second year master programme - science.uu.nl project csgdelde102/soac numerical methods.pdf ·...

http://www.staff.science.uu.nl/~delde102/StudyAdvice_MPOC-master.htm

Second year master programme

Simulation of Ocean, Atmosphere and Climate (7.5 ec)Making, Analysing and Interpreting Observations (7.5 ec)

Thesis Research Project (45 ec) (periods 2-4)

http://www.staff.science.uu.nl/~delde102/SOAC.htm

Simulation of Ocean, Atmosphere and Climate (SOAC)

week 36: lectures, invited talks & exercises

week: 37, 38, 39: project under supervision of IMAU-staf-member

(in couples)

week 40Thu. 2 Oct.: oral presentation of project

write a report (<3000 words)

Schedule: SOAC 2014 (http://www.staff.science.uu.nl/~delde102/SOAC.htm) Monday 1 September (MIN 025) 09:15-9:30: Aarnout van Delden: Introduction to the “research” year of the master program 09:30-10:15: Aarnout van Delden: Numerical Fluid Dynamics 1 10:30-12:15: Lars Tijssen: Introduction to Python 12:15-13:00: Aarnout van Delden: Introduction to Exercises 1(weather and climate of a simple recursive model) and 2 (Lagrangian model of the vertical motion of a buoyant fluid parcel) Afternoon: Working on exercise 1 and 2 Tuesday 2 September (MIN 205) Hand in answers to exercise 1 (individually) 09:15-09:45: Aarnout van Delden: Discussion of exercise 1 09:45-10:30: Aarnout van Delden: Numerical Fluid Dynamics 2 Rest of the day: Working on exercise 2 Wednesday 3 September (MIN 205) 09:15-10:00: Aarnout van Delden: Numerical Fluid Dynamics 3 10:15-11:00: Michael Kliphuis: Computer hardware and climate models 11:00-12:45 Sander Tijm: Hydrostatic and non-hydrostatic limited area models for weather prediction 13:30-14:15: Aarnout van Delden: Introduction to Exercise 3 (Solving the advection equation with different numerical schemes) Afternoon: Working on exercise 2 and 3 Thursday 4 September (MIN 205) Hand in answers to exercise 2 (individually) 09:15-10:15: Dewi Le Bars, Wim Ridderinkhof/Niels Alebregtse, Carles Penades/ Huib de Swart, Willem-Jan van den Berg, Anna von der Heydt, Rianne Giesen, Claudia Wieners and Aarnout van Delden: Overview of the projects 10:30-10:50: Aarnout van Delden: Discussion of exercise 2 10:50-11:15: Lars Tijssen: presentation (an extension of exercise 2) 11:30-13:15 Jordi Vila: Large Eddy Simulation Afternoon: Working on exercise 3 Friday 5 September (MIN 205) Hand in answers to exercise 3 (individually) 9:15-10:00: Aarnout van Delden: Discussion of exercise 3 10:15-11:15: Rein Haarsma: The atmosphere in EC-Earth 11:30-13:15: Dewi Le Bars: The ocean in CESM-climate model Finally: Who is who with the projects (couples)? Thursday 2 October (HFG 611AB) 9:15-12:00: Presentations of the results from the projects

Programming languages

FortranC/C++Pascal

MATLABPython

…

Programming languages

Advantages of Python over MATLAB:1) Python code is more compact and readable than Matlab code.

2) The Python world is free and open (in several senses).

3) Like C/C++, Java, Perl, and most other programming languages other than Matlab, Python conforms to certain de facto standards, including zero-based indexing and the use of square brackets rather than parentheses for indexing..

4) Python makes it easy to maintain multiple versions of shared libraries

5) Python offers more choice in graphics packages and toolsets

Pythonhttps://www.python.org/

http://www.numpy.org/

http://matplotlib.org/index.html

http://www.johnny-lin.com/pyintro/

http://www.staff.science.uu.nl/~delde102/StudyAdvice_MPOC-master.htm

Master thesisFrom November/December you will need to find a

thesis project. Decide what you find interesting. Talk to potential thesis supervisors. It is also possible to do a

thesis project at KNMI, NIOZ or any other (foreign)university. However, next to the daily

supervisor at the other institute, you must always have a staff member of IMAU as second supervisor.

