Scientific programming in Earth's Sciences Fortran90 and Matlab applications

1
Scientific programming in Earth's
SciencesFortran90 and Matlab applications
J.W. Goethe University. Frankfurt

• Dr Guillaume RICHARD
• Institüt für Geowissenchaften 1.232
• richard_at_geophysik.uni-frankfurt.de

2
Overview

• 12 x 45min Lectures / 45min training classes
  (praktical)
• Basics and introduction to the problem (Earth
  Mantle Convection ). Historical Background, State
  of the Art
• Model (ODE, PDE)
• Finite differencing Finite volume (goudunov
  formalism)
• Finite elements
• )
• )
• )
• )
• e
• d
• D

3
Conservation Laws
Definition Balance equation relating the rate of
change of some physical quantity to its flux
through the region boundary
The only way to change the amount of a quantity
A (unit of A/unit volume) within this volume with
time is to flux (FA) it through the boundary or
create it within the volume (QA).
4
Ordinary Differential equations (O.D.E)
Note that all information about the spatial
variation of within the volume V, the fluxes or
the sources has been lost. If the volume is
large, we may call it a box model.
5
Representative Volume Element

• If were interested in spatial variations as well
  as time, we need to consider small volume of
  reference.
• This concept of the Representative Volume Element
  (RVE) is the base of continuum mechanics. It has
  to be large enough and small enough.
• The RVE is defined by
• A is relatively constant on a scale comparable
  to the RVE.
• The average A of for each contiguous RVE varies
  smoothly (differentiable)

Note It implies that we assume that any
variation smaller than this scale does not change
the gross behaviour of the problem.
6
Partial Differential equations (P.D.E)
7
Conservation equations

• Mass conservation (continuity equation)
• Momentum conservation (force balance)
• Energy conservation

8
Continuity equation
Uncompressible fluid Velocity field V is said to
be solenoidal In 2D
9
Force balance equation
Continuous form of Newtons second law (Law of
acceleration)
10
Stress Tensor (Surficial Forces)
The stress tensor for an isotropic and
incompressible fluid (viscous fluid rheology)
P is the pressure (hydrostatic stress tensor)
Where h is the shear viscosity
11
What are the effects of the viscous forces
(constant viscosity) ?
12
Energy conservation (Heat)
Where k is the thermal conductivity
Boussinesq approximation The fluid densities are
effectively constant r0 except in the body force
terms where they drive most of the flow.
k constant and Boussinesq approximation
Where k is the thermal diffusivity (k k/rCp)
13
Conservation equations

• Mass conservation (continuity equation)
• Momentum conservation (force balance)
• Energy conservation

14
Equations of state
For thermal convection the density of the fluid
is temperature dependent
KS Module of uncompressibility (KS 0)
Anelastic approximation
Anelastic approximation The fluid is weakly
compressible. r depends on the hydrostatic
pressure Ph Sound waves are neglected
15
Conservation equations (Boussinesq and anelastic
approximations)
16
Scaling Dimensionless numbers
To determine the various magnitudes of each of
the terms for the problem and to compare
different experiments, it is useful to scale
everything to the caracteristics variables of the
problem. Length scale H Thickness of the
fluid layer Time scale r0CpH2/k Thermal
diffusive time scale Temperature
scale dT Temperature gradient
(Ts-Tb) Pressure Scale kh/ r0CpH2 Density r
0 Density of reference
17
Scaling Dimensionless numbers
Ra is the Rayleigh Number Pr is the Prandtl
Number
18
Dimensionless equations
19

• Bibliography
• Landau Lifchitz, Fluid Mechanics, 2d Ed. Vol.
  6 of Course of Theoretical Physics, Reed Educ.
  Pro. Publishing, 1987
• Turcotte Schubert, Geodynamics, 2d Ed.,
  Cambridge Univ. Press. 2002
• Furbish, Fluid physics in Geology, Oxford Univ.
  Press, 1997
• Ranalli, Rheology of the Earth, Allen Unwin
  inc., 1987