Meteorology 2603
Severe and Unusual Weather
Solutions to Problem Set #4

Distributed Monday, 23 April 2001
Due Wednesday, 2 May 2001

1.  Describe the principal differences between sprites and jets.

About 7 years ago, scientists verified the existence of electrical discharges that travel upward from the tops of thunderstorms.  Known as sprites, these large, red, lightning-like emissions shoot upward as high as 60 miles above cloud top and have the appearance of tall mushrooms.  They seem to occur simultaneously with cloud-to-ground lightning below, and are believed to result from the intense heating of ionized air above the cloud. 

Studies of sprites led to the discovery of another phenomenon, known as blue jets, that shoot vertically upward from thunderstorms at speeds 300 times the speed of sound.   These narrow cones of blue light can extend 25 miles above cloud top.  A third emission, known as elves, are ring-like haloes of light above cloud top that expand to 200 miles in diameter in less than a thousandth of a second. 

2.  If you see a lighting flash and hear thunder 12 seconds
later, approximately how far away is the lightning?  What
sorts of things could affect the thunder's time of arrival
at your location?

On average, thunder travels 1 mile in 5 seconds.  Thus,
if you divide by 5 the number of seconds between a
lightning flash and its associated thunder, you'll have
a rough estimate of the distance between you and the
lightning.

In the present case, the thunder took 12 seconds to arrive,
so the lightning was approximatley 12/5 = 2.4 miles away. 

3.  Plot the following sounding on a thermodynamic
diagram and use it to find the Lifted Index, Showalter
Index, Total Totals Index, and K-Index.  Using this
information, determine the type of weather that would
result if air parcels were lifted to their condensation
level. Are the results of each index's prediction
consistent?  Why or why not?

Severe weather indices can be computed by simply finding
the appropriate data on the thermodynamic diagram and
plugging them into the associated formulas. 

Lifted Index:  Lift a parcel from the surface to its LCL
(from the diagram, at about 860 mb altitude)
and from there up to 500 mb (see diagram).  The Lifted
Index is the temperature of the environment at 500 mb
minus the temperature of the parcel, in this case
approximately -28 - (-9) = -19 degrees Celsius.  This
would suggest monstrous thunderstorms!

Showalter Index:  This is identical to the Lifted Index,
except we begin lifting the parcel at 850 mb rather than
the ground.  In this case, we have -28 - (-20) - 9 degrees
Celsius.  This value would suggest tornadoes.

Total Totals Index:  This equals the temperature at 850 mb
plus the dew point at 850 mb minus twice the temperature at
500 mb.  From the diagram and table, this turns out to be
approximately 11 + 1 - 2(-28) = 68 degrees Celsius.  This
value would suggest tornadoes.

K-Index:  This equals the temperature at 850 mb
plus the dew point at 850 mb minus (temperature at 700 mb
minus the dew point at 700 mb) minus the temperature at
500 mb.  Performing this computation gives 11 + 1
- [-4 - (-25)] - (-28) = 19 degrees Celsuis. This value
would suggest scattered thunderstorms.

Only the K-Index is inconsistent with the weather
predicted by the other indices, mostly because it seeks
to identify warm, moist air near the ground, dry air at
mid-levels (700 mb), and cold air aloft.  The air in this
sounding at 700 mb is not terribly dry, and thus the K-
Index doesn't suggest sufficient instability to obtain
strong thunderstorms (this in fact is probably the most
representative index -- the others are simpler).  Further,
this example points to the fact that forecasters must
evaluate a wide variety of information, especially the
vertical wind profile (e.g., via a hodograph), which is
not included here.  Bottom line:  You can't only look at
the sounding to predict the type of weather that is likely
to occur.

4.  Suppose the tangential winds in a developing hurricane
are 20 kts at a radius from the hurricane center of 150 km.
Assuming that angular momentum is conserved, find the
speed of the tangential winds if the air at 150 km radius
spirals inward to a radius of 30 km.  [See pages 424 and
286 in the text.]  Assume that you're following a single
air parcel that has a mass of 1.0 kg.

The conservation of angular momentum states that the
product of the tangential velocity and radial distance
from the center of rotation is a constant following the
motion of an air parcel (the book multiplies V and R
by the mass).  In equation form:

V(1)R(1)m(1) = V(2)R(2)m(2)

where 1 and 2 refer to the first and second positions,
respectively, and m is the mass.  We assume here that
m(1)=m(2) in this case, so it can be cancelled from
both sides of the equation.

We're given in this problem that V(1)=20, R(1)=150,
and R(2)=30 in their appropriate units.  Thus, we can
solve for V(2) as

V(2) = V(1)R(1)/R(2) = (20)(150)/(30) = 100 kts.