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13.4: Spin-up of Cyclonic Rotation - Geosciences

13.4: Spin-up of Cyclonic Rotation - Geosciences


Cyclogenesis is associated with: (a) upward motion, (b) decreasing surface pressure, and (c) increasing vorticity (i.e., spin-up). Let us start with vorticity.

The equation that forecasts change of vorticity with time is called the vorticity tendency equation. We can investigate the processes that cause cyclogenesis (spin up; positive-vorticity increase) and cyclolysis (spin down; positive-vorticity decrease) by examining terms in the vorticity tendency equation. Mountains are not needed for these processes.

The change of relative vorticity ζr over time (i.e., the spin-up or vorticity tendency) can be predicted using the following equation:

( egin{align} ag{13.9}end{align})

Positive vorticity tendency indicates cyclogenesis.

Vorticity Advection: If the wind blows air of greater vorticity into your region of interest, then this is called positive vorticity advection (PVA). Negative vorticity advection (NVA) is when lower-vorticity air is blown into a region. These advections can be caused by vertical winds and horizontal winds (Fig. 13.23a).

Stretching: Consider a short column of air that is spinning as a vortex tube. Horizontal convergence of air toward this tube will cause the tube to become taller and more slender (smaller diameter). The taller or stretched vortex tube supports cyclogenesis (Fig. 13.23b). Conversely, horizontal divergence shortens the vortex tube and supports cyclolysis or anticyclogenesis. In the first and second lines of eq. (13.9) are the stretching terms for Earth’s rotation and relative vorticity, respectively. Stretching means that the top of the vortex tube moves upward away from (or moves faster than) the movement of the bottom of the vortex tube; hence ∆W/∆z is positive for stretching.

Beta Effect: Recall from eq. (11.35) in the General Circulation chapter that we can define β = ∆fc/∆y. Beta is positive in the N. Hemisphere because the Coriolis parameter increases toward the north pole (see eq. 13.2). If wind moves air southward (i.e., V = negative) to where fc is smaller, then relative vorticity ζr becomes larger (as indicated by the negative sign in front of the beta term) to conserve potential vorticity (Fig. 13.23c).

Tilting Terms: (A & B in eq. 13.9) If the horizontal winds change with altitude, then this shear causes vorticity along a horizontal axis. (C in eq. 13.9) Neighboring up- and down-drafts give horizontal shear of the vertical wind, causing vorticity along a horizontal axis. (A-C) If a resulting horizontal vortex tube experiences stronger vertical velocity on one end relative to the other (Fig. 13.23d), then the tube will tilt to become more vertical. Because spinning about a vertical axis is how we define vorticity, we have increased vorticity via the tilting of initially horizontal vorticity.

Turbulence in the atmospheric boundary layer (ABL) communicates frictional forces from the ground to the whole ABL. This turbulent drag acts to slow the wind and decrease rotation rates (Fig. 13.23e). Such spin down can cause cyclolysis.

However, for cold fronts drag can increase vorticity. As the cold air advances (black arrow in Fig. 13.23f), Coriolis force will turn the winds and create a geostrophic wind Vg (large white arrow). Closer to the leading edge of the front where the cold air is shallower, the winds M are subgeostrophic because of the greater drag. The result is a change of wind speed M with distance x that causes positive vorticity.

All of the terms in the vorticity-tendency equation must be summed to determine net spin down or spin up. You can identify the action of some of these terms by looking at weather maps.

Fig. 13.24 shows the wind vectors and absolute vorticity on the 50 kPa isobaric surface (roughly in the middle of the troposphere) for the case-study cyclone. Positive vorticity advection (PVA) occurs where wind vectors are crossing the vorticity contours from high toward low vorticity, such as highlighted by the dark box in Fig. 13.24. Namely, higher vorticity air is blowing into regions that contained lower vorticity. This region favors cyclone spin up.

Negative vorticity advection (NVA) is where the wind crosses the vorticity contours from low to high vorticity values (dark oval in Fig. 13.24). By using the absolute vorticity instead of relative vorticity, Fig. 13.24 combines the advection and beta terms.

Fig. 13.25 shows vertical velocity in the middle of the atmosphere. Since vertical velocity is near zero at the ground, regions of positive vertical velocity at 50 kPa must correspond to stretching in the bottom half of the atmosphere. Thus, the updraft regions in the figure favor cyclone spin-up (i.e., cyclogenesis).

In the bottom half of the troposphere, regions of stretching must correspond to regions of convergence of air, due to mass continuity. Fig. 13.26 shows the divergence field at 85 kPa. Negative divergence corresponds to convergence. The regions of low-altitude convergence favor cyclone spin-up.

Low-altitude spin-down due to turbulent drag occurs wherever there is rotation. Thus, the rotation of 10 m wind vectors around the surface low in Fig. 13.13 indicate vorticity that is spinning down. The tilting term will also be discussed in the Thunderstorm chapters, regarding tornado formation.

Above the boundary layer (and away from fronts, jets, and thunderstorms) the terms in the second line of the vorticity equation are smaller than those in the first line, and can be neglected. Also, for synoptic scale, extratropical weather systems, the winds are almost geostrophic (quasi-geostrophic).

These weather phenomena are simpler to analyze than thunderstorms and hurricanes, and can be well approximated by a set of equations (quasi-geostrophic vorticity and omega equations) that are less complicated than the full set of primitive equations of motion (Newton’s second law, the first law of thermodynamics, the continuity equation, and the ideal gas law).

As a result of the simplifications above, the vorticity forecast equation simplifies to the following quasi-geostrophic vorticity equation:

( egin{align} frac{Delta zeta_{g}}{Delta t}=-U_{g} frac{Delta zeta_{g}}{Delta x}-V_{g} frac{Delta zeta_{g}}{Delta y}-V_{g} frac{Delta f_{c}}{Delta y}+f_{c} frac{Delta W}{Delta z} ag{13.10}end{align})

where the relative geostrophic vorticity ζg is defined similar to the relative vorticity of eq. (11.20), except using geostrophic winds Ug and Vg:

( egin{align} zeta_{g}=frac{Delta V_{g}}{Delta x}-frac{Delta U_{g}}{Delta y} ag{13.11}end{align})

For solid body rotation, eq. (11.22) becomes:

( egin{align} zeta_{g}=frac{2 cdot G}{R} ag{13.12}end{align})

where G is the geostrophic wind speed and R is the radius of curvature.

The prefix “quasi-” is used for the reasons given below. If the winds were perfectly geostrophic or gradient, then they would be parallel to the isobars. Such winds never cross the isobars, and could not cause convergence into the low. With no convergence there would be no vertical velocity.

However, we know from observations that vertical motions do exist and are important for causing clouds and precipitation in cyclones. Thus, the last term in the quasi-geostrophic vorticity equation includes W, a wind that is not geostrophic. When such an ageostrophic vertical velocity is included in an equation that otherwise is totally geostrophic, the equation is said to be quasi-geostrophic, meaning partially geostrophic. The quasi-geostrophic approximation will also be used later in this chapter to estimate vertical velocity in cyclones.

Within a quasi-geostrophic system, the vorticity and temperature fields are closely coupled, due to the dual constraints of geostrophic and hydrostatic balance. This implies close coupling between the wind and mass fields, as was discussed in the General Circulation and Fronts chapters in the sections on geostrophic adjustment. While such close coupling is not observed for every weather system, it is a reasonable approximation for synoptic-scale, extratropical systems.

Sample Application

Suppose an initial flow field has no geostrophic relative vorticity, but there is a straight north to south geostrophic wind blowing at 10 m s–1 at latitude 45°. Also, the top of a 1 km thick column of air rises at 0.01 m s–1, while its base rises at 0.008 m s–1. Find the rate of geostrophic-vorticity spin-up.

Find the Answer

Given: V = –10 m s–1, ϕ = 45°, Wtop = 0.01 m s–1, Wbottom = 0.008 m s–1, ∆z = 1 km.

Find: ∆ζg/∆t = ? s–2

First, get the Coriolis parameter using eq. (10.16): fc = (1.458x10–4 s–1)·sin(45°) = 0.000103 s–1

Next, use eq. (13.2):

(eta=frac{Delta f_{c}}{Delta y}=frac{1.458 imes 10^{-4} mathrm{s}^{-1}}{6.357 imes 10^{6} mathrm{m}} cdot cos 45^{circ})

Use the definition of a gradient (see Appendix A):

(frac{Delta W}{Delta z}=frac{W_{t o p}-W_{b o t t o m}}{z_{t o p}-z_{b o t t o m}}=frac{(0.01-0.008) mathrm{m} / mathrm{s}}{(1000-0) mathrm{m}})

= 2x10–6 s–1

Finally, use eq. (13.10). We have no information about advection, so assume it is zero. The remaining terms give:

(frac{Delta zeta_{g}}{Delta t}=-(-10 mathrm{m} / mathrm{s}) cdotleft(1.62 imes 10^{-11} mathrm{m}^{-1} cdot mathrm{s}^{-1} ight) + (0.000103s_{-1})cdot (2x10^{-6}s^{-1}))

spin-up ( quad) ( quad)( quad)beta( quad)( quad)( quad)( quad)( quad)( quad)( quad)( quad)( quad) stretching

=(1.62x10–10 + 2.06x10–10 ) s–2 = 3.68x10–10 s–2

Check: Units OK. Physics OK.

Exposition: Even without any initial geostrophic vorticity, the rotation of the Earth can spin up the flow if the wind blows appropriately.

An idealized weather pattern (“toy model”) is shown in Fig. 13.27. Every feature in the figure is on the 50 kPa isobaric surface (i.e., in the mid troposphere), except the L which indicates the location of the surface low center. All three components of the geostrophic vorticity equation can be studied.

