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ENVIRONMENTAL SCIENCE: PHYSICAL PRINCIPLES

AND APPLICATIONS

A. Speranza, Department of Mathematics and Computer Science, University of

Camerino, Italy.

V. Lucarini, Department of Mathematics and Computer Science, University of

Camerino, Italy.

PACS:

02.50.-r Probability theory, stochastic processes, and statistics

02.60.-x Numerical approximation and analysis

02.70.-c Computational techniques

05.45.-a Nonlinear dynamics and nonlinear dynamical systems

47.27.-I Turbulent flows, convection, and heat transfer

47.32.-y Rotational flow and vorticity

92.10.-c Physics of the oceans

92.40.-t Hydrology and glaciology

92.60.-e Meteorology

92.70.-j Global change

Links to other articles: Computer simulation techniques, Numerical approximation

and analysis, Nonlinear dynamics and nonlinear dynamical systems

Accepted for publication as article of the Encyclopedia of Condensed Matter Physics, Elsevier, Eds: F.

Bassani, J. Liedl, P. Wyder

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Introduction

Environmental science almost invariably proposes problems of extreme complexity,

typically characterized by strongly nonlinear evolution dynamics. The systems under

investigation have many degrees of freedom - which makes them complicated – and

feature nonlinear interactions of several different components taking place on a vast

range of time-space scales – which makes them complex. Such systems evolve under

the action of macroscopic driving (typically the solar heating) and modulating (e.g.

the Earth’s rotation and gravitation) agents. The most comprehensive example is the

entire climatic system. In its most rigorous definition, the climatic system is

constituted by four intimately interconnected sub-systems, atmosphere, hydrosphere,

cryosphere, and biosphere (figure 1), and is powered by the electromagnetic radiation

emitted by the Sun (figure 2). These subsystems are by themselves complicated and

complex and interact nonlinearly with each other on various time-space scales.

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Figure 1: Components of the climatic system; the main interactions are also indicated. From

IPCC 2001.

Moreover, environmental sciences are not usually provided with real

laboratories where theories can be tested against experiments, since phenomena often

take place only once and cannot be reproduced.

Theories can often be tested only against observational data from the past,

which may feature problems of various degrees of criticality, essentially because of

the physical extension of the systems under analysis. In many cases, the available

observations feature a relatively low degree of reciprocal synchronic coherence and

individually present problems of diachronic coherence, due to changes in the

strategies of data gathering with time. The presence of strong variability of both the

outputs of theoretical models and of the real systems contributes to blur the line

between a failed and a passed test, in terms of model reliability.

It is important to emphasize that the theoretical models unavoidably suffer of

two well-distinct kinds of uncertainties. The uncertainties in the initial conditions of

the systems, termed uncertainties of the first kind, are problematic because of the

effects of chaos. These uncertainties may be partially dealt with using well-suited

Monte Carlo approaches on the initial conditions.

The uncertainties of the second kind are structural uncertainties, due to the

simplifications adopted in the models for the description of the processes and of the

feedbacks characterizing the system under investigation. These latter uncertainties

seem to be somehow more fundamental – the dynamic framework is uncertain - and

harder to deal with.

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Figure 2: Energy fluxes through the atmosphere - in units of Wm-2. From IPCC 2001.

Therefore, it is conceptually incorrect to expect that the essentially

phenomenological theories which have been developed for the environmental

sciences could provide answers having comparable precision and similar structure to

those provided by theories relevant for the study of systems having lower degree of

complexity.

In view of the above difficulties, it is not surprising that the historical

development of Environmental Science took place essentially along two different

main paths: on the one side the descriptive, basically qualitative, approach of

naturalistic tradition, on the other the quantitative formulation of physical-

mathematical tradition. Linneo’s classification of living beings and Laplace tidal

equations are classical examples. The synthesis between the two approaches took

place in the nineteenth and twentieth centuries mostly through applications of fluid

dynamics and/or thermodynamics to specific systems. However, a certain separation

(sometimes even contradiction) between the two approaches occasionally still

emerges today. Meteorological maps, interpreted on the basis of the analogies of the

so-called synoptic Meteorology, are examples of the modern evolution in the

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descriptive tradition, while numerical weather forecasting exemplifies the quantitative

one.

