Beach modelling III: Morphodynamic models – Description

Adonis F. VelegrakisAdonis F. VelegrakisDept Marine SciencesDept Marine Sciences

University of the AegeanUniversity of the Aegean

Synopsis

1 Beach morphodynamic models1.1 Purpose 1.2 ‘Static’ and dynamic ‘bottom-up’ models

2 Static models2.1 The Bruun (1962, 1988) model (rule)2.2 The Edelman (1972) model 2.3 The Dean (1991) model

3 Dynamic models 3.1 Basic structure3.2 Hydrodynamic models 3.3 Sediment transport modules3.4 Morphological module

4 Beach retreat 1-D models 4.1 The SBEACH model4.2 The Leont’ yev model 4.3 The Boussinesq Model

1.1 Beach morphodynamic models: Purpose

Diagnosis/prediction of beach response to (wave forcing) and sea level changes

Basic principle: when the sea level and/or the wave forcing changes the beach profile is forced to change to a new profile

Beach erosion/retreat models can be differentiated into

• ‘static’ models • ‘dynamic’ / ‘bottom-up’ models

1.2 Beach morphodynamic models: ‘Static’ models

In static models, beach erosion/retreat is assessed through the solving of one or of a system of equations

In these models, hydrodynamic and sediment dynamic processes are not (fully) considered

Therefore, (most of) static models are used to predict effects of long-term sea level rise (ASLR) on the cross-shore beach profile; thus, these models are 1-D models

Here, we will considered 3 of such models, i.e. the models Bruun (1962, 1988), Edelman (1972) and Dean (1991)

1.2 Beach morphodynamic models: Dynamic ‘bottom-up’ models

The basic ‘ingredient’ of the dynamic beach morphodynamic models is the coupling of

• hydrodynamic and

• sediment transport models

The results of the coupled models are then used to determine morphological changes using e.g. some form of the sediment continuity equation (see below)

2.1 Static models: The Bruun (1962, 1988) model (rule)

Bruun model assumptions:

• The active cross-shore beach profile attains an ‘equilibrium’ profile • Under ASLR, the equilibrium profile migrates onshore, causing erosion in the sub-

aerial and deposition in the sub-marine beach • Deposition occurs between the (new) shoreline and the closure depth and • Seabed elevation increase due to deposition equals sea level rise

Governing expression:

where s, coastal retreat; l, distance to the closure depth; hc, closure depth; α, sea level rise; and Bh, berm elevation

Note: There is no control by sediment size or wave characteristics, except by the most energetic waves of the year that define the closure depth (see Presentation 2)

Much has been written for and against the validity of assumptions of the Bruun model (e.g. Pilkey et al., 1993; Cooper και Pilkey, 2004; Zhang et al., 2004)

hc Bh

als

Fig.1 Schema showing the parameters of the Bruun model. Key: S, coastal retreat; l, distance to the closure depth; hc the closure depth; α, sea level rise; Bh, berm elevation; ht, hc + B (after Slott, 2003).

2.2 Static models: The Edelman (1972) model

)(ln)(

tBh

Bhwts

hb

obb

This model can be used for more realistic profiles and higher and shorter-term sea level rises (e.g. storm surges)

According to the model, beach profiles maintain their basic morphology. The governing expression is:

Where s, beach retreat; α, sea level rise; Β(t), instantaneous height of the profile above the current level; and hb and wb depth at wave breaking and width of the active beach inshore of wave breaking, respectively.

Thus, beach retreat is controlled by the wave parameters

Replacing and integrating leads to:

Where Βo, initial berm elevation.

)(tBh

w

dt

d

dt

ds

hb

b

2.3 Static models: The Dean (1991) model

The Dean (1991) model uses the beach equilibrium profile defined by h = A xm, where Α a parameter controlled by the beach sediment grain size.

Beach retreat is given by:

Where hb the depth at wave breaking; Hb, wave height at breaking; Wb, width of the surf zone defined as wb = (hb/A)3/2, where Α is a scale parameter (Α = 2.25 (ws2/g)1/3) controlled by sediment grain size (ws, sediment settling velocity).

