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ARTICLE IN PRESSG ModelQUE-1761; No. of Pages 12

Aquacultural Engineering xxx (2014) xxx–xxx

Contents lists available at ScienceDirect

Aquacultural Engineering

jo ur nal home p ag e: www.elsev ier .com/ locate /aqua-onl ine

arrying capacity for finfish aquaculture. Part I—Near-fieldemi-analytic solutions

ohn F. Middletona,b,∗, Mark Doubell a

Aquatic Sciences, South Australian Research and Development Institute, 2 Hamra Ave., West Beach, South Australia 5024, AustraliaSchool of Mathematical Sciences, University of Adelaide, Adelaide, South Australia 5000, Australia

r t i c l e i n f o

rticle history:eceived 13 January 2014ccepted 11 July 2014

eywords:dvection–diffusion equationlushing time scalesage nutrient concentrationsinfish carrying capacity

a b s t r a c t

The ecosystem carrying capacity for aquaculture cage farming in South Australia is based on guidelinesthat the maximum feed rates (and farmed fish biomass) be determined such that the concentration cof a given dissolved nutrient does not exceed a prescribed value (say cP). The problem then is one ofrelating the nutrient flux F, due to feeding, to the tracer concentration c. To this end the evolution ofconcentration is modelled using the depth-averaged advection–diffusion equation for a constant sourceflux F over a finite area cage (or lease) and for both constant and time dependent (tidal) velocities. Thedivergence theorem is applied to this equation to obtain a new scale estimate of the relation betweenthe flux F and the maximum concentration cmax of a nutrient in the cage region: cmax ≈ F·T*, where T* is atime scale of cage “flushing” that involves both advection and diffusion. The maximum allowed nutrientflux F (and carrying capacity of fish biomass) can then be simply estimated from: F ≈ cP/T*. New semi-

analytic solutions of the advection–diffusion equation for a finite (cage) source are then derived to explorethe physics of concentration evolution for constant and tidally varying currents, and to show that theestimate cmax ≈ F·T* is surprisingly robust and generally within 40% of the exact values for a wide set ofadvective/diffusive parameters. The results generally should find application in other finite source fluxproblems in the coastal oceans including desalination plants and waste water outfalls.

Crown Copyright © 2014 Published by Elsevier B.V. All rights reserved.

. Introduction

The world’s coastal oceans are subject to increased exploita-ion through developments such as finfish aquaculture and oceanutfalls including those associated with waste water, industrialffluent and desalination. In Spencer Gulf (Fig. 1), cage aquaculturef Southern Bluefin Tuna (SBT) and Yellowtail Kingfish (YTK) repre-ent a farm gate value of $228M for 2010/2011 (Econsearch, 2012),ith plans for significant expansion in the coming years. Withinefined aquaculture zones (Fig. 1), finfish aquaculture is under-aken through smaller leases that typically consist of 6 or so cagesach of 50 m diameter. The leases themselves typically occupy an

Please cite this article in press as: Middleton, J.F., Doubell, M., Carryingsolutions. Aquacult. Eng. (2014), http://dx.doi.org/10.1016/j.aquaeng.

rea of 600 m × 600 m. Along with most Australian capital cities aajor desalination plant has also been constructed on the Adelaide

oast.

∗ Corresponding author at: Aquatic Sciences, South Australian Research andevelopment Institute, 2 Hamra Ave., West Beach, South Australia 5024, Australia.el.: +61 882075449; fax: +61 882075481.

E-mail address: john.middleton@sa.gov.au (J.F. Middleton).

ttp://dx.doi.org/10.1016/j.aquaeng.2014.07.005144-8609/Crown Copyright © 2014 Published by Elsevier B.V. All rights reserved.

In Australia and elsewhere, the management of these “outfalls”often invokes the concept of ecosystem carrying capacity. For fin-fish aquaculture, carry capacity of farmed fish biomass is guidedby the concentration of nutrients discharged and limited to beless than a government prescribed maximum concentration cP,(ANZECC/ARMCANZ, 2000) or an environmental quality standard(e.g. Stigebrandt, 2011). This in turn limits the feeding and stock-ing rates, (tonnes of fish) and subsequent nutrient fluxes (F) into theaquaculture regions to maximum values of “carrying capacity”. Theprescribed maximum concentrations are themselves determinedso as to maintain ecosystem health. Gecek and Legovic (2010) referto this as ecosystem carrying capacity.

Within temperate estuarine and coastal systems nitrogen is gen-erally considered the primary cause of eutrophication (Howarthand Marino, 2006). In Spencer Gulf, at least 80% of the wasteproducts associated with the supplementary feeding of finfish arereleased as dissolved inorganic nitrogen into the water column

capacity for finfish aquaculture. Part I—Near-field semi-analytic2014.07.005

(Fernandes et al., 2007). However, the relation between nutrientloading and its impacts on the pelagic ecosystem in the gulf remainslargely unknown. For this reason, nutrient concentrations recom-mended by the Australian and New Zealand guidelines for water

ARTICLE ING ModelAQUE-1761; No. of Pages 12

2 J.F. Middleton, M. Doubell / Aquacultura

Fig. 1. Map of Spencer Gulf showing the location of; the 6 aquaculture zones (brownshaded regions), 2010/2011 field survey sites (black crosses), tuna (red triangles) andYellowtail Kingfish finfish (blue triangles) aquaculture leases, waste water treat-ment plants (pink circles) and the Onesteel steel works (blue square). The location(S.W. gulf) of the 2005 R4a mooring is indicated by the black circle. The locationoi

qaitwt(

toRmoctbeair

clgtmrts

f aquaculture leases is plotted for the 2010/11 period. The 10 m, 20 m and 40 msobaths are plotted.

uality (ANZECC/ARMCANZ, 2000) are typically adopted to providen indicator value for assessing the state of a water body and todentify potential eutrophication related risks on marine ecosys-em health. ANZECC/ARMCANZ (2000) trigger values for marineaters in south-central Australia sets dissolved nutrient concen-

rations at 50 �g N L−1 for oxides of nitrogen (NOx) and ammoniumNH4

+) and 1 �g Chl a L−1 for phytoplankton.Concentration values of nutrients will vary in time and space

hrough advection and diffusion and also due to the distributionf cages, feed-rates and timing. Since 2005, the South Australianesearch and Development Institute (SARDI) have provided infor-ation to government on carrying capacity feed rates at the scale

f the government prescribed zones (Fig. 1). These zones canonsist of 10–20 lease regions. The carrying capacity estimateshen provide a basis for the estimation of feed rates and fishiomass through the use of a Food Conversion Ratio (Collingst al., 2007). The S.A. Government uses this information to licensequaculture lease holders on fish biomass stocking rates. Thendustry in turn reports to government on monthly average feedates.

One model used by SARDI to relate the carrying capacity flux F toP has been a simple flushing model based on a mean speed U andength of the zone D, whereby the nutrient flux (and feed rates) areiven by the carrying capacity limit F = cP/Ta where Ta = D/U is theime taken to flush the zone (Collings et al., 2007). Limited current

Please cite this article in press as: Middleton, J.F., Doubell, M., Carryingsolutions. Aquacult. Eng. (2014), http://dx.doi.org/10.1016/j.aquaeng.

etre data (2–8 weeks) has been used to estimate U for variousegions in Spencer Gulf. This simple approach was necessitated byhe lack of data and more realistic models. The information at thecale of the zone (∼10 km) says little about concentrations at the

PRESSl Engineering xxx (2014) xxx–xxx

scale of the cage, typically 50 m. The nutrient concentrations at thecage level are important since if very large, they may impact on thedownstream health of the ecosystem through algal and bacterialgrowth and biological cycling.

