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Estuarine, Coastal and Shelf Science 77 (2008) 679e687www.elsevier.com/locate/ecss

A dynamic equation for the potential energy anomaly foranalysing mixing and stratification in estuaries and coastal seas

Hans Burchard*, Richard Hofmeister

Baltic Sea Research Institute Warnemunde, Seestrasse 15, D-18119 Rostock, Germany

Received 16 July 2007; accepted 31 October 2007

Available online 6 November 2007

Abstract

In this paper, a time-dependent dynamic equation for the potential energy anomaly, f, is rigorously derived from dynamic equations for po-tential temperature and salinity, the continuity equation and the equation of state for sea water. The terms locally changing f are (A) the f-advection, (B) the depth-mean straining, (C) the non-mean straining, (D) the vertical advection, (E) the vertical mixing, (F) surface and bottomdensity fluxes, (G) inner sources of density e.g. due to absorption of solar radiation and the non-linearity of the equation of state, and (H) hor-izontal divergence of horizontal turbulent density fluxes. In order to derive the equation in concise form, a vertical velocity (linearly varying withdepth) with respect to depth-proportional vertical coordinates had to be defined. The evaluation of the terms in the f-equation is then carried outfor a one-dimensional tidal straining study and a two-dimensional estuarine circulation study. Comparisons to empirical estimates for these termsare made for the one-dimensional study. It is concluded that the f-equation provides a general reference for empirical bulk parameterisations ofstratification and mixing processes in estuaries and coastal seas and that it is a tool for complete analysis of the relevant terms from numericalmodels.� 2007 Elsevier Ltd. All rights reserved.

Keywords: mixing; stratification; estuaries; potential energy anomaly; tidal straining; differential advection

Z h

1. Introduction

Processes of stratification and destratification play a majorrole in the physics of shallow coastal seas. It is therefore es-sential to define a suitable measure for the stability of the wa-ter column, which can be easily quantified from fieldobservations as well as from numerical models. As a conve-nient measure, the potential energy anomaly f has been de-fined by Simpson (1981) as the amount of mechanicalenergy (per m3) required to instantaneously homogenise thewater column with a given density stratification:

f¼ 1

D

Z h

�H

gzðr� rÞ dz¼�1

D

Z h

�H

gz~r dz ð1Þ

with the depth-mean density

* Corresponding author.

E-mail address: hans.burchard@io-warnemuende.de (H. Burchard).

0272-7714/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ecss.2007.10.025

r¼ 1

D �H

r dz; ð2Þ

the deviation from the depth-mean density, ~r ¼ r� r, themean water depth H, the sea surface elevation h, the actualwater depth D ¼ h þ H, and the gravitational acceleration g.

Due to its tremendous relevance to the marine ecosystem,the understanding of stratification and de-stratification pro-cesses in shelf seas has been intensively studied during thelast decades. In their key paper on fronts in the Irish Sea,Simpson and Hunter (1974) made an important step towardsquantifying processes determining stratification in shelf seas.Simpson et al. (1977) first suggested to use the potential den-sity of the water column derived from continuous density pro-files as a measure for stratification, and Simpson (1981) finallydefined the potential energy anomaly as shown in eq. (1).

Several authors have derived dynamic equations for f un-der idealised conditions. Simpson (1981) assumed vanishinghorizontal density gradients and divergence-free depth-mean

680 H. Burchard, R. Hofmeister / Estuarine, Coastal and Shelf Science 77 (2008) 679e687

horizontal currents and considered surface heat fluxes as theonly source of stratification, and from these assumptions de-rived a f-equation including f-advection, surface heat fluxand vertical mixing. The f-advection term has been neglectedby van Aken (1986), but he added the effects of horizontaldensity gradients (constant over the vertical). This processwas denoted as differential advection and includes effects oftidal straining and estuarine circulation. Recently, in order toinvestigate the processes of stratification and mixing in anidealised Rhine outflow model, de Boer et al. (submitted forpublication) derived a dynamic equation for f under the as-sumptions of constant water depth and zero surface and bot-tom density fluxes, as well as vanishing horizontal turbulenttransport and sea surface elevation and water depth gradients.

