Monsoons and Mumbai

 

Every year towards the end of May, the city of Mumbai begins to change. People start to prepare for the coming of the Monsoon, which is itself derived from the word “mausam” in Arabic which roughly translates as a season. Perhaps in no other place than Mumbai, the might of monsoon is felt by such a large and varied section of people. All the residents, in all the areas of Mumbai, irrespective of their class do prepare for the annual coming of the monsoon. In Mumbai, everywhere you would see temporary structures being erected, plastic sheets of many colours and the most prominent blue tarpaulins being used to cover myriad of houses. It seems that the city itself is getting ready to greet the incoming monsoon. The various preparations done by people of all classes reminds one of festive preparation that families do. Some buy waterproofing for their bodies, some for their homes and shops and some for their vehicles. Some, who are very rich, even buy waterproofing for their pets. Government bodies like the BMC clean up the gutters and nallahs, which were freshwater rivers once, hard to believe when you look at their present states. You can see piles of cleaned up garbage and debris stacked neatly alongside the gutters manholes and nallahs. They say it helps in preventing floods. So, every year, like a seasonal change, you will see people cleaning them, and stacking the filth accumulated in these water bodies, both above the ground and underground in neat stacks.

Business establishments erect skeletons of bamboos and timber around the areas which are not covered. This happens in the case of hotels which have open-air seating areas. Then on these skeletons, they apply sheaths of water-proofing materials. At the end of it, the area is ready for use even in the heaviest rains. Shops which would sell, clothes and other stuff in other seasons, have umbrellas in their display. Even the street and traffic signal vendors start selling them. The umbrellas come in a variety of sizes, colours and designs. Males mostly carry black ones, while the females carry ones with a variety of colours. The umbrella is the protection of the commons against the rain. In the crowded local trains and buses, it is not seen kindly if you wear a raincoat and enter and transfer your wetness to others in the process. Umbrellas, on the other hand, can be folded and kept inside your bag. These are the kind of umbrellas, which are compact are the ones which are most common. The price range generally indicates the number of seasons that an umbrella will last. Also, the cobblers, who double up as umbrella repairers can be seen fixing broken ribs, handles and mechanisms of old umbrellas. Then there are the large ones, which you need to carry in hand, for they are larger than any bags you would usually carry. I myself carry one of the largest sizes available. Because when it really pours, the compact ones, though easy to pack and carry, aren’t going to protect you from getting wet in the Mumbai monsoon.

Then there is the rain footwear. The idea is that in the rains leather (even faux leather) will get damaged. Hence one should wear something made of rubber or plastic which is not affected by the rains. All the branded footwear companies have a stock of rainwear, which people buy as a preparation for the rain. But this is not for the real elite, they will wear their suedes even in the rains. From the by-lanes of Kurla, there emerge cheap replicas of the designer rainwear which are sold at less than 10th of the prices of the originals. They are sold outside the stations, on the streets, and in the shops. Each year, the designs, patterns and colours change depending on what is in vogue in the market.

People also buy protection for their mobile phones and wallets. Just before the rains pavement sellers, who sell a plethora of mobile trinkets and accessories, also start with selling waterproofing for the mobiles. Same is the case with the bags that working people and school children carry. Those not well off use covers of plastic to cover the valuables inside their bags. Good quality plastics are always in demand for such things. But now with the ban, we don’t know what will happen. Others purchase rain covers for their bags. Even with all these precautions mobiles and stuff inside the bags do get wet. And they get damaged.

They say when you are already anticipating something, the shock value isn’t that much as you would expect. But in the case of Mumbai rains, it is not so. Amongst the places I have lived in, the longest has been in Mumbai. Even then, I consider myself alien to this city, an outsider. I have seen and explored parts of it, yet I do not consider myself as a Mumbaikar. There are two primary reasons for this. The first is that even though after staying for more than a decade, I don’t use public transport for my daily commute. Neither the train nor the bus I use regularly. The two major forms of transport in Mumbai. So far, I have not stayed very far away from my place of work. Hence I do not have to suffer [safar?] daily travel. The other reason is that I have not yet made myself at home with the rains. It is not that I do not enjoy the rains, I do, but only for the first few weeks. Then it becomes torture. A melancholy if you will. The sunless skies for weeks on end are depressing enough, and then you have to prepare for the wetness. No matter how hard you try, the rains will get you. There hasn’t been a season where I haven’t got completely drenched. I have given up.

In case, I am caught unawares, without umbrella or any other protection. I just let the rain do its thing. I don’t fight it. But the city itself is fully prepared for it. Unless it is a literal flood, the city continues its routine as if, nothing has happened. All the supply lines will be working as they are supposed to. Waterlogging will produce delays in traffic and trains, but that is about it. If half the amount of rain that lashes Mumbai every season pours into any other metro it will come to a standstill. But Mumbai has made sure that the services operate despite this amount of rain. It is because of anticipation and will of the people to work.

