Storm Active: July 22-23

Around July 12, a tropical wave moved off the coast of Africa. It was among the first of the season to be seriously monitored for cyclone development, but it traversed the Atlantic basin for the following week without incident. A portion of the wave axis took a northern route, passing north of the Caribbean islands and approaching the Bahamas by July 21. Stable dry air in the region made progress difficult for the disturbance, but it managed to spin up a small area of convection driven by very warm ocean waters. This led to a tiny circulation and the system strengthened into Tropical Depression Three on July 22 over the western Bahamas.

Soon after, the depression began to feel the influence of an approaching cold front and turned northward on July 23. The center passed just offshore of east Florida, but its small size meant that only a few showers and occasional gusty winds impacted land. By the late morning, the system had already lost its identity and dissipated as it combined with the front.

Even though the tropical depression formed over very warm water, it succumbed quickly to dry mid-level air.

Tropical Depression Three was a small and short-lived system with minimal land impacts.

## Monday, July 22, 2019

## Thursday, July 11, 2019

### Hurricane Barry (2019)

Storm Active: July 11-14

During the second week of July, a trough of over the southeast United States drifted slowly south-southeastward, producing some scattered afternoon thunderstorms as it went. A few days later, on July 9, this anomalous motion brought it into the extreme northeastern Gulf of Mexico, where more consistent convection began to flare up. Weak steering currents allowed the system to meander west-southwestward and it gradually organized, developing a broad circulation. By July 10, a clear low-level center had formed, but it was displaced well to the northeast of the mid-level circulation. Moreover, the strongest thunderstorms were actually located over southeast Louisiana, where significant flooding occurred before the tropical disturbance had even been classified.

Finally, on July 11, improvements in organization prompted the naming of Tropical Storm Barry, located now nearly due south of the Mississippi delta. Even after naming, however, dry continental air pushing in from the other restricted cloud cover to the southern portion of the tropical storm, and several small low-level vortices were evident on satellite imagery. This disorganized hampered Barry's intensification. Nevertheless, the pressure fell appreciably over the next day and aircraft reconnaissance indicated that the system's maximum winds steadily increased to strong tropical storm strength by July 12.

Meanwhile, Barry took a turn north of west around the edge of the mid-level steering ridge and began to move toward Louisiana. Even by the afternoon of July 12, however, the northern semicircle remained very dry, so few effects were felt over land even with the system less than 100 miles offshore. Despite its unconventional structure, Barry steadily strengthened through landfall. It peaked as a category 1 hurricane with 75 mph winds and a pressure of 993 mb on July 13 as it crossed the central Louisiana coastline around noon local time. The slow movement of the system resulted in only a gradual weakening trend and prolonged heavy rainfall, especially just east of the landfall point. Nevertheless, Barry weakened to a tropical storm shortly after landfall and a tropical depression on July 14 as the center of circulation pushed further inland. Upper level winds out of the north kept most of the precipitation over water even as the center moved away, sparing inland areas from more severe flooding. Soon after, the storm became extratropical over the midwest.

The above image shows Hurricane Barry near landfall, with most of the northern half of the circulation exposed.

Barry originated from a non-tropical disturbance over the southeast U.S.

During the second week of July, a trough of over the southeast United States drifted slowly south-southeastward, producing some scattered afternoon thunderstorms as it went. A few days later, on July 9, this anomalous motion brought it into the extreme northeastern Gulf of Mexico, where more consistent convection began to flare up. Weak steering currents allowed the system to meander west-southwestward and it gradually organized, developing a broad circulation. By July 10, a clear low-level center had formed, but it was displaced well to the northeast of the mid-level circulation. Moreover, the strongest thunderstorms were actually located over southeast Louisiana, where significant flooding occurred before the tropical disturbance had even been classified.

Finally, on July 11, improvements in organization prompted the naming of Tropical Storm Barry, located now nearly due south of the Mississippi delta. Even after naming, however, dry continental air pushing in from the other restricted cloud cover to the southern portion of the tropical storm, and several small low-level vortices were evident on satellite imagery. This disorganized hampered Barry's intensification. Nevertheless, the pressure fell appreciably over the next day and aircraft reconnaissance indicated that the system's maximum winds steadily increased to strong tropical storm strength by July 12.

Meanwhile, Barry took a turn north of west around the edge of the mid-level steering ridge and began to move toward Louisiana. Even by the afternoon of July 12, however, the northern semicircle remained very dry, so few effects were felt over land even with the system less than 100 miles offshore. Despite its unconventional structure, Barry steadily strengthened through landfall. It peaked as a category 1 hurricane with 75 mph winds and a pressure of 993 mb on July 13 as it crossed the central Louisiana coastline around noon local time. The slow movement of the system resulted in only a gradual weakening trend and prolonged heavy rainfall, especially just east of the landfall point. Nevertheless, Barry weakened to a tropical storm shortly after landfall and a tropical depression on July 14 as the center of circulation pushed further inland. Upper level winds out of the north kept most of the precipitation over water even as the center moved away, sparing inland areas from more severe flooding. Soon after, the storm became extratropical over the midwest.

The above image shows Hurricane Barry near landfall, with most of the northern half of the circulation exposed.

Barry originated from a non-tropical disturbance over the southeast U.S.

Labels:
2019 Storms

## Monday, May 20, 2019

### Subtropical Storm Andrea (2019)

Storm Active: May 20-21

As the third week of May began, a frontal boundary moved off of the U.S. east coast. The southern end of the front stalled north of Hispaniola and formed a trough of low pressure. There the system found a relatively favorable atmosphere and marginally warm ocean temperatures, supporting some scattered storm development. Before long, a low-pressure center had developed. By May 20, the storm had a small convective shield displaced to the north and east and aircraft reconnaissance measured gale-force winds. Since the circulation was still interacting with an upper-level low to its southwest and the gale force winds were spread out from the center, the storm was classified Subtropical Storm Andrea, the first named storm of the 2019 Atlantic hurricane season.

The system moved northward that day, but began to slow down and veer eastward by the the afternoon of May 21. Meanwhile, the convection associated with the system dissipated, leaving behind just a swirl of low-level clouds to mark the center of circulation. As a result, Andrea was downgraded to a subtropical depression. Early that evening, it degenerated further into a remnant low and these remnants dissipated the next day as a new front approached. Moisture that had been associated with Andrea brought some rain showers to Bermuda on the 22nd. The formation of Andrea marked the 5th consecutive year during which a named storm formed prior to the official start of the hurricane season on June 1, surpassing the record set in 1951-4. However, short-lived weak systems such as Andrea may very well have been missed prior to the era of satellite observation.

This image shows Subtropical Storm Andrea on May 20. Also visible is the upper-level low to its southwest which helped to weaken the system.

Andrea formed in the far western Atlantic, one of the typical areas for early season cyclogenesis.

As the third week of May began, a frontal boundary moved off of the U.S. east coast. The southern end of the front stalled north of Hispaniola and formed a trough of low pressure. There the system found a relatively favorable atmosphere and marginally warm ocean temperatures, supporting some scattered storm development. Before long, a low-pressure center had developed. By May 20, the storm had a small convective shield displaced to the north and east and aircraft reconnaissance measured gale-force winds. Since the circulation was still interacting with an upper-level low to its southwest and the gale force winds were spread out from the center, the storm was classified Subtropical Storm Andrea, the first named storm of the 2019 Atlantic hurricane season.

