The Physics of Tornadoes and Hurricanes

Archimedes Principle

Angular Momentum

Rotational Force

Coriolis Force

Tornadoes and hurricanes are weather phenomena that are examples of physical vortexes.

A tornado is a violent windstorm with a twisting, funnel shape cloud and is usually spawned by thunderstorms when cool air and warm air meet, forcing warm air to rise quickly. Damage from tornadoes are due to high speed winds and flying debris.

A hurricane is tropical storm which also has a twisting, funnel shaped form but is much larger than a tornado. In the centre of a hurricane is a calm area called the "eye". High winds, torrential rain, floods, and storm surges caused when hurricanes come near or over land result in significant damage over large areas.

We'll examine a few of the physics aspects of how hurricanes and tornadoes form.

Archimedes Principle

Archimedes's Principle says that a body immersed in a fluid is buoyed up by a force equal to the weight of the displaced fluid.

The relative density of the object compared to the density of the fluid determines whether it will sink or float.

The pressure due to the fluid is equal to pgh (density times acceleration due to gravity, times height) and when applied to an area, gives the force.

Therefore the force at the top of the object is:

P₁A = p_fluidgh₁A

and the force at the bottom of the object is:

P₂A = p_fluidgh₂A

so the buoyant force pushing up on the object is:

F_b = F₂ – F₁

= p_fluidgA (h₂-h₁)

= p_fluidgAh

= p_fluidgV

which is the density of the fluid, times acceleration due to gravity, times the volume of the fluid displaced.

The formula: F_b = p_fluidgV

applies not only to a rectangular object, but can be generalized to an object with any shape.

If the buoyant force is more than the weight of the object, this means that the object's density is less than the fluid's density. This results in the object rising or floating up.

If the buoyant force is less than the weight of the object, this means that the object's density is more than the fluid's density. This results in the object sinking.

Also when objects are heated, they expand.

density formula: p=m/V, where p is density, m is mass, and V is volume

When a pocket of air is heated, it expands and the volume increases, thus density decreases. And if the density of the air pocket is less than the density of the surrounding cooler air it will rise.

Rising air in the centre meets less resistance because it is surrounded by air that is also rising.

Rising air creates a vacuum causing cooler air from the sides to move in to replace the rising air. This causes a wind which further pushes the rising air.

Angular Momentum

Ordinary linear momentum is a measure of an object's tendency to move at constant speed along a straight path. Linear momentum depends on speed and mass.

When objects moved in curved paths, we can generalize the idea of linear momentum to something called angular momentum, an object's tendency to spin.

Like linear momentum, the total angular momentum of an isolated object is conserved.

Imagine an object with mass, rotating around an axis. Here, the angular momentum is the product of its mass, velocity (tangential), and radius (the distance from the axis point around which the object is spinning around).

L = mvr

Note that this applies to an object that has its mass at the distance r from the axis.

However, we can generalize this to an object whose mass is distributed all along the distance r and not just at the end of that distance r. For example, the body of a figure skater has most of its mass near the axis of rotation, and the mass of his or her arms and hands at a farther distance out.

Let us assume all the mass is at the distance r from the axis of rotation.

Conservation of angular momentum then explains why figure skaters or divers spin faster when they bring in their arms or tuck themselves in a roll.

For angular momentum to conserve in a spin, the angular momentum before must equal the angular momentum after.

The mass of the figure skater doesn't change. So if the radius decreases (by the skater bringing in their arms), then the velocity increases.

L_before = L_after

mv_ir_i = mv_fr_f

Rotational Force

Acceleration is the change of velocity in a short period of time:

a=Δv/Δt

When an object is rotating around something, the acceleration force to keep it rotating is:

F = ma = m v² / r

The acceleration causes the direction of velocity to continually change to keep it rotating.

When you swing a ball on a string, you exert a force on the string, causing a tension, which then exerts a force on the ball.

At the same time, the ball exerts an equal amount of force in the opposite direction. The ball exerts a force on the string, causing a tension, which then exerts a force on your hand.

Coriolis Force

The •Coriolis Force is an inertial force that was described by the French engineer-mathematician Gustave Gaspard Coriolis in 1835.

In a rotating frame of reference, it is an inertial force acting to the right of the direction of movement when rotating counter-clockwise and to the left of the direction of movement when rotating clockwise.

It occurs because the Earth rotates eastward and it rotates faster (tangentially) as you approach the Equator and slower at the poles.

The Coriolis force is very, very weak and plays an insignificant role in the spinning of water in a sink or a toilet. The way the water spins is more likely due to the oval shape of the bowl or the off centre drain.

On the scale of large storms and hurricanes, the Coriolis force causes air to rotate around the centre in a cyclonic direction (counter clockwise in the northern hemisphere, clockwise in the southern hemisphere).

Tornadoes

The funnel of a tornado is visible due to the condensation of water vapor from the pressure of the spinning wind.

They have damage paths that can be 1.5 km wide and up to 75 km long.

