Published on *METEO 300: Fundamentals of Atmospheric Science* (https://www.e-education.psu.edu/meteo300)

Meteorologists devote their lives to one purpose: to forecast the weather. But many other scientists work to build our understanding of the atmosphere, which is the basis for better prediction that meteorologists can use. Atmospheric science is the foundation upon which all meteorology is based.

Just as with all other physical sciences, mathematics is the language of atmospheric science. If you had our first-year meteorology course, you learned about weather forecasting and some of the physical basis behind the forecasts. This course will introduce you to even more physical concepts of atmospheric science and some of the mathematics that describe and quantify those physical concepts.

You have learned some of the mathematics that you need for this course and will be learning more about vectors and vector calculus soon. In this lesson, we will practice some of the mathematics. You will also prepare an Excel workbook that will help you with some of this course’s assessment problems.

By the end of this lesson, you should be able to:

- correctly use significant figures, dimensions, and units
- solve simple problems using integral and differential calculus
- prepare and use a course Excel workbook for course calculations

If you have any questions, please post them to the Course Questions discussion forum. I will check that discussion forum daily to respond. While you are there, feel free to post your own responses if you, too, are able to help out a classmate.

The atmosphere is amazing, awe-inspiring, frightening, deadly, powerful, boring, strange, beautiful, and uplifting – just a few of thousands of descriptions. So much of our lives depend on the atmosphere, yet we often take it for granted.

Credit: W. Brune © Penn State is licensed under CC BY-NC-SA 4.0 [1]

Atmospheric science attempts to describe the atmosphere with physical descriptions using words, but also with mathematics. The goal is to be able to write down mathematical equations that capture the atmosphere’s important physical properties (predictability) and to use these equations to determine the atmosphere’s evolution with time (prediction). Predicting the weather has long been a primary focus, but, increasingly, we are interested in predicting climate.

We know quite a lot about the atmosphere. It has taken decades, if not centuries, of careful observation and insightful theory that is based on solid physical and chemical laws. We have more to learn. You could help to advance the understanding of the atmosphere, but you must first understand the physical concepts and mathematics that are already well known. That is a primary purpose of this course – to give you that understanding.

What follows, below, is a series of pictures and graphical images. Each one depicts some atmospheric process that will be covered in this course. Look at these images; you will see them again, each in one of the next ten lessons. Of course, in each observation there are many processes going on simultaneously. In the last lesson, you will have the opportunity to look at an observation and attach the physical principles and the mathematics that describe several processes that are causing the phenomena that you are observing.

We have offered Meteo 300 online several times but use of the Canvas assessment platform is relatively recent. There may still be errors despite our best efforts.

As we go through the course, if you find an error or typo, post it in the **Finding Errors Discussion Forum** in Canvas. If you are the first to find an error, you will be awarded 0.1 additional points on your final grade. If you aren't the first one to post, you receive no credit for your find, so it's a good idea to read each lesson at the beginning of each week. A student can earn up to 1 point total (i.e., by being the first to find an error ten times). Now, I hope that we have very few errors, but ...

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.

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 45^{o} relative to the east. Therefore, the eastward motion of the squall line is just the velocity times the cosine of 45^{o}. 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:

$\text{time}=\frac{17\text{km}}{\text{(}57\text{km/h)}\cdot \mathrm{cos}\left(\mathrm{45\xba}\right)}=0.42178\text{hours}=25.3070\text{minutes}$[2]@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:

- Non-zero numbers (1,2,3,4,5,6,7,8,9) are ALWAYS significant.
- Zeroes are ALWAYS significant:
- between non-zero numbers
- SIMULTANEOUSLY to the right of the decimal point AND at the end of the number
- to the left of a written decimal point and part of a number $\ge $[2]@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedmvETj2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9Gqpi0dc9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyyzImlaaa@2FDF@ 10

- 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.
- 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 to the right of the decimal point as the number with the least number of significant figures to the right of the decimal point.
- The number of significant figures is unchanged by trigonometric functions. For other special functions:,
- Logarithms: When taking log of a number with X significant figures, the resulting value should have X number of digits to the right of the decimal point. We can generally apply the same rule for natural logarithms (ln).
- Anti-logarithms: When taking the anti-logarithm (eg. 10
^{value}) of a number with X values to the right of the decimal point, the resulting value should have X significant figures. We can generally apply the same rule for exponentials (e).