The approach here is limited to predicting the maximum nutri-ent concentrations that may arise from finfish aquaculture. In theanalysis below it is assumed that the nutrients are conserved andthat biological cycling and uptake is relatively small. In Section 5this assumption will be shown to be reasonable over the spatialscales of the lease and time scales of flushing (1–10 h).

An additional application of the results below is the determina-tion of the “carrying capacity” of salt discharged into the ocean bydesalination plants. The carrying capacity for the S.A. Port Stanvacplant has been set by the S.A. Environmental Protection Authoritythrough a condition that the average salinity (over a 24 h period)and at any point 100 m from the diffusers should not exceed theambient value by more than 1.3 ppt. In other studies of desalinationplant discharges, carrying capacity has not been defined althoughthe focus of concentration prediction has been on adjacent beachesand/or far from the diffuser sites (Purnama and Al-Barwani, 2006;Al-Barwani and Purnama, 2008).

In this study, concentrations at the scale of the cage input sourcewill be considered and the focus will be on applications to aqua-culture and carrying capacity. To this end, semi-analytic integralsolutions for the depth-averaged advection–diffusion equationswill be determined for the case of a finite source (the cage) withtime-dependent tidal velocities.

There is an enormous literature on solutions to the twodimensional advection–diffusion equation in the context of fluiddynamics (Crank, 1975), coastal engineering (Fischer et al.,1979), desalination and aquaculture impacts (the Elsevier journals“Desalination” and “Aquaculture”) and the oceans and atmosphere(Csanady, 1973). An extensive review of these texts and the webwas made and it was found that analytic solutions to the problemof a flux source with time-dependent velocities do exist but only foran infinite point source (Kay, 1990, 1997; Mustafa et al., 1996). Ana-lytic solutions for a finite source were not found although Csanady(1973) points out how such solutions might be obtained from thosefor a point source.

In the following, a hierarchy of solutions to the one and thentwo-dimensional advection–diffusion equations are presentedwith a focus on determining carrying capacity for idealized, con-stant nutrient fluxes that are uniformly applied over some region.This includes regions at the scale of the cage (50 m), lease (600 m)or zone (10 km). We assume that the nutrient tracer is conserved,the ocean depth is constant and the currents depend only on time(including tidal variations). Coastal boundaries will not be consid-ered but their effects are briefly outlined in Section 5

In Section 2, the divergence theorem is applied to theadvection–diffusion equation to derive a new and simple scale esti-mate linking the input cage flux F to the maximum prescribedconcentration cP and to the time scales of both advection and dif-fusion.

A brief discussion is also made on the estimates of the diffusiveand advective time scales that are needed to relate F and cmax forthe Spencer Gulf region. The discussion includes the effects of tidalresonance and the “dodge” tide. The dodge tide is an extreme formof the neap tide where tidal current speeds vanish every 15 days orso (see below).

In Section 4, the numerical solutions of the integral equationsfor the concentration c(x, y, t) are obtained and for a wide varietyof advective/diffusive regimes so as to build intuition regarding thephysics and also to determine the range of applicability of the scaleestimates found in Section 2.

capacity for finfish aquaculture. Part I—Near-field semi-analytic2014.07.005

In Section 5, a summary and discussion is presented and anoutline of related and future research given.

IN PRESSG ModelA

ultural Engineering xxx (2014) xxx–xxx 3

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2.05

Fig. 2. Normalized (1-D) nutrient concentration c(x, t)/FTa as a function of distancefrom the source that lies between x = −D/2 and D/2 – the vertical lines. Solutions are

p =Td

(2.9)

ARTICLEQUE-1761; No. of Pages 12

J.F. Middleton, M. Doubell / Aquac

. Maximum concentration estimates, carrying capacitynd time scales

First consider a source region of nutrients defined by the rect-ngle centred upon the origin:

−D

2< x <

D

2,

−W

2< y <

W

2

}(2.1)

We will focus on results at the scale of the cage (D = W = 50 m)lthough the formalism below applies to any flux region (lease andone). We assume that a constant flux F [kg/(m3 s)] over this regions applied at the surface and after time t = 0. Formally, this flux couldlso include that due to re-suspension (if it could be estimated). Theepth of the ocean h is assumed constant and the flux at the surface

s related to the concentration c′(x, y, z, t) by

=(

Kv

h

)c′

z at z = 0 (2.2)

here Kv is the vertical diffusivity (assumed constant) and sub-cripts of t, x, y and z denote partial derivatives. The units of F areoncentration c′ divided by time. The equation for the evolution ofhe depth-averaged concentration c is then

t + (uc)x + (vc)y = (K1cx)x + (K2cy)y + F(x, y, t) (2.3)

here K1 and K2 are the constant diffusivities in the x- and y-irections respectively. Initial and boundary conditions are that c =0t t = 0 and c → 0 as |x| → ∞. The velocity field u(t) is assumed to bendependent of space but not time.

.1. An advective model for cmax

One model that has been used by SARDI is based on the flushingy a constant current U (Collings et al., 2007). If the length of theource region in the direction of U is D and water depth is h, then for

given input rate of nutrients F [e.g. kilograms of nitrate/(m3 s)],he maximum concentration of nutrient is estimated as

max = FTa (2.4)

here

a = D/U (2.5)

is the time taken to flush the cage length D by the con-tant current U. The estimates (2.4) and (2.5) have been used byARDI to estimate maximum nutrient input F or feed rates for aiven maximum level of allowed nutrient concentration under theNZECC/ARMCANZ (2000) guidelines.

No derivation for (2.4) was given by Collings et al. (2007). How-ver, if a balance exists between the advective flux (Uc)x ∼ Ucmax/Dnd the input flux F in (2.3), then (2.4) follows: diffusive effects aregnored. Exact solutions can be obtained using the method of char-cteristics. After several time scales Ta, the concentration variesinearly from zero at x = −D/2 to cmax at x = D/2. Such a result ishown by the exact solutions to the advection–diffusion Eq. (2.3)btained below and presented in Fig. 2. Moreover, as this state ischieved the concentrations downstream of the cage are asymp-otic to cmax.

.2. An advective-diffusive model for cmax

To extend the advective scale estimate (2.4) to include diffusion, steady state (ct = 0) and a constant mean velocity U are assumed.n average over a volume V = hWD is taken of (2.3) which may beewritten as

Please cite this article in press as: Middleton, J.F., Doubell, M., Carryingsolutions. Aquacult. Eng. (2014), http://dx.doi.org/10.1016/j.aquaeng.

∫∇ · (Uc)dv

Ia

= K

∫∇ · (∇c)dv + FV

Ib

(2.6)

presented at times t equal to (0.05, 1.05, 2.05, 3.05, . . ., 19.05) Ta and for p = 0.02 areprincipally driven by advection. The constant velocity is U = 0.05 m/s and Ta = 1000 s.The input flux is denoted by F and the advective flux by Ucx .

where the integral is over the rectangular volume V and a constantdiffusivity in both the x- and y-directions has been assumed. Usingthe divergence theorem, the volume integrals of the first two termsin (2.6) may be rewritten as integrals over the outer surfaces of thesource. The first term is

Ia =∫

(Uc) · n da

where the integral is over the surface defined by V, n is a unit vec-tor for each surface that is directed out of the volume and U = (U,0).The only non-zero term is then Ia = hW[Uc]x=D/2 = hWUcmax, sincefollowing the advective solutions in Fig. 2, c is assumed to be zeroat x = −D/2.