Geostrophic and gradient winds are parallel to the height contours. The trough axis is a region of cyclonic (counterclockwise) curvature of the wind, which yields a large positive value of geostrophic vorticity. At the ridge is negative (clockwise) relative vorticity. Thus, the advection term is positive over the L center and contributes to spin-up of the cyclone because the wind is blowing higher positive vorticity into the area of the surface low.

For any fixed pressure gradient, the gradient winds are slower than geostrophic when curving cyclonically (“slow around lows”), and faster than geostrophic for anticyclonic curvature, as sketched with the thick-line wind arrows in Fig. Examine the 50 kPa flow immediately above the surface low. Air is departing faster than entering. This imbalance (divergence) draws air up from below. Hence, W increases from near zero at the ground to some positive updraft speed at 50 kPa. This stretching helps to spin up the cyclone.

The beta term, however, contributes to spindown because air from lower latitudes (with smaller Coriolis parameter) is blowing toward the location of the surface cyclone. This effect is small when the wave amplitude is small. The sum of all three terms in the quasi-geostrophic vorticity equation is often positive, providing a net spin-up and intensification of the cyclone.

In real cyclones, contours are often more closely spaced in troughs, causing relative maxima in jet stream winds called jet streaks. Vertical motions associated with horizontal divergence in jet streaks are discussed later in this chapter. These motions violate the assumption that air mass is conserved along an “isobaric channel”. Rossby also pointed out in 1940 that the gradient wind balance is not valid for varying motions. Thus, the “toy” model of Fig. 13.27 has weaknesses that limit its applicability.

HIGHER MATH • The Laplacian

A Laplacian operator ( abla^{2}) can be defined as

( abla^{2} A=frac{partial^{2} A}{partial x^{2}}+frac{partial^{2} A}{partial y^{2}}+frac{partial^{2} A}{partial z^{2}})

where A represents any variable. Sometimes we are concerned only with the horizontal (H) portion:

( abla_{H}^{2}(A)=frac{partial^{2} A}{partial x^{2}}+frac{partial^{2} A}{partial y^{2}})

What does it mean? If ∂A/∂x represents the slope of a line when A is plotted vs. x on a graph, then ∂2A/∂x2 = ∂[ ∂A/∂x ]/∂x is the change of slope; namely, the curvature.

How is it used? Recall from the Atm. Forces & Winds chapter that the geostrophic wind is defined as

(U_{g}=-frac{1}{f_{c}} frac{partial Phi}{partial y} quadquad quad V_{g}=frac{1}{f_{c}} frac{partial Phi}{partial x})

or

( egin{align} zeta_{g}=frac{1}{f_{c}} abla_{H}^{2}(Phi) ag{13.11b}end{align})

This illustrates the value of the Laplacian — as a way to more concisely describe the physics.

For example, a low-pressure center corresponds to a low-height center on an isobaric sfc. That isobaric surface is concave up, which corresponds to positive curvature. Namely, the Laplacian of |g|·z is positive, hence, ζg is positive. Thus, a low has positive vorticity


13.4: Spin-up of Cyclonic Rotation - Geosciences

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Contents

QuikSCAT was launched on 19 June 1999 with an initial 3-year mission requirement. QuikSCAT was a "quick recovery" mission replacing the NASA Scatterometer (NSCAT), which failed prematurely in June 1997 after just 9.5 months in operation. QuikSCAT, however, far exceeded these design expectations and continued to operate for over a decade before a bearing failure on its antenna motor ended QuikSCAT's capabilities to determine useful surface wind information on 23 November 2009. The QuikSCAT geophysical data record spans from 19 July 1999 to 21 November 2009. While the dish could not rotate after this date, its radar capabilities remained fully intact. It continued operating in this mode until full mission termination on October 2, 2018. Data from this mode of the mission was used to improve the accuracy of other satellite surface wind datasets by inter-calibrating other Ku-band scatterometers.

QuikSCAT measured winds in measurement swaths 1,800 km wide centered on the satellite ground track with no nadir gap, such as occurs with fan-beam scatterometers such as NSCAT. Because of its wide swath and lack of in-swath gaps, QuikSCAT was able to collect at least one vector wind measurement over 93% of the World's Oceans each day. This improved significantly over the 77% coverage provided by NSCAT. Each day, QuikSCAT recorded over 400,000 measurements of wind speed and direction. This is hundreds of times more surface wind measurements than are collected routinely from ships and buoys.

QuikSCAT provided measurements of the wind speed and direction referenced to 10 meters above the sea surface at a spatial resolution of 25 km. Wind information cannot be retrieved within 15–30 km of coastlines or in the presence of sea ice. Precipitation generally degrades the wind measurement accuracy, [1] although useful wind and rain information can still be obtained in mid-latitude and tropical cyclones for monitoring purposes. [2] In addition to measuring surface winds over the ocean, scatterometers such as QuikSCAT can also provide information on the fractional coverage of sea ice, track large icebergs (>5 km in length), differentiate types of ice and snow, and detect the freeze–thaw line in polar regions.

While the rotating dish antenna can no longer spin as designed, the rest of the instrument remains functional and data transmission capabilities remain intact, although it cannot determine the surface vector wind. It can, however, still measure radar backscatter at a fixed azimuth angle. QuikSCAT is being used in this reduced mode to cross-calibrate other scatterometers in hopes of providing long-term and consistent surface wind datasets over multiple on-orbit scatterometer platforms, including the operational European Organisation for the Exploitation of Meteorological Satellites (EUMETSAT) Advanced Scatterometer (ASCAT) on MetOp-A and MetOp-B, India's Oceansat-2 scatterometer operated by the Indian Space Research Organization (ISRO), and the China's HaiYang-2A (HY-2A) scatterometer operated by China's National Satellite Ocean Application Service, as well as future NASA scatterometer missions in development. A NASA Senior Review panel in 2011 endorsed the continuation of the QuikSCAT mission with these modified objectives through 2018. QuikSCAT was declared fully decommissioned on October 2, 2018.

SeaWinds used a rotating dish antenna with two spot beams that sweep in a circular pattern. The antenna consists of a 1-meter diameter rotating dish that produces two spot beams, sweeping in a circular pattern. [3] It radiates 110 W microwave pulses at a pulse repetition frequency (PRF) of 189 Hz. QuikSCAT operates at a frequency of 13.4 GHz, which is in the Ku-band of microwave frequencies. At this frequency, the atmosphere is mostly transparent to non-precipitating clouds and aerosols, although rain produces significant alteration of the signal. [4]

The spacecraft is in a sun-synchronous orbit, with equatorial crossing times of ascending swaths at about 06:00 LST ±30 minutes. Along the equator, consecutive swaths are separated by 2,800 km. QuikSCAT orbits Earth at an altitude of 802 km and at a speed of about 7 km per second.

Wind measurement accuracy Edit

Measurement principles Edit

Scatterometers such as QuikSCAT emit pulses of low-power microwave radiation and measure the power reflected back to its receiving antenna from the wind-roughened sea surface. Gravity and capillary waves on the sea surface caused by the wind reflect or backscatter power emitted from the scatterometer radar primarily by means of a Bragg resonance condition. The wavelengths of these waves are roughly 1 cm and are usually in equilibrium with the local surface wind. Over water surfaces, the microwave backscatter is highly correlated with the surface wind speed and direction. The particular wavelength of the surface waves is determined by the wavelength of the microwave radiation emitted from the scatterometer's radar.

QuikSCAT consists of an active microwave radar that infers surface winds from the roughness of the sea surface based on measurements of radar backscatter cross section, denoted as σ0. σ0 varies with surface wind speed and direction relative to the antenna azimuth, incidence angle, polarization, and radar frequency. QuikSCAT uses a dual-beam, conically scanning antenna that samples the full range of azimuth angles during each antenna revolution. Backscatter measurements are obtained at fixed incidence angles of 46° and 54°, providing up to four views of each region of the surface at different incidence angles.

Standard processing of the QuikSCAT measurements yields a spatial resolution of about 25 km. A higher spatial resolution of 12.5 km is also achieved through special processing, but has significantly more measurement noise. An even higher spatial resolution of 5 km is also produced, but only for limited regions and special cases.

The σ0 observations are calibrated to the wind speed and direction of the wind at a reference height of 10 meters above the sea surface.

In 1996, the NASA Scatterometer (NSCAT) was launched aboard the Japanese Advanced Earth Observing Satellite (ADEOS-1). This satellite was designed to record surface winds over water across the world for several years. However, an unexpected failure in 1997 led to an early termination of the NSCAT project. Following this briefly successful mission, NASA began constructing a new satellite to replace the failed one. They planned to build it and have it prepared for launch as soon as possible to limit the gap in data between the two satellites. [5] In just 12 months, the Quick Scatterometer (QuikSCAT) satellite was constructed and ready to be launched, faster than any other NASA mission since the 1950s. [6]

The QuikSCAT project was originally budgeted at $93 million, including the physical satellite, the launch rocket, and ongoing support for its science mission. [7] A series of rocket failures in November 1998 grounded the Titan (rocket family) launcher fleet, delayed the launch of QuikSCAT, and added $5 million to this initial cost. [7]

A new instrument, the SeaWinds scatterometer, was carried on the satellite. The SeaWinds instrument, a specialized microwave radar system, measured both the speed and direction of winds near the ocean surface. It used two radars and a spinning antenna to record data across nine-tenths of the oceans of the world in a single day. It recorded roughly four hundred thousand wind measurements daily, each covering an area 1,800 kilometers (1,100 mi) in width. [6] Jet Propulsion Laboratory and the NSCAT team jointly managed the project of construction of the satellite at the Goddard Space Flight Center. Ball Aerospace & Technologies Corp. supplied the materials to construct the satellite.

