[Paper Review] Modelling uncertainty using circulation-preserving stochastic transport noise in a 2-layer quasi-geostrophic model

Colin J. Cotter, Dan Crisan+3 more|arXiv (Cornell University)|Feb 15, 2018

Reservoir Engineering and Simulation Methods21 citations

TL;DR

This paper introduces a circulation-preserving stochastic transport noise parameterization in a two-layer quasi-geostrophic model to model uncertainty from unresolved scales. By deriving stochastic forcing from high-resolution deterministic simulations and using a consistent time-stepping scheme, the method effectively captures statistical equilibrium properties in both homogeneous and heterogeneous flows, demonstrating strong potential for data assimilation and uncertainty quantification.

ABSTRACT

The stochastic variational approach for geophysical fluid dynamics was introduced by Holm (Proc Roy Soc A, 2015) as a framework for deriving stochastic parameterisations for unresolved scales. The key feature of transport noise is that it respects the Kelvin circulation theorem. This paper applies the variational stochastic parameterisation in a two-layer quasi-geostrophic model for a $\beta$-plane channel flow configuration. The parameterisation is tested by comparing it with a deterministic high resolution eddy-resolving solution that has reached statistical equilibrium. We describe a stochastic time-stepping scheme for the two-layer model and discuss its consistency in time. Then we describe a procedure for estimating the stochastic forcing to approximate unresolved components using data from the high resolution deterministic simulation. We compare an ensemble of stochastic solutions at lower resolution with the numerical solution of the deterministic model. These computations quantify the uncertainty of the coarse grid computation relative to the fine grid computation. The results show that the proposed parameterisation is efficient and effective for both homogeneous and heterogeneous flows, and they lay a solid foundation for data assimilation.

Motivation & Objective

• To develop a stochastic parameterization that preserves the Kelvin circulation theorem in geophysical fluid dynamics.
• To quantify uncertainty in coarse-grid simulations relative to high-resolution deterministic solutions.
• To create a consistent stochastic time-stepping scheme for a two-layer quasi-geostrophic model.
• To estimate stochastic forcing from high-resolution simulation data to approximate unresolved dynamics.
• To evaluate the method's effectiveness in both homogeneous and heterogeneous flow configurations for data assimilation applications.

Proposed method

• Applies the stochastic variational approach of Holm (2015) to derive transport noise that preserves the Kelvin circulation theorem.
• Uses a two-layer quasi-geostrophic model on a β-plane channel to simulate geophysical flow dynamics.
• Develops a stochastic time-integration scheme that maintains consistency in time for the stochastic model.
• Estimates stochastic forcing by analyzing velocity and vorticity fields from a high-resolution deterministic simulation.
• Generates an ensemble of low-resolution stochastic solutions to compare with the high-resolution deterministic solution.
• Compares statistical properties of the stochastic ensemble with the deterministic solution to assess uncertainty quantification.

Experimental results

Research questions

• RQ1 Can circulation-preserving stochastic transport noise effectively represent unresolved scale dynamics in a two-layer quasi-geostrophic model?
• RQ2 How well does the stochastic parameterization reproduce the statistical equilibrium of a high-resolution deterministic simulation?
• RQ3 Does the method maintain consistency and accuracy in both homogeneous and heterogeneous flow regimes?
• RQ4 Can the estimated stochastic forcing from high-resolution data reliably approximate unresolved components in coarse-grid models?
• RQ5 To what extent does the stochastic model improve uncertainty quantification for potential data assimilation applications?

Key findings

• The stochastic parameterization successfully preserves the Kelvin circulation theorem, ensuring physical consistency in the stochastic model.
• The stochastic ensemble captures the statistical equilibrium properties of the high-resolution deterministic solution with high fidelity.
• The method demonstrates robust performance in both homogeneous and heterogeneous flow configurations, indicating broad applicability.
• The estimated stochastic forcing accurately reflects unresolved dynamics derived from high-resolution simulation data.
• The consistent time-stepping scheme ensures numerical stability and reliability in long-term stochastic simulations.
• The results establish a strong foundation for using this approach in data assimilation frameworks due to its accuracy and physical consistency.