4.1 The Basis
99
to the centre of the Cartesian coordinates system defining f o . Note that y becomes
negative for locations south of this centre.
Horizontal divergence of geostrophic flow is given by:
∂u geo
∂ x
+
∂v geo
∂ y
= −
β
f
v geo
(4.8)
On small spatial scales <100 km, this divergence is negligibly small. As a consequence of this, barotropic geostrophic flow tends to follow bathymetric contours –
a feature referred to as topographic steering. On larger scales of ocean basins
(∼1,000 km), meridional (north–south) flow creates a small divergence that plays
an important role in the wind-driven circulation of the oceans balancing the windinduced flow divergence in the surface Ekman layer. This balance is known as the
Sverdrup balance.
4.1.3 Scaling
The temporal Rossby number, defined by Eq. (2.7), compares the inertial period
with the time scale of a time-variable process. Coriolis effects become important for
a temporal Rossby number of values less than unity or, in other words, if the time
scale of a process exceeds the inertial period. Stationary processes, on the other
hand, are controlled by the Coriolis force if the Rossby number (without prefix) is
small compared with unity. The Rossby number is defined by:
Ro =
U
f L
(4.9)
where U is characteristic flow speed, f is the Coriolis parameter, and L is a characteristic length scale. The relevant time scale is given by L/U which is the time it
takes for flow to carry a parcel over a characteristic distance.
Calculations of Rossby numbers for previous exercises reveals that the Coriolis
force could be neglected in most instances. The focus of the following exercises is
placed on stratified processes that are influenced or even controlled by the Coriolis
force.
4.1.4 Conservation of Potential Vorticity
Vorticity is the ability of flow to produce rotation. Frictionless flow that is almost in
a geostrophic balance is called quasi-geostrophic flow. For a layered ocean, it can
be shown that quasi-geostrophic flow conserves a quantity called potential vorticity.
This statement is valid along flow trajectories and reads (Cushman-Roisin, 1994):
f + ξ i
h i
= constant
(4.10)
99
to the centre of the Cartesian coordinates system defining f o . Note that y becomes
negative for locations south of this centre.
Horizontal divergence of geostrophic flow is given by:
∂u geo
∂ x
+
∂v geo
∂ y
= −
β
f
v geo
(4.8)
On small spatial scales <100 km, this divergence is negligibly small. As a consequence of this, barotropic geostrophic flow tends to follow bathymetric contours –
a feature referred to as topographic steering. On larger scales of ocean basins
(∼1,000 km), meridional (north–south) flow creates a small divergence that plays
an important role in the wind-driven circulation of the oceans balancing the windinduced flow divergence in the surface Ekman layer. This balance is known as the
Sverdrup balance.
4.1.3 Scaling
The temporal Rossby number, defined by Eq. (2.7), compares the inertial period
with the time scale of a time-variable process. Coriolis effects become important for
a temporal Rossby number of values less than unity or, in other words, if the time
scale of a process exceeds the inertial period. Stationary processes, on the other
hand, are controlled by the Coriolis force if the Rossby number (without prefix) is
small compared with unity. The Rossby number is defined by:
Ro =
U
f L
(4.9)
where U is characteristic flow speed, f is the Coriolis parameter, and L is a characteristic length scale. The relevant time scale is given by L/U which is the time it
takes for flow to carry a parcel over a characteristic distance.
Calculations of Rossby numbers for previous exercises reveals that the Coriolis
force could be neglected in most instances. The focus of the following exercises is
placed on stratified processes that are influenced or even controlled by the Coriolis
force.
4.1.4 Conservation of Potential Vorticity
Vorticity is the ability of flow to produce rotation. Frictionless flow that is almost in
a geostrophic balance is called quasi-geostrophic flow. For a layered ocean, it can
be shown that quasi-geostrophic flow conserves a quantity called potential vorticity.
This statement is valid along flow trajectories and reads (Cushman-Roisin, 1994):
f + ξ i
h i
= constant
(4.10)