# A Subtidal Box Model Based on the Longitudinal Anomaly of Potential Energy for Narrow Estuaries. An Application to the Guadalquivir River Estuary (SW Spain)

## Abstract

The objective of the present study is to demonstrate the informative capacity of the longitudinal anomaly of potential energy (LAPE) in the analysis of the magnitude and spatiotemporal variability of estuarine processes. For this purpose, a LAPE balance equation is formulated. The LAPE integrates and varies with the vertical and longitudinal density distribution. The formulation is applied on a subtidal scale to each box or stretch of the Guadalquivir River estuary, a narrow, highly turbid, weakly stratified, and strongly anthropized estuary. Data recorded by a large network of monitoring stations in 2008 and 2009 are used to quantify advective transports as well as the transports associated with longitudinal dispersion and vertical turbulent mixing in different hydraulic regimes. In low-river flow conditions, (river flows *Q*<40 *m*^{3}*s*^{−1}), the magnitude of LAPE transports decreases upstream and varies locally, depending on neap-spring tidal cycles. The direction of the net LAPE transport creates convergence zones that are particularly consistent with maximum levels of estuarine turbidity. During high-river flows (*Q*>400 *m*^{3}*s*^{−1}), this convergence disappears and the maximum longitudinal density gradient moves towards the mouth. More specifically, tidal pumping -induced LAPE governs during these conditions and manages to compensate the sum of the mean nontidal and dispersive and differential advective LAPE transports. However, during the post-riverflood period, the mechanisms controlling recovery downstream from the mouth are the longitudinal dispersive and differential advective LAPE transports. Furthermore, the convergence zone reappears with a longitudinal gradient of the net LAPE transport that is even greater than in low-river flow conditions.

## Key Points

- A balance equation of the longitudinal anomaly of the potential energy (LAPE) was derived
- The balance was applied in boxes to the GRE and explains the salt intrusion and turbidity maxima
- The dominant LAPE transports in the GRE vary spatially and also depend on the hydraulic regime

## 1 Introduction

### 1.1 Knowledge Integration and Simplification for Estuary Management

Human action is the main reason why many estuaries have experienced significant changes in recent decades. Such changes include (i) modification of their hydrographic basins because of water diverted for irrigation, and the damming of the main water course and its tributaries; (ii) industrial and farming activities on their banks; and (iii) changes in estuary geometry because of depth and maintenance dredging (e.g., Hobbie, 2000; Ruiz et al., 2015; Reyes-Merlo et al., 2017). The anthropization of an estuary alters its biochemical and physical dynamics by modifying nutrient mixing as well as the sediment and oxygen transport in water bodies adjacent to rivers and seas (Officer, 2013; Simpson & Sharples, 2012).

In certain estuaries, such as the Guadalquivir River Estuary (SW Spain), these interventions are reflected in changes in hydromorphodynamics all along the estuary as well as in different hydraulic regimes of the fluvial agents (low-river flow, high-river flow, and postriverflood recovery) and marine agents (neap and spring tides) (Ruiz et al., 2015). From an estuary management perspective, it is a considerable challenge to integrate this scientific evidence in simple, manageable models such as box models that are capable of quantifying the effects of such changes on estuarine dynamics. Box models are generally used to describe processes and to establish balances based on volumetric averages of the main oceanographic and estuarine variables with a suitable spatiotemporal scale (Austin, 2002; Díez-Minguito et al., 2014; Hamilton et al., 1985; Rice et al., 2008; Officer, 1980; Sun et al., 2017). This type of model can facilitate decision-making based on knowledge of the sustainable use of the estuary (Leschine et al., 2003).

Such decisions should be regulated by a legal framework such as the Water Framework Directive 2000/60/EC (WFD), which establishes the need to adopt measures to protect and improve transitional water bodies, such as estuaries, whose quality depends on their ecological status (Ganju et al., 2016). The WFD describes the ecological status of an estuary in terms of the values of hydromorphological parameters, such as the current velocity, salinity, temperature, turbidity and mixing characteristics, among others.

### 1.2 Longitudinal Anomaly of Potential Energy (LAPE)

*ϕ*

_{S}(Jm

^{−3}), quantifies the water column stratification and is defined as the amount of mechanical energy per m

^{3}needed to instantaneously homogenize the water column (Simpson, 1981). This is mathematically expressed as follows:

*g*is the gravity constant,

*t*is the time,

**x**defines the horizontal position,

*z*represents the vertical coordinate,

*ρ*(

**x**,

*z*,

*t*) is the water density, and is the reference density in complete mixing conditions, both of which include the suspended sediment density;

*d*is the total depth that is the sum of

*η*(

**x**,

*t*), the elevation over mean sea level and

*h*(

**x**), the mean depth. From the time when it was first proposed in the mid 1970s (Simpson & Hunter, 1974), the potential energy anomaly has been widely used to identify physical processes that produce water exchanges in shelf seas (Hofmeister et al., 2009; Simpson et al., 1990; Yang et al., 2017), regions influenced by freshwater (De Boer et al., 2008; Simpson, 1997), estuaries (Garvine & Whitney, 2006; Rice et al., 2008; Sun et al., 2017), and lakes (Zhao et al., 2018).

*x*

_{0}, located downstream on the continental shelf instead of the mean density in the water column (equation 1). The definition of

*ϕ*in GW2006 is henceforth referred to as the Longitudinal Anomaly of Potential Energy (LAPE) and is formulated as follows:

*ρ*

_{0}is the mean density of slope water beyond the shelf at the location

**x**

_{0}. This density is the reference density in complete mixing conditions, which are in this case longitudinal. GW2006 represents the characteristics of the system with a single box to represent the estuary, which is directly connected to the continental platform at the subtidal scale. They considered the change rate in the LAPE as induced by freshwater inflow, water outflow to the shelf, tidal and wind mixing, and surface heat flux. However in relatively shallow estuaries that are weakly stratified or well mixed, in addition to gravitational circulations, it is also necessary to consider other contributions to the LAPE, such as those associated with longitudinal dispersion, tidal pumping, and differential advection (Burchard & Hetland, 2010; Becherer et al., 2016; van Aken, 1986). Differential advection includes two terms. The first term is associated with depth-mean straining due to the vertical mean horizontal density gradient strained by the deviation from the depth-mean velocity vector (depth-mean straining). The second term is associated with the deviations of the longitudinal velocities and the gradient of density deviations (non-depth-mean straining) (Burchard & Hofmeister, 2008).