There is enough force to pick up cars and rip homes to shreds, turning the debris into potentially lethal missiles.

There are an average of 1000 per year reported in the US and result in around 80 deaths and 1500 injuries (NOAA).

More tornadoes form in the mid western United States than any where else in the world. They usually occur in the fall and spring.

_{Steps of formation:}

_{Warm, moist winds flow up (and a little east due
to the Coriolis Force) from the Gulf of Mexico.}
_{Air becomes unstable because warm air rises and
continues to rise due to the heat from the sun.}
_{The air cools as it rises higher and condenses
into clouds.}
_{A cold front containing cooler, drier air from
the north in Canada flows south focusing the rising air.}
_{The Rockies also help divert northern winds
eastward towards the flat plains in central US.}
_{Interactions between air at different altitudes
causes storms, lightning, rain, air circulation, and strengthening the
rotating updraft.}
_{A column of spinning air can form, which narrows
and spins faster and extends higher into the storm.}

Fujita Scale

The Fujita Scale classifies tornadoes according to the damage they cause. This is because the size of a tornado is not an indication of its intensity and so there can be large but weak tornadoes or small but powerful ones.

It does try to link wind speeds with damage. However, there are some problems with this measurement system:

can only be measured after the tornado is gone and damage is assessed
tornadoes cannot be measured if it results in little damage

note: No F6 tornadoes have been officially reported as the winds are highly unlikely and they would probably be classified as strong F5's. F6 would be difficult to determine as there would be no objects left to study.

Scale	Wind Speed (km/h)	Expected Damage
F0	64-116	MINIMAL DAMAGE: Some damage to chimneys, TV antennas, roof shingles, trees, and windows.
F1	117-180	MODERATE DAMAGE: Automobiles overturned, carports destroyed, trees uprooted.
F2	181-253	MAJOR DAMAGE: Roofs blown off homes, sheds and outbuildings demolished, mobile homes overturned.
F3	253-331	SEVERE DAMAGE: Exterior walls and roofs blown off homes. Metal buildings collapsed or are severely damaged. Forests and farmland flattened.
F4	332-418	DEVASTATING DAMAGE: Few walls, if any, standing in well-built homes. Large steel and concrete missiles thrown far distances.
F5	419-510	INCREDIBLE DAMAGE: Homes leveled with all debris removed. Schools, motels, and other larger structures have considerable damage with exterior walls and roofs gone. Top stories demolished.
F6	511+	INCONCEIVABLE DAMAGE: Probably nothing left to measure.

Below is are pictures before and after of an F4 tornado hitting a factory in Roanoke (July 13, 2004)

Hurricanes

Hurricanes form in the tropics where the air and water are warm and moist.

Ocean water is heated and evaporates, which takes in energy from the ocean. The warm air rises quickly as it is heated and forms a low pressure system over a large area, creating tropical storms.
Like in tornadoes, as the warm air rises, a vacuum forms from the low pressure and to replace the rising air, air from the perimeter is drawn into the centre.
The incoming winds are curved due to Coriolis force and prevailing winds. The rising air, saturated with water, cools and condenses to form clouds.
As the water rises, it cools and condenses, releasing latent heat energy to the surrounding air, causing it to warm further.
At the hurricanes core, wind sinks from being expelled at the top, preventing clouds from forming and creating a calm "eye"
This feedback mechanism continues to intensify as long as there is warm water from which to draw energy.
Once a hurricane moves over land, the large energy supply form the ocean is no longer available and the storm begins to lose its strength and eventually dissipates.

Demos

Conservation of Angular Momentum

A person sits in a rotating office chair hold their arms out at full reach. The person is also carrying a dumbbell around 10 lbs or so in each hand. He or she starts spinning the chair and during the spin, the person quickly brings in their arms towards their chest. This will cause them to spin faster. The person can then quickly bring their arms out and that will quickly slow down the spin.

Vortex Bottles

These are essentially two 2L pop bottles with a special joiner. One is filled with water and food coloring and/or sparkles which make it more visible. The other bottle is kept empty.

Flip the bottle so that the colored water is at the top. Spin the bottle at the top end in either direction then let it go and a vortex will form as the water drains to the bottom empty bottle.

Hurricane in a Box

A box is made with dimensions around 12 inches wide and 30 inches high, with the back, left side, and bottom made of plywood and the front and right sides made of Plexiglas. The back and left side wood is painted black for better visibility.

On each of the sides of the box, a 1 inch slit is carved out about 2 inches from the right of the edge. These slits provide the rotational spin to the column.

A metal plate of water is brought to boil and soon a column of steam is visible. As the air above the water is heated and as water evaporates, the air rises. Cooler air is brought in through the slits of the box and the column of air and steam starts spinning due to the offset slits.

Liquid nitrogen was put into the boiling water to make the funnel very visible. I think the effect is quite impressive.

The hurricane box was based off the plans from tornadoproject.com (see Link and Resources section)