Strong Internal Tides in the Strait of Gibraltar:

Measurements and Modelling

E. G. Morozov

1 a

and M. G. Velarde

Shirshov Institute of Oceanology, Russian Academy of Sciences, 36 Nakhimovsky prospekt, Moscow, Russia

Instituto Pluridisciplinar, 1 Paseo Juan XXIII, Madrid, Spain

Keywords: Strait of Gibraltar, Internal Waves, Numerical Modeling.

Abstract: We analyze moored current measurements in the Strait of Gibraltar. Internal waves are extremely strong in

the strait. The vertical displacements of water particles with a semidiurnal frequency sometimes exceed 200

m. These displacements are associated with forced tidal internal waves over the Camarinal Sill, which

crosses the strait. The amplitudes of the waves decrease with the distance from the sill, and at a distance of

50 km from the sill, the amplitudes are three times smaller than over the sill. Numerical modelling shows

that the lower current in the strait directed from the Mediterranean Sea to the ocean has a significant

influence on internal tides. The effect of internal waves propagating in a hydraulic flow leads to the

formation of internal bore, followed by a wave packet of shorter internal waves.

https://orcid.org/0000-0002-0251-3454

1 INTRODUCTION

The Strait of Gibraltar (Fig. 1) has been known from

the ancient ages. It is characterized by a two-layer

system of opposite flows resulting from the

difference in water density between the Atlantic

Ocean and the Mediterranean Sea. A strong surface

current of relatively fresher water from the ocean

compensates for intensive evaporation in the

Mediterranean. A deep-water current of more saline

Mediterranean water flows into the ocean. A

barotropic tidal wave is imposed on this system with

velocities in the range 70-80 cm/s, about the same

order of magnitude as the mean velocities of the

currents between the ocean and the sea, and thus, an

unsteady flow is formed. The tide generates a strong

tidal internal wave, when the tidal currents flow over

uneven topography in the strait.

The estimate for the Mediterranean water

outflow is equal to -0.68 Sv (1 Sv = 10

/s), while

the flow of Atlantic waters is 0.72 Sv, which equals

to the sum of the outflow and net evaporation that is

approximately 52 cm of the sea level per year

(Bryden et al., 1994; Morozov et al., 2002, 2003).

Figure 1: Chart of the Strait of Gibraltar and locations of

moorings (black dots).

2 MOORED MEASUREMENTS

Moored measurements in the strait reveal very

strong internal waves whose double amplitude

(vertical displacements of particles) can be as high

as 200 m over a depth of 400 m. The displacement

of 13C isotherm gives the best illustration of such a

displacement. The vertical displacement ranges from

100 to 300 m (Fig. 2).

To estimate the wavelength of the semidiurnal

waves we used the antenna method developed in

354

Morozov, E. and Velarde, M.

Strong Internal Tides in the Strait of Gibraltar: Measurements and Modelling.

DOI: 10.5220/0007840603540357

In Proceedings of the 5th International Conference on Geographical Information Systems Theory, Applications and Management (GISTAM 2019), pages 354-357

ISBN: 978-989-758-371-1

Figure 2: Depth variation of 13C isotherm from July 2 to

July 31, 1986.

seismology and applied for oceanic waves by Barber

(1963) assuming arbitrary position of the wave

sensors. The method is based on the calculation of

the cross spectra for each pair of the possible

combinations of sensors with further convolution at

the semidiurnal frequency of the waves. The

amplitude and phase cross-characteristics of the

oscillation are used to calculate the spatiotemporal

spectra at the wave frequency and estimate the

components of the horizontal wave number. The

method basically accounts for the statistical phase

difference between each pair of wave sensors.

The wave propagating to the east of the sill was

estimated on the basis of 16 different combinations

of moorings. The wave propagates to the east in the

azimuth interval between 90 and 120 degrees, while

the wavelength ranges between 90 and 140 km.

3 NUMERICAL MODELING

In order to model the wave propagation we use the

fully nonlinear non-hydrostatic model of the

baroclinic tides developed by (Vlasenko, 1992;

Morozov et al., 2002). We consider a two

dimensional (x,z) flow in a continuously stratified

rotating ocean of variable depth.

