A One‐Dimensional Model for Turbulent Mixing in the Benthic Biolayer of Stream and Coastal Sediments

Grant, Stanley B.; Gomez‐Velez, Jesus D.; Ghisalberti, Marco; Guymer, I.; Boano, Fulvio; Roche, Kevin; Harvey, J. W.

doi:10.1029/2019wr026822

Cited by 13 publications

(20 citation statements)

References 63 publications

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“…The 1D diffusivity framework, on the other hand, lumps these geomorphic processes into a surficial dispersion coefficient and an inverse decay length scale that can be directly calculated from the aforementioned advective model parameters (see Equations and ). These formulae also predict that the surficial dispersion coefficient for bedform pumping increases with the dimensionless permeability Reynolds number, consistent with diffusivities measured for turbulent exchange across flat streambeds (Grant et al, 2020; Voermans et al, 2018) and streambeds with bedforms (Grant, Gomez‐Velez, & Ghisalberti, 2018; Grant et al, 2012; O'Connor & Harvey, 2008). Efforts are currently underway to extend these analytical solutions to open systems (e.g., stream networks), bedform turnover, unsteady flows, and the nonlinear reactions that drive nutrient cycling in the benthic biolayer of streams.…”

Section: Discussionsupporting

confidence: 74%

“…The turbulent and dispersive diffusivities increase with the permeability Reynolds number,

{Re}_{K} = u_{*} \sqrt{K} / υ

(‐), a dimensionless ratio of a permeability length scale (

\sqrt{K}

[L]) and the viscous length scale that governs turbulence at the surface of the streambed (ratio of the kinematic viscosity of water υ [L 2 T −1 ] and the shear velocity u * [L T −1 ]) (Voermans et al, 2017, 2018). For turbulent mass transfer across a flat SWI, and accounting for the exponential decay of diffusivity with depth, the surficial effective diffusivity exhibits different permeability Reynolds number scaling behavior in the dispersive (

D_{eff, 0} \propto {Re}_{K}^{2.5}

,0.01 < Re K < 1) and turbulent diffusive (

D_{eff, 0} \propto {Re}_{K}^{1}

, Re K > 1) regimes (Grant et al, 2020). Our formula linking 2D advective and 1D dispersive descriptions of bedform pumping (Equation ) implies that dispersive mixing by bedform pumping also increases with the permeability Reynolds number,

E_{0} \propto {Re}_{K}^{2}

.…”

Section: Discussionmentioning

confidence: 99%

“…From the results presented above, a set of explicit solutions can be derived for solute concentration in the water column and interstitial fluids of a closed system (Grant et al, 2020): Figure 4. Scaling behavior for mass remaining in the streambed after an impulsive injection of mass at the SWI at time t diff ¼ 0.…”

Section: Solute Concentration In the Water And Sediment Columns Of CLmentioning

confidence: 99%

“…Grant et al (2020) demonstrated that Equation is a reasonable descriptor of vertical solute transport by turbulent dispersion and turbulent diffusion through the benthic biolayer of a flat streambed, provided that the diffusion coefficient declines exponentially with depth through the sediment bed. In this paper we hypothesize that a similar result applies to bedform pumping, but with the effective diffusivity replaced by an exponentially declining dispersion coefficient, D eff ( y ) = E ( y ) = E 0 e − ay .…”

Section: Diffusive Model Of Hyporheic Exchange By Bedform Pumpingmentioning

confidence: 99%

“…As detailed in Grant et al (2020), we can link the mass balance equations above (Equation ) and below (Equation ) the SWI and thereby account for two‐way coupling across the SWI, using Duhamel's theorem, an analytical approach for solving the diffusion equation in cases where the forcing function at one boundary is a piece‐wise continuous function of time (Guerrero et al, 2013). Duhamel's theorem allows us to express the evolution of interstitial solute concentration in the sediment bed as a convolution of the water column concentration

C_{w} ((), {\overset{t}{true}}_{diff})

and a so‐called auxiliary function

C_{s}^{A} ((),, {\overset{y}{true}}_{diff}, {\overset{t}{true}}_{diff})

where v is a dummy integration variable (Guerrero et al, 2013):

