, they can fail to reproduce the correct dynamics, even if they account for 99% of the variance. This is particularly true in systems with bursty regime transitions, where low variability modes can be crucial in forcing the system between metastable states [Crommelin and Majda, 2004]. An alternative is to consider Optimal Persistence Patterns (OPPs) [DelSole, 2001]. In this case the basis is chosen to optimise persistence measures: either the integrated autocorre- lation function or square integrated autocorrelation. This is a natural basis if one is aiming for long term predictive skill. Another choice are Principal Interaction Patterns (PIPs) [Kwasniok, 1996]. These take account of the dynamics of the sys- tem and so are a natural choice, although their calculation is more complicated. Basically, one minimises the integrated difference between the full system and the low dimensional system up to some final time to calculate the projection operator. Expressions for the gradient of the error can be calculated to facilitate the minimi- sation [Kwasniok, 1997]. A problem with the approach is that the calculation of PIPs can be sensitive to the final time chosen.

Crommelin and Majda [2004] calculated EOFs, OPPs and PIPs for the barotropic model on the beta-plane: the much studied model of Charney and De Vore [1979]. They studied the six dimensional truncated model and assessed the ability of the dimension reduction strategies to reproduce the regime transitting behaviour. They found that, even with five variables, the EOFs were not able to simulate the regime switching. OPPs also failed to produce the chaotic nature of the regime switching. Instead the five dimensional model produced periodic behaviour. The authors conclude that short time scale behaviour must be important to produce

regime switching, which OPPs fail to retain. The PIP models are able to reproduce the regime switching. However, the behaviour of the reduced model was noted to depend upon the final time for the “training”. If a short time is used then PIP models can fail to reproduce the climate statistics; a long time and the variability of the reduced system can be too low. These problems with PIPs, applied to a semi-realistic model of the atmosphere, were noted by Kwasniok [2004].

Another related technique are Principal Oscillation Patterns (POPs) [Von Storch et al., 1995]. POPs are the normal modes of the linearised system and correspond to the unstable modes calculated from linear stability analysis. POP analysis includes both stages of the model reduction with the closure problem already solved by the resulting linear system. POP analysis can be used for prediction but the linearity of the model means extended forecasts have little skill.

In this thesis we will focus on the second stage of the model reduction: the closure problem. We focus on the type of problems that apply to LFV. We use the term climate to refer to the resolved modes of the system and occasionally refer to the fast modes as weather variables in agreement with terminology used in the literature.

As mentioned in the Introduction a characteristic of the climate system is its variability on multiple time scales. One way of explaining this variability has been to find external forcing factors driving the system at a range of frequencies such as some unknown solar forcing. In 1976 Hasselmann, with his seminal papers [Hasselmann, 1976; Frankignoul and Hasselmann, 1977], initiated a field of research aiming to explain this variability as part of the internal dynamics of the system. He considered the slow changes in climate to be the integrated response of rapid fluctuations in weather similar to the way a Brownian particle integrates the many collisions with faster moving fluid particles. The idea was to treat the fast deterministic motion as a stochastic process and then average the equations to leave an effective equation for the slow climate variables. This has been the starting point for much research into stochastic climate models. In this chapter we review some of this work but first we introduce some of the mathematical language as summarised by Arnold [2001].

Consider the full description of the climate given by the vectorz. A climate

model starts with a set of deterministic equations

dz

dt =h(z).

Hasselmann considered the case where there exist separate components of z

that evolve on different time scales. In this casexcould represent climate variables

with characteristic timeτxandycould be weather variables with characteristic time

τy. In the atmosphereτy would be the order of one day andτx could be on the scale

of weeks to months representing the intraseasonal variability associated with large scale teleconnections. To represent different response times we introduce a small

scaling parameterand write the (non-dimensionalised) system as

dx dt =f(x,y) ,x0 =X ∈R n dy dt = 1 g(x,y) ,y0 =Y ∈R m

such that τy ≈ τx ≈1. We would like an approximate equation with solution

ut ∈[0, T] such that lim→0xt = ut. The simplest case where g(x,y) = g(y) can

be treated by the classical method of Averaging, which Hasselmann refers to as a

Statistical Dynamical Model. In this case the forcing term forxis averaged overy

giving the approximate equation

du dt =F(u), where F(x) = lim T→∞ 1 T Z T 0 f(x,y_t)dt= Z f(x,y)µ(dy).

Here,µis the unique invariant measure foryand ergodicity is assumed. Calculating

F(x) is known as the closure problem. The next step is to consider the error in this

approximation. It was shown by Khasminskii [1966] that if the fast variables are a

stochastic process then on the interval t∈[0, T], there is a Central Limit Theorem

(CLT) such that

ξ_t = √1

(x

t−ut)

has a limiting Gaussian distribution as → 0. Over longer time periods there are

many phenomena that are not captured by the Method of Averaging or the Central

Limit Theorem. These could include xt hopping between stable attractors of the

system. This could be described as aLarge Deviationphenomena. The concepts

of Averaging (a Law of Large Numbers), the CLT and Large Deviations are three fundamental concepts in asymptotic probability theory. In this thesis we will be focussing on approximations using the CLT.

At this point we have only considered the classical case whereg(x,y) =g(y).

Arnold [2001] calls the generalisation “Hasselmann’s case”. Now the slow and fast variables are coupled and the situation is now much more complicated. The fast

dynamics now have invariant measuresµx(dy) which depend uponx. If we consider

xto be frozen, the solution operator ofymaps the initial conditiony₀ =Y forward

in time: (t,Y)→φx

t(Y). Then we can write the averaged forcing as

Fµx(x) = Z Rm f(x,y)µx(dy) = lim T→∞ 1 T Z T 0 f(x, φx_t(Y))dt .

Then we have the approximate equation

du

dt =Fµut(ut) ,u0=X

and lim→0xt(X,Y) =ut(X).

Averaging for ODEs is known asAnosov’s theorem[Pavliotis and Stuart,

2008]. For the results to follow it is sufficient that the fast dynamics are a hyperbolic system [Kifer, 2001]. In this case one can also say something about the deviations from the average system. Kifer [1995] proved that the deviations from the averaged system are a Gaussian diffusion process. The problem is that the ergodicity and fast mixing assumption often fails for ODEs. It is easier to work with an SDE where there is a stochastic term entering into the equation for the fast dynamics as

dx dt =f(x,y), x(0) =x0, dy dt = 1 g(x,y) + 1 √ β(x,y) dV dt , y(0) =y0, (3.1)

whereV is a standard Brownian motion. Given certain conditions on the coefficients

g(x,y) andβ(x,y) it can be shown that the invariant measures foryhave a density

with respect to Lebesgue measure,µx(dy) =ρx(y)dy. In simple cases this density

is known explicitly.