Sunday, December 9, 2018

2018 Season Summary

The 2018 Atlantic Hurricane Season had above-average activity, with a total of

16 cyclones attaining tropical depression status,
15 cyclones attaining tropical storm status,
8 cyclones attaining hurricane status, and
2 cyclones attaining major hurricane status.

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

18 cyclones attaining tropical depression status,
16 cyclones attaining tropical storm status,
8 cyclones attaining hurricane status, and
4 cyclones attaining major hurricane status.

The average number of named storms, hurricanes, and major hurricanes for an Atlantic hurricane season (over the 30-year period 1981-2010) are 12.1, 6.4, and 2.7, respectively. The 2018 season was somewhat above average in these categories, with the exception of the number of major hurricanes. The formation of many short-lived subtropical storms inflated the named storm total, but the Accumulated Cyclone Energy (ACE) value of 127 for the season was still above average. This value accounts for the duration and intensity of tropical cyclones as well as their number.

As usual, the ENSO oscillation was a major player in tropical cyclone activity this year. As hurricane season progressed into autumn, ocean temperatures of the equatorial Pacific trended higher than normal, signaling the advent of an El Niño event. Typically, such an event causes higher wind shear over the Atlantic and suppresses tropical cyclone activity, but it arose later in the year than anticipated, mitigating its effects.

The increase in wind shear during El Niño is most pronounced in the Caribbean Sea. Indeed, this region was a "graveyard" for tropical cyclones during 2018, as indicated by the above map of all the season's tracks. Every storm that entered the eastern Caribbean dissipated shortly thereafter due to unfavorable atmospheric winds. Ocean temperatures in the tropical Atlantic east of the Caribbean were also fairly cool for much of the season. This setup prevented long-track hurricanes from forming, with one notable exception: Hurricane Florence. Florence took a highly unusual route farther north but still pushed westward into the U.S. east coast. Overall, my predictions were slightly higher than the actual season activity, but they did correctly indicate the risk to the east coast.

The two most notable storms of the season were Hurricane Florence and Hurricane Michael. Florence made landfall in North Carolina, where it stalled and brought over 30 inches of rain to some areas. The record-breaking rainfall caused unprecedented flooding and extensive damage. Michael brought torrential rain to central America as it was forming and then went on to strengthen right up until landfall in the Florida Panhandle. With a pressure of 919 mb at landfall, Michael was at the time the 3rd most intense cyclone ever to make landfall in the United States. Some other notable facts and records from the 2018 Atlantic season include:
  • The 2018 season had seven storms that were at some point subtropical, a new record
  • On September 12, Florence, Helene, Isaac, and Joyce all coexisted in the Atlantic. This was the first time four named storms existed simultaneously since 2008
  • Hurricane Leslie took a highly unusual track over the far eastern Atlantic near the end of its lifetime. As a result, the first tropical storm warning on record was issued for Madeira Island southwest of Portugal; Leslie became post-tropical just before landfalling in Portugal itself
Overall, the 2018 season was only a little above average but nevertheless featured two devastating major hurricanes.


Saturday, October 27, 2018

Hurricane Oscar (2018)

Storm Active: October 26-31

On October 23, an area of low pressure a few hundred miles east-northeast of the Lesser Antilles began to produce some shower and thunderstorm activity. Over the next few days, the disturbance moved slowly northward and atmospheric conditions for development improved. As westerly shear diminished, convection persisted closer to the center of the developing low. Late on October 26, the low-level center had become well-defined. However, due to its interaction with a nearby upper-level low, the system was classified Subtropical Storm Oscar. This was the seventh named storm in 2018 to be subtropical during some part of its lifetime, making 2018 the first known season for such an occurrence.

Interaction with the upper-level low caused Oscar to turn sharply westward on the 27th. Meanwhile, significant deepening took place, indicating that Oscar's maximum winds had increased. Oscar's structure evolved throughout that day until the cyclone possessed a small core of deep convection and maximum winds close to the center. As a result, it was reclassified as a tropical storm. A ridge pushed the system south of west on the 28th and favorable conditions allowed an eye feature to begin forming. Around the same time, Oscar strengthened into a hurricane. The trend of gradual intensification persisted on October 29 and the system became a category 2 that evening, reaching its peak intensity of 105 mph winds and a pressure of 970 mb. Meanwhile, a mid-latitude frontal system approaching from the west began to sheer the cyclone toward the north.

By the 30th, Oscar was picking up speed toward the north and north-northeast and began to encounter cooler waters as it passed east of Bermuda. As a result, deep convection near the center waned, the maximum winds dropped, and the system began to take on extratropical characteristics. Nevertheless, it remained a potent cyclone and brought rough surf to Bermuda that day. Around midday on October 31, Oscar transitioned to a hurricane-strength extratropical storm as it sped north-northeastward over the open Atlantic. The post-tropical low deepened over the north Atlantic during the following days, reaching a minimum pressure of 950 mb on November 2. It dissipated a few days later near Iceland.

This image shows the small hurricane Oscar at peak intensity as a category 2 hurricane.

Oscar did not directly affect any land areas during its time as a tropical or subtropical cyclone.

Tuesday, October 9, 2018

Tropical Storm Nadine (2018)

Storm Active: October 9-12

A late-season tropical wave entered the Atlantic ocean on October 6 and began to show signs of development. Waters in the eastern Atlantic were still fairly warm and shear was low, so organization proceeded fairly quickly. By October 8, convection had wrapped nearly around the disturbance, but it still lacked a closed circulation. The next day, it formed into Tropical Depression Fifteen. Within a few hours, satellite intensity estimates supported its upgrade into Tropical Storm Nadine. Nadine formed unusually late in the season for a system so far east in the tropical Atlantic.

The cyclone was fairly small, and hence prone to rapid changes in intensity. Over the next day, it took advantage of a favorable environment and strengthened quickly to its peak intensity as a strong tropical storm with 65 mph winds and a pressure of 997 mb on October 10. However, wind shear sharply increased that night and displaced all of Nadine's convection to the east of the center by October 11. As a result, the storm decayed rapidly as it moved northwest. The next day, Nadine dissipated over the central Atlantic.

Nadine was a small cyclone that quickly succumbed to unfavorable atmospheric conditions a few hours after formation.

The short-lived Nadine did not affect any land areas, but was an unusually late-season storm to form in the central tropical Atlantic.

Sunday, October 7, 2018

Hurricane Michael (2018)

Storm Active: October 7-12

During the first few days of October, a broad area of low pressure developed in the southwestern Caribbean, with associated showers and thunderstorms extending from central America all the way to Jamaica and Haiti. Such systems are common in this region in the autumn, and are known as Central American Gyres (CAGs). CAGs tend to bring heavy rainfall to a wide area of central America, which held true in this case. In addition, they can sometimes spawn tropical cyclones. Nevertheless, the large circulation of a CAG takes time to consolidate, and the system only slowly organized as it moved northwestward. By October 6, the center of the low was just north of Honduras and the region of strong upper-level winds that had been affecting the system retreated to the north. This allowed further organization, and a flare up of organized thunderstorm activity led to the classification of Tropical Depression Fourteen early on October 7.

Even immediately after formation, the storm had an impressive satellite signature, with very cold cloud tops to the east of the center of circulation. Soon after, satellite and aircraft data indicated an immense radius of gale force winds, extending over 200 miles from the center in some quadrants, and the depression was upgraded to Tropical Storm Michael. Under the influence of some westerly shear, the center of Michael underwent some reformations toward the east that day, but the large cyclone strengthened steadily into the evening as it moved slowly northward. Already, the outer rainbands were hitting the southern tip of Florida, even though the center was still just east of the Yucatan Peninsula. By the morning of October 8, Michael had strengthened into a hurricane.

