Migration of sand waes in shallow shelf seas Attila A. Németh, Suzanne J.M.H. Hulscher and Huib J. de Vriend 3,,3 Uniersity of Twente Department of Ciil Engineering, Integrated Modelling P.O.BOX 7, 75 AE Enschede, the Netherlands Fax: (3) () 53 489 44 Email: A.A.Nemeth@sms.utwente.nl S.J.M.H.Hulscher@sms.utwente.nl 3 H.J.deVriend@sms.utwente.nl http://www.sms.utwente.nl/akgr/cit/mnu_mics.htm Abstract Sand waes form a prominent regular bed pattern, which can be found on the seabed of shallow seas. The position of these sand waes is not fixed. They are assumed to migrate in the direction of the residual current. We are interested in what the physical mechanisms are, causing sand waes to migrate and at what rate. We assumed that sand waes eole as free instabilities of the system. Within this study, a linear stability analysis has been performed on a DV morphological model describing the interaction between ertically arying water moement and an erodible bed in a shallow sea. Here we break the symmetry of the basic flow by choosing it as a constant current on top of a sinusoidal tidal motion. This constant current can be generated by two different physical mechanisms: by a wind stress applied at the sea surface or by a pressure gradient. As the ertical structure in these cases is different, we hae inestigated both. The results show that it is possible for sand waes to eole under these flow conditions and that they are able to migrate in the direction of the residual flow. The migration aspect introduced in this paper has not been alidated yet. Before we want to commence the alidation, we want to gain insight into the mechanisms crucial for migration with this model. Introduction Large parts of shallow seas, as for instance the North Sea (Figure ), are coered with bed features, which are fascinatingly regular. Sand waes form a prominent bed pattern haing crests spaced about 5 metres apart. Their heights can lead up to seeral metres. They are obsered at a water depth of the order of 3 metres. This means that the relatie sand wae height is ery significant. It is largely assumed that their crests are oriented perpendicular to the principal current [Johnson et al., 98], [Langhorne, 98], [Tobias, 989]. On a theoretical basis, Hulscher [996] found that the sand wae crests can be rotated slightly anti-clockwise with respect to the principal current, for up to. Figure : Bathymetry measurements made in the North Sea near the Eurogeul [Source data: North Sea Directorate; Created by: Knaapen (Uniersity of Twente)]
Migration has been inestigated in the past and has already been connected to studies done into bedform dynamics in riers. Although high elocities can be found in shallow seas like the North Sea, the long-term aeraged amount of bed load is often still small. This is attributed to the fact that the residual current in a tidal enironment is much smaller compared with unidirectional currents found in riers. The migration elocities of sand waes are therefore one to two orders of magnitude smaller than the elocities attained by dunes in riers [Allen, 98]. Fredsøe and Deigaard [993] described the behaiour of finite amplitude dunes under a unidirectional current. To model sand waes, they applied their dune model in a tidal enironment by assuming that the time-dependence can be neglected. Huthnance [98] was the first to look at a system of sandbanks as being a dynamically coupled system, consisting of seawater and an erodible seabed. Within this framework, one can inestigate whether certain regular patterns get excited, being free instabilities of the system. Hulscher [996] used this principle to explain the formation of sand waes due to symmetrical tidal motion. In this model sand waes do not migrate. The predictie ability of this model for sand wae occurrence has been shown in [Hulscher and Van den Brink, 999]. Blondeaux et al. [999] introduced forcing due to surface waes, which changes the elocity distribution oer the ertical. Migration is indicated without focussing on the mechanisms leading to this behaiour. We can conclude that the mechanisms leading to migration of sand waes in a tidal enironment are not understood yet. The research question used within this paper can be described as follows: What are the physical mechanisms causing sand waes in shallow shelf seas to migrate and at what rate? In order to gain insight into the physical mechanisms inoled in the migration of sand waes, an analytical model has been set up, based on the model presented by Hulscher [996]. The model includes tidal motion together with a residual current. This residual current is assumed to be responsible for the migration of sand waes. This assumption is based on the results obtained by Fredsøe and Deigaard [993]. First a description of the idealised model is gien. It is based on the two dimensional ertical shallow water equations together with a simple sediment transport formula, describing bed load transport. Next, the method of scaling the different ariables is presented. The scaling method results into the usage