Heterogeneity Matters: Aggregation Bias of Gas Transfer Velocity Versus Energy Dissipation Rate Relations in Streams

Abstract The gas transfer velocity, k, modulates gas fluxes across air‐water interfaces in rivers. While the theory postulates a local scaling law between k and the turbulent kinetic energy dissipation rate ε, empirical studies usually interpret this relation at the reach‐scale. Here, we investigate how local k(ε) laws can be integrated along heterogeneous reaches exploiting a simple hydrodynamic model, which links stage and velocity to the local slope. The model is used to quantify the relative difference between the gas transfer velocity of a heterogeneous stream and that of an equivalent homogeneous system. We show that this aggregation bias depends on the exponent of the local scaling law, b, and internal slope variations. In high‐energy streams, where b>1, spatial heterogeneity of ε significantly enhances reach‐scale values of k as compared to homogeneous settings. We conclude that small‐scale hydro‐morphological traits bear a profound impact on gas evasion from inland waters.

: where the shape parameter E  and the scale parameter E  constrain the moments of E S. In particular, E  is the inverse of the square of the coefficient of variation of the slope, that is, . The longitudinal pdf of the kinetic energy dissipation rates, ( ) L E p  can be obtained from Equations 4 and 5 as: Likewise, the volumetric pdf of E  can be obtained combining the relation between V E p and L E p and Equation 6 as: These pdfs express the probability distribution functions of the total kinetic energy dissipation rate within the focus reach, as a function of four parameters: two morphological parameters that define the internal heterogeneity of the slope ( E  and E ), and two hydrodynamic parameters ( 1 E  and E d).

Upscaling Total Kinetic Energy Dissipation Rates
Let us consider the 1D stream reach of length E L depicted in Figure 1. The total power dissipated within the reach, E P, can be expressed by integrating along the reach the product between the dissipation rate ( ) E x  , and the related water mass per unit length, that is,: where E  is the water density. The spatial patterns in the local energy dissipation rate ( ) E x  originate a reachwise equivalent counterpart, which is here denoted as eq E  . The equivalent energy dissipation rate, eq E  , should provide the same total energy losses induced by the assortment of local ( ) E x  experienced by the flow along the reach. The total dissipated power can be written in terms of eq E  as: where E A   is the average cross sectional area of the stream reach. By equaling Equations 8 and 9, the equivalent energy dissipation rate can be written as: which corresponds to a weighted average of ( ) E x  over the stream length, the weights being set by the volu- A more explicit formulation for eq E  can be obtained by inserting Equations 3 into 10 and recalling that U x , that is,: where 1 E Q A    is the mean water velocity measured by means of tracer injections.

Upscaling Gas Transfer Velocity
The concentration of a conservative gas in water is governed by the equation: where x E D is the hydrodynamic dispersion, E C is the gas concentration in water, a E C is the corresponding equilibrium concentration (the concentration that equilibrate with the atmosphere) and ( ) E k x is the gas transfer velocity. If the dispersion term is neglected (i.e., 0 x E D  ), as typically done in experimental tracer studies (e.g., Benson et al., 2014;Heilweil et al., 2016;Ulseth et al., 2019), the general solution of Equation 12 reads: . This equality leads to the following expression for eq E k : Equation 14 shows that the equivalent mass transfer velocity is a spatial longitudinal average of the local gas transfer velocities along the stream.

