Untangling Complex Systems, page 67
y x + ∫ d
x y. Suppose we need to calculate either the indefinite integral
∫ yd
x or the definite integral l 2
∫ ydx . The solution will be ∫ ydx = x ⋅ y − ∫ xdy and l 2
∫ ydx = ( x ⋅ y) − ( x ⋅ y
l 2
) − ∫ xdy,
l 1
l 1
l 2
l 1
l 1
respectively. This method of integration by parts makes sense when the solution of the integral ∫ xd
y is known.
The Emergence of Chaos in Time
333
the system. Using the Fourier’s law and assuming the thermal conductivity k constant, the second
integral of [10.26] becomes:
dP*
2
2
1
k T T
= 2 k∇
∂
T
∫
dV
2
[10.27]
2
dV
∫
dt
= −
∇
∂
t
∂ T
T
t
∂
V
V
The continuity (or balance) equation for the internal energy of the fluid (Anderson Jr. 2009) con-
sidering (I) heat advection, (II) heat conduction, (III) viscous heating and (IV) cooling by volume
expansion is
∂
2
(ρ u) = −ρ v∇u + k∇ T +τ∇ v − P∇ v [10.28]
∂ t
In [10.28], ρ is the density, v the velocity, u the internal energy per unit of mass, τ the viscous stress, and P the pressure. If we express u = cvT , where c is the specific heat at constant volume, and we v
assume that c is constant in the fluid, we get (Ozawa and Shimokawa 2014)
v
T
∂
ρ c
)
2
v
= −ρ vcv∇( T − P∇ v +τ∇ v + k∇ T [10.29]
t
∂
It is now evident that the temporal change of T depends on the cooling induced by the heat advection
and volume expansion, and on the heating due to the viscous force and conduction. The contribu-
tion of heat conduction, k∇2 T , can be expressed as a function of the other terms of equation [10.29]: 2
T
k∇ T =
∂
ρ c
ρ
)
v
+ vcv∇( T + P∇ v − ∇
τ v [10.30]
t
∂
We can insert this definition of k∇2 T into [10.27] and achieve
dP*
T
∂ 1
=
∂
2
ρ c
ρ
)
v
+ vcv∇( T + P∇ v − ∇
τ v
∫
dt
t
∂
dV [10.31]
t
∂ T
V
In the absence of convection, v = 0, equation [10.31] reduces to
dP*
c
2
ρ v T
= −
∂
2
dV 0 [10.32]
dt
T 2
≤
∫ t∂
V
Equation [10.32] tells us that P* tends to decrease until it reaches a minimum in its final steady
state, when (∂ T ∂ t) = 0. This is the theorem of Minimum Entropy Production that we have learned
in Chapter 3.
What happens in the presence of convection?
dP*
k T
2
T
k T
2
P
τ
k
2
= −
∇
∂
2
dV
2
v
T
v
∇ v +
∇ T dV [10.33]
dt
T 2 t
∂
= −
∇
− ∇
2
( )−
∇ +
∫
∫ T
ρ cv
ρ cv
ρ v
c
V
V
If we assume that the cooling due to volume expansion and the viscous heating are negligible with
respect to terms describing the conduction and heat advection, equation [10.33] reduces to [10.34]:
dP*
k T
2
k
2
= −
∇
2
− v∇
∫
[10.34]
2
( T )+
∇ T dV
dt
T
ρ cv
V
334
Untangling Complex Systems
When the thermal gradient Δ T is so large that convection is the most effective mechanism of heat trans-
fer, the Nusselt number is larger than 1, and v∇ T
( ) ≥ ( k / ρ c
2
2
v ) ∇ T ≥ 0 or v ∇ T
( ) ≤ ( k / ρ cv)∇ T ≤ .
0
In both cases, it derives that
dP*
≥ 0 [10.35]
dt
In the nonlinear regime, the Entropy Production always grows until it maximizes after fixing the
contour conditions. Typically, in very far-from-equilibrium conditions, there are many available
states. According to the most recent investigations, it seems there exists a selection criterion that
is useful to predict the evolution of such systems. And this is the Maximum Entropy Production
principle (MaxEP) that has been proved in some circumstances, such as fluid turbulence, crystal
growth morphology, biological evolution (Dewar and Maritan 2014). The out-of-equilibrium
system evolves spontaneously to the state that maximizes Entropy Production (Martyushev and
Seleznev 2006). The theoretical basis of MaxEP is a subject of open debate. One hypothesis is
that MaxEP has a statistical explanation (Dewar 2009): when a system is forced very far-from-
equilibrium, the MaxEP condition is selected simply because it is by far the most probable one.
