Numerically Modelling Stochastic Lie Transport in Fluid Dynamics

Cotter, Colin J.; Crisan, Dan; Holm, Darryl D.; Pan, Wei; Shevchenko, I. V.

doi:10.48550/arxiv.1801.09729

Cited by 8 publications

(24 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Equation ( 12) will play a role in deriving the Kelvin circulation theorem, itself, and thereby interpreting the solution behaviour of the fluid motion equation, derived below from Hamilton's principle. In the next section, we will show how passing from the Itô representation of the Lagrangian trajectory in (4) to its equivalent Stratonovich representation in (6) enables the use of variational calculus to derive the equations of stochastic fluid motion via the approach of stochastic advection by Lie transport (SALT), based on Hamilton's variational principle using Stratonovich calculus, [14]. The resulting equations will raise the issue of fictitious forces and this issue will be resolved by elementary considerations.…”

Section: Stochastic Kelvin Circulation Dynamicsmentioning

confidence: 99%

“…Following [14] we apply Hamilton's principle δS = 0 with the following action integral S = T 0 ℓ(u L t , D, b) dt whose fluid Lagrangian ℓ(u L t , D, b) depending on drift velocity u L t , buoyancy function b(x, t) and the density D(x, t)d 3 x for (x, t) ∈ R 3 × R. We constrain the variations to respect the stochastic advection equations with transport velocity dx L t given in (6),…”

Section: Salt Derivation Of Stochastic Euler-boussinesq (Seb) 21 Hami...mentioning

confidence: 99%

“…The observation of the persistence of the Kelvin form (1) for Stratonovich stochastic fluid trajectories has led to the SALT algorithm for uncertainty quantification and data assimilation for stochastic fluid models. 1 The SALT algorithm proceeds from data acquisition, to coarse graining, to uncertainty quantification by using stochastic fluid dynamical modelling, and finally to uncertainty reduction via data assimilation based on machine learning via particle filtering methods [5,6]. Problem statement.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Stochastic Modelling in Fluid Dynamics: Itô vs Stratonovich

Holm

2019

Preprint

Self Cite

View full text Add to dashboard Cite

Suppose the observations of Lagrangian trajectories for fluid flow in some physical situation may be modelled sufficiently accurately by a spatially correlated Itô stochastic process obtained from data which is taken in fixed Eulerian space. Suppose we also want to apply Hamilton's principle to derive the stochastic fluid equations for this situation. Now, the variational calculus for applying Hamilton's principle requires the Stratonovich process, so we must transform from Itô noise in the data frame to the equivalent Stratonovich noise. However, the transformation from the Itô process in the data frame to the corresponding Stratonovich process shifts the drift velocity of the transformed Lagrangian fluid trajectory out of the data frame into a non-inertial frame obtained from the Itô correction. The question is, "Will fictitious forces arising from this transformation of reference frames make a difference in the interpretation of the solution behaviour of the resulting stochastic equations?" This issue will be resolved by elementary considerations.

show abstract

Section: Stochastic Kelvin Circulation Dynamicsmentioning

confidence: 99%

Section: Salt Derivation Of Stochastic Euler-boussinesq (Seb) 21 Hami...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Stochastic Modelling in Fluid Dynamics: Itô vs Stratonovich

Holm

2019

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…The domain we consider in this paper is the two-dimensional torus. The vector fields ξ i are time-independent and divergence-free and can be associated with uncertainty induced by missing physics or incomplete data (see [24], [25]). The processes W i , i ≥ 1, are independent Brownian motions and δ is the aspect ratio of the domain (that is the ratio between vertical and horizontal length scales).…”

Section: Introductionmentioning

confidence: 99%

“…Notwithstanding, we do this by proving a set of specific inequalities for the operator L (see Lemma 18) which are based on some smoothness and summability assumptions for the vector fields ξ i (see Assumptions 3). These assumptions would be of particular interest when using this model as a signal process in stochastic filtering applications: see for instance [24], [25], [26] in the case of the stochastic 2D Euler equation.…”

Section: Introductionmentioning

confidence: 99%

Local well-posedness for the great lake equation with transport noise

Crisan,

Lang

2020

Preprint

Self Cite

View full text Add to dashboard Cite

This work is a continuation of the authors' work in [8]. In [8] the equation satisfied by an incompressible fluid with stochastic transport is analysed. Here we lift the incompressibility constraint. Instead we assume a weighted incompressibility condition. This condition is inspired by a physical model for a fluid in a basin with a free upper surface and a spatially varying bottom topography (see [20]). Moreover, we assume a different form of the vorticity to stream function operator that generalizes the standard Biot-Savart operator which appears in the Euler equation. These two properties are exhibited in the physical model called the great lake equation. For this reason we refer to the model analysed here as the stochastic great lake equation. Just as in [8], the deterministic model is perturbed with transport type noise. The new vorticity to stream function operator generalizes the curl operator and it is shown to have good regularity properties. We also show that the initial smoothness of the solution is preserved. The arguments are based on constructing a family of viscous solutions which is proved to be relatively compact and to converge to a truncated version of the original equation. Finally, we show that the truncation can be removed up to a positive stopping time.

show abstract

Comparison of Stochastic Parametrization Schemes Using Data Assimilation on Triad Models

Lobbe,

Crisan,

Holm

et al. 2023

Mathematics of Planet Earth

View full text Add to dashboard Cite

In recent years, stochastic parametrizations have been ubiquitous in modelling uncertainty in fluid dynamics models. One source of model uncertainty comes from the coarse graining of the fine-scale data and is in common usage in computational simulations at coarser scales. In this paper, we look at two such stochastic parametrizations: the Stochastic Advection by Lie Transport (SALT) parametrization introduced by Holm (Proc A 471(2176):20140963, 19, 2015) and the Location Uncertainty (LU) parametrization introduced by Mémin (Geophys Astrophys Fluid Dyn 108(2):119–146, 2014). Whilst both parametrizations are available for full-scale models, we study their reduced order versions obtained by projecting them on a complex vector Fourier mode triad of eigenfunctions of the curl. Remarkably, these two parametrizations lead to the same reduced order model, which we term the helicity-preserving stochastic triad (HST). This reduced order model is then compared with an alternative model which preserves the energy of the system, and which is termed the energy preserving stochastic triad (EST). These low-dimensional models are ideal benchmark models for testing new Data Assimilation algorithms: they are easy to implement, exhibit diverse behaviours depending on the choice of the coefficients and come with natural physical properties such as the conservation of energy and helicity.

show abstract

Numerically Modelling Stochastic Lie Transport in Fluid Dynamics

Cited by 8 publications

References 0 publications

Stochastic Modelling in Fluid Dynamics: Itô vs Stratonovich

Stochastic Modelling in Fluid Dynamics: Itô vs Stratonovich

Local well-posedness for the great lake equation with transport noise

Comparison of Stochastic Parametrization Schemes Using Data Assimilation on Triad Models

Contact Info

Product

Resources

About