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Use of Mathematics in Modelling Climate : By Zhang Chenxi

Zhang Chenxi

Use of Mathematics in Modelling Climate : By Zhang Chenxi

To most people, they might be only familiar with the more well-known fraternal twin, pure Mathematics, characterised by its abstract, computational nature. Indeed, the solving of Math problems at a pre-university level is limited to the grueling lucubrations of calculus equations, graphing of hyperbolic functions, or the ironic calculations of imaginary numbers that only exists in unreality, to the extent that Math becomes perceived as an abstract subject divorced from reality. Contrary to popular belief, the study of Applied Mathematics is in fact very much relevant to the solving of salient, real-life problems of today’s society. In my essay today, I shall discuss the burgeoning corpus of Mathematical models employed to model the climate on Mother Earth.

There is a plethora of numerical models used to simulate the interactions and dynamics of the Earth’s system, and the derivations obtained from the models can be used to extrapolate future behaviours of Earth, therein influencing the attitudes multiple stakeholders take towards tackling climate change. Whilst most of the models hinge on persipacious intuition of physical processes, assumptions are often adopted to simplify the complex systems into quantifiable variables and parameters which can be modelled by mathematical equations. Examples of such assumptions include modelling the ocean as an incompressible fluid[1] in the predictions of tsunamis.

Firstly, box models are used in the study of spatial processes in the earth’s atmosphere to model how a quantity changes with time. The change per unit time step can be intuitively expressed as the production or destruction of the variable[2] in the box itself, as well as the change in the variable as a result of convection from the neighbouring boxes. This allows us to derive systems of first order differential equations, therein deriving numerical solutions for how these measured variables changes with time. For instance, if the carbon cycle is monitored to model how the carbon dioxide levels in atmosphere changes with time, the box model[3] factors in the increase of carbon dioxide from the plants, soils, human activity, and its decrease due to various chemical processes which absorb carbon dioxide. Another example of how the box model can be used to derive more elaborate systems are general circulation models, where both space and time are broken up into infinitesimal components.

General Circulation Models study only the atmospheric and oceanic processes, notably using the Navier-Stokes equations to model the dynamics of fluids in these spheres, in terms of its surface pressure, velocity, temperature, and various other factors. Due to the simplified nature of box models, they are useful in decomposing complex physical systems into components that can be easily analyzed to obtain valuable insights. Of course, just as it is lauded for its ability to elucidate the physical processes in an accessible manner, box models are criticised for assuming spatial homogeneity and negating how various quantities may vary with location. As a result, it is often complemented with more complex and larger-scale models like the Earth System Models.

These aforementioned models differ from conventional climate models by seeking to simulate all the aspects of Earth system, including plant ecology, human activities, chemical processes which also have a sizeable impact on the concentration of greenhouse gases. The increase in scope and the interplay between the interdependent biological, chemical and physical systems. These models are computationally expensive, requiring much manpower and state-of-the-art research to maintain. One of the most widely known Earth System Model used is the Community Earth System Model developed by National Center for Atmospheric Research to come up with a comprehensive prediction[4] of the Earth’s climate. The new model enabled scientists to consider the influences of greenhouse gases on marine ecosystems, which opened the new window to a much wider variety of applications[5]. This also allowed more literature on such cross-disciplinary processes, since previous models tend to only consider the Earth’s climate in separate components.

While the models mentioned above take into account the spatial factors of Earth, the energy balance models are another viable conceptual tool employed to provide insight into climate change. By focusing only on the Earth’s near-surface air temperature averaged over the entire surface, it eliminates considerations of spatial factors like the ocean currents, and the Earth’s rotation[6], with much of the model being largely governed by laws of thermodynamics to monitor the rate of change of thermal energy entering and leaving Earth. Some source of these change in energy include black body radiation, reflection of heat from Sun, greenhouse gases which absorb the thermal energy in the atmosphere[7], and similar factors that add a layer of complexity to the model. While they might not represent the atmospheric processes in a realistic way, this model is still widely used in climate modelling literature for its quick, accessible nature[8], and how the model responds to a small perturbation in the system can also inspire the derivation of higher order, more complex models.

As one major function of these models are to make predictions on future behaviours of the climate, there has to be uncertainties in the measured variables, as well as the errors from the quantitative models. Since contemporary climate models are nonlinear and tough to understand, statistical models are necessary to assess these uncertainties[9] and make more accurate, comprehensive, probabilistic forecasts. These are developed based on the analysis of the patterns in past weather behaviors to derive all the possible outcomes with randomness present, and measuring these against the current state of the climate. New methods are developed to model the anomalies and turning points of the climate, to better analyse these crucial statistical points and obtain a better understanding of the climate.

In conclusion, while there is contemporary models covers extensive details about the current climate, there’s still components of the climate not addressed by these models, for instance aerosol composition which can change the Earth’s climate sensitivity[10] and the inability to simulate clouds accurately. Retrospectively, there’s still much room for improvements in terms of addressing the uncertainties caused by the assumptions made in the model, though we can be optimistic about the improving the complexities of the model with larger computing power in the future.

References:

  1. Ali Abdolali, Usama Kadri, James T.Kirby, (2019), Effect of Water Compressibility, Sea-floor Elasticity, and Field Gravitational Potential on Tsunami Phase Speed, https://www.nature.com/articles/s41598-019-52475-0
  2. Dyminikov VP, Mathematical Models for Proving Climate Change, Mathematical Matters of Life Support Systems, Volume 1
  3. R. Avenhaus, S.Fenyi, H.Frick(1979), Box models for the CO2 cycle of the earth, Environment International Volume 2, Issues 4-6, https://ocw.nagoya-u.jp/files/524/chapter7.pdf
  4. James W Hurrell, M.M. Holland, P.R. Gent, S.Ghan, Jennifer E.Kay, P.J. Kushner, J.F. Lamarque, W.G.Large, D.Lawrence, K.Lindsay, W.H.Lipscomb, M.C. Long, N.Mahowald, D.R.Marsh, R.B.Neale, P.Rasch, S.Vavrus, M.Vertenstein, D.Bader, W.D Collins, J.J.Hack, J.Kiehl and S.Marshall(2013), The Community Earth System Model, A Framework for Collaborative Research, American Meteorological Society, https://www.osti.gov/servlets/purl/1565081
  5. NCAR(2010), The Community Earth System Model, NCAR’s latest step in the quest to model Earth‘s climate, https://news.ucar.edu/2746/community-earth-system-model
  6. Gerrif Lohmann, Temperatures from energy balance models: the effective heat capacity matters, https://esd.copernicus.org/articles/11/1195/2020/
  7. Daniel Flath, Hans G.Kaper, Frank Watterberg, Esther Widiasih(2012), Energy Balance Models, http://dimacs.rutgers.edu/archive/MPE/Energy/DIMACS-EBM.pdf
  8. T S Ledley(2003), Energy Balance Model, Surface, Cambridge https://curry.eas.gatech.edu/Courses/6140/ency/Chapter9/Ency_Atmos/Energy_Balance_Model_Surface.pdf
  9. Peter F.Craigmile, The Role of Statistics in Climate Research, Chance, https://chance.amstat.org/2017/11/climate-research/
  10. Chad Small(2022), What’s wrong with these climate? Bulletin of the Atomic Scientists. https://thebulletin.org/2022/12/whats-wrong-with-these-climate-models/#:~:text=While%20the%20global%20climate%20models,sensitivity%2C%20and%20consequently%20ocean%20temperatures.

By Zhang Chenxi (Raffles institution)