The MaxEP state can be microscopically obtained in an overwhelmingly greater number of
ways than any other non-equilibrium state. The MaxEP principle needs further confirmations
and studies to become a general evolutive criterion. However, it is worthwhile stressing that
the discovery of general principles in nonlinear regime is a tough task. In fact, in the nonlinear
regime, the events are often unique and unreproducible. They are irreversibly and make history.
Hence, history is not a prerogative of humans but is also made by inanimate physical and chemi-
cal systems.
10.7 THE “BUTTERFLY EFFECT”
10.7.1 The comPlexiTy of convecTion in The TerresTrial aTmosPhere
As anticipated in paragraph 10.1, the phenomenon of convection is widespread in nature. For
example, it is the mechanism of heat transfer in the stars, like our sun. In our planet, convection
is the motive force for the slow migration of the continents and the vast oceans currents. Also,
the global circulation of the atmosphere depends on convective flows. The UV-Visible radia-
tion emitted by the sun and reaching the biosphere is absorbed mainly by the terrestrial crust,
and it is re-emitted as infrared. Such thermal radiation having small frequencies is absorbed
by the lowest layers of the atmosphere due to the presence of water, carbon dioxide, and other
gaseous compounds. A vertical thermal gradient originates, and small-scale convective flows
are induced. Other more significant convective flows are generated by the temperature gradients
existing between the warm tropics and the cold poles. An accurate theoretical analysis of these
natural convective phenomena is not straightforward, at all. The Rayleigh’s model studied in
paragraph 10.5 assumes that only the density of the fluid changes with temperature; the other
physical properties are considered to be constant. Of course, this is an approximation that in real
fluids is quite coarse. In reality, also viscosity and thermal diffusivity change with temperature.
Moreover, fluids are compressible, especially if they are gaseous, and pressure is a significant
variable, which in turn affects density and other properties. The variables are quite tangled. For
example, viscous drag dissipates kinetic energy in the form of heat, and so it raises the tempera-
ture. The local heating determines a reduction in viscosity. A theory that explicitly considers all
the relations and the positive and negative feedback actions among the variables becomes too
cumbersome. Therefore, we must seek a compromise between the natural complexity and the
complexity of our theories.
The Emergence of Chaos in Time
335
10.7.2 The lorenz’s model
In the 1960s, an idealized dissipative hydrodynamical model for the convective flows in the
atmosphere, formulated for pursuing weather forecasts, consisted in the following system of
three nonlinear differential equations, developed by Edward Lorenz (1963), who started from the
Navier-Stokes equations:8
dX
= σ ( Y − X) [10.36]
dt
dY
= rX − Y − XZ [10.37]
dt
dZ
= XY − bZ [10.38]
dt
In these equations, there are only two nonlinear terms, which are the products XZ in [10.37] and XY
in [10.38]. The variable X is proportional to the velocity of the convective motion, and Y is propor-
tional to the temperature difference between the ascending and descending currents. The variable
Z is proportional to the deviation of the vertical temperature profile from linearity (the linearity
is verified before convection starts, when the heat is transferred only by conduction) and a posi-
tive value indicates that the strongest gradients are close to the boundaries. The parameter σ is the
Prandtl number,6 r = ( Ra) / ( Ra )
c , and b is related to the aspect ratio of the rolls. The steady state
solutions are ( X , Y , Z )
), (
−1
−1
−1 ) (labelled as S
s
s
s = (0, 0, 0) (labelled as S 0
b( r
), b( r
), ( r
)
+),
(− b( r− )1,− b( r− )1,( r− )1) (labelled as S−) when r > 1, and just S 0= (0, 0, 0) when r < 1. The solution S represents the absence of convection, whereas the other two solutions, S
0
+ and S−, dif-
fering just in the signs of X and Y , represent left- and right-turning convection rolls. The stability s
s
of the solutions can be investigated by considering the evolution of small perturbations and apply-
ing the linear stability analysis we learned in Chapter 3. If we indicate with x ( t) = ( X ( t) − X ) s ,
y( t) = ( Y ( t) − Y )
( ) ( ( )
)
s and z t = Z t − Zs the distance of the variables from their steady-state values,
they evolve in time according to the following linearized equations:
x( t) −σ
σ
0 x ( t)
y
( t) = r − Z
y( t)
s
−1
− Xs
[10.39]
z
( t)
Y
z( t)
s
Xs
− b
For the steady-state solution S , the characteristic equation of [10.39] is
0
( b + λ)λ2 + λ(1+σ )+σ (1− r)
= 0 [10.40]
Equation [10.40] has three real negative roots when 0 < r < 1, and two negative and one positive
when r > 1. This result confirms that when 0 < r < 1, the solution S is stable and the fluid transfers 0
heat only by conduction and not by convection. On the other hand, when r > 1, convection starts and
S is not any more stable.