In light of the record-setting construction time, engineers who worked on the project were given the American Electronics Achievement Award. This was only achieved due to the new type of contract made specifically for this satellite. Instead of the usual year given to select a contract and initiate development, it was constrained to one month. [8]

The newly constructed satellite was set to launch on a Titan II rocket from Vandenberg Air Force Base in California. The rocket lifted off at 7:15 pm PDT on 19 June 1999. Roughly two minutes and thirty seconds after launch, the first engine was shut down and the second was engaged as it moved over the Baja California Peninsula. A minute later, the nose cone, at the top of the rocket, separated into two parts. Sixteen seconds later, the rocket was re-oriented to protect the satellite from the sun. For the next 48 minutes, the two crafts flew over Antarctica and later over Madagascar, where the rocket reached its desired altitude of 500 mi (800 km). [9]

At 59 minutes after launch, the satellite separated from the rocket and was pushed into its circular orbit around Earth. Shortly after, the solar arrays were deployed and connection was established with the satellite at 8:32 pm PDT with a tracking station in Norway. For the next two weeks, the shuttle used bursts from its engine to fine-tune its location and correct its course to the desired motion. On July 7, eighteen days after take-off, the scatterometer was turned on and a team of 12 personnel made detailed reviews of function of QuikSCAT. A month after entering orbit, the team completed the checks, and QuikSCAT began collecting and transmitting backscatter measurements. [9]

Weather Forecasting Edit

Many operational numerical weather prediction centers began assimilating QuikSCAT data in early 2002, with preliminary assessments indicating a positive impact. [10] The U.S. National Centers for Environmental Prediction (NCEP) and the European Centre for Medium-Range Weather Forecasts (ECMWF) led the way by initiating assimilation of QuikSCAT winds beginning, respectively, on 13 January 2002 and 22 January 2002. QuikSCAT surface winds were an important tool for analysis and forecasting at the U.S. National Hurricane Center since becoming available in near–real time in 2000. [11]

QuikSCAT wind fields were also used as a tool in the analysis and forecasting of extratropical cyclones and maritime weather outside the tropics at the U.S. Ocean Prediction Center [12] and the U.S. National Weather Service. [10] [13]

Data was also provided in real-time over most of the ice-free global oceans, including traditionally data-sparse regions of the ocean where few observations exist, such as in the Southern Ocean and the eastern tropical Pacific Ocean.

QuikSCAT observations are provided to these operational users in near-real-time (NRT) in binary universal form for the representation of meteorological data (BUFR) format by the National Oceanic and Atmospheric Administration/National Environmental Satellite, Data, and Information Service (NOAA/NESDIS). [14] The data latency goal is 3 hours, and almost all data are available within 3.5 hours of measurement. To meet these requirements, the QuikSCAT NRT data processing algorithms combine the finest-grained backscatter measurements into fewer composites than the science data algorithms. Otherwise the QuikSCAT NRT processing algorithms are identical to the science data algorithms.


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Contents

Among the ancient Greeks, several of the Pythagorean school believed in the rotation of Earth rather than the apparent diurnal rotation of the heavens. Perhaps the first was Philolaus (470–385 BCE), though his system was complicated, including a counter-earth rotating daily about a central fire. [4]

A more conventional picture was supported by Hicetas, Heraclides and Ecphantus in the fourth century BCE who assumed that Earth rotated but did not suggest that Earth revolved about the Sun. In the third century BCE, Aristarchus of Samos suggested the Sun's central place.

However, Aristotle in the fourth century BCE criticized the ideas of Philolaus as being based on theory rather than observation. He established the idea of a sphere of fixed stars that rotated about Earth. [5] This was accepted by most of those who came after, in particular Claudius Ptolemy (2nd century CE), who thought Earth would be devastated by gales if it rotated. [6]

In 499 CE, the Indian astronomer Aryabhata wrote that the spherical Earth rotates about its axis daily, and that the apparent movement of the stars is a relative motion caused by the rotation of Earth. He provided the following analogy: "Just as a man in a boat going in one direction sees the stationary things on the bank as moving in the opposite direction, in the same way to a man at Lanka the fixed stars appear to be going westward." [7] [8]

In the 10th century, some Muslim astronomers accepted that Earth rotates around its axis. [9] According to al-Biruni, Abu Sa'id al-Sijzi (d. circa 1020) invented an astrolabe called al-zūraqī based on the idea believed by some of his contemporaries "that the motion we see is due to the Earth's movement and not to that of the sky." [10] [11] The prevalence of this view is further confirmed by a reference from the 13th century which states: "According to the geometers [or engineers] (muhandisīn), the Earth is in constant circular motion, and what appears to be the motion of the heavens is actually due to the motion of the Earth and not the stars." [10] Treatises were written to discuss its possibility, either as refutations or expressing doubts about Ptolemy's arguments against it. [12] At the Maragha and Samarkand observatories, Earth's rotation was discussed by Tusi (b. 1201) and Qushji (b. 1403) the arguments and evidence they used resemble those used by Copernicus. [13]

In medieval Europe, Thomas Aquinas accepted Aristotle's view [14] and so, reluctantly, did John Buridan [15] and Nicole Oresme [16] in the fourteenth century. Not until Nicolaus Copernicus in 1543 adopted a heliocentric world system did the contemporary understanding of Earth's rotation begin to be established. Copernicus pointed out that if the movement of Earth is violent, then the movement of the stars must be very much more so. He acknowledged the contribution of the Pythagoreans and pointed to examples of relative motion. For Copernicus this was the first step in establishing the simpler pattern of planets circling a central Sun. [17]

Tycho Brahe, who produced accurate observations on which Kepler based his laws of planetary motion, used Copernicus's work as the basis of a system assuming a stationary Earth. In 1600, William Gilbert strongly supported Earth's rotation in his treatise on Earth's magnetism [18] and thereby influenced many of his contemporaries. [19] Those like Gilbert who did not openly support or reject the motion of Earth about the Sun are called "semi-Copernicans". [20] A century after Copernicus, Riccioli disputed the model of a rotating Earth due to the lack of then-observable eastward deflections in falling bodies [21] such deflections would later be called the Coriolis effect. However, the contributions of Kepler, Galileo and Newton gathered support for the theory of the rotation of Earth.

Empirical tests Edit

Earth's rotation implies that the Equator bulges and the geographical poles are flattened. In his Principia, Newton predicted this flattening would occur in the ratio of 1:230, and pointed to the pendulum measurements taken by Richer in 1673 as corroboration of the change in gravity, [22] but initial measurements of meridian lengths by Picard and Cassini at the end of the 17th century suggested the opposite. However, measurements by Maupertuis and the French Geodesic Mission in the 1730s established the oblateness of Earth, thus confirming the positions of both Newton and Copernicus. [23]

In Earth's rotating frame of reference, a freely moving body follows an apparent path that deviates from the one it would follow in a fixed frame of reference. Because of the Coriolis effect, falling bodies veer slightly eastward from the vertical plumb line below their point of release, and projectiles veer right in the Northern Hemisphere (and left in the Southern) from the direction in which they are shot. The Coriolis effect is mainly observable at a meteorological scale, where it is responsible for the opposite directions of cyclone rotation in the Northern and Southern hemispheres (anticlockwise and clockwise, respectively).

Hooke, following a suggestion from Newton in 1679, tried unsuccessfully to verify the predicted eastward deviation of a body dropped from a height of 8.2 meters , but definitive results were obtained later, in the late 18th and early 19th century, by Giovanni Battista Guglielmini in Bologna, Johann Friedrich Benzenberg in Hamburg and Ferdinand Reich in Freiberg, using taller towers and carefully released weights. [n 1] A ball dropped from a height of 158.5 m departed by 27.4 mm from the vertical compared with a calculated value of 28.1 mm.

The most celebrated test of Earth's rotation is the Foucault pendulum first built by physicist Léon Foucault in 1851, which consisted of a lead-filled brass sphere suspended 67 m from the top of the Panthéon in Paris. Because of Earth's rotation under the swinging pendulum, the pendulum's plane of oscillation appears to rotate at a rate depending on latitude. At the latitude of Paris the predicted and observed shift was about 11 degrees clockwise per hour. Foucault pendulums now swing in museums around the world.

True solar day Edit

Earth's rotation period relative to the Sun (solar noon to solar noon) is its true solar day or apparent solar day. [ citation needed ] It depends on Earth's orbital motion and is thus affected by changes in the eccentricity and inclination of Earth's orbit. Both vary over thousands of years, so the annual variation of the true solar day also varies. Generally, it is longer than the mean solar day during two periods of the year and shorter during another two. [n 2] The true solar day tends to be longer near perihelion when the Sun apparently moves along the ecliptic through a greater angle than usual, taking about 10 seconds longer to do so. Conversely, it is about 10 seconds shorter near aphelion. It is about 20 seconds longer near a solstice when the projection of the Sun's apparent motion along the ecliptic onto the celestial equator causes the Sun to move through a greater angle than usual. Conversely, near an equinox the projection onto the equator is shorter by about 20 seconds . Currently, the perihelion and solstice effects combine to lengthen the true solar day near 22 December by 30 mean solar seconds, but the solstice effect is partially cancelled by the aphelion effect near 19 June when it is only 13 seconds longer. The effects of the equinoxes shorten it near 26 March and 16 September by 18 seconds and 21 seconds , respectively. [25] [26]

Mean solar day Edit

The average of the true solar day during the course of an entire year is the mean solar day, which contains 86 400 mean solar seconds . Currently, each of these seconds is slightly longer than an SI second because Earth's mean solar day is now slightly longer than it was during the 19th century due to tidal friction. The average length of the mean solar day since the introduction of the leap second in 1972 has been about 0 to 2 ms longer than 86 400 SI seconds . [27] [28] [29] Random fluctuations due to core-mantle coupling have an amplitude of about 5 ms. [30] [31] The mean solar second between 1750 and 1892 was chosen in 1895 by Simon Newcomb as the independent unit of time in his Tables of the Sun. These tables were used to calculate the world's ephemerides between 1900 and 1983, so this second became known as the ephemeris second. In 1967 the SI second was made equal to the ephemeris second. [32]

The apparent solar time is a measure of Earth's rotation and the difference between it and the mean solar time is known as the equation of time.