The area of Mumbai is about 600 \, km^{2}, and the average rainfall in Mumbai is about 2200 \,mm. This essentially means that we have rainfall equivalent to about 2.2 \, m water present on the entire surface of about 600 \, km^{2}. So the total volume of water that Mumbai receives each year is

V_{Rain} = 600 \times 10^{6} \, m^{2} \cdot 2.2 \, m = 1.3 \times 10^{9} \, m^{3}

As a first approximation we can consider the rain drops to be of uniform size, and the diameter of the drop is about 1 \,mm = 10^{-3} m. So the volume of the drop is about
V_{Drop} = \frac{4}{3} \,\pi \,(0.5 \times 10^{-3} m)^{3} = 5.2 \times 10^{-10} \,m^{3}

Hence in each year the number of raindrops on Mumbai would be

N = \frac{V_{Rain}}{V_{Drop}} = 2.5 \times 10^{18}

As a better approximation we can take into account the fact that not all raindrops have the same size. We can then make a distribution of the raindrops according to their size. We can have a distribution of the raindrops according to their size as follows:

Type of
Shower
Diameter
of drops
(mm)
Percentage Volume m^{3}
Drizzle 0.5 15 6.5 \times 10 ^{-11}
Normal 1 70 5.2 \times 10 ^{-10}
Thunder 3 15 8.1\times 10 ^{-9}

So the total volume of our rail gets distributed according to the above table. Now we calculate the number of drops for each type of shower:

V_{Drizzle} = \frac{15}{100} \times 1.3 \times 10^{9} \, m^{3} = 1.9 \times 10^{8} \, m^{3}

Hence the number of drops in drizzle are:

N_{Drizzle} = \frac{1.9 \times 10^{8} \, m^{3} }{6.5 \times 10 ^{-11}} = 2.9\times 10^{18}

Similarly for normal shower we get

V_{Normal} = \frac{70}{100} \times 1.3 \times 10^{9} \, m^{3} = 9.1 \times 10^{8} \, m^{3}

Hence the number of drops in normal shower are:

N_{Normal} = \frac{9.1 \times 10^{8} \, m^{3} }{5.2 \times 10 ^{-10}} = 1.75\times 10^{18}

Similarly for thunder shower we get

V_{Thunder} = \frac{15}{100} \times 1.3 \times 10^{9} \, m^{3} = 1.9 \times 10^{8} \, m^{3}

Hence the number of drops in normal shower are:

N_{Thunder} = \frac{9.1 \times 10^{8} \, m^{3} }{8.1 \times 10 ^{-9}} = 1.1\times 10^{17}

So if we add all these up we get the total number of drops:

N = N_{Thunder} + N_{Normal} + N_{Drizzle} = 1.1\times 10^{17}+ 1.75\times 10^{18}+ 2.9\times 10^{18} = 4.7\times 10^{18}

This is not very different from our first rough estimate.

Humans as Fermions

Humans as Fermions

* The Fermions

  Fermions are one set of fundamental particles and the other one are
  bosons. The distinguishing factor between bosons and fermions is
  that the fermions have half integral spins, whereas the boson have
  integral spins. Their names suggest that the bosons were discovered
  by S N Bose, an Indian physicist and fermions by E Fermi. Now
  another this is that the fermions follow what is known as the Pauli
  exclusion principle. That is to say you cannot have two fermions
  which have all the quantum numbers same.

The Pauli exclusion principle is a quantum mechanical principle formulated by the Austrian physicist Wolfgang Pauli in 1925. In its simplest form for electrons in a single atom, it states that no two electrons can have the same four quantum numbers; that is, if n, l, and ml are the same, ms must be different such that the electrons have opposite spins. More generally, no two identical fermions (particles with half-integer spin) may occupy the same quantum state simultaneously. A more rigorous statement of this principle is that for two identical fermions, the total wave function is anti-symmetric.

http://en.wikipedia.org/wiki/Pauli_exclusion_principle

And electrons are fermions It is this principle which decides the electronic
  configuration in atoms. The filling up principle or the aufbau
  principle works according to the exclusion principle. So when near
  to each other the electrons will tend to have different quantum
  numbers. If all the quantum numbers are same for a given pair of
  electrons, then they must have the spins opposite. But now if a
  third electron is to be arranged in the same orbit, it simple cannot
  be accommodate; it has to go in a different orbit. So that the
  electrons behave, as if they do not like the proximity of each
  other.

* Local trains
  Now when observing humans when they are in a crowded environment
  like a local train in Mumbai, I feel that the humans do behave
  exactly like fermions. That is to say that they do not like the
  proximity of each other, just like the electron do not like
  proximity of each other in the electronic orbits. I have observed
  this many a times in the local trains. Usually the trains are very
  crowded. Even to get a position to stand comfortably is a privilege,
  especially in the peak hours.

  When you board the train at the starting station like the VT, then
  what follows is closely analogous to filling up of the electronic
  orbitals in the atom. The seats that are usually taken first are the
  window seats. In the atom it would correspond to the first filling
  of the principal quantum number. In the window seats also the
  preference is to the seats for the windows which face the incoming
  air, that is facing towards the direction of travel.