Upscaling Gas Transfer Velocity Versus Total Kinetic Energy Dissipation Rate Relations
Theoretical analysis and empirical results have long suggested the existence of a power-law relation between E k and E , there is no consensus on the value of the scaling exponent and empirical data suggest that bubble-mediated transport process could significantly increase E b up to 1.2 . Although these data mostly refer to reach-scale estimates, they likely reflect the fact that the local scaling exponent might be higher than the theoretical value of 0.25 in case of air entrainment. For these reasons, here we treat E b as a model parameter. Regardless of the specific value of the scaling exponent, the ( ) E k  relation has an inherent local nature implied by the mechanistic link among energy dissipation, turbulence, and gas exchange at the interface.
Nevertheless, such scaling laws have been empirically analyzed at the scale of entire reaches, including highly heterogeneous river segments. Using Equation 14, the equivalent mass transfer rate for a reach with length E L, eq E k , can be written as: In Equation 15, the coefficient E a-which depends on the features of the underlying eddy dynamics-was assumed to be spatially uniform as we expect the turbulent structures to be weakly heterogeneous along the path. Equation 15 quantifies the equivalent gas transfer rate of an heterogeneous stream with a mean value of the total kinetic energy dissipation rate equal to V E 10.1029/2021GL094272 6 of 12 slope is spatially uniform within the reach). The resulting aggregation bias, E e is quantified as the relative difference between the reach scale gas transfer velocity of an internally heterogeneous stream (Equation 15) and the gas transfer velocity of an equivalent homogeneous stream with the same mean turbulent kinetic energy dissipation rate ( b V E a    ). Under the assumptions made, the following analytical expression for E e can be obtained (see Supporting Information): Interestingly, Equation 16 shows that the bias does not depend on the discharge, E Q, and the friction coefficient, E , but is only a function of the following independent parameters: (a) the exponent of the local scaling law that links the mass transfer rate to the total kinetic energy dissipation rate, E b; (b) the hydraulic exponent E d, which quantifies the relation between the hydraulic radius and slope; and (c) the parameter E , which quantifies the relative variations of the slope within the focus reach ( In Figure 2, e exceeds 10% only when S E CV is above 1, whereas for 0.5 E b  the magnitude of the bias is usually negligible. Interestingly, for b  1 4 / (which is the expected scaling exponent according to the theory), the bias is negative up to 1.5 S E CV  . The behavior shown in Figure 2 is generated by the combined action of two independent components that are responsible for generating the overall aggregation bias. The first component is termed

Discussion
The main goal of this study is to clarify analogies and differences between local and reach-scale ( ) E k  relations. To this aim, a simple hydrodynamic model was proposed in which the spatial variations of E  and E k are linked to the heterogeneity of the slope. While the formulation is quite general, the suitability of the proposed model to describe the water flow in mountain reaches needs to be carefully assessed case-by-case. In these settings, in fact, the flow can be tortuous and irregular, thereby challenging the use of a simplified 1D framework. Several studies suggested the use of an updated version of the classical Gauckler-Manning equation to model water flows in small mountain streams in which the Manning coefficient varies with the hydraulic radius (see Marcus et al., 1992). The proposed model could incorporate the effect of this dependence of the roughness on H E R by properly tuning the coefficients E  and E  in Equation 1. Furthermore, local energy dissipations with enhanced gas evasion could be observed in correspondence of hydraulic jumps (which are not described by our model), owing to the transition from supercritical and subcritical flow. While we acknowledge that 2 E D and 3 E D formulations might be more flexible in describing the spatial heterogeneity of complex flow fields typically observed in high-energy streams (including those observed in presence of, for example, step and pools, hydraulic jumps, and abrupt planar discontinuities), we propose that the concept of aggregation bias introduced in this paper is quite general. In particular, our study indicates that-regardless of the specific features of the flow field-internal heterogeneity of the energy dissipation processes generates a potential bias in spatially integrated ( ) E k  laws. The magnitude of the bias is eventually driven by the local scaling exponent E b and the underlying pdfs of the turbulent kinetic energy dissipation rate in the focus reach (Equations 10 and 14), which could be evaluated on a case-by-case basis. Therefore, the proposed framework not only provides a first-order assessment of the impact of hydrodynamic heterogeneity on the outgassing of streams with variable slope, but offers a robust conceptual basis for evaluating the aggregation bias in many other settings, in which the major assumptions of the hydraulic model developed here are not fulfilled.