THE EJECTION-SWEEP CYCLE OVER GENTLE HILLS COVERED WITH BARE AND FORESTED SURFACES Davide Poggi1,2, and Gabriel Katul1

1Nicholas School of the Environment and Earth Sciences, Duke University, Durham, North Carolina, USA2Dipartimento di Idraulica, Trasporti ed Infrastrutture, Civili, Politecnico di Torino, Torino, Italy

*contact: gaby@duke.edu

AbstractProgress on practical problems such as linking biological sources and sinks with turbulent scalar

flux measurements in ecosystems situated over non-flat terrain are now requiring fundamental understanding of how topography modulates the basic properties of turbulence. In particular, how hilly terrain alter the ejection-sweep cycle, which is the main coherent transporting motion, remains a problem that received surprisingly little theoretical and experimental attention. Here, we investigate how boundary conditions, including canopy and gentle topography, alter the properties of the ejection-sweep cycle and whether it is possible to quantify their combined impact using simplified models. Towards this goal, we conducted two new flume experiments that explore the higher order turbulence statistics above a train of gentle hills. The first set of experiments were conducted over a bare surface while the second set were conducted over a modeled vegetated surface composed of tall and densely arrayed rods. Using this data, the connections between the ejection-sweep cycle and the higher order turbulence statistics across various positions above the hill surface were investigated. We showed that ejections dominate momentum transfer for both surface covers at the top of the inner layer. However, within the canopy and near the canopy top, sweeps dominate momentum transfer irrespective of the longitudinal position. Ejections remain the dominant momentum transfer mode in the entire inner region over the bare surface. These findings were well reproduced using an incomplete cumulant expansion and the measured profiles of the second moments of the flow. This agreement partly stems from the fact that the imbalance in the flux-transport terms, needed in the incomplete cumulant expansion, were well modeled using "local" gradient-diffusion principles. This finding suggests that in the inner layer, the higher-order turbulence statistics appear to be much more impacted by their relaxation history towards equilibrium rather than the advection-distortion history from the mean flow. Hence, we showed that it is possible to explore how various boundary conditions, including canopy and topography, alter the properties of the ejection-sweep cycle by quantifying their impact on the gradients of the second moments only.

IntroductionUnderstanding the modulation of organized motion by complex terrain near boundaries, bare or vegetated, remains a central research topic now being stimulated by a diverse set of applications. Particularly, how complex terrain modulates the ejection-sweep cycle, which is the main coherent motion responsible for the bulk turbulent transport, remains a problem that received surprisingly little attention. In contrast, studies on how hills modify the mean flow properties received significant theoretical and experimental attention. While the mean flow properties are strongly forced by the topography and can be plagued by 'discontinuities' such as separation or re-circulation regions, it is not yet clear how the turbulence is impacted by complex terrain. The main problem lies in the fact that turbulence 'remembers' a significant portion of the strain rate history injected by the mean flow, and the energy-containing eddies remember a significant portion of their relaxation history towards equilibrium. The relative importance of these two 'memories' can profoundly affect higher-order statistics across the hill and quantifying their interplay remains a vexing problem, the subject of this work.

The Experiment

Discussion• The mixing length is well approximated by constant inside the canopy and linear in the inner layer (Figure 1).

•For the canopy case, we found that the mean flow experiences a re-circulation region at the leeside. Furthermore, the mean flow inside the canopy is highly variable longitudinally but not vertically. The converse is true at the top of the canopy (Figure 2-top). For the bare surface case, the mean flow is both longitudinally and vertically variable. The longitudinally averaged value is approximately logarithmic (Figure 2-bottom).

•For the canopy case, the turbulent shear stress is highly variable vertically but not longitudinally inside the canopy. The opposite in the inner layer above the canopy is observed (Figure 3-top). For the bare surface case, a thin viscous sub-layer near the flume floor is observed (Figure 3 – bottom). The turbulent stress is highly variable longitudinally in all regions of the inner layer (Figure 3 – bottom).

•For the canopy case, the ejection-sweep cycle statistic is well reproduced by all 3 models (CEM, ICEM, and gradient-diffusion). Sweeps dominate momentum transfer inside and near the canopy top at all longitudinal positions on the hill. The thickness of the equilibrium layer (here refers to the layer when sweeps balance ejections) is longitudinally variable across the hill (Figure 4). Ejections dominate momentum transfer for the bare surface case (Figure 5). Furthermore, the performance of gradient-diffusion models is much worse near the hill top (Figure 5).

Results

Theory

Figure 2: Mean Velocity Distribution Figure 4: Ejections and Sweeps with Canopy Figure 5: Ejections and Sweeps for the Bare Surface

Hill Properties:Four hill modulesHill Height (H) = 0.08 mHill Half Length (L) = 0.8 m

Canopy PropertiesCanopy Height = 0.1 mRod diameter = 0.004 mRod density = 1000 rods/m2

Flow Properties:Water Depth = 0.6 mBulk Re > 1.5 x 105

Velocity MeasurementsLaser Doppler AnemometerSampling Frequency = 3000 HzSampling Period = 300 s

ObjectiveThe compass of this work is restricted to the connection between the ejection-sweep cycle and the higher order statistics across various positions above a train of gentle hills with a 'lens' on the flux-transporting terms, the terms most impacted by ejections and sweeps in the TKE budget. The tall forested experiment is designed such that the topographic variations remain comparable to the canopy height so that the interplay between organized eddy motion dominating canopy turbulence and the terrain variability is amplified. The bare surface experiment are carried out over an almost smooth surface to permit a sufficiently deep inner layer.

Figure 3: Turbulent Stress Distribution

1) Quadrant analysis

4) Gradient-Diffusion Approximation

2) Third-Order Cumulant Expansion (CEM)

3) Incomplete Cumulant Expansion (ICEM)

Figure 1: Measured Mixing Length

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AcknowledgementsSupport was provided by the Department of Energy (DOE) through the FACE-FACTS and Terrestrial Carbon Processes (TCP) programs, by the National Institute of Global Environmental Change (NIGEC) through the Southeast Regional Center at the University of Alabama, Tuscaloosa (DOE cooperative agreement DE-FC030-90ER61010) and by the SERC-NIGEC RCIAP Research Program, and through the National Science Foundation (EAR and DMS).

The broader impact of this study can be summarized as follows: If gradient-diffusion closure schemes for the triple moments do capture the statistical properties of the ejection-sweep cycle shown here, then these ejection-sweep properties must be in local equilibrium with the local gradients of the second moments (even within a re-circulation region). Given that advection remains significant in the mean momentum equation, the only way this local equilibrium can be maintained here is when the time-scales responsible for the production of ejections and sweeps are sufficiently shorter than the advection distortion time scale by the mean velocity within the inner layer.

Conclusion

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