a. Mixing relation in an estuary bounded by isohalines

Here, we will derive the equation of estuarine mixing in a volume bounded by an isohaline surface, where we impose a temporal change of the water and the salt content within this estuarine volume.

Let V(S) be an estuarine volume bounded by an isohaline surface of salinity S, the sea surface, the sea bottom, and a river boundary, where a freshwater runoff is prescribed (see an illustration in Fig. 1a). Averaged over a prescribed time interval Δt, the budgets of water volume and salt within these boundaries are closed as follows (Hieronymus et al. 2014; Burchard 2020):

$\begin{array}{cc}{V}_{\text{stor}}=-Q+Q_r;& S_{\text{stor}}=-SQ-F^{\mathit{s}},\end{array}$(1)

where

$\begin{array}{l}{V}_{\text{stor}}(S)=\frac{1}{\mathrm{\Delta }t}\{V(S,t+\mathrm{\Delta }t)-V(S,t)\}\text{ \hspace{1em}and}\\ S_{\text{stor}}(S)=\frac{1}{\mathrm{\Delta }t}\left\{{\displaystyle {\int }_{V(S,t+\mathrm{\Delta }t)}}s d V-{\displaystyle {\int }_{V(S,t)}}s d V\right\}\end{array}$(2)

are the storage terms for water volume and salt (change in volume and volume-integrated salinity during the time interval Δt), and

$Q(S)=\langle {\displaystyle {\int }_{A(S)}}{u}^{\text{dia}} dA\rangle \hspace{0.17em}\text{\hspace{0.5em}and\hspace{0.5em}}\hspace{0.17em}{F}^{\mathit{s}}(S)=\langle {\displaystyle {\int }_{A(S)}}-K_n{\partial }_{n}s dA\rangle$(3)

are the diahaline water transport and the diahaline turbulent salinity transport across the isohaline S, respectively, with the outward-pointing diahaline velocity u^dia = (u − u^s) ⋅ n, the velocity vector u, the velocity of a point moving with the isohaline surface u^s, the unit vector n, which is orthogonal to the isohaline and pointing toward higher salinity, the diahaline diffusivity K_n (which is a weighted average of the horizontal diffusivity K_h and the vertical diffusivity K_υ; see Burchard et al. 2021), the diahaline salinity gradient ∂_ns, the isohaline area A(S), and the time averaging operator ⟨⋅⟩.

(a) Volume-integrated budgets of water and salt within an estuarine volume V(S) (blue shading) bounded by an isohaline of salinity S (thick green line). The terms Q(S) and F^s(S) are the diahaline water transport and diahaline salt transport across this isohaline, respectively. On the isohaline surface, there are subareas (thick red line) with inward diahaline water transport Q_in(S), and other subareas (thick blue line) with outward diahaline water transport Q_out(S). At the river end of the estuary, runoff with salinity S = 0 and a rate of Q_r is entering the volume. (b) The lower part of one single water column at location x with isohaline surfaces of salinity S at two adjacent time steps used in the derivation of the vertical projection of the entrainment velocity w_e. With $l^{>}(x,S)=z(x,S)+H$ denoting the distance of isohaline S from the bottom (z = −H), Q^x>(x, S) represents the horizontal water transport across l^>(x, S). The vertical projection of the entrainment velocity w_e(x,S) = u^dia(x,S)/(n ⋅ k) can be calculated according to (14), with u^dia being the outward-pointing diahaline velocity.

Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPO-D-21-0292.1

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(a) Volume-integrated budgets of water and salt within an estuarine volume V(S) (blue shading) bounded by an isohaline of salinity S (thick green line). The terms Q(S) and F^s(S) are the diahaline water transport and diahaline salt transport across this isohaline, respectively. On the isohaline surface, there are subareas (thick red line) with inward diahaline water transport Q_in(S), and other subareas (thick blue line) with outward diahaline water transport Q_out(S). At the river end of the estuary, runoff with salinity S = 0 and a rate of Q_r is entering the volume. (b) The lower part of one single water column at location x with isohaline surfaces of salinity S at two adjacent time steps used in the derivation of the vertical projection of the entrainment velocity w_e. With $l^{>}(x,S)=z(x,S)+H$ denoting the distance of isohaline S from the bottom (z = −H), Q^x>(x, S) represents the horizontal water transport across l^>(x, S). The vertical projection of the entrainment velocity w_e(x,S) = u^dia(x,S)/(n ⋅ k) can be calculated according to (14), with u^dia being the outward-pointing diahaline velocity.

Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPO-D-21-0292.1

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(a) Volume-integrated budgets of water and salt within an estuarine volume V(S) (blue shading) bounded by an isohaline of salinity S (thick green line). The terms Q(S) and F^s(S) are the diahaline water transport and diahaline salt transport across this isohaline, respectively. On the isohaline surface, there are subareas (thick red line) with inward diahaline water transport Q_in(S), and other subareas (thick blue line) with outward diahaline water transport Q_out(S). At the river end of the estuary, runoff with salinity S = 0 and a rate of Q_r is entering the volume. (b) The lower part of one single water column at location x with isohaline surfaces of salinity S at two adjacent time steps used in the derivation of the vertical projection of the entrainment velocity w_e. With $l^{>}(x,S)=z(x,S)+H$ denoting the distance of isohaline S from the bottom (z = −H), Q^x>(x, S) represents the horizontal water transport across l^>(x, S). The vertical projection of the entrainment velocity w_e(x,S) = u^dia(x,S)/(n ⋅ k) can be calculated according to (14), with u^dia being the outward-pointing diahaline velocity.

Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPO-D-21-0292.1

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In (7), m_∞ is the impact of freshwater transport from rivers, and $\tilde{m}$ is the unsteadiness due to estuarine storage of salt and water volume. For sufficiently long temporal averaging (vanishing storage term), the universal law of estuarine mixing m = 2SQ_r is retrieved (Burchard 2020). However, the averaging time may differ between estuaries according to their external forcings and internal time scales. The tidal variation of $\tilde{m}(S)$ might be significant compared to m_∞ if a large amount of water and salt exchange occur on the isohaline surface. Nevertheless, previous studies have shown that $\tilde{m}$ converges to zero when averaged over several spring–neap cycles in relatively simple estuaries (Burchard 2020; Burchard et al. 2021). In this study, we will explore the convergence properties of $\tilde{m}$ in an estuary with more complex geometry.

b. Water-column consideration

Apart from the temporal variation of salinity mixing, salinity mixing and diahaline transport also show high spatial variations on the isohaline surface.

a. Pearl River Estuary

The Pearl River Estuary (PRE) is a multioutlet estuary located at the northern reach of the South China Sea (Figs. 2a,b). Freshwater from three main tributaries (West River, North River, and East River) passes through a highly complex river network and then discharges into the coastal ocean through eight outlets (short red lines in Fig. 2b). A large amount of freshwater, sediments, and nutrients are thus discharged to the northern shelf of the South China Sea, affecting the dynamics of the shelf and the various biogeochemical processes in the region (Harrison et al. 2008).

(a) Mesh grid over the model domain shown in every fifth grid line. (b) Bathymetry of the Pearl River Estuary. Short red lines indicate the location of the eight outlets of the PRE. (c) Enlarged view of the numerical grid inside the estuarine area, superimposed with the composite RGB image based on Landsat-8 satellite image collected on 18 Sep 2016 (downloaded from earthexplorer.usgs.gov). (d) Enlarged view of the numerical grid in the river network area.

Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPO-D-21-0292.1

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(a) Mesh grid over the model domain shown in every fifth grid line. (b) Bathymetry of the Pearl River Estuary. Short red lines indicate the location of the eight outlets of the PRE. (c) Enlarged view of the numerical grid inside the estuarine area, superimposed with the composite RGB image based on Landsat-8 satellite image collected on 18 Sep 2016 (downloaded from earthexplorer.usgs.gov). (d) Enlarged view of the numerical grid in the river network area.

Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPO-D-21-0292.1

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(a) Mesh grid over the model domain shown in every fifth grid line. (b) Bathymetry of the Pearl River Estuary. Short red lines indicate the location of the eight outlets of the PRE. (c) Enlarged view of the numerical grid inside the estuarine area, superimposed with the composite RGB image based on Landsat-8 satellite image collected on 18 Sep 2016 (downloaded from earthexplorer.usgs.gov). (d) Enlarged view of the numerical grid in the river network area.

Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPO-D-21-0292.1

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The river discharge shows substantial seasonal variations, with ∼80% of the river discharge occurring in the wet season (April–September). The total river discharge of all the tributaries reaches 20000 m³ s⁻¹ in summer and drops to 4000 m³ s⁻¹ in winter, roughly half of which enters the PRE (Dong et al. 2004). The wind in the PRE is dominated by two main patterns: northeasterly monsoon in winter and southwesterly monsoon in summer. The dynamics in the PRE exhibit a strong seasonal variation due to the large differences of runoff and wind direction between the wet season and the dry season (Dong et al. 2004). During summer, the plume is spreading cross shelf due to the upwelling-favorable wind as well as the high runoff (Pan et al. 2014). During winter, the plume is confined along the coast due to the downwelling-favorable wind and the low runoff (Lai et al. 2016). M₂ and K₁ are the two dominant tidal constituents in the PRE, making the PRE a mixed semidiurnal tidal regime (Mao et al. 2004). The tidal range is ∼1 m at the mouth of the PRE (near the red box in Fig. 2b) and is amplified to 1.7 m at the head of the PRE (near the green box in Fig. 2b). The observed water level at Makou station (for location, see Fig. 2a) shows that the tidal waves can progress far upstream into the river network (Lai et al. 2015). In the PRE, there are two deep zonal-oriented channels (the West Channel and the East Channel, see Fig. 2b), while the rest of the estuarine area consists of shoals and islands. In the shallow regions, the water depth varies from 2 to 10 m. Around the islands, the depth increases, reaching up to 30 or 40 m at the outer islands.

The presence of the islands disturbs the tidal flow and leads to the generation of strong horizontal density gradients in their vicinity. Additionally, vortex-like structures can emerge at the lee side of the islands. Such structures are a quite common feature in the PRE, as can be seen by available satellite images (Fig. 2c). Island wakes have been extensively studied due to the so-called island mass effect—the enhancement of biological productivity in the vicinity of the islands. Besides, island wakes play a crucial role in the cascade of mesoscale energy toward dissipation and turbulent mixing (Heywood et al. 1990; Hasegawa et al. 2004; Chang et al. 2013; Liu and Chang 2018; Chang et al. 2019). We observe a similar feature: an enhanced near-surface mixing ${\chi }_{\text{sur}}^s$ around the islands in the PRE.

b. Model description and simulation setup

In this study, we performed the high-resolution simulations of the PRE using the General Estuarine and Transport Model (GETM; see Burchard and Bolding 2002). GETM is a coastal ocean model supporting curvilinear horizontal and vertically boundary-following coordinates. It has a nonlinear free surface with robust drying-and-flooding capability and an efficient subdomain decomposition for massively parallel computations on high performance computing (HPC) systems. More details and the governing equations can be found in Klingbeil and Burchard (2013) and Klingbeil et al. (2018).

A curvilinear grid with 918 cells along the estuary and 742 cells across the estuary (Fig. 2a) was constructed using the Delft3D grid generation tool (https://oss.deltares.nl/web/delft3d/get-started) and then applied to this study. One challenge of building a structured numerical grid for the PRE is to resolve the highly complex river network. Based on the analysis of model results from the unstructured-grid Finite-Volume Community Ocean Model (FVCOM) PRE circulation model, Lai et al. (2015) found that the amount of freshwater discharged through each of the eight outlets in the PRE is controlled by the upstream river runoff, the tidal energy entering the river network, and the complex water exchange at the intersections inside the river network. Therefore, it is crucial to resolve the geometry of the river network to reproduce the freshwater dynamics. In contrast to previous estuarine studies on salinity mixing (e.g., Burchard 2020; Burchard et al. 2021) that have used either idealized or low-complexity topographies, we use a realistic topography. This enables us to also investigate the potential effects of the deep channels and the multichannel river network on the spatial distribution of estuarine mixing. The model domain starts from the hydrologic stations (Gaoyao, Shijiao, and Boluo) of the three major tributaries, resolves the complex river network and the downstream estuaries, and then fans out to cover the shelf of northern South China Sea to the 4000-m isobath (Fig. 2a).

The horizontal resolution varies from 60 to 800 m in the river network (Fig. 2d), from 150 to 300 m inside the estuary (Fig. 2c), from 300 to 800 m in the coastal ocean area, and decreases to 9000 m at the southern open boundary. The horizontal grid size around the islands is approximately one-tenth of the scale of the eddies, suggesting the ability to resolve the eddies around the islands. In the vertical, we applied 40 terrain-following σ layers. Since the depth inside the PRE is mostly below 10 m, this vertical resolution is fine enough to resolve the vertical structure of the river plume. We used a baroclinic time step of 10 s and a barotropic time step of 1 s for the temporal discretization.