You must make clear arrangements about supervision and about what is expected of you. Independence

and originality are very much appreciated! At the start you have to fill in a “research application form”

Introduction to numerical fluid dynamics for geophysical flows

Anna von der HeydtAarnout van Delden

a.j.vandelden@uu.nl BBL 615

This visualization shows early test renderings of a global computational model of Earth's atmosphere based on data from NASA's Goddard Earth Observing System Model, Version 5 (GEOS-5). This particular run, called 7km GEOS-5 Nature Run (7km-G5NR), was run on a supercomputer, spanned 2 years of simulation time at 30 minute intervals, and produced Petabytes of output. The model uses a 7.5 km cube-sphere parameterization. Geographic coordinate output volumes from the model are 5760 x 2881 x 72 voxels per time step. For each voxel numerous physical parameters are available such as temperature, wind speed and direction, pressure, humidity, etc. This visualziation uses a combination of the CLOUD and TAUIR parameters.The visualization spans a little more than 7 days of simulation time which is 354 time steps. The time period was chosen because a simulated category-4 typhoon developed off the coast of China. The frames were rendered using Renderman. Brickmap volumes generated for each time step are about 2.6 Gigabytes. This short visualization referenced nearly a terabyte of brickmap files. The 7 day period is repeated several times during the course of the visualization.This animation was presented at SIGGRAPH 2014 during the Dailies session. (http://svs.gsfc.nasa.gov/vis/a000000/a004100/a004180/)

http://www.gfdl.noaa.gov/visualization

climate modellinghttps://www.youtube.com/watch?v=ADf8-rmEtNg

High resolution climate model outputhttps://www.youtube.com/watch?v=Cxsg7uvVSBE

Uncertainty in climate modelshttps://www.youtube.com/watch?v=4AjCeXl5tE0

Partial differential equationshttps://www.youtube.com/watch?v=fYVMmEykiMw

Cloud model outputhttps://www.youtube.com/watch?v=mlvLX7YvI88

Ocean modellinghttps://www.youtube.com/watch?v=B-TSwthjPYE

GFDL visualizationhttp://www.gfdl.noaa.gov/visualization

Animations and lectures

Earth system models

NCAR Community Earth System Model (CESM)

Atmosphere grid box: wind vectors, humidity, clouds, temperature, chemical species.

Surface: ground temperature, water, energy, momentum, CO2 fluxes.

Ocean grid box: current vectors, temperature, salinity.

General circulation models (GCMs)

Resolution in IPCC simulations(IPCC 1990)

(IPCC 1996)

(IPCC 2001)

(IPCC 2008)

Spin up•  Spin up time =

integration time, the model needs to reach a (statistical) equilibrium.

• Depends on the slowest component of the modelled climate system:

•  Atmosphere ~ 15 years.•  Ocean ~ 3000 years.•  Ice sheets ~ even longer.

How to build a numerical model for a fluid system?���

(air or water)Recommended books:

Benoit Cushman-Roisin, Jean-Marie Beckers, Introduction to Geophysical Fluid Dynamics - Physical and Numerical Aspects. 2nd Edition, Academic Press (2011) (chapters 5 & 6)

Dale R. Durran, Numerical Methods for Fluid Dynamics – With applications to geophysics, 2nd Edition, Springer (2010)

SHALLOW-WATER EQN’S(the “e-coli” of Geophysical Fluid Dynamics)

characteristics

solution

what about boundary conditions?

ADVECTION EQUATION

FINITE DIFFERENCE

stencil

forward

backward

central

ACCURACYTaylor expansion

truncation error

the lowest order of in the truncation error determines the accuracy

finite difference approximation exact

ACCURACYtruncation error

the lowest order of in the truncation error determines the accuracy

Lecture 2

Grading

Exercises 1-3 + attendance first week: 20% of grade

Oral presentation: 40%Written report: 40%

ADVECTION EQUATION

truncation errorcomputer sets this to zero

notation:

this scheme is first-order accurate in time and space

ADVECTION EQUATION

truncation error

numerical diffusion numerical dispersion

∂ 2u∂t2

= −c ∂∂t

∂u∂ x

= −c ∂∂ x

∂u∂t

= −c2 ∂2u

∂ x2

STABILITYVon Neumann’s Method

insert Fourier series to represent the discretized solution

at

at

note that

amplification factor

stable if

(page 96 Durran)

Analysis is restricted to linear equations, implying that amplification does not vary from time step to time step

STABILITYinsert

rewriting

for every

most unstable modealways unstable

euler forward / downwind

STABILITYinsert

rewriting

for every

most unstable mode

euler forward / upwind

stable if Courant Frederich Lewy condition

CFL CONDITION

euler forward / upwindeuler forward / downwind

always unstable Courant Frederich Lewy condition

unstablestable

IMPLICIT SCHEMEinsert

euler backward / upwind

unconditionally stable!

advantage: we can take large time steps

but 1: accuracy determines time step

but 2 : implicit schemes are harder / impossible to solve

in general form Euler forward Euler backward

DIFFUSION EQUATION

The finite difference approximation for the second derivative

DIFFUSION EQUATION

euler forward / central difference

accuracy and

stability

condition

MATSUNO SCHEMETime differencing:

€

C * jn+1 −Cj

n

Δt= u0

∂C∂x% & '

( ) * j

n

€

Cjn+1 −C j

n

Δt= u0

∂C *∂x

% & '

( ) * j

n+1

Step 1

Step 2

€

∂C∂t

= −u0∂C∂x

Lax-Wendroff schemeA well-known finite difference scheme:

Taylor series:

Since€

Cjn+1 =Cj

n +Δt ∂C∂t

$ % &

' ( ) j

n+Δt2 ∂

2C∂t2$

% &

'

( ) j

n

+ ...