During that day, the inner core gradually became more organized and the cyclone steadily intensified as it passed near the western tip of Cuba. A large eyewall struggled to surround the center throughout the day, but coverage increased during the evening. The storm became a category 2 early on October 9. The system gained some forward speed toward the north that day and the warm waters of the Gulf of Mexico supported extremely intense convection. Shear also lessened, and the eyewall became complete early that evening, bring Michael to category 3 strength. The outer bands of the storm were now crossing the Gulf coastline, but proximity to land did nothing to slow the system's intensification. A symmetric eye cleared out on satellite imagery overnight and Michael rocketed to category 4 status, deepening even when the center was within 50 miles of landfall. The powerful cyclone reached a peak intensity of 155 mph winds and a pressure of 919 mb when it slammed into the Florida Panhandle around 1:00pm local time on October 10. In terms of pressure, Michael became the third strongest hurricane ever on record to make landfall in the United States, behind only Hurricane Camille of 1969 and the Labor Day Hurricane of 1935. At the time, it also broke the top 10 overall list for strongest landfall recorded for an Atlantic hurricane. Furthermore, it was the only category 4 ever known to hit the Florida panhandle.

The storm surge that Michael brought to the coastline was unprecedented and record-braking in some areas and the wind damage was catastrophic, though the worst of it was confined to quite a small area where the eyewall made landfall. However, the rainfall was not especially severe, as the system accelerated northeastward as it moved inland and did not linger. In fact, the system entered southwestern Georgia before losing major hurricane status, thus becoming the first major hurricane to impact the state since 1898. Nevertheless, the core did rapidly deteriorate once inland, and Michael weakened to a tropical storm early on October 11 over central Georgia. Later that day, it moved through the Carolinas, bringing heavy rain and wind to regions inundated by Florence's rains the previous month. Fortunately, the storm was moving so fast that the flooding impacts were not as severe as they otherwise would have been. The cyclone emerged over the Atlantic near the border of North Carolina and Virginia and became post-tropical early on October 12. The system crossed the Atlantic and eventually brought some rain and wind to western Europe on October 15.

The above image shows Hurricane Michael making landfall in the Florida panhandle at strong category 4 intensity. This was among the strongest landfalls ever recorded for an Atlantic hurricane.

Michael's speedy development amid only marginally favorable conditions and rapid strengthening prior to landfall were very unexpected and demonstrate how far there is to go in modeling intensity changes in tropical cyclones.

Sunday, September 23, 2018

Hurricane Leslie (2018)

Storm Active: September 23-25, September 28-October 13

On September 18, an extratropical system associated with the remnants of Hurricane Florence moved away from the U.S. east coast over the tropical Atlantic. A new low formed along the frontal boundary around September 22 in the central subtropical Atlantic. Over the next day, the low developed spiral banding and lots its frontal nature. By the morning of September 23, it had transitioned into Subtropical Storm Leslie. This was the sixth subtropical storm of the 2018 season, setting a new record.

At the time of formation, Leslie was drifting westward, but steering currents were quite weak and it turned southward and ultimately eastward over the next day. The system had never had much in the way of deep convection, but what was there diminished further by September 25. Meanwhile, a new front was approaching from the west and interacting with Leslie, elongating its circulation. By late that morning, the system had become extratropical. Upon transition, it underwent a rapid burst of the strengthening and was producing hurricane force winds by the 26th. Since it was non-tropical, however, it was not designated a hurricane.

At the same time it continued to turn toward the north and then back west. Conditions were still fairly favorable for tropical cyclone development so it began to transition back the next day. On September 28, enough deep convection had reappeared near the center for Leslie to again be classified as subtropical. However, its maximum winds had subsided back to around 50 mph, so this was the initial intensity. The system moved slowly southwest over the next few days and gradually developed more banding features south and east of the circulation center. Leslie transitioned to a fully tropical storm for the first time on September 29. Sea surface temperatures increased and wind shear decreased along the storm's path, leading to some slow strengthening over the next few days as thunderstorms finally wrapped entirely around the center.

By October 2, Leslie was approaching hurricane strength and had dipped in latitude to below 30° N due to its unusual southwestward motion. A ragged eye formed that evening and the system was upgraded to a hurricane for the first time. Overnight, the cyclone became stationary around 500 miles east-southeast of Bermuda. It also peaked in intensity at maximum sustained winds of 80 mph and a central pressure of 975 mb. Due to the influence of an upper-level low pressure system to the north, Leslie began to move northward on October 3. This motion took it over cooler water, and convection waned again, with a shallow ring of convection around the center separated from the outer bands by a "moat" of dryer air. Leslie began to weaken as a result and soon was a tropical storm again.

Although the system was still quite distant from any landmasses, the large size of the circulation generated significant ocean swells that led to rough surf in Bermuda and even the east coast of North America. Leslie stalled again about 450 miles northeast of Bermuda on October 5, and began to feel the influence of the mid-latitude westerlies. The cyclone turned sharply eastward that day. Meanwhile, the structure of the storm had changed quite a bit; a central area of strong thunderstorms had replaced the large eye, and a large area of convection persisted to the north of the center. Leslie began to separate from a trough to its north and turned south of east on October 7. The storm accelerated southeastward over the next day, bringing it over warmer waters, and it began to restrengthen.

The cyclone developed a central dense overcast on October 8 and approached hurricane strength on the 9th, achieving category 1 status that evening a week after doing so the first time. Leslie turned due south for a little while on the 10th, reaching a southernmost latitude of 27.8 ° N. However, another trough moving to its north turned the system east-northeast and began to accelerate it toward the far eastern Atlantic. The inner core structure fluctuated a great deal in organization during the following day, but overall it became a bit better defined and Leslie strengthened somewhat. Late on October 11, Leslie reached its peak intensity as a top-end category 1 hurricane with 90 mph winds and a pressure of 969 mb.

The system picked up even more speed the next day and colder waters weakened the storm's convection. A tropical storm warning was issued for the island of Madeira, located southwest of Portugal. This was the first ever warning issued for the island and Leslie was the first known tropical cyclone ever to affect it in modern history. The center passed north of Madeira later on the 12th. Finally, on October 13, Leslie transitioned to an extratropical low just before making landfall in northern Portugal. This transition did not prevent the cyclone from bringing hurricane force winds gusts and heavy rain to the Iberian Peninsula. The low finally dissipated inland a few days later.

This image shows Leslie during its second and final stint as a hurricane, moving east-northeastward toward Europe.

Leslie's convoluted track included some highly unusual southward dips over the central Atlantic. Just after becoming extratropical, it moved over the Iberian Peninsula, though this is not shown above.

Saturday, September 22, 2018

Tropical Storm Kirk (2018)

Storm Active: September 22-24, 26-

On September 21, a strong tropical wave entered the Atlantic. It began to travel westward at quite a low latitude, far to the south of the Cabo Verde Islands. It organized quickly and developed a closed circulation the next day. Satellite measurements indicated gale force winds at that time, so the system was named Tropical Storm Kirk on September 22. At the time of naming, Kirk's latitude was 8.3 °N, quite close to the equator for tropical cyclone genesis in the Atlantic. In fact, no north Atlantic system on record had reached tropical storm strength so far south since 1902.

Shortly after formation, a ridge to the north of Kirk began to accelerate it westward. Due to its fast forward motion and its proximity to the equator, the system had difficulty acquiring much spin, although some evidence of curved banding started to appear on September 23. After that time though, it struggled to maintain deep convection. In addition, its forward speed continued to increase to over 20 mph. On the 24th, this caused the circulation to fall apart and Kirk dissipated.

The remnants continued quickly westward and sea surface temperatures increased while upper-level winds remained fairly favorable. As a result, the system began to reorganize. On September 26, a well-defined center reformed and Kirk regained tropical storm status. It even underwent some strengthening that day to its peak intensity of 60 mph winds as measured by hurricane hunter aircraft. However, its window of favorable conditions was short-lived. By early on September 27, it had entered the area of wind shear that dominated the east Caribbean and its immediate vicinity for most of the season. This quickly exposed the center of circulation as convection was displaced eastward. Nevertheless, the center passed over the Lesser Antilles that day and deep convection continued to flare up in the eastern semicircle, bringing heavy rains and gale force winds to the islands, some of which were still recovering from the passage of Hurricane Maria the previous year.

