### UNIVERSIT `

### A DI PISA

### Dipartimento di Fisica

### Corso di Laurea Magistrale in Scienze Fisiche

### Academic Year 2014-2015

### Numerical study of a low order model for the dynamics

### of the El Ni˜

### no-Southern Oscillation (ENSO)

### phenomenon

### Candidate

### Andrea Oliveri

### Supervisors

### Prof. Riccardo Mannella

**Introduction**

This thesis deals with the problem of giving a statistical description of the El Niño and
La Niña (ENSO) as much similar as possible to the available data. El Niño-Southern
Oscillation (ENSO) is the most important and studied phenomenon aﬀecting the
cli-mate variability at interannual timescales. As the name suggests, ENSO consists of
two related aspects which ﬂuctuate periodically in an irregular way. The former, El
Niño, is associated to an increase in the sea surface temperature (SST) along the coasts
of Ecuador and Peru, whereas the latter one, the Southern Oscillation, is related to
changes in the east-west pressure gradient leading to the movement of air masses in
the atmosphere along the Paciﬁc Equator line (Walker’s circulation). The ENSO cycle
is the consequence of slow feedbacks in the ocean-atmosphere system acting alongside
the strong air-sea interaction processes in the tropics that allow the growth of small
disturbances to the large-scale ocean state. Over the years, a number of models have
been developed in order to fully understand this phenomenon both qualitatively and
quantitatively. These models usually rely on more or less strong approximations of the
ﬂuid dynamic equations which regulate the atmospheric and oceanic dynamics. These
approximations are based on the fact that the ENSO phenomenon involves a very thin
layer of the ocean along the Paciﬁc Equator, i.e. a strip of water 15000km long, few
*hun-dreds km width and with an average depth of some tens of meters. This very elongated*
strip of water, behaves like a river that has the source at the extreme east, the sink at
the west boundary, and it is driven by the rotation of the Earth and by the easterly
trade winds. This simpliﬁed scheme of the ocean dynamics, along the Paciﬁc Equator,
leads to a reduced set of equations in which one or more stochastic terms are introduced.
The stochastic terms are used to simulate the ﬂuctuations in periodicity and intensity of
the phenomenon, generated by the interaction between the ocean and the atmosphere
(Recharge oscillator models, ROM). Recent works [1], based on projection techniques,
allows to couple a generic system under examination (in this case the oceanic variables)
with a deterministic system that correctly reproduces the contribution of the stochastic
forcing. This technique, applied to a model which reproduces the ENSO phenomenon,
allows us to derive a partial diﬀerential equation which describe the statistics of two
*ENSO relevant variables: the anomalies of the average thermocline depth (h) and the*
*eastern Paciﬁc SST (T ), respectively. This last, in particular, is considered the main*

the study and the realisation of a software that allows the numerical resolution of the
Fokker-Planck equation for the ENSO phenomenon. This software has made it possible
to obtain the probability density function (PDF) for the stationary state in the
*two-dimensional space (h, T ) and a reduced form of it for the sole variable T . This reduced*
PDF was directly compared to the analytical results obtained in the fundamental work
[1]. These theoretic results are particularly important since allow a direct and
immedi-ate comparison with the data obtained from the observation of the ENSO phenomenon.
Nevertheless, these result are obtained by adopting an hypothesis ("the ansatz") which
is assumed to be approximately correct: one of the aim of the present thesis is to verify,
through the numerical solution of the Fokker-Planck equation, the validity or not of this
hypothesis. Subsequently it was performed the numerical integration of the stochastic
equations associated to the previously PDE in order to calculate the quantities related
to the periodicity and the intensity of extreme El Niño events. To this end it was
cal-culated the mean ﬁrst passage time (MFPT) corresponding to El Niño events of great
intensity and this result was compared with the available observations for the ENSO
phenomenon and with the analytical results stemming from standard MFPT techniques
applied to the reduced model obtained in [1].

This thesis work is structured in seven chapters:

**• Chapter 1: In this chapter we introduce the physical phenomenon of ENSO**
giving a general description of it and its importance in the Earth’s general
circula-tion. After this introductory section we describe, in details, El Niño and La Niña
showing their characteristics and statistics.

**• Chapter 2: Here we introduce the projective method in this version developed**
by Dr. Marco Bianucci for a generic deterministic system and we focus on the
necessary assumptions. Finally we give the general expression for the
Fokker-Planck-like equation which we will implement in the speciﬁc case of this thesis
work.

**• Chapter 3: We introduce the low order model (LOM) used in literature to **
simu-late the ENSO statistics and its main properties and, shortly, how this model can
be simply derived. After we use the projective method, developed in Chapter 2,
on the LOM model and we derive the Fokker-Planck equation, core of the thesis.
**We obtain some simple original results about the moments of the PDF, solution**
*of the associated Fokker-Planck equation and we introduce the ansatz.*

**• Chapter 4: In this chapter we introduce the numerical methods used to integrate**
the Fokker-Planck equation and the associated stochastic equations. We also focus

on the technique necessary in crossing barrier problems to correct the ﬁniteness of the sampling rate of the relevant stochastic trajectories.

**• Chapter 5: Here we use the techniques explained in the previous chapter to**
**integrate the FP equation. This chapter contains completely original results.**
First we discuss the parameter used in the Fokker-Planck integration. Then we
show and comment the equilibrium PDF and its reduced form. Finally we test
the validity of the ansatz.

**• Chapter 6: This chapter introduces the completely original results obtained by**
the integration of the stochastic equation associated with the FPE. We illustrate
the integration parameters choice. We calculate the mean ﬁrst passage times and
*duration for diﬀerent T barriers, which represent strong El Niño events, and we*
discuss the results comparing them with the available experimental data.

**• Chapter 7: Here we summarize the already exposed results and we pose new**
questions to be solved in a possible future work.

**Chapter 1**

**ENSO**

*In this chapter we introduce the physical phenomenon of ENSO. First, we give a general*
*description of the ENSO and its importance in the Earth’s general circulation. After this*
*introductory section we describe the mean state of the tropical Paciﬁc region necessary to*
*understand the ENSO process and its two phases, El Niño and La Niña, described in the next*
*section. Finally we illustrate the characteristics of the statistic of the ENSO process and its*
*characteristics.*

**1.1 An overview on ENSO process**

El Niño-Southern Oscillation (ENSO) is the most important and studied phenomenon
of interannual climatic variability of the equatorial Paciﬁc ocean. ENSO is composed of
two correlated aspects: El Niño1 _{and the Southern Oscillation. El Niño, the former, is}

an anomalous increase of the surface temperature of the Paciﬁc ocean along the coasts
of Peru and Ecuador. The latter, Southern Oscillation, is a variation in the motion of
air masses in the Walker circulation, an atmospheric zonal circulation on the tropical
Paciﬁc ocean, due to the variation in the east-west gradient of atmospheric pressure.
The upper layer of the Paciﬁc ocean (mix layer), responsible of energetic and
momen-tum exchange with the atmosphere, is a long strip of water, alike a, with an average
*depth of some tens of meters, some hundreds of kilometers width, and almost 10.000*
km length which ﬂows over the deeper and colder ocean water, from the south
Amer-ica coasts to the warmer Asian coasts, mainly driven by the easterly tradewinds. This
motion accumulates warm water in the western Paciﬁc and, because of the closed East
surface boundary, it forces the colder and deeper water to rise up in the eastern Paciﬁc
generating a temperature gradient between the two sides of the ocean. The
tempera-ture gradient, in turn, reinforces the easterly tradewinds, thus this process is in part
self-sustaining. Notice that the engine of this process is the Coriolis Force, as better

1_{El Niño is the name used in Spanish language for Christ child because the phenomenon appears}

illustrated in the next Paragraph. Inside this shallow river we can have the formation of planetary waves which travel eastward and westward. When a warm wave, started from the western Paciﬁc, reaches the south America coast, the temperature gradient which sustains the shallow water ﬂow, can be perturbed generating an anomalous increase of surface temperature in the eastern Paciﬁc and starting an El Niño event. El Niño has a counterpart, La Niña, in which the surface temperature of the ocean along the South America coasts has an anomalous decrease, due to the start of an increased number of westward waves.

