2.5: Just For Fun: Hurricanes

Figure 29: Satellite photo of Hurricane Ivan as of September 8, 2004. Note the roughly symmetric rain bands circulating in towards the center and the small but clearly defined “eye”.

Hurricanes are of great interest, at least in the Southeast United States where every fall several of them (on average) make landfall somewhere on the Atlantic or Gulf coast between Texas and North Carolina. Since they not infrequently do billions of dollars worth of damage and kill dozens of people (usually drowned due to flooding) it is worth taking a second to look over their Coriolis dynamics.

In the northern hemisphere, air circulates around high pressure centers in a generally clockwise direction as cool dry air “falls” out of them in all directions, deflecting west as it flows out south and east as it flow out north.

Air circulates around low pressure centers in a counterclockwise direction as air rushes to the center, warms, and lifts. Here the eastward deflection of north-travelling air meets the west deflection of south-travelling air and creates a whirlpool spinning opposite to the far curvature of the incoming air (often flowing in from a circulation pattern around a neighboring high pressure center).

If this circulation occurs over warm ocean water it picks up considerable water vapor and heat. The warm, wet air cools as it lifts in the central pattern of the low and precipitation occurs, releasing the energy of fusion into the rapidly expanding air as wind flowing out of the low pressure center at high altitude in the usual clockwise direction (the “outflow” of the storm). If the low remains over warm ocean water and no “shear” winds blow at high altitude across the developing eye and interfere with the outflow, a stable pattern in the storm emerges that gradually amplifies into a hurricane with a well defined “eye” where the air has very low pressure and no wind at all.

Figure 32 shows a “snapshot” of the high and low pressure centers over much of North and South America and the Atlantic on September 8, 2004. In it, two “extreme low” pressure centers are clearly visible that are either hurricanes or hurricane remnants. Note well the counterclockwise circulation around these lows. Two large high pressure regions are also clearly visible, with air circulating around them (irregularly) clockwise. This rotation smoothly transitions into the rotation around the lows across boundary regions.

(North−−Counter Clockwise)

Hurricane

N

Eye

Southward trajectory

North to South deflects west

direction of rotation

Falling deflects east

S

Figure 30:

Figure 31: Coriolis dynamics associated with tropical storms. Air circulating clockwise (from surrounding higher pressure regions) meets at a center of low pressure and forms a counterclockwise “eye”.

As you can see the dynamics of all of this are rather complicated – air cannot just “flow” on the surface of the Earth – it has to flow from one place to another, being replaced as it flows. As it flows north and south, east and west, up and down, pseudoforces associated with the Earth’s rotation join the real forces of gravitation, air pressure di erences, buoyancy associated with di erential heating and cooling due to insolation, radiation losses, conduction and convection, and moisture accumulation and release, and more. Atmospheric modelling is di cult and not terribly skilled (predictive) beyond around a week or at most two, at which point small fluctuations in the initial conditions often grow to unexpectedly dominate global weather patterns, the so-called “butterfly e ect”76 .

In the specific case of hurricanes (that do a lot of damage, providing a lot of political and economic incentive to improve the predictive models) the details of the dynamics and energy release are only gradually being understood by virtue of intense study, and at this point the hurricane models are quite good at predicting motion and consequence within reasonable error bars up to five or six days in advance. There is a wealth of information available on the Internet77 to any who wish to learn more. An article78 on the Atlantic Oceanographic and Meteorological Laboratory’s Hurricane Frequently Asked Questions79 website contains a lovely description of the structure of the eye and the inflowing rain bands.

Atlantic hurricanes usually move from Southeast to Northwest in the Atlantic North of the

76Wikipedia: http://www.wikipedia.org/wiki/Butterfly E ect. So named because “The flap of a butterfly’s wings in Brazil causes a hurricane in the U.S. some months later.” This latter sort of statement isn’t really correct, of course – many things conspire to cause the hurricane. It is intended to reflect the fact that weather systems exhibit deterministic chaotic dynamics – infinite sensitivity to initial conditions so that tiny di erences in initial state lead to radically di erent states later on.

