JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 109, C05011, doi:10.1029/2003JC002057, 2004
Numerical modeling of the semidiurnal tidal exchange through
the Strait of Gibraltar
G. Sannino, A. Bargagli, and V. Artale
Ocean Modeling Unit, Special Project Global Climate, ENEA C. R. Casaccia, Ente per le Nuove Technologie, l’Energia
e l’Ambiente, Rome, Italy
Received 21 July 2003; revised 22 February 2004; accepted 5 March 2004; published 7 May 2004.
[1] A three-dimensional sigma coordinate free surface model is used to investigate the
semidiurnal tidal exchange through the Strait of Gibraltar. The model makes use of a
coastal-following, curvilinear orthogonal grid that includes the Gulf of Cadiz and the
Alboran Sea, with very high resolution in the strait (<500 m). A lock-exchange initial
condition is used: the western part of the model domain is filled with Atlantic water,
whereas the eastern part is filled with Mediterranean water. The model is forced at the
open boundaries through the specification of the semidiurnal (M2 and S2) tidal surface
elevation. The model is run over a spring neap cycle (fortnightly period), and the results
are compared with available observed data. Simulated cotidal maps of the M2 and S2 tidal
elevation components are in quantitative and qualitative good agreement with observed
data as well as with the simulated major and minor axis of tidal ellipse. The model
reproduces the generation and the subsequent propagation of internal bores both eastward
and westward, showing that they are always generated during the fortnightly period.
However, the principal aim of this work is to quantify the effects of tidal forcing on mean
quantities, entrainment, and transport of Atlantic and Mediterranean water along the strait.
Model results reveal that the contribution of the semidiurnal tidal component (M2) to the
transport is relevant over Camarinal Sill, whereas it is negligible at the eastern end of
the strait. Model results indicate, also, that the effect of the semidiurnal tide is to increment
INDEX TERMS: 4255
the mean transport by about 30% both for the inflow and the outflow.
Oceanography: General: Numerical modeling; 4560 Oceanography: Physical: Surface waves and tides (1255);
4512 Oceanography: Physical: Currents; KEYWORDS: Gibraltar, tides, POM
Citation: Sannino, G., A. Bargagli, and V. Artale (2004), Numerical modeling of the semidiurnal tidal exchange through the Strait of
Gibraltar, J. Geophys. Res., 109, C05011, doi:10.1029/2003JC002057.
(Roma) Horrenda late nomen in ultimas extendat oras, qua medius
liquor secernit Europen ab Afro, qua tumidus rigat arva Nilus.
(Rome) Feared everywhere, let her extend her name to the uttermost
shores, where the midway water separates Europe from Africa,
where the swollen Nile irrigates the fields.
[Horatio, 8 B.C., pp. 45 – 47]
1. Introduction
[4] As 2000 years ago so now the Strait of Gibraltar
separates Europe from Africa and connects the Atlantic
Ocean to the Mediterranean Sea. It is 60 km long and
20 km wide with a minimum width of less then 15 km
near the contraction of Tarifa Narrows and a shallow sill
located near Camarinal (west of Tarifa) with a minimum
depth of less than 300 m (Figure 1).
[5] The excess of evaporation over precipitation and river
runoff in the Mediterranean basin represents, together with
the conservation of mass and salt, the main driving force
of a mean circulation through the strait. This circulation,
Copyright 2004 by the American Geophysical Union.
0148-0227/04/2003JC002057$09.00
generally called inverse estuarine, is characterized by two
counter flowing currents: in the upper layer warm and
relatively fresh Atlantic water, with a salinity of 36.2 practical salinity units (psu), flows eastward, spreading into the
Mediterranean Sea, and in the lower-layer cold Mediterranean water, with a salinity of 38.5 psu, flows westward
toward the Atlantic Ocean [Lacombe and Richez, 1982]. As
initially suggested by Bryden and Stommel [1984], the mean
circulation of the strait can be described as a two-layer system
hydraulically controlled at Camarinal Sill. Many other papers
have subsequently dealt with the applicability of the hydraulic control theory to the Strait of Gibraltar, and the number
and possible locations of such controls within the strait (see,
for example, Armi and Farmer [1985], Farmer and Armi
[1986], Bryden and Kinder [1991], and more recently, for a
modeling study, Sannino et al. [2002] (hereinafter referred to
as SBA02)).
[6] Various processes, at different timescales, modify the
mean flow through the Strait of Gibraltar. The mean flow
shows seasonal [Garrett et al., 1990] and interannual
variability, weekly modifications driven by the wind and
by atmospheric pressure differences between the Atlantic
and Mediterranean Sea [Candela et al., 1989; Garcı́a
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Figure 1. Chart of the Strait of Gibraltar showing the principal geographic features referred to in the
text. Locations of current meter moorings deployed during the Gibraltar Experiment (October 1985 –
1986) and during the Canary Islands Azores Gibraltar Observations (CANIGO) observations (October
1995 –April 1996) are also shown with red and blue solid circles, respectively.
Lafuente et al., 2002], semidiurnal variations due to strong
tides and finally, on very short timescales, modifications
due to internal bores (internal wave reaching amplitudes of
up to 150 m [Richez, 1994]).
[7] The tidal forcing in the strait has been extensively
studied and analyzed in the past. On the basis of data
collected during the Gibraltar Experiment during 1985 –
1986 [Bryden and Kinder, 1988]. Candela et al. [1990]
(hereinafter referred to as CA90) and Bryden et al. [1994]
described the structure of the barotropic M2 tide and of the
tidal transport through the strait, respectively, Bruno et al.
[2000] have described the vertical structure of the semidiurnal tidal current at Camarinal Sill, while Wang [1993] used a
numerical model to study tidal flows, internal tide as well as
fortnightly modulation. Recently, others studies have been
carried out, based on direct observations collected during the
Canary Islands Azores Gibraltar Observations (CANIGO)
project (1995 – 1996) [Parrilla et al., 2002]: Tsimplis [2000]
has described the vertical structure of tidal currents at
Camarinal Sill, Tsimplis and Bryden [2000] (hereinafter
referred to as TB00) have estimated the water transports
through the strait, Garcı́a Lafuente et al. [2000] have
analyzed in detail the tide at the eastern section of the strait,
and Baschek et al. [2001] (hereinafter referred to as BA01)
have estimated the transport with a tidal inverse model.
[8] To estimate the effect of tidal forcing on mean flow,
Farmer and Armi [1986] included tides into their hydraulic
theory by using a quasi-steady approximation in which the
steady solution is verified at each time of a tidal cycle.
However, Helfrich [1995] showed that this approach is not
valid for dynamically long straits, i.e., straits having a
length greater than the distance traveled by an internal wave
during a tidal cycle, which is precisely the situation that
occurs in the Strait of Gibraltar. Both theories assert that the
exchanged flows increase with the strength of the barotropic
tidal forcing, but the quasi-steady theory always predicts
more flow than the time-dependent theory.
[9] The purpose of this work is to implement a threedimensional (3-D) high-resolution, primitive equation, freesurface numerical model, of the circulation in the strait
region and to use it to: (1) reproduce the semidiurnal tides
within the Strait of Gibraltar, (2) estimate the water transports through the strait and (3) evaluate the effect of tidal
forcing on the mean exchanges and entrainment.
[10] The paper is organized as follows. Section 2 contains a description of the model used to simulate the tide
in the strait. In section 3, model results are compared with
available data of surface elevation, currents and internal
bores measurements. Section 4 is devoted to the study of
the tidal effect on water transports and entrainment
through the strait, while summary and conclusions complete the paper.