Thus, beach retreat is controlled by both the sediment size and the wave parameters

bh

bb hB

wHas

068.0

3.1 Dynamic models: Basic structure

They perform calculations at different locations (nodes) of the beach (profile) and They perform calculations at different locations (nodes) of the beach (profile) and simulate its evolution in the desired time stepsimulate its evolution in the desired time step. .

They consist of the following sub-models (modules)They consist of the following sub-models (modules)::

The hydrodynamic sub-model (module)The hydrodynamic sub-model (module) which estimates beach which estimates beach hydrodynamic conditions (waves and wave-induced currents) with input hydrodynamic conditions (waves and wave-induced currents) with input parameters the seabed morphology (bathymetry), the offshore wave parameters the seabed morphology (bathymetry), the offshore wave conditions, and the sediment characteristics (as bed friction control)conditions, and the sediment characteristics (as bed friction control)

The sediment dynamic sub-model (module)The sediment dynamic sub-model (module), which estimates sediment transport due to waves, wave-induced currents (and their interaction) on currents (and their interaction) on the basis of the hydrodynamci conditions estimated by the hydrodynamic the basis of the hydrodynamci conditions estimated by the hydrodynamic module module

The morphological sub-model (module)The morphological sub-model (module) that estiamates the new that estiamates the new morphology on the basis of the sediment transport patterns estimated by morphology on the basis of the sediment transport patterns estimated by the sediment dynamic model the sediment dynamic model

3.2 Hydrodynamic models I

.

Objective: to provide a synoptic picture, in various temporal scales, of the coastal hydrodynamics of the study area

They use numerical analysis, which solves through approximations complex mathematical problems

The solving method is called algorithm, and its suitability depends on 2 criteria

(i) speed and (ii) accuracy

Two types of potential errors:

• Those originating from inaccuracies in the input information (e.g. coastal bathymetry inaccuracies) and

• Those inherent in the algorithm

A hydrodynamic (circulation or wave) model requires:

• Bathymetric information of the best possible resolution • Information on the model forcing (e.g. wind, tides, water density etc)• Information on the bed type (sediment texture and forms)

On the basis of the above, a model is constructed using (a) spatial and temporal discretisation techniques and (b) Resolution techniques

Spatial discretisation refers to the division of the area into boxes or meshes; the hydrodynamic equations are numerically solved in 3, 2 or 1 coordinates with the models being 3D, 2DH (depth-averaged), 2DV (longitudinal) και 1DH.

Temporal discretisation refers to the time-step of the solution and depends on the process to be modelled (e.g. waves, tides etc)  

The resolution techniques refer to the type of the mesh (finite differences-finite elements).

3. 2 Hydrodynamic models II

3. 2 Hydrodynamic models III

In order to estimate flows, hydrodynamic models solve a system of equations, i.e.:

• the momentum equations (Navier Stokes) and• the mass conservation (continuity) equation

Their requirements are good bathymetric data, and good information on the forcing (winds, density, tides etc).

Problems with open domain boundaries. Boundary conditions must be established, which can be acquired by wider domain models

Using numerical analysis, they can define flow vectors within the domain

Several accomplished hydrodynamic models ((3-D, 2-D etc) are available (e.g. POM)

3. 2 Hydrodynamic models IV

Coastal wave models have a different construction (see below)

The ultimate forcing is the wind, which generates offshore waves that they are driven inshore changing by the seabed friction

Although waves in the open sea transfer only energy (not mass), inshore waves can generate currents (i.e. mass transport), generating wave-induced coastal circulation (flow)

Waves and wave-induced currents can interact with natural and/or artificial structures inducing secondary circulation

3.3 Sediment dynamic modules

Τhe modules use as inputs the hydrodynamic model results and estimate sediment transport for each spatial step.

Sediment transport can take place as bedload, suspended load and, under particular conditions, as sheet flow.

Estimations can consider total sediment transport in a wave period (time-averaged), or sediment transport in shorter (intra-wave) temporal scales.

They can describe sediment mobility and/or sediment transport patterns i.e. coastal sediment circulation (flow)

There are issues with the non-linearity of sediment transport and particularly with the complex transfer function linking hydrodynamics to sediment mobility and sediment transport rate

Sediment transport due to wave - current interaction

Wave current interaction may change significantly sediment transport in a non-linear way.

