Infinity: The Cardinality of the Continuum

Before reading this post, make sure you have read Infinity: The First Transfinite Cardinal, and Infinity: Countable Sets.

In the previous posts of this series, it was established that the sets of natural numbers, integers, rational numbers, and even algebraic numbers have an equivalent cardinality: aleph-zero. However, not all real numbers fall under the umbrella of algebraic numbers. All of the numbers that are real but non-algebraic are irrational, and are specifically known as transcendental. Numbers such as e and π are transcendental.

To determine the cardinality of the real numbers, this problem can be again simplified to a problem involving ordered n-tuplets. This is done by considering the construction of an arbitrary real number. The general real number has a finite whole number part, followed by an infinite decimal expansion. For example, the real number π has a whole number part of 3, and a decimal expansion of .14159265... It is simpler to just ignore the whole number part, and focus on the real numbers on the interval (0,1). All of these are defined uniquely (almost, as we will see below) by their infinite decimal expansion. Therefore, each of these numbers is defined by an ordered n-tuplet, with n being infinite, and of the form

(a1,a2,a3...)

Since all values in this sequence are place values, each must be 0, 1, 2, 3, 4, 5, 6, 7, 8 or 9. However, no clarity is lost if these real numbers are converted to binary, and each number still as a unique infinite decimal expansion, this time only incorporating 1's and 0's. We have the just simplified the problem to determining whether the set of all infinite sequences consisting of 1's and 0's is countable, i.e. whether it has a cardinality of aleph-zero.

Georg Cantor was the first to devise this method, and through infinite binary sequences found a very elegant way to find the cardinality of the real numbers, through proof by contradiction. It is called the diagonal argument. To understand this argument, consider all possible infinite binary sequences as making up a set, called S. Each element Sn is then an infinite binary sequence. If the cardinality of the real numbers is aleph-zero, then each element in the set can be numbered Sn, with n being a natural number. The first few elements of the set are shown below:



The actual ordering of this set is arbitrary, since if the cardinality of S is aleph-zero, all sequences with be covered eventually. For the next part of the proof, consider a sequence Sx, which is constructed by taking the nth element of each Sn and reversing it, i.e. 0 becomes 1, and 1 becomes 0. This is illustrated below:



The nth element of every nth sequence is bolded, and for each bolded element, the opposite one is placed in the sequence Sx. The resulting sequence is clearly an infinite binary sequence, and therefore is a member of the set S. However, by definition, it is different from any sequence in the set (S1,S2,S3...) because for any sequence Sn, with n a natural number, the nth element of the sequence is different from that of Sx. Therefore, assigning a natural number to each infinite binary sequence does not cover all such sequences, and the function from the natural numbers to S is not a bijection. Therefore, the set S has a cardinality greater then aleph-zero. Sets with cardinalities greater than aleph-zero are also known as uncountable.

When extending this back to our real number problem, there are a few slight glitches in this system, one of which being that an infinite binary decimal expansion such as .001111... with infinite 1's is actually equal to another, namely .010000... Therefore, each real number does not quite have a unique decimal expansion. However, this problem can be resolved.

Only numbers with terminating decimal expansions can be expressed in the two ways shown above, and in binary, the only such numbers are those whose denominators involve only a power of 2. For example, 1/2=.1000...=.0111..., and 5/8=.101000...=.100111... These numbers can be collected into a set of their own, called A, the first few members of which are {1/2, 1/4, 3/4, 1/8, 3/8,...}.

This is a subset of the rational numbers, and therefore has cardinality aleph-zero. The set of infinite binary sequences with infinite 0's or infinite 1's (which starts {.1000...,.0111...,.01000...,.00111...,...}) merely has two elements for each element of set A, and still has a cardinality of aleph-zero. (just as the sets {1,2,3...} and {1,-1,2,-2...} have the same cardinality, even though there are two elements in the latter whose absolute values correspond to the former) Because of this, a bijection can be set up between them, corresponding .1000... to 1/2, .0111... to 1/4, and so on. Adding this to the original set S, one finds that each real number on (0,1) now corresponds to a single unique infinite binary sequence.

Clearly, the cardinality of the entire set of real numbers must be greater than or equal to that of the real numbers on (0,1), and so we have now proved that

The cardinality of the real numbers, known as the cardinality of the continuum and denoted by , is strictly greater then the cardinality of the natural numbers, aleph-zero. In other words

> ℵ₀

More number systems still remain, including the complex numbers. However, it is fairly easy to see that the complex numbers still have the cardinality of the continuum, as any complex number can be defined uniquely as an ordered pair of two real numbers (a,b). Also, through a similar method that was used for integral ordered n-tuplets that is not detailed here, it can be proved that sets of ordered n-tuplets of real numbers or of complex numbers both have the cardinality of the continuum. This result also seems intuitively correct from previous examinations of ordered n-tuplets.

Using Cantor's diagonal argument, it was established that the cardinality of the set of real numbers was greater than that of natural numbers, in other words that there is no bijection between them. However, it has not been found exactly what this cardinality of the continuum actually is, or how to relate these quantities to each other in any way.

In order to accomplish this, one must first understand the concept of a power set. A power set is denoted P(S), where S is any ordinary set. Every set has a corresponding power set, and the above statement says that the power set of S is called P(S).

