Hilbert problems for the geosciences in the 21st century

Abstract. The scientific problems posed by the Earth's fluid envelope, and its atmosphere, oceans, and the land surface that interacts with them are central to major socio-economic and political concerns as we move into the 21st century. It is natural, therefore, that a certain impatience should prevail in attempting to solve these problems. The point of this review paper is that one should proceed with all diligence, but not excessive haste: "festina lente," as the Romans said two thousand years ago, i.e. "hurry in a measured way." The paper traces the necessary progress through the solutions to the ten problems:

1. What is the coarse-grained structure of low-frequency atmospheric variability, and what is the connection between its episodic and oscillatory description?

2. What can we predict beyond one week, for how long, and by what methods?

3. What are the respective roles of intrinsic ocean variability, coupled ocean-atmosphere modes, and atmospheric forcing in seasonal-to-interannual variability?

4. What are the implications of the answer to the previous problem for climate prediction on this time scale?

5. How does the oceans' thermohaline circulation change on interdecadal and longer time scales, and what is the role of the atmosphere and sea ice in such changes?

6. What is the role of chemical cycles and biological changes in affecting climate on slow time scales, and how are they affected, in turn, by climate variations?

7. Does the answer to the question above give us some trigger points for climate control?

8. What can we learn about these problems from the atmospheres and oceans of other planets and their satellites?

9. Given the answer to the questions so far, what is the role of humans in modifying the climate?

10. Can we achieve enlightened climate control of our planet by the end of the century?

A unified framework is proposed to deal with these problems in succession, from the shortest to the longest timescale, i.e. from weeks to centuries and millennia. The framework is that of dynamical systems theory, with an emphasis on successive bifurcations and the ergodic theory of nonlinear systems. The main ideas and methods are outlined and the concept of a modelling hierarchy is introduced. The methodology is applied across the modelling hierarchy to Problem 5, which concerns the thermohaline circulation and its variability.

Key words. Climate dynamics, nonlinear systems, numerical bifurcations, mathematical geophysics