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Coastal wastewater-discharged effluents contain a mixture of pollutants with decay rates that vary with water depth. Analytical models using a two-dimensional advection-diffusion equation are presented to study the effects of a cross-stream sudden depth change and decay on mixing and dispersing steady discharge of effluents through a sea outfall. The solutions are illustrated graphically by plotting contours of concentration, resembling snapshots of discharged effluent plumes in the far-field. Different shapes of effluent plumes are observed due to the variability of length of the step seabed, and the concentration at the step seabed is formulated to measure how much has discharged effluents dispersed into or out of the shallow coastal waters.

Steady discharge of wastewater effluents through marine outfall systems into the sea includes (treated) municipal wastewaters [

One factor affecting the dilution and spreading of wastewater effluent discharge in coastal waters is the seabed depth profiles [

Modeling studies of the effects of a step increase in depth and decay that decreases or increases with water depth in dispersing steady coastal discharged effluents from a sea outfall in the far-field is investigated using a two-dimensional advection-diffusion equation with a point source (see for example [

Some seabed depth profiles are extremely flat such that the variations in water depth become insignificant. Therefore, as a reference, we introduce first a highly simplified model of the flat seabed with a constant depth, and for simplicity, the other complexities, such as tidal motions, density and temperature, are ignored. The shoreline, which in this case, is a continuation of the rocky sea cliffs, is assumed to be straight, and the effluent is discharged at a steady rate Q from a point at ( x = 0 , y = α h 0 ) , where h = h 0 is an arbitrary reference depth. The (drift) longshore current with speed U 0 is assumed to be in the x-direction at all times. The decay rate μ 0 with a typical value up to 0.5 day^{−1} is used to represent decay of faecal in recreational coastal waters [

The two-dimensional advection-diffusion equation for the concentration c ( x , y ) , incorporating a first-order decay for the non-conservative discharged effluent, is given by

h 0 μ 0 c + h 0 U 0 ∂ c ∂ x − h 0 D 0 ∂ 2 c ∂ y 2 = Q δ ( x ) [ δ ( y − α h 0 ) + δ ( y + α h 0 ) ] , (1)

where the Dirac delta function δ ( * ) is used to represent the position of a source at ( x = 0 , y = α h 0 ) , and an imaginary source at ( x = 0 , y = − α h 0 ) has been added to satisfy the no-slip boundary condition at y = 0 . Note that, the first decay term can be eliminated from Equation (1) by rewriting c = c * exp ( − μ 0 x / U 0 ) .

For the graphical representation of solutions, we define dimensionless quantities

y = Y h 0 , x = X h 0 and c * ( x , y ) = C * ( X , Y ) Q / U 0 h 0 2 ,

Equation (1) reduces to

∂ C * ∂ X − 1 λ ∂ 2 C * ∂ Y 2 = δ ( X ) [ δ ( Y − α ) + δ ( Y + α ) ]

and the solution for X ≥ 0 is given by

C = C * exp ( − γ X ) = λ 4 π X [ exp { − λ ( Y − α ) 2 4 X } + exp { − λ ( Y + α ) 2 4 X } ] exp ( − γ X ) , (2)

where the parameter λ = U 0 h 0 / D 0 represents discharged effluent plumes elongation in the x-direction, and γ = μ 0 h 0 / U 0 represents the loss rate of discharged effluents.

To investigate the uncertainty and variability of sea conditions in these parameters,

The contours of concentration Equation (2) are plotted in

800 m respectively for h 0 in the range of 5 - 10 m. We observed that, due to loss of discharged effluents, the contours ( γ > 0 ) are smaller than that of no decay γ = 0 , plotted in black. As shown in the right column when λ = 0.2 , the discharged effluent plumes are more elongated (in the flow direction).

