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A drifting buoy set in motion by strong westerly winds in the Baltic Sea in July 1969. When the wind has decreased the uppermost water layers of the oceans tend to follow approximately inertia circles due to the Coriolis effect. This is reflected in the motions of drifting buoys. In the case there are steady ocean currents the trajectories will become cycloides. The inertia circles are not eddies; a set of buoys close to each other would be co-moving, rather than revolve around each other
Rotating planet
Schematic representation of inertial oscillation. The pattern of motion of the buoy that is being tracked is due to a pattern of dynamics that is present only in the case of a rotating planet. Animation 2 shows a preview. I will first discuss a simpler case, and then I will proceed to how it works for the Earth as a whole.
Also available: the java applet inertial oscillation, a 3D simulation of the physics of inertial oscillation.
Understanding of the physics of inertial oscillation is the key to understanding how the rotation of the Earth affects the oceanic and atmospheric dynamics. Inertial oscillation is the simplest, purest case. Usually the motion of air mass is affected by both pressure gradient and Coriolis effect. Inertial oscillation shows how water mass and air mass move when there is no pressure gradient to begin with, and no buildup of pressure gradient in the course of the motion.
Simpler case as physical model: parabolic dish The dynamics of terrestrial inertial oscillations can be understood by noting the parallels with a simpler case: motion over the surface of a dish with a parabolic cross section.
In the fluid dynamics lab of the MIT Earth, Atmosphere and Planetary sciences faculty students can set up various demonstrations, and one of them is the construction of a parabolic turntable. A platform with a diameter of one meter, with a rim of a couple of centimeters, rotating at a constant angular velocity of about 30 revolutions per minute, is filled with a resin that takes several hours to set. Because the platform is rotating, the resin gets redistributed in such a way that in the final state the surface has a parabolic shape; the center is a centimeter or so deeper than the perimeter. The equilibrium state of the rotating fluid is called 'solid body rotation'. The formal name of the solid that is formed is 'paraboloid of revolution'. After the resin has set the surface is sanded to a smooth finish.
For demonstrations a small disk of dry ice is placed on the parabolic dish. The evaporating carbondioxide formes an air cushion, resulting in very low friction. Also, in demonstrations the parabolic dish must be rotating at exactly the same angular velocity as when it was manufactured.
For demonstrations a small disk of dry ice is placed on the parabolic dish. The evaporating carbondioxide formes an air cushion, resulting in very low friction. Also, in demonstrations the parabolic dish must be rotating at exactly the same angular velocity as when it was manufactured.
The red arrow represents the force of gravity. The green arrow represents the normal force. The blue arrow represents the resultant force. When it was still liquid the resin was in solid body rotation, hence at every distance to the center of rotation the inclination of the surface is such that the resultant force of gravity and the normal force provides the amount of centripetal force that is necessary to co-rotate with the dish. The strength of this centripetal force is exactly proportional to the distance to the axis of rotation.
If the puck is released in such a way that it has a small velocity relative to the dish then it will follow (to a first approximation) the trajectory that is shown on the left side of animation 4. The shape of the trajectory is an ellipse, and the motion is rather like an orbit. The right side depicts the motion as seen from a point of view that is co-rotating with the dish.
Energy conversions: doing work Harmonic oscillation; the restoring force is proportional to the distance to the center. The case of inertial oscillation on a shallow parabolic dish and the rotational-vibrational coupling discussed in the previous article are perfectly analogous. In the case of inertial oscillations the potential energy reaches its maximum at the extremal points. As the object is being drawn closer to the center of rotation the centripetal force is doing work, converting potential energy to kinetic energy. The kinetic energy reaches its maximum at the points where the object is at its closest to the center of rotation
Dissipation of energy I will discuss dissipation of energy now, for that will reveal some interesting aspects. The natural process is: as long as a system can dissipate energy, it will. Consider a situation where the only way to dissipate energy is friction between the surface and the object that is supported by the surface. In the case of inertial oscillation there is dissipation of energy as long as there is a velocity with respect to the rotating parabolic dish. Friction reduces the eccentricity of the orbit. When the orbit has become circular there is still a lot of potential and kinetic energy, but as there is no friction there is no opportunity for energy dissipation.
Animation 8 depicts the state that the system will evolve to. Interestingly, it is an example of equipartition of energy. It is a common property of many kinds of dynamical systems that when the system has reached a final state (no more opportunity for energy dissipation) then the system's total energy will be divided evenly over the available degrees of freedom. Here, the available degrees of freedom are gravitational potential energy and rotational kinetic energy. The final state can be understood as energized degrees of freedom in a state of equilibrium with each other. In the final state the ratio of rotational kinetic energy to gravitational potential energy is 1:1 at every distance to the center of rotation. During manufacture of the parabolic dish, this is the state of equilibrium that the still liquid resin ended in when it reached a state of solid body rotation.