To construct the power set for any set S, take each unique combination of the elements in S, and put it into its own set. Then compile all of these sets and place them within another set. This is the power set, P(S). For example, the set {1,2,3} includes eight unique combinations of the elements contained within it, namely: {}, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, and {1,2,3}. (the first of these is included as a choice of zero elements from the set) All of these are then included in a larger set, yielding {{}, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}}. In conclusion:

P({1,2,3})={{}, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}}

If one compares the cardinality of these two sets, where each subset of the latter is a single element, it is easy to see that they are 3 and 8, respectively. One also notices that 2^3=8. This is no coincidence. The total number of combinations of n elements is (2^n)-1, and when one adds the empty set to this total, 2^n. Expressed in formula form, with the cardinality of a set S written as |S|:



Applying this knowledge to what we know about the set of natural numbers, it is easy to identify the set of all real numbers as the set of combinations of natural numbers, using the technique of binary sequences that was used previously. Combinations of natural numbers can be interpreted at the decimal expansion of real numbers through binary sets. For example, one can draw the combination {1,4,7} from the set of natural numbers {1,2,3...} and take it to mean .1100111000... which is a decimal sequence in which the binary expansions of 1 4, and 7 are joined, and then followed by zeros. There are also infinite subsets of the natural numbers, such as the even numbers {2,4,6...}, which will provide an infinite decimal expansion. This is not a precise mathematical way of doing this, but it serves as an intuitive glimpse into the cardinality of the continuum.

One can also choose the first element of the subset to represent the whole number part, which is always a finite natural number for all positive reals. Using this method, every real number can be generated from a subset (finite or infinite) of the natural numbers, and the real set is the power set of the natural numbers. This proves that



In other words, the cardinality of the continuum is two to the power of aleph-zero. But how can one even comprehend arithmetic operations with cardinal numbers, and infinite ones at that? What other properties does aleph-zero have? The answers can be defined through sets, and are discussed in the next post.

Sources: http://en.wikipedia.org/wiki/Cardinality_of_the_continuum, http://en.wikipedia.org/wiki/Cantor's_diagonal_argument

Infinity: Countable Sets

Before reading this post, read Infinity: The First Transfinite Cardinal.
At the end of the previous post, it was stated that any infinite set or subset of a number system defined by integral ordered n-tuplets, where n is a natural number, has a cardinality of aleph-zero. This statement is more easily understood with examples.

We have already seen that natural numbers can be put in one-to-one correspondence with any set of n-tuplets that contains only integers. When n=1, the resulting number system is simply the integers themselves: {...-3,-2,-1,0,1,2,3...} Furthermore, the above guarantees that any infinite subset of integers will have equivalent cardinality. The even numbers {2,4,6...}, multiples of 10, {10,20,30...}, and even the powers of 2 {1,2,4,8,16...} all can be accessed from the natural numbers through a simple bijection (in this case the functions y=2x, y=10x, and y=2^x, respectively) and therefore have the same cardinality, aleph-zero.
However, the above theorem states that a number system defined by integral ordered n-tuplets for any finite n are also allowed. First, consider the ordered pairs. How does one define a number system with the general integral pair (a,b)? One way, is to view these numbers as the solutions to polynomial equations with coefficients a and b, namely
ax+b=0

Solving for x, one obtains x=-b/a. The general solutions of these equations are the rational numbers, any numbers that can be formed by the ratio of two integers. Admittedly, using the spiral method to pair each natural number with an ordered pair does not hold up well when converted into rational numbers. For instance, the ordered pairs (-2,1) and (-4,2) both produce the same solution, namely 1/2, and pairs such as (0,1) are not defined at all! Therefore, the function from natural numbers to rational numbers through the spiral method is not a bijection.
These problems can be resolved, however. One way is to simply discard duplicates and undefined ordered pairs. The natural numbers corresponding to unique rational numbers will henceforth be known as unique ordered pair numbers, or UOPN's for short. The first few are

2 -> (1,0) -> 0
3 -> (1,1) -> -1
5 -> (-1,1) -> 1
10 -> (2,-1) -> 1/2
12 -> (2,1) -> -1/2
14 -> (1,2) -> -2

In the above expression, the first number of each row is a natural number, followed by the corresponding ordered pair defined by the spiral rule, and the final number is the rational number that results using the polynomial method on the ordered pair. Since it is clear that there are an infinite number of rational numbers accessed by the above series, one can set up a bijection pairing each natural number n to the nth UOPN. This would change the set {1,2,3,4...} into {2,3,5,10...}. The UOPN's then have the same cardinality as the natural numbers, and therefore the rational numbers do as well.
The result just established is remarkable. Despite there being infinite rational numbers between the natural numbers 0 and 1 alone, the cardinality of both of these sets are identical. But this still isn't the end of it. The theorem also deals with numbers defined by ordered n-tuplets. Continuing the theme of using ordered n-tuplets to define integral polynomial equations, the solutions of the resulting equation give the value of the ordered n-tuplet. For example, the ordered quadruplet (1,0,0,-2) corresponds to
x^3+0x^2+0x-2=x^3-2=0,

the real solution of which is the cube root of two. However, with higher degree polynomials, there may be multiple solutions. Consider the polynomial graph below.


