5 European Semi-enclosed Seas
141
where the subscript i = 1, 2 is the layer number, u i is the layer speed, g = ggρ/ρ is
the reduced gravity, ρ is the typical density, ρ is the density difference between the
layers and h i is the layer depth. The flow is called supercritical if a Froude number
exceeds 1 and subcritical if its value is <1. Supercritical flows become subcritical
by way of hydraulic jumps, associated with a large level of turbulence adjusting the
flow to a subcritical state. One special case is the so-called maximal exchange which
occurs (Armi 1986):
G
2
= F
2
1 + F
2
2 = 1.
(5.2)
One can understand the above balance and relate it to a continuously stratified fluid
in terms of the relative propagation speed of long internal waves: a control occurs
where a long internal wave is arrested. Thus subcritical flow propagates in both
directions, while changes from one basin cannot propagate in both directions when
the flow is supercritical.
Farmer and Armi (1988) and Armi and Farmer (1988) found two controls in
the Strait of Gibraltar supporting the concept of maximal exchange. In most of the
real straits friction drives the flow towards a critical state. This effect is similar to
decreasing the width or increasing the bottom height of the strait in pushing the flow
towards critical conditions (Garrett 2004). Under this type of dynamics a control
point is shifted towards the location where it would be without friction. This happens
in longer straits, e.g., the Dardanelles (Pratt 1986). According to Pratt (1986) the
ratio of friction to the inertial term is
C d u 2 /h 2
u 2 /l
=
C d l
h 2
,
(5.3)
where l is the along-channel distance over which layer thickness and velocity
change significantly, and C d is the drag coefficient. Obviously bottom friction
should be important in long, shallow straits but not in short and deep ones. Pratt
(1986) estimated C d l/ h 2 ≈ 1 in the Bosporus and ≈0.1 in Gibraltar. With very
large drag coefficient the flow cannot be hydraulically controlled and will be limited
by friction.
One important question concerning straits’ dynamics is to what extent straits and
sills regulate the thermohaline circulation in the inter-connected basins. This issue is
relevant to how the signals are routed from basin to basin and needs to be addressed
accounting for the rotational effects (a review can be found in Gill 1977; Whitehead
et al. 1974; Pratt and Lundberg 1991). The problem becomes ‘rotational’ when the
first mode baroclinic Rossby radius is of the same order or smaller than the strait
width.
An interesting case are the straits connecting the Baltic Sea with the North Sea
(see Fig. 5.7 and Fig. 2.2 in Chap. 2). Two of them, the Little Belt and Øresund (also
known as Sound) are narrow (at their narrowest sections the Øresund is 4 km and
Little Belt 1 km wide). However, the Great Belt (width between 16 and 32 km) and
the straits’ extensions in the adjacent seas (e.g., the Kattegat) could be rotationally
dominated, provided the friction does not dominate the inertial forces. In the Strait
of Gibraltar, which is 14 km wide at its narrowest section and has a sill depth of less
141
where the subscript i = 1, 2 is the layer number, u i is the layer speed, g = ggρ/ρ is
the reduced gravity, ρ is the typical density, ρ is the density difference between the
layers and h i is the layer depth. The flow is called supercritical if a Froude number
exceeds 1 and subcritical if its value is <1. Supercritical flows become subcritical
by way of hydraulic jumps, associated with a large level of turbulence adjusting the
flow to a subcritical state. One special case is the so-called maximal exchange which
occurs (Armi 1986):
G
2
= F
2
1 + F
2
2 = 1.
(5.2)
One can understand the above balance and relate it to a continuously stratified fluid
in terms of the relative propagation speed of long internal waves: a control occurs
where a long internal wave is arrested. Thus subcritical flow propagates in both
directions, while changes from one basin cannot propagate in both directions when
the flow is supercritical.
Farmer and Armi (1988) and Armi and Farmer (1988) found two controls in
the Strait of Gibraltar supporting the concept of maximal exchange. In most of the
real straits friction drives the flow towards a critical state. This effect is similar to
decreasing the width or increasing the bottom height of the strait in pushing the flow
towards critical conditions (Garrett 2004). Under this type of dynamics a control
point is shifted towards the location where it would be without friction. This happens
in longer straits, e.g., the Dardanelles (Pratt 1986). According to Pratt (1986) the
ratio of friction to the inertial term is
C d u 2 /h 2
u 2 /l
=
C d l
h 2
,
(5.3)
where l is the along-channel distance over which layer thickness and velocity
change significantly, and C d is the drag coefficient. Obviously bottom friction
should be important in long, shallow straits but not in short and deep ones. Pratt
(1986) estimated C d l/ h 2 ≈ 1 in the Bosporus and ≈0.1 in Gibraltar. With very
large drag coefficient the flow cannot be hydraulically controlled and will be limited
by friction.
One important question concerning straits’ dynamics is to what extent straits and
sills regulate the thermohaline circulation in the inter-connected basins. This issue is
relevant to how the signals are routed from basin to basin and needs to be addressed
accounting for the rotational effects (a review can be found in Gill 1977; Whitehead
et al. 1974; Pratt and Lundberg 1991). The problem becomes ‘rotational’ when the
first mode baroclinic Rossby radius is of the same order or smaller than the strait
width.
An interesting case are the straits connecting the Baltic Sea with the North Sea
(see Fig. 5.7 and Fig. 2.2 in Chap. 2). Two of them, the Little Belt and Øresund (also
known as Sound) are narrow (at their narrowest sections the Øresund is 4 km and
Little Belt 1 km wide). However, the Great Belt (width between 16 and 32 km) and
the straits’ extensions in the adjacent seas (e.g., the Kattegat) could be rotationally
dominated, provided the friction does not dominate the inertial forces. In the Strait
of Gibraltar, which is 14 km wide at its narrowest section and has a sill depth of less