2. Model Description
[11] The numerical model used for this study was implemented in SBA02, where it was used to investigate the
mean exchange through the Strait of Gibraltar. The model
was only forced by the density contrast between the
Alboran Sea and the Gulf of Cadiz, without any other
forcing, such as tides, wind or atmospheric pressure. The
main differences introduced in the present model regard the
treatment of open boundary conditions, forcing and vertical
resolution. In the following we only focus on the principal
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model characteristics and on the main differences with
respect to the model implemented in SBA02.
2.1. Model Grid and Bathymetry
[12] The region covered by our model includes the
Strait of Gibraltar and the two adjacent subbasins
connected to it: the Gulf of Cadiz and the Alboran Sea.
The horizontal model domain is discretized by a curvilinear orthogonal grid made by 306 53 grid points (see
Figure 2 in SBA02). The resolution in the strait is much
higher (500 m) than in the eastern (8 – 15 km) and
western ends (10 – 20 km), so that the dynamics in the
strait will be well resolved. The vertical grid is made of
32 sigma levels, logarithmically distributed at the surface
and at the bottom, and uniformally distributed in the rest
of the water column. The model topography has been
obtained by merging the high-resolution (<1 km) topographic data set of the Strait of Gibraltar provided by the
Laboratoire d’Oceanographie Dynamique et de Climatologie with the relatively low-resolution (5 min) U.S. Navy
Digital Bathymetric Database-5 data set (available from
U.S. Naval Oceanographic Office, Bay St. Louis, Mississippi, at https://128.160.23.42/dbdbv/dbdbv.html) for
the Alboran Sea and the Gulf of Cadiz. In an attempt
to reduce the well-known pressure gradient error produced by sigma coordinate grids in regions of steep
topography [Haney, 1991] an additional smoothing has
been applied where dH/H > 0.2, as suggested by Mellor
et al. [1994]. In order to estimate the residual pressure
gradient error, we have integrated the model for one year
without initial horizontal density gradient, i.e., with salinity and temperature fields only varying with depth,
with no open boundary applied, i.e., closed domain, and
without any other external forcing. In this integration the
maximum intensity of erroneous currents introduced by
the sigma coordinates is of about 2 cm s1. Since the
expected baroclinic velocities are up to 1 m s1 this error
seems to be tolerable. The resulting model topography in
the region of the strait, with the minimum depth of the
shelf set to 25 m, is shown in Figure 2. The dominant
topographic features of the strait (from west to east) are
clearly recognizable: Spartel Sill (Sp), Tangier basin,
Camarinal Sill (Cm) with a minimum depth of 284 m
and Tarifa Narrows.
2.2. Boundary, Initial, and Forcing Conditions
[13] Near the eastern and western ends of the computational domain two open boundaries are defined, where
values of velocity, temperature, and salinity must be
specified. In order to minimize the contamination of the
interior model solution due to wave reflection at the
boundaries, an Orlanski radiation condition [Orlanski,
1976] is used for the depth-dependent velocity at both
boundaries. A forced Orlanski radiation condition [Bills
and Noye, 1987] is used for the surface elevation at the
western and eastern boundaries:
nþ12
zi
¼
1
n
n1
zTi 2 þ zMi ðCr=2Þzi 2 þ Crzni1
1 þ Cr=2
;
ð1Þ
where zni represents the surface elevation at the i grid point
of the open boundary at time step n, Cr = cDt/(2Dx) is a
is the
Courant number defined in the x direction, zn1
Ti
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forcing tide elevation at the grid point i and time step n 1,
and zMi is the time-independent mean elevation at the grid
point i, which is set to about 12 cm at the western open
boundary and to 0 cm at the eastern open boundary.
Condition equation (1) incorporates a radiation mechanism
that allows the undesired transients to pass through the open
boundaries, going out of the model basin, without
contaminating the desired forced solution [Arnold, 1987].
A zero gradient condition is used for the depth-integrated
velocity.
[14] The time-independent mean elevation values used at
the open boundaries (zM) are obtained running the model in
barotropic mode. This model, as the baroclinic version, has
at the eastern and western ends of the computational domain
two open boundaries where values of barotropic velocity
and surface elevation must be specified. For the surface
elevation an Orlansky radiation condition [Orlanski, 1976]
was used at the western boundary while a clamped to zero
condition was used for the eastern end. For the barotropic
velocity a zero gradient condition was used at both ends. In
this way the barotropic model was able to freely adjust the
western surface elevation, after 180 days of simulation, to
about 12 cm.
[15] Temperature and salinity are specified on the open
boundaries by using an upwind advection scheme that
allows the advection of temperatures and salinities into
the model domain under inflow conditions. As in SBA02
the normal velocities are set to zero along coastal
boundaries, at the bottom, adiabatic boundary conditions
are applied to temperature and salinity and a quadratic
bottom friction, with a prescribed drag coefficient, is
applied to the momentum flux. This is calculated by
combining the velocity profile with the logarithmic law
of the wall:
CD ¼ max 2:5 103 ; k 2 lnðDzb =z0 Þ ;
ð2Þ
where k is the Von Karman constant, z0 is the roughness
length, set to 1 cm, and Dzb is the distance from the bottom
of the deepest velocity grid point.
[16] For the initial condition we have used the same
lock-exchange condition as in SBA02, i.e., we have filled
the model with two water masses, horizontally uniform
and vertically stratified, separated by an imaginary dam in
the middle of the strait (longitude 5420W) that is removed
at the initial time. Initial temperature and salinity fields for
the Alboran basin have been obtained from a horizontal
average of the spring MODB data (available at http://
modb.oce.ulg.ac.be/modb), while the spring Levitus [1982]
data set has been used to set initial values over the Gulf of
Cadiz (see Figure 5 in SBA02). As in SBA02 and
Napolitano et al. [2003], we have used the Smolarkiewicz
upstream-corrected advection scheme [Smolarkiewicz,
1984, 1990], in order to simulate correctly the free flow
adjustment to the density gradient within the strait after the
dam is removed.
[17] The model is forced at the open boundaries
through the specification of the surface tidal elevation.
Candela et al. [1990] and more recently Tsimplis [2000]
have found that 75% of the current variability in the strait
is due to the semidiurnal tide, so we have limited our
modeling study to the semidiurnal component, forcing the
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Figure 2. (top) Model bathymetry, computational grid, and transects for the presentation of model
results within the Strait of Gibraltar. The gray levels indicate the water depths. The points Cm and Sp
mark the points where Spartel Sill and Camarinal Sill are located, respectively. (bottom) Bathymetry
along the longitudinal section E.
model with only the M2 tide, with period of 12.42 hours,
and the S2 tide, with period of 12.00 hours:
zT ð y; t Þ ¼
2
X
An ð yÞ cosðsn t jn ð yÞÞ;
ð3Þ
boundary (ywm) during the neap tide ranges from 48 to
+75 cm, while during the spring tide ranges from 128 to
+140 cm. Owing to the strong velocities generated by the
tidal forcing short external and internal time steps of 0.1 and
6 s need be used in the simulation.