Stellar and sidereal day Edit

Earth's rotation period relative to the International Celestial Reference Frame, called its stellar day by the International Earth Rotation and Reference Systems Service (IERS), is 86 164.098 903 691 seconds of mean solar time (UT1) (23 h 56 m 4.098 903 691 s , 0.997 269 663 237 16 mean solar days ). [33] [n 3] Earth's rotation period relative to the precessing mean vernal equinox, named sidereal day, is 86 164.090 530 832 88 seconds of mean solar time (UT1) (23 h 56 m 4.090 530 832 88 s , 0.997 269 566 329 08 mean solar days ). [33] Thus, the sidereal day is shorter than the stellar day by about 8.4 ms . [35]

Both the stellar day and the sidereal day are shorter than the mean solar day by about 3 minutes 56 seconds . This is a result of the Earth turning 1 additional rotation, relative to the celestial reference frame, as it orbits the Sun (so 366.25 rotations/y). The mean solar day in SI seconds is available from the IERS for the periods 1623–2005 [36] and 1962–2005 . [37]

Recently (1999–2010) the average annual length of the mean solar day in excess of 86 400 SI seconds has varied between 0.25 ms and 1 ms , which must be added to both the stellar and sidereal days given in mean solar time above to obtain their lengths in SI seconds (see Fluctuations in the length of day).

Angular speed Edit

The angular speed of Earth's rotation in inertial space is (7.292 115 0 ± 0.000 000 1) × 10 ^ −5 radians per SI second . [33] [n 4] Multiplying by (180°/π radians) × (86,400 seconds/day) yields 360.985 6°/day , indicating that Earth rotates more than 360° relative to the fixed stars in one solar day. Earth's movement along its nearly circular orbit while it is rotating once around its axis requires that Earth rotate slightly more than once relative to the fixed stars before the mean Sun can pass overhead again, even though it rotates only once (360°) relative to the mean Sun. [n 5] Multiplying the value in rad/s by Earth's equatorial radius of 6,378,137 m (WGS84 ellipsoid) (factors of 2π radians needed by both cancel) yields an equatorial speed of 465.10 metres per second (1,674.4 km/h). [38] Some sources state that Earth's equatorial speed is slightly less, or 1,669.8 km/h . [39] This is obtained by dividing Earth's equatorial circumference by 24 hours . However, the use of the solar day is incorrect it must be the sidereal day, so the corresponding time unit must be a sidereal hour. This is confirmed by multiplying by the number of sidereal days in one mean solar day, 1.002 737 909 350 795 , [33] which yields the equatorial speed in mean solar hours given above of 1,674.4 km/h or 1040.0mph .

The tangential speed of Earth's rotation at a point on Earth can be approximated by multiplying the speed at the equator by the cosine of the latitude. [40] For example, the Kennedy Space Center is located at latitude 28.59° N, which yields a speed of: cos(28.59°) × 1674.4 km/h = 1470.2 km/h. Latitude is a placement consideration for spaceports.


4. A Feedback Dynamo

[16] As a feedback dynamo we define a situation where an ambient field in which the dynamo is embedded is generated by the dynamo action itself (cf. Figure 1). The origin of such a self-generated ambient field can be the interaction of the planetary magnetic field with the solar wind plasma. As the dynamo is influenced by the ambient field and this field depends on the dynamo generated field a feedback situation emerges.

[18] Of particular interest are those situations where the dynamo number is close to the critical dynamo number. At this critical dynamo number we have nominal convection conditions, that is convection with magnetic field generation in the absence of any ambient magnetic field. For such nominal conditions a self-regenerative kinematic dynamo can exist [e.g., Parker, 1970 ].

[19] As Figure 2 indicates there are apparently always two situations where one can meet such nominal convection conditions. One occurs at a relatively weak core magnetic field, the other at large magnetic fields. The feedback dynamo discussed here has two stationary solutions, a weak field and a strong field one. In both cases the convection system does not need to be altered significantly and is described by its critical dynamo number, determined by differential rotation and cyclonic motions.

[20] The two situations differ in the following way. The strong planetary field situation causes, according to equation (4), a large magnetopause distance. In turn, this implies a weak Chapman-Ferraro field in the planetary core much as observed at Earth. The dynamo operates in the absence of an ambient field. The terrestrial dynamo is a realization of such a solution.

[21] The other solution, that one with a weak planetary field, can be explained as follows (Figure 3): differential rotation causes toroidal field generation out of the primary poloidal field with this field being regenerated via the α-effect. The primary poloidal field causes the generation of a magnetosphere whose Chapman-Ferraro currents generate a secondary poloidal field in the differentially rotating liquid core. As this secondary poloidal field is oriented in the opposite direction of the primary poloidal field the secondary toroidal field opposes the primary toroidal magnetic field. This in turn leads to a smaller regenerative poloidal field. We conclude that the proposed feedback dynamo offers two stationary solutions, a Hermean-type with a weak planetary magnetic field, and an Earth-like solution with a significantly stronger field.

[22] Planet Mercury is a suitable candidate for the operation of a Hermean-type dynamo. From our kinematic dynamo model we infer that its field strength should be of the order of a few hundred nT. This corresponds remarkable well with the actually measured field at Mercury.


1. Introduction

[2] Laboratory modeling can be considered an effective tool for studies of the dynamics of the Black Sea in addition to field experiments and numerical simulations. Geographical features of the Black Sea allow one to model it in a small-scale laboratory experiment observing the similarity of a number of important dimensionless governing parameters. The Black Sea is a nearly enclosed basin (Figure 1) having limited exchange with the Mediterranean through the narrow Bosphorus Strait. The net outflow from the Black Sea through the Bosphorus is approximately 300 km 2 yr −1 [ Unluata et al., 1990 ] which constitutes 0.06% of the total volume of the Black Sea. The sea is elongated in the East-west direction with the aspect ratio of approximately 3:1. The Black Sea is relatively narrow in its central part between the Crimean peninsula and Anatolian coast where the minimum distance is 260 km. The limited extent of the sea in the latitudinal direction allows us in most cases to exclude the effect of the variation of the Coriolis parameter with latitude from consideration of the dynamical features of the circulation. The Black Sea is a deep basin (maximum depth 2246 m, average depth 1300 m) with a narrow shelf and a steep continental slope except in the northwestern part of the sea where the shelf occupies a very broad region. In the northwestern part of the Black Sea the shelf is over 200 km wide with the depth ranging from 0 to 100 m. In other parts of the sea it has a depth of less than 100 m and a width of 2.2 to 15 km. Near the Caucasian and Anatolian coasts the shelf is only a narrow intermittent strip.

[3] The hydrological structure of the Black Sea is characterized by a very stable stratification. A simplified description of the stratification can be summarized as follows. A thin upper layer of 40–50 m and of a low salinity of approximately 18 psu is due to the continuous supply of fresh water by major rivers in the western part of the sea. Most of the inflow comes from four major rivers namely the Danube, the Dniester, the Dnieper and the Southern Bug. The average total annual discharge into the Black Sea is 300 km 3 per year [ Unluata et al., 1990 ], which constitutes approximately 0.06% of the total volume of the Black Sea. The stratification is especially pronounced along the coast in the boundary current [e.g., Oguz and Besiktepe, 1999 ] called Rim Current [ Oguz et al., 1992 ] transporting fresh water around the sea in a cyclonic direction. This basin-scale current is also referred to as the Main Black Sea Current in Russian literature.

[4] The results of numerical modeling by Kourafalou and Stanev [2001] also indicate that a so-called Coastal Low Salinity Band is formed because of combined river input. The rest of the sea can be effectively considered as a denser water mass of salinity in excess of 20 psu. This two-layer stratified system allows one to use a straightforward approach to modeling the hydrological structure of the Black Sea in the laboratory.