{\overset{true¯}{C}}_{s} ((),, {\overset{y}{true}}_{diff}, {\overset{t}{true}}_{diff}) = \int_{0}^{{\overset{t}{true}}_{diff}} {\overset{true¯}{C}}_{s}^{A} ((),, {\overset{y}{true}}_{diff}, {\overset{true¯}{t}}_{diff} - v) \frac{d}{dv} ([], {\overset{true¯}{C}}_{w} ((), v) H ((), v)) italicdv

The auxiliary function is a solution to the same system of equations (Equations –), but with the inhomogeneous term replaced by the Heaviside function (cf.…”

Section: Diffusive Model Of Hyporheic Exchange By Bedform Pumpingmentioning

confidence: 99%

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Unifying Advective and Diffusive Descriptions of Bedform Pumping in the Benthic Biolayer of Streams

Grant

Monofy

Boano

et al. 2020

Water Resources Research

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Many water quality and ecosystem functions performed by streams occur in the benthic biolayer, the biologically active upper (~5 cm) layer of the streambed. Solute transport through the benthic biolayer is facilitated by bedform pumping, a physical process in which dynamic and static pressure variations over the surface of stationary bedforms (e.g., ripples and dunes) drive flow across the sediment-water interface. In this paper we derive two predictive modeling frameworks, one advective and the other diffusive, for solute transport through the benthic biolayer by bedform pumping. Both frameworks closely reproduce patterns and rates of bedform pumping previously measured in the laboratory, provided that the diffusion model's dispersion coefficient declines exponentially with depth. They are also functionally equivalent, such that parameter sets inferred from the 2D advective model can be applied to the 1D diffusive model, and vice versa. The functional equivalence and complementary strengths of these two models expand the range of questions that can be answered, for example, by adopting the 2D advective model to study the effects of geomorphic processes (such as bedform adjustments to land use change) on flow-dependent processes and the 1D diffusive model to study problems where multiple transport mechanisms combine (such as bedform pumping and turbulent diffusion). By unifying 2D advective and 1D diffusive descriptions of bedform pumping, our analytical results provide a straightforward and computationally efficient approach for predicting, and better understanding, solute transport in the benthic biolayer of streams and coastal sediments. Plain Language Summary How far and fast pollutants travel downstream is often conditioned on what happens in a thin veneer of biologically active bottom sediments called the benthic biolayer. However, before a pollutant can be removed in the benthic biolayer, it must first be transported across the sediment-water interface and through the interstitial fluids of these surficial sediments. In this paper we demonstrate that one important mechanism for transporting solutes to, and through, the benthic biolayerbedform pumping-can be interchangeably represented as either a two-dimensional advective process or a one-dimensional dispersion process. The complementary nature of these models expands the range of benthic biolayer processes that can be studied and predicted with the end goal of improving coastal and stream water quality.

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Section: Discussionsupporting

confidence: 74%

“…The turbulent and dispersive diffusivities increase with the permeability Reynolds number,

{Re}_{K} = u_{*} \sqrt{K} / υ

(‐), a dimensionless ratio of a permeability length scale (

\sqrt{K}

D_{eff, 0} \propto {Re}_{K}^{2.5}

,0.01 < Re K < 1) and turbulent diffusive (

D_{eff, 0} \propto {Re}_{K}^{1}

E_{0} \propto {Re}_{K}^{2}

.…”

Section: Discussionmentioning

confidence: 99%

Section: Solute Concentration In the Water And Sediment Columns Of CLmentioning

confidence: 99%

Section: Diffusive Model Of Hyporheic Exchange By Bedform Pumpingmentioning

confidence: 99%

C_{w} ((), {\overset{t}{true}}_{diff})

and a so‐called auxiliary function

C_{s}^{A} ((),, {\overset{y}{true}}_{diff}, {\overset{t}{true}}_{diff})

where v is a dummy integration variable (Guerrero et al, 2013):

{\overset{true¯}{C}}_{s} ((),, {\overset{y}{true}}_{diff}, {\overset{t}{true}}_{diff}) = \int_{0}^{{\overset{t}{true}}_{diff}} {\overset{true¯}{C}}_{s}^{A} ((),, {\overset{y}{true}}_{diff}, {\overset{true¯}{t}}_{diff} - v) \frac{d}{dv} ([], {\overset{true¯}{C}}_{w} ((), v) H ((), v)) italicdv

The auxiliary function is a solution to the same system of equations (Equations –), but with the inhomogeneous term replaced by the Heaviside function (cf.…”