Both the anomalous increase of the oceanic surface temperature and the diﬀerent
atmo-spheric pressure gradient during an El Niño event have a direct impact on the climate
of the continental regions bordering on the tropical Paciﬁc ocean [2][3]: Australia suﬀers
from a reduction in rainfalls as well as the eastern Paciﬁc while in the South America
we have the opposite situation with strong rainfalls and a generally wet climate. During
a La Niña event we have the specular situation: a strong drought in south America
and an anomalous increase of precipitation in the Southeastern Asia. The ENSO events
alters not only the tropical Paciﬁc regions in which they directly operate but also aﬀects
distant regions of the planet. This is possible because the atmospheric system is
char-acterized by teleconnections2 _{between some regions of the planet: variations in these}

regions are mutually correlated and inﬂuenced, even if they are spatially distant. An example of teleconnection is the correlation between air pressure at sea level in Darwin and in Tahiti (measured by the SOI index explained later). The reasons of this tele-connections remain unknown and seem related both to planetary scale motions of air masses in atmosphere and of water currents in the oceans.

**1.2 The mean state of the tropical Paciﬁc region**

Before outlining the mean state of the tropical Paciﬁc region, necessary to explain correctly the ENSO anomalies, we introduce some concepts and deﬁnitions that permit to better characterize the system of interest. With regard to the thin upper layer of water, introduced in the previous section, and represented as a river above the deep ocean, we need to give better deﬁnition of its boundaries. While its upper boundary is simply represented by the sea surface, for its lower boundary we have to introduce the deﬁnition of thermocline. The thermocline is the local depth of the isothermic surface at 20 ◦C and divides vertically the ocean in two regions: the mix layer and the deep

waters. The mix layer is basically the layer of ﬂuid close the surface and it is responsible for the energy and momentum exchange between ocean and atmosphere. As can be seen from Figure 1.1 the temperature in the mix layer varies very quickly with the depth. On the contrary, the temperature of the deep waters remains approximately

2_{Teleconnection in atmospheric science refers to climate anomalies being related to each other at}

1.2. THE MEAN STATE OF THE TROPICAL PACIFIC REGION

constant with the variation of depth, in fact, this ocean layer does not take part in the energy and momentum exchange with atmosphere. Another quantity, fundamental to

Figure 1.1: White line: temperature of the ocean versus depth. Note the great thermal excursion in the mix layer. The wavy top part of the ﬁgure represent the region which directly interacts with the atmosphere and determines the sea surface temperature.[4] the characterization of the mix layer is the Sea Surface Temperature (SST) (Figure 1.5) which is the local temperature of the top of the mix layer.

Now we illustrate the mean state of the tropical Paciﬁc region due to the interaction
between the ocean itself and the tropical atmosphere. In the paciﬁc tropical atmosphere
there are two main planetary scale motions: the Hadley cells (Figure 1.2) and the Walker
circulation (Figure 1.4). The Intertropical Convergence Zone (ITCZ) is a band region
along the Equator in which there is an upward ﬂow of warm air from the lower layers of
the atmosphere. The uplifted masses proceed towards the poles and reach the tropical
band. Here, the masses cool, sink in the low atmosphere and push new air masses from
the tropical regions to the ITCZ. The Coriolis force deviates the motion from the tropics
to the ITCZ rightwards in the northern hemisphere and leftwards in the southern. This
*deﬂected motion of air is called easterly trade winds and globally this air circulation is*
called Hadley Cell. As we can see in Figure 1.2, there are two Hadley Cells: one in the
northern hemisphere and one in the southern and both contribute to the global system
of atmospheric cells of the Earth. The easterly trade winds push masses of warm water

Figure 1.2: Hadley cells, note the trade winds at the boundaries of the ITCZ.[5]
from east to west and form an accumulation of warm water along the coasts in the
*western Paciﬁc: the warm pool, while in the equatorial East Paciﬁc, due to the closed*
boundary with the coast, the surface displacement of water pulls cold water up from
*the deep and creates the cold tongue. Thus in the West region the SST is higher than*
the mean SST value while along the coasts of Peru in the eastern Paciﬁc, the SST is
lower than the mean SST. The greater quantity of warm water in the warm pool pushes
the thermocline deeper while in the cold tongue the upwelling of cold water rises up the
thermocline so it shows an upward trend from west to east (Figure 1.4). Furthermore,
the balance between gravity and the easterly trade winds creates an east-west gradient
(opposite to the thermocline gradient) in the sea level height approximately of 60 cm
between the American and Asian coasts, distant more than 10.000km each other (Figure
1.3).

Figure 1.3: The sea level and the thermocline in the tropical Paciﬁc region. Note the opposite east-west direction of the two gradients.[6]

The east-west SST gradient starts the second important circulation in this area: the Walker circulation (Figure 1.4). The air masses heated on the warm pool rise reaching the high layers of the troposphere and here they are pushed to east, until they meet the cold tongue and sink. This motion establishes a pressure gradient between the warm pool (low pressure) and the cold tongue (high pressure) that increases the intensity of

1.2. THE MEAN STATE OF THE TROPICAL PACIFIC REGION the easterly trade winds.

Figure 1.4: The mean state of the tropical Paciﬁc region. The gray arrows represent the Walker circulation.[7]

Figure 1.5: A typical distribution of the Sea Surface Temperature in the tropical Paciﬁc region with no ENSO events in progress. Note the temperature diﬀerence between the west Paciﬁc (the warm pool) and the eastern Paciﬁc (the cold tongue).[8]

**1.3 El Niño event**

The ENSO cycle is the consequence of slow feedbacks in the ocean-atmosphere system
acting alongside the strong air-sea interaction processes in the tropics that allow the
growth of small disturbances to the large-scale ocean state. An El Niño event can start
*in several (unpredictable) ways. For example, through the action of westerly windbursts*
(short-lived storm-like events in the Western Paciﬁc) that disturb the balance maintained
by easterly winds, or through slow evolution of the ocean thermocline as a consequence
of a strong Madden Julian Oscillations3_{. These quite stochastic perturbations generates,}

in the thin equatorial mixed layer, cycles of waves from East to West (Kelvin waves4_{)}

and from East to West, after the reﬂection on the Peruvian coast (Rossby waves5_{) [9],}

that, modifying the thermocline depth, decrease the temperature gradient of the SST
along the Equator. The decreased temperature gradient of the SST, in turn, mitigates
the intensity of the westerly tradewinds, or even interrupts them, increasing further, in
the cold tongue, the SST and the depth of the thermocline (Bjerkness feedback). When
conditions are favorable, this positive feedback generates an El Niño event. The reduced
SST in the warm pool does not permit anymore the upward ﬂow of the air masses while
*the reduced strength of easterly trade winds permits the formation of the westerly wind*

*burst. These winds push the air masses from the warm pool to the east up to reach the*

cold tongue which now has an increased SST temperature.

Here the air masses rise up and form a sort of Walker circulation centered on the eastern Paciﬁc ocean which causes an increase in precipitations along the coasts of America and drought in the western Paciﬁc.

3_{The Madden Julian Oscillations is a traveling pattern of anomalous rainfall which propagates}

eastward at approximately 4 to 8 m/s through the atmosphere above the warm parts of the Indian and Paciﬁc oceans.

4_{Kelvin waves are a type of low-frequency gravity waves in the ocean that balances the Earth’s}

Coriolis force against a topographic boundary such as a coastline, or in our case, a waveguide such as the equator. This type of waves are able to move only eastward from Asia to America in about three to four months and are responsible for the decrease in the height gradient of the sea surface.

5_{The Rossby waves are a consequence of the balance between the gravity and the Coriolis force}

which is not constant but depends on the distance from the equator along a meridian. They move only westward and are slower than the Kelvin waves: for them, it can take from nine months to four years to reach the Asian coasts starting from America.

1.4. LA NIÑA EVENT

Figure 1.6: The state of the tropical Paciﬁc region when an El Niño event occurs.[7]

Figure 1.7: A typical distribution of the Sea Surface Temperature Anomaly (excursion from the mean value calculated on historical data) in the tropical Paciﬁc region when an El Niño event occurs. Note the warm pool which extends up to the south America.[10]

**1.4 La Niña event**

La Niña is characterized by unusually cold ocean temperatures in the Equatorial Paciﬁc, with a stronger East-West SST gradient, thus it is the opposite of El Niño. A La Niña event arise, as El Niño, as a positive feedback after the onset of some perturbations. The starting disturbance can be, for example, the oscillating regression toward normal conditions of a previous strong El Niño event, some strong atmospheric perturbation that generates Kelvin waves, that tend to increase the upwelling of cold water. This fact increases the strength of the Walker circulation while the thermocline rises up along the Peruvian coasts. Stronger easterly trade winds push more air masses on the warm pool

thus heavy precipitations strike the Australia and the south Asia, while in America we have severe drought.