n¼1
where An (y) and jn (y) are the prescribed surface elevation
amplitude and phase of the nth tidal constituent and sn is its
frequency. The M2 and S2 surface tidal elevation amplitudes
and phases have been obtained from the global tidal model
of Kantha [1995] and Kantha et al. [1995]. The resulting zT
(ywm, t) applied at the middle point of the western open
3. Model Results
[18] The model was run for 360 days without tidal forcing
(zT ( y, t) = 0) in order to achieve a steady two-layer
exchange system. The steady exchange obtained is characterized by an inflow (toward the Mediterranean) and an
outflow (toward the Atlantic Ocean) of 0.62 and 0.51 Sv
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Table 1. Comparison Between Observed and Predicted Amplitudes A and Phases P of M2 Tidal Elevationa
Observed M2
Location
Gibraltar
Pta. Gracia
Tarifa
Pta. Cires
Pta. Carnero
DN
DS
SN
SS
DW
TA
AL
CE
DP5
Latitude North
Longitude West
36080
05210
3605.40
3600.20
3554.70
3604.30
35580
35540
36030
35500
35530
36010
36080
35530
36000
A, cm
Predicted M2
P, deg
Difference (Pre Obs)
A, cm
P, deg
A, cm
A, %
P, deg
Tsimplis et al. [1995]
29.8
46.0
29.7
46.0
0.1
0.3
+0.0b
0548.60
0536.40
0528.80
0525.70
Garcı́a Lafuente [1986]c
64.9 ± 0.2
49.0 ± 0.5
41.5 ± 0.2
57.0 ± 0.5
36.4 ± 0.2
46.5 ± 0.5
31.1 ± 0.2
47.5 ± 0.5
64.9
40.5
33.6
29.1
51.0
46.3
50.1
43.8
+0.0
0.8
2.6
1.8
0.0
1.9
7.1
5.8
+1.5
+10.2
+3.1
3.2
05460
05440
05430
05430
05580
05360
05260
05180
05340
Candela et al. [1990]
60.1
51.8
54.0
61.8
52.3
47.6
57.1
66.8
78.5
56.1
41.2
41.2
31.0
48.0
29.7
50.3
44.4
47.6
56.2
51.4
50.1
58.0
73.3
41.0
28.6
27.5
38.2
53.9
61.6
48.2
65.3
58.4
47.3
46.0
47.3
43.9
3.9
2.6
2.2
+0.9
5.2
0.2
2.4
2.2
6.2
6.4
4.8
4.2
1.5
6.6
0.4
7.7
7.4
13.9
+2.1
0.2
+0.6
1.5
+2.3
+6.1
2.0
3.0
3.8
a
Station locations are shown in Figure 1.
Calibration.
c
± indicates standard errors.
b
at the Camarinal Sill section, and of 0.69 and 0.58 Sv at
the Gibraltar-Ceuta section (1 Sv = 106 m3/s1).
[19] Transports were computed integrating the alongstrait velocity vertically from the bottom up to the depth
where the along-strait reverts its direction for the outflow,
and from this depth up to the surface for the inflow, and then
meridionally, across the Camarinal Sill and Gibraltar-Ceuta
sections (sections C and D in Figure 2):
Z
North
INð xÞ ¼
Z
0
uð x; y; zÞdzdy
South
hð x;yÞ
North Z hð x;yÞ
ð4Þ
Z
OUTð xÞ ¼
uð x; y; zÞdzdy;
South
bottom
where u is the along-strait velocity, h is the depth of the
interface, and x is the longitude. The computed transports
are about 15% less than the estimation carried out in
SBA02. This difference mainly depends on the value of
mean elevation (zM) used in the open boundary condition
equation (1) and on the better vertical resolution implemented in this present work.
[20] In order to achieve a stable time-periodic solution, the
model was run for further 29 days, forced only by the two
principal semidiurnal tidal components. Finally, after reaching the stable time-periodic regime, the model was run for a
further fortnightly period and the least squares harmonic
analysis was applied to the surface elevation and currents.
3.1. Tidal Elevation
[21] In Tables 1 and 2 the observed [Tsimplis et al., 1995;
Garcı́a Lafuente, 1986; Candela et al., 1990] and simulated
amplitudes (A) and phases (P) are compared, for the M2 and
S2 tidal elevation, respectively. A good agreement between
observed and predicted values is found; the maximum
Table 2. Comparison Between Observed and Predicted Amplitudes A and Phases P of S2 Tidal Elevationa
Observed S2
Location
Gibraltar
Latitude North
Longitude West
36080
05210
A, cm
Predicted S2
P, deg
Tsimplis et al. [1995]
10.7
72
Difference (Pre Obs)
A, cm
P, deg
A, cm
A, %
P, deg
10.5
72.0
!0.2
1.8
+0.0
b
Pta. Gracia
Tarifa
Pta. Cires
Pta. Carnero
DN
DS
SN
SS
DW
TA
AL
CE
DP5
3605.40
3600.20
3554.70
3604.30
0548.60
0536.40
0528.80
0525.70
Garcı́a Lafuente [1986]
22.3 ± 0.2
74.0 ± 1.0
14.2 ± 0.2
85.0 ± 1.5
14.1 ± 0.2
74.0 ± 1.0
11.5 ± 0.2
71.0 ± 1.0
20.3
14.7
13.1
10.6
77.9
69.8
76.7
68.6
1.8
0.3
0.8
0.7
8.1
2.0
5.7
6.9
+2.9
13.7
+1.7
1.4
35580
35540
36030
35500
35530
36010
36080
35530
36000
05460
05440
05430
05430
05580
05360
05260
05180
05340
Candela et al. [1990]
22.5
73.8
21.1
83.3
18.5
73.4
20.6
92.3
29.0
82.2
14.7
67.9
11.1
73.9
11.4
75.6
16.1
73.9
20.3
18.3
18.1
21.0
26.6
15.1
10.2
9.6
14.0
77.9
87.3
74.2
90.0
81.8
70.7
71.2
74.8
69.1
2.2
2.8
0.4
+0.4
2.4
+0.4
0.9
1.8
2.1
9.7
13.2
2.1
1.9
8.2
2.7
8.1
15.7
13.0
+4.1
+4.0
+0.8
2.3
0.4
+2.8
2.7
0.8
4.8
a
Station locations are shown in Figure 1.
± indicates standard errors.
b
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Figure 3. Cotidal charts of the (a) M2 and (b) S2 surface tides. Solid lines are phase contours in degrees;
dashed lines are amplitude contours in cm.
differences do not exceed 6.2 cm in amplitude (with a
maximum error of about 15%) and 13 in phase. The
maximum differences are confined to coastal points as
Ceuta (CE), Algesiras (AL), Tarifa and Pta. Cires, since
our model grid is not coastal fitted.
[22] In Figure 3 are also shown the computed cotidal
charts for the strait region, for the simulated M2 and S2
surface tidal waves. The M2 chart is in good qualitative
agreement with the empirical cotidal chart presented by
CA90. The only difference is in the Camarinal Sill area,
where the cotidal lines (lines of constant phase) undergo a
deviation toward North. The principal features to be noted
on this chart are the reduction (more than 50%) of the
amplitude in the along-strait direction, the invariability of
the amplitude in the cross-strait direction (except for the
eastern part of Tarifa narrow), and the southwestward phase
propagation, more evident east of Camarinal Sill as far as
the eastern entrance of the strait. The same features are also
present on the S2 cotidal chart even if the cotidal lines
exhibit a greater deviation toward North over the Camarinal
Sill. In agreement with CA90, the ratios and phase differences between the M2 and S2 components remain quite
constant throughout the strait; the amplitude ratio is confined between 2.6 and 2.8 and the phase difference
decreases from west to east of only 2 degrees between
24 to 26.