As stated above, the dynamics in the PRE exhibit a strong seasonal variation due to the large differences in runoff and wind direction between the wet and dry seasons. Therefore, we conducted two idealized forcing simulation scenarios: the summer simulation (high runoff) and the winter simulation (low runoff). The configurations of these two simulations differ only in the river discharge and the wind: the summer simulation has a total steady runoff of 15 600 m³ s⁻¹ and a constant southwesterly wind of 5 m s⁻¹. In contrast, the winter simulation has a total steady runoff of 3550 m³ s⁻¹ and a constant northeasterly wind of 5 m s⁻¹. These two different regimes are typical for their seasons (Mao et al. 2004). According to their climatological mean, we specified the runoff data at three hydrologic stations (Gaoyao, Shijiao, and Boluo) of the three major tributaries. In both simulations, the open ocean boundary conditions are driven by simplified tidal water level from the TPXO tidal datasets (Egbert and Erofeeva 2002), with only four harmonic constituents (M₂, S₂, K₁, and O₁) to reflect the spring–neap variation. Since density is mainly controlled by salinity in the estuarine region of the PRE, we neglected temperature in the model simulations. Although evaporation and precipitation have an impact on estuarine mixing (Lorenz et al. 2021), we do not consider the air–sea exchange of heat or freshwater. These simplifications help to bind the salinity variance of an estuarine volume to an isohaline surface enclosed by the river mouth and the open ocean. The initial sea surface elevation and currents are zeros, while the initial salinity distribution is from the GLBv0.08 dataset of the HYCOM global ocean forecasting results (www.hycom.org/data/glbv0pt08/expt-57pt2). Salinities at the open boundaries are set to 35 g kg⁻¹. Both simulations were run for 12 months and started to exhibit a repeating spring–neap cycle after 5 months. To ensure the analysis of a quasi-steady state, we only analyzed the last month of the year-long simulation.

Vertical turbulent diffusion is calculated through the coupling with the turbulence module of the General Ocean Turbulence Model (GOTM; see Burchard and Bolding 2001) by solving the k–ϵ two-equation turbulence closure model. Horizontal turbulent diffusion is parameterized following Smagorinsky (1963) with a Smagorinsky constant of 0.28 and a turbulent Prandtl number of 2.0 for salinity. For advection of velocities and salinity, the TVD-Superbee scheme is employed in directional-split mode due to its minimal numerical mixing (Klingbeil et al. 2014). For the isohaline analysis, a discrete salinity increment of ΔS = 0.1 g kg⁻¹ is adopted. The salinity binning is computed during each baroclinic time step of the model as proposed by Lorenz et al. (2019). Sensitivity tests show that the results are only weakly dependent on ΔS.

a. Mixing

The analysis of mixing per salinity class shows that the universal law holds for both the high-runoff simulation and the low-runoff simulation when averaged over 14 days (Fig. 4). Due to the numerical discretization and the binning procedure for calculating properties per salinity class, the numerical representations of the relations (7) and (8) are not identical (e.g., Fig. 4a). However, both methods still closely follow the theoretical prediction of m(S) = 2SQ_r. Note here that for water with higher salinities (e.g., S > 22 g kg⁻¹ in Fig. 4a), the mixing per salinity class m deviates from the theoretical linear curve since the isohaline volume interacts with the numerical open boundary.

Properties related to the salinity mixing and the diahaline transport as a function of salinity S calculated from long-term averaged model data from (left) the high-runoff simulation and (right) the low-runoff simulation, respectively. (a),(b) Total [red line, Eq. (8)], physical (green line), and numerical (blue line) mixing per salinity class m(S), in comparison to another method $m=m_{\infty }+\tilde{m}$ [brown line, Eq. (7)] as well as to the universal law under equilibrium condition m = 2SQ_r (dashed gray line). (c),(d) The total diahaline transport (red line), the inflow (green line), and the outflow (blue line) of the diahaline transport as a function of salinity S. The brown line represents the magnitude of the exchange flow expressed as (−Q_in + Q_out)/2.

Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPO-D-21-0292.1

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Properties related to the salinity mixing and the diahaline transport as a function of salinity S calculated from long-term averaged model data from (left) the high-runoff simulation and (right) the low-runoff simulation, respectively. (a),(b) Total [red line, Eq. (8)], physical (green line), and numerical (blue line) mixing per salinity class m(S), in comparison to another method $m=m_{\infty }+\tilde{m}$ [brown line, Eq. (7)] as well as to the universal law under equilibrium condition m = 2SQ_r (dashed gray line). (c),(d) The total diahaline transport (red line), the inflow (green line), and the outflow (blue line) of the diahaline transport as a function of salinity S. The brown line represents the magnitude of the exchange flow expressed as (−Q_in + Q_out)/2.

Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPO-D-21-0292.1

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Properties related to the salinity mixing and the diahaline transport as a function of salinity S calculated from long-term averaged model data from (left) the high-runoff simulation and (right) the low-runoff simulation, respectively. (a),(b) Total [red line, Eq. (8)], physical (green line), and numerical (blue line) mixing per salinity class m(S), in comparison to another method $m=m_{\infty }+\tilde{m}$ [brown line, Eq. (7)] as well as to the universal law under equilibrium condition m = 2SQ_r (dashed gray line). (c),(d) The total diahaline transport (red line), the inflow (green line), and the outflow (blue line) of the diahaline transport as a function of salinity S. The brown line represents the magnitude of the exchange flow expressed as (−Q_in + Q_out)/2.

Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPO-D-21-0292.1

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For both simulations, the physical mixing m^phy dominates the total mixing m, whereas the numerical mixing m^num contributes at most one-third of the total mixing. The higher contributions of the numerical mixing to the total mixing in summer could be related to the seaward spreading of the river plume pushed by the stronger runoff into the region with coarse resolution.

b. Diahaline exchange flow

When horizontally integrated over all locations where a given isohaline surface with salinity S occurs, Q_out(S) and Q_in(S) can be regarded as the exchange flow with respect to this isohaline surface. The magnitude of the outflow of the diahaline water transport Q_out equals Q_r for S = 0, while the magnitude of the inflow of the diahaline water transport Q_in equals 0 for S = 0 (Figs. 4c,d). For S > 0 they both increase with salinity. Although the net diahaline water transport $Q=Q_{\text{out}}+Q_{\text{in}}$ equals the constant runoff Q_r when averaged over sufficiently long time, the strength of diahaline exchange flow $Q_{\text{st}}=(-Q_{\text{in}}+Q_{\text{out}})/2$ increases with salinity. For smaller salinities, Q_st is less than the runoff Q_r. However, for larger salinities, the strength of diahaline exchange flow becomes larger than the runoff. Take the winter (low-runoff) scenario for example, the strength of diahaline exchange flow Q_st can be more than 3 times the runoff Q_r at high salinities (S > 25 g kg⁻¹ in Fig. 4d).

c. Vertical distribution of diahaline exchange flow

To analyze the vertical distribution of diahaline exchange flow, the local diahaline water transport Q(x, y, S) calculated as (13) was mapped into σ coordinates with Δσ = 0.025 g kg⁻¹ (Fig. 5). The mapping process from salinity space to σ space is presented in the appendix of this paper. For both seasons, the long-term averaged diahaline water transport Q is pointing to lower salinity in the bottom layers and pointing to higher salinity in the surface layers, which is very similar to the classical exchange flow. In general, the magnitude of the diahaline water transport inflow Q_in is zero at the bottom (σ = −1), and then gradually increases to its maximum in the near-bottom layer, followed by the decrease to zero again at middle depth. Please note that we do not exactly have the bottom (σ = −1) value. Since there is no water transport through the bottom and Q_in approaches zeros toward the bottom (e.g., σ ≤ −0.95), we set the bottom (σ = −1) values to zero. Influenced by the buoyant runoff, the outflow of diahaline water transport Q_out generally is largest in surface layers and then decreases to zero at that middle depth. Because of the relations (9) and (18), the magnitudes of local mixing per salinity class m_lo (or in other words, the diffusive diahaline salinity flux $f_{\text{diff}}^s$) will reach its maximum at this middle depth when u^dia = 0 for any single water column. The upper part of the isohaline surface would span a large estuarine area in the form of the surface plume when the runoff is large or when the tides are weak (especially during neap tide in summer). In this case, the domain-integrated diahaline water transport for high salinities may not be simply monotonously increasing or decreasing along with the σ coordinates due to the complexity of the geometry and dynamics in the PRE. An example for such an exception can be seen occurring around S = 20 in Figs. 5e and 5f, which will be further explained in section 4e.