- Exact numbers never limit the number of significant figures in the result of a calculation and therefore can be considered to have an infinite number of significant figures. Common examples of exact numbers are whole numbers and conversion factors. For example, there are exactly 4 sides to a square and exactly 1000 m in a km.
- For multi-step calculations, any intermediate results should keep at least one extra significant figure to prevent round-off error. Calculators and spreadsheets will typically keep these extra significant figures automatically.
- 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 |

$25\xb70.3$ | 8 | 1 | $25\xb70.3=7.5$ , round to 8 because 0.3 has only 1 significant figure |

$1.5\left({10}^{3}\right)+3.24\left({10}^{2}\right)$ | $1.8\left({10}^{3}\right)$ | 2 | $1.5\left({10}^{3}\right)+0.324\left({10}^{3}\right)=1.824\left({10}^{3}\right)$ , then drop 2 to get $1.8\left({10}^{3}\right)$ |

$1.5\left({10}^{3}\right)+3.86\left({10}^{2}\right)$ | $1.9\left({10}^{3}\right)$ | 2 | $1.5\left({10}^{3}\right)+3.86\left({10}^{2}\right)=1.886\left({10}^{3}\right)$ , round up then drop 2 to get $1.9\left({10}^{3}\right)$ |

$\frac{\left(57.3+6.41\right)}{15.6}$ | 4.08 | 3 | $\frac{63.71}{15.6}=4.0840$ , trim to 3 significant figures to get 4.08 |

$200\left(3.142\right)$ | 600 | 1 | 200. has 3 significant figures; 200 (no decimal point) has 1 but is ambiguous |

$152\left({e}^{-.52}\right)$ | 90 | 2 | number in exponent has only 2 significant figures |

Check out this video (11:23): Unit Conversions & Significant Figures for a brief (1 minute) explanation of those rules! Start watching at 9:14 for the most relevant information. Note a minor error starting at 9:50 in which "60" should actually have a decimal point following the zero.

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

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

**L**, length**T**, time**Θ**, temperature**M**, mass**I**, electric current

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

The numbers associated with most variables have units. The system of units we will use is the International System (SI, from the French Système International), also known as the MKS (meter-kilogram-second) system, even though English units are used in some parts of meteorology.

We will use the following temperature conversions:

$K{=}^{o}C+273.15$

$\left(\frac{5}{9}\right)\left({}^{o}F-32\right){=}^{o}C$

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

Type | Variable | Dimensions | MKS Units | Common Unit Name |
---|---|---|---|---|

Scalar | length (x or ...) | L | m | |

area (A) | L^{2} |
m^{2} |
||

volume (V) | L^{3} |
m^{3} |
||

speed (u, v, w) | L/T | m/s | ||

energy (E) | ML^{2}/T^{2} |
kg m^{2}/s^{2} |
J = Joule | |

power (P) | ML^{2}/T^{3} |
kg m^{2}/s^{3} |
W = Watt | |

density (ρ) | M/L^{3} |
kg/m^{3} |
||

pressure (p) | M/LT^{2} |
kg/ms^{2} |
Pa = Pascal | |

electrical potential | ML^{2}/T^{3}A |
kg m^{2}/s^{3}A |
V = Volt | |

temperature (T) | Θ | K | ||

Vectors | velocity (v) |
L/T | m/s | |

momentum (mv) |
ML/T | kg m/s | ||

acceleration (a) |
L/T^{2} |
m/s^{2} |
||

force (F) |
ML/T^{2} |
kg m/s^{2} |
N = Newton |

**p = (normal force)/area = (mass x acceleration)/area = ML/T ^{2}L^{2} = M/LT^{2}**

**1 Pa = 1 kg m ^{–1} s^{–2}; 1 hPa = 100 Pa = 1 mb = 10^{–3} bar (hPa = hecto-Pascal)**

**1013.25 hPa = 1.01325 x 10 ^{5} Pa = 1 standard atmospheric pressure = 1 atm**

The knot (kt) is equal to one nautical mile (approximately one minute of latitude) per hour or exactly 1.852 km/hr. The mile is nominally equal to 5280 ft and has been standardized to be exactly 1,609.344 m.

