Circulation in harbours and embayments
Jüri Elken
elken@phys.sea.ee
Tallinn University of Educational Sciences, Chair of Geophysics
Estonian Marine Institute,
Department of Marine Physics
Contents
Introduction
Basic forces and dynamical balances
Coriolis force
Hydrostatic balance
Geostrophic balance
Forcing from the open sea
Shelf conditions
Water level
Wind waves
Estuaries
Types of estuaries
Water and salt budget
Stratification of estuaries and fjords
Estuarine circulation
Barotropic response of a semienclosed basin (harbour resonance)
Nearshore processes
Transformation of waves in the nearshore zone
Some applied examples
Practical tips
Lecture notes by Jüri Elken
Page 1
Introduction
Hydrodynamics follow the Second Newton Law:
momentum change
local change (time derivative)
advective change
Coriolis acceleration
=
sum of applied forces
gravity
pressure gradient force
frictional forces
Mass conservation leads to coninuity of velocity field
Temperature, salinity and other constituents follow
advection-diffusion equation.
Water density depends on temperature and salinity. It forms horizontal
pressure gradients together with sea level.
Turbulence is formed by irregular eddies that act like viscous friction
and diffusion
In general, forcing of circulation is done by
wind
pressure gradients in water that come from
sea level
density
due to differential heating
due to fluxes of freshwater and saline water
Coastal embayments have restricted connection to the open sea.
This allows to develop their own internal circulation that is still
greatly affected by open sea conditions. Forcing of embayment
circulation is also done by
 freshwater discharge and adjacent sea water salinity
 water level oscillations
 transformation of wind waves
Lecture notes by Jüri Elken
Page 2
Coriolis force




becomes evident due to Earth rotation
deflects motion to the right in the Northern Hemisphere
allows geostrophic balance
causes inertial oscillations
It is expressed by Coriolis parameter f  2 sin  ,
 - angular velocity of Earth,  - geographic latitude
Momentum equations for inertial oscillations
u
 fv  0
t
v
 fu  0
t
u  A sin ft, v  A cos ft
that has solution
This means that current vector rotates clockwise with a period
Tf 
2
f =14 h
at latitude 600 N
Inertial period T f is frequently dominating in current observations in
the interior of large basins
Lecture notes by Jüri Elken
Page 3
Hydrostatic balance
Hydrostatic balance holds for motions with time scale > hour
vertical pressure gradient = gravity force per unit volume
p
 g
z
Pressure at depth z (counted here downwards from undisturbed surface) is
calculated by integrating hydrostatic equation from sea level elevation
 (positive upwards)
z
p  pa  g    x, y, z, t  dz  g 0  x, y, t 
0
Horizontal pressure gradient at a given depth depends on the
gradients of water density  and sea level  .
If density is constant, then p   g z    and pressure gradient
p



g
is determined by sea level gradient x
x .
Barotropic motions have no horizontal density gradient (vertical
density gradient is allowed) and all the pressure gradient comes from
sea level
Baroclinic motions have both the density and sea level contribution
to the pressure gradient
Lecture notes by Jüri Elken
Page 4
Geostrophic balance
In geostrophic motions horizontal pressure gradient force is
balanced by Coriolis force
 fv  
1 p
 x
fu  
1 p
 y
Geostrophic flow is along the lines of constant pressure. In the
Northern Hemisphere the flow is to the right from the pressure
gradient (from high to low pressure).
p
1
y
mp
mp
v
v
H
p2
mp
L
v
p2 >p1
x
Typical cross-section of sea level and density in oceanic eddies:
cyclone (left) and anticyclone (right)
z
0
ξ
ρ
zR
With increasing depth, density gradient works usually against the
sea level gradient that deep waters are in "no motion". In the ocean
the level of "no motion" z R is taken frequently 1500 m.
Lecture notes by Jüri Elken
Page 5
Shelf conditions
Steady frictional drift currents form in the upper layer Ekman
transport that is deflected by 900 to the right from wind direction
on the Northern Hemisphere under the influence of Coriolis force.
Along-shore wind may blow the surface water offshore. Near the
coast deeper waters below the pycnocline (density jump layer) rise to
the surface to replace the offshore transported surface waters. This
process is called upwelling.
Upwelling is a time-dependent phenomenon. It modifies the
temperature, salinity, nutrients and other water properties that are
transported to the coastal embayments. Largest changes in the nearshore water take place when the pycnocline outcrops to the surface.
Lecture notes by Jüri Elken
Page 6
Water level
Equations
Small water level oscillations are well described by simplified
shallow water equations
u~
  x   bx
 fv~   g

