1 Introduction

Climate change is causing surface water temperatures to increase in many water bodies (both marine and freshwater), driving increased duration and intensity of summer stratification. Increased stratification can limit the transport of dissolved oxygen (DO) from surface to bottom waters, exacerbating hypoxia in stratified water bodies (Bryant et al. 2010; Jenny et al. 2016; Birt et al. 2021; Jane et al. 2021). Much attention has been drawn to sediment oxygen uptake (\({J}_{{O}_{2}}\)) when managing or studying hypoxia in stratified water bodies, as \({J}_{{O}_{2}}\) is the major sink of DO in hypolimnia. \({J}_{{O}_{2}}\) also has important implications for biological activity, release of nutrients from sediment, carbon cycling, and water-quality in water bodies, with microbial activity in upper sediments often being greater than in the water column (Glud 2008). Globally, the availability of DO in hypolimnia affects atmospheric oxygen, carbon dioxide (CO2), and methane (CH4) concentrations and thus has profound impacts on future climate (Berner and Canfield 1989; Archer and Maier-Reimer 1994).

Due to the negative consequences of hypolimnetic hypoxia, some lake and reservoir managers have implemented strategies to increase oxygen in hypolimnia, including hypolimnetic aeration and oxygenation systems (Ashley 1985) such as bubble plume diffusers (McGinnis and Little 2002) and airlift devices (Burris et al. 2002). These techniques have been shown to be effective for increasing hypolimnetic oxygen concentrations and suppressing the flux of reduced species out of sediment (e.g., iron and manganese) (Zaw and Chiswell 1999; Bryant et al. 2011a; Gerling et al. 2014) and have been verified in long-term case studies (Gantzer et al. 2019; Preece et al. 2019), but caution is needed when adding DO to the hypolimnion to avoid increasing sediment oxygen uptake (\({J}_{{O}_{2}}\)), which can be induced by oxygenation itself (Gantzer et al. 2009; Bryant et al. 2011b).

\({J}_{{O}_{2}}\), the main oxygen sink in the hypolimnion of relatively shallow lakes and reservoirs (Nürnberg 1987), is usually modeled in hydrodynamic and water-quality models as a DO sink in the bottom of water bodies using literature or calibrated values, for example, in the commonly used lake model GLM-AED2 (Hipsey et al. 2019) or empirical expressions (Moore et al. 1996; Beutel 2003) when oxygenation systems are deployed.

\({J}_{{O}_{2}}\) is influenced by both the complex oxygen consuming processes on the sediment side of the sediment–water interface (SWI) and the characteristics of the bottom boundary layer (BBL) on the water side of the SWI (see Fig. 1). The BBL is defined as the region of the water column directly affected by the SWI (Boudreau and Jørgensen 2001). The BBL comprises a turbulent boundary layer (TBL), a viscous sublayer, and a diffusive boundary layer (DBL), as shown in Fig. 1. In the TBL, ~ 1 m above the SWI, momentum transport is governed by turbulent eddies, and solutes are transported at a much faster rate than that by molecular diffusion (Boudreau and Jørgensen 2001). As the SWI is approached, turbulence gradually decreases until the vertical transport of momentum is governed primarily by molecular diffusion in the viscous sublayer (~ 1 cm above the SWI). Solute transport at the SWI is controlled by molecular diffusion in the DBL directly above the SWI, which is a millimeter-scale layer with a thickness governed by the turbulence in the BBL (Beutel 2003). In modeling practice, it is common to adopt a simplified two-layer BBL with the DBL immediately below the TBL (Brand et al. 2009; Bryant et al. 2010).

Fig. 1
figure 1

A representative dissolved oxygen profile obtained in Carvins Cove Reservoir including the turbulent boundary layer, viscous sublayer, diffusive boundary layer (DBL), and the sediment–water interface (SWI). The upper bound of the viscous sublayer and diffusive boundary layer are indicated by dashed lines. For clarity, the dissolved oxygen profile is shown above the sediment–water interface in a log scale and below the sediment–water interface in a linear scale

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On the sediment side of the SWI, \({J}_{{O}_{2}}\) is influenced by various oxygen consuming processes including organic matter mineralization and chemical cycling within the upper sediment, as well as upward diffusing reduced chemical species from the deep sediment (Müller et al. 2012).

