3.2: Music of the Renaissance
Characteristics of the Renaissance Music include: steady beat, balanced phras- es (the same length), polyphony (often imitative), increasing interest in text-music relationships, Petrucci and the printing of music, and a growing merchant class singing/playing music at home. Word painting was utilized by Renaissance composers to represent poetic images musically. For example, an ascending me- lodic line would portray the text “ascension to heaven.” Or a series of rapid notes would represent running.
Art music in the Renaissance served three basic purposes: (1) worship in both the Catholic and burgeoning Protestant Churches, (2) music for the entertainment and edification of the courts and courtly life, and (3) dance music. Playing musical instruments became a form of leisure and a significant, valued pastime for every educated person. Guests at social functions were expected to contribute to the eve- ning’s festivities through instrumental performance. Much of the secular music in the Renaissance was centered on courtly life. Vocal music ranged from chansons (or songs) about love and courtly intrigue to madrigals about nymphs, fairies, and, well, you name it. Both chansons and madrigals were often set for one or more voices with plucked-string accompaniment, such as by the lute, a gourd-shaped instrument with frets, raised strip on the fingerboard, somewhat similar to the modern guitar.
A madrigal is a musical piece for several solo voices set to a short poem. They originated in Italy around 1520. Most madrigals were about love. Madrigals were published by the thousands and learned and performed by cultured aristocrats. Similar to the motet, a madrigal combines both homophonic and polyphonic tex- tures. Unlike the motet, the madrigal is secular and utilizes unusual harmonies and word painting more often. Many of the refrains of these madrigals utilized the text “Fa La” to fill the gaps in the melody or to possibly cover risqué or illicit con- notations. Sometimes madrigals are referred to as Renaissance Fa La songs.
A volume of translated Italian madrigals were published in London during the year of 1588, the year of the defeat of the Spanish Armada. This sudden public interest facilitated a surge of English madrigal writing as well as a spurt of other secular music writing and publication. This music boom lasted for thirty years and was as much a golden age of music as British literature was with Shakespeare and Queen Elizabeth I. The rebirth in both literature and music originated in Italy and migrated to England; the English madrigal became more humorous and lighter in England as compared to Italy.
Renaissance music was mostly polyphonic in texture. Comprehending a wide range of emotions, Renaissance music nevertheless portrayed all emotions in a bal- anced and moderate fashion. Extreme use of and contrasts in dynamics, rhythm, and tone color do not occur. The rhythms in Renaissance music tend to have a smooth, soft flow instead of a sharp, well-defined pulse of accents.
Composers enjoyed imitating sounds of nature and sound effects in their com- positions. The Renaissance period became known as the golden age of a cappella choral music because choral music did not require an instrumental accompaniment.
Instrumental music in the Renaissance remained largely relegated to social purposes such as dancing, but a few notable virtuosos of the time, including the English lutenist and singer John Dowland, composed and performed music for Queen Elizabeth I, among others.
Dowland was a lutenist in 1598 in the court of Christian IV and later in 1612 in the court of King James I. He is known for composing one of the best songs of the Renaissance period, Flow, my Teares. This imitative piece demonstrates the melancholy humor of the time period. Dowland’s Flow, My Teares may be heard at https://www.youtube.com/watch?v=jkRrzAo9Wl4 . For more information on Dowland, and lyrics to Flow My Tears , go to genius.com/John-dowland- flow-my-tears-annotated.
The instruments utilized during the Renaissance era were quite diverse. Local availability of raw materials for the manufacture of the instrument often deter- mined its assembly and accessibility to the public. A renaissance consort is a group of renaissance instrumentalists playing together. A whole consort is an ensemble performing with instruments from the same family. A broken consort is an ensem- ble comprised of instruments from more than one family.
Instruments from the Medieval and Renaissance may be found at: www. music.iastate.edu/antiqua/instrumt.html.
3.4.1 style overview
Medieval Music
- Mainly monophony
- Majority of the music’s rhythm comes from the text
- Use of perfect intervals such as fourths, fifths, and octaves for cadences
- Most music comes from the courts or church
- Music instruction predominantly restricted to the church and patron’s courts
Renaissance Music
• Mainly polyphony (much is imitative polyphony/overlapped repetition—please see music score below)
• Majority of the music’s rhythms is indicated by musical notation
• Growing use of thirds and triads
• Music – text relationships increasingly important with the use of word painting
• Invention of music publishing
• Growing merchant class increasingly acquires musical skills


7.19: Data Warehousing and Data Mining
- Differentiate between data warehousing and data mining
All of the RIS we have discussed so far have one major thing in common—an underlying database to store their unique data. Through the process of mergers and acquisitions, most large retailers inherit duplicative systems that continue to exist independent of each other due to the large cost of consolidation. With data “everywhere,” retailers turn to the latest IT techniques.
Data warehouses (DW) are created to bring related information from disparate databases to one large database so that it can be easily analyzed.
In computing, a data warehouse (DW or DWH) is a system used for reporting and data analysis, and is considered a core component of business intelligence. DWs are central repositories of integrated data from one or more disparate sources.
Once the data has been migrated to the DW, data scientists can begin to provide retail management with meaningful information through the practice of data mining. Data mining is the process of discovering patterns in large data sets and involves methods at the intersection of machine learning, statistics, and database systems.
With the mining of information in the data warehouse, management can gain valuable insights as to how best to run the business. This is usually accomplished through queries and reporting.
Queries are business questions translated into code to bring results from the DW. What is our best-selling product line? What is the profit margin on our private brand versus the name brand products? Who are our best customers? How do our online sales affect our inventory position for our stores?
Business reporting is simply scheduling the most common or requested queries at regular intervals and pushing the information out to the organizations information consumers on a regular basis.
One of the most notable data warehouse success stories comes from the healthcare industry in the 1990’s. A large national health management company had more than nine regional centers, each operating semi-independently. Each regional center had its own management and business infrastructure, including information technology.
The company’s top medical experts noticed that the care being delivered for its diabetes patients was inconsistent across the regions. Some regions claimed that certain treatments were more effective, but came at a higher cost to the business. But the real problem was that the clinical data needed to understand what was the most effective treatment was locked up in 10 different databases, many of which were using different database software.
A data warehouse was constructed, pooling the data from all of the regional warehouses and providing access for the first time for national clinical research analysis. Out of hundreds of different treatment programs, three were found to be the most effective, and of those, two were found to be the least expensive. These data warehouse results were a win for the patients, doctors and the business.
Contributors and Attributions
- Data Warehousing and Data Mining. Authored by : Bob Danielson. Provided by : Lumen Learning. License : CC BY: Attribution
- Data Mining. Authored by : Arbeck. Provided by : Wikimedia Commons. Located at : https://commons.wikimedia.org/wiki/File:Data_Mining.svg . License : CC BY: Attribution


Section 2: Complete truth tables
The truth-value of sentences which contain only one connective are given by the characteristic truth table for that connective. In the previous chapter, we wrote the characteristic truth tables with ‘T’ for true and ‘F’ for false. It is important to note, however, that this is not about truth in any deep or cosmic sense. Poets and philosophers can argue at length about the nature and significance truth , but the truth functions in SL are just rules which transform input values into output values. To underscore this, in this chapter we will write ‘1’ and ‘0’ instead of ‘T’ and ‘F’. Even though we interpret ‘1’ as meaning ‘true’ and ‘0’ as meaning ‘false’, computers can be programmed to fill out truth tables in a purely mechanical way. In a machine, ‘1’ might mean that a register is switched on and ‘0’ that the register is switched off. Mathematically, they are just the two possible values that a sentence of SL can have.
Here are the truth tables for the connectives of SL, written in terms of 1s and 0s.
| \(\mathcal{A}\) | ¬\(\mathcal{A}\) |
|
1 0 |
0 1 |
| \(\mathcal{A}\) | \(\mathcal{B}\) | \(\mathcal{A}\) & \(\mathcal{B}\) | \(\mathcal{A}\)∨\(\mathcal{B}\) | \(\mathcal{A}\)→\(\mathcal{B}\) | \(\mathcal{A}\)↔\(\mathcal{B}\) |
|
1 1 0 0 |
1 0 1 0 |
1 0 0 0 |
1 1 1 0 |
1 0 1 1 |
1 0 0 1 |
Table 3.1: The characteristic truth tables for the connectives of SL.
The characteristic truth table for conjunction, for example, gives the truth conditions for any sentence of the form (\(\mathcal{A}\)&\(\mathcal{B}\)). Even if the conjuncts \(\mathcal{A}\) and \(\mathcal{B}\) are long, complicated sentences, the conjunction is true if and only if both \(\mathcal{A}\) and \(\mathcal{B}\) are true. Consider the sentence (\(H\)&\(I\)) → \(H\). We consider all the possible combinations of true and false for \(H\) and \(I\), which gives us four rows. We then copy the truth-values for the sentence letters and write them underneath the letters in the sentence.
| \(H\) | \(I\) | (\(H\) & \(I\))→\(H\) |
|
1 1 0 0 |
1 0 1 0 |
1 1 1 1 0 1 0 1 0 0 0 0 |
Now consider the subsentence \(H\)&\(I\). This is a conjunction \(\mathcal{A}\)&\(\mathcal{B}\) with \(H\) as \(\mathcal{A}\) and with \(I\) as \(\mathcal{B}\). \(H\) and \(I\) are both true on the first row. Since a conjunction is true when both conjuncts are true, we write a 1 underneath the conjunction symbol. We continue for the other three rows and get this:
| \(H\) | \(I\) | (\(H\) & \(I\))→\(H\) |
|
1 1 0 0 |
1 0 1 0 |
\(\mathcal{A}\) & \(\mathcal{B}\) 1 1 1 1 1 0 0 1 0 0 1 0 0 0 0 0 |
The entire sentence is a conditional \(\mathcal{A}\)→\(\mathcal{B}\) with (\(H\)&\(I\)) as \(\mathcal{A}\) and with \(H\) as \(B\). On the second row, for example, (\(H\)&\(I\)) is false and \(H\) is true. Since a conditional is true when the antecedent is false, we write a 1 in the second row underneath the conditional symbol. We continue for the other three rows and get this:
| \(H\) | \(I\) | (\(H\) & \(I\))→\(H\) |
|
1 1 0 0 |
1 0 1 0 |
\(\mathcal{A}\) → \(\mathcal{B}\) 1 1 1 0 1 1 0 1 0 0 1 0 |
The column of 1s underneath the conditional tells us that the sentence (\(H\)&\(I\)) → \(I\) is true regardless of the truth-values of \(H\) and \(I\). They can be true or false in any combination, and the compound sentence still comes out true. It is crucial that we have considered all of the possible combinations. If we only had a twoline truth table, we could not be sure that the sentence was not false for some other combination of truth-values.