t
x
H
v~
  y   by
 fu~   g

t
y
H
  ~  ~
 Hu  Hv  0
t x
y
momentum balance x
momentum balance y
mass balance (continuity)
0
0
1
~  v dz are depth-averaged currents,  is water level,
u
dz
v
Here u~ 
,

H H
H
 x ,  y  bx ,  by are the wind and bottom stress components, both due to friction.
Free oscillations in a narrow channel of constant depth
are governed by equations
u~


u~
 g
 H
t
x ,
t
x
Eliminating water level we obtain hyperbolic wave equation
 2 u~
 2 u~
 gH 2  0
t 2
x
which solution is
u~  u~0 exp ikx   t   u~01 coskx   t   u~02 coskx   t  ,
where  
2
2
is wave frequency, T - period, k 
- wavenumber,  - wavelength.

T
Long non-rotational gravity waves have
dispersion relation   gH k , with phase speed
Lecture notes by Jüri Elken
c  gH
Page 7
Seiches
The channel closed at two end sets up the boundary conditions
u~0, t   u~L, t   0
that are satisfied if the specific solution is u~  u~0 cos  t sin k n x
n
,
n  1, 2...
and the wavenumber has discrete values k n 
L
In the result we will get discrete modes of standing waves or
seiches.
Along-channel velocity modes
Water level modes
1
3
3
1
2
The longest period of seiche is
2
T1 
2L
gH . Variable wind is
inducing motions that have resonance at seiche periods.
Lecture notes by Jüri Elken
Page 8
Kelvin waves
For 2D transient gravitational long waves Coriolis force becomes


2
2
2
2
important. Dispersion relation is   f  gH k  l .
If the long waves are periodic in both directions, that is
k 2  0; l 2  0; k 2  l 2  0 then minimum frequency is   f .
For lower frequencies or longer wave periods T  T f there must
2
2
be k  l  0 . It is possible for Kelvin waves that have exponential
profile in one direction, having maximum wave displacement at
the coast.
In a channel Kelvin waves propagate with    gH k and
c

k
  gH . The wave decay across the channel is dependent on
the barotropic Rossby deformation radius Rde 
gH
f .
Kelvin wave in a channel. The wall of maximum wave
displacement depends on the wave direction
c
x
z
y
H
L
Lecture notes by Jüri Elken
Page 9
Tides, amphidromic systems
Tides are generated by periodically acting gravitational forces by Sun
and Moon. Dominating tide periods are around 12 h and 24 h.
At periods longer than the inertial period, the excited waves form
amphidromic systems, both for seiches and tides.
The phase of Kelvin wave progresses anticlockwise along the coast of
the basin as given by co-tidal lines. The value of highest or lowest
water level deviation is given by co-range lines. The range for the
Kelvin wave is highest near the coast, but may vary from one coast
position to another. Co-tidal lines converge at amphidromic points,
where the wave amplitude vanishes.
Lecture notes by Jüri Elken
Page 10
Surface Waves
Equations
For rapid processes like wind waves we have to use
non-hydrostatic equations
3 momentum components
u
1 p

t
 x
v
1 p

t
 y
w
1 p

g
t
 z
continuity
u v w
 
0
x y z
we consider deviations from hydrostatic pressure
p  pa  g z  px, y, z, t 
Boundary conditions at the surface are

 w z    0
kinematic
t
p  g 
dynamic p  pa z   
or
Introducing the velocity potential