Thus, **1 m/s = 3.6 km/hr ≈** **1.944 kt **and** 1 kt ≈ 1.151 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**

**Kelvin (K) must be used** in all physical and dynamical meteorology calculations. **Surface temperature is typically reported in ^{o}F (or ^{o}C for METARS) and in ^{o}C for upper air soundings**.

$$\text{w=}\frac{{\text{massH}}_{2}\text{O}}{\text{massdryair}}$$[2]@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 [4]. View the following video (2:42).

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 : ).

- 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. - 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!

Calculus is an integral part of a meteorologist’s training. The ability to solve problems with calculus differentiates meteorologists from weather readers. You should know how to perform both indefinite and definite integrals. Brush up on the derivatives for variables raised to powers, logarithms, and exponentials. We will take many derivatives with respect to time and to distance.

Visit the Khan Academy website that explains calculus with lots of examples [6], practice problems, and videos. You can start with single variable calculus, but may find it useful for more complicated calculus problems.

1.

$$\begin{array}{l}\frac{da}{dt}=-ka\\ \frac{da}{a}=-kdt\\ {\displaystyle \underset{{a}_{o}}{\overset{{a}_{1}}{\int}}\frac{da}{a}}=-{\displaystyle \underset{{t}_{o}}{\overset{{t}_{1}}{\int}}kdt}\\ \mathrm{ln}({a}_{1})-\mathrm{ln}({a}_{0})=-k\left({t}_{1}-{t}_{0}\right)\\ \mathrm{ln}({a}_{1}/{a}_{0})=-k\left({t}_{1}-{t}_{0}\right)\\ {a}_{1}/{a}_{0}={e}^{\left(-k\left({t}_{1}-{t}_{0}\right)\right)}=\mathrm{exp}\left(-k\left({t}_{1}-{t}_{0}\right)\right)\\ {a}_{1}={a}_{0}{e}^{\left(-k\left({t}_{1}-{t}_{0}\right)\right)}={a}_{0}\mathrm{exp}\left(-k\left({t}_{1}-{t}_{0}\right)\right)\end{array}$$[2]@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hHeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=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@B6C6@

2. $p={p}_{o}{e}^{\left(-z/H\right)}\text{\hspace{1em}};\text{\hspace{1em}}{\displaystyle \underset{0}{\overset{\infty}{\int}}pdz=\text{\hspace{0.17em}}\text{?(Dothedefiniteintegral.)}}$[2]@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedmvETj2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9Gqpi0dc9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaiabg2da9iaadchadaWgaaWcbaGaam4BaaqabaGccaWGLbWaaWbaaSqabeaadaqadaqaaiabgkHiTiaadQhacaGGVaGaamisaaGaayjkaiaawMcaaaaakiaaywW7caGG7aGaaGzbVpaapehabaGaamiCaiaadsgacaWG6bGaeyypa0JaaGjbVlaab+dacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabIcacaqGebGaae4BaiaabccacaqG0bGaaeiAaiaabwgacaqGGaGaaeizaiaabwgacaqGMbGaaeyAaiaab6gacaqGPbGaaeiDaiaabwgacaqGGaGaaeyAaiaab6gacaqG0bGaaeyzaiaabEgacaqGYbGaaeyyaiaabYgacaqGUaGaaeykaaWcbaGaaGimaaqaaiabg6HiLcqdcqGHRiI8aaaa@615B@

$${{\displaystyle \underset{0}{\overset{\infty}{\int}}p\text{\hspace{0.17em}}dz=-H{p}_{o}{e}^{-z/H}|}}_{0}^{\infty}=-H{p}_{o}\left(0-1\right)={p}_{o}H$$[2]@5@5@+=faaagCart1ev2aqaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hHeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpi0dc9GqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaapehabaGaamiCaiaaykW7caWGKbGaamOEaiabg2da9maaeiaabaGaeyOeI0IaamisaiaadchadaWgaaWcbaGaam4BaaqabaGccaWGLbWaaWbaaSqabeaacqGHsislcaWG6bGaai4laiaadIeaaaaakiaawIa7aaWcbaGaaGimaaqaaiabg6HiLcqdcqGHRiI8aOWaa0baaSqaaiaaicdaaeaacqGHEisPaaGccqGH9aqpcqGHsislcaWGibGaamiCamaaBaaaleaacaWGVbaabeaakmaabmaabaGaaGimaiabgkHiTiaaigdaaiaawIcacaGLPaaacqGH9aqpcaWGWbWaaSbaaSqaaiaad+gaaeqaaOGaamisaaaa@55F1@

