http://www.uvm.edu/~mrfrank/thesis.html
Thesis Defense is Loading Data......
Please Wait.
The purpose of this webpage is to enjoy an INTERACTIVE thesis
defense experience because papers should be interactive. Please go to http://www.uvm.edu/~mrfrank/thesis.html
and follow along!
Standing Swells Surveyed Showing Surprisingly Stable Solutions for the Lorenz '96 System
Morgan R. Frank
Department of Mathematics & Statistics
Vermont Complex Systems Center
Computational Story Lab
Vermont Advanced Computing Center
University of Vermont
The purpose of this webpage is to enjoy an INTERACTIVE thesis
defense experience because papers should be interactive. Please go to http://www.uvm.edu/~mrfrank/thesis.html
and follow along!
Motivation
-
scientists are interested in improving prediction techniques. e.g.
- weather forecasting
- atmospheric forecasting
-
the Lorenz '96 system (L96) is ideal for testing prediction
- tunable dynamics
- tunable dimensionality
- computationally tractable
- how hard is it to predict L96 for different parameter choices?
|
|
The Lorenz '96 System
-
Originally,
\begin{equation}
\frac{dx_{i}}{dt}=x_{i-1}(x_{i+1}-x_{i-2})-x_{i}+F
\end{equation}
where $i=1,2,\dots,I$ and $F$ is the forcing parameter. Also, $x_{i+I}=x_{i-I}=x_{i}$.
- Now
\begin{equation}
\frac{dx_{i}}{dt}=x_{i-1}(x_{i+1}-x_{i-2})-x_{i}+F-\frac{hc}{b}\displaystyle\sum_{j=1}^{J}y_{(j,i)}
\end{equation}
\begin{equation}
\frac{dy_{(j,i)}}{dt}=cby_{(j+1,i)}(y_{(j-1,i)}-y_{(j+2,i)})-cy_{(j,i)}+\frac{hc}{b}x_{i}
\end{equation}
where $j=1,2,\dots,J$.
- $b$ & $c$ indicate difference in timescales in solutions for eq (1) and eq (2)
- $h$ is the coupling parameter
- We fix $b=c=10$ and $h=1$ (unless otherwise noted).
The Lorenz '96 System
|
\begin{equation}
\frac{dx_{i}}{dt}=x_{i-1}(x_{i+1}-x_{i-2})-x_{i}+F-\frac{hc}{b}\displaystyle\sum_{j=1}^{J}y_{(j,i)}
\end{equation}
\begin{equation}
\frac{dy_{(j,i)}}{dt}=cby_{(j+1,i)}(y_{(j-1,i)}-y_{(j+2,i)})-cy_{(j,i)}+\frac{hc}{b}x_{i}
\end{equation}
|
- blue dots represent values of slow oscillators
- distance from the origin indicates the difference in the slow oscillators state and the lowest state of the trajectory
- the red curve is a cubic spline interpolation of the slow oscillator states
- the green curve is a cubic spline interpolation of the fast oscillator states
|
|
Example Dyanmics by Dimensionality
$I=4$, $J=8$, $F=14$
 |
 |
 |
$I=10$, $J=5$, $F=14$
 |
 |
 |
Lyapunov Exponents
Integration:
- Pick $I$, $J$, & $F$
- Generate random initial state
- Integrate the system 500 model time units using Runge-Kutta method
of order 4 with timesteps of size $10^{-3}$ without analysis
- Integrate for 500 more model time while performing analysis
Calculating
Lyapunov Exponents:
\begin{equation}
L_{i}(\vec{v})\approx\frac{1}{\Delta time_{total}}\displaystyle\sum_{n=1}^{N} \ln(|f(\vec{v}_{i}^{(n)})|)
\end{equation}
Calculating
Lyapunov Dimension:
\begin{equation}
L=D+\frac{1}{|L_{D+1}(\vec{v})|}\displaystyle\sum_{d=1}^{D}L_{d}(\vec{v})
\end{equation}
This demo loads a heat map according to user specification above. Mouse over the heat map and select a cell to being an integration of a L96 system.
Picking large $\bf I$ or $\bf J$ will lead to slower demos due to the more complicated integrations required.
Furthermore, longer runtimes will be required to resolve transient activity.
F=
You Selected I=, J=
What's With the Stable Regions?
Periodograms
- frequencies can be found in Fourier space
- we use the FFT algorithm to accomplish this quickly
Is the Stability Prevalent?- A Frequency Analysis
Bifurcation Diagrams by F and J
Bifurcation Diagrams by F and J
What Are the Effects of J?
Recall:
\begin{equation}
\frac{dx_{i}}{dt}=x_{i-1}(x_{i+1}-x_{i-2})-x_{i}+F-\frac{hc}{b}\displaystyle\sum_{j=1}^{J}y_{(j,i)}.
\end{equation}
How does the coupling parameter, $h$, relate to the effects of $J$?
($J$=50, $F$=14)
|
from earlier....
|
We see that increasing $h$ produces a similar figure to increasing $J$
Conclusions & Future Work
- our sampling of parameter space can inform prediction scientists
- a range of choices of $J$ leads to system-wide dampening of dyanmics, similar to increasing $h$
- future work includes an analytic explanation for this relationship
- system stability presents itself as standing waves among the slow oscillators
Special Thanks to:
- MITRE Corporation for funding
- Mathematics and Climate Research Network for funding
- my advisor, Prof. Danforth
- my other advisor, Prof. Dodds
- my former advisor (but current friend), Prof. Dinitz
- my father, Stephen Frank
- my mother, Ellen Frank
- my brother, Justin Frank
- my fiance, Catherine Westbom
- my dog, Mocha Caffiene Plus
|
|
