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
1El 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
2Teleconnection 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.
3The 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.
4Kelvin 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.
5The 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 0g�(x) ˙ξ = F (ξ, π, −Δ1h(x)) ˙π = Q (ξ, π, −Δ1h(x)) (2.1) (2.2) 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 −Δ1h(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)W1℘(ξ, π; t) (2.4)
where W1 is the contribute at the first order in K.
We explicit the form of the Liouvillian operator of the entire system:
L ≡ La+ Lb+ LI (2.5)
we have already defined Lbin Equation 2.4 as the Liouvillian operator of the unperturbed
booster, while La is the Liovillian operator of the unperturbed system of interest and it
is given by La ≡ V�(x) ∂ ∂v − v ∂ ∂x + Γ ∂ ∂vv (2.6)
and finally LI is the Liouvillian of the interaction between the booster and the system
of interest and it can be expressed as LI ≡ Δ0g�(x)ξ
∂
∂v + Δ1h(x)W1 (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 LI, of the interaction obtaining
[19, 20, 21]: ∂ ∂tσ(x, v; t) = Laσ(x, v; t) + � 1 ℘eq,b(ξ, π)PLI ℘eq,b(ξ, π) + 1 ℘eq,b(ξ, π)PLI � t 0 du(1 − P)e L0uL I℘eq,b(ξ, π)e−L0u � σ(x, v; t) + 1 ℘eq,b(ξ, π)PLI eL0t(1 − P)ρ(x, v, ξ, π; 0) + 1 ℘eq,b(ξ, π)PLI � t 0 du(1 − P)e L0uL Ie−L0u(1 − P)eL0tρ(x, v, ξ, π; 0) (2.11) with L0 = La+ Lb.
2.3 Correlation function and booster response to
perturbations
We introduce now the booster correlation function as
ϕ(t) = �ξe Lbtξ� b �ξ2�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) + � ∂ ∂vΔ 2 0�ξ2�bg�(x) � t 0 du �ξeLbuξ� b �ξ2�b eLau � g�(x) ∂ ∂v � e−Lau + ∂ ∂vΔ0Δ1g �(x)� t 0 du�ξe LbuW 1�b eLauh(x)e−Lau � 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(K2) (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) = −�ξeLbtW1�
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) + � ∂ ∂vΔ 2 0�ξ2�bg�(x) � t 0 du ϕ(u) e Lau � g�(x) ∂ ∂v � e−Lau + ∂ ∂vΔ0Δ1g �(x)� t 0 du χ � ∂ ∂uc(u) � eLauh(x)e−Lau � 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) = −Δ0Δ1χ g�(x)h(x) − Δ0Δ1χ g�(x) � ∞ 0 du c(u) � ∂ ∂vh(xa(x; −u)) � V�(x) + Δ0Δ1χ g�(x) � ∞ 0 du c(u) �� ∂ ∂xh(xa(x; −u)) � − Γ � ∂ ∂vh(xa(x; −u)) �� v (2.22) and xa(x; −u) ≡ �
e−L+aux�, 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) = Δ20�ξ2�bg�(x) � ∞ 0 du e −Γuϕ(u) (g�◦ x a)(x; −u) � ∂ ∂xxa(x; −u) � B(x) = −Δ20�ξ2�bg�(x) � ∞ 0 du e −Γuϕ(u) (g�◦ x a)(x; −u) � ∂ ∂vxa(x; −u) � (2.23)
2.4. THE GENERALIZED FOKKER-PLANCK EQUATION with (s ◦ xa)(x; −u) ≡ s(xa(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
1With red noise we mean a Wiener stochastic process W (t) which is characterized by the power
spec-trum S(f) =���[FW (t)](f)���2 = f0
f2. The probability density function of a Wiener process is obtainable as solution of the pure diffusion Fokker-Planck equation ∂tp(x, t) = 12∂x2p(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 �ξ2T� = 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), Δ0g�(x, v) → ΔG(T), Γ → λ,
V�(x) → ω2h e Δ1 = 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 ) = Δ2�ξ2� G(T ) ×� ∞ 0 du e −λuϕ(u)(G ◦ T a)(h, T ; −u) � ∂ ∂hha(h, T ; −u) � B(h, T ) = −Δ2�ξ2� G(T ) × � ∞ 0 du e −λuϕ(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 2u � hcos(Ω u) − � λ 2h− ωT �sin(Ω u) Ω � Ta(h, T ; −u) = e λ 2u � T cos(Ω u) − � ωh− λ2 T � sin(Ω u) Ω � (3.3)
having defined an actual frequency for the unperturbed system Ω ≡ �ω2− (λ
2)2. We
substitute (3.3) in (2.23) so we have a polynomial form for the diffusion coefficients
A(h, T ) = A0+ βA1h+ βA2T + β2A3hT + β2A4T2
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:
A0 = Δ2�ξ2�(τ − λη2) + O(τ3) (3.5) A1 = −Δ2�ξ2�ω η2+ O(τ3) (3.6) A2 = A4+ A0 = Δ2�ξ2�(2τ − λη2) + O(τ3) (3.7) A3 = A1 (3.8) A4 = Δ2�ξ2�τ + O(τ3) (3.9) B0 = −Δ2�ξ2�η2+ O(τ3) (3.10) B1 = O(τ3) (3.11) B2 = 2B0 (3.12) B3 = 0 (3.13) B4 = B0 (3.14) hence A= Δ2�ξ2� (1 + βT )�τ(1 + βT ) − η2(λ + βωh)�+ O(τ3) (3.15) B = −Δ2�ξ2� (1 + βT )2η2+ O(τ3) (3.16)
where η2 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(τ2) (3.17)
with D ≡ Δ2�ξ2� 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 � Jiσ � + 1 2∂i∂j � Kijσ � (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 � ∂T2 (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 − 2β2D)�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 − 2β2D) ˙�T � + ω2�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 − 2β2D)2− 4ω2
and a decay time of
τ = 2
λ− 2β2D (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: ˙ �h2� = −2ω �hT � ˙ �hT � = ω�h2� − (λ − 2β2D)�hT� + 2βD�h� ˙ �T2� = 2ω�hT � − 2(λ − 3β2D)�T2� + 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: �h2�∞ = D λ− 3β2D + 4β2D2 ω2 �hT �∞ = 0 �T2�∞ = D λ− 3β2D (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�hk−1T� ˙ �hk−1T� = −(k − 1)ω�hk−2T2� + (ω − λ + 2β2D)�hk−1T� + 2βD�hk−1� ˙ �hk−sTs� = −ω(k − s)�hk−s−1Ts+1+ Ds(s − 1)�hk−sTs−2�+ +�ω− λ + 2β2D+ Dβ2(s − 1) � s�hk−sTs� + 2Dβs�hk−sTs−1� ˙ �Tk� = � ω− λ + 2β2D+ (k + 1)Dβ2 � k�Tk� + 2Dβk2�Tk−1� + Dk(k − 1)�Tk−2� (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
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 limh→±∞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 > 3β2D). 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/(Dβ2) > 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 = x0 + iΔx, on y is yj = y0 + jΔy and the time t = t0 + 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 |u0i,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 consistent1.
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
unj,l = ξneikxjΔxeikylΔy (4.1)
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
1A 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
∂2u
∂x2 (4.2)
We discretize the previous equation with an integrator obtained by averaging an explicit integrator with an implicit2 one.
uni+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(kl2) 1 + 2D Δt (Δx)2 sin2(kl2) (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 ∂2u ∂x2 + ∂2u ∂x2 (4.5)
2An 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 δx2 and δy2 on the solution uni,j as ut→ uni,j+1− un i,j Δt ≡ δt[uni,j] uxx → un i+1,j− 2uni,j + uni−1,j (Δx)2 ≡ δx2[uni,j] uyy → un
i,j+1− 2uni,j + uni,j−1
(Δy)2 ≡ δy2[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 2δx2− r 2δy2 unij+1 = 1 + r 2δx2+ r 2δ2y unij (4.7)
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+ (Δt)2� 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 N2 × N2 However, to solve the linear problem (4.7) in un+1 we need to invert
them. 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(M2) or even O(M3).
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+ (Δt)2�, but also computationally
efficient.
Let us start to notice that the application of (Δt)2
4 δx2δy2δt[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 2δ2x− r 2δ2y+ r 2δx2 r 2δy2 unij+1 = 1 + r 2δx2+ r 2δ2y+ r 2δx2 r 2δy2 unij (4.9) factorizing it we obtain 1 − r 2δy2 1 − r 2δx2 unij+1 = 1 + r 2δx2 1 + r 2δ2y unij (4.10)
4.2. INTEGRATOR FOR STOCHASTIC DIFFERENTIAL EQUATIONS (SDE) which can be written as
1 − r 2δx2 u∗ij = 1 + r 2δy2 unij 1 − r 2δy2 unij+1 = 1 + r 2δ2x u∗ij (4.11)
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 δ2
x, δy2 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(t0) + � t
t0f(x(s), s)ds +
� t