Although the model is two dimensional, we

introduce the equation for the V-component of

velocity normal to the x,z plane to account for the

effects of rotation. However, the V-component is

considered constant. For convenience, the equation

of density diffusion has been used instead of the

equations of heat and salt diffusion.

The boundary conditions at the surface located at

z = 0 are zero for the density gradient, vorticity, and

stream function; hence, no tangential stresses are

considered. We also consider zero vertical motion

and no heat and salt transport through the surface.

At the bottom, no heat, salt, and mass transports

exist. The boundary condition for vorticity at the

bottom is calculated using equation  =  with the

value of the stream function field



obtained at the

previous time step.

The wave perturbations of vorticity, stream

function, and density are assumed zero at the lateral

boundaries located far from the bottom irregularities

at the submarine ridge. The calculations start from a

state of rest when the fluid at the bottom is

motionless, and the isopycnals are horizontal; hence:

at t=0: =0, =0, =0. We stop the calculations,

when the wave perturbations reach the lateral

boundaries. The phase velocity of the perturbations

does not exceed 2-3 m/s, which allows us to

continue the calculations for a suitable number of

time steps.

The bottom topography was introduced in the

model from the digital databases of bottom

topography. We specify a density field unperturbed

by internal waves corresponding to the vertical

distribution of the Brunt-Väisälä frequency N(z)

from observations.

A semi-implicit numerical scheme utilizes a

rectangular grid with second order approximations

to the spatial derivatives and first order

approximation of the temporal derivatives in every

temporal semi-layer. At each time step, the implicit

system, which is a tri-diagonal matrix, is solved

using standard techniques.

We model the following physical phenomenon.

A long barotropic tidal wave propagates from the

open ocean to the continental slope or submarine

ridge. The tidal currents flow over the topographic

obstacles and obtain a vertical component.

Periodically oscillating vertical components with a

tidal period displace water particles; thus a tidal

internal wave is generated. The input parameters of

the model are stratification, bottom topography, and

stream function of the tidal current. The model

outputs the fields of density and velocity over the

domain of calculations.

We have chosen a domain 300 km long with a

horizontal step of 200 m and 20 vertical levels. The

horizontal size of the domain significantly exceeds

the size of the strait, but allows us to analyze the

processes in the middle of the domain before the

perturbations reach the lateral boundaries. The time

step was approximately equal to 7 seconds. These

parameters satisfy the Courant-Friedrichs-Levy

condition. The coefficients of the horizontal eddy

viscosity and density diffusivity were fixed to 60

/s over the ridge and 4 m

/s beyond the ridge over

the flat bottom in the model. Strong motion at the

Strong Internal Tides in the Strait of Gibraltar: Measurements and Modelling

355

slopes of the sill requires great values of the

coefficients. The coefficients of vertical turbulent

viscosity and density diffusion were set to 0.001

/s.

The main objective of the model investigation is

to study the influence of the currents on internal tide

propagation in the Strait. In the first version of

model calculations we make the approximation of

the model with zero mean current. We introduce

only a periodical barotropic tidal flow by periodical

increasing and decreasing the value of the stream

function at the surface, while the value of the stream

function at the bottom is specified equal to zero. The

maximum value of the stream function was chosen

to provide the maximum horizontal tidal velocities

equal to 80 cm/s. The period of reciprocating flow is

12.4 hours.

Figure 3: Perturbations of the density field induced by

propagating internal tide. The contour lines of density are

shown with an interval of 0.00025 g/cm

. Grey coloured

pattern shows bottom topography. The density

perturbations develop on the background of different

currents in the strait:

(a) Zero mean current

(b) Westward barotropic current

The periodical changes in the horizontal flow

induce an internal wave propagating in both

directions from the sill positioned in the middle of

the computation area. The perturbations of the

density field induced by propagating internal tide are

shown in Fig. 3a. This snapshot of the density field

is depicted after four tidal periods of calculation.

