On the migration of tidal free bars

PHYSICS OF FLUIDS 18, 096601 �2006�

On the migration of tidal free barsValeria Garotta,a� Michele Bolla Pittaluga,b� and Giovanni Seminarac�

Department of Environmental Engineering, Genova University, Via Montallegro 1, 16145 Genova, Italy

�Received 6 June 2005; accepted 23 May 2006; published online 13 September 2006�

We derive and employ a depth-averaged model for the formation of free bars in infinitely long tidalchannels in order to investigate the mechanism whereby tidal bars may experience a net migrationover a tidal cycle. The flux of the suspended sediment is modeled by means of an analyticalrelationship derived by Bolla Pittaluga and Seminara �M. Bolla Pittaluga and G. Seminara, WaterResour. Res. 39, 5 �2003�� for slowly varying flows. The model is validated by performing a linearstability analysis of flow and bed topography in a rectangular channel with an erodible bed, subjectto the propagation of a symmetric tidal wave of small amplitude. The results of the present depthaveraged model show a fairly satisfactory agreement with previous results based on athree-dimensional model �G. Seminara and M. Tubino, J. Fluid Mech. 440, 49 �2001��. We theninvestigate the role of overtides, showing that a flood or ebb asymmetry of the basic flow gives riseto a net migration of bars. The mechanism is due to the nonlinearity of the dependence of sedimentflux on bottom stress. This phenomenon is somewhat similar to processes occurring in various fieldsof fluid mechanics, such as steady streaming in an oscillatory boundary layer �N. Riley, Annu. Rev.Fluid Mech. 33, 43 �2001�� or acoustic streaming. The present investigation also bears somerelevance to the problem of nonlinear development of tidal bars, as it suggests that a depth averagedapproach may be adequate to its treatment, while definitely requiring a computational effort muchsmaller than treatments based on 3D formulations. © 2006 American Institute of Physics.�DOI: 10.1063/1.2221346�

I. INTRODUCTION

The bed topography of tidal channels is characterized bythe presence of a wide variety of bedforms of spatial scalesfalling in the range of a few centimeters to some kilometers.Tidal bars are sediment waves with wavelengths scaling withchannel width. They arise from an instability of the interfacebetween the erodible bed and the fluid phase. Dalrymple andRhodes1 classified estuarine bars into “repetitive barforms”�alternate, point and braid bars�, “elongated tidal bars” �typi-cally observed at locations where there is strong, rectilinear,tidal flow, i.e., at the outer part of the macrotidal estuaries,but also at the mouth of estuaries with smaller tidal ranges�,and “delta-like bodies” �located at points of flow expansion,typically at the end of channels�. Our attention is focusedhere on the first class of estuarine bars, in particular we in-vestigate the formation of alternate free bars in straight tidalchannels �Fig. 1�. Bars are also typically observed in rivers,where they may occur in single rows �alternate bars� or inmultiple rows �Fig. 2� depending on the value of the aspectratio �width/depth� of the channel being relatively small orlarge. The problem of formation and evolution of free barshas been widely investigated in the fluvial context �Colom-bini et al.2; Schielen et al.3; Repetto et al.4; Federici andSeminara5� being considered as a key factor controlling sev-eral important fluvial processes, like river meandering andbraiding. Recently, Seminara and Tubino6 have analyzed theproblem of formation of tidal bars in long straight channels

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by means of a three-dimensional hydrostatic model showingthat, unlike river bars, tidal bars forming under a symmetri-cal small amplitude tide are nonmigrating features; more pre-cisely, in the absence of a mean residual current and over-tides, tidal bars migrate alternatively forward and backwardin a symmetric fashion, displaying a vanishing net migrationin a tidal cycle. In order to evaluate the effect of overtides onbar migration, we employ a depth averaged model for barformation. This goal can be pursued, accounting for thethree-dimensional nature of the suspended load, followingthe recent approach proposed by Bolla Pittaluga andSeminara7 to deal with the transport of suspended sedimentsin slowly varying contexts. Essentially Bolla Pittaluga andSeminara7 take advantage of the assumed minor importanceof advective effects relative to vertical settling and turbulentdiffusion, the ratio between the former effect and the latterones being measured by a small parameter �. They then ex-pand the solution for the sediment flux in powers of thisparameter; at leading order the classical “Rouse” balance be-tween vertical settling and turbulent diffusion leads to thewell known “equilibrium” relationship for the suspendedsediment flux; at the next order advection gives rise to cor-rections proportional to the spatial and temporal derivativesof the average concentration. The inclusion of such correc-tions allows one to describe the settling lag effect, wherebyparticle settling occurs at some distance from the locationwhere particles have been entrained. The importance of thelatter effect strongly depends on the length scale of spatialvariations of the hydrodynamics. For large length scales, set-tling lag is negligible. On the contrary, it will be shown that,
at the scale of bars, settling lag is crucially important, being
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096601-2 Garotta, Bolla Pittaluga, and Seminara Phys. Fluids 18, 096601 �2006�

responsible for the damping of high wavenumber bar pertur-bations. Hence, through the inclusion of the O�� correctionfor the suspended sediment flux, it will appear that thepresent analysis captures the main mechanism responsiblefor the formation of bars in tidal channels, a result whichencourages one to use depth averaged models of the presenttype to investigate more complex problems, like the nonlin-ear development of tidal bars.

Having validated our depth averaged approach we thenproceed to investigate bar instability in the case of asymmet-ric tides. The analysis is made simpler by expanding thesolution for bar perturbations in powers of the small param-eter �. Results show that a flood or ebb asymmetry of thebasic flow, induced by the presence of overtides, gives rise toa net migration of bars which depends on the aspect ratio ofthe channel as well as on the dimensionless parameters con-trolling the hydrodynamics and sediment transport. Typi-cally, the net migration falls in the range of a fraction of barwavelength per day.