The system moved northward that day, but began to slow down and veer eastward by the the afternoon of May 21. Meanwhile, the convection associated with the system dissipated, leaving behind just a swirl of low-level clouds to mark the center of circulation. As a result, Andrea was downgraded to a subtropical depression. Early that evening, it degenerated further into a remnant low and these remnants dissipated the next day as a new front approached. Moisture that had been associated with Andrea brought some rain showers to Bermuda on the 22nd. The formation of Andrea marked the 5th consecutive year during which a named storm formed prior to the official start of the hurricane season on June 1, surpassing the record set in 1951-4. However, short-lived weak systems such as Andrea may very well have been missed prior to the era of satellite observation.

This image shows Subtropical Storm Andrea on May 20. Also visible is the upper-level low to its southwest which helped to weaken the system.

Andrea formed in the far western Atlantic, one of the typical areas for early season cyclogenesis.

Labels:
2019 Storms

## Sunday, May 19, 2019

### Professor Quibb's Picks – 2019

My personal prediction for the 2019 North Atlantic hurricane season (written May 19, 2019) is as follows:

15 cyclones attaining tropical depression status,

14 cyclones attaining tropical storm status,

6 cyclones attaining hurricane status, and

3 cyclones attaining major hurricane status.

Following a fairly average hurricane season in 2018 (which nevertheless featured two devastating major hurricanes), I predict that the 2019 season will see a comparable number of cyclones, albeit with rather different areas to watch. Note that the average Atlantic hurricane season (1981-2010 average) has 12.1 tropical storms, 6.4 hurricanes, and 2.7 major hurricanes. As with any season, our prediction begins with a look at the El Niño Southern Oscillation (ENSO) index, a measure of equatorial sea temperature anomalies in the Pacific ocean that have a well-documented impact on Atlantic hurricane activity. These anomalies are currently positive, corresponding to an El Niño state, and have been since last fall. The image below (click to enlarge) shows model predictions for the ENSO index through the remainder of 2019.

In comparison to the last several years, the situation is more static: no significant change of state is expected during this year's hurricane season (though there is, of course, significant uncertainty). This state of affairs tends to suppress hurricane activity and increase the chance of cyclones in the subtropical Atlantic curving away from the North American coastline (unlike, for example, the unusual track of Hurricane Florence last year).

This is fortunate, because all indications are the subtropical Atlantic will continue to churn out named storms as it did last season. Sea surface temperatures continue to run high in the region, and El Niño effects are not as pronounced there, partially explaining why my prediction still features an above average number of storms. Other factors also somewhat offset the El Niño: ocean temperatures in the tropical Atlantic (the birthplace of most long-track hurricanes) are slightly above normal this year, a trend expected to persist over the next several months. The atmosphere has also been less dry in the region, with less Saharan dry air than in 2018 and the beginning of the 2017 season to quash developing tropical waves. Expect the tropics to be less hostile to long-track hurricane formation than last year, when all cyclones taking the southerly route dissipated upon entering the Caribbean.

My estimated risks on a scale from 1 (least risk) to 5 (most risk) for different specific parts of the Atlantic are as follows:

U.S. East Coast: 3

Though the subtropical Atlantic will be active, I predict less of a risk to the U.S. coastline, with a smaller chance of a Florence-like system this year. Though there may be a few hurricanes passing offshore, most should recurve out over open water. Bermuda, however, is at higher risk.

Yucatan Peninsula and Central America: 2

These regions may benefit the most from a persistent El Niño, with wind shear making the development of an intense hurricane in the western Caribbean difficult. Further, I expect tracks to curve northward more often than striking Central America directly. Later season cyclones originating in the monsoonal gyre near Panama may pose the primary threat, and these tend to be principally rainmakers.

Caribbean Islands: 4

With the main development region (MDR) of the tropical Atlantic more favorable this year, the Caribbean is unlikely to continue the reprieve last year that followed arguably its worst season of all time (2017). Early season storms are still likely to fizzle out due to El Niño-related shear, but a wetter atmosphere suggests that tropical disturbances will have to be watched carefully. This includes a greater possibility of tropical cyclogensis in the Caribbean itself.

Gulf of Mexico: 3

Sea temperatures are consistently higher in the Gulf this year than they have been recently, especially near the Florida gulf coastline, but conditions here overall are a mixed bag. A strong jet stream across the continental U.S. will support more severe thunderstorms over land this summer, but this actually may work against cyclones thriving in the region. Balancing these factors yields an average risk, though this overall rating is a combination of a higher-than-normal risk in the eastern Gulf and a lower-than-normal risk farther west.

Overall, I expect the 2019 hurricane season to feature close-to-average activity. Nevertheless, this is just an informal forecast. Individuals in hurricane-prone areas should always have emergency measures in place. For more on hurricane safety sources, see here. Remember, devastating storms can occur even in otherwise quiet seasons.

Sources: https://www.cpc.ncep.noaa.gov/products/analysis_monitoring/lanina/enso_evolution-status-fcsts-web.pdf, https://www.cpc.ncep.noaa.gov/products/CFSv2/CFSv2seasonal.shtml, https://www.ospo.noaa.gov/Products/ocean/sst/anomaly/

15 cyclones attaining tropical depression status,

14 cyclones attaining tropical storm status,

6 cyclones attaining hurricane status, and

3 cyclones attaining major hurricane status.

Following a fairly average hurricane season in 2018 (which nevertheless featured two devastating major hurricanes), I predict that the 2019 season will see a comparable number of cyclones, albeit with rather different areas to watch. Note that the average Atlantic hurricane season (1981-2010 average) has 12.1 tropical storms, 6.4 hurricanes, and 2.7 major hurricanes. As with any season, our prediction begins with a look at the El Niño Southern Oscillation (ENSO) index, a measure of equatorial sea temperature anomalies in the Pacific ocean that have a well-documented impact on Atlantic hurricane activity. These anomalies are currently positive, corresponding to an El Niño state, and have been since last fall. The image below (click to enlarge) shows model predictions for the ENSO index through the remainder of 2019.

In comparison to the last several years, the situation is more static: no significant change of state is expected during this year's hurricane season (though there is, of course, significant uncertainty). This state of affairs tends to suppress hurricane activity and increase the chance of cyclones in the subtropical Atlantic curving away from the North American coastline (unlike, for example, the unusual track of Hurricane Florence last year).

This is fortunate, because all indications are the subtropical Atlantic will continue to churn out named storms as it did last season. Sea surface temperatures continue to run high in the region, and El Niño effects are not as pronounced there, partially explaining why my prediction still features an above average number of storms. Other factors also somewhat offset the El Niño: ocean temperatures in the tropical Atlantic (the birthplace of most long-track hurricanes) are slightly above normal this year, a trend expected to persist over the next several months. The atmosphere has also been less dry in the region, with less Saharan dry air than in 2018 and the beginning of the 2017 season to quash developing tropical waves. Expect the tropics to be less hostile to long-track hurricane formation than last year, when all cyclones taking the southerly route dissipated upon entering the Caribbean.