3.2. Tidal Currents
[23] A direct comparison between the predicted fields of
major and minor axes of tidal ellipse and data are difficult
because of the lack of data in most part of the strait, with the
exception of Camarinal Sill (see CA90) and of the eastern
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Figure 4. Comparison between observed and simulated semimajor axis components of tidal ellipses.
Observed data M1, M2, M3, M7, M8, M9, and F3 are from Candela et al. [1990], and N, C, and S are
from Garcı́a Lafuente et al. [2000].
entrance [see Garcı́a Lafuente et al., 2000]. Thus in order to
quantitatively compare the model results with observed data,
a linear regression between predicted and observed semimajor axis, in only ten different locations, was performed
(Figure 4). The mean errors and the root mean square errors
are shown in Table 3. The errors are limited to 4.0 cm s1 and
7.5 cm s1 for the S2 and 5.9 cm s1 and 7.9 cm s1 for the
M2, except for the stations M3 and F3 where the mean error
reaches the value of 24.7 cm s1 and the root mean square
reaches 31.9 cm s1. These differences are mainly due to an
overestimation of the simulated lower-layer currents.
[24] Figures 5 and 6 show a complete semidiurnal tidal
cycle simulated by the model during spring tide at the
Gibraltar-Ceuta and Camarinal Sill sections, respectively. It
is apparent from Figure 5 that the lower-layer flow, at the
eastern section D, is periodically reversed by tidal currents
toward the Mediterranean Sea (also during neap tide, not
showed). The typical currents range from 60 to 30 cm s1
during spring tide and from 40 to 30 cm s1 during neap
tide. On the contrary, the upper layer is always directed
toward the Mediterranean Sea, indicating a clear weakness of
the tidal amplitude in comparison with the mean upper layer
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Table 3. Mean and Root-Mean-Square Error of the Simulated
Semimajor Axis
M2
Station
M1
M2
M3
M7
M8
M9
F3
N
C
S
Mean Error
S2
RMS Error
Mean Error
RMS Error
3.4
0.9
24.7
0.8
1.0
5.9
24.5
Candela et al. [1990]
3.9
0.8
5.5
4.0
29.8
3.5
1.9
0.1
1.0
3.3
5.9
2.3
31.9
0.4
3.3
7.5
7.2
0.3
3.3
2.6
0.8
2.7
3.9
0.1
Garcı́a Lafuente [1986]
7.9
0.9
7.9
0.9
1.1
0.5
1.7
1.5
0.8
flow, that is too strong to be reversed. Here the upper layer,
the currents range from 80 to 140 cm s1 during spring tide
and from 60 to 110 cm s1 during neap tide. These results are
in good agreement with BA01, who showed very similar
results for the M2 component, computed with an inverse
model at the eastern entrance of the strait.
[25] At Camarinal Sill, the tidal signal is so strong to
always reverse the currents, both in the upper and lower
layers, for a part of each semidiurnal tidal cycle, except for
the neap tide where the Mediterranean layer is not reversed
completely (for the spring tidal cycle see Figure 6). To
discriminate between upper and lower-layer velocities we
superimposed to the velocity contours the depth of the
37.25 isohaline, that, as suggested by SBA02, can be
considered as an interface between the two layers. Using
this method, it is possible to see that velocity in the upper
layer ranges from 130 to 200 cm s1 during spring tide and
from 100 to 130 cm s1 during neap tide. For the lower
layer, velocity ranges from 230 to 150 cm s1 during
spring tide and from 190 to 70 cm s1 during neap tide.
[26] Figures 7 – 10 show the simulated M2 and S2 tidal
amplitude and phase of the along-strait velocity at Camarinal Sill and Gibraltar-Ceuta cross-strait sections. Looking at
Figures 7a and 8a it is clear that there is a drastic decrease in
the M2 amplitude (more then 70%) going from Camarinal
Sill to the eastern entrance of the strait. At Camarinal Sill
the amplitude constantly increases from 100 cm s1 at the
surface up to 140 cm s1 at a depth of about 220 m and then
decreases in the vicinity of the bottom due to the influence
of friction. On the other hand, in good agreement with
BA01, at the eastern entrance of the strait the amplitude
increases from 8 cm s1 at the surface to 42 cm s1 in the
lower layer. The main increase is in the upper layer:
amplitude reaches the value of 34 cm s1 in the first
200 m, and remains rather constant in the rest of the water
column. It is also evident a meridional variation of the
amplitude from the southern part (40 cm s1) to the
northern part (18 cm s1) of the strait. Another point to
highlight is that the phase at Camarinal Sill (Figure 8b) is
quite constant from the upper layer to the lower layer; there
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is only a difference of 20, i.e., a difference of 40 min
between the appearing of the maximum velocity in the
upper layer and the appearing of the maximum velocity in
the lower layer. This difference goes up to 60(2h) at the
eastern entrance (Figure 7b), where the phase decreases
from about 210 in the upper layer to 150 in the lower
layer.
[27] The S2 tidal current amplitude also decreases of more
than 70% from Camarinal Sill to the eastern entrance
(Figures 10a and 9a, respectively). At the eastern entrance
the amplitude increases with depth, from the surface to about
250 m, of only 2 cm s1, remaining constant at 11 cm s1 as
far as the bottom on the southern side. S2 tidal current phase
(Figure 9b) decreases from 170 to 130 in the first 200 m
and increasing up to 150 at about 350 m, remaining
constant below 350 m to the bottom. At Camarinal Sill the
S2 tidal current amplitude increases from surface to 90 m of
about 14 cm s1, with an increment that is not uniform along
the cross section (maximum values of about 42 cm s1 are
concentrated on the south and north sides), while below 150
m the amplitude decreases going toward the bottom. Phase
(Figure 10b) is constant (150) from the surface to the
bottom for nearly the whole section.
3.3. Internal Bore
[28] One of the most important features of the dynamics in
the strait is the presence of internal bores which are generated
over Camarinal Sill and propagate both eastward and westward [Armi and Farmer, 1988; Farmer and Armi, 1988]. In
Figure 11 we present six sequential snapshots of longitudinal
salinity sections which cover the overall spring tidal period.
Here one can see that, in good agreement with the twodimensional, two-layer, hydrostatic model of Izquierdo et al.
[2001], the generation of the eastward propagating internal
bore begins with the formation of an interfacial depression
over the western edge of Camarinal Sill, approximately
1.5 hours before high tide at Tarifa, i.e., as soon as the
westward barotropic forcing over Camarinal Sill starts
weakening and the interface located upstream of Camarinal
Sill is not sustained any more. Subsequently, about 30 min
before high tide at Tarifa, the internal bore is released from
Camarinal Sill and starts to travel eastward. The bore is
released when the upper layer starts to move toward east
while the lower layer continues to move westward. Its initial
length scale, in the along-strait direction, is about 3 km and
its travel times from Camarinal Sill to Tarifa, Pta. Cires and
Gibraltar sections are 2, 4 and 6 hours, respectively. It
follows that, always in agreement with the two dimensional
model of Izquierdo et al. [2001], the speed of the bore is
about 1.7 m s1 between Camarinal Sill and Tarifa sections,
2.5 m s1 between Tarifa and Pta. Cires sections, and
1.5 m s1 between Pta. Cires and Gibraltar sections.
[29] In agreement with Armi and Farmer [1988], a much
weaker westward propagating internal bore is also released
from Camarinal Sill, just 30 min before the eastward propagating bore reaches Gibraltar-Ceuta section, i.e., 40 min
before the low tide at Tarifa. The amplitude of the eastward
Figure 5. (a – f ) Simulated sections of the along-strait current (cm s1) showing several phases of a semidiurnal (M2 + S2)
tidal cycle during spring tide at the Gibraltar-Ceuta section. The time difference between the single sections is 2 hours.