[5] The general circulation of the Black Sea is characterized therefore by a basin-scale cyclonic boundary current [e.g., Neumann, 1942 Bogatko et al., 1979 ]. Two interconnected large-scale cyclonic gyres are formed in the interior of the western and eastern parts of the basin because of the narrowing in the central part of the sea. Persistent or recurrent features of this basin-wide circulation are also anticyclonic eddies along the coast. Some “taxonomy” of these features of the surface circulation of the Black Sea is provided by Oguz et al. [1993] . Some features are particularly worth noting in the context of the experimental results reported herein. One large anticyclonic eddy occupying the southeastern corner of the Black Sea is called Batumi Eddy. It is formed by the recirculation to the right of the Rim Current. The current separates from the coast in this region and crosses the sea toward the northern shore at approximately 40°E. Another persistent anticyclonic eddy (Sevastopol Eddy) is located west of the tip of the Crimean peninsula. It is similarly located to the right of the main jet of the Rim current which separates from the coast following the continental slope at this region. According to Ginzburg et al. [2002b] the Sevastopol Eddy often migrates slowly in the southwestern direction (arrow in Figure 1). Both of these eddies exhibit significant variability during the year. It can be clearly seen in the geostrophic velocity maps deduced from the TOPEX/Poseidon altimeter data [see Korotaev et al., 2001 , Figure 9 Korotaev et al., 2003 , Figures 7 and 12–13] that the Batumi Eddy starts to appear in winter and is most strong in spring while in summer and fall it is undistinguishable. Moreover, in summer and fall the separation of the jet from the northern shore rather than from the southern shore can be observed. This can result in a reversal of the circulation in the region of location of the Batumi Eddy although the reversed circulation is weaker than the circulation associated with the Batumi Eddy. The origin of the Batumi Eddy is still a subject of debate and its existence is sometimes attributed to the surface freshwater forcing associated with the intensification of precipitation toward the eastern coast of the Black Sea. Herein we will argue that the Batumi Eddy is a transient feature of the circulation occurring as a result of the separation of the boundary current due to variations of the intensity of the general circulation. The Sevastopol Eddy is subject to even stronger variability. In fact the identity of this eddy is artificial to large extent. Satellite imagery of this region of the sea often reveals two smaller anticyclones or even anticyclones accompanied by cyclonic eddies [ Ginzburg et al., 2000 ]. This combination constitutes a vortex dipole or mushroom-like current. Numerous recent observations [e.g., Fedorov and Ginzburg, 1992 Ginzburg, 1994 , 1995 Ginzburg et al., 2000 Afanasyev et al., 2002 Ginzburg et al., 2002a , 2002b ] using satellite imagery demonstrate significant mesoscale variability of the entire circulation system. The observed mesoscale features of the circulation include meanders, closely spaced anticyclonic and cyclonic eddies, dipoles and filaments. These hydrodynamical structures are also typical features of any quasi-two-dimensional turbulent flow [e.g., Voropayev and Afanasyev, 1994 ]. It is especially worth noting that regular arrays of meanders are often observed along the southern coast between 30°E and 40°E [e.g., Oguz and Besiktepe, 1999 , Figure 8 Ginzburg et al., 2000 , Figure 1 Afanasyev et al., 2002 , Figure 5]. These meanders are likely due to the baroclinic instability of the Rim Current modified by the coastal features in this region. Similar instability is also observed at the northern shore of the Black Sea. The temporal and spatial characteristics of the mesoscale variability in the Black Sea as well as dynamical processes involved are yet to be fully understood. It is important to study these flows not only because of their many practical connections such as the understanding of exchange between the shelf region and the interior of the sea, but also because such flows often reveal the influence of fundamental hydrodynamics interactions.

[6] The major mechanisms that force cyclonic circulation in the Black Sea are buoyancy flux due to river input and wind stress. The importance of the relative contributions of these two mechanisms is not completely clear. The buoyancy flux alone can provide the required cyclonic circulation according to the theoretical and experimental analysis by Bulgakov et al. [1996a , 1996b] . Recent estimates [ Efimov and Shokurov, 2002 Efimov et al., 2002 ], however, show that the seasonal variability of the circulation system is correlated with the vorticity of the wind over the sea. The wind fields are of a larger spatial extent than that of the Black Sea. The annual cycle of wind vorticity follows a simple harmonic law such that the cyclonic vorticity prevails in winter while the anticyclonic vorticity is persistent in summer. The amplitudes of the cyclonic and anticyclonic peaks are slightly different such that the annual average value is positively signed (cyclonic) and is approximately 10 −6 s −1 . The magnitude of peaks reaches 3–4 times of the average value. It is also worth noting that river discharge also follows an annual cycle with the maximum value occurring in May and the minimum in October. The seasonal variability of the discharge is significant. The difference between the peak and the mean value constitutes approximately 75% of the mean [ Kourafalou and Stanev, 2001 ]. The overall intensity of the circulation in the Black Sea also exhibits a strong seasonal variation. It attenuates in summer to fall and intensifies in winter to spring [ Korotaev et al., 2001 ]. The Rim Current jet is clearly observed in winter and spring while during summer months the mesoscale eddy variability is so large that it masks the Rim Current such that it can only be defined in terms of statistically defined transport. The temporal evolution of kinetic energy of the surface geostrophic circulation derived from the TOPEX/Poseidon altimeter data [see Korotaev et al., 2001 , Figure 10] is characterized by a strong peak in January/February which can be associated with atmospheric forcing and weaker peak in May which is probably due to the intensified river discharge. The variability of the kinetic energy can be as high as 70% of the mean value.

[7] In recent years numerous attempts have been made to model different dynamical aspects of the circulation of the Black Sea in numerical simulations [e.g., Oguz and Malanotte-Rizzoli, 1996 Staneva et al., 2001 Stanev and Staneva, 2000 Beckers et al., 2002 Stanev and Beckers, 1999 ]. Laboratory experiments on this subject however are relatively few. Bulgakov et al. [1996b] considered the process of development of quasi-stationary circulation induced by buoyancy fluxes in a rotating basin. It was demonstrated that the slow injection of fresh water at the surface and saltier water at an intermediate depth results in the formation of a cyclonic circulation at the surface with a countercurrent at an intermediate depth. A similarity of a number of control parameters including the Rossby and Ekman numbers was achieved in the experiments. The formation of the stationary circulation was observed on a significant timescale of several hours which was mainly controlled by the timescale of viscosity. The addition of a simple coastline in the form of two convex plates in the circular tank allowed the authors to show the formation of two macro gyres. The issue of variability of the general cyclonic circulation was considered in laboratory experiments by Zatsepin et al. [2002] . The cyclonic circulation in a circular tank was forced by a freshwater source at the surface. It was observed that the flow was stable initially when the value of the Burger number which was defined as the baroclinic Rossby deformation radius normalized by the characteristic size of the basin, was high enough. After some significant time period the upper layer was formed over the entire surface area and the value of the Burger number became lower. The flow began meandering eventually filling with eddies the entire area of the basin.

[8] In the new series of experiments reported herein, we follow in general the approach proposed by Bulgakov et al. [1996b] and Zatsepin et al. [2002] , which includes forcing the circulation by a buoyancy source. The present study is focused on the modeling of the unstable peripheral boundary current within a scaled model of the Black Sea. We will show that the laboratory model correctly reproduces the main features of the circulation system including meanders and mesoscale eddies occurring because of the finite amplitude development of the baroclinic instability of the boundary current.

[9] Seasonal variations of river inflow or wind stress cause variations of the intensity of the circulation. We reproduce these transient effects by varying the rotation rate of the platform in the second series of our experiments. The slow down of the background rotation corresponds to the intensification of the cyclonic circulation in the laboratory basin while the spin-up of the platform models the attenuation of the circulation. The results of these experiments demonstrate the occurrence of the transient features typical for the circulation in the Black Sea. We believe that specific features of the geometry of the coastline are important for the formation of some distinct features of the circulation, namely the big hook-shaped meanders, Batumi and Sevastopol eddies. The laboratory experiments described in this paper demonstrate in particular that Batumi and Sevastopol eddies are transient features which occur because of the separation of the boundary layer in the eastern part of the sea and at the tip of the Crimean peninsula. Complete dynamical similarity of laboratory flows and the flows in the Black Sea was satisfied in our experiments with respect to the normalized Rossby deformation radius, the Rossby number and the Ekman number.

[10] In the following sections of this paper, the laboratory setup, visualization and measuring techniques that we employ in our experiments are described in section 2, while section 3 contains the results and analysis of the experiments. Section 3 also contains an analysis of the main control parameters of the flow and the dynamical similarity between the laboratory flows and the flows in the Black Sea. We conclude with a discussion of our results.


3 Results and Discussion

3.1 A Cyclonic Eddy-Induced Large Kuroshio Intrusion Event

Vélez-Belchí et al. [ 2013 ] showed that there was a cyclonic eddy-induced large Kuroshio intrusion event in October 2008 (their Figure 6). To highlight the intrusion, we designated September as the period before eddy impact and October as the period during eddy impact. Both the AVISO-derived and HYCOM-simulated monthly mean surface velocity fields showed that the KC northeast of Taiwan (Figure 2) turned from northeastward in September to nearly northward in October and intruded further onto the midshelf. During this period, the KC interacted with a mesoscale cyclonic eddy propagated from the open ocean, and meandered offshore east of Taiwan. Trajectories data from 18 drifters that passed through the I-lan Ridge transect depicted paths of the KC during our study period (Figure 3A). All drifters flowed along the slope in September. Then in October most drifters moved directly north once they flowed over the I-lan Ridge and penetrated to the 200 m isobath, consistent with the simulated drifters' trajectories from the AVISO and HYCOM data (Figures 3B and 3C).

Mean geostrophic velocity (A) derived from AVISO sea surface height data and HYCOM model-simulated mean 15 m velocity (B) in September (top) and October (bottom). Gray contours are the 200 m isobaths.

(A) Trajectories of observed surface drifters, (B) simulated drifters with the surface geostrophic velocity from AVISO data, and (C) the 15 m velocity from HYCOM simulation results in September (top) and October (bottom). Dots represent the starting locations of drifters. Six drifters were observed in September and 12 in October. The simulated drifters were released every other day in each month and tracked by the fourth-order Runge-Kutta integration scheme.

During the same period, low Kuroshio transport was reflected by both tidal gauge and HYCOM-simulated sea level differences between Ishigaki and Keelung (Figure 4A). Our 7 month long ADCP mooring deployment northeast of Taiwan took concurrent measurements during this eddy-Kuroshio interaction period (Figures 4B and 4C).