My estimated risks on a scale from 1 (least risk) to 5 (most risk) for different specific parts of the Atlantic are as follows:

U.S. East Coast: 3

Though the subtropical Atlantic will be active, I predict less of a risk to the U.S. coastline, with a smaller chance of a Florence-like system this year. Though there may be a few hurricanes passing offshore, most should recurve out over open water. Bermuda, however, is at higher risk.

Yucatan Peninsula and Central America: 2

These regions may benefit the most from a persistent El Niño, with wind shear making the development of an intense hurricane in the western Caribbean difficult. Further, I expect tracks to curve northward more often than striking Central America directly. Later season cyclones originating in the monsoonal gyre near Panama may pose the primary threat, and these tend to be principally rainmakers.

Caribbean Islands: 4

With the main development region (MDR) of the tropical Atlantic more favorable this year, the Caribbean is unlikely to continue the reprieve last year that followed arguably its worst season of all time (2017). Early season storms are still likely to fizzle out due to El Niño-related shear, but a wetter atmosphere suggests that tropical disturbances will have to be watched carefully. This includes a greater possibility of tropical cyclogensis in the Caribbean itself.

Gulf of Mexico: 3

Sea temperatures are consistently higher in the Gulf this year than they have been recently, especially near the Florida gulf coastline, but conditions here overall are a mixed bag. A strong jet stream across the continental U.S. will support more severe thunderstorms over land this summer, but this actually may work against cyclones thriving in the region. Balancing these factors yields an average risk, though this overall rating is a combination of a higher-than-normal risk in the eastern Gulf and a lower-than-normal risk farther west.

Overall, I expect the 2019 hurricane season to feature close-to-average activity. Nevertheless, this is just an informal forecast. Individuals in hurricane-prone areas should always have emergency measures in place. For more on hurricane safety sources, see here. Remember, devastating storms can occur even in otherwise quiet seasons.

Sources: https://www.cpc.ncep.noaa.gov/products/analysis_monitoring/lanina/enso_evolution-status-fcsts-web.pdf, https://www.cpc.ncep.noaa.gov/products/CFSv2/CFSv2seasonal.shtml, https://www.ospo.noaa.gov/Products/ocean/sst/anomaly/

Labels:
Hurricane Stats

## Wednesday, May 15, 2019

### Hurricane Names List – 2019

The name list for tropical cyclones forming in the North Atlantic basin for the year 2019 is as follows:

Andrea

Barry

Chantal

Dorian

Erin

Fernand

Gabrielle

Humberto

Imelda

Jerry

Karen

Lorenzo

Melissa

Nestor

Olga

Pablo

Rebekah

Sebastien

Tanya

Van

Wendy

This list is the same as the list for the 2013 season, with the exception of Imelda, which replaced the retired name Ingrid.

Andrea

Barry

Chantal

Dorian

Erin

Fernand

Gabrielle

Humberto

Imelda

Jerry

Karen

Lorenzo

Melissa

Nestor

Olga

Pablo

Rebekah

Sebastien

Tanya

Van

Wendy

This list is the same as the list for the 2013 season, with the exception of Imelda, which replaced the retired name Ingrid.

Labels:
Hurricane Stats

## Tuesday, May 7, 2019

###
The *abc* Conjecture: Applications and Significance

This is the third part of a three-part post concerning the

The first post in this series presented some explanation as to why the

The

Again, the radical

In 2012, the Japanese mathematician Shinichi Mochizuki shocked the mathematical community by publishing, out of the blue, what he claimed was a proof of the

Moreover, he refused to publicly lecture on the new material, instead only working with a few close colleagues. The combination of the length and inscrutability of the proof with his unwillingness to elucidate it discouraged people from attempting to understand it. In the years since the proof was published, skepticism has mounted concerning the proof's validity. While a small group of mathematicians defend it, a majority of the mathematical community thinks it is unlikely that the proof is valid. For now, the

Nevertheless, it is certain that attempts to prove the conjecture will continue. It has a number of useful applications that would solve a myriad of other mathematical problems, should it be true. To illustrate the power of the

One of the first equations we considered in this series was

So let us assume that we have (somehow) proven the

On the other hand, applying the conjecture to this triple, we have that for ε > 0,

in all but finitely many cases. Since we can choose ε to be any positive number, we can make it small enough so that 3 + 3ε < 4 (e.g. if ε = 0.1). Then if

So the

Of course, the

Conjecture: Every natural number

For example, the special case

These examples start to indicate how important the

Sources: http://projectwordsworth.com/the-paradox-of-the-proof/, Shinichi Mochizuki: Inter-Universal Teichmüller Theory I: Construction of Hodge Theaters, http://mathworld.wolfram.com/PillaisConjecture.html, https://rlbenedetto.people.amherst.edu/talks/abc\_intro14.pdf, Brian Conrad: The

*abc*conjecture. For the first, see here.The first post in this series presented some explanation as to why the

*abc*conjecture seems like a reasonable attempt to mathematically codify a big idea. This idea is that the prime factorization of a sum of two numbers should not really relate to those of the individual numbers. Equivalently, it says that if we see an equation like 3 + 5^{3}= 2^{7}, we should think of it as a "rare event" or "coincidence" that big powers of small primes are related in this way. The second post provided some examples and numerical evidence rigorous version of the conjecture. To review, this states thatThe

*abc*Conjecture: For any ε > 0, no matter how small, for all but finitely many equations of the form*a*+*b*=*c*where*a*and*b*are relatively prime,*rad*(*abc*)^{1 + ε}>*c*.Again, the radical

*rad*(*n*) of an integer*n*is the product of its distinct prime factors. However, none of what has been discussed so far constitutes a mathematical proof that the*abc*conjecture is true or false.In 2012, the Japanese mathematician Shinichi Mochizuki shocked the mathematical community by publishing, out of the blue, what he claimed was a proof of the

*abc*conjecture. However, the initial excitement at this announcement was quickly replaced by confusion; almost no one was able to decipher the tools used in the proof, which totaled over 500 pages in length! Mochizuki, working in isolation for years, had built up a brand new mathematical formalism which he called "Inter-Universal Teichmüller Theory" that was bizarre and unfamiliar to other researchers. The language and notation (an sample of which is provided in the screenshot below) seemed alien, even to mathematicians!Moreover, he refused to publicly lecture on the new material, instead only working with a few close colleagues. The combination of the length and inscrutability of the proof with his unwillingness to elucidate it discouraged people from attempting to understand it. In the years since the proof was published, skepticism has mounted concerning the proof's validity. While a small group of mathematicians defend it, a majority of the mathematical community thinks it is unlikely that the proof is valid. For now, the

*abc*conjecture remains effectively open.Nevertheless, it is certain that attempts to prove the conjecture will continue. It has a number of useful applications that would solve a myriad of other mathematical problems, should it be true. To illustrate the power of the

*abc*conjecture, we give one famous example of an application: Fermat's Last Theorem.One of the first equations we considered in this series was

*x*^{2}+*y*^{2}=*z*^{2}, which relates the side lengths of right triangles. This equation has infinitely many solutions, namely 3^{2}+ 4^{2}= 5^{2}, 5^{2}+ 12^{2}= 13^{2}, etc.**Fermat's Last Theorem**states that if we raise the exponents from 2 to any higher power, there are no solutions in the positive integers. That is,*x*^{3}+*y*^{3}=*z*^{3},*x*^{4}+*y*^{4}=*z*^{4}, and so on are not satisfied by any*x*,*y*, and*z*> 0. Famously claimed by Pierre de Fermat in the 17th century, this problem remained unsolved for centuries. In 1985, when the*abc*conjecture was first stated, it remained open.So let us assume that we have (somehow) proven the

