Vorticity in the
Ocean
Principles of
Oceanography
Introduction
Definitions of Vorticity
Conservation of Vorticity
Influence of Vorticity
Vorticity and Ekman pumping
Consequence
Definitions of vorticity
Vorticity is analagous to angular
momentum.
Vorticity is a conserved quantity
(Conservation of Vorticity)
Two types of vorticity
Planetary Vorticity f
Rotation imparted by Earth
Relative Vorticity z
Due to currents in the ocean
2D flow assumption
Order z << f
Planetary Vorticity f and Relative
Vorticity z
Planetary Vorticity: Every object on earth has a vorticity given to it by the rotation of the earth (except an object on the
equator). This vorticity is dependent on latitude.
Relative Vorticity: The ocean and rotate at exactly the same rate as earth. They haveatmosphere do not some rotation
relative to earth due to currents and winds. Relative vorticity ζ is the vorticity due to currents in the ocean. Mathematically it is:
y k u x
j v x w z
i u z v y
w w
v u
z y
x
k j
i
u
ζ is usually much smaller than f, and it is greatest at the edge of fast currents such as the Gulf Stream. To
obtain some understanding of the size of ζ, consider the edge of the Gulf Stream off Cape Hatteras where the velocity decreases by 1 m/s in 100 km at the
boundary. The curl of the current is approximately (1m/s)/(100 km) = 0.14 cycles/day = 1 cycle/week.
Hence even this large relative vorticity is still almost seven times smaller than f. A more typical values of relative vorticity, such as the vorticity of eddies, is a cycle per month.
Absolute Vorticity
The sum of the planetary and relative vorticity
Equations of Motion for frictionless flow
Derivation of Vorticity equation
Fx
x fv p Dt
Du
1
Fy
y fu p Dt
Dv
1
0 0
x fv p y
v u x u u t u Dt
Du
) 1 ( ) (
y fu p y
v v x u v t v Dt
Dv
) 1 ( ) (
y f v yv f y x
p y
v u y u y v y x u u x u y u y u
t
2
2 2
2 1
) ( )
(
x f u xu f y x
p y
x v v y v x v x u v x v x u x v
t
2 2
2
2 1
) ( )
(
yv u f x f y v x f u y
u x v y v y u x v x u y u x v v y
y u x v u x
y u x v
t
0
v
y u f x f y v x f u y v x u v y
u x
tz z z z
0
v
y f y v x f u
Dt
Dz z
0
0 0
0
Dt
Df y v x f u
Dt
Dz z
Potential Vorticity
As a conserved quantity potential vorticity is a valuable tool in
studying ocean dynamics. The potential vorticity is defined as the
product of the absolute vorticity and the stratification.
P is conserved along a fluid trajectory
Barotropic, geostrophic flow in an ocean with depth H(x, y, t)
Fig 1.1
Integrate the continuity equation
Vertical velocity
Conservation of Vorticity
Conservation of z in a spinning ocean
Transfer of momentum between two bodies
Friction is essential
Air-sea boundary Ekman layer transfer momentum
Sea-bottom boundary Ekman layer transfer momentum
Sides of subsea mountains friction pressure differences form drag
In the vast interior of the ocean frictionless vorticity is conserved
Conservation of the angular momentum
The angular momentum of any isolated spinning body is conserved
Transfer of angular momentum between two bodies
Need not be in physical contact; gravitational forces can transfer momentum between bodies in space
Ocean Surface
Ocean bottom
A parcel of water moves east
(constant latitude) in N.Hemis.
As the parcel hits the bump, H
decreases. We know that (f +
ξ )/H=Constant. So if H decreases,
(f + ξ ) must decrease. If f
decreases, the parcel moves
equatorward. If ξ decreases the
parcel spins clockwise.
An example of
conservation of
vorticity when H
doesn’t stay constant
H
Bump in bottom
H
What happens when the
parcel leaves the bump?
North
South
A parcel of water moves east
(constant latitude) in N.Hemis.
As the parcel hits the bump, H
decreases. We know that (f + ξ
)/H=Constant. So if H decreases, (f
+ ξ ) must decrease. If f decreases,
the parcel moves equatorward. If
ξ decreases the parcel spins
clockwise. Or a combination.
An example of
conservation of
vorticity when
H doesn’t stay
constant
Bump in bottom
H
H
From ABOVE
Parcel Moves Equatorward
Conservation of Potential
Vorticity
The conservation of potential vorticity couples changes in depth, relative vorticity, and changes in latitude. All three interact:
Changes in the depth H of the flow causes changes in the
relative vorticity. The concept is analogous with the way figure skaters decreases their spin by extending their arms and legs.
The action increases their moment of inertia and decreases their rate of spin
Changes in latitude require a corresponding change in ζ. As a column of water moves equatorward, f decreases, and ζ must increase.
Influence of Vorticity
The concept of conservation of potential vorticity has far reaching consequences, and its application to fluid flow in the ocean gives a deeper understanding of ocean currents.
f >> z f / H = constant
The flow in an ocean of constant depth be zonal
Depth is not constant, but in general, currents tend to be east-west rather than north south
Wind makes small changes in z, leading to a small meridional component to the flow
Figure 11.3
Influence of Vorticity
Barotropic flows are diverted by seafloor features
Figure 1.4: a flow encounters a subsea ridge, Friction along the sides of sub-sea mountains leads to pressure differences on either side of the mountain which causes another kind of drag called form drag.
