Periodic Reporting for period 1 - GeoCascade (Elucidating the bidirectional energy cascade of geophysical turbulence in time, space, and scale)
Período documentado: 2023-09-18 hasta 2025-09-17
Turbulence—the chaotic motion of fluids—governs how energy and momentum move through the atmosphere and oceans, shaping jets, storm tracks and mixing, and therefore affecting climate and extreme events. Because models cannot resolve all turbulent scales, they rely on closures built largely on idealised three dimensional turbulence, where energy cascades downscale. Real geophysical flows, however, are rotating, stratified and often confined in thin layers. Under these conditions, turbulence can split its energy simultaneously upscale and downscale (a bidirectional cascade), with a sharp onset as control parameters (e.g. rotation or aspect ratio) change. There is no quantitative theory for when this transition occurs, how the energy split depends on parameters, or which mechanisms drive it. This limits our ability to represent mixing, dissipation and the emergence of large scale structures in environmental models.
This project set out to: (i) identify mechanisms responsible for the onset of bidirectional cascades; (ii) characterise their spatial and spectral signatures across scales; and (iii) develop a simple, quantitative model for the bidirectional cascade that can inform improved closures. Two complementary lenses were combined:
- Spatial perspective: track how structures stretch, split, thin or merge across scales, to quantify their contributions to forward (downscale) and inverse (upscale) transfer.
- Spectral perspective: analyse how interactions among Fourier modes depend on their phases; sustained flux requires phase organisation (“synchronisation”) across triads of interacting modes.
High performance simulations (reduced “shell” models, two dimensional turbulence, and a large wall bounded flow used for a deviation study) enabled unprecedented sampling of noisy, nonlocal triad interactions, and provided the data needed to test theory. The key novelty is elevating triad phase dynamics from anecdotal evidence in extreme events to a predictive, scale by scale framework that links directly to energy flux.
Pathway to impact: (a) open diagnostics and datasets to detect and quantify the balance between inverse and forward transfer; (b) theory informed parameterisations of energy flux direction and magnitude for weather, ocean and climate models; (c) practical guidance on when inverse cascade effects are expected and how to adjust closures; and (d) open source tools that lower the barrier to analysing cascade physics.
This project set out to: (i) identify mechanisms responsible for the onset of bidirectional cascades; (ii) characterise their spatial and spectral signatures across scales; and (iii) develop a simple, quantitative model for the bidirectional cascade that can inform improved closures. Two complementary lenses were combined:
- Spatial perspective: track how structures stretch, split, thin or merge across scales, to quantify their contributions to forward (downscale) and inverse (upscale) transfer.
- Spectral perspective: analyse how interactions among Fourier modes depend on their phases; sustained flux requires phase organisation (“synchronisation”) across triads of interacting modes.
High performance simulations (reduced “shell” models, two dimensional turbulence, and a large wall bounded flow used for a deviation study) enabled unprecedented sampling of noisy, nonlocal triad interactions, and provided the data needed to test theory. The key novelty is elevating triad phase dynamics from anecdotal evidence in extreme events to a predictive, scale by scale framework that links directly to energy flux.
Pathway to impact: (a) open diagnostics and datasets to detect and quantify the balance between inverse and forward transfer; (b) theory informed parameterisations of energy flux direction and magnitude for weather, ocean and climate models; (c) practical guidance on when inverse cascade effects are expected and how to adjust closures; and (d) open source tools that lower the barrier to analysing cascade physics.
New numerical capabilities and datasets
- Shell models (2D/3D and 2D MHD): a flexible suite enabled hundreds of simulations for phase statistics and parameter sweeps through a bidirectional cascade transition; curated triad phase datasets were produced.
- 2D turbulence on GPU: a Python/CuPy solver achieved up to ~1000x speed up over a CPU baseline and collected on the fly statistics for thousands of nonlocal triads and ensemble runs—essential for converging noisy triad signals.
- Wall bounded turbulence (deviation): seven unprecedented large domain, high Reynolds number simulations of Waleffe flow were completed on a European supercomputer, systematically varying a large scale drag.
Spectral analysis and diagnostics
- Efficient routines were developed to identify and sample thousands of interacting triads across scales in steady, forced dissipative turbulence.
- Shell models (2D/3D): comprehensive measurements revealed weak inter triad coupling and robust, scale dependent triad phase statistics; “phase only” variants isolated the dynamical role of phases in setting flux direction and magnitude.
- 2D turbulence: first steady state quantification of triad phase synchronisation across thousands of triads, with clear links to inverse energy transfer and, in some regimes, detectable scale dependence that challenges standard inertial range assumptions.
Mathematical model linking phases to flux
- A reduced statistical model treated each triad phase as a self interaction plus noise (a noisy Adler equation). Via Fokker–Planck analysis, stationary phase distributions were derived with a single fitted noise parameter, yielding an analytic prediction for the scale wise average of cos(phase) that enters the exact energy flux expression.
- Validation in shell models showed excellent agreement of phase PDFs and correct prediction of cascade direction without fitting, and explained a long standing failure of an “exact 2D” shell model limit as a consequence of phase dynamics.
- Extension to 2D turbulence demonstrated that measured triad phase PDFs agree with theory for the majority of triads; combining these statistics with classical inertial range arguments provided, for the first time, a dynamical proof that energy flows upscale while enstrophy flows downscale.
Bidirectional cascade in a minimal two field setting (2D MHD shell model)
- First shell model to reproduce a sharp onset of inverse transfer as a control parameter varies, mirroring full geophysical systems.