There have been many suggestions made for parameterisa-tions of the source and sink terms in the f-equation. Based onconstant bulk mixing efficiencies and drag coefficients, Simp-son (1981) and Simpson and Bowers (1981) derived expres-sions for wind and tidal mixing. Since indeed the mixingefficiency is highly variable (zero for vanishing stratification),Simpson and Bowers (1981) suggested mixing efficiencies de-pending on f itself. Nunes Vaz et al. (1989) parameterised themixing effect of convective surface cooling by means of a con-stant mixing efficiency for this process. As further mixing pro-cess, Wiles et al. (2006) parameterised the effect of surfacewave breaking as function of significant wave height andwave period. Straining of vertically constant horizontal den-sity gradients by tides and estuarine circulation has been para-meterised by Simpson et al. (1990) on the basis of velocityprofiles from analytical theory for tidal flow and for estuarinecirculation. Wiles et al. (2006) furthermore suggested anempirical expression for the effect of absorption of short-wave radiation in the water column and (for shallow water)at the sea bed.

The potential energy anomaly has been used in numerousstudies for quantifying the relative contributions of differentprocesses of stratification and de-stratification in coastal seasand estuaries. Early studies concentrated on the positionsand movements of fronts in shelf seas, see e.g. Simpsonet al. (1977), Simpson (1981) and Simpson and Bowers(1981) who analysed field data from the Irish Sea and the Brit-ish Channel, and van Aken (1986) who studied frontal dynam-ics in the Southern North Sea. Mixing by surface cooling andre-stratification by dense water overflows have been studied byRippeth and Simpson (1996) for the Clyde Sea. The competi-tion between tidal straining and vertical mixing has beenintensively investigated for the Liverpool Bay by Simpsonet al. (1990), Sharples and Simpson (1995), Rippeth et al.(2001), and Scott (2004). For the shallow Limfjord in Den-mark in which competition between the stratificational effectsof estuarine circulation and surface heating and the mixing dueto wind, waves and surface cooling leads to episodic stratifica-tion in summer with potentially hazardous consequences forbenthic filter feeder populations, Wiles et al. (2006) studiedthe balance of empirical source terms in the f-equation. Thedynamics of Intermittently Closed and Open Lakes andLagoons (ICOLLs) has been intensively studied by means of

analysing the empirical f-equation terms by Ranasinghe andPattiaratchi (1999) and Gale et al. (2006)

No study however has so far rigorously derived the f-equation. Effects of sloping sea beds, surface slopes, verticalvariations of horizontal density gradients, vertical advection,internal heating due to absorption of short-wave radiation,non-linear effects of the equation of state for sea water andthe divergence of horizontal turbulent density fluxes havenot yet been considered.

It is therefore the aim of this study to rigorously derivea time-dependent dynamic equation for f, based on thedynamic equations for potential temperature and salinity, thecontinuity equation and an equation of state for the potentialdensity. With these equations given, no further approximationswill be made. A f-equation which is fully consistent with thedynamic equations discretised in numerical models will pro-vide a tool for numerically quantifying all terms relevant forgeneration and destruction of stratification. It will furtherhelp to improve parameterisations for empirical f-equationsas they have been extensively used for studying the dynamicsof estuaries and coastal seas.

This paper is organised as follows. First, the time-dependentdynamic equation for f is derived in Section 2 and the result-ing terms in the f-equation are discussed. Afterwards, the dy-namics of f is evaluated for idealised vertically-resolvingmodel studies (Section 3). The models applied for this purposeare briefly introduced in Section 3.1. In Section 3.2, a one-dimensional study of tidal straining and wind mixing is de-scribed, and the resulting source terms in the f-equation arecompared to empirical estimates in Section 3.3. In Section3.4, a two-dimensional estuarine study is presented, with theevaluation of f-terms at two locations, in the periodically strat-ified area of the outer estuary and further upstream in the per-manently stratified salt wedge area. Some conclusions aboutimplications of the f-equation will be given in Section 4.

2. Dynamic equation for f

The dynamic equation for f will be obtained by combiningthe definition of f from eq. (1) with a dynamic equation for thepotential density r, see van Aken (1986). In oceanography, thismay be derived from the dynamic equations for potential tem-perature q and salinity S:

vtqþu$Vhqþwvzq�vzðKvvzqÞ�VhðKhVhqÞ¼ 1

r0Cp

vzI; ð3Þ

vtSþ u$VhSþwvzS� vzðKvvzSÞ �VhðKhVhSÞ ¼ 0; ð4Þ

with the horizontal velocity vector u and the vertical velocityw, jointly fulfilling the continuity equation