The rest of the paper proceeds as follows: The hydrody-namic and morphodynamic formulations are presented inSec. II. In Sec. IV we perform a linear stability analysis ofthe basic tidal solution, derived in Sec. III, with respect toperturbations of the bar type. Results, concerning in particu-lar the net migration of bars, are reported in Sec. V, which isfollowed by some concluding remarks in Sec. VI.

II. FORMULATION OF THE PROBLEM

A. Hydrodynamics

We consider an infinitely long straight channel with rect-angular cross section of constant width 2B*, connected at oneend to a tidal sea �Fig. 3�. We assume that the channel banksare nonerodible while the bed is cohesionless. The sedimentlying on the bed is assumed to have uniform grain size ofdiameter ds

* small enough to be suspended by tidal currentsthroughout most of the tidal cycle. Note that, hereafter, a starwill denote a dimensional quantity. Bar length is typically ofthe order of a few channel widths; this suggests that theproblem of bar formation can be investigated using the shal-low water equations. These well known equations are readilyderived performing a depth average of the Reynolds equa-

FIG. 1. A three-dimensional view of the bed elevation of a tidal bar ob-served in the experiments of Tambroni et al.25 Note the presence of smallerscale bedforms superimposed on bars. The x coordinate measures the dis-tance from the inlet.

tions. With the help of the kinematic and dynamic boundary


conditions at the bottom and at the free surface, one ends upwith the set of equations reported below. Note that the hy-drostatic approximation is embodied into the shallow waterassumption; in other words, the mean pressure �averagedover turbulence� is hydrostatically distributed, which allowsone to get rid of the vertical component of the momentumequations, expressing the mean pressure in terms of the freesurface elevation. The hydrodynamical problem in dimen-sionless form then reads

�0U,t + UU,x + VU,y + H,x + ��x

D= 0, �1a�

�0V,t + UV,x + VV,y + H,y + ��y

D= 0, �1b�

�0D,t + �UD�,x + �VD�,y = 0. �1c�

In Eqs. �1a�–�1c� x and y denote the longitudinal andlateral Cartesian coordinates, respectively, with x alignedwith the channel axis; t is time; U and V are longitudinal andlateral components of the depth averaged velocity; H is the

FIG. 2. Multiple row bars in the Waimakariri River, New Zealand �courtesyof Bianca Federici�.

free surface elevation referred to the still water level; D is


096601-3 On the migration of tidal free bars Phys. Fluids 18, 096601 �2006�

flow depth, while �x and �y are the longitudinal and lateralcomponents of the bottom stress. The dimensional quantitieshave been scaled as follows:

�x*,y*� = B*�x,y� , �2a�

t = �t*, �2b�

�U*,V*� = Ur*�U,V� , �2c�

�H*,D*� = Dr*�Fr

2H,D� , �2d�

�* = �Ur*2� , �2e�

where � is the angular frequency of the tidal wave, Ur* is the

peak velocity of the basic tidal wave in the channel reachunder investigation, Dr

* is the mean water depth and Fr2 is the

square of a characteristic Froude number. Note that the scal-ing adopted herein arises from the observation that perturba-tions of free surface elevation scale typically with the quan-tity Ur

*2 /g. Hence, in the dimensionless problem �1� threeparameters arise, namely the square of the Froude numberFr

2, the aspect ratio of the channel �, and the ratio �0 be-tween the time required for the average flow to travel along areach of length B* and the tidal period. They read

Fr2 =

Ur*2

gDr* , � =

B*

Dr* , �0 =

�B*

Ur* , �3�

where g is gravity. Both Fr2 and �0 are typically small pa-

rameters. In fact, with typical values of the channel width B*

falling in the range of 10−103 m and Ur* about 1 m/s, the

value of �0 for a semidiurnal tide ��=1.4�10−4 s−1� isfound to fall in the range 10−3−10−1; moreover, with flowdepth of the order of 10 m, Fr

2 typically takes values about10−2. Note that the problem of bar formation can be studiedemploying the shallow water equations provided that thechannel half width B* is much larger than the mean flowdepth Dr

*, an assumption which, written in mathematicalform, reads

� � 1. �4�

The governing equations �1a�–�1c� must be associated with aclosure relationship for bottom stress which can be written in
the dimensionless form

��x,�y� = Cf�U,V��U2 + V2, �5�

where the friction coefficient Cf, in the case of an undis-turbed plane bed, can be given the classical logarithmic form

Cf = �6 + 2.5 ln� D

2.5ds�−2

. �6�

Here, ds is a measure of the relative roughness ds* /Dr

*, havingassumed the value 2.5ds

* for the absolute roughness �fromEngelund and Hansen8�. The boundary conditions at the sidewalls to be associated with the differential system �1� imposevanishing fluid flux through the walls, hence

V = 0 �y = ± 1� . �7�

B. Evolution equation of the bed interface

In order to describe the evolution of the bed interface,the hydrodynamics must be coupled with an overall state-ment of mass conservation for the solid phase. Denoted by q*

and c the local and instantaneous values of the volume fluxof the sediment and of the volume concentration of solidparticles in the flowing mixture, respectively, the equation ofmass conservation for the solid phase may be written in theform

c,t* + � · q* = 0. �8�

The boundary conditions associated with �8� impose

• no flux through the free surface

�q* · n − Vnh* c�z*=H* = 0, �9�

• the flux through the bed interface is driven by entrainmentof bed material due to the motion of the interface, hence

�q* · n − Vn* �c − cM��z*=* = 0, �10�

being denoted by n the outward normal unit vector and byVnh

* and Vn* the normal components of the speeds of the free

surface and of the bed interface, respectively; furthermore,* represents the bed elevation �H*−D*� and z* is theCartesian coordinate normal to the x*−y* plane. Finally, cM

FIG. 3. Sketch of the channel andnotations.

is the packing concentration of the solid phase in the granu-



lar medium and is often written as �1− p� with p as the voidfraction.

