EOSC 512 2019

4 Rotation Coordinate Systems and the Equations of Motion

4.1 Rates of change revisited

We have now derived the Navier-Stokes equations in an inertial (non-accelerating) frame of reference for which

Newton’s third law is valid. However, in oceanography and meteorology it is more natural to put ourselves in

an earth-fixed coordinate frame that is rotating with the planet (Only natural – this is the frame from which we

observe motions in the ocean and the atmosphere). However, rotating reference frames are not inertial. It is

therefore nexessary to examine how the equations of motion must be altered to take this into account.

Consider two observers. one fixed in inertial space and one fixed in our rotating frame. Consider how the rate of

change of a vector fixed in length will appear to both observers if the vector rotates with the rotation of the Earth.

Where ⌦ is the angular velocity of the rotating frame. The earth has a rotational period of one day, so ⌦ =

7.29⇥ 10�5 s�1.

This introduces three interesting questions:

1. What rate of change of i (DiDt ) is seen by the inertial observer? By the rotating observer?

2. Now let i be an arbitrary vector ~A = Aj ij . How are the rates of change of this vector seen by the inertial and

rotating observer related? That is, how does⇣

D ~ADt

⌘

inertialrelate to

⇣D ~ADt

⌘

rotating?

3. Now choose ~A to be the velocity seen in the inertial frame (~u)inertial. How is the rate of change of the inertial

velocity as seen in inertial space related to the acceleration as seen in the rotating coordinate frame?

We will answer these questions indiviually below:

1. Consider the motion of the unit vector i in a time �t. It will swing around the rotation axis under the

unfluence of the rotation, moving at right andles to itself and to the rotation vector as shown:

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The di↵erence between the two vectors, �~i can be approximated – for small �t – as the chord generated in

that time between the perpendicular distance from the rotation axis to the tip of the unit vector at t = t0 &

the perpendicular distance from the rotation axis to the tip of the unit vector at t = t0+�t. Top-down view:

Here, the radius of the circle traced out by the tip of the unit vector is r = sin����i��� and the angle of swing is

the product of the angular velocity of rotation and the “swinging time”, ✓ = ⌦�t. Thus, the chord length is

r✓ =���i��� sin�⌦�t.

The direction of �~i is perpendicular to both the unit vector and the rotation vector since the change can only

alter its direction and not its length. Thus, the change in the unit vector in time �t is:

~�i =���i��� sin�⌦�t

~⌦⇥ i���~⌦⇥ i���

(4.1)

Where���i��� sin�⌦ =

���~⌦⇥ i��� by the definition of the cross product. This leaves ~�i = �t⌦⇥ i. Dividing by �t

and taking the limit �t ! 0, Di

Dt

!

inertial

= ~⌦⇥ i (4.2)

This is the rate of change of the unit vector i seen by the inertial observer! Notice that if the observer is

rotating with ⌦, the see no change at all, so:

Di

Dt

!

rotating

= 0 (4.3)

2. Let ~A be represented as three components along the coordinate axes with unit vectors i1, i2, and i3. Assuming

the coordinate axes are still rotating with angular velocity ⌦.

The individual component of ~A along each coordinate axis is the shadow of the vector cast along that axis. It

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is a scalar whose value and rate of change is seen the same by both the inertial and rotating observers. The

inertial observer also sees the rate of change of the unit vectors. Thus, D ~A

Dt

!

inertial

=DAi

Dtii +Ai

Di

Dt

=DAi

Dtii +Ai

⇣⌦⇥ i

⌘

=DAi

Dtii + ⌦⇥ ~A

(4.4)

Physically, DAiDt ii is the rate of change of ~A seen by the rotating obeserver. He/she is only aware of the increase

or decrease of the individual scalar coordinates along the coordinate unit vectors that the observer sees as

constant. This can be interpreted as a stretching of the components of ~A. ⌦⇥ ~A is an additional component

of the rate of change of ~A seen only by the inertial observer. This is the swinging of ~A due to the rotation.