This is the graph of x^3-2x^2-x+1, the ordered quadruplet for which would be (1,-2,-1,1). In this case, there are three solutions to the polynomial equation x^3-2x^2-x+1=0 on the real number line, i.e. the three intersections of the graph with the x-axis. How then does one avoid ambiguity? Which of the three solutions does (1,-2,-1,1) represent? The solution lies in specifying an interval on which the solution is found. For example, the first solution, to the left of the origin, lies on the interval [-1,0], and with this constraint, the unique solution can be specified.
Generally, given a ordered n-tuplet of the form (A1,A2,A3...,An-4,B1,B2,C1,C2), one specifies the corresponding number to be a zero of the polynomial

(A1)x^(n-4)+(A2)x^(n-3)+...(An-3)x+An-4
on the interval whose endpoints are the rational numbers defined by the ordered pairs (B1,B2) and (C1,C2).
It is now clear that any solution of any polynomial equation of integral coefficients has an accompanying natural number. Since these solutions can be set up in a one-to-one correspondence with some subset of the natural numbers, one can be set up with the natural numbers themselves. As before, this implies an identical cardinality, i.e. the set of the solutions to integral polynomial equations has a cardinality of aleph-zero.
Just what are these numbers? We know that they are the general solutions of integral polynomial equations, but what form do they take? This specific set of numbers are called the algebraic numbers, and they include square roots, cube roots, and for that matter any nth root. They also include more complicated sets of nested roots, such as numbers of the form



for integer a, b, and c. Any number with nested roots such as this is algebraic. Specifically, algebraic numbers encompass all rational numbers along with many irrational numbers, but not all real numbers are algebraic. For example, π and e are not algebraic, and cannot be expressed as the solutions of polynomials of any finite degree.
All of the above sets have a cardinality the same as that of the natural numbers, and they are therefore denoted countable sets, named after the ability to count natural numbers. All of the rational numbers, and even some irrational numbers are countable, but one hurdle remains: the real numbers. All of the surprising discoveries above suggest that the idea of real numbers being countable is not an implausible notion. The answer is revealed in the next post.
Sources: http://en.wikipedia.org/wiki/Cardinal_number

Infinity: The First Transfinite Cardinal

In mathematics, infinity, often denoted ∞, is defined as exceeding all natural numbers, or, conversely, as the limiting value as a variable n increases without bound. ∞ has always been regarded as a sort of mystical quantity, ever out of reach from most mathematical concepts and calculations. However, through set theory, insights into infinity, in fact multiple infinities, can be gained.

A set is a series of elements, such as {1,2,3} or {12,58,-1,4}. The number of elements in a set is known as its cardinality. For example, {1,2,3} has a cardinality of 3, and {12,58,-1,4} has a cardinality of 4. Any number that can represent the cardinality of a set is known as a cardinal number. 3 and 4, as demonstrated above, are examples of cardinal numbers.

In fact, every natural number 1, 2, 3... is a cardinal number, and even 0 is a cardinal number, as it measures the number of elements in the empty set {}, also written Ø.

Now, it is useful to define functions on sets, namely rules for changing one set into another. The most important of these are known as bijections, and they are defined as functions of sets that preserve the cardinality of a set.



The above diagram is a pictorial representation of a bijection, defined as a function that maps each point in set X to exact one point in set Y, in other words a one-to-one correspondence. It is clear that such a function, when applied to a set, will preserve its cardinality. For example, the function y=x-1 maps the set {2,3,4} to the set {1,2,3}, each of the sets having a cardinality of exactly 3. Since this function preserves the cardinality of all sets in the same matter, it is a bijection.

It is also possible to define an infinite set, or a set with an infinite number of elements. The simplest of these is the set of all natural numbers, namely {1,2,3,4...}. This set has infinite cardinality, and the cardinal number representing this set is the first so-called transfinite cardinal. It is denoted aleph-zero, or ℵ₀.

Aleph-zero is not contained within the normal number system that we think of, but rather describes the size of the set containing all natural numbers. One could then wonder whether adding another element, for example 0, to the set would result in a cardinality of ℵ₀+1. This intuitively seems reasonable, but it is not the case. It was earlier shown that the function y=x-1 is a bijection, so that, when applied to the above set X, will preserve its cardinality of aleph-zero. The result is as follows:

X={1,2,3,4...}
Y=X-1={0,1,2,3...}

Therefore, the set containing all whole numbers, which include both the natural numbers and 0, also has a cardinality of aleph-zero.

At first glance, the above result appears paradoxical. It seems that by subtracting all the terms by 1, an element is added to the beginning of the set, but taken off the end. This is certainly true for finite sets of natural numbers. For any finite natural number n, if

X={1,2,3...,n-1,n}, then
Y=X-1={0,1,2...,n-1}

However, as n increases without bound towards ∞, n-1 is ultimately indistinguishable from n, as ∞-1 is still ∞. In addition to this, when one considers any natural number in the infinite set {1,2,3,4...}, it is decreased by 1 when the function is applied, but there is always another number to take its place. When one takes these points into consideration, the result begins to make sense.

One can easily use the functions y=x-2, y=x-3... etc. to incorporate the elements -1, -2, etc. with the same logic as before, generalizing the above statement to include any finite number of negative integers.