The description of the macroscopic dynamics of environmental systems is

based on the systematic use of dominant balances derived on a phenomenological

basis in order to specialize the dynamical equations. Such balances are suitable

classes of approximate solutions of the evolution equations which represent a

reasonably good approximation to the actual observed fields when sufficiently large

spatial or temporal averages are considered. Actually, different balances have to be

considered depending on the time and space scales we are focusing our interest on.

Such an approach reflects the fundamentally heuristic-inductive nature of the

scientific research in environmental sciences, where the traditional reductionistic

scientific attitude is not always effective. In order to exemplify this procedure, we

consider the very relevant case of the motion of the fluids that permit the existence of

life on the Earth, air and water: the so-called geophysical fluids.

Geophysical fluids and phenomenological balances

Geophysical fluid systems are very complex in microphysical structure and

composition and evolve under the action of macroscopic driving (solar heating) and

modulating (Earth’s rotation and gravitation) agents. The complexity essentially

depends on the presence of nonlinear interactions and feedbacks between the various

parts of the system, which couple very different time and space scales. In many cases,

the dynamics of such systems is chaotic (the self-correlation of the fields vanishes

within a finite time domain) and is characterized by a large natural variability on

different time scales.

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The dynamics of a geophysical fluid for an observer in a uniformly rotating

frame of reference is described by the Navier-Stokes equation:

(1) Fpu

dt

ud +Φ∇−∇−=×Ω+

ρρρ

2

where is the density of the fluid,

(

)

w,v,uu = is the velocity vector,

Ω

is the Earth’s

rotation vector, p is the pressure, is the geopotential (essentially coincident with the

gravitational potential), F is the frictional force per unit mass, and the total derivative

is expressed as:

(2) ∇⋅+

∂

∂

≡u

t

dt

d,

thus including the nonlinear advection term.

Through the use of suitable approximations, the equations of motion can be

focused upon the desired components of the flow. The filtering process is the

introduction of a set of mathematical approximations into the Navier-Stokes equation

having the scope of filtering out (excluding) solutions corresponding to the physical

processes that are heuristically assumed to contribute only negligibly to the dynamics

of the system at the time and space scale under examination. The magnitudes of

various terms in the governing equations for a particular type of motion are estimated

using the so-called scale analysis technique.

Hydrostatic balance

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As a relevant example of these procedures, we consider the hydrostatic

approximation. In a local Cartesian coordinate system where z is perpendicular to the

surface an obvious fixed point (stationary solution) of the Navier-Stokes equation can

be derived from the time-independent classical hydrostatic equilibrium equations:

(3)

(

)

0

0,0,0

=

∂

∂

−−

=

z

p

g

u

h

h

ρ

where the functions

(

)

tz,y,x,

h

ρ

and

(

)

tz,y,x,ph satisfying the above balance (the

subscript h stands for “hydrostatic”) depend only on the spatial variable z. Three

points related to the above described balance are worth mentioning explicitly:

• the fluid velocity is assumed negligible everywhere in the system;

• the condition of hydrostatic equilibrium does not determine univocally the

thermodynamic fields

(

)

z

h

ρ

and

(

)

zph but only their relationship; other

information is requested in order to determine univocally the profiles of density,

pressure, temperature;

• the balance is observable (and practically useful!) only if it is stable.

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Now, the interesting point is that quite different geophysical flows (atmosphere,

ocean, surface and, ground water, etc.) can be represented as being in near hydrostatic

equilibrium when suitable spatial and temporal averaging process is considered.

Figure 3: Typical temperature vertical profile of the atmosphere. The main sections of the

atmosphere are indicated and the small extent of the quasi-constant temperature sections is

emphasized.

In general terms, hydrostatic equilibrium is realized when the portions of a

fluid with lower specific entropy are positioned below those with higher specific

entropy, where directions are defined by the local gravity force vector. When this

condition is broken because of an external forcing, the stratification is not stable and

the fluid rearranges so that hydrostatic condition is re-established. Therefore,

violations of hydrostatic equilibrium exist only on short time and space scales and

often are not easily observable. Typical models are formulated in such a way that

local non-hydrostatic conditions are quickly eliminated and the equilibrium condition

that is recovered is parameterized in terms of variables explicitly represented on the

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numerical grids. In a later section we will briefly describe the general scope of the

parameterization procedures.

