Mathematical description
59
3.1.5. Ratio of scales
Ratios of time scales provide a priori insight into the importance of dynamical
processes in the flow. In a stationary flow, this ratio provides information as to
whether the dynamical processes represented in both time scales are able to balance. In time-dependent flows, the ratio provides information on which process
contributes most to the tendency of certain quantities in the flow.
Dimensionless parameters can always be written as the ratio of length scales or
the ratio of time scales. For example, the (local) Rossby number ǫ is the ratio of
τ f and τ a , according to
ǫ =
τ f
τ a
=
U
f 0 L
.
(3.22)
If ǫ ≪ 1, then the inertial acceleration is much smaller than the Coriolis acceleration, i.e., τ a ≫ τ f . In the vorticity view the relative vorticity is much smaller than
the planetary vorticity. For motions on a length scale much larger than L = U/f 0
the effect of inertia is not important with respect to Coriolis effects.
Another parameter that is important for the large-scale ocean flows in the dimensionless quantity β defined by
β =
τ a
τ β
=
β 0 L 2
U
.
(3.23)
If β ≫ 1, then the effect of the inertial acceleration on the motion is much smaller
than that due to the gradient in the Coriolis acceleration. In other words, the relative vorticity is much smaller than the gradient of planetary vorticity; for motions
on a length scale larger than
U/β 0 , effects of inertia can be neglected with
respect to effects due to variations in the Coriolis acceleration.
The Ekman numbers E H and E V ,definedby
E H =
A H
fL 2 =
τ f
τ H
w
; E V =
A V
fD 2 =
τ f
τ D
w
,
(3.24)
measure the ratio of the acceleration due to friction and the Coriolis acceleration.
If E V ≪ 1, then the vorticity input due to ‘diffusive’ vertical momentum transport
is not important compared to that due to the Coriolis acceleration. As an example,
E H can also be written as E H =( L w /L) 2 , where L w =
A H /f is a frictional
length scale. For motions on a much larger length scale than L w , the effects of
friction can be neglected.
Stratification introduces a dimensionless parameter, the Burger number S,that
can be represented as ratio of length and time scales as,
S =(L D /L)
2 =
N 2 D 2
f 2 L 2 =
τ 2
f δ 2
τ 2
s
.
(3.25)
59
3.1.5. Ratio of scales
Ratios of time scales provide a priori insight into the importance of dynamical
processes in the flow. In a stationary flow, this ratio provides information as to
whether the dynamical processes represented in both time scales are able to balance. In time-dependent flows, the ratio provides information on which process
contributes most to the tendency of certain quantities in the flow.
Dimensionless parameters can always be written as the ratio of length scales or
the ratio of time scales. For example, the (local) Rossby number ǫ is the ratio of
τ f and τ a , according to
ǫ =
τ f
τ a
=
U
f 0 L
.
(3.22)
If ǫ ≪ 1, then the inertial acceleration is much smaller than the Coriolis acceleration, i.e., τ a ≫ τ f . In the vorticity view the relative vorticity is much smaller than
the planetary vorticity. For motions on a length scale much larger than L = U/f 0
the effect of inertia is not important with respect to Coriolis effects.
Another parameter that is important for the large-scale ocean flows in the dimensionless quantity β defined by
β =
τ a
τ β
=
β 0 L 2
U
.
(3.23)
If β ≫ 1, then the effect of the inertial acceleration on the motion is much smaller
than that due to the gradient in the Coriolis acceleration. In other words, the relative vorticity is much smaller than the gradient of planetary vorticity; for motions
on a length scale larger than
U/β 0 , effects of inertia can be neglected with
respect to effects due to variations in the Coriolis acceleration.
The Ekman numbers E H and E V ,definedby
E H =
A H
fL 2 =
τ f
τ H
w
; E V =
A V
fD 2 =
τ f
τ D
w
,
(3.24)
measure the ratio of the acceleration due to friction and the Coriolis acceleration.
If E V ≪ 1, then the vorticity input due to ‘diffusive’ vertical momentum transport
is not important compared to that due to the Coriolis acceleration. As an example,
E H can also be written as E H =( L w /L) 2 , where L w =
A H /f is a frictional
length scale. For motions on a much larger length scale than L w , the effects of
friction can be neglected.
Stratification introduces a dimensionless parameter, the Burger number S,that
can be represented as ratio of length and time scales as,
S =(L D /L)
2 =
N 2 D 2
f 2 L 2 =
τ 2
f δ 2
τ 2
s
.
(3.25)