3.12 The Coriolis Force
47
where the constant η o can be determined from the requirement that the total volume
of flui contained in the tank has to be conserved (if the tank is void of leaks). The
tank’s rotation leads to a parabolic shape of the flui surface and it is essentially
gravity (via the hydrostatic balance) that operates to balance out the centrifugal
force. The latter balance is valid for all flui parcels in the tank. The pathways of
flui parcels are circles in the fi ed frame of reference. The observer in the rotating
system, however, will not spot any movement at all.
3.12.5 Motion in a Rotating Fluid as Seen in the Fixed Frame
of Reference
With reference to a f xed frame of reference, flui parcels in the rotating tank
exclusively feel the centripetal force provided by the pressure-gradient force. In
the absence of relative motion, flui parcels describe circular paths. How does the
trajectory of a flui parcel look like, if we give it initially a push of a certain
speed into a certain direction? The momentum equations governing this problem are
given by:
dU
dt
= −Ω
2
X and
dV
dt
= −Ω
2
Y
(3.37)
where (X, Y ) refers to a location and (U, V ) to a velocity in the f xed coordinate system. On the other hand, the location of our flui parcel simply changes
according to:
d X
dt
= U and
dY
dt
= V
(3.38)
Owing to rotation, velocities in the fi ed and rotating reference systems are not
the same. Instead of this, it can be shown that they are related according to:
U = u − Ω y and V = v + Ω x
(3.39)
where (x, y) refers to a location and (u, v) to a velocity in the rotating coordinate
system.
3.12.6 Parcel Trajectory
Before reviewing the analytical solution, we employ a numerical code (see below)
to predict the pathway of a flui parcel in a rotating flui tank as appearing in the
fi ed frame of reference. To this end, we consider a flui tank, 20 km in diameter,
rotating at a rate of Ω = −0.727×10
−5
s
−1
, which corresponds to clockwise rotation
with a period of 24 h.
47
where the constant η o can be determined from the requirement that the total volume
of flui contained in the tank has to be conserved (if the tank is void of leaks). The
tank’s rotation leads to a parabolic shape of the flui surface and it is essentially
gravity (via the hydrostatic balance) that operates to balance out the centrifugal
force. The latter balance is valid for all flui parcels in the tank. The pathways of
flui parcels are circles in the fi ed frame of reference. The observer in the rotating
system, however, will not spot any movement at all.
3.12.5 Motion in a Rotating Fluid as Seen in the Fixed Frame
of Reference
With reference to a f xed frame of reference, flui parcels in the rotating tank
exclusively feel the centripetal force provided by the pressure-gradient force. In
the absence of relative motion, flui parcels describe circular paths. How does the
trajectory of a flui parcel look like, if we give it initially a push of a certain
speed into a certain direction? The momentum equations governing this problem are
given by:
dU
dt
= −Ω
2
X and
dV
dt
= −Ω
2
Y
(3.37)
where (X, Y ) refers to a location and (U, V ) to a velocity in the f xed coordinate system. On the other hand, the location of our flui parcel simply changes
according to:
d X
dt
= U and
dY
dt
= V
(3.38)
Owing to rotation, velocities in the fi ed and rotating reference systems are not
the same. Instead of this, it can be shown that they are related according to:
U = u − Ω y and V = v + Ω x
(3.39)
where (x, y) refers to a location and (u, v) to a velocity in the rotating coordinate
system.
3.12.6 Parcel Trajectory
Before reviewing the analytical solution, we employ a numerical code (see below)
to predict the pathway of a flui parcel in a rotating flui tank as appearing in the
fi ed frame of reference. To this end, we consider a flui tank, 20 km in diameter,
rotating at a rate of Ω = −0.727×10
−5
s
−1
, which corresponds to clockwise rotation
with a period of 24 h.