of two time scales (the tidal period and the morphological time scale). The morphological changes are calculated oer a longer time scale than the water moement. This makes it possible to aerage the bottom eolution oer the tidal period. Subsequently a linear stability analysis is performed. The basic state consists of a constant current, together with tidal moement (M ). This constant current is either induced by a wind stress applied at the sea surface or induced by a pressure gradient. The initial behaiour of the system is then inestigated by the feedback of small amplitude sand waes. Lastly, the results will shortly be discussed.. Description of the analytical model We will start from the analytical model [Hulscher, 996] based on the three-dimensional model used to describe sand waes and tidal sandbanks. It will be used to look at the driing factors behind migration.. Flow model The three-dimensional study [Hulscher, 996] has shown that the sand wae crests are oriented almost perpendicular with respect to the tidal motion. Furthermore, the Coriolis force only slightly affects sand waes. The behaiour of sand waes can therefore also be based described with the help of the two-dimensional Naier-Stokes equations. Making the shallow water approximation results in the two-dimensional shallow water equations (See also Figure ): ζ + u + w = g + A w + = () ()
Here u and w are the elocities in respectiely the x- and z- directions. t and g are respectiely the time and the acceleration due to graity. z=ζ is the free surface eleation and A is the ertical iscosity. The horizontal iscosity is neglected. Komaroa and Hulscher [999] hae shown that this term does not introduce any new mechanism and does not change the results significantly. They furthermore showed that the shallow water approximation does not change the mechanisms or results significantly.. Boundary conditions and assumptions The boundaries in the horizontal plane are located infinitely far away. The boundary conditions at the surface are defined as follows, whereby τ w describes the wind induced stress at the sea surface: ζ ζ A = τ w ; + u = w (3) The horizontal flow components at the bottom are described with the help of a partial slip condition (with S the resistance parameter). The ertical component is described by a kinematic condition: A = Su ; + u = w (4).3 Sediment transport and seabed behaiour The sediment transport model only consists of bed load transport. This mode of transport is considered to be dominant in the tidal offshore regimes we are looking at. As the flow elocities in the ertical direction are being calculated explicitly, bed load transport can be modelled here as a direct function of the bottom shear stress. The following general bed load formula is used: b Sb = α τ b τ b λ (5) The leel of the seabed is denoted by h with respect to the undisturbed water depth H. S b is the olumetric sediment transport. b is the power of transport and a dimensionless parameter. is the scale factor for the bed slope mechanism, taking into account that sand is transported more easily downwards than upwards. τ b is the bottom shear stress. The effect of a critical shear stress is not explicitly studied in the model at this moment. The bed shear stress is defined as: τ b = A (6) z= bottom The net inflow of sediment is assumed to be zero. This results in the following sediment balance, which couples the flow model with the sediment transport model: Sb + = (7) z=(x,t) z= z x z=-h z=-h+h(x,t) Figure : Illustration framework model
. Scaled set of equations. Scaling The ariables in the equations are scaled as follows making the study of the behaiour of sand waes possible: ( u, w ) = ( Uu *, εuw * ) ; ( ) x, z, h = δx*, δz*, δh* (8) ε ULσ t = t* σ ; ζ * ;[ τ ] ν / δ g = U (9) Time (t) is scaled with the tidal frequency ( - ) because tidal moement is assumed to be the main forcing mechanism of these large-scale bed forms. For the time scale of the eolution of the bottom a longer time scale is used (T long ). The horizontal elocity (u) is scaled with the tidal elocity amplitude (U) being in the order of m s -. The ertical co-ordinate (z) and the amplitude of the bottom perturbation (h) are scaled with the Stokes depth (), as was done in Komaroa and Hulscher [999]. is an expression related to the thickness of the boundary layer: ν δ = () σ The shear stress ( b ) is scaled with the iscosity together with the amplitude of the tidal elocity diided by the Stokes depth. This is analogous to the definition of shear stress. The water leel is scaled with the length oer which the tidal wae aries (L). The last two scales to be discussed here are the scale for the ertical elocity () and the horizontal length scale (x). The parameter is introduced here making it possible to scale the ariables with physical releant scales combined with a correct order of magnitude. The alue of (order of magnitude ) is obtained by looking at the balance between the shear stress and the slope term: τ b λ and τ b λ τ () b With: 3Θ c g( s ) d λ ~ = ; λ γ tan φ = () tanφ. Set of equations The equations describing the moement of water can now be presented in non-dimensional form Lσ ζ + Ru + Rw = R + E U w + = The boundary conditions at the free surface become: and at the bottom: s (3) (4) Lσ ζ ULσ ζ = τˆ w ; + u = w (5) gε gδ E = Su ˆ ; + u = w R This results furthermore in the following bottom eolution equation: (6) T m b = τ b τ ˆ b λ (7)