Efforts to understand sediment DO transport are often limited by the subjective and time-consuming characterization of the DBL thickness (\({\delta }_{DBL}\)). In the common “effective DBL” approach (Jørgensen and Revsbech 1985), \({\delta }_{DBL}\) is obtained by extrapolating the DO gradient from the SWI to the DO value of the bulk water. This widely adopted approach is practical because the DBL upper boundary may be ambiguous in some DO profiles. In this approach, the SWI is determined by visual inspection (Bryant et al. 2010; Bierlein et al. 2017; Man et al. 2020). However, in some circumstances, the DBL lower boundary (i.e., the SWI) may also be ambiguous, especially for permeable sediments and at high energy dissipation rates (Riedl et al. 1972). Bryant et al. (2010) compared the water-side and sediment-side approaches to evaluate \({J}_{{O}_{2}}\) and \({\delta }_{DBL}\), concluding that \({J}_{{O}_{2}}\) and \({\delta }_{DBL}\) on both sides are influenced by the location of the SWI. Determining the SWI by visual inspection may be inaccurate and may adversely impact the quality of sediment/DBL modeling studies.

To overcome these challenges, a computational procedure is introduced in this study to determine the SWI and the lower DBL boundary objectively and automatically. More than 300 DO profiles obtained in the laboratory and in situ are examined using both the visual inspection and computational methods. Previously, there have been some DO microprofile datasets (Hoffmann et al. 2020), but few of them include in situ freshwater DO microprofile studies. Laboratory microprofiling is much more cost-effective than in situ data collection, but without the ability to replicate the DBL, the laboratory microprofiling results are not comparable to in situ results. This study attempts to resolve this issue by creating a DBL in the laboratory by diffused aeration. In addition, modeled profiles referring to Brand et al. (2009) are also analyzed by the computational procedure to more conclusively validate the procedure.

2 Materials and methods

2.1 Study sites

The study sites are Falling Creek Reservoir (FCR) and Carvins Cove Reservoir (CCR), both in southwestern Virginia, USA, and Lake Hallwil (LH), located north of Lucerne in Switzerland. Table 1 provides the surface area, volume, maximum water depth, and elevation for the three study sites.

Table 1 Physical characteristics of the study sites
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Falling Creek Reservoir (Chen et al. 2017) is a shallow, eutrophic reservoir which is equipped with a hypolimnetic oxygenation system. One sediment core was extracted in the shallow epilimnion of the reservoir on 9 November 2020. The sediment core was stored in a fridge set to 4 °C for 1 day and then placed in an ice-water bath during microprofile measurements.

Carvins Cove Reservoir (Bierlein et al. 2017) is the main drinking water supply for the Roanoke region, Virginia, USA. CCR is a relatively shallow and eutrophic reservoir, whose width and length are ~ 600 m and ~ 8000 m, respectively. Two parallel line-source diffusers located in the hypolimnion add oxygen employing a wide range of oxygen flow rates in the stratified period. Both FCR and CCR are managed by Western Virginia Water Authority.

Lake Hallwil (Bierlein et al. 2017) is a deep, medium-sized and eutrophic natural lake, which is 8.3 km long and 1.4 km wide. Six diffusers are arranged in a circle in the deepest region to add oxygen to the hypolimnion.

Seasonal anoxia occurs in LH while CCR maintains better DO level and hypolimnion anoxia only happens in summers (Holzner et al. 2012; Doubek et al. 2018). Both LH and CCR are dimictic, and the inflow conditions are weak. According to Wang et al. (2023), the first-order internal seiche mode in CCR and LH have a period of 10.5 h and 14 h, relatively. These internal motions lead to oscillating BBL but the BBL turbulence still shows quasi-steady structures given the long seiche time scales.

