## Geocentric Rankine Vortex

This post examines the proposition that the Earth is at the centre of a Rankine Vortex that generates the centripetal force that is commonly known as Gravity.

The Rankine vortex model is an attempt to describe the velocity profile through vortices in real, viscous, fluids.

It is named after its creator, William John Macquorn Rankine.

A swirling flow in a viscous fluid is characterized by a forced vortex in the central core, surrounded by a free vortex.

The Rankine vortex best describes this phenomenon.

http://en.wikipedia.org/wiki/Rankine_vortex

Atmospheric vortices typically consist of an outer circulatory whirl of irrotational (or potential motion) enclosing a very small inner core of solid rotation.

The presence of the solid rotational core is physically necessary because a pure circulatory or potential vortex would require a velocity V of infinity at the core where r = 0, and so, in nature, some adjustment is necessary.

This problem of infinity is avoided by a core having solid-like rotation, since then the velocity drops to zero at the central axis where r = o as required.

The two flow systems together are often called a Rankine combined vortex

The outer circulatory whirl of an irrotational “free vortex” enclosing a very small inner core of a solid-like rotational “forced vortex” can be clearly seen in the following images of Rankine vortices.

North pole Rankine vortex on the planet Saturn

The air flow from the wing of an agricultural aircraft

Forced Rotational Vortex

In a forced vortex the fluid rotates as a solid body [there is no shear].

The motion can be realized by placing a dish of fluid on a turntable rotating at ω radian/s; the fluid has vorticity of 2ω everywhere, and the free surface (if present) is a paraboloid

Free Irrotational Vortex

When fluid is drawn down a plug-hole, one can observe the phenomenon of a free irrotational vortex. The tangential velocity [v] varies inversely as the distance [r] from the centre of rotation, so the angular momentum [rv] is constant.

In non-technical terms, the fluid near the centre of the vortex circulates faster than the fluid far from the centre. The speed along the circular path of flow is held constant or decreases as you move out from the centre. At the same time the inner streamlines have a shorter distance to travel to complete a ring.

Imagine a leaf floating in a free irrotational vortex.
The leaf’s tip points to the centre and the blade straddles multiple streamlines.
The outer flow is slow in terms of angle traversed and it exerts a backwards tug on the base of the leaf while the faster inner flow pulls the tip forwards.
The drag force opposes rotation of the leaf as it moves around the circle.

Vortex Centripetal Force

Vortices are renowned for generating centripetal force.

The Wikipedia introductory paragraph to vortices [in 2008] described the ability of a vortex “to suck everything within the fluid toward its centre”.

A vortex [plural vortices] is a rapidly spinning, circular or spiral flow of fluid around a central axis. The swirling motion tends to suck everything within the fluid toward its centre.

In 1966 Arthur C. Schouw patented a centripetal technique for separating particles from a liquid.

CENTRIPETAL SEPARATION METHOD AND APPARATUS
Arthur C. Schouw

Robert L Mehlberg further describes the use of centripetal separation in his patent application for an “Apparatus for Separating Solids from Gas”.

The most common method of separating particulate solids from a gas stream uses centripetal separation. Centripetal separators are well known and operate by imparting a tangential velocity to gases containing entrained solid particles that forces the heavier solids particles outwardly away from the lighter gases for upward withdrawal of gases and downward collection of solids.

Cyclones for separating particulate material from gaseous materials are well known to those skilled in the art of FCC processing. Cyclones usually comprise an inlet duct that is tangential to the outside of a cylindrical barrel that forms an outer wall of the cyclone.

In the operation of the cyclone, the inlet duct and the inner surface of the barrel cooperate to create a spiral flow path of the gaseous materials and catalyst that establishes a vortex in the cyclone.

Apparatus for Separating Solids from Gas
Robert L. Mehlberg – 2010
http://www.faqs.org/patents/app/20100025305

Wikipedia seems to be very bashful regarding vortices and centripetal acceleration.

However, the centripetal force generate by a Forced Rotational Vortex can be very easily discovered for yourself at home:

https://malagabay.wordpress.com/2012/09/19/forced-vortex-demonstration/

Earth’s Forced Rotational Vortex

One of the most fascinating [and frequently overlooked] aspects of the Earth’s atmosphere is that it corotates with the Earth as the planet spins on its axis.