This is only the beginning. Next, consider the function defined below.

If x is odd, then y=(x-1)/2
If x is even, then y=-x/2

The above is not a usual function that can be defined in simple operators of x. However, it is still a bijection when applied to the set of natural numbers, as each element of the set {1,2,3,4...} is transferred to a unique number in a second set. This set is {0,-1,1,-2,2-3,3...}. Remarkably, the output of this function covers all integers! Again, it is easy to see that for any integer one could choose, there is always a natural number that produces it with the above function. Therefore, the total set of integers still has the same cardinality, namely aleph-zero, as the natural numbers.

Nor does the fun stop there! Next consider the mapping of a set to a set of ordered pairs, namely assigning a set of two integers to each natural number. This can be done in the following way:

1 -> (0,0), 2 -> (1,0), 3 -> (1,1), 4 -> (0,1), 5 -> (-1,1), 6 -> (-1,0), 7 -> (-1,-1), 8 -> (0,-1), 9 -> (1,-1), 10 -> (2, -1)...

The exact pattern of these ordered pairs can take several different forms, the above being one of these. Initially, the above sequence seems to have no clear pattern, but it does have a clear geometric significance.



The above pattern lists all the integral ordered pairs (white circles) reached as one follows a rectangular counterclockwise spiral beginning at the origin (the black path). Clearly, by following this path for a sufficient distance, we will visit any ordered pair of integers (a,b) that we could choose!

Before exploring the implications of the above statement, we must make one more logical step. Consider an ordered triplet of the form (a,b,c), again with a, b, and c integers. One can use a similar system to the one above to set up a one-to-one correspondence between the natural numbers and these triplets. The first few terms are listed below:

1 -> (0,0,0), 2 -> (0,0,1), 3 -> (0,1,1), 4 -> (1,1,1), 5 -> (1,0,1), 6 -> (1,-1,1), 7 -> (0,-1,1), 8 -> (-1,-1,1), 9 -> (-1,0,1), 10 -> (-1,1,1)...

Just as before, this can be visualized as a rectangular spiral in the three dimensional coordinate system, where each point is given coordinates (x,y,z). In fact, this pattern, and its geometric interpretation, continue for any order n-tuplets, each consisting of elements (a1,a2,a3...an) and representing a spiral in the n-dimensional Cartesian system.

But what does this mean in terms of the number systems? Clearly, the set of all integral n-tuplets, for any finite n, has a cardinality of aleph-zero. From this, we draw the similar result that

Any infinite set or subset of a number system whose members can be represented by integral ordered n-tuplets, with n a natural number, has the same cardinality as the set of natural numbers, namely aleph-zero.

The implications of the above statement are explored in the next post.

Sources: http://en.wikipedia.org/wiki/Cardinal_number

Juno

Juno is a NASA spacecraft whose mission is to orbit Jupiter and gain further insight to its composition and formation. It is named for the goddess Juno, wife of Jupiter in Roman mythology.

The spacecraft launched on August 5, 2011 to start its six year mission, culminating in a Jupiter arrival in 2016. The probe's trajectory included a flyby of Earth designed to conserve fuel. Unlike previous missions to the outer Solar System, Juno's energy will come only from solar panels, despite the relative dimness of the Sun at Jupiter's orbit.

Juno's trajectory from launch in 2011 to arrival at Jupiter in 2016.

In 2012, Juno executed several deep-space-maneuvers that prepared the probe for its flyby of Earth. Next, in October 2013, Juno completed its Earth flyby, assuming a trajectory directly toward Jupiter.

On July 4, 2016, the spacecraft executed an engine burn that inserted it into orbit around Jupiter. The probe assumed a highly elliptical orbit that took it past the north and south poles of Jupiter with every revolution.

The image above shows Juno's orbits around Jupiter over time, beginning with the orbital insertion on July 4.

This image, Juno's first acquired from orbit, shows the gas giant as well as three of the four Galilean moons, Io, Europa, and Ganymede (from left to right).

After its initial insertion burn, the Juno spacecraft spent over two months completing an elongated orbit that took it far away from the Solar System's largest planet. The first of 37 science flyby took place on August 27 and brought Juno over the north pole of Jupiter, capturing the first ever image of this polar region (see below).



The polar region is very different in appearance than the midlatitudes and equatorial region of Jupiter. The latter regions have characteristic colored bands of red, white, and orange, as well as prominent storm features. The poles are bluer, and lack these storm features. Juno's initial orbit was 53.4 days in duration. At its second closest approach to Jupiter on October 19, a maneuver was planned that would reduce the orbit to 14 days. However, the spacecraft entered safe mode just before the flyby when the onboard computer found conditions to be awry and neither data collection nor orbital maneuvering occurred on the 19th. Juno was later found to be functioning normally.

After two more successful flybys on December 11, 2016 and February 2, 2017, mission directors decided to not risk the reduction maneuver and maintain Juno in its 53-day orbit indefinitely. The main impediment to the function of the probe was the radiation belts near Jupiter's poles, which would gradually deteriorate Juno's functioning with every flyby. Since this radiation is only significant at closest approach, the longer orbit will not prevent the spacecraft from making the planned number of flybys. However, it did reduce their frequency by a factor of almost 4. Originally, 33 total orbits were planned in less than 1.5 years. The adapted budget plan covered only 12 orbits through July 2018, a span of two years.