The diffusive flux term may be written as

Ib =∫

(K∇c) · n da

The only non-zero terms here are Kcy at y = ±W/2 since cx = 0 atx = ±D/2. This means we can (below) replace K with that for the y-direction and that Ib = −2hDK2 cy. The only available scale for cy is−cmax/W so that Ib = −2hDK2 cmax/W. Eq. (2.6) may then be writtenas

hWUcmax = −2hDK2cmax

W+ FV

so that

cmax = FT∗ (2.7)

with

T∗ = Ta

1 + p(2.8)

Ta

capacity for finfish aquaculture. Part I—Near-field semi-analytic2014.07.005

and

Ta = D

Uand Td = D2

2K2(2.10)

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ARTICLEQUE-1761; No. of Pages 12

J.F. Middleton, M. Doubell / Aquac

For convenience we take W = D. The time scale Td = D2/(2K2) rep-esents the time taken for a spot to diffuse over a distance W ands well known in the literature (e.g., Fischer et al., 1979). For smallalues of p, the maximum concentration is determined largely bydvection and given by cmax = F·Ta. For large values of p, Ta Tdnd diffusive processes dominate the flushing. In this case (2.7)ecomes cmax = F·Td and since Td Ta the maximum concentrationsbtained are much smaller than indicated by the advective scaling.imply, the diffusion and advection flush the cage more quicklyhan where (weak) advection acts alone.

The implications for carrying capacity of a source region aremportant since the flux of nutrients F is related to feed rates (andsh biomass) and these can be adjusted (up or down) so as toptimize carrying capacity of a cage (or lease). The allowed fluxan differ by an order of magnitude depending on the relativemportance of advection and diffusion. Further comment is maden Section 2.3.

As far as we can tell, the carrying capacity results (2.7)–(2.10) areew. As will be shown below, these scale results provide reasonablestimates for the exact maximum concentrations obtained fromhe advection–diffusion equation, both with and without strongidal velocities and for a wide range of advective and diffusive timecales.

.3. Estimation of the time scales – Spencer Gulf

Estimation of the advective scale Ta = D/U requires a knowledgef the mean current velocity U. This might be obtained from a cur-ent metre mooring and/or a hydrodynamic model.

In Part II of this study, Middleton et al. (2014) used a validatedydrodynamic model of Spencer Gulf to estimate the time scale Ta

t every 600 m grid cell and as a seasonal and depth average. Therid resolution here is about the scale of the leases for Spencer Gulf.or winter, Ta was found to range from 3 h in shallow water, where

was relatively large (0.05 m/s), to 17 h in deeper water (h > 20 m),here U is smaller (∼0.01 m/s).

In general, estimation of the diffusive time scale Td = D2/(2 K)s more problematic since the eddy diffusion coefficient K is essen-ially a Lagrangian quantity. It could be directly measured by drifterr dye release experiments. However, in many coastal regionsither vertical or horizontal current shear can give rise to enhancedhear dispersion (Fischer et al., 1979) with coefficient values Ks thatan greatly exceed the eddy diffusion values K.

For Spencer Gulf, the largest currents arise from a 3/4 wave res-nance of the semi-diurnal tides (Easton, 1978; Middleton and Bye,007). Observations and the hydrodynamic model of Middletont al. (2014) indicate the r.m.s tidal velocities (U*) to be much largerhan the vector mean speed U and to vary from 0.4 m/s (lower gulf)o 1.0 m/s (mid to upper gulf).

Middleton et al. (2014) adopted a parabolic model for the verti-al shear of horizontal currents and applied the shear dispersionodel from Fischer et al. (1979) to provide seasonal and depth

veraged estimates of KS and Td at each model grid cell. Near theoast the associated shear diffusivity KS was of order 10–20 m2/s,hile in the mid to upper gulf the values were large and of order

0–80 m2/s.The associated diffusive time scale Td was found to be about 10 h

long the coast and in the lower gulf and of order 1 h in the mid topper gulf. The ratio of time scales p = Ta/Td is about one along theoast and in the lower gulf and of order 5–10 in the mid to upper gulfhere diffusive flushing is largest and the diffusive limit pertains.

Please cite this article in press as: Middleton, J.F., Doubell, M., Carryingsolutions. Aquacult. Eng. (2014), http://dx.doi.org/10.1016/j.aquaeng.

hese estimates refer to a region W = D, the size of the lease or gridell (600 m). For regions the size of the cage (50 m) p is 10 timesarger and the diffusive limit applies to both the coastal and deeper

ater regions.

PRESSl Engineering xxx (2014) xxx–xxx

The larger values of p in the near shore regions have importantimplications for carrying capacity. The flux input at the scale of thelease or cage may be written as

F = cP(1 + p)Ta

so that larger feed rates can be maintained where p is largerand for the same prescribed concentration maxima cP. Larger feedrates, fish growth and biomass may lead to an increased financialreturn to the farmer. While this might be offset by increased costsin farming operation, it might be used to better inform industry onoptimal feed rates.

In addition to tidal resonance, a second feature of Spencer Gulf isthe existence of the dodge tide whereby the amplitudes and phasesof the M2 and S2 tides are approximately equal so that their summay be written as

uT (t) = Ud(t) sin (ωt) (2.11)

where Ud (t) = 2UT cos (ωd·t), ω = (ωM + ωS)/2 is a frequency witha period close to 12 h while ωd = (ωM − ωS)/2 is the differencefrequency with a period of about 30 days. The tide is then “semi-diurnal” but with an amplitude that is close to zero every 15 days:the “dodge” tide (Easton, 1978).

For simplicity, the response to a single semi-diurnal tide is con-sidered:

uT (t) = UT sin(ωt) (2.12)

The effect on carrying capacity of the neap (dodge) tides andspring tides that arise in Spencer Gulf will be addressed by consid-ering cases where UT is relatively large (springs) and small (neaps).In Section 3, exact integral solutions for the advection diffusionequations are obtained for such time-dependent tidal velocitiesand constant diffusion coefficients. A more precise analysis for thedodge tide would need to incorporate time dependent shear dis-persion coefficients.

3. Exact integral solutions to the advection–diffusionequations

Exact integral solutions are next obtained for theadvection–diffusion problem defined by (2.1)–(2.3). Resultswill be obtained first for the one dimensional case where c(x,t) is independent of y and the velocity field is time-dependentbut spatially constant: u = (u(t), 0). These are then generalizedto two dimensions for a (general) time-dependent velocity fieldu = [u(t), v(t)]. Conditions for bounded and unbounded solutions areobtained. Solutions for time dependent sources and diffusivitiesare summarized at the end of this section.