In this example, we have not repeated all of the entries in every successive table. When actually writing truth tables on paper, however, it is impractical to erase whole columns or rewrite the whole table for every step. Although it is more crowded, the truth table can be written in this way:
| \(H\) | \(I\) | (\(H\) & \(I\))→\(H\) |
|
1 1 0 0 |
1 0 1 0 |
\(\mathcal{A}\) → \(\mathcal{B}\) 1 1 1 1 1 1 0 0 1 1 0 0 1 1 0 0 0 0 1 0 |
Most of the columns underneath the sentence are only there for bookkeeping purposes. When you become more adept with truth tables, you will probably no longer need to copy over the columns for each of the sentence letters. In any case, the truth-value of the sentence on each row is just the column underneath the main logical operator of the sentence; in this case, the column underneath the conditional.
A complete truth table has a row for all the possible combinations of 1 and 0 for all of the sentence letters. The size of the complete truth table depends on the number of different sentence letters in the table. A sentence that contains only one sentence letter requires only two rows, as in the characteristic truth table for negation. This is true even if the same letter is repeated many times, as in the sentence [(\(C\) ↔ \(C\)) → \(C\)]&¬(\(C\) → \(C\)). The complete truth table requires only two lines because there are only two possibilities: \(C\) can be true or it can be false. A single sentence letter can never be marked both 1 and 0 on the same row. The truth table for this sentence looks like this:
| \(C\) | [(\(C\)↔\(C\) )→\(C\) ] & ¬(\(C\)→\(C\) ) |
|
1 0 |
1 1 1 1 1 0 0 1 1 1 0 1 0 0 0 0 0 0 1 0 |
Looking at the column underneath the main connective, we see that the sentence is false on both rows of the table; i.e., it is false regardless of whether \(C\) is true or false.
A sentence that contains two sentence letters requires four lines for a complete truth table, as in the characteristic truth tables and the table for (\(H\)&\(I\)) → \(I\).
A sentence that contains three sentence letters requires eight lines. For example:
| \(M\) | \(N\) | \(P\) | \(M\) & (\(N\) ∨\(P\)) |
|
1 1 1 1 0 0 0 0 |
1 1 0 0 1 1 0 0 |
1 0 1 0 1 0 1 0 |
1 1 1 1 1 1 1 1 1 0 1 1 0 1 1 1 0 0 0 0 0 0 1 1 1 0 0 1 1 0 0 0 0 1 1 0 0 0 0 0 |
From this table, we know that the sentence \(M\) &(\(N\)∨\(P\)) might be true or false, depending on the truth-values of \(M\), \(N\), and \(P\).
A complete truth table for a sentence that contains four different sentence letters requires 16 lines. Five letters, 32 lines. Six letters, 64 lines. And so on. To be perfectly general: If a complete truth table has \(\mathcal{n}\) different sentence letters, then it must have 2 n rows.
In order to fill in the columns of a complete truth table, begin with the rightmost sentence letter and alternate 1s and 0s. In the next column to the left, write two 1s, write two 0s, and repeat. For the third sentence letter, write four 1s followed by four 0s. This yields an eight line truth table like the one above. For a 16 line truth table, the next column of sentence letters should have eight 1s followed by eight 0s. For a 32 line table, the next column would have 16 1s followed by 16 0s. And so on.


The doors we open and close each day decide the lives we live.
~Flora Whittemore
When you envision yourself as a college student, what do you see?
What will your daily life be like?
Jane McGonigal: Gaming Can Make A Better World (TED Talk).
Video link:
https://youtu.be/dE1DuBesGYM
After watching the Jane McGonigal’s TED Talk, think about the following questions:
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What are gamers good at?
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What is the importance of “10,000” hours?
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Are gamers goal oriented?
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How do gamers feel about tests and being measured?
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What happens when a gamer makes a mistake?
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How do gamers handle frustration?
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How do gamers feel about change?
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Can the skills of a gamer be applicable to the skills a college student will need?
College is constant change. Not just in terms of studying and learning new material, but also in terms of how it is structured. If the college is on the quarter system, a student’s classes, teachers, and the hours a student needs to be on campus will change every 11-12 weeks – 4 times a year. Semesters divide an academic year into thirds and may have short intensive sessions in between the main semesters. People sometimes use the words quarters and semesters as if they are synonyms because both divide up a school year, but they represent different units of time.
Dividing up the academic year provides an opportunity for varied learning and developing specialties, but it also means new faces in classes, unknown expectations from new teachers, and juggling a new schedule. It means you may have new routes to travel on campus as you make your way to a different building if your college has a large campus. If a student is working along with going to college, it may mean negotiating new work hours with a boss and coworkers. All of these changes can feel like chaos that comes in like a tidal wave. Every term it can feel like starting over, especially for students who are not in a specific program yet. The beginning of a college experience can seem blurry to a new student trying to navigate the system.
“There’s no blinking light to say, hey, look over here, this changed!”
~Amber McCoy, Lane Community College Student
Many students come to college with at least some high school experience and expect college to be similar. After all, many classes have similar names: Biology, Algebra, Writing, Chemistry, and so on. However, the expectations that accompany those titles may be very different. College classes tend to cover course material at a faster pace and expect students to carry more of the burden of learning the material on their own outside of classroom activities.
Compared to college, high school has a straightforward curriculum. High school is segmented and chronological. Students generally go to school at the same time each morning and finish at a similar time in the afternoon. Students are assigned counselors to guide them. High school students usually don’t have to buy textbooks for their classes. There are clear deadlines and the teacher monitors progress and potentially shares progress with parents. The academic benchmarks of quizzes, tests, and projects are concrete indicators of progress. Teachers may monitor students’ use of smart phones in class and help students maintain focus on classroom materials. The high school a student attends is picked for him or her, either by geographic location or their parent’s choice.
College is about choice. Initially, the choice is where to go to school. The student has to find the right “fit” on his or her own and figure out the process of college admission. There are forms to fill out, submit, and process. Students may have to learn the steps for admission and enrollment for more than one college, and the process can vary from school to school. Students are expected to be able to complete the application process on their own. Students must determine if college placement tests are required and if so, when they must be taken.
The next choice for the student as part of the enrollment process is what to study in terms of declaring a major. The major a student declares may impact financial aid awards. If a student is unsure of what to study and doesn’t choose a major, financial aid may not be given to the student.
A student can choose to attend classes part-time or full-time. College class times try to accommodate a variety of student needs and my can occur during the day, evening, online, or a combination of classroom and online (Hybrid). Unless the student has someone to be accountable to, probably no one will check to see if attendance happens or if a student cruises the Internet or social media while in class.
Monitoring of time and its use will be student driven. Understanding the workload associated with a college schedule can be a surprise to the new college student. The first year of college can have a steep learning curve of time management and self-responsibility. For the first time college student, starting college can feel like pushing a big rock up a steep hill all alone.
How much time do you have in your life for school?
What is Considered Half-time or Full-time Status?
The answer to this question may vary from college to college. Lane Community College’s website uses the following definitions:
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Full-Time Status
: 12 or more credits per term (limit of 18 per term)
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3/4 Time Status:
9-11 credits per term
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Half-Time Status:
6-8 credits per term
An average student full-time credit load is between 15-18 credits. This means that a student will be in the classroom 1 hour per credit. Based on the 15-credit schedule, a student would be in the classroom 15 hours/week. Students mistakenly think that is all there is to it. A schedule requiring a student to be in class 15 hours/week sounds much easier than high school where students typically attend 6-7 hours a day or 30-35 hours/week. College has hidden expectations for students in terms of outside of class “homework.” What does that mean? College classes expect 2-3 hours of homework, and sometimes more, per credit. That means for 1 hour in class, a student can expect to spend 2-3 hours on homework or more. A 15-credit load expects a student to put in 30-45 hours outside of class each week on homework.
What does this mean in terms of your life?
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Activity
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Hours Required/Week
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168 hours in a week
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Full-time attendance
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15 in class
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-15
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Homework
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30 plus hours
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-30 (minimum)
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Sleeping
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6hrs/ day x 7 days
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-42
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Eating
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1.5hrs /day x 7
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-10.5
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Work
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20hrs/week
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-20
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Subtotal
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117.5 hrs
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168-117.5 = 50.5hrs
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Fill in the blanks with what else you would need to do each week
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How many hours will each item take to complete?
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Add the hours into the spaces below
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Total hours
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50.5- _______=_______
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Many students enter college with uninformed expectations. First-generation college students are at a disadvantage without family to help them understand the context of college, what to expect as a college student, and what college life is like. As a result, first-generation college students may be less prepared to handle the challenges they encounter. Students tend to be idealistic in their expectations of college. Pre-college characteristics and experiences play a role in shaping expectations.
Things to think about:
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How prepared are you to go back to school?
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How much time can you devote to college?
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How would you rate your time management skills?
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How do you feel about reading/homework?
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How are your technology skills?
-
What kind of support do you have for going to college?
-
Who is your support system?
-
Make of a list of the resources you have to support your college lifestyle.
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What strengths do you bring with you that will help you succeed in college?
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What skills will you need to improve?
-
What tips did you gain from watching the video?
How do you know if you are academically ready for college?
If you are accepted into college, does that mean you are ready?
College readiness is not clearly defined. Traditionally, completing high school was viewed as preparation for college, but course completion in high school does not guarantee college readiness. For example, English classes in high school may focus more on Literature where as entry-level college courses may stress expository reading and writing skills. If you have gone the route of getting your GED, did you work to dig deeper into the subjects and develop your skills, or just try to pass the tests as soon as possible? How did you handle attending classes and participating in classroom activities?
Another measure of college readiness has been standardized test scores. The problem with using a standardized test to determine readiness is its inability to measure the
soft skills
college courses require. Soft skills include qualities like accepting feedback, adaptability, dealing with difficult situations, critical thinking, effective communication, meeting deadlines, patience, persistence, self-direction, and trouble-shooting, to name a few. Meeting deadlines, for example, is a key to college success. The skills and behaviors needed to thrive in college may be different from those it takes to be admitted. Being accepted into college does not necessarily mean you are ready to face the challenges and frustrations that might lie between you and your goal.
