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Corso di Laurea in Fisica - UNITS
ISTITUZIONI DI FISICA PER IL
SISTEMA TERRA
Geosphere
FABIO ROMANELLI
Department of Mathematics & Geosciences
University of Trieste
[email protected]
http://moodle2.units.it/course/view.php?id=887
Earth system: geosphere
Earth layers
Primarily silica
plus light
metallic
elements
Primarily
silica plus iron
and
magnesium
Primarily iron
and nickel
Composition
crust
mantle
Mechanical Characteristics
lithosphere
asthenosphere
brittle solid
solid (but
nearly
liquid)
mesosphere
solid
outer core
liquid
inner core
solid
core
Vp Vs
ρ P
Kμ
Seismological
model
Seismic tomography
Mean density
Moment of inertia
Equation of state
A-W equation
Birch’s law
Bullen parameter
Bulk composition
(T,P) phase diagrams
Vp Vs
ρ P
Kμ
Seismological
model
Seismic tomography
Mean density
Moment of inertia
Equation of state
A-W equation
Birch’s law
Bullen parameter
Compositional
model
Mineralogy
Assemblage
Chemistry
Vp Vs
ρ P
Kμ
Seismological
model
Seismic tomography
Mean density
Moment of inertia
Equation of state
A-W equation
Birch’s law
Bullen parameter
Heat flux
Anchoring points
Mantle viscosity
Adiabatic gradient
Thermal
model
T
Bulk composition
Mineralogy
Assemblage
Chemistry
Compositional
model
(T,P) phase diagrams
Vp Vs
ρ P
Kμ
Seismological
model
Seismic tomography
Mean density
Moment of inertia
Equation of state
A-W equation
Birch’s law
Mantle viscosity
Heat flux
Bullen parameter
Adiabatic gradient
Anchoring points
Thermal
model
T
Ground motion animation: time scales...
Courtesy of Takashi Furumura
Tsunami animation: time scales...
http://outreach.eri.u-tokyo.ac.jp/eqvolc/201103_tohoku/eng/
http://supersites.earthobservations.org/honshu.php
http://eqseis.geosc.psu.edu/~cammon/Japan2011EQ/
“Earthquake Research Institute, University of Tokyo, Prof. Takashi
Furumura and Project Researcher Takuto Maeda”
Tsunami animation - NOAA
Tsunami signature in the ionosphere
By dynamic coupling with the atmosphere, acousticgravity waves are generated
Traveling Ionospheric Disturbances (TID) can be
detected and monitored by high-density GPS networks
Tsunami signature in the ionosphere
Tsunami-generated IGWs and the response of the ionosphere to
neutral motion at 2:40 UT.
Normalized vertical velocity
Perturbation in the ionospheric plasma
Tsunami signature in the ionosphere
The TEC (Total Electron Content) perturbation induced by tsunami-coupled IGW
is superimposed on a broad local-time (sunrise) TEC structure.
Mathematical reference: PDE
Classification of Partial Differential Equations (PDE)
Second-order PDEs of two variables are of the form:
∂ f( x, y )
∂ f( x, y)
∂ f (x,y )
∂f( x, y)
∂f (x,y)
a
+b
+c
+d
+e
= F (x,y)
2
2
∂x
∂x∂y
∂y
∂x
∂y
2
2
2
b2 − 4ac < 0
elliptic
b − 4ac = 0
parabolic DIFFUSION equation
b2 − 4ac > 0
hyperbolic WAVE equation
2
LAPLACE equation
Elliptic equations produce stationary and energy-minimizing solutions
Parabolic equations a smooth-spreading flow of an initial disturbance