3.1. One dimensional solutions c(x, t) for a time dependentcurrent u(t)

We first obtain the Green’s Function solution (e.g., Haberman,1987) for a point source located at x = 0:

F(x) = ı(x) (3.1)

In this case, (2.3) may be rewritten as

Gt + u(t)Gx = KGxx + ı(x) (3.2)

and for convenience, we drop the sub-script on the diffusivity:

capacity for finfish aquaculture. Part I—Near-field semi-analytic2014.07.005

again the sub-scripts t, x and y denote partial derivatives. The initialand boundary conditions are

G = 0 at t = 0 and G → 0 as |x| → ∞ (3.3)

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ARTICLEQUE-1761; No. of Pages 12

J.F. Middleton, M. Doubell / Aquac

Solutions are obtained by taking a Fourier Transform in x of (3.2),ith the transform of G is defined by

T (k, t) = (2�)−1∫ ∞

−∞G(x, t)eikxdx

The resultant differential equation can be integrated in time andhe inverse transform taken to obtain the Green’s Function:

(x, t) = (2�)−1∫ t

0

[�

K�

]1/2exp

[−(x − X0)2

4K�

]dϑ (3.4)

where

0 =∫ t

t−�

u(�) · d� (3.5)

The solution (3.4) is just that given by Kay (1990, 1997) with thehange of variable � = t − t0 and ti = 0. For a point source at (x*, t*),3.4) is rewritten as

(x − x∗, t) = (2�)−1∫ t

0

[�

K�

]1/2exp

[−(x − x ∗ −X0)2

4K�

]dϑ (3.6)

The solution for c(x, t) for the finite source region (2.1) is thenbtained from

(x, t) =∫ ∞

−∞F(x∗, t)G(x − x∗, t)dx∗ (3.7)

which can be rewritten as

(x, t) =∫ t

0

J(x, �)d� (3.8a)

with

(x, �) = F(2�)−1∫ D

2

− D2

[�

K�

]1/2exp

[−(x − x∗ − X0)2

4K�

]dϑ (3.8b)

Finally, using a change of variable � = (x − x* − X0), (3.8b) may bentegrated to yield

(x, q) = 12

× F[erf (ϕ1) − erf (ϕ2)] (3.9a)

where

1 = x − X0 + D/2

(4K1�)1/2(3.9b)

2 = x − X0 − D/2

(4K1�)1/2(3.9c)

and K has been relabeled as K1.The solution for c(x, t) is then

(x, t) = 12

F

∫ t

0

[erf(1) − erf(2)]d� (3.10)

and represents an integral solution that can be readily evaluatedsing numerical integration.

The Green’s Function solutions can also be used to readily deter-ine under what conditions the solutions for c(x, t) will be bounded

r not. This is important as it goes to the heart of the conceptf carrying capacity. Now, first examine the case where u(t) = U aonstant. The Green’s Function (3.4) at x = 0 is then

(0, t) = (2�)−1∫ t

0

[�

K1�

]1/2

exp

[−(U�)2

(4K�)

]dϑ (3.11)

Please cite this article in press as: Middleton, J.F., Doubell, M., Carryingsolutions. Aquacult. Eng. (2014), http://dx.doi.org/10.1016/j.aquaeng.

With a change of variable = �1/2 this can be integrated to showhat

→ 1/U as t → ∞ (3.12)

PRESSl Engineering xxx (2014) xxx–xxx 5

and is finite and bounded. The concentration c(x, t) will also bebounded as its results from the Green’s Function through the spatialintegral (3.7).

In the case where U = 0

G(0, t) = (2�)−1∫ t

0

[�

K1�

]1/2dϑ = t1/2(K1�)−1/2 (3.13)

so that as t → ∞, G → ∞, and is unbounded. Thus, for one dimen-sional advection–diffusion, a non-zero mean velocity U is needed tokeep solutions bounded. This implies that the diffusive flux termsalone cannot balance the constant source term F.

3.2. Two dimensional solutions c(x, t) for a time dependentcurrent u(t)

The Green’s Function in this case, and for a point source at x = 0,can be shown to be

G(x, t) = (2�)−2∫ t

0

[�

K�

]exp

[−(x − X0)2

(4K1�)

]exp

[−(y − Y0)2

(4K2�)

]dϑ

(3.14)

where K = (K1K2)1/2 and

[X0, Y0] =∫ t

t−ϑ

[u(�), v(�)]d�

Again we consider two cases to determine if G and c are bounded.If X0 = 0, then at x = 0, (3.14) can be integrated to yield

G(0, t) = ln(t)�K

→ ∞ as t → ∞ (3.15a)

so that G and c are unbounded if u = 0. On the other hand, if [X0,Y0] = [U, U] � is non-zero, and assuming K = K1 = K2, then (3.14) canbe written as

G(0, t) =∫ t

0

(4�Kϑ)−1 exp[−˛�]d� (3.15b)

where ̨ = U2/(2K). From Speigel (1968), as t → ∞, G(0, t) → (4�K)−1

so that G and c are bounded. Thus, for both one and two dimensionalcases a non-zero constant velocity will lead to bounded solutionsfor concentration: further verification of this is given below.

The two dimensional solutions for c(x, t) may also be obtainedin an analogous manner to those above and written as:

c(x, t) = 14

F

∫ t

0

[erf(1) − erf(2)] · [erf(3) − erf(4)]d� (3.16)

where ϕ1 and ϕ2 are as defined by (3.9b) and (3.9c) and

ϕ3 = y − Y0 + W/2

(4K2�)1/2(3.17a)

ϕ4 = y − Y0 − W/2

(4K2�)1/2(3.17b)

Finally it is noted that the above can be extended to includetime-dependent diffusivities (see also Kay, 1997) and also step-wise time-dependence for the nutrient flux through the Green’sfunction solution to Eq. (3.2). The effects of coastal boundaries arebeyond the scope of this work and also have not been detailed. Theymay be included in (3.16) above by using the method of images(Haberman, 1987).

capacity for finfish aquaculture. Part I—Near-field semi-analytic2014.07.005

4. Numerical solutions

Here solutions for the concentration c(x, t) will be obtained from(3.16) so as to determine the validity of the scale estimate (2.7) for

IN PRESSG ModelA

6 ultural Engineering xxx (2014) xxx–xxx

coudtfc

ftfia

eui(

mFifr

10 10 10 100

0.5

1

1.5

2

p

c max

/ F

T*

Fig. 3. Summary of the scaled maximum concentrations cmax/FT* from the numericalexperiments and as a function of p = Ta/Td (from Table 1). Results for the one dimen-sional cases have been scaled by FTa and are indicated by the crosses. Results for the

TDon

ARTICLEQUE-1761; No. of Pages 12

J.F. Middleton, M. Doubell / Aquac

max and help build understanding of when and where such maximaccur. We will consider first the case of a constant mean velocity

= (U, 0) with one dimensional diffusion in the x-direction and theniffusion in both the x- and y-directions. The effects of includingidal advection are considered. As will be shown, the estimate (2.7)or the maximum concentration is surprisingly robust for the casesonsidered.

To assist in the discussion, a summary of the parameters adoptedor each of the solution “experiments” is given in Table 1 along withhe numerically determined value of cmax/FT*. Except where speci-ed, it has been assumed that U = 0.05 m/s, D = W = 50 m, Ta = 1000 snd K1 = K2 = K with a value specified by the ratio p adopted.

In order to span a wide range of the parameter p = Ta/Td, thequations were put in non-dimensional form using x′ = x/D, t′ = t/Ta,′ = u/(D/Ta) = U, and K′ = K/(DU). The number of time steps was var-

ed (up to 10,000) to ensure that the solutions for large timest/Ta 1) were stable and accurate.