Answering the question about being academically prepared for college is tough. Test scores and grades are indicators of readiness, but don’t guarantee success in college courses. Soft skills are important to college success, but without basic academic skills, soft skills alone won’t be enough. Most colleges use some type of placement test to try to place students into courses that will be appropriate for their skill levels. Usually, colleges have minimum placement test scores in Reading, Math, and Writing, requiring students to demonstrate they are able to handle the minimal expectations of college courses in terms of basic content areas. The degree or certificate associated with the student’s goal also influences the academic readiness required for success. Recognizing the importance of balancing the academic and soft skills, and how that relates to student goals is essential for college success and beyond.


5.5: Empirical Laws Paradigm
-
- Last updated
- Save as PDF
Theories in the Empirical Laws Paradigm approach Communication from the perspective that there are universal laws that govern how we communicate . Other names for Empirical Laws include: hard science, the positivist approach, the covering-laws approach, and the classical approach. Natural scientists look for universal laws to understand and explain our world.
Using our example of gravity, we know that objects fall to the earth 100% of the time when we drop them. This is a universal law. As Chapter 2 showed, in the late 1950’s scholars began studying human communication using approaches developed in the natural sciences (aka the Scientific Method). Thus, early proponents of Empirical Laws theories studied communication to see if there were universal communication laws similar to those in the natural world.
Laws and Communication
Natural laws at work in our world influence every moment of our lives. Every time you fly in an airplane or cross a bridge you trust that the people who designed and built the plane and bridge followed the physical laws that allow a plane to fly and a bridge to span a distance without collapsing. Every time you press the brakes on a car you trust them to slow you down based on the laws explaining how long a mass, traveling at a certain speed, takes to stop. Even if you do not understand all of these laws, you live by them and believe the laws themselves hold true 100% of the time.
Are there laws you follow about communication with this kind of regularity? Are they applicable 100% of the time, in all situations, and with all people? What happens if someone breaks one of these laws? Are the consequences similar to when you break physical laws? For example, is the consequence for calling someone by the wrong name comparable to that of hitting your brakes and them not working?
Those who approach communication from an empirical laws perspective believe there are laws that govern human communication. The premise of this approach can be stated as a simple equation of causation: If X, then Y. For example, if I greet a person with “Hi, how are you?” then I anticipate a response, “Fine, how are you?” It’s likely that you conduct much of your communication using this equation. However, does that mean that it works all of the time?
Communication Theory Then
The Empirical Revolution (1950–1970): Theory In A Test Tube
Speech departments in the 1950s promoted the ancient rhetorical wisdom that persuasive discourse was a matter of an ethical speaker using logical arguments—‘‘the good man speaking well.”11 But younger faculty with training in the social sciences were no longer willing to accept this ‘‘truth” by faith. Armed with a scientific skepticism and new methods to assess attitudes, they put rhetorical principles to the test…
Aristotle, for example, wrote that ethos was a combination of a speaker’s intelligence, character, and goodwill toward the audience. Empirically oriented speech researchers subsequently discovered that audience rankings of “communicator credibility” did indeed include factors of competence (intelligence) and trustworthiness (character).12 But they found no evidence that audiences regarded goodwill or positive intentions as traits separate from character.
Scholars interested in this kind of study adopted the media-effects term communication research to distinguish their work from the historical-critical textual analysis of rhetoricians. In 1950 a group of communication researchers founded what is now the International Communication Association (ICA) as a science-based professional organization to rival the Speech Association of America, which was grounded in the humanities. Traditional speech teachers of this era often accused communication researchers of succumbing to ‘‘the law of the hammer.” This was a not-so-subtle dig at those who would pound away with newly acquired statistical tools no matter what the job required.
But irony did little to slow the radical transformation within the communication discipline….The empiricists continued to borrow their core ideas from other disciplines—especially social psychology. Indeed, five of the thirty-three communication theories in this book come from that specialized branch of psychology. Their common methodology and unity of world view gave social scientists in the communication field a greater impact than their numbers alone would indicate. In 1969, the SAA changed its name to the Speech Communication Association (SCA). The term communication in the title was tacit evidence that the scientific approach now dominated the discipline. At the start of the 1960s few departments that taught speech had the word communication as part of their title. By the mid 1970s there were few that didn’t.
www.afirstlook.com/archive/talkabout.cfm?source=archther
There are three characteristics that help us understand empirical laws theories: causation, prediction, and generalization (Infante, Rancer & Womack). Causation states that there is a “cause and effect” relationship for all actions . In the physical world, if someone drops a pen it will fall. In human communication, if someone says “hello” to someone, that person responds. Prediction suggests that once someone determines a particular law is at work, they will use it to predict outcomes of communication situations . Have you ever rehearsed how you will ask someone out on a date and tried to predict the outcome?
What evidence did you use to make your prediction? In this example, you are using the “if X, then Y” equation to predict the outcome of the interaction. Generalization suggests that if a prediction shows that a behavior produces a certain outcome, we can generalize our predictions to include a wide variety of people, situations, and contexts . We make generalization such as, “If I’m friendly to others, they will be friendly to me” based on our past experiences with this type of behavior. However, this does not account for scenarios in which the person might not hear you, might be having a bad day and do not wish to respond, or assume you are talking to another person so they choose not to acknowledge you.
In the physical sciences, laws are absolute. This is comforting because it allows us to make informed decisions based on what we know about the laws that govern the world around us. In our example of gravity, we know that dropping an object will produce the same result every time. We could spend the rest of our lives testing this theory, but we don’t have to. We know what the result will be without having to continuously drop an object. Now, imagine what it would be like to always know what the outcome would be of every communication situation! Would that be comforting to you, or make your life boring?
Unlike the physical world, laws that govern human communication are not absolute and are most often bound by culture and context. Empirical laws theories are generally approached from the perspective of probability rather than absoluteness (Miller & Berger). Probability states that under certain conditions it is highly likely that we can predict communication outcomes . For example, when you greet someone with “hello” it is probable, not absolute, they will respond back with a greeting of their own. If they do not, you might run through a variety of reasons why the other person did not respond in accordance with the “laws” that govern greetings in our culture. Even though empirical laws theories do not produce absolutes about communication, we still use them in our everyday interactions with one another. Businesses, advertisers, schools, and other organizations use this approach to predict consumer, educational, and behavioral habits of particular demographic groups. While their approaches never produce a 100% cause-effect relationship, the information they gather helps them determine what actions to take to be successful in their communicative behaviors.
Empirical Laws in Action
Empirical laws theories are a relatively new approach for understanding communication. We have only been developing empirical laws theories of communication for the past 100 or so years. To date, none of this research has come to the conclusion that, given a certain circumstance, a particular communicative behavior will ALWAYS produce a particular outcome. However, working under an empirical laws approach that accepts probability, we have many research examples that demonstrate probable laws that govern human communication.
Communication Theory Then
Hovland, Janis, and Kelley (1953) were some of the first to use empirical laws theories to explain communication. Their interests in mass communication and propaganda during World War II led them to study effective persuasion in mass communication and propaganda campaigns. They theorized that the more attractive a communicator, along with other traits, the more likely people would be persuaded. Their empirical laws theories still influence how a great deal of mass media is produced today. Think about movies, television shows, and advertisements you see. Are most of the people you watch in these mediums considered attractive and intelligent? Those who produce mass media use tremendous resources to research probabilistic empirical laws of human behavior before making decisions about what and who to include in their messages. On a smaller scale, we all use probabilistic empirical laws to govern, predict, and control our communication with others.
The area of leadership in group and organizational communication has a body of well-established empirical laws theories called the trait approaches . These theories suggest that there are certain physical, personality, and communicative characteristics that make one person more likely to be a leader over another (Northouse). Trait theories propose that people in western societies who are physically tall, charismatic, intelligent, white, and male are more likely to be leaders, be perceived as leaders, be placed in more leadership positions, and make better leaders than those who don’t exhibit these characteristics. You may be thinking, “But what about people like Mother Theresa, Mohatma Ghandi, Martin Luther King, Jr., Caeser Chavez, or Barack Obama?” This question brings up two important points. First, it shows that communication theories are not absolute. Second, it shows that some theoretical viewpoints may work to promote a certain worldview of those in positions of power, an idea we’ll explore more thoroughly when we look at the Critical Theories Paradigm. Despite feeling uncomfortable with some of the assumptions of trait theories, if you look at those in the highest levels of leadership in the U.S., the vast majority have characteristics described in trait theories.
Strengths
A particular strength of empirical laws theories is that they help us determine cause and effect relationships in our communication with others. Understanding communication using these theories helps us predict the outcomes of our interactions with others. While we know that not all outcomes can be determined with 100% reliability, prediction and control allows us to more easily navigate our encounters. Think about the number of encounters you have each day in which you quickly predict and control your interaction with others. While not 100% conclusive, it’s comforting that a great number of our interactions have a certain level of probable outcomes.
Weaknesses
A criticism of empirical laws theory is that while it is useful for understanding relatively simple interactions, it can oversimplify or fail to explain situations where a number of variables exist. Your classroom environment serves as a good example. While there are certain predictions you can make about how communication will occur in your classes, why is it that each classroom experience is unique? In your classrooms, it is impossible to predict, control, and generalize how a class will go with 100% accuracy because it is impossible to replicate classes in exactly the same ways. This approach does not account for the variety of human choices and behaviors that are brought into every communication context. It operates under the assumption that, given the same context, people bring the same things to the context each time. Obviously, this is not the case. Human behaviors are complex and cannot be predicted at a 100% accuracy rate. However, empirical laws theories work well for showing us patterns of behavior that guide our communication.
Contributions and Affiliations
- Survey of Communication Study. Authored by : Scott T Paynton and Linda K Hahn. Provided by : Humboldt State University. Located at : en.wikibooks.org/wiki/Survey_of_Communication_Study/Preface. License : CC BY-SA: Attribution-ShareAlike


16.2: Prokaryotic Gene Regulation
Skills to Develop
- Describe the steps involved in prokaryotic gene regulation
- Explain the roles of activators, inducers, and repressors in gene regulation
The DNA of prokaryotes is organized into a circular chromosome supercoiled in the nucleoid region of the cell cytoplasm. Proteins that are needed for a specific function, or that are involved in the same biochemical pathway, are encoded together in blocks called operons . For example, all of the genes needed to use lactose as an energy source are coded next to each other in the lactose (or lac ) operon.