Vhuþ vzw¼ 0: ð5Þ

Here, Vh denotes the horizontal gradient operator, such thatVh $ u is the horizontal flow divergence. In eqs. (3) and (4),Kv and Kh denote the vertical eddy diffusivity and the horizon-tal eddy diffusivity, respectively, resulting from down-gradient

681H. Burchard, R. Hofmeister / Estuarine, Coastal and Shelf Science 77 (2008) 679e687

parameterisations of vertical and horizontal turbulent fluxes.For simplicity, higher order turbulence closure approaches par-tially resulting in counter-gradient parameterisations are notconsidered here, but may be included in this approach ina straight-forward way. The term on the right-hand side ofthe temperature equation is the heating due to absorption ofsolar radiation, with the constant reference density r0, theheat capacity Cp (here assumed to be constant) and the shortwave radiation I, the gradient of which strongly depends onthe light attenuation properties of the water, see Paulson andSimpson (1977).

For the derivation of a dynamic equation for potential den-sity based on the temperature and salinity equation, it is essen-tial to consider the non-linear equation of state for sea water(see e.g. Jackett et al., 2006):

r¼ rðq;S;p0Þ 0

Vhr¼�raVhqþ rbVhS;

vzr¼�ravzqþ rbvzS;

vtr¼�ravtqþ rbvtS;

8>>>><>>>>:

ð6Þ

with the constant reference pressure p0, the thermal expansioncoefficient a ¼ �(1/r)vqr and the haline contraction coeffi-cient b ¼ (1/r)vSr.

By combining eqs. (3) and (4) under consideration of eq.(6), we obtain

vtrþ u$Vhrþwvzr� vzðKvvzrÞ �VhðKhVhrÞ ¼ Q ð7Þ

with the source term for density,

Q¼ � ra

r0Cp

vzIþKvvzq vzðraÞ �KvvzS vzðrbÞ

þ KhVhq$VhðraÞ �KhVhS$VhðrbÞ:ð8Þ

For the derivation of a dynamic f-equation, we furthermoreneed to define the depth mean horizontal velocity vector

u¼ 1

D

Z h

�H

u dz: ð9Þ

It is furthermore convenient to define a vertical velocity w,which together with u fulfils a continuity equation:

Vh$uþ vzw¼ 0: ð10Þ

From eq. (10), it is clear that w is linear and integration of eq.(10) gives

w¼�u$VhHh� z

Dþ ðvthþ u$VhhÞzþH

D: ð11Þ

Here, we have used the vertically integrated incompressibilityequation,

vth¼�Vh$ðDuÞ; ð12Þ

and we have applied a kinematic boundary condition for w atthe bottom,

wð�HÞ ¼ �u$VhH: ð13Þ

With this, w is the vertical velocity which would result fromkinematic boundary conditions (alignment of the flow velocityvector with bottom and surface) and the incompressibility con-dition for the case of vertically homogeneous horizontal flowvelocity. For this case, the flow would be parallel to depth-pro-portional coordinates (so-called s coordinates, see e.g. Blum-berg and Mellor (1987)). With this definition of the linearlyvarying vertical velocity w, a dynamic equation for f can bederived in a concise form (see below).

The deviation from the depth-mean horizontal velocity vec-tor is defined as ~u ¼ u� u and the deviation from the linearvertical velocity as ~w ¼ w� w. Surface volume fluxes dueto precipitation and evaporation are neglected here for simplic-ity, but may be included in a straight-forward way.

Combining eqs. (1), (2), (7), (9) and (11), we obtain a time-dependent dynamic equation for f:

vtf¼�VhðufÞ|fflfflfflfflffl{zfflfflfflfflffl}ðAÞ

þ g

DVhr$

Z h

�H

z ~u dz|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}ðBÞ

� g

D

Z h

�H

�h�D

2� z

�~u �Vh~r dz|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

ðCÞ

� g

D

Z h

�H

�h�D

2� z

�~wvz~r dz|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

ðDÞ

þ r0

D

Z h

�H

Pb dz|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}ðEÞ

�r0

2

�Ps

bþPbb

�|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}

ðFÞ

þ g

D

Z h

�H

�h�D

2� z

�Qdz|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

ðGÞ

þ g

D

Z h

�H

�h�D

2� z

�VhðKhVhrÞdz|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

ðHÞ

; ð14Þ

with the vertical buoyancy flux

Pb ¼g

r0

Kvvzr; ð15Þ

and the surface buoyancy flux Psb and the bottom buoyancy

flux Pbb. For the derivation of eq. (14), we have applied the ki-

nematic boundary conditions

wðhÞ ¼ vthþ uðhÞ$Vhh; wð�HÞ ¼ �uð�HÞ$VhH ð16Þ

as well as the Leibniz rule for derivatives of integrals withmoving limits,

vs

Z bðsÞ

aðsÞf ðs; xÞdx¼

Z bðsÞ

aðsÞvs f ðs;xÞdxþ vsbðsÞf ðs;bðsÞÞ

� vsaðsÞf ðs;aðsÞÞ:ð17Þ

682 H. Burchard, R. Hofmeister / Estuarine, Coastal and Shelf Science 77 (2008) 679e687

In eq. (14) the terms on the right-hand side have the followingmeaning:

A: f-advection due to the vertical mean horizontal velocityvector, u, including also density advection with the linearlyvarying vertical velocity w as defined in eq. (11)dthe con-sideration of f-advection has already been suggested bySimpson (1981), who however assumed a non-divergentdepth-mean flow (Vh$u ¼ 0) for better estimating themovements of fronts in shelf seas;B: depth-mean straining, based on the vertical mean hori-zontal density gradient strained by the deviation from thedepth-mean velocity vector, ~u; this term has first been de-rived by Bowden (1981) and has been denoted as the differ-ential advection term by van Aken (1986);C: non-mean straining, based on straining of the devia-tion from the vertical mean horizontal density gradient;for the case of no mean horizontal density gradient(Vhr ¼ 0), the non-mean straining term may still changethe stratification. If in the lower half of the water column(h � D/2 � z > 0) the velocity anomaly is in the direc-tion of decreasing density (~u$Vh~r < 0), then the watercolumn is stabilised (vtf > 0), and vice versa, see Fig. 1;

A

B

C

D

Fig. 1. Sketch explaining the principle of how the kinematic terms AeD in equation

for flat (upper left) or sloping (upper right) sea bed. (B) Depth-mean straining. (C

D: vertical advection, based on the deviation from the linearvertical velocity, ~w; for a stably stratified two-layer flow thevalue of f is largest when the interface between the twolayers is at mid-depth. Therefore, for stable stratification(vz~r < 0), an upward displacement of isopycnals (~w > 0)in the lower half of the water column (h � D/2 � z > 0) in-creases f and vice versa, see Fig. 1;E: vertical mixing of density expressed as the integratedvertical buoyancy flux. By using the dynamic equationfor the turbulent kinetic energy, this term may be expressedby means of the vertically integrated shear production anddissipation, see e.g. van Aken (1986);F: surface and bottom buoyancy fluxes, with negative(downward, e.g. surface warming) buoyancy fluxes both in-creasing f;G: inner sinks or sources of potential density, with a densitydecrease (Q < 0, e.g. solar radiation) in the upper half ofthe water column (h � D/2 � z < 0) increasing f andvice versa;H: divergence of horizontal turbulent transport. The struc-ture of this term is in analogy to term G, i.e. horizontal den-sity flux divergence (VhðKhVhrÞ < 0) in the upper half ofthe water column has a stabilising effect.

eq. (14) change the potential energy anomaly f. (A) Horizontal advection of f

) Non-mean straining. (D) Vertical advection.

683H. Burchard, R. Hofmeister / Estuarine, Coastal and Shelf Science 77 (2008) 679e687

The kinematic terms AeD are graphically sketched inFig. 1.

3. Applications to estuarine flow

3.1. Model description

−4

−2

0

Density , contour interval: 0.5 kg m–3

The turbulence closure model which is applied for the verti-cally resolving simulations of the present study is the k-3 modelwith transport equations for the turbulent kinetic energy (TKE),k, and the turbulence dissipation rate, 3. As second-moment clo-sure, the model suggested by Cheng et al. (2002) is used.

In several applications, this model has proven to agreequantitatively well with turbulence observations in coastaland shelf sea waters, see e.g. the investigations by Burchardet al. (2002), Simpson et al. (2002), and Arneborg et al.(2007).

The turbulence closure schemes described above have beenimplemented into the Public Domain water column modelGOTM (General Ocean Turbulence Model, see http://www.gotm.net and Umlauf et al., 2005), which is used herefor the one-dimensional studies described in Section 3.2.