By way of a summary of the results presented below, the scaledaximum concentrations cmax/FT* in Table 1 are also presented in

ig. 3 as a function of p = Ta/Td. The symbols in the figure caption

Please cite this article in press as: Middleton, J.F., Doubell, M., Carryingsolutions. Aquacult. Eng. (2014), http://dx.doi.org/10.1016/j.aquaeng.

ndicate the cases considered which include strong and weak dif-usion, constant and tidally varying velocity fields. The success inepresenting the maximum concentrations by the scaling cmax/FT*

able 1etails of the numerical experiments for both one and two dimensional advection–dif u = [U + ut(t), 0] and u = [U, vt(t)]. In all cases except where noted, D = W = 50 m, K = K1

umerical solution at x/D = (0.5, 0). The time scale T* is given by Ta for the one dimension

Exp. no. 1D/2D U (m/s) Tide (m/s)

1D – Constant advection u = (U, 0) and diffusion1 1 0 0

2 1 0.05 0

3 1 0.05 0

4 1 0.05 0

2D – constant advection u = (U, 0) and diffusion5 2 0.05 0

6 2 0.05 0

7 2 0.05 0

2D – Tidal advection = (U + uT(t), 0) and diffusion8 2 0.05 1

9 2 0.05 0.01

10 2 0.05 0.40

11 2 0 0

12 2 0.05 0.4

13 2 0.05 0.05

15 2 0.05 1

14 2 0.05 0.05

16 2 0.05 0.4

17 2 0.05 0.05

18 2 0.05 1

U = 0: 2-D and 1-D tides and diffusion19 2 0 1

21 2 0 1

22 1 0 1

23 1 0 1

2D Advection u = (U, vT(t)) and diffusion24 2 0 0.4

25 2 0.05 1.0

26 2 0.05 0.05

27 2 0.05 0.4

28 2 0.05 0.05

29 2 0.05 0.4

31 2 0.05 0.05

two dimensional cases for constant K and U (the circles), tidal advection u = (U + uT (t),0) (the stars) and the two dimensional velocity u = (U, vT (t)) (the squares).

above is notable given it was derived assuming a constant currentU in the x-direction and diffusion only in the y-direction.

4.1. One dimensional solutions for c(x, t) and with u = (U, 0)

capacity for finfish aquaculture. Part I—Near-field semi-analytic2014.07.005

Consider first the solutions in the x-direction alone. For the caseU = 0, (exp. 1), the solutions were found to be unbounded as indi-cated by the result (3.13).

ffusion. The first 7 experiments assume UT = 0. The remainder include the cases= K2 and Ta = 1000 s. The maximum concentration cmax is that obtained from the

al cases and (2.8) for the two dimensional cases.

p cmax/FT* Comment

2 Unbounded0.02 1 Steady >20 Ta

2 120 0.95 Steady >80 Ta

0.2 0.98 Steady >20 Ta

2 0.88 Steady >80 Ta

20 1.14 Steady >80 Ta

2 0.612 0.922 0.85 Spring tide model

2 unbounded c ∼ ln(t)

10 0.8710 1.0510 0.69

2 1.19 U = UT: neap tide model

0.4 1.080.4 1.900.4 0.69

2 Unbounded10 Unbounded

10 Unbounded c ∼ t2 unbounded c ∼ t

2 0.612 0.532 0.80

10 0.6810 0.92

0.4 0.730.4 0.90

ARTICLE IN PRESSG ModelAQUE-1761; No. of Pages 12

J.F. Middleton, M. Doubell / Aquacultural Engineering xxx (2014) xxx–xxx 7

−3 −2 −1 0 1 2 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

times=0.05 to 19.05

c(x,

t)/(

FT

a)

x/D

0.05

1.05

2.05

3.05

Fig. 4. Normalized (1-D) nutrient concentration c(x, t)/FTa as a function of distancefrom the source that lies between x = −D/2 and D/2 – the vertical lines. Solutions arepresented at times t equal to (0.05, 1.05, 2.05, 3.05, . . ., 19.05)Ta and for p = 2 wherea

aFdsan

veeF{cstas

sttt

4

dcccFe

totpa

−3 −2 −1 0 1 2 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

times=0.05 to 79.05

c(x,

t)/(

FT

a)

x/D

Fig. 5. Normalized (2-D) nutrient concentration c(x, 0, t)/FTa as a function of distancefrom the source that lies between x = −D/2 and D/2 – the vertical lines. Solutions

allows for the more rapid set-up found.Finally, results were also obtained with both x- and y-diffusion

and with U = 0. As expected from (3.15a), the results are unboundedand at large times t/Ta 1, the concentration c increases as ln(t).

−3 −2 −1 0 1 2 30

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

c(x,

t)/(

FT

a)

x/D

Fig. 6. Normalized (2-D) nutrient concentration c(x, 0, t)/FTa as a function of distancefrom the source that lies between x = −D/2 and D/2 – the vertical lines. Solutions are

dvection and diffusive effects are comparable.

For the other one dimensional cases below that include diffusionnd U /= 0, the scale for cmax is exactly or very close to FTa and notTd. The reason for this was outlined in Section 2.1 above: in oneimension, the advective flux (Uc)x ∼ Uc/D balances the input flux Fo cmax = FD/U = FTa. Thus, the scaling for the one dimensional casedopts the advective time scale in Table 1 and Fig. 3 even if p ison-zero.

For experiment 2, p = 0.02 and the solutions correspond toery strong advection and the results (Fig. 2) were discussedarlier. For experiment 3, p = 2, so that diffusive and advectiveffects are comparable. Results for c(x, t)/FTa are presented inig. 4 for {−3 < x/D < 3} and for 20 time steps in the interval0.05 ≤ t/Ta ≤ 19.5}. The source region is indicated by the verti-al lines at x/D = ±0.5. Comparing the results with those for verytrong advection (Fig. 2), it is clear that diffusion acts to smearhe concentration in both directions. Nonetheless, the solutions aresymptotic to the estimated maximum value at x ∼ D/2 and over aeveral advective time periods Ta.

Solutions obtained for very strong diffusion p = 20 (not pre-ented) show the concentration field to be much more diffuse thanhose in Fig. 4 where p = 2. A steady solution is achieved, by a time/Ta = 80 and the numerical value of cmax/FTa is about 0.95 and closeo unity.

.2. Two dimensional solutions for c(x, t) and with u = (U, 0)

Consider now solutions with diffusion in both the x- and y-irections. The scale for cmax is that given by (2.7)–(2.9) wheremax = FT* and T* = Ta/(1 + p). In order to illustrate the reduction inoncentration due to the diffusion in the y-direction, results for(x, 0, t) scaled by the advection estimate FTa and are presented inig. 5 for p = 2 and Fig. 6 for p = 20. The horizontal dashed line is thestimate cmax/FT* < cmax/FTa where again T* = Ta/(1 + p).

The numerical results for p = 2 (Fig. 5) show that the scaled solu-ion cmax/FTa near x/D ∼ 0.25 is, as predicted close to about 1/(1 + p)r 1/3rd of that based on the advective scaling Ta. More precisely, at

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/Ta = 79.05, and x/D = 0.52, cmax = 0.88 FT* (see Table 1). Results (notresented) show that the values of concentration are quasi-steadyfter times of 20–30 Ta.

are presented at times t equal to (0.05, 1.05, 2.05, 3.05, . . ., 79.05)Ta and for p = 2where advective and diffusive effects are comparable. The horizontal dashed linecorresponds to the scale estimate of maximum concentration FT*.

Unlike the one dimensional results (Figs. 2 and 4), the concen-tration does not asymptote to a constant for x > D since diffusion inthe y-direction acts to reduce c.