In order to derive a differential equation governing themotion of the bed interface one must first average �8�–�10�over turbulence and then integrate the resulting equationsover the flow depth. Taking advantage of the assumptionthat, typically, in estuaries and lagoons the flowing mixture issufficiently dilute for its average concentration to be at least2 orders of magnitude smaller than cM �which is typically anorder 1 quantity�, one finally obtains the following globalstatement of mass conservation for the solid phase:

�1 − p�,t** + �h

* · Q* = 0. �11�

Q* is the total flux of sediment transported per unit channelwidth, hence

Q* = �Qx*,Qy

*� = *

H*

�qx*,qy

*�dz*, �12a�

�h* � �

�x* ,�

�y*� �12b�

and the symbol � �, hereafter omitted, denotes average overturbulence. Equation �11� is commonly known as the 2DExner9 equation. In dimensionless form it reads

�1 − p�,t +q0

�0�h · Q = 0, �13�

where the sediment flux per unit width Q has been scaledusing the classical Einstein10 scale ��s−1�gds

*3� and q0 isthe following dimensionless parameter

q0 =��s − 1�gds

*3

Ur*Dr

* , �14�

with s as the relative particle density �typically 2.65�. Notethat q0 is extremely small. In fact, the size of the sedimentsin tidal environments is about 0.1 mm, while the flow depthis typically of the order of some meters. Hence the value ofq0 is about 10−6. Note that, since q0 /�0 is very small, Eq.�12� suggests that the time scale of the bed evolution is muchlarger than the tidal period; in other words bar growth arisesfrom the integrated effect of bed evolution over several tidalperiods. Finally we impose the boundary conditions requir-ing that the sidewalls be impermeable to sediments and write

Qy = 0 �y = ± 1� . �15�

In order to complete the formulation of the problem an ap-propriate closure relationship for the sediment flux Q isneeded.

C. Sediment transport

The transport of sediments in open channels may occurthrough different mechanisms. The interested reader mayfind it useful to refer to the brief summary contained inSeminara11. It suffices here to state that, in those portions ofestuaries and lagoons where sediments are noncohesive,transport occurs typically in the form of bed load and sus-
pended load. The distinction between these two mechanisms

is not purely conventional; it is motivated by the role playedby turbulent coherent structures acting near the bed interface.

Bed load occurs through particles sliding, rolling andmostly saltating within a layer adjacent to the bed of thick-ness of the order of few �typically 2–3� grain diameters. Un-der equilibrium conditions �supply equal transport capacity�the average entrained flux of sediments is balanced by thedepositing flux, the average being taken over time. Further-more entrainment is activated by sweeps sufficiently intenseto mobilize clouds of particles which move downstream afinite length until they stop waiting for a new sweep able tomobilize them again �Drake et al.12�.

Suspended load occurs through particles small enough tobe entrained into the bulk of the flow, i.e., able to overcomethe “logarithmic barrier,” which is then transported by theflow as nearly passive tracers except for their tendency tosettle due to their excess weight. Coherent turbulent struc-tures responsible for particle entrainment in suspensionare sufficiently energetic bursts �Sutherland13 and Klineet al.14�. Under equilibrium conditions the average entrain-ment balances again the average deposition.

We now briefly outline some of the available knowledgeon the tools developed to quantify bedload and suspendedload.

Bed load

Let us define an important dimensionless parameterwhich is a measure of the ratio between the scale of desta-bilizing forces acting on sediment particles and the stabiliz-ing effect of gravity, the Shields parameter �Ref. 15�. Interms of the bottom stress �0 it reads

=�0

��s − 1�gds* , �16�

where � is the fluid density. A wide literature on the incep-tion of bedload transport starting from the work of Shieldshas established that incipient transport in a statistical senseoccurs as exceeds some critical value c. Under uniformflow conditions, at equilibrium, the dimensionless bed loadflux per unit width, denoted by Qb, is aligned with the bot-tom stress �, and its intensity, denoted by �, is a monotoni-cally increasing function of the excess Shields stress�−c�. A well established semiempirical relationship em-ployed to evaluate the function � is due to Meyer-Peter andMüller16 and reads

� = 8� − 0.047�3/2. �17�

On sloping beds the gravitational tendency of particles tomove downhill modifies the average particle trajectory. Onweakly sloping beds the deviation angle is linearly propor-tional to the lateral bed slope, hence the bed load flux perunit width can be written in the dimensionless form

Qb = ��

��− R,y j , �18�

where R is an empirical parameter which has been shown to17 18
depend on �Parker and Talmon et al. � in the form


R = 0.56/�� 19�

and j is the unit vector in the y direction. Below we neglectthe effects of the longitudinal bed slope, typically smallerthan its lateral component.

Suspended load

Incipient transport in suspension is also associated withsome threshold value N of the Shields parameter dependingon the particle Reynolds number Rp. The function N�Rp� hasbeen experimentally investigated by various authors �e.g.,Van Rijn19�. However, note that the parameter usually em-ployed to define the threshold conditions for transport in sus-pension is the so-called Rouse number Z, defined as Ws

* /ku*,with Ws

* the settling speed, u* the friction velocity, and k theVon Karman constant. It is easy to show that Z is simply afunction of the Shields stress and of the particle Reynoldsnumber Rp. For values of Z below N�Rp�, under uniformequilibrium conditions, a vertical distribution of the meanconcentration C0 is established as a result of an exact averagebalance between upward turbulent diffusion and downwardvertical settling. This distribution is characterized by an“equilibrium concentration” Ce evaluated at some small con-ventional reference distance zR

* from the bed �Rouse20�. Sys-tematic experimental investigations �Van Rijn19� show thatthe value of Ce depends on the uniform Shields stress 0 forgiven particle Reynolds number Rp �i.e., particle size�, whilethe value of zR

* is a function of the bottom roughness. Theclosure relationships for Ce and zR

* employed for the presentcalculations are reported in Appendix A.

Under uniform equilibrium conditions the flux of sus-pended sediment per unit width Qx0

s* is aligned with the di-rection x of the uniform bottom stress. Expressing the uni-form logarithmic velocity distribution in the form Ur

*F��with � dimensionless vertical coordinate scaled by Dr

*, theintensity of the suspended flux per unit width reads

Qx0s* = �Ur

*Dr*�

�R

1

C0��F��d� = �Ur*Dr

*� 0, �20�

or, in dimensionless form

q0Qx0s = 0�0,Rp� . �21�

The function 0 has first been evaluated by Rouse20 and canbe written in the form

0 = C0I��R,Z� , �22�

with I an integral function of the Rouse number, reported inAppendix A, and C0 the depth averaged value of the meanconcentration.

Transport in suspension under nonuniform and unsteadyconditions is affected by additional ingredients, namely ad-vection as well as temporal variations of the mean concen-tration. Fortunately, these further ingredients, when evalu-ated at the spatial scale of the bars and at the temporal scaleof the tides, turn out to be “slowly varying” both in space
and time. This is readily shown comparing the orders of

magnitude of the advective term and vertical settling in theadvection-diffusion equation. Denoted by L0

* the length ofthe scale of bars, one finds

�u*C,x*�

�Ws*C,z*�

� O�Ur*Dr

*

Ws*L0

*� = O� 1

2�

Ur*�

Ws*�� , �23�

where u* is the local longitudinal velocity and � is the di-mensionless bar wavenumber defined as �=2�B* /L0

*. A par-ticle with a diameter of 0.1 mm, settles with a speed rangingabout 1 cm/s. With Ur

*�1 m/s, ��0.1, ��5, it followsthat advective effects are an order of magnitude smaller thenvertical settling.

Similarly

�C,t*�

�u*C,x*�� O� L0

*

Ur*T� , �24�

with T as the tidal period.With L0

*�103 m and the above value of Ur*, we find that

for a semidiurnal tide the term involving the time derivativeof the concentration in the advection-diffusion equation is anorder of magnitude smaller than advective terms. Let us de-fine the dimensionless parameters �1 and � in the form

�1 =�Dr

*

Ws* , �25a�

� =Ur

*

Ws*

Dr*

B* . �25b�

Note that the definition �25b� involves the channel widthrather than the bar wavelength, as �23� would suggest. Thischoice has been made in order to avoid an uncomfortabledependence of the small parameter � on a variable quantity,namely the bar wavenumber. We formally assume that � is asmall parameter, recalling that this assumption can safely beemployed for sufficiently large bar wavelengths. Moreover,we also assume that �1��. These assumptions have allowedBolla Pittaluga and Seminara7 to develop a depth-averagedmodel for the transport of suspended sediment in slowlyvarying flows, based on a rational perturbation method. It isworth noting that the approach of these authors retains in aformal fashion all the relevant information arising from thethree-dimensional nature of the advection-diffusion equation.The output of their analysis is a rational perturbation expan-sion for the suspended flux per unit width of the form

q0�Qxs,Qy

s� = D�U,V�� 0 + � 1 + O��2�� . �26�

In fact, at O�� a correction C1 for the mean concentrationarises from the effects of advection, hence the correction 1

for the suspended flux reads

1 = �R

1

C1F��d� = �D�U�C0

�x+ V

�C0

�y� , �27�

with � an integral function of �R and Z reported in
Appendix A.


III. BASIC FLOW

The basic flow felt at the scale of bars reduces, at theleading order of approximation, to a uniform unsteady flow.This is readily shown. In fact, if we set

�U,V,H,D� = �U0,0,H0,D0� �28�

the hydrodynamic equations take the simpler form

�0U0,t + U0U0,x + H0,x + �Cf0U0�U0�

D0= 0, �29a�

�0D0,t + U0D0,x + D0U0,x = 0, �29b�

where Cf0, in the case of the plane bed, is obtained from �6�setting D=D0. Moreover, at the spatial scale of bars and atthe time scale of bar growth, the average bed profile may betaken to be horizontal; having chosen the still water level asa reference elevation, we may then set H0= �D0−1� /Fr

2. Wecan readily show that the basic solution can be given a per-turbative structure of the form

U0 = U0�t� + x · O�Fr2�Cf0,�0Fr

2� , �30a�

D0 = D0�t� + x · O�Fr2�Cf0,�0Fr

2� . �30b�

Hence, as Fr2�10−2, ��10, Cf0�10−2−10−3, �0�10−1

−10−3, flow depth and average speed can be assumed spa-tially uniform at the leading order of approximation. Thestructure of the functions U0�t� and D0�t� cannot be deter-mined on a local basis; they are the solutions of the fullcontinuity and De Saint Venant equations describing thepropagation of the tidal wave with appropriate boundaryconditions forced at the channel ends. In the following, weanalyze tidal waves forced by sea oscillations of small am-plitude a0

*, such that we can write

D0 = 1 + O�� 31�

with � defined as the small ratio a0* /Dr

*. We then investigatevarious forms of the basic velocity, characterized by differentharmonic contents.

IV. LINEAR STABILITY OF TIDAL BARS

Let us analyze the stability of the basic flow discussed inthe previous section with respect to perturbationsU1 ,V1 ,H1 ,D1 of infinitesimal amplitude �. Since we employthe formulation of Bolla Pittaluga and Seminara7 for the sus-pended flux, it is appropriate to expand the perturbed flowconfiguration in the form

�U,V,H,D� = �U0�t�,0,0,1� + ��U1,V1,H1,D1� , �32�

� 0, 1� = � 00,0� + �� 01, 11� , �33�

and further expand linear perturbation in powers of the smallparameter �, hence

�U1,V1,H1,D1, 01, 11� = �U10,V10,H10,D10, 010, 110�

+ ��U11,V11,H11,D11, 011, 111� .