Nevertheless, valuable data and images continued to pour in. The image below is a color-enhanced view of Jupiter's south pole, highlighting the massive swirling storms circling the pole.

In addition to images of the top of the atmosphere, Juno's instruments provided clues about deeper layers of the Jovian clouds. By the middle of 2017, enough Data had been collected from the Juno Microwave Radiometer, which detects thermal radiation from different depths in the atmosphere, to conclude that Jupiter's equatorial belts penetrated down to a great depth. In contrast, belts and storms at higher latitudes are relatively "shallow," with other structures appearing at increasing depth.

Nevertheless, the "weather layer" of Jupiter, which contains all the belts and cyclones, penetrates much further in depth than the analogous atmospheric layer on Earth. That is, the patterns of atmospheric movement (e.g. the spinning of Jupiter's great storms) persist down from the top portion of the atmosphere for a few thousand miles. In March 2018, four papers were published concerning Jupiter's atmospheric structure using Juno data. Among the results were the discovery of persistent circumpolar cyclones, as shown below.

The above two images are (false-color) computer generated composites of data from Juno's Jovian Infrared Auroral Mapper (JIRAM) instrument. The top image is the north polar region, showing a central cyclone surrounded by eight satellite cyclones. The south pole is similar, but has only five surrounding cyclones. Despite being in close proximity, these storms are much more persistent through time than those seen on Earth.

Another result published in this set of papers analyzed how Jupiter rotates below the weather layer. Precise gravitational measurements from the spacecraft indicate that a few thousand miles down into the atmosphere, the planet orbits approximately as a rigid body. That is, any deviations from steady rotation (such as the jet streams, belts, and storms) have a much, much smaller magnitude in this deeper layer.

In June 2018, NASA approved an extension of Juno's mission through 2021. Later in the year, Juno completed its global mapping of Jupiter with its 16th flyby of the planet. Each flyby had taken place at a different longitude (separated from the previous by 22.5°), allowing imaging of the entirety of the giant planet. By this time, 16 additional passes had been planned, offset from the first set to provide a composite total picture with finer resolution. During the first half of 2019, the Juno team had compiled enough measurements of Jupiter's magnetic field to determine that the field had changed measurably with time. Indeed, the measurements differed slightly but significantly from those of the Pioneer and Voyager spacecraft decades before. Nowhere other than the Earth had a changing magnetic field previously been detected. Experts expect that the variations stem from Jupiter's strong atmospheric winds, which move around material even in the deep layers containing molten metal. These inner layers drive the magnetic field just as on Earth.

A few months later, Juno executed a creative solution to a long-foreseen but very serious problem. As Juno explored different regions of Jupiter, it stood to reason that one flyby or another would bring the spacecraft into the giant planet's shadow. This posed a major problem for a solar-powered spacecraft, however, because only the batteries would prevent the temperature of various instruments from dropping too low. An unmodified orbital path on November 3, 2019, would have taken Juno into Jupiter's shadow for 12 hours, long enough to drain the batteries and possibly compromise the mission. Therefore, on September 30, the probe executed a 10.5 hour-long burn of its engines that allowed it to "jump the shadow" during the flyby a month afterward.

On its way to the December 26, 2019 flyby of Jupiter, Juno took images of Ganymede's north pole, shown below. The polar orbit of Juno let it see the poles more completely than previous space missions. At these poles were ice formations shaped by impacting charged particles directed toward the poles by Ganymede's magnetic field.

Sources: https://www.missionjuno.swri.edu/news/juno_spacecraft_in_orbit_around_mighty_jupiter, https://www.nasa.gov/feature/jpl/nasa-s-juno-spacecraft-sends-first-in-orbit-view, http://www.nytimes.com/2016/07/05/science/juno-enters-jupiters-orbit-capping-5-year-voyage.html?_r=0, https://www.nasa.gov/feature/jpl/jupiter-s-north-pole-unlike-anything-encountered-in-solar-system, https://www.nasaspaceflight.com/2016/09/juno-closest-approach-jupiter-readies-for-primary-science-mission/, https://www.nasa.gov/press-release/nasa-s-juno-mission-to-remain-in-current-orbit-at-jupiter, https://www.nasa.gov/press-release/a-whole-new-jupiter-first-science-results-from-nasa-s-juno-mission, https://www.nasa.gov/feature/jpl/nasa-juno-findings-jupiter-s-jet-streams-are-unearthly, https://www.nature.com/articles/nature25775, https://www.nasa.gov/feature/jpl/nasas-juno-mission-halfway-to-jupiter-science, https://www.missionjuno.swri.edu/news/Juno-Finds-Changes-in-Jupiters-Magnetic-Field, https://www.missionjuno.swri.edu/news/jun_prepares_to_jump_jupiters_shadow, https://www.nasa.gov/feature/jpl/nasa-juno-takes-first-images-of-jovian-moon-ganymedes-north-pole

2011 Unnamed Tropical Storm

Storm Active: August 31-September 2

*This cyclone was classified as a tropical storm during the 2011 postseason analysis. It therefore received no name, despite being tabulated in the number of tropical depressions and tropical storms of the 2011 season.