As a matter of fact, non-hydrostatic models are nowadays available and are

currently used. However, for any practical purpose, sufficiently detailed initial and

boundary conditions are not available and the above expressed methodological

problems reappear in connection with their definition.

Geostrophic balance

Another instructive example is that of time-independent purely horizontal balanced

flows where the horizontal components of the pressure gradient force and the Coriolis

force cancel out in the Navier-Stokes equation. Such flows are termed geostrophic,

which, etymologically, means Earth-turning. The structure parameter determining

qualitatively the goodness of such approximation is the Rossby number Ro=U/f0L

where U is a characteristic horizontal velocity of the fluid, L is a characteristic

horizontal extension of the fluid and f0 is twice the value of the projection of Earth

angular velocity vector on the plane tangent to the surface of the Earth at the

considered latitude. This approximation holds for relatively large scale extratropical

flows in the in the regions where friction is not important – i.e. away from the

boundaries of the fluid. Geostrophic equations are obtained as a zeroth-order Ro

expansion of the Navier-Stokes equation. The fluid motion is introduced by

considering small perturbations breaking the translation symmetry of the baseline

purely hydrostatic density and pressure fields. This is achieved by assuming that the

actual density and pressure fields are given by

(

)

(

)

(

)

zzy,x,zy,x, hg

ρρρ

+= and

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(

)

(

)

(

)

zpzy,x,pzy,x,p hg += respectively. Geostrophic equations thus result to be the

following:

(4)

z

p

g

y

p

vf

y

p

uf

g

g

g

h

g

h

∂

∂

−=

∂

∂

=

∂

∂

−=

ρ

ρ

ρ

0

0

Geostrophic and hydrostatic balance constrain atmospheric motions so that the fluid

flows are uniquely determined by the pressure field, since the currents are parallel

(rather then perpendicular as in non-rotating fluids) to the isobars at a given

geopotential height. The time-independent nature of these equations implies that the

non-geostrophic terms, although relatively small, are important for the time-evolution

of the flow.

Figure 4: Geostrophic balanced flow; pressure gradient force (upper arrow) and Coriolis force

(lower arrow) cancel out and the flow (horizontal arrow) is parallel to the isobars.

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A system or flow that evolves slowly in time compared to 1/f0 can be

described using the so called quasi-geostrophic theory. This is a perturbative theory

obtained as a first-order Ro filtering of the Navier-Stokes equation and analyzes

departures from geostrophic balance. The quasi-geostrophic approximation is used in

the analysis of extratropical systems, in which currents can be closely approximated

by their geostrophic values. The use of the quasi-geostrophic approximation

effectively filters out solutions corresponding to high-speed atmospheric gravity

waves. This approximation is not accurate in situations in which non-geostrophic

currents play an important advective role, e.g., around the frontal discontinuities.

Although modern computers allow a fully non-geostrophic description of most

geophysical fluids, quasi-geostrophic approximation remains a fundamental tool of

theoretical research and is used for practical applications in everyday life: every time

we read an atmospheric isobaric map in terms of wind flowing along the isobars we

are, in fact, using the notion of geostrophic balance.

Applications of Geophysical Fluid Dynamics: weather and climate

Given the nature and the impact on human society of their research, environmental

scientists have had access to powerful computers since the early days of numerical

computing. Actually, in the late ‘40s, the first large-scale application of automatic

computing consisted in the first numerical weather forecast, based on greatly

simplified equations, which was proposed by Von Neumann and mainly devised by

Charney. Since the late ‘50s, the US technical services have been using computer-

assisted numerical integration of relatively accurate equations descriptive of the

physics of the atmosphere to routinely produce publicly available weather forecasts.

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Geophysical fluids numerical models are usually divided into limited area and

global models, the fundamental difference being that the former require lateral

boundary conditions (at the boundaries of the integration domain) that have to be

continuously provided by either observations or, more typically, larger-scale models,

while the latter essentially perform quasi-autonomous integrations - obviously, still

there is external forcing. In geophysical fluids model, the three-dimensional fields are

discretized on a lattice, and the temporal evolution is discretized through the adoption

of a time-step. The adoption of a finite spatial and temporal resolution implies that the

physical processes occurring on a space and/or time scale smaller than the actual

resolution of the model can be taken care of only with approximate parameterizations

relating such processes to coarser grain information. The parameterization of the so-

called subgrid processes is usually heuristic and devised ad-hoc by statistical

interpretation of observed or model-generated data.

