Francois question I

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Francois question I
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Francois question II
Why does the lapse-rate increase in the
stratosphere? Lapse-rate in stratosphere is due
to UV absorption due to Ozone Lapse rate in
troposphere is partly due to water vapor
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Concepts of Climate modeling
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Spinup
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Flux adjustments
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Model uncertainties
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Model uncertainties
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Model uncertainties
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Numerical techniques
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Introduction into spectral and grid-point methods

• Eulerian/Lagrangian grid point methods
• Spectral methods
• Derivatives are formed using differences over
  finite space and time -gt finite difference
  methods-gt algebraic equations
• -gtsolvable on computer

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Solving a 1st order ODE
Taylor (EQ1)
Euler scheme
Taylor (EQ2)
Centered Diff (EQ1-EQ2)/(2dt).
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Solving a 1st order ODE
Continuous
Discretization with Euler forward
Dt -gt 0, a-gt 8
Discretized solution converges to
analytical solution
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Solving a 1st order ODE
For dt1/A and dt2/A wrong results
The Euler scheme is the simplest and most
inaccurate 1st order scheme. dy/dx f(x,y(x))
is evaluated only for t, t?t Through evaluation
of f(x,y) for further points with t,t?t, we
can achieve a higher order accuracy gt Runge
Kutta scheme of 4th order
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Solving a 1st order ODE
RUNGE KUTTA SCHEME of 4th order
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Classification of partial differential equations
2nd order PDEs
a,b,c,d,e,f are functions of x,y
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Classification of partial differential equations
PDE
Linear
Nonlinear
Homogenous
Inhomogenous
parabolic
elliptic
hyperbolic
Parabolic source
Elliptic source
Hyperbolic source
Laplace equation
Wave equation
Heat, diff. Equation
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Stability analysis
Final goal Convergence
The solution of the approximate difference
equation (ADE_ approaches that of the PDE for ?t,
?x gt 0 difficult because analytical solution
often unknown
1st step Consistency
The equation (not the solution) of an ADE
approaches PDE, as ?t, ?x gt 0
2nd step Stability
The ADE gas an upper linit as tgt 8, to the
growth of the solution or the truncation of the
errors inroduced by roundoff etc
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Stability analysis
Lax equvalence theorem Convergence ltgt
Consistency Stability
Methods for stability analysis - Matrix
method - Von Neumann Method - Energy
method We will only discuss the Von Neumann
method, because it is most convenient
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Analytical solution of linear partial
differential equations
Galerkin
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Analytical solution of linear partial
differential equations
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Analytical solution of linear partial
differential equations
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HOMEWORK
Homework I, produce movie of the solution in
matlab for fdelta-function for Laplacian friction
Homework II, find analytical solution for
biharmonic friction produce movie of the
solution in matlab for fdelta-function for
biharmonic friction
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Solving PDEs
As the dynamic system becomes more complicated,
analytical solutions will be no longer
available. We have two options 1. Reduce the
complexity of the dynamics, such that the
simplified equations can be solved
analytically 2. Keep the original complicated
dynamics and find its approximate solution using
numerical techniques There are two ways to solve
PDEs A Galerkin Method B Finite Difference
Method
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Galerkin Method
Main Idea Represent the solution in terms of
basis functions, coefficients
will be function of time PDE-gt ODE for
time-evolution coefficients (solvable e.g. with
Runge Kutta)
L is differential operator, u is dependent
variable and f is specified forcing function, BC
at xa and altxltb
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Galerkin Method
Error
Choose an(t) such that eN is orthogonal to each
basis function
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Galerkin Method
The above N algebraic equations can be used to
solve the N unknown coefficients a(t)
Spectral method
Finite element method
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Galerkin Method Example again diffusion equation
Time-discretization
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Galerkin Method Example again diffusion equation
Approximate
Example K1, n10, ?t0.1
Sign alternating solution Rubbish
Approximate
Exact
Numerical instability has to be avoided but
how????
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Von Neumann stability
A solution to a linear equation can be expressed
in terms of a Fourier series, where each harmonic
solution is also a solution We can test the
stability of a single harmonic solution
stability of all harmonics can then be a
necessary conditions for stability of the
scheme assume at some time (t0) that 1.) A
Fourier expansition of the initial field f(x) can
be made 2.) A serparation of time and space
variables can be made, such that at time t a
simple term in the Fourier series is
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Von Neumann stability
Ensures numerical stability
4. Requiring
Is required physically
5. Oftentimes
Example 1-d diffusion equation, again and again
and again
Assume forward differencing in time and finite
differencing in space
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Von Neumann stability
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Solving partial differential equations
Hyperbolic wave equation Initial value (Cauchy)
problem
parabolic diffusion equation Initial value
problem
elliptic Poisson equation Boundary value problem
From a numerical point of this distinction is not
too important
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Solving PDEs
Initial value problem, defined by the anwers to
the following questions - What are the dependent
variables to be propagated in time? - What is the
time-evolution equation for each variable? - What
is the highest time derivative for each
variable? - What boundary conditions govern the
evolution in time? (e.g. Dirichlet Bcs
boundary values as function of time, Neumann
conditions specify normal gradient at the
boundary - Most crucial aspect is the stability
of the numerical scheme!!!
Boundary value problem, defined by the anwers to
the following questions - What are the
variables? - What equations are satisfied in the
interior of the domain? - What equations are
satisfied by points on the boundary region of
interest? - Most schemes are pretty stable - BC
must be satisifed simultanously -gt linear matrix
equations
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Example 1-d wave equation
Dispersion-free movement of initial
wave-package Initial value problem, perturbation
moving to the left
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Example 1-d wave equation
Search for equation whose solution can propagate
both to the left and right
Classical wave equation
With
Solution of
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Example 1-d wave equation
Traveling wave moving to positive x (to the
right) Characteristics tx/c
Discretization xm?x, m0,1,2,.....
tn?t, m0,1,2,..... u(x,t)
u(m?x,n?t)umn Using forward differences in
time and centered in space (FTCS)
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Example 1-d wave equation Stability
P
Unconditionally unstable scheme
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Example 1-d wave equation Stability
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Example 1-d wave equation
Discretization xm?x, m0,1,2,.....
tn?t, m0,1,2,..... C(x,t)
C(m?x,n?t)Cmn Using centered differences in
time and space CTCS
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Example 1 d wave equation
Called leap-frog scheme, doesn't work for the
first two time steps use Euler forward (forward
in time, centered in space, FTCS) for first two
steps
T
N1
N
N-1
M
X
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Term projects
Francois Jets in a nonlinear eddy-resolving
shallow-water model Thomas West-Antarctic
ice-sheet collapse, sea level rise Denise
Drake passage closure in BARBIE model Pierre
Bifurcation analysis Laurie Ice-sheet model
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Term project Francois
1. Implementation of Eddy-resolving 1.5 layer
shallow water model 2. Configuration for Pacific
basin (wind-stress curl idealized) 3. Jet
Diagnostic for different friction parameters 4.
Eddy mean-flow interactions
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Term project Thomas
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Term project Pierre
1. Learn about basic bifurcation theory 2.
install and run tutorial for AUTO 3. learn how to
run AUTO for PDEs 4. Nonlinear diffusion equation
with AUTO etc....
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Term project Denise
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Term project Laurie
1. Numerical scheme for ice-sheet model 2.
Sensitivity to ablation parameterization 3.
Coupling with simple ocean 4. Stationary-wave
feedback (Roe and Lindzen)