# Atmospheric and Oceanic Physics: Climate Modelling

## Using Observations to Accurately and Efficiently Model Turbulent Flows: Parameter Recovery, Sensitivity Analysis, Nonlinear Data Assimilation Algorithms, and a Real-World Implementation.

One of the challenges of the accurate simulation of turbulent flows is that initial data is often incomplete. Data assimilation circumvents this issue by continually incorporating the observed data into the model. A new approach to data assimilation known as the Azouani-Olson-Titi algorithm (AOT) introduced a feedback control term to the 2D incompressible Navier-Stokes equations (NSE) in order to incorporate sparse measurements. The solution to the AOT algorithm applied to the 2D NSE was proven to converge exponentially to the true solution of the 2D NSE with respect to the given initial data. In this talk, we present our tests on the robustness, improvements, and implementation of the AOT algorithm, as well as generate new ideas based off of these investigations. First, we discuss the application of the AOT algorithm to the 2D NSE with an incorrect parameter and prove it still converges to the correct solution up to an error determined by the error in the parameters. This led to the development of a simple parameter recovery algorithm, whose convergence we recently proved in the setting of the Lorenz equations. The implementation of this algorithm led us to provide rigorous proofs that solutions to the corresponding sensitivity equations are in fact the Fréchet derivative of the solutions to the original equations. Next, we present a proof of the convergence of a nonlinear version of the AOT algorithm in the setting of the 2D NSE, where for a portion of time the convergence rate is proven to be double exponential. Finally, we implement the AOT algorithm in the large scale Model for Prediction Across Scales - Ocean model, a real-world climate model, and investigate the effectiveness of the AOT algorithm in recovering subgrid scale properties.

### Speaker Biography

Elizabeth Carlson, is a homeschooler turned math PhD! She grew up in Helena, MT, USA, where she also graduated from Carroll College with a Bachelor's in mathematics and minor in physics. She became interested in fluid dynamics as an undergraduate, and followed this interest through her graduate work at the University of Nebraska - Lincoln in Lincoln, NE, USA, where she just earned my PhD in May 2021. Her research focus is in fluid dynamics, focusing on the well-posedness of systems of partial differential equations and numerical computations and analysis in fluid dynamics. In her free time, she enjoys hiking, playing piano, reading, and martial arts.

Read more about Elizabeth Carlson on our PIMS Medium blog here.

## Data accuracy for risk management in changing climate

The decade of the 2010s was the hottest yet in more than 150 years of global mean temperature measurements. The key climate change signatures include intensifying extreme events such as widespread droughts, flooding and heatwaves, severe impacts on human health, food security, ecology, and species biodiversity. Climate has been changing from ice-age and is expected to change in future, yet the rate of change is alarming. Data plays a crucial role in developing risk management, mitigation and adaptation strategies under changing climate conditions. This talk focuses on uncertainties in hydrological data and the subsequent effect on extreme events like floods, droughts and heatwaves. Projected changes along with apparent biases in the global climate models, tools available for understanding future climate, are discussed. Importance of understanding uncertainties in observations and simulations and the need to probabilistically evaluate simulations to identify those that agree with observations is emphasized. Finally, the effect of data accuracy and incorporating uncertainty in informed decisions and risk management strategies is highlighted through a case study.

### Speaker Biography

Chandra Rajulapati is a GWF-PIMS PDF, working with Dr. Simon Papalexiou at the Global Institute for Water Security (GIWS), University of Saskatchewan, on the Global Water Futures (GWF) project. She obtained her doctoral degree from the Indian Institute of Science (IISc) Bangalore, India, under the supervision of Prof. Pradeep Mujumdar. Her research focuses on understanding historical and future changes in hydroclimatic variables like precipitation and temperature at different scales, estimating risk due to extreme events like floods, droughts and heatwaves, and developing sustainable water management systems, risk assessment, adaptation and mitigation strategies.

## 2016 Graduate Mathematical Modelling in Industry Workshop

This gallery contains photos from the 2016 Graduate Mathematical Modelling in Industry Workshop. See the event webpage for more information.

## The long road to 0.075: a statistician’s perspective of the process for setting ozone standards

The presentation will take us along the road to the ozone standard for the United States, announced in Mar 2008 by the US Environmental Protection Agency, and then the new proposal in 2014. That agency is responsible for monitoring that nation’s air quality standards under the Clean Air Act of 1970. I will describe how I, a Canadian statistician, came to serve on the US Clean Air Scientific Advisory Committee (CASAC) for Ozone that recommended the standard and my perspectives on the process of developing it. I will introduce the rich cast of players involved including the Committee, the EPA staff, “blackhats,” “whitehats,” “gunslingers,” politicians and an unrevealed character waiting in the wings who appeared onstage only as the 2008 standards had been formulated. And we will encounter a couple of tricky statistical problems that arose along with approaches, developed by the speaker and his coresearchers, which could be used to address them. The first was about how a computational model based on things like meteorology could be combined with statistical models to infer a certain unmeasurable but hugely important ozone level, the “policy related background level” generated by things like lightning, below which the ozone standard could not go. The second was about estimating the actual human exposure to ozone that may differ considerably from measurements taken at fixed site monitoring locations. Above all, the talk will be a narrative about the interaction between science and public policy - in an environment that harbors a lot of stakeholders with varying but legitimate perspectives, a lot of uncertainty in spite of the great body of knowledge about ozone and above all, a lot of potential risk to human health and welfare.

