Friday, February 12, 2016

The Projective Plane: A Visual Introduction

The statement "any two distinct lines intersect in a point" is almost true in normal plane geometry. The exception, of course, is the case of two parallel lines. However, from real experience we know from the rules of perspective that two parallel lines "converge" very far away, even if we know that they in fact maintain the same distance apart.

From this, we naturally comes the intuition that "parallel lines intersect at infinity." Certainly this tidies up our intersection statement because it provides a way for even parallel lines to intersect. But what does "at infinity" mean? Is there really a "point" there? The notion of projective space makes these ideas explicit and rigorous.

We focus on the (real) projective plane, the extension of the normal plane to include these "points at infinity" where parallel lines intersect. The set of points in the projective plane is defined, somewhat enigmatically, as "the set of lines through the origin in three-dimensional space." Defining each point to be a line in a different space seems extremely confusing at first, but there are multiple ways to visualize this concept.

The first method of visualization illustrates how the projective plane is related to the ordinary plane (sometimes called the affine plane). Consider three-dimensional space with ordinary coordinates x,y, and z as shown. The plane labeled z = 1 contains all points for which the z coordinate is 1, namely all those of the form (x,y,1). Clearly this plane is just like the ordinary two-dimensional plane (under the correspondence (x,y,1) → (x,y)), only embedded in three dimensions, like a flat sheet of paper in our world (but infinite). The dotted line shown that passes through the origin intersects the plane at the particular point (a,b,1). Remembering that the projective plane is meant to be an extension of the affine plane, we identify the dotted line with the point where it intersects the plane. Clearly, for each point in the z = 1 plane, there exists exactly one line through the origin and the given point. This shows how the ordinary plane is a subset of projective space (the set of lines through the origin)!

However, not every line through the origin intersects the plane z = 1. For instance, the x-axis, the y-axis (both shown), and any other line in the plane of these two axes only contain points for which the z coordinate is 0 and can never intersect the plane z = 1 (to be clear, the three-dimensional space considered here does not have its own points at infinity!). Therefore, these lines cannot correspond to points on the ordinary plane. These special lines, in fact, are the points at infinity in the projective plane.

The above visualization illustrates the connection between the projective plane and the affine plane. It also indicates that there are many points at infinity, one for each line through the origin "lying flat" in the xy-plane. However, it fails to indicate how points at infinity are truly the intersections of parallel lines. For this, we use another visualization that chooses different representative points.

Using a sphere (or a hemisphere, to be more precise) to represent the projective plane is just as legitimate as using a plane: all that matters is that there is one point for each line through the origin. It does not matter which points we choose.

In fact, nearly every person is intimately familiar with this representation of projective space! Imagine that it is a clear night and you go out to look at the stars. You catch sight of the familiar constellation Orion, the hunter. The stars marking Orion's shoulders are Betelgeuse and Bellatrix, which we perceive to be neighboring stars that connect to form the figure of Orion. In fact, however, Betelgeuse is between two and three times as distant as Bellatrix. When we look up at the sky, we do not perceive the true three-dimensional space but points of light etched into the inner surface of the celestial sphere passing overhead. Stars in similar directions, regardless of their distances, are projected onto nearby points. This is why the result of treating all points along a line through the origin as equivalent is known as projective space.

It is clear, however, that every line through the origin intersects the sphere at exactly two points, while there can only be one representative for a point of projective space. Thus, by convention we consider only intersections with the upper hemisphere (just as in our example of the night sky - one cannot see stars looking downward!). This leaves only the "horizontal" lines intersecting the equator of the sphere twice. For these, we choose the points of intersection for positive y-values (the area colored dark green above) and finally the x-axis is represented by the dark red point of positive x. The projective plane is therefore the union of the yellow upper hemisphere, the dark green semicircle, and the dark red point. The latter two parts are the points at infinity.

The above image shows how the affine plane (and our first visualization) relate to our second visualization of the projective plane as (part of) a sphere. Lines through the origin (O) and a point in the upper hemisphere intersect the plane to form a one-to-one correspondence. As we would expect, points at infinity correspond to lines through the sphere's equator that are parallel to the plane and are therefore not part of our original affine plane.

Finally, the sphere illustrates how the projective plane solves the motivating problem of parallel lines than began this post.

Two parallel lines in the plane correspond to precisely the same lines in our first visualization, which indeed embeds a "copy" of the affine plane in three-dimensional space. When these parallel line are transferred to the sphere in the same manner that the point was above (remember: each transferred point represents a line through the origin and "transferring" a point is merely choosing a different representative), the figure above is the result. However, it is evident that the resulting arcs on the sphere intersect at the equator (green circle) and we know the equator contains the points at infinity! Though there appear to be two intersections, recall that points diametrically opposite from one another are on the same line through the center, so that these points are identified as one in the projective plane. We have our desired result: two parallel lines intersect in exactly one point.

The next post (coming soon) provides an algebraic description of the projective plane and explores more of its properties.

Sources: Algebraic Curves: An Introduction to Algebraic Geometry by William Fulton,,,,

Friday, January 22, 2016

Applications of Ion Propulsion

This is the second part of a two-part post. The first post, describing the function of ion propulsion engines, may be found here.

Ion thrusters have many advantages over other forms of propulsion. In comparison with traditional chemical propellent, they are roughly 10 times more efficient, lightening the load of space-traveling craft and saving massive amounts of fuel for launch. This efficiency originates in part from the higher exhaust velocity of the xenon propellant particles, which are ejected from the spacecraft at speeds of 20-50 km/s! In addition, the electric power required to run ion engines is relatively small, on the order of a few kilowatts. In comparison, a typical microwave oven consumes 1.1 kW, and the power consumption is significantly less than that of a typical automobile. Solar panels can meet this power demand during flight, allowing ion thrusters to create smooth and continuous acceleration over the entire duration of a mission.

However, this efficiency comes at a price: ion propulsion produces very small thrusts. The first model of ion engine actually used in spaceflight, the NSTAR engine, produced a thrust of around 90 millinewtons. This is the same force that your hand would experience from a single piece of paper on Earth by gravity, a barely detectable force! However, this minute force can operate continuously, adding up to a significant acceleration over time in the frictionless environment of space. In comparison, space probes which operate on chemical propellant may exert thrusts in the hundreds or thousands of newtons (the equivalent of a couple hundred pounds at Earth's surface), but only for very short times.

The small thrust of ion engines pales even more in comparison to that of rockets that launch payloads from Earth's surface. To escape Earth's gravity, such rockets must exert a force greater than the force of gravity on the often huge rockets. For example, the Saturn V rocket (shown above) that launched humans to the Moon generated an astounding 34,500,000 newtons of thrust at launch. For this reason, ion engines cannot be used to launch spacecraft.

The idea of electric rock propulsion dates to the 1930's and the first test of ion engines in space came during the 1960's. However, no operational mission utilized ion thrusters until NASA's Deep Space 1 probe, launched in 1998. This spacecraft performed a flyby of the asteroid 9989 Braille and the comet 19P/Borrelly. To meet the acceleration requirements of the mission extension to comet Borrelly, Deep Space 1 changed its velocity by 4.3 km/s using less than 74 kilograms of xenon. The Hayabusa probe, launched by the Japan Aerospace Exploration Agency (JAXA) in 2003, was another important demonstration of ion thruster technology. This probe used ion thrusters on its mission to land on the near-Earth asteroid 25143 Itokawa, collect samples, and return them to Earth for analysis. In 2010, the probe successfully returned the sample to Earth, completing the first ever asteroid sample return mission. Hayabusa 2, another asteroid sample return mission, launched in 2014 with similar objectives.

An even greater demonstration of ion propulsion technology began in 2007 with the launch of the Dawn spacecraft. This spacecraft carried three ion engines, operating alternately throughout the mission. Thrusting frequently throughout its eight-year journey, Dawn followed a spiral outward from the Earth, past Mars, into orbit of the asteroid Vesta, and subsequently into orbit of the asteroid Ceres.

The above image shows Dawn's outward spiral as well as the intervals during which one of its ion engines was in operation. With the exception of the gravitational assist at Mars, Dawn thrusted almost continuously, moving outward under its own power. Its total velocity change exceeded 10 km/s, the greatest yet for a spacecraft under its own power. When it successfully entered orbit around Ceres in April 2015, Dawn became the first spacecraft to orbit two different extraterrestrial targets. This feat would not have been possible using traditional methods of chemical propulsion.

Meanwhile, advances continued in ion thruster technology. NASA's Evolutionary Xenon Thruster (NEXT) performed a 48000 hour test in a vacuum chamber lasting from 2004 to 2009 to demonstrate its successful operation. Using power at a higher rate but also providing somewhat greater thrust, the NEXT engine (shown above) is at least 30% more efficient than its predecessors and can operate for a longer time. Continued development of ion propulsion technology promises to provide the foundation necessary for more ambitious interplanetary space missions.


Wednesday, January 13, 2016

Hurricane Alex (2016)

Storm Active: January 13-15

On January 7, a low pressure system situated along a front northeast of the Bahamas began deepening, producing a large area of strong winds over the western Atlantic. Though strong upper-level winds and cool ocean temperatures precluded immediate development into a tropical or subtropical cyclone, the National Hurricane Center began to monitor the disturbance. The low moved eastward, remaining frontal in nature, but strengthened even more, producing maximum winds to hurricane force on January 10. Over the next few days, shower activity increased modestly near the system's center as it took a southeast heading into the far eastern Atlantic. By January 12, bands of shallow convection surrounded a well-defined center. The next day, despite marginal sea surface temperatures, thunderstorm activity increased near the center. Due to the relatively shallow convection and gale force winds associated with the system, it was classified Subtropical Storm Alex that afternoon. Alex was only the fourth known tropical or subtropical system to form in the north Atlantic basin in January.

By the time of its formation, Alex had turned toward the northeast and was headed in the direction of the Azores. Meanwhile, despite marginal sea surface temperatures, convection continued to deepen and Alex developed a well-defined eye feature. At the same time, the upper-level low situated over Alex moved away, allowing the cyclone to transition to a tropical cyclone. Since the eyewall now had hurricane force winds, Alex was upgraded to a hurricane during the morning of January 14. It became the first hurricane in January since 1955, and the first to form during the month since 1938. By the afternoon, the outer bands began to affect the Azores Islands. The same evening, it reached its peak intensity of 85 mph winds and a central pressure of 981 mb. The next morning, the center of circulation passed among the central Azores, bringing hurricane-force winds to the region as it sped northward. Meanwhile, the eyewall disintegrated and the convective structure became lopsided as Alex began extratropical transition and weakened slightly. The system became extratropical that afternoon.

The above image shows Alex at peak intensity less than a day before it passed over the Azores. The hurricane developed a remarkable eye feature highly unusual for an off-season storm.

The track of Alex includes several days during which the system was an extratropical system producing winds near hurricane force.

Friday, January 1, 2016

Introduction to Ion Propulsion

In physics, ion propulsion is a type of electric propulsion used by spacecraft. As with any traditional method of rocket propulsion, ion propulsion depends on Newton's Third Law: for every action, there is an equal and opposite reaction.

A typical rocket engine uses internal mechanisms to accelerate some type of exhaust away from the rocket. Since this constitutes a force on the exhaust, the engine experiences a force in the opposite direction. Crucially, propulsion requires that mass be lost from the rocket to exhaust. Other vehicles, such as cars, use friction between wheels and road to provide a force and therefore do not need to expel mass. Operating in space or the atmosphere in which friction is minimal (there is nothing to "push off" of), rockets instead carry extra mass to accelerate. As the name suggests, ion propulsion works by accelerating ions.

The above schematic illustrates the function of a gridded electrostatic ion thruster (which is usually what is meant by "ion propulsion"). On the left side, neutral atoms of the propellant move from storage tanks (not shown) into the ionization chamber. Simultaneously, an electrode fires electrons into the chamber with high velocity. These electrons knock other electrons out of the neutral propellant atoms to create ions. As a result, the ionization chamber becomes filled with electrons and positive ions.

At the other end of the chamber are two grids. They are connected to a voltage source that maintains a static positive charge on the inner grid and an equal and opposite negative charge on the outer one. Two effects combine to remove many of the free electrons from the ionization chamber. First, the positively charged plate attracts electrons, conducting them out of the chamber. Second, the contents of the chamber are very hot. Since electrons are much lighter than the positive ions, they move faster with the same amount of thermal energy and have a greater chance of collecting on the grid. Soon, positively ionized gas (plasma) builds up in the chamber.

This closeup on a gap in the positive grid shows how positive ions eventually escape the chamber. Eventually, so many ions accumulate in the plasma layer that the repulsive force between ions exceeds the force pushing this ions away from the positive grid. Ions then pass through gaps in the grid. Once they reach the other side, the repulsive forces from both the grid and the other plasma accelerate them outward. Returning to the larger diagram, the negative grid focuses the beam of ions so that they all proceed in roughly the same direction. Finally, another electrode fires electrons at the escaping ions, preventing them from losing velocity due to the influence of the negative grid and preventing a buildup of net charge in the engine.

Typically, the type of propellant used for ion propulsion is xenon gas. The use of xenon, which is element 54 on the periodic table, has several advantages. First, it is a noble gas so it is inert and does not react chemically with other parts of the engine. Further, it is the heaviest (non-radioactive) noble gas, so the thermal effect removing electrons from the gaseous plasma is enhanced.

NASA tested the ion thruster shown above in the early 1990's. The blue glow originates from charged xenon particles.

The next post describes the applications of the ion thruster and its impact on space travel.


Tuesday, December 22, 2015

2015 Season Summary

The 2015 Atlantic hurricane season had below-average activity, with a total of

12 cyclones attaining tropical depression status,
11 cyclones attaining tropical storm status,
4 cyclones attaining hurricane status, and
2 cyclones attaining major hurricane status.

Before the beginning of the season, I predicted that there would be

9 cyclones attaining tropical depression status,
7 cyclones attaining tropical storm status,
3 cyclones attaining hurricane status, and
1 cyclone attaining major hurricane status.

The season was in fact slightly below average, since the average numbers of tropical storms, hurricanes, and major hurricanes are 12.1, 6.4, and 2.7, respectively. However, the actual totals exceeded my predictions in all categories. Nonetheless, a majority of the cyclones that did form were short-lived, a fact which is quantitatively confirmed by the low value of the Accumulated Cyclone Energy. This value, which was 59 for the 2015 season, takes into account both the intensity and longevity of each storm. 59 is only 67% of the thirty year median and is lower than that of the 2014 season, though it featured a smaller number of cyclones.

The greatest influence on cyclone development of the 2015 season was the strong El Nino event. Indicated by anomalously warm ocean temperatures over the equatorial East Pacific region, this event began in 2014 but strengthened significantly through 2015, becoming one of the three strongest events in recorded history and at least the strongest since 1997-8. As a result, strong upper-level winds prevailed across the Caribbean sea and Gulf of Mexico (as well as other parts of the tropical Atlantic). Notably, no tropical storms formed in the Caribbean during the season, and the two that entered it, Danny and Erika, were torn apart by wind shear shortly after doing so.

In contrast, high sea surface temperatures supported tropical cyclone formation. The above map, showing global temperature anomalies during September 2015 (the heart of hurricane season) show a band of record warm ocean temperatures over the subtropics, extending from the Bahamas and southeastern U.S. coast eastward. This band supported the formation of Hurricane Joaquin toward the end of September and early October, which became the strongest storm of the season with 155 mph winds and a minimum pressure of 931 mb. Record sea temperatures fueled this system in spite of the El Nino; the comparable El Nino events of 1997, 1982, and 1972 had no hurricanes even close to this strength. The season included a few tropical storm landfalls in the United States, but Joaquin's impact on the Bahamas was by far the most devastating.

Some other notable facts and statistics concerning this season include:
  • Tropical Storm Ana made the second-earliest U.S. landfall in the Carolinas on May 10, behind only the unusual Groundhog Day storm of 1952
  • Hurricane Fred was named at 18.9° W, one of only 4 storms to be named this far east; it also was the first storm in a century to affect the Cape Verde Islands as a hurricane
  • Hurricane Joaquin was the strongest cyclone in the Atlantic basin since Igor of 2010

Overall, the season was quiet, with Joaquin as the only notable cyclone in terms of intensity and damage.


Monday, November 9, 2015

Hurricane Kate (2015)

Storm Active: November 8-12

Kate originated from a trough of low pressure that moved across the tropical Atlantic during the first week of November. The system moved west-northwest across the northeastern Caribbean, passing near Puerto Rico on November 7. Early on November 8, a low pressure center appeared on the south end of the trough just north of Hispaniola and thunderstorm activity became better organized. The low became well-defined enough to be classified Tropical Depression Twelve late that evening. The next morning, the depression was upgraded to Tropical Storm Kate as deep convection increased near the center. At that time, the storm was moving northeast over the Bahamas, but its small size limited rainfall totals.

After the center briefly became exposed on the south side of the convective canopy, organization began to improve, with banding features appearing. Kate strengthened as it moved northward away from the Bahamas during the afternoon. The system meanwhile began to accelerate toward the north and eventually northeast on November 10. Despite diminishing sea surface temperatures, the storm continued to intensify through November 11, when it strengthened into a hurricane. Hurricane Kate reached its peak intensity of 75 mph winds and a pressure of 985 mb that morning, at which time it was already north of Bermuda, speeding toward the northeast at 40 mph. The cyclone was already showing signs of extratropical transition that afternoon as it moved over the cold northern Atlantic. By November 12, the circulation had become quite elongated and the remaining convection warm and asymmetric, so Kate was declared extratropical. The still powerful system continued north and east across the Atlantic before merging with another low a few days later.

The above image shows Kate at peak strength moving rapidly out to sea on November 11. The cyclone was already exhibiting some asymmetry and diminished central convection, two signs of extratropical transition.

Hurricane Kate affected the Bahamas as a tropical storm before recurving away from the United States coastline.

Monday, September 28, 2015

Hurricane Joaquin (2015)

Storm Active: September 27-October 8

On September 25, a trough of low pressure situated over the tropical western Atlantic began to exhibit some signs of organization as it moved slowly toward the north-northwest. The next day, a surface low formed in association with the system and thunderstorm activity became more concentrated. On September 27, the low deepened and became better defined. Persistent deep convection near the center of circulation therefore led to the designation of Tropical Depression Eleven late that evening.

Immediately after formation, Eleven already faced increasing shear out of the north-northwest that exposed the center of circulation. Situated to the south of a high pressure system over the western Atlantic, the cyclone moved slowly and erratically over the next day, heading generally westward. Late on September 28, the cyclone's center suddenly moved rapidly southwestward, coming closer to the deep convection. As a result, organization increased, and the system was upgraded to Tropical Storm Joaquin. Shear lessened on September 29 while the cyclone was situated over very warm waters. As a result, Joaquin began a period of fairly rapid strengthening. Deep convection slowly became more symmetric about the center, and the system reached strong tropical storm intensity that afternoon. Meanwhile, the ridge over the northeast Atlantic grew stronger, pushing Joaquin southwest toward the central Bahamas. Early on September 30, healthy outflow was established in the northern semicircle and the beginnings of an eye appeared on satellite imagery. As a result, the system was upgraded to a category 1 hurricane.

Joaquin's central pressure continued to plummet that day. Even though a stable eye had yet to appear on infrared imagery, cloud tops in the eyewall cooled significantly and the system became more symmetric overall. The outer bands of the system affected the central Bahamas during that morning as the storm approached. During the evening, Joaquin exploded in intensity to become a major hurricane, the most powerful of the season. By the morning of October 1, the center of the storm entered the central Bahamas. The cyclone made a little more progress southwestward that day, walloping the Bahamas with high winds, a sustained period of torrential rain, and substantial storm surge in some locations. That afternoon, Joaquin achieved category 4 status, reaching its peak intensity intensity of 130 mph winds and a minimum pressure of 931 mb. This was the lowest pressure observed in an Atlantic hurricane since Hurricane Igor in 2010. Early on October 2, Joaquin began its long anticipated turn toward the north and then northeast as a trough moving eastward over the United States began to influence its motion. Meanwhile, though Joaquin never directly affected the U.S. east coast, tropical moisture from the hurricane interacted with a frontal system moving eastward to bring immense amounts of rain to the Carolinas.

Joaquin finally moved away from the Bahamas later that day. The eye clouded over but reappeared again on October 3, causing some fluctuations between category 3 and 4 intensity. For a brief period that afternoon, the eye of Joaquin became very well-defined and winds were estimated to reach 155 mph, just under category 5 strength (these were the strongest winds by an Atlantic hurricane since Igor). However, the hurricane did not beat its previous minimum pressure. After this secondary peak, wind shear began to weaken Joaquin. Meanwhile, acceleration toward the north and east continued, and the system began to affect Bermuda as a category 2 hurricane on October 4. During the evening, around 8:00 pm EDT, Joaquin made its closest approach to Bermuda, passing about 65 miles to the west-northwest. The hurricane was close enough to bring tropical storm force winds and hurricane gusts to the island.

Wind shear abated while the system was still over marginally warm water overnight and through the morning of October 5, halting weakening at category 1 status. Joaquin continued to maintain a well-defined eye even as it gained latitude. On October 6, the hurricane turned northeastward and accelerated toward the open northern Atlantic. That night, the system began to lose tropical characteristics as it moved north of the Gulf Stream into colder waters and convection became asymmetric. It weakened to a tropical storm on October 8 and became post-tropical late that evening.

Joaquin was an unusually powerful hurricane for the El Nino conditions in which it formed. Anomalously warm ocean temperatures contributed to its intensity.

Hurricane Joaquin was very difficult to predict early in its lifetime, leading to large forecast errors in the extent of its southwestward motion and timing of its northeastward turn.

Friday, September 18, 2015

Tropical Storm Ida (2015)

Storm Active: September 18-27

Yet another tropical wave entered the eastern Atlantic on September 13, bearing generally westward south of the Cape Verde Islands. A low pressure center appeared in association with the system soon after. By the 15th, the disturbance assumed a more west-northwestward heading as the ridge to its north remained somewhat weak. Convection remained vigorous throughout the next few days, but the low was unable to take advantage of mainly favorable conditions. However, thunderstorm activity became much more concentrated near the center early on September 18, and the formation of Tropical Depression Ten followed later that morning. The depression strengthened into Tropical Storm Ida that night.

By the morning of September 19, the newly named tropical storm began to experience wind shear out of the northwest. Though banding features and deep convection remained healthy on the south and east quadrants, the center of circulation became exposed to the northwest. Over the next day, Ida continued to lose organization. During the afternoon of September 20, the cyclone's forward speed suddenly increased, but this trend was to reverse almost immediately as the storm approached an area of weak steering currents by that evening. Meanwhile, shear relaxed and Ida's convection came roaring back: a huge area of intense convection appeared and covered the center. As a result, the system began to strengthen.

However, this new trend was stopped by a new feature in the environment - a pronounced increase in mid-level shear. Throughout the day of September 21, the shower and thunderstorm activity slowly separated from the center. In addition, the lower- and upper-level centers themselves became separated, making the system difficult to pinpoint. By September 22, the influence of a large trough to the north of Ida caused it to cease progress towards the west-northwest and instead drift in the entirely opposite direction, toward the east-southeast. The havoc caused by the wind shear caused the system's circulation to fragment somewhat, with multiple low-level vortices evident as cloud swirls on satellite imagery west of the convective canopy. Overnight, as it hovered around minimal tropical storm strength, Ida even took an unusual southward turn. On September 23, however, the though began to carry Ida on a more steady path toward the east.

Though the trough began to move away from the system, the destructive influence of dry air and wind shear did not abate, causing Ida to weaken to a tropical depression on September 24. Shortly afterward, a ridge to the system's north forced it to switch directions again, this time to the north-northwest. The cyclone continued to persist in the face of unfavorable conditions, pulsating periodically with new convection. Therefore, it maintained tropical depression status into September 26. Ida's motion turned more toward the west that day and began to degrade further as bursts of shower activity became less frequent. By the afternoon of September 27, Ida had been devoid of convection so long that it no longer met the criteria for a tropical cyclone and degenerated into a remnant low. The low continued generally northwest for several days, even showing some signs of organization near the end of September. However, no redevelopment occurred and the system eventually merged with a frontal boundary.

The above image shows the disorganized tropical storm Ida over the central Atlantic.

Ida did not approach any landmass throughout its lifetime as a tropical cyclone.

Thursday, September 17, 2015

Tropical Depression Nine (2015)

Storm Active: September 16-19

On September 10, a tropical wave began to produce some shower activity as it moved off of the coast of Africa. Atmospheric conditions were initially unfavorable for any further development, but the system's environment improved on September 12, allowing it to organize. Meanwhile, it passed well south of the Cape Verde Islands, moving on a west-northwestward trajectory. On September 14, dry air interrupted the system's development and convection collapsed. The low associated with the system continued to persist and deepen, however, as upper-level winds remained low. After making a turn to the northwest, the system rebounded on September 16 and was classified Tropical Depression Nine.

Almost immediately after becoming a tropical cyclone, Nine entered an area of strong upper-level winds as it moved closer to an upper-level low to the northwest. Despite its convection being displaced well to the east of the center, Nine still managed a little strengthening early on September 17. In the midst of powerful wind shear associated with the strong El Nino event, the depression did little besides generate limited convection east of the center throughout that day and the next. By September 19, the system had become devoid of convection for the final time and the circulation became elongated along the southeast-northwest axis. Shortly afterward, the system lost tropical cyclone status. Its remnants continued generally northwestward until losing their identity.

Tropical Depression Nine was at its most organized immediately after formation (as shown above) while it remained in an environment that supported development.

The above image shows the track of Nine.

Wednesday, September 9, 2015

Tropical Storm Henri (2015)

Storm Active: September 8-11

On September 7, a low pressure system formed near the tail of a stationary front situated over the central Atlantic, well to the southeast of Bermuda. Despite inhibitive upper-level winds, the system began to produce a concentrated area of shower and thunderstorm activity while it remained nearly stationary. The low became better defined on September 8 and though convection was largely confined to the eastern side of the circulation, it had acquired enough organization that night to be classified Tropical Depression Eight. At the time of its classification, Eight had a large area of maximum winds and some other subtropical characteristics, but was more tropical than subtropical.

Though the satellite presentation remained extremely disorganized, with a broad circulation barely in contact with thunderstorm activity to the east, the system's maximum winds did increase somewhat over the next day and Eight was upgraded to Tropical Storm Henri late on September 9. Soon after, the cyclone began to move northward as the ridge holding it place slowly eroded. It passed well east of Bermuda during the morning of September 10, accelerating toward the north as it did so. Wind shear relaxed later that day, but the poorly organized system, still battling incursions of dry air, was unable to take advantage and remained at minimal tropical storm strength. The circulation became elongated by September 11 with multiple low-level swirls evident on satellite imagery. Henri lost its well-defined center during that afternoon and degenerated into a trough. Its remnants merged with a frontal boundary the next day.

Never in its lifetime did Henri develop any convective structure that circumnavigated the center of circulation. This contributed to its inability to significantly intensify.

Henri (and its progenitor system) remained stationary for a few days due to a ridge to the north before an incoming front allowed it to accelerate poleward.