### 1.3 Objectives and Organization

The objective of this research was to evaluate the joint spatiotemporal variability of state variables in the WFD, such as density. This includes the contribution of the suspended solids concentration and the flow velocity throughout the estuary and in different fluvial regimes (i.e., low-river flow, high-river flow, and post-riverflood recovery) and tidal regimes (neap and spring tides). This study led to the formulation of a subtidal-scale dynamic equation that is able to evaluate the LAPE, and which integrates the previously mentioned state variables and the transports that control them. This equation includes the impact of intratidal processes on the subtidal scale by means of the tidal decomposition method (Becherer et al., 2016; Burchard et al., 2018; Díez-Minguito et al., 2013; Scully & Friedrichs, 2007). It also extends the work of Garvine and Whitney (2006) by including the influence of the following: (i) longitudinal dispersion, (ii) tidal pumping associated with cross-correlations between density, elevations, and currents, and (iii) differential advection due to depth-mean straining, and non-depth-mean straining. These terms are particularly relevant to narrow estuaries (Burchard & Hetland, 2010; Jay & Musiak, 1994).

The LAPE general balance equation was applied to the dynamics of the Guadalquivir River Estuary (GRE). Since the GRE is a narrow, weakly stratified, and highly anthropized estuary (Díez-Minguito et al., 2012, 2013, 2014; Ruiz et al., 2015), these same characteristics made it possible to simplify the balance equation and extend Garvine and Whitney (2006) by segmenting the GRE in *N* boxes. This box model was applied to the GRE over a 1-year period with all the previously mentioned hydraulic regimes. To better explain the spatiotemporal variability of the LAPE, we analyzed a high-resolution data set regarding salinity, suspended solids, tidal currents and elevations, and fluvial discharges, which had been recorded in 2008–2009. These data were used to evaluate the LAPE and the different LAPE transport terms in each box.

The rest of this paper is organized as follows. Section 2 presents the theoretical derivation of LAPE balance equation. In section 3, the study zone is described, along with the monitoring network and the configuration of the box model. Section 4 analyzes the processes observed and relates them to the LAPE and the resulting transports of the model. After section 5 discusses the results and the validity of the model, section 6 presents the conclusions derived from this research.

## 2 Formulation of the LAPE

This study focused on a relatively narrow estuary in which the Coriolis effect is negligible and the river flow is transversally uniform (Hansen & Rattray, 1966; Valle-Levinson, 2008, 2010). The system of coordinates (*x*, *z*) has the origin at the estuary mouth, lying over the mean sea level. The positive *x*-axis follows the thalweg and the *z*-axis points upwards; *d*=*η*+*h* is the total depth, *h*(*x*) the mean depth of the bed, and *η*(*x*, *t*) is the water surface that varies over time, *t*.

### 2.1 Interpretation of the LAPE

*ϕ*, is the mechanical energy per m

^{3}needed to instantaneously homogenize the water mass between sections

*x*

_{0}and

*x*(equation 2). In weak stratification conditions, the LAPE can be simplified as follows:

*x*

_{0}<

*x*, where

*x*

_{0}is located beyond the shelfbreak (Garvine & Whitney, 2006). As can be observed in equation 3, calculating

*ϕ*(

*x*,

*t*) is the same as calculating the potential energy of the semidifference of densities between

*x*and

*x*

_{0}. For a single box, equation 3 establishes that if the longitudinal density gradient is zero [ ], then

*ϕ*is zero. This is the same as finding conditions of complete longitudinal mixing, as can be observed, for example, in tidal river reaches. In normal conditions, density decreases upstream (Officer, 2013) and

*ϕ*>0. However, in inverse estuaries or in regions where the profile of

*ρ*(

*x*) is locally inverted, density increases upstream and

*ϕ*<0.

### 2.2 Development of the LAPE Balance Equation

*ρ*(

*x*,

*z*,

*t*), is governed by the balance equation or the dynamic density equation (van Aken, 1986; Burchard & Hofmeister, 2008) defined as

**D**is the density equation defined as

*∂*

_{χ}

*ζ*denotes the derivative of the generic variable,

*ζ*, in regard to the independent variable

*χ*. The terms

*k*

_{x}and

*k*

_{z}denote the longitudinal and vertical eddy diffusivities that result from the parameterizations of turbulent fluxes (Burchard & Hofmeister, 2008). The term

**S**(

*x*,

*z*,

*t*), encompasses the local source and sink terms, which includes the nonlinear effects that appear from the equation of state for sea water (Burchard & Hofmeister, 2008), the incident solar radiation, and the suspended sediment:

*α*

_{w}, is the water expansion coefficient;

*c*

_{p}is the heat capacity of the water;

*ε*is the light absorption coefficient;

*ρ*

_{s}is the sediment density;

*w*

_{s}is the settling velocity; and

*C*is the suspended sediment concentration. Since the values for precipitation, evaporation, and water diverted for irrigation are relative small compared to the fluvial discharges, they are not taken into account. The first term of the RHS in equation 6 can be usually neglected because it is several orders of magnitude smaller than its counterpart in equation 5. Assuming weakly stratified to well-mixed conditions, the second term of the RHS of equation 6 can also be neglected. This simplification is similar to considering a linear equation of state for seawater.