(g) Time moments referred to the surface elevation at Tarifa. The contour interval is 10 cm s1. Red and blue shadows
highlight outflow and inflow currents, respectively. Yellow lines represents the depth of the 38.1 isohaline.
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Figure 5
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Figure 6. Same as Figure 5, but for the Camarinal Sill section. Yellow lines represent the depth of the
37.25 isohaline.
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Figure 7. M2 tidal constituent of the along-strait velocity at the eastern section D. (a) Amplitude in
cm s1; the contour interval is 2.0 cm s1. (b) Phase relative to the Moon transit at Greenwich in degrees;
the contour interval is 10.
propagating bore diminishes progressively from about 100 m
on the western edge of Camarinal Sill to about 50 m at the
Gibraltar section. Initially the bore is characterized by two
large and steep internal waves that during the eastward
propagation seem to be subject to an amplitude dispersion.
What happen actually is that the bore, during its eastward
propagation, disintegrates into a train of internal solitary
waves [Artale and Levi, 1990; Artale et al., 1990; Brandt et
al., 1996]. The model is not able to reproduce these internal
solitary waves since nonhydrostatic effects are neglected and
the horizontal model resolution is lower in the eastern part of
the domain; however the final effect is the same, since the
bore is in any case dispersed. The model shows also that the
bores are always released from Camarinal Sill in the course
of the fortnight period, even during neap tides.
4. Transport
4.1. Effect of Tidal Forcing on Transport and Mean
Quantities
[30] Recent estimates of transport based on direct measurements over Camarinal Sill have been carried out by
TB00. They considered two methods of estimating the mean
transport across the sill: in the first one they used the timeaveraged along-strait velocities, fixing the interface depth at
147 m, while in the second method they produced 30 min
time series of transport by finding the depth of the interface
for each measurement. In the first case the inflow transport
was estimated to be 0.46 Sv, while in the second case the
average over the time series gave an estimated transport of
0.78 Sv.
[31] At the eastern entrance of the strait other direct
measurements have been carried out by BA01. They report
an inflow of 0.81 Sv, estimated by using an inverse model to
predict every instant the interface displacement and the
along-strait velocities, while using an interface at constant
mean depth they estimated a transport higher than 7%
respect to the nonstationary interface case.
[32] As initially argued by Bryden at al. [1994] and more
recently by TB00 the contribution of the fluctuating terms in
velocity and interface depth represents the main difference
between the two methods of computation. To better explore
the effect of these fluctuating terms on the mean flow along
the strait, we analyze numerical results of both experiment
Figure 8. M2 tidal constituent of the along-strait velocity at the Camarinal Sill section B. (a) Amplitude
in cm s1; the contour interval is 10.0 cm s1. (b) Phase relative to the Moon transit at Greenwich in
degrees; the contour interval is 10.
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Figure 9. S2 tidal constituent of the along-strait velocity at the eastern section D. (a) Amplitude in cm
s1; the contour interval is 2.0 cm s1. (b) Phase relative to the Moon transit at Greenwich in degrees; the
contour interval is 10.
(with and without tidal forcing (hereinafter TE and NTE)),
in a two-dimensional two-layer formulation; in particular
we have integrated the model results in the cross-strait
direction choosing, as in SBA02, the 37.25 psu as interface
isohaline between the two layers. In this case the momentum and continuity equations for the upper layer can be
written as
@u @ u2
r @ ðh he Þ
þ 0g
þ
¼0
r1
@t @x 2
@x
ð5Þ
@h @ ðhuÞ
þ
¼ 0;
@t
@x
ð6Þ
where u is the velocity, h is the thickness of the layer, h is
the surface elevation, he is the equilibrium potential, r0 is the
density of surface water, and r1 is the mean density of the
layer. Moreover we decompose each model variable (c) in a
mean term, plus the two semidiurnal components (M2 and
S2) and plus a residual component that also includes the
internal bore:
ð xÞ þ c
~ M2 ð xÞ cos wM2 t þ jcM2 ð xÞ þ c
~ S2
cð x; t Þ ¼ c
h
i
c
ð xÞ cos wS2 t þ jS2 ð xÞ þ c
^ ð x; t Þ;
ð7Þ
represents the mean component, c
~ M2, c
~ S2 are the
where c
amplitudes of the semidiurnal components, wM2, wS2 and
, jSc
are the frequencies and phases of the semidiurnal
jMc
2
2
^ represents the residual
components, respectively, and c
component, which includes the internal bore. Time-averaging the continuity equation (6), we obtain the following
transport equation for the upper layer:
~
~
@
uM ~
uS ~
hM
hS
u
h þ 2 2 cos juM2 jhM2 þ 2 2
2
2
@x
!
^ ¼ 0:
cos ju jh þ ^
uh
S2
S2
Figure 10. S2 tidal constituent of the along-strait velocity at the Camarinal Sill section B. (a) Amplitude
in cm s1; the contour interval is 10.0 cm s1. (b) Phase relative to the Moon transit at Greenwich in
degrees; the contour interval is 10.
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ð8Þ
Figure 11. Evolution of salinity perturbations during a tidal period. Contours are shown with an interval of 0.5 psu. The
snapshots are plotted at an interval of 2 hours. The time moments are referred to the surface elevation at Tarifa (insets).
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Figure 12. Along-strait total upper layer transport in the case with (A) and without (B) tidal forcing,
and single component of upper layer transport in the case of tidal forcing: mean component (C), M2 and
S2 components (D, E), and residual component (F).
In Figure 12 all terms of equation (8) are plotted: the mean
transport (C), the transport due to the M2 and S2
components (D, E) and the residual transport (F). Also
plotted are the total upper layer transport (A) and the
transport computed for NTE (B). This figure reveals that
the contribution of the semidiurnal tidal component M2
((~uM2 ~hM2 cos(juM2 jhM2 ))/2) is relevant over Camarinal
Sill whereas it is negligible at the eastern end of the strait.
In practice, while in the eastern region of the strait the
mean current nearly determines the whole transport, it only
contributes about 60% of the total transport near
Camarinal Sill, in agreement with the results of TB00 and
BA01. Contributions of the S2 component and of the bore are
less than 4%, but, whereas the S2 component has its
maximum effect near Camarinal Sill, the bore is more
effective in the eastern region. Deviations from the
conservation relation equation (8) are mainly due to the
large entrainment of Mediterranean water near the hydraulic
jump, just west of Camarinal Sill, whereas in the eastern part
diffusion of salinity moves the interface toward 38.1 psu
(see BA01, see also section 4.2 of this paper), implying an
underestimation of transport in this zone when the upper
layer is limited to 37.25 psu.
[33] In order to investigate the effect of the tidal forcing
on the mean currents of the layers we have plotted in
Figure 13 (for the upper layer only) the mean currents u
(A), half of the M2 current amplitude (1/2 ~uM2) (C), half of
uS2) (D), the mean quadratic
the S2 current
pffiffiffiffiffi amplitude (1/2 ~
residual ( ^u2 ) (E), and the current for NTE (B). It is evident
that the mean current (A) shows a local minimum in the
region where the amplitude of the M2 current (C) has its
maximum value, whereas in the case without tidal forcing
the current (B) shows a local maximum in the same region.
Residual (E) and S2 (D) terms appear negligible.