The three-dimensional General Estuarine Transport Model(GETM, see Burchard and Bolding, 2002; Burchard et al.,2004) which is used for the two-dimensional estuarine studyin Section 3.4 combines the advantages of bottom-followingcoordinates with the turbulence module of GOTM.

−8

−6z / m

3.2. One-dimensional strain-induced periodicstratification

1.0−10

0.0 0.2 0.4 0.6 0.8

−5

0

5

10

15

20

25

0 0.2 0.4 0.6 0.8 1

/ Jm

–3

−5−4−3−2−1

012

0 0.2 0.4 0.6 0.8 1t / T

t / (10

-3W

m

-3)

Tendency ( t

)Vertical mixing (E)Depth-mean Straining (B)

Fig. 2. Idealised one-dimensional simulation of tidal straining during one tidal

cycle. Upper panel: time series of density profiles; middle panel: time series of

potential energy anomaly f; lower panel: time series of the tendency (vtf),

vertical mixing (E) and depth-mean straining (B) terms in the f-equation.

There is ebb tide for t/T < 0.5 and flood tide for t/T > 0.5.

Here, the dynamics of strain-induced periodic stratification(SIPS, see Simpson et al., 1990) is studied with the one-di-mensional model by prescribing a rectilinear tide interactingwith a constant wind stress and a constant density gradient,both acting in the direction of the flow. For simplicity, Earthrotation is neglected. With this, the dynamic momentum equa-tion reads as

vtu� vzðAvvzuÞ ¼ zg

r0

vxr� gpgðtÞ; ð18Þ

where the first term on the right-hand side represents the effectof the internal pressure gradient and pg is the surface slope os-cillating with period T chosen in such a way that the depth-averaged transport is sinosodial:

uðtÞ ¼ umaxcos�

2pt

T

�; ð19Þ

with the vertical mean velocity amplitude umax (see Burchard,1999 for details). In eq. (18), Av denotes the vertical eddyviscosity.

In this one-dimensional case, the transport equation fordensity is given as

vtrþ uvxr� vzðKvvzrÞ ¼ 0; ð20Þ

with zero density fluxes through surface and bottom.

For the simulation discussed here, the following parametersare used: constant water depth D ¼ H ¼ 10 m; tidal periodT ¼ 44,714 s (same as for M2 tide); tidal velocity amplitudeu max ¼ 0.5 m s�1, constant horizontal density gradient vxr

¼ �5� 10�4 kg m�4; surface wind stress ts ¼ 0.1 N m�2

(equivalent to a 10 m wind speed of the order of 5 m s�1).Fig. 2 shows the resulting density structure, potential en-

ergy anomaly f and the balance of the f-equation after 30tidal periods. After flood (t/T ¼ 0 and t/T ¼ 1), the water col-umn is vertically fully homogenised, resulting in vanishing f.With the onset of the ebb current (0 � t/T � 0.5), less densewater is sheared over denser water, with an increasingly pos-itive straining term, not fully balanced by vertical mixing,such that f is growing. At the end of the ebb tide (t/T ¼ 0.4), the vertical mixing is ceasing, leading to a maximumincrease of f. After the onset of flood (t/T > 0.5), straining isreversed with the effect of decreasing f. With full flood (t/T ¼ 0.75) vertical mixing is supporting the erosion of stratifi-cation, resulting in a maximum decrease of f at t/T ¼ 0.8, andslightly unstable stratification (f < 0). With unstable

684 H. Burchard, R. Hofmeister / Estuarine, Coastal and Shelf Science 77 (2008) 679e687

stratification, however, mixing is increasing f, such that forthe last phase of the flood tide (0.8 < t/T < 1) destabilisingstraining and stabilising mixing are in balance.

3.3. Comparison to empirical estimates

–3

–2

–1

0

1

2

0 0.2 0.4 0.6 0.8 1t/T

Vertical mixing (1D)Vertical mixing (emp)

Depth-mean Straining (1D)Depth-mean Straining (emp)

t / (10

-3W

m

-3)

Fig. 3. Comparison between forcing terms in the f-equation as calculated from

the one-dimensional model discussed in Section 3.2 (full lines) and as esti-

mated from the empirical model suggested by Simpson et al. (1990) (dashed

lines). The bold lines show the vertical mixing term (E) and the thin lines

show the depth-mean straining term (B).