(A) Observed and modeled sea level anomaly difference (unit: m) between Ishigaki and Keelung. Both time series are 36 h low-pass filtered. Period H (15 August to 15 September) is defined as the high (before eddy interaction) Kuroshio transport phase, and Period L (1 October to 31 October) is defined as the low (during eddy interaction) Kuroshio transport phase. (B) Depth-averaged (36 h low-pass filtered) along and cross-shelf velocities (unit: m s −1 ) measured by the mooring. Positive values are northeastward (shoreward) for the along-shelf (cross-shelf) velocities. Dashed line is the mean value (0.21 m s −1 ) of the along-shelf velocity. (C) Bottom temperature (36 h low-pass filtered, unit: °C) measured by the mooring. The dashed line indicates the mean temperature (18.45°C).

The depth-averaged subtidal current (36 h low-pass filtered) was decomposed into along-shelf (45° from north) and cross-shelf (315° from north) components. Mean and maximum along-shelf velocities were 0.21 and 0.56 m s −1 , respectively. Spectrum analyses showed a ∼2–5 day period in along and cross-shelf velocity, and an 11 day period in cross-shelf velocity and bottom temperature. A high correlation between velocity and sea surface wind over a ∼2–4 day period was found over the ECS continental slope northeast of Taiwan [Chuang and Liang, 1994 ]. The 11 day oscillation can be attributed to the Kuroshio baroclinic instability waves that were observed near the ECS slope [James et al., 1999 ]. With a 20 day low-pass filter, the correlation coefficient between along and cross-shelf velocities was −0.68, the correlation coefficient between along-shelf velocity and the bottom temperature was 0.67, and the correlation coefficient between cross-shelf velocity and the bottom temperature was −0.45. Thus, the along-shelf velocity indicated the Kuroshio intrusion, which raised the bottom temperature by delivering warmer water.

There was a persistent strong along-shelf flow in October, with a mean of 0.35 m s −1 , twice as fast as during the other 6 month observations (0.18 m s −1 ). Observed bottom temperature also showed a significant increase (mean 1.3°C, maximum 3.5°C) in October. Together, the velocity and temperature observations provide direct evidence for the Kuroshio intrusion. Although the eddy-induced strong intrusion only lasted for 1 month, it accounted for more than 30% of the total transport observed over the 7 month period, representing a significant contribution to cross-shelf water exchange.

In addition, there was another strong subtidal current from December to February (Figure 4B), which should be associated with the KBC induced by surface cooling [Oey et al., 2010 ]. The KVT also decreased and was associated with the impact of another weaker cyclonic eddy located southeast of Ishigaki (not shown here). Therefore, the Kuroshio intrusions in December–February were potentially initiated by both surface cooling and cyclonic eddy effects.

(1) where is the local Coriolis parameter, is the velocity vector, and is the viscosity coefficient. The surface boundary condition is (2) where is the constant density (1025 kg m −3 ) and is the surface wind stress. Here is 0.01 m 2 s −1 according to Price and Sundermeyer [ 1999 ]. After removing the Ekman current, the depth-averaged current was taken as the barotropic current, and the remaining current was taken as the baroclinic current (Figure 5). Compared with the barotropic and baroclinic currents, the Ekman currents were very small (e.g., ∼5 cm/s at 14 m) and thus negligible. Except in November, the magnitude of baroclinic current was only about 5–20% of that of barotropic current. This quasi-barotropic current should be largely related to stronger mixing associated with wind and surface cooling in late fall to early spring [e.g., Oey et al., 2010 ].

Vertical profiles of (A) observed along-shelf sub-tidal current velocity (color, unit: m s −1 ), and (B) its barotropic and (C) baroclinic components. All three time series are 36 h low-pass filtered.

To fill the spatial and temporal gaps in mooring observations and satellite data, the high-resolution data-assimilative HYCOM outputs for this period were used for further diagnostic analyses. At the sea surface, comparisons of the surface current (Figure 2), drifter trajectories (Figure 3) and sea level anomaly difference (Figure 4A) showed good performance by HYCOM. The deficiency was that both the extent of onshore intrusion and its westward migration along the 200 m isobath were weaker than observed (Figure 6A). However, the simulated bottom temperature at the mooring station matched well with the observation (Figure 6B). In general, the HYCOM outputs resolved the cyclonic eddy and its interaction with Kuroshio reasonably.

Comparisons between mooring observations and HYCOM simulation results at two adjacent grid points (124.5°E, 26°N) and (124.5°E, 26.5°N). (A) U and V are the depth-averaged along (45° from north) and cross (315° from north) shelf velocity (unit: m s −1 ), respectively. Uadcp and Vadcp are from mooring observations. U26N and V26N are model-simulated depth-averaged velocity at (124.5°E, 26°N), and U26.5N and V26.5N are model-simulated depth-averaged velocity at (124.5°E, 26.5°N). (B) Tadcp is the bottom temperature (unit: °C) from mooring observations. T26N and T26.5N are the model-simulated bottom temperature at (124.5°E, 26°N) and (124.5°E, 26.5°N), respectively. All time series are filtered by the 10 day running mean.

To focus on the eddy's impact, Period H (15 August to 15 September) was defined as the high (before eddy impact) Kuroshio transport phase, and Period L (1 October to 31 October) was defined as the low (during eddy impact) Kuroshio transport phase (Figure 4A). The P transect and the 200 m isobath northeast of Taiwan (see Figure 1) were selected to estimate the variability of both Kuroshio and its intrusion in the subsurface ocean, respectively. The mean normal velocities and their differences at the two locations are shown in Figure 7, and the volume transports across the two transects are shown in Table 1. With the impact of the cyclonic eddy, the onshore volume transport across the 200 m isobath from 122°E to 124°E decreased by 41.7%. It increased (37.9%) west of 122.6°E and decreased (73%) east of 122.6°E, showing a distinct westward migration from 123°E to 122°E–122.5°E. A weakened Kuroshio and an intensified (31.3%) northeastward current on the outer shelf can be seen along the P transect. The southwestward flow at the bottom along the slope disappeared completely. There was a remarkable shoreward axis migration, especially in the upper layer. The temperature, Brunt-Väisälä frequency, and their differences at the two transects are shown in Figure 8. Usually, the cyclonic eddy traps cold water and propagates westward. The arrival of the cyclonic eddy uplifted isopycnals in the offshore direction and depressed isopycnals around the slope region. As a result, in the main intrusion region (west of 123.25°E), the vertical stratification was weakened (enhanced) in the upper (lower) layer around the shelf break region and enhanced (weakened) in the upper (lower) layer in the deep ocean side. So the cyclonic eddy changed the thermocline tilting across the 200 m isobath northeast of Taiwan and weakened cross-shelf PV gradient in the upper layer, which played the similar role with the anomalous positive wind stress curl and surface cooling as shown in Wang and Oey [ 2014, 2016 ].

Profiles of mean normal velocities (unit: m s −1 ) at the 200 m isobath northeast of Taiwan (positive values are onshore, top) and at the P transect (positive values are northeastward, bottom) during (A) Period H, (B) Period L, and (C) the difference (B-A). The contour interval is 0.1 m s −1 .

Profiles of mean Brunt-Väisälä frequency (color, unit: 10 −3 s −1 ) at the 200 m isobath northeast of Taiwan (top) and P transect (bottom) during (A) Period H, (B) Period L, and (C) the difference (B-A). Contours are potential density (unit: kg m −3 ) during Period H (black) and Period L (green).

200 m Isobath P Transect
Time Period 122°E–122.6°E 122.6°E–124°E 122°E–124°E 25.7°N–26.5°N
Period L 2.95 Sv 37.9% 1.47 Sv −73% 4.42 Sv −41.7% 3.27 Sv 31.3%
Period H 2.14 Sv 5.44 Sv 7.58 Sv 2.49 Sv
Anticyclonic Eddies 2.03 Sv −4.7% 4.75 Sv 13.4% 6.78 Sv 7.3% 1.96 Sv −10.1%
JA 2.13 Sv 4.19 Sv 6.32 Sv 2.18 Sv
Cyclonic Eddies 2.82 Sv 30.6% 1.63 Sv −47.8% 4.45Sv −15.7% 3.0 Sv 31%
SO 2.16 Sv 3.12 Sv 5.28 Sv 2.29 Sv

3.2 Dynamical Interpretations of the Cyclonic Eddy-Induced Kuroshio Intrusion Variability

(3) where and is a volume transport vector with being the velocity is the JEBAR term with is the wind stress is the bottom stress, is the drag coefficient, and represents the averaged-velocity in the bottom layer (10 m above the seafloor) and two vectors, and , are the nonlinear advection term and the vertically intergrated horizontal diffusion term, respectively. The JEBAR term is the only term related to the density field, which was used to diagnose the contribution of the vertical stratification change to the Kuroshio intrusion across the ECS shelf break [Mertz and Wright, 1992 ].

The mean values of each PV term in equation 3 during Periods H and L were estimated along the 200 m isobath northeast of Taiwan. All other terms were smaller (not shown here) and negligible, except the APV (advection of the geostrophic potential vorticity, second term in the left-hand side of equation 3) and JEBAR terms (Figures 9A and 9B). The positive and negative APV terms correspond to onshore and offshore transport, respectively. In both periods, the JEBAR term basically balanced the APV term along the 200 m isobath, and the variability of the two terms coincided well with that of the Kuroshio intrusion. These two terms increased west of 122.6°E and decreased east of 122.6°E from Period H to Period L. These results suggest that JEBAR is the main term determining the cross-shelf advection. In other words, the change in local stratification, which appears as the JEBAR term, is responsible for the Kuroshio intrusion variability. A similar explanation was proposed by Oey et al. [ 2010 ] and Wang and Oey [ 2014 ], who attributed enhanced Kuroshio intrusion to an increased JEBAR term along the ECS shelf break introduced by strong sea surface wind curl and cooling.