�34�

We then focus our attention on a class of linear perturbations


which admits a Fourier representation both in the longitudi-nal and in the transverse direction, thus performing a normalmode analysis, and write

�U1k,V1k,H1k,D1k�

= �u1k�t�Sm,v1k�t�Cm,h1k�t�Sm,d1k�t�Sm�E + c.c.,

�35�k = 0 .. 1;

where m denotes the transverse Fourier mode, c .c. denotesthe complex conjugate of a complex number and Sm ,Cm ,Eread

Sm = sin�m�

2y, Cm = cos�m

�

2y, �m odd� �36a�

Sm = cos�m�

2y, Cm = sin�m

�

2y, �m even� �36b�

E = exp�i�x� . �36c�

The lateral structure of alternate bars corresponds to themode m=1. The mode m=2 describes central bars, whilehigher modes correspond to multiple row bars. On substitut-ing from �34� into the governing system and performing thelinearization, at the leading order we find the differentialproblem

�37�

where L is the linear differential operator reported in Appen-dix B. We now reduce the homogeneous problem �37� to findan equation for the leading order perturbation of the bedelevation 10= �Fr

2h10−d10� which reads:

10,t = C�t�10, �38�

which admits the solution

10�t� = A exp 0

t

C��d� , �39�

with A as an arbitrary constant and the function C�� re-ported in Appendix B� depending on the coefficients of thedifferential operator L. Note that the present analysis reducesto the fluvial �steady� case when the function C takes a con-stant value. At the next order of approximation a nonhomo-geneous differential system is obtained, whose homogeneouspart is identical with the differential system at O��0�. It
reads


�40�

where F10 is a forcing term depending on the solution at theprevious order, which reads

F10 = − i�U0�t� 110 = i�U0�t�G�t�10, �41�

with G�t� a function of time reported in Appendix B. Again,the problem is readily reduced and leads to the followingequation for the O�� correction of bed perturbation:

11,t = C�t�11 +F10�t�

�1 − p��0, �42�

which admits the solution

11�t� = 10�t�i�

�1 − p��0

0

t

U0��G��d� . �43�

The bottom perturbation then reads

1 = 10 + �11 = A exp 0

t

�C�� + �P��d� , �44�

where P is a function which satisfies the integral relationship

0

t

P��d� =1

�ln�1 + �

i�

�1 − p��0

0

t

U0��G��d� .

�45�

Once the bottom perturbation is known, the other variablesare readily derived �see Appendix B�.

Note that in the present approach we have neglected lo-cal inertia effects. In fact, perturbations of the hydrodynam-ics are driven by perturbations of the bottom, a processwhich occurs on a time scale much larger than the tidal pe-riod. As will be mentioned in Sec. V, local inertia of pertur-bations is also negligible.

The mechanism of bar instability and bar migration

Marginal stability conditions are identified by the valuesof the parameters � and � for which perturbations do neitherexperience a net growth nor a net decay in a tidal cycle T. Itis then convenient to express the perturbation of bed eleva-tion in the form

1 = A exp 0

t

��d� exp�− i�0

t

s��d� , �46�

where ��t� and s�t� are the instantaneous values of thegrowth rate and migration speed of bars, respectively. Theyread

��t� = R�C�t�� + �R�P�t�� , �47a�


s�t� = −1

��I�C�t�� + �I�P�t�� , �47b�

where the symbols R and I denote the real and imaginarypart of a complex number, respectively. Neutral stability con-ditions are then determined by setting

t

t+T

��d� = 0. �48�

Similarly, the net displacement of bars over a tidal cycle �xcan be defined as

�x = t

t+T

s��d� . �49�

In order to give a simple physical interpretation of bar insta-bility and bar migration, we follow the lead of Seminara andTubino21 and write

�u1,v1,h1,d1,qx1b ,qy1

b ,qx1s ,qy1

s �

= 1��1,�2,�3,�4,�5,�6,�7,�8�

· exp�− i��1,�2,�3,�4,�5,�6,�7,�8�� , �50�

where �i and �i �i=1· · 8� are periodic functions of timewhich denote the amplitude and phase lag of the correspond-ing quantity relative to bottom elevation, respectively. Forexample, u1 reads

u1 = cu1, �51�

where cu is a complex periodic function in the form

cu =�1

�Fr2�3 − 1�

, �52�

with �1 and �3 complex periodic functions reported inAppendix B. Using the Euler relationship, the complex co-efficient cu can be posed in the form

cu = �1e−i�1. �53�

Note that since �1 and �1 are periodic functions, they do notaffect the growth rate � or the migration s, but they onlydetermine a different temporal behavior of the velocity per-turbation u1 with respect to the bottom perturbation 1.

Moreover, we have defined:

�Qx1b ,Qy1

b ,Qx1s ,Qy1

s � = E�qx1b Sm,qy1

b Cm,qx1s Sm,qy1

s Cm� + c.c.,

�54�

where the subscript 1 refers to the solution corrected atO��. For example the longitudinal velocity perturbation U1

can be expressed in the form

U1 = A�1Sm exp 0

t

��d�

· exp�i��x − 0

t

s��d�� − �1� + c.c. �55�

On substituting from �50� into the sediment continuity equa-
tion it turns out that the instantaneous growth rate ��t� and


migration speed s�t� can be written as sums of four distinctcontributions:

� = �i=1

4

�i, �56�

s = �i=1

4

si, �57�

where �i and si �i=1· ·4� are functions of time which read

��1,�2,�3,�4� = −q0

A�1 − p��0��5 sin �5,

− M�6 cos �6,��7 sin �7,

− M�8 cos �8� , �58�

�s1,s2,s3,s4� =q0

A�1 − p��0��5 cos �5,

M

��6 sin �6,�7 cos �7,

M

��8 sin �8� , �59�

and

M = �− 1��m−1�m�

2. �60�

Hence bar instability and migration depend on the am-plitudes and phase lags of the longitudinal and transversalcomponents of sediment transport �bedload and suspendedload�. A discussion of the implications of �57�–�59� is re-ported in the next section.