*ϕ*is based on the expressions of

**D**(equation 5), evaluated in

*x*and

*x*

_{0}, and the equation of the longitudinal anomaly of the potential energy (equation 2). When

**D**(

*x*

_{0},

*z*,

*t*) is depth-averaged, taking into account the continuity equation,

*∂*

_{x}

*u*+

*∂*

_{z}

*w*=0, and the dynamic and kinematic boundary conditions (see Appendix A), the result is the following:

_{η}and

_{−h}indicate the respective surfaces on which the terms (

*z*=

*η*and

*z*=−

*h*) are evaluated. The chain rule is applied to the derivatives and Leibniz's rule to integrals with variable limits. To calculate the average of , each dependent variable

*ζ*is decomposed in the superposition of the depth-averaged value , and a deviation,

*ζ*

_{v}. The same operation is performed with the term . By substituting the results obtained in equations 7 and A5, the following dynamic equation is obtained for the LAPE:

The first term *t*_{0} is the time variation of *ϕ* (eq. 6, De Boer et al., 2008). The term *t*_{1} represents the horizontal advection of the longitudinal density difference in relation to the average density. The term *t*_{A} is the longitudinal advection of the potential energy anomaly as defined by Simpson (1981), *ϕ*_{S} (equation 1), due to the vertical average of the longitudinal velocity; *t*_{B} is the depth-mean straining, based on the vertical mean horizontal density gradient strained by the deviation from the depth-mean velocity vector (Bowden, 1981), also called the differential advection term by van Aken (1986) (see also Burchard & Hofmeister, 2008); *t*_{C} is the nonaveraged differential advection based on the deviations of the longitudinal velocities and the gradient of the density deviations. As for the potential energy anomaly, *t*_{B} and *t*_{C} include velocity deviations *u*_{v} and their longitudinal gradients. The term *t*_{D} is the vertical advection due to the deviations in regard to the vertical average of the vertical velocity. The term *t*_{E} is an estimation of the turbulent flux of density that depends on the eddy diffusivity coefficient (Simpson & Bowers, 1981; Simpson & Sharples, 2012; van Aken, 1986). The surface and bottom buoyancy transports are included in the term *t*_{F}. The source and sink terms related to heat transport on the free surface (
, (first term) and the bed sediment resuspension (second term) are indicated in *t*_{G}. Finally, the mean turbulent transport divergence throughout the estuary is defined by *t*_{H} whereas *t*_{I} is the divergence of the mean turbulent transport through the boundaries *z*=*η* and *z*=−*h*. To facilitate their tracking, these terms were renamed as follows:
for *ζ*=*η*,−*h*.

#### 2.2.1 Application to Weakly Stratified Estuaries

*ρ*

_{v}≃

*ρ*

_{0,v}≃0. The term

*t*

_{E}cannot be neglected since it is the mechanism that sustains mixing in the water column and determines the vertical mass exchange in weakly stratified estuaries (Jay & Smith, 1990). Assuming this hypothesis, equation 8 is simplified as follows:

In the case of other estuaries, different hypotheses can be adopted that simplify the general equation (equation 8).

### 2.3 Box Model of the LAPE

*i*(Figure 1), equation 9 is integrated with respect to a

*x*between two sections,

*x*

_{i}and

*x*

_{i}+

*L*

_{i}. When the tidal decomposition of the resulting variables is substituted, this gives the LAPE equation for the box at the subtidal scale:

*ζ*, is the integral in the length of a box,

*L*

_{i}; and, is the time average in a tidal cycle , which permits the decomposition of , where

*ζ*

*′*is the deviation in regard to the average. At the same time, Δ

_{t}

*ζ*represents the time difference of

*ζ*evaluated in

*t*and and Δ

_{x}

*ζ*represents the spatial difference of

*ζ*evaluated in

*x*and

*x*+

*L*. The functions

*F*

_{1}and

*F*

_{2}are given by

In equation 10,
(J·m^{−1}) represents the LAPE of the box. The term *τ*_{0} is the variation experienced by Φ from the beginning to the end of the tidal cycle. The term *τ*_{1} includes the superposition of the mean transport [
], the Stokes transport [
], and three tidal pumpings related to correlations between *η*+*h*, *u**′* and
[
;
;
]. The term *τ*_{2} considers the depth-averaged advection density, spatiotemporal depth changes, and mean vertical mass transports. The term *τ*_{3}, which is obtained from (*t*_{E}) in equation 8, is empirically related to tidal and wind mixing (Simpson, 1981; van Aken, 1986). According to Simpson (1981), Simpson and Bowers (1981), and Simpson et al. (1991), the mixing term is described by the cube of the tidal velocity
and the wind velocity
. The constant parameters *ϵ*, *δ*,*C*_{d}, and *k*_{s} represent the effective surface drag force, bottom drag force, tidal current mixing, and wind mixing, respectively. The term *ρ*_{a} denotes the air density at sea level. The surface solar radiation is *τ*_{4}. Finally, *τ*_{5} describes the turbulent horizontal transport difference between *z*=*η* and *z*=−*h*. The LAPE transports due to tidal pumping (*τ*_{1,4}), differential advection (*τ*_{2}) and longitudinal dispersion (*τ*_{5}) were not contemplated by Garvine and Whitney (2006) in their box model.

In what follows, all variables are averaged in a tide cycle, although for the sake of simplicity, the lower bar in the equations indicating this has been eliminated.

## 3 Application to the Guadalquivir River Estuary

### 3.1 Study Zone

The Guadalquivir River Estuary (GRE) is located in the southeast of Spain (36°43*′*N to 37°32*′*N, 5°56*′*W to 6°30*′*W). It is a transition zone between the river basin of the same name and the Gulf of Cadiz (Figure 2). The estuary comprises the last 110 km, 85 of which are navigable as far as Seville. The Alcalá del Río dam is the upstream tidal limit. The estuary is weakly convergent: channel widths vary from 800 m near the mouth to 150 m at the head. The channel mean depth is approximately equal to 7 m, artificially maintained in many stretches by dredging to favor navigation.

The GRE is a positive estuary, dominated by the flood tide and mesotidal range (tidal range of <4 m at the mouth), where the most significant tidal constituents are *M*_{2}, *S*_{2}, and *N*_{2}. The GRE is generally dominated by tidal processes and is weakly stratified when fluvial discharges are not greater than 40 *m*^{3}*s*^{−1}, which occurs over 75% of the year (Díez-Minguito et al., 2012). Mean density deviations from the vertical density average are less than 1.42 kg·m^{−3}. The estuary number is less than 0.05, the densimetric Froude number is smaller than 0.014, and the Richardson number of the estuary is less than 0.3 (Díez-Minguito et al., 2013).