*abc*conjecture, and were interested in Fermat's Last Theorem. The first thing to note about the equation*x*^{n}+*y*^{n}=*z*^{n}is that if we had a solution for this equation, we could always find one for which*x*^{n}and*y*^{n}were relatively prime. This is because if they have a common prime factor, so must*z*^{n}, and we can cancel this factor (raised to the*n*th power) from both sides. Therefore, we have arrived at a situation in which we can apply the*abc*conjecture. The radical of*x*^{n}, for any*n*, is at most*x*since multiplying*x*by itself does not introduce any more prime factors that were not already there. Hence*rad*(*x*^{n}*y*^{n}*z*^{n}) =*rad*(*x*^{n})*rad*(*y*^{n})*rad*(*z*^{n}) ≤*xyz*<*z*^{3}. Therefore, for ε > 0, we have that*rad*(*x*^{n}*y*^{n}*z*^{n})^{1 + ε}< (*z*^{3})^{1 + ε}=*z*^{3 + 3ε}.On the other hand, applying the conjecture to this triple, we have that for ε > 0,

*rad*(*x*^{n}*y*^{n}*z*^{n})^{1 + ε}>*z*^{n}in all but finitely many cases. Since we can choose ε to be any positive number, we can make it small enough so that 3 + 3ε < 4 (e.g. if ε = 0.1). Then if

*n*≥ 4, the two inequalities above directly contradict each other. Since the top one always holds and the bottom holds in all but finitely many cases, we conclude that there can be at most finitely many exceptions to Fermat's Last Theorem when*n*≥ 4.So the

*abc*conjecture does not quite imply Fermat's Last Theorem, but it comes very close. If, in addition, we knew just a bit more about how the exceptional*abc*triples behaved, we could manually verify that there are no counterexamples to Fermat's Last Theorem for*n*≥ 4. Interestingly, this argument does not say anything about the*n*= 3 case, that is, about the non-existence of solutions to*x*^{3}+*y*^{3}=*z*^{3}. This special case, however, had already been proven by Euler in the mid-1700s.Of course, the

*abc*conjecture remains unproven, while Fermat's Last Theorem was finally proven by Andrew Wiles in 1995. This was done by entirely different means. Nevertheless, this serves as a relatively simple example of how the conjecture can prove results about Diophantine equations without invoking very difficult mathematics. Another example of a consequence is the following statement, sometimes called Pillai's conjecture:Conjecture: Every natural number

*k*occurs only finitely many times as the difference of two perfect powers.For example, the special case

*k*= 1 is the subject of Catalan's conjecture, and states that*x*^{p}-*y*^{q}= 1 has only one solution: 3^{2}- 2^{3}= 1. This was proven by Preda Mihăilescu in 2002 (again by very different means from those above and from Wiles' methods), but the general case remains unsolved. If we knew for a fact that the*abc*conjecture were true, we would be able to prove this result by a very similar argument to the one given above for Fermat's Last Theorem (the reader is encouraged to try this!). Note that Pillai's conjecture also implies that the original equation that motivated the*abc*conjecture, namely*y*^{2}=*x*^{3}+*k*, also has only finitely many solutions (for fixed*k*). This is the result David Masser and Joseph Oesterlé sought on their way to first formulating the statement.These examples start to indicate how important the

*abc*conjecture is to the study of Diophantine equations; if it were proven, it would resolve many different problems that are currently treated separately in a single stroke. Even reproving known results in a new and simple way would be greatly beneficial to the theory, since a set of tools that could prove*abc*would help to unify disparate parts of number theory. As a result, mathematicians will doubtlessly continue work toward solving the conjecture and probing the most fundamental structure of numbers.Sources: http://projectwordsworth.com/the-paradox-of-the-proof/, Shinichi Mochizuki: Inter-Universal Teichmüller Theory I: Construction of Hodge Theaters, http://mathworld.wolfram.com/PillaisConjecture.html, https://rlbenedetto.people.amherst.edu/talks/abc\_intro14.pdf, Brian Conrad: The

*abc*Conjecture, 12 sep 2013.
Labels:
Mathematics

## Tuesday, April 16, 2019

###
The *abc* Conjecture: *abc* Triples

This is the second post in a series about the

In the last post, we defined the radical of an integer

Conjecture 1: For all but finitely many equations of the form

However, as mentioned at the end of the previous post, this is in fact false. To prove this, we have to exhibit an

Claim: For any prime number

Proof: This family is infinite because there are infinitely many prime numbers

Since

In fact the situation is even worse than this. Since the radical is less than 2

All of this shows that we cannot correct our conjecture 1 by adding a multiplicative factor to our inequality. The next reasonable thing one might try is a power law. Perhaps

The

The variable ε could for example be 1, and then we recover the

This is another measure of how large

The

Let's see if our conjecture seems plausible from the numerical data. One possible way to do this is to come up with many triples and see how large the quality

In the above diagram (click to enlarge), the

In fact, there are well over a hundred known with higher quality, a list of which may be found here. Currently, the highest known quality belongs to the triple (2,6436341,6436343) = (2,3

Of course, no matter how many examples we check, we are no closer to proving that the

Sources: http://www.math.leidenuniv.nl/~desmit/abc/, Greg Martin and Winnie Miao:

*abc*conjecture. For the first post, see here.In the last post, we defined the radical of an integer

*n*, namely the product of distinct prime factors of*n*. We suspected in the last post that for most equations*a*+*b*=*c*where*a*and*b*are relatively prime,*rad*(*abc*) >*c*. This is because this inequality expresses our hypothesis that there should not be too many high powers of primes in the factorizations of*a*,*b*, and*c*. As a result, we made the following conjecture:Conjecture 1: For all but finitely many equations of the form

*a*+*b*=*c*where*a*and*b*are relatively prime,*rad*(*abc*) >*c*.However, as mentioned at the end of the previous post, this is in fact false. To prove this, we have to exhibit an

*infinite family*of equations*a*+*b*=*c*with*rad*(*abc*) ≤*c*. Any triple (*a*,*b*,*c*) of numbers satisfying this property is called an**. The only example we've seen so far is (1,8,9), or in equation form, 1 + 8 = 9. In terms of this new definition, we are trying to show that there are infinitely many***abc*triple*abc*triples. The following claim gives the desired result.Claim: For any prime number

*p*grater than 2, the triple (*a*,*b*,*c*) = (1,2^{p(p-1)}- 1,2^{p(p-1)}) is an*abc*triple.Proof: This family is infinite because there are infinitely many prime numbers

*p*. The proof depends on a fact in elementary number theory known as the**Euler-Fermat Theorem**. This theorem may be used to show that*b*= 2^{p(p-1)}- 1 is divisible by*p*^{2}. This is significant because we now know that the radical of*b*cannot be greater than*b*divided by*p*; this is because taking the radical of*b*"forgets" about at least one of the factors of*p*. Of course,*rad*(1) = 1 and*rad*(2^{p(p-1)}) = 2 soSince

*p*> 2, this last value is less than*c*, so that we do in fact have an infinite family of*abc*triples.In fact the situation is even worse than this. Since the radical is less than 2