On August 29, a circulation took shape in an area of convection north of Bermuda, some of which had been associated with Tropical Storm Jose a few days previously. The resulting trough organized over the next few days, increasing in shower activity. Late on August 31, a closed low formed on the southeastern edge of the shower activity, and the system became a tropical depression (although it was not then recognized as such). Convection increased markedly on September 1, and gale force winds were recorded, suggesting that the cyclone at this time became a tropical storm.

A banding feature to the southwest of the center formed the same day, and the cyclone strengthened overnight, reaching its peak intensity of 45 mph winds and a pressure of 1002 mb early on September 2. By this time, an approaching front had begun to push the system northeast, away from the U.S. east coast. The proximity of the front caused the unnamed storm to lose definition during the day of September 2, and the system became extratropical that evening. The remnants continued to move north-northeastward, and were fully absorbed on September 2. This event marked the first time since 2006 that a tropical storm was added in postseason analysis.



The unnamed tropical storm weakening on September 2.



Track of the unnamed tropical storm.

2011 Season Summary

The 2011 Atlantic hurricane season was an above average season, with

20 cyclones attaining tropical depression status
19 cyclones attaining tropical storm status*
7 cyclones attaining hurricane status†
and 3 cyclones attaining major hurricane status

*In the NHC postseason analysis, an additional unnamed tropical storm was identified to have formed during the month of September. This means, that although 2011 only reached the letter "S" in tropical cyclone names, and 2010 reached "T", both seasons had the same number of tropical cyclones form.

†Nate was upgraded from a tropical storm to a hurricane during the postseason analysis.

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

20 cyclones attaining tropical depression status
19 cyclones attaining tropical storm status
10 cyclones attaining hurricane status
and 6 cyclones attaining major hurricane status

The tropical depression and tropical storm predictions happened to be exactly correct, although there was a lower number of hurricanes and major hurricanes than I predicted. As with the 2010 season, the 2011 season was tied for third in overall number of tropical storms with 19. This was caused by an ongoing La Nina event that actually intensified towards the latter part of the season. However, many of these storms were short-lived, and this reflects the abundance of favorable conditions for formation, but not for intensification. These conditions included high wind shear over much of the Caribbean for long periods of time, and also large pockets of dry air associated with anticyclones, which worked their way into many developing systems.

Some notable cyclones and facts about the season include:

• Hurricane Ophelia, the strongest storm of the season, attained Category 4 status at an unusually high latitude of 32.5° N
• 2011 was the first season in which none of the first eight tropical storms (Arlene through Harvey) became a hurricane
• Hurricane Irene, the first hurricane and major hurricane of the season, was also the first cyclone of hurricane strength to make landfall in the U.S. since Ike of 2008
• An unnamed tropical storm formed in early September, the first cyclone to be recognized only in the postseason analysis since 2006
• Nate was upgraded from a tropical storm to a hurricane in postseason analysis, the first such instance since 2007
• Hurricane Philippe was the longest lived storm in the Atlantic basin since 2008, but, despite its longevity, it affected no land.

Overall, the 2011 season was one of numerous, but weak, storms. The U.S. was affected much more than it had been in the previous two years with one hurricane and two tropical storm landfalls, but the damage associated with these systems was not severe.

Tropical Storm Sean (2011)

Storm Active: November 8-11

On November 4, a frontal boundary moved off of the U.S. east coast. A low pressure system along the front deepened as it moved off the coast of North Carolina later that day. The section of the front to the north of the low had most of the cloud cover associated with it, but the convection moved closer to the circulation of November 6, as the system drifted southeast. After temporarily losing definition, the low strengthened again on November 7. By this time, gale force winds occupied a region around the low, extending hundreds of miles in each direction.

Over the following day, the southern extension of the frontal boundary degenerated, devolving into a banding feature expanding clockwise from the low. Meanwhile, the remainder of the front had moved away to the east, and the leftover moisture became entrenched in the circulation of the cyclone. Early on November 8, convection had circumnavigated the center, and the system was upgraded to Subtropical Storm Sean.

The cyclone continued to increase in organization that afternoon, the eye contracting, and the surface circulation becoming better defined. The movement of the circulation into the lower levels of the atmosphere merited a reclassification of Sean into a tropical storm. Throughout the day, Sean remained nearly stationary, initially revolving around a broader cyclonic center, and later adopting a slow westward motion. Convection developed in earnest during the morning of November 9, and the system intensified into a strong tropical storm, also forming an eye feature.

By this time, Sean had entered the steering currents of the west Atlantic and began to accelerate northward. On November 10, the system curved to the northeast, also reaching its peak intensity of 65 mph winds and a pressure of 983 mb. Late that night, the windfield of Sean enveloped Bermuda, causing tropical storm force winds on the island, along with periods of heavy rain.

As it moved away from Bermuda on November 11 winds shear drastically increase and extratropcal transition began as Sean cam into close proximity with a front. By that evening, the center had elongated, and convective bands associated with the circulation ere stripped away. As a result, the system became extratropical that night. Sean caused minor damage and one fatality in Bermuda.



Tropical Storm Sean near peak intensity on November 10. The Outer Banks of North Carolina are visible on the upper left.



Erratic track of Sean through the Western Atlantic.

