**LMS Research School**

**Mathematics of Climate**

**8-12 July, 2019**

**University of Reading, Room Met 1L61.**

**Organised in partnership with London Mathematical Society
**

Research lying at the intersection between mathematics and geoscience has been gaining more and more prominence in recent years, e.g. the worldwide Year of Mathematics for Planet Earth 2013 which incited a lot of activities. The Planet Earth is an excellent example of a forced, dissipative non-equilibrium system dominated by nonlinear processes and featuring multi-scale interactions. The challenges posed by this system have been and will be an inspiration for mathematicians and geoscientists alike. The synergy between mathematics and geoscience is exemplified by the scientific research areas like chaos theory, fractals, extreme events, and data assimilation, which stem or essentially developed through such an interaction. More recently, interdisciplinary exchange with geosciences has led to important advances in further crossover areas such as geometrical mechanics and optimal transport theory with geophysical fluid dynamics, coarse graining and model reduction techniques.

This School aims to educate and motivate young talented PhD students and early career scientists through three specialist lecture courses (supplemented by tutorials) and guest lectures that will provide a panorama of some of the most promising research areas within this interdisciplinary field. The school is organised in partnership with the London Mathematical Society and with the support of the EPSRC-funded CDT Mathematics of Planet Earth (Imperial/Reading).

To register, please use the link above. For questions and further details, please contact the organisers at t.kuna-at-reading.ac.uk for details. (do not use this email address for registering).

You can download a printable poster for the meeting here: Mathematics of Climate 2019.

**Dates**:

– Deadline for registration including referee forms: 8 May 2019

– Deadline for application for support : 24 May 2019

**Registration information:**

– All participants, including local ones, must fill the application form nominating a referee. Please register here. Note that the number of participants is limited to 40 and so applying to attend does not guarantee a place. Successful applicants will be notified by 10 May 2019.

– The fee for research students is £150 and for early career researchers it is £250. This fee includes the cost of accommodation (non local participants only) and meals but not travel, which participants must pay for themselves. Accommodation can be provided in university halls. Other accommodations cannot be reimbursed.

– Limited financial support towards travel expenses and the fees is available for research students. You may apply for this shortly after your application was accepted.

**Speakers **(See abstracts at the bottom of this page):

– Course 1: *Hakima Bessaih* (University of Wyoming): Random vortex methods for 3D fluids

– Course 2: *Alberto Carrassi* (NERSC, Norway): Mathematical Theory of Data Assimilation with Applications

– Course 3: *Darryl Holm* (Imperial College, London): Variational principles for Stochastic Fluid Dynamics

**Guest Lectures**:

– *Beatrice Pelloni* (Heriot-Watt): Mathematical aspects of the semigeostrophic system

– *Peter Ashwin* (Exeter): Response of the Pleistocene ice-ages to astronomical forcing

**Tutors**:

Course 1: Giulia Carigi (Reading); Course 2: Colin Grudzien (Assistant Professor of Statistics, University of Nevada, Reno); Course 3: Erwin Luesink (Imperial)

**Speaker Abstracts**:

*Hakima Bessaih* (University of Wyoming): Random vortex methods for 3D fluids

Vortex eddies and filaments are quite new objects. Their theory is still at the beginning and several issues are still unsolved or poorly understood. Nevertheless, they look promising for the purpose of describing turbulent fluids and our aim is to understand them as deeply as possible. There is an extensive literature related to the two dimensional case where the vorticity can be defined as a measure-valued field supported by N point vortices and the theory is well studied when N goes to infinity. Less is known for the 3D case when vortices are replaced by curves. The collection of N smooth curves commonly called vortex filaments can define a flow when a Biot-Savart formula is used. Moreover, their mean field limit (for N large) solves some nonlinear vector-valued partial differential equations of Euler type. We think that there is some potential to make more progress towards 3D Euler equation and potentially 3D Navier-Stokes equations. These models are important for the understanding of 3D turbulent fluids.