*c*/*p*(as shown in the proof), it is not enough to replace the hypothesis*rad*(*abc*) >*c*with 2*rad*(*abc*) >*c*, or any higher multiple. We can make 2/*p*arbitrarily small by increasing*p*so that the radical is smaller than*c*by an arbitrarily large factor. For example, taking*p*= 5 gives the*abc*triple (1,1048575,1048576). Note that 5^{2}= 25 divides*b*= 1048575, as claimed. Our proof guarantees that*rad*(*abc*) ≤ 2*c*/5. In fact the radical of this product is 419430. This is indeed less than 2/5 of*c*.All of this shows that we cannot correct our conjecture 1 by adding a multiplicative factor to our inequality. The next reasonable thing one might try is a power law. Perhaps

*rad*(*abc*)^{2}>*c*for all but finitely many equations, or something similar. This, in fact, is the correct idea. However, the choice of 2 as the exponent again seems arbitrary. We know already that the statement is false when the power is 1, so let's try increasing it just a little. This leads us to the actual*abc*conjecture.The

*abc*Conjecture: For any ε > 0, no matter how small, for all but finitely many equations of the form*a*+*b*=*c*where*a*and*b*are relatively prime,*rad*(*abc*)^{1 + ε}>*c*.The variable ε could for example be 1, and then we recover the

*rad*(*abc*)^{2}>*c*inequality. However, ε could also be very close to 0, giving an exponent of 1 + ε very close to 1. Crucially, any function*x*^{1 + ε}with ε > 0 eventually increases faster than any constant multiple of*x*, for example*x*^{1.1},*x*^{1.00001}, etc. Therefore, this conjecture gets around the counterexample to conjecture 1. Nevertheless, the*abc*conjecture in some sense says that conjecture 1 is*really*close to being true. All we needed to do was increase the exponent by any positive amount. These concepts may become a little clearer with a new concept, called the**quality**of a triple (*a*,*b*,*c*). The formula for the quality, denoted*q*(*a*,*b*,*c*) isThis is another measure of how large

*c*is compared to*rad*(*a*,*b*,*c*). In fact,*rad*(*a*,*b*,*c*)^{q(a,b,c)}=*c*. For example,*q*(13,22,35) is about 0.386, and*q*(1,8,9) is close to 1.226. This allows a more succinct description of the conjecture: for most triples,*q*≤ 1. It follows from our definition that*abc*triples are those for which*q*> 1. Finally, the*abc*conjecture is equivalent to the following.The

*abc*Conjecture (Second Formulation): For any ε > 0, no matter how small, for all but finitely many equations of the form*a*+*b*=*c*where*a*and*b*are relatively prime,*q*(*a*,*b*,*c*) < 1 + ε.Let's see if our conjecture seems plausible from the numerical data. One possible way to do this is to come up with many triples and see how large the quality

*q*is for each.In the above diagram (click to enlarge), the

*x*-axis is our variable*c*. For each*c*between 2 and 2000, the plot goes through*all*possible relatively prime values*a*and*b*adding to*c*, finds the triple among these with the highest quality, and plots a corresponding point there. Therefore, all points are already among the highest quality triples. Even among these,*abc*triples (those that lie above the horizontal*q*= 1 line) are rare. Furthermore, they seem to get even more rare as*c*increases. In terms of diagrams of this sort, the conjecture states that only finitely many dots lie above a given horizontal line*q*= 1 + ε for any ε > 0. The highest quality*abc*triple that appears on the plot is (3,125,128) = (3,5^{3},2^{7}), with a quality of 1.426. Are there any higher quality triples out there?In fact, there are well over a hundred known with higher quality, a list of which may be found here. Currently, the highest known quality belongs to the triple (2,6436341,6436343) = (2,3

^{10}109,23^{5}), with*q*= 1.6299. Even assuming the*abc*conjecture does not answer the question of whether this triple is really the highest quality there is. All it says is that examples of this sort must eventually die out as we approach infinity. For instance, there may very well be no triples at all with*q*≥ 2, meaning that*c*≤*rad*(*abc*)^{2}may hold in all cases with no exceptions.Of course, no matter how many examples we check, we are no closer to proving that the

*abc*conjecture holds. In the last post, we will discuss attempts to prove it, as well as the applications of the statement, should it be true.Sources: http://www.math.leidenuniv.nl/~desmit/abc/, Greg Martin and Winnie Miao:

*abc*Triples; Arxiv:1409.2974v1 [math.NT] 10 sep 2014, Brian Conrad: The*abc*Conjecture, 12 sep 2013.
Labels:
Mathematics

## Tuesday, March 26, 2019

###
The *abc* Conjecture: Motivation

Some of the earliest problems in mathematics asked about the integer solutions to simple polynomial equations. For instance, what are the possible right triangles with whole number side lengths? The solution dates at least back to the Ancient Greeks; the side lengths are related by Pythagoras' famous formula

In general, a

Historically, different types of Diophantine Equations were typically solved by ad hoc methods, as they come in many different varieties. However, one general observation that connects many of these equations is that

The

Simply put, we want the answer to our motivating question to be "it doesn't." Somehow, the additive and multiplicative structures of the integers should be independent of one another. This is in some ways a deep statement, and not at all intuitively clear, but we'll begin with this assumption. In other words, for an equation

Definition: Two numbers are

For example, 34 and 45 are relatively prime, but 24 and 63 are not, because they are both divisible by 3. Here is how we will express our independence hypothesis: for any equation of the form

2

in which all three numbers are divisible by arbitrarily high powers of 2. This doesn't represent some special connection between addition and multiplication - all we've done is multiplied the equation 1 + 1 = 2 by 2

Definition: The

For example,

Now we return to our equation

"Almost" Conjecture: For equations of the form

We're close! The equation

Conjecture 1: For all but finitely many equations of the form

Finally, a real conjecture! Unfortunately, it's false. In other words, there are

Sources: https://rlbenedetto.people.amherst.edu/talks/abc\_intro14.pdf, Brian Conrad: The

*x*^{2}+*y*^{2}=*z*^{2}. The 7th century Indian mathematician Brahmagupta studied integer solutions to the equation*x*^{2}- 2*y*^{2}= 1 as well as the same formula with 2 replaced by a general integer*n*(called Pell's equation). Many other similar equations have been studied for centuries or millennia.In general, a

**Diophantine equation**is a polynomial equation for which we are interested in integer solutions. Counterintuitively, some questions about solving these in the integers may be more difficult than considering all types of solutions. For example, the fundamental theorem of algebra states that any polynomial in a single variable has a root over the complex numbers (e.g.*x*^{3}- 4*x*^{2}+ 17*x*+ 20 = 0 is true for some complex number). However, there is often no integer solution to such equations.Historically, different types of Diophantine Equations were typically solved by ad hoc methods, as they come in many different varieties. However, one general observation that connects many of these equations is that

*they state something about the factorization of a sum of two numbers*. Pythagoras' equation says something special about the sum of two squares, namely that it is another square! Similarly, Pell's equation says that one plus some number multiplied by a square has the property that it too is square. Our motivating question may then be taken to be:*How does the factorization of a sum of two numbers relate to the factorizations of the individual numbers?*The