Vertical distribution of the (negative) diahaline transport $\langle {\int }_{A_z(S)}{w}_{e}dA\rangle$ per σ (a),(b) averaged over 14 days and averaged over (c),(d) spring tide and (e),(f) neap tide. Left panels represent results from the high-runoff simulation, right panels represent the low-runoff simulation. In each panel, the (negative) diahaline transport at different salinities is shown in different gray levels. The blue and red shading in (b) helps to illustrate the definition of outflow (Q_out) and inflow (Q_in) of diahaline water transport. σ = −1 means bottom, and σ = 0 means surface.

Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPO-D-21-0292.1

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Vertical distribution of the (negative) diahaline transport $\langle {\int }_{A_z(S)}{w}_{e}dA\rangle$ per σ (a),(b) averaged over 14 days and averaged over (c),(d) spring tide and (e),(f) neap tide. Left panels represent results from the high-runoff simulation, right panels represent the low-runoff simulation. In each panel, the (negative) diahaline transport at different salinities is shown in different gray levels. The blue and red shading in (b) helps to illustrate the definition of outflow (Q_out) and inflow (Q_in) of diahaline water transport. σ = −1 means bottom, and σ = 0 means surface.

Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPO-D-21-0292.1

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Vertical distribution of the (negative) diahaline transport $\langle {\int }_{A_z(S)}{w}_{e}dA\rangle$ per σ (a),(b) averaged over 14 days and averaged over (c),(d) spring tide and (e),(f) neap tide. Left panels represent results from the high-runoff simulation, right panels represent the low-runoff simulation. In each panel, the (negative) diahaline transport at different salinities is shown in different gray levels. The blue and red shading in (b) helps to illustrate the definition of outflow (Q_out) and inflow (Q_in) of diahaline water transport. σ = −1 means bottom, and σ = 0 means surface.

Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPO-D-21-0292.1

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d. Lateral distribution of mixing and entrainment

The spring–neap averaged spatial distributions of the mixing per salinity class and the entrainment velocity at S = 20 g kg⁻¹ are presented in Fig. 6. The horizontal distribution of the salinity mixing on isohalines m_lo(S) are similar to the horizontal distribution of the depth-averaged mixing χ^s presented in other studies (e.g., Warner et al. 2020). In this study, the S = 20 g kg⁻¹ isohaline is a good representative among different isohaline surfaces to show the spatial distribution of salinity mixing. In Figs. 6a and 6b, areas with nonzero m are those where isohaline surface with S = 20 g kg⁻¹ occurs during the spring–neap cycle. Comparing Fig. 6a and Fig. 6b, the isohaline with salinity S = 20 g kg⁻¹ covers a much larger area in summer than in winter. The major difference occurs near the surface layers (e.g., −0.2 < σ < 0). As mentioned above, the size and shape of the river plume are mainly controlled by the river discharge and the wind field. The isohaline areas seaward of the σ = −0.2 iso-sigma contour are the part of the isohaline area which are very close to the sea surface. For salinity S = 20 g kg⁻¹, this upper part of the isohaline surface spans in summer a major part of the isohaline area due to the combined effects of buoyancy and wind stress. In contrast, it only covers a minor part in winter.

Spatial distribution of (a),(b) local salinity mixing m_lo(S) and (c),(d) diahaline velocity w_e(S) at S = 20 g kg⁻¹ averaged over 14 days from (left) the high-runoff simulation and (right) the low-runoff simulation, respectively. Green contours represent the σ layer height, starting from σ = −0.8 (near bottom, lightest green) to σ = −0.2 (near surface, darkest green), with an interval of 0.2. Positive w_e means upward. The thick green line in (a) and (c) represents the transect studied in Fig. 7. The distance along the transect (not along the x axis) is shown in a dark red line, just for better comparison with Fig. 7.

Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPO-D-21-0292.1

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Spatial distribution of (a),(b) local salinity mixing m_lo(S) and (c),(d) diahaline velocity w_e(S) at S = 20 g kg⁻¹ averaged over 14 days from (left) the high-runoff simulation and (right) the low-runoff simulation, respectively. Green contours represent the σ layer height, starting from σ = −0.8 (near bottom, lightest green) to σ = −0.2 (near surface, darkest green), with an interval of 0.2. Positive w_e means upward. The thick green line in (a) and (c) represents the transect studied in Fig. 7. The distance along the transect (not along the x axis) is shown in a dark red line, just for better comparison with Fig. 7.

Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPO-D-21-0292.1

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Spatial distribution of (a),(b) local salinity mixing m_lo(S) and (c),(d) diahaline velocity w_e(S) at S = 20 g kg⁻¹ averaged over 14 days from (left) the high-runoff simulation and (right) the low-runoff simulation, respectively. Green contours represent the σ layer height, starting from σ = −0.8 (near bottom, lightest green) to σ = −0.2 (near surface, darkest green), with an interval of 0.2. Positive w_e means upward. The thick green line in (a) and (c) represents the transect studied in Fig. 7. The distance along the transect (not along the x axis) is shown in a dark red line, just for better comparison with Fig. 7.

Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPO-D-21-0292.1

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In summer, intensified salinity mixing occurs in the surface layers, seaward the Lantau Island and around many other small islands (Fig. 6a). As we have shown in Figs. 5a and 5b, the entrainment velocities are generally upward in the bottom layers and downward in the surface layers (Figs. 6c,d). However, there are also upward entrainment velocities in the surface layers in summer, mainly seaward of Lantau Island (Fig. 6c). In summer, the islands in the PRE play an essential role in controlling the salinity mixing and entrainment velocity (see Fig. 3 for the spreading of the plume). At the same time, in winter, the zones of intensification correspond more to the two deep channels. Negative w_e shows the areas where freshwater leaves the estuarine volume bounded by the isohaline surface. The patchy distribution of intensified w_e in summer suggests that the water exchange through an isohaline surface can be spatially highly variable.

e. Detailed view along a transect

We analyzed a vertically resolved transect to examine the vertical structure of the salinity mixing and the entrainment velocity in the PRE under high-runoff conditions. We placed it in the deep channel to the southwest of the Lantau Island (thick green line in Fig. 6a), oriented along the main direction of surface currents in summer (see Figs. 3a,c). The water along this transect is strongly stratified during the entire spring–neap cycle in summer, with a thin plume covering roughly the upper 3 m of the water column (Fig. 7a). In the context of the following analysis, the slope of the isohalines does not need to be considered for the distinction between w_e and u^dia, since the n ⋅ k in (9) can roughly be neglected here.

Vertical distribution of (a) salinity, (b) entrainment velocity (positive means upward), (c) mixing χ^s, and (d) local mixing per salinity class m_lo along the transect shown in Fig. 6 averaged over 14 days from the high-runoff simulation. The thick green line in (a) represents the depth-averaged salinity $\overline{S}$. The thick blue line in (c) represents the depth-integrated mixing $\int {\chi }^{s}dz$. The thick blue line in (d) represents the local salinity mixing m_lo(S) at S = 20 g kg⁻¹. The dashed magenta, black, and blue lines in each panel correspond to the time-averaged positions of isohalines S = 4, 20, and 34.5 g kg⁻¹, respectively.

Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPO-D-21-0292.1

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Vertical distribution of (a) salinity, (b) entrainment velocity (positive means upward), (c) mixing χ^s, and (d) local mixing per salinity class m_lo along the transect shown in Fig. 6 averaged over 14 days from the high-runoff simulation. The thick green line in (a) represents the depth-averaged salinity $\overline{S}$. The thick blue line in (c) represents the depth-integrated mixing $\int {\chi }^{s}dz$. The thick blue line in (d) represents the local salinity mixing m_lo(S) at S = 20 g kg⁻¹. The dashed magenta, black, and blue lines in each panel correspond to the time-averaged positions of isohalines S = 4, 20, and 34.5 g kg⁻¹, respectively.

Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPO-D-21-0292.1

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Vertical distribution of (a) salinity, (b) entrainment velocity (positive means upward), (c) mixing χ^s, and (d) local mixing per salinity class m_lo along the transect shown in Fig. 6 averaged over 14 days from the high-runoff simulation. The thick green line in (a) represents the depth-averaged salinity $\overline{S}$. The thick blue line in (c) represents the depth-integrated mixing $\int {\chi }^{s}dz$. The thick blue line in (d) represents the local salinity mixing m_lo(S) at S = 20 g kg⁻¹. The dashed magenta, black, and blue lines in each panel correspond to the time-averaged positions of isohalines S = 4, 20, and 34.5 g kg⁻¹, respectively.

Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPO-D-21-0292.1

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The relationship between entrainment velocity, salinity mixing, and diffusive salt flux can be inferred from Figs. 7b–d. For each single water column, saltwater from the bottom and freshwater from the surface are brought into the inner layers due to entrainment. Note that for each of the isohaline surfaces, there are both subareas with downward w_e and other subareas with upward w_e (see Fig. 5). The S = 20 g kg⁻¹ isohaline roughly separates the downward and upward w_e, especially in the outer plume region (seaward of 22 km in Fig. 7b). However, in the near-surface region (σ > −0.2), this isohaline still crosses some nonzero regions of w_e, for example at 25 km along the transect. This explains the positive w_e for S = 20 to the southwest of Lantau Island in Fig. 6c.

According to (4), χ^s is equal m_lo times the salinity gradient ∂_nS. Equation (9) further shows that the local salinity mixing m_lo is directly related to the diffusive salt flux $f_{\text{diff}}^s$. The vertical structure of local salinity mixing m_lo (or in other words, $-2f_{\text{diff}}^s$) shown in Fig. 7d is similar to the distribution of mixing χ^s shown in Fig. 7c. This pattern reflects the strong correlation between the salinity mixing χ^s and the diffusive salt flux $f_{\text{diff}}^s$.

When averaged over a sufficiently long time, the diffusive salinity transport F^s through any isohaline surface must be balanced by the advective diahaline salinity transport SQ. However, due to the presence of the horizontal salinity flux divergence (the first two terms on the RHS of expression 16), the diffusive salt flux $f_{\text{diff}}^s$ in a single water column is not necessarily in local equilibrium with the advective diahaline salt flux u^diaS. Instead, the local diffusive salt flux $f_{\text{diff}}^s$ is balanced by ${\int }_{s>S}{u}^{\text{dia}}ds$ [Eq. (18)].

f. Spring–neap and intratidal variations

Whereas the strong dependence of salinity mixing on the runoff in a steady state can be illustrated by the universal law of estuarine mixing, drastic variations of estuarine mixing can occur during a spring–neap cycle or even within single tidal cycles. To illustrate this, Fig. 8 shows the temporal variation of salinity mixing m (scaled by 2SQ_r such that the temporally averaged value should be unity) for salinities in the estuary within nine days from neap tide to spring tide. The resemblance between Figs. 8a and 8c as well as between Figs. 8b and 8d indicates that the results from the two methods [Eqs. (7) and (8)] calculating m are consistent. The salinity mixing m oscillates around its long-term averaged value m_∞ = 2SQ_r throughout the flood–ebb tidal cycles. In the high-runoff simulation, the temporal variation of the normalized salinity mixing m/2SQ_r does not show distinct differences in magnitude and phase between different salinities (Figs. 8a,b). Therefore, the total mixing $M(S)={\int }_{s\le S}m(s)ds$ within the estuarine volume V(S) and the salinity mixing m(S) on the isohaline surface of S = 20 g kg⁻¹ show very similar temporal variability (Fig. 9). Nevertheless, the normalized magnitude m/2SQ_r indicates some discrepancy between high salinity and low salinity for low runoff (e.g., ∼343.5 day in Fig. 8b).

Temporal variation of mixing per salinity class m calculated by two methods: (a),(b) $m=m^{\text{phy}}+m^{\text{num}}$ and (c),(d) $m=m_{\infty }+\tilde{m}$ [Eq. (7)]. Results from the (left) high-runoff simulation and (right) the low-runoff simulation. In all panels m is normalized by m_∞ = 2SQ_r. Capital letters N and S represent the time of neap tide and spring tide, respectively.

Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPO-D-21-0292.1

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Temporal variation of mixing per salinity class m calculated by two methods: (a),(b) $m=m^{\text{phy}}+m^{\text{num}}$ and (c),(d) $m=m_{\infty }+\tilde{m}$ [Eq. (7)]. Results from the (left) high-runoff simulation and (right) the low-runoff simulation. In all panels m is normalized by m_∞ = 2SQ_r. Capital letters N and S represent the time of neap tide and spring tide, respectively.

Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPO-D-21-0292.1

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Temporal variation of mixing per salinity class m calculated by two methods: (a),(b) $m=m^{\text{phy}}+m^{\text{num}}$ and (c),(d) $m=m_{\infty }+\tilde{m}$ [Eq. (7)]. Results from the (left) high-runoff simulation and (right) the low-runoff simulation. In all panels m is normalized by m_∞ = 2SQ_r. Capital letters N and S represent the time of neap tide and spring tide, respectively.

Citation: Journal of Physical Oceanography 52, 9; 10.1175/JPO-D-21-0292.1

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