Fundamentals of Atmospheric Science

1.2 You won’t believe what you can do with math!

You’ve been told many times that meteorology is a math-intensive field. It is. But for this course, you already know much of the math, and what you haven’t seen, you will see in vector calculus. To get ready for the meteorology and atmospheric science in this course, you will need to refresh your ability to solve simple math problems, including solving simple problems in differential and integral calculus. At the same time, we will remind you about the importance of correctly specifying significant figures and units in your answers to the problems. The goal of this first lesson is to boost your confidence in the math you already know.

How many figures should be in my answer?

Suppose you are asked to solve the following word problem:

In the radar loop, a squall line is oriented in the north-south direction and is heading northeast at 57 km hr-1. In the last frame of the loop, the line is 17 km west of the Penn State campus. You are out running and know that you can make it back to your apartment in 25 minutes. Will you get back to your apartment before you get soaked?

You reason that the line is moving northeast, and thus, at an angle of 45o relative to the east. Therefore, the eastward motion of the squall line is just the velocity times the cosine of 45o. That gives you the eastward speed. You decide to divide the distance by the eastward speed to get the amount of time before the line hits campus. You plug the numbers into your calculator and get the following result:

time= 17 57cos( 45 ) =25.3070 minutes MathType@MTEF@5@5@+=faaagCart1ev2aqaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hHeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpi0dc9GqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaabshacaqGPbGaaeyBaiaabwgacqGH9aqpdaWcaaqaaiaaigdacaaI3aaabaGaaGynaiaaiEdacqGHflY1ciGGJbGaai4BaiaacohadaqadaqaaiaaisdacaaI1aaacaGLOaGaayzkaaaaaiabg2da9iaaikdacaaI1aGaaiOlaiaaiodacaaIWaGaaG4naiaaicdacaqGGaGaaeyBaiaabMgacaqGUbGaaeyDaiaabshacaqGLbGaae4Caaaa@501D@

According to your calculation, you will make it back with 0.3 minutes (18 seconds) to spare. But can you really be sure that the squall line will strike in 25.3070 minutes? Maybe you should figure out how many significant figures your answer really has. To do that, you need to remember the rules:

Significant Figures Rules

  1. Non-zero numbers (1,2,3,4,5,6,7,8,9) are ALWAYS significant.
  2. Zeroes are ALWAYS significant:
    1. between non-zero numbers
    2. SIMULTANEOUSLY to the right of the decimal point AND at the end of the number
    3. to the left of a written decimal point and part of a number MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedmvETj2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9Gqpi0dc9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyyzImlaaa@2FDF@ 10
  3. In a calculation involving multiplication or division, multiply numbers as you see them. Then the answer should have the same number of significant figures as the number with the fewest significant figures.
  4. In a calculation involving addition and subtraction, the number of significant figures in the answer depends on the number of significant figures to the right of the decimal point when all the added or subtracted numbers are put in terms of the same power-of-ten. Add or subtract all the numbers. The answer has the same number of significant figures as the number with the least number of significant figures to the right of the decimal point.
  5. When rounding, numbers ending with the last digit > 5 are rounded up; numbers ending with the last digit < 5 are rounded down; numbers ending in 5 are rounded up if the preceding digit is odd and down if it is even.
Number(s) Answer Number of Significant Figures Reason
25 + .3 25 2 25 has only 2 significant figures
25 * 0.3 8 1 25 x 0.3 = 7.5, round to 8 because 0.3 has only 1 significant figure
1.5 x 103 + 3.24 x 102 1.8 x 103 2 1.5 x 103 + 0.324 x 103 = 1.824 x 103, then drop 2 to get 1.8 x 103
1.5 x 103 + 3.86 x 102 1.9 x 103 2 1.5 x 103 + 0.386 x 103 = 1.886 x 103, round up then drop 2 to get 1.9 x 103
(57.3 + 6.41) / 15.6 4.08 3 63.71 / 15.6 = 4.0840, trim to 3 significant figures to get 4.08
200 x 3.142 628 3 200. has 3 significant figures; 200 (no decimal point) has 1 but is ambiguous
152 x e-.52 90 2 number in exponent has only 2 significant figures

Check out this video: Unit Conversions & Significant Figures for a brief (1 minute) explanation of those rules! Start watching at 9:14 for the most relevant information.

Click for Transcript of Unit Conversions & Significant Figures

What are the typical types of variables?

There are two types of variables – scalars and vectors. Scalars are amount only; vectors also have direction.