Together, D ~A

Dt

!

inertial

=

D ~A

Dt

!

rotating

+ ⌦⇥ ~A (4.5)

3. Taking the previous equation and using ~x as the position of the fluid element,

✓D~x

Dt

◆

inertial

=

D ~Ax

Dt

!

rotating

+ ⌦⇥ ~x (4.6)

The derivative of position with respect to time is velocity, so:

~uinertial = ~urotating + ⌦⇥ ~x (4.7)

Similarly, if we choose ~A to be the velocity seen in the inertial frame ~uinertial,✓D~uinertial

Dt

◆

inertial

=

✓D~uinertial

Dt

◆

rotating

+ ~⌦⇥ ~uinertial

=

✓D

Dt

⇣~urotating + ~⌦⇥ ~x

⌘◆

rotating

+ ~⌦⇥

⇣~urotating + ~⌦⇥ ~x

⌘

=

✓D~urotating

Dt

◆

rotating

+ ~⌦⇥

✓D~x

Dt

◆

rotating

+ ~⌦⇥ ~urotating + ~⌦⇥

⇣~⌦⇥ ~x

⌘

✓D~uinertial

Dt

◆

inertial

=

✓D~urotating

Dt

◆

rotating| {z }1

+2~⌦⇥ ~urotating| {z }2

+ ~⌦⇥

⇣~⌦⇥ ~x

⌘

| {z }3

(4.8)

1 ⌘ The acceleration that an observer in the rotating frame would see.

2 ⌘ The coriolis acceleration.

3 ⌘ The (familiar?) centripetal accerlation. This can be written as the gradient of a scalar, ~⌦⇥

⇣~⌦⇥ ~x

⌘=

�12r (⌦⇥ ~x)2. This is the acceleration an object feels resulting from the change of direction (but not the

change in speed!) associated with uniform circular motion. Has magnitude |⌦|2r where r is the distance

between the rotation axis and the tip of the vector. It is directed toward the centre of the circle (hence

centripetal!).

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4.2 The Coriolis acceleration/force

We have just seen that the acceleration in the inertial frame is related to the acceleration that an observer would see

in the rotating frame with two additional contributions: The centripetal accerlation (acceleration directed towards

the centre of the circle traced out by the tip of the rotating vector) & the Coriolis acceleration.

The so-called Coriolis acceleration acts perpendicular to the velocity as seen in the rotating frame. It represents

the acceleration due to the swinging of the velocity vector by the rotation vector. A factor of two enters because

the rotation vector swings the velocity ~urotating around, but also the velocity ~⌦⇥~x which is not seen in the rotating

frame. This term will increase if the position vector ~x increases, giving rise to a second factor of ~⌦⇥ ~urotating in the

acceleration.

As we will see, the Coriolis acceleration is a dominant (lead order) term in the dynamics of large-scale, low frequency

motion in the atmospheric and oceans. This includes general banded atmospheric circulations such as trade winds

and westerlies and large-scale ocean gyre circulations.

The Coriolis acceleration, like all accelerations, is produced by a force in the direction of the acceleration.

First we assume we have uniform velocity seen in the rotating system (which you will note is counterclockwise

rotation). This implies that there will be no acceleration perceived in the system. In order for the flow to continue

on a straight path as in the figure above, a force must be applied from the right.

If that force is removed, the fluid element can no longer continue in a straight line and will veer to the right. This

gives rise to the sensation that a “force” is acting to push the fluid to the right. Of course, since there are no forces

in the inertial frame, then there are no acceleration in the inertial frame. For this reason, both the centripedal force

and the Coriolis force are sometimes called pseudoforces as they are experienced only in the frame of reference of

the body undergoing circular motion.

It should be emphasized that there is nothing fictitious about these forces or accelerations – they are real – they

are just not recognized as acceleration by an observer in the rotating frame.

An alternative, more heurisitic way to think of the Coriolis e↵ect is the tendency for objects in motion (viewed in

a rotating frame of reference) to follow curved paths. They deflect to the right for counter-clockwise rotation.

Consider an observer on Earth watching the motion of a fluid element in the atmosphere.

Because the Earth is rotating, points on the Earth’s surface travel eastward with a speed given by the Earth’s

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angular velocity. Further, because the Earth is spherical, the distance a point travels in one day at low latitudes

is bigger compared to higher latitudes. As a results, the eastward velocity of a point on the Earth’s surface varies

with latitude: Fast at the equator and slower as you move towards the poles.

Example 4.1. If a fluid parcel at the equator has a northward inertial velocity and no other forces acting, it also

has an eastward component of velocity of approximately 444 m/s. To a rotating observer at 49°, it also has an

eastward component of velocity due to di↵erence in the angular velocity between the equator and 49°.

