A Summary of Some Important Points about the Coriolis Force/Mass Introduction Newton s Second Law applied to fluids (also called the Navier-Stokes Equation) in an inertial, or absolute that is, unaccelerated, reference frame is: (1) D a U a -------------- 1 -- p + g+ F ρ fr where D a U a / is the acceleration of a fluid parcel as observed in the absolute frame of reference, and the three terms on the right-hand side (rhs) are the pressure-gradient force/mass, the force/mass of the earth s gravity (also misleadingly called the acceleration of gravity), and the force/mass due to friction. (Remember that the acceleration here is defined as the rate at which the parcel s absolute velocity, U a,is observed to change with respect to time, where U a is the parcel s velocity as observed in the absolute frame of reference.) Eq. (1) says simply that if the (vecto forces acting on a parcel are added up, the net force acting on each unit of the parcel s mass equals the observed acceleration of the parcel. That was Newton s great insight, embodied in his Second Law. Now, the forces acting on a fluid parcel are independent of what any observer of the parcel s motion might happen to be doing. There might be no observer at all, or the observer might be in free fall, or standing nearby watching the fluid, or moving through space with the center of mass of the universe (whatever that means) the forces would still be doing exactly the same thing: the pressure-gradient force would be pushing just as hard toward lower pressure; gravity would be pulling just as hard toward the center of the earth s mass; and friction would still be doing it s thing (more on that later, probably). In contrast, the observed acceleration of the parcel would depend on whether the observer were accelerating or not. In fact, as a special case, if the observer and the parcel were accelerating in exactly the same way, the observer wouldn t observe any acceleration at all! (Similarly, if the observer and the parcel had the same velocity, then the observer wouldn t observe any motion, either.) It follows that, in a noninertial frame of reference (that is, one in which the observer is accelerated), the rhs of Eq. (1) would remain unchanged, but the left-hand side (lhs), which we d then write as /, would not be the same. Obviously, Eq. (1) would no longer be an equation. Page 1
This state of affairs is unacceptable to us! We want to be able to use Newton s Second Law to explain observed accelerations and predict future motions. To satisfy the spirit of Newton s Second Law, in which the sum of forces/mass equals the observed acceleration, we have to add one or more fictitious or apparent forces to the rhs of Eq. (1) to account for the apparent discrepancy: (2) -------- 1 -- p + g+ F ρ fr + [Note that not only / but also U Dr/ (where r is the vector position of the parcel) is observed from the perspective of the new reference frame.] The Apparent Forces in a Frame of Reference Rotating with the Earth The apparent force(s)/mass depend on the acceleration of the reference frame. In the case of a frame of reference rotating with the earth, that acceleration is easy to specify it s just uniform circular motion, because every part of the rotating earth moves in a circle at a steady speed (albeit at different linear speeds depending on distance from the axis of rotation). The acceleration of anything in uniform circular motion is centripetal that is, toward the center of the circle, or in the case of a rotating body such as the earth, toward the axis of rotation. The magnitude of the acceleration is Ω 2 R, where Ω is the angular speed of rotation and R is the distance from the axis of rotation. (Note that since R differs from one part of the earth to another, the magnitude of the acceleration of the reference frame differs from place to place, as does the direction.) Bluestein, Volume I stated that in a rotating reference frame, for any vector s, (3) D a s ---------- Ds ------- + Ω s That is, the rate at which the vector s is observed to change w/r/t time in an absolute reference frame is equal to the rate at which it is observed to change in a rotating reference frame plus a term that accounts for the rotation of the reference frame. Since s can be any vector, Eq. (3) would apply to the position vector of a fluid parcel, r: (4) D a r --------- Dr ------ + Ω r apparent force(s)/mass But D a r/ is U a, the parcel s velocity in an absolute reference frame; Dr/ is U, the parcel s velocity relative to the earth; and Ω r is just the velocity of the frame Page 2
