Following article on "Colourful Optics of Nature" in the present series
on natural phenomena, basic science is discussed for natural vertices ranging from bathtub drainage to cyclone and tornado .
Vortex is a generic term for swirling movement. It describes various circular forces that exist in the universe. The drainage of water from a bathtub, kitchen sink or industrial reservoir creates a natural phenomenon of hydraulic vortex (pl. vortices) in the drain.
Vortex similar to the bathtub situation is often encountered in nature on a much larger scale such as atmospheric vertices in cyclone/hurricane/typhoon and tornado.
Physics
The basic science of vortex is discussed for water flow in bathtubs and air circulation in cyclones and tornadoes.
Primary factors influencing vortex movement
Earth’s rotation
The ground in the northern hemisphere rotates counter clockwise as the Earth spins eastward beneath our feet. Everything around us including any fluid (air or water) gyrates at the same time aided by angular momentum. The daily rotation of Earth causes vortices, specially the atmospheric types, to spiral anti-clockwise in the northern hemisphere and clockwise in the southern hemisphere. Does this mean that a storm would have its rotation changed if it manages to cross over the equator? Perhaps, the air currents will not allow the storm to cross over equatorial lines.
Coriolis force
If the Earth did not rotate, air would simply move across the globe blowing straight from the tropics to the poles, or vice versa. But this does not happen. Winds and storms follow curved paths because of the Earth’s daily spin on its invisible north-south axis. This phenomenon is called the Coriolis effect.
If we consider an object on the equator completing a circuit around the Earth in one day, it would have to travel Earth’s circumference at an average speed of ~1000 miles/h. If the object starts from a point longitudinally away from the equator, both the speed and distance covered would decrease and eventually become negligible at the pole.
The development of differential speed by moving away from the equator would affect movement of a fluid that would be deflected from its path as seen by an observer on Earth. Such deflection of the inertia (power of resistance to change of motion) of fluid due to Earth’s rotation is called the Coriolis force which is quite small because of Earth’s very slow one complete rotation per day.
The Coriolis force despite being weak is noticeable for largish events such as atmospheric vortices. However, any deflection of fluid disappears at the equator where the Coriolis force due to Earth’s rotation is negligible.
Rosby number (Ro)
Rosby number is the ratio of inertia to Coriolis force. It determines the forces at play in a given situation whether it is water spinning down a drain or air swirling in a cyclone. A large Ro (>> 1) signifies a system where inertial and centrifugal forces dominate as in tornadoes (Ro~10^3). However, for Ro << 1, motion of a fluid is affected by Coriolis force i.e. rotation of the system.
1. 2. 3. 4. 5. 6. L.Kantha and C.Clayson, Numerical Models of Oceans and Oceanic Processes, Academic Press,2000. 7.Corlios Myths, Am.J.Phys.Vol.62(12), December, p.1063,1994
Specific cases
Vortex in bathtub
Bathtub vortex is characterized by intense axial down-flow. When the cross-section area of the drain hole relative to the bathtub is sufficiently small, the angular momentum dominates. The speed and rate of rotation of water increase as it approaches the hole centre, but decrease progressively with distance from the centre.
At first, the water particles running towards the drain are pushed off to one side because of momentum gathered by other water particles rushing toward the drain at the same time. This deflection along with the principle of conservation of angular momentum sets a chain reaction with water particles spiralling down the plughole. The strong rotating effect ultimately creates a vortex.
For low Ro (~0.1) and bathtub vortex length of 0.1 meters, the Coriolis force is estimated to be very small (< 10^-6) . Being very weak, the Coriolis force can affect the direction of vortex only if the motion of water in the bathtub is less than Earth’s daily rotation of 0.00001 per second. This apparently unrealistic ideal condition has also to be supplemented by elimination of thermal current, vibration and any other external disturbances.
8. 9. 10. 11. 12.
Vortex in Cyclone/Hurricane/Typhoon and Tornado
Terminology
Cyclones, hurricanes and typhoons are all similar weather phenomena of violent storms. They are called differently in different parts of the world. In the South Pacific and Indian oceans including the Bay of Bengal, the term “cyclone” is used (affecting Australia and Bangladesh for examples). The same type of climatic disturbance in the Atlantic and Northeast Pacific is called “hurricane” (occurring in the U.S.), or “typhoon” in the Northwest Pacific (threatening Japan and Philippines).
In contrast, tornado is a violently rotating column of air in contact with the surface. It is often visible as a funnel cloud. For a vortex to be classified as tornado, it must be literally in contact with the ground and the cloud base.
