Coriolis and Centrifugal Forces

Date : 12 August 1998

This feature I begin discussing dynamics analysis, beginning with apparent forces because we live on a rotating oblate spheroid. Thus, meteorologists speak of differences among inertial and rotating reference frames. I don't necessarily know why they are called frames, but the former is motionless (inert), as if the earth were not rotating, as the latter implies. Though we don't often notice rotational effects in our atmosphere, even considering storm movement and local winds, 2 apparent forces - centrifugal and coriolis - are important on Earth and in our atmosphere. Neither are true forces, but they are called such because they behave as if they were (with this understood).

Centrifugal Force

Centrifugal force is what you experience on a merry-go-round - the tendency for you to continue forward (and off the edge unless you hold on) rather than around. It is a component of Earth's gravity, which is the combined effect of gravitational force and the much smaller centrifugal force. An equation describing gravity is :

g =  Fg + W2 R

Fg : Gravitational force
W : angular (rotation) speed of Earth = 7.292116 × 10-5 radians/sec

as illustrated - the figures to right & equations being from the text Atmospheric Thermodynamics. Boldface variables indicate vector quantities. You may notice that the value for W means that the Earth rotates once every 23 hr, 56 min, 4.09 sec (86164.09 sec) - orbit around the sun accounting for the additional time of a 24-hour day. 2p radians is once around the globe (circle), so :

2p radians/ 86164.09 sec = 7.292116 × 10-5 radians/sec

You may recall that Fg at a location is the sum of gravtitations from all objects according to the Law of Gravitation, close and massive objects influencing this much more than those lighter and further away. Comparing magnitudes of the 2 terms of the above equation for the gravity component perpendicular with Earth's rotation axis (no centripetal acceleration parallel with it) at 35 °N (mks units - please notice that a radian is a (Earth) radius - about 6369900 m) :

gx  =  (9.79747) (cos 35°)  =  8.02562 m/sec2
W2 R  =  (7.292116 × 10-5)2 (6369900) (cos 35°)
          =  2.77462 × 10-4 m/sec2 = .0277462 m/sec2
Fgx  =  gx  - W2 R  =  8.02562 -  .02775  =  7.99787 m/sec2

indicates that the centrifugal term is very small. I.e., spacing of red & blue lines on the diagram above is quite exaggerated.

Coriolis Force

As indicated above, Earth's rotation speed varies latitudinally. Because of this and momentum conservation, horizontally-moving objects slightly curve. A simplified way of thinking of this is if an object moves such that its angular momentum is increased from that of Earth's below, it moves to a location of larger Earth angular momentum, and vice-versa. For example, an object moving northeastward in the Northern Hemisphere increases its angular momentum and is moving toward a region with less angular momentum (planet moves slower toward Pole). Thus it turns southeastward (right) and rises (larger angular momentum further from Earth's center).

Mathematical Description

Perhaps the best description of these apparent forces is mathematical. This (present) section paraphrases that in Holton's An Introduction to Dynamic Meteorology. You can read pages of descriptions and illustrations of the Coriolis Effect which can be stated using a few vector equations as he does. The gist of the mathematical argument is that a (simple) cross product of Earth's rotational axis (vector) and an object's velocity (vector) describes Earth's rotational affects.

The total derivative for an arbitrary vector A in an inertial reference frame (subscript a) is :

daA/dt  =  i dAx/dt  +  j dAy/dt  +  k dAz/dt

and in a rotating frame (no subscript, but primed axis unit vectors and components) is :

dA/dt  =  i' dA'x/dt  +  j' dA'y/dt  +  k' dA'z/dt

dAa/dt  =  dA/dt  +  (di'/dt A'x  +  dj'/dt A'y  +  dk'/dt A'z)
      1               2                                       3

relates these equations. Change of A in the inertial frame (1) equals its change in the rotating frame (2) plus rotation's affect (3). If you imagine the i' unit vector rotating (di'/dt) along the equator (unit vector pointing eastward), you may be able to envision that it turns left as Earth rotates, the cross product W × i' mathematically describing this. Similarly for the j' & k' unit vectors. Thus, the above equation can be written:

dAa/dt  =  dA/dt  +  W × A

which is a general expression relating total derivatives of a property in inertial and rotating reference frames.