Hyperbolic equations a propagating disturbance
Fabio Romanelli
IFST
Boundary and Initial conditions
Initial conditions: starting point for
propagation problems
Boundary conditions: specified on
domain boundaries to provide the
interior solution in computational
domain
∂R
R
s
⎧
n
⎪(i) Dirichlet condition : u = f on ∂R
⎪
∂u
∂u
⎪
= f or
= g on ∂R
⎨(ii) Neumann condition :
∂
n
∂
s
⎪
∂u
⎪
(iii)
Robin
(mixed)
condition
:
+
ku
=
f
on
∂
R
⎪⎩
∂n
Fabio Romanelli
IFST
Elliptic PDEs
Steady-state two-dimensional heat conduction
equation is prototypical elliptic PDE
Laplace equation - no heat generation
∂ T ∂ T
+
=
0
2
2
∂x
∂y
2
2
Poisson equation - with heat source
∂ T ∂ T
+
=
f
(
x
,
y
)
2
2
∂x
∂y
2
Fabio Romanelli
2
IFST
Heat Equation: parabolic PDE
Heat transfer in a one-dimensional rod
x=0
x=a
g1(t)
g2(t)
2
∂u
∂ u
=d
, 0 ≤ x ≤ a, 0 ≤ t ≤ T
2
∂t
∂x
u(x, 0) = f (x)
⎧ u(0, t) = g1(t)
B.C.s ⎨
⎩ u(a, t) = g 2 (t)
I.C.s
Fabio Romanelli
0≤ x≤a
0≤ t ≤T
IFST
Wave Equation: hyperbolic PDE
b2 - 4ac = 0 - 4(1)(-c2) > 0 : Hyperbolic
2
∂ u
∂t
2
=v
2
2
∂ u
∂x
2
, 0 ≤ x ≤ a, 0 ≤ t
⎧ u(x, 0) = f 1(x)
I.C.s ⎨
0≤ x≤a
⎩ u t (x, 0) = f 2 (x)
⎧ u(0, t) = g1(t)
B.C.s ⎨
t>0
⎩ u(a, t) = g 2 (t)
Fabio Romanelli
IFST
Coupled PDE
Navier-Stokes Equations
⎧ ∂u ∂v
+
=0
⎪
⎪ ∂x ∂y
2
2
⎪
⎛∂ u ∂ u⎞
∂u
∂u
1 ∂p
⎪ ∂u
⎟
+u
+v
=−
+ ν ⎜⎜ 2 +
⎨
2 ⎟
∂x
∂y
ρ ∂x
∂y ⎠
⎝ ∂x
⎪ ∂t
2
2
⎪ ∂v
⎛∂ v ∂ v⎞
∂v
∂v
1 ∂p
+v
=−
+ ν ⎜⎜ 2 + 2 ⎟⎟
⎪ +u
∂
t
∂
x
∂
y
ρ
∂
y
∂y ⎠
⎪
⎝ ∂x
⎩
Fabio Romanelli
IFST
What is a wave?
Small perturbations of a
stable equilibrium point
Linear restoring
force
Harmonic
Oscillation
Repulsive Potential ∝ 1/rm
Total Potential
Attractive Coulombic
Potential ∝ 1/r
Fabio Romanelli
IFST
What is a wave? - 2
Small perturbations of a
stable equilibrium point
Linear restoring
force
the disturbances can
propagate
Coupling of
harmonic oscillators
A
B
0
Fabio Romanelli
Harmonic
Oscillation
0
IFST
What is a wave? - 2
Small perturbations of a
stable equilibrium point
Coupling of
harmonic oscillators
Linear restoring
force
Harmonic
Oscillation
the disturbances can
propagate, superpose and
stand
Normal modes of the system
Fabio Romanelli
IFST
What is a wave? - 3
Small perturbations of a
stable equilibrium point
Linear restoring
force
Harmonic
Oscillation
the disturbances can
propagate, superpose and
stand
Coupling of
harmonic oscillators
WAVE: organized propagating imbalance,
satisfying differential equations of motion
2
∂ y
2
∂x
=
2
1 ∂ y
v
2
2
∂t
General form of LWE
Fabio Romanelli
IFST
What is a wave? - 4
Small perturbations of a
Linear restoring
stable equilibrium point
force
Harmonic
Oscillation
the disturbances can propagate,
Coupling of
harmonic oscillators
superpose, stand and be dispersed
WAVE: organized propagating imbalance,
satisfying differential equations of motion
Organization can be destroyed,
when interference is destructive
non linearity
Turbulence
Solitons
Fabio Romanelli
strong
scattering
Diffusion
Exceptions
Phonons
IFST
Alcuni Links
http://www.physicalgeography.net/fundamentals/contents.html