0 For the other two steady-state solutions, i.e., S+ and S−, the characteristic equation becomes
λ3 + λ2 (σ + b +1) + λ b( r +σ ) + σ
2 b( r −1) = 0 [10.41]
8 The Navier-Stokes equations are the fundamental equations that describe how fluids behave, and they are based on the momentum’s conservation.
336
Untangling Complex Systems
Z
Z
→
n∧
v
as time goes on
V 0
V
X
t
X
Y
Y
FIGURE 10.13 Time evolution for the volume V of the Lorenz’s phase space. shrinks to after the time 0
V 0
Vt
interval ( t – t ).
0
It has been demonstrated (Marsden and McCracken 1976) that when 1 < r < rc = σ (σ + b + 3) /
(σ − b − )1 equation [10.41] has one real negative and two complex conjugate roots. S+ and S− are stable fixed points and they are surrounded by unstable limit cycles. This behavior means that r = rc
is a subcritical Hopf bifurcation (see Chapter 4). For r > rc, the fixed points are unstable, and there are no attractors in the neighborhood. Nevertheless, the dynamical trajectories cannot diverge to
infinity. In fact, the Lorenz system is dissipative: volumes in the three-dimensional phase space
contract as we demonstrate, right now.
Let us consider a portion of the Lorenz’s phase space having volume V and surface at time
0
A 0
t (see Figure 10.13). We may conceive all the points laying on the surface as initial conditions.
0
A 0
What happens to the volume if they evolve in time? Let dA be an infinitesimal portion of the surface
A , and n be its outward normal. If v = dX dt + dY dt + k dZ dt is the punctual velocity, the 0
î ( / ) ĵ ( / )
(
/ )
product v * n represents the projection of the velocity onto the outward normal vector. The infini-
tesimal volume swept in the infinitesimal time interval dt is dV = ( v n)( dt )( dA). Hence, the total volume swept in the time interval by all the points of the initial surface A is:
0
dV
= v ndA
∫ [10.42]
dt
A
Applying the divergence theorem, equation [10.42] transforms in
dV
= ∇ v dV
∫ [10.43]
dt
V
The divergence of v is
dX
∂ dY
∂ dZ
∇ v = ∂
σ 1 b 0 [10.44]
∂
+
+
= − − − <
X dt ∂ Y dt ∂ Z dt
Therefore, [10.43] becomes
dV
= −(σ +1+ b) V [10.45]
dt
After separating the variables and integrating from time t up to time
0
t, we obtain
V t
V e (σ 1 b)( t− t
( ) = − + + 0)
0
[10.46]
The Emergence of Chaos in Time
337
20
10
0
X( t) −10
−20
0
10
20
30
40
50
60
70
30
15
0
Y( t) −15
−30 0
10
20
30
40
50
60
70
45
30
15
Z( t)
0
0
10
20
30
40
50
60
70
Time
FIGURE 10.14 Time evolutions for X, Y and Z when σ = 10, b = 8/3 and r = 28. The horizontal straight lines that are present in each plot represent the solutions S+ and S−.
Equation [10.46] tells us that the initial volume of the phase space, V , shrinks (as shown in