In prokaryotic cells, there are three types of regulatory molecules that can affect the expression of operons: repressors, activators, and inducers. Repressors are proteins that suppress transcription of a gene in response to an external stimulus, whereas activators are proteins that increase the transcription of a gene in response to an external stimulus. Finally, inducers are small molecules that either activate or repress transcription depending on the needs of the cell and the availability of substrate.
The trp Operon: A Repressor Operon
Bacteria such as E. coli need amino acids to survive. Tryptophan is one such amino acid that E. coli can ingest from the environment. E. coli can also synthesize tryptophan using enzymes that are encoded by five genes. These five genes are next to each other in what is called the tryptophan ( trp ) operon (Figure \(\PageIndex{1}\)). If tryptophan is present in the environment, then E. coli does not need to synthesize it and the switch controlling the activation of the genes in the trp operon is switched off. However, when tryptophan availability is low, the switch controlling the operon is turned on, transcription is initiated, the genes are expressed, and tryptophan is synthesized.
A DNA sequence that codes for proteins is referred to as the coding region. The five coding regions for the tryptophan biosynthesis enzymes are arranged sequentially on the chromosome in the operon. Just before the coding region is the transcriptional start site . This is the region of DNA to which RNA polymerase binds to initiate transcription. The promoter sequence is upstream of the transcriptional start site; each operon has a sequence within or near the promoter to which proteins (activators or repressors) can bind and regulate transcription.
A DNA sequence called the operator sequence is encoded between the promoter region and the first trp coding gene. This operator contains the DNA code to which the repressor protein can bind. When tryptophan is present in the cell, two tryptophan molecules bind to the trp repressor, which changes shape to bind to the trp operator. Binding of the tryptophan–repressor complex at the operator physically prevents the RNA polymerase from binding, and transcribing the downstream genes.
When tryptophan is not present in the cell, the repressor by itself does not bind to the operator; therefore, the operon is active and tryptophan is synthesized. Because the repressor protein actively binds to the operator to keep the genes turned off, the trp operon is negatively regulated and the proteins that bind to the operator to silence trp expression are negative regulators .
Link to Learning
Watch this video to learn more about the trp operon.
Catabolite Activator Protein (CAP): An Activator Regulator
Just as the trp operon is negatively regulated by tryptophan molecules, there are proteins that bind to the operator sequences that act as a positive regulator to turn genes on and activate them. For example, when glucose is scarce, E. coli bacteria can turn to other sugar sources for fuel. To do this, new genes to process these alternate genes must be transcribed. When glucose levels drop, cyclic AMP (cAMP) begins to accumulate in the cell. The cAMP molecule is a signaling molecule that is involved in glucose and energy metabolism in E. coli . When glucose levels decline in the cell, accumulating cAMP binds to the positive regulator catabolite activator protein (CAP) , a protein that binds to the promoters of operons that control the processing of alternative sugars. When cAMP binds to CAP, the complex binds to the promoter region of the genes that are needed to use the alternate sugar sources (Figure \(\PageIndex{2}\)). In these operons, a CAP binding site is located upstream of the RNA polymerase binding site in the promoter. This increases the binding ability of RNA polymerase to the promoter region and the transcription of the genes.
The lac Operon: An Inducer Operon
The third type of gene regulation in prokaryotic cells occurs through inducible operons , which have proteins that bind to activate or repress transcription depending on the local environment and the needs of the cell. The lac operon is a typical inducible operon. As mentioned previously, E. coli is able to use other sugars as energy sources when glucose concentrations are low. To do so, the cAMP–CAP protein complex serves as a positive regulator to induce transcription. One such sugar source is lactose. The lac operon encodes the genes necessary to acquire and process the lactose from the local environment. CAP binds to the operator sequence upstream of the promoter that initiates transcription of the lac operon. However, for the lac operon to be activated, two conditions must be met. First, the level of glucose must be very low or non-existent. Second, lactose must be present. Only when glucose is absent and lactose is present will the lac operon be transcribed (Figure \(\PageIndex{3}\)). This makes sense for the cell, because it would be energetically wasteful to create the proteins to process lactose if glucose was plentiful or lactose was not available.
Art Connection
In E. coli , the trp operon is on by default, while the lac operon is off. Why do you think this is the case?
If glucose is absent, then CAP can bind to the operator sequence to activate transcription. If lactose is absent, then the repressor binds to the operator to prevent transcription. If either of these requirements is met, then transcription remains off. Only when both conditions are satisfied is the lac operon transcribed (Table \(\PageIndex{1}\)).
| Glucose | CAP binds | Lactose | Repressor binds | Transcription |
|---|---|---|---|---|
| + | - | - | + | No |
| + | - | + | - | Some |
| - | + | - | + | No |
| - | + | + | - | Yes |
Link to Learning
Watch an animated tutorial about the workings of lac operon here.
Summary
The regulation of gene expression in prokaryotic cells occurs at the transcriptional level. There are three ways to control the transcription of an operon: repressive control, activator control, and inducible control. Repressive control, typified by the trp operon, uses proteins bound to the operator sequence to physically prevent the binding of RNA polymerase and the activation of transcription. Therefore, if tryptophan is not needed, the repressor is bound to the operator and transcription remains off. Activator control, typified by the action of CAP, increases the binding ability of RNA polymerase to the promoter when CAP is bound. In this case, low levels of glucose result in the binding of cAMP to CAP. CAP then binds the promoter, which allows RNA polymerase to bind to the promoter better. In the last example—the lac operon—two conditions must be met to initiate transcription. Glucose must not be present, and lactose must be available for the lac operon to be transcribed. If glucose is absent, CAP binds to the operator. If lactose is present, the repressor protein does not bind to its operator. Only when both conditions are met will RNA polymerase bind to the promoter to induce transcription.
Art Connections
Figure \(\PageIndex{3}\): In E. coli , the trp operon is on by default, while the lac operon is off. Why do you think that this is the case?
- Answer
-
Tryptophan is an amino acid essential for making proteins, so the cell always needs to have some on hand. However, if plenty of tryptophan is present, it is wasteful to make more, and the expression of the trp receptor is repressed. Lactose, a sugar found in milk, is not always available. It makes no sense to make the enzymes necessary to digest an energy source that is not available, so the lac operon is only turned on when lactose is present.
Glossary
- activator
- protein that binds to prokaryotic operators to increase transcription
- catabolite activator protein (CAP)
- protein that complexes with cAMP to bind to the promoter sequences of operons that control sugar processing when glucose is not available
- inducible operon
- operon that can be activated or repressed depending on cellular needs and the surrounding environment
- lac operon
- operon in prokaryotic cells that encodes genes required for processing and intake of lactose
- negative regulator
- protein that prevents transcription
- operator
- region of DNA outside of the promoter region that binds activators or repressors that control gene expression in prokaryotic cells
- operon
- collection of genes involved in a pathway that are transcribed together as a single mRNA in prokaryotic cells
- positive regulator
- protein that increases transcription
- repressor
- protein that binds to the operator of prokaryotic genes to prevent transcription
- transcriptional start site
- site at which transcription begins
- trp operon
- series of genes necessary to synthesize tryptophan in prokaryotic cells
- tryptophan
- amino acid that can be synthesized by prokaryotic cells when necessary


4.1: Introduction
Three Laws of Thermodynamics describe the flow and transfer of energy in the universe.
1. Energy can neither be created nor destroyed.
2. Universal entropy (disorder) is always increasing .
3. Entropy declines with temperature -as temperatures approach absolute zero, so goes entropy
In living systems, we do not have to worry about the third law because equations for energy exchange in living systems already reflect the temperature dependence of entropy changes during reactions. Here we look at how we came to understand basic thermodynamic principles and how they apply to living systems. First, we will look at different kinds of energy and at how redox reactions govern the flow of energy through living things. Next, we will try to understand some simple arithmetic statements of the Laws of Thermodynamics for closed systems and then at how they apply to chemical reactions conducted under standard conditions . Finally, since there really is no such thing as a closed system , we look at the energetics of reactions occurring in open systems. For an excellent discussion of how basic thermodynamic principles apply to living things, see Lehninger A. (1971) Bioenergetics: The Molecular Basis of Biological Energy Transformation. Benjamin Cummings, San Francisco.
Learning Objectives
When you have mastered the information in this chapter, you should be able to:
1. explain the difference between energy transfer and energy transduction.
2. compare and contrast potential vs. kinetic as well as other categories of energy (e.g.,
mass, heat, light…, etc .).
3. explain the reciprocal changes in universal free energy and entropy.
4. derive the algebraic relationship between free energy , enthalpy and entropy.
5. state the difference between exothermic , endothermic , exergonic , and endergonic
reactions
6. predict changes in free energy based on changes in the concentrations of reactants and products in closed systems and open systems.
7. explain how the same reaction can be endergonic but will (under appropriate conditions) release free energy.
8. predict whether a biochemical reaction will release free energy if it is exothermic, and if so, under what conditions (you should be able to do this after working some sample problems of closed system energetics).
9. distinguish between the equilibrium and steady-state of reactions and explain how an endergonic reaction could also be spontaneous (i.e., could release free energy).


15.4.2: Buoyancy
Following Olivia's thoughts, we assume that an object will displace an amount of water \(V_W\) such that the weight of water displaced is equal to the weight of the object. Let's test her first statement that an object with density equal to water, \(\rho_O/\rho_W=1\), will displace a volume equal to the object. \(V_W/V_O=1\). To do this
- Select the "Intro" simulation and choose a custom object.
- Set the object's mass and volume to equal values so that \(\rho_O=m_O/V_O = 1 kg/L\). Note this is the same density as water.
- Does your observation agree with Olivia?
Test her other hypotheses.
- \(\rho_O/\rho_W=0.5 \rightarrow V_W/V_O = 0.5\)
- \(\rho_O/\rho_W=1.5 \rightarrow V_W/V_O = 1.5\)
Do you draw the same conclusions as Olivia?


2.6.5: Water and Life on Mars
Learning Objectives
By the end of this section, you will be able to:
- Describe the general composition of the atmosphere on Mars
- Explain what we know about the polar ice caps on Mars and how we know it
- Describe the evidence for the presence of water in the past history of Mars
- Summarize the evidence for and against the possibility of life on Mars
Of all the planets and moons in the solar system, Mars seems to be the most promising place to look for life, both fossil microbes and (we hope) some forms of life deeper underground that still survive today. But where (and how) should we look for life? We know that the one requirement shared by all life on Earth is liquid water. Therefore, the guiding principle in assessing habitability on Mars and elsewhere has been to “follow the water.” That is the perspective we take in this section, to follow the water on the red planet and hope it will lead us to life.