## Conference on the Mathematics of Sea Ice

Interesting mathematics arises in many areas of the study of sea ice and its role in climate. Partial differential equations, numerical analysis, dynamical systems and bifurcation theory, diffusion processes, percolation theory, homogenization and statistical physics represent a broad range of active fields in applied mathematics and theoretical physics which are relevant to important issues in climate science and the analysis of sea ice in particular.

## Climate Change – does it all add up?

Climate change has the potential to affect all of our lives. But is it really happening, and what has maths got to do with it?

In this talk I will take a light hearted view of the many issues concerned with predicting climate change and how mathematics and statistics can help make some sense of it all. Using audience participation I will look at the strengths and weaknesses of various climate models and we will see what the math can tell us about both the past and the future of the Earth's climate and how mathematical models can help in our future decision making.

## Reconstructing carbon dioxide for the last 2000 years: a hierarchical success story

Knowledge of atmospheric carbon dioxide (CO2) concentrations in the past are important to provide an understanding of how the Earth's carbon cycle varies over time. This project combines ice core CO2 concentrations, from Law Dome, Antarctica and a physically based forward model to infer CO2 concentrations on an annual basis. Here the forward model connects concentrations at given time to their depth in the ice core sample and an interesting feature of this analysis is a more complete characterization of the uncertainty in "inverting" this relationship. In particular, Monte Carlo based ensembles are particularly useful for assessing the size of the decrease in CO2 around 1600 AD. This reconstruction problem, also known as an inverse problem, is used to illustrate a general statistical approach where observational information is limited and characterizing the uncertainty in the results is important. These methods, known as Bayesian hierarchical models, have become a mainstay of data analysis for complex problems and have wide application in the geosciences. This work is in collaboration with Eugene Wahl (NOAA), David Anderson (NOAA) and Catherine Truding.

## Mathematics and the Planet Earth: a Long Life Together II

When Colombus left Spain in 1492, sailing West, he knew that the Earth was round and was expecting to land in Japan. Seventeen centuries earlier, around 200 BC, Eratosthenes had shown that its circumference was 40,000 km, just by a smart use of mathematics, without leaving his home town of Alexandria. Since then, we have learned much more about Earth: it is a planet, it has an inner structure, it carries life , and at every step mathematics have been a crucial tool of discovery and understanding. Nowadays, concerns about the human footprint and climate change force us to bring all this knowledge to bear on the global problems facing us. This is the last challenge for mathematics: can we control change?

This is a two-part lecture, investigating how our idea of the world has influenced the development of mathematics. In the first lecture on July 15, I will describe the situation up to the twentieth century, in the second one on July 17 I will follow up to the present time and the global challenges humanity and the planet are facing today.

## Mathematics and the Planet Earth: a Long Life Together I

When Colombus left Spain in 1492, sailing West, he knew that the Earth was round and was expecting to land in Japan. Seventeen centuries earlier, around 200 BC, Eratosthenes had shown that its circumference was 40,000 km, just by a smart use of mathematics, without leaving his home town of Alexandria. Since then, we have learned much more about Earth: it is a planet, it has an inner structure, it carries life , and at every step mathematics have been a crucial tool of discovery and understanding. Nowadays, concerns about the human footprint and climate change force us to bring all this knowledge to bear on the global problems facing us. This is the last challenge for mathematics: can we control change?

This is a two-part lecture, investigating how our idea of the world has influenced the development of mathematics. In the first lecture (July 15), I will describe the situation up to the twentieth century, in the second one (July 17) I will follow up to the present time and the global challenges humanity and the planet are facing today.

## Warming Caused by Cumulative Carbon Emissions: the Trillionth Tonne

The eventual equilibrium global mean temperature associated with a given stabilization level of atmospheric greenhouse gas concentrations remains uncertain, complicating the setting of stabilization targets to avoid potentially dangerous levels of global warming. Similar problems apply to the carbon cycle: observations currently provide only a weak constraint on the response to future emissions. These present fundamental challenges for the statistical community, since the non-linear relationship between quantities we can observe and the response to a stabilization scenario makes estimates of the risks associated with any stabilization target acutely sensitive to the details of the analysis, prior selection etc. Here we use ensemble simulations of simple climate-carbon-cycle models constrained by observations and projections from more comprehensive models to simulate the temperature response to a broad range of carbon dioxide emission pathways. We find that the peak warming caused by a given cumulative carbon dioxide emission is better constrained than the warming response to a stabilization scenario and hence less sensitive to underdetermined aspects of the analysis. Furthermore, the relationship between cumulative emissions and peak warming is remarkably insensitive to the emission pathway (timing of emissions or peak emission rate). Hence policy targets based on limiting cumulative emissions of carbon dioxide are likely to be more robust to scientific uncertainty than emission-rate or concentration targets. Total anthropogenic emissions of one trillion tonnes of carbon (3.67 trillion tonnes of CO2), about half of which has already been emitted since industrialization began, results in a most likely peak carbon-dioxide induced warming of 2○C above pre-industrial temperatures, with a 5-95% confidence interval of 1.3-3.9○C.