Conservation of angular momentum – the Coriolis force
Bill Watterson Conservation of momentum in the ocean – Navier Stokes equation dv F / m (Newton) dt
dv - 1/r 흏p/흏z + horizontal component dt + Coriolis force + g vertical vector + tidal forces + friction
2 Attention:
Newton‘s law only valid in absolute reference system.
I.e. in a reference system which is at rest or in constant motion.
What happens when reference system is rotating? Then „inertial forces“ will appear.
A mass at rest will experience a centrifugal force.
A mass that is in motion will experience a Coriolis force.
3 Which reference system do we want to use?
z
y
x
4 Stewart, 2007 Which reference system do we want to use?
We like to use a Cartesian coordinate system at the ocean surface
convention: x eastward y northward z upward
This reference system rotates with the earth around its axis.
Newton‘s 2. law does only apply, when taking an „inertial force“ into account.
5 Coriolis ”force” - fictitious force in a rotating reference frame
Coriolis force on a merrygoround …
https://www.youtube.com/watch?v=_36MiCUS1ro
https://www.youtube.com/watch?v=PZPSfv_YssA 6 Coriolis ”force” - fictitious force in a rotating reference frame
Plate turns. View on the plate from outside. View from inside the plate.
7 Disc world (Terry Pratchett)
Angular velocity is the same everywhere w = 360°/24h
Tangential velocity varies with the distance from rotation axis: w NP 푣푇 = 2휋 ∙ 푟 /24ℎ r
vT (on earth at 54°N: vT= 981 km/h)
Eq
8 Disc world
Tangential velocity varies with distance from rotation axis:
푣푇 = 2휋 ∙ 푟 /24ℎ NP
r
Eq vT
9 Disc world
Tangential velocity varies with distance from rotation axis:
푣푇 = 2휋 ∙ 푟 /24ℎ NP
Hence vT is smaller near the North Pole than at the equator.
r Eq
@ equator: vT=1669 km/h
10 Disc world
푣푇 = 2휋 ∙ 푟 /24ℎ
NP If an object at the egde of the disc gets momentum towards the center, it maintains its tangential momentum from the edge and keeps moving eastwards at
the same speed vT = 1669 km/h.
Eq
11 Disc world
Tangential velocity
푣푇 = 2휋 ∙ 푟 /24ℎ
If an object at the egde of the disc gets momentum towards the center, it NP keeps its tangential momentum from the edge and keeps moving eastwards
at the same speed vT = 1669 km/h.
Thereby it gets into regions which move eastward at smaller tangential Eq speed.
12 Disc world
NP
Hence the object moves eastward, diverted to the right by a „fictitious“ force.
Eq It does not move straight, but along a curve.
13 Disc world
NP
… or it will turn westward in case it moves from the center to the rim; Eq Again diverted to the right.
In the northern hemisphere all moving objects (currents, wind, trains, …) are deflected to the right, i.e. they rotate clockwise.
14 Shift between coordinate systems
When transforming the Navier Stokes equation from an absolute coordinate system into one moving together with a point at the earth surface, a new term emerges: Coriolis force per mass! FCor 2 v
angular velocity of earth rotation v velocity of water parcel relative to the earth surface 15 Earth is not a disc but a sphere
2 / d 5 FCor 2 v ( 7.310 / s)
y x z Coordinate system at ocean surface
angular velocity is perpendicular to ocean surface only at poles!
16 Earth is a sphere e e e x y z FCor 2 v FCor 2 0 cos sin u v w
Fcor x w2 cos v 2sin w< Fcor y u 2sin Fcor z u 2 cos Fcor_z<< g 17 Coriolis force on earth FCor 2 v Horizontal component of Coriolis force contains only the vertical component of the angular velocity. F v 2sin v f CorH Coriolisparameter f increases with latitude Coriolis force is largest at poles and zero at equator. 18 Effect of Coriolis force on earth Léon Foucault 1851: „You are invited to take a look at the rotation of the earth …“ Foucault pendulum time lapse Pendulum plane seems to be Instead: the earth rotates under rotating … the pendulum plane. Literature: Umberto Eco „Foucault's Pendulum“ 19 Coriolis force (Fcor) - deflects objects that move relative to a rotating coordinate system. - The acceleration is perpendicular to the direction of motion, i.e. does not change the amount but the direction of motion. - In the northern hemisphere all objects in motion (ocean particles (currents), atmospheric particles (wind), trains, rockets, …) are diverted to the right (clockwise). - In the southern hemisphere all objects in are diverted to the left (anticlockwise). 20 Conservation of momentum in the ocean – Navier Stokes equation dv F / m (Newton) dt dv - 1/r 흏p/흏z + horizontal component dt + f u horizontal vector + g vertical vector + tidal forces + friction What happens if only the Coriolis force is acting? 21 Conservation of momentum in the ocean dv F / m (Newton) dt Acceleration of a water parcel = dv pressure gradient force (per mass) dt + f v + centrifugal forces of earth rotation constant with time + gravity weight + tidal forces + friction g Fgrav Fcentrif g: gravity Equation of motion (Navier Stokes equations) 22 Result of rotation shape of earth is a rotational ellipsoid Earth radius r = 6 356 km r = 6 378 km 23 Inertial oscillations in the Baltic Sea water parcel with momentum without exposure to force: v in an absolute system: duabs / dt 0 dvabs / dt 0 u in a rotating system: du/ dt fv dv/ dt fu solving the DEq: u(t) V sin( f t) v(t) V cos(f t) Dietrich et al., 1975 one more integration: 푥 푡 = −푟 ∙ cos 푓 ∙ 푡 푦 푡 = 푟 ∙ sin 푓 ∙ 푡 24 Little exercise: Are the observed oscillations really „inertial oscillations“ caused by the the Coriolis force? To check this, compare whether the Coriolis parameter meets the period of the oscillations. 26 Little exercise: Since the Coriolis force has an impact on all moving masses on earth: Please give a quantitative estimate whether or not a soccer player should take Coriolis force into account when planning a shot from a position far away from the goal. 27