v   ; u 
x
; v

y
; w
z   

z
we reach the Laplace equation
 2  2  2


0
x 2 y 2 z 2
with boundary conditions

z   H 
0
z
 2


g
0
z
t 2
Lecture notes by Jüri Elken
 z  0
Page 11
Wave properties
The wave solution gives
dispersion relation
  gk tanh kH 
phase speed
c

k

g
tanh kH 
k
Wave properties have two extreme cases:
1) short wave limit kH  1 then tanh kH  1
  gk , c 
g

k
g
2
short waves are dispersive = phase speed depends on wavelength
2) long wave limit kH  1 then tanh kH  kH
  gH k
, c  gH
long waves are non-dispersive
the expressions are the same as derived from shallow water equations
Water particles perform in the wave field orbital motions
which amplitude decreases by depth
Large depth / short wave
Lecture notes by Jüri Elken
Small depth / long wave
Page 12
Wave superposition, group speed
Actual waves may be described as a superposition (sum)
of elementary waves
Superposition of two waves which wavenumber and frequency are
slightly different is mathematically given as
x dk  t d 
 x dk  t d  
 cos kx   t 

2
2

 

 x, t   a coskx   t   cosk  dk x    d t   2a cos
The wave is modulated by Ax, t   2a cos x dk  t d  .

2

Resulting wave packet moves with the group speed c g 
d
dk
A superposition of 5 elementary waves at two different time instances
1
c

Group speed of surface waves: g 2

g
2kH 

tanh kH  1 


k
sinh
2
kH


Group speed of short waves is half of the phase speed
Long waves have the same group speed as phase speed
they are non-dispersive
Lecture notes by Jüri Elken
Page 13
Excitation of wind waves
Wave growth is limited by wind energy input. Mechanisms for steady
wave regimes are:
 moderate winds - energy input by wind gets minimal, wave speed
gets equal to wind speed by increasing the wavelength
 stronger winds - wind energy is lost into turbulent mixing by
breaking waves
Studies like JONSWAP have shown, that wind-generated wave
properties depend on wind speed VW and fetch F , the angle-average
length the wind is blowing over the water.
F
h

0
.
0016
V
s
W
Significant wave height
g
Wave period (spectral pike)
V F 
T  0.286 W 2 
 g 
1
3
Besides the dominating wave period, also other periods are excited
close to the spectral pike.
Spectra of wind waves at different wind speeds
Lecture notes by Jüri Elken
Page 14
Other wave properties may be calculated from the linear wave theory,
for example:
Wavelength derived from the period by the dispersion relation
gT 2
 2H 

tanh 

2



Maximum orbital speed of wave motions near the bottom
u max 
2 hs
 2H 
T sinh 




Fetch (km) for given wind direction
in the Gulf of Riga
Frequency (%) during a year
that near-bottom wave velocity
exceeds 10 cm/s
Wind
Lecture notes by Jüri Elken
Page 15
Types of estuaries
Lecture notes by Jüri Elken
Page 16
Water and salt budget
Knudsen formula for steady state water exchange
Water budget:
Q R  Q B  QS
Salt budget:
S R QR  S B QB  S S QS
inflow (volume transport) from the open sea
QB 
Q R S S  S R 
SB  SS
Knudsen formulae works well, if
 salinity of the basin is clearly different from the adjacent sea water
 all the variables are steady or slowly changing
 inflow and outflow are well separated
Residence time (flushing time) for water, basin volume V
TR 
V
QS
Resolving the Knudsen budgets on a monthly scale, changes of
volume and total salt amount have to be added. Changes in salt
amount are in many cases statistically rather uncertain.
Lecture notes by Jüri Elken
Page 17
Stratification of estuaries and fjords
It is necessary to resolve budget equations for salt and heat
rate of change of total amount = advective fluxes + mixing fluxes
model types:
 bulk (Knudsen)
integrated over basin
 fixed layers/boxes
integrated over boxes
 1D vertical resolution
horizontally integrated
Ai
 Si
t