The following parameters can be obsered in the equations shown aboe (3)-(7): T m = αˆt ; A E = ; σδ εu R = ; δσ + b ε ν U ˆ α = α σδ δ S S ˆ = ; ˆ εδ λ = λ (8) σδ ν σt long U δ ; ˆ τ w = τ (9) w UA E can be seen as a measure for the influence of the iscosity on the water moement (by definition the tidal moement) in the water column. Ŝ is a parameter describing the resistance at the bottom. R can be physically interpreted as the square root of the Reynolds number times..3 Typical alues for a North Sea location The tidal elocity amplitude (U) is of the order m s -. The iscosity ( equal to A ) are set at.. The frequency of the M -tide in the North Sea (σ) is about.4-4 s -. The proportionality constant α can be computed from Van Rijn. It is set at a alue of about.3 sm -. The downhill sediment transport coefficient (λ) was set at two and b is set at /. The aerage depth (H) is in the order of 3 metres [Hulscher, 996]. For a wind speed W the wind-induced stress at the sea surface can be approximated by [Prandle, 997]: τ w =.3 W () 3. Linear stability analysis The solution of the problem can formally be presented by the ector = (u,w,,h). Starting from an exact solution of the problem, a certain basic state can be perturbed by a small amplitude ( < ) perturbation. The solution can be expanded as follows: ψ = ψ + γψ + γ ψ + γ ψ 3 +... () For < and = i () the successie terms decrease in magnitude. This means that the one but largest contribution is fully gien by the linear term. Therefore, the instability of the basic state can be tested by determining the initial behaiour of. Amplification of in time means that is unstable and decay means that is stable. 3. Basic state The basic state describes a tidal current together with a constant current oer a flat bottom (horizontally uniform flow). A Taylor expansion in the parameter γ enables transferring the boundary condition from z = - + h to z = -. The following set of equations arises: With the boundary condition at the free surface: And the boundary conditions at the bottom: Lσ = R U ζ + E = τˆ E = S ˆu w 3 w ; = The elocity in the horizontal direction for the basic state can be formulated as follows: () (3) ; w = (4) u = u + u sin t u cos t (5) r s + c
Firstly, a description for the constant current (u r (z)) is deried. Two possible types of constant currents will be discussed here. These are a wind drien (I) current and a current induced by a pressure gradient (II): Wind stress (I) Pressure gradient (II) E ur = ˆ τ w+ + z = E ur P z S S With: εl P = ζ (6) δ The shear stress at the bottom is therefore equal to the wind stress in case (I) and equal to P ( 5 times the pressure gradient) in case (II). In figure 3 the elocity distribution oer the ertical ( denotes the sea surface and the sea bed) can be found for the wind and pressure gradient induced currents. They are set at a depth-aeraged elocity of. m s - in the figure. The elocities at the seabed are not zero in this case and in the case of tidal moement. Models, which consist of a z-independent iscosity formulation, oerestimate the bottom shear stress. The shear stress determines directly the amount of sediment transported. Therefore, the slip parameter S needs to be finite, if the model is to produce realistic results. E Figure 3: elocity distribution oer the ertical for a wind stress and pressure induced current Tidal moement (with an amplitude of m s - ) can be described by (See (5)) [Hulscher, 996]: u s = Im( uˆ ) ; Re( uˆ ) i = α ˆ u u c cosh αe α Sˆ ( αz) cosh( α ) = (7) αe sinhα (8) Sˆ sinhα coshα α = ( + i) (9) 3. Perturbed state The stability of the basic state can be tested by determining the initial behaiour of. Subtracting the basic state ()-(4) and the smaller order terms leaes: E Lσ = R U ζ + Ru + Rw w + = + E (3) (3)
The boundary conditions at the free surface are and at the bottom: ˆ S = u E u = Sˆ + h E w u h The unknowns are Fourier transformed as follows with ( ) (3) (33) w = u (34) ψ = u, w, ζ, h [Hulscher, 996]: ~ ikx ψ ψ ( t) e dk + c.. (35) = c in which c.c. means complex conjugate and k is the wae number of the way bottom perturbation. Harmonic truncation in time is applied. This means that the perturbation is restricted to a finite number of tidal components. In the case of a unidirectional tide, the following truncation will contain the dominant physical processes [Hulscher, 996]. u~ ~ trunc ( z, t) = h[ ia + as sin t + ac cost] (36) w~ ~ trunc ( z, t) = h[ c + ics sin t + icc cost] (37) ~ ~ ζ t = h d + id sint id cost (38) trunc ( ) [ ] s + The bottom ariable will hardly ary on a tidal time scale. The eolution of the seabed can therefore at best be described by aeraging the sediment fluxes oer the tidal period. The solution for the bottom eolution equation can now be written with h the initial small amplitude and ω describing the behaiour of the bed: ~ h = h cos c ( kx T ) ω T e r m ω (4) i m The real and imaginary part of ω can be written as follows, with the brackets denoting the tidal aerage: r ( b + ) a '( ) ω = + ˆ (4) b b k τ b k λ τ b b b ( b + ) a '( ) τ sin t k( b + ) a '( ) cost ω = k (4) i s b c τ b The Fourier transformed functions can now be soled numerically together with the boundary conditions. The shear stress can then be calculated, which is then to be imported in the bottom eolution equation. 