Second year master programme

Transcription

1 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)

2 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 40 Thu. 2 Oct.: oral presentation of project write a report (<3000 words)

3 Schedule: SOAC 2014 ( 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

4 Programming languages Fortran C/C++ Pascal MATLAB Python

5 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

6 Python

7 Master thesis From 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

8 Introduction to numerical fluid dynamics for geophysical flows Anna von der Heydt Aarnout van Delden BBL 615

9 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. (

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11 Animations and lectures climate modelling High resolution climate model output Uncertainty in climate models Partial differential equations Cloud model output Ocean modelling GFDL visualization

12 Earth system models NCAR Community Earth System Model (CESM)

13 General circulation models (GCMs) Surface: ground temperature, water, energy, momentum, CO2 fluxes. Atmosphere grid box: wind vectors, humidity, clouds, temperature, chemical species. Ocean grid box: current vectors, temperature, salinity.

14 Resolution in IPCC simulations (IPCC 1990) (IPCC 2001) (IPCC 1996) (IPCC 2008)

15 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.

16 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. 2 nd Edition, Academic Press (2011) (chapters 5 & 6) Dale R. Durran, Numerical Methods for Fluid Dynamics With applications to geophysics, 2 nd Edition, Springer (2010)

17 SHALLOW-WATER EQN S (the e-coli of Geophysical Fluid Dynamics)

18 ADVECTION EQUATION characteristics solution what about boundary conditions?

19 FINITE DIFFERENCE forward backward central stencil

20 ACCURACY Taylor expansion truncation error finite difference approximation exact the lowest order of in the truncation error determines the accuracy

21 ACCURACY truncation error the lowest order of in the truncation error determines the accuracy

22 Lecture 2

23 Grading Exercises attendance first week: 20% of grade Oral presentation: 40% Written report: 40%

24 ADVECTION EQUATION notation: truncation error computer sets this to zero this scheme is first-order accurate in time and space

25 ADVECTION EQUATION 2 u t = c u 2 t x = c x truncation error u t = 2 u c2 x 2 numerical diffusion numerical dispersion

26 STABILITY (page 96 Durran) Von Neumann s Method note that insert Fourier series to represent the discretized solution at amplification factor at Analysis is restricted to linear equations, implying that amplification does not vary from time step to time step stable if

27 STABILITY insert euler forward / downwind rewriting for every most unstable mode always unstable

28 STABILITY insert euler forward / upwind rewriting for every most unstable mode stable if Courant Frederich Lewy condition

29 CFL CONDITION euler forward / downwind always unstable euler forward / upwind Courant Frederich Lewy condition stable unstable

30 IMPLICIT SCHEME insert 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

31 DIFFUSION EQUATION The finite difference approximation for the second derivative

32 DIFFUSION EQUATION euler forward / central difference accuracy stability and condition

33 Time differencing: C t = u 0 C x MATSUNO SCHEME Step 1 C * j n+1 C j n Δt % C( = u 0 ' * & x ) n j Step 2 C j n+1 C j n Δt % C *( = u 0 ' * & x ) n+1 j

34 A well-known finite difference scheme: C t = u 0 C x Lax-Wendroff scheme Taylor series: C n+1 j = C n $ j + Δt C ' & ) % t ( n j + Δt 2 $ 2 C' & % t 2 ) ( n j +... Since 2 C t 2 = u C x 2 C n+1 j = C n % j u 0 Δt C ( ' * & x ) n j + u 0 2 Δt 2 2 % 2 C( ' & x 2 * ) n j +...

35 C t = u 0 C x Lax-Wendroff scheme C n+1 j = C n % j u 0 Δt C ( ' * & x ) n j + u 0 2 Δt 2 2 % 2 C( ' & x 2 * ) n j +... which becomes: C j n+1 C j n Δt % n n C u j+1 C ( j 1 0 ' & 2Δx * + u 0 2 Δt % ) 2 ' & n C j+1 2C n n j + C j 1 2Δx ( * ) central difference central difference

36 Spectral method Set of uniformly spaced grid points in one dimension x j = jδx, j =1,2,..., N, where NΔx = L. 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 k exp( 2πikx j / L) C x j N = C k exp 2πikjΔx / NΔx = C k exp 2πikj / N k=1 N k=1 ( ( )) C k are the Fourier coefficients. The inverse of this equation yields the complex Fourier coefficients from the grid point values: N k=1 ( ) C k = 1 N N ( ) C x j j=1 exp( 2πikj / N) (2)

37 Spectral method C t = u 0 C x (1) For the periodic domain, L, the time-dependent distribution of C can be expressed as N C( x,t) = C k ( t)exp( 2πikx / L) k=1 Substituting this equation into the linear advection equation (1) yields N independent first order ordinary differential equations : dc k dt = 2πiu 0k L C k for k =1,2,..., N (3) 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

38 Lecture 3

39 1D SHALLOW-WATER EQN S assume wave solution dispersion relation phase velocity group velocity

40 WAVE ON A GRID: aliasing long wave short wave limit aliasing

41 1D SHALLOW-WATER EQN S (linear, non-rotational, inviscid) only spatial discretization assume wave solution dispersion relation phase velocity long wave short wave limit group velocity c g * = dω dk = ± gh cos(kδx)

42 LEAP-FROG SCHEME centered in time & centered in space stencil 2 decoupled grids grid waves 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,h u,h u,h

43 LEAP-FROG SCHEME centered in time & centered in space stencil staggered grid no grid waves two times faster (but also coarser) h u h u h u h u h h u h u h u h u h h u h u h u h u h

44 Numerical phase speeds c u * = ω k = u = unstaggered grid Dispersion relation c kδx sin(kδx) c * s = ω k = 2c kδx sin(kδx 2 ) s = staggered grid

45 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

46 2D SHALLOW-WATER EQN S (linear, inviscid) assume wave solution dispersion relation long waves are inertial oscillations short waves are gravity waves

47 STAGGERED GRIDS Arakawa A unstaggered Arakawa B v* Arakawa C h u h u* h u* h v v* interpolation

48 STAGGERED GRIDS Arakawa A unstaggered Arakawa B Arakawa C h u v u h* u h* u v* interpolation

49 STAGGERED GRIDS fine Arakawa B grid Arakawa C - grid medium True solution coarse