Hurricane Rina (2011)

Storm Active: October 23-28

During the day of October 18, a cold front swept across the Gulf coast. Heading unusually far south, the front continued to moved over the Gulf, crossing the Yucatan Peninsula during the afternoon of October 19. Simultaneously, a tropical wave, moving through the Caribbean, degenerated into a trough, though it still produced showers and thunderstorms. On October 20, these systems merged off of the coast of Nicaragua and a low pressure system formed.

Over the next day, the low drifted south and then west, all the while increasing in organization. Thunderstorm activity became concentrated just off the coast of Nicaragua during the evening of October 21, However southwesterly wind shear displaced thunderstorm to the west of the low pressure center, causing some heavy rainfall in Central America the following day. On October 23, now moving northward off of the coast of Honduras, the low experienced a reformation, with the new center within the southern edge of the convection. That afternoon, sufficient organization had been attained to classify the system as Tropical Depression Eighteen.

An increase in the symmetry of Eighteen later that night merited an upgrade to Tropical Storm Rina. A series of increases in sturutal organization followed early on October 24, with the outflow imporving in all quadrants. Despite this, the eyewall itself did not undergo any major changes until the early afternoon, at which time a contraction caused a well-defined eyewall, as well as a radiating band feature, to form. Rina was therefore upgraded directly from 45 mph to a hurricane.

Shortly afterward, Rina turned to the west and decreased in speed due to the influence of a ridge over the Gulf of Mexico. Meanwhile, Rina's internal structure continued to improve, and an eye appeared on visible, but not infrared images. Therefore, early on October 25, the cyclone underwent another burst of strengthening and became a Category 2 hurricane. Further fluctuations in the strength of the eyewall pushed Rina to its peak intensity that night, achieving winds of 110 mph and a pressure of 966 mb.

The hurricane maintained this intensity through the morning, but a tongue of dry air invaded from the north. This, coupled with increasing wind shear caused rapid deteriorated of the circulation, reducing Rina's intensity to that of a Category 1 hurricane. As the cyclone began to turn north, the area of deep convection associated with the system shrunk. By October 27, interaction with the Yucatan Peninsula to the west and the exposure of the center of circulation weakened Rina to a tropical storm. Later that day, Rina made landfall near Cancun, causing heavy rainfall and a small area of tropical storm force winds.

By the morning of October 28, the system emerged into the Yucatan Channel and weakened into a tropical depression. All convection had been displaced to the north by this time, and Rina degenerated into a remnant low that afternoon. It was absorbed by a front the next day. Due to its rapid degeneration, the cyclone caused only minimal damage in Central America.



Hurricane Rina at peak intensity, with an eye feature apparent on visible imagery.



Track of Rina.

Hurricane Philippe (2011)

Storm Active: September 24-October 8

A well-defined low pressure area formed west of the African nation of Guinea on September 23. The low deepened rapidly later that day, already having developed large rain bands spread out evenly around the circulation. Convection decreased that night, but became very concentrated near the center, and the system became closed early the next morning. It was then declared Tropical Depression Seventeen. Slow organization followed later in the day as Seventeen moved west-northwest, and the system soon intensified into Tropical Storm Philippe.

A weakness in the ridge to Philippe's north allowed it to turn northwestward and decelerate into the day of September 25, when it reached an intensity of 60 mph winds and a pressure of 997 mb. The tropical storm maintained this strength into September 26, by which time it began to encounter higher wind shear and cooler waters. A gradual weakening ensued, as the circulation became exposed on September 27. That afternoon, Philippe's forward motion slowed to 5 mph and its shallow circulation was steered back to the west-northwest with a restrengthening of the subtropical ridge to its north.

That night, despite the fact that Philippe was a minimal tropical storm with little convection near the center, the outflow remained fairly robust, and thunderstorm activity reappeared just north of the center on September 28. The cyclone even strengthened a little that day. On September 29, the center reformed to the north of its previous position, and once again was completely enveloped in the convection associated with Philippe. The exact center position of Philippe remained difficult to identify overnight, but the circulation became more well-defined on September 30, and the cyclone strengthened further.

As the storm moved farther to the northwest, it encountered very strong shear associated with the outflow of Ophelia, but the tropical storm once again demonstrated resilience to strong upper-level winds, and maintained its intensity, deep convection even increasing near the center of circulation overnight. An analysis of the windfield of Philippe revealed it to be stronger than expected, with winds of 65 mph. Late that night, more intensification occurred, bringing Philippe to the verge of hurricane strength early on October 2.

However, the unfavorable conditions near the cyclone finally began to take their toll that morning, exposing the center and weakening the system. As before, however, Philippe recovered its convection, as Ophelia moved north, and conditions became less hostile. The next day, winds were found to be stronger than previously estimated, and the cyclone was a strong tropical storm yet again. Meanwhile, the center reformed farther to the south than before, and Philippe moved west-southwestward during the day of October 3.

After navigating around the periphery of the ridge steering it, the tropical storm began to turn north on October 4. Recurvature occurred quickly during the following days, and shear gradually decreased. On October 5, the center remained displaced from the deepest convection, but, on October 6, strengthening finally began, bringing Philippe to hurricane strength after over 12 days as a tropical cyclone. An eye feature formed later that evening, particularly visible on infrared imagery, and the hurricane reached its peak intensity of 90 mph winds and a pressure of 976 mb that evening.

