# Variational principles for stochastic parameterisations in geophysical fluid dynamics

Lead Research Organisation:
Imperial College London

Department Name: Dept of Mathematics

### Abstract

Our proposal is inspired by the clear and present need for understanding statistical variability of weather and climate.

Dynamical weather prediction stems from the deterministic laws of mechanics and thermodynamics, established by the mid-19th century. With the advent of digital computers in the second half of the 20th century, these ideas led to operational Numerical Weather Prediction (NWP) and shortly thereafter, with the advent of satellite observations, to numerical experiments that explored the atmosphere's general circulation. The new type of scientific exploration via numerical simulations soon raised the issue of limits of predictability of atmospheric dynamics, due to uncertainty in the initial state, unresolved scales of motion, and the extreme sensitivity of the numerical output to these uncertainties. This sensitivity was famously popularised as the Butterfly Effect. The recognition of the loss of predictability for NWP summoned research into a stochastic approach in designing simulators for NWP. NWP cannot be entirely deterministic, but must also involve a form of randomness, or noise. A new approach to NWP arose, which coupled randomness and probability with determinism. Parallel processing methods in the early 1990's and improved operational forecasting systems, in both simulator physics and data assimilation methods, have led to more reliable forecasts produced by modern operational stochastic dynamic Ensemble Prediction Systems (EPS) now used at ECMWF, and the UK Met Office.

Yet it still remains to determine the most appropriate way to introduce stochastic dynamics into the simulator, so as to couple data assimilation with ensemble forecasting and to determine the number of samples in the ensemble sufficient for a required reliability. Current work continues to explore these avenues with great vigour.

This project addresses the remaining challenge of Stochastic Dynamics for NWP, by taking an integrated approach to data-driven mathematical modelling, compatible numerics and model-driven data assimilation.

The mathematical modelling uses an optimal, systematic method of introducing stochasticity into Geophysical Fluid Dynamics (GFD). The method is based on a stochastic version of the family of variational principles whose critical points yield the entire sequence of deterministic equations of motion for ideal GFD at each level of approximation. The levels of approximation are obtained from asymptotic expansion of the unapproximated variational principle that yields the fundamental Euler equations for a rotating, stratified, incompressible fluid.

Stochasticity is introduced into the variational principle by using resolved spatial correlations of data obtained from observations of fluctuating tracer paths. In turn, the stochastic variational principle generates the equations of motion for the fluid flow carrying these tracers along their fluctuating paths.

The proposed mathematical research on these new equations of motion will be integrated with numerical simulations and data assimilation methods, aiming to create an implementable modelling approach of significance for the mathematical foundations of NWP, climate science, and other highly unstable fluid dynamics applications. For this, we adopt a Bayesian perspective in blending the newly developed SPDEs with data completely integrated with its modelling and simulation efforts with connections as shown in Figure 1. Likewise, the numerical algorithms will be informed by the mathematical analysis. Once the numerical simulations are developed and performed, the subsequent data assimilation will produce the posterior distribution of the current state of the model via particle filtering methods.

Dynamical weather prediction stems from the deterministic laws of mechanics and thermodynamics, established by the mid-19th century. With the advent of digital computers in the second half of the 20th century, these ideas led to operational Numerical Weather Prediction (NWP) and shortly thereafter, with the advent of satellite observations, to numerical experiments that explored the atmosphere's general circulation. The new type of scientific exploration via numerical simulations soon raised the issue of limits of predictability of atmospheric dynamics, due to uncertainty in the initial state, unresolved scales of motion, and the extreme sensitivity of the numerical output to these uncertainties. This sensitivity was famously popularised as the Butterfly Effect. The recognition of the loss of predictability for NWP summoned research into a stochastic approach in designing simulators for NWP. NWP cannot be entirely deterministic, but must also involve a form of randomness, or noise. A new approach to NWP arose, which coupled randomness and probability with determinism. Parallel processing methods in the early 1990's and improved operational forecasting systems, in both simulator physics and data assimilation methods, have led to more reliable forecasts produced by modern operational stochastic dynamic Ensemble Prediction Systems (EPS) now used at ECMWF, and the UK Met Office.