### 3.2 Data Collection

The data for the study were collected in real time from 2008 until 2010 with a remote monitoring system installed by the Institute of Marine Sciences of Andalusia (ICMAN-CSIC) (Navarro et al., 2011). Figure 2 shows the instrumentation at the estuary, namely, six environmental-quality monitoring stations, six water level sensors, six acoustic Doppler velocimeters, and a meteorological station, represented by *α*, *β*, *γ*, and M.S., respectively. The environmental quality station recorded temperature (*T*), salinity (*S*), and suspended solids concentration every 30 min at four different depths (*z*=−1,−2,−3 and −4 m). These data were used to calculate the fluid density *ρ*(*S*,*T*,*C*,*p*) (sea water and suspended sediment mixture) as *ρ*=*ρ*_{w}+(1−*ρ*_{w}/*ρ*_{s})*C*, in which the water density, *ρ*_{w}(*S*,*T*,*p*) was obtained from the equation of state for seawater. The sediment density, *ρ*_{s}, was considered to be equal to 2,650 kg·m^{−3} (Díez-Minguito et al., 2014). Water level sensors recorded the position of the free water surface every 10 minutes, and these data were used to calculate the tidal range. Current data were measured every 15 min by means of acoustic current profilers. At the head of the estuary, daily data pertaining to the discharges from the Alcalá del Río dam were obtained from the Regional Water Management Agency (Agencia Andaluza del Agua, Junta de Andalucá). Wind velocity and direction and solar radiation were measured every 10 min in the meteorological station at the mouth of the GRE.

### 3.3 Configuration of the Model

The previously described LAPE formulation was applied to the GRE from 24 July 2008 to 24 July 2009. Figure 3f shows the following hydraulic regimes that were observed during this 1-year period (Díez-Minguito et al., 2012): (1) low-river flow, (2) high-river flow, and (3) an intermediate riverflood recovery period. These regimes are discussed at length in the sections that follow.

The spatial domain extended from the mouth, *x*=0 km, to *x*=57.6 km upstream, the average position reached by the 2-psu isohaline (Monismith et al., 2002), which characterizes salt intrusion. The upper stretch of the estuary, which normally has the characteristics of a tidal river, was not considered. The GRE was divided into five boxes, whose length, width, and depth are defined in Table 1. These boxes are representative of estuarine dynamics regarding tidal wave propagation, salt transport, and suspended particulate material transport along the GRE.

Box |
Geometry |
Density variability |
||||||
---|---|---|---|---|---|---|---|---|

x_{i} |
L_{i} |
h_{i} |
A_{i} |
A_{s,i} |
(5%, 95%) | ρ_{v} (5%, 95%) |
||

No. | Color | (km) | (km) | (m) | (10^{2} m^{2}) |
(10^{6} m^{2}) |
kg·m^{−3} |
kg·m^{−3} |

1 | Cyan | 0 | 17.3 | 7.26 | 58.39 | 12.91 | 1,023.75 (−4.30, 3.12) | 0.41 (−0.31, 0.60) |

2 | Green | 17.3 | 8.9 | 7.13 | 43.82 | 5.22 | 1,012.64 (−7.01, 6.82) | 0.21 (−0.04, 0.24) |

3 | Orange | 26.2 | 9.1 | 7.05 | 37.80 | 4.54 | 1,007.01 (−4.48, 4.70) | 0.11 (−0.11, 0.44) |

4 | Red | 35.3 | 11.8 | 6.95 | 32.51 | 4.90 | 1,004.22 (−3.12, 5.31) | 1.42 (−1.20, 0.50) |

5 | Purple | 47.1 | 10.5 | 6.85 | 26.72 | 3.56 | 1,001.67 (−1.99, 3.31) | 0.07 (−0.05, 0.24) |

- | - | 57.6 | - | 6.80 | 22.45 | - | 1,000.97 (−2.54, 3.37) | 0.25 (−0.14, 0.20) |

Total | - | 57.6 | - | - | 31.13 | - | - |

*Note*. Each box is defined by its initial downstream position*x*_{i}, its final upstream position*x*_{i+1}, length*L*_{i}, and the limiting cross-sections of areas*A*_{i+1}and*A*_{i}(Figure 1). The surface area of each box is defined by*A*_{s,i}. Also shown are typical vertical averaged density values and their deviations in regard to the vertical average value,*ρ*_{v}, in each box. Besides the density values, the percentiles 5 and 95 of the time series of density deviations are shown in brackets.

Equation 3 was used to calculate *ϕ* in the five boxes, using the data observed (Figure 3). In each box, the reference density, *ρ*_{0,i}, is the density at the point farthest downstream, *ρ*_{i} is the density at the point farthest upstream, and *h*_{i} is the average depth for *i* =1, 2, …, 5. Typical average density values at these points are also shown in Table 1. The equation 10 evaluates the transports that control the LAPE, which are derived in section 2.2.1, also based on the data observed.

*ϕ*

_{e}, as though it were a single box. This was obtained with the following expression:

*j*, as follows:

## 4 Results

### 4.1 Observational Background

This section analyzes the observations in order to ascertain the influence of the suspended solids concentration, water temperature, and salinity on water density for the different fluvial regimes during the study period. This analysis clarifies the interpretation of the spatiotemporal variability of the LAPE. In Figure 3, panels (a), (b), (c), and (d) show, respectively, the tidal-averaged time series of the vertically averaged suspended solids concentration, salinity, water temperature, and density for different locations in the estuary (colored curves). Panel (e) shows the tidal range at the estuary mouth, and panel (f), the fluvial discharge during the study period.

During the summer of 2008 (1), the estuary remained in a low-river flow regime with fluvial discharges of less than 40 m^{3}s^{−1}. In these conditions, tidal currents determine estuarine dynamics. The suspended particulate material throughout the estuary scarcely reached 2 kg·m^{−3} during the summer. In autumn and the beginning of the winter in that same year, pulsed discharges reached 100 m^{3}s^{−1} and lasted for various days (blue arrows in Figure 3f). This produced a marked decrease in salinity and density (Δ*ρ*≃ 5 kg·m^{−3}).

In the middle of winter (2), intense rains made it necessary to open the floodgates of the dam upstream. The discharge reached the threshold of extreme fluvial conditions in the GRE (*Q*> 400 m^{3}s^{−1}). Density dropped sharply to freshwater levels in the entire estuary, except at the mouth, where the density reduction was 15 kg·m^{−3}. The fluvial discharge caused the salt to be transported seaward from the estuary. It also increased the quantity of sediments suspended from the bed, which led to a turbidity peak approaching 10 kg·m^{−3} (Figure 3a). During this period, the mouth showed the maximum horizontal density gradient of the estuary.