Figure 1.8: The state of the tropical Paciﬁc region when a La Niña event occurs.[7]

Figure 1.9: A typical distribution of the Sea Surface Temperature Anomaly (excursion from the mean value calculated on historical data) in the tropical Paciﬁc region when a La Niña event occurs. Note the cold tongue which extends up to the center of the tropical Paciﬁc region.[10]

1.4. LA NIÑA EVENT

Figure 1.10: Schematic representation of the Kelvin and Rossby waves propagation in the tropical Paciﬁc region.[11]

Figure 1.11: Time evolution of the sea surface temperature, and its anomaly, in the Equatorial Paciﬁc Ocean in function of the geographical position.[12]

**1.5 ENSO characteristics**

In order to perform a quantitative characterization of the ENSO process we have to introduce some conventions and indexes commonly used in literature. First of all we deﬁne the NINO3.4 band, a geographical region in the tropical Paciﬁc in which we measure the quantities of interest. This zone is comprised between 120W-170W, 5N-5S which is, approximately, in the middle of the Paciﬁc ocean, over the Equator, between south America and Asia. The excursion of the spatially mediated SST on the NINO3.4 band from its historical data is called the NINO3.4 index anomaly and it is used as a ENSO indicator. When the Paciﬁc is in its "normal state" the NINO3.4 index anomaly is 0◦C while, by deﬁnition, we have an El Niño event when the NINO3.4 index anomaly

is greater than 0.5◦C, while when it is lesser than -0.5◦C is deﬁned as a La Niña event.

For the sake of completeness we cite here another index, not utilized in this work, which is used to identify and characterize the atmospheric component of ENSO: the Southern Oscillation Index (SOI). This index is the monthly average of the daily diﬀerence of the atmospheric pressure at sea level between Tahiti and Darwin and is assumed as an indicator of the current atmospheric gradient of pressure on the Paciﬁc Ocean which is one of the two component of ENSO [2]

The Figure 1.13 shows the NINO3.4 index historical data from 1950 to 2016. ENSO events start approximately around the boreal summer and reach the maximum intensity around the end of the year. The ENSO events are not periodic: in literature [2] is common use to deﬁne an "ENSO band" as a time interval between 2 and 7 years in which it is more probable to have an ENSO event. There are also large time intervals (as the 30’s of the 20th century) in which no ENSO events are recorded [2]. As we can see in Figure 1.12 relative to NINO3.4 data averaged over three months, the El Niño events are more frequent than La Niña events. Furthermore, the distribution has a power tail in the El Niño interval (positive anomalies) and a cut-oﬀ in the La Niña interval: in this work we try to reproduce these peculiarities, without however, exploring the causes which remain unknown. The power tail in the ENSO intensity distribution permits to have rare, but extreme in intensity, El Niño events (1982-1983, 1997-1998, 2015-2016), fact which does not happen for La Niña. So, as in literature, we deﬁne an extreme El Niño event when the NINO3.4 index is greater than +2.0◦C and, accordingly, as extreme

La Niña event when is smaller than -1.5◦C (1973-1974, 1975-1976, 1988-1989).

It still remains to explain how and why the SST temperature in the warm pool rises up (or goes down) starting an ENSO event. The causes are, at the moment, unknown: an important source of variability seems to be the Madden-Julian Oscillation [13, 14, 13, 15]. Another source of variability is the Sun [2]: it is the climate "engine" of the Earth system and a variation of the terrestrial energetic balance (ocean + atmosphere) due to solar activity can explain part of the variability of the period and intensity of the ENSO events.

1.5. ENSO CHARACTERISTICS 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 T

Figure 1.12: Density of the mean value of the measured SST anomaly in the region NINO3.4, mediated over 3 months. Data obtained from NOAA [16]

Figure 1.13: Measured anomaly of the NINO3.4 index av eraged ov er 3 mon ths from 1950 to 2016.

**Chapter 2**

**From a deterministic system to a**

**generalized Fokker-Planck equation**

*In this chapter we introduce the projective method in the version developed in [17]. First,*
*we give a general description of the projective method from a general point of view. Then*
*we summarize the main step necessary to apply the technique on a generic system and we*
*focus on the necessary assumptions. Finally we give the general expression for the *
*Fokker-Planck-like equation which we will implement in the speciﬁc case of this thesis work in the*
*next chapter.*

**2.1 The projective method**

In this chapter, we summarize some recent works [17, 18] which are the basis of this thesis work. In a modern approach, based on the chaotic dynamic, the interconnection between statistical mechanics and hamiltonian deterministic mechanics can be put in relation to microscopical mixing phenomena which permit to deﬁne an equilibrium statistical distribution of the system. We are interested in systems separable in two parts: a part in a "full chaos" state, which satisﬁes the mixing condition needed to develop a thermodynamics (the booster), and a second part, the system of interest, coupled to the ﬁrst one, for which we aim to obtain a generalized Fokker-Planck equation. The solution of this equation is the canonical distribution of the system of interest and permits to obtain its thermodynamics. The booster in full chaos condition replaces the thermal bath of the classical approach to statistical mechanics but we do not introduce ad-hoc assumptions as the booster temperature, necessary in a classical approach to the canonical derivation of the thermodynamics. With a small number of hypothesis, it is possible to demonstrate [17] that a booster composed of a small number of variables in a full chaos state coupled to a small and ﬁnite system of interest is able to correctly produce the thermodynamics of it. An interesting feature of this approach is related to the linear response of the booster to an external perturbation: we require that only

the mean value of the variables of the booster respond linearly to perturbation, and not the variables themselves. This feature allows us to use a wider class of physical systems as boosters. The obtained generalized Fokker-Planck equation permits to read the deterministic system of interest in a stochastic way: if we ﬁne-tune the coupling constant between the booster and the system of interest it is possible [17, 18] to obtain the markovianicity of the stochastic process associated with the generalized Fokker-Planck equation. We stress also that we do not require an hamiltonian booster or an hamiltonian coupling: this approach remains valid also for non-hamiltonian systems.

**2.2 The Zwanzig projection**

The systems we are going to study are supposed to be separable in two sub-parts weakly
*interacting with each other. One of them is called the system of interest, for which we*
will derive an equation for the probability density in the phase space. The other part
*is called the booster, which represents the part of the system composed of the chaotic*
variables and substitutes the external thermal bath. In the rest of the thesis we will use
the bold font for vector quantities.

*˙x = v*
*˙v = −V*�* _{(x) − Γv − ξΔ}*
0

*g*�

**(x)***1*

**˙ξ = F (ξ, π, −Δ***h*

**(x))***1*

**˙π = Q (ξ, π, −Δ***h*(2.1) (2.2)

**(x))**

**Equations 2.1 represents the system of interest in variables x ≡ (x, y), while Equations**

**2.2 is a generic booster of arbitrary dimension in variables (ξ, π). If we assume an***unitary mass for the system of interest −Γv represents the intrinsic friction of the system*

*of interest and V*�

*(x) as a force. Obviously if we set to zero the (weak) coupling constants*

Δ0, Δ1*, the interaction between the two sub-parts of the system, mediated by ξ, vanishes*

and the two sub-parts decouple.