Simpson et al. (1990) suggested an empirical model for es-timating some of the forcing terms in the f-equation. Theirmodel is valid for scenarios with flow and gradients only inone horizontal direction (which is here aligned with the x-di-rection, with the x-component of the flow velocity denotedas u). For the depth-mean straining term (B) they overlaidan empirical profile for unstratified tidal flow (see Bowdenand Fairbairn, 1952) with a profile for estuarine circulationderived from analytical theory (see Officer, 1976) as estimatefor ~u. With these assumptions, the empirical depth-meanstraining term is of the following form:

ðvtfÞB¼�

auþ bgD3

Avr0

vxr

�gDvxr ð21Þ

with the empirical parameters a ¼ 0.031 and b ¼ 0.0031, andthe depth mean eddy diffusivity

Av ¼ gjujD; ð22Þ

see Simpson et al. (1990). In eq. (22), g is an empirical param-eter which is estimated here as g ¼ 0.00333 (see also NunesVaz et al. (1989)) by taking the depth average of the parabolicempirical eddy viscosity formulation for channel flow (see e.g.Fischer et al. (1979)):

Av ¼ ku�ð � zÞ�

1þ z

D

�¼ kc

1=2d jujð � zÞ

�1þ z

D

�ð23Þ

with the bed friction coefficient cd ¼ 2.5 $ 10�3, the von Kar-man number k ¼ 0.4 and the bed friction velocity, u*. In orderto prevent the estuarine circulation part of eq. (21) from in-creasing towards infinity for slack tides, a lower backgroundlimit for Av has been prescribed as one tenth of the maximumvalue.

The mixing term has been estimated by Simpson et al.(1990) as

ðvtfÞE¼�cdGr0

ju3jD

ð24Þ

with the bulk mixing efficiency G. Here we use G ¼ 0.04,which means that 4% of the turbulent kinetic energy producedin the entire water column is used for vertical mixing (increas-ing the potential energy) and 96% are dissipated into heat.Given that a typical local mixing efficiency is 20% for strati-fied flow (Osborn, 1980) and that unstably or neutrally strati-fied situations (with negative or zero mixing efficiencies) arepresent at times, a bulk mixing efficiency of 4% seems reason-able. This is by a factor of 10 larger than the value chosen bySimpson et al. (1990) who used estimates from mixing in shelfseas which are substantially deeper than the 10 m deep testcase considered here and where mixing is probably is much

less efficient due to the typically well-mixed bottom boundarylayer. Thus, the bulk mixing efficiency may be considered asan adjustable parameter which may be estimated by meansof comparing ðvtfÞE from eq. (24) and the mixing term E ineq. (14).

Wind mixing, although included in our simulations dis-cussed in Section 3.2, is not considered for this comparison,since it is here hardly contributing to changes in f. Nonethe-less, in order to roughly separate mixing in the surface (SBL)and in the bottom boundary layer (BBL), we have estimatedthe height of the SBL by finding at each time the highest localminimum of the turbulent kinetic energy (TKE) and definingthis location as the lower bound of the SBL. The tidal mixingpart of term B in the f-equation has then been defined as theintegral of the buoyancy flux G from the bottom to the lowerbound of the SBL, thus excluding wind mixing.

Fig. 3 shows a comparison between the straining and mix-ing terms of the f-equation computed from the one-dimen-sional model in Section 3.2 and as estimated from theempirical model given in the equations (21) and (24). It canbe seen that there is good agreement for the straining term be-tween the empirical and the resolved model only during lateflood when stratification is weak. During full ebb (t/T ¼ 0.25) and full flood (t/T ¼ 0.75) straining is underesti-mated by the empirical model, because the modification ofthe velocity profiles due the tidal mixing asymmetry (en-hanced mixing during flood and suppressed mixing duringebb due to tidal straining, see Jay and Musiak, 1994), is notconsidered. The mixing terms of the resolved and the empiri-cal model are in quite good agreement for ebb (0 < t/T < 0.5).However for flood, the agreement is poor, in the first half ofthe flood (0.5 < t/T < 0.75) due to a time lag between mixingand flow velocity represented only in the resolved model, andaround the end of the flood (0.8 < t/T < 0.95) due to the factthat most of the mixing acts on unstable stratification in the re-solved model, such that the effective bulk mixing efficiencybecomes negative. The mixing term may be parameterised ina more realistic way by using a formulation with the mixingefficiency varying with the value of f (see Simpson andBowers, 1981).