of reference at the spot where the fluid parcel happens to be located. In other words, (5) U a U + Ω r Hence, when we apply Eq. (3) to U a, we have to substitute Eq. (5) for U a to get the information we really want, which is the rate at which the velocity, observed in the rotating frame, itself changes with respect to time in the rotating frame: (6) D a U a -------------- a ----------- + Ω U a D( U + Ω -------------------------------- + Ω ( U + Ω D( Ω -------- + --------------------- + Ω U + Ω ( Ω -------- DΩ + -------- r + Ω ------ Dr + Ω U + Ω ( Ω The last step above invoked the product rule of calculus applied to a vector cross-product. Since the earth s angular velocity doesn t change direction or magnitude in either frame of reference, it follows that the third term on the rhs above is zero. Also, since Dr/ by definition is just U, we have: (7) D a U a -------------- -------- + Ω U + Ω U + Ω ( Ω -------- + 2Ω U + Ω ( Ω Ω r is the velocity of the rotating frame of reference itself at the location of the parcel. This velocity is oriented parallel to the earth s surface, eastward, which is perpendicular to both the earth s axis of rotation and r. Hence, Ω ( Ω must lie in the plane perpendicular to the axis of rotation but directed toward the axis, perpendicular to Ω r. If R is a position vector pointing from the axis of rotation to the parcel, perpendicular to the axis of rotation, then Ω ( Ω can be written as ( R R)Ω 2 R ( R R)Ω 2 rcosϕ, where R/R is a unit vector pointing outward Page 3
from the axis of rotation toward the parcel, and ϕ is the latitude. Hence, (8) D a U a -------------- -------- + 2Ω U R --- 2 Ω rcosϕ R Substituting Eq. (8) into Eq. (1) and solving for / tells us what the apparent forces in Eq. (3) have to be for the frame of reference rotating with the earth: (9) -------- 1 R -- p g F ρ fr 2Ω U --- 2 + + + Ω rcosϕ R This is Newton s 2nd Law applied in a frame of reference rotating with the earth. The apparent force/mass ( R R)Ω 2 rcosϕ is a centrifugal force/mass, directed outward from the axis of rotation. Its magnitude depends only on the angular speed of the earth, Ω; the distance of the parcel from the center of the earth, r; and the parcel s latitude, ϕ. The apparent force/mass 2Ω U is the Coriolis force. It is directed perpendicular to both the parcel s velocity relative to the earthand to the axis of rotation (which means that it lies in a plane perpendicular to the axis of rotation). It s magnitude depends on the angular velocity of the earth and on the magnitude of the component of the parcel s velocity lying in a plane perpendicular to the earth. (The component of velocity that doesn t contribute to the Coriolis force is parallel to the axis of rotation.) Interpretations of the Coriolis Force Given that the Coriolis force applies only to the component of velocity lying in a plane perpendicular to the axis of rotation, we can give the Coriolis force a more physically intuitive feel by considering two types of earth-relative motion separately: motion toward or away from the axis, and motion in the same sense as the motion of parts of the earth itself, namely east/west (also called tangential motion). (1) Motion toward/away from the earth s axis of rotation. Consider an object (such as a fluid parcel) moving away from the axis of rotation but not east/west relative to the earth (that is, not tangentially). (Of course, in an absolute frame of reference, this object would be moving tangentially, because the earth s surface at the object s location does and the object isn t moving Page 4
tangentially relative to the earth s surface). A short time later, the object has moved farther from the axis of rotation, to a spot where the frame of reference itself is moving faster tangentially because the speed of uniform circular motion which is what the frame of reference is doing increases linearly with distance from the axis of rotation. However, the absolute tangential motion of the object hasn t changed, so it falls behind the rotating frame of reference and appears to be moving westward relative to the rotating frame. Since the object had no tangential motion relative to the rotating frame initially, clearly the object s relative velocity has changed. Newton s Second Law requires that a force must act to cause the observed change. This force is the Coriolis force, which appears to deflect the object westward when it was initially moving radially outward from the axis of rotation. The opposite happens when the object moves toward the axis of rotation it finds itself moving eastward relative to the rotating frame of reference because the tangential motion of the frame itself decreases with decreasing distance from the axis. As a result, from the point of view of the rotating frame, the object appears to deflect eastward as a result of its motion toward the axis of rotation. These behaviors are captured by the principle of conservation of angular momentum in the absence of torques (that is, forces that act in the tangential direction), the product of the absolute tangential velocity and the distance from the axis of rotation is conserved (that is, remains constant): D a ( Ru a ) 0 where u a is the absolute tangential component of velocity. In turn, u a can be written as the sum of the speed of the rotating reference frame (which moves entirely tangentially) and the tangential component of velocity relative to the rotating reference frame: u a ΩR + u. Hence: D a ( Ru a ) ---------------------- R D a u a D a R ------------- + u a ----------- R D a ------------------------------ ( ΩR + u) ( ΩR + u) D a + ----------- R RΩ D a ----------- R R D a ---------- u ( ΩR + u) D a + + ----------- R RΩ D a ----------- R R Du ------- ( ΩR + u) D a + + ----------- R 0 Note that the rate at which u changes w/r/t time will be the same in a rotating reference frame as in an absolute one, which is why we can replace D a u/ with Du/ in the last step above. Solving for Du/, the rate at which the relative tangential Page 5