The results for p = 20 are shown in Fig. 6 and the numericalresults show a large reduction in maximum concentration that isclose to the scale estimate 1/(1 + p) ∼ 0.045. The solutions for c(x, 0,t) at x/D = 0.52, (not presented) show that a steady state is achievedafter 30–40 time scales Ta and that at this x-location, cmax = 1.14 FT*.These results show a more rapid convergence in time than wherediffusion was only allowed in the x-direction (Fig. 4). Presumablythe reason for this is that the additional diffusion in the y-direction

capacity for finfish aquaculture. Part I—Near-field semi-analytic2014.07.005

presented at times t equal to (0.05, 1.05, 2.05, 3.05, . . ., 79.05) Ta and for p = 20 wherediffusive effects are dominant. The horizontal dashed line corresponds to the scaleestimate of maximum concentration FT*.

ARTICLE ING ModelAQUE-1761; No. of Pages 12

8 J.F. Middleton, M. Doubell / Aquacultura

0.5 1 1.5 2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

c(t)

/(F

T* )

t/TT

Fig. 7. Normalized (2-D) nutrient concentration c(x, 0, t)/FT* at x = D/2 and as afunction of non-dimensional time t/TT , where TT is the tidal period. Solutions arepresented for p = 2 (thick black curves) where advective and diffusive effects arecht

40

dv

u

Tta

(s

abpntti

fstaAattcTst

ls

omparable and with U = 0.05 m/s and a tide of amplitude UT = 0.4 m/s. The darkorizontal line is included to help determine if the solutions are quasi-periodic. Thehin black curve denotes non-dimensional tidal displacement.

.3. Two dimensional solutions for c(x, t) and with u = (U + UT(t),)

Consider now solutions with diffusion in both the x- and y-irections and with a simple model for the semi-diurnal tidalelocity:

T (t) = UT sin(2�t/TT ) (4.1)

For numerical convenience, we take the tidal period TT to be 42a (11.7 h). Numerical solutions here are presented for a variety ofidal amplitudes and values of p. Again, U = 0.05 m/s, D = W = 50 mnd Ta = 1000 s.

Consider first the spring tide case for the coastal gulf regionsSection 2.3) where the tidal velocity UT = 0.4 m/s is large, the meanpeed U = 0.05 m/s is small and p = 2.

Time series of numerical results (experiment 10) for c(D/2, 0, t)re presented in Fig. 7 and have been normalized by FT* (the solidlack curves). To aid interpretation, the non-dimensional tidal dis-lacement is also indicated by the thin black curve. The results areow presented as a function of tidal period t/TT. At times t/TT > 2.5,he solutions are quasi-periodic as indicated by the solutions andhe negligible changes in the peak heights compared with the hor-zontal line.

In the absence of a tide (uT = 0), the maximum concentrationsound for this case were cmax = 0.88 FT* (exp. 6; Fig. 5). Now con-ider the solutions obtained with the tide (Fig. 7). At a time t = 3 TT,he tidal displacement is zero and the maximum concentration isbout 0.85 FT* and close to that of 0.88 FT* obtained without a tide.fter this time, the tidal velocity and displacement are positive andct to enhance the advective flux (uc)x leading to smaller concen-rations over the next 1/3rd of a tidal period. During this time theidal velocity becomes small and then zero at t = 3.5 TT. The con-entration at this time increases to a local maxima of c/FT* ∼ 0.6.he increase results from the fact that the advective flux is now toomall compared to the source flux F to maintain the low concen-

Please cite this article in press as: Middleton, J.F., Doubell, M., Carryingsolutions. Aquacult. Eng. (2014), http://dx.doi.org/10.1016/j.aquaeng.

rations.After a time 3.5 TT, the tidal velocities become negative and bring

ower concentration water (mixed by diffusion) back towards theource site. The concentration shown in Fig. 7 rapidly drops to a

PRESSl Engineering xxx (2014) xxx–xxx

local minimum at t = 3.75 TT. After this time, the source flux againdominates and acts to increase the concentration until at a time t =4 TT the concentration has again reached its maximum value of 0.85FT*. We note that at all times the estimate of cmax = FT* provides anupper bound for the concentration.

To see how concentration evolves in space and time, plots of c(x,0, t) as a function of x and for several quarters of the tidal cycle arepresented in Figs. 8–10 below. [The solutions are symmetric in y.]To assist interpretation, a small circle is shown that illustrates thedisplacement of a fluid column due to the velocity U + uT(t).

At a time just after the introduction of the flux (Fig. 8a), the con-centration close to the source has a value of around 0.5 FT*. A quarterof a tidal period later, (Fig. 8b), the total velocity has increased to itsmaximum value (u = 0.45 m/s) and the concentration in the sourceregion has dropped since the advective flux has exceeded the sourceflux F. The fluid initially near the source has moved to x ∼ 3 km anda local concentration maxima of 0.08 FT* is found near this point(Fig. 8b).

After this time and up to a half tidal period later (Fig. 9a), thetide decelerates to near zero but the flow is still in the positivex-direction due to the mean flow U = 0.05 m/s. The local maximain concentration has now moved to x ∼ 6.5 km and has reduced inmagnitude due to diffusion. The concentration near the source hasincreased up to 0.62 FT* since the flux associated with advection hasbeen reduced over this phase of the tide. This peak corresponds tothat shown in Fig. 7 at t/TT = 0.5.

At a time near 3/4 of the tidal period (t/TT = 0.76), the tide hasreversed and the total flow velocity (u = −0.35 m/s) is large and neg-ative (Fig. 9b). A secondary local maxima in concentration has nowformed on the negative side of the source where the concentrationhas again dropped (see also Fig. 7).

After a full tidal period, the tidal velocity is zero and the dom-inance of the applied flux F has led to a large increase in theconcentration in the source region of about 0.8 FT* (Fig. 10a). Notethis value slightly exceeds that shown in Fig. 7 for x/D = 0.5, sincethere are small variations on c(x, y, t) across the source region.

Finally the results for a time close to 1.25 tidal periods are shownin Fig. 10b. The velocity is again a maximum and the concentrationin the source region has dropped back to a minimum. The localconcentration maximum for x < 0, is now being absorbed into thesource region and a second maxima is about to be formed for x > 0.

The succession of maxima formed by tidal advection has beennoted by Kay (1997, 1990) and Purnama and Al-Barwani (2006).The former used asymptotic methods to estimate peak concen-trations. Here, we note that the localized peak concentrations areabout equal to the source concentration minima since both occurand are formed at the same time. A second inference from thesesolutions is that the maximum concentrations do occur close tothe source region and generally on the downstream edge of thesource (in the sense of the mean current U).

Increasing the tidal amplitude to UT = 1.0 m/s with p = 2 (exp.8), makes only a quantitative difference to the solutions. Results(not presented) show the peaks in c to be sharper since the advec-tion due to the tides is larger. The troughs in c are closer to zerosince the concentration of water returned to the source region hasbeen more diluted by diffusion during the larger tidal displacement.The maximum concentrations of 0.61 FT* are again bounded by thescale estimate FT* but smaller for the larger tidal velocity consid-ered here. The reason is again that flushing by the tides is largerand that the concentration of water returned to the source regionis more diluted by diffusion over the larger tidal displacement.