Numerical modeling options strongly rely on the available computer power, so

that the continuous improvements in both software and hardware have permitted a

large increase in the performances of the models and at the same time an impressive

widening of their horizons. On one side, the adoption of finer and finer resolutions has

allowed a more detailed description of the large scale features of the dynamics, and,

more critically, a more direct physical description of a larger set of processes, thus

limiting the need for parameterization procedures, which, where needed, have become

more accurate. On the other side, it has been possible to implement and then refine the

coupling between models pertaining to different systems having a common boundary,

such as the atmosphere and the ocean. The main purpose of such procedure is to study

in detail the time-space scales of the interactions between the different systems and

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reframe such interactions in the more intrinsic terms of internal feedbacks of the

compounded system.

Today, computer simulations probably constitute the best laboratory

instruments for environmental scientists, since on one side permit the testing of

theories, at least in the previously exposed weak sense, and on the other side provide

stimulations for formulating new hypotheses. It is notable that the present most

powerful computer of world has been engineered in order to simulate geophysical

fluid dynamics with the particular scope of providing the most detailed simulation of

the present and past climate.

When considering complex systems like those investigated by the

environmental sciences, there is a clear distinction between providing efficient

descriptions of the local and of the global properties in the phase space of the system.

The local properties of the phase space are addressed by considering dynamic

equations able to represent the short-time deterministic evolution of the system. The

study of the global properties of the phase space entails considering dynamic

equations whose goal is to represent the statistical properties of the system. The

efficient description of such properties requires very long integrations of the

equations. In order to clarify these well-distinct - albeit intersecting - perspectives, we

will specifically discuss the problem of weather forecast and of climate simulation.

Weather forecast

The task of weather forecast is to provide the best description of the short-time

evolution of the relevant fields descriptive of the state of the atmosphere – wind,

temperature, pressure, humidity, precipitation – as represented on a 3D lattice.

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Weather forecast is addressed to describing and predicting the instantaneous values of

the main atmospheric variables. Weather forecast models use in the equations quite a

number of ad hoc assumptions and parameterizations that have empirically proved

their reliability for that purpose. Since the aim of these models is to be as precise as

possible in the local sense, they do not necessarily need to obey global constraints,

e.g. energy conservation, which are obviously fundamental in terms of Physics, but

not necessarily relevant when a small neighborhood of the phase space is considered.

Figure 5: Scheme of the loss of predictability for ensemble forecasting. The initial Gaussian

distribution of the ensemble (1) is distorted (2), and then loses its Gaussian shape (3); eventually

the ensemble points populate the attractor of the system.

The time range of such forecasts extends up to the deterministic predictability horizon

of the atmosphere, which can be estimated in about 7-10 days. It is important to

emphasize that the actual spread of the model predictions can be estimated by using

Monte Carlo techniques on the initial state of the system. Such approach is termed

ensemble forecasting and relies on sampling the phase space of system by integrating

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forward in time a set of initial conditions. The initial conditions are suitably selected

in order to represent the unavoidable uncertainty on the initial state of the system.

Climate simulation

The climate consists of a set of statistical properties - in the time and/or space

domains - of quantities that describe the structure and the behavior of the various parts

of the climatic system.

The goal of climate models is to represent as well as possible such statistical

properties, thus aiming at a precise picture of the main properties of the phase space

attractor of the system. In the case of past or present climate change studies, models

try to assess how such statistical properties change with time. In the case of climate

modeling, the relevant uncertainties are not related to the initial conditions of the

system but rather to the very structure of the model. Since ab-initio formulations

based on first principles are not possible and since we are anyway using with

relatively coarse-grain discretization (the best available models have average

resolutions of 100 Km in the horizontal and 200 m in the vertical directions), the

presence of structural uncertainties, which are equivalent to uncertainties in the

attractor’s shape, is unavoidable.

A critical point in climate modeling is that, depending on the time scale of

interest and on the problem under investigation, the relevant active degrees of

freedom, which need the most careful modelization, change dramatically. For

relatively short time scales (1-100 years) the atmospheric degrees of freedom are

active while the other sub-systems can be considered essentially frozen. For longer

time scales (100-1000 years) the ocean dominates the dynamics of climate, while for

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even longer time scales (1000-10000 years) the ice sheet changes are the most

relevant factors of variability. Therefore the very structure of climate models is

gauged with respect to the purpose of the specific study and, depending on this, the

models may be formulated in totally different ways.