After entering the Caribbean, Kirk began to lose organization more quickly. Late on September 28, the system dissipated in the eastern Caribbean.

Friday, September 21, 2018

Tropical Depression Eleven (2018)

Storm Active: September 21-23

On September 18, a disturbance developed in the tropical Atlantic well east of the Windward Islands. It moved generally west-northwestward over the following view days and began to exhibit a small but organized canopy of thunderstorm activity. A weak low pressure center appeared on September 20, but wind shear increased significantly around the same time and the atmosphere was quite dry as the system approached the Caribbean. Despite unfavorable conditions, sheared convection persisted near the surface circulation center long enough the next day for the system to be classified Tropical Depression Eleven.

Before long, upper level winds out of the west increased even further, and any bursts of deep convection from Eleven were swiftly blown away. The center moved erratically the next day and began to lose definition, and by the morning of September 23, the system was downgraded to a remnant low. This low dissipated soon after.

Tropical Depression Eleven never achieved much organization during its brief lifetime.

The area of high wind shear over the Caribbean was persistent during the 2018 season, and claimed Eleven as another victim.

Thursday, September 13, 2018

Tropical Storm Joyce (2018)

Storm Active: September 12-18

On September 11, a non-tropical low formed along a frontal boundary situated over the north central Atlantic, well west of the Azores. The low drifted generally southward or southwestward over the next day. By September 12, it was producing gale force winds and displayed some organized convective banding. However, the surface low was still colocated with an upper-level low, not the high found for tropical cyclones, so it was designated Subtropical Storm Joyce. Existing contemporaneously with Florence, Helene, and Isaac, the new system was one of four named storms simultaneously occupying the Atlantic basin. This was the first time this had occurred since 2008. Also, Joyce was the fifth subtropical cyclone of the season, the first time that had happened since 1969.

Shortly after formation, the cyclone felt the influence of the much larger Tropical Storm Helene to its southeast. Steered around the periphery of its circulation, Joyce moved west-southwest and then turned south around Helene's left side as it reached the same latitude. Meanwhile, on September 13, Joyce transitioned into a tropical storm. The next day, it strengthened slightly to a moderate tropical storm and turned eastward in the wake of Helene. This intensification was short-lived, however, as increasing wind shear out of the southwest stripped away the little convection that formed in bursts near the center of circulation. On September 16, Joyce weakened to a tropical depression.

The shallow system was left behind by now ex-Helene and instead followed the boundary of a mid-level high located in the subtropical Atlantic. This caused the depression to turn south of due east by September 17 and then south by the 18th. Late on September 18, Joyce had ceased to produce deep convection and was finally downgraded to a remnant low. The low moved slowly southwestward until dissipation.

The above image shows the small Tropical Storm Joyce as well as the edge of the larger Tropical Storm Helene to the east, which greatly influenced Joyce's motion.

Joyce did not affect any landmasses during its journey through the northeast Atlantic.

Friday, September 7, 2018

Hurricane Isaac (2018)

Storm Active: September 7-15

At the beginning of September, a tropical wave moved into the Atlantic. Though it was producing some showers and thunderstorms, but the disturbance remained disorganized for several days as it traveled westward. By September 5, the low had become better defined, but convection was quite limited near the center of circulation. On September 7, the disturbance was classified Tropical Depression Nine. That day, it also became nearly stationary as steering currents collapsed and it felt the pull of newly formed Tropical Depression Eight (which would become Helene) to its east. The center was nearly exposed at first due to shear, but the system's organization increased considerably on September 8. This prompted an upgrade to Tropical Storm Isaac.

The storm was very small, with tropical storm force winds extending only a few dozen miles from the center. Such storms are subject to rapid changes in intensity, and Isaac did gain strength quickly over the next day. Meanwhile, it finally picked up some forward speed toward the west. During the evening of September 9, it was upgraded to a hurricane. There was little change to the system over the next day, despite generally quite favorable conditions. The banding structure and core of Isaac struggled to improve, even with low wind shear. Soon, the satellite presentation became more ragged in appearance and the system was downgraded back to a tropical storm.

Early on September 12, an upper-level trough north of Isaac caused a sudden increase in shear on the system, quickly stripping convection away from the center. The cyclone began to rapidly weaken as a result. Nevertheless, it caused scattered heavy rains and tropical storm force winds as it passed among the Leeward Islands and entered the Caribbean during the morning of September 13. Thunderstorm activity started to make a comeback near the circulation center later that day, but the circulation itself was ill-defined and showed signs of becoming elongated. Isaac weakened to a tropical depression on September 14. Soon after, all traces of a closed circulation vanished and the storm dissipated. The remnants of Isaac brought scattered thunderstorms to Jamaica a few days later.

This image shows the small Tropical Storm Isaac moving over the open Atlantic.

Isaac dissipated shortly after entering the eastern Caribbean. This is a quite common event and this region is often referred to as a "tropical cyclone graveyard" by meteorologists.

Hurricane Helene (2018)

Storm Active: September 7-16

On September 6, an extremely large tropical wave entered the Atlantic basin from west Africa, to which it had brought torrential rains over the past few days. By the time it hit water, the system already possessed a clear spin, and only a lack of central convection lay between it and tropical cyclone designation. This was immediately remedied on September 7 as more concentrated thunderstorms developed near the center. That afternoon, it was designated Tropical Depression Eight, only a few hundred miles west of Senegal. The massive system consolidated fairly quickly for its size and strengthened into Tropical Storm Helene shortly after.

There was some Saharan dry air to the north of Helene, but convection still managed to wrap around the center on September 8 and the storm continued to intensify. That evening, the center passed well south of the Cabo Verde islands, bringing some rain due to its large size, but sparing them from worse impacts. The next day, Helene was upgraded to a hurricane as it moved west away from the islands and it closed off a very large eye that evening. The cyclone turned toward the west-northwest by September 10. The eye also became better defined and Helene intensified into a category 2 hurricane. The next day, Helene reached its peak intensity of 110 mph sustained winds and a minimum central pressure of 966 mb.

By this time, a weakness in the subtropical ridge to the north of Helene (caused by a low over the northeast Atlantic) allowed the system to turn northwest and then toward the north. As it gained latitude, the cyclone encountered lower water temperatures and a drier atmosphere. It weakened to a category 1 on September 12, and to a tropical storm the next day.

Helene began to appear less tropical as it accelerated northward since it became increasingly asymmetric. Nevertheless, it maintained its status as a strong tropical storm into September 14. Soon after, rain bands in its eastern semicircle began to sweep across the Azores Islands. The center of circulation passed just west of the islands on September 15 and veered northeastward. Helene picked up more forward speed and transitioned into an extratropical cyclone over the cold northeastern Atlantic on September 16. The system eventually brought some stormy conditions to Ireland and the UK a few days later.

Helene was a large system, and developed an unusually large eye when it intensified into a hurricane.

Though it did not affect any large landmasses, Helene did have impacts in both the Cabo Verde and Azores Islands.

Monday, September 3, 2018

Tropical Storm Gordon (2018)

Storm Active: September 3-8

During the final days of August, a tropical wave whose axis extended from the Caribbean south of Hispaniola northward to the adjacent Atlantic waters began to produce increased thunderstorm activity throughout the region. However, upper level winds were quite strong over the wave. This prevented further development for a few days as it moved toward the west-northwest. The system passed over the southern Bahamas on September 2 and its satellite presentation improved markedly, though it still lacked a surface circulation. Only on September 3 did Florida radar detect a well-defined surface low, prompting the naming of Tropical Storm Gordon. At the time of naming, Gordon's center was over southern Florida. It moved over the Gulf of Mexico shortly afterward but continued to bring rain to the southern half of the peninsula for the remainder of the day.

