Periodic Reporting for period 3 - STUOD (Stochastic Transport in Upper Ocean Dynamics)
Reporting period: 2023-03-01 to 2024-08-31
STUOD (Stochastic Transport in Upper Ocean Dynamics) project looks at changing regimes of uncertainty in ocean fluid transport.
Our approach accounts for transport on scales that are currently unresolvable in computer simulations, yet are observable by satellites, drifters and floats. Our research is: driven by data and new methods for its analysis, informed by mathematical modelling, quantified in concert with computer simulation.
Novel methodologies, and/or inter-disciplinary developments over the past two years include: (i) A comprehensive approach to stochastic climate modelling has developed for the example of an idealised Atmosphere-Ocean model that rests upon Hasselmann's paradigm for stochastic climate models; (ii) A new mathematical approach to stochastic wave-current interaction has been developed based on assuming that the wave motion is evolving in the frame of the current motion; (iii) Similar to our Stochastic Advection by Lie Transport approach based on stochastic transport which preserves circulation of the fluid, the new Stochastic Forcing by Lie Transport approach based on stochastic forcing preserves fluid energy; and (iv) We prove the existence of martingale solutions for the stochastic Navier-Stokes equations in Location Uncertainty approach, which preserves the energy of transported quantities.
The following studies recently investigated go beyond the state of the art: (i) composition of maps (CoM) approach for wave-current interactions using Stochastic Advection by Lie Transport extends the ongoing series of applications of stochastic geometric mechanics in multiscale, multiphysics continuum dynamics to the case of the interaction of fluid waves and currents; (ii) The Stochastic Forcing by Lie Transport approach is a stochastic geometric mechanics and is complementary to the Stochastic Advection by Lie Transport approach in that Stochastic Forcing by Lie Transport forgoes conservation of circulation properties in order to exactly preserve total integrated energy, while Stochastic Advection by Lie Transport preserves circulation properties but forgoes exact conservation of total integrated energy; (iii) The Holm-Mémin paradigm provides stochastic parameterisations that model uncertainty generated by the reduction of high-resolution solutions to coarser scale. (iv) For the Location Uncertainty setting we proved that if the noise intensity goes to zero, these solutions converge, up to a subsequence in dimension 3, to a solution of the deterministic Navier-Stokes equation (in 2D it converges toward the unique solution). By the end of the project, we will provide solutions for collecting, coordinating and interpreting data, which will be made widely available for oceanographic community research and benefit society by providing forecast capabilities for ocean science.
Our approach accounts for transport on scales that are currently unresolvable in computer simulations, yet are observable by satellites, drifters and floats. Our research is: driven by data and new methods for its analysis, informed by mathematical modelling, quantified in concert with computer simulation.
Novel methodologies, and/or inter-disciplinary developments over the past two years include: (i) A comprehensive approach to stochastic climate modelling has developed for the example of an idealised Atmosphere-Ocean model that rests upon Hasselmann's paradigm for stochastic climate models; (ii) A new mathematical approach to stochastic wave-current interaction has been developed based on assuming that the wave motion is evolving in the frame of the current motion; (iii) Similar to our Stochastic Advection by Lie Transport approach based on stochastic transport which preserves circulation of the fluid, the new Stochastic Forcing by Lie Transport approach based on stochastic forcing preserves fluid energy; and (iv) We prove the existence of martingale solutions for the stochastic Navier-Stokes equations in Location Uncertainty approach, which preserves the energy of transported quantities.
The following studies recently investigated go beyond the state of the art: (i) composition of maps (CoM) approach for wave-current interactions using Stochastic Advection by Lie Transport extends the ongoing series of applications of stochastic geometric mechanics in multiscale, multiphysics continuum dynamics to the case of the interaction of fluid waves and currents; (ii) The Stochastic Forcing by Lie Transport approach is a stochastic geometric mechanics and is complementary to the Stochastic Advection by Lie Transport approach in that Stochastic Forcing by Lie Transport forgoes conservation of circulation properties in order to exactly preserve total integrated energy, while Stochastic Advection by Lie Transport preserves circulation properties but forgoes exact conservation of total integrated energy; (iii) The Holm-Mémin paradigm provides stochastic parameterisations that model uncertainty generated by the reduction of high-resolution solutions to coarser scale. (iv) For the Location Uncertainty setting we proved that if the noise intensity goes to zero, these solutions converge, up to a subsequence in dimension 3, to a solution of the deterministic Navier-Stokes equation (in 2D it converges toward the unique solution). By the end of the project, we will provide solutions for collecting, coordinating and interpreting data, which will be made widely available for oceanographic community research and benefit society by providing forecast capabilities for ocean science.
Over the first two years of the project we have made good progress towards the realisation of several of the objectives listed in the proposal, as well as on related issues. The project is advancing at a pace that is inline with the initial objectives. No reorientation of the project has been necessary. Below we summarise the main scientific achievements related to the Work Packages (WP) identified in the proposal:
WP-1 Multi-modal ocean data compilation, analysis and interpretation
The following studies have been performed to analyse data in view of the setting of the Holm-Mémin paradigm of ocean models:
-Multi-modal datasets have been assembled.
-A methodology to obtain fast simulations to perform and test ensembles of wave-propagation predictions, is now being considered to account for surface wave propagation at both slow (resolved) rapid (unresolved) currents.
WP-2 Uncertainty representation and stochastic parameterization
We have systematically explored stochastic formulations of ocean dynamics models expressed within the Holm-Mémin setting (LU and SALT paradigms). We explored stochastic Hamiltonian LU formulations for linear or weakly nonlinear water waves. A LU formulation of the rotating shallow water model as well as different multi-layers quasigeostrophic models have been proposed. The analysis of mathematical models developed through the Holm-Mémin paradigm (Stochastic Advection for Lie Transport, Location Uncertainty) is by now well-developed.