We observed that from

A plot of effluent flux can be used to illustrate how a discharged effluent plume is dispersing and spreading out from a point source by drawing an arrow at each grid point ( x , y ) , where the direction of the arrow indicates the direction of motion of the discharged effluent, and the length of the arrow is proportional to the magnitude of the flux at that point. Since longitudinal dispersion has been neglected, the flux in the x-direction is due to advection, ( Q / h 0 ) F x = h 0 U 0 c ; and the flux in the y-direction is due to dispersion, ( Q / h 0 ) F y = − h 0 D 0 ∂ c / ∂ y . From Equation (2), the dimensionless form of the effluent flux are

F x = λ 4 π X [ exp ( − λ ( Y − α ) 2 4 X ) + exp ( − λ ( Y + α ) 2 4 X ) ] exp ( − γ X )

and

F y = λ 4 π X [ ( Y − α ) 2 X exp ( − λ ( Y − α ) 2 4 X ) + ( Y + α ) 2 X exp ( − λ ( Y + α ) 2 4 X ) ] exp ( − γ X ) .

Due to diffusion process,

plume spreads towards the shoreline, and reaches the shoreline at a relatively short downstream distance, then due to the no-slip condition, the plume continues to move in direction of the flow. By moving the point source further away from the shoreline at α = 20 (

Substituting Y = 0 in Equation (2), the concentration at the shoreline

C ( X , 0 ) = λ π X exp ( − λ α 2 4 X − γ X ) . (3)

As shown in

By differentiating, it is straightforward to obtain the maximum value

C max = 1 + 1 + 4 λ γ α 2 π α 2 exp ( − 1 + 4 λ γ α 2 2 ) ,

which occurs at X max = λ α 2 / ( 1 + 1 + 4 λ γ α 2 ) . For no decay γ = 0 , C max = 2 / π e α 2 and X max = λ α 2 / 2 . This suggests that, since C max is inversely proportional to α , the concentration build-up close to the shoreline can be avoided by building a long sea outfall. The maximum concentration is about 0.0242 for a short point source α = 20 , and for large values of α , it decreases to 0.006 for a long point source α = 80 . Similarly, due to loss of discharged effluents at a slightly higher rate of γ = 0.0003 , C max decreases slightly to 0.024 for α = 20 ; and decreases significantly to 0.0053 for α = 80 .

For a long point source at α = 80 , due to loss of discharged effluents, the position of the maximum concentration is smaller than that of no decay γ = 0 . As the decay rate γ increases, X max decreases from 480 for γ = 0 to about 389

for γ = 0.0003 . Thus, we concluded that the induced effect of the no-slip condition at the shoreline y = 0 can be ignored for discharging effluents from a long point source at α = 80 .

The sandbags landfill is one of the popular methods to restore and protect the beach erosion from constant wave attack. Due to the pillow shape of sandbags, the beach (face) makeover of piling sandbags creates a profile of steps seabed along the shoreline. In the oceanography textbooks, going further seaward from the shore, the first submerged region is termed continental shelf. The seaward limits of the shelf are determined by the distinct change in depths between the shelf and its adjacent continental slope. Thus, a seabed depth profile is typically depicted as a shallow depth flat seabed coming in contact with a deeper one. As a first attempt to study the effect of variations in water depth, we consider, as shown in

h ( y ) = { h 0 , 0 ≤ y < l h 0 h 1 , y > l h 0 ,

where the sudden cross-stream water depth change occurs at a discontinuity line y = l h 0 ( l > 0 ) and r = h 1 / h 0 > 1 is the ratio of water depths ( h 1 > h 0 ). Note that if r = 1 (and l = 0 ), there is no depth change, and a seabed depth profile is exactly that of a simple flat seabed of depth h = h 0 (and h = h 1 ).

For a turbulent shallow-water flow over a smooth bed, the variations in the y-direction of current U 1 and dispersivity D 1 are assumed as the power functions only of water depth h [

discharged effluents that varies with depth can be illustrated according to the values of σ .