The Earth’s surface is not buffeted at the equator by winds travelling at 1,674.4 kilometres per hour [465.1 metres per second] as the Earth [with an equatorial radius of 6,378.1 kilometres] performs its daily spin around its axis.

Instead, the atmosphere corotates with the planet so that a person standing on the equator will generally experience still, calm air. Local weather systems may generate winds but the speed of the wind is always measured [and experienced] in the context the rotating Earth [and its corotating atmosphere].

The corotation of the atmosphere is observed at an altitude 16 kilometres [the maximum altitude of the Subtropical Jet Stream] where the wind speeds are still measured within the context of the rotating Earth. Hence, the maximum of the quoted range of speeds, 92 to 398 kilometres per hour, for the Jet Streams is a long way short of the Equatorial rotation speed of 1,674.4 kilometres per hour.

Image credit: Wikipedia http://en.wikipedia.org/wiki/Jet_stream

The corotation of the Earth’s atmosphere is an important factor influencing the atmospheric drag [orbital decay] experienced by satellites. During the 1970s there were several papers published that examined atmospheric corotation up to an altitude of [around] 300 kilometres. Interesting, the atmosphere experiences “super-rotation” [above 200 kilometres] which is 20 to 30 percent faster that the rotational speed of the terrestrial planet.

From observations of satellite orbits it has been deduced that the atmosphere above about 200 km altitude rotates 20–30% faster than the Earth, so that there exists a net west-to-east wind of order 100 m/s. This “super-rotation” has not yet been satisfactorily explained.

Rotation of the Variation of Upper Atmosphere
H. RISHBETH – 1971
Radio and Space Research Station, Slough, Buckinghamshire
http://www.nature.com/nature/journal/v229/n5283/abs/229333a0.html

However, the “super-rotation” effect dissipates with increased altitude and by [about] 300 kilometres the rotation of the atmosphere is once again synchronised with the rotation of the terrestrial planet.

The Rotational Speed of the Upper Atmosphere
determined from orbital inclinations of Interkosmos satellites
L. Sehnal – 1975
Astronomical Institute of the Czechoslovak Academy of Sciences, Ondrejov

The corotation of the Earth’s atmosphere extends into the plasmasphere and it is generally agreed that the corotation eventually breaks down at the plasmapause.

Corotation cannot extend to arbitrarily large distances from the planet but must ultimately break down as the result either of external forces or of the inertia of the corotating plasma itself. In the case of earth’s magnetosphere, external stresses imposed by the solar wind impede corotation beyond the plasmapause at about 5 earth radii distance [e.g., Brice, 1967].

Inertial Limit on Corotation
T. W. HILL – 1979
http://www.igpp.ucla.edu/public/mkivelso/refs/PUBLICATIONS/Hill%2079%20Inert%20lim%20corot%20JA084iA11p06554.pdf

Therefore, it can be argued that the Earth’s atmosphere forms a Forced Rotational Vortex [that extends to the plasmapause] which generates a centripetal force which is commonly called Gravity.

IMAGE Extreme Ultraviolet Imager
http://euv.lpl.arizona.edu/euv/

Earth’s Rankine Vortex Gravity

There is basic observational agreement regarding the effects of terrestrial Gravity:

This means that, ignoring air resistance, an object falling freely near the Earth’s surface increases its velocity by 9.81 m/s (32.2 ft/s or 22 mph) for each second of its descent.

Image credit: Wikipedia http://en.wikipedia.org/wiki/Gravity

However, the Wikipedia example of a ball falling down in a straight line is not quite the whole picture because the ball is falling within the context of the rotating Earth.

At the Equator the ball would accelerate downwards at 9.81 m/s2 while also moving laterally at 465 m/s. Thus, as illustrated below, a ball dropped out a window 122.25 metres above the ground [from a Singapore tower block] at the Equator would take 5 seconds to hit the surface and the trajectory of the ball in predominately lateral [because the Earth is rotating at 465 m/s].

Therefore, the ball [in these examples] does not a follow a Newtonian straight line descent. Instead the ball actually follows a curved trajectory towards the Earth surface where the final angle of descent [at impact] is determined by the height of the fall.