While the core intermittently showed signs of an eyewall forming on radar, the overall satellite presentation of Gordon was lackluster as it moved through the Gulf over the next day. The radius of tropical storm force winds remained very small and heavy rain did not extend much beyond it. Nevertheless, maximum winds at the center reached strong tropical storm intensity by September 4. The waters of the Gulf of Mexico were quite warm, but fortunately Gordon's forward speed remained fairly high due to consistent steering patterns, and it moved quickly northwest. Late in the day, the cyclone reached its peak intensity of 70 mph winds and a pressure of 997 mb just before making landfall along the Gulf coast near the border of Mississippi and Alabama. Mississippi was spared most of Gordon's flooding rain impacts since almost all rain was to the east of the center. Some hurricane force gusts were also reported during landfall.

The system quickly weakened inland and became a tropical depression on September 5. Its movement slowed considerably once over land and it gradually moved northward through the midwest over the following couple of days, bringing heavy rainfall to a wide swath of the central U.S. before being absorbed.

The above image shows Gordon just before landfall on the Gulf coast.

Gordon's slow movement after landfall contributed to flooding rains throughout the central U.S.

Friday, August 31, 2018

Hurricane Florence (2018)

Storm Active: August 31-September 17

On August 30, a vigorous tropical wave entered the Atlantic from the west African coastline. While the system was initially large and disorganized, its center became much better defined over the next couple of days. The low passed just south of the Cape Verde Islands on the 31st, bringing some heavy rain and winds. Around the same time, it was classified Tropical Depression Six. Ocean temperatures were only marginal for development, but wind shear was very low and the air surrounding the system quite humid (any Saharan dry air was well to the north). Because of this, the cyclone strengthened as it moved west-northwestward away from the islands, becoming Tropical Storm Florence early on September 1.

Some more intensification occurred that evening, but wind shear out of the southwest suddenly exposed the center of circulation on September 2 and the system weakened slightly. As Florence continued on its west-northwest heading over the open central Atlantic, sea surface temperatures began to warm beneath it. Overnight, the system developed an inner core and strengthened into a strong tropical storm. However, even as temperatures warmed, wind shear increased also and the satellite presentation became a bit ragged. Nevertheless, Florence held its own on the 3rd and approached hurricane strength when a central dense overcast appeared. In defiance of almost all models and forecasts, the system continued to intensify on September 4 and became a hurricane. It turned a bit toward the northwest that evening and an eye became apparent on satellite imagery. The next morning, the eye became better defined and cleared out further. Florence was then upgraded to a category 3 hurricane, the first major hurricane of the 2018 season.

That afternoon, the cyclone peaked as a category 4 hurricane with winds of 130 mph and a minimum pressure of 953 mb before wind shear out of the southwest finally began to take its toll. The southern eyewall thinned and then broke under the shear, the inner core became disrupted, and Florence lost major hurricane status on September 6. Rapid weakening continued throughout the day, and the center nearly became exposed from time to time overnight as the system weakened back to a tropical storm. Meanwhile, mid-level ridging to the north of Florence turned the shallower system back toward the west on September 7. The weakening trend halted that afternoon as shear began to abate and convection gradually worked its way back over the center. But while waters were warm and upper-level winds were more favorable, the circulation had ingested some dry air that it struggled to mix out through the day of September 8. Nevertheless, it approached hurricane strength once again by that evening.

The next day, deeper convection blossomed near the center and hints of an eye reappeared. Florence regained category 1 status that morning and began a process of rapid intensification. This process really accelerated when a symmetrical eye appearing late on September 9. By the afternoon on the 10th, the system was a dangerous category 4, surpassing its earlier peak intensity to reach a new peak of 140 mph winds and a central pressure of 939 mb. Meanwhile, a very strong subtropical ridge began building to its north and Florence turned west-northwest with a greater forward speed. Overnight, the core underwent an eyewall replacement as the small inner eyewall collapsed in favor of a broader outer one. This took some time to consolidate, leaving Florence with a lopsided appearance on infrared imagery. However, the new eyewall eventually closed off and the system maintained category 4 strength. Moreover, eyewall replacement cycles usually are accompanied by a broadening of the wind field; this held true for Florence. Its tropical storm force and hurricane force wind radii increased markedly by September 12.

By September 13, the maximum winds had decreased to category 2 strength, but the system had grown considerably. Its outer bands began to sweep across the coast of southeastern North Carolina that morning. Florence's speed slowed during the day and it stalled off the coastline. That evening, it turned toward the west and ultimately made landfall overnight as a category 1 hurricane. Even as the storm weakened, torrential rains continued to fall over much of coastal North Carolina and pushed into South Carolina on September 14. Florence became a tropical storm that day and the center crossed the border into South Carolina over land. As the circulation spun down, the core lost much of its strength, but the portion of the storm still over water to the northeast of the center was still dragging immense amounts of tropical moisture over coastal North Carolina on September 15. Along with the flooding rains came intermittent tornadoes reported in the northeast part of the circulation.

Overnight, Florence weakened to a tropical depression and finally began to pick up some speed inland toward the west and then north through Appalachia. By September 17, the system turned northeast under the influence of the mid-latitude westerlies and rains spread eastward into the mid-Atlantic, though nowhere near as severe as they had been in the Carolinas. The low pressure center of Florence became elongated later that day and the system finally transitioned to post-tropical near West Virginia. Soon, the remnants moved eastward over the Atlantic. The extratropical successor to Florence eventually spawned a new low in the subtropical Atlantic that would become Leslie. Even well after the storm had passed, river flooding continued for the Carolinas. Florence's devastating rainfall, totaling over 30 inches in parts of coastal North Carolina, made it among the costliest tropical cyclones ever recorded.

The above image shows Florence near peak intensity as a category 4 hurricane.

Florence's track was highly unusual. In a vast majority of instances, cyclones that traversed the Atlantic at the latitude that Florence did would recurve out to sea before hitting land. However, very strong subtropical ridges steered the storm into the Carolinas.

Sunday, August 19, 2018

Tropical Storm Ernesto (2018)

Storm Active: August 15-18

On August 12, a low pressure area formed over the subtropical Atlantic well to the southeast off the coast of Nova Scotia. Over the following few days it moved generally toward the east and then southeast. In the meantime, very warm waters in the region allowed it to organize as atmospheric conditions improved. By August 15, the low's center had become well-defined, though the surface center was still situated under an upper-level low and the radius of maximum winds was rather broad. In light of these features, the system was classified Subtropical Depression Five that day.

The depression turned toward the north that day and strengthened into Subtropical Storm Ernesto. Marginal sea surface temperatures allowed a slight bit of intensification the next day and a transition to a fully tropical storm with convection closer to the center. At the same time, the storm began to feel the influence of the mid-latitude westerlies and accelerated northeast and then east-northeast. Ocean temperatures under the storm plummeted on August 17, but humid and unstable air in the region allowed Ernesto to maintain tropical cyclone status for somewhat longer than originally expected. Early on August 18, the system transitioned into a post-tropical storm west of Ireland. Nevertheless, it brought areas of heavy rain and gusty winds to northern Ireland and the United Kingdom as it merged with a front later that day. In both origin and track, Ernesto was very similar to its predecessor, Debby. In addition, it was the fourth system to become a subtropical storm in the 2018 season. This was the first such occurrence since 1969.

The above image shows Ernesto as a subtropical storm shortly after formation.

While Ernesto did not affect land as a subtropical or tropical cyclone, its remnants did impact the United Kingdom and Ireland.

Wednesday, August 8, 2018

Tropical Storm Debby (2018)

Storm Active: August 7-9

During the first few days of August, a non-tropical low meandered over the north central Atlantic. By the 4th, it was producing gale force winds, but had very little thunderstorm activity associated with it. Over the next couple days, it drifted southeastward, moving over water that was a tad warmer. The system again became stationary and then started to move back to the north, but by this time it acquired more significant convection. On August 7, the system was classified as Subtropical Storm Debby due to the spread of tropical storm force winds and outer banding from the center.