We introduce now a fundamental assumption which permits to deﬁne the Liouvillian operators and associate a phase space distribution probability density to the determin-istic system of interest:

**Assumption 1 We postulate the existence of inﬁnite copies of the system (booster +**

*system of interest) so, at each time, we can describe the state of the system with a*
**probability distribution σ(x, ξ, π) in phase space.**

The interaction term −Δ1*h (x) in the irrelevant part of the system (booster) may be*

2.2. THE ZWANZIG PROJECTION

allows to write an expression for the probability distribution of the booster dynamics:

*∂*

*∂t℘ (ξ, π; t) = M(ξ, π, −K(t))℘(ξ, π; t)* (2.3)

where M is the Liouvillian operator of the perturbed booster . If we suppose that the
*perturbation K(t) is small we can expand the Liouvillian operator in power series of K:*

*∂*

*∂t℘ (ξ, π; t) ≈ Lb℘(ξ, π; t) − K(t)W*1

*℘*(2.4)

**(ξ, π; t)**where W1 *is the contribute at the ﬁrst order in K.*

We explicit the form of the Liouvillian operator of the entire system:

L ≡ L*a*+ L*b*+ L*I* (2.5)

we have already deﬁned L*b*in Equation 2.4 as the Liouvillian operator of the unperturbed

booster, while L*a* is the Liovillian operator of the unperturbed system of interest and it

is given by
L*a* *≡ V*�*(x)*
*∂*
*∂v* *− v*
*∂*
*∂x* + Γ
*∂*
*∂vv* (2.6)

and ﬁnally L*I* is the Liouvillian of the interaction between the booster and the system

of interest and it can be expressed as
L*I* ≡ Δ0*g*�**(x)ξ**

*∂*

*∂v* + Δ1*h (x)W*1 (2.7)

Now we can introduce the Zwanzig projector [17] operator which projects the phase space of the entire system onto the subspace of the variables of the system of interest:

**P · · · = ℘**eq,b**(ξ, π)**

�

**dξdπ**_{· · ·} (2.8)

*where ℘eq,b* is the equilibrium probability distribution of the unperturbed booster. This

*operator permits to rewrite σ(x, v; t), which is the probability distribution of the system*
of interest, as

*σ(x, v; t) ≡* 1

*℘eq,b (ξ, π)Pρ(x, ξ, π) =*

�

* dξdπρ(x, ξ, π)* (2.9)

If we use the projector onto the evolution equation of the probability density of the entire system

and we consider a weak interaction between the two sub-parts of the system, we can
re-tain only terms up to the second order in the Liouvillian L*I*, of the interaction obtaining

[19, 20, 21]:
*∂*
*∂tσ(x, v; t) = Laσ(x, v; t) +*
�
1
*℘eq,b (ξ, π)*

**PL**

*I*

*℘eq,b*+ 1

**(ξ, π)***℘eq,b*

**(ξ, π)****PL**

*I*�

*t*0

*du*L0u

**(1 − P)e**_{L}

*I℘eq,b*−L0

**(ξ, π)e***u*�

*σ(x, v; t)*+ 1

*℘eq,b*

**(ξ, π)****PL**

*I*

*e*L0t

*+ 1*

**(1 − P)ρ(x, v, ξ, π; 0)***℘eq,b*

**(ξ, π)****PL**

*I*�

*t*0

*du*L0u

**(1 − P)e**_{L}

*Ie*−L0u

*L0t*

**(1 − P)e***ρ*(2.11) with L0 = L

**(x, v, ξ, π; 0)***a*+ L

*b*.

**2.3 Correlation function and booster response to**

**perturbations**

We introduce now the booster correlation function as

*ϕ(t) = �ξe*
L*bt _{ξ}*�

*b*

*�ξ*2�

*b*(2.12) characterized by

*τ*≡� ∞ 0

*ϕ(t)dt < ∞*(2.13)

*where the average �. . .�b* is calculated on the probability distribution of the unperturbed

booster. If we introduced some assumptions we could remove the inhomogeneous term from the Equation 2.11, and generalize the technique discarding the dependence on the initial conditions of the booster:

**Assumption 2 The correlation function of the variable ξ of the unperturbed booster**

*has a ﬁnite correlation time.*

**Assumption 3 The observation time scale of the system is much larger than the decay**

*time scale of the correlation function of the variable ξ of the unperturbed booster (which*
*is equal to the decay time of the equivalent stochastic system).*

2.3. CORRELATION FUNCTION AND BOOSTER RESPONSE TO PERTURBATIONS

*If we also redeﬁne ξ so that �ξ�b* = 0 we can rewrite (2.11) as

*∂*
*∂tσ(x, v; t) = Laσ(x, v; t)*
+
�
*∂*
*∂v*Δ
2
0*�ξ*2�*bg*�* (x)*
�

*t*0

*du*

*�ξe*L

*bu*�

_{ξ}*b*

*�ξ*2�

*b*

*e*L

*au*�

*g*�

**(x)***∂*

*∂v*�

*e*−L

*au*+

*∂*

*∂v*Δ0Δ1

*g*�

*�*

_{(x)}*t*0

*du�ξe*L

*bu*

_{W}1�

*b*

*e*L

*auh*−L

**(x)e***au*�

*σ(x, v; t)*(2.14)

*If a perturbation K(t) is applied at time t = 0, the average value of ξ, for t > 0, can be*

*approximated by a linear function of the perturbation K(t), through a time convolution*

*with a response function S(t) :*

*�ξ(t)�K* =

� *t*

0 *S(u)K(t − u)du + O(K*

2_{)} _{(2.15)}

In the special case where the external perturbation, abruptly applied to the booster
*at time t = 0, is constant, Equation 2.15 yields*

**Assumption 4**

*�ξ(t)�K* *= Kχ(t) + O(K*2) (2.16)

*where the function χ(t) deﬁned by*

*χ(t) ≡*

� *t*

0 *S(u)du* (2.17)

*is called susceptibility.*

*As a consequence of its deﬁnition, the susceptibility vanishes at t = 0, and we call*

*χ _{≡ χ(∞) its stationary asymptotic value χ(∞). Assuming that χ exists, it is natural}*

*to introduce another function, c(t), deﬁned by*

*χ(t) = (1 − c(t)) χ* (2.18)

*It is evident that c(0) = 1 and c(∞) = 0. From Equations 2.17 and 2.18 we have*

*S(t) =* *∂*

*∂tχ(t) = −χ*
*∂*

*∂tc(t)* (2.19)

It is important to point out that the trajectories of a chaotic booster that satisfy the assumption 3, with a correlation time equal to the decay time of the stochastic process, generally does not respond linearly to a perturbation. This phenomenon is not in contrast with assumption 4: the condition relates to the mean value of the coupling

*variable ξ of the booster and not the individual trajectories ξ(t).*

*Using the ﬁrst order expansion of Equation 2.4 in Equation 2.16, we get S(t) = −�ξe*L*bt*W_{1}�

*b*.

*Inserting this expression for S(t) in Equation 2.14, and exploiting Equation 2.12, we have*

*∂*
*∂tσ(x, v; t) = Laσ(x, v; t)*
+
�
*∂*
*∂v*Δ
2
0*�ξ*2�*bg*�* (x)*
�

*t*0

*du ϕ(u) e*L

*au*�

*g*�

**(x)***∂*

*∂v*�

*e*−L

*au*+

*∂*

*∂v*Δ0Δ1

*g*�

*�*

_{(x)}*t*0

*du χ*�

*∂*

*∂uc(u)*�

*e*L

*au*−L

_{h}**(x)e***au*�

*σ(x, v; t)*(2.20)

**2.4 The generalized Fokker-Planck equation**

With the use of the notion of Lie derivative it is possible to demonstrate [17]

*∂*
*∂tσ(x, v; t) =*
�
− *∂*
*∂xv+ V*
�_{(x)}*∂*
*∂v* + Γ
*∂*
*∂vv* +
*∂*
*∂vG(x, v)*
+ *∂*
*∂vA(x, v)*
*∂*
*∂v* +
*∂*
*∂vB(x, v)*
*∂*
*∂x*
�
*σ(x, v; t)* (2.21)
where
*G (x) = −Δ*0Δ1

*χ g*�

*− Δ0Δ1*

**(x)h(x)***χ g*�

*�*

**(x)**_{∞}0

*du c(u)*�

*∂*

*∂vh*�

**(x**a**(x; −u))***V*�

*(x)*+ Δ0Δ1

*χ g*�

*�*

**(x)**_{∞}0

*du c(u)*��

*∂*

*∂xh*� − Γ �

**(x**a**(x; −u))***∂*

*∂vh*��

**(x**a**(x; −u))***v*(2.22)

*�*

**and x**a**(x; −u) ≡***e*−L+*au x*�, i.e. the backward evolution.

*G (x) is a drift term for the velocity containing the average forces stemming from the*

interaction between the system of interest and the booster, while the diﬀusion coeﬃcients

*A(x, v) and B(x, v) are*
*A (x) = Δ*2

_{0}

*�ξ*2�

*bg*�

*�*

**(x)**_{∞}0

*du e*

*−Γu*�

_{ϕ}_{(u) (g}

_{◦ x}*a*�

**)(x; −u)***∂*

*∂xxa*�

**(x; −u)***B*2

**(x) = −Δ**_{0}

*2*

_{�ξ}_{�}

*bg*�

*�*

**(x)**_{∞}0

*du e*

*−Γu*�

_{ϕ}_{(u) (g}

_{◦ x}*a*�

**)(x; −u)***∂*

*∂vxa*� (2.23)

**(x; −u)**2.4. THE GENERALIZED FOKKER-PLANCK EQUATION
**with (s ◦ x**a**)(x; −u) ≡ s(x**a**(x; −u)) for a generic function s(x).**

*Generally the term B(x, v) is not equal to zero. This means that the Equation 2.21*
represents a second order hyperbolic partial diﬀerential equation. The theory produces
hyperbolic equation, not a parabolic one, as is the classical Fokker-Planck equation.
*However, from Equation 2.23 it is easy to see that the coeﬃcient B vanishes when there*
*is a very large separation between the decay time of ϕ(t) and the typical time scale of*
the unperturbed system of interest. In this case we reobtain a parabolic equation.