77Wikipedia: http://www.wikipedia.org/wiki/Tropical Cyclone.

78http://www.aoml.noaa.gov/hrd/tcfaq/A11.html

79http://www.aoml.noaa.gov/hrd/weather sub/faq.html

Figure 32: Pressure/windfield of the Atlantic on September 8, 2004. Two tropical storms are visible

– the remnants of Hurricane Frances poised over the U.S. Southeast, and Hurricane Ivan just north of South America. Two more low pressure “tropical waves” are visible between South America and Africa – either or both could develop into tropical storms if shear and water temperature are favorable. The low pressure system in the middle of the Atlantic is extratropical and very unlikely to develop into a proper tropical storm.

equator until they hook away to the North or Northeast. Often they sweep away into the North Atlantic to die as mere extratropical storms without ever touching land. When they do come ashore, though, they can pack winds well over a hundred miles an hour. This is faster than the “terminal velocity” associated with atmospheric drag and thereby they are powerful enough to lift a human or a cow right o their feet, or a house right o its foundations. In addition, even mere “tropical storms” (which typically have winds in the range where wind per se does relatively little damage) can drop a foot of rain in a matter of hours across tens of thousands of square miles or spin down local tornadoes with high and damaging winds. Massive flooding, not wind, is the most common cause of loss of life in hurricanes and other tropical storms.

Hurricanes also can form in the Gulf of Mexico, the Carribean, or even the waters of the Pacific close to Mexico. Tropical cyclones in general occur in all of the world’s tropical oceans except for the Atlantic south of the equator, with the highest density of occurrence in the Western Pacific (where they are usually called “typhoons” instead of “hurricanes”). All hurricanes tend to be highly unpredictable in their behavior as they bounce around between and around surrounding air pressure ridges and troughs like a pinball in a pinball machine, and even the best of computational models, updated regularly as the hurricane evolves, often err by over 100 kilometers over the course of just a day or two.

Homework for Week 2

Problem 1.

Physics Concepts: Make this week’s physics concepts summary as you work all of the problems in this week’s assignment. Be sure to cross-reference each concept in the summary to the problem(s) they were key to, and include concepts from previous weeks as necessary. Do the work carefully enough that you can (after it has been handed in and graded) punch it and add it to a three ring binder for review and study come finals!

Problem 2.

A block of mass m sits at rest on a rough plank of length L that can be gradually tipped up until the block slides. The coe cient of static friction between the block and the plank is µs; the coe cient of dynamic friction is µk and as usual, µk < µs.

a)Find the angle θ0 at which the block first starts to move.

b)Suppose that the plank is lifted to an angle θ > θ0 (where the mass will definitely slide) and the mass is released from rest at time t0 = 0. Find its acceleration a down the incline.

c)Finally, find the time tf that the mass reaches the lower end of the plank.

Problem 3.

m1 r

v

m2

A hockey puck of mass m1 is tied to a string that passes through a hole in a frictionless table, where it is also attached to a mass m2 that hangs underneath. The mass is given a push so that it moves in a circle of radius r at constant speed v when mass m2 hangs free beneath the table. Find r as a function of m1, m2, v, and g.

Problem 4.

θ

m

F

M

A small square block m is sitting on a larger wedge-shaped block of mass M at an upper angle θ0 such that the little block will slide on the big block if both are started from rest and no other forces are present. The large block is sitting on a frictionless table. The coe cient of static friction between the large and small blocks is µs. With what range of force F can you push on the large block to the right such that the small block will remain motionless with respect to the large block and neither slide up nor slide down?