Vortex in cyclone
Cyclones, as the deepest of all low-pressure weather systems, are developed over warm tropical ocean near the equator. Air heated by the sun rises swiftly creating low pressure voids. Cool air rushing in to fill the voids bends inwards because of Earth’s constant turning on its axis (Coriolis force). The inward movement of the wind spirals upwardly faster and faster forming a huge circle that would lead to atmospheric vortex. Water vapour and droplets are sucked up from the ocean surface into the vortex which ends up with devastating effect once enough power is acquired.
Air flowing around a cyclone spins counter-clockwise in the northern hemisphere and clockwise in southern hemisphere (as does the Earth itself). If the Earth did not rotate, the air would flow directly towards the low pressure centre where a calm, cloudless atmosphere called the "eye" exists with light winds and no rain.
Because the Coriolis force initiates and maintains cyclone rotation, such storm could rarely form or move near the equator within ~ 5 degrees where the Coriolis effect is weakest. Cyclones thus chiefly occur in the mid-latitude belts of both hemispheres.
13. 14. 15. 16
Vortex in tornado
Tornadoes begin at low-pressure areas along Earth’s surface that would draw in cooler high-pressure air around them. Once brought in, the cooler air pushes the low-pressure air to higher altitude where it gets hotter and is still forced upwards by the air behind it. This results in the cooler surrounding air, at pressure of as much as 10 % higher than inside a tornado, to rush into creating a tornado even faster, causing the air to rotate even more. Consequently, a tornado picks up speed forming cylindrical vortex. A classic example of tornado development is the collision between the warm moist air from the Gulf of Mexico and the cooler, drier air from the northern Plains and the Rockies.
Tornadoes apparently contain enormous amount of energy and power around 500 million horsepower, mostly as kinetic energy of rotation of air moving upward inside a tornado. They look like giant vortex-funnels that actually descend from the cloud above them, rather than the optical illusion of starting from the ground.
17. 18. 19.
Maths of vortex formation
Vector analysis
It is difficult to convince people of the rotation of bathtub vortex. One mathematical approach would be to consider the equations of fluid flow on a rotating Earth and compute numerically by feeding in all details of flow in a bathtub. This kind of numerical simulation lacks physical insight.
A vector analysis representing magnitude and direction of forces can provide a simplistic way of estimating the relative strengths of forces that act on a fluid forming vortex. Let us denote the rotational velocity of water in the bathtub and Earth’s rotating velocity as ? and ? respectively. The primary forces forming a vortex would include pressure force (Fx) due to the pressure gradient in the fluid, the Coriolis force (Fy) from Earth’s rotation, and the centrifugal force (Fz) due to the circular motion of the fluid in the vortex.
Both Fx and Fz, unlike Fy, are independent of the position of vortex on the Earth. They will point inward (Fx) and outward (Fz) in relation to the vortex. This means that any asymmetry between the behaviour of a vortex in the two hemispheres must be due to the Coriolis effect (Fy).
Based on vector analysis, both Fy and Fz, unlike Fx pointing toward the vortex, would always direct away (outward) from the vortex. Fx needs to be balanced by Fy or Fz or by both to restore balance of forces,
The ratio of the strengths of the Coriolis (Fy) and centrifugal (Fz)forces can be shown to be of the order of the ratio of the rotation rate of the Earth to the rotation rate of the vortex in the bathtub i.e. ?/?. Since the Earth rotates once a day and if it is assumed that the vortex rotates once per second, ?/? would be 0.00001. This implies that the Coriolis force is much smaller than other forces operating on the vortex. It also follows from Fy/Fz ~ ?/? that both these forces would be of the same order of magnitude when the rotation rate of water in the bathtub is slow enough to be equal to Earth’s rotation.
Advection-diffusion equation (ADE)
ADE describes a broad range of natural phenomena, and will be mentioned here very briefly because of its highly complex mathematical nature. ADE depending on application can also be called convection-diffusion equation. It describes mixing of water within a vortex that would involve convection (advection) and diffusion of water. Advection by circulating flow generates spirals of water which are so tightly wound that molecular diffusion becomes an important issue. The interaction between advection and diffusion is subtle and can only be understood through mathematical and computational analysis of ADE.
20. R.Sneider, Mathematical Methods for the Physical Sciences, CambridgeUniversity Press,2004 21. 22. D.Dan et.al.,Solving the advection-diffusion equations, Phys.Rev., Vol.E72, 041909, 2005
Final comments
Vortices in all different scales and intensities are ubiquitous in the Earth’s atmosphere. Although it is a common phenomenon, its structure, formation and dynamics are still not completely understood despite years of intense fascination. Vortices generated in both cyclones and tornadoes are based on a similar principle as bathtub vortex, except air instead of water being the fluid, and, of course, occurring with a much larger motion of the fluid.