This episode was followed by a recovery period (3) in which the fluvial discharge of the dam again decreased with values below the threshold of 40 m^{3}s^{−1}. The salt that had been transported towards the mouth by the discharge began to move upstream again (Figure 3b). These conditions persisted until the beginning of the summer of 2009, when salinity levels and the sediment concentration were once more typical of low river flows. The summer was characterized by a slight linear reduction in density detected by all sensors. This was caused by the increase in water temperatures because of short-wave radiation (Padilla et al., 2016).

### 4.2 Evaluation of the LAPE

#### 4.2.1 Longitudinal Variation of *ϕ*(*x*)

Figure 4a shows the results of equation 3 at several sections representative of the conditions at the boxes defined in Table 1, where *ϕ*_{i} indicates the longitudinal anomaly of the potential energy (LAPE or *ϕ*) at *x*_{i}. The first third of the estuary (Boxes 1 and 2) shows the highest values of *ϕ*, where the maximum average longitudinal density gradients occur as well as the greatest time variability. The LAPE for Box 1 during the study period is *ϕ*_{1}= 395.04 ± 122.68 J m^{−3}, where the first term is the mean value and ±*X* is the standard deviation. The amplitude of the fortnightly modulation of *ϕ*_{i} and its mean value decrease upstream (*ϕ*_{5}≃ 90 ± 45 J m^{−3}). In Box 1, *ϕ* oscillates out of phase with the fortnightly constituent of the tidal range with minimums of *ϕ* during spring tides and maximums during neap tides. Upstream, just the opposite occurs. The reason of the fortnightly change of *ϕ* will be discussed in the next section. The dependence of *ϕ* on the neap-spring-tide cycles was no longer visible from October 2008 until February 2009 because the intermediate fluvial pulsed discharges became more numerous (≃100 m^{3}s^{−1}) (Figure 4d). The 2-psu isohaline (Figure 4c) reaches Box 5, where *ϕ*_{i} remains almost constant during low riverflow conditions. However, it retreats to *x*= 40 km with the pulsed discharges observed at the end of autumn, which is manifested in a reduction of *ϕ* in Boxes 4 and 5. Then, it moves to Box 1 during the flood (2), causing a marked decrease in *ϕ* in all boxes. On a yearly scale, at the mouth, positive temperature increases generated negative increases in *ϕ*. Upstream, the temperature oscillation is in phase with *ϕ*.

Generally, the value of *ϕ*_{e} remains more or less constant in low-river flow conditions (1), which is to be expected because of the small time variations in the density observed at the estuary mouth and head (Figure 3d). During the high-river flow regime (2), *ϕ*_{e} decreases by half because of the seaward movement of the salt intrusion. The dependence of the salt intrusion increases with the freshwater flow (Reyes-Merlo et al., 2013). The salt intrusion during (2) moves upstream to kilometer 15. During the subsequent recovery period (3), *ϕ*_{e} oscillates out of phase *π* with the tides. The mean value and variability of *ϕ*_{e}≃ 218.82 ±36.89 J m^{−3} are in accordance with the values obtained by Garvine and Whitney (2006) in the Delaware estuary where the mean estuary depth is 8 m. The amplitude of the oscillation triples during the recovery period ranging from a little more than 15 J m^{−3}, corresponding to what is observed in normal conditions, to almost 50 J m^{−3}, mostly conditioned by the box of the estuary mouth.

#### 4.2.2 Variation of *ϕ*(*x*) Depending on Fluvial Regime

Figure 5 shows the profiles of *ρ*(*x*) and *ϕ*(*x*) for the different regimes (low-river flow, high-river flow, and postriverflood recovery). In all cases, it was observed that the variability in regard to that mean density profile (orange line, Figure 5) decreases upstream.

In normal conditions (1) (Figure 5a), the mean density in each box decreases from the mouth upstream. This decrease is linear to kilometer 40. The LAPE decreases more gradually upstream. During the high-river flow regime (2) (Figure 5b), almost the entire estuary is composed of freshwater, which gives values of *ϕ*≃0 except for the mouth. In Boxes 3 and 5, *ϕ* is negative, which means that the density is locally inverted (*ρ*_{3}>*ρ*_{2} and *ρ*_{5}>*ρ*_{4}).

In the recovery period (3), the along-channel density (Figure 5c, upper panel) shows a similar profile to that observed in normal conditions, where the variability in *ρ*(*x*) as well as *ϕ*(*x*) is maximum during recovery. The negative values of *ϕ* persist until the density observed in normal conditions is recovered because the values approach the estuarine turbidity maximum (ETM) (Díez-Minguito et al., 2014).

### 4.3 Evaluation of LAPE Transports

#### 4.3.1 Longitudinal Variation of Advective Transports of *ϕ*

Figure 6 shows the main advective transport terms in each box, calculated in accordance with equation 10. Terms *τ*_{1,1}, *τ*_{1,2}, and *τ*_{1,4} represent the average nontidal advective *ϕ* transport, the LAPE transport due to Stokes, and the tidal pumping *ϕ* transport induced by the correlation between currents and densities. These terms represent more than 99% of the advective transport of *ϕ*. The terms *τ*_{1,3} and *τ*_{1,5} are not shown since they are various orders of magnitude lower than the latter ones (≃ 10^{2} W m^{−1}). At the estuary mouth, the mean nontidal advective LAPE transport, *τ*_{1,1}, is negative ≃−1.5 ·10^{4} W m^{−1} (blue curve in Figure 6a), which indicates that the nontidal residual currents flow seaward. Upstream, this mean value becomes positive (green curve), then negative (purple curve), and then positive (red curve) again, which indicates the convergence and divergence zones of *τ*_{1,1} all along the estuary. This longitudinal *ϕ* transport variation is also observed in Stokes-induced and tidal pumping transport processes.

During the low-river flow regime (1), *ϕ* transports due to Stokes (*τ*_{1,2}) and tidal pumping (*τ*_{1,4}) are in phase with the tidal range. At the estuary mouth, the dependence of *τ*_{1,1} on the tidal range is not as clear. The term, *τ*_{1,1} has a greater magnitude than *τ*_{1,4} and *τ*_{1,2}. More specifically, *τ*_{1,2} is approximately one-fourth of *τ*_{1,1}.

The nonidal mean advective *ϕ* transport and tidal pumping are primarily responsible for the fortnightly change in density during the low-river flow (≃ 2· 10^{4} W m^{−1}, in absolute value). Transport magnitude decreases upstream except in Box 3 where it increases. Upstream transports (Boxes 4 and 5) become negligible.