2.2 Microprofile measurements

A microprofiler (MP4; Unisense A/S) was used to measure oxygen microprofiles in the FCR sediment core in the laboratory (over a period of 4 days in 2020) and in situ (over a period of 5 days in CCR in 2013 and 4 days in LH in 2012). The microprofiler was equipped with a Clarke-type oxygen microsensor (Unisense OX-100) calibrated using the lower sediment as the zero reading and aerated water as the atmospheric reading. Microprofile measurements were made in triplicate at a sampling rate of 1 Hz. After estimating the location of the SWI, measurements were made at approximately the following vertical spatial resolutions: 10-mm resolution from 95 to 15 mm above the estimated SWI, 1-mm resolution from 14 to 5 mm above the SWI, and 0.1-mm resolution from 4.9 mm above the SWI to 5 mm below the SWI. The protocol measures a complete DO profile roughly every hour (Bierlein 2015).

2.3 Aeration/oxygenation schedule

At each study site, the aeration/oxygenation rate varied and the data collected at different aeration/oxygenation rates were analyzed to evaluate the performance of the computational procedure. Due to the inherent differences between the laboratory and in situ experiment setups, different terms are used to describe the DO addition into the water. “Aeration” is used for the FCR core where air is added using a bubbler in the laboratory sediment core, whereas “oxygenation” is used for CCR/LH, where concentrated oxygen gas is added and stratification is retained. Table 2 presents the aeration/oxygenation schedule for the three study sites. In the laboratory experiment, a mini-diffuser was used to add DO to the water, simulating a hypolimnetic aeration system in the field. The aeration rates (see Table 2) at FCR (core) were measured by an Alicat flow controller. The mini-diffuser (2 cm height, 1 cm diameter) was placed in the water above the sediment core at approximately 2 cm above the sediment. Figure 2 presents a schematic of the setup for the FCR laboratory experiment.

Table 2 Schedule of the laboratory and field (Rezvani 2016) campaigns
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Fig. 2
figure 2

The laboratory experiment setup used to measure microprofiles in the Falling Creek Reservoir sediment core

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2.4 The computational procedure

Figure 3 presents a typical DO profile with major components involved in the computational procedure (i.e., the fitted curve, SWI, gradient at the SWI, preliminary and final DBL upper bounds) identified in the profile. Figure 3a presents the entire DO profile from the measurement, and Fig. 3b presents the part of the DO profile closer to the SWI. The computational procedure applied to each DO profile to determine the SWI is:

  1. I.

    First, the monotonic part of the DO profile curve (in Fig. 3b, 95.5–100 mm lander position) is characterized with a best-fit polynomial x = f(DO) using MATLAB Curve Fitting Toolbox (The MathWorks Inc 2022) (the blue fit curve in Fig. 3b). The best-fit polynomial refers to the fit with the smallest least-square error.

  2. II.

    The second derivative (\(\frac{{d}^{2}x}{d{DO}^{2}}\)) of the fitted curve is calculated to determine the SWI location and a preliminary DBL upper bound (the brown line in Fig. 3b) as the first and last points where the sign of the second derivative changes, respectively.

Fig. 3
figure 3

A typical profile and the diffusive boundary layer (DBL) determined by the computational procedure. Y-axis indicates the downward distance the sensor traveled from the start of the profile. a The whole profile. b The part of profile closer to the sediment–water interface (SWI)

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Regarding the order of the best-fit polynomial in the first step, a 4th order polynomial is the lowest order polynomial applicable to the computational procedure because the first and last points where the sign of the second derivative changes are used to identify the preliminary DBL bounds. If the second derivatives are of a linear relationship with the DO concentration (i.e., the best-fit polynomial is 2nd or 3rd order), there is only one point where the sign of the second derivative changes. After testing from 4 to 10th order polynomials, a 6th order polynomial was adopted in this study to ensure fitting accuracy (with R2 > 0.90 for more than 90% profiles) and to avoid overfitting. Users of this computational procedure can easily adjust the polynomial order, or criteria/thresholds based on their experimental conditions and requirements. For DO profiles at steady state, users may start with the 4th order polynomials, and for DO profiles with noisy signal or at non-steady state, the 5th or 6th order polynomial is recommended.

When adopting a visual inspection method, the DO profile is examined carefully to determine the SWI as the point where the concentration gradient decreases due to the porosity difference between water and the sediment (Bryant et al. 2010; Man et al. 2020).