WP-3 Numerical schemes for stochastic fluid transport and diffusion
Our numerical methods have been tailored to the stochastic fluid models we have designed. Recent progress is listed below:
-We explored efficient numerical schemes for advection terms in ocean dynamics models based on weighted essentially non-oscillatory (WENO) schemes.
-We studied the setting of computationally efficient higher-order temporal discrete schemes for LU SPDEs, in which the Levy area has been assumed to vanish.
WP-4 Multiscale Ensemble Data Assimilation and forecasting methods
To ensure compatibility of our derived stochastic models and the multimodal available data, we have explored new data assimilation techniques:
-We proposed new techniques for ensemble Kalman filtering techniques based on Girsanov transformation.
-We devised a technique coupling reproducing kernel Hilbert space and ensemble methods to estimate Koopman eigenpairs.
WP-1 Multi-modal ocean data compilation, analysis and interpretation
The following studies have been performed to analyse data in view of the setting of the Holm-Mémin paradigm of ocean models:
-Multi-modal datasets have been assembled.
-A methodology to obtain fast simulations to perform and test ensembles of wave-propagation predictions, is now being considered to account for surface wave propagation at both slow (resolved) rapid (unresolved) currents.
WP-2 Uncertainty representation and stochastic parameterization
We have systematically explored stochastic formulations of ocean dynamics models expressed within the Holm-Mémin setting (LU and SALT paradigms). We explored stochastic Hamiltonian LU formulations for linear or weakly nonlinear water waves. A LU formulation of the rotating shallow water model as well as different multi-layers quasigeostrophic models have been proposed. The analysis of mathematical models developed through the Holm-Mémin paradigm (Stochastic Advection for Lie Transport, Location Uncertainty) is by now well-developed.
WP-3 Numerical schemes for stochastic fluid transport and diffusion
Our numerical methods have been tailored to the stochastic fluid models we have designed. Recent progress is listed below:
-We explored efficient numerical schemes for advection terms in ocean dynamics models based on weighted essentially non-oscillatory (WENO) schemes.
-We studied the setting of computationally efficient higher-order temporal discrete schemes for LU SPDEs, in which the Levy area has been assumed to vanish.
WP-4 Multiscale Ensemble Data Assimilation and forecasting methods
To ensure compatibility of our derived stochastic models and the multimodal available data, we have explored new data assimilation techniques:
-We proposed new techniques for ensemble Kalman filtering techniques based on Girsanov transformation.
-We devised a technique coupling reproducing kernel Hilbert space and ensemble methods to estimate Koopman eigenpairs.
Crisan and Holm have developed a new framework that extends fluid dynamical models with stochastic transport to transport on geometric rough paths. Another paper has been published in the Journal of Functional Analysis. This second paper analyses the solutions of the 3D Euler equations on geometric rough paths using the new method. We regard this development as a significant breakthrough, because it provides the basis for extending the capabilities of the STUOD project to more general scope in mathematical modelling and analysis, as well as bringing new approaches for data analysis and data assimilation.
Other most significant achievements include:
Importantly, we have shown that the 2D&3D Navier-Stokes equation in the LU setting can be truly interpreted as large-scale representation of the flow, as they converge in the vanishing noise limit (in a weak probabilistic sense in 3D and strongly in 2D) toward a weak solution of the deterministic Navier-Stokes solution in 3D, and toward a unique solution in 2D.
We published a paper in J. of Physical Oceanography showing that the LU setting provides a generalisation of the so-called Stokes drift. This statistical drift represents the action of the small-scale inhomogeneity to shape larger scales of the currents.
We demonstrated Well-posedness Properties for a Stochastic Rotating Shallow Water Model and performed theoretical and computational analysis of the thermal quasi-geostrophic model.
A generalization of Holm [2015] has been accomplished that sets out a wide class of stochastic parameterizations by removing that memoryless constraint on the stochastic noise and justifies the semi-martingale property of the pressure term in stochastic models.
A new methodology to incorporate unresolved small-scale effects in GFD models has been proposed through variational principles for fluid dynamics on rough paths.
We devised a novel particle filter designed to handle high-dimensional systems.
Five stochastic ocean dynamics models have been derived (STUOD WP2 Stochastic modelling and mathematical analysis). Twelve papers reporting work on these five models have been published in peer-reviewed scientific journals.
Other most significant achievements include:
Importantly, we have shown that the 2D&3D Navier-Stokes equation in the LU setting can be truly interpreted as large-scale representation of the flow, as they converge in the vanishing noise limit (in a weak probabilistic sense in 3D and strongly in 2D) toward a weak solution of the deterministic Navier-Stokes solution in 3D, and toward a unique solution in 2D.
We published a paper in J. of Physical Oceanography showing that the LU setting provides a generalisation of the so-called Stokes drift. This statistical drift represents the action of the small-scale inhomogeneity to shape larger scales of the currents.
We demonstrated Well-posedness Properties for a Stochastic Rotating Shallow Water Model and performed theoretical and computational analysis of the thermal quasi-geostrophic model.
A generalization of Holm [2015] has been accomplished that sets out a wide class of stochastic parameterizations by removing that memoryless constraint on the stochastic noise and justifies the semi-martingale property of the pressure term in stochastic models.
A new methodology to incorporate unresolved small-scale effects in GFD models has been proposed through variational principles for fluid dynamics on rough paths.
We devised a novel particle filter designed to handle high-dimensional systems.
Five stochastic ocean dynamics models have been derived (STUOD WP2 Stochastic modelling and mathematical analysis). Twelve papers reporting work on these five models have been published in peer-reviewed scientific journals.