- Discovery of a “front” of magnetic phase synchronisation appearing once the magnetic to kinetic energy ratio crosses a threshold; the front advances towards larger scales near criticality. Magnetic synchronisation disrupts velocity phase organisation and suppresses kinetic energy flux—consistent with observations in full MHD turbulence. Amplitude only experiments confirmed amplitudes matter but are insufficient, motivating coupled phase–amplitude models.
Large scale rollers in wall bounded turbulence (deviation)
- Streamwise rollers were observed in stress free Waleffe flow. Their presence, size and number depend sensitively on large scale drag: absent at very low drag, relaminarisation at very high drag, robust in between. This shows boundary layers are not required and provides a clean, tuneable system to develop instability based models of very large scale motions.
- Shell models (2D/3D and 2D MHD): a flexible suite enabled hundreds of simulations for phase statistics and parameter sweeps through a bidirectional cascade transition; curated triad phase datasets were produced.
- 2D turbulence on GPU: a Python/CuPy solver achieved up to ~1000x speed up over a CPU baseline and collected on the fly statistics for thousands of nonlocal triads and ensemble runs—essential for converging noisy triad signals.
- Wall bounded turbulence (deviation): seven unprecedented large domain, high Reynolds number simulations of Waleffe flow were completed on a European supercomputer, systematically varying a large scale drag.
Spectral analysis and diagnostics
- Efficient routines were developed to identify and sample thousands of interacting triads across scales in steady, forced dissipative turbulence.
- Shell models (2D/3D): comprehensive measurements revealed weak inter triad coupling and robust, scale dependent triad phase statistics; “phase only” variants isolated the dynamical role of phases in setting flux direction and magnitude.
- 2D turbulence: first steady state quantification of triad phase synchronisation across thousands of triads, with clear links to inverse energy transfer and, in some regimes, detectable scale dependence that challenges standard inertial range assumptions.
Mathematical model linking phases to flux
- A reduced statistical model treated each triad phase as a self interaction plus noise (a noisy Adler equation). Via Fokker–Planck analysis, stationary phase distributions were derived with a single fitted noise parameter, yielding an analytic prediction for the scale wise average of cos(phase) that enters the exact energy flux expression.
- Validation in shell models showed excellent agreement of phase PDFs and correct prediction of cascade direction without fitting, and explained a long standing failure of an “exact 2D” shell model limit as a consequence of phase dynamics.
- Extension to 2D turbulence demonstrated that measured triad phase PDFs agree with theory for the majority of triads; combining these statistics with classical inertial range arguments provided, for the first time, a dynamical proof that energy flows upscale while enstrophy flows downscale.
Bidirectional cascade in a minimal two field setting (2D MHD shell model)
- First shell model to reproduce a sharp onset of inverse transfer as a control parameter varies, mirroring full geophysical systems.
- Discovery of a “front” of magnetic phase synchronisation appearing once the magnetic to kinetic energy ratio crosses a threshold; the front advances towards larger scales near criticality. Magnetic synchronisation disrupts velocity phase organisation and suppresses kinetic energy flux—consistent with observations in full MHD turbulence. Amplitude only experiments confirmed amplitudes matter but are insufficient, motivating coupled phase–amplitude models.
Large scale rollers in wall bounded turbulence (deviation)
- Streamwise rollers were observed in stress free Waleffe flow. Their presence, size and number depend sensitively on large scale drag: absent at very low drag, relaminarisation at very high drag, robust in between. This shows boundary layers are not required and provides a clean, tuneable system to develop instability based models of very large scale motions.
Beyond the state of the art
- Predictive theory: a validated, dynamical link from triad phase statistics to energy flux sign (no fitting) and magnitude (single parameter), providing the first mechanistic explanation of cascade directions in 2D turbulence and resolving a decades old shell model puzzle.
- First steady state, large sample measurements of triad phase synchronisation in fully turbulent 2D flow, revealing non random phase organisation and, in some regimes, scale dependence.
- Minimal two field testbed for critical cascade transitions (2D MHD), uncovering a phase synchronisation front that suppresses kinetic flux.
- Definitive simulations showing large scale rollers without boundary layers, establishing Waleffe flow as a clean canonical system for mechanism and model development.
- Open tools and datasets (GPU solver, shell model suite, curated statistics; Waleffe flow data to follow) enabling replication and uptake.
Impacts and uptake needs
- Impacts: methods for quantifying triad phase synchronization in fully turbulent flows; theory informed closures and diagnostics for regimes with inverse transfer; computational efficiency via GPU native tools and minimal models.
- Key needs: extend the phase framework to multi field and 3D anisotropic flows
- Predictive theory: a validated, dynamical link from triad phase statistics to energy flux sign (no fitting) and magnitude (single parameter), providing the first mechanistic explanation of cascade directions in 2D turbulence and resolving a decades old shell model puzzle.
- First steady state, large sample measurements of triad phase synchronisation in fully turbulent 2D flow, revealing non random phase organisation and, in some regimes, scale dependence.
- Minimal two field testbed for critical cascade transitions (2D MHD), uncovering a phase synchronisation front that suppresses kinetic flux.
- Definitive simulations showing large scale rollers without boundary layers, establishing Waleffe flow as a clean canonical system for mechanism and model development.
- Open tools and datasets (GPU solver, shell model suite, curated statistics; Waleffe flow data to follow) enabling replication and uptake.
Impacts and uptake needs
- Impacts: methods for quantifying triad phase synchronization in fully turbulent flows; theory informed closures and diagnostics for regimes with inverse transfer; computational efficiency via GPU native tools and minimal models.
- Key needs: extend the phase framework to multi field and 3D anisotropic flows