Atmosphere and Clouds on Mars
The atmosphere of Mars today has an average surface pressure of only 0.007 bar, less than 1% that of Earth. (This is how thin the air is about 30 kilometers above Earth’s surface.) Martian air is composed primarily of carbon dioxide (95%), with about 3% nitrogen and 2% argon. The proportions of different gases are similar to those in the atmosphere of Venus (see Table \(10.3.1\)), but a lot less of each gas is found in the thin air on Mars.
While winds on Mars can reach high speeds, they exert much less force than wind of the same velocity would on Earth because the atmosphere is so thin. The wind is able, however, to loft very fine dust particles, which can sometimes develop planet-wide dust storms. It is this fine dust that coats almost all the surface, giving Mars its distinctive red color. In the absence of surface water, wind erosion plays a major role in sculpting the martian surface (Figure \(\PageIndex{1}\)).
Figure \(\PageIndex{1}\) Wind Erosion on Mars. These long straight ridges, called yardangs, are aligned with the dominant wind direction. This is a high-resolution image from the Mars Reconnaissance Orbiter and is about 1 kilometer wide.The issue of how strong the winds on Mars can be plays a big role in the 2015 hit movie The Martian in which the main character is stranded on Mars after being buried in the sand in a windstorm so great that his fellow astronauts have to leave the planet so their ship is not damaged. Astronomers have noted that the martian winds could not possibly be as forceful as depicted in the film. In most ways, however, the depiction of Mars in this movie is remarkably accurate.
Although the atmosphere contains small amounts of water vapor and occasional clouds of water ice, liquid water is not stable under present conditions on Mars. Part of the problem is the low temperatures on the planet. But even if the temperature on a sunny summer day rises above the freezing point, the low pressure means that liquid water still cannot exist on the surface, except at the lowest elevations. At a pressure of less than 0.006 bar, the boiling point is as low or lower than the freezing point, and water changes directly from solid to vapor without an intermediate liquid state (as does “dry ice,” carbon dioxide, on Earth). However, salts dissolved in water lower its freezing point, as we know from the way salt is used to thaw roads after snow and ice forms during winter on Earth. Salty water is therefore sometimes able to exist in liquid form on the martian surface, under the right conditions.
Several types of clouds can form in the martian atmosphere. First there are dust clouds, discussed above. Second are water-ice clouds similar to those on Earth. These often form around mountains, just as happens on our planet. Finally, the CO2 of the atmosphere can itself condense at high altitudes to form hazes of dry ice crystals. The CO2 clouds have no counterpart on Earth, since on our planet temperatures never drop low enough (down to about 150 K or about 125 °C) for this gas to condense.
The Polar Caps
Through a telescope, the most prominent surface features on Mars are the bright polar caps, which change with the seasons, similar to the seasonal snow cover on Earth. We do not usually think of the winter snow in northern latitudes as a part of our polar caps, but seen from space, the thin winter snow merges with Earth’s thick, permanent ice caps to create an impression much like that seen on Mars (Figure \(\PageIndex{2}\)).
Figure \(\PageIndex{2}\) Martian North Polar Cap. (a) This is a composite image of the north pole in summer, obtained in October 2006 by the Mars Reconnaissance Orbiter. It shows the mostly water-ice residual cap sitting atop light, tan-colored, layered sediments. Note that although the border of this photo is circular, it shows only a small part of the planet. (b) Here we see a small section of the layered terrain near the martian north pole. There is a mound about 40 meters high that is sticking out of a trough in the center of the picture.The seasonal caps on Mars are composed not of ordinary snow but of frozen CO 2 (dry ice). These deposits condense directly from the atmosphere when the surface temperature drops below about 150 K. The caps develop during the cold martian winters and extend down to about 50° latitude by the start of spring.
Quite distinct from these thin seasonal caps of CO 2 are the permanent or residual caps that are always present near the poles. The southern permanent cap has a diameter of 350 kilometers and is composed of frozen CO 2 deposits together with a great deal of water ice. Throughout the southern summer, it remains at the freezing point of CO 2 , 150 K, and this cold reservoir is thick enough to survive the summer heat intact.
The northern permanent cap is different. It is much larger, never shrinking to a diameter less than 1000 kilometers, and is composed of water ice. Summer temperatures in the north are too high for the frozen CO 2 to be retained. Measurements from the Mars Global Surveyor have established the exact elevations in the north polar region of Mars, showing that it is a large basin about the size of our own Arctic Ocean basin. The ice cap itself is about 3 kilometers thick, with a total volume of about 10 million km3(similar to that of Earth’s Mediterranean Sea). If Mars ever had extensive liquid water, this north polar basin would have contained a shallow sea. There is some indication of ancient shorelines visible, but better images will be required to verify this suggestion.
Images taken from orbit also show a distinctive type of terrain surrounding the permanent polar caps, as shown in Figure \(\PageIndex{2}\). At latitudes above 80° in both hemispheres, the surface consists of recent layered deposits that cover the older cratered ground below. Individual layers are typically ten to a few tens of meters thick, marked by alternating light and dark bands of sediment. Probably the material in the polar deposits includes dust carried by wind from the equatorial regions of Mars.
What do these terraced layers tell us about Mars? Some cyclic process is depositing dust and ice over periods of time. The time scales represented by the polar layers are tens of thousands of years. Apparently the martian climate experiences periodic changes at intervals similar to those between ice ages on Earth. Calculations indicate that the causes are probably also similar: the gravitational pull of the other planets produces variations in Mars’ orbit and tilt as the great clockwork of the solar system goes through its paces.
The Phoenix spacecraft landed near the north polar cap in summer (Figure \(\PageIndex{3}\)). Controllers knew that is would not be able to survive a polar winter, but directly measuring the characteristics of the polar region was deemed important enough to send a dedicated mission. The most exciting discovery came when the spacecraft tried to dig a shallow trench under the spacecraft. When the overlying dust was stripped off, they saw bright white material, apparently some kind of ice. From the way this ice sublimated over the next few days, it was clear that it was frozen water.
Figure \(\PageIndex{3}\) Evaporating Ice on Mars. We see a trench dug by the Phoenix lander in the north polar region four martian days apart in June 2008. If you look at the shadowed region in the bottom left of the trench, you can see three spots of ice in the left image which have sublimated away in the right image. (credit: modification of work by NASA/JPL-Caltech/University of Arizona/Texas A&M University)Example \(\PageIndex{1}\): Comparing the Amount of Water on Mars and Earth
It is interesting to estimate the amount of water (in the form of ice) on Mars and to compare this with the amount of water on Earth. In each case, we can find the total volume of a layer on a sphere by multiplying the area of the sphere (\(4 \pi R^2\)) by the thickness of the layer. For Earth, the ocean water is equivalent to a layer 3 km thick spread over the entire planet, and the radius of Earth is 6.378 × 10 6 m (see Appendix F). For Mars, most of the water we are sure of is in the form of ice near the poles. We can calculate the amount of ice in one of the residual polar caps if it is (for example) 2 km thick and has a radius of 400 km (the area of a circle is \(\pi R^2\)).
Solution
The volume of Earth’s water is therefore the area \(4 \pi R^2\)
\[ 4 \pi \left( 6.378 \times 10^6 \text{ m} \right)^2=5.1 \times 10^{14} \text{ m}2 \nonumer\]
multiplied by the thickness of 3000 m:
\[ 5.1 \times 10^{14} \text{ m}^2 \times 3000 \text{ m} = 1.5 \times 10^{18} \text{ m}^3 \nonumber\]
This gives 1.5 × 10 18 m 3 of water. Since water has a density of 1 ton per cubic meter (1000 kg/m 3 ), we can calculate the mass:
\[1.5 \times 10^{18} \text{ m}^3 \times 1 \text{ ton/m}^3 = 1.5 \times 10^{18} \text{ tons} \nonumber\]
For Mars, the ice doesn’t cover the whole planet, only the caps; the polar cap area is
\[ \pi R^2= \pi \left( 4 \times 10^5 \text{ m} \right)^2 = 5 \times 10^{11} \text{ m}^2 \nonumber\]
(Note that we converted kilometers to meters.)
The volume = area × height, so we have:
\[ \left( 2 \times 10^3 \text{ m} \right) \left( 5 \times 10^{11} \text{ m}^2 \right) = 1 \times 10^{15} \text{ m}^3=10^{15} \text{ m}^3 \nonumber\]
Therefore, the mass is:
\[10^{15} \text{ m}^3 \times 1 \text{ ton/m}^3=10^{15} \text{ tons} \nonumber\]
This is about 0.1% that of Earth’s oceans.
Exercise \(\PageIndex{1}\)
A better comparison might be to compare the amount of ice in the Mars polar ice caps to the amount of ice in the Greenland ice sheet on Earth, which has been estimated as 2.85 × 10 15 m 3 . How does this compare with the ice on Mars?
- Answer
-
The Greenland ice sheet has about 2.85 times as much ice as in the polar ice caps on Mars. They are about the same to the nearest power of 10.
Channels and Gullies on Mars
Although no bodies of liquid water exist on Mars today, evidence has accumulated that rivers flowed on the red planet long ago. Two kinds of geological features appear to be remnants of ancient watercourses, while a third class—smaller gullies—suggests intermittent outbreaks of liquid water even today. We will examine each of these features in turn.
In the highland equatorial plains, there are multitudes of small, sinuous (twisting) channels—typically a few meters deep, some tens of meters wide, and perhaps 10 or 20 kilometers long (Figure \(\PageIndex{4}\)). They are called runoff channels because they look like what geologists would expect from the surface runoff of ancient rain storms. These runoff channels seem to be telling us that the planet had a very different climate long ago. To estimate the age of these channels, we look at the cratering record. Crater counts show that this part of the planet is more cratered than the lunar maria but less cratered than the lunar highlands. Thus, the runoff channels are probably older than the lunar maria, presumably about 4 billion years old.
The second set of water-related features we see are outflow channels (Figure \(\PageIndex{4}\)) are much larger than the runoff channels. The largest of these, which drain into the Chryse basin where Pathfinder landed, are 10 kilometers or more wide and hundreds of kilometers long. Many features of these outflow channels have convinced geologists that they were carved by huge volumes of running water, far too great to be produced by ordinary rainfall. Where could such floodwater have come from on Mars?