V. RESULTS

We first analyze the case of a monochromatic �semidiur-nal� tide. Hence, the basic state is imposed in the form

U0 = cos t , �61a�

D0 = 1, �61b�


0 = rCf0

Cfr

U02, �61c�

where r and Cfrrepresent the values of the Shields stress

and friction coefficient, at the instant of maximum velocity.In Fig. 4 we report the marginal stability curve for alternatebars derived at the leading order of approximation and theone corrected to order O��. Note that the higher order cor-rections play a crucial role as bars shorten, being responsiblefor the damping of the high wave numbers. In fact, as the barwavenumber � increases, the spatial derivatives of 1

�see �27�� give an increasing contribution. The minimum ofthe marginal curve occurs for values of � much larger thanone, hence the depth-averaged approach proposed herein canbe safely employed. Also, note that, for high values of thebar wavenumber, a further unstable region appears at orderO��. For those values of � and �, the theory is likely tofail, so the actual physical meaning of these modes will re-quire further investigations. At this point it may be of someinterest to discuss how each term of �56� contributes to bargrowth. Figure 5�a� shows the temporal distribution of �i

�i=1· · 4� throughout the tidal cycle. Figure 5�b� shows the

FIG. 4. Marginal stability curves for tidal alternate bars at the leading orderof approximation O��0� and at the first order O��. Values of the relevantparameters are: Rp=4, ds=2·10−5, r=0.75.

FIG. 5. The instantaneous contribu-tions to bar growth �a� and the corre-sponding integral values �b� are plot-ted as functions of time. The curvesare calculated for �=5.5 and�=0.062, i.e., close to the minimum ofthe marginal curve. Values of therelevant parameters are: Rp=10,ds=5·10−5, r=2.



corresponding integral values, calculated along the maxi-mum growth rate curve for values of � and � close to thecritical ones. Note that, under these conditions, the largestcontribution to bar instability comes from the longitudinalcomponent of the suspended sediment flux. Lateral bedloadtransport has a comparable stabilizing effect ��2 is negative�which is driven by gravity �Eq. �17��. Note that Eq. �18�suggests that the stabilizing role of gravity is inversely pro-portional to � hence, for large values of �, instability invari-ably occurs �see, for details, Seminara and Tubino21�. Minoreffects are associated with the longitudinal component ofbedload transport and with the lateral component of sus-pended flux. These effects are closely related to the phase lagbetween sediment transport and bottom elevation. In particu-lar, in Fig. 6 we have plotted the temporal evolution of thephase lag between longitudinal suspended flux and bottomelevation �7 �see �58�� at the leading order of approximationO��0� and at first order O�� for two different values of thebar wavenumber �. Recalling Eq. �58�, it turns out that thelongitudinal suspended sediment flux is destabilizing �stabi-

FIG. 6. The phase lag �7 between the longitudinal component of the sus-pended sediment flux and bed elevation is plotted as a function of time fortwo different values of the bar wavenumber. A comparison between thesolutions obtained at the leading order approximation O��0� �thin line� andat the first order O�� thick line� is also shown. Values of the relevantparameters are: Rp=4, ds=2·10−5, r=0.75, �=5.5.


lizing� if �7 belongs to the third/fourth �first/second� quad-rants. The plot shows two opposite scenarios: for values ofthe bar wavenumber close to criticality ��=0.164�, the sus-pended sediment flux is destabilizing. On the contrary, atlarger values of the bar wavenumber ��=0.88� suspensionhas a stabilizing effect. Such a stabilizing effect does notoccur at the leading order of approximation O��0�, the mar-ginal curve at the leading order �see Fig. 4� being flat forhigh wave numbers. Figure 7 shows a comparison betweenthe present results and the three-dimensional, yet hydrostatic,model of Seminara and Tubino.6 The plot reports three mar-ginal stability curves obtained for different values of thepeak Shields stress r. In agreement with Seminara and Tu-bino’s findings the present model predicts a destabilizing roleof suspended load at low wave numbers, displayed by thedecrease of the marginal value of � associated with increas-ing Shields stress; high wave numbers show an oppositetrend, with an increasing stabilizing contribution associatedwith increasing suspended load. Hence, it appears that thedepth averaged model proposed herein is able to capture themain mechanisms underlying the formation of tidal bars.Some quantitative discrepancy appears to be justified by the

FIG. 7. Marginal stability curves foralternate bars for different values ofthe peak Shields stress r at the firstorder O�� Rp=4, ds=2·10−5�. Onthe left are reported the curves ob-tained by the three-dimensionalmodel.

FIG. 8. The value of the parameter � calculated along the marginal stabilitycurve is plotted as a function of �, for three different values of the peakShields stress r �Rp=4, ds=2·10−5�.



asymptotic nature of the present treatment. In fact, the per-turbative parameter � is not small enough to allow perfectagreement with the three-dimensional hydrostatic model, assuggested by Fig. 8. Note that the parameter � plotted in Fig.8 has been calculated choosing the effective spatial scale ofthe concentration filed, i.e., the bar wavelength, rather thanthe channel half width B*. Figures 9 and 10 show the depen-dence of the critical values for bar instability �c �Fig. 9� and�c �Fig. 10� on the peak Shields stress r, for three differentvalues of the roughness parameter ds.

We have also checked that local inertia is effectivelynegligible for the perturbations, by including at first order a�0 correction. Results concerning the marginal stabilitycurves are almost indistinguishable.

Figure 11 shows the marginal stability curves derived atorder O�� for the first three modes, namely the alternatebar mode m=1, the central bar mode m=2, and the multiplebar mode m=3. As expected, higher order modes are increas-ingly stable, and the critical values of the width-to-depthratio and of the bar wavenumber for the m-th mode are mtimes the critical values for the first mode

FIG. 9. The critical value for bar instability �c is plotted as a function of thepeak Shields stress �Rp=10�.

FIG. 10. The critical value of the bar wavenumber �c is plotted as a function
of the peak Shields stress �Rp=10�.

