It stores heat, greenhouse gases and gives back at a measured rate. It is the World’s Ocean. As a player in our understanding and predicting the climate on Earth, it has had a supporting role to the atmosphere. But things have changed and it is moving to center-stage.
To a great extent, our understanding of the climate comes from massive mathematical models. There are only a few around the world that make the cut in the Intergovernmental Panel on Climate Change’s list (see www.ipcc.ch ).
The idea of a full climate model, now often known as an Earth system model, is to create a mathematical replica of the Earth. We have only one Earth and so experimentation in the normal sense is impossible on the full climate system. But the computational output of a climate model can tell what is likely to happen under a particular scenario, for instance from a certain pattern of greenhouse gas emissions in the future.
The primary concern in climate change is the heating of the air that sustains life on Earth. We know the ocean is involved as heating leads to ice melting and that will lead to sea level rise. A myth is that the Ocean is a passive partner in this process. Its role in regulating the climate and moderating its change is enormous. But, in the twentieth century, climate models treated the ocean as a slab. If it is so important, how did they get away with such a crude representation? The answer lies in what the models were then used for: to test the prediction that our climate would be warming over the long term, upwards of fifty years. On such time scales, the variability of the ocean is not that important.
The climate community accepts that the case for climate change, as well as its causes in anthropogenic emissions, is now closed. The focus has shifted to prediction on shorter time-scales, driven by an interest in what will happen in five, ten, or twenty years. The thinking is that we are committed to considerable climate change already and we need to know how it will impact us in specific ways. On this time scale, the ocean will play a critical and active role.
It is a myth is that we know everything about the ocean from taking measurements. Of course, we know a lot more than we did in the past. This has been made possible through international efforts, such as the ARGO project ( www.argo.net ), and the availability of new autonomous vehicles taking measurements. These so-called gliders can be viewed as waterborne drones. But the ocean is vast and deep and what we observe is only a small fraction of the whole. So, how do we fill in all the missing information? The answer is again to use a mathematical model. We know how the ocean must work from its underlying physics. It all comes from classical mechanics and the mathematical equations were formulated over 200 years ago. Although new simplifications are coming along all the time, the basic set of equations used in the models have remained the same.
How does this help us fill in all the missing information? The picture to have here is of a massive three dimensional grid filling the ocean, both in breadth and depth. A full description of the ocean would consist of an assignment of physical variable values at each node of this grid. The physical variables will include: flow velocity, temperature, pressure, and density among others. The observations, coming from all these measuring instruments, will tell us the values of these quantities at some of the nodes. It is a myth that the remaining values can be found just by interpolation; there are simply too few nodes reporting and too many possibilities for how the interpolation might work. This is where the mathematical equations come in as they can tell us, through computation of the model, what constrains the physical variables across the entire grid. This process of merging data and model is known as data assimilation (DA).
It is hard to underestimate the importance of DA. While much work is devoted to big data, it is often not recognised that data comes from different sources and that fact influences how we can effectively tease out the most accurate information. DA works as follows: the model gives an estimate of the system state, which is in our case an assignment of physical variable values at each grid node, and is evolving in time in a way that is provided by solving the equations on a large computer. At observation times, two things should happen: first, the estimate of the system state needs to be adjusted in light of the observational data. Inevitably, the inaccuracies in the model will have driven the estimate off track and there would be little hope of a match with observations. Secondly, certain parameters set in the model will need adjustment in the light of observations coming from the real world. This is a way in which the model can learn from observations.
This all sounds simple enough: somebody just goes in and switches out a few values in the model. But to ensure physical consistency, we cannot perform this task arbitrarily and the whole process ends up being as complex as the underlying model itself. The issues are further exacerbated by the fact that the observations are often not of the physical variables of the model, but something related to them by a further model!
There are a plethora of mathematical challenges in data assimilation, particularly for the ocean. The hardest, and arguably most important, is simultaneously dealing with the inherent nonlinearity in the system and its high dimension. The high dimension comes from the fact that we have a number of physical variables at each grid node and the dimension is the product of the number of grid nodes, of which there will usually be a million or more, and the number of variables. We have good DA methods for nonlinear systems in low dimensions as well as methods that work in high dimensions, but these involve linear approximations that may be broken by an underlying nonlinear system. This issue of dealing with nonlinearity versus dimension is, in my mind, one of the greatest mathematical challenges of our day. We will not get the ocean correct until we perform data assimilation much better, and we cannot predict the climate on decadal time-scales until we get the ocean right.
Christopher Jones
Guthridge Distinguished Professor
RENCI and Department of Mathematics
University of North Carolina at Chapel Hill
Director
Mathematics and Climate Research Network
Department of Mathematics, University of North Carolina at Chapel Hill
Tel: +1 919 923 3569
ckrtj@amath.unc.edu