685H. Burchard, R. Hofmeister / Estuarine, Coastal and Shelf Science 77 (2008) 679e687

3.4. Two-dimensional estuarine dynamics

For studying the impact of f-advection (term A in eq. (14)),the non-mean straining (term C in eq. (14)) and verticaladvection (term D in eq. (14)), which are all not included inthe one-dimensional simulations of Section 3.2, we carry outa two-dimensional simulation along an idealised estuary.This scenario is a slight modification of the idealised estuarineused by Warner et al. (2005) for comparing various turbulenceclosure schemes. The length of the estuary is 100 km, thedepth is linearly decreasing from the open boundary at theriver mouth at x ¼ 0 km (depth: 15 m) to the riverine boundaryat x ¼ 100 km (depth: 10 m). Earth rotation is neglected. Thehorizontal resolution is Dx ¼ 500 m, and 20 equidistant s

layers are used in the vertical. The timestep is dt ¼ 20 s forthe barotropic mode (explicitly calculated free surface) aswell as for the baroclinic mode, including the rate of changeof potential energy and its contributions. As initial conditions,we chose the same setup as Warner et al. (2005), with a verti-cally homogeneous salt distribution, with a salinity of 30 forx � 30 km, a salinity of 0 for x � 80 km and decreasing line-arly from 30 to 0 in between. As boundary conditions, we pre-scribe a salinity of 30 at x ¼ 0 and a freshwater runoff at theriver boundary with a velocity of 0.01 m s�1. At the openboundary, we further prescribe a sinosodial semi-diurnal tidewith a sea surface elevation amplitude of 1 m, resulting intoa horizontal velocity amplitude of about 1 m s�1 near theopen boundary. Meteorological forcing is not considered. Inorder to allow for complete destabilisation of the water columndue to tidal straining at the end of the flood in parts of the es-tuary, the present scenario deviates from the Warner et al.(2005) idealised estuary by increased the tidal forcing anddepth and decreased freshwater inflow.

The simulation was carried out for 8 days. After about5 days, a periodic steady state is reached, with a salt wedgemoving up and down the estuary with the tide, see the snap-shot at the end of ebb in Fig. 4.

The dynamics of the potential energy anomaly is examinedat two locations with different characteristics: at x ¼ 38 km(position A), where the SIPS mechanism fully destabilisesthe water column during flood and at x ¼ 80 km (positionB), where the water column is strongly stratified during thewhole tidal period.

0 10 20 30 40−15

−10

−5

0

1416182022242628

A

x /

z / m

Fig. 4. Salinity distribution in the idealised estuary near the end of ebb. The section

the f-equation have been extracted, see Fig. 5.

At position A, the dynamics is comparable to the situationsimulated with the one-dimensional model in Section 3.2. Theonly terms playing a significant role are the depth-mean strain-ing and the vertical mixing, see Fig. 5. After the end of ebb, f

has a maximum, which is then steeply eroded by both, depth-mean straining and mixing, such that at full flood the watercolumn becomes unstably stratified. Here, in contrast to theone-dimensional study, a local maximum of f is visiblearound the slack water after flood, showing that here estuarinecirculation is more effective than in the one-dimensionalmodel. A similar intensification of the stratification maximumafter slack tide flood has been observed by Burchard et al. (inpress) for a three-dimensional model simulation in the WaddenSea if horizontal density gradients are considered. A smallcontribution of f-advection is also visible for location A.

At location B, the balance of the terms in the f-equation iscompletely different. Here, the major balance is between thef-advection, the vertical advection and the depth-mean strain-ing. Smaller contributions come from vertical mixing andnon-mean straining. After ebb, f has a minimum value, sincethe salt wedge, the maximum extent of which is always up-stream of location B, has a minimum thickness at minimumdensity. After the onset of flood, f-advection strongly forcesstabilisation of stratification, however, depth-mean straining isopposing to it, similarly to the situation at location A. With in-creasing flood current, the vertical advection is stabilising theflow as well, due to a rise of the main halocline. At slack tideafter flood, f has a maximum value, which is then mainly re-duced by f-advection, again opposed by depth-mean straining.The significant role of the non-mean straining is explained hereby the fact that the horizontal density gradient is substantial in-side the salt wedge, but negligible in the near-surface waters.

4. Conclusions

With the time-dependent dynamic equation for the potentialenergy anomaly, f, which has been rigorously derived herefrom the dynamic equations for potential temperature and sa-linity, the continuity equation and the equation of state for seawater, we have now provided a complete reference solution forall empirical model parameterisations. The physical basiswhich has been used here is sufficiently general for includingmost of the processes relevant for estuaries and coastal seas.