Rotating planet
Schematic representation of inertial oscillation. The pattern of motion of the buoy that is being tracked is due to a pattern of dynamics that is present only in the case of a rotating planet. Animation 2 shows a preview. I will first discuss a simpler case, and then I will proceed to how it works for the Earth as a whole.
Also available: the java applet inertial oscillation, a 3D simulation of the physics of inertial oscillation.
Understanding of the physics of inertial oscillation is the key to understanding how the rotation of the Earth affects the oceanic and atmospheric dynamics. Inertial oscillation is the simplest, purest case. Usually the motion of air mass is affected by both pressure gradient and Coriolis effect. Inertial oscillation shows how water mass and air mass move when there is no pressure gradient to begin with, and no buildup of pressure gradient in the course of the motion.
Simpler case as physical model: parabolic dish The dynamics of terrestrial inertial oscillations can be understood by noting the parallels with a simpler case: motion over the surface of a dish with a parabolic cross section.
In the fluid dynamics lab of the MIT Earth, Atmosphere and Planetary sciences faculty students can set up various demonstrations, and one of them is the construction of a parabolic turntable. A platform with a diameter of one meter, with a rim of a couple of centimeters, rotating at a constant angular velocity of about 30 revolutions per minute, is filled with a resin that takes several hours to set. Because the platform is rotating, the resin gets redistributed in such a way that in the final state the surface has a parabolic shape; the center is a centimeter or so deeper than the perimeter. The equilibrium state of the rotating fluid is called 'solid body rotation'. The formal name of the solid that is formed is 'paraboloid of revolution'. After the resin has set the surface is sanded to a smooth finish.
For demonstrations a small disk of dry ice is placed on the parabolic dish. The evaporating carbondioxide formes an air cushion, resulting in very low friction. Also, in demonstrations the parabolic dish must be rotating at exactly the same angular velocity as when it was manufactured.
For demonstrations a small disk of dry ice is placed on the parabolic dish. The evaporating carbondioxide formes an air cushion, resulting in very low friction. Also, in demonstrations the parabolic dish must be rotating at exactly the same angular velocity as when it was manufactured.
The red arrow represents the force of gravity. The green arrow represents the normal force. The blue arrow represents the resultant force. When it was still liquid the resin was in solid body rotation, hence at every distance to the center of rotation the inclination of the surface is such that the resultant force of gravity and the normal force provides the amount of centripetal force that is necessary to co-rotate with the dish. The strength of this centripetal force is exactly proportional to the distance to the axis of rotation.
If the puck is released in such a way that it has a small velocity relative to the dish then it will follow (to a first approximation) the trajectory that is shown on the left side of animation 4. The shape of the trajectory is an ellipse, and the motion is rather like an orbit. The right side depicts the motion as seen from a point of view that is co-rotating with the dish.
Energy conversions: doing work Harmonic oscillation; the restoring force is proportional to the distance to the center. The case of inertial oscillation on a shallow parabolic dish and the rotational-vibrational coupling discussed in the previous article are perfectly analogous. In the case of inertial oscillations the potential energy reaches its maximum at the extremal points. As the object is being drawn closer to the center of rotation the centripetal force is doing work, converting potential energy to kinetic energy. The kinetic energy reaches its maximum at the points where the object is at its closest to the center of rotation
Dissipation of energy I will discuss dissipation of energy now, for that will reveal some interesting aspects. The natural process is: as long as a system can dissipate energy, it will. Consider a situation where the only way to dissipate energy is friction between the surface and the object that is supported by the surface. In the case of inertial oscillation there is dissipation of energy as long as there is a velocity with respect to the rotating parabolic dish. Friction reduces the eccentricity of the orbit. When the orbit has become circular there is still a lot of potential and kinetic energy, but as there is no friction there is no opportunity for energy dissipation.
Animation 8 depicts the state that the system will evolve to. Interestingly, it is an example of equipartition of energy. It is a common property of many kinds of dynamical systems that when the system has reached a final state (no more opportunity for energy dissipation) then the system's total energy will be divided evenly over the available degrees of freedom. Here, the available degrees of freedom are gravitational potential energy and rotational kinetic energy. The final state can be understood as energized degrees of freedom in a state of equilibrium with each other. In the final state the ratio of rotational kinetic energy to gravitational potential energy is 1:1 at every distance to the center of rotation. During manufacture of the parabolic dish, this is the state of equilibrium that the still liquid resin ended in when it reached a state of solid body rotation.
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