**Chapter 3**

**The projective method applied to**

**the LOM model**

*We introduce the low order model (LOM) used in literature to simulate the ENSO statistics,*
*the main properties of this last and, shortly, how this model can be derived. Then we use*
*the projective method, introduced in Chapter 2, to the LOM model and we derive the *
**Fokker-Planck equation, core of the thesis. We obtain some simple original results about the***moments of the PDF, solution of the associated Fokker-Planck equation and we introduce the*
*theoretical ansatz used to obtain the reduced PDF in the variable T .*

**3.1 LOM and ROM models**

Many models have been developed in order to simplify the ﬂuid dynamics equations which govern the atmospheric and oceanic processes, and to facilitate the study of ENSO and its statistics [22, 23, 24, 25]. Generally these simpliﬁcations lead to diﬀerential systems easier to solve, the so-called Low-Order Models (LOM) which contains the subclass of the Recharge Oscillator Models (ROM). The idea behind ROM models is that the heat content, namely the warm water volume over the entire tropical Paciﬁc, tends to accumulate gradually over time and during an El Niño event it is discharged. After this phase the tropical Paciﬁc becomes cold (La Niña events) and then warm water slowly starts to accumulate again. Up to now various triggers, which should kick oﬀ the heat release and therefore the El Niño phase, have been identiﬁed: recent studies [26, 27, 28] consider the Madden-Julian Oscillation (MJO) and the Westerly Wind Burst (WWB), already introduced in the Chapter 1, as possible triggers of the process. In this thesis we focus on a ROM model obtained for the ﬁrst time in [25] and later developed in details in [29] [13], which considers the MJO and WWB as possible triggers.

**3.2 The reference model**

As we have already said the ENSO process involves a thin layer of water, comprised
between the sea surface and the thermocline, straddling the Equator. The variables
*relevant in our description are the mean equatorial thermocline depth h and the mean*
*eastern sea surface temperature T .*

In order to address the problem in a rigorous way, we should start from the Navier-Stokes
equation in approximation of shallow water introducing some assumptions. This process
is described in details in [13, 29]: the various steps involve a lot of phenomenological
assumptions. Essentially, since El Niño and La El Niña cause small perturbations on
*the variables h and T (±1-2* ◦C on 20-27 ◦C) all the relations between the variables

are linearized and the proportional coeﬃcients are obtained ﬁtting the available data. The most relevant considerations are the following: we suppose the existence of a linear dependence between the temperature variation and the thermocline depth. This is possible because, in shallow water approximation, there is a linear dependency between the heat capacity and the sea surface temperature. We suppose also that an increase of the sea surface temperature causes an instant increase in equatorial winds which in turn increase linearly the depth of the thermocline. As we have already said Kelvin and Rossby waves transport heat and energy from side to side of the equatorial Paciﬁc ocean, and their not perfect reﬂection at the boundaries can be seen as one of the mechanism which dissipate the energy of the system. So we can write our model, based on the work [15]:

*˙h = −ωT*

*˙T = ωh − λT + Δ ξG(T)* (3.1)

The phenomenological parameters of the model are obtained ﬁtting the available
*data [30] obtaining ω = 2π/48 months*−1 *e λ = 1/6 months*−1.

*Δ ξG(T ) is a stochastic term, with intensity Δ, which forces the system and represents*
the MJO and WWB triggers, which randomly increase (or decrease) the SST, starting
*the ENSO events. The form of the function G(T ) determines the modulation of the*
*atmospheric stochastic processes and it is assumed to be G(T ) = 1 + βT . This choice of*

*G(T ) provides a multiplicative noise to the system which is regulated by the parameter*
*β* *= 0.2. In the original work of Jin et al. the variable ξ is a "red noise"*1 which try

to reproduce the randomness of the MJO and WWB processes but in our model in
*order to maintain a more generic approach the ξ variable is a source of "white noise".*
This choice permit us to include more triggers than the only MJO and WWB in the

1_{With red noise we mean a Wiener stochastic process W (t) which is characterized by the power }

*spec-trum S(f) =*��* _{�[FW (t)](f)}*��

_{�}2 =

*f*0

*f*2. The probability density function of a Wiener process is obtainable
*as solution of the pure diﬀusion Fokker-Planck equation ∂tp(x, t) =* 1_{2}*∂x*2*p(x, t).*

3.3. THE PROJECTIVE APPROACH APPLIED TO THE ROM SYSTEM

*stochastic force. Moreover, this allows us to consider the variable ξ as a component of*
a deterministic chaotic system representing the MJO and WWB processes whereas, in
the original approach, they are rendered with a stochastic process. In this way, keeping
in mind the theory developed in Chapter 2, we consider the ROM system as the system
of interest, for which we want to derive the probability density function, weakly coupled
*with a booster represented by the MJO and WWB processes through the variable ξ.*
The booster aﬀects the dynamics of the ROM system but we suppose that the dynamics
of the booster is not altered by the ROM one (Δ1 = 0 in Equation 2.7). Furthermore,

*our approach introduced by Jin et al. avoids the ﬁctitious request �ξ*2_{T}_{� = 0 which}

allows them to obtain a closed system of equation for the ﬁrst and second moments of the ROM. We can obtain then the stationary probability density function of the ROM system solving the Fokker-Planck equation, which we will derive in the next section, associated to the ROM system itself.

**3.3 The projective approach applied to the ROM**

**system**

*Following we proceed substituting (x, v) → (h, −ωT), Δ*0*g*�*(x, v) → ΔG(T), Γ → λ,*

*V*�*(x) → ω*2* _{h}* e Δ

_{1}

*= 0 into Equation 2.23 we obtain the diﬀusion coeﬃcients A(h, T ) e*

*B(h, T ) (while G(h, T ) is null) of the associated Fokker-Planck equation:*
*A(h, T ) = Δ*2* _{�ξ}*2

*×� ∞ 0*

_{� G(T )}*du e*

*−λu*

_{ϕ}(u)(G ◦ T*a)(h, T ; −u)*�

*∂*

*∂hha(h, T ; −u)*�

*B(h, T ) = −Δ*2

*�ξ*2

*� G(T )*× �

_{∞}0

*du e*

*−λu*

_{ϕ}_{(u)(G ◦ T}*a)(h, T ; −u)*�

*∂*

*∂Tha(h, T ; −u)*� (3.2)

The integration of the backward evolution of the unperturbed variables of a harmonic
*oscillator in h and T , Ta(h, T ; −u) ha(h, T ; −u) gives:*

*ha(h, T ; −u) = e*
*λ*
2*u*
�
*hcos(Ω u) −*
�
*λ*
2*h− ωT*
�* _{sin(Ω u)}*
Ω
�

*Ta(h, T ; −u) = e*

*λ*2

*u*�

*T*

*cos(Ω u) −*�

*ωh*−

*λ*

_{2}

*T*�

*sin(Ω u)*Ω � (3.3)

having deﬁned an actual frequency for the unperturbed system Ω ≡ �*ω*2− (*λ*

2)2. We

substitute (3.3) in (2.23) so we have a polynomial form for the diﬀusion coeﬃcients

*A(h, T ) = A*0*+ βA*1*h+ βA*2*T* *+ β*2*A*3*hT* *+ β*2*A*4*T*2

We do not write the expanded expressions for the coeﬃcients (they are available in [1])
because we are going to simplify them. Without losing the main statistical features of
*the system, as shown in [1], we consider the case in which the time scale τ of the MJO*
*and WWB is much smaller than the time scales of the ROM system 1/ω, 1/λ so the*
*coeﬃcients can be approximated to the second order in τ:*

