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)

Tornado

CM1 is the most recent model that has revolutionized tornado research