In the high-river flow regime (2), *τ*_{1,1} is inverted and the transport in Boxes 2 and 3 drops to ≃−2.5·10^{4} W m^{−1}, whose absolute value exceeds that of the first box. Nontidal mean advective and Stokes *ϕ* transports at the estuary mouth are in the upstream direction in contrast to what occurs in the following box. The *τ*_{1,2} and *τ*_{1,4} transports intensify with the discharge. This means that tidal pumping (*τ*_{1,4}) is now the dominant process in the first third of the estuary, moving seaward at the mouth (≃−15·10^{4} W m^{−1}) and upstream in the following box (≃5·10^{4} W m^{−1}).

During the recovery period (3), the mean transport regains its values in low-river flow conditions and when it is in phase with the tide. The Stokes-induced LAPE transport (*τ*_{1,2}) recovers the values observed in (1). In the case of tidal pumping, maximums observed during the discharge progressively attenuated until the total recovery of the estuary in May 2009.

#### 4.3.2 Longitudinal Variability of *ϕ* Transports for Different Fluvial Regimes

This section jointly analyzes, in terms of LAPE transport, the longitudinal variability of the dominant processes observed in the GRE during periods of neap and spring tides in normal conditions, recovery conditions, and during the high-river flow regime. Figure 7 shows the median values ( ) of the LAPE transport results.

During (1), nontidal mean, Stokes-induced, and tidal pumping *ϕ* transports at the mouth are seaward. This behavior is inverted upstream (*x* = 26.2 km), where the value is of the same order, but positive. From here upstream, these *ϕ* transports decrease and show a negative value in *x* = 47.5 km. This variability causes various points of energy transport convergence and divergence to appear throughout the estuary. More specifically, near kilometer 15, there is a divergence point of *τ*_{1,1}, *τ*_{1,2}, and *τ*_{1,4}. Near *x* = 35 km, another convergence point for the same transports is also identified. In spring tides, the mean LAPE transport value is greater near the mouth and decreases after the divergence point. The term *τ*_{1,1} varies considerably along the estuary, due to phase changes between the longitudinal velocity and density. Although the longitudinal transport profile intensifies during spring tides in the estuary recovery period, in neap tides, this same value visibly decreases in *x*= 17.3 km. It is important to highlight how the longitudinal profile of *τ*_{1,1} is inverted during the fluvial discharge (2). Moreover, as can be observed, the net LAPE transport in spring tides (Figure 7j) is positive at the mouth and negative, upstream, whereas in neap tides, it is practically zero in the entire estuary.

The Stokes-induced *ϕ* transport (Figure 7b) shows a similar pattern during flow conditions (1) and (2) except for the fact that it becomes practically negligible upstream from the divergence point and during the recovery phase. In any case, the Stokes-induced *ϕ* transport has a magnitude that is slightly less than that obtained for the nontidal mean advection. The upstream decrease in the Stokes-associated *ϕ* transport is also controlled by the phase lag between elevations and tidal currents. This phase lag approached *π*/2 upstream because of the partial standing nature of tidal wave propagation due to the tide reflection in the dam at the head of the estuary (Díez-Minguito et al., 2012). Tidal pumping *τ*_{1,4} (Figure 7c) also has the same profile with the absolute maximum value of its magnitude at the estuary mouth. During the high-river flow regime (2), *τ*_{1,4} maintains the same longitudinal profile and significantly increases so as to become the dominant process in these conditions.

As can be observed in Panels (d), (e), and (i) of Figure 7, the LAPE transports due to differential advection and dispersion also have an important role in the balance. Transport values due to longitudinal dispersion are greater near the mouth and decrease upstream. In low-river flow conditions, the transport median value approaches 1.3·10^{4} W m ^{−1} in spring tides and is about half that value in neap tides. This shows that dispersive transport is very important in the bottom third of the estuary with a contribution that in certain conditions reaches 30% of the total transport. Transport induced by differential advection of the average terms is the second dominant process in magnitude during (2), and is comparable to the dispersive transport, *τ*_{5}. After kilometer 35.3, transport induced by the differential advection of the mean terms and dispersion practically disappears.

The contribution of the surface solar radiation and mixing terms to *ϕ* transport (Panels (f), (g), and (h) of Figure 7) is not significant. In the Delaware estuary studied in Garvine and Whitney (2006) and Rice et al. (2008), the solar radiation and tidal and wind mixing *ϕ* transports are significant buoyancy terms. On the other hand, in the GRE case, the contribution of these terms are not significant because they depend on the surface area, and the boxes considered in the model (see Table 1) are two order of magnitude smaller than the surface area of the Delaware estuary (2.1·10^{9} m^{2}, Table 2 in GW2006).

Figure 8 shows the most significant net LAPE transports and their relation to the 2-psu isohaline in different hydraulic regimes. The magnitude of *τ*_{i} decreased upstream because the longitudinal density distribution depends to a great extent on the salinity of the estuary. This behavior is similar to that of the salt transport observed in Díez-Minguito et al. (2013). As can be observed, salt intrusion due to the river flow (Reyes-Merlo et al., 2013) is linked to LAPE transports. Although the relation between both variables is different for each transport, certain similarities can be observed. Firstly, in normal conditions (1), the absolute transport value does not reach 10^{4} W m ^{−1} and the salt intrusion remains between kilometers 63 and 70. In high-river flow conditions (2), the salt intrusion travels approximately to kilometer 12. Insofar as their absolute value, transports are maximum although the nontidal mean advective moves seaward the same as the tidal pumping, Stokes transport, and differential advectives move upstream. LAPE transport due to longitudinal dispersion is a special case because during (2), its magnitude increases in absolute value. Nevertheless, its dependence on the spring and neap tide cycles is important and dispersion increases its variability and maintains its mean value close to zero. During the recovery stage (3), the salt intrusion moves from the estuary mouth upstream and transports regain their values in (1).