[34] Always from Figure 13 one can note that in most part
of the strait, and in particular in the eastern part, the mean
current is higher for TE (A) respect to NTE (B). Most of tidal
energy is dissipated toward smaller scales, but we suppose
that part of this energy can be transferred also to the mean
flow. This supposition is based on the fact that the only
difference between the two experiments is the tidal forcing,
and so the difference between the mean currents can only be
caused by this forcing. It is plausible that everywhere within
the strait, with exception at Camarinal Sill and surroundings,
tidal fluxes interact with mean motion enhancing it. At
Camarinal Sill, dissipation processes (bottom friction, mixing) and energy transfer to internal bore generation drop
energy probably from both tidal and mean motion.
[35] The effects of tide on the interface depth are shown
in Figure 14. Plotted are the mean depth of interface (A),
its range of variation due to the M2 component (Bmin,
Bmax), its range of variation due to the residual term
(Cmin, Cmax), and the interface depth for NTE (D). The
variation due to the residual term is mainly due to the bore
and it is of the same order of that associated with the M2
component. In the presence of tidal forcing the mean depth
of the interface rises of about 20 m just west of Camarinal
Sill up to about 40 m over the sill respect to the depth of
interface of NTE. The minimum difference of about 13 m
is limited at Tarifa Narrow. This interface rising is probably
related to the increased mixing between upper and lower
layer introduced by tidal forcing. However, in spite of this
reduction of the upper layer thickness the transport
increases for TE, indicating that the effect of a stronger
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Figure 13. Along-strait current amplitudes for the mean component u (A), the Mp
(1/2 ~
uM2)
2 component
ffiffiffiffiffi
uS2) (D), the mean quadratic residual component ( ^
u2 ) (E), and for the
(C), the S2 component (1/2 ~
experiment without tidal forcing (B).
mean current, together with that of tidal transport prevail
on the effect of depth reduction.
[36] Mean surface elevation is another quantity that shows
an unexpected change due to tidal forcing. It is well known
that there is a gradient of elevation between Atlantic Ocean
and Mediterranean Sea that compensates for the different
densities of the two seas. In the presence of tide, there is a
strong gradient of elevation just west of Camarinal Sill with
an extra 1.9 cm of gradient between the two seas. The equation for the mean elevation is analogous to equation (8);
fM h
~
in the presence of tide there are terms like 1/2UA
2 M2
h
cos(jUA
M2 jM2 ) (where UA represents the barotropic current), that act to modify the mean elevation with respect to the
case without tide. This quantity has its maximum value near
Camarinal Sill, in coincidence with the maximum value of the
fM .
tidal amplitude of the barotropic current UA
2
[37] The mean salinity field within the strait is also
modified by tidal forcing. This appears particularly clear
in Figure 15, where are shown the along-strait (section E)
difference between the salinity field obtained for NTE and a
fortnightly average of the tidally forced salinity field. This
figure shows strong differences in the mean profiles of
salinity: the yellow spot (+0.6 psu) east of Camarinal Sill is
an evidence of the increased entrainment of Mediterranean
water in the upper layer due to the effect of tide; while the
long blue patch (0.3 psu) is due to an increment of
entrainment of the Atlantic water in the denser layer. The
hydraulic jump is characterized by strong mixing also in the
case without tidal forcing, and for this reason in the region,
west of Camarinal Sill, the effect of tide is less evident.
4.2. Effect of Tidal Forcing on Entrainment
[38] In order to estimate entrainment and detrainment
fluxes, we have analyzed model results in a two-dimensional
four layers framework. This was accomplished integrating
model results in the across-strait direction and then choosing
the following three separating isohalines: 36.8, 37.5 and 38.2
psu. Figure 16 shows the thickness of the four layers and the
depth of the three interface isohalines for NTE and TE; here
the layers are numbered, starting from the upper one, from 1
to 4 (hereinafter L1, L2, L3 and L4), while arrows represent
volume fluxes between adjacent layers, i.e., entrainment and
detrainment.
[39] Including to volume and salt conservation equations
terms representing intrusion of volume flux from one layer
into an adjacent one, it is possible to calculate entrainment
and detrainment fluxes:
Dx
Dx
@ ðBk hk Þ
þ rH ðBk hk uk Þ ¼ Fupkþ1 Fdwk Fupk þ Fdwk1
@t
ð9Þ
@ ðBk hk Sk Þ
þ rH ðBk hk uk Sk Þ ¼ Skþ1 Fupkþ1 Sk Fdwk
@t
Sk Fupk þ Sk1 Fdwk1 ;
ð10Þ
where k indicates the number of the layer, hk and Bk are the
thickness and the width of the kth layer, Dx is the
longitudinal distance between two adjacent grid point
(600 m), and Fupk and Fdwk represent upward and
downward volume flux of the kth layer through the BkDx
surface, respectively.
[40] The resulting time averaged (on a fortnight period)
upward and downward fluxes, for NTE and TE, are
shown in Figure 17. For NTE entrainment increases just
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Figure 14. Along-strait mean depth of interface (37.25 psu) (A), its range of variation due to the M2
(Bmin, Bmax), its range of variation due to the residual term (Cmin, Cmax), and interface depth for the case
without tidal forcing (D).
west of
location
mass is
0.06 Sv
Camarinal Sill (Figure 17a), i.e., in the same
of the stationary hydraulic jump. Here water
exchanged prevalently between L3 and L4:
of L4 water intrudes into L3, while 0.04 Sv of
water flows from L2 to L3. Weaker entrainment is also
evident at the western entrance of Tarifa Narrow. Here,
water mass is exchanged prevalently between L1 and L2:
0.013 Sv of L1 water intrudes into L2, while 0.01 Sv of
Figure 15. Along-strait (section E) salinity difference between the field obtained in the case without
tidal forcing and the field obtained by averaging the tidally forced salinity field on 15 days.
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Figure 16
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Figure 17. Along-strait time-averaged (on a fortnight
period) entrained and detrained volume fluxes between
layers (the same as Figure 16) for the case (a) without and
(b) with tidal forcing. Positive values (solid lines) indicate
upward volume flux, while negative value (dashed lines)
represent downward volume fluxes. Positive red, blue, and
green lines represent entrainment from L2 to L1, from L3 to
L2, and from L4 to L3, respectively. Negative red, blue, and
green lines represent entrainment from L1 to L2, from L2 to
L3, and from L3 to L4, respectively.
water flows from L2 to L1. 0.01 Sv of water are also
exchanged from L2 to L3.
[41] TE shows an increased entrainment along whole the
strait respect to NTE (Figure 17b). Strong exchange between
the first and second layer as well as the second and third layer
are located at Camarinal Sill and within whole Tarifa Narrow,
while exchange between the fourth and third layer seems to
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show a behavior similar to NTE. West of Camarinal Sill the
most active layers are the second, third and fourth: 0.075 Sv
of L4 water are entrained into L3 and 0.08 Sv of L3 water into
L2, while 0.07 Sv of L2 water are detrained downward L3
(see peaks number 1 in Figure 17b). East of Camarinal Sill
the most active layers are the first and second: 0.12 Sv are
exchanged from L1 to L2 and a slightly more is exchanged
from L2 to L1 (peaks n. 2), while 0.08 are exchanged from L3
to L2. Entrainment between layers decreases from Camarinal
Sill as far as Tarifa, from here to the eastern entrance of the
strait entrainment increments again showing two relative
maximum (peaks n. 4 and n. 5). Within Tarifa Narrow the
most active layers are the first, second and third: in the
western maximum, 0.08 Sv of water are exchanged between
L1 and L2 in both direction, 0.05 Sv of water flows from L3
to L2, 0.025 Sv are exchanged from L2 to L3, while only
0.01 Sv flows from L4 to L3. The second maximum shows a
behavior similar to the first one, except for the amplitudes
that are reduced of about 25% for layers 1, 2 and 3.
