Transcript Διαφάνεια 1

Beach modelling III: Morphodynamic
models – Description
Adonis F. Velegrakis
Dept Marine Sciences
University of the Aegean
Synopsis
1 Beach morphodynamic models
1.1 Purpose
1.2 ‘Static’ and dynamic ‘bottom-up’ models
2 Static models
2.1 The Bruun (1962, 1988) model (rule)
2.2 The Edelman (1972) model
2.3 The Dean (1991) model
3 Dynamic models
3.1 Basic structure
3.2 Hydrodynamic models
3.3 Sediment transport modules
3.4 Morphological module
4 Beach retreat 1-D models
4.1 The SBEACH model
4.2 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 ‘bottomup’ 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 subaerial 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:
l a
s
hc  Bh
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)
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
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:

wb
ds d 



dt dt  hb  Bh (t ) 
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:
 hb  Bo

s(t )  wb ln 

h

B


(
t
)
h
 b

Where Βo, initial berm elevation.
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:
wb
s  a  0.068H b 
Bh  hb
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
3.1 Dynamic models: Basic structure
They perform calculations at different locations (nodes) of the beach (profile) and
simulate its evolution in the desired time step.
They consist of the following sub-models (modules):
The hydrodynamic sub-model (module) which estimates beach
hydrodynamic conditions (waves and wave-induced currents) with input
parameters the seabed morphology (bathymetry), the offshore wave
conditions, and the sediment characteristics (as bed friction control)
The sediment dynamic sub-model (module), which estimates sediment
transport due to waves, wave-induced currents (and their interaction) on
the basis of the hydrodynamci conditions estimated by the hydrodynamic
module
The morphological sub-model (module) that estiamates the new
morphology on the basis of the sediment transport patterns estimated by
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
3. 2 Hydrodynamic models II
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 differencesfinite elements).
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 waveinduced 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
4.1 The SBEACH model (Larson and Κraus, 1989)
Governing expressions
Wave module
ΕF: wave energy flow
kw
dEF
  E F  E Fs 
dx
h
ΕFs: constant wave energy flow
Ks: empirical coefficient of sediment
transport rate ,
Sediment transport module
q   s ( De  Deq 
kw:empirical coefficient of wave
dissipation
 dh
 s dx
De: Energy dissipation ,
)
Deq: energy dissipation in equilibrium qiη
ενέργεια διάχυσης σε ισορροπία
ε: coefficiet related to the sediment
transport rate for the bed slope term
4.2 The Leont’ yev model: Hydrodynamic module
It is based on the energetics approach of Battjes and Janssen (1978), i,e.
on the assumption that cross-shore changes in wave energy flow in
each profile location equal wave energy losses due to bottom fiction
φ: wave angle

 Ew  c g  cos
x
  D
e
Ew: wave energy
cg: wave group celerity and
De: wave energy dissipation
4.2 The Leont’ yev model: Sediment transport module
The beach profile is divided into zones, with sediment transport varying along the
profile as (Leont’yev 1996):
Wave refraction zone: qR = 0 και q = qW
Surf 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 qW, in the refraction
zone is:
Bedload/suspended sediment load


 ws d 
b
3
2
~
~
qW 
f w  u cos  3u U d   s Fe  Be 
 
2 tan
 U d x 
f w : friction factor
ws: sediment settling velocity
φ: angle of approach
εs: effectiveness coefficient
Fe , Be: energy losses due to bed friction and turbulence, respectively
1
4.2 The Leont’ yev model: Sediment transport module
Sediment transport in the swash zone:
  1  x / xm 