If σ = − 1 / 2 , then μ 1 = μ 0 , and the decay rate is independent of water depth. Radioactive decay or consumption by bacteria (at a rate unaffected by sunlight or turbidity) is examples of constant temporal decay. For σ < − 1 / 2 , the decay rate μ 1 decreases with depth. For example, if σ = − 1 then μ 1 = μ 0 r − 1 / 2 represents the decay rate that includes consumption by bacteria which are killed by sunlight at the surface, and air-water exchange (evaporation) at the surface [

On writing the concentration, in dimensionless form, as

C ( X , Y ) = { C 0 = C 0 * exp ( − γ X ) , 0 ≤ Y < l C 1 = C 1 * exp ( − γ r σ X ) , Y > l ,

the solution of a two-dimensional advection-diffusion equation for discharging effluents from a point source at ( X = 0 , Y = α ) on the step seabed can be obtained using the method of image [

However, as there can be no discontinuities in either the concentration or its gradient across the line Y = l , the additional matching conditions are

C 0 = C 1 and ∂ C 0 ∂ Y = r 5 / 2 ∂ C 1 ∂ Y .

The discontinuity line serves as a “gate” for discharging effluent plumes to cross over and spread into or out of the shallow nearshore region. Thus, to separate the effect of a sudden depth change (at the line Y = l ) to that of the no-slip condition at the shoreline (at Y = 0 ), we consider two values of l : a short step seabed with l ≤ 15 (to represent a sandbags landfill depth profile), where the presence of shoreline induces the effect of the no-slip condition in dispersing effluent discharged plumes; and a long step seabed with l ≥ 70 (to represent a continental shelf depth profile), where a discharged effluent plume most likely does not feel the presence of shoreline.

Older sea outfalls are typically short and some discharges are made through open channel at the shoreline, where higher build-up of effluents is expected in coastal waters close to and along the shoreline. For a relatively short step seabed with l ≤ 15 , we consider a source at ( X = 0 , Y = α ) where α > l , that is sufficiently close to the discontinuity line Y = l . The concentration in the shallow region C 0 ( X , Y ) = C 0 * exp ( − γ X ) is obtained due to a virtual source at ( X = 0 , Y = β 1 ) discharging at a rate b 1 , and satisfies the no-slip boundary condition at Y = 0 . Thus, in dimensionless form, the advection-diffusion equation for C 0 * ( X , Y ) is given by

∂ C 0 * ∂ X − 1 λ ∂ 2 C 0 * ∂ Y 2 = b 1 δ ( X ) [ δ ( Y − β 1 ) + δ ( Y + β 1 ) ]

and the solution for X ≥ 0 is

C 0 = b 1 λ 4 π X [ exp { − λ ( Y − β 1 ) 2 4 X } + exp { − λ ( Y + β 1 ) 2 4 X } ] exp ( − γ X ) . (4)

The concentration in the deeper region C 1 ( X , Y ) = C 1 * exp ( − γ r σ X ) is obtained due to a point source at ( X = 0 , Y = α ) and an imaginary source at ( X = 0 , Y = 2 l − α ) discharging at a different rate a 1 . Thus, the advection-diffusion equation for C 1 * ( X , Y ) is given by

∂ C 1 * ∂ X − r λ ∂ 2 C 1 * ∂ Y 2 = 1 r 3 / 2 δ ( Y ) [ δ ( Y − α ) + a 1 δ ( Y − 2 l + α ) ] ,

and the solution for X ≥ 0 is

C 1 = 1 2 r 2 λ π X [ exp { − λ ( Y − α ) 2 4 r X } + a 1 exp { − λ ( Y − 2 l + α ) 2 4 r X } ] exp ( − γ r σ X ) . (5)

The matching conditions at Y = l are required for calculating a 1 , b 1 and β 1 , and thus, we obtain

[ r 2 + 1 + { r 2 + β 1 + l β 1 − l } exp ( − λ l β 1 X ) ] a 1 = [ r 2 − 1 + { r 2 − β 1 + l β 1 − l } exp ( − λ l β 1 X ) ]

and

b 1 exp ( − γ X ) [ r 2 + 1 + { r 2 + β 1 + l β 1 − l } exp ( − λ l β 1 X ) ] = 2 exp ( − γ r σ X ) .

We noted numerically that for X < 5.6 , exp ( − λ l β 1 / X ) < 0.001 , and thus,

a 1 = r 2 − 1 r 2 + 1 and b 1 exp ( − γ X ) = 2 r 2 + 1 exp ( − γ r σ X ) .