Figure \(\PageIndex{4}\) Runoff and Outflow Channels. (a) These runoff channels in the old martian highlands are interpreted as the valleys of ancient rivers fed by either rain or underground springs. The width of this image is about 200 kilometers. (b) This intriguing channel, called Nanedi Valles, resembles Earth riverbeds in some (but not all) ways. The tight curves and terraces seen in the channel certainly suggest the sustained flow of a fluid like water. The channel is about 2.5 kilometers across. (credit a: modification of work by Jim Secosky/NASA; credit b: modification of work by Jim Secosky/NASA)As far we can tell, the regions where the outflow channels originate contained abundant water frozen in the soil as permafrost. Some local source of heating must have released this water, leading to a period of rapid and catastrophic flooding. Perhaps this heating was associated with the formation of the volcanic plains on Mars, which date back to roughly the same time as the outflow channels.
Note that neither the runoff channels nor the outflow channels are wide enough to be visible from Earth, nor do they follow straight lines. They could not have been the “canals” Percival Lowell imagined seeing on the red planet.
The third type of water feature, the smaller gullies , was discovered by the Mars Global Surveyor (Figure \(\PageIndex{5}\)). The Mars Global Surveyor’s camera images achieved a resolution of a few meters, good enough to see something as small as a truck or bus on the surface. On the steep walls of valleys and craters at high latitudes, there are many erosional features that look like gullies carved by flowing water. These gullies are very young: not only are there no superimposed impact craters, but in some instances, the gullies seem to cut across recent wind-deposited dunes. Perhaps there is liquid water underground that can occasionally break out to produce short-lived surface flows before the water can freeze or evaporate.
Figure \(\PageIndex{5}\) Gullies on the Wall of Garni Crater. This high-resolution image is from the Mars Reconnaissance Orbiter. The dark streaks, which are each several hundred meters long, change in a seasonal pattern that suggests they are caused by the temporary flow of surface water. (credit: NASA/JPL-Caltech/University of Arizona)The gullies also have the remarkable property of changing regularly with the martian seasons. Many of the dark streaks (visible in Figure \(\PageIndex{5}\)) elongate within a period of a few days, indicating that something is flowing downhill—either water or dark sediment. If it is water, it requires a continuing source, either from the atmosphere or from springs that tap underground water layers (aquifers.) Underground water would be the most exciting possibility, but this explanation seems inconsistent with the fact that many of the dark streaks start at high elevations on the walls of craters.
Additional evidence that the dark streaks (called by the scientists recurring slope lineae ) are caused by water was found in 2015 when spectra were obtained of the dark streaks (Figure \(\PageIndex{6}\)). These showed the presence of hydrated salts produced by the evaporation of salty water. If the water is salty, it could remain liquid long enough to flow downstream for distances of a hundred meters or more, before it either evaporates or soaks into the ground. However, this discovery still does not identify the ultimate source of the water.
Figure \(\PageIndex{6}\) Evidence for Liquid Water on Mars. The dark streaks in Horowitz crater, which move downslope, have been called recurring slope lineae. The streaks in the center of the image go down the wall of the crater for about a distance of 100 meters. Spectra taken of this region indicate that these are locations where salty liquid water flows on or just below the surface of Mars. (The vertical dimension is exaggerated by a factor of 1.5 compared to horizontal dimensions.) (credit: NASA/JPL-Caltech/University of Arizona)Ancient Lakes and Glaciers
The rovers ( Spirit , Opportunity , and Curiosity ) that have operated on the surface of Mars have been used to hunt for additional evidence of water. They could not reach the most interesting sites, such as the gullies, which are located on steep slopes. Instead, they explored sites that might be dried-out lake beds, dating back to a time when the climate on Mars was warmer and the atmosphere thicker—allowing water to be liquid on the surface.
Spirit was specifically targeted to explore what looked like an ancient lake-bed in Gusev crater, with an outflow channel emptying into it. However, when the spacecraft landed, it found that the former lakebed had been covered by thin lava flows, blocking the rover from access to the sedimentary rocks it had hoped to find. However, Opportunity had better luck. Peering at the walls of a small crater, it detected layered sedimentary rock. These rocks contained chemical evidence of evaporation, suggesting there had been a shallow salty lake in that location. In these sedimentary rocks were also small spheres that were rich in the mineral hematite, which forms only in watery environments. Apparently this very large basin had once been underwater.
The small spherical rocks were nicknamed “blueberries” by the science team and the discovery of a whole “berry-bowl” of them was announced in this interesting news release from NASA.
The Curiosity rover landed inside Gale crater, where photos taken from orbit also suggested past water erosion. It discovered numerous sedimentary rocks, some in the form of mudstones from an ancient lakebed; it also found indications of rocks formed by the action of shallow water at the time the sediment formed (Figure \(\PageIndex{7}\)).
Even today there is evidence of large quantities of ice just below the surface of Mars. In the mid-latitudes, high-resolution photos from orbit have revealed glaciers covered with dirt and dust. In some cliffs, the ice is observed directly (see Figure \(\PageIndex{7}\)). These glaciers are thought to have formed during warm periods, when the atmospheric pressure was greater and snow and ice could precipitate. They also suggest readily available frozen water that could support future human exploration of the planet.
Figure \(\PageIndex{7}\) Gale Crater and Underground Ice Deposits. (a) This scene, photographed by the Curiosity rover, shows an ancient lakebed of cracked mudstones. (b) Geologists working with the Curiosity rover interpret this image of cross-bedded sandstone in Gale crater as evidence of liquid water passing over a loose bed of sediment at the time this rock formed. (c) Ice bands a hundred meters tall are visible in blue in a cliff-face on Mars, suggesting large deposits of frozen water buried just a few meters below the surface. Note that the blue color has been exaggerated in this photo, taken by the Mars Reconnaissance Orbiter spacecraft. (credit a: modification of work by NASA/JPL-Caltech/MSSS; credit b: modification of work by NASA/JPL-Caltech/MSSS; credit c: modification of work by NASA/JPL-Caltech/UA/USGS)astronomy and pseudoscience: the "face on Mars"
People like human faces. We humans have developed great skill in recognizing people and interpreting facial expressions. We also have a tendency to see faces in many natural formations, from clouds to the man in the Moon. One of the curiosities that emerged from the Viking orbiters’ global mapping of Mars was the discovery of a strangely shaped mesa in the Cydonia region that resembled a human face. Despite later rumors of a cover-up, the “Face on Mars” was, in fact, recognized by Viking scientists and included in one of the early mission press releases. At the low resolution and oblique lighting under which the Viking image was obtained, the mile-wide mesa had something of a Sphinx-like appearance.
 A band of “true believers” grew around the face and tried to deduce the nature of the “sculptors” who made it. This group also linked the face to a variety of other pseudoscientific phenomena such as crop circles (patterns in fields of grain, mostly in Britain, now known to be the work of pranksters).
Members of this group accused NASA of covering up evidence of intelligent life on Mars, and they received a great deal of help in publicizing their perspective from tabloid media. Some of the believers picketed the Jet Propulsion Laboratory at the time of the failure of the Mars Observer spacecraft, circulating stories that the “failure” of the Mars Observer was itself a fake, and that its true (secret) mission was to photograph the face.
The high-resolution Mars Observer camera (MOC) was reflown on the Mars Global Surveyor mission, which arrived at Mars in 1997. On April 5, 1998, in Orbit 220, the MOC obtained an oblique image of the face at a resolution of 4 meters per pixel, a factor-of-10 improvement in resolution over the Viking image. Another image in 2001 had even higher resolution. Immediately released by NASA, the new images showed a low mesa-like hill cut crossways by several roughly linear ridges and depressions, which were misidentified in the 1976 photo as the eyes and mouth of a face. Only with an enormous dose of imagination can any resemblance to a face be seen in the new images, demonstrating how dramatically our interpretation of geology can change with large improvements in resolution. The original and the higher resolution images can be seen in Figure \(\PageIndex{8}\).
Figure \(\PageIndex{8}\) Face on Mars. The so-called “Face on Mars” is seen (a) in low resolution from Viking (the “face” is in the upper part of the picture) and (b) with 20 times better resolution from the Mars Global Surveyor.After 20 years of promoting pseudoscientific interpretations and various conspiracy theories, can the “Face on Mars” believers now accept reality? Unfortunately, it does not seem so. They have accused NASA of faking the new picture. They also suggest that the secret mission of the Mars Observer included a nuclear bomb used to destroy the face before it could be photographed in greater detail by the Mars Global Surveyor .
Space scientists find these suggestions incredible. NASA is spending increasing sums for research on life in the universe, and a major objective of current and upcoming Mars missions is to search for evidence of past microbial life on Mars. Conclusive evidence of extraterrestrial life would be one of the great discoveries of science and incidentally might well lead to increased funding for NASA. The idea that NASA or other government agencies would (or could) mount a conspiracy to suppress such welcome evidence is truly bizarre.
Alas, the “Face on Mars” story is only one example of a whole series of conspiracy theories that are kept before the public by dedicated believers, by people out to make a fast buck, and by irresponsible media attention. Others include the “urban legend” that the Air Force has the bodies of extraterrestrials at a secret base, the widely circulated report that UFOs crashed near Roswell, New Mexico (actually it was a balloon carrying scientific instruments to find evidence of Soviet nuclear tests), or the notion that alien astronauts helped build the Egyptian pyramids and many other ancient monuments because our ancestors were too stupid to do it alone.
In response to the increase in publicity given to these “fiction science” ideas, a group of scientists, educators, scholars, and magicians (who know a good hoax when they see one) have formed the Committee for Skeptical Inquiry. Two of the original authors of your book are active on the committee. For more information about its work delving into the rational explanations for paranormal claims, see their excellent magazine, The Skeptical Inquirer , or check out their website at www.csicop.org/.
Climate Change on Mars
The evidence about ancient rivers and lakes of water on Mars discussed so far suggests that, billions of years ago, martian temperatures must have been warmer and the atmosphere must have been more substantial than it is today. But what could have changed the climate on Mars so dramatically?
We presume that, like Earth and Venus, Mars probably formed with a higher surface temperature thanks to the greenhouse effect. But Mars is a smaller planet, and its lower gravity means that atmospheric gases could escape more easily than from Earth and Venus. As more and more of the atmosphere escaped into space, the temperature on the surface gradually fell.
Eventually Mars became so cold that most of the water froze out of the atmosphere, further reducing its ability to retain heat. The planet experienced a sort of runaway refrigerator effect , just the opposite of the runaway greenhouse effect that occurred on Venus. Probably, this loss of atmosphere took place within less than a billion years after Mars formed. The result is the cold, dry Mars we see today.
