Another blockbuster disaster film, 2004’s The Day After Tomorrow concerns the lead up and aftermath to a series of extreme weather events that lead to a new ice age. After attempts to understand what’s happening, the characters end up trying to save as many lives as possible, both at a global and local scale.
You can see the original trailer here:
Math Scenario 1 – Mass Exodus
One of the more memorable scenes in the movie is US residents storming the Mexican border to get away from the cold coming down from the north. The US population is about 320 million and the Canadian population is about 35 million. Evacuating 355 million people in a short period of time would have to make good use of boats, trains, planes and every known source of transport. There would be massive chaos, but we’re going to look at just the logistics of how many people you could get out with various transport types.
Let’s say they had 10 days to do the mass evacuation.
From this site, there were about 718 million passengers in 2016 over the entire year – or about 1.97 million per day. Let’s say in an emergency, they could double this number per day:
Number evacuated by plane = number per day * 10 days
Number evacuated by plane = 2 * 1970000 * 10
Number evacuated by plane = 39400000
Over 39 million – not too bad for a start.
According to this site, there are about 12 million registered boats in the United States.
Let’s assume half of these are near oceans or major rivers leading to oceans and hence able to help in the evacuation. The vast majority are probably small boats – let’s say that can carry an average of 5 people for one trip.
There are also likely at least 50 large cruise ships (we’ll let some of those number represent the larger number of smaller, moderate-sized ships available) that could probably cram on 10000 people each, and do several return trips – let’s say 2 trips each.
Number evacuated by ship = small boats + large boats
Number evacuated by ship = 12000000 * 5 + 2 * 50 * 10000
Number evacuated by ship = 61000000
Over 60 million. We’re getting there.
Let’s say there are 4 suitable border crossings, which can handle 50 trains per day. Each train, when crammed full, can take 5000 people.
Number evacuated by train = #trips per crossing per day * #crossings * people per train * days available
Number evacuated by train = 50 * 4 * 5000 * 10
Number evacuated by train = 10000000
Another 10 million by train. Let’s tally up:
Total so far = 110400000
That’s 110 million – less than a third of the number we need to evacuate. Some will get out by car (although roads will likely get traffic locked very quickly) and some near the border will be able to walk.
In the movie, the authorities make the decision to advise everyone in the northern part of the country to stay put, as they will not likely be able to make it to shelter before the cold hits.
Math Scenario 2 – Ride out the Storm or Make a Run For It
Halfway through the movie, some of the survivors are sheltering in the New York Public Library. They face a tough decision – wait out the increasing cold storm in the library, or strike out south for the safety of warmer countries. The library has more than 50 million items – including books that can be burnt for warmth.
Stay Put Option
The storm is predicted to last days. Let’s say its duration is 5 days. If they’re burning 10 books a minute, how many books will they need to source to last the full 5 days?
Number of books = books per minute * time duration
Number of books = 10 * 5 days
Number of books = 10 * 5 days * 24 hrs / day * 60 minutes / hour
Number of books = 72000 books
Walk to Safety
It’s about a 3200 kilometre journey to the nearest Mexican border town at Laredo, from the library. A realistic speed for walking on snowshoes is about 3 km/hr . The party of 20 that sets out can walk for 14 hrs a day. The population of the group over time as the cold takes it toll can be described by the function:
n = Nstart * 0.95^d
where Nstart is the starting number of the group, and d is the day number
How many people will likely make it?
First we can work out the entire walk duration.
Total days walking = total distance / distance covered per day
Total days walking = 3200 / (14 * 3)
Total days walking = 76.19 days
Then we can use the group size formula:
n = Nstart * 0.95^d
n = 20 * 0.95^76.19
n = 0.4016
The “expected” number people to make it to the end of the journey is less than one – so it doesn’t look good for the group.
What if they discovered snowmobiles and food that raised their average distance covered per day to 90 km?
Total days = total distance / distance covered per day
Total days = 3200 / 90
Total days walking = 35.56 days
Again we can use the group size formula:
n = Nstart * 0.95^d
n = 20 * 0.95^35.56
n = 3.228
The odds are still rather grim, but with faster transport, some people in the group might make it.
Math Scenario 3 – Redistributing Populations
About 90% of the world’s population currently lives in the northern hemisphere. In 2016 this population was about 7.4 billion people.
The land area in the southern hemisphere (where safety lies in the movie) is about 50 million square kilometres. If the entire world population was evenly spread throughout this region, we can calculate the population density per square kilometre:
Population density = number of people / total area
Population density = 7400000000 / 50000000
Population density = 148 people / square kilometre
The current population density of Australia is about 3 people per square kilometre – so this change would increase the density by a factor of about 50!
Real Life Example – Ice Ages
There have been 5 major ice ages, where ice covered a significant part of the Earth’s surface, starting more than 2 billion years ago, with the last major ice age currently in progress. Within an ice age, there are individual “glacial periods” which are particularly cold and “interglacials” which are warmer periods. We’re currently in an interglacial period.
Here is a graphical example of temperature cycles over the past 400,000 years, sourced from Wikipedia. You can see that there is a significant variation in temperature of up to 12 degrees in range, with huge implications for conditions and ice coverage over the surface of the earth.
This figure was produced by Robert A. Rohde from publicly available data and is incorporated into the Global Warming Art project. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled GNU Free Documentation License.