*A*0 = Δ2*�ξ*2*�(τ − λη*2*) + O(τ*3) (3.5)
*A*1 = −Δ2*�ξ*2*�ω η*2*+ O(τ*3) (3.6)
*A*2 *= A*4*+ A*0 = Δ2*�ξ*2*�(2τ − λη*2*) + O(τ*3) (3.7)
*A*3 *= A*1 (3.8)
*A*4 = Δ2*�ξ*2*�τ + O(τ*3) (3.9)
*B*0 = −Δ2*�ξ*2*�η*2*+ O(τ*3) (3.10)
*B*1 *= O(τ*3) (3.11)
*B*2 *= 2B*0 (3.12)
*B*3 = 0 (3.13)
*B*4 *= B*0 (3.14)
hence
*A*= Δ2* _{�ξ}*2

*�*

_{� (1 + βT )}*τ(1 + βT ) − η*2

*(λ + βωh)*�

*+ O(τ*3) (3.15)

*B*= −Δ2

*2*

_{�ξ}*2*

_{� (1 + βT )}*η*2

*+ O(τ*3) (3.16)

*where η*2 _{is deﬁned as the ﬁrst moment of the autocorrelation function ϕ(t) of the}

*variable ξ which permits the coupling between the LOM and the chaotic atmospheric*
booster.

*The B coeﬃcient is diﬀerent from zero so the Equation 2.21, written in h e T , is*
a hyperbolic partial diﬀerential equation. If we consider only the terms containing the
*ﬁrst order in τ, the B coeﬃcients is zero and the coeﬃcient A becomes:*

*A= D(1 + βT )*2*+ O(τ*2) (3.17)

*with D ≡ Δ*2* _{�ξ}*2

_{� τ. Now the Equation 2.21 is a parabolic diﬀerential equation, a}Fokker-Planck with non constant coeﬃcients. Summarising: if we have a large time
scale separation between ENSO and the atmospheric forcing processes, the equation,
*core of the thesis work, which describe the evolution of the probability density σ(h, T ; t)*
*of the variable h and T relative to the ROM system which represents the ENSO process*
is:
*∂*
*∂tσ(h, T ; t) =*
*ωT* *∂*
*∂h* *− ωh*
*∂*
*∂T* *+ λ*
*∂*
*∂TT* +
*∂*
*∂TD(1 + βT )*
2 *∂*
*∂T*
*σ(h, T ; t)*
(3.18)

3.4. EQUIVALENT FORMS OF THE FOKKER-PLANCK EQUATION

**3.4 Equivalent forms of the Fokker-Planck equation**

In this section we rewrite the Equation 3.18 into diﬀerent forms which will be used later.
*We recall the deﬁnition of the diﬀusion coeﬃcient A(T ):*

*A= D(1 + βT )*2 (3.19)

We rewrite the equation in the canonical form of a Fokker-Planck equation and, from this
*one, we obtain the other forms (the indexes i and j represent the variable h and T , we*
also assume the Einstein convenction on repeated indices and we omit the dependence
*of σ(h, T ; t) on time and variables):*

*∂*
*∂tσ* *= −∂i*
�
J*iσ*
�
+ 1
2*∂i∂j*
�
K*ijσ*
�
(3.20)
having deﬁned the drift vector J and the diﬀusion matrix K

J =
�
*−ωT*
*ωh _{− λT + 2βD(1 + βT )}*
�
(3.21)
K =
�
0 0
0

*2D(1 + βT )*2 � (3.22) We can also deﬁne the operator L as

*L = ωT ∂h+ ∂T*
�
*− ωh + λT − 2βD (1 + βT )*
�
*+ ∂*2
*T*
�
*D* *(1 + βT )*2
�
(3.23)
and its adjoint

L† *= −ωT ∂h*+
�
*ωh _{− λT + 2βD (1 + βT )}*
�

*∂T*+ �

*D*

*(1 + βT )*2 �

*∂T*2 (3.24)

so that the Equation 3.18 can be rewritten in an operator form:

*∂tσ= Lσ* (3.25)

Furthermore we can associate [31] at the Fokker-Planck a stochastic diﬀerential equation obtained with the Ito integral calculus:

*d�x= � _{A(h, T )dt + B �}dw* (3.26)

with A = J and B = √K, since K is diagonal, B is the matrix with diagonal element equal to the root square of the diagonal element of K.

**3.5 Time evolution of the ﬁrst and second moment**

**of the distribution**

We recall that one of the reasons for which we applied the technique of the Chapter
2, to the ROM is the attempt to remove the ﬁctitious hypothesis of the cutoﬀ on the
moments introduced in [14]. In this section we will demonstrate that in our approach
*the moments of order k of the probability distribution can be obtained in a closed form*
as a combination of the moment with lower orders.

*The moments of order k function of time are obtainable, formally, solving the diﬀerential*
*equation obtained by applying the evolution operator (3.24) to monomials of degree k*
*of the variables h and T . Now we apply the evolution operator to h and T in order to*
obtain their average values (over the probability density function) in function of time:

*∂t�h(t)� = L*†*�h(t)�*

*∂t�T (t)� = L*†*�T (t)�*

(3.27) substituting the expression for (3.24) we obtain

˙

*�h(t)� = − ω �T (t)�*
˙

*�T (t)� =ω�h(t)� − (λ − 2β*2*D)�T(t)� + 2βD* (3.28)

*This is a closed linear ﬁrst order diﬀerential system for the average values of h and T ,*
*namely they do not depend on moments in h and T of higher degrees. If we substitute an*
equation in the other one (after a derivation in time) we can obtain a linear diﬀerential
*equation of the second order in T (for clarity, we omit the dependence on time):*

¨

*�T � + (λ − 2β*2*D*) ˙*�T � + ω*2*�T � = 0* (3.29)
We can identify it as the diﬀerential equation of a damped harmonic oscillator: this fact
should not surprise us because indeed the ROM system is a damped harmonic oscillator
coupled to a stochastic force, the booster.

*For some reasons, that will be explained later, we ﬁx D = 0.11 months so the harmonic*
oscillator is not overdamped and it has an eﬀective frequency of Ω = �*(λ − 2β*2_{D}_{)}2* _{− 4ω}*2

and a decay time of

*τ* = 2

*λ _{− 2β}*2

*D*(3.30)

In Figure 3.1 we can see the vector ﬁeld generated by the ﬁrst order diﬀerential system
of the ﬁrst moments. The diﬀerential ﬂux converge, spiraling, to a ﬁxed point in the
*(h, T )-plane which is the mean value of the equilibrium distribution. If we want to ﬁnd*

3.5. TIME EVOLUTION OF THE FIRST AND SECOND MOMENT OF THE DISTRIBUTION

this equilibrium point we can set to zero the time derivative in Equations 3.28 and solve the resulting system of algebric equations:

*�h�*_{∞}= −*2βD*

*ω*

*�T �*_{∞}= 0

(3.31) Quantities with an inﬁnity subscribe are equilibrium quantities.

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 -3 -2 -1 1 2 3 T h

Figure 3.1: Vector ﬁeld generated by the set of Equations 3.28. The black point in the chart is the ﬁxed point where the diﬀerential system converge for large times.

The case of higher moments (greater than 1) is complicated to solve because the diﬀerential equation, obtained combining the ﬁrst order diﬀerential equations of the system, requires for its resolution the roots of an high order polynomial and it is in general not analitically computable

*If we apply the adjoint operator to the three monomials of order two in h and T we*
can obtain the expression for the variance and covariance of the variables in function of
time:
˙
*�h*2*� = −2ω �hT �*
˙
*�hT � = ω�h*2*� − (λ − 2β*2*D)�hT� + 2βD�h�*
˙
*�T*2*� = 2ω�hT � − 2(λ − 3β*2*D)�T*2*� + 2βD + 8βD�T �*
(3.32)
We will solve this system via numerical integration and, as already done for the ﬁrst

moments, we calculate the stationary value for the second moments:
*�h*2�∞ =
*D*
*λ _{− 3β}*2

*D*+

*4β*2

*2*

_{D}*ω*2

*�hT �*

_{∞}= 0

*�T*2�

_{∞}=

*D*

*λ*2

_{− 3β}*D*(3.33)

The results obtained for ﬁrst and second moments, considering only the lower order
*in τ for the coeﬃcients A (see Equations 3.5 - 3.9), can be obtained, in a approximated*
*form, also without stopping to the ﬁrst order in τ as demonstrated in [1]. It is important*
to note that when the distribution reaches its stationary value, as well as its moments,
*the variables h and T have cross covariance equal to zero. Zero covariance could mean*
*that h and T are uncorrelated. This is the base hypothesis of an important assumption*
which we will explain in the next section of this chapter.