Upper-atmospheric conditions rapidly deteriorated on October 7, causing weakening as the cyclone accelerated to the east-northeast. The center quickly became exposed that afternoon, and Philippe's brief period as a hurricane was over. A front encouraged extratropical transition on October 8, and absorbtion completed in that afternoon. Philippe was the longest lived tropical cyclone in the Atlantic since Bertha of 2008, lasting over two weeks. Despite this, it affected no land.



Hurricane Philippe over the open Atlantic.



Track of Philippe.

Hurricane Ophelia (2011)

Storm Active: September 20-October 3

On September 17, a broad area of showers and thunderstorms formed in association with a large low pressure system in the eastern Atlantic. Three low pressure centers existed in close proximity near the area during the day of September 18, oriented west to east near the 10ºN parallel. Over time, the central low became stronger and dissipated the others, and convection increased. The circulation also increased in definition over the following days, but numerous reformations of the center kept the circulation open. However, late on September 20, thunderstorms became concentrated near a single circulation center, and the low was upgraded directly to Tropical Storm Ophelia.

As the tropical storm moved west over the central Atlantic, shear steadily increased, but despite unfavorable conditions, winds in Ophelia's rain bands increased significantly, and strengthening ensued. On September 21, convection was displaced to the east of the center, but notwithstanding any satellite data to the contrary, winds were reported that suggested a strong tropical storm intensity. By the morning of September 22, Ophelia had reached an intensity of 65 mph winds and a pressure of 994 mb.

Wind shear began to finally take its toll on the system later that day, and the cyclone weakened. By the morning of September 23, Ophelia, had weakened to a minimal tropical storm, and all remaining convection was pushed northeast into a rain band that extended several hundred miles. Meanwhile, Ophelia took a turn to the west-northwest. Later on September 23, shear temporarily relaxed, and shower activity re-ignited near the center of the cyclone, causing it to unexpectedly strengthen that evening.

However, this increase in intensity did not persist, as strong upper-level winds resumed early on September 24. Ophelia weakened once again during that day, and, as a result, turned back to the west. As the system's circulation continued to become shallower, a due west motion was assumed, and by the morning of September 25, Ophelia was a minimal tropical storm. A large area of convection was over 150 miles east of the center during the day, with the center itself bare but for a few showers to the northeast. The Lesser Antilles, to Ophelia's southwest, experienced some gusty winds during the afternoon, as the center of the circulation became more elongated. Subsequently, the system was observed to lack tropical characteristics, and was downgraded to a remnant low.

The exposed center quickly dissipated, but a new swirl quickly became evident in the thunderstorm activity to the east. During the day of September 26, upper-level winds relaxed, and the circulation of the newly formed low became much better defined that evening. As the system continued to organize overnight, it drifted eastward and southward, stalling close to the Leeward Islands. On September 27, moderate rainfall occurred over these islands, as convection increased further during the afternoon, and the low became organized enough to be redesignated as Tropical Depression Ophelia.

Shear still abounded over the region overnight, and strengthening was gradual, bringing Ophelia to tropical storm strength during the morning of September 28. By this time, the cyclone had assumed a definite north-northwestward motion, and it moved away from the Caribbean. Outflow improved throughout the same day, allowing the cyclone to hold its own despite marginally favorable conditions. Further strengthening followed that night, and the system was a powerful tropical storm by September 29. An impressive banding feature formed to the east of the circulation later that afternoon, causing Ophelia to be upgraded to Category 1 hurricane status.

During the same evening, Ophelia's eyewall solidified, and the eye itself became more consistent in its appearance on satellite imagery. Following these structural changes, the cyclone rapidly strengthened. By the morning of September 30, Ophelia was a Category 2 hurricane! However, the strengthening was by no means over, as the hurricane assumed a more rounded appearance, the outflow improving and the eye broadening further. By that afternoon, Ophelia was a Category 3 hurricane. Meanwhile, the storm turned north and started to accelerate, as a large trough exited the U.S. east coast and pushed it farther poleward. Late that night, the system finally stabilized in intensity at 120 mph winds and a pressure of 956 mb.

Early on October 1, the pressure dropped slightly to 952 mb, and the outer bands of Ophelia began to impact Bermuda. During the day, gusty winds and scattered heavy squalls affected the island, but the hurricane passed well to the east, making its closest approach late that afternoon. Just after passing Bermuda, Ophelia unexpectedly underwent rapid intensification, bringing it to its peak intensity as a Category 4 hurricane with 140 mph winds and a pressure of 940 mb. During the morning of October 2, Opheila finally began to weaken as it turned to the north-northeast, reaching a forward speed of 30 mph north of 35ºN latitude. The circulation slowly lost definition as it raced towards Newfoundland ahead of a cold front to its west.

That night, Ophelia lost most of its remaining tropical characteristics, weakening to a tropical storm during the morning of October 3. Any central convection still associated with the system vanished as Ophelia made landfall in the Avalon Peninsula, and the system became extratropical. The cyclone continued northeastward until dissipating on October 5. Minimal damage was the only effect of Ophelia, with no fatalities occurring.



Hurricane Ophelia nearing peak intensity as a Category 4 hurricane.



Track of Ophelia, including time spent as an non-tropical system before reforming.