Problem 5.

m1

m 2

A mass m1 is attached to a second mass m2 by an Acme (massless, unstretchable) string. m1 sits on a table with which it has coe cients of static and dynamic friction µs and µk respectively. m2 is hanging over the ends of a table, suspended by the taut string from an Acme (frictionless, massless) pulley. At time t = 0 both masses are released.

a)What is the minimum mass m2,min such that the two masses begin to move?

b)If m2 = 2m2,min, determine how fast the two blocks are moving when mass m2 has fallen a height H (assuming that m1 hasn’t yet hit the pulley)?

Problem 6.

A car of mass m is rounding a banked curve that has radius of curvature R and banking angle θ. Find the speed v of the car such that it succeeds in making it around the curve without skidding on an extremely icy day when µs ≈ 0.

Problem 7.

A car of mass m is rounding a banked curve that has radius of curvature R and banking angle θ. The coe cient of static friction between the car’s tires and the road is µs. Find the range of speeds v of the car such that it can succeed in making it around the curve without skidding.

Problem 8.

v

m

You and a friend are working inside a cylindrical new space station that is a hundred meters long and thirty meters in radius and filled with a thick air mixture. It is lunchtime and you have a bag of oranges. Your friend (working at the other end of the cylinder) wants one, so you throw one at him

~

at speed v0 at t = 0. Assume Stokes drag, that is F d = −b~v (this is probably a poor assumption depending on the initial speed, but it makes the algebra relatively easy and qualitatively describes the motion well enough).

a)Derive an algebraic expression for the velocity of the orange as a function of time.

b)How long does it take the orange to lose half of its initial velocity?

Problem 9.

R

m v

A bead of mass m is threaded on a metal hoop of radius R. There is a coe cient of kinetic friction µk between the bead and the hoop. It is given a push to start it sliding around the hoop with initial speed v0. The hoop is located on the space station, so you can ignore gravity.

a)Find the normal force exerted by the hoop on the bead as a function of its speed.

b)Find the dynamical frictional force exerted by the hoop on the bead as a function of its speed.

c)Find its speed as a function of time. This involves using the frictional force on the bead in Newton’s second law, finding its tangential acceleration on the hoop (which is the time rate of change of its speed) and solving the equation of motion.

All answers should be given in terms of m, µk, R, v (where requested) and v0.

Problem 10.

m2

m1

A block of mass m2 sits on a rough table. The coe cients of friction between the block and the table are µs and µk for static and kinetic friction respectively. A mass m1 is suspended from an massless, unstretchable, unbreakable rope that is looped around the two pulleys as shown and attached to the support of the rightmost pulley. At time t = 0 the system is released at rest.

a)Find an expression for the minimum mass m1,min such that the masses will begin to move.

b)Suppose m1 = 2m1,min (twice as large as necessary to start it moving). Solve for the accelerations of both masses. Hint: Is there a constraint between how far mass m2 moves when mass m1 moves down a short distance?

c)Find the speed of both masses after the small mass has fallen a distance H. Remember this answer and how hard you had to work to find it – next week we will find it much more easily.

Problem 11.

1

m

2

m

θ

Two blocks, each with the same mass m but made of di erent materials, sit on a rough plane inclined at an angle θ such that they will slide (so that the component of their weight down the incline exceeds the maximum force exerted by static friction). The first (upper) block has a coe cient of kinetic friction of µk1 between block and inclined plane; the second (lower) block has coe cient of kinetic friction µk2. The two blocks are connected by an Acme string.

Find the acceleration of the two blocks a1 and a2 down the incline:

a)when µk1 > µk2;

b)when µk2 > µk1.

Advanced Problem 12.

ω

θR

m

A small frictionless bead is threaded on a semicircular wire hoop with radius R, which is then spun on its vertical axis as shown above at angular velocity ω.

a)Find an expression for θ in terms of R, g and ω.

b)What is the smallest angular frequency ωmin such that the bead will not sit at the bottom at θ = 0, for a given R.

Optional Problems

The following problems are not required or to be handed in, but are provided to give you some extra things to work on or test yourself with after mastering the required problems and concepts above and to prepare for quizzes and exams.

No optional problems (yet) this week.