Mean APV and JEBAR terms (unit: s −2 ) in a meridional range of 1/3° centered on the 200 m isobath northeast of Taiwan during (A) Period H and (B) Period L. (C) Depth-averaged PV (unit: s −1 m −1 ) in the upper 100 m at the P transect during Period H and L. Mean PV (color, unit: s −1 m −1 ) profiles during (D) Period H, (E) Period L, and (F) the difference (E-D). Contours are potential density (unit: kg m −3 ) during Period H (black) and Period L (green).

In addition to the impact of the cyclonic eddy, surface heat flux also has an effect on the density field, which may account for part of the JEBAR term. It is hard to separate the contribution of surface heat flux to the JEBAR term. The mean net heat flux (from the ERA-interim data set) on the shelf (122°E–124°E, 25.5°N–27°N) was 48.88 and −26.78 W m −2 during Periods H and L, respectively. Oey et al. [ 2010 ] produced increased onshore velocity of about 30% in winter with a seasonal heat flux of ±400 W m −2 . If we use this result as a reference, the onshore velocity anomaly induced by the surface heat flux during Period L will be about 3% and thus negligible. Therefore, the local ocean vertical stratification change during Period L was mainly caused by the cyclonic eddy.

The above diagnostic analyses indicate that the strong Kuroshio intrusion in October 2008 is mainly due to the large JEBAR term from eddy-induced local density field change. But how could the density change help the Kuroshio overcome the topographic PV barrier in the slope region? The constraint of bottom topography on the current in the upper layer is determined by the vertical structure of density field. In the case of a barotropic fluid, the current can fully feel the topography variation when it collides with a slope. However, the baroclinicity of the fluid could reduce the bottom topography PV constraint to the upper layer. Taking the ocean as a two-layer fluid and assuming that fluid in the lower layer is nearly stagnant, the topography variation in the lower layer does not affect the upper layer current. As mentioned earlier, in the lower ocean layer around the shelf-break, the vertical stratification was enhanced and would eliminate the pressure gradient, which made the southwestward flow at the bottom disappear completely during Period L (Figures 7 and 8). This weakens the bottom topography PV constraint and leads the upper layer Kuroshio to intrude into the shelf region.

The PV during the two periods (Figures 9D and 9E) was calculated using the equation , where is PV, f is the local Coriolis parameter, is relative vorticity, and is density. In general, due to the difference of vertical stratification, the PV in the upper layer is largest on the shelf and decreases gradually offshore, which generates a strong PV gradient across the slope and blocks the Kuroshio water intrusion. During the strong intrusion Period L, the cyclonic eddy carried positive PV flux and increased PV in the offshore direction by lifting the isopycnal layers (Figure 9E). The depth-averaged PV gradient in the upper 100 m across the slope (around 25.5°N) was weakened from 2.89 × 10 −9 s −1 m −1 to 0.34 × 10 −9 s −1 m −1 per degree (Figure 9C).

So far, based on mooring observations and HYCOM assimilative data, the large intrusion has mainly been attributed to the eddy-induced vertical stratification change and PV flux. Are these features and interpretations universal and applicable during other cyclonic eddy events? And how about anticyclonic eddy events? In the following section, we will discuss these issues by using long-term HYCOM reanalysis data.

3.3 Kuroshio Intrusion Variability Induced by Mesoscale Eddies

Using the trajectories of mesoscale eddies detected by Chelton et al. [ 2011 ], we traced open ocean eddies that arrived eat of Taiwan (122°E–125°E, 22°N–25°N) from 2004 to 2012. There were 20 anticyclonic eddies and 24 cyclonic eddies. The arrival of anticyclonic (cyclonic) eddy could rise up (bring down) the sea level at the east side of Kuroshio and then enhance (weaken) the KVT across the I-Lan Ridge transect. Therefore, the KVT anomaly was used as an indicator of the impact of mesoscale eddies. The extreme positive KVT anomaly represents the influence of an anticyclonic eddy, while the extreme negative KVT anomaly represents the influence of a cyclonic eddy.

Based on the HYCOM reanalysis data from 2004 to 2012, the mean daily KVT across the I-Lan Ridge was 26.54 Sv (anomalies are shown in Figure 10A). The KVT was filtered with a 2 years high-pass filter, and the extreme KVT was defined as events exceeding 1.2 standard deviations (std) of the demeaned time series. Over 9 years, summer and autumn stand out as the seasons having the greatest number of extreme KVT events (Figure 10B). The total time of the extreme high KVT events in July and August (JA) was 165 days (red stars), and the total time of the extreme low KVT events in September and October (SO) was 122 days (blue stars). There were nine extreme high KVT events in JA with a mean KVT anomaly of 8.44 Sv, while there were seven extreme low KVT events in SO with a mean KVT anomaly of −9.46 Sv. In contrast, the KVT anomaly in JA and SO climatology (composited by the percentage of each month) was only 3.09 Sv and −1.99 Sv, respectively. In this study, we focused on the extreme KVT events in JA and SO to investigate the impact of mesoscale eddies on the Kuroshio and its intrusion.

(A)Two years high-pass filtered KVT anomalies (unit: Sv) derived from HYCOM simulation results. Dashed lines are ±1.2 std (5.69 Sv) of KVT anomalies. The red stars correspond to the KVT anomalies that are higher than 1.2 std in JA, while the blue stars correspond to the KVT anomalies that are lower than −1.2 std in SO. (B) Monthly distribution of the extreme KVT anomalies which exceed the 1.2 std ranges of the demeaned time series. (C) Trajectories of nine eddies. Red and blue lines correspond to anticyclonic and cyclonic eddies, respectively. “+” and “o” corresponds to the position of eddy's generation and end, respectively. (D) Composite fields of extreme high KVT events in JA and (E) low KVT events in SO: mean velocity at 15 m depth (arrows) and sea level anomaly (color, unit: m).

We traced mesoscale eddies east of Taiwan for each of the 16 KVT anomaly events (Figure 10C). Among these eddies, the mean amplitude, rotation speed, and radius of the nine anticyclonic eddies were 20.9 cm, 39.7 cm s −1 , and 139 km, respectively. The mean amplitude, rotation speed, and radius of the seven cyclonic eddies were 30.2 cm, 48.3 cm s −1 , and 139.1 km, respectively. Most of these eddies were generated in the STCC region near the western boundary only one anticyclonic eddy was tracked back to east of 150°E.

HYCOM reanalysis fields during these extreme high KVT events in JA and low KVT events in SO were averaged to form composite velocity fields at 15 m and sea surface height anomalies to describe the impact of anticyclonic and cyclonic eddies (Figures 10D and 10E), respectively. Similar to Gawarkiewicz et al. [ 2011 ], we find that during the impact of anticyclonic eddies, the sea surface height composite showed a positive anomaly east of Taiwan and a weak negative anomaly at the southern edge of the ECS shelf. The Kuroshio was intensified, but the onshore intrusion northeast of Taiwan was inconspicuous. In contrast, during the impact of cyclonic eddies, the sea surface height composite showed a negative anomaly east of Taiwan and a larger positive anomaly at the southern edge of the ECS shelf. The Kuroshio was weaker but had a larger meander northeast of Taiwan. The onshore intrusion reached the midshelf, consistent with the statistical results of 20 year drifter tracks [Vélez-Belchí et al., 2013 , Figure 8]. In addition, the directions of the flow through the Kerama Gap in the two composites were opposite, which should be associated with the impact of mesoscale eddies east of Taiwan.

The mean normal velocities and their anomalies during the impact of eddies at the 200 m isobaths and P transect were calculated, and the volume transports across them are shown in Table 1. In contrast to the results of JA climatology, it shows an intensified Kuroshio and a weakened (10.1%) northeastward current on the outer shelf with the impact of anticyclonic eddies (Figure 11). In addition, the southwestward flow at the bottom along the slope was enhanced. The onshore volume transport across the 200 m isobath from 122°E to 124°E increased by 7.3%. However, it decreased (4.7%) west of 122.6°E and increased (13.4%) east of 122.6°E. With the impact of cyclonic eddies, a weakened Kuroshio and an intensified (31%) northeastward current on the outer shelf were shown compared with the results of SO climatology (Figure 12). The southwestward flow at the bottom along the slope almost disappeared. There was a significant shoreward axis migration, especially in the upper layer. The onshore volume transport across the 200 m isobath from 122°E to 124°E decreased by 15.7%. However, it increased (30.6%) west of 122.6°E and decreased (47.8%) east of 122.6°E, which shows a conspicuous westward migration of Kuroshio intrusion.

Profiles of mean normal velocities (unit: m s −1 ) at the 200 m isobath northeast of Taiwan (positive values are onshore, top) and P transect (positive values are northeastward, bottom) during (A) the impact of anticyclonic eddies, (B) JA climatology, and (C) the difference (A-B). The contour interval is 0.1, 0.1, and 0.05 m s −1 in A, B, and C, respectively.

Profiles of mean normal velocities (unit: m s −1 ) at the 200 m isobath northeast of Taiwan (positive values are onshore, top) and P transect (positive values are northeastward, bottom) during (A) the impact of cyclonic eddies, (B) SO climatology, and (C) the difference (A-B). The contour interval is 0.1, 0.1, and 0.05 m s −1 in A, B, and C, respectively.

