Results are presented from an experimental study of shallow flow in a channel partially obstructed by an array of circular cylinders. The cylinder array is a model for emergent vegetation in an open channel, but also represents a simple sparse porous medium. A shear layer with regular vortex structures forms at the edge of the array, evolving downstream to an equilibrium width and vortex size. The vortices induce nearly periodic oscillations with a frequency that matches the most unstable linear mode for a parallel shear flow. The shear layer is asymmetric about the array interface and has a two-layer structure. An inner region of maximum shear near the interface contains a velocity inflection point and establishes the penetration of momentum into the array. An outer region, resembling a boundary layer, forms in the main channel, and establishes the scale of the vortices. The vortex structure, educed by conditional sampling, shows strong crossflows with sweeps from the main channel and ejections from the array, which create significant momentum and mass fluxes across the interface. The sweeps maintain the coherent structures by enhancing shear and energy production at the interface. A linear stability analysis is consistent with the experimental results and demonstrates that the instability is excited by the differential drag between the channel and the array.

[1] The shear layer at the top of a submerged canopy generates coherent vortices that control exchange between the canopy and the overflowing water. Unlike free shear layers, the vortices in a canopy shear layer do not grow continuously downstream but reach and maintain a finite scale determined by a balance between shear production and canopy dissipation. This balance defines the length scale of vortex penetration into the canopy, d e , and the region of rapid exchange between the canopy and overflow. Deeper within the canopy, transport is constrained by smaller turbulence scales. A two-box canopy model is proposed on the basis of the length scale d e . Using diffusivity and exchange rates defined in previous studies, the model predicts the timescale required to flush the canopy through vertical exchange over a range of canopy density and height. The predicted canopy retention times, which range from minutes to an hour, are consistent with canopy retention inferred from tracer observations in the field and comparable to retention times for some hyporheic regions. The timescale for vertical exchange, along with the in-canopy velocity, determines the minimum canopy length for which vertical exchange dominates water renewal. Shorter canopies renew interior water through longitudinal advection. Finally, canopy water retention influences longitudinal dispersion through a transient storage process. When vertical exchange controls canopy retention, the transient storage dispersion increases with canopy height. When longitudinal advection controls water renewal, dispersion increases with canopy patch length.

[1] This paper presents a method for predicting the distributions of velocity and shear stress in shallow channels with a boundary of emergent vegetation. Experiments in a laboratory channel with model vegetation show that the velocity profile exhibits a distinct two-layer structure, consisting of a rapidly varying shear layer across the vegetation interface and a more gradual boundary layer in the main channel. In addition, coherent vortices are observed which span both layers, and are the dominant contributors to lateral momentum fluxes. From these observations, we propose a model for the vortex-induced exchange and find expressions for the width of momentum penetration into the vegetation, the velocity and shear stress at the vegetation edge, and the width of the boundary layer in the main channel. These variables, along with a momentum balance in the main channel, comprise a modeling framework which accurately reproduces the observed velocity and shear stress distributions. The predictions for the velocity and shear stress can provide a basis for modeling flood conveyance, overbank sediment transport, and scalar residence time in the vegetated layer.

This paper theoretically describes and experimentally verifies two mechanisms leading to longitudinal dispersion of a passive tracer in a random array of circular cylinders. We focus on moderate Reynolds numbers of order 10-1000, specifically the range characterized by unsteady cylinder wakes. In this regime, two mechanisms contribute to dispersion, each associated with a distinct region of the cylinder wakes: (i) the unsteady recirculation zone close to each cylinder, and (ii) the velocity defect behind each cylinder, which extends downstream of the cylinder over a distance of the order of the cylinder spacing. The first mechanism, termed vortex-trapping dispersion, is due to the entrainment of tracer into the unsteady recirculation zone, where it is momentarily trapped and then released. A theoretical expression for this dispersive mechanism is derived in terms of the residence time and size of the recirculation zone. The second mechanism is due to advection through the random velocity field created by the random distribution of the wake velocity defect. We derive an expression for the defect behind an average cylinder, and show that it decays owing to array drag over a length scale called the attenuation length, which is of the order of the cylinder spacing. The superposition of the wake defect behind each cylinder creates the random velocity field. Theoretical predictions for dispersion agree very well with observations of tracer transport in a laboratory cylinder array, correctly capturing the dependence on array density and Reynolds number. The laboratory studies also document a transition in small-scale mixing at cylinder Reynolds number ≈ 200. Below this limit, individual filaments of tracer remain distinct, producing significant fluctuations in the local concentration field. At higher Reynolds number, cylinder wakes contribute sufficient turbulence to erase the filament signature and smooth the tracer distribution.

Machine learning (ML) provides novel and powerful ways of accurately and efficiently recognizing complex patterns, emulating nonlinear dynamics, and predicting the spatio-temporal evolution of weather and climate processes. Off-the-shelf ML models, however, do not necessarily obey the fundamental governing laws of physical systems, nor do they generalize well to scenarios on which they have not been trained. We survey systematic approaches to incorporating physics and domain knowledge into ML models and distill these approaches into broad categories. Through 10 case studies, we show how these approaches have been used successfully for emulating, downscaling, and forecasting weather and climate processes. The accomplishments of these studies include greater physical consistency, reduced training time, improved data efficiency, and better generalization. Finally, we synthesize the lessons learned and identify scientific, diagnostic, computational, and resource challenges for developing truly robust and reliable physics-informed ML models for weather and climate processes. This article is part of the theme issue ‘Machine learning for weather and climate modelling’.