�cm = m�c1, �cm = m�c1. �62�

Let us finally discuss the main result of the present work,concerning the role of overtides on the migration of freebars. Hence, we write the basic flow velocity in the form

U0 = cos t + �M4 cos�2�t − �M4�� . �63�

In Fig. 12 we have plotted the marginal stability curvefor the case of a tide of the type �63�. Note that the marginalconditions are not affected by the presence of overtides, pro-vided the peak value of the Shields stress is kept constant. Inthe unstable region of the ��−�� plane we have reportedsome isomigration curves over a tidal cycle. The plot clearlysuggests that an overtide induces a net migration of bars,which can be positive or negative, depending on the valuesof � and �.

In agreement with the analysis of Seminara and Tubino,6

a basic flow consisting of a purely semidiurnal tide, givesrise to steady bedforms in the mean, i.e., bedforms charac-terized by no net migration over a tidal cycle. On the con-trary, we show that a flood or ebb dominance, namely thefact that the flood �ebb� peak velocity is higher than its ebb

FIG. 11. Marginal stability curves at the first order O�� for differentmodes �Rp=4, ds=1·10−4, r=0.75�.

FIG. 12. Marginal stability curve and isomigration curves over a tidal cycle−5
�Rp=10, ds=5·10 , r=2, �M4
=0.2, �M4=0�.



�flood� value, affects the amplitude and phase of perturba-tions, and leads to an asymmetric forward-backward move-ment of bed forms throughout the tidal cycle. We have re-cently been made aware that a similar mechanism has beenpointed out by Besio et al.22 for sand wave migration. Inorder to quantify the flow asymmetry, we define a parameter� in the form

� =�Uflood� − �Uebb�

max��Uflood�, �Uebb��, �64�

where Uflood and Uebb denote the flood and ebb peak velocity,respectively. In Fig. 13 we have plotted the net migration ofbars over a tidal cycle as well as the parameter � as a func-tion of the phase lag �M4 of the M4 component. Note that thenet migration of bars displays the same pattern as the � pa-rameter, being maximum when the overtide is in phase withthe dominant harmonic. Figure 14 shows the role of each of

FIG. 13. The dimensionless net bar displacement over a tidal cycle �x andthe flow asymmetry parameter � are plotted as functions of the phase lag�M4 of the diurnal component of the tide relative to its semidiurnal compo-nent ��=0.062, �=5.5, Rp=10, r=2, ds=5·10−5, �M4

=0.2, �M4=0�.


the quantities affecting bar migration �Eqs. �59�� for valuesof � and � close to the minimum of the marginal curve. Inthe case under consideration the largest contribution to themigration speed arises from the lateral component of the sus-pended sediment flux �s4�.

Figure 15 shows that the O�� correction has a signifi-cant role in the net bar displacement. At order O��, thetidal wave investigated herein gives rise to a dimensionlessnet migration �x�0.03, i.e., the bar migrates a distanceranging about 0.015 channel widths in a tidal cycle.

Figure 16 shows the dependence of the net displacementof bars over a tidal cycle on the peak Shields stress r, forthree different values of the roughness parameter ds. The netmigration is shown to be strongly enhanced by increasingvalues of the Shields stress and of the relative roughness.The curves have been calculated for values of the bar wave-number close to criticality and for given values of the ampli-tude and phase of the M4 overtide. Note that, as long as thepeak value of the Shields stress is kept constant, the ampli-tude of the quarter diurnal tide plays a minor role on bar

FIG. 14. The contributions s1, s2, s3,and s4 to the migration speed of bars�a� and the corresponding integral val-ues �b� at order O�� see Eq. �59��are plotted as functions of time��=0.062, �=5.5, Rp=10, r=2,ds=5·10−5, �M4

=0.2, �M4=0�.

FIG. 15. The dimensionless net displacement of the most unstable bar wave-number is plotted as a function of time at the leading order and at orderO�� =5.5, �=0.062, Rp=10, ds=5·10−5, r=2, �M4

=0.2, �M4=0�.



migration, the major effect coming from the phase lag of thelatter constituent with respect to the dominant harmonic, asalready shown in Fig. 13.

VI. CONCLUDING REMARKS

Let us first briefly summarize our achievements. Wehave shown that the problem of bar formation can be fairlysatisfactorily solved through a depth averaged model em-ploying the analytical relationship derived by Bolla Pittalugaand Seminara7 for suspended sediment flux; the marginalstability curves predicted by our analysis are consistent withthose derived by the three-dimensional model of Seminaraand Tubino6 and the role of suspended load is confirmed tobe crucial in the high wavenumber range. A depth averagedmodel has also the advantage of allowing for further non-linear developments with a relatively small computationaleffort.

Next we have shown that a flood or ebb asymmetrygives rise to a net migration of bars in a tidal cycle which canbe positive or negative depending on the phase lag of theovertides relative to the dominant harmonic. Under the ex-amined conditions, the length of bar migration turns out to beof the order of a percent of channel width per cycle.

Various developments of the present work will deserveattention in the near future. In particular, the present resultssuggest that further mechanisms leading to the occurrence ofa flood or ebb dominance may also cause a net migration oftidal bars in a cycle; one such mechanism is definitely relatedto finite amplitude effects acting on the basic state, wherebyflood or ebb dominance may be generated even under a pureharmonic forcing. Moreover, tidal channels are typicallylandward convergent, a feature which is likely to affect thespatial and temporal development of bars. The finite lengthof tidal channels in coastal lagoons adds further constraints:the basic equilibrium state is typically aggrading landwardand develops a shore at the inner end �Lanzoni andSeminara23�. The latter constraint is quite important as it

FIG. 16. The dimensionless net displacement of bars over a tidal cycle �x isplotted as a function of the peak Shields stress r, for three different valuesof the roughness parameter ds �Rp=10, �M4

=0.2, �M4=0�.

gives rise to a slowly varying spatial distribution of the


aspect ratio of the channel; this feature also awaits to beexplored, along with the role of tidal flats adjacent to thechannel.