[42] Results shown for TE are representative of a complete
fortnight period, however for a complete understanding of the
tidal entrainment along the strait it is necessary to investigate
also the single ebb (toward Gulf of Cadiz) and flood (toward
Alboran Sea) tidal periods both for spring and neap tide.
[43] During ebb tide (not showed) only peaks n. 1 are
present, that is entrainment is located only west of Camarinal Sill where 0.18 Sv of water is exchanged from L4 to L3
and from L3 to L2 for spring tide and 0.12 Sv for neap tide,
while 0.15 Sv are exchanged between L3 and L4 for spring
tide and 0.09 for neap tide.
[44] During flood tide (not showed) peaks n. 1 are not
present and entrainment is located only at east of Camarinal
Sill and within Tarifa Narrow. East of Camarinal Sill, during
spring tide, the water exchanges principally from L1 to L2
and from L2 to L1 at a rate of 0.4 Sv, and from L3 to L2 and
from L2 to L3 at a rate of 0.21 Sv and 0.11 Sv, respectively.
During neap tide the active layers are always L1, L2 and L3
with weaker values for peaks n. 3, 4, and 5 while peaks n. 2
are totally absent. The only possible cause of a such behavior
is that the dynamical mechanisms generating peaks n. 2 are
activated only when intensity of tidal currents exceed a
threshold value. Observing that at Camarinal Sill only during
neap tide the entire water column it is not reversed completely, we can assume that peaks n. 2 appear only when both
the upper Atlantic layer and the lower Mediterranean layer
flow simultaneously in the same direction.
4.3. Transport Estimates
[45] The simple and intuitive method of computation of
inflow and outflow volume transport introduced in section 3
is strictly related to the existence of an internal surface of
zero along-strait velocity, used as an interface between
Atlantic and Mediterranean water. However, this method
cannot be used to determine the volume transport when tidal
forcing is included, since, as described in section 3.2, the
semidiurnal tidal signal is so strong to reverse the inflow or
the outflow during part of each tidal cycle, obscuring the
two-layer character of the mean flow. Another way of
Figure 16. Along-strait time-averaged (on a fortnight period) thickness of four layers (L1, L2, L3, and L4) separated by
three isohalines (36.8, 37.5, and 38.2 psu) for the experiment (a) without and (b) with tidal forcing. Arrows represent
volume fluxes (Fup and Fdw) between adjacent layers, i.e., entrainment and detrainment.
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Figure 18. Internal surface salinity interface between the upper Atlantic layer and lower Mediterranean
layer.
defining the interface between upper and lower layer is by
using an isohaline. For example, Bryden et al. [1994] and
Candela et al. [1989] used the 37.0 and 37.5 isohalines,
respectively, to define the exchange interface over Camarinal
Sill, while BA01 used the 38.1 isohaline at the eastern
entrance of the strait. The choice of different values for the
separating isohaline has to be ascribed, as argued in the
previous section, to the strong entrainment developing along
the strait: in particular, along Tarifa Narrow the inflowing
Atlantic water entrains denser water and west of Camarinal
Sill the outflowing Mediterranean water entrains part of the
inflowing Atlantic water [Bray et al., 1995; SBA02].
[46] Thus it emerges that it is incorrect to use a single
isohaline as an interface for the whole strait. For this reason,
an alternative definition is used in this paper. We define as
interface the fortnightly averaged internal salinity surface
associated with the internal surface where fortnightly averaged along-strait velocity zero occurs. The internal salinity
surface obtained is shown in Figure 18; here it is possible to
note that the salinity contrast between upper- and lower-layer
changes from 37.25 psu at Camarinal Sill up to 38.1 at the
east entrance of the strait. This salinity surface is then used to
find the time-dependent depth of the internal surface interface
between the two layers. Now we are able to calculate the
instantaneous upper (ULT) and lower-layer transport (LLT)
in the whole strait by using the following equations:
Z
north
Z
0
ULTð x; t Þ ¼
Z
uð x; y; z; t Þdzdy
south
north
Z
hð x;y;t Þ
hð x;y;t Þ
LLTð x; t Þ ¼
ð11Þ
uð x; y; z; t Þdzdy;
south
bottom
where u is the along-strait velocity and h is the timedependent depth of the interface. In Figure 19, the computed
upper and lower-layer transports are shown for a complete
fortnight cycle at three different cross-strait section over
Camarinal Sill (section B) (Figure 19a), at Tarifa (section C)
(Figure 19b), and at the east entrance of the strait (section D)
(Figure 19c). In agreement with CA90 the largest amplitude
of the instantaneous transport occurs in the upper layer at the
sill, and in the lower layer at the eastern section. The behavior
of tidal currents noted in 3.2 is apparent in the transports: it is
clear that the upper currents have decreasing amplitudes
going eastward and reverse their directions only as far as
Tarifa, while the lower currents increase eastward and reverse
their direction everywhere in the strait.
[47] Red lines in Figure 20 show the mean along-strait
transports, obtained averaging over the fortnight period the
ULT and LLT. From west to east the upper layer transport
ranges from 0.68 Sv to 0.9 Sv, while lower-layer transport
ranges between 0.5 Sv to 0.75 Sv. At Camarinal Sill the
transports are 0.85 Sv and 0.70 Sv for the upper and lower
layer, respectively, while at the east entrance they are 0.9 Sv
and 0.75 Sv.
[48] At Camarinal Sill the most accurate estimates of
transports from direct measurements are the ones given by
Bryden et al. [1994] and, more recently, by TB00. In their
computation they considered the vertical movement of the
interface and determined the transport of the upper layer to
be 0.72 ± 0.16 Sv and 0.78 Sv, respectively, and the
transport of the lower layer to be 0.68 ± 0.15 Sv and
0.67 Sv, respectively. At the eastern entrance of the strait
the last most accurate estimates of transports are from
BA01. They calculated the transports using an inverse
model to predict for every instant the depth of the isohaline
38.1 obtaining an upper layer transport of 0.81 ± 0.07 Sv
and a lower-layer transport of 0.76 ± 0.07 Sv. The results
of the present study are in reasonable agreement with all
these transport estimates since they lie within the error bars.
[49] Also plotted in Figure 20 (blue lines) are the transports computed for the experiment without tidal forcing
(equation (4)). Comparison with the results with tidal forcing
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Figure 19. Sixteen days of computed upper (blue) and lower (red) layer transport at three different
cross-strait sections: (a) over Camarinal sill, (b) at Tarifa, and (c) at the east entrance of the strait.
shows that the tidal forcing increases transport, both in the
upper and in the lower layer. It is also interesting to note that
the increment is different between upper and lower layer; in
particular, at Camarinal Sill is 37% for the upper layer and
34% for the lower layer, while at the eastern entrance the
increment is 28% and 29% for the upper and lower layer,
respectively.
[50] With the purpose of verifying our transport estimation, a comparative experiment using particle tracking has
been carried out. A particle was released every 60 min from
each grid point of two cross-strait sections located at the
western (l ’ 5500) and eastern (l ’ 5180) end of
the strait, both with and without tidal forcing. In Figure 21
the total number of particles arrived at the west (east)
section starting from the east (west) section, in the cases
with and without tidal forcing, are showed. It is evident that
tidal forcing increases the particle fluxes in both directions.