3. $p={p}_{0}{e}^{\left(-\frac{z}{H}\right)};\frac{1}{p}\frac{dp}{dz}=?$[2]@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedmvETj2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9Gqpi0dc9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaiabg2da9iaadchadaWgaaWcbaGaaGimaaqabaGccaWGLbWaaWbaaSqabeaadaqadaqaaiabgkHiTmaalaaabaGaamOEaaqaaiaadIeaaaaacaGLOaGaayzkaaaaaOGaai4oamaalaaabaGaaGymaaqaaiaadchaaaWaaSaaaeaacaWGKbGaamiCaaqaaiaadsgacaWG6baaaiabg2da9iaac+daaaa@3F8A@

$$\frac{dp}{dz}=-\frac{1}{H}{p}_{0}{e}^{\frac{-z}{H}}=-\frac{1}{H}p;\frac{1}{p}\frac{dp}{dz}=-\frac{1}{H}$$[2]@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedmvETj2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8FesqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9Gqpi0dc9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGKbGaamiCaaqaaiaadsgacaWG6baaaiabg2da9iabgkHiTmaalaaabaGaaGymaaqaaiaadIeaaaGaamiCamaaBaaaleaacaaIWaaabeaakiaadwgadaahaaWcbeqaamaalaaabaGaeyOeI0IaamOEaaqaaiaadIeaaaaaaOGaeyypa0JaeyOeI0YaaSaaaeaacaaIXaaabaGaamisaaaacaWGWbGaai4oamaalaaabaGaaGymaaqaaiaadchaaaWaaSaaaeaacaWGKbGaamiCaaqaaiaadsgacaWG6baaaiabg2da9iabgkHiTmaalaaabaGaaGymaaqaaiaadIeaaaaaaa@49A9@

4. $\begin{array}{l}\frac{d\mathrm{ln}(ax)}{dt}=?\text{}\frac{d\mathrm{ln}(ax)}{dt}=\text{}\frac{1}{ax}\frac{d(ax)}{dt}=\text{}\frac{1}{ax}\frac{adx}{dt}=\frac{1}{x}u\text{,whereu=velocity}\\ \end{array}$[2]@5@5@+=faaagCart1ev2aqaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hHeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=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@6C7F@

5. $d(\mathrm{cos}(x))=?\text{}d(\mathrm{cos}(x))=-\mathrm{sin}(x)dx\text{}$[2]@5@5@+=faaagCart1ev2aqaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hHeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpi0dc9GqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadsgacaGGOaGaci4yaiaac+gacaGGZbGaaiikaiaadIhacaGGPaGaaiykaiabg2da9iaac+dacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaWGKbGaaiikaiGacogacaGGVbGaai4CaiaacIcacaWG4bGaaiykaiaacMcacqGH9aqpcqGHsislciGGZbGaaiyAaiaac6gacaGGOaGaamiEaiaacMcacaWGKbGaamiEaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaaaa@59D0@

Often in meteorology and atmospheric science you will need to manipulate equations that have variables raised to powers. Sometimes, you will need to multiply variables at different powers together and then rearrange your answer to simplify it and make it more useful. In addition, it is very likely that you will need to invert an expression to solve for a variable. The following rules should remind you about powers of variables.