For both the computational and visual inspection methods, the upper bound of the DBL is characterized by the following steps:

  1. I.

    The data points between the SWI and the preliminary DBL upper bound are used in an ordinary least squares regression to obtain the DO gradient (\(\frac{\partial C}{\partial x}\)) (the black line in Fig. 3b) in the DBL region.

  2. II.

    The final DBL upper bound is determined by extrapolating \(\frac{\partial C}{\partial x}\) to the DO concentration in bulk water (\({C}_{bulk}\)) of the water column (indicated by the cyan line in Fig. 3a and b), which is calculated as the average DO concentration of the first nine measurements (that is, the first nine data points from the top in Fig. 3a). These nine measurements correspond to 15 mm to 95 mm lander positions with a 10-mm interval.

  3. III.

    The \({\delta }_{DBL}\) is then determined as the distance between the SWI and the final DBL upper bound.

2.5 DBL and \({J}_{{O}_{2}}\) modeling

In the modeling approach described here, \({\delta }_{DBL}\) is determined by Eq. (1) (Brand et al. 2009):

$${\delta }_{DBL}=\left(\frac{{C}_{bulk}}{{C}_{swi}}-1\right)\frac{{D}_{w}}{\sqrt{{k}_{1}{D}_{s}}}$$
(1)

where \({C}_{bulk}\) is the DO concentration in bulk water, \({C}_{swi}\) is the DO concentration at the SWI, \({k}_{1}\) is the first-order kinetic rate constant (day−1), and \({D}_{s}\) is the diffusion coefficient of oxygen in the sediment (\({D}_{s}\) = \(\mathrm{\varnothing }{D}_{w}\), where \(\mathrm{\varnothing }\) is the porosity). k1 considers all the chemical/biological reactions in the upper sediment and assumes the lumped-up reaction has first-order kinetic. \(\mathrm{\varnothing }\) is obtained by analyzing the sediment at 0 ~ 1 cm depth from SWI (Azcue et al. 1997) after the microprofiler measurement for the FCR core and from the literature value for CCR/LH (Bierlein et al. 2017). \({k}_{1}\) is obtained by fitting a wide range of kinetic rate constants to first-order kinetic models (refer to Man et al. 2020). The best-fit \({k}_{1}\) value is the one that minimizes the root-mean square error (RMSE) of modeled DO profiles.

To simplify the model, for modeled profiles at each study site, \({C}_{bulk}\) is set to the average measured \({C}_{bulk}\) value. In addition, the transition from the bulk water to the DBL via the viscous sublayer is not considered. Instead, the DO concentration is assumed constant above the elevation of the modeled \({\delta }_{DBL}\) and decreases linearly from \({C}_{bulk}\) at the upper bound of the modeled \({\delta }_{DBL}\) to \({C}_{swi}\) at the SWI. Below the SWI, DO is transported by molecular diffusion and consumed by kinetic reaction and the upward reduced species fluxes (Man et al. 2020). The reduced species flux in FCR is estimated by sediment sample analysis based on the linear relationship between the total organic carbon mass accumulation rate (\(\text{TOC-MAR}\)) and the reduced flux (Steinsberger et al. 2017). The total organic carbon mass accumulation rate is calculated by Eq. (2):

$$\text{TOC-MAR}=SR\times {\rho }_{dry}\times (1-\varnothing )\times TOC\times 10$$
(2)

where \({\text{TOC}}\) is the sediment organic matter (mg g−1), SR is the literature sedimentation rate (0.1 cm year−1) (Gray et al. 2009), and \({\rho }_{dry}\) is the density of the sediment (g cm−3, obtained after evaporating water at 110 °C) (Azcue et al. 1997). \({\text{TOC}}\) was estimated as the loss of weight after heating the dried sediment sample to ~ 400 °C for 24 h (Schumacher 2002). \(\varnothing\) is calculated by Eq. (3):

$$\varnothing \left(\%\right)=100-\frac{{\rho }_{Bulk}}{{\rho }_{dry}}\times 100$$
(3)

where \({\rho }_{Bulk}\) is the density of the sediment before evaporating water (g cm−3). Due to the lack of sediment characteristic data, literature values of reduced species flux are adopted for CCR and LH (Steinsberger et al. 2020). The literature sedimentation rate is a sedimentation rate value from RESSED – an online reservoir sedimentation survey database for the United States (Gray et al. 2009). The sedimentation rate in FCR is not available in the database, so the value for a reservoir near Roanoke was adopted.