An expression for how Earth's rotation affects velocity in the inertial frame is desired. Considering a position vector r and the above equation,

dar/dt  =  dr/dt  +  W × r

Because velocity V (Holton used U for this) is time rate of change of position, dr/dt,

Va  =  V  +  W × r

This mathematically states that velocity the inertial frame equals that in the rotating frame plus rotation affects. Similarly as for r above,

dVa/dt  =  dVa/dt  +  W × Va

Substituting for Va from the equation directly above it,

dVa/dt  =  d/dt(V + W × r)  +  W × (V + W × r)
              =  dV/dt  +  d/dt(W × r)  +  W × V  +  W × (W × r)

You may notice that dW/dt = 0 (Earth rotation is very nearly constant), so using the calculus' chain rule,

d/dt(W × r)  =  dW/dt ×  r  +  W × dr/dt  =  (0) r  +  W × V

You can also calculate (or use the right hand rule ) to see that :

W × (W × r)  =  W × (W × R)  =  - W2 R

as R is defined further above. Thus,

dVa/dt  =  dV/dt  +  2 W × V  -  W2 R
      1               2                3             4

the sought expression. This mathematically expresses that velocity (e.g., wind direction and speed) change in an inertial reference frame (1) equals velocity change in a rotating reference frame (2) plus the Coriolis Effect (3) plus centripetal acceleration (4) (which is negative because it acts the opposite direction of R). Thus, the difference of velocity changes (i.e., turning of moving objects) in these reference frames is the Coriolis Effect and Centrifugal forces, though the absolute velocities greatly differ because our planet rotates @ 1040 mph at the Equator.

Interpretation

Considering the above equation, you should envision that centripetal acceleration always acts directly toward and perpendicular with Earth's rotation axis. Regarding the cross product for Coriolis Effect, because Earth's rotation axis is naturally chosen as the vertical coordinate axis, only the 2 terms involving it are non-zero :

2 W × V = 2 W (- i Vy + j Vx)

This equation is not extremely useful for analysis on Earth's curved surface, but it does clearly illustrate that Coriolis force only turns wind components in or parallel with the Earth's equatorial plane (x-y plane in the inertial frame), none parallel with Earth's rotation axis, as Hess mentions; and that it acts only perpendicular with winds, and does so both horizontally and vertically at a specific location on Earth (where horizontal is tilted with respect to the inertial frame mentioned here). Soon I hope to describe the most typically used meteorological coordinates, which illustrate the idea that the Coriolis force turns winds to the right in the Northern Hemisphere and left in the Southern Hemisphere, as illustrated above - maximum at the Poles, minimum at the Equator, proportional with horizontal wind speed, and much more relevant for large than small-scale wind regimes.

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A quite useful mathematical operation for meteorological analysis is the total derivative. My previous discussion describes advection, which is part of this (and cross products). The total derivative is the time rate of change of a property (T) of an air parcel. Using Cartesian coordinates, this is :

dT/dt  =  T/t  +  T/x dx/dt  +  T/y dy/dt  +  T/z dz/dt

Thus,

dT/dt  =  (T/t)  +  (Vx T/x  +  Vy T/y  +  Vz T/z)
    1                2                                           3

If you read the relevant feature, you may recognize :

Vx T/x  +  Vy T/y  +  Vz T/z  =  V • Ñ T

as advection of T. So another way of writing this equation is :

dT/dt  =  T/t  +  V • Ñ T
    1              2               3

T/t is the local rate of change of T. So if T were temperature, for example, the equation is simply a way of mathematically describing that the total temperature change you experience at a location (1) equals local changes (2), which may be because of solar or infrared energy exchange for example, plus advection (3), which is air transport from another location. If T is a vector quantity, such as wind (V) in this feature's example, total derivative is calculated for each of the vector's scalar components :

dVx/dt  =  Vx/t  +  V • Ñ Vx
dVy/dt  =  Vy/t  +  V • Ñ Vy
dVz/dt  =  Vz/t  +  V • Ñ Vz

because the gradient of a vector is not meaningful.


Text and embedded images are copyright of Joseph Bartlo, though may be used with proper crediting.

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