q R  q R 
 1  x R / xm 
3/ 2
x R  x  xm

qR  qR expc3 x  xR / H o  x  x R
 : maximum sediment transport
qR
c3=0.2-0.3
Η0 : offshore wave height
4.2 The Leont’ yev model: Morphological module
Beach morphological change is defined by the sediment
continuity equation
Sediment porosity is also considered
4.3 Boussinesq model
This state-of-the-art model is not included in the training tool as
(a) It is very heavy (a day to run)
(b) It requires extensive expertise and
(c)
It is still under development
Nevertheless, as it is used for model intercalibration (see Presentation 4), a brief
description is given
The model, that has been initially developed by Karambas and Koutitas (2002),
includes:
A wave model that is based on Boussinesq equations for dispersive, nonlinear waves
A sediment transport module, that calculates:
1.
2.
3.
bedload and sheet flow using the expressions of Dibajnia et al. (2001),
suspended sediment load through the energetics approach
sediment transport at the swash zone using an expression based on MeyerPeter & Muller
4. 3 Boussinesq hydrodynamic module
Continuity equation
  Ud 

0
t
x
U: horizontal velocity
ζ: sea level rise,
h: normal sea level
d=h+ζ
τb: bed shera stress
Β i= 1/15
Εv:eddy coefficient
Μu:effect of the non-normal velocity distribution
Mom. equations
U 1 M u 1 Ud 
 h 2  2h  3U
 2U

 U
g

 hx d
2
t d x
d
x
x
3
xt
x t
  3U U  2U 
  2U
 2U
 U
U 3 


h

hh
U

h
x
x
x x 2 
x x 2
x t
x 2
 x

U  
2


U


3
3
  2U
 2  U 

U


 2

x


2

h


  Bi h  2  g 3 
 2 Bi hhx  xt  g x 2
xt  x 
x t
x
x 2





h2

3
 b
   Ev
 d
4. 3 Boussinesq sediment dynamic module
The expression for the estimation of the sediment transport sheet flow qb due to
non-monochrmatic waves is:
bedload
qb
u cTc  c  t   ut Tt  t  c 
 0.0038
ws d 50
Tc  Tt  s  gd50
uc , u t
Tc ,, Tt
: velocities under crest and trough
: the associated durations
ws
: settling velocity
 c  t : percentages of sediment that are directly transported
c, t : percentages of sediment that remain after the half period
4. 3 Boussinesq sediment dynamic module
Suspended load
1 b s DeU b
qs 
as ws
De
Ub
b
s
: mean wave energy diffusion due to breaking
: current velocity
: parameter that relates nera bed value to mean value
De
: effectiveness coefficient
4. 3 Boussinesq sediment dynamic module
Sediment transport rate ate the swash zone is estimated by a
modified Meyer-Peter & Muller model:
bedload
QR 
qb 1   
s

 1gd503
QR : non dimensional transport rate

Cr
U
32
tan 
U
1
tan
 : Shields parameter
 : porosity
 : angle of repose
t an  : bed slope
C r : parameter that takes different value whne the water moves
onshore
4. 3 Boussinesq morphological module
It estimates profile changes using:
Numerical solution of the continuity equation
An additional gravity term to take into account the bed slope
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
If time < wished
new morphology
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)
1.4
1.3
1.2
1.1
700
1
y (m)
0.9
600
0.8
0.7
0.6
500
0.5
0.4
400
400
0.3
500
600
700
800
x (m)
900
1000
1100
1200
0.2
0.1
0
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),
y (m)
700
600
500
400
400
500
600
700
800
900
1000
1100
1200
x (m)
2 m/s
Εξέλιξη μορφολογίας
Αρχική μορφολογία
10
9
700
8
y (m)
7
600
6
5
4
500
3
2
400
400
1
500
600
700
800
900
1000
1100
1200
0
x (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).
Kt 
Ht
Hi
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
85000
80000
Sand Transport Rates
(kg/m/tidal cycle)
25
10
5
75000
2
1.75
1.5
1.25
70000
1
0.75
0.5
0.25
65000
0
355000
360000
365000
370000
375000
380000
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)
Sediment continuity equation
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
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.
Swash zone
Surf zone
>
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.