Dynamics II: Sheet 2

Lecture: April 19. (Monday), 14:00 Prof. Dr. Gerrit Lohmann

Tutorial: April 19. (Monday),

Justus Contzen, Lars Ackermann

 
 

before April 19:

Watch videos for Rayleigh-Bénard convection and the Lorenz system

Experiments:

trailer Cellules de Bénard (1min),
Rayleigh–Bénard convection made with mix of cooking oil and small aluminium particles (5 min),
Was haben Benard-Zellen mit Kochen zu tun? (3 min, German)

Simulations:

Rayleigh Benard Thermal Convection with LBM (5 min),
Rayleigh Benard Thermal Convection 3D Simulation (2 min)
Sketch,
Clouds,
Cartoon

Bifurcations

Bifurcation youtube (20 min)
Bifurcation Khan academy (13 min) [Bifurcation Khan academy](https://www.youtube.com/watch?v=ovJcsL7vyrk

 
 

April 19, 14:00: Lecture 2 (online G. Lohmann, 45 min)

klick here for the zoom session

 

Second lecture (link here)

Content in the script: Rayleigh-Bénard convection, Lorenz system, nonlinear dynamics, bifurcations, multiple equilibria

 

After the lecture: Read the script about Fluid-dynamical Examples and Stability Theory (Chapter 2)

Reading/learning (the sections with a star are voluntary). It might take 120 min.

 
Reading Bifurcation theory

 

April 19, Tutorial (online 30 min)

klick here for the zoom session

 

Exercise 2 introduced, questions to the exercise (15 min)

Further stuff with RStudio (15 min)

 

Homework 1: Solve Exercise 2

This might take 2 h.

 

 

Literature:

Holton, J.R., and Hakim, G. J., 2013: Introduction to Dynamical Meteorology, Academic Press, Oxford (UK). —Fifth edition / Gregory J. Hakim. ISBN 978-0-12-384866-6 pdf

Marchal, J., Plumb, R. A., 2008. Atmosphere, Ocean and Climate Dynamics: An Introductory Text. Academic Press, 344 pp; videos pdf

Lohmann, G., 2020: Climate Dynamics: Concepts, Scaling and Multiple Equilibria. Lecture Notes 2020, Bremen, Germany. (pdf of Chapter 2) (pdf of the full script)

R Core Team (2013). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. URL http://www.R-project.org/. An Introduction to R derived from an original set of notes describing the S and S-PLUS environments written in 1990–2 by Bill Venables and David M. Smith when at the University of Adelaide. Online document at https://cran.r-project.org/doc/manuals/r-release/R-intro.html.

Torfs, P., and & Brauer, C., 2014: A (very) short introduction to R

Fieguth, P., An Introduction to Complex Systems Society, Ecology, and Nonlinear Dynamics. Publisher textbook page at Springer ISBN 978-3-319-44605-9 1st ed. 2017, XII, 346 p. 243 illus., 178 illus. in color. link

Lorenz model

r=24
s=10
b=8/3
dt=0.01
x=0.1
y=0.1
z=0.1
vx<-c(0)
vy<-c(0)
vz<-c(0)
for(i in 1:10000){
x1=x+s*(y-x)*dt
y1=y+(r*x-y-x*z)*dt
z1=z+(x*y-b*z)*dt
vx[i]=x1
vy[i]=y1
vz[i]=z1
x=x1
y=y1
z=z1
}
plot(vx,vy,type="l",xlab="x",ylab="y",main="LORENZ ATTRACTOR")

r=5000/150000
#r=1/10
Bev=85000000
K=1
dt=0.01

N=150000/Bev

vN=c(0); vNp=c(0); vt=c(0)
vN[1]=N; vNp[1]=0; vt[1]=0

for(i in 2:100000){
N1=N+r*N*(1-N/K)*dt
vNp[i]=r*N*(1-N/K)
vN[i]=N1
vt[i]=i*dt
N=N1
}

plot(vt,vN,type="l",xlab="time [days]",ylab=" ",main="Logistic growth Corona: N(t)", lwd=2, lty="solid", cex.main=3, cex.lab=3, cex.axis=3,cex.lab=3)

plot(vt,vNp*Bev/100,type="l",xlab="time [days]",ylab=" ",main="New infections/100: intensive care medicine",cex.main=3, cex.lab=3,cex.axis=3,cex.lab=3)

max(vNp[]*Bev/100)

## [1] 7083.333

 

 

Overview of Dynamics II

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