The storm slowed a bit overnight and turned toward the north-northeast on August 8. At the same time, the maximum winds increased somewhat and thunderstorms became more concentrated close to the center of circulation. As a result, Debby was reclassified as a tropical storm that morning. The cyclone also reached its peak intensity of 45 mph winds and a pressure of 1003 mb. Soon, however, the system began to weaken over the cold north Atlantic. Debby transitioned to a post-tropical storm during the afternoon of August 9 as it accelerated to the northeast, far from any land.

Debby was a small and short-lived cyclone that did not have any land impacts.

The above image shows Debby's track over the north Atlantic.

Saturday, July 7, 2018

Hurricane Chris (2018)

Storm Active: July 6-12

During the first couple days of July, an disturbance formed in the subtropical Atlantic southeast of Bermuda. Over the next few days, it moved slowly northwestward and environmental conditions gradually improved for development. On July 5, the system acquired a weak low-pressure center. This became someone better defined that day, but thunderstorm activity remained quite limited. It increased on July 6, however, and the system was classified Tropical Depression Three well offshore of the Carolinas.

That night, it turned toward the north and became somewhat more organized over the warm waters. Surface pressure were still high, however, and the system's maximum winds increased only slowly. Meanwhile, steering currents collapsed and the depression moved very little on July 7. Most thunderstorm activity was displaced south and southeast of the center, keeping coastal North Carolina, which was not far to the northwest, dry. A reformation of the surface circulation to the south allowed the system to organize further and strengthen into Tropical Storm Chris by early on July 8. The next day saw gradual strengthening as the circulation tightened, but dry air intrusion prevented Chris from closing off an eyewall. By that evening, the system was on the verge of hurricane strength and finally began to move slowly toward the northeast. Though it began to move away from land, high surf continued to pound the coastline.

Since it had been stationary for days, Chris had caused cold waters to upwell underneath it (due to its strong winds). Though this decrease in temperature was mitigated somewhat by the steady flow of warm Gulf stream waters, it slowed the cyclone's intensification. However, once it started moving, the system rapidly intensified. Late on July 10, it reached its peak intensity as a category 2 hurricane with 105 mph winds and a pressure of 970 mb. As it accelerated northeast, it began to encounter cooler water, weaken, and become less symmetric. Soon, it weakened to a tropical storm and was quickly transitioning to an extratropical system. Chris became fully extratropical on July 12 as it raced northeast over cold north Atlantic waters. The post-tropical cyclone made landfall in southeastern Newfoundland that night, bringing wind gusts to near hurricane force and brief but heavy rains.

The above image shows Hurricane Chris strengthening off the U.S. east coast. Chris was the earliest second hurricane to develop in the Atlantic since 2005.

Chris did not affect land directly as a tropical cyclone, but made landfall in Newfoundland as a post-tropical cyclone.

Thursday, July 5, 2018

Hurricane Beryl (2018)

Storm Active: July 5-8, 14-15

At the beginning of July, a tropical wave emerged off of the coast of Africa. This tropical wave was among the first of the season to develop significant convection over the central tropical Atlantic. Despite anomalously cool waters, conditions were still favorable enough in the deep tropics to support development. By July 4, the disturbance was quite well organized, with low shear in its environment and an evident circulation. Nevertheless, the low did not yet appear closed. The next day, organization increased further, and the system was classified as Tropical Depression Two. The depression was quite small and moving fairly quickly toward the west.

It is not uncommon for small cyclones to change rapidly in intensity, and the system strengthened quickly that evening and into July 6, becoming Tropical Storm Beryl just a few hours after formation. A minuscule pinhole eye appeared on satellite imagery and the maximum winds shot up to hurricane strength, making Beryl the first hurricane of the 2018 season by early on the 6th. The system strengthened a little bit more during that day, reaching a peak intensity of 80 mph winds and a pressure of 994 mb. Beryl turned a bit toward the west-northwest during the evening and into July 7. As it did so, it began to encounter increased wind shear. Quickly, convection was displaced toward the southeast and the center was exposed. As a result, Beryl began to quickly weaken and became a tropical storm. Convection flared up near the center from time to time over the following day but the low-level center all but disappeared as the cyclone accelerated west-northwest toward the Windward Islands. As shear continued to increase, Beryl quickly dissipated into a tropical wave.

Nevertheless, Beryl's remnants passed over the easternmost Caribbean islands on July 8, bringing heavy rains and strong winds out of the east. By the 9th, these rains had begun to move over Puerto Rico and the Virgin Islands. Overnight, the heavy rains moved over Hispaniola as the wave proceeded west-northwestward. The wave began to encounter more favorable conditions over the Bahamas, though the main impact to these areas was still locally heavy rain. After that, the disturbance turned northward and then northeastward, passing to the west of Bermuda.

Only on July 13 did the remnants of Beryl begin to show signs of reorganization, with a well-defined circulation developing. Despite marginal sea surface temperatures and atmospheric conditions, thunderstorm activity reappeared near the low-pressure center. On July 14, a full six days after dissipation, Beryl reformed. Though its windfield was concentrated near the center (typical for a tropical cyclone), the system lay under an upper-level trough. Therefore, it was deriving its energy in a manner uncharacteristic of a tropical cyclone. As a result, forecasters classified it as Subtropical Storm Beryl north of Bermuda.

The system slowed its forward motion and meandered somewhat over the next day. Vertical shear was increasing, but Beryl maintained enough convection in the southeastern quadrant to hang on to subtropical storm status. Even this was short-lived, however, and the system finally weakened into a remnant low late on July 15. This low approached Newfoundland before dissipating.

Beryl was one of the smallest hurricanes ever recorded; winds of hurricane force extended only 10 miles from the center. Note also the minuscule eye on satellite imagery.

A majority of Beryl's impacts occurred while the system was a wave moving through the Caribbean (triangle points).

Friday, May 25, 2018

Subtropical Storm Alberto (2018)

Storm Active: May 25-30

Beginning around May 20, a trough of low pressure located in the western Caribbean produced widespread thunderstorm activity as it interacted with an upper-level low. Over the next several days, the disturbance tracked generally northwestward. In the mean time, abundant moisture in the area caused sporadic rainfall from portions of Honduras to the Yucatan Peninsula to western Cuba. Even after a surface low formed, the system remained quite disorganized due to land interaction with the Yucatan and strong upper-level winds out of the west. Despite fairly hostile conditions, the low became better defined during the day of May 24. By the morning of the 25th, the surface low had emerged over water adjacent to the northeast Yucatan Peninsula with a large area of strong thunderstorms to the north and east. In addition, buoy and ship reports suggested the presence of winds to gale force. Since the low was situated under an upper-level trough, and not the upper-level high associated with traditional tropical storms, the system was designated Subtropical Storm Alberto late that morning.

During that day, the surface circulation of Alberto was far removed from the thunderstorm activity to the north and east. In fact, the overall circulation appeared to me moving northeast while the low-level swirl drifted just south of east. Nevertheless, heavy rains continued over much of Cuba and the outer bands began to affect southern Florida. Overnight, upper level winds lessened considerably, and limited convection finally appeared near the surface center. The center also turned north and accelerated early on May 26, essentially "catching up" with the rest of the circulation. As a result, Alberto's satellite presentation improved considerably. A further reformation of the center took place later that day, this time to the northeast of the previous position. This and the system's generally northward movement brought Alberto into the eastern Gulf of Mexico, not too far from the west coast of Florida. However, this coast was saved from the heavier rainfall by a dry slot in the eastern semicircle; it was now the other side that had most of the convection.

That evening and overnight, Alberto's pressure dropped considerably, its center became better defined, and it began to take on some more tropical characteristics. The storm's maximum winds increased in turn during the day of May 27. The storm also turned toward the northwest briefly under the influence of an upper-level low. Despite organization improvements, dry air was taking its toll on Alberto, invading via the western side and eroding deep convection away from the center. Situated over relatively cold eastern Gulf waters, the system also did not develop the deep warm core needed to be classified as a tropical storm. Nevertheless, Alberto reached its peak intensity of 65 mph winds and a pressure of 991 mb that evening as it approached the Florida panhandle.