Topographic steering:
H ζ f the flow is turned toward the equator
Topographic blocking
If the change in depth is sufficiently large, no change in latitude will be sufficient to conserve potential vorticity, and the flow will be unable to cross the ridge
Fig.1.4
Influence of Vorticity
An alternate explanation for the existence
of western boundary currents
Figure 12.5
Wind blow negative z
Eastern boundary southward flow f
providing positive z conservation
Western boundary northward flow f
providing negative z conservation?!!
A strong source of positive vorticity is provided by the current shear in the western boundary current as the current rubs against the coast causing the northward velocity to go to zero at the coast
Convergence/Divergence
This idea is nothing more then the piling up or moving of water away from a region.
Conservation of VOLUME: (du/dx+dv/dy+dw/dz=0)
Rearranging... du/dx + dv/dy = -dw/dz
If water comes into the box (du/dx + dv/dy)>0 there is a velocity out of the box: dw/dz < 0 DOWNWARD
Vorticity and Ekman Pumping
First consider flow in a fluid with constant rotation
Secondly, how vorticity constrains the flow of a fluid with rotation that varies with latitude
Fluid dynamics on the f Plane: the Taylor-Proudman Theorem
f-plane, constant rotation f = f0
Slowly varying flow in a homogeneous (constant density ρ0), rotating, inviscid fluid
Geostrophic equations
Continuity equation
Vorticity and Ekman Pumping
Taking the z derivative and using gives;
Similarly, for the u-component of velocity
The vertical derivative of the horizontal velocity field must be zero
Because w = 0 at the sea surface and at the sea floor, if the bottom is level, there can be no vertical velocity on an f–plane.
Vorticity and Ekman Pumping
(Fluid dynamics on the f Plane: the Taylor-Proudman Theorem)
Implication
Rotation greatly stiffens the flow
Cannot expand or contract in the vertical direction
As rigid as a steel bar
Geostrophic flow cannot go over a seamount, it must go around it
w(z=0) = 0 + w(z=H) = 0 + w/z = 0 w(z) = 0
Vorticity and Ekman Pumping (Fluid
Dynamics on the Beta Plane)
Ekman pumping
b-plane
Consider, f = f0 + b y
G: geostrophic flow
Using the continuity equation, and recalling that β y ≪ f0
Thus the variation of Coriolis force with latitude allows vertical velocity gradients in the geostrophic interior of the ocean, and the vertical velocity leads to north-south currents.
This explains why Sverdrup and Stommel both needed to do their calculations on a β-plane.
In the central region of the jet, some streamlines that are not purely horizontal are also visible. They correspond to beads trapped in the bottom or top Ekman layers and show net fuid exchange from source to sink.
Vorticity and Ekman Pumping
Winds at the sea surface drive Ekman transports to the right of the wind in this northern hemisphere example (bold arrows in shaded Ekman layer). The converging Ekman transports driven by the trades and westerlies drives a downward geostrophic flow just below the Ekman layer (bold vertical arrows), leading to downward bowing constant density surfaces i. Geostrophic currents associated with the warm water are shown by bold arrows.
Vorticity and Ekman Pumping
An example of how winds produce geostrophic currents running upwind. Ekman
transports due to winds in the north Pacific (Left) lead to Ekman pumping (Center), which sets up north-south pressure gradients in the upper ocean. The pressure gradients are balanced by the Coriolis force due to east- west geostrophic currents (Right). Horizontal lines indicate regions where the curl of the zonal wind stress changes sign. AK: Alaskan Current, NEC: North Equatorial Current, NECC: North
Equatorial Counter Current
Ocean Surface
Mixed Layer
Ocean bottom
A parcel of water moves into an
area of downwelling. It
becomes shorter (and fatter).
f/H must be
conserved!
We know that (f + ξ )/H= Constant. So if H
decreases, (f + ξ ) must decrease. I gave
examples before that either f or ξ could
change. But in this process; it is f that
decreases. f can only decrease by the parcel
moving equatorward.
With DOWNWELLING, the
vertical velocity is downward.
This pushes on the column of
water, making it shorter (and
fatter). What happens when a
column of water gets short and
fat (Vorticity must be
conserved).
H H
Ekman Convergence
Ekman transport creates convergence and
divergence of upper waters.
Convergence
Convergence
Divergence
Divergence
Divergence
Sea Surface Height and Mean Geostrophic Ocean Circulation
Important concepts
Vorticity strongly constrains ocean dynamics.
Vorticity due to Earth's rotation is much greater
than other sources of vorticity.
Taylor and Proudman showed that vertical
velocity is impossible in a uniformly rotating flow.
The ocean is rigid in the direction parallel to the
rotation axis. Hence Ekman pumping requires
that planetary vorticity vary with latitude. This
explains why Sverdrup and Stommel found that
realistic oceanic circulation, which is driven by
Ekman pumping, requires that f vary with latitude
Important concepts (cont.)
The curl of the wind stress adds relative vorticity to central gyres of each ocean basin. For steady state circulation in the gyre, the ocean must lose vorticity in western boundary currents.
Positive wind stress curl leads to divergent flow in the Ekman layer. The ocean's interior geostrophic
circulation adjusts through a northward mass transport.
Conservation of absolute vorticity in an ocean with constant density leads to the conservation of potential vorticity. Thus changes in depth in an ocean of
constant density requires changes of latitude of the current.