z
Ai K iV
 3D resolution
 Si
z


z
Ai wi Si qi½ Si½ qi½ Si½
full differential equation
S
S
S 
S 
S  S
S
 u  v  w  s  s   s
t
x
y
z x x y
y z z
advection/currents are calculated from hydrodynamics,
mixing is parameterised
(= effect is expressed by model state variables)
Lecture notes by Jüri Elken
Page 18
Estuarine circulation
Fujiwara et al. (1997) have studied residual circulation in a gulf type
ROFI that’s width is larger than internal Rossby deformation radius
Upward entrainment produces flow divergence in the upper layer
Lecture notes by Jüri Elken
Page 19
Total circulation (right) is composed from
1) anticyclonic flow that assumes no divergence but only vorticity of
the flow (right)
2) estuarine flow of the narrow basin that assumes no vorticity but
only divergence (centre)
Resulting flow is anticyclonic near the river mouth and seaward
sheared flow further on. These two flow regimes are separated by
stagnation point (dot). Pycnocline slope is in a geostrophic balance
with the sheared seaward flow.
Lecture notes by Jüri Elken
Page 20
An example of anticyclonic circulation near the river mouth is given
from the Gulf of Riga, Baltic Sea (Elken and Raudsepp).
a) Salinity, May 1993
b) Currents, May 1993
58.6
58.6
58.4
58.4
58.2
58.2
58.0
58.0
57.8
57.8
57.6
57.6
57.4
57.4
57.2
57.2
57.0
57.0
21.5
22.0
22.5
23.0
23.5
24.0
24.5
c) Salinity, November 1993
21.5
58.6
58.4
58.4
58.2
58.2
58.0
58.0
57.8
57.8
57.6
57.6
57.4
57.4
57.2
57.2
57.0
57.0
22.0
22.5
23.0
23.5
10
22.0
22.5
23.0
23.5
24.0
24.5
d) Currents, November 1993
58.6
21.5
cm/s
24.0
24.5
21.5
cm/s
20
22.0
22.5
23.0
23.5
24.0
24.5
Observed surface salinity distribution (a,c) and modelled monthly
mean surface currents superimposed on depth contours (b,d) during
May (a,b) and November (c,d) 1993. Mean wind stress has been (0.074;-0.035) dyn/cm2 in May and (-0.56; 0.63) dyn/cm2 in
November.
Lecture notes by Jüri Elken
Page 21
Barotropic response of a semienclosed basin
(harbour resonance)
Helmoltz or pumping mode of water level oscillations (zero mode
in term of seiches of the closed basin) where the sea level executes
spatially uniform oscillation becomes evident in short and relatively
deep harbours, embayments and estuaries which are connected to the
open sea by a narrow channel.
Consider a basin of constant depth with an area A that is connected
to the open sea by a channel with depth H and width B . If the flow
speed in channel is u , then basin water level  i is subject to mass
d i
 uHB  0
conservation (continuity) A
dt
 i   e 
du

g
 ru
The momentum equation
dt
L
is derived from
u

g
 ru
t
x
by integrating along the channel of
length L , with  e being the water level on the open sea side of the
channel and r the linear friction coefficient.
Resulting equation for water level is
d 2 i
d i gHB
gHB

r



e
i
dt
AL
AL
dt 2
that is exactly equal to forced oscillations of frictional pendulum.
If the external water level forcing is absent  e  0 then
embayment water level performs damped oscillations
2
 r   gHB r 
i  i 0 exp   t  cos

t

4 
 2   AL
If the external water level is periodic then amplification of interior
embayment water level oscillations takes place near
resonant frequency  
Lecture notes by Jüri Elken
gHB r 2