*abc*Conjecture provides a partial answer to this question. Its name comes from the fact that we are considering equations of the form*a*+*b*=*c*and asking how the factorizations of the three numbers relate. Mathematicians David Masser and Joseph Oesterlé first made the conjecture in 1985 while studying integer points on what are called**elliptic curves**, in this case given by the equation*y*^{2}=*x*^{3}+*k*(where*k*is a fixed integer). This is yet another example of a sum having special factorization properties. Throughout the rest of this post, we will see how thinking about the motivating question might lead you to formulating the*abc*conjecture.Simply put, we want the answer to our motivating question to be "it doesn't." Somehow, the additive and multiplicative structures of the integers should be independent of one another. This is in some ways a deep statement, and not at all intuitively clear, but we'll begin with this assumption. In other words, for an equation

*a*+*b*=*c*, if all three numbers satisfy some special factorization properties (e.g. being cubes, etc.) it should in some sense*be a coincidence*. Our next task is to make this progressively less vague. First, we need a definition.Definition: Two numbers are

**relatively prime**if they share no common prime divisors.For example, 34 and 45 are relatively prime, but 24 and 63 are not, because they are both divisible by 3. Here is how we will express our independence hypothesis: for any equation of the form

*a*+*b*=*c*, where*a*and*b*are relatively prime, if*a*and*b*are divisible by high powers of primes,*c*almost always is not. This is in keeping with our theme because "divisible by high powers of primes" is special factorization property. That is, most prime factorizations should look more like 705 = 3*5*47 and not 768 = 2^{8}*3. The assumption that*a*and*b*are relatively prime exists to rule out silly equations like2

^{n}+ 2^{n}= 2^{n + 1},in which all three numbers are divisible by arbitrarily high powers of 2. This doesn't represent some special connection between addition and multiplication - all we've done is multiplied the equation 1 + 1 = 2 by 2

^{n}. If we assert that*a*and*b*are relatively prime, then the prime factors of each of the three numbers are distinct, and we eliminate the uninteresting examples. Next, we require a mathematical notion that measures "divisibility by high prime powers".Definition: The

**radical**of a number*n*, denoted*rad*(*n*), is the product of the distinct prime powers of*n*. Also define*rad*(1) = 1.For example,

*rad*(705) = 3*5*47 = 705 (since the factors 3, 5, and 47 are distinct) but*rad*(768) = 2*3 = 6. The radical function forgets about any powers in the prime factorization, keeping only the primes themselves. Notice that the radical of a number can be as large as the number itself, but it can also be much smaller. The amount by which*rad*(*n*) is smaller than*n*can be taken as a measure of to what extent*n*is divisible by large prime powers.Now we return to our equation

*a*+*b*=*c*(where we will now consistently assume the relatively prime hypothesis). A reasonable way to test for high prime power divisibility for all three of these numbers is to calculate*rad*(*abc*) =*rad*(*a*)*rad*(*b*)*rad*(*c*) (the reader may wish to prove this last equation). Since*rad*(*abc*) could be as large as*abc*itself, it seems likely that*rad*(*abc*) would usually be much larger any of the individual numbers, the largest of which is*c*. For example, consider 13 + 22 = 35. In this case,*rad*(*abc*) =*rad*(13*22*35) = 13*2*11*7*5 = 10010, which is*much*larger than*c*= 35. However, this property does not always hold true. Consider another example, 1 + 8 = 9. Now we have*rad*(*abc*) =*rad*(1*8*9) = 2*3 = 6, and 6 < 9 =*c*. Notice that this anomaly reflects something weird going on; the equation can also be written 1 + 2^{3}= 3^{2}, so one plus a cube is a square. Testing different values of*a*,*b*, and*c*gives the impression that equations of the second sort are rare. Therefore, we make an almost mathematical conjecture:"Almost" Conjecture: For equations of the form

*a*+*b*=*c*where*a*and*b*are relatively prime,*rad*(*abc*) is almost always greater than*c*.We're close! The equation

*rad*(*abc*) > c is a bona fide mathematical condition that we can check. However, we have yet to render "almost always" into mathematical language. Clearly there are infinitely many*a*+*b*=*c*equations to look at. What does it mean to say that "most of them" behave in some way? We know from our 1 + 8 = 9 example that there are at least some exceptions. Maybe we could assert that there are less than 10 total exceptions, or less than 100. However, these numbers seem arbitrary, so we'll just guess that there are only*finitely many*exceptions. That is, all but at most*N*of these equations, for some fixed finite number*N*, satisfy our hypothesis. In conclusion, we conjecture that:Conjecture 1: For all but finitely many equations of the form

*a*+*b*=*c*where*a*and*b*are relatively prime,*rad*(*abc*) >*c*.Finally, a real conjecture! Unfortunately, it's false. In other words, there are

*infinitely*many such equations for which*rad*(*abc*) ≤*c*. Don't worry! It's rare in mathematics to come up with the correct statement on the first try! In the next post, we'll prove our conjecture 1 false and see how to correct it.Sources: https://rlbenedetto.people.amherst.edu/talks/abc\_intro14.pdf, Brian Conrad: The

*abc*Conjecture, 12 sep 2013.
Labels:
Mathematics

## Tuesday, March 5, 2019

### The Casimir Effect

The idea of the

Maxwell's theory is still an essential backbone of physics today. Nevertheless, the introduction of quantum mechanics in the early 20th century introduced new aspects of electromagnetism. Perhaps most importantly, it was discovered that light comes in discrete units called

Naturally, QED predicted new phenomena that classical electromagnetism had not. One especially profound change was the idea of

In these diagrams, the loops represent the evanescent

The setup of the Casimir effect involves two conducting metal plates placed parallel to one another. The fact that the plates are conducting is important because the electric field vanishes inside conducting materials. Now, the vacuum energy between the plates can be calculated as a sum over the possible wavelengths of the fluctuations of the electromagnetic field. However, unlike the free space vacuum, the possible wavelengths are limited by the size of the available space: the longest wavelength contributions to the vacuum energy do not occur between the plates (this is schematically illustrated in the image above). A careful subtraction of the vacuum energy density inside the plates from outside yields that there is more energy outside. Remarkably, this causes an attractive force between these plates known as the

Around 50 years after first being postulated, the effect was finally measured experimentally with significant precision. The primary issue was that for the Casimir force to be large enough to measure, the metal plates would have to be put

In the figure above, the squares indicate data points from the experiment and the curve is the theoretical model (including the corrections mentioned). The distance on the

The existence of the Casimir effect would seem to vindicate the rather strange predictions of QED with respect to the quantum vacuum, suggesting that it is indeed full of energy that can be tapped, if indirectly. However, others have argued that it is possible to derive the effect without reference to the energy of the vacuum, and therefore the experiment does not necessarily mean that vacuum energy is "real" in any meaningful way. Continued study into the existence of vacuum energy may help to explain the accelerating expansion of the universe since some mysterious "dark energy" is believed to be the source. In the mean time, the Casimir effect is an important experimental verification of QED and could someday see applications in nanotechnology, since the force would be relatively large on small scales.