The set of Equations 3.32 depends only on the second moments themselves and on lower
*moments (ﬁrst order here). This dependence for moments of order k from moments of*
*order k themselves and lower moments derives from the peculiar form of the adjoint*
*operator: indeed it contains always, at the least, a derivative of order p times the same*
*variables at the same power p. This particular form of the operator does not permit*
to increase the order of the monomial, on which it is applied, beyond the order of the
*monomial itself. To demonstrate this fact we consider all the monomials of degree k*
*relative to the moment of order k of the distribution, and we applied the adjoint operator*
*to each one. We distinguish four cases (with 2 ≤ s ≤ k − 1):*

˙
*�hk _{� = −kω�h}k*−1

_{T}_{�}˙

*�hk*−1

*−2*

_{T}_{� = −(k − 1)ω�h}k*2*

_{T}*2*

_{� + (ω − λ + 2β}*−1*

_{D}_{)�h}k*−1*

_{T}_{� + 2βD�h}k_{�}˙

*�hk−s*+1

_{T}s_{� = −ω(k − s)�h}k−s−1_{T}s*+ Ds(s − 1)�hk−s*−2

_{T}s_{�+}+�

*ω− λ + 2β*2

*D+ Dβ*2

*(s − 1)*�

*s�hk−s*−1

_{T}s_{� + 2Dβs�h}k−s_{T}s_{�}˙

*�Tk*

_{� =}�

*ω*2

_{− λ + 2β}*D+ (k + 1)Dβ*2 �

*k*2

_{�T}k_{� + 2Dβk}*−1*

_{�T}k*−2*

_{� + Dk(k − 1)�T}k_{�}(3.34)

*We can see that every term in the equations contain variables up to order k, which*

*means that the evolution of every moments of order k is deﬁned by the evolution of the*

*moments of order k themselves and by the evolution of the moments of lower orders.*This result permit us to have a "closed" form for moments of every order in terms of moments already calculated without the introduction of an artiﬁcial "cut" as in the original approach in [30].

*3.6. THE ANSATZ*

**3.6 The ansatz**

**3.6 The ansatz**

In this section we try to obtain an exact solution of the Fokker-Planck equation in
*the variable T only, having integrated it on h. This reduced solution allows a better*
comparison between data and the model. We start integrating Equation 3.18 in variable

*h*, obtaining:
*∂*
*∂tσ(T ; t) = −*
*∂*
*∂Tω* *�h�(T ; t) +*
*λ*
*∂*
*∂TT* +
*∂*
*∂TD(1 + βT )*
2 *∂*
*∂T*
*σ(T ; t)* (3.35)

with the reasonable assumption lim*h*→±∞*σ(h, T ; t) = 0. This equation in T depends on*

*the mean value of the variable h in function of T which is not known. So, if we want*
to obtain an useful result we should introduce an assumption on the analytical form of
*the mean value of h in function of T . As we have noticed in the previous section of this*
*chapter the variables h and T have a zero cross correlation at the equilibrium, and we*
note now that the Equations 3.28 and 3.32 are the same that we can obtain for a forced
*damped harmonic oscillator (this is true if λ > 3β*2_{D}_{). These considerations lead us to}

*formulate an ansatz for the analytical form of �h�(T) in an equilibrium condition where*
moments are constants:

**Ansatz 1**
*�h�*_{∞}*(T ) = �h�*∞ *σ*∞*(T ) = −2β*
*D*
*ωσ*∞*(T )* (3.36)
*with �h�*_{∞}*(T ) =*�∞
∞ *h σ(h, T )*∞*dh* *and �h�*_{∞}=�∞∞*h σ(h, T )*∞*dh dT*.

*This ansatz binds the equilibrium mean value of h in function of T to the product of*
*the equilibrium mean value of h mediated also in T and the equilibrium distribution*
*integrated on h. Another target of the numerical simulations will be to understand if*
the ansatz is true or false and which is the approximation introduced.

*We have a formal expression for �h�*∞*(T ) and we can obtain a diﬀerential equation for*

*σ*∞*(T ) inserting the ansatz in (3.35) and removing the time derivative:*

*D(1 + βT )*2 *∂*

*∂Tσ*∞*(T ) + (2βD + λT )σ*∞*(T ) = 0,* (3.37)

This is a ﬁrst order diﬀerential equation with non constant coeﬃecient easy to
inte-grate:
*σ*_{∞}*(T ) ∝ exp*
�
*2 − µ*
*1 + βT*
�
*|1 + βT |−µ* *for T > −1/β*
*σ*_{∞}*(T ) = 0* *for T ≤ −1/β*
(3.38)
*with µ ≡ λ/(Dβ*2_{) > 3. The last condition, combined with the attempt to optimize}

*its width is a consequence of the strong dependence of the distribution width on the β*
parameter which governs the intensity of the multiplicative part of the stochastic forces.
*If β → 0 the distribution becomes a normal distribution instead if β �= 0 it has a power*
*law tail which permits big ﬂuctuations for positive T (El Niño events), while the cutoﬀ*
*in T = −*1

**Chapter 4**

**Numerical methods**

*In this chapter we introduce the numerical methods, based on [32, 33, 31], used to integrate*
*the Fokker-Planck equation and the associated stochastic equations. In the ﬁrst section we*
*introduce the notation and a theoretical framework to check the stability properties of the*
*method used to integrate the FPE. Then we introduce a simple one-dimensional integrator*
*which will be used as a base for the development of the two-dimensional integrator used in*
*this thesis work. Finally we introduce brieﬂy the theory of the SDE, the integration method*
*used and the technique necessary in crossing barrier problems to correct the ﬁniteness of the*
*sampling rate of the relevant stochastic trajectories.*

**4.1 Integrators for parabolic partial diﬀerential **

**equa-tion**

The integrator we are going to discuss in this section is the one that will be used to
integrate the partial diﬀerential Equation 3.18 which is a parabolic partial diﬀerential
*equation, in time and in two dimensions in the space (h, T ) with Dirichlet’s initial*
conditions.

**4.1.1 Notation and the von Neumann Linear Stability Analisys**

Before going into the description of the integrator, we introduce the notation used as
well as some useful concepts. We use the following notation: we discretize the
*coordi-nate space, here (x, y), with a square matrix of size N × N, on which the discretized*
*position on x is xi* *= x*0 *+ iΔx, on y is yj* *= y*0 *+ jΔy and the time t = t*0 *+ nΔt.*

The exact solution of the diﬀerential problem will be indicated with the capital letter

*Un*

We introduce now the following deﬁnitions:

**Deﬁnition 1 Convergence: A ﬁnite diﬀerence approximation converges to the solution**

*of a partial diﬀerential equation on 0 < t < T if |Un*

*i,j−uni,j| → 0 as Δt → 0, Δx, Δy → 0,*

*n* _{→ ∞ and nΔt < T for every i and j.}

This deﬁnition is equivalent to request that the solution of the ﬁnite diﬀerence approx-imation approaches the true solution of the PDE when the mesh is reﬁned. This is the main feature that we require to our integrator because it guarantees to obtain meaningful results.

