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# Weather Forecasting with Calculus

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## Courtney Laudick

on 22 May 2014

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#### Transcript of Weather Forecasting with Calculus

Storm forecasting has always been a tricky subject for meteorologists to predict but due to advancements in mathematical analysis and computer power they have been able to make recent advancements.
Primitive Equations are used to approximate global atmospheric flow.
Primitive Equations are a set of six nonlinear differential equations.
There are three main sets of primitive equations:
Conservation of Momentum
Thermal energy Equation
Continuity Equation
Numerical Weather Forecasting: Primitive Equations
These primitive equations are able to be simplified and made linear to create this analytical solution:
Using Primitive Equations to Predict Weather
Forecasting Storms
Alex Maurer, Courtney Laudick, and Eva Heydrich
Weather Forecasting with Calculus
History of Calculus in Weather
How Weather is Predicted Today
Weather Prediction in the Future
Meteorologists must use weather instruments such as barometers to measure air pressure, thermometers for temperature, and anemometers to measure wind speed,
Weather balloons are used to measure all of this in the troposphere.
Satellites track cloud patterns and radar measures precipitation
All of this data collected is put into Numerical Weather Predictions computers that use the calculus stated earlier to help predict the weather
Already in 2014 there is a weather tracker called weather wiz that measures up to 88 percent accuracy the weather beyond ten days, but it is still being improved on
Improvements in software and observing tools continue to be made, helping forecasters to spot local and regional weather developments and track their progress in more detail than ever
Meteorologists develop increasingly complex models and manipulate them with more powerful supercomputers that can compute the calculus behind weather to even greater accuracy
The Main Variables
Almost all forms of Primitive Equations relate these variables to their evolution of space and time.
u is the zonal velocity
v is the meridional velocity
w is the vertical velocity in isobaric coordinates
T is the temperature
W is the water that can form a precipitate
The Basic Equations
Storms including Hurricanes, tornadoes, and even cyclones use calculus to predict their direction and intensity.
Vector Calculus
Vector Calculus plays a vital role in predicting tornadoes:
It can display pressure and humidity as scalar fields and wind as a vector field
These elements conspire to produce a moist mass of rotating air, which meteorologists have taken to calling a "supercell."
The Basics of Vector Calculus
Vectors are used to represent quantities that have both a direction and magnitude E.g. Wind in Tornadoes
They don't impart about where the quantity is implied only about the magnitude and direction, thus when they are graphed, line segments are used to display the vector
Modern Technology
Thousands of computers are linked across the globe and their data is pooled
Wind gauges, thermometers, etc. are strapped out at sea to buoys
Satellites photograph Earth's weather from their orbit in space, while balloons monitor upper-air data over a particular location.
All of these sensors and gauges produce more than 1 million weather-related observations every day.
Super computers calculate millions of calculations a minute
These computers are located in the National Centers for Environmental Protection in Camp Springs, Md.
Horizontal Momentum Equations
Vertical Momentum Equation
Thermodynamic Equation
Continuity Equation
Sources:
science.howstuffworks.com/nature/climate-weather
theweatherwiz.com
howstuffworks.com/weatherforecasting
http://derecho.math.uwm.edu/classes/NWP/intro.ppt
http://www.mathmotivation.com/science/weather.html
Unfortunately these simplifications do not all correspond to conditions in the atmosphere, so many more equations are used besides this one.
What are Analytical Solutions and how they are found.
Analytical Expressions are equations that use well-known operations making them easy to calculate.
To simplify the primitive equations into an analytical solution Reynolds Averaging is used.
Before the 1900s scientists relied on pressure, temperature, and sky conditions as well as their own experiences with weather to predict future weather forecasts
In 1901 Abbe proposed that the atmosphere is governed by laws similar to that of gases and liquids
In 1904 Bjerknes took the theory a step further by saying that physical laws including laws of thermodynamics and conservation could be used
Atlas! A Solution!
British Scientist Lewis Fry Richardson was the first to put the numerical weather prediction to use
It took him 6 weeks to finish the calculations to predict the weather pattern over a 6-hour period
Numerical Weather prediction was at a standstill until 1948 when a group of Meteorologists at New Jersey's Institute for Advanced Study developed the first computer that was used to complete the mathematical equations
The computer was known as the Electronic Numerical Integrator and Computer (ENIAC)