Wave current interaction also changes sediment transport direction. Generally:

(i) For bedload transport, the transport direction is controlled by the (non-linear) combination of the magnitudes/directions of the shear stresses (force per unit area) due to currents (τc) and waves (τw)

(ii) For suspended sediment transport, transport direction is controlled by the current direction

3.4 Morphological module

Dynamic models of beach retreat can be generally differentiated into (Roelvink and Brøker, 1993):

(i) profile development models and

(ii) process-based models

In most models, beach morphological development is estimated using the sediment continuity (conservation) equation (analogous to water continuity equation)

4 Beach retreat 1-D dynamic models

For the purpose of the present (RiVAMP) training, 1-D dynamic models have been selected, as they are more manageable

It must, however, be understood that such models diagnose/predict beach morphological changes without taking into account lateral sediment transport i..e. longshore and/or oblique sediment transport

Beach response to sea level changes is a non-linear process, depending mainly on:

• the rate of ASLR and the magnitude/duration of storm surges • the coastal slope/morphology• the impinging (and generated-infragravity) wave energy and • the nature (texture/composition) of beach sediments

Note: Our knowledge on costal erosion processes is still incomplete and, thus, predictions are associated with a large degree of uncertainty

)(dx

dhDDq

seqes

FsFwF EEh

k

dx

dE

Wave module Wave module Wave module Wave module

Sediment transport moduleSediment transport moduleSediment transport moduleSediment transport module

kkww:empirical coefficient of wave dissipation

ΕΕFF: wave energy flow

ΕΕFsFs: constant wave energy flow

KKss: empirical coefficient of sediment transport rate ,

DDee: Energy dissipation ,

DDeqeq: energy dissipation in equilibrium qiη ενέργεια διάχυσης σε ισορροπία

εε: coefficiet related to the sediment transport rate for the bed slope term

4.1 The SBEACH model (Larson and Κraus, 1989)

Governing expressions

4.2 The Leont’ yev model: Hydrodynamic modulemodule

It is based on the energetics approach of Battjes and JanssenIt is based on the energetics approach of Battjes and Janssen (1978) (1978), i,e. , i,e. on the assumption that cross-shore changes in wave energy flow in on the assumption that cross-shore changes in wave energy flow in each profile location equal wave energy losses due to bottom fiction each profile location equal wave energy losses due to bottom fiction

e

gw Dx

cE

cos

φ: wave angle

Ew: wave energy

cg: wave group celerity and

De: wave energy dissipation

The beach profile is divided into zones, with sediment transport varying along the profile as ((Leont’yev 1996)Leont’yev 1996): :

Wave refraction zone: qR = 0 και q = qW

Surf zoneSurf zone: : q = qW + qR

Swash zone: qW = 0 και q = qR

4.2 The Leont’ yev model: Sediment transport module

Sediment transport rate due to wave-current interaction Sediment transport rate due to wave-current interaction qqWW, , in the refraction in the refraction zone iszone is::

1

23 ~3cos~tan2

x

d

U

wBFUuufq

d

seesdw

bW

Bedload/suspended sediment load Bedload/suspended sediment load

: friction factor

ws: sediment settling velocity

φ: angle of approach

εs: effectiveness coefficient

Fe , Be: energy losses due to bed friction and turbulence, respectively

wf

4.2 The Leont’ yev model: Sediment transport module

Sediment transport in the swash zoneSediment transport in the swash zone::

mRmR

mRR xxx

xx

xxqq

2/3

/1

/1

oRRR Hxxcqq /exp 3 Rxx

: maximum sediment transport

c3=0.2-0.3

Η0 : offshore wave height

Rq

4.2 The Leont’ yev model: Sediment transport module

Beach morphological change is defined by the sediment Beach morphological change is defined by the sediment continuity equation continuity equation

Sediment porosity is also considered Sediment porosity is also considered

4.2 The Leont’ yev model: Morphological module

This state-of-the-art model is not included in the training tool as This state-of-the-art model is not included in the training tool as (a)(a) It is very heavy (a day to run) It is very heavy (a day to run) (b)(b) It requires extensive expertise and It requires extensive expertise and (c)(c) It is still under development It is still under development