  Then the seats are filled in the order of least occupancy. People
  want to sit at the seats which are least occupied. Normally the
  seats can take 3 people, and 4 with a bit of difficulty. But the
  norm is that 4 people are seated on a single seat. Once all the seats
  are filled up to 4 occupants, then people tend to stand in between
  the seats. The analogy does not extend to the people who are
  standing at the doors, there it is more like an ensemble of free
  particles, which are jumping in and out of the compartments.

  So coming back to the seating arrangements what I have observed is
  that once the seats are filled with 4 occupants. That is the maximum
  that our ‘seat’ orbital can take. The rest are occupied in between
  states. They are like virtual states, ready to jump into the empty
  seats as soon as one gets empty.

* The Law of 3 
  Lets assume that the people standing in between are like the
  electron sea in metals. Now lets assume a situation in which there
  are a few people who are standing in between seats and all the seats
  are seated by 4 people. Now lets see what happens when one of the
  person who is sitting stands up to get off the train. As soon as the
  seat gets empty, one of the persons who is standing goes to fill in
  the empty seat. As more and more people get off, the people who are
  standing take up their seats. Finally we reach a state when there
  are no more people who left are standing. Now all the seats have
  four seated occupants. Now if a single person gets up. There is one
  seat with just three people, but people don’t tend to move to that
  seat. It just not worth the effort, by going from a 4 seated seat
  again to a 4 seated seat, you don’t gain much. So you remain seated
  where ever you are. But if you are one of the people who are seated
  on the seat where the person just left from, you surely feel
  relieved.
 
  Now let us try to visualize the situation if 2 people from a single
  seat leave off. Two people leaving from 2 different seats will not
  help. It has to be 2 people who were seated on the same seat. After
  this what we have is that, there is a seat where only 2 people are
  seated and rest of the seats have 4 people seating on them. As soon
  as this happens, a person from a 4 seater, will try to get to the 2
  seater seat. This results in two 3 seater seats, whereas the rest
  are 4 seaters. Even more if 3 people from the same seat go away, the
  resulting changing of seats by people results in maximizing the
  number of 3 seater seats. This is the law of behavior of people in a
  local train ;). I call it the Law of 3. This just also touches on
  the idea of what is called in psychology as personal space. We
  are comfortable only within a certain distance from each other. And
  make it a point to bring this into existence we make the movements.

  Well this is just a vague analogy, to the actual behavior of the
  fermions is much more involved, but nonetheless the analogy is worth
  observing.

Mumbai Locals…

Well Mumbai locals are the life line of the city. But ever wondered how many people can one local train carry? Here I try to estimate the carrying capacity of the local train.

We first want to make an order of magnitude guess for the carrying capacity of the
local train. First let us take the dimensions of one coach of the train.
Let us take the width of the coach to be ~ 3 m or 10 ft. We consider the length of the coach to be
of the order of ~50 ft. Then the floor area that we have in each coach is about 500
sq. ft. We neglect the actual seating arrangement in the local, and consider the
floor area only. We make an assumption that all the people are standing in the coach to
get an upper limit on the carrying capacity of the coach. The passengers are standing
as close to each other as possible. Now we make an estimate of how much area one
person requires to stand. One person would require about 1 sq. ft. area to stand.
Thus in a coach of about 500 sq. ft, about 500 people can stand. Actually there are
9 coaches, and their configuration is as follows. In the Central Railways , a 3-coach
unit is classified as 76, 70, or 72, where 76 is the leading motor coach, 70 is the motor
coach with a pantograph, and 72 is the trailer coach. So a nine-coach train has three
units in the following sequence (for the details and lot of other interesting information about Indian Railways visit here):

(76 -70 – 72)(72 – 70 – 76)(72 -70 – 76)

So in out of 9 coaches some space is lost to the motor coach [3 nos.], the driver
coach [2 nos.] and the eeffective area of the train is reduced. The motor coach has an
area of about 10 ft. and the driver coach of about 5 ft, so about 40 ft is reduced. So
the eeffective number of coaches are 8. Since each coach can hold about 500 people,
8 eeffective coaches will have about 4000 people. We have given about 1 sq. ft. for
one person to stand, but in reality especially in the peak hours the rush is much more
than that, so this estimate will have to be increased. We consider that about 1.5
people can stand in 1 square foot of area. Also the presence of the seats and partitions
in the coaches will reduce the eeffective area usable for standing so we assume that
about 10 % of the entire area is lost in furniture. So the number of people in one coach
450*1:5 = 675. So that in 8 coaches 675*8 = 5400 people can stand. But since not
all people can stand we also have to make a correction for this. About 100 people can
sit in a coach, who effectively take about 2 sqf ft. So about about 150 sq. ft. is taken
by them. So out of the 450 sq we are left with 300 sq ft, so eeffectively 300*1:5 = 450
people are standing. So the total number of people per coach is 450 + 100 = 550. So
that total number of people per train is 550* 8 = 4400. The figures that we get from
Wikipedia show that about 4500-5000 people travel in the local trains during the
peak hours.

So our guess is near about correct!!

This method of analysis is known as solving problem the Fermi way and the problems are Fermi problems. Named after the 20th century physicist Enrico Fermi, such problems typically involve making justified guesses about quantities that seem impossible to compute given limited available information. Fermi was known for his ability to make good approximate calculations with little or no actual data, hence the name.