Continued dry air intrusion and proximity to land decreased the storm's winds gradually as bands of heavy rain swept across the Gulf coast early on May 28. The center of Alberto made landfall that afternoon in the Florida panhandle, bringing heavy rains and localized flooding to parts of the southeast U.S. At landfall, the storm had maximum winds of 50 mph. That night, it weakened to a subtropical depression over land as it continued northward over Alabama. Curiously, the system completed its transition to a tropical cyclone (becoming a tropical depression) over Tennessee late that evening. The circulation maintained its identity and continued to cause rainfall even into May 30, when it finally became extratropical over Michigan. Alberto marked an early start to the Atlantic hurricane season for the 4th consecutive time, only the 2nd known time this has occurred (after 1951-4).

The above image shows Alberto in the eastern Gulf of Mexico on May 27.

Alberto was subtropical most of its life (square points) but transitioned over land to a tropical depression (blue circular points) and maintained this status remarkably far north.

Wednesday, May 16, 2018

Professor Quibb's Picks – 2018

My personal prediction for the 2018 North Atlantic hurricane season (written May 16, 2018) is as follows:

18 cyclones attaining tropical depression status,
16 cyclones attaining tropical storm status,
8 cyclones attaining hurricane status, and
4 cyclones attaining major hurricane status.

In the wake of the especially devastating 2017 season, it is difficult to predict with any certainty the outcomes for this year. Once again, models indicate that the El Niño Southern Oscillation Index (or ENSO index) will be near zero or slightly positive during this hurricane season. This index, which is a certain quantitative measure of sea surface temperature anomalies in the tropical Pacific Ocean, has some ability to predict Atlantic hurricane activity. A positive index indicates an El Niño event, which tends to correlate with higher wind shear across the Atlantic basin and less tropical cyclone development. This effect is especially pronounced in the Gulf of Mexico and Caribbean Sea. The image below shows the ENSO forecast for this season (image from the International Research Institute for Climate and Society):

However, last year's forecast was qualitatively similar, but the index ended up dipping back negative and leaving very favorable conditions for hurricane formation. Though consideration of the ENSO index alone would lead to the prediction of an average hurricane season, there is significant uncertainty. Overall, I consider the ENSO to mainly a neutral factor this year.

Present ocean temperatures in the Atlantic are slightly above average in the Gulf of Mexico and Caribbean, and significantly above average in the subtropical Atlantic and near the U.S. east coast. However, there is a large area of below average temperatures in the tropical Atlantic which is forecast by long-term models to possibly persist for a few months. The tropical Atlantic has also been dry and stable, in contrast to elevated storm activity in the Caribbean and Gulf of Mexico. These trends also show some signs of persisting into the beginning of hurricane season. I therefore expect a slow start to the season in the main development region of the tropical Atlantic (extending from Africa to the Caribbean) and a corresponding lack of Cape Verde or long-track hurricanes, though these could appear more in late September and October. There is significant potential for formation in areas closer to land, so I expect some shorter lived hurricanes in the northern Caribbean/Gulf of Mexico and U.S. east coast regions.

My estimated risks for different parts of the Atlantic basin are as follows (with 1 indicating very low risk, 5 very high, and 3 average):

U.S. East Coast: 5

The jet stream over the U.S. has been weaker than usual so far this season, and the Bermuda high stronger. However, with a weak El Nino possibly developing, long hurricane tracks westward into the Gulf still seem unlikely. The east coast, in contrast, is at a greater risk. Ocean temperatures offshore are anomalously warm and region will be very moist, suggesting a fairly high probability of tropical cyclone impacts.

Yucatan Peninsula and Central America: 3
The western Caribbean shows some signs of being a fertile area for cyclonogenesis, but with prevailing upper-level patterns as they are, it is difficult to see strong system taking due westward tracks into central America. Compared to the last few years, strong hurricanes are less of a threat, though the potential for flooding rains may be equal or greater.

Caribbean Islands: 2
As discussed above, the main development region may remain quiet for at least the first half of hurricane season. This would insulate the Caribbean islands from the approach of Cape Verde hurricanes to the west, but does not preclude development occurring locally. Nevertheless, it is somewhat more likely this year that the islands will receive a break from intense hurricane landfalls, especially the easternmost islands.

Gulf of Mexico: 3
Factors in the Gulf point in different directions. Ocean waters are warm and will likely continue to be so, particularly in eddies originating in the northern Caribbean (which also happens to be a likely source of Gulf hurricanes). On the other side, if an El Niño does develop, the Gulf of Mexico is where the suppression of hurricane activity would be most felt. Putting this together suggests a near-average risk this year.

Overall, the 2018 season is expected to be a bit above average; it should not be a repeat of the devastating 2017 season, but many areas such as the U.S. east coast may still be at high risk. Further, this is just an informal forecast and uncertainty in the outcome remains significant. Everyone in hurricane-prone areas should still take due precautions as hurricane season approaches. Dangerous storms may still occur even in overall quiet seasons.


Tuesday, May 15, 2018

Hurricane Names List – 2018

The name list for tropical cyclones forming in the North Atlantic basin for the year 2018 is as follows:


This list is the same as the list for the 2012 season, with the exception of Sara, which replaced the retired name Sandy.

Monday, May 7, 2018

Goodstein's Theorem and Non-Standard Models of Arithmetic

This is the final post in a four-part series on logic and arithmetic, with a focus on Goodstein's Theorem. For the first post, see here.

In the previous post, Goodstein's Theorem, a statement about the properties of certain sequences of natural numbers, was proven using infinite ordinals. The use of a method "outside" arithmetic makes it reasonable that this proof cannot be encoded in the language of Peano Arithmetic (PA), the formal logical system for discussing the natural numbers. A stronger statement is also true: there is no proof of Goodstein's Theorem in PA because it cannot be deduced from the axioms of PA.

But how does one go about proving something unprovable? Certainly it is intractable to check every possible method, as the diversity of such attempts could be infinite. Mathematicians take a different approach, using tools from what is called model theory. In mathematical logic, a model of a collection of axioms is a specific structure within which the axioms (and all theorems derived from them) are interpreted to be true. Recall that the axioms of PA mentioned five specific objects, that were assumed to be given from the start: a set N, a specific member, 0, a function S from N to itself, and two binary operations on N, + and *. Of course, to actually do arithmetic we interpret N as the set of natural numbers, 0 as the number 0, S as the "successor" function taking in a number n and returning n+1, and + and * as the usual addition and multiplication. Until we provide an interpretation of to what these objects refer, namely a model, they are just symbols! We may prove statements about them, such as the fact that S(0) and S(S(0)) are distinct members of N, but this is just a mathematical sentence resulting as the end product of a series of formal deductive rules.

Any collection A = (NA,0A,SA,+A,*A) of a set NA, a member 0A of the set, a function SA:NANA, and two binary operations +A and *A that satisfies the axioms is a model of PA. Of course, we know fairly well what we mean by "natural numbers", namely {0,1,2,...} with 0 the first element, S sending 0 to 1, 1 to 2, etc, and the usual addition and multiplication. The entire point of selecting axioms for PA is to study ℕ = (N,0,S,+,*), the standard natural numbers. A natural question (called the question of categoricity) arises: is the standard model the only type of model for PA, or are there others? The answer is no; there are other, non-standard models A that still satisfy every axiom of PA. These were first discovered by Norwegian mathematician Thoralf Skolem in 1934. To be clear, they are not the natural numbers, at least, not as we intend them to be. Their existence exemplifies another limitation of first-order logic: axiom systems often fail to specify structures uniquely and hence fail to capture some features of the field to be studied.