component of velocity changes w/r/t time, and replacing D a R/ with the symbol U R (the component of velocity normal to the axis of rotation, which is the same in both the absolute and rotating reference frames) gives: Du ------- uu R ----------- 2ΩU R R This says that if the object has any component of its motion toward or away from the axis of rotation (that is, U R 0 ), then the object s tangential (east/west) velocity relative to the rotating frame of reference will change. In particular, if the object isn t already moving eastward or westward (u 0) and U R is positive (that is, away from the axis of rotation), then u will decrease (that is, it will begin moving westward). The opposite statements apply if U R is negative (i.e., toward the axis of rotation) the object will begin moving eastward. The second term on the rhs above is the Coriolis force/mass. In both cases, the Coriolis force appears to act like a deflecting force: radially outward moving objects will appear to deflect westward (from an absolute point of view, slowing down tangentially to conserve angular momentum); radially inward-moving objects will appear deflect eastward ( speeding up tangentially to conserve angular momentum). Sense of rotation U R Coriolis force (tries to deflect object eastward) (radially inward) U R (radially outward) Coriolis force (tries to deflect object westward) As viewed from rotating reference frame. Page 6
(2) Tangential motion relative to the rotating reference frame. Consider an object moving tangentially relative to the earth but not radially (that is, moving east/west but not toward or away from the axis of rotation). In this case, u 0 that is, the object is moving tangentially faster or slower than the frame of reference itself is (depending on the sign of u: u > 0 that is, eastward means faster; u < 0 that is, westward means slower than the earth itself at that point). Uniform circular motion of radius R requires a centripetal force with constant magnitude. To move in a circle with given radius, R, it takes a centripetal force of 2 magnitude u a R. For an object not moving tangentially relative to the rotating reference frame (that is, u 0), u a RΩ, and so the centripetal force/mass 2 2 required is u a R Ω R, whereas for an object moving tangentially relative to the rotating reference frame, ua RΩ + u, and so the centripetal force/mass that would be required is Ω 2 R + 2Ωu + u 2 R. This means that there may be either too much or too little centripetal force/mass (depending on the sign and magnitude of u) actually provided to keep the object moving in a circle of radius R. For example, if u > 0, then the object moves tangentially faster than the frame of reference, and more centripetal force/mass than is actually present would be required to keep it moving in a circle of radius R. The inadequate centripetal force acting on the object would cause the object to move in a circle of somewhat greater radius than R, which means that it would move outward radially relative to the rotating frame of reference. If it had no radial motion to start with, this would certainly appear as a change in it s velocity in particular, there would be a non-zero radial acceleration, R /. To satisfy Newton s Second Law, this radial acceleration would require a radial force to account for it. From the point of view of the rotating frame: R ------------ C e + Ω 2 R + 2Ωu + u 2 R where C e Ω 2 R is the centripetal force/mass that keeps the frame of reference in uniform circular motion. (In the case of the earth, gravity moderated by any normal force exerted by the earth to oppose part of gravity, provides the necessary centripetal force). The second term on the rhs is a centrifugal force associated with the rotation of the frame of reference, an apparent force that appears exactly to balance the centripetal acceleration for stationary objects. The third and fourth terms can be though of as additional centrifugal forces/mass due to the object s tangential motion relative to the earth they, like the second term, are apparent forces that we have to invent to satisfy Newton s Second Law in the rotating reference frame. Page 7
The third term on the rhs is the Coriolis force/mass. It says that if an object moves eastward relative to the earth (u > 0), the object will experience an apparent force directed radially outward. The opposite applies to an object that moves westward (u < 0). In both cases, the Coriolis force appears to act like a deflecting force: eastward-moving objects will appear to deflect outward from the axis of rotation due to an excess of centrifugal force (beyond what is needed to keep it moving in a circle of radius R); westward-moving objects will appear to fall inward toward the axis of rotation, due to a deficit of centrifugal force. Sense of rotation ui (westward) ui (eastward) Coriolis force (tries to deflect object radially inward) Coriolis force (tries to deflect object radially outward) As viewed from rotating reference frame. Page 8