U. Bethers, J. Sennikovs

Laboratory for mathematical modeling of environmental and technological processes, University of Latvia, 8 Zellu, Riga, Latvia

 A longshore load transport near Latvian harbour Ventspils is investigated by means of mathematical modeling. A set of models, including description of wave field, near-shore hydrodynamics, bed and suspended load transport, and seabed dynamics, is developed. The models are verified comparing model's predictions and surveys' results for particular storm period. The impact of different storm events is analyzed. The artificial forcing data sets are synthesized from meteorological data. These data sets are used for model forecasting of sedimentation and erosion rates in typical and critical seasons.

 1 Introduction

A longshore load transport is a typical feature of about 500 km long, almost sandy Latvian shoreline (see Fig.1). The southwestern winds prevail at the Latvian coast of Baltic proper, especially during autumn and winter storm periods. The breaking waves produce longshore currents causing about 1 million m³ integral northward sand transport yearly. Reasonable processes of the depth redistribution occur besides the transit load transport. These processes of sedimentation and erosion are the most developed in the interaction of the load transport with the hydroengineering constructions and the sea entrance channels of seaports. The largest Latvian harbour is Ventspils (see Fig.1) with approximately 16 m deep and 120 m wide sea entrance channel. The sedimentation causes reduction of the navigation safety and raises the maintenance costs there. The reconstruction of this seaport started 1996 including the widening (up to 140 m) and deepening (up to 17.7 m) of the sea entrance channel. Thus, seaport faced also the problems of (i) prediction of the increase of sedimentation volume after reconstruction, (ii) estimation of the efficiency of the overdredging additional areas to support continuous navigation.
 
 

Figure 1. The coastline of Latvia. Major harbours and rivers included.	Figure 2. Scheme of physical processes nearby harbour.

2 Physical processes

The principal scheme of the physical processes governing sedimentation in a sea entrance channel of seaport is shown on Fig.2.

• Wave vectors are almost parallel to the wind in the far sea. Wave fronts transforming in a shallow coastal area, reach the breaking zone non-parallel to the isobaths.
• Part of the wave energy transforms into (energetic) longshore current with maximum discharge at about 3 to 6 m depth. The longshore current carries material load, partially in suspension. It inclines seawards (to the greater depths) before wave-breakers.
• The longshore current decelerates at the greater depths and becomes oversaturated. Sedimentation is a consequence of the above.
• Oversaturation of the longshore current becomes even more expressed in the artificial sea entrance channel. Bed load transport stops there, whilst the depletion of the suspension depends on the width of the channel.
• The longshore current is undersatured and decelerated passing the seaport. It restores a pre-harbour velocity almost as close as waves are able to diffract into the shadow zone after the downwind wave-breaker.
• Bottom erosion downwind from the harbour is caused by the restoring of saturation load capacity of the longshore current.
• The decrease of the depths in the upwind side of harbour shifts the wave-breaking zone seawards. This results in growth of the beach there, whilst opposite trends are typical for the downwind side of the harbour.

3 Mathematical model

The two-dimensional wave-field model is developed assuming that (i) waves are linear and monochromatic, (ii) wave field is quasi-steady-state, (iii) wave energy in the breaking zone is dependent only on water depth, (iv) the calculation of wave direction can be separated from the calculation of wave energy. We assume the dispersion relation [1] w²=gkth(kh), where w is an angular frequency of monochromatic wave, g is acceleration due to the gravity, k is an absolute value of the wave vector k, h is a water depth. The quasi-stationarity of the wave field allows to simplify the kinematical relation [1]:

(1)

Thus, the wave direction can be calculated independently by means of stationaring the equations (similarly to [2])

,

(2)

where respective absolute values of wave phase velocity C=w₀/k and group velocity C_g=0.5C[1+2kh/sh(2kh)] are given as in [1].
    The quasi-steady-state equation for wave energy stands as

(3)

here H is waveheight, r is water density. We assume that in the breaking zone waveheight is dependent only on the water depth . Eq. (3.2) is used instead of transfer equation (3.1) to calculate the wave energy under condition of breaking waves. The field of the wave energy calculated by (3) determines the tensor of the radiation stresses [3,4]

(4)

    The setting of wave energy E and direction of the wave vector k are necessary on the inlets (kn<0) of the calculation domain (see Fig. 3). On the far-sea boundary we assume k being parallel to the wind vector W and employ empirical relationships H=H(W) and w ₀=w₀(H) from the statistical analysis of 4 years' wave and 19 years wind observations. The solution of one-dimensional equations (1-3) is given as a boundary condition for k and E on the one of the boundaries (inlet) that is orthogonal to the coastline.

Figure 3. Boundary conditions.

    The calculation of the non-steady coastal hydrodynamics is performed under shallow water approximation, accounting for wave and wind forces, water level fluctuations, bottom friction and turbulent momentum exchange, but neglecting the gradients of water density and air pressure, and moving boundaries of the calculation domain. The shallow water equations under above assumptions then stand as

	(5)

here q is water flux vector, z is water elevation, v is vertically averaged velocity, m and C_w are, respectively, turbulent exchange and wind drag coefficients. The Chezy coefficient of bottom friction is given by [5]

(6)

where d stands for average grain diameter. The boundary conditions for (5) see on Fig.3; q=0 on coastline, q is parallel to boundary on far-sea. One and two Dirichlet conditions are supplied for, respectively, downwind () and upwind (q_x, q_y) side boundaries, from the one-dimensional solution of (1-5).