**Deﬁnition 2 Stability: A ﬁnite diﬀerence approximation is stable if exists a positive**

*constant C such that |un*

*i,j| ≤ C |u*0*i,j| when Δt → 0, Δx, Δy → 0, n → ∞ and nΔt < T*

*for every i and j.*

This deﬁnition is equivalent to request that the numerical errors decay as the computa-tion proceeds from one step to the next. Now we can introduce the following fundamental theorem which permits to bind this two deﬁnition and permits, with the aid of the Von Neuman Stability, to demonstrate the convergence of our integrator:

**Theorem 1 Lax equivalence theorem: Given a diﬀerential problem and a ﬁnite **

*diﬀer-ences approximation of it, the approximation is convergent to the exact solution if, and*
*only if, it is stable and consistent*1_{.}

To satisfy the hypothesis of the stability of the numerical method, it will be used the Von Neumann Stability Analysis. This technique, used for our work in Chapter 5, allows us to determine if the chosen integration method is able to ”correctly” integrate each of the spatial Fourier components, and therefore, thanks to the superposition principle possessed by our diﬀerential problem, we can guarantee the stability for the entire solution. Let us then consider the two-dimensional wave function

*un _{j,l}*

*= ξneikxjΔx*(4.1)

_{e}ikylΔy*with kx* *e ky* *wave numbers, whereas ξn* *= ξ(kx, ky*) complex number. The application

of the scheme of integration to this wave produces a wave with the same wave number
*but with an amplitude ξn*+1. If the absolute value of the amplitude ratio *ξn*+1

*ξn* , known

as ampliﬁcation factor, is greater than one, the integrator will cause the exponential growth of the wave amplitude as the iterations increase in number, therefore it will not integrate it correctly. On the other hand, if the ampliﬁcation factor is smaller than 1, the integrator damps the wave as the number of iterations increases: again, it does not

1_{A ﬁnite diﬀerence approximation is considered consistent if, by reducing the mesh and time step}

4.1. INTEGRATORS FOR PARABOLIC PARTIAL DIFFERENTIAL EQUATION correctly reﬂect the behavior of the continuous diﬀerential problem, however this is not a problem if we want to obtain the equilibrium solution of the diﬀerential problem. The optimal behavior of the integrator is thus achieved when the ampliﬁcation factors is approximately 1 at the intersting scales for our problem, while still not exceeding 1 at the other ones.

**4.1.2 The ADI method**

To introduce the two-dimension integrator that will be used in the present work, we need to introduce an additional one-dimensional integrator which will be used as a guide for the development of our 2D method.

We consider the equation for a one-dimensional pure diﬀusion

*∂u*
*∂t* *= D*

*∂*2*u*

*∂x*2 (4.2)

We discretize the previous equation with an integrator obtained by averaging an explicit
integrator with an implicit2 _{one.}

*un _{i}*+1

*− un*

*i*

*Δt*=

*D*2

*(u*

*n*+1

*i*+1

*− 2uni*+1

*+ uni*−1+1

*) + (uni*+1

*− 2uni*

*+ uni*−1)

*(Δx)*2 (4.3)

We use the Von Neumann stability analysis to demonstrate that such integrator, called
*Crank-Nicolson, is stable for every value of Δt and Δx we chose. Replacing un*

*l* *= ξneiklΔx*

in (4.3) with simple calculation, we can obtain the ampliﬁcation factor

�
�
�
�
�
�
*ξn*+1
*ξn*
�
�
�
�
�
�=
*1 − 2D* *Δt*
*(Δx)*2 sin2(*kl*_{2})
*1 + 2D* *Δt*
*(Δx)*2 sin2(*kl*_{2})
(4.4)
*which is clearly less than 1 for every k, l, Δt and Δx. This method requires only to*
invert, on each integration step, a tridiagonal matrix, which is a relatively fast procedure
*O(N) in the matrix size.*

Now we can imagine to simply extend the Crank-Nicolson integrator to the case of a two-dimensional pure diﬀusion equation

*∂u*
*∂t* *= D*
*∂*2*u*
*∂x*2 +
*∂*2*u*
*∂x*2
(4.5)

2_{An explicit methods calculates the solution at the time n + 1 from the solution at the current time}

*n, while an implicit methods ﬁnds a solution at the time n + 1 by solving an equation involving both*

*and substitute the operators δt* *δx*2 *and δy*2 *on the solution uni,j* as
*ut*→
*un _{i,j}*+1

_{− u}n*i,j*

*Δt*

*≡ δt[uni,j*]

*uxx*→

*un*

*i+1,j− 2uni,j*

*+ uni−1,j*

*(Δx)*2

*≡ δx*2

*[uni,j*]

*uyy*→

*un*

*i,j*+1*− 2uni,j* *+ uni,j*−1

*(Δy)*2 *≡ δy*2*[uni,j*]

(4.6)

*having deﬁned r = DΔt and supposing, without loss of generality that Δx = Δy. The*
simplest extension we can imagine for the two-dimensional Crank-Nicolson integrator
for the Equation 4.5 is

1 − *r*
2*δx*2−
*r*
2*δy*2
*un _{ij}*+1 =
1 +

*r*2

*δx*2+

*r*2

*δ*2

*y*

*un*(4.7)

_{ij}It is possible to show [34], using the Von Neumann stability analysis extended to the
two-dimensional case, that the method of order O�*(Δx)*2* _{+ (Δy)}*2

*2�*

_{+ (Δt)}_{just introduced}

is stable thus it is optimal for the integration of a two-dimensional parabolic PDE.
Nonetheless this applies just for a "theoretical" approach because this integrator would
be very ineﬃcient. The combinations of operators shown in brackets in (4.7) are matrices
*of size N*2 * _{× N}*2

*+1 we need to invert*

_{However, to solve the linear problem (4.7) in u}nthem. For tridiagonal matrices there are algorithms that permit the inversion in a total
*number of elementary operations equal to O(M) (with M × M size of the matrix). In*
the case of generic matrices (or band matrices) it is not known an algorithm with such
*eﬃciency and the only ones known have an eﬃciency equal to O(M*2* _{) or even O(M}*3

_{).}

So we try to obtain a new integrator that generalizes the Crank-Nicolson, making it
stable and with an error of order O�*(Δx)*2* _{+ (Δy)}*2

*2�*

_{+ (Δt)}_{, but also computationally}

eﬃcient.

Let us start to notice that the application of
*(Δt)*2

4 *δx*2*δy*2*δt[uni,j*] (4.8)

gives a quantity of order of the integrator (4.7) so adding it the order of the integrator does not change, and we obtain:

1 −*r*
2*δ*2*x*−
*r*
2*δ*2*y*+
*r*
2*δx*2
*r*
2*δy*2
*un _{ij}*+1 =
1 +

*r*2

*δx*2+

*r*2

*δ*2

*y*+

*r*2

*δx*2

*r*2

*δy*2

*un*(4.9) factorizing it we obtain 1 −

_{ij}*r*2

*δy*2 1 −

*r*2

*δx*2

*un*+1 = 1 +

_{ij}*r*2

*δx*2 1 +

*r*2

*δ*2

*y*

*un*(4.10)

_{ij}4.2. INTEGRATOR FOR STOCHASTIC DIFFERENTIAL EQUATIONS (SDE) which can be written as

1 − *r*
2*δx*2
*u*∗* _{ij}* =
1 +

*r*2

*δy*2

*un* 1 −

_{ij}*r*2

*δy*2

*un*+1 = 1 +

_{ij}*r*2

*δ*2

*x*

*u*∗

*(4.11)*

_{ij}This integrator is called Alternating Direction Implicit (ADI) method. This name
de-rives from the fact that each integration step can be divided in two phases: during the
ﬁrst phase, of the ﬁrst half step, we multiply a tridiagonal operator and the solution.
*This is equivalent to making N products matrix-vector between a tridiagonal matrix, of*
*size N ×N which operates onto elements of the grid at ﬁxed x, and a "slice" of the spatial*
*grid relative to the same ﬁxed x. In the second phase of the ﬁrst half step we solve the N*
*implicit systems in the x direction using the implicit x operator onto the "slices" of the*
*grid at ﬁxed y, and so we obtain the intermediate solution u*∗

*ij*. In the second half step,

we apply to this last the same procedure but we exchange operators (matrix product
*with ﬁxed y, solving with ﬁxed x). The fact that on each phase we have to solve N*
systems of tridiagonal linear equations permits us to use the rapid tridiagonal matrix
solving algorithms and to parallelize the code on a parallel CPU architecture. Obviously
this integrator works for every diﬀerential problem discretized to the ﬁrst order in the
*time step, with operator δ*2

*x, δy*2 of second order in their respective increment which can

be posed in a tridiagonal form. This permit us to use the integrator developed in this section also for our diﬀerential problem (3.18) on the condition that we descretized it respecting the above prescriptions.

**4.2 Integrator for stochastic diﬀerential equations**

**(SDE)**

We introduce in this section the integrator that will be used in the following part of this work to integrate the system of stochastic diﬀerential equations equivalent to the Fokker-Planck. First order stochastic diﬀerential equations has the form

*dx= f(x, t)dt + g(x, t)dW (t)* (4.12)

*with W (t) a Wiener process.*

We can formally integrate the (4.12) and obtain

*x(t) = x(t*0) +
� *t*

*t0f(x(s), s)ds +*

� *t*