Now consider results incorporating a simple model for the neaps

capacity for finfish aquaculture. Part I—Near-field semi-analytic2014.07.005

of the dodge tide (exp. 14). We take p = 2 and UT = U = 0.05 m/s sothat u > 0. The concentrations c(D/2, 0, t) are illustrated in Fig. 11and the maximum values exceed FT* by 19% or so: note the verticalaxis extends from 0 to 2. Quasi-periodic solutions are obtained after

Please cite this article in press as: Middleton, J.F., Doubell, M., Carrying capacity for finfish aquaculture. Part I—Near-field semi-analyticsolutions. Aquacult. Eng. (2014), http://dx.doi.org/10.1016/j.aquaeng.2014.07.005

ARTICLE IN PRESSG ModelAQUE-1761; No. of Pages 12

J.F. Middleton, M. Doubell / Aquacultural Engineering xxx (2014) xxx–xxx 9

−6 −4 −2 0 2 4 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

u = 0.115 t/TT =0.025

U (m/s)=0.05 UT (m/s)=0.4

x (km)

a b

c(x,

0,t)

/FT

*

−6 −4 −2 0 2 4 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

u = 0.45 t/TT =0.2631

U (m/s)=0.05 UT (m/s)=0.4

x (km)

c(x,

0,t)

/FT

*

Fig. 8. (a) The concentration c (x, 0, t) normalized by FT* for the case shown in Fig. 6, but now plotted as a function of x (km) and for the fraction of the tidal period (t = 0.025TT )shown and velocity u (m/s) = U + UT sin(2�t/TT ), U = 0.05 m/s and UT = 0.4 m/s and p = 2. The black circle denotes the location of a fluid parcel advected by the velocity field.(b). As in Fig. 8a, but for the time t = 0.26TT .

−6 −4 −2 0 2 4 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

u = 0.045 t/TT =0.50119

U (m/s)=0.05 UT (m/s)=0.4

x (km)

a b

c(x,

0,t)

/FT

*

−6 −4 −2 0 2 4 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

u = −0.35 t/TT =0.7631

U (m/s)=0.05 UT (m/s)=0.4

x (km)

c(x,

0,t)

/FT

*

Fig. 9. (a) As in Fig. 8a, but for the time t = 0.50TT . (b). As in Fig. 8a, but for the time t = 0.76TT .

−6 −4 −2 0 2 4 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

u = 0.055 t/TT =1.0012

U (m/s)=0.05 UT (m/s)=0.4

x (km)

a b

c(x,

0,t)

/FT

*

−6 −4 −2 0 2 4 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

u = 0.45 t/TT =1.2631

U (m/s)=0.05 UT (m/s)=0.4

x (km)

c(x,

0,t)

/FT

*

Fig. 10. (a) As in Fig. 8a, but for the time t = 1.00TT . (a). As in Fig. 8a, but for the time t = 1.26TT .

ARTICLE ING ModelAQUE-1761; No. of Pages 12

10 J.F. Middleton, M. Doubell / Aquacultura

0.5 1 1. 5 2 2. 5 3 3. 5 40

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

c(t)

/(F

T* )

t/TT

Fig. 11. Normalized (2-D) nutrient concentration c(x, 0, t)/FT* at x = D/2 and as afunction of non-dimensional time t/TT , where TT is the tidal period Solutions arepresented for p = 2 (thick black curves) where advective and diffusive effects arecomparable and with U = 0.05 m/s and a tide of amplitude UT = 0.05 m/s. The darkhc

oncmtsi

mnUsbocT6f

clfet

a1wm0v

4

sac

advection dominates and T* = Ta and cmax is determined by the

orizontal line is included to see if the solutions are quasi-periodic. The thin blackurve denotes non-dimensional tidal displacement.

ne tidal period TT. The dual peaks and troughs found in Fig. 7 areo longer found as u is always positive or zero and mixed low con-entration water cannot be brought back to the source region. Theaximum concentrations 1.19 FT* while still bounded, are larger

han for the spring tide case where cmax = 0.85 FT* and p = 2, anduggest that significant increases in concentration may result dur-ng neap tides.

Indeed, during neap tides, the shear dispersion diffusivitiesay be expected to be small and mean advection may domi-

ate. To investigate this, results were obtained for p = 0.4 and = UT = 0.05 m/s. In this case (experiment 17), the solutions (nothown) are a smeared version of those shown in Fig. 11, areounded and have a maximum concentration of 1.90 FT*. The valuef T* is 0.71Ta in this case and implies a maximum concentration ofmax = 1.36FTa. For the spring tide example above cmax = 0.85 FT* and* = 0.25Ta so cmax = 0.21FTa (with Ta = 1000 s in both cases). Thus, a-fold increase in the maximum concentration might be expectedrom the spring to the neap phases of the dodge tide.

The results for experiment 17 also show that the maximum con-entration is 1.9 FT*, almost twice the scaled estimate of FT* andarger than all other results obtained (Fig. 3; Table 1). The reasonor this is thought to be due to (a) the weak diffusion and (b) theffect of advective flushing is zero or small when the tide opposeshe constant velocity.

Results were also obtained for case where diffusion dominatesnd p = 10. The results for the tidal amplitudes UT = 0.05, 0.4 and.0 m/s are qualitatively very similar to those presented abovehere p = 2. The solutions are quasi-periodic after 100Ta and theaximum concentrations vary between 0.61 FT* (for UT = 1 m/s),

.88 FT* (for UT = 0.4 m/s) and 1.05 FT* (for UT = U = 0.05 m/s): thesealues are listed in Table 1.

.4. Solutions for c(x, t) with U = 0 and a tide: u = (uT(t), 0)

It was shown above that solutions in both one and two dimen-

Please cite this article in press as: Middleton, J.F., Doubell, M., Carryingsolutions. Aquacult. Eng. (2014), http://dx.doi.org/10.1016/j.aquaeng.

ions will be bounded provided U is non-zero. We have not beenble to establish analytically if the presence of a tide alone is suffi-ient to bound the solutions. To this end, the analytical model has

PRESSl Engineering xxx (2014) xxx–xxx

been re-run in both one and two dimensions with U = 0 and thenon-zero tidal velocity (4.1).

In the one dimensional case the solutions (not shown) wereobtained for a strong tide (UT = 1.0 m/s) and for both Td = 1000 s and200 s. The solutions c(D/2, t) appear to grow linearly with time fort ε [60,180] Td and appear to be unbounded. The linear growth isunexpected and must in some way be related to the inclusion ofthe tide: the Green’s function solution (3.13) would suggest thatgrowth in c be proportional to t1/2.

For the two dimensional case, these parameters were alsoadopted (experiments 19 and 21). The results (not presented) sug-gest that the solutions become quasi-periodic after 2.5 tidal cycles.However, integrating the solutions out to 8.5 tidal cycles shows alinear increase in concentrations of about 5% over this time.

The trend is much weaker than in the one dimensional case.A qualitative explanation for this is that in the two dimensionalcase, water is advected away from the source region where it isthen diffused in both the x- and y-directions. This diluted wateris then returned to the source region by the tide and only a weaktrend results. In the one dimensional case, mixing in the y-directioncannot occur so the concentrations in the source region grow morequickly.

4.5. Numerical solutions – advection in the x- and y-directions:u(t) = (U, vT(t)) only

Now consider solutions where the mean current U and tidalvelocity

vT (t) = VT sin(

2�t

TT

)(4.2)

are in orthogonal directions. Such a scenario might result froma low frequency “mean” current U that is directed along coastalisobaths while the tidal currents are directed across isobaths.

Since there is now diffusion and advection in both directions,the maximum concentrations may be expected to be smaller thanin the unidirectional cases above. Indeed, this was found to be thecase for a variety of parameters adopted (Table 1). The results werealso found to be qualitatively similar to those in Section 4.3.

An exception is illustrated by the case (exp. 24) where p = 2,U = 0.05 m/s and the tidal velocity is large VT = 0.4 m/s. The solutions(Fig. 12) show that a quasi-periodic solution is rapidly achievedafter one tidal period TT or so. The maximum solutions are alsobounded by 0.61 T*.

