2 Topography, Hydrography, Circulation and Modelling of the Baltic Sea
47
The basic equation for the inertial motion is obtained from the horizontal equations
of motion. The external forcing and friction terms are zero and the flow field is
assumed to be spatially homogeneous. Thus the equations of motion are reduced to
dq
dt
= −if q.
(2.5)
The multiplication of q by the imaginary unit −i on the right-hand side of Eq. (2.5)
(the operation −iq) means that the presence of the Coriolis acceleration results
in a permanent inclination of the velocity vector q by 90 ◦ to the right from its
instantaneous direction. This first-order equation can be directly integrated to reveal
that
q = q 0 e
−if t .
(2.6)
Therefore, the current speed |q| = |q 0 | is constant and the direction of the velocity
rotates by an angular velocity −f (clockwise in the northern hemisphere where
f > 0). Each resulting cycle draws the so-called inertial circle with a radius R I and
period T I :
R I =
|q 0 |
f
,
T I =
2π
f
.
(2.7)
In the Baltic Sea the inertial period is 13.2–14.5 hours (13.8 hours at latitude
60 ◦ ). If the initial velocity is 10 cm/s, the radius of the circle is about 800 m. In
reality the size of the circle reduces in time due to frictional forces. This can be
modelled in a simple way by adding a damping term −rq on the right hand side of
Eq. (2.5) where r is a friction coefficient. The solution is then q = q 0 e −rt e −if t . The
quantity 1/r is obviously the relaxation time 6 of the system. The data of Gustafsson
and Kullenberg (1936) suggest that 1/r ≈ 1 week in summer conditions. The inertial motion is in different phases above and below the thermocline. Near the coasts
it is quickly damped due to friction.
2.3.5 Ekman Drift
The Swedish oceanographer V.W. Ekman explained theoretically the formation of
wind-driven surface-layer currents already more than 100 years ago (Ekman 1905).
The wind induces a shear stress onto the sea surface. The vertical turbulent friction
transfers the momentum of the wind downwards while the Coriolis acceleration
turns the flow direction to the right in the northern hemisphere. The Ekman equations describe the steady state of this process in a horizontally homogeneous ocean.
6 In a spin-down or spin-up process, during the relaxation time (also called e-folding time scale)
the speed (or any other suitable measure of the process) changes e times compared to its original
value.
47
The basic equation for the inertial motion is obtained from the horizontal equations
of motion. The external forcing and friction terms are zero and the flow field is
assumed to be spatially homogeneous. Thus the equations of motion are reduced to
dq
dt
= −if q.
(2.5)
The multiplication of q by the imaginary unit −i on the right-hand side of Eq. (2.5)
(the operation −iq) means that the presence of the Coriolis acceleration results
in a permanent inclination of the velocity vector q by 90 ◦ to the right from its
instantaneous direction. This first-order equation can be directly integrated to reveal
that
q = q 0 e
−if t .
(2.6)
Therefore, the current speed |q| = |q 0 | is constant and the direction of the velocity
rotates by an angular velocity −f (clockwise in the northern hemisphere where
f > 0). Each resulting cycle draws the so-called inertial circle with a radius R I and
period T I :
R I =
|q 0 |
f
,
T I =
2π
f
.
(2.7)
In the Baltic Sea the inertial period is 13.2–14.5 hours (13.8 hours at latitude
60 ◦ ). If the initial velocity is 10 cm/s, the radius of the circle is about 800 m. In
reality the size of the circle reduces in time due to frictional forces. This can be
modelled in a simple way by adding a damping term −rq on the right hand side of
Eq. (2.5) where r is a friction coefficient. The solution is then q = q 0 e −rt e −if t . The
quantity 1/r is obviously the relaxation time 6 of the system. The data of Gustafsson
and Kullenberg (1936) suggest that 1/r ≈ 1 week in summer conditions. The inertial motion is in different phases above and below the thermocline. Near the coasts
it is quickly damped due to friction.
2.3.5 Ekman Drift
The Swedish oceanographer V.W. Ekman explained theoretically the formation of
wind-driven surface-layer currents already more than 100 years ago (Ekman 1905).
The wind induces a shear stress onto the sea surface. The vertical turbulent friction
transfers the momentum of the wind downwards while the Coriolis acceleration
turns the flow direction to the right in the northern hemisphere. The Ekman equations describe the steady state of this process in a horizontally homogeneous ocean.
6 In a spin-down or spin-up process, during the relaxation time (also called e-folding time scale)
the speed (or any other suitable measure of the process) changes e times compared to its original
value.