€

∂2C∂t2

= u02 ∂

2C∂x2

€

Cjn+1 =Cj

n − u0Δt∂C∂x% & '

( ) * j

n+u02Δt2

2∂2C∂x2%

& '

(

) * j

n

+ ...

€

∂C∂t

= −u0∂C∂x

Lax-Wendroff scheme

€

Cjn+1 =Cj

n − u0Δt∂C∂x% & '

( ) * j

n+u02Δt2

2∂2C∂x2%

& '

(

) * j

n

+ ...

which becomes:

€

Cjn+1 −Cj

n

Δt≈ −u0

Cj+1n −Cj−1

n

2Δx

%

& ' '

(

) * * +

u02Δt2

Cj+1n − 2Cj

n +Cj−1n

2Δx

%

& ' '

(

) * *

€

∂C∂t

= −u0∂C∂x

central difference central difference

Spectral method

€

x j = jΔx, j =1,2,...,N, where NΔx = L.

Set of uniformly spaced grid points in one dimension

The Fourier series of C, whoose values are given only at N grid points, requires N Fourier coefficients. The Fourier expansion of C is

€

C x j( ) = Ck exp 2πikx j /L( )k=1

N∑ = Ck exp 2πikjΔx / NΔx( )( )

k=1

N∑ = Ck exp 2πikj /N( )

k=1

N∑

Ck are the Fourier coefficients. The inverse of this equation yields the complex Fourier coefficients from the grid point values:

€

Ck =1N

C x j( )j=1

N∑ exp −2πikj /N( ) (2)

Spectral methodFor the periodic domain, L, the time-dependent distribution of C can be expressed as

Substituting this equation into the linear advection equation (1) yields N independent first order ordinary differential equations :

At initial time, t=0, we need to compute the complex Fourier coefficients from eq. 2 and then integrate the system (3) in time, implying that we need only to numerically approximate the time derivative. Note that since the Fourier coefficients are complex, (3) represents a system of 2N equations

€

C x,t( ) = Ck t( )exp 2πikx /L( )k=1

N∑

€

dCkdt

= −2πiu0kL

Ck for k =1,2,...,N

€

∂C∂t

= −u0∂C∂x

(1)

(3)

Lecture 3

1D SHALLOW-WATER EQN’S

assume wave solution

dispersion relation

phase velocity

group velocity

WAVE ON A GRID: aliasing

long wave

short wavelimit

aliasing

1D SHALLOW-WATER EQN’S

only spatial discretization

assume wave solution

dispersion relation

phase velocity

group velocity

long wave short wave limit

(linear, non-rotational, inviscid)

�

cg* =

dωdk

= ± gH cos(kΔx)

LEAP-FROG SCHEME

centered in time & centered in spacestencil

grid waves2 decoupled grids

u,h

u,h

u,h

u,h u,h u,h u,h

u,h u,h u,h

u,h u,h

u,h

u,h

u,h

u,h

u,h

u,h u,h u,h

u,h

u,h

u,h

u,h

u,hu,h

u,h

LEAP-FROG SCHEME

centered in time & centered in spacestencil

no grid wavesstaggered grid

h h

u u u u

u u u

h h

u

h

h

h

h

u u u

h

u

h

hh

hh h

two times faster (but also coarser)

Dispersion relation• Numerical phase speeds

�

cu* =

ωk

=c

kΔxsin(kΔx)

�

cs* =

ωk

=2c

kΔxsin(

kΔx2

)

u = unstaggered grid s = staggered grid

2D SHALLOW-WATER EQN’S(linear, inviscid)

assume wave solution

dispersion relation

spatial discretization

again: short waves have too small phase velocity and a group velocity in the wrong direction

2D SHALLOW-WATER EQN’S(linear, inviscid)

assume wave solution

dispersion relation

short waves are gravity waves

long waves are inertial oscillations

STAGGERED GRIDSArakawa Aunstaggered

Arakawa B Arakawa C

h hh

v

interpolation

u* u*

v*

v*

uh

STAGGERED GRIDSArakawa Aunstaggered

Arakawa B Arakawa C

h

u

v

interpolation

u u uh* h*

v*

STAGGERED GRIDS

medium

coarse

fineArakawa B grid

Arakawa C - grid

True solution