Averaged *ϕ* transports, *τ*_{e} in equation 13 were compared with values obtained by GW2006 in the Delaware estuary. In normal conditions, the mean LAPE transport in the GRE, ranged from −1 to 1·10^{4} W m ^{−1}. In fact, all individual transport modes were found to have similar magnitude ∼10^{4} W m^{−1}. During the high-river flow regime, it is observed that the values of individual transports do depend significantly on the flow conditions even though the order of magnitude of *τ*_{e} hardly differs from its value in normal conditions. Furthermore, the order of magnitude of the mean transport coincides with the estimates made by Garvine and Whitney (2006) for high-river flows in the Delaware. This occurs because in our model, the transport associated with tidal pumping compensates the transport associated with longitudinal dispersion and differential advection. This first term was not considered in Garvine and Whitney (2006).

## 5 Discussion

This study adopted the definition of the longitudinal anomaly of the potential energy in GW2006 and extended it to the zone of tidal river reaches by segmenting the estuary in stretches. The basic physics premises used led to the derivation of a general balance equation (equation 8). In realistic scenarios such as this one, balances are determined by only a few terms that can be different in time and space (Burchard & Hofmeister, 2008; Díez-Minguito et al., 2013). The initial hypotheses of a weakly stratified or well-mixed estuary of limited width allowed us to simplify the LAPE balance equation in the GRE for a box and average it in a tidal cycle (equation 10). The model includes the major intratidal processes in the GRE. The nonlinear terms that appeared from the equation of state for sea water were neglected because their influence under weakly stratified to well-mixed conditions is limited. In addition, adiabatically corrected temperature and density (i.e., potential temperature and density, respectively), do not play a role in the GRE, where mean depths are about 7 m. However, these terms could be important in stratified estuaries. It is worth to note that the vertical mixing of density is expressed in terms of eddy diffusivity, and it is simplified with empirical mixing terms at a subtidal scale (Simpson, 1981; Simpson & Bowers, 1981; Simpson et al., 1990; van Aken, 1986). A more rigorous definition of the mixing terms would require the analysis of high resolution spatial and temporal measurements that is out of the scope of this study.

The longitudinal variability of LAPE and of *ϕ* transports in other fluvial regimes is evident as reflected in the previous analyses. In the low-river flow regime (1), the time variability of the density in neap and spring tides modulated the LAPE field and its transports all along the estuary. Although in our study *τ* includes mass anomaly fluxes, the results show alterations similar to those observed in mass fluxes in other estuaries (Vaz et al., 1989), where the alterations were caused by the turbulence modulation induced by tidal currents, basically in fortnightly cycles. The longitudinal variability of *τ* produced convergence regions that corresponded to at least one of the ETMs, more specifically, to the one located near ∼17 km (Díez-Minguito et al., 2014). The net estuary transport was positive in Box 1 and negative in 2, 3, and 4 during spring tides, whereas the opposite occurred during neap tides. Nevertheless, the magnitude of the net transport decreased considerably in neap tides in the box of the estuary mouth compared to its value in spring tides. It is striking that the same did not occur in high riverflows (2), where the net transport was directed upstream. Nor did it occur in the postriverflood recovery (3), where the transport at the estuary mouth had the same magnitude in spring tides and neap tides, which mutually compensate each other.

After intermediate fluvial discharges (i.e., flows that are neither low nor high) of flow *Q*≃100 m^{3} s^{−1}, there was a slight increase in the contribution of the *ϕ* transport associated with tidal pumping. If those pulsed discharges continued during the fortnightly semicycle, the magnitude of the Stokes-induced transport significantly decreased in the lower stretch of the estuary (Boxes 1 to 3). This reduction in magnitude was apparently caused by the movement of the salt intrusion towards the estuary mouth (Reyes-Merlo et al., 2013). The phase lag between elevations and tidal currents, which to a great extent controls the Stokes transport, was not altered by these intermediate discharges.

During the high-river flow episodes (2), the longitudinal profile of *ρ* was constant inside the estuary and approximately equal to the freshwater density. As a result, the LAPE was minimum in these conditions except at the estuary mouth where it was at its maximum. This situation mainly intensified the magnitudes of the nontidal mean advective transport and dispersive transport, which was expected. However, it also significantly increased the LAPE transports associated with tidal pumping and differential advection. For this reason, these contributions cannot be ignored in the GRE in high-river flow conditions. These results widen the vision of the GRE and extend it to other hydraulic regimes since in previous studies (Díez-Minguito et al., 2013, 2014), only transports in low-river flow conditions were considered.

During the initial stages of the postriverflood recovery phase (3), differential advection terms and longitudinal dispersion terms were positive (upstream) near the estuary mouth. This may suggest that there are mechanisms that control the recovery of the LAPE in the estuary, such as the differential advection or the strain-induced periodic stratification (Simpson et al., 1990). However, more research is needed to confirm this possibility.

The influence of sediment suspension was also relevant in the post-riverflood recovery phase. The sediment that was suspended with the discharge remained so because of the turbulence generated by the tidal currents (Losada et al., 2017). The high concentration of suspended solids during the recovery phase was reflected in greater time fluctuations of the LAPE and its transports. The larger difference between neap and spring tides also increased the landward transport of the tidal pumping (Scully & Friedrichs, 2007) in Boxes 2 and 3.

In addition, the increase in the nontidal mean *ϕ* transport in the same boxes could be promoting salt transport towards the estuary. Erosion and suspension of the bed sediment favored tidal suspension (Van Leussen & Dronkers, 1988). Along with this, *ϕ* transport due to longitudinal dispersion became significant during the recovery phase. At the intratidal scale, these conditions created periods during which stratification-induced turbulence decreased, which indicated an increase in the capacity to retain suspended sediment (Geyer, 1993; Jay & Musiak, 1996).

As for the applicability of the model, it should be mentioned that in the theoretical development of the equation 9, the estuary is assumed to be weakly stratified. The observations show that the vertical density variability, whether in low-river or high-river flow conditions, ranges from −1.20 to 0.60 kg· m^{−3} (Table 1), which justifies this hypothesis. The second hypothesis is that transversal processes are negligible. This hypothesis is quantified in the Kelvin (*K*_{e}) and Ekman (*E*_{k}) numbers parameter space (Valle-Levinson, 2008). The channel width is related to the internal Rossby radius of deformation through the Kelvin number whereas the Ekman number establishes the relation between the viscosity and Coriolis forces. For the mean value of the vertical eddy viscosity coefficient, typical of the GRE ∼3.3·10^{−4} m^{2}s^{−1} (Reyes-Merlo et al., 2013), values, *K*_{e}∼−2 and *E*_{k}∼−2.5, clearly signal a vertically-sheared exchange. Near the estuary mouth, where channel widths are greater and where the values of the vertical eddy viscosity coefficient are ∼1.2·10^{−3} m^{2}s^{−1} (*K*_{e}∼−2 and *E*_{k}∼−1.25), a certain degree of lateral variability is expected.