However a quantitative estimate of the effect of tidal forcing
on particles flux is better deduced looking at ratios of
particles arrived with and without tide; taking into account
also other intermediate sections at the longitudes 5150,
5200, 5250. After 15 days, these ratios indicate that the
inflow transport increment is in the range 40%/65%, whereas the outflow increment is in between 15%/35%.
5. Summary and Conclusions
[51] In this paper we have presented a 3-D numerical
model that is capable to reproduce reasonably well the main
aspects of the semidiurnal tidal cycle in the Strait of
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Figure 20. Variation of the eastward (positive values) and westward (negative values) transports along
the strait computed for the cases without tidal forcing (blue) and with tidal forcing (red). Dashed lines
represent net water flow.
Gibraltar and, also, to provide new water transport
estimates.
[52] Differently from previous modeling works, which
have concentrated on specific aspects of the phenomenology
related to tidal forcing, this numerical experiment tried to
reproduce all the principal effects of the tide. For example
many 2-D models were developed to study the generation
and propagation of internal waves within the strait [Pierini,
1989; Hibiya, 1990; Longo et al., 1992; Brandt et al., 1996;
Izquierdo et al., 2001; Morozov et al., 2002], another 2-D
model was instead developed to investigate the semidiurnal
surface tides in the strait [Tejedor et al., 1999], and two 3-D
models were developed, in the last 14 years, to study tidal
flows, internal tide as well as fortnightly modulation [Wang,
1989, 1993]. However, none of these models, except partly
Brandt et al. [1996], was able to estimate water transports
Figure 21. Total number of particles that arrived at (a) the west section starting from the east section
and at (b) the east section starting from the west section. Solid lines represent the experiment with tidal
forcing, while dashed lines represent the experiment without tidal forcing.
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along the whole strait and, at the same time, to provide an
estimation of the impact of tidal forcing on the mean flow
exchange.
[53] The 3-D model described in this work can be
considered as a natural improvement of Wang’s models.
It is based on the model developed by SBA02: it makes
use of a curvilinear grid and terrain-following vertical
grid, with mean horizontal resolution, within the strait, of
500 m, it is forced at the open boundaries through the
specification of the M2 and S2 surface elevation. The
validity of our numerical model has been tested by
applying it to the description of tidal elevation, tidal
currents and internal bores. It has been shown that results
of our model are in good agreement with the observed
tidal elevation amplitudes and phases. The model reproduces all of the known features of the spatial structure of
the M2 and S2 tidal waves: a decrease of more than 50%
in amplitudes and slight variations in phases along the
strait, a prevailing propagation of phases southwestward,
and nearly constant amplitude ratios and phase differences
between the M2 and S2 tidal elevations throughout the
strait. At the same time the model has revealed a
distribution of amplitude and phase in the region of
Camarinal Sill (both for the M2 and S2) that is different
from the empirical cotidal chart presented by CA90. The
predicted semimajor axis as well as amplitude and phase
of the along-strait velocities are quantitatively and qualitatively in good agreement with all available observed
data. The simulated eastward and westward internal bores
are also in agreement with available data as well as the
internal bore speeds in different sections of the strait
coincide with those estimated by Izquierdo et al. [2001],
who used a completely different model.
[54] However, the principal aim of this work was to
quantify the effects of tidal forcing on transport of
Atlantic and Mediterranean water along the strait. To this
end, initial conditions were produced by the stationary
experiment, where about 12 cm of mean sea level
difference was set between west and east boundaries.
With this setting the stationary experiment simulated an
inflow of 0.62 and 0.69 Sv at Camarinal Sill and
Gibraltar-Ceuta sections, respectively, whereas the outflow
at the same locations was 0.51 and 0.58 Sv, with a
mean net flow toward the Mediterranean sea of about
0.1 Sv. In presence of the semidiurnal tidal forcing (M2 +
S2), the Atlantic inflow is of 0.85 and 0.90 Sv at the
same locations, whereas the Mediterranean outflow is of
0.70 and 0.75 Sv. This implies a mean increment
of 1.32 and 1.31 for the inflow and the outflow, respectively, with respect to the stationary experiment.
[55] Helfrich [1995] studied exchange flows through a
two-dimensional variable geometry strait forced by a periodic barotropic (tidal) flow. He showed that the exchange
flow is a function of two nondimensional parameters: the
dynamic length of the strait and the tidal forcing strength
(qb0). Moreover he indicated that for the Strait of Gibraltar,
with a qb0 = 0.6 (corresponding to a barotropic velocity over
Camarinal Sill of about 1.5 m s1), transport increases of
1.2 respect to the unforced case, but he also suggested that
this increment should be compensated by the increased
diffusion caused by the tidal mixing. Applying to the
quasi-steady theory by Farmer and Armi [1986] the same
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transport reduction (20%) assumed by Helfrich [1995], their
estimated transport increment reduces from 1.6 to 1.3, i.e., a
value very similar to that simulated by our model.
[56] Experiments with particles seem to show an increment for the upper layer tidal transport (1.4/1.65) greater
than the ones calculated in the Eulerian way. However, this
result must be evaluated with care, observing that the
particle count does not depend on the arriving depth of
particles. For example, Atlantic particles are counted also if
they arrive at the Mediterranean section at a depth characterized by salinity greater than 38.1 psu, that is inside the
part of upper layer that entrains the lower layer. In order to
evaluate the increased entrainment due to tidal forcing, we
have compared the number of Atlantic particles arrived at
the Mediterranean section, below the upper layer, for the
case with and without tidal forcing. What we have observed
is that for the case without tide only 2% of Atlantic particles
arrived below 150 m, while for the case with tide this value
increased up to 9%, so that the increased entrainment of
Atlantic water in the lower layer can be estimated as 7%.
Applying this value of increased entrainment to reduce the
upper layer tidal transport computed with the particle
tracking method, we obtained an increment in transport
limited between 1.30 and 1.53, values that are closer to
those resulting from our direct Eulerian calculation. Moreover, if we reduce the upper layer tidal transport computed
by Farmer and Armi [1986] and Helfrich [1995] of the
same factor (7%.), their tidal transport increase of 1.49 and
1.12, respectively, positioning our result in between of
them.
[ 57 ] The study of calculated entrainment fluxes
(section 4.2) reveals that this phenomenon is effective
only at precise positions along the strait, in particular at
Camarinal Sill and within Tarifa Narrow, i.e., the two zones
where the flow has higher probability to become critical.
However a more theoretical treatment is needed to justify
that higher values of entrainment fluxes will be associated
with critical transition of the flow.
[58] Acknowledgments. We thank many colleagues, particularly
Roberto Iacono, Adriana Carillo, and Paolo Ruti, for discussion and
Emanuele Lombardi and Antonio Iaccarino for informatic support. The
reviewers made helpful and knowledgeable suggestions. This work was
done using climatological data supplied by S. Levitus and the MODB
project partners. This work was supported by the National project Ambiente
Mediterraneo – SINAPSI.
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V. Artale, A. Bargagli, and G. Sannino, Ocean Modeling Unit, Special
Project Global Climate, ENEA C. R. Casaccia, Ente per le Nuove
Technologie, l’Energia e l’Ambiente, SP 91 Via Anguillarse 301, S.M. di
Galeria, I-00060 Rome, Italy. ([email protected])
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