Content uploaded by Valerio Lucarini
Author content
All content in this area was uploaded by Valerio Lucarini on Nov 01, 2017
Content may be subject to copyright.
1
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
2
3
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.
4
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.
5
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
6
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.
7
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
8
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.
9
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
10
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
11
(
)
(
)
(
)
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.
12
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.
13
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
14
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.
15
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
16
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
17
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
18
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.
19
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.
20
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
21
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.:
22
(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.
23
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.
25
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