The trajectory becomes even more interesting if we drop an object from about 129,530 kilometres above the surface. The object will now take 24 hours to reach the surface and if it rotates within context then it will slowly circle the Earth for 23 hours before finally plunging [almost vertically] towards Earth during the final hour of the day. The resulting spiral [below] is simply generated by combining the inverse square law with contextual rotation.

The implication of a theoretical [unrestrained] Geocentric Rankine Vortex is that the vortex has a distinct “boundary” at about 130,000 kilometres [above Earth’s surface] above which the Earth’s “gravitational” force becomes [primarily] a circular vector with only a very slight gradient that slowly spirals inwards towards Earth.

Trying to determine whether a 130,000 kilometres circular “boundary” is a realistic proposition becomes difficult [using mainstream references such as Wikipedia] due to knowledge fragmentation, topic segmentation, natural variability, gaps and vagueness, approximations and a reluctance to openly acknowledge electro-magnetic effects.

However, there is a group of 130 scientists [studying the Earth’ magnetic field] that believe the Earth’s outermost boundary [the bow shock] is at an altitude of between 95,671 and 127,562 kilometres.

Earth’s Bow Shock: Origin of Ion Beams Revealed
The first outermost boundary is formed in a distance of about 15-20 Earth radii- it is called the Earth bow shock. Within this layer the solar wind rapidly decelerates, the interplanetary magnetic field and plasma density increases, and strong currents are formed. A very prominent feature at the Earth’s bow shock is the presence of back streaming accelerated ions.
http://www.spacedaily.com/news/earth-magnetic-04e.html

Bow Shock
The bow shock is a shock wave formed at a distance of 3-4 Earth radii or so in front of the nose of the magnetopause by the encounter of the supersonic solar wind with the “obstacle” to its flow presented by the Earth’s magnetic field. Passing through the shock, which ranges in thickness from roughly 100 km to 2 Earth radii, the solar wind is slowed, compressed, and heated. The region downstream of the bow shock, between the shock and the magnetopause, that is occupied by the shocked solar wind plasma is known as the magnetosheath.

The bow shock is so named by analogy to the wave formed by the bow of a ship as it passes through water. Unlike the shock wave that precedes an airplane flying at supersonic speeds, the bow shock is a stationary (i.e., non-propagating) shock. Moreover, it differs fundamentally from the familiar shocks in the Earth’s atmosphere in that it is a “collisionless” shock. That is, the bow shock occurs in a medium–the solar wind–that is so tenuous that collisions among the charged particles that make up the solar wind plasma are exceedingly rare and have no significant influence on the formation of the shock and the dissipation of the solar wind’s kinetic energy that occurs there. In this collisionless regime, wave-particle interactions take over the role played by particle collisions in a collisional shock.

… … … …

The turbulent foreshock region
Some of the inflowing solar wind particles are reflected from the shock instead of being transmitted through it. These reflected particles–together with some magnetosheath particles that have “leaked” back across the shock–travel upstream along the interplanetary magnetic field lines and populate a region upstream of the shock and magnetically connected to it. This region is known as the foreshock. There are, in fact, two foreshock regions: one populated preferentially by reflected energetic electrons (the electron foreshock) and one populated preferentially by suprathermal ions (the ion foreshock). (This spatial distribution results from the velocity differences between the faster electrons and the slower ions.) The foreshock is characterized by extensive wave activity, which results from the interaction of the backstreaming particles with the inflowing solar wind. This interaction gives rise to instabilities, which in turn excite ultra-low-frequency (ULF) magnetohydrodynamic waves, ion acoustic waves, and electron plasma oscillations.

http://pluto.space.swri.edu/image/glossary/bow_shock.html

Therefore, the theoretical 130,000 kilometres circular “boundary” [of the Geocentric Rankine Vortex] appears to be a realistic possibility when the Solar Wind is weak. However, compression of this boundary to around 100,000 kilometres [the limit of the observed geocorona] also seems highly likely when the Solar Wind is strong.

Furthermore, it is not inconceivable that the Earth’s “bow shock” is where the [primarily] circular motion of the Geocentric Rankine Vortex causes an abrupt drop in the speed of the solar wind.