2.6 Validate the computational procedure using modeled profiles

The modeled DO profiles (refer to the above section) are used to validate the performance of the computational procedure. The evaluation addresses the practical problem that the “correct” SWI and DBL upper boundary positions are unknown for measured profiles. Brand et al.’s (2009) model was adopted to generate more realistic DO profiles rather than setting arbitrary or random \({\delta }_{DBL}\). The modeled profiles are presented in the supplementary information (Fig. S2) for the readers’ information, but the mechanism/quality of the model is beyond the scope of this manuscript.

3 Results and discussion

3.1 SWI and DBL characterized by the computational procedure

In Fig. 4, the computational (blue dots) and visual SWI positions determined by two independent inspectors (brown square for inspector 1; red triangle for inspector 2) are shown.

Fig. 4
figure 4

The computational and visual sediment–water interfaces (SWI) (from two inspectors) in Lake Hallwil. The discontinuity of data at 15 h is due to the required repositioning of the microsensor to prevent it from shifting out of the measurement range as the base of the microprofiler slowly settled into the soft sediment. Sediment–water interface positions are relative to the lander’s initial position (lander position 0 mm) with downward being positive

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The methodology of the visual inspection method attempts to identify a linear part in the DO profile as the DBL (i.e., where \(\frac{dDO}{dx}\) is a constant and \(\frac{{d}^{2}DO}{d{x}^{2}}\) is zero). The methodology is simple and reflects the physics of the DBL, where molecular diffusion dominates. However, a plot of \(\frac{dDO}{dx}\) and \(\frac{{d}^{2}DO}{d{x}^{2}}\) of a typical profile presented in supplementary information (Fig. S1) does not show clear signal of DBL or SWI, which indicates the visual inspection method may not be reliable for real-world DO profiles.

According to the observations of the data presented in Fig. 4, the SWI positions indicate a sinking microprofiler, and the sinking rate is decreasing smoothly during each deployment period. The minor fluctuation of the SWI positions is probably due to the noise in the DO signal and the non-steady character of the DO profiles. It supports the use of our computational procedure for characterizing the DBL and SWI given that our procedure is applicable to non-steady DO profiles.

The two inspectors are two co-authors of this paper who are both experienced in visually inspecting sediment DO profiles. Although very similar results are obtained independently, the two inspectors gave slightly different visually characterized SWIs, which is dependent on the inspector’s subjective judgment. In contrast, the computationally characterized SWI is objective. To ensure consistency, the visual inspection results presented below are all based on visual inspection 2.

The comparison between the computational and visual SWIs at all three study sites is provided in Fig. 5. The computational SWI is generally close to the visual one, with the average variation between them in each water body in the range of 0.4–0.8 mm. The corresponding variation in DO flux can be up to 10.8 mmol m2 day−1 estimated by simple film theory, while the DO flux in FCR measured by a sediment chamber was 14.0 \(\pm\) 3.2 mmol m2 day−1 (Krueger et al. 2020). The computational SWI for CCR and LH shows the same trend as the visual one, with the SWI position becoming closer to the microprofiler zero position. This trend is due to the unavoidable downward movement of the microprofiler when it is left at the bottom of the lake. This downward settling was corrected daily in CCR/LH (i.e., in situ studies) to prevent the microsensor from shifting out of the microprofiler measurement range. The corrections are visible in the gaps between data (e.g., at 20 h and 60 h in Fig. 5b). In FCR, the microprofiler was also repositioned once or twice per day for the same purpose as the in situ deployments. The microprofiler was not deployed at night in FCR (core) due to the limitation of battery capacity, which leads to the blanks between the profiles of two consecutive days (e.g., between 8 and 22 h in Fig. 5a). Compared to the amount of time required by the visual inspection method (usually days for hundreds of profiles), the computational method characterizes SWI automatically within a few seconds with equivalent accuracy. Referring to the testing results using modeled profiles of the computational procedure to be presented in Assessment section (Table 4), it appears that the average difference between the visual and computational SWI (0.4–0.8 mm in the three study sites) is much larger than the average difference between the computational and known SWI (0.1–0.3 mm), which supports adopting the described computational procedure.

