Ocean currents I Ocean currents are driven by surface effects such as the surface motion driven by the drag interaction of wind and water and the vertical motion driven by density and salinity gradients in the deeper ocean layers. These motions couple with the Coriolis force, which results from the influence of Earth’s rotation on a northward (or southward) moving bodies with velocity v.

The directional forces Wind-water interaction The drag forces

Circular pattern of ocean currents

The ocean currents run clockwise on the northern hemisphere and turn counter-clockwise on the southern hemisphere!

Earth rotates counter-clockwise

The Coriolis Force The Coriolis force is a result of classical mechanics on a moving object in a rotating frame of reference! The Coriolis force pushes a moving object to the right for counterclockwise rotation of the reference frame or to the left for clockwise rotation. The force is a vector cross product between velocity vector of the moving object and the orbital momentum vector of the rotation.

transportofdirectionvectordefines

directionlarperpendicuinpoints

termsvectorinproductcross

amf

latitude as withparameterCoriolissa

v and componts, of direction the givethey

vectors unit are v,zf

vzF

vzF

vz

vzF

c

c

c

ˆ~

ˆ~

ˆ

sin2

ˆˆ

1

The Coriolis force represents a vector cross product

Coriolis Force on Earth The earth rotates with angular velocity around its axis. A body located on earth with mass m has a tangential linear velocity of vx=r in the external reference frame . The angular momentum is:

If the body moves pole-wards with a velocity v=vy , the angular momentum L changes, generating the torque :

sin2 vmFc

2rmL

latitudeis

vdt

dy

dt

dr

Frdt

drrm

dt

dL

dt

rmd

dt

dL

c

sinsin

2

2

R

r

m

The Coriolis force accelerates a body moving south on the southern hemisphere to the west or or a body moving

north on the northern hemisphere to the east.

150

10292.72360

sin2

sin2

sday

rad

day

va

vmF

amF

c

c

cc

A motor yacht crosses the 20o N latitude moving northwards in the Northern Atlantic with a constant

speed of 10 knots (1knot=1.852 km/h). What is the strength of the Coriolis force Fc acting on the boat, if the yacht has a mass of 20 tons?

NFSfor

Ns

mkgF

s

mskgF

c

c

c

127.520

127.5127.5

20sin3600

1085.110292.71022

0

2

04

154

The boat needs one full minute to accelerate from zero to its 10 knots speed. Calculate the force Fm, the engine has to generate for bringing the boat up to the speed of 10 knots.

Ns

s

m

kgF

min

knotskgamF

m

m

171360

14.5

102

1

10102

4

4

The Coriolis force is relatively small but becomes larger at higher latitudes. Calculate the Coriolis force acting on the boat when it crosses the 60o N latitude on its way to Iceland?

Ns

mkgF

s

mskgF

c

c

1398.12

60sin3600

1085.110292.71022

2

04

154

Coriolis force on air or water mass forms

inertial cycles! Air or water moving with speed v without any external forces acting upon them, except for the Coriolis force Fc, will take a circular clockwise flow trajectory on the northern hemisphere, and a circular counter-clockwise flow trajectory on the southern hemisphere, balanced by the centripetal force Fcf.

sin525.0sin263.02sin2

sin2

11

2

h

v

h

vvR

FvmR

vmF ccf

The Gulf Stream is the fastest ocean current in the world with peak velocities near 3m/s. What is the curvature at 35o north latitude?

kmm

s

s

m

R

s

s

m

h

vR

36825,35

35sin1

1046.1

3

sin1

1046.1

3

sin525.0

04

41

Only small eddies form at that low flow speed!

Reality is more complex because of additional forces due to friction and temperature effects, which add to the eddy formation phenomenon!

Reality is more complex because of additional forces due to the drag forces provided by the atmospheric wind circulation and by the friction forces exerted by deeper water layers!

Winds as pressure phenomenon Air moves; less dense (higher temperature) air pockets rise to higher altitude, cools and sinks back towards earth surface. Airflow rushes toward low pressure regions and away from high pressure zones. As larger the pressure gradient, as larger is the force acting upon an air mass m, increasing the wind velocity.