Non-standard models often serve as an essential tool in independence proofs. First, we know from the previous post that the standard model ℕ of PA does satisfy Goodstein's Theorem (the standard model has the properties the natural numbers possess within the larger field of set theory, the methods of which were used in the proof). This means that the negation of Goodstein's Theorem cannot be a theorem of PA, since there is a model satisfying both the axioms and the theorem. If one could find a model of PA in which the negation of Goodstein's Theorem were true, then this would prove independence, because there would be models in which it is true and others in which it is false! Kirby and Paris used precisely this method in their 1982 proof of the result.

But what do non-standard models of natural numbers actually look like? First, we may infer what they have in common. PA axiom 1 guarantees the existence of a number 0. Axiom 2 gives it successors S(0), S(S(0)), etc. Axiom 3 says that S(n) = S(m) implies m = n. Therefore, all the successors generated from 0 are distinct from one another. This means that any model A has a set of natural numbers NA containing the analogues of 0, 1, 2, and so on. The set of standard natural numbers N is thus contained in NA for every A. The difference is that non-standard models have extra numbers!

At first brush, having additional "non-standard" numbers seems to contradict the Peano axioms, specifically the fifth, the axiom schema of induction. It states that if 0 has some property and that any n having the property implies that n + 1 does as well, then all natural numbers have the property. The spirit of this axiom schema, if not the letter, is that beginning at 0 and knocking down the inductive dominoes will eventually reach every natural number. If we could choose the property to be "is in the set {0,1,2,...} (the standard natural numbers N)" then this would immediately rule out nonstandard models: 0 is this set, and for any n in the set, its successor is also standard, so all of NA is contained in {0,1,2,...} and hence we would have NA = {0,1,2,...}. Unfortunately, it is impossible to define the set {0,1,2,...} inside of the first-order logic formulation. It is also impossible to simply add an axiom "there are no other numbers besides 0, 1, 2, etc." for the same reason. Both approaches require infinitely long logical sentences to formulate, which are forbidden in the finitary system of first-order logic.

Though the axioms of PA cannot rule out non-standard natural numbers, they are forced by the axioms to satisfy some strange conditions. Any nonstandard number c must be greater than all standard numbers. Further, PA can prove that 0 is the only number without a successor, so a "predecessor" to c, which we may call c - 1, must exist. Similarly, c - 2, c - 3, etc. must exist, as must, of course, c + 1, c + 2, etc. These must all be new non-standard numbers. Therefore, the existence of one non-standard number guarantees the existence of a whole non-standard "copy" of the integers: {...,c - 2,c - 1,c,c + 1,c + 2,...}. However, it gets much, much worse. The operation of addition is part of Peano Arithmetic, so there must be a number c + c, that may be proven to be greater than all numbers c + 1, c + 2, and so on. From here, we get another new infinite collection of non-standards {...,c + c - 2,c + c - 1,c + c,c + c + 1,c + c + 2,...}. A similar story occurs for c + c + c = c*3 and larger numbers as well, but we can also go in reverse. One can prove in PA that every number is either even or odd; that is, for any n, there is an m satisfying either m + m = n (if n is even), or m + m + 1 = n (if n is odd). This theorem means that c is even or odd, so there must be a smaller non-standard d with d + d = c or d + d + 1 = c. This d has its own infinite set of non-standard neighbors. The reader may continue this type of exercise and eventually derive the type of picture illustrated above: any non-standard model of natural numbers must contain the standard numbers plus (at least) an infinite number of copies of the integers, ℤ, one for each member of the set of rational numbers, ℚ.

As strange as these models are, they cannot be ruled out in PA, nor is there a natural addition to the axioms that may do so. Rather than being just a defect of first-order logic however, non-standard models are a useful tool for examining the structure of different theories. Now that we have a non-standard model at our disposal, it seems reasonable that Goodstein's Theorem should fail for some non-standard models: "Goodstein sequences" beginning at non-standard natural numbers do not seem likely to terminate at zero. After all, they have infinitely many copies of the integers to move around in! These sequences often cannot be computed explicitly, but using other logical machinery, one can prove the fact that they do not necessarily terminate. This establishes the independence of the theorem from PA.

Goodstein sequences, interesting in their own right for their rapid growth, allow an interesting perspective on Peano Arithmetic and its limitations. The questions of independence and non-standard models arise frequently in the foundations of mathematics, as we seek to define precisely the scope of our mathematical theories.


Monday, April 16, 2018

Proving Goodstein's Theorem and Transfinite Methods

This is the third part of a post series on Goodstein's theorem. For the first part, see here.

The previous post introduced the reader to Peano arithmetic (PA), the archetypical example of an axiomatic system introduced to standardize the foundations of mathematics. Despite having tremendous expressive power in formulating and proving theorems about natural numbers, the system is not without limitations. Gödel's Incompleteness Theorem guarantees the existence of statements out of the reach of formal proofs in PA. Goodstein's Theorem, which states that every Goodstein sequence terminates at 0, is an example. Further, it is a "natural" example in the sense that it was not contrived to demonstrate incompleteness. In fact, Goodstein proved the theorem in 1944, decades before Laurence Kirby and Jeff Paris discovered that it is independent of the axioms of PA in 1982.

To see why this independence holds, we consider the proof of Goodstein's Theorem. As mentioned, it is not provable in PA, so the proof makes use of tools outside of arithmetic: in particular, infinite ordinal numbers. A more thorough discussion of ordinal numbers may be found elsewhere on this blog. For our purposes, the key property of ordinal numbers is that they represent order types of well-ordered sets.

A well-ordered set is simply a set of elements and an ordering relation (often called "less than" and denoted by "<") such that any two elements are comparable (each is less than, equal to, or greater than) the others, and every subset has a minimal element. The set may be finite or infinite. Here are some examples:

  • The set of natural numbers itself, {0,1,2,3,...}, with the relation "less than" is well-ordered because every two elements are comparable and every subset has a smallest number
  • The set of natural numbers with the "greater than" relation is not well-ordered: with this relation, "minimal" elements are really largest elements, and the subset {3,4,5,...}, for example, has no greatest element
  • The set {A,H,M,R,Z} with the relation "comes before in the alphabet" is well-ordered because we can compare any two letters to see which comes first in the alphabet, and any subset has a first letter
  • The set of all integers {...-2,-1,0,1,2,...} is not well-ordered by either less than or greater than relations

The gist of the definition is that all set elements are listed in a particular order so that we can always tell which of a pair comes first, and that infinite ascending sequences are acceptable while decreasing ones are not. To understand order type, we need a notion of when two well-ordered sets are "the same". For example, the set {1,2,3,4,5} with the less than relation and {A,H,M,R,Z} with the alphabet relation are quite similar. Using the one-to-one relabeling 1 → A, 2 → H, 3 → M, 4 → R, 5 → Z, we can move from one set to the another and preserve the ordering relation. That is, 1 < 2 in the first set and their images satisfy A < H in the second set, and so on. If there is a relabeling like the one above between two sets, they are said to be of the same order type.

The purpose of ordinal numbers is to enumerate all possible order types for well-ordered sets; there is one ordinal for each order type. To make thinking about ordinals simpler, we often say that an ordinal is a specific set of the given order type, a particularly nice set. Specifically, we choose the set of all smaller ordinals. Since the sets in the example of the previous paragraph have five elements, their order type is the ordinal 5 = {0,1,2,3,4} (the reader may wish to show that any well-ordered five element set in fact has this order type). In fact, for finite sets, there is simply one ordinal for each size set. For infinite sets, matters become more interesting.

The ordinal corresponding to the order type of the natural numbers is called ω. Using the canonical choice of representative set, ω = {0,1,2,3,...} (where we now view the elements as ordinals). The next ordinal is ω + 1, the order type of {0,1,2,3,...,ω} or in general the order type of a well-ordered set with an infinite ascending collection plus one element defined to be greater than all others. One can go on to define ω + n for any finite n and ω*2, the order type of {0,1,2,3,...,ω,ω + 1,ω + 2,...} (two infinite ascending chains stuck together). The precise details do not concern us here, but ω2, ωω, ωωω, and so on may be defined as well. What matters is that these ordinals exist and that the set of all ordinals expressible with ordinary operations on ω (for example, (ωω)*4 + (ω3)*2 + ω*5 + 7) is a well-ordered set. In fact, the set of such ordinals is itself a larger ordinal called ε0.