AL
4
Page 22
Otsmann, Suursaar and Kullas (1999) have constructed the system
of Helmoltz resonators for the Gulf of Riga, Baltic Sea that is
connected to the open sea by a system of straits and buffering
Väinameri basin
Hari St.
Baltic
ESTONIA
Proper
Väinameri
Ristna
Suur St.
Soela St.
58ºN
Gulf of
Riga
Irbe St.
LATVIA
40 km
23ºE
Daugava
River
Configuration of the model as a system of Helmoltz resonators
Lecture notes by Jüri Elken
Page 23
The Helmoltz resonator model gives very good results compared to
the current observations in a narrow strait where the flow is oscillating
but at any time moment is uni-directional.
Forcing functions of the model are:
1) water level in the open sea
2) winds above the channels
3) constant river discharge
Measured and calculated velocities in the Suur Strait,
January-March 1995
u (cm/s)
120
(a)
calculated +60cm/s
80
40
0
-40
measured
-80
0
15
30
45
60
75
90
Time (days)
days
Lecture notes by Jüri Elken
Page 24
Nearshore processes
Transformation of incident wind waves induces residual wave
currents. Wave action is mathematically expressed as wave stress
that enters as a forcing term (similar to wind stress) into the
circulation model.
Waves approaching the coast by an angle induce longshore current.
Rip currents transport water seaward
Waves approaching the coast elevate the water level that supports
longshore transport
Lecture notes by Jüri Elken
Page 25
Transformation of waves in the nearshore zone
Reflection and diffraction at vertical walls
Ideal wave reflection follows the rules of reflected light.
Diffraction will spread the wave around the corner of wall (cape,
quay etc). It acts as secondary wave point source from where the
waves propagate in circular wave patterns with smaller amplitude.
Refraction takes place at variable bottom depth that controls the phase
speed of waves. Curvilinear bottom contours act as positive or
negative lenses for the light, focusing or defocusing the wave rays
Lecture notes by Jüri Elken
Page 26
When waves propagate to smaller depths, wave steepness increases
and they will break
In realistic basins and structures the wave transformation can be
calculated by a model based on Boussinesq wave equations that
1) calculates 2D time-dependent behaviour of water level and
horizontal velocity components that are expressed by the velocity
potential
2) accounts for dispersive effects on variable depth, i.e. dependence
of wave phase speed on wavelength (dispersion is not accounted in
traditional shallow water circulation models)
3) accounts for non-linear wave effects like increase of steepness by
increase of wave height, accounts the effects of wave breaking
4) allows calculation of wave stress that is forcing the residual
currents
Currents also modify the waves.
Lecture notes by Jüri Elken
Page 27
Some applied examples
Wave transformation near the harbour entrance of Ruhnu,
Estonia.
Calculations by Uno Liiv with the model MIKE 21BW (DHI) . The model grid
step is 2m. Incident waves are prescribed from the JONSWAP spectrum.
Lecture notes by Jüri Elken
Page 28
Coastal structures modify wave-induced transport considerably
From Aelbrecht and Denot, 1999.
Numerical simulation of the laboratory experiments on the effects of
L-shaped breakwater. From Pan et al., 1999.
Lecture notes by Jüri Elken
Page 29
Influence of circulation and sediment flow by land reclamation in the
Rotterdam harbour area. From de Kok et al., 1999.
Regulating the residual current loops that trap cohesive sediments by
Current Deflection Wall. From Crowder et al., 1999
Tidal river
Tidal river
Current
Deflection
Wall
Lecture notes by Jüri Elken
Page 30
Practical tips
 Try Knudsen budget for estimating the steady water exchange if the
salinity of the embayment is clearly different from the adjacent sea
 Try Helmholtz resonator model for water-level induced in- and
outflows
When negotiating with professional modellers, remember:
 There is optimum complexity of the model, too complex models
are very difficult to be "tuned"
 If the embayment is vertically well mixed, 2D shallow water
equations are usually enough adequate. Otherwise use 3D model
but it is much more costly for setup, calibration and computation
time
 wave-induced currents are important in the nearshore zone
Suggested reading:
 Waves, tides and shallow-water processes. G.Bearman (editor),
Pergamon Press, 1989. (easy reading with few equations)
 Fluid Mechanics for Marine Ecologists. S.R.Massel. Springer,
1999. (textbook)
 Hydrodynamics and transport for water quality modeling.
J.L.Martin and S.C.McCutcheon, Lewis Publishers, 1998.
(textbook, handbook of available models)
 Coastal Engineering and Marina Developments. C.A.Brebbia and
P.Anagnostopoulus (editors). WIT Press, 1999. (state-of-the-art
collection of papers)
Lecture notes by Jüri Elken
Page 31