Further, for no decay γ = 0 (and σ = 0 ), a 1 + b 1 = 1 . From the previous section, for r > 1 , l < β 1 < α , and a 1 increases from 0.6 for r = 2 to 0.8 for r = 3 .

To investigate the effect of decay that increases with depth, contours of concentration for discharged effluents from a point source at α = 18 on a short step seabed with l = 15 are plotted in

The effect of a step increase in depth on dispersing discharged effluent plumes with γ = 0.0003 and σ = 1 / 2 is shown in

Again, a plot of effluent flux is used to investigate the movement of discharged effluent plumes passes through the discontinuity line Y = l . In dimensionless form, the effluent flux on either side of the discontinuity line Y = l are in the shallow region 0 ≤ Y < l , similar to that of the flat seabed, but with β 1 instead of α :

F x = b 1 λ 4 π X [ exp ( − λ ( Y − β 1 ) 2 4 X ) + exp ( − λ ( Y + β 1 ) 2 4 X ) ] exp ( − γ X )

and

F y = b 1 λ 4 π X [ ( Y − β 1 ) 2 X exp ( − λ ( Y − β 1 ) 2 4 X ) + ( Y + β 1 ) 2 X exp ( − λ ( Y + β 1 ) 2 4 X ) ] exp ( − γ X ) .

Within the deeper region Y > l , the flux in the x-direction is given by ( Q / h 0 ) F 1 x = h 1 U 1 c and in the y-direction is ( Q / h 0 ) F 1 y = − h 1 D 1 ∂ c / ∂ y . In the dimensionless form,

F 1 x = λ 4 π r X [ exp ( − λ ( Y − α ) 2 4 r X ) + a 1 exp ( − λ ( Y − 2 l + α ) 2 4 r X ) ] exp ( − γ X )

and

F 1 y = λ 4 π r X [ ( Y − α ) 2 X exp ( − λ ( Y − α ) 2 4 r X ) + a 1 ( Y − 2 l + α ) 2 X exp ( − λ ( Y − 2 l + α ) 2 4 r X ) ] exp ( − γ X ) .

As shown in

The concentration levels at the discontinuity line Y = l play an important role in determining the portions of discharged effluent plumes that crossing over and dispersing in the shallow region. By putting Y = l in Equations (4) and (5), we obtain the concentration at the discontinuity line

C 0 ( X , l ) = C 1 ( X , l ) = 1 + a 1 2 r 2 λ π X exp { − λ ( α − l ) 2 4 r X } exp ( − γ r σ X ) . (6)

We note from

a very short downstream distance, and thus, a 1 = ( r 2 − 1 ) / ( r 2 + 1 ) . By differentiating, it has a maximum value

C 1 max = 1 r 2 + 1 1 + 1 + 4 λ γ r σ − 1 ( α − l ) 2 π ( α − l ) 2 / r exp ( − 1 + 4 λ γ r σ − 1 ( α − l ) 2 2 ) ,

which occurs at X 1 max = λ ( α − l ) 2 / r [ 1 + 1 + 4 λ γ r σ − 1 ( α − l ) 2 ] . Note that the maximum value only depend on algebraic power of ( α − l ) . For no decay γ = 0 , C 1 max = 2 r / π e ( α − l ) 2 / ( r 2 + 1 ) and X 1 max = λ ( α − l ) 2 / 2 r , which shows that C 1 max is inversely proportional to r 3 / 2 , and the position of maximum concentration X 1 max is inversely proportional to r.

Due to small portions of discharged effluent plumes entering and dispersing in the shallow region, the maximum concentration that passes through the discontinuity line decreases from 0.0608 for r = 1.5 to 0.0279 for r = 3 , a decrease of more than 54%. In comparison with discharging effluents from a point source on the flat seabed, the maximum concentration at the shoreline is greatly reduced as a result of a step increase in depth at l = 15 . Finally, we note that as l → 0 , the concentration at the shoreline is

C 0 ( X , 0 ) = 1 1 + r 2 λ π X exp ( − λ α 2 4 r X − γ r σ X ) , (7)

which is 1 / ( 1 + r 2 ) smaller than that of the flat seabed (see Equation (3)), and it has a maximum value given by

C 1 max = 1 1 + r 2 1 + 1 + 4 λ γ r σ − 1 α 2 π α 2 / r exp ( − 1 + 4 λ γ r σ − 1 α 2 2 ) .