Pm

Fp

The force points from a higher pressure towards a lower pressure range

gdz

dP Upwards and downwards motion is determined by the hydrostatic equation leading to the lapse rate.

Wind direction points perpendicular to isobars (lines of constant pressure)

Isobars indicate lines of constant pressure P=0!

Global isobar map Presence of large continental land areas in the northern hemisphere affects the ‘ideal’ isobar patterns observed at southern latitudes. During the summer land is warmer than ocean causing the development of low pressure areas, which change northward/southward winds patterns.

During winter land is colder than ocean water, causing the development of low pressure areas over the ocean regions.

The emergence of localized weather zones is further enhanced by the geographical affects from lakes or mountains.

Geostrophic winds on non-rotating earth

Wind convection cells form because of the equatorial heating effects and the polar cooling effects. The ITCZ (intertropical convergence zone) shows only a negligible horizontal wind component, and is also called the doldrums, where the seasonal lack of wind often delayed a sailing ship’s journey.

Friction effects are neglected Wind direction would be defined entirely by the latitudinal isobars

The intertropical convergence zone Area of cloud formation because of extended condensation and uprising air masses.

Also origin of thunderstorms and cyclones if a rotational friction component is added!

The effects of rotation, Coriolis force Wind continues to move perpendicular to isobars, but air masses are subject to the Coriolis force, which prevents winds to reach the poles, forming three circulation patterns. At 30o latitude the air cools and sinks, moving back to equator in an inertial cycle in north easterly directions (the trade winds). Winds that continue to move north are deflected eastwards by increasing Coriolis force (prevailing westerly winds).

Radius of circular wind patterns

km

s

s

m

h

vR 1194

35sin1

1046.1

100

sin525.0 041

sin525.0sin2 1h

vvR

The trade winds blow persistently westward and toward the Equator as part of the geostrophic wind pattern from the subtropical high-pressure belts toward the intertropical convergence zone (ITCZ). At high altitude the average speed over the oceans is about 50-100 m/s. On low altitude the speed slows down due to friction. The trade winds were named by the crews of sailing ships that depended on the winds during westward ocean crossings.

Reality shows larger radii (2500 km) due to the impact of pressure gradients!

trade winds (southeasterly)

trade winds (northeasterly)

Westerlies

Hurricanes

Cyclones

Westerlies

trade winds (southeasterly)

trade winds (northeasterly)

Westerlies

doldrums doldrums doldrums

Trade routes for Guano supply

Coriolis force on low pressure area This low-pressure system spins counter-clockwise due to the balance between the Coriolis force and the pressure gradient force. The air tends to flow towards it, but is deflected perpendicular to its velocity by the Coriolis force. A system of equilibrium can then establish itself creating circular movement, or a cyclonic flow.

Fc

Fp

Wind-Water surface interaction Water motion at the surface of the ocean (mixed layer) is driven by wind effects. Friction causes drag effects on the water, transferring momentum from the atmospheric winds to the ocean surface water.

The drag force Wind generates vertical and horizontal motion in the water, triggering convective motion, causing turbulent mixing down to about 100m depth, which defines the isothermal mixed layer. The drag force FD on the water depends on wind velocity v:

waves!ofemergencethelyparticular

androughness,surfaceondepending

areasectionalcrossACninteractiowaterwindfor

constantessdimensionltcoefficiendragCvACF

D

DaDD

,002.0

2

The Beaufort Scale is an empirical measure describing wind speed based on the observed sea conditions (1 knot = 0.514 m/s = 1.85 km/h)!

s

mBv 2/3836.0

A strong breeze of B=6 corresponds to wind speed of v=39 to 49 km/h at which long waves begin to form and white foam crests become frequent. The drag force can be calculated to:

2

2

2

3

3

2

188.0188.05.122.1001.0

001.05.12452.1

mNAForNAs

mAm

m

kgF

Cs

m

h

kmv

m

kgvACF

DD

DaaDD

For a strong gale (B=12), v=35 m/s, the drag stress on the water will be:

0025.0352.1

68.3352.10025.0/

3

2

2

3

Da

D

Cs

mv

m

kg

mNs

m

m

kgAF

Conversion from scale to wind velocity: