vorticity measure of angular momentum for a fluid tendency of a parcel to rotate two components of...

Vorticity

• Measure of angular momentum for a fluid

• Tendency of a parcel to rotate

• Two components of vorticity

– relative (angular momentum in rotating frame)

– planetary (rotation of the frame)

• Important for understanding western boundary currents

Relative Vorticity

Positive

Negative

• Relative vorticity, , is driven by shears in the flow field

Relative Vorticity

Positive

Negative

Negative Positive

anti-cyclonic cyclonic

The Sign of Vorticity

Negative Positive

anti-cyclonic cyclonic

Relative Vorticity

Positive

Negative

• Relative vorticity is defined as = v/x - u/y

East

x or u direction

North

y or v direction

Example of Relative Vorticity

• Northward velocity increases as a function of x distance (@ 34oN)

• Relative vorticity is positive

East

x or u direction

North

y or v direction

10 cm/s

500 km

Relative Vorticity• Relative vorticity is defined as = v/x - u/y = v/x

• Change in v is 0.1 m/s for x = 500 km

• Relative vorticity () = v/x = (0.1 m / s) / (500x103 m) = 2x10-7 s-1

Another Example• Eastward velocity decreases as a function of y (north)

distance

East

x or u direction

North

y or v direction

10 cm/s

500 km

Relative Vorticity• Relative vorticity is defined as = v/x - u/y = - u/y

• Change in u is 0.1 m/s for y = 500 km

• Relative vorticity () = - u/y = - (- 0.1 m / s) / (500x103 m) = 2x10-7 s-1

Relative Vorticity

• Relative vorticity, = v/x - u/y

v/x > 0 -> > 0 u/y < 0 -> > 0

+ +

cyclonic vorticity

Relative Vorticity

• Relative vorticity, = v/x - u/y

v/x < 0 -> < 0 u/y > 0 -> < 0

- -

anti-cyclonic vorticity

Planetary Vorticity

• The planet also rotates about its axis

• Objects are affected by both planetary & relative vorticity components

• Planetary vorticity = 2 sin (= f)

2 @ north pole

0 on equator

- 2 @ south pole

Example for Planetary Vorticity

• Planetary vorticity = 2 sin (= f)

• At 34oN, f = 2 sin 34o = 8.2x10-5 s-1

• Previous examples -> = 2x10-7 s-1

• Ratio of || / f = (2x10-7 s-1)/(8.2x10-5 s-

1) = 0.0025

• Relative vorticity is small compared with f except near equator (Rossby number)

Total Vorticity

• Only the total vorticity (f + ) is significant

• For flat bottom ocean with uniform & no friction, total vorticity (f + ) is conserved

– Coffee cup example…

• Water transported north will decrease its to compensate for changes in f

• Water advected south will increase its

Potential Vorticity

• Potential vorticity = (f + ) / D

Potential Vorticity

• Potential vorticity = (f + ) / D

• PV is conserved except for friction

• If f increases, a water mass can spin slower (reduce ) or increase its thickness

• Typically, PV is approximated as f/D ( <<

f)

• Used to map water mass distributions & assess topographic steering

Potential Vorticity

WOCE SalinityP16150oW

Potential Vorticity

WOCE PVP16150oW

PV~f/D

Potential Vorticity

PV on = 25.2

• Potential vorticity = (f + ) / D ~ f / D

• Uniform zonal flow over a ridge

• Let D decreases from 4000 to 2000 m

• If PV = constant, f must decrease by 2, leading to a equatorward deflection of current

• This is topographic steering

Topographic Steering

U D

Plan view (NH)

Topographic Steering

Topographic Steering

• A factor of two reduction in f

• For 30oN, f = 7.29x10-5 s-1

• f/2 = 3.6x10-5 s-1 which corresponds to a latitude of 14.5oN

• Displacement = (30-14.5o)*(111 km/olat) = 1700 km

• Water column is really stratified which reduces the changes of D & thereby f

Topographic Steering

Bascially f/H

Vorticity

• Measure of the tendency of a parcel to rotate

• Relative (= rotation viewed from Earth frame)

• Planetary (= f rotation of the frame)

• Total ( + f) & potential vorticity ( + f) / D are relevant dynamically

• Important for diagnosing water mass transport & western intensificaiton...

Western Intensification

• Subtropical gyres are asymmetric & have intense WBC’s

• Western intensification is created by the conservation of angular momentum in gyre

• Friction driven boundary current is formed along the western sidewall

• Maintains the total vorticity of a circulating water parcel

Wind Driven Gyres

Symmetric gyre

Wind Driven Gyres

Wind Torque in Gyres

Need process to balance the constant addition of negative

wind torque

Curl of the wind stress…

• Model of steady subtropical gyre

• Includes rotation and horizontal friction

f = constant

f = 2 sin

Stommel’s Experiments

Stommel’s Experiments

• Stommel showed combination of horizontal friction & changes in Coriolis parameter lead to a WBC

• Need to incorporate both ideas into an explanation of western intensification

Western Intensification

• Imagine a parcel circuiting a subtropical gyre

• As a parcel moves, it gains negative vorticity (wind stress curl)

• Gyre cannot keep gaining vorticity or it will spin faster and faster

• Need process to counteract the input of negative vorticity from wind stress curl

Western Intensification

• Conservation of potential vorticity (f + )/D

Assume depth D is constant (barotropic ocean)

Friction (i.e., wind stress curl) can alter (f + )

• In the absence of friction

Southward parcels gain to compensate reduction in f

Northward parcels lose to compensate increase in f

Symmetric Gyre

Western Intensification

• Friction plays a role due to

wind stress curl (input of -)

sidewall friction (input of +)

+

+

WBC EBC

Western Intensification

• In a symmetric gyre,

Southward: wind stress input of - is balanced

+ inputs by ’s in latitude & sidewall friction

Northward: ’s in latitude result in an input of -

along with the wind stress input of -

This is NOT balanced by + by sidewall friction

Need an asymmetric gyre to increase sidewall friction in the northward flow!!

Symmetric Gyre

Western Intensification

• In a symmetric gyre,

Southward: wind stress input of - is balanced

+ inputs by ’s in latitude & sidewall friction

Northward: ’s in latitude result in an input of -

along with the wind stress input of -

This is NOT balanced by + by sidewall friction

Need an asymmetric gyre to increase sidewall friction in the northward flow!!

Potential Vorticity

Western Intensification

• In a asymmetric gyre,

Southward: wind stress input of - is balanced +

inputs by ’s in latitude & sidewall friction

Northward: ’s in latitude result in an input of -

along with the wind stress input of -

This IS balanced by LARGE + from sidewall friction

Total vorticity balance is satisfied & we have an asymetric gyre

Potential Vorticity

Role of Wind Stress Curl

• Spatial ’s in wind stress control where Ekman transports converge

• Where changes in w = 0, the convergence

of Ekman transports = 0

• This sets the boundaries of gyres

• My = 1/(f/y) curl w = (1/) curl w

-> Sverdrup dynamics

Munk’s Solution

Currents

Western Intensification• Intense WBC’s create a source of positive

vorticity that maintains total vorticity balance

• Creates asymmetric gyres & WBC’s

• Boundary currents are like boundary layers

• Wind stress curl & ’s in Coriolis parameter with latitude are critical elements

• Can be extended to quantitatively predict water mass transport (Sverdrup theory)