Sources: https://www.scientificamerican.com/article/what-is-the-casimir-effec/, https://arxiv.org/pdf/hep-th/0503158.pdf,

**electromagnetic field**is essential to physics. Dating back to the work of James Clark Maxwell in the mid-1800s, the classical theory of electromagnetism posits the existence of certain electric and magnetic fields that permeate space. Mathematically, these fields assign vectors (arrows) to every point in space, and their values at various points determine how a charged particle moving in space would behave. For example, the magnetic field generated by a magnet exerts forces on other nearby magnetic objects. Crucially, the theory also explains light as an electromagnetic phenomenon: what we observe as visible light, radio waves, X-rays, etc. are "waves" in the electromagnetic field that propagate in space.Maxwell's theory is still an essential backbone of physics today. Nevertheless, the introduction of quantum mechanics in the early 20th century introduced new aspects of electromagnetism. Perhaps most importantly, it was discovered that light comes in discrete units called

**photons**and behaves in some ways both as a wave and a particle. Though electromagnetism on the human scale still behaves largely as the classical theory predicts, at small scales there are quantum effects to account for. Around the middle of the century, physicists Richard Feynman, Shinichiro Tomonaga, Julian Schwinger, and many others devised a new theory of**quantum electrodynamics**(or QED) that described how light and matter interact, even on quantum scales.Naturally, QED predicted new phenomena that classical electromagnetism had not. One especially profound change was the idea of

**vacuum energy**. For most purposes, "vacuum" is synonymous with "empty space". As is typical of quantum mechanics, however, a system is rarely considered to be in a single state, but rather in a superposition of many different states simultaneously. These different states can have different "weights" so that the system is "more" in one given state than another. This paradigm applies even to the vacuum. Certain pairs of particles may appear and disappear spontaneously in many of these states and even exchange photons. Some of the possible interactions are illustrated below with**Feynman Diagrams**.In these diagrams, the loops represent the evanescent

**virtual particle**pairs described above. Wavy lines represent the exchange of photons. Each of the six diagrams represents a possible vacuum interaction, and there are many more besides (infinitely many, in fact!). The takeaway is that the QED vacuum is*not*empty, but rather a "soup" of virtual particle interactions due to quantum fluctuations. Further, these interactions have energy, known as**vacuum energy**. This, at least, is the mathematical description. There are some curious aspects to this description, because the vacuum energy calculation in any finite volume yields a divergent series. In other words, there is theoretically an infinite amount of vacuum energy in any finite volume! Because of this, physicists devised a process called**renormalization**that cancels out these infinities in calculations describing the interaction of real particles. This process in fact gives results that have been confirmed by experiment. Nevertheless, it does not follow that the infinite vacuum energy exists in any "real" sense or is accessible to measurement. One possible way in which it is, however, is the**Casimir Effect**.The setup of the Casimir effect involves two conducting metal plates placed parallel to one another. The fact that the plates are conducting is important because the electric field vanishes inside conducting materials. Now, the vacuum energy between the plates can be calculated as a sum over the possible wavelengths of the fluctuations of the electromagnetic field. However, unlike the free space vacuum, the possible wavelengths are limited by the size of the available space: the longest wavelength contributions to the vacuum energy do not occur between the plates (this is schematically illustrated in the image above). A careful subtraction of the vacuum energy density inside the plates from outside yields that there is more energy outside. Remarkably, this causes an attractive force between these plates known as the

**Casimir force**. The force increases as the distance between plates is decreased. Precisely, the magnitude of the force*F*is proportional to 1/*d*^{4}, where*d*is the distance between the plates. As a result, if the distance is halved, the force goes up by a factor of sixteen! The initial calculation of this effect was due to H.G.B Casimir in 1948.Around 50 years after first being postulated, the effect was finally measured experimentally with significant precision. The primary issue was that for the Casimir force to be large enough to measure, the metal plates would have to be put

*very*close to one another, less than 1 micrometer (0.001 mm). Even then, very sensitive instruments are necessary to measure the force. One landmark experiment took place in 1998. Due to the practical difficulty of maintaining two parallel plates very close to one another, this experiment utilized one metal plate and one metal sphere with a radius large compared to the separation (so that it would "look" like a flat plate close up). The authors of the experiment also added corrections to Casimir's original equation accounting for the sphere instead of the plane and the roughness of the metal surfaces (at the small distances of the experiment, microscopic bumps matter). They obtained the following data for the force as it varies with distance:In the figure above, the squares indicate data points from the experiment and the curve is the theoretical model (including the corrections mentioned). The distance on the

*x*-axis is in nanometers and the smallest distance measured was around 100 nm, hundreds of times smaller than the width of a human hair. Even at these minuscule distances, the force only reached a magnitude of about 1*10^{-10}Newtons, a billion times smaller than the weight of a piece of paper. Nevertheless, the results confirmed the presence of the Casimir force to high accuracy.The existence of the Casimir effect would seem to vindicate the rather strange predictions of QED with respect to the quantum vacuum, suggesting that it is indeed full of energy that can be tapped, if indirectly. However, others have argued that it is possible to derive the effect without reference to the energy of the vacuum, and therefore the experiment does not necessarily mean that vacuum energy is "real" in any meaningful way. Continued study into the existence of vacuum energy may help to explain the accelerating expansion of the universe since some mysterious "dark energy" is believed to be the source. In the mean time, the Casimir effect is an important experimental verification of QED and could someday see applications in nanotechnology, since the force would be relatively large on small scales.

Sources: https://www.scientificamerican.com/article/what-is-the-casimir-effec/, https://arxiv.org/pdf/hep-th/0503158.pdf,

*The Quantum Vacuum: An Introduction to Quantum Electrodynamics*by Peter W. Milonni, http://web.mit.edu/kardar/www/research/seminars/PolymerForce/articles/PRL-Mohideen98.pdf
Labels:
Astronomy and Physics,
Forces

## Tuesday, February 12, 2019

### Ocean Currents and the Thermohaline Circulation

Ocean currents are ubiquitous and familiar. Beach goers are wary of tidal currents, as well as those caused by weather systems. Currents caused by tides and weather are constantly changing and chaotic. However, under this noise exists a larger-scale and more orderly system of circulation. By averaging over long time periods (in effect screening out the noise of short-term fluctuations), larger currents such as the Gulf Stream, which brings warm water northward along the east coast of the United States, appear. Another example is the California current, which brings the cold waters of the north Pacific down along the west coast. But why do these currents exist? Some patterns may be seen if we expand our view to the world as a whole.

The above image shows major surface ocean currents around the world. Note that despite geographical differences, some currents in each ocean in each hemisphere follow the same general pattern, flowing east to west in the tropical latitudes, toward the poles on the western edge of ocean basins, west to east at mid-latitudes, and finally toward the equator on the eastern edges. These circular currents are known as

The prevailing winds at the Earth's surface fit into a larger dynamic atmospheric pattern. The greater heating of the tropics as compared to the polar regions and the rotation of the Earth lead to the formation of three atmospheric cells in each hemisphere. The winds in these cells rotate due to the Coriolis effect (in essence the fact that straight paths appear to curve from the viewpoint of an observer on a rotating planet), producing east to west winds in the tropics and polar regions, and west to east winds in the mid-latitudes. Look back at the subtropical gyres on the map of currents. Notice that the currents labeled "equatorial" follow the trade winds, and the north Pacific, north Atlantic, etc. currents follow the prevailing westerlies. The west and east boundary currents then "complete the circle" and close the flow. This is no coincidence. It is friction between air and water that drives subtropical gyres: the force of wind tends to make water flow in the same direction. Another important example is the Antarctic circumpolar current, the largest in the world. Since there are no landmasses between roughly 50°S and 60°S latitude, the westerlies drive a current unimpeded that stretches all the way around the continent of Antarctica. Many other currents in the global diagram are responses to these main gyres, or are connected to prevailing winds in more complicated ways.