Conditions a few meters below the martian surface, however, may be much different. There, liquid water (especially salty water) might persist, kept warm by the internal heat of Mars or the insulating layers solid and rock. Even on the surface, there may be ways to change the martian atmosphere temporarily.
Mars is likely to experience long-term climate cycles, which may be caused by the changing orbit and tilt of the planet. At times, one or both of the polar caps might melt, releasing a great deal of water vapor into the atmosphere. Perhaps an occasional impact by a comet might produce a temporary atmosphere that is thick enough to permit liquid water on the surface for a few weeks or months. Some have even suggested that future technology might allow us to terraform Mars—that is, to engineer its atmosphere and climate in ways that might make the planet more hospitable for long-term human habitation.
The Search for Life on Mars
If there was running water on Mars in the past, perhaps there was life as well. Could life, in some form, remain in the martian soil today? Testing this possibility, however unlikely, was one of the primary objectives of the Viking landers in 1976. These landers carried miniature biological laboratories to test for microorganisms in the martian soil. Martian soil was scooped up by the spacecraft’s long arm and placed into the experimental chambers, where it was isolated and incubated in contact with a variety of gases, radioactive isotopes, and nutrients to see what would happen. The experiments looked for evidence of respiration by living animals, absorption of nutrients offered to organisms that might be present, and an exchange of gases between the soil and its surroundings for any reason whatsoever. A fourth instrument pulverized the soil and analyzed it carefully to determine what organic (carbon-bearing) material it contained.
The Viking experiments were so sensitive that, had one of the spacecraft landed anywhere on Earth (with the possible exception of Antarctica), it would easily have detected life. But, to the disappointment of many scientists and members of the public, no life was detected on Mars. The soil tests for absorption of nutrients and gas exchange did show some activity, but this was most likely caused by chemical reactions that began as water was added to the soil and had nothing to do with life. In fact, these experiments showed that martian soil seems much more chemically active than terrestrial soils because of its exposure to solar ultraviolet radiation (since Mars has no ozone layer).
The organic chemistry experiment showed no trace of organic material, which is apparently destroyed on the martian surface by the sterilizing effect of this ultraviolet light. While the possibility of life on the surface has not been eliminated, most experts consider it negligible. Although Mars has the most earthlike environment of any planet in the solar system, the sad fact is that nobody seems to be home today, at least on the surface.
However, there is no reason to think that life could not have begun on Mars about 4 billion years ago, at the same time it started on Earth. The two planets had very similar surface conditions then. Thus, the attention of scientists has shifted to the search for fossil life on Mars. One of the primary questions to be addressed by future spacecraft is whether Mars once supported its own life forms and, if so, how this martian life compared with that on our own planet. Future missions will include the return of martian samples selected from sedimentary rocks at sites that once held water and thus perhaps ancient life. The most powerful searches for martian life (past or present) will thus be carried out in our laboratories here on Earth.
planetary protection
When scientists begin to search for life on another planet, they must make sure that we do not contaminate the other world with life carried from Earth. At the very beginning of spacecraft exploration on Mars, an international agreement specified that all landers were to be carefully sterilized to avoid accidentally transplanting terrestrial microbes to Mars. In the case of Viking, we know the sterilization was successful. Viking’s failure to detect martian organisms also implies that these experiments did not detect hitchhiking terrestrial microbes.
As we have learned more about the harsh conditions on the martian surface, the sterilization requirements have been somewhat relaxed. It is evident that no terrestrial microbes could grow on the martian surface, with its low temperature, absence of water, and intense ultraviolet radiation. Microbes from Earth might survive in a dormant, dried state, but they cannot grow and proliferate on Mars.
The problem of contaminating Mars will become more serious, however, as we begin to search for life below the surface, where temperatures are higher and no ultraviolet light penetrates. The situation will be even more daunting if we consider human flights to Mars. Any humans will carry with them a multitude of terrestrial microbes of all kinds, and it is hard to imagine how we can effectively keep the two biospheres isolated from each other if Mars has indigenous life. Perhaps the best situation could be one in which the two life-forms are so different that each is effectively invisible to the other—not recognized on a chemical level as living or as potential food.
The most immediate issue of public concern is not with the contamination of Mars but with any dangers associated with returning Mars samples to Earth. NASA is committed to the complete biological isolation of returned samples until they are demonstrated to be safe. Even though the chances of contamination are extremely low, it is better to be safe than sorry.
Most likely there is no danger, even if there is life on Mars and alien microbes hitch a ride to Earth inside some of the returned samples. In fact, Mars is sending samples to Earth all the time in the form of the Mars meteorites. Since some of these microbes (if they exist) could probably survive the trip to Earth inside their rocky home, we may have been exposed many times over to martian microbes. Either they do not interact with our terrestrial life, or in effect our planet has already been inoculated against such alien bugs.
More than any other planet, Mars has inspired science fiction writers over the years. You can find scientifically reasonable stories about Mars in a subject index of such stories online. If you click on Mars as a topic, you will find stories by a number of space scientists, including William Hartmann, Geoffrey Landis, and Ludek Pesek.
Key Concepts and Summary
The martian atmosphere has a surface pressure of less than 0.01 bar and is 95% CO 2 . It has dust clouds, water clouds, and carbon dioxide (dry ice) clouds. Liquid water on the surface is not possible today, but there is subsurface permafrost at high latitudes. Seasonal polar caps are made of dry ice; the northern residual cap is water ice, whereas the southern permanent ice cap is made predominantly of water ice with a covering of carbon dioxide ice. Evidence of a very different climate in the past is found in water erosion features: both runoff channels and outflow channels, the latter carved by catastrophic floods. Our rovers, exploring ancient lakebeds and places where sedimentary rock has formed, have found evidence for extensive surface water in the past. Even more exciting are the gullies that seem to show the presence of flowing salty water on the surface today, hinting at near-surface aquifers. The Viking landers searched for martian life in 1976, with negative results, but life might have flourished long ago. We have found evidence of water on Mars, but following the water has not yet led us to life on that planet.


4.2: Preliminary Research Strategies
Evaluate preliminary research strategies
As we have discussed, all research is based upon your research question. Having a well-defined and scoped question is essential to a good research strategy. If your question is not specific enough, or if it lacks boundaries (i.e., it is not well-scoped), your subsequent strategy will be difficult to maintain.
Steely Library discusses developing a good research question in the video below:
Let’s return to Martha’s case. We can recall that her research question was,
“Is The Human Fund’s work helping homeless families in downtown Chicago?”
If we first break her question down into its sub-parts, developing a research strategy will be much easier. Her question asks,
- Is The Human Fund’s work — i.e., what The Human Fund does — its actions
- helping — i.e., we must define “helping” in relation to…
- …the homeless families…
- …in downtown Chicago?
From her question, we know that we will need sources that,
- Outline The Human Fund’s activities
- Define how charities and government help the homeless in their cities
- Help to define and understand “homeless”
- Are geographically bound to downtown Chicago
With the above in mind, any secondary source that does not specifically address a part of the question above—and how it is broken down—will be off topic or out of scope.
We will also recall that Martha conducted background reading (i.e., secondary source reading) before determining the type of primary source material (i.e., fieldwork and interviews with the homeless) she would use. This can be confusing; when we research, we do background or secondary source reading before determining what primary source material might still be needed. You will not typically see a research process that advocates doing primary source research when there is already secondary source material available on a given topic because it is not efficient. It is also important to note that if secondary source material sufficiently addresses your research question, consider this to be a win; this means that the much slower and much more elaborate primary source research process is no longer required. Your report will be that much faster to compile. If Martha, for example, had recent accounts of interviews with homeless people in downtown Chicago about The Human Fund’s work, she would not need to conduct her own interviews.Contributors and Attributions


16: Stream Function, Vorticity Equations
Stream function
A streamline at time \(t\) is defined as the curve whose tangent is everywhere parallel to the velocity vector. With \(d \mathbf{x}\) along the tangent, we have
\[\nonumber \mathbf{u} \times d \mathbf{x}=0 \nonumber \]
and with \(\mathbf{u}=(u, v, w)\) and \(d \mathbf{x}=(d x, d y, d z)\) , the cross product yields the three equations
\[v d z=w d y, \quad u d z=w d x, \quad u d y=v d x \nonumber \]
or equivalently,
\[\frac{d x}{u}=\frac{d y}{v}=\frac{d z}{w} \nonumber \]
Streamlines have the following two properties. They cannot intersect except at a point of zero velocity, and as streamlines converge the fluid speed increases. The latter is a consequence of the incompressibility of the fluid: as the same flow rate of fluid passes through a smaller cross-sectional area, the fluid velocity must increase.
Related to streamlines is the stream function. We specialize here to a two dimensional flow, with
\[\mathbf{u}=(u(x, y), v(x, y), 0) \nonumber \]
The incompressibility condition becomes
\[\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0 \nonumber \]
which can be satisfied by defining the scalar stream function \(\psi=\psi(x, y)\) by
\[u(x, y)=\frac{\partial \psi}{\partial y}, \quad v(x, y)=-\frac{\partial \psi}{\partial x} \nonumber \]
Now, the differential of the stream function \(\psi(x, y)\) satisfies
\[\begin{aligned} d \psi &=\frac{\partial \psi}{\partial x} d x+\frac{\partial \psi}{\partial y} d y \\ &=-v d x+u d y \end{aligned} \nonumber \]
which from (16.1) is equal to zero along streamlines. Thus the contour curves of constant \(\psi\) represent the streamlines of the flow field, and can provide a good visualization of a fluid flow in two dimensions.
Vorticity
The vector vorticity field is defined from the vector velocity field by
\[\omega=\nabla \times \mathbf{u} \nonumber \]
The vorticity is a measure of the local rotation of the fluid as can be seen from an application of Stokes’ theorem:
\[\begin{aligned} \int_{S} \omega \cdot \hat{\mathbf{n}} d S &=\int_{S}(\boldsymbol{\nabla} \times \mathbf{u}) \cdot \hat{\mathbf{n}} d S \\ &=\oint_{C} \mathbf{u} \cdot d \mathbf{r} . \end{aligned} \nonumber \]
Flows without vorticity are called irrotational, or potential flow, and vorticity is sometimes called swirl.