The load-transport model is developed under the assumptions that:

• the bed and suspended load transport can be calculated separately;
• the uniform grain size dispersion is assumed with d» 0.2 mm;
• the parameterization of the unidirectional flow can be used for load transport;
• the infinitely thick layer of sand material is available on the seabed.

(7)

(8)

(9)

(10)

(11)

4 Numerical solution

The numerical solution of the developed mathematical model is performed by means of finite element method. The approximations of the wave refraction equation (2) and steady wave energy transfer equation (3) are done by Petrov-Galerkin method with streamline upwinding [7] against C_g direction. The non-steady hydrodynamical problem (5) is solved similarly to [8], using Bubnov-Galerkin method [9] for the spatial discretization, splitting the time step, and employing Lagrangian integration of convective terms along streamlines [10]. The non-steady suspended load transport equation (10) is solved by implicit two-layer time scheme, using Petrov-Galerkin spatial discretization. The seabed dynamics (11) is calculated explicitly.

The quasi-steady-state of the wave field is assumed for 6 hour long time periods. It corresponds to the characteristic observation frequency for both wind and waves. Thus, wave field parameters and hydrodynamics are calculated in the outer region (see Fig.3) reaching the steady-state conditions. Then these fields are recalculated in more details as non-steady-state problem within the embedded inner region using the outer domain results as boundary conditions. The problems of load transport and seabed dynamics are solved only in the inner domain. The back-impact of changing bathymetry on the hydrodynamics is accounted for this region.
 

5 Characteristic storm events

The calculation series for (quasi-)steady-state wave field and hydrodynamics are performed to investigate the hydrology for different storm events. The flow patterns for two wind directions see on Fig.4. The basic conclusions regarding the character of physical processes for different storms stand as follows.

• The load transport near Ventspils is governed by the energetic longshore currents. Wind currents and gradiental longshore currents are much less important.
• The wave breaking against southern wave-breaker in the case of western winds, and against the northern wave-breaker in the case of northern winds (for this situation see lower Fig.4) produces the recirculation oriented against the main longshore current. Thus, for these situations the longshore current is decelerated before crossing the sea entrance channel, and shifted seawards.
• The northwestern wind causes the similar effect. It produces recirculation patterns with flow direction away from sea entrance channel along both wave breakers but the weak longshore current is pushed more than 500 m seawards. Thus, this type of storm events causes mainly raise of the water level via wave set-up.
• The southwestern winds (see upper Fig.4) would be the most dangerous from the point of view of sedimentation. The wave breaking against the southern wave breaker raises the transport capacity of longshore current. It flows across the channel directly along the port entrance, contrary to the cases of N and W winds.

Figure 4. Flow patterns near Ventspils harbour during SW(left) and N(right) storms.

6 Verification of model

The verification of the model is performed via the comparison of model predictions with surveys’ results. A particular winter storm period from 27th of December, 1991 until 19th of January, 1992 is selected. This period is characteristic with enlarged changes in bathymetry because of

• high storm frequency, i.e. winds above 7 m/s was registered in 43.5% of this 23 days long time period;
• high storm variability, i.e. 42.5%, 15%, 17.5%, and 25% of storm events were in, respectively, SW, W, NW, and N sectors;
• no maintenance dredgeworks affecting the natural processes.

Figure 5. Measured and calculated depth differences after particular storm period (from 27-Dec-91 to 19-Jan-92)

7 Prognostic calculations

The analysis of the meteorological (forcing) data was done to perform the model forecasts. The statistical analysis of wind data for 19 years long observation period allowed to create the artificial forcing data sets for characteristic and critical seasons. We assume four seasons for a year (each 3 months long). The repetitiveness and the sequence of storm events in the artificial data sets are close to real observation series. The winters are characteristic with enlarged southern wind repetitiveness. The cold winters are usually calm, whilst mild (Atlantic) winters are stormy. The most important storm events occur in SW or N sector. The average wind speed for typical winter is 4.7 m/s (storm probability above 8 m/s is 14%) but for critical winter 6.1 m/s (storm probability is 30%). The autumns are even more stormy with high storm probability in all directions from open sea. The average wind speed is 5.0 m/s (storm probability 18%) for the characteristic but 6.9 m/s (storm probability 41%) for the critical autumn.

The forecasts of the depth redistribution are performed during above eight seasons (four critical and four typical) for (i) existing configuration of sea entrance channel of Ventspils harbour, (ii) reconstructed (widened and deepened) channel, (iii) reconstructed channel with additional 12 m deep overdredged areas (see Fig.6 for configuration).

Figure 6. Configuration of additionaly overdredged area near sea entrance channel.

Conclusions

The prognostic calculations lead to following conclusions concerning the sedimentation in the sea entrance channel of Ventspils harbour:

1. The navigation regime cannot be maintained without dredgeworks during critical autumn and winter seasons.
2. The total amount of sedimentation in the overdredged areas is 1.1 million m³ per year. It would raise 1.4 times after the reconstruction. The total sedimentation volume is greater than the integral northward load transport because channel traps the sand from both directions during particular storm events.
3. The seasonal distribution of sedimentation in the typical year is 46% during autumn, 33% during winter, 17% in summer but only 4% in spring.
4. Southwestern winds cause up to 79% but northern winds about 19% of the total sedimentation in the sea entrance channel. The role of western and northwestern wind is almost negligible.
5. The efficiency of the better of additional overdredged areas (see Fig.6, configuration 4) does not exceed 11%, i.e. it reduces the sedimentation in the channel by 80,000 m³ trapping 680,000 m³ of sand itself.

This investigation was funded by Ventspils Port Authority under contract 0396/1-b. The acknowledge also the Latvian Science Council for its support.
 

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