Lately, a relevant strategy, which may be termed statistical ensemble

forecasting, has been proposed for dealing with the delicate issue of structural

uncertainties, especially in the context of the study of the future projections of climate

change, even if in principle this is not its only application. The main idea is to

consider a climate model and apply Monte Carlo techniques not only on the initial

conditions, but also on the value of some key uncertain parameters characterizing the

climate machine. The resulting statistics (and change of the statistics) is expressed as

a set of probability distributions rather than as a set of values.

Figure 6: Evolution of the complexity of climate models in terms of representation of the main

processes; the red lines indicate the integration of specific models into climate models. From

IPCC 2001

Statistical inference

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Another prominent approach to the problem of providing an efficient description of

the main properties of the environmental systems consists in trying to infer

information directly from observations. Such an approach constitutes a natural and

necessary complement to the theoretical and model-based analyses, which

unavoidably suffer of the above discussed uncertainties. A classical way of gathering

information on the investigated system from the actual measurements consists in

applying techniques that fall into the wide chapter of statistical inference.

Environmental Science is rich of such applications, some of which are quite well

known to a wide public. In the last two decades, the outstanding example is that of

Global Warming. In statistical terms, the problem consists in trying to detect signs of

anthropogenic changes (expressed in terms of trends) of local as well as global

measures of relevant climatic variables. In particular, the average global surface

temperature is – somewhat anthropocentrically - usually considered as the variable

descriptive of such change. The general problem is quite abundantly documented in

the reports of the International Panel on Climate Change (IPCC).

What is the concrete appearance of surface temperature time-series? The

presence of variability on a vast range of time scales, or, in other terms, of a rather

composite spectrum, implies that suitable filtering procedures are needed in order to

detect changes with time of the signal. We next present an example of how statistical

inference can be applied to characterize the climatology of a system.

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Figure 7: Daily records (1951-2000) of maximum temperature for northern (a) and southern (b)

Italy. The signature of the seasonal cycle is evident.

In figure 7 we show the 1951-2000 daily maximum temperature records for

two synthetic stations representative of northern and southern Italy. These data have

been obtained after a suitable homogenization process of few dozens of real

meteorological stations in each of the two parts of Italy. Apart from the signature of

the seasonal cycle, the signal features an evident noisy structure. When the higher

frequency components are removed and the inter-annual components only are

retained, we may observe an increasing trend for both records, which is consistent

with the large scale picture for the last 50 years. The observed warming trend can be

proved to be statistically significant. But, instead of focusing on the long-term

changes in the value of the signal, we are often interested in detecting changes in the

properties of the seasonal cycle, defined in terms of phase and amplitude of the (1

year)-1 component of the spectrum. Such a problem may be approached by providing a

time-dependent estimate of the spectrum, obtained with a shifting window Fourier

analysis, and then performing a statistical analysis of the various estimates of the

locally defined (in time) seasonal signal.

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Figure 8: Amplitude of the seasonal signal

{

}

S/N

max

TA for the maximum temperature of northern

(a) and southern (b) Italy. No statistically significant trend can be detected in either case.

In the case of the records shown in figure 7, we have that the amplitude and the phase

of the signal, albeit characterized by a relatively large variability, do not have

statistically significant trends. In figure 8 we present the results for the amplitude of

the signal: the linear trend does not have a statistically well-defined sign for both

records. Similar situation occurs for the phase of the signals. From these results we

can infer that the seasonal cycle has not undergone relevant long-term changes. Apart

from the analysis of the variability of the signals under examination, we can infer

from the average properties some basic climatic differences between the two systems,

which in this case are defined in terms of geography only. First we have, quite

trivially, that northern Italy is colder, since its average temperature is notably smaller

(figure 7). This is mostly due to latitudinal effects. Second, northern Italy has a more

continental climate, since its seasonal cycle has larger average amplitude (figure 8).

As a first approximation, we can interpret this property an effect of the smaller

effectiveness of the air-sea thermal exchanges in smoothing out the effect of the

annual changes in the intensity of the incoming solar radiation.