Vortex is a generic term for swirling movement. It describes various circular forces that exist in the universe. The drainage of water from a bathtub, kitchen sink or industrial reservoir creates a natural phenomenon of hydraulic vortex (pl. vortices) in the drain.
Vortex similar to the bathtub situation is often encountered in nature on a much larger scale such as atmospheric vertices in cyclone/hurricane/typhoon and tornado.
Physics
The basic science of vortex is discussed for water flow in bathtubs and air circulation in cyclones and tornadoes.
Primary factors influencing vortex movement
Earth’s rotation
The ground in the northern hemisphere rotates counter clockwise as the Earth spins eastward beneath our feet. Everything around us including any fluid (air or water) gyrates at the same time aided by angular momentum. The daily rotation of Earth causes vortices, specially the atmospheric types, to spiral anti-clockwise in the northern hemisphere and clockwise in the southern hemisphere. Does this mean that a storm would have its rotation changed if it manages to cross over the equator? Perhaps, the air currents will not allow the storm to cross over equatorial lines.
Coriolis force
If the Earth did not rotate, air would simply move across the globe blowing straight from the tropics to the poles, or vice versa. But this does not happen. Winds and storms follow curved paths because of the Earth’s daily spin on its invisible north-south axis. This phenomenon is called the Coriolis effect.
If we consider an object on the equator completing a circuit around the Earth in one day, it would have to travel Earth’s circumference at an average speed of ~1000 miles/h. If the object starts from a point longitudinally away from the equator, both the speed and distance covered would decrease and eventually become negligible at the pole.
The development of differential speed by moving away from the equator would affect movement of a fluid that would be deflected from its path as seen by an observer on Earth. Such deflection of the inertia (power of resistance to change of motion) of fluid due to Earth’s rotation is called the Coriolis force which is quite small because of Earth’s very slow one complete rotation per day.
The Coriolis force despite being weak is noticeable for largish events such as atmospheric vortices. However, any deflection of fluid disappears at the equator where the Coriolis force due to Earth’s rotation is negligible.
Rosby number (Ro)
Rosby number is the ratio of inertia to Coriolis force. It determines the forces at play in a given situation whether it is water spinning down a drain or air swirling in a cyclone. A large Ro (>> 1) signifies a system where inertial and centrifugal forces dominate as in tornadoes (Ro~10^3). However, for Ro << 1, motion of a fluid is affected by Coriolis force i.e. rotation of the system.
1. 2. 3. 4. 5. 6. L.Kantha and C.Clayson, Numerical Models of Oceans and Oceanic Processes, Academic Press,2000. 7.Corlios Myths, Am.J.Phys.Vol.62(12), December, p.1063,1994
Specific cases
Vortex in bathtub
Bathtub vortex is characterized by intense axial down-flow. When the cross-section area of the drain hole relative to the bathtub is sufficiently small, the angular momentum dominates. The speed and rate of rotation of water increase as it approaches the hole centre, but decrease progressively with distance from the centre.
At first, the water particles running towards the drain are pushed off to one side because of momentum gathered by other water particles rushing toward the drain at the same time. This deflection along with the principle of conservation of angular momentum sets a chain reaction with water particles spiralling down the plughole. The strong rotating effect ultimately creates a vortex.
For low Ro (~0.1) and bathtub vortex length of 0.1 meters, the Coriolis force is estimated to be very small (< 10^-6) . Being very weak, the Coriolis force can affect the direction of vortex only if the motion of water in the bathtub is less than Earth’s daily rotation of 0.00001 per second. This apparently unrealistic ideal condition has also to be supplemented by elimination of thermal current, vibration and any other external disturbances.
8. 9. 10. 11. 12.
Vortex in Cyclone/Hurricane/Typhoon and Tornado
Terminology
Cyclones, hurricanes and typhoons are all similar weather phenomena of violent storms. They are called differently in different parts of the world. In the South Pacific and Indian oceans including the Bay of Bengal, the term “cyclone” is used (affecting Australia and Bangladesh for examples). The same type of climatic disturbance in the Atlantic and Northeast Pacific is called “hurricane” (occurring in the U.S.), or “typhoon” in the Northwest Pacific (threatening Japan and Philippines).
In contrast, tornado is a violently rotating column of air in contact with the surface. It is often visible as a funnel cloud. For a vortex to be classified as tornado, it must be literally in contact with the ground and the cloud base.