.The strong tidal velocity vT(t) and constant mean current U are

now orthogonal and do not cancel each other out at any stage ofthe tide so that the maxima and minima are all of equal ampli-tude. In contrast the comparable solutions shown in Fig. 7, whereu = (U + uT(t),0), show local minima and maxima of different ampli-tudes depending on the phase of the tide.

5. Summary and discussion

Using the divergence theorem, a simple scale estimate has beenderived for two dimensional advection and diffusion that relatesthe maximum (depth averaged) concentration cmax near the source(cage or lease) to a constant surface nutrient source flux F through

cmax = FT∗ (5.1)

Here, T* = Ta/(1 + Ta/Td) is a new time scale involving both advec-tion/flushing (Ta = D/U) and diffusion (Td = D2/(2K)). For Ta Td,

capacity for finfish aquaculture. Part I—Near-field semi-analytic2014.07.005

flushing time scale. For Ta Td diffusive processes dominate andcmax = F·Td. Since Td Ta the maximum concentrations obtained aremuch smaller than indicated by the advective scaling FTa. Simply,

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0.5 1 1. 5 2 2. 5 3 3. 5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

c /F

T*

t/TT

Fig. 12. Normalized (2-D) nutrient concentration c(x, 0, t)/FT* at x = D/2 and as afunction of non-dimensional time t/TT , where TT is the tidal period. Solutions arepresented for p = 2 (thick black curves) where advective and diffusive effects arechb

ta

a(tba

fdsaTc

cwaf(

otdb

admo

Tozrail

omparable and with U = 0.05 m/s and a strong tide of amplitude VT = 0.4 m/s. Theorizontal dark line is included to see if the solutions are quasi-periodic. The thinlack curve denotes non-dimensional tidal displacement.

he diffusion and advection flush the zone more quickly than wheredvection acts alone.

The implications for carrying capacity of a cage or zone regionre important since the flux of nutrients F is related to feed ratesand fish biomass) and these can be adjusted (up or down) so aso optimize carrying capacity F = cP/T*. The allowed flux can differy an order of magnitude depending on the relative importancedvection and diffusion.

To validate this scale relation, exact semi-analytic solutionsor c(x, t) were determined from the depth-averaged advectioniffusion equations with time dependent velocities. These newolutions were obtained for a wide parameter range of diffusionnd advection including simplified tidal velocity representations.hey showed that the exact maximum concentrations were typi-ally within 40% of that given by the scale estimate cmax = FT*.

The solutions also revealed some important aspects of the con-entration field for a variety of flows. In the one dimensional case, itas shown that the concentration is ultimately dominated by mean

dvection (U) and that the maximum concentration cmax = FTa isound near and beyond the downstream edge of the source regionlength D).

In both the one and two dimensional cases, the solutionsbtained with a tidal velocity but with the mean current U equalo zero indicate unbounded growth in concentration. In the twoimensional case and with strong tides, the growth was found toe weaker than in the one dimensional case.

In the two-dimensional case, diffusion occurs in both the x-nd y-directions and either the diffusive or advective fluxes canominate depending on the value of U and the diffusivity K. Theaximum concentrations were found near the “downstream” edge

f the source region.Solutions were also detailed for the case of tidal advection.

hese results illustrate that an increased load of nutrient is peri-dically shed into the water column. When the tidal velocity isero the advective flux is small and the concentration in the source

Please cite this article in press as: Middleton, J.F., Doubell, M., Carryingsolutions. Aquacult. Eng. (2014), http://dx.doi.org/10.1016/j.aquaeng.

egion is largest. As the flow accelerates, this nutrient load is movedway from the source where the concentration drops due to thencreased advection. In this manner, secondary and then tertiaryocal maxima are generated and moved away from the source. This

PRESSl Engineering xxx (2014) xxx–xxx 11

mechanism was described also by Kay (1997) although the rela-tionship between the concentration at the source and tidal velocitywas not detailed since the solutions were obtained for an infinitepoint source.

In a companion analysis, (Middleton et al., 2014), the seasonallyaveraged time scales of advection and diffusion are estimated froma hydrodynamic model of Spencer Gulf so that the maximum con-centrations can be readily estimated from the scale estimate (5.1)above. The diffusive time scale was determined using estimates ofthe shear dispersion coefficients KS that arise from the very strongtidal currents of the region. In other aquaculture regions aroundthe world, shear dispersion may not be as dominant as in SpencerGulf. For such regions, the horizontal diffusivity and associated timescale might have to be estimated from Lagrangian drifter data orturbulence closure schemes.

In a subsequent analysis, we also plan to examine the impacts oftime-dependent shear diffusivities and nutrient fluxes: the formerwill arise from the 30 day period dodge tide while the latter mayarise from morning and evening feeding of cultured fish. The effectsof coastal boundaries a distance L from the source region have notbeen detailed here either. As a simple rule of thumb, (5.1) shouldbe approximately valid provided L is much larger than the width ofthe source region itself.

Finally, it has been assumed that the nutrients are conservedand that biological cycling and uptake is relatively small. To showthis it is first noted that in Spencer Gulf, the daily dissolved inor-ganic nitrogen input for one Southern Bluefin Tuna lease can beup to 1000 kg per day. Nutrients will of course be taken up phyto-plankton and cycled through the food web by biological activity.Maximum daily primary productivity measured in the gulf hasbeen measured to be approximately 900 mg C m−2 d−1 (Van Ruthand Doubell, 2013). Assuming an instantaneous response by phy-toplankton, primary production would uptake about 2 kg/per dayof nitrogen at the scale of the lease and much smaller than the1000 kg/d input flux due to feeding. Typically, the phytoplanktonbiomass response following enhanced nutrient supply will take 3–7days and will be realized downstream at some distance away fromthe point source (Olsen and Olsen, 2008). However, near the leasewhere the maximum concentrations are found and over the timescale of flushing (1–10 h), the nutrients may well be regarded asconservative. A second limitation to the results obtained is thatoxygen concentrations cannot be estimated using the approachadopted here due to the strong relation between dissolved oxygenlevels and biological processes such as microbial and planktonicrespiration and fish metabolism. These issues will be dealt with inthe biogeochemical modelling study of Doubell et al. (2013).

Acknowledgements

John Middleton and Mark Doubell are supported by MarineInnovation Southern Australia (MISA). Our thanks to Shaun Byrnesfor assisting with the references and to Anthony Kay and CraigStevens for some useful comments.

References

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John F. Middleton has made significant contributions tounderstanding shelf and slope circulation through ana-lytical and numerical models. He has demonstrated theimportance of coastal trapped waves and bottom frictionto upwelling. Notable contributions include progress inunderstanding the weather band circulation around seastraits as well as the circulation along Australia’s southernshelves, slopes and Gulfs. This includes the role of Sver-drup transport in driving downwelling in the central GreatAustralian Bight. He has attracted over $10M in externalfunding and leads the Southern Australian Marine Observ-ing mooring facility.

Mark Doubell’s research interests combine aquaticmicrobial ecology and biological oceanography. In par-ticular, his research focusses on understanding thephysical and biological factors that influence the spa-tial and temporal heterogeneity of phytoplankton at theocean’s micro-scale, and the consequences of interactionsbetween individuals for aquatic ecosystems. Mark’s workhas been the first to demonstrate a critical measurementscale below which fluorescence based measurements of

capacity for finfish aquaculture. Part I—Near-field semi-analytic2014.07.005

phytoplankton are often distributed non-randomly. Morerecently, he has led the development of an ecosystem

model for Spencer Gulf in South Australia.