This system of ocean currents has profound impacts on weather and climate. Due to the Gulf Stream, north Atlantic current, and its northern extension, the Norwegian current, temperatures in northwestern Europe are several degrees warmer than they would otherwise be.

The above image illustrates one of the influences of ocean currents on weather. It shows all tropical cyclone tracks (hurricanes, typhoons, etc.) worldwide from 1985-2005. Since tropical cyclones need warm ocean surface waters to develop, the cold California current helps to suppress eastern Pacific hurricanes north of 25° N or so. In contrast, the warm Kuroshio current in the western Pacific allows typhoons to regularly affect Japan, which is at a higher latitude. Note also the presence of cyclones in the southwest Pacific and the lack of any formation in the southeast Pacific (though very cold surface waters are only one of several factors in this).

Despite their vast impact, which goes well beyond the examples listed, all of the currents considered so far are

The above image gives a very schematic illustration of the global three-dimensional circulation of the oceans, known as the

The above graphic illustrates the

One climatological influence of this phenomenon is the ocean's increased ability to take up carbon dioxide. Most of the carbon dioxide emitted by human industry since the late 1800s has dissolved in the oceans. Since deep water takes so long to circulate, increased CO

The network of mechanisms driving ocean currents and the thermohaline circulation is quite intricate, and we have only touched on some of them here: weather systems, prevailing winds, differences in density, etc. There are many more subtleties as to why the ocean circulates the way it does. The study of these nuances is essential for fully understanding the Earth's weather and climate.

Sources: Atmosphere, Ocean, and Climate Dynamics: An Introductory Text by John Marshall and R. Alan Plumb, https://www.britannica.com/science/ocean-current, http://www.seos-project.eu/modules/oceancurrents/oceancurrents-c01-p03.html

The above image shows major surface ocean currents around the world. Note that despite geographical differences, some currents in each ocean in each hemisphere follow the same general pattern, flowing east to west in the tropical latitudes, toward the poles on the western edge of ocean basins, west to east at mid-latitudes, and finally toward the equator on the eastern edges. These circular currents are known as

**subtropical gyres**. For example, the Gulf Stream is the western poleward current in the north Atlantic subtropical gyre and the California current the eastern current toward the equator in the north Pacific tropical gyre. These exist largely due to the Earth's prevailing winds.The prevailing winds at the Earth's surface fit into a larger dynamic atmospheric pattern. The greater heating of the tropics as compared to the polar regions and the rotation of the Earth lead to the formation of three atmospheric cells in each hemisphere. The winds in these cells rotate due to the Coriolis effect (in essence the fact that straight paths appear to curve from the viewpoint of an observer on a rotating planet), producing east to west winds in the tropics and polar regions, and west to east winds in the mid-latitudes. Look back at the subtropical gyres on the map of currents. Notice that the currents labeled "equatorial" follow the trade winds, and the north Pacific, north Atlantic, etc. currents follow the prevailing westerlies. The west and east boundary currents then "complete the circle" and close the flow. This is no coincidence. It is friction between air and water that drives subtropical gyres: the force of wind tends to make water flow in the same direction. Another important example is the Antarctic circumpolar current, the largest in the world. Since there are no landmasses between roughly 50°S and 60°S latitude, the westerlies drive a current unimpeded that stretches all the way around the continent of Antarctica. Many other currents in the global diagram are responses to these main gyres, or are connected to prevailing winds in more complicated ways.

This system of ocean currents has profound impacts on weather and climate. Due to the Gulf Stream, north Atlantic current, and its northern extension, the Norwegian current, temperatures in northwestern Europe are several degrees warmer than they would otherwise be.

The above image illustrates one of the influences of ocean currents on weather. It shows all tropical cyclone tracks (hurricanes, typhoons, etc.) worldwide from 1985-2005. Since tropical cyclones need warm ocean surface waters to develop, the cold California current helps to suppress eastern Pacific hurricanes north of 25° N or so. In contrast, the warm Kuroshio current in the western Pacific allows typhoons to regularly affect Japan, which is at a higher latitude. Note also the presence of cyclones in the southwest Pacific and the lack of any formation in the southeast Pacific (though very cold surface waters are only one of several factors in this).

Despite their vast impact, which goes well beyond the examples listed, all of the currents considered so far are

**surface currents**. Typically, these currents exist only in the top kilometer of the ocean, and the picture below this can look quite different.The above image gives a very schematic illustration of the global three-dimensional circulation of the oceans, known as the

**thermohaline circulation**. The first basic fact about this circulation, especially the deep ocean circulation, is that it is*slow*. Narrow, swift surface currents such as the Gulf Stream have speeds up to 250 cm/s. Even the slower eastern boundary currents often manage 10 cm/s. In contrast, deep ocean currents seldom exceed 1 cm/s. Their tiny speed and remoteness makes them extremely difficult to measure; in fact, rather than directly charting their course, the flow is inferred from quantities called "tracers" in water samples. Measurements of the proportion of certain radioactive isotopes, for example, are used to calculate the last time a given water sample "made contact" with the atmosphere.The above graphic illustrates the

**age**of deep ocean water around the world. The age (in years) is how long it has been since a given water parcel came to equilibrium with the surface. Note that the thermohaline circulation occurs on timescales of over 1000 years. This information indicates that**deep water formation**(when water from the surface sinks) takes place in the North Atlantic but*not*the North Pacific, as indicated in the first graphic. This is because all the deep waters of the Pacific are quite "old". Deep water formation also occurs in the Southern Ocean, near Antarctica. In both cases, the mechanism is similar: exposure to frigid air near the poles makes the surface waters very cold, and therefore dense. Further, in winter, sea ice forms in these cold waters, leaving saltier water behind (since freshwater was "taken away" to form sea ice). This salty, cold water is denser than the ocean around it and it sinks. The newly formed deep water can flow near the bottom of the ocean for hundreds of years before coming back to the surface.One climatological influence of this phenomenon is the ocean's increased ability to take up carbon dioxide. Most of the carbon dioxide emitted by human industry since the late 1800s has dissolved in the oceans. Since deep water takes so long to circulate, increased CO

_{2}levels are only now beginning to penetrate the deep ocean. Most ocean water has not "seen" the anthropogenic CO_{2}so it will continue to take up more of the gas for hundreds of years. Without this, there would much more carbon dioxide in the atmosphere, and likely faster global warming.The network of mechanisms driving ocean currents and the thermohaline circulation is quite intricate, and we have only touched on some of them here: weather systems, prevailing winds, differences in density, etc. There are many more subtleties as to why the ocean circulates the way it does. The study of these nuances is essential for fully understanding the Earth's weather and climate.

Sources: Atmosphere, Ocean, and Climate Dynamics: An Introductory Text by John Marshall and R. Alan Plumb, https://www.britannica.com/science/ocean-current, http://www.seos-project.eu/modules/oceancurrents/oceancurrents-c01-p03.html

Labels:
Meteorology

Subscribe to:
Posts (Atom)