Fig. 5
figure 5

The computational and visual characterized sediment–water interface (SWI) for the three study sites: a Falling Creek Reservoir (core), b Carvins Cove Reservoir, and c Lake Hallwil. In Falling Creek Reservoir (core), the microprofiler was not deployed at night

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Figure 6 presents four sample profiles at different aeration rates in FCR (core). It can be observed by visual inspection that for all four DO profiles presented here the computational procedure reasonably identifies the SWI (i.e., the DBL lower bounds). In these profiles, the DO concentration gradient is approximately constant above the computationally determined SWI and the DO concentration gradient becomes non-linear and smaller below the SWI than that in the DBL, which reflects the dominant DO diffusion process above the SWI and the complex DO consumption processes below the SWI, respectively.

Fig. 6
figure 6

Four typical profiles and the diffusive boundary layer (BDL) determined by the computational procedure in Falling Creek Reservoir. Measurement time and aeration rates are noted below each profile

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Regarding the DBL upper bound, the final upper bound determined by the proposed procedure appears to be more accurate than the preliminary one. The final upper bounds are located at the transition region between the bulk water and the diffusion dominated region, but the preliminary upper bounds are still in the diffusion dominated regions in the DO profiles presented in Fig. 6b and d. To conclude, for most of the DO profiles under consideration, the proposed computational procedure correctly identifies the DBL as the part of the DO profile where the gradient is approximately constant.

Although similar results may be obtained by visual inspection and the computational procedure, the latter approach ensures that all DO profiles are obtained consistently and efficiently. It should be noted that this computational procedure does not yield perfect results for all DO profiles due to the ambiguity of the DBL upper and lower bounds. Effort has been made to avoid any subjective judgment in the computational procedure. For future applications, visual inspection may be adopted to guide the computational procedure for individual profiles that clearly deviate from the norm, especially for anoxic profiles where the DBL is more ambiguous than oxic profiles and the profiles whose fitted curves have low R2 values.

Table 3 compares the computational and visual average \({\delta }_{DBL}\) at the three study sites. The computational \({\delta }_{DBL}\) has lower standard deviations than the visual \({\delta }_{DBL}\) for LH and FCR. For CCR, although the two methods have equal standard deviation, the computational \({\delta }_{DBL}\) still yields more stable results since it has a larger average \({\delta }_{DBL}\) than the visual method (i.e., has lower coefficient of variation). Based on the discussion above, the computational \({\delta }_{DBL}\) is adopted in sections below for further analysis.

Table 3 The date, number of profiles, and average computational and visual \(\delta\) DBL with one standard deviation in Falling Creek Reservoir, Carvins Cove Reservoir, and Lake Hallwil
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\({\delta }_{DBL}\) and aeration/oxygenation rate.

Figures 7, 8, and 9 present the relationship between \({\delta }_{DBL}\) and aeration/oxygenation rates in FCR, CCR, and LH, respectively. Subplots (a) provide the \({\delta }_{DBL}\) and aeration/oxygenation rate for all profiles, and subplots (b) provide the average \({\delta }_{DBL}\) at each aeration/oxygenation rate.

Fig. 7
figure 7

a Computationally characterized \({\delta }_{DBL}\) and aeration rates in Falling Creek Reservoir. b average \({\delta }_{DBL}\) at each aeration rate in Falling Creek Reservoir

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Fig. 8
figure 8

a Computationally characterized \({\delta }_{DBL}\) and aeration rates in Carvins Cove Reservoir. b average \({\delta }_{DBL}\) at each aeration rate in Carvins Cove Reservoir

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Fig. 9
figure 9

Computationally characterized \({\delta }_{DBL}\) for bubble plume diffusers on and off in Lake Hallwil presented in time sequence. The microprofiler was 30 m from the diffusers

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In the FCR sediment core, \({\delta }_{DBL}\) responds as expected to the aeration rate changes, with the higher aeration rates corresponding to smaller \({\delta }_{DBL}\). In Fig. 7a, most \({\delta }_{DBL}\) less than 2 mm correspond to the high aeration rate (0.04 slpm) and the average \({\delta }_{DBL}\) in Fig. 7b has a clear negative relationship with the aeration rate.