ACKNOWLEDGMENTS

This work has been supported by CORILA �Consorzioper la Gestione del Centro di Coordinamento delle Attivita’di Ricerca inerenti il Sistema Lagunare di Venezia� �SecondResearch Program, 2004-2007, Linea 3.14, Processi diErosione e Sedimentazione nella Laguna di Venezia� andby the Italian Ministry of University and Research under theNational Project “Idrodinamica e morfodinamicadell’evoluzione a lungo termine di ambienti lagunari”�M. Marani coordinator�. This work is also part of thePh.D. thesis of V. G. submitted to the University of Genova.

APPENDIX A: SEDIMENT TRANSPORTRELATIONSHIPS

The values of �R�=zR* /Dr

*� and Ce were calculated em-ploying Van Rijn’s19 relationships

Ce = 0.015ds

*

zR* ��

c− 1�1.5

Rp−0.2, �A1a�

�R =�l

Dr* ��l � 0.01D*� , �A1b�

�R = 0.01 ��l � 0.01D*� , �A1c�

where � is the effective Shields stress acting on bedloadparticles, �l is an effective roughness accounting for the ef-fect of dunes, and c is the critical Shields stress for sedi-ment motion evaluated by means of the Brownlie24 relation-ship

c = 0.22Rp−0.6 + 0.06 exp�− 17.77Rp

−0.6� . �A2�

The function I appearing in the leading order expression forthe flux of suspended sediment reads

I =�Cf0

k� I2

I1+ �1�1 − �R� , �A3�

where



• Cf0 is the friction coefficient Cf calculated at the leadingorder of approximation;

• �1 is the parameter

�1 = 0.777 +k

�Cf0

; �A4�

• I1 and I2 are the following integral functions:

I1��R,Z� = �R

1 � 1 − �

1 − �R

�R

��Z

d� , �A5�

following relationships:


I2��R,Z� = �R

1

�ln � + 1.84�2 − 1.56�3�� 1 − �

1 − �R

�R

��Z

d� .

�A6�

Parameter � appearing in �27� is also an integral function of�R and Z which has been calculated numerically and is re-ported in �Bolla Pittaluga and Seminara7� with a slightly dif-ferent notation.

APPENDIX B: THE L OPERATOR AND THE FUNCTIONS C„�… AND G„�…

L = �a11 0 a13 a14

0 a22 a23 0

a31 a32 0 a34

a41 a42 − Fr2�1 − p��0d/dt + a43 �1 − p��0d/dt + a44

� , �B1�

with

a11 = i�U0 + �Cf0q1�U0� , �B2a�

a13 = i� , �B2b�

a14 = �Cf0U0�U0��q2 − 1� , �B2c�

a22 = i�U0 + �Cf0�U0� , �B2d�

a23 = M , �B2e�

a31 = i� , �B2f�

a32 = − M , �B2g�

a34 = i�U0, �B2h�

a41 = − i�q0� f1�0

�U0�+ 00�1 + g1� , �B2i�

a42 = Mq0� �0

�U0�+ 00 , �B2j�

a43 = − M2Fr2q0�0R , �B2k�

a44 = − i�q0��0f2U0

�U0�+ U0 00�1 + g2� + M2q0�0R .

�B2l�

Furthermore q1, q2, f1, f2, g1 and g2 are expressed by the

q1 =2

1 − CT, �B3a�

q2 =CD

1 − CT, �B3b�

f1 = q1�T, �B3c�

f2 = �D + q2�T, �B3d�

g1 = q1 0T, �B3e�

g2 = 0D + q2 0T, �B3f�

where

CD =1

Cf0� �Cf

�D�

, �B4a�

CT =0

Cf0� �Cf

��

D, �B4b�

�D =1

�0� ��

�D�

, �B4c�

�T =0

�0� ��

��

D, �B4d�

0D =1

00� � 0

�D�

, �B4e�

0T =0

00� � 0

��

D. �B4f�

The function C�� has the form:



C�� = −�a41�1 + a42�2 + a43�3 + a44�

�1 − p��0�1 − Fr2�3�

, �B5�

where

�1 = −a13a22a34 + a14a23a32

a11a23a32 + a22a31a13,

�B6�

�2 =a23�a31a14 − a11a34�

a11a23a32 + a22a31a13, �3 = − �2

a22

a23.

The function G�� reads

G�� = i��0Ce0I10

�1 − �R0��1t1 + U0t2��1 − Fr

2�3�. �B7�

The coefficients t1 and t2 are expressed by the followingrelationships:

t1 = �CeT + I1��RT +�R0

1 − �R0�RT −

I1Z

2�q1, �B8a�

t2 = �CeD + I1��RD +�R0

1 − �R0�RD� +

q2

q1t1, �B8b�

where CeT, CeD, �RT, �RD, I1� and I1Z are defined as

CeT =0

Ce0� �Ce

��

D, �B9a�

CeD =1

Ce0� �Ce

�D�

, �B9b�

�RT =0

�R0� ��R

��

D, �B9c�

�RD =1

�R0� ��R

�D�

, �B9d�

I1� =�R0

I10� �I1

��R�

Z, �B9e�

I1Z =Z0

I10� �I1

�Z�

�R

, �B9f�

and �0, Ce0, I10 and �R0 are �, Ce, I1 and �R calculated at theleading order of approximation.

The variables d1�t�, u1�t�, v1�t�, and d1�t� are related tothe bottom perturbation 1�t� in the form

d1�t� =1

�Fr2�3 − 1�

1�t� , �B10a�

u1�t� =�1

�Fr2�3 − 1�

1�t� , �B10b�


v1�t� =�2

�Fr2�3 − 1�

1�t� , �B10c�

h1�t� =�3

�Fr2�3 − 1�

1�t� . �B10d�

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On the migration of tidal free bars

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