4. Results The obtained solution is complex. The real and imaginary parts represent respectiely the initial growth and migration rates of the sand waes. As was found in preious research positie growth rates appear for arious waelengths. These are taken into account by the real part of ω. This means that sand waes can eole due to the interaction between water moement and seabed morphology. Furthermore, migration of the sand waes has been found. These displacements of the bottom perturbations are described by the imaginary part of ω. They can be considered as phase shifts of the sinuses describing the sand waes. These phase shifts are caused by the introduced asymmetry in the water moement. They depend on the nature and magnitude of the asymmetry in the water moement. If the constant current is set to zero, the water
moement is symmetrical. In this case, no migration is found since the time aerage of the imaginary part (between brackets) is then zero. An estimate of the order of magnitude of the migration elocity of the sand waes can be deduced from ω i. A typical waelength for a sand wae is 5 meters. The morphological time scale (T m ) is about 6 years. A basic state is taken consisting of tidal moement (M ) with an amplitude of m s - (based on the results from [Hulscher, 996]) together with a wind drien current with a depth aeraged elocity of. m s -. With this basic state an estimate of the term within brackets can be obtained. The first term has a alue of. and the second one is two orders of magnitude smaller. a s (-) is determined based on the results obtained by Hulscher [996] and has an order of magnitude of. The second term with a c (-) is also much smaller. The second term can therefore be left out of this first estimate. A migration rate in the order of about 6 meters per year is now obtained. In the case of a pressure gradient induced constant current with a depth aeraged elocity of. m s -, the shear stress would be smaller at the seabed. This would then result in a slightly smaller migration rate of the sand waes. Acknowledgements This abstract is based on work funded by the Technology Foundation STW under contract TCT.4466. References Allen, J.R.L., 98, Sand wae immobility and the internal master bedding of sand wae deposits, Geol. Mag., 7 [5], pp. 347-446 Blondeaux, P., Brocchini, M., Drago, M., Ioenitti, L., Vittori, G., 999, Sand waes formation: Preliminary comparison between theoretical predictions and field data, IAHR symposium on Rier Coastal and Estuarine Morphodynamics, Genoa, Italy, Proceedings, Volume, pp.97-6 Fredsøe, J. and Deigaard, R., 99, Mechanics of coastal sediment transport, Institute of Hydrodynamics and Hydraulic Engineering, Technical Uniersity of Denmark, pp. 6-89 Hulscher, S.J.M.H. and Van den Brink, G.M., 999, Comparison between predicted and obsered sand waes in the North Sea, Uniersity of Twente, The Netherlands, submitted to Journal of Geophysical Research Hulscher, S.J.M.H., 996, Tidal induced large-scale regular bed form patterns in a three dimensional shallow water model, Journal of Geophysical Research, (C9), pp., 77-, 744 Hulscher, S.J.M.H., De Swart, H.E. and De Vriend, H.J., 993, The generation of offshore tidal sand banks and sand waes, Institute for Marine and Atmospheric Research Utrecht and Delft Hydraulics, Continental Shelf Research, Vol. 3, No.. pp. 83-4 Huthnance, J.M., 98, On one mechanism forming linear sand banks, Estuarine and shelf science, 4, pp. 79-99 Johnson, M.A., Stride, A.H., Belderson, R.H. and Kenyon, N.H., 98, Predicted sand-wae formation and decay on a large offshore tidal-current sand-sheet; in: Spec. Publs. Int. Ass. Sediment, 5, pp. 47-56 Komaroa, L. and Hulscher, S.J.M.H., 999, Linear instability mechanisms for sand wae formation, Submitted to Journal of Fluid Mechanics Langhorne, D.N., 98, An ealuation of Bagnold s dimensionless coefficient of proportionality using measurements of sand wae moements, Marine Geology, 43, pp. 49-64 Németh, A.A., 999, Modelling sand wae dynamics in shallow shelf seas, Practical releance, Uniersity of Twente, in preparation Prandle, D., 997, The influence of bed friction and ertical eddy iscosity on tidal propagation, Continental Shelf Research, 7 (), pp. 367-374 Tobias, C.J., 989, Morphology of sand waes in relation to current, sediment and wae data along the Eurogeul, North Sea; report GEOPRO 989.; Dep. Of Physical Geography, Uniersity of Utrecht, The Netherlands