In CCR, there is also a general correlation between \({\delta }_{DBL}\) and oxygenation rate. In Fig. 8a, most profiles have a \({\delta }_{DBL}\) less than 2 mm at high oxygenation rates (equal to or more than 473 slpm) and some profiles have a \({\delta }_{DBL}\) more than 4 mm at low oxygenation rates (less than 473 slpm). When the oxygenation rate is equal to or lower than 473 slpm, the average \({\delta }_{DBL}\) results in Fig. 8b suggest a negative relationship with the oxygenation rate. When the oxygenation rate is higher than 473 slpm, increasing aeration rate does not necessarily reduce \({\delta }_{DBL}\).

In LH, a negative relationship of \({\delta }_{DBL}\) with the aeration rate change is also observed (Fig. 9). When the diffusers are turned off, there is a distinct increase in \({\delta }_{DBL}\), and the average \({\delta }_{DBL}\) increases from 1.4 to 1.8 mm. In Fig. 9, the aeration rate is not presented because there are only two conditions: “aeration on” and “aeration off.” The aeration rate has been presented in Table 2, which is 2838 slpm. It should be noted that the experimental setup in LH adjusted both the oxygenation rate and the distance from the microprofiler to the diffusers, but the \({\delta }_{DBL}\) results with the microprofiler placed at 300 m and 3700 m from the diffusers are not presented here. The reason is that the diffusers were turned on all the time for the two distances and thus no comparison between diffusers on and diffusers off can be made. Furthermore, Wang et al. (2023) suggest bubble plume diffusers only impact near-field BBL turbulence structure.

To conclude, \({\delta }_{DBL}\) results from all three study sites appear reasonable by referring to field studies linking small \({\delta }_{DBL}\) to high turbulence dissipation rates in the BBL (Lorke et al. 2003; Bryant et al. 2010). The enhanced oxygenation may also lead to a thin DBL given that the hypolimnetic oxygenation/aeration systems have proved efficient in adding DO to the hypolimnion (Chen et al. 2018).

3.2 Verify the computational procedure with modeled profiles

Figure S1 presents several typical laboratory and modeled DO profiles for FCR sediment. In this section, the computational procedure was applied to the modeled DO profiles to further evaluate its performance. Table 4 presents the average variations of the DBL upper and lower bound positions between the computational determined and known values for the modeled DO profiles in the three study sites. As discussed in the Assessment section, the computational procedure determines the upper and lower DBL bounds successfully. The average difference in FCR is the least among the three study sites, supporting that the laboratory condition is better controlled than in situ. Another interesting point is that the DBL upper bounds are determined with zero difference from the known values. This finding supports using all the data points between two preliminary bounds for a linear regression for the purpose of obtaining the DO gradient in DBL, rather than adopting several data points near the SWI as existing studies usually do.

Table 4 Variations of the diffusive boundary layer upper and lower bound positions between the computational and known values for the modeled profiles, with one standard deviation
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4 Conclusions

Current \({J}_{{O}_{2}}\) predictions in stratified water bodies are empirical rather than mechanistic. Determining the DBL is essential for sediment and DBL modeling studies to predict \({J}_{{O}_{2}}\) in a mechanistic way. However, both the upper and lower bounds of the DBL can be ambiguous especially for permeable sediments and at high turbulence. In this study, a computational procedure is introduced to determine the SWI and the DBL objectively and automatically. More than 300 DO profiles obtained in laboratory and in situ from three study sites with different aeration/oxygenation rates are examined. With a microprofiler, the laboratory experiment was able to replicate the DBL similar to field conditions. The proposed computational procedure can characterize the DBL well for sediment DO profiles in all three study sites under various turbulence-inducing oxygenation conditions and modeled DO profiles. This procedure can save time and improve consistency in determining DBL in future sediment/DBL modeling studies.