Nevertheless, as it is used for model intercalibration (see Presentation 4), a brief Nevertheless, as it is used for model intercalibration (see Presentation 4), a brief description is given description is given

The model, that has been initially developed by The model, that has been initially developed by Karambas and Koutitas (2002)Karambas and Koutitas (2002), , includesincludes::

A wave model that is based on A wave model that is based on Boussinesq Boussinesq equations for dispersive, non-equations for dispersive, non-linear waves linear waves

A sediment transport module, that calculatesA sediment transport module, that calculates::

1.1. bedload and sheet flow using the expressions of Dbedload and sheet flow using the expressions of Dibajnia et al. (2001), ibajnia et al. (2001), 2.2. suspended sediment load through the energetics approach suspended sediment load through the energetics approach 3.3. sediment transport at the swash zone using an expression based on sediment transport at the swash zone using an expression based on Meyer-Meyer-

PeterPeter && Muller Muller

4.3 Boussinesq model

tx

Udh

tx

Uhh

xg

x

UdU

dx

M

dt

Ux

u

2

2

32

3

211 Mom. equationsMom. equationsMom. equationsMom. equations

t

U

xh

x

UUhh

x

U

xh

x

U

x

U

x

UU

hxx

2

2

2

2

2

2

3

32

3

2

2

3

3

2

32

2

x

x

UU

xg

tx

UhB

x

U

txh i

vb

xi Edx

gtx

UhhB

2

22

2

U: horizontal velocity ζ: sea level rise, h: normal sea level d = h + ζτb: bed shera stress Β i= 1/15 Εv:eddy coefficient Μu:effect of the non-normal velocity distribution

0

x

Ud

t

Continuity equationContinuity equationContinuity equationContinuity equation

4. 3 Boussinesq hydrodynamic module

The expression for the estimation of the sediment transport sheet flow The expression for the estimation of the sediment transport sheet flow qqbb due to due to non-monochrmatic waves isnon-monochrmatic waves is: :

5050

0038.0gdsTT

TuTu

dw

q

tc

cttttccc

s

b

bedloadbedload

, : velocities under crest and trough , : the associated durations ,

: percentages of sediment that are directly transported , : percentages of sediment that remain after the half period

: settling velocity

c tcT tTcu

sw

tu

c t

4. 3 Boussinesq sediment dynamic module

Suspended loadSuspended load

s

bes

ss w

UDb

aq

1

: mean wave energy diffusion due to breaking : current velocity

: parameter that relates nera bed value to mean value

: effectiveness coefficient

eDbU

s

beD

4. 3 Boussinesq sediment dynamic module

Sediment transport rate ate the swash zone is estimated by a modified Meyer-Peter & Muller model:

bedloadbedload

U

UC

gds

qQ rbR

23

350

tan

tan11

1

: non dimensional transport rate : Shields parameter : porosity : angle of repose : bed slope : parameter that takes different value whne the water moves onshore

RQ

tan

rC

4. 3 Boussinesq sediment dynamic module

It estimates profile changes usingIt estimates profile changes using::

Numerical solution of the continuity equation Numerical solution of the continuity equation

An additional gravity term to take into account the bed slope An additional gravity term to take into account the bed slope

4. 3 Boussinesq morphological module

Thank you!!See you later

Fig. 2 Edelman (1972) model. Key:, Key: S, beach retreat; α, sea level rise; Β(t) the instantaneous height of the beach profile above the current level; and hb και wb the breaking wave zone depth and the width of the active beach inshore of the wave breaking, respectively (after CEM, 2008).

initial morphology

Offshore wave conditions

sediment size

hydrodynamics sediment transport

new morphology

If time < wished

final morphology

Fig. 3 Generalised flow diagram of dynamic models

Fig. 3 Sketch showing models with different spatial discretisation

Fig. 4 Sketch showing nested models of different spatial discretisation. The output of the larger scale-small discretisation area is used for the boundary conditions at the open boundaries of the coastal models.