To the observer at 49°, in some time �t, the parcel has moved north, but also eastwards by an amount (444 �

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300) m/s�t. That is, the observer at 49° sees the parcel travel a curved path that is deflected to the right of its

initial trajectory northwards.

We can heuristically think of this as arising because the parcel is moving eastward faster than the ground over

which it is flying as it moves northward (in a sense, it gets ahead of the planet).

You can extend this argument to arrive at the rule of thumb results that due to the Coriolis e↵ect:

• In the northern hemisphere, an object in motion will get deflected to the right.

• In the souther hemisphere, an object in motion will get deflected to the left.

You can remember this with the following picture. ⇥ denotes the starting location, the dashed arrow in the initial

trajectory and the solid arrows are the curved trajectories observed in the rotating frame. It is this e↵ect that

results in the deflection of the large-scale winds on Earth from N-S (that would results from the di↵erential heating

at the equator vs the poles in the case of a non-rotating planet) to the E-W associated with the NE/SE trade winds,

and the mid-latitude westerlies and the polar easterlies. In the absence of rotation:

When the earth starts to rotate:

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4.3 The Navier-Stokes Equations in a rotating frame

We are now in a position to rewrite the Navier-Stokes equations in a rotating frame and we will rarely look back.

Going forward, we will always use the rotating frame, and (unless otherwise stated) the velocities we use will

be those observed from the rotating frame. And so we can dispense with the cumbersome (quantity)rotating

demarcation. We we ever wish to use an inertial frame, we need only to set ⌦ = 0 in our set of rotating equations.

We will return to the general vector(-ish) representation of the Navier-Stokes equations for a fluid (equation 3.44).

We will substitute D~uDt (remembering that this is now equivalent to our clunky

⇣D~uinertial

Dt

⌘

inertial) in terms of the

position, velocity, and acceleration vectors, seen in the rotation frame. This will require a bit of massaging.

⇢D~u

Dt+ ⇢ (2⌦⇥ ~u) = ⇢

0

B@~F +r

⇣~⌦⇥ ~x

⌘2

2

1

CA

2

�rP + µr2~u| {z }a

+ (�+ µ)r (r · ~u)| {z }b

+ (r�) (r · ~u)| {z }c

+ ii2eij@µ

@xj| {z }d

(4.9)

Where D~uDt is the acceleration that an observer in the rotating frame would see, 2⌦⇥ ~u is the Coriolis acceleration,

andr(~⌦⇥~x)2

2 is the centripedal acceleration moved to the RHS. Some notes:

• The spatial gradient are the same in both frames. Notice that the pressure gradient in unaltered!

• The viscous terms a , b , c , & d are each proportional to gradient of the rate of strain tensor or the

divergence of the velocity –each of which is independent of rotation. So these terms are also invariant between

the two frames. We often assume that viscosity is constant, so we can set c = d = 0.

• We’ve moved the centripetal acceleration to the RHS where it can now be interpreted as the cenripetal force.

This is convenient because the body force we most frequently encounter is due to gravity, which can also be

written as the gradient of a scalar (~Fg = �r� where � is the gravitational potential).

Assuming this is the only body force acting (a good assumption for the types of flows we will consider) and assuming

constant viscosity coe�cients (also usually a good assumption), the Navier-Stokes equations in the rotating frame

simplify to:

⇢D~u

Dt+ ⇢ (2⌦⇥ ~u) = �⇢r

✓��

1

2

⇣~⌦⇥ ~x

⌘2◆�rP + µr2~u+ (�+ µ)r (r · ~u) (4.10)

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Where the combination of the gradient of the gravitation and centrifual potential is what we experience as gravity.

(So ~g = �r

✓��

12

⇣~⌦⇥ ~x

⌘2◆). This gives us a working form of the Navier-Stokes equations valid for an observer

in the rotating frame!

⇢D~u

Dt+ ⇢ (2⌦⇥ ~u) = ⇢~g �rP + µr2~u+ (�+ µ)r (r · ~u) (4.11)

We find that this will be applicable to most atmospheric and oceanic flows. Note that our equation for the

conservation of mass is the same in the rotating frame, since it involves only the divergence of the velocity and the

rate of change of a scalar.

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