Finally, this study highlights the dependence of the LAPE on the following: (i) origin of the forcing agent, (ii) along-estuary position, (iii) neap and spring tide cycles, and (iv) hydraulic regime. A snapshot of the LAPE along the estuary provides information about the hydrodynamic and mixing conditions of the estuary and could be useful for the management and regulation of the freshwater flow in the head dam. Its use would allow estuary management at the subtidal and local scale (e.g., freshwater discharges in certain stretches of the estuary by taking advantage of the irrigation network).

## 6 Conclusions

The objective of this study was to analyze the spatiotemporal evolution of estuarine processes within the framework of a LAPE box-averaged approach. For this purpose, we developed a dynamic equation of the longitudinal anomaly of the potential energy at the subtidal scale. This equation includes the nontidal mean advective and Stokes-induced LAPE transport, as well as empirical approximations of the tidal vertical turbulent mixing and the mixing due to wind and solar radiation. In addition, this research extends previous work by including the influence of (i) longitudinal dispersion, (ii) tidal pumping associated with cross-correlations between density, elevations, and currents, and (iii) differential advection due to depth-mean straining, and non-depth-mean straining. The general balance equation was applied to the GRE, whose characteristics allowed for simplification and segmentation in five boxes. The analyzed data pertains to the salinity distribution, temperature, suspended solids, tidal elevation and currents, and fluvial discharges recorded along the estuary over a 1-year period (2008–2009) in which low-river flow, high-river flow, and postriverflood recovery conditions were found. These observations were used to evaluate the LAPE and transports throughout the GRE. The results can help to better understand the processes in a narrow and shallow water estuary under different hydraulic regimes.

The time variability of LAPE transport in low-river flows was modulated with neap and spring tides. At the estuary mouth, they were in phase with each other, and further upstream, they were out of phase. Time variability in regard to the mean value was similar in spring tides and neap tides (±10.65%). The only exception was the Stokes-induced LAPE transport, whose magnitude significantly increased in spring tides. The LAPE transports of all processes in the GRE changed sign on various occasions at approximately kilometers 17 and 35. This gave rise to transport convergence and divergence zones, respectively. The transport convergence region was consistent with the location of the ETM in previous studies. Transports due to dispersive and differential advection moved upstream from the estuary mouth, whereas the others went seaward.

In high-river flows, the longitudinal profiles of the LAPE transport were inverted. The opposite was true for tidal pumping, longitudinal dispersion and differential advection of the deviation in regard to the mean terms, which became considerably greater. The convergence and divergence zones of the nontidal mean advective LAPE transport disappeared as well as those of the Stokes-induced, and differential advection transport because of the mean terms. Tidal pumping exported the LAPE through the estuary mouth and was thus balanced with the transport associated with the longitudinal dispersion and differential advection because of the deviation terms that import it.

During the postriverflood recovery, all transport processes intensified except for the Stokes-induced transport. In the vicinity of the tidal river reach, the magnitude of all LAPE transports was no longer significant. The differential advection and longitudinal dispersion terms moved upstream in the estuary mouth, which suggests that they are the terms that recover the LAPE of the estuary. The time variability in these conditions was considerable in the LAPE transport because of the longitudinal dispersion (±19.49% of the mean value). The greater difference between neap and spring tides increased the landward transport of tidal pumping in the middle stretch of the estuary. In addition to this, there was a significant increase in the nontidal mean LAPE transport. In these conditions, the longitudinal gradient of LAPE transport was amplified, which caused the convergence and divergence zones to reappear.

## Acknowledgements

This research was partially funded by the Campus de Excelencia Internacional del Mar (Cei-MAR) and the Spanish Ministry of Economy and Competitiveness, Project PIRATES (CTM2017-89531-R). It was also supported by AQUACLEW. Project AQUACLEW is part of ERA4CS, an ERA- NET initiated by JPI Climate, and funded by FORMAS (SE), DLR (DE), BMWFW (AT), IFD (DK), MINECO (ES), ANR (FR) with co-funding by the European Commission. We would like to thank two anonymous reviewers for the thoughtful contribution that has signicantly improved the quality of the paper. Datasets for this research are available in http://doi.org/10.5281/zenodo.3459610. This study is a tribute to the memory of Richard W. Garvine, whose research was the inspiration for our work.

## Appendix A: Derivation of the LAPE Equation

**D**(

*x*,

*t*) in one reference position

*x*and another

*x*

_{0}and substituting the values in the following expression:

**D**(

*x*,

*t*) is defined by equation 5, (repeated below to clarify the derivation):

**S**includes the density variation due to heat transfer by the surface layer and the resuspension of sediment from the estuary bottom as

*α*

_{w}is the water expansion coefficient,

*c*

_{p}is the heat capacity of the water,

*ε*is the light absorption along the vertical profile,

*w*

_{s}is the sedimentation velocity of the particulated material, assumed to be constant for a given grain size, and

*C*is the suspended solids concentration. The parameterization of heat transfer follows the Lambert-Beer Law where

*ε*is the attenuation coefficient (Ruiz et al., 2013). In shallow basins, a significant portion of solar radiation can reach the bottom where it is transformed into heat (Wiles et al., 2006). By calculating

*t*

_{G}of equation 8 is obtained.

The processes related to transversal circulation are negligible in narrow low-friction channels (Valle-Levinson, 2008) and thus dependent on *x*, *z*, and time *t*.

*w*

_{η}=

*∂*

_{t}

*η*+

*u*

_{η}·

*∂*

_{x}

*η*, for a fixed bed,

*w*

_{−h}=0:

_{η}and

_{−h}indicate that the terms are evaluated in

*z*=

*η*and

*z*=−

*h*. Inserting equations 7 and A5 in the equation A1,

A general variable is composed of the sum of a depth-averaged value and its deviation. When this variable is applied in the equation, equation 8 is obtained. As in Burchard and Hofmeister (2008) and De Boer et al. (2008), no hypothesis has been formulated regarding the form of the longitudinal velocity profile.