The temperature, the Brunt-Väisälä frequency, and their anomalies at the two transects during eddy impact were also calculated. During the impact of anticyclonic eddies, isopycnals were uplifted around the shelf-break and depressed in the offshore region (Figure 13). The stratification was enhanced (weakened) in the upper (lower) ocean around the shelf-break. In the offshore region, the stratification was weakened in the upper layer and almost remained unchanged in the lower layer. The impact of cyclonic eddies on the stratification is opposite comparing to that of anticyclonic eddies (Figure 14).

Profiles of mean Brunt-Väisälä frequency (color, unit: 10 −3 s −1 ) at the 200 m isobath northeast of Taiwan (top) and P transect (bottom) during (A) the impact of anticyclonic eddies, (B) JA climatology, and (C) the difference (A-B). Contours are potential density (unit: kg m −3 ) during JA climatology (black) and the impact of anticyclonic eddies (green).

Profiles of mean Brunt-Väisälä frequency (color, unit: 10 −3 s −1 ) at the 200 m isobath northeast of Taiwan (top) and P transect (lower panels) during (A) the impact of cyclonic eddies, (B) SO climatology, and (C) the difference (A-B). Contours are potential density (unit: kg m −3 ) during SO climatology (black) and the impact of cyclonic eddies (green).

We also performed PV budget analyses with the HYCOM reanalysis data to examine the role of the density field in the Kuroshio intrusion during four periods: the impact of anticyclonic eddies, JA climatology, the impact of cyclonic eddies, and SO climatology. All other PV terms in equation 3 are smaller (not shown here) and negligible along the 200 m isobath northeast of Taiwan, except the APV and JEBAR term. In the four periods, the JEBAR term basically balances with the APV term along the 200 m isobath (Figure 15), and the variability of the two terms coincides well with the Kuroshio intrusion. These two terms decreased (increased) west of 122.6°E and increased (decreased) east of 122.6°E with the impact of anticyclonic (cyclonic) eddies, revealing a large westward migration of the maximum with cyclonic eddies. These results suggest that the change in local vertical stratification is responsible for the intrusion variability. Both the surface net heat flux (from the ERA-interim data set) and its horizontal gradient during the eddy periods showed little change (Figure 16), so the contribution of surface heat flux forcing to the variation of JEBAR term is negligible. Thus, we concluded that the variability of ocean vertical stratification and Kuroshio intrusion during these extreme KVT events was mainly determined by mesoscale eddies.

Mean APV and JEBAR terms (unit: s −2 ) in a meridional range of 1/3° centered on the 200 m isobath northeast of Taiwan during (A) the impact of anticyclonic eddies, (B) JA climatology, (C) the impact of cyclonic eddies, and (D) SO climatology.

Mean sea surface net heat flux (unit: W m −2 ) during (A) the impact of anticyclonic eddies, (B) JA climatology, (D) the impact of cyclonic eddies, and (E) SO climatology. Mean sea surface net heat flux anomaly during the impact of (C) anticyclonic eddies (A-B) and (F) cyclonic eddies (D-E). The contour interval is 10 W m −2 . The thick black contours are the coastline, and the gray contours are the 200 m isobaths.

The dynamical interpretations of the Kuroshio intrusion variability, proposed from a previous single cyclonic eddy event, were also verified. During the impact of anticyclonic eddies, a strengthened PV constraint of bottom topography on the upper layer current from the joint effect of weakened vertical stratification and enhanced southwestward flow in the lower layer at the shelf break can be derived. The anticyclonic eddy inputs additional PV flux during its interaction with the Kuroshio, decreasing PV offshore (Figure 17C). As a result, the depth-averaged PV gradient in the upper 50 m across the slope (around 25.5°N) is increased from 2.18 × 10 −9 s −1 m −1 to 2.94 × 10 −9 s −1 m −1 per degree (Figure 18A), which inhibits the Kuroshio onshore intrusion. During the impact of cyclonic eddies, a weakened PV constraint of bottom topography on the upper layer current from the joint effect of intensified vertical stratification and weakened southwestward flow in the lower layer at the shelf-break region can also be derived. The cyclonic eddy inputs additional PV flux during its interaction with the Kuroshio, increasing PV offshore (around 24.6°N in Figure 17F). As a result, the depth-averaged PV gradient in the upper 100 m across the slope (around 25.5°N) is significantly reduced from 1.19 × 10 −9 s −1 m −1 to 0.48 × 10 −9 s −1 m −1 per degree (Figure 18B), enabling the Kuroshio to overcome the PV barrier and intrude onto the midshelf.

Mean PV (color, unit: s −1 m −1 ) profiles during (A) the impact of anticyclonic eddies, (B) JA climatology, (D) the impact of cyclonic eddies, and (E) SO climatology along the P transect. Mean PV anomaly during the impact of (C) anticyclonic eddies (A-B) and (F) cyclonic eddies (D-E). Contours are the potential density (unit: kg m −3 ).

Depth-averaged PV (unit: s −1 m −1 ) in the upper 50 m during the impact of anticyclonic eddy and JA climatology (A), and in the upper 100 m during the impact of cyclonic eddy and SO climatology (B) at the P transect.

In this section, based on the 9 year HYCOM assimilative data, the variability of the Kuroshio, the onshore intrusion, and surrounding ocean vertical stratification were investigated through nine anticyclonic (JA) and seven cyclonic (SO) eddy events. The variability induced by multiple cyclonic eddy events is consistent with that of the single cyclonic eddy event in October 2008. The variability induced by anticyclonic eddies was generally opposite to that of cyclonic eddies, and the onshore intrusion variability with cyclonic eddies was larger in this study. The dynamical interpretations on how the current in the upper layer overcomes the ECS slope PV constraint are applicable to multiple strong eddy events.


Rotating Thermal Flows in Natural and Industrial Processes

Rotating Thermal Flows in Natural and Industrial Processes provides the reader with a systematic description of the different types of thermal convection and flow instabilities in rotating systems, as present in materials, crystal growth, thermal engineering, meteorology, oceanography, geophysics and astrophysics. It expressly shows how the isomorphism between small and large scale phenomena becomes beneficial to the definition and ensuing development of an integrated comprehensive framework. This allows the reader to understand and assimilate the underlying, quintessential mechanisms without requiring familiarity with specific literature on the subject.

Topics treated in the first part of the book include:

  • Thermogravitational convection in rotating fluids (from laminar to turbulent states)
  • Stably stratified and unstratified shear flows
  • Barotropic and baroclinic instabilities
  • Rossby waves and Centrifugally-driven convection
  • Potential Vorticity, Quasi-Geostrophic Theory and related theorems
  • The dynamics of interacting vortices, interacting waves and mixed (hybrid) vortex-wave states
  • Geostrophic Turbulence and planetary patterns.

The second part is entirely devoted to phenomena of practical interest, i.e. subjects relevant to the realms of industry and technology, among them:

  • Surface-tension-driven convection in rotating fluids
  • Differential-rotation-driven (forced) flows
  • Crystal Growth from the melt of oxide or semiconductor materials
  • Directional solidification
  • Rotating Machinery
  • Flow control by Rotating magnetic fields
  • Angular Vibrations and Rocking motions

Covering a truly prodigious range of scales, from atmospheric and oceanic processes and fluid motion in "other solar-system bodies", to convection in its myriad manifestations in a variety of applications of technological relevance, this unifying text is an ideal reference for physicists and engineers, as well as an important resource for advanced students taking courses on the physics of fluids, fluid mechanics, thermal, mechanical and materials engineering, environmental phenomena, meteorology and geophysics.


Planned Papers

The below list represents only planned manuscripts. Some of these manuscripts have not been received by the Editorial Office yet. Papers submitted to MDPI journals are subject to peer-review.

Title: Inversion and general invariance in Space-Time
Authors: John Williamson Martin van der Mark
Affiliation: University of Glasgow Quicycle
Abstract: A general formula for division in a relativistic Clifford-Dirac algebra is derived. Where division is undefined turns out, in many cases, to correspond to invariants of dynamical significance, such as the light cone, the general invariant quantities in electromagnetism, and to the basis set of quantities in the Dirac equation. Apart from such areas, where there has already been significant development in science, new sets of inter-related quantities, involving the spin and the total energy for example, are suggested as possible areas for further investigation and development. The scope, and hence the abstract for "Invariance, inversion and inter-action" is still under development and will follow later.

Title: The geometrical meaning of spinors as a key to make sense of quantum mechanics
Authors: Gerrit Coddens
Affiliation: Laboratoire des Solides Irradiés
Abstract: This paper aims at explaining that the key to understanding quantum mechanics (QM) is a perfect geometrical understanding of the spinor algebra that is used in its formulation. Spinors occur naturally in the representation theory of certain symmetry groups. The spinors that are relevant for QM are those of the homogeneous Lorentz group SO(3,1) in Minkowski space-time R4 and its subgroup SO(3) of the rotations of three-dimensional Euclidean space R3. In the three-dimensional rotation group the spinors occur within its representation SU(2). We will provide the reader with a perfect intuitive insight about what is going on behind the scenes of the spinor algebra. We will then use the understanding acquired to derive the free-space Dirac equation from scratch proving that it is a description of a statistical ensemble of spinning electrons in uniform motion, completely in the spirit of Ballentine’s statistical interpretation of QM. This is a mathematically rigorous proof. Developing this further we allow for the presence of an electromagnetic field. We can consider the result as a reconstruction of QM based on the geometrical understanding of the spinor algebra. By discussing a number of problems in the interpretation of the conventional approach, we illustrate how this new approach leads to a better understanding of QM.

Title: Iterants and the Dirac Equation
Authors: Louis H. Kauffman
Affiliation: Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago Department of Mechanics and Mathematics, Novosibirsk State University
Abstract: -


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