Once these preliminaries are established, the proof of Goodstein's Theorem is rather simple, and even clearer when considered intuitively. For any Goodstein sequence, the members are represented in hereditary base-n notation at every step: the first member is put into base 2, the next base 3, and so on. The idea is to take each member of the sequence and replace the base with ω to obtain a sequence of ordinals. For example, the sequence G4 generates a sequence of ordinals H4 in the following way:

G4(1) = 4 = 1*21*21H4(1) = ωω (the 2's are replaced with ω's),
G4(2) = 26 = 2*32 + 2*31 + 2 → H4(2) = (ω2)*2 + ω*2 + 2 (3's are replaced with ω's),
G4(3) = 41 = 2*42 = 2*41 + 1 → H4(3) = (ω2)*2 + ω*2 + 1 (4's are replaced by ω's),
and so on.

Note that the multiplication by coefficients has been moved to the other side of the ω's for technical reasons and that some of the 1's have been removed for clarity. One may precede in this matter to get a sequence of ordinals, the key property of which is that the sequence is strictly decreasing. In the above example, ωω > (ω2)*2 + ω*2 + 2 > (ω2)*2 + ω*2 + 1 and this downward trend would continue if we were to list more terms. This is because the H sequences "forget" about the base: it is always replaced by ω. The only change is caused by the subtraction of 1 at each step, which slowly reduces the coefficients. Intuitively, this is the point of the proof: by forgetting about the base, we replace the extreme growth of Goodstein sequences with a gradual decline. The units digit of the H sequence decreases by 1 every step. When it reaches 0, on the next step the ω coefficient is reduced by 1 and the units digit is replaced by the current base minus 1 (the highest allowed coefficient). These may become quite large, but they always reach zero eventually. Reasoning this way, it is clear that Goodstein's Theorem should be true.

In formal terms, the set of ordinals is well-ordered, so the set consisting of all members of an H sequence must have a minimal element, i.e., it cannot decrease forever. The only way that it can stop decreasing is if the G sequence stops, and Goodstein sequences only terminate at 0. Therefore, every Goodstein sequence terminates at 0 after a finite number of steps. We've proved Goodstein's Theorem!

Bringing in infinite ordinals to prove a statement about natural numbers is strange. So strange in fact that the argument is not formalizable in PA; there is simply no way to even define infinite ordinals in this language! This indicates why the given proof does not go through in PA, but does not settle the matter as to whether there is no possible proof of Goodstein's Theorem within PA. It leaves the possibility that there is a different clever approach that can succeed without infinite ordinals. A discussion of why this in fact does not occur may be found in the final post of this series (coming May 7).


Monday, March 26, 2018

Goodstein's Theorem and Peano Arithmetic

This is the second part of a post series on Goodstein's theorem. For the first part, see here.

We saw last post that Goodstein sequences are certain sequences of positive integers defined using base representations. Despite their simple definition, they grow extraordinarily large. However, Goodstein's Theorem states that no matter how large the starting value is, the sequence will eventually terminate at 0 after some finite number of steps. Before discussing the proof to this remarkable theorem, we move in a completely different direction and define the axioms of Peano arithmetic.

Increasing standards of rigor were a hallmark of late 19th and early 20th century mathematics. With this movement came a need to precisely define the basic building blocks with which a given branch of mathematics worked. This was even true for simple arithmetic! By 1890, the mathematician Giuseppe Peano had published a formulation of the axioms (fundamental assumptions) of arithmetic, which are used nearly unchanged to this day. Written in simple english, the axioms state the following about the collection N of natural numbers (nonnegative integers), a function S known as the successor function, a function + known as addition, and a function * known as multiplication:

1) There exists a natural number called 0 in N
2) For every natural number n in N there is a successor S(n) in N (commonly written
n + 1)
3) There is no natural number whose successor is 0
4) For any two natural numbers m and n in N, if S(m) = S(n), then m = n
5) For any natural number n, n + 0 = n
6) For any natural numbers n and m, n + S(m) = S(m + n)
7) For any natural number n, n*0 = 0
8) For any natural numbers n and m, n*S(m) = n*m + n
9) For any "well-behaved" property P, if P(0) holds and for each n in N, P(n) implies
P(n + 1), then P is true for every natural number in N

The first two axioms simply say that you can start from 0 and count upwards forever through the natural numbers. The third says that there is no natural number below 0 (this is of course false for larger sets of numbers such as integers, but Peano's axioms are only for properties of the natural numbers). The fourth shows that the successor of a natural number is unique. The fifth through eighth axioms state the common properties of addition and multiplication. The ninth axiom is actually a large collection of axioms (an axiom schema) in disguise, called the "axiom schema of induction."

The idea of induction may be familiar to readers who recall their high school mathematics: one often wishes to prove that a given property holds for every natural number. To do so, it is sufficient to show that it is true in the "base case," that is, true for 0, and that the statement being true for a number means that it is also true for the next. Then since the property is true for 0, it is true for 1. Since it is true for 1, it is true for 2, and so on. The figure above illustrates the idea of induction, where each "statement" is that the given property is true for some number. The final axiom simply codifies the fact this type of reasoning works; we get that the property is true for all natural numbers. In this context, a "well-behaved" property is one expressible in the semantics of the variety of logic being used (first-order logic in this case).

Remarkably, the above assumptions are all that is needed to perform arithmetic. In principle, though formalizing complicated proofs would be quite lengthy, true statements such as "17 is prime" and "every number is the sum of four squares" become theorems in Peano arithmetic. All of these proofs would take the form of a chain of statements beginning from axioms and concluding with the desired result, such as "17 is prime," rendered appropriately in the formal mathematical language. The progression from each statement to the next would also follow a collection of well-defined deductive rules. In principle, almost all theorems concerning natural numbers could be proven this way, beginning from just a small collection of axioms! However, Peano arithmetic still runs afoul of a result that vexes many axiom systems of first-order logic: Gödel's Incompleteness Theorem.

Gödel's (First) Incompleteness Theorem, originally proven by the mathematician Kurt Gödel in 1931, dashed the hopes of those who imagined that formal logical systems would provide a complete description of mathematics. It states, informally, that for any first-order logical system powerful enough to encode arithmetic (Peano arithmetic is of course such a system), there exist statements in the language of the system that are neither provable nor refutable from the axioms. Further, there are explicit sentences in the logical system that are true (under the intended interpretation of the theory, more on this later) but unprovable. The above diagram illustrates the situation: there will always be things we know to be true or false that are beyond the reach of the axioms to formally prove or refute.

Some questions may spring to mind at this unintuitive result. What is the distinction between "true" and "provable"? How do we define "true" in mathematics if not as the end result of a proof? What do these unprovable statements look like, and what do they say?

The answer to the first of these depends on there being something "we mean by" the term "natural numbers". In other words, there is an intended interpretation of what natural numbers should be that the logical system fails to completely capture. Consequently, there are statements that we know to be true using methods outside the formal system but are unprovable within it. Bringing in additional assumptions does not simply resolve the incompleteness theorem, however. For each outside axiom added, the theorem guarantees the existence of a new unprovable statement. And if the system ever does become complete by the addition of more axioms, it also becomes inconsistent, that is, able to prove a contradiction (and loses all validity as a mathematical system).

As for the final question, the first known unprovable statements were those constructed in the proof of Gödel's theorem; these are known as Gödel sentences. They are highly contrived for the proof, however, and do not have any intuitive meaning. In the years following the original proof, the concern remained whether, for example, any statement that "naturally" arises in the study of natural numbers would be unprovable and unrefutable from the axioms of Peano arithmetic. Amazingly, such statements exist! In fact, a great example is Goodstein's Theorem. No proof exists, beginning from the Peano axioms, that has it as a conclusion. To read more about how it can be proven and why it is not a theorem of Peano arithmetic, see the next post.