SEE6212/8213: Additional details on project
topics
November 12, 2020
Here are some additional details on the recommended topics. More
straightforward topics are in italics.
1 Applications
1. Point-vortex dynamics (fluid dynamics, ODEs)
Simulating turbulent flow is computationally expensive. One way of
mimicking the behaviour of turbulent fluids is to consider a collection
of interacting point vortices. This is also a simple model of so-called
oceanic eddies and intense atmospheric storms.
For this topic, you should integrate the equations of motion for different
numbers of vortices in a 2-D periodic domain and relate the behaviour
to what’s described in papers and textbooks. If you’re ambitious you
can examine other things (e.g. unequal vortex strengths, background
shear, scaling with respect to the number of vortices, N).
2. Large-scale atmospheric winds in different regions of the atmosphere∗
(atmospheric science, finite differences)
In class we reviewed the so-called geostrophic approximation for large-
scale atmospheric winds.
For this topic, you should examine the accuracy of this approximation
using atmospheric data. In particular, you should consider the accu-
racy of this approximation as a function of height, season and averaging
period. You should also discuss the effect of higher-order approxima-
tions.
1
This project is not recommended for PhD students.
3. Effect of spatial and temporal interpolation on offline advection∗ (in-
terpolation, ODEs, air pollution/atmospheric science)
In class we described the equations for offline advection of particles.
For this topic, you will write a numerical code that works with real
atmospheric data. To make things interesting, you should integrate
∼ 10000 particles for at least 72 hours. You should do so on at least
one vertical level. To keep things simple, you can limit yourself to 2-D
calculations. The curvature of the Earth needs to be taken into account.
You should examine the sensitivity to temporaal sampling frequency,
temporal interpolation and spatial interpolation. Your results should
be compared to those from HYSPLIT.
4. Offline chemical transport modelling (ODEs, atmospheric chemistry)
In class we described simple chemical box models. They’re useful for
understanding the chemistry, but rather limited when it comes to real-
world applications.
Chemical transport models, which solve for dynamics and chemistry
simultaneously, are complicated. Offline chemical transport models
solve the chemical equations given a prescribed velocity field.
For this topic, you can consider a prescribed velocity field, e.g.
(u, v) = (sin(y), cos(x)). (1)
on the domain [0, 2pi] × [0, 2pi]. For simplicity, you may want to focus
on the 1-D periodic case,
u = sin(x) (2)
on [0, 2pi]. If you’re ambitious, you can look at real reactions but this
is not crucial: a simple bimolecular reaction is fine. For simplicity,
focus on the limit of fast chemistry, in which the reaction rates are fast
compared to advection. You should discuss what happens when this
restriction is relaxed.
5. Lagrangian stochastic modelling∗ (ODEs, stochastic differential equa-
tions, air pollution/atmospheric science)
The advection equations reviewed in class are incomplete. They don’t
include the effect of diffusion or small-scale turbulence. This effect can
be incorporated through the inclusion of a diffusion or noise term.
2
There are different versions of this project. (i) You will implement
the diffusion term in an offline advection model and apply it to at-
mospheric data. The basic setup should follow that of Project 1.3
(ii) You will compare the equations to the solution of a diffusion (or
advection-diffusion) equation. You need to determine how to determine
the diffusion coefficient.
For more on the underlying mathematics, you can look for references on
Brownian Motion or stochastic differential equations. This is a useful
Python tutorial.
6. Master Chemical Mechanism (ODEs, atmospheric chemistry)
Many chemical mechanisms exist for tropospheric chemistry. A well-
known one is the so-called Master Chemical Mechanism.
Write Python code that works with MCM data. Consider different
reactions sets (e.g. 10 or 100 reactions).
Note: this project is more suitable for PhD students in air quality/at-
mospheric chemistry.
7. Anisotropic diffusion (linear systems, PDEs)
For some physical problems, diffusion is represented using a diffusivity
tensor rather than a constant diffusivity, e.g. ∇ · ∇κθ = κxx ∂2θ∂x2 +
κyy
∂2θ
∂y2
+ 2κxy
∂θ2
∂y∂y
.
In this problem, you will solve the 2-D diffusion equation for isotropic
and anisotropic diffusion. For simplicity, 2-D periodic boundary con-
ditions can be assumed but this isn’t crucial. A classical problem is
diffusion through an array. The results should be related to a physi-
cal problem. The time-dependent problem is more interesting but you
can also look at the steady-state problem if you can find a relevant
application
Note: this problem may be relevant to students interested in environ-
mental or chemical engineering.
2 Numerical methods
1. Spectral methods
We have emphasised finite differences methods in this course. However,
there are other methods. The spectral methods works extremely well
for certain types of problems.
3
Solve a 2-D PDE using the spectral method. For simplicity, you can
consider the diffusion equation in 2-D periodic domain. Compare your
results to a finite-difference or iterative solution.
This project can be done by MSc students so long as they are comfort-
able with Fourier analysis.
2. Finite element methods
The finite element method is another alternative to finite differences.
It’s closely related but has some important advantages. For example,
it’s more suitable for complex geometries or boundary conditions.
Solve a 2-D PDE using the finite element method. For simplicity, you
can consider the diffusion equation in complex 2-D domain (see Topic
1.7).
This project is recommended only for ambitious PhD students.
3. Volume or mass-preserving integration algorithms
For certain problems, it’s desirable that the volume (area) or mass be
preserved. There are many integration algorithms that do this. They
are usually referred to a geometric or symplectic integrators.
Apply these integrators to a long integration of a set of ODEs. An
interesting set is the so-called three-body problem in astronomy. Al-
ternatively you may apply it to advection of tracer particles by a time-
dependent velocity field (suggestions can be provided upon request).
This project recommended only for PhD students who are interested
in physics or numerical methods.
4. Stiff ODE solvers for atmospheric chemistry
We mentioned in class that most atmospheric chemistry models use
some sort of stiff solver. We did not, however, discuss how they work.
In this project, you will implement a stiff solver and compare it to RK4
for a set of stiff chemical reactions of your choice.
This project may be appropriate for PhD students working on atmo-
spheric chemistry or air pollution.
5. Parallel/GPU computing∗
Modern scientific computing uses multiple processes. This can greatly
improve the performance of many algorithms.
4
The numba package can be used to general parallel code. Use it to
develop a parallel version of one of the algorithms discussed in class.
An obvious choice is our simple algorithm for particle advection. Your
parallel code can be applied to the basic problem described in Problem
1.3.
This project is intended for students who are interested in high-performance
scientific computing.
3 Atmospheric data (i.e. topics with aster-
isks)
For Problems 1.2, 1.3, 1.5 and 2.5, atmospheric data (e.g. winds) are re-
quired. This data can be obtained from this site. For the wind data, look
for u, v components output four times daily. The accompanying Python pro-
gram, netcdfread.py, converts a downloaded netcdf file into a text file. You
can modify the program to suit your needs.
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