The fluctuations of the density field are not

symmetrical with respect to the position of the

Camarinal Sill because the bottom topography is not

symmetrical. Camarinal Sill is strongly abrupt at the

eastern slope, while the bottom topography is more

corrugated and the mean inclination of the slope is

smaller west of the sill. A calculation, when the

bottom topography was specified only by a

symmetric sill, gave a symmetric pattern of

propagating fluctuation in both directions from the

sill.

Internal bore is formed on the trailing edge of the

wave, which is steeper than the leading edge. A

packet of shorter internal waves follows the bore.

The structure of the westerly propagating wave is

the same except for the fact that the bore and wave

train of shorter internal waves are less intensive in

the western part compared to the eastern part of the

basin.

In the second version of model calculation we

introduce a steady barotropic westward current in

the entire water column of the Strait by specifying

the permanent value of the stream function at the

surface and zero at the bottom so that the velocity of

the mean current is equal to 30 cm/s at the western

boundary. At the eastern boundary the velocity is

equal to 18 cm/s. The mean velocity over the sill is

equal to 41 cm/s. We superimpose a periodical

barotropic tidal current on the mean current as done

in the previous step. At the western boundary, the

maximum amplitude of the barotropic tidal velocity

is 80 cm/s, the same as it was without the mean

current.

The westward flow changes the internal wave

field (Fig. 3b). In the eastern part, where the internal

tide propagates opposite to the current, we observe a

well pronounced internal bore followed by a train of

short period internal waves. The mean current

opposite to the wave makes the wavelength shorter,

and hence the slopes of the internal tide are steeper,

which leads to wave breaking and formation of a

packet of shorter internal waves. For a uniform

westerly current the bore is observed in the entire

water column.

West of Camarinal Sill the internal tide

propagates in the same direction as the outflowing

current. The current increases the wavelength. The

internal bore is formed, but only a weak packet of

shorter waves appears.

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In the third version of the model calculation we

have analyzed the internal tide developing in the

strait with two opposite currents modeling the real

situation. The eastward flow with mean velocities of

50 cm/s occupies the upper 250-300 m layer and the

lower current with a vertically average velocity of

25 cm/s occupies the rest of the water column. These

two opposite flows yield a strong shear at 200 m

depth. The two oppositely directed currents in the

stream were introduced in the model by specifying

two different vertical distributions of density in the

eastern and western parts. In this version of

calculations the computational area was increased in

the eastern and western directions. In the beginning

of the calculations the water was set to flow free

under the influence of different density distributions

in the eastern and western parts. After the

adjustment and formation of a two-layer flow in the

central part of the computational area we

superimpose the same tidal flow as in the previous

versions of calculations. The wavelength of the

easterly propagating wave is 90 km. The leading

edge of the wave is flatter than the trailing edge,

which is very steep. This leads to the formation of

internal bore. A train of shorter internal waves

follows a sharp depression of density contour lines.

The westerly propagating wave is shorter. Its

wavelength is 60 km, as the depth west of Camarinal

Sill is smaller. The structure of the wave is

approximately the same except for the fact that

internal bore is less intensive in the western part than

in the eastern part of the basin.

The introduction of two opposite currents with a

shear intensifies the internal bore in the upper layer

at depths of 100-200 m, while in the deeper water it

becomes less apparent (Fig. 3c). Intensification of

the internal bore and associated short period internal

waves in the upper layer in the eastern part obtained

in the model calculations confirms the observations

at the surface made from satellites, airplanes, and

coastal radars that surface manifestations of internal

tide are clearly seen in the eastern part of the strait.

4 CONCLUSIONS

The main results of this research are the following:

Internal tidal oscillations observed in the Strait

are mostly generated over Camarinal Sill.

The observations analyzed depict the generation

of internal tides with peak-to-peak amplitudes

exceeding 200 m. The waves propagate in both

directions from the sill loosing energy while

propagating. The major motion of internal tide is

associated with the semidiurnal M2 frequency.

ACKNOWLEDGEMENTS

This research was performed within the framework

of the state assignment of Russia (theme no. 0149-

2019-0004) and supported in part by the Russian

Foundation for Basic Research) (project no. 17-08-

00085).

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