The governing equation for vorticity may be found by taking the curl of the Navier-Stokes equation; that is,
\[\boldsymbol{\nabla} \times\left\{\frac{\partial \mathbf{u}}{\partial t}+(\mathbf{u} \cdot \boldsymbol{\nabla}) \mathbf{u}\right\}=\boldsymbol{\nabla} \times\left\{-\frac{1}{\rho} \boldsymbol{\nabla} p+v \nabla^{2} \mathbf{u}\right\} \nonumber \]
Computing term-by-term, we have
\[\begin{aligned} \nabla \times\left\{\frac{\partial \mathbf{u}}{\partial t}\right\} &=\frac{\partial}{\partial t}(\nabla \times \mathbf{u}) \\ &=\frac{\partial \omega}{\partial t} \end{aligned} \nonumber \]
And because the curl of a gradient is zero,
\[\boldsymbol{\nabla} \times\left\{-\frac{1}{\rho} \boldsymbol{\nabla} p\right\}=0 . \nonumber \]
Also,
\[\begin{aligned} \boldsymbol{\nabla} \times\left\{v \nabla^{2} \mathbf{u}\right\} &=v \nabla^{2}(\boldsymbol{\nabla} \times \mathbf{u}) \\ &=v \nabla^{2} \omega \end{aligned} \nonumber \]
The remaining term to compute is the curl of the convection term in the Navier-Stokes equation. We first consider the following equality (where the subscript \(i\) signifies the \(i\) -th component of the vector):
\[\begin{aligned} \{\mathbf{u} \times(\boldsymbol{\nabla} \times \mathbf{u})\}_{i} &=\epsilon_{i j k} u_{j} \epsilon_{k l m} \frac{\partial u_{m}}{\partial x_{l}} \\ &=\epsilon_{k i j} \epsilon_{k l m} u_{j} \frac{\partial u_{m}}{\partial x_{l}} \\ &=\left(\delta_{i l} \delta_{j m}-\delta_{i m} \delta_{j l}\right) u_{j} \frac{\partial u_{m}}{\partial x_{l}} \\ &=u_{m} \frac{\partial u_{m}}{\partial x_{i}}-u_{l} \frac{\partial u_{i}}{\partial x_{l}} \\ &=\frac{1}{2} \frac{\partial}{\partial x_{i}} u_{m} u_{m}-u_{l} \frac{\partial u_{i}}{\partial x_{l}} \end{aligned} \nonumber \]
Therefore, in vector form,
\[\mathbf{u} \times(\boldsymbol{\nabla} \times \mathbf{u})=\frac{1}{2} \boldsymbol{\nabla}\left(\mathbf{u}^{2}\right)-(\mathbf{u} \cdot \boldsymbol{\nabla}) \mathbf{u} \nonumber \]
This identity allows us to write
\[(\mathbf{u} \cdot \boldsymbol{\nabla}) \mathbf{u}=\frac{1}{2} \boldsymbol{\nabla}\left(\mathbf{u}^{2}\right)-\mathbf{u} \times(\boldsymbol{\nabla} \times \mathbf{u}) \nonumber \]
Taking the curl of both sides and making use of the curl of a gradient equals zero and \(\boldsymbol{\nabla} \times \mathbf{u}=\boldsymbol{\omega}\) , results in
\[\begin{aligned} \boldsymbol{\nabla} \times\{(\mathbf{u} \cdot \boldsymbol{\nabla}) \mathbf{u}\} &=-\boldsymbol{\nabla} \times(\mathbf{u} \times \omega) \\ &=\boldsymbol{\nabla} \times(\boldsymbol{\omega} \times \mathbf{u}) \end{aligned} \nonumber \]
Combining all the above terms, we have thus obtained the vorticity equation
\[\frac{\partial \omega}{\partial t}+\nabla \times(\omega \times \mathbf{u})=v \nabla^{2} \omega \nonumber \]
An alternative form of the vorticity equation rewrites the convection term to explicitly include the substantive derivative. We have
\[\begin{aligned} \{\boldsymbol{\nabla} \times(\omega \times \mathbf{u})\}_{i} &=\epsilon_{i j k} \frac{\partial}{\partial x_{j}} \epsilon_{k l m} \omega_{l} u_{m} \\ &=\epsilon_{k i j} \epsilon_{k l m} \frac{\partial}{\partial x_{j}}\left(\omega_{l} u_{m}\right) \\ &=\left(\delta_{i l} \delta_{j m}-\delta_{i m} \delta_{j l}\right) \frac{\partial}{\partial x_{j}}\left(\omega_{l} u_{m}\right) \\ &=\frac{\partial}{\partial x_{m}}\left(\omega_{i} u_{m}\right)-\frac{\partial}{\partial x_{l}}\left(\omega_{l} u_{i}\right) \\ &=u_{m} \frac{\partial w_{i}}{\partial x_{m}}-\omega_{l} \frac{\partial u_{i}}{\partial x_{l}} \end{aligned} \nonumber \]
where to obtain the last equality we have used both \(\partial u_{m} / \partial x_{m}=0\) and \(\partial \omega_{l} / \partial x_{l}=0 .\) Therefore, in vector form,
\[\nabla \times(\omega \times \mathbf{u})=(\mathbf{u} \cdot \nabla) \omega-(\omega \cdot \nabla) \mathbf{u} \nonumber \]
The vorticity equation can then be rewritten as
\[\frac{\partial \omega}{\partial t}+(\mathbf{u} \cdot \nabla) \omega=(\omega \cdot \nabla) \mathbf{u}+v \nabla^{2} \omega \nonumber \]
Compared to the Navier-Stokes equation, there is an extra term, called the vortex stretching term, on the right-hand-side of (16.13).
Two-dimensional Navier-Stokes equation
We have already seen that in two dimensions, the incompressibility condition is automatically satisfied by defining the stream function \(\psi(\mathbf{x}, t)\) . Also in two dimensions, the vorticity can be reduced to a scalar field. With \(\mathbf{u}=(u(x, y), v(x, y))\) , we have
\[\begin{aligned} \omega &=\nabla \times \mathbf{u} \\ &=\left|\begin{array}{ccc} \hat{\mathbf{x}} & \hat{\mathbf{y}} & \hat{\mathbf{z}} \\ \partial / \partial x & \partial / \partial y & \partial / \partial z \\ u(x, y) & v(x, y) & 0 \end{array}\right| \\ &=\hat{\mathbf{z}}\left(\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y}\right) \\ &=\omega(x, y) \hat{\mathbf{z}}, \end{aligned} \nonumber \]
where we have now defined the scalar field \(\omega\) to be the third component of the vector vorticity field. Making use of the stream function, we then have
\[\begin{aligned} \omega &=\frac{\partial v}{\partial x}-\frac{\partial u}{\partial y} \\ &=-\frac{\partial^{2} \psi}{\partial x^{2}}-\frac{\partial^{2} \psi}{\partial y^{2}} \end{aligned} \nonumber \]
Therefore, in vector form, we have
\[\nabla^{2} \psi=-\omega \nonumber \]
where
\[\nabla^{2}=\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}} \nonumber \]
is the two-dimensional Laplacian.
Now, with \(\omega=\omega(x, y) \hat{\mathbf{z}}\) , the third component of the vorticity equation (16.13) becomes
\[\frac{\partial \omega}{\partial t}+(\mathbf{u} \cdot \nabla) \omega=v \nabla^{2} \omega \nonumber \]
where the vortex stretching term can be seen to vanish. We can also write
\[\begin{aligned} \mathbf{u} \cdot \nabla &=u \frac{\partial}{\partial x}+v \frac{\partial}{\partial y} \\ &=\frac{\partial \psi}{\partial y} \frac{\partial}{\partial x}-\frac{\partial \psi}{\partial x} \frac{\partial}{\partial y} \end{aligned} \nonumber \]
The vorticity equation in two dimensions then becomes
\[\frac{\partial \omega}{\partial t}+\left(\frac{\partial \psi}{\partial y} \frac{\partial \omega}{\partial x}-\frac{\partial \psi}{\partial x} \frac{\partial \omega}{\partial y}\right)=v \nabla^{2} \omega \nonumber \]
For a stationary flow, this equation becomes the Poisson equation,
\[\nabla^{2} \omega=\frac{1}{v}\left(\frac{\partial \psi}{\partial y} \frac{\partial \omega}{\partial x}-\frac{\partial \psi}{\partial x} \frac{\partial \omega}{\partial y}\right) \nonumber \]
We have thus obtained for a stationary flow two coupled Poisson equations for \(\psi(x, y)\) and \(\omega(x, y)\) given by \((16.14)\) and \((16.18)\) .


3.1.17: Langston Hughes (1902-1967)
Selected Poems American Modernism / Harlem Renaissance A leading figure of the Harlem Renaissance in the United States, Langston Hughes developed an international reputation for his poetry. Hughes spent his childhood in the Midwest; he was born in Joplin, Missouri, but he also lived in Lincoln, Illinois and Cleveland, Ohio. As a young man, he began a college education at Columbia University, but withdrew to travel as a merchant seaman. He eventually completed his education at Lincoln University. Hughes is particularly known for his perceptive portrayals of black life in America from the twenties through the sixties. He wrote prolifically and in a variety of genres--poems, plays, short stories, and novels. A significant feature of his work is the influence of jazz on his poetry, particularly in Montage of a Dream Deferred (Holt, 1951). Hughes also mentored other young poets and writers like Ralph Ellison. In 1926, he articulated the purpose of young black writers and poets in "The Negro Artist and the Racial Mountain": "The younger Negro artists who create now intend to express our individual dark-skinned selves without fear or shame. If white people are pleased we are glad. If they are not, it doesn't matter. We know we are beautiful . . . If colored people are pleased we are glad. If they are not, their displeasure doesn't matter either. We build our temples for tomorrow, strong as we know how, and we stand on top of the mountain free within ourselves." Donald B. Gibson noted in the introduction to Modern Black Poets: A Collection of Critical Essays (Prentice Hall, 1973) that Hughes "differed from most of his predecessors among black poets ... in that he addressed his poetry to the people, specifically to black people." Hughes considered himself to be, indeed, a "people's poet" who elevated the black aesthetic while confronting racism and stereotypes in his work. Consider while reading:
- In the poem "The Weary Blues," what connection does Hughes suggest about the relationship between blues music and the experience of African Americans?
- In the poem "The Negro Speaks of Rivers," which four rivers are named? In what way is each of these rivers significant?
- What connections can you draw between the experiences of the speaker in "Theme for English B" and the young Olaudah Equiano?
Written by Anita Turlington