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Weather Forecasting

All of a sudden, there was this big explosion.

Hail was like the size of oranges.

I was flying up and down.

Trailers went flying past us. Trees went flying past us.

A car stood wrecked with a large piece of wood piercing the windshield right where the driver should sit.

All around us were chaos and destruction, pain, sorrow and death as people picked themselves up in total shock and dismay from this dreadful event.

How could this happen? Is this real?

These were some of the eyewitness accounts of the Pine Lake Tornado near Red Deer, Alberta on July 14, 2000, which caused 12 deaths and injured 140 people.

A tornado is one of the most violent, severe storms nature produces. Though beautiful and fascinating to look at, tornadoes can be extremely dangerous. The sheer power of tornadoes with wind speeds of 400 kilometers per hour or more is capable of tremendous destruction: lifting houses off foundations, tossing cars into the air. Damage paths of large tornadoes can be in the scale of 1 kilometer wide and 80 kilometers long. They can travel with speeds from 30 km/h to over 100 km/h and last an hour or more. Since 1950, when the U.S. government started keeping records, there have been an average of 800 tornadoes in America every year, resulting in an average of 80 tornado-related deaths and over 1500 injuries annually. Canada is second to the U.S. in the total number of tornadoes per year, with an average if 80 tornadoes, 2 deaths and 20 injuries annually. Southern Ontario has the highest frequency of tornadoes within Canada, followed by the Prairie Provinces: Alberta, Saskatchewan and Manitoba.

If meteorologists ever realize their dream of forecasting tornadoes in precise locations, experts agree that key contributions will come from mathematical techniques for tracking the creation of severe storms and handling the massive computations that appear unavoidable in numerical weather prediction. In early weather prediction, mathematical modeling proved invaluable in understanding storm formation and dynamics. However, the computations necessary to obtain a numerical solution are so intensive that they were not feasible until the 1970s, when computer power became sufficient to begin to tackle the three-dimensional computation.

Mathematically speaking, a tornado is vector calculus with a vengeance. Severe storms are composed of a vector field representing wind and two scalar fields: pressure and humidity. These elements combine to produce a moist mass of rotating air, which meteorologists call a "supercell." The term was coined in the 1960s by Keith Browning, who proposed a theory describing how updrafts become tilted and start to rotate in what is called the mesocyclone. Roughly speaking, wind near the ground tends to come from the southeast; moving upward, the wind shifts, coming from the south at an altitude of one kilometer and from the east at two kilometers.

It is the rotational shear that creates the possibility of tornadoes. The equations that describe storm dynamics are not unduly complicated. For the most part, they represent the conservation laws for mass and energy, with angular momentum taking center stage. What makes tornado equations more difficult to deal with is the scope of the problem. A supercell can be tens of kilometers across; the tornado itself can dwarf (not to mention destroy) a manufacturing plant. And unlike a manufacturing plant, which has been engineered to a high degree of stability, severe weather is inherently turbulent.

These features are severe challenges for numerical storm modeling. Lack of data is another problem. In any numerical simulation, computation of wind speed, pressure, etc., is performed only in an arbitrarily confined region. While those values inside the region can be obtained by solving the tornado equations, the values on the boundary need to be initialized, which is often done by using real data. However, as weather stations are still widely scattered, it is difficult to accurately initialize a numerical model based on a fine grid. Furthermore, observations are often made at different times and may contain different types of information, so it is a daunting task just to assimilate the data that do exist.

No model is yet able to forecast a specific tornado, or even to predict precisely where a storm will be most severe. However, scientists and mathematicians continue to make progress by using more advanced detection equipment, more sophisticated mathematical models, more accurate and efficient numerical simulation algorithms, and more powerful supercomputers. The hope is to understand tornadoes well enough so that we can accurately predict when they will occur, and protect the lives of people unfortunate enough to be in the destructive path of a tornado.

Examples of ongoing projects in Computational Mathematics