If $a={b}^{x}$ , then raise both sides to the exponent $\frac{1}{x}$ to move the exponent to the other side: ${a}^{\frac{1}{x}}={\left({b}^{x}\right)}^{\frac{1}{x}}={b}^{\frac{x}{x}}=b$

If ${a}^{x}{b}^{y}$ , and you want to get an equation with *a* raised to no power, then raise both sides to the exponent $\frac{1}{x}$ : ${\left({a}^{x}{b}^{y}\right)}^{\frac{1}{x}}={\left({a}^{x}\right)}^{\frac{1}{x}}{\left({b}^{y}\right)}^{\frac{1}{x}}=a{b}^{\frac{y}{x}}=\text{newconstant}$

This brief video (7:42) sums up these important rules:

Are you ready to give it a try? Solve the following problem on your own. After arriving at your own answer, click on the link to check your work. Here we go:

What does y equal?

$$\begin{array}{l}{x}^{1/b}={\left(a{y}^{b}\right)}^{1/b}={a}^{1/b}{\left({y}^{b}\right)}^{1/b}={a}^{1/b}y\\ y={x}^{1/b}/{a}^{1/b}={\left(\frac{x}{a}\right)}^{1/b}\end{array}$$[2]@5@5@+=faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbItLDhis9wBH5garmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hHeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=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@5F4C@

Now it's time to to take another quiz. Again, I *highly* recommend that you begin by taking the Practice Quiz before completing the graded Quiz, since it will make you more competent and confident to take the graded Quiz : ).

- Go to the Canvas and find
**Practice Quiz 1-2**. 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. - When you feel you are ready, take
**Quiz 1-2**. 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!

Meteorologists and atmospheric scientists spend much of their time thinking deep thoughts about the atmosphere, the weather, and weather forecasts. But to really figure out what is happening, they all have to dig into data, solve simple relationships they uncover, and develop new ways to look at the data. Much of this work is now done by programming a computer. Many of you haven’t done any computer programming yet, and for those of you who have, congratulations – put it to good use in this class. For those who are programming novices, we can introduce you to a few of the concepts of programming by getting you to use Excel or another similar spreadsheet program.

To help you learn and retain the concepts and skills that you will learn in this course, you will solve many word problems and simple math problems. For several activities, we give you the opportunity to practice solving particular types of problems enough times until you gain confidence that you can solve those same types of problems on a quiz. That means that you will be solving some types of problems several times and only the numbers for the variables will change. The simplest way for you to solve these problems is to program a spreadsheet to do that repetitive math for you.

Credit: Image developed by W. Brune © Penn State with Excel [8] 2019 and licensed under CC BY-NC-SA 4.0 [9]

Click for a text description of the spreadsheet screenshot.

Screenshot shows an Excel Spreadsheet

A text box says "put activity number in row 1" and an arrow points to cell A1.

A second text box says "put variable names in row 2" with an arrow pointing to cells A2 and B2

A third text box says "start numbers for variables in row 3" with an arrow pointing to cells A3 and B3

A final text box says "calculations follow variable numbers" with an arrow pointing to C3.

Let’s do a simple example. Suppose we have several boxes, some with different shapes and sizes, and we want to calculate the volume of the boxes and find the total volume. I have put in the names of the variables (with units!) and then the numbers for the length, width, and height of each box type and the total number of each box. To calculate the volume of each box, click on E3 and put an “= a3*b3*c3” in the equation line. Hit enter and it will do the calculation and put the answer in E3. A small square will appear in the lower right corner of E3. Click on this square with the mouse and pull down over the next three rows. Excel will automatically do the calculations for those rows. To calculate the total volume, go to F3 and enter “=d3*e3,” and hit “enter.” Grab the small box and pull down to get the total volume of each type of box. To get the total volume, click on F7, click on “Formulas,” and then “AutoSum,” and finally “Sum.” Excel will show you which cells it intends to sum. You can change this by adjusting the edges of the box it shows. Finally, pay attention to significant digits in your calculated volumes. You can adjust the number of decimal places by highlighting a cell or group of cells and then by clicking on the appropriate icon on the Excel tool bar. **Note that an incorrect number of significant figures is displayed in some of the answers--see if you can figure out where.**

Credit: Image developed by W. Brune © Penn State with Excel [8] 2019 and licensed under CC BY-NC-SA 4.0 [9]

Click for a text description of the spreadsheet example part 1.

length (m) | width (m) | height (m) | number of boxes | box volume (m3) | total volume (m3) |
---|---|---|---|---|---|