50 60 70 80 90 100

24681012

B

km

s A and B indicate the two positions at which time series of f and the terms in

1T 2T 3T

0

5

10

/ Jm

-3

A

−3

−2

−1

0

1

2

x 10−3

t

t

/ W

m-3

t / T

1T 2T 3T45

50

55

60

B

−1

−0.5

0

0.5

1

1.5x 10−3

t / T

3

depth–mean straining mixing −advection vertical advection non−mean straining:

Fig. 5. Time series of f (upper panels) and the terms in the f-equation (lower panels) for the locations indicated in Fig. 4. The terms shown in the lower panel are

the depth-mean straining (term B in eq. (14)), vertical mixing (term E in eq. (14)), f-advection (term A in eq. (14)), vertical advection (term D in eq. (14)), and

non-mean straining (term C in eq. (14)). The vertical dotted lines roughly indicate the end of the ebb flow.

686 H. Burchard, R. Hofmeister / Estuarine, Coastal and Shelf Science 77 (2008) 679e687

The major approximation on which the dynamic equations forpotential temperature and salinity are based on is the Reynoldsaveraging of state variables leading to the definition of turbu-lent fluxes. The down-gradient approximation for these turbu-lent fluxes applied here has been included for conveniencesince it is used in most coastal models, but a general formula-tion of the turbulent salt and heat fluxes could be considered ina straight-forward way. A number of thermodynamic approx-imations has been made as well, commonly summarised as theBoussinesq approximation. Some of these idealisations couldalso be relaxed and included into the f-equation. With thef-equation, we have now a tool for completely quantifyingthe processes of mixing and stratification from numericalmodel simulations.

In realistic estuarine and coastal scenarios, all terms in thef-equation will be present, but typically the major local bal-ance will be based on a few terms only. It is however clearfrom the two-dimensional estuarine scenario discussed in Sec-tion 3.4 that completely different regimes may be spatially lo-cated close to each other, each with a different balance ofterms. Regimes may also change in time (e.g. seasonal cycle,spring-neap cycle) and with this also the balance of terms inthe f-equation.

Great effort has been made by many studies to find empir-ical estimates for various processes leading to depth-meanstraining (term B) and vertical mixing (term E), for whichthe f-equation provides only one term each. Depth-meanstraining is divided into estuarine circulation (asymmetric)and tidal straining (symmetric for ebb and flood), see equa-tion eq. (21). It should in principle be possible in numericalmodel studies to separate tidal flow velocity profiles intosymmetric and asymmetric parts in order improve the empir-ical estimates for depth-mean straining, but wind, Earth

rotation and non-rectilinear tides would complicate this. Al-though rotational effects are not directly included in the f-equation, they have an indirect impact on the stability ofthe water column through the advection and straining terms.This has been demonstrated by Rippeth et al. (2001) in theirfield observations of transverse straining in Liverpool Baydue to Coriolis acceleration; see also the numerical simula-tions by Simpson et al. (2002).

Due to non-linear interactions, similar problems wouldarise for numerically separating influences of wind, tides, sur-face cooling and waves on vertical mixing. An attempt to doso has been demonstrated in Section 3.3 where wind mixingwas assumed to be the part of mixing which takes place inthe surface mixing layer, defined by the depth range abovethe highest TKE minimum in the water column. This workedfine for constant wind stress, but for highly varying wind stressand simultaneous impact of surface waves and surface cooling,these impacts can practically not be separated. This means thatfor realistic flow situations empirical estimates of mixing willbe inaccurate and that the model-based mixing impact on strat-ification as given by the potential energy anomaly has to belimited to term E, the vertically integrated vertical buoyancyflux.

Acknowledgements

The authors are grateful to John Simpson (Bangor, Wales),Charita Pattiaratchi (Crawley, Australia), Rainer Feistel (War-nemunde, Germany), Julie Pietrzak and Gerben de Boer(Delft, The Netherlands) for their valuable comments on themanuscript. We are further indebted to Karsten Bolding (Bar-ing, Denmark) for keeping GOTM and GETM up and running

687H. Burchard, R. Hofmeister / Estuarine, Coastal and Shelf Science 77 (2008) 679e687

and to Anja Bachmann (Warnemunde, Germany) for her sup-port in setting up the two-dimensional idealised estuary.

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