Alberto Carrassi (NERSC, Norway): Mathematical Theory of Data Assimilation with Applications

State estimation theory in geosciences is commonly referred to as data assimilation. This term encompasses the entire sequence of operations that, starting from the observations of a system, and from additional statistical and/or dynamical information (such as an evolution model), provides an estimate of its state. Data assimilation is common practice in numerical weather prediction but its application is becoming widespread in many other areas of climate, atmosphere, ocean and environment modelling. The course will provide first the formulation of the problem from a Bayesian perspective and will then present the two popular families of Gaussian based approaches, the Kalman-filter/-smoother and the variational methods. Ensemble based methods will then be considered, starting from the well-known Ensemble Kalman filter, in its stochastic or deterministic formulation, and then the state-of-the-art ensemble-variational methods, as well as particle filters. The course will focus on the specific challenges that data assimilation has encountered to deal with high-dimensional chaotic systems, such as the atmosphere and ocean, and the countermeasures that have been taken and which have driven the recent dramatic development of the field.

Darryl Holm (Imperial College, London): Variational principles for Stochastic Fluid Dynamics

The lectures will be based on Hamilton’s principle for ideal fluids and its infinite dimensional symmetries for temporally stochastic evolution of fluids under spatially smooth invertible maps. This means that the Lagrangian particle trajectories exist as stochastic paths generated by Eulerian stochastic vector fields. Thus, Hamilton’s principle retains its original gauge symmetry under relabelling of Lagrangian particles. Because this gauge symmetry persists, Noether’s theorem delivers Kelvin’s theorem for circulation dynamics of the stochastic flow in the same form as for the deterministic case, but now Kelvin’s circulation loop moves with the fluid along a stochastic path. The corresponding nonlinear stochastic PDEs for the fluid motion provide a probabilistic estimation of model error, based on the observed spatial correlations of uncertainties in the fluid transport. Thus, this variational approach yields a data-driven model of stochastic flow for probabilistic estimates of Variability in Geophysical Fluid Dynamics (GFD). Details and examples for GFD may be found in: D.D. Holm, Proc Roy Soc A, 471: 20140963 (2015)

**Programme:**

Monday: July 8

- 9:00-10:00 Course 1
- 10:00-11:00 Course 1
- 11:00-11:30 Coffee break
- 11:30-12:30 Course 2
- 12:30-14:00 Lunch
- 14:00-15:00 Course 2
- 15:00-16:00 Course 3
- 16:00-16:30 Tea Break
- 16:30-17:30 Course 3
- 17:30-18:30 Tutorial 1

Tuesday: July 9

- 9:00-10:00 Course 2
- 10:00-11:00 Course 2
- 11:00-11:30 Coffee break
- 11:30-12:30 Course 3
- 12:30-14:00 Lunch
- 14:00-15:00 Course 3
- 15:00-16:00 Course 1
- 16:00-16:30 Tea Break
- 16:30-17:30 Course 1
- 17:30-18:30 Tutorial 1

Wednesday: July 10

- 9:00-10:00 Course 3
- 10:00-11:00 Course 3
- 11:00-11:30 Coffee break
- 11:30-12:30 Tutorial 3
- 12:30-14:00 Lunch
- 14:00- Social event

Thursday: July 11

- 9:00-10:00 Course 1
- 10:00-11:00 Course 1
- 11:00-11:30 Coffee break
- 11:30-12:30 Course 2
- 12:30-14:00 Lunch
- 14:00-15:00 Course 2
- 15:00-16:00 Tutorial 1
- 16:00-16:30 Tea Break
- 16:30-17:30 Tutorial 2
- 17:30-18:30 Tutorial 3

Friday: July 12

- 9:00-10:00 Guest Lecture 1
- 10:00-11:00 Guest Lecture 2
- 11:00-11:30
*Coffee break* - 11:30-12:30 Tutorial 1
- 12:30-14:00
*Lunch* - 14:00-15:00 Tutorial 2
- 15:00-16:00 Tutorial 3

**Organisers**:

Valerio Lucarini,Tobias Kuna, Jennifer Scott, Dan Crisian

Email the organizers at t.kuna-AT-reading.ac.uk.

**Funding**:

The school is in partnership with the London Mathematical Society (LMS). and supported by the EPSRC Centre of Doctoral Training Mathematics of Planet Earth. The LMS is the UK’s learned society for mathematics. Registered charity no. 252660 (www.lms.ac.uk)