0.72 | 0.52 | 0.52 | 3 | =A3*B3*C3 | |

0.52 | 0.52 | 0.52 | 4 | ||

1.2 | 0.85 | 1.01 | 1 | ||

0.62 | 0.62 | 0.95 | 2 |

Credit: Image developed by W. Brune © Penn State with Excel [8] 2019 and licensed under CC BY-NC-SA 4.0 [9]

Click for a text description of the spreadsheet example part 2.

length (m) | width (m) | height (m) | number of boxes | box volume (m3) | total volume (m3) |
---|---|---|---|---|---|

0.72 | 0.52 | 0.52 | 3 | 0.19 | 0.58 |

0.52 | 0.52 | 0.52 | 4 | 0.14 | 0.56 |

1.2 | 0.85 | 1.01 | 1 | 1.03 | 1.03 |

0.62 | 0.62 | 0.95 | 2 | 0.37 | 0.73 |

Hopefully this example is a refresher for most of you. For those who are totally unfamiliar with Excel, please click on the question mark in the upper right of the screen and type in the box “creating your first workbook.” You can also visit Microsoft's help page for additional step-by-step instructions for how to Use Excel as Your Calculator [10]. The best way to learn, after the introduction, is by doing. The Keynote Support website [11] also lists helpful summaries of instructions.

Please follow the instructions above for setting up an Excel workbook. You will be using this workbook to do calculations, plot graphs, and answer questions on quizzes and problems for the rest of the course.

Use the same headings and the same numbers in the first four columns. Enter your own equations and find the box volumes and total volumes.

This assignment is worth 15 points. Your grade will mostly depend upon showing that you set up the workbook, but some additional points will be assigned contingent upon how well you follow the instructions. When your Excel workbook is complete, please do the following:

- Make sure that the file for your workbook follows this naming convention:
**Workbook_your last name (i.e., Smith)_your first name_(i.e., Eileen).xlsx**. - In Canvas, find
**Activity 1-3: Setting up your Meteo 300 Excel workbook**. Upload your Excel workbook there.

There is a very good reason that you are taking this class and I am teaching it – all of us are fascinated by the weather, awed by the atmosphere’s power, and passionate about learning more about it. Quite honestly, I can’t imagine a more rewarding career than the one that you are embarking upon or the one that I have. Nothing could be more rewarding than saving lives by making the atmosphere more predictable or by making the perfect prediction. Nothing.

But, do you know what? The best forecasters are the ones who can not only read weather maps, but who also know physically what the atmosphere is doing. The best forecasters know how to translate the physics into mathematics so that hand-waving can be turned into usable numbers. This course will start to make all of these connections between observations and physical cause-and-effect and help us find numerical solutions to questions.

For those of you who are in related disciplines, this course will give you a solid basic understanding of the atmosphere that you can apply in your studies and career, whether it be civil engineering, mechanical engineering, environmental engineering, chemistry, hydrology, or many other fields.

We have now reviewed some important concepts like significant figures and dimensions and units. You will continue to gain confidence in using the differential and integral calculus that you already know. As you go through the course, I want you to look back at the pictures of the atmosphere and imagine which equations are governing the processes that are causing your observations.

You have reached the end of Lesson 1! Make sure that you have completed all of the tasks in Canvas.

**Links**

[1] https://creativecommons.org/licenses/by-nc-sa/4.0/deed.ml

[2] mailto:MathType@MTEF

[3] https://www.youtube.com/channel/UCX6b17PVsYBQ0ip5gyeme-Q

[4] http://en.wikipedia.org/wiki/Mars_Climate_Orbiter

[5] https://www.youtube.com/channel/UCbRTUq5LGqbn4ngTTi9AnMQ

[6] https://www.khanacademy.org/math/calculus-1

[7] https://www.youtube.com/channel/UCUDlvPp1MlnegYXOXzj7DEQ

[8] https://www.microsoft.com/en-us/microsoft-365/excel

[9] https://creativecommons.org/licenses/by-nc-sa/4.0/

[10] https://support.office.com/en-au/article/Use-Excel-as-your-calculator-df26bd14-e1a8-4618-b411-92349dd777a7

[11] http://www.keynotesupport.com/excel-basics/excel-math-formulas.shtml#basic