Sea outfalls should be sufficiently long to take the full benefit of stronger current and more depth to dilute discharged effluents. For simplicity, we assume that a source at ( X = 0 , Y = α ) is located sufficiently close to the line Y = l , where 70 < α ≤ 80 for discharging effluents in the deeper region and 60 ≤ α < 70 for discharging effluents in the shallow region, or else very small parts of the effluent plumes will be able to cross over the discontinuity line at Y = l . As previously mentioned, no imaginary source is needed to satisfy the boundary condition at Y = 0 .

The solutions of the advection-diffusion equations are similar to that of discharging effluents on a short step seabed, but with different discharge rates of a 2 and b 2 instead of a 1 and b 1 , respectively. For discharging effluents from a point source at ( X = 0 , Y = α ) where α > l , the concentration C 0 ( X , Y ) = C 0 * exp ( − γ X ) is obtained due to a virtual source at ( X = 0 , Y = β 1 ) discharging at a rate b 1 (see Equation (4)). Thus, in dimensionless form, the solution in the shallow region is

C 0 = b 2 λ 4 π X exp { − λ ( Y − β 1 ) 2 4 X } exp ( − γ X ) ,

and in the deeper region (see Equation (5)) is

C 1 = 1 2 r 2 λ π X [ exp { − λ ( Y − α ) 2 4 r X } + a 2 exp { − λ ( Y − 2 l + α ) 2 4 r X } ] exp ( − γ r σ X ) .

The matching conditions at Y = l are required for calculating a 2 , b 2 and β 1 , and thus, we obtain

β 1 = l + α − l r , a 2 = r 2 − 1 r 2 + 1 and b 2 exp ( − γ X ) = 2 r 2 + 1 exp ( − γ r σ X ) ,

and for example, the solution in the shallow region can be rewritten as

C 0 = 1 r 2 + 1 λ π X exp { − λ ( Y − β 1 ) 2 4 X } exp ( − γ r σ X ) ,

which can be interpreted as the portions of discharged effluent plumes that cross over the line Y = l and dispersing within the shallow nearshore region. We note that for r > 1 , a 2 < 1 , b 2 > 0 and l < β 1 < α . It is also easy to verify that, for no decay γ = 0 (and σ = 0 ), then a 2 + b 2 = 1 ; and if there is no depth change r = 1 (and l = 0 ), then β 1 = α , a 2 = 0 and b 2 = 1 .

In comparison with discharging effluents on the flat seabed (right of

The effect of variations in water depth on dispersing the discharged effluent plumes is shown in

The concentration levels at the discontinuity line Y = l play an important role in determining the portions of discharged effluent plumes that crossing over and dispersing in the shallow region. By putting Y = l , we obtain the concentration at the discontinuity line

C 0 ( X , l ) = C 1 ( X , l ) = 1 r 2 + 1 λ π X exp { − λ ( α − l ) 2 4 r X } exp ( − γ r σ X ) ,

which is exactly that of Equation (6), and it has a maximum value C 1 max that occurs at X 1 max .

The concentration at the discontinuity line for discharging effluents from a point source at α = 80 on a long step seabed with l = 70 is plotted in

For discharging effluents from a point source at ( X = 0 , Y = α ) , where α < l , the concentration C 1 ( X , Y ) is obtained due to a virtual point source at ( X = 0 , Y = β 2 ) discharging at a rate b 3 . Thus, in dimensionless form, the advection-diffusion equation for C 1 * ( X , Y ) is given by

∂ C 1 * ∂ X − r λ ∂ 2 C 1 * ∂ Y 2 = b 3 r 3 / 2 δ ( X ) δ ( Y − β 2 ) ,

and the solution for X ≥ 0 is

C 1 = b 3 2 r 2 λ π X exp { − λ ( Y − β 2 ) 2 4 r X } exp ( − γ r σ X ) . (8)

