Research topic: The mathematics of climate change and modeling
Research Questions:
How to model the climate and how to solve it? Many models are run using linear systems and by integrating the differential equations. This can be difficult to solve analytically however due to the number of variables. They divide the earth’s atmosphere into a finite number of boxes (grid cells). Assuming that each variable has the same value throughout the box and write a budget for each box, defining the changes within the box, and the flows between the boxes. When we identify the variables, we have to understand the equation. Models of the earth’s climate are based on laws of physics: Ideas gas law; Conservation of energy; Conservation of momentum; Conservation of mass. We want to solve for the values of the variables described by these equations over time. Essentially we have seven (or more) variables described by the same number of equations that describe change with respect to time. (T, p, ρ, u, v, w, ρ(water), ρ(ice), etc.). So we should be able to solve for the values of the variables through time.
Climate change is a case of modeling where the next state relies on the previous state, so Markov chains are one way of creating a model. These Markov chains can then be approximated using differential equations, but the same difficulty arises with solving the equations analytically.
Methodology:
First, we will be looking into how to design a climate model. There are different kinds of models such as box models are simplified versions of complex systems, reducing them to boxes linked by fluxes and zero-dimensional models, higher-dimension models etc. This research project is about a model for climate change. The Global Climate Models ( GCMs) where we divide the Earth’s atmosphere into a finite number of boxes (grid cells). Assume that each variable has the same value throughout the box. GCMs use mathematical equations to describe the behavior of factors of the Earth system that impact climate. These factors include dynamics of the atmosphere, oceans, land surface, living things, and ice, plus energy from the sun.
And then we will be looking at other researches and papers that are done for the climate modeling. We will be going over those researches and papers to get the idea of what people are looking at and how they are designing the climate model themselves as well as how mathematics involved in this process.
We will be looking at mathematical equations that are being used to develop climate models. GCMs apply the discrete equations for fluid motion and integrate these forward in time. There are equations for mass continuity, motion, thermodynamic, chemical continuity. We will be looking at these equations closely.
Expect to find:
We are trying to put together a general climate model that provides good indicators as to how our climate changes. This can help us to understand more about how global climate varies and how those changes will affect the local climate. Most GCMs can provide a reasonable representation of regional climatic features but they tend to have larger spatial scales. When we look at a very local area, the GCMs may be inappropriate.
For global climate, we expect to obtain a comprehensive model which hopefully includes not only changes in the atmosphere but also changes in hydrosphere, cryosphere, biosphere and even in land surface and solar inputs. For local climate, on the other hand, we still need to start with GCMs and then downscale to a certain region. The model should be able to produce at least temperature, precipitation, wind speed and direction in a certain area and we will be able to compare these results with data from observations.
Resources:
https://ocw.mit.edu/courses/sloan-school-of-management/15-023j-global-climate-change-economics-science-and-policy-spring-2008/lecture-notes/lec3.pdf
（this is the main source you will be using）
https://pdfs.semanticscholar.org/cdd5/f9267ec4fb3ed30a2d26940b6c146f05d0a5.pdf
https://issm.jpl.nasa.gov/documentation/
https://www.ams.org/notices/200806/tx080600695p.pdf
https://plus.maths.org/content/taxonomy/term/868
https://scied.ucar.edu/longcontent/climate-modeling
Comprehensive
analysis
of
the
findings
of
your
research.
What
implications
do
your
findings
have?
What
are
the
main
points?
You
should
specifically
address
and
answer
your
research
questions
here.
4
page,
double
space,
please
read
all
the
instructions
below
and
especially
read
all
the
resources
provided.
Correctly
cite
all
the
source
you
all
be
using.
You
need
to
answer
all
questions
from
‘
research
questions
’
session
and
focuses
on
improving
‘
expect
to
find
’
session
with
findings.
The
content
of
‘
methodology
’
session
is
just
for
information
but
you
should
not
include
this
in
the
submission.
Research
topic:
The
mathematics
of
climate
change
and
modeling
Research
Questions:
How
to
model
the
climate
and
how
to
solve
it?
Many
models
are
run
using
linear
systems
and
by
integrating
the
differential
equations.
This
can
be
difficult
to
solve
analytically
however
due
to
the
number
of
variables.
They
divide
the
earth
’
s
atmosphere
into
a
finite
number
of
boxes
(grid
cells).
Assuming
that
each
variable
has
the
same
value
throughout
the
box
and
write
a
budget
for
each
box,
defining
the
changes
within
the
box,
and
the
flows
between
the
boxes.
When
we
identify
the
variables,
we
have
to
understand
the
equation.
Models
of
the
earth
’
s
climate
are
based
on
laws
of
physics:
Ideas
gas
law;
Conservation
of
energy;
Conservation
of
momentum;
Conservation
of
mass.
We
want
to
solve
for
the
values
of
the
variables
described
by
these
equations
over
time.
Essentially
we
have
seven
(or
more)
variables
described
by
the
same
number
of
equations
that
describe
change
with
respect
to
time.
(T,
p,
?
,
u,
v,
w,
?
(water),
?
(ice),
etc.).
So
we
should
be
able
to
solve
for
the
values
of
the
variables
through
time.
Climate
change
is
a
case
of
modeling
where
the
next
state
relies
on
the
previous
state,
so
Markov
chains
are
one
way
of
creating
a
model.
These
Markov
chains
can
then
be
approximated
using
differential
equations,
but
the
same
difficulty
arises
with
solving
the
equations
analytically.
john snowman
writinginstruction.docx
0