Diagnostics

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Another relevant area of matching between quantitative and qualitative approaches to

the study of environmental systems is that of diagnostics of the complex space-time

structure of fields based on indicators derived from measure theory. It is well known

that, e.g., the definition of the structure of precipitation, land cover, etc. can be given

in terms of estimators of the fractal measure derived from observations. In fact, the

very concept of fractal dimension has been elaborated also in the effort of measuring

territorial systems, with the classical example given by Mandelbrot for the evaluation

of the length of the southern coast of England.

Let us consider standard precipitation, which is defined as the quantity of

atmospheric water reaching the Earth’s surface in unit time. Such a quantity is the

final result of an extremely long and complex chain of events, virtually covering all

the space-scales, from molecular water to the global scale of cyclonic areas, and the

time-scales from the fractions of second of enucleation to the life-time of weather

perturbations (days). Of course, other modulations exist at longer time scales (like the

seasonal oscillation), but these are to be considered variation of external forcing rather

then part of the precipitation process in itself. Therefore, any series of precipitation

observations shows structure virtually at all scales. One way of measuring such

structure is to make use of the so called box counting, which is very instructive since

it makes reference to a basic definition of dimension in terms of the statistics of

occupation of the considered parameter space. When investigating the statistical

properties of homogeneous, stationary, and isotropic rainfall, a more appropriate

definition of structure functions can be given through the moments of the integral

measures of precipitation. Let us consider a positive random field

(

)

xP ,

(

)

0xP >,

defined in the set

[

]

L,0x∈. Without loss of generality, P(x) is normalized to 1, i.e.:

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(5)

( )

=

LdyyP

01

for any realization. Our primary interest is to discuss the scaling properties of the

structure function:

(6)

( ) ( )

+≡

rx

xxrdyyP

µ

(

)

xP is said to display anomalous scaling properties if:

(7)

(

)

[

]

( )

q

q

xr~r

ξ

µ

,

where (q) is a nonlinear function of q. The scaling exponents

(

)

q

ξ

are also referred

to as multi-fractal exponents of

(

)

xP . The random field is assumed to be ergodic, so

that in the previous equation stands both for x-average (x being either a space or

time coordinate) and ensemble average. It follows that we must require also

stationarity and homogeneity. Furthermore, if the field is embedded in a space of

dimension d>1, the field is also assumed to be isotropic, i.e. the scaling properties

depend only on the modulus of r.

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Figure 9: Time-structure of the precipitation at a rain-gauge in the area of Rome (1 minute

interval of sampling from 1992 to 1998). On the left, autocorrelation. On the right, sequence of

moments in increasing order from bottom up. The time scale in abscissa is in powers of 2

minutes.

In figure 9 we present some statistical results for the precipitation in Rome.

We learn from correlation (the station of Rome is quite typical in this respect) that

precipitation is characterized by a limited (with respect to other atmospheric variables

like, for example, the surface temperature considered above) autocorrelation. The

scale of decay of correlation is of the order of hours. A simple fractal structure would

be represented in the logarithmic diagram of moments by a straight line. Apart from

modulations due to sampling, we see that the higher moments show exponent

dependence on scale and, therefore, indicate a multi-fractal distribution. The meaning

of the inflection around 28 m ~ 4 h is that organized dynamical processes (like the

formation of rain bands) are operating at larger time scales, while the incoherent,

micrometeorological processes dominate at smaller scales. This kind of analysis can

be performed also in space (although this requires an extensive observational

coverage that is rarely available) and produces similar results. Other statistical

approaches are possible, in particular for what concerns the parametric distributions

that are used in order to represent the above shown type of behavior. As we have seen,

such an analysis can identify statistical features revealing aspects of the structure and

24

the dynamics of the systems in question. Another use is for operations of upscaling

and downscaling the observed fields.

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Further Readings:

Holton JR (1992), An Introduction to Dynamic Meteorology, Academic Press

IPCC, (2001), Climate change 2001: the scientific basis, Intergovernmental Panel on

Climate Change

Lorenz EN (1967), The nature and theory of the general circulation of the

atmosphere, World Meteorological Organization

Mandelbrot B (1977), Fractal Geometry of Nature. W.H. Freeman

Pedlosky J (1998), Geophysical Fluid Dynamics, Springer

Peixoto JP and Oort AH (1992), Physics of Climate, American Institute of Physics

Saltzmann B (2002), Dynamic Paleoclimatology, Academic Press