Fig. 5 Modelled currents in the Gera Gulf, Lesbos, E. Med (Stagonas, 2004)

Fig. 6 Depth-integrated 2-D tiadal flow model for a region of the English Channel Bastos et al., 2003)

Fig. 7. 2-D numerical model results for wave heights (a) and wave-induced currents (b) at the Negril beach. Conditions: Offshore wave height (Hrms) = 2.8 m, Tp=8.7 s. Waves approach from the northwest. Note the diminishing wave heights and changed nearshore flow patterns at the lee of the shallow coral reefs (RiVAMP, 2010)

400 500 600 700 800 900 1000 1100 1200400

500

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400 500 600 700 800 900 1000 1100 1200

x (m )

400

500

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y (m

)

0

0 .1

0.2

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0.4

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0.9

1

1.1

1.2

1.3

1.4

Fig. 8. Wave height contours Ηs (m) at beach with the breakwater (Karambas et al., 2007)

Fig. 9. 3-D schema of wave transmission over a submerged breakwater, breaking and wave run-up on the beach from a pseudo 3-D Boussinesq model (Koutsouvela, 2010),

400 500 600 700 800 900 1000 1100 1200400

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x (m )

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y (m

)

2 m /s

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0

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Εξέλιξη μορφολογίαςΑρχική μορφολογία

400 500 600 700 800 900 1000 1100 1200

x (m )

400

500

600

700

y (m

)

Fig.10 East Lesbos beach, E. Mediterranean. Wave generated currents and seabed morphology development under weighed N, NE and E waves, in the presence of a breakwater. Solid line, initial bathymetry; strippled line, final bathymetry (Karambas et al., 2007).

Fig. 11 Velocity field and morphological change (red, initial bathymetry and grey, final bathymetry) for wave transmission coefficient Kt = Ht/Hi = 0.8. Step = 1 m (Koutsouvela, 2010)

Initial distance of structure from coastline =150 m,

LG, structure spacing = 120 m,

LB, breakwater length = 120 m. Monochromatic waves, Ηο =1.5 m, T = 8s

Note the development of salients (accretion) behind the structures, but alos the very strong (and dangerous currents) between the structures

tt

i

HK

H

355000 360000 365000 370000 375000 380000

65000

70000

75000

80000

85000

0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

5

10

25

S a n d T ra n sp o rt R a tes(k g /m /tid a l cy c le )

Fig. 12 Sediment transport due to tidal currents in a region of the northern coast of the English Channel (Bastos et al., 2003).

Fig 13. Potential resuspension time under: (a, c, e) tidal currents alone (spring tides); and (b, d, f) tidal currents (spring tides) and waves approaching from the west (significant wave height Hs = 1 m, Period = 8 s). Potential resuspension time has been estimated for: particles sizes 0.040 mm (a, b); 0.100 mm (c, d); and 0.200 mm (e, f).

Seabed mobility under: (g) spring tidal currents alone and (h) spring tidal currents and waves.

Both potential resuspension and seabed mobility times are expressed as percentages of the total tidal cycle time, during which bed shear stress exceeds the critical shear stress for the initiation of resuspension/movement. Origin (0, 0) of grid is at 49.293 N0, 2.363 W0.(Velegrakis et al., 1999)

In all sedimentary environments, if there is a difference between the sediment input and output through a control volume, then this should represent either bed deposition or erosion.

If ix is the sediment input, ix + δx the sediment output and q the sediment that settles through the water column, then for a time δt :

(ix - ix+δx) δx δt + q δx δx δt = δz δx δx

Dividing by δx δx δt

(ix - ix+δx)/δx + q = δz/δt

And if δx → 0 and δt → 0 then

-I / x + q = z / t

Sediment continuity equation

The SBEACH model (Larson and Κraus, 1989)

Fig. 14 Discretisation of the SBEACH model (after CEM, 2008). It approximates the sediment continuity equation with finite differences and a step-mesh of discretisation.. Vertical changes in the water depth h are defined by the horizontal gradients of sediment transport rate q.

Fig. 15 Coastal wave zones. Longshore transport in the coastal zone occurs mainly in the surf and swash (wave run up) zones (After SEPM, 1996). Key: h, water depth; H, wave height; L, wave length.

Surf zone

Swash zone

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