The concentration C 0 ( X , Y ) is obtained due to a point source at ( X = 0 , Y = α ) and an imaginary source at ( X = 0 , Y = 2 l − α ) discharging at a different rate a 3 . The advection-diffusion equation for C 0 * ( X , Y ) is given by

∂ C 0 * ∂ X − 1 λ ∂ 2 C 0 * ∂ Y 2 = δ ( X ) [ δ ( Y − α ) + a 3 δ ( Y − 2 l + α ) ] ,

and the solution for X ≥ 0 is

C 0 = λ 4 π X [ exp { − λ ( Y − α ) 2 4 X } + a 3 exp { − λ ( Y − 2 l + α ) 2 4 X } ] exp ( − γ X ) .

From the matching conditions at Y = l we obtain

β 2 = l − r ( l − α ) , a 3 = 1 − r 2 1 + r 2 and b 3 r 2 exp ( − γ r σ X ) = 2 1 + r 2 exp ( − γ X ) ,

and thus, Equation (8) can be rewritten as

C 1 = 1 1 + r 2 λ π X exp { − λ ( Y − β 2 ) 2 4 r X } exp ( − γ X ) ,

which is independent of σ , i.e. there are no effects of variability of decay with depth, and it can be interpreted as the portions of discharged effluent plumes that escaping out to the deeper offshore region. We noted that, for no decay γ = 0 , then a 3 + b 3 = 1 ; and if there is no depth change r = 1 (and l = 0 ), then β 2 = α , a 3 = 0 and b 3 = 1 . Also for r > 1 , a 3 < 0 , b 3 < 1 and α < β 2 < l .

As plotted in

Next, by putting Y = l , we obtain the concentration at the discontinuity line

C 0 ( X , l ) = C 1 ( X , l ) = 1 r 2 + 1 λ π X [ exp { − λ ( l − α ) 2 4 X } ] exp ( − γ X ) .

By differentiating, the maximum value is

C 2 max = 1 r 2 + 1 1 + 1 + 4 λ γ ( l − α ) 2 π ( l − α ) 2 exp ( − 1 + 4 λ γ ( l − α ) 2 2 ) ,

which occurs at X 2 max = λ ( l − α ) 2 / [ 1 + 1 + 4 λ γ ( l − α ) 2 ] . Again, the maximum value only depend on algebraic power of ( l − α ) . For no decay γ = 0 , C 2 max = 2 / π e ( l − α ) 2 / ( r 2 + 1 ) and X 2 max = λ ( l − α ) 2 / 2 . This shows also that

C 2 max is inversely and quadratically proportional to r; however, the position of maximum concentration X 2 max is independent of r, as shown in

As shown in

C 0 ( X , 0 ) = 1 1 + r 2 λ π X exp ( − λ α 2 4 X − γ X ) .

Mathematical models are presented using a two-dimensional advection-diffusion equation with a point source to study the effects of a sudden water depth change and decay in mixing and dispersion of steady discharged effluents through a sea outfall in coastal waters. Due to the small nature of decay rates, for shorter sea outfalls, the results show that it is difficult to overcome the slow mixing processes in the nearshore region due to the no-slip condition at the shoreline. A large increase in water depth for more than three times that of the shallow region is needed to suppress the effect of the no-slip condition.

For steady discharges from a point source in the deeper region, the results show that in comparison to that of a flat seabed, the maximum concentration at the shoreline can be greatly reduced by the presence of a step increase in water depth. However, for steady discharges in the shallow region, the reduction is inversely proportional to the quadratic of depth ratio.

The model extensions to account for multiple point source discharging effluents on a long step seabed have been published [

The authors declare no conflicts of interest regarding the publication of this paper.

Al-Muqbali, A.A. and Purnama, A. (2022) The Effect of a Step Increase in Depth and Decay upon Dispersion of Coastal Effluent Discharges. Applied Mathematics, 13, 37-55. https://doi.org/10.4236/am.2022.131004