Vortex in cyclone
Cyclones, as the deepest of all low-pressure weather systems, are developed over warm tropical ocean near the equator. Air heated by the sun rises swiftly creating low pressure voids. Cool air rushing in to fill the voids bends inwards because of Earth’s constant turning on its axis (Coriolis force). The inward movement of the wind spirals upwardly faster and faster forming a huge circle that would lead to atmospheric vortex. Water vapour and droplets are sucked up from the ocean surface into the vortex which ends up with devastating effect once enough power is acquired.
Air flowing around a cyclone spins counter-clockwise in the northern hemisphere and clockwise in southern hemisphere (as does the Earth itself). If the Earth did not rotate, the air would flow directly towards the low pressure centre where a calm, cloudless atmosphere called the "eye" exists with light winds and no rain.
Because the Coriolis force initiates and maintains cyclone rotation, such storm could rarely form or move near the equator within ~ 5 degrees where the Coriolis effect is weakest. Cyclones thus chiefly occur in the mid-latitude belts of both hemispheres.
13. 14. 15. 16
Vortex in tornado
Tornadoes begin at low-pressure areas along Earth’s surface that would draw in cooler high-pressure air around them. Once brought in, the cooler air pushes the low-pressure air to higher altitude where it gets hotter and is still forced upwards by the air behind it. This results in the cooler surrounding air, at pressure of as much as 10 % higher than inside a tornado, to rush into creating a tornado even faster, causing the air to rotate even more. Consequently, a tornado picks up speed forming cylindrical vortex. A classic example of tornado development is the collision between the warm moist air from the Gulf of Mexico and the cooler, drier air from the northern Plains and the Rockies.
Tornadoes apparently contain enormous amount of energy and power around 500 million horsepower, mostly as kinetic energy of rotation of air moving upward inside a tornado. They look like giant vortex-funnels that actually descend from the cloud above them, rather than the optical illusion of starting from the ground.
17. 18. 19.
Maths of vortex formation
Vector analysis
It is difficult to convince people of the rotation of bathtub vortex. One mathematical approach would be to consider the equations of fluid flow on a rotating Earth and compute numerically by feeding in all details of flow in a bathtub. This kind of numerical simulation lacks physical insight.
A vector analysis representing magnitude and direction of forces can provide a simplistic way of estimating the relative strengths of forces that act on a fluid forming vortex. Let us denote the rotational velocity of water in the bathtub and Earth’s rotating velocity as ? and ? respectively. The primary forces forming a vortex would include pressure force (Fx) due to the pressure gradient in the fluid, the Coriolis force (Fy) from Earth’s rotation, and the centrifugal force (Fz) due to the circular motion of the fluid in the vortex.
Both Fx and Fz, unlike Fy, are independent of the position of vortex on the Earth. They will point inward (Fx) and outward (Fz) in relation to the vortex. This means that any asymmetry between the behaviour of a vortex in the two hemispheres must be due to the Coriolis effect (Fy).
Based on vector analysis, both Fy and Fz, unlike Fx pointing toward the vortex, would always direct away (outward) from the vortex. Fx needs to be balanced by Fy or Fz or by both to restore balance of forces,
The ratio of the strengths of the Coriolis (Fy) and centrifugal (Fz)forces can be shown to be of the order of the ratio of the rotation rate of the Earth to the rotation rate of the vortex in the bathtub i.e. ?/?. Since the Earth rotates once a day and if it is assumed that the vortex rotates once per second, ?/? would be 0.00001. This implies that the Coriolis force is much smaller than other forces operating on the vortex. It also follows from Fy/Fz ~ ?/? that both these forces would be of the same order of magnitude when the rotation rate of water in the bathtub is slow enough to be equal to Earth’s rotation.
Advection-diffusion equation (ADE)
ADE describes a broad range of natural phenomena, and will be mentioned here very briefly because of its highly complex mathematical nature. ADE depending on application can also be called convection-diffusion equation. It describes mixing of water within a vortex that would involve convection (advection) and diffusion of water. Advection by circulating flow generates spirals of water which are so tightly wound that molecular diffusion becomes an important issue. The interaction between advection and diffusion is subtle and can only be understood through mathematical and computational analysis of ADE.
20. R.Sneider, Mathematical Methods for the Physical Sciences, CambridgeUniversity Press,2004 21. 22. D.Dan et.al.,Solving the advection-diffusion equations, Phys.Rev., Vol.E72, 041909, 2005
Final comments
Vortices in all different scales and intensities are ubiquitous in the Earth’s atmosphere. Although it is a common phenomenon, its structure, formation and dynamics are still not completely understood despite years of intense fascination. Vortices generated in both cyclones and tornadoes are based on a similar principle as bathtub vortex, except air instead of water being the fluid, and, of course, occurring with a much larger motion of the fluid.
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