Dimensions and units are your friends.

Most variables have dimensions. The ones used in meteorology are:

  • L, length
  • T, time
  • Θ, temperature
  • M, mass
  • A, ampere (electric current)

Some constants such as π have no units, but most do.

The numbers associated with most variables have units. We will use MKS units, even though English units are used in some parts of meteorology.

We will use the following temperature conversions:

K = Co + 273.15

(5/9) x (Fo - 32) = Co

We will use the following variables frequently. Note the dimensions of the variables and the MKS units that go with their numbers.

Variables With Associated Dimensions and MKS Units
Type Variable Dimensions MKS Units Common Unit Name
Scalar length (x or ...) L m
area (A) L2 m2
volume (V) L3 m3
speed (u, v, w) L/t m/s
energy (E) ML2/t2 kg m2/s2 J = Joule
power (P) ML2/t3 kg m2/s3 W = Watt
density (ρ) M/L3 kg/m3
pressure (p) M/Lt2 kg/ms2 Pa = Pascal
electrical potential ML2/t3A kg m2/s3A V = Volt
temperature (T) Θ K
Vectors velocity (v) L/t m/s
momentum (mv) ML/t kg m/s
acceleration (a) L/t2 m/s2
force (F) ML/t2 kg m/s2 Nt = Newton

Pressure is used for many applications.

p = (normal force)/area = m a/ L2 = mL/t2L2 = m/Lt2

1 Pa = 1 kg m-1 s-2; 1hPa = 100 Pa = 1 mb = 10-3 bar (hPa = hecto-Pascal)

1013 hPa = 1.013x105 Pa = 1 standard atmospheric pressure

Wind speed is another frequently used variable.

speed is in m/s or in knots, where 1 m/s = 1.94 kt = 3.6 km/hr. 1 kt = 1.15 mph.

surface winds are typically 10 kts ~ 5 m/s

500 mb winds are ~50 kts ~ 25 m/s

250 mb winds are ~100 kts ~ 50 m/s

Temperature is a third frequently used variable.

Kelvin (K) must be used in all physical and dynamical meteorology calculations. Surface T is reported in Fo or (Co for METARS) and in Co for upper air soundings.

Water vapor mixing ratio is another frequently used variable.

w =  mass H 2 O mass dry air MathType@MTEF@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hHeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpi0dc9GqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaabEhacaqGGaGaaeypaiaabccadaWcaaqaceaadbGaaeyBaiaabggacaqGZbGaae4CaiaabccacaqGibWaaSbaaSqaaiaaikdaaeqaaOGaae4taaqaaiaab2gacaqGHbGaae4CaiaabohacaqGGaGaaeizaiaabkhacaqG5bGaaeiiaiaabggacaqGPbGaaeOCaaaaaaa@47B5@

Usually the units for water vapor mixing ratio are g kg-1. In the summer w can be 10 g kg-1; in the winter, it can be 1-2 g kg-1.

Dimensions truly are your friend. Let me give you an example. Suppose you have an equation ax + b = cT, and you know the dimension of b, x (a distance), and T (a temperature), but not a and c. You also know that each term in the equation – the two on the left-hand side and the one on the right-hand side must all have the same units. Therefore, if you know b, you know that the dimensions of a must be the same as the dimensions of b divided by L (length) and the dimensions of c must be the same as the dimensions of b divided by Θ.

Also, if you invert a messy equation and you're not sure that you didn’t make a mistake, you can check the dimensions of the individual terms and if they don’t match up, it’s time to look for your mistake. Or, if you have variables multiplied or divided in an exponential or a logarithm, the resulting product must have no units.

Always write units down and always check dimensions if you aren’t sure. That way, you won’t crash your spacecraft on the back side of Mars. View the following video (2:42).

Click for Transcript of When NASA Lost a Spacecraft Because it Didn't Use Metric

Quiz 1-1: Significant figures, dimensions, and units.

Now it's time to to take a quiz. I highly recommend that you begin by taking the Practice Quiz before completing the graded Quiz. Practice Quizzes are not graded and do not affect your grade in any way (except to make you more competent and confident to take the graded Quizzes : ).

  1. In Canvas, find Practice Quiz 1-1. You may complete this practice quiz as many times as you want. It is not graded, but it allows you to check your level of preparedness before taking the graded quiz.
  2. When you feel you are ready, take Quiz 1-1. You will be allowed to take this quiz only once. This quiz is timed, so after you start, you will have a limited amount of time to complete it and submit it. Good luck!