2013, A Rotating Framework, 02-Coriolis

7/30/2019 2013, A Rotating Framework, 02-Coriolis

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Ocean Modeling - EAS 8803

We model the ocean on a rotatingplanet

Rotation effects are consideredthrough the Coriolis and

Centrifugal Force

The Coriolis Force arises becauseour reference frame (the Earth) isrotating

The Coriolis Force is the sourceof many interesting geophysicalprocesses

Chapter 2

rotation


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A rotating framework - The coordinates

-

6

1

M

M

y

x

X

r

Y i

j J

I

M

6

1

-tFigure 2-1 Fixed (X, Y) and rotating

(x, y) frameworks of reference.

x = + X cost + Y sint

y = X sint + Y cost.

Fixed reference

Rotatingreference

Angular Velocity


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A rotating framework - The velocity (1st derivative)

dx

dt= +

dX

dtcost +

dY

dtsint

+y

X sint + Y cost

dy

dt=

dX

dtsint +

dY

dtcost X cost Y sint

xAbsoluteVelocity

Relative Velocitychange of thecoordinate relative tothe moving frame

u =dx

dt i +dy

dt j U =dX

dt I +dY

dt J


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dx

dt= +

dX

dtcost +

dY

dtsint

+y

X sint + Y cost

dy

dt=

dX

dtsint +

dY

dtcost X cost Y sint

x

U

V

u

v

U = uy, V = v + x.

Relation between absolute and relative velocity

!y

!"x

entraining velocitydue to rotation

relativevelocity

+absolutevelocity

=

A rotating framework - The velocity (1st derivative)


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A rotating framework - The acceleration (2nd derivative)

d2x

dt2=

d2X

dt2cost +

d2Y

dt2sint

+ 2

dX

dtsint +

dY

dtcost

V

2 (X cost + Y sint)

x

d2

ydt2

=

d2

Xdt2

sint + d2

Ydt2

cost

2dXdt

cost + dYdt

sint

U

2 ( X sint + Y cost)

y

.

du

dt

dU

dt 2!V

! "2x

dv

dt

dV

dt !2"U

!"2y


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d2x

dt2=

d2X

dt2cost +

d2Y

dt2sint

+ 2

dX

dtsint +

dY

dtcost

V

2 (X cost + Y sint)

x

du

dt

dU

dt 2!V

! "2x

U = u y, V = v + x.

use this equality:

Relation between absolute and relative velocity


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Coriolisacceleration

relativeacceleration

+absolute

acceleration=

Centrifugalacceleration

+

U = u y, V = v + x.

use this equality:

dU

dt=

du

dt ! 2"v !"2

x

dV

dt

=

dv

dt

! 2"u !"2y


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+ 2 u + ( r)VectorNotation

Coriolis acceleration Centrifugal acceleration

r =

x

y

!

"#

$

%&

du

dt

define vectors: u =u

v

!

"#

$

%&

d

dt+! !

time derivative inrotating frame:


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Coriolis vs. Centrifugal Force

Centrifugal Forcedepends only on location

Coriolis Force is active

only when things move

dU

dt=

du

dt ! 2"v !"2

x

dV

dt

=

dv

dt

! 2"u !"2y


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Centrifugal Force is unimportant for motions

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...................................... ...... ....... ........ ......... ......... .................. ......... ......... ........ ............. ............

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S

N

Net

Centrifugal

Grav

ity

Local

vertic

al

Figure 2-2 How the flattening of the

rotating earth (grossly exaggerated in

this drawing) causes the gravitational

and centrifugal forces to combine into

a net force aligned with the local verti-

cal, so that equilibrium is reached.

geoidan equipotential surface


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Free Motion on a rotating frame

Coriolis Force is active

only when things move

dU

dt=

du

dt ! 2"v !"2

x

dV

dt

=

dv

dt

! 2"u !"2y


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f = 2

The general solution to this system of linear equations is

u=

Vsin(

f t+

)

, v=

Vcos(

f t+

)

Free Motion on a rotating frame

dudt

2v = 0, dvdt

+ 2u = 0.

NOTE: the speed does not change with timeyet u and vdo change with time!

changes in u and vimply change in direction.

Inertial Oscillations


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Trajectory of inertial oscillations

x = x0 V

f cos(ft + )

y = y0 +V

fsin(ft + )

(x x0)2 + (y y0)

2 =

V

f

2combine and take

the square

R = Vf

...........................................................................................................................................................

......................

......................

V

(x0, y0)

this is the equation ofa circle with radius

R = Vf

1

Tp = 2/f

period of a completecircle is calledinertial period

2


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Coriolis acceleration in 3D - observations of inertial motions

du

dt fv = 0

dv

dt+ fu = 0

equation of inertial oscillations

these describe the unforced motion


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Discretizing the intertial oscillation equations

du

dt fv = 0

un+1 un

t fvn = 0

dv

dt+ fu = 0

vn+1 vn

t+ fun = 0

Euler Method

un+1 un

t fvn+1 = 0

vn+1 vn

t+ fun+1 = 0

Euler Method Implicit

when rigth-hand side isat future time

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2013, A Rotating Framework, 02-Coriolis