Yet it still remains to determine the most appropriate way to introduce stochastic dynamics into the simulator, so as to couple data assimilation with ensemble forecasting and to determine the number of samples in the ensemble sufficient for a required reliability. Current work continues to explore these avenues with great vigour.

This project addresses the remaining challenge of Stochastic Dynamics for NWP, by taking an integrated approach to data-driven mathematical modelling, compatible numerics and model-driven data assimilation.

The mathematical modelling uses an optimal, systematic method of introducing stochasticity into Geophysical Fluid Dynamics (GFD). The method is based on a stochastic version of the family of variational principles whose critical points yield the entire sequence of deterministic equations of motion for ideal GFD at each level of approximation. The levels of approximation are obtained from asymptotic expansion of the unapproximated variational principle that yields the fundamental Euler equations for a rotating, stratified, incompressible fluid.

Stochasticity is introduced into the variational principle by using resolved spatial correlations of data obtained from observations of fluctuating tracer paths. In turn, the stochastic variational principle generates the equations of motion for the fluid flow carrying these tracers along their fluctuating paths.

The proposed mathematical research on these new equations of motion will be integrated with numerical simulations and data assimilation methods, aiming to create an implementable modelling approach of significance for the mathematical foundations of NWP, climate science, and other highly unstable fluid dynamics applications. For this, we adopt a Bayesian perspective in blending the newly developed SPDEs with data completely integrated with its modelling and simulation efforts with connections as shown in Figure 1. Likewise, the numerical algorithms will be informed by the mathematical analysis. Once the numerical simulations are developed and performed, the subsequent data assimilation will produce the posterior distribution of the current state of the model via particle filtering methods.

### Planned Impact

The proposed research will forge new bonds between two previously disjoint mathematical fields; namely, stochastic analysis and variational principles with symmetry. The proposed research will generate a systematic and rigorously justified approach for introducing and analysing stochasticity in the GFD equations of motion and performing Bayesian data assimilation within this framework.

We will aim to reformulate the fundamental implications of noisy perturbations in stochastic transport in fluid dynamics and generate a shift in emphasis towards high dimensional problems within the mathematical sciences community. As these problems arise in many applications of national and international importance, the need to make such a shift is apparent. The research will create new pathways for impact of mathematics and new challenges for mathematical investigations.

The results of our work will primarily impact academic-based mathematical sciences researchers. Particular areas of expected impact are Stochastic Partial Differential Equations (SPDEs), Stochastic Variational Principles and Numerical Modelling, in connection with Complex Nonlinear Systems, Applied Analysis, Numerical Analysis, Geophysical Fluid Dynamics (GFD) and Turbulence.

The broad applicability of fluid dynamics also implies significant potential for impact on research in climate sciences and weather variability. The advances in basic research we propose here and their use in modelling nonlinear effects of stochasticity in GFD using stochastic variational principles, and in designing numerical algorithms compatible with these principles are likely have significant effects in collaborations with people who are indeed responsible for numerical weather prediction, particularly in the UK Met Office Gung Ho project, which includes one of the co-PIs, as well as ECMWF and the German Weather Service, through other close connections.

In addition to the impact on the scientists directly involved (the PI and co-PIs, the two PDRAs and the six named visitors), the research will benefit other members other members of the Department of Mathematics at Imperial College London. At least 30 members of the Department have directly relevant research interests. These include members of the Stochastic Analysis and the Geometric Mechanics groups (staff, PhD students, postdocs). Our colleagues will benefit through natural collaborations and discussions, the talks and mini-courses presented by the project's visitors, and attendance at the yearly workshops.

The PI and both Co-PI's play leading roles together in the EPSRC Centre for Doctoral Training in the Mathematics of Planet Earth (MPE CDT). The MPE CDT is a major avenue of impact of the proposed research, and a focus of its coordination. Indeed the theme of the research is in perfect alignment with the research themes covered by the Centre. Over 80 academics are involved in the Centre, and they will benefit from the proposed research through the interactions and results dissemination ensured in the manner explained above. Perhaps more importantly, the PhD students enrolled in the Centre, will benefit from being exposed and hopefully contributing to the research. The two PDRAs will not only contribute to achieving the research objectives, but will also be full partners in the MPE CDT experience and will help build our PhD cohorts.

The overall impact will thus be felt on a variety of time-scales: immediate impact on academic mathematical sciences within the duration of the grant; building of research understanding and mutual communication between academic mathematical scientists and applied researchers from both governmental and academic institutes, again within the duration of the grant; and on a longer time scale the potential for significant impact on issues of UK societal importance, such as numerical weather prediction and predictive oceanography and, further afield, in traffic flow and population demographics.

We will aim to reformulate the fundamental implications of noisy perturbations in stochastic transport in fluid dynamics and generate a shift in emphasis towards high dimensional problems within the mathematical sciences community. As these problems arise in many applications of national and international importance, the need to make such a shift is apparent. The research will create new pathways for impact of mathematics and new challenges for mathematical investigations.

The results of our work will primarily impact academic-based mathematical sciences researchers. Particular areas of expected impact are Stochastic Partial Differential Equations (SPDEs), Stochastic Variational Principles and Numerical Modelling, in connection with Complex Nonlinear Systems, Applied Analysis, Numerical Analysis, Geophysical Fluid Dynamics (GFD) and Turbulence.

The broad applicability of fluid dynamics also implies significant potential for impact on research in climate sciences and weather variability. The advances in basic research we propose here and their use in modelling nonlinear effects of stochasticity in GFD using stochastic variational principles, and in designing numerical algorithms compatible with these principles are likely have significant effects in collaborations with people who are indeed responsible for numerical weather prediction, particularly in the UK Met Office Gung Ho project, which includes one of the co-PIs, as well as ECMWF and the German Weather Service, through other close connections.

In addition to the impact on the scientists directly involved (the PI and co-PIs, the two PDRAs and the six named visitors), the research will benefit other members other members of the Department of Mathematics at Imperial College London. At least 30 members of the Department have directly relevant research interests. These include members of the Stochastic Analysis and the Geometric Mechanics groups (staff, PhD students, postdocs). Our colleagues will benefit through natural collaborations and discussions, the talks and mini-courses presented by the project's visitors, and attendance at the yearly workshops.

The PI and both Co-PI's play leading roles together in the EPSRC Centre for Doctoral Training in the Mathematics of Planet Earth (MPE CDT). The MPE CDT is a major avenue of impact of the proposed research, and a focus of its coordination. Indeed the theme of the research is in perfect alignment with the research themes covered by the Centre. Over 80 academics are involved in the Centre, and they will benefit from the proposed research through the interactions and results dissemination ensured in the manner explained above. Perhaps more importantly, the PhD students enrolled in the Centre, will benefit from being exposed and hopefully contributing to the research. The two PDRAs will not only contribute to achieving the research objectives, but will also be full partners in the MPE CDT experience and will help build our PhD cohorts.

The overall impact will thus be felt on a variety of time-scales: immediate impact on academic mathematical sciences within the duration of the grant; building of research understanding and mutual communication between academic mathematical scientists and applied researchers from both governmental and academic institutes, again within the duration of the grant; and on a longer time scale the potential for significant impact on issues of UK societal importance, such as numerical weather prediction and predictive oceanography and, further afield, in traffic flow and population demographics.

### Publications

Alonso-Orán D
(2020)

*Modelling the Climate and Weather of a 2D Lagrangian-Averaged Euler-Boussinesq Equation with Transport Noise*in Journal of Statistical Physics
Arnaudon A
(2018)

*A Geometric Framework for Stochastic Shape Analysis*in Foundations of Computational Mathematics
Arnaudon A
(2018)

*String Methods for Stochastic Image and Shape Matching*in Journal of Mathematical Imaging and Vision
Arnaudon A
(2018)

*The stochastic energy-Casimir method*in Comptes Rendus Mécanique
Bendall T
(2018)

*Statistical properties of an enstrophy conserving finite element discretisation for the stochastic quasi-geostrophic equation*in Geophysical & Astrophysical Fluid Dynamics
Cotter C
(2019)

*Numerically Modeling Stochastic Lie Transport in Fluid Dynamics*in Multiscale Modeling & Simulation*Numerically Modeling Stochastic Lie Transport in Fluid Dynamics*in Multiscale Modeling & Simulation

Cotter CJ
(2017)

*Stochastic partial differential fluid equations as a diffusive limit of deterministic Lagrangian multi-time dynamics.*in Proceedings. Mathematical, physical, and engineering sciences*Stochastic partial differential fluid equations as a diffusive limit of deterministic Lagrangian multi-time dynamics.*in Proceedings. Mathematical, physical, and engineering sciences

Crisan D
(2018)

*Wave breaking for the Stochastic Camassa-Holm equation*in Physica D: Nonlinear PhenomenaDescription | The stochastic fluid velocity decomposition results of [1,2] show that the principles of transformation theory and multi-time homogenisation can be used to lay the foundations for a physically meaningful, data-driven and mathematically-based approach for decomposing the fluid transport velocity into its drift and stochastic parts. This approach can be applied immediately to the class of continuum flows whose deterministic motion is based on fundamental variational principles. Two related papers [3,4] have recently used this approach to develop a New Methodology to implement the velocity decomposition of [1,2] for uncertainty quantification in computational simulations of fluid dynamics. The new methodology was tested numerically and found to be suitable for coarse graining in two separate types of problems based on discretisations using either finite elements, or finite differences. Specifically, uncertainty quantification tests using this velocity decomposition were performed by comparing ensembles of coarse-grid realisations of solutions of the resulting stochastic partial differential equation with the ``true solutions" of the deterministic fluid partial differential equation, computed on a refined grid. The time discretisation used for approximating the solution of the stochastic partial differential equation was shown to be consistent. Comprehensive numerical tests confirmed the non-Gaussianity and quantified the uncertainty of the stream function, velocity and vorticity fields for incompressible 2D Euler fluid flows in a bounded domain using finite elements [3] and for 2-layer quasi-geostrophic flows in a 2D periodic channel using finite differences [4]. References: [1] D. D. Holm. Variational principles for stochastic fluid dynamics. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 471(2176), 2015. [2] C. J. Cotter, G. A. Gottwald, and D. D. Holm. Stochastic partial differential fluid equa- tions as a diffusive limit of deterministic Lagrangian multi-time dynamics. Proc. Roy. Soc. A, 473:20170388, 2017. [3] ColinJ.Cotter,DanCrisan,DarrylD.Holm,WeiPan,andIgorShevchenko.Numerically modelling stochastic lie transport in fluid dynamics. arXiv:1801.09729, 2018. [4] Colin J. Cotter, Dan Crisan, Darryl D. Holm, Wei Pan, and Igor Shevchenko. Mod- elling uncertainty using circulation-preserving stochastic transport noise in a 2-layer quasi- geostrophic mode. arXiv:1802.05711, 2018. |

Exploitation Route | Our findings can be used to lay the foundations for a physically meaningful, data-driven and mathematically-based approach for decomposing the fluid transport velocity into its drift and stochastic parts. This approach can be applied immediately to the class of continuum flows whose deterministic motion is based on fundamental variational principles. In particular, our findings can be applied as a model of turbulent transport for numerical weather prediction and climate science. |

Sectors | Aerospace, Defence and Marine,Education,Energy,Environment |