Flow and transport through aquatic vegetation is characterized by a wide range of length scales: water depth ($H$), plant height ($h$), stem diameter ($d$), the inverse of the plant frontal area per unit volume (${a}^{\ensuremath{-} 1} $) and the scale(s) over which $a$ varies. Turbulence is generated both at the scale(s) of the mean vertical shear, set in part by $a$, and at the scale(s) of the stem wakes, set by $d$. While turbulence from each of these sources is dissipated through the energy cascade, some shear-scale turbulence bypasses the lower wavenumbers as shear-scale eddies do work against the form drag of the plant stems, converting shear-scale turbulence into wake-scale turbulence. We have developed a $k$–$\varepsilon $ model that accounts for all of these energy pathways. The model is calibrated against laboratory data from beds of rigid cylinders under emergent and submerged conditions and validated against an independent data set from submerged rigid cylinders and a laboratory data set from a canopy of live vegetation. The new model outperforms existing $k$–$\varepsilon $ models, none of which include the $d$ scale, both in the emergent rigid cylinder case, where existing $k$–$\varepsilon $ models break down entirely, and in the submerged rigid cylinder and live plant cases, where existing $k$–$\varepsilon $ models fail to predict the strong dependence of turbulent kinetic energy on $d$. The new model is limited to canopies dense enough that dispersive fluxes are negligible.

This work is motivated by a case study in which cooling water effluent is discharged from a line source diffuser onto the shallow southern shelf of a large lake. The effluent is discharged with high momentum to the north, and the plume influences the surrounding flow for hundreds of meters before becoming a passive tracer. We plan to use a three-dimensional hydrodynamic model to examine the impact of this effluent on water residence time within the southern shelf, but in order to accomplish this task, we require a method for incorporating the effluent-driven flow into the hydrodynamic model, which by default does not predict the correct rate of spreading of the plume or the correct rate of lateral entrainment of ambient water. In this paper we develop a method for incorporating the effects of high-momentum line source discharges into any three-dimensional or two-dimensional (depth-averaged) hydrodynamic model that employs a horizontal eddy viscosity and diffusivity. Our approach is to modify horizontal eddy viscosity and diffusivity to enforce the correct rate of lateral entrainment of the plume. We validate the new method using three test cases relevant to our case study: a neutrally buoyant jet, a cylindrical gravity current, and a negatively buoyant jet.

Setup, testing, and application of a 2-dimensional longitudinal-vertical hydrothermal/transport model (the transport submodel of CE-QUAL-W2) was documented for Cayuga Lake, New York, where the Rossby radius is on the order of the lake's width. The model was supported by long-term monitoring of meteorological and hydrologic drivers and calibrated and validated using in-lake temperature measurements made at multiple temporal and spatial scales over 16 years. Measurements included (1) temperature profiles at multiple lake sites for 10 years, (2) near-surface temperatures at one end of the lake for 16 years, (3) high frequency temperature at multiple depths for 2 years, and (4) seasonal measurements of a conservative passive tracer. Seiche activity imparted prominent signatures within these measurements. The model demonstrated excellent temporal stability, maintaining good performance in uninterrupted simulations over a period of 15 years. Performance was improved when modeling was supported by on-lake versus land-based meteorological measurements. The validated model was applied through numeric tracer experiments to evaluate various features of transport of interest to water quality issues for the lake, including (1) residence times of stream inputs within the entire lake and a smaller region defined bathymetrically as a shallow shelf, (2) transport and fate of negatively buoyant streams, and (3) the extent of transport from the hypolimnion to the epilimnion. This hydrothermal/transport model is appropriate to serve as the transport submodel for a forthcoming water quality model for this lake and for other high aspect (length to width) ratio lacustrine systems for which the internal Burger number is order one or greater.

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