**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, 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.