Notes & Problems for GEOL 1520:

Ocean Circulation and Climate

Baylor Fox-KemperBrown University

Dept. of Earth, Environmental, and Planetary Sciences

January 30, 2015

Page ii, Section 0.0, January 30, 2015 Version

Contents

Syllabus vii0.1 Course Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii0.2 Contacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii0.3 Getting Help! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii0.4 Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii0.5 Meetings and Places . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

0.5.1 Calendar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix0.6 Canvas and Websites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix0.7 Textbooks and Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix0.8 Assignments and (lack of) Exams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

0.8.1 Why papers? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi0.8.2 Peer review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi0.8.3 What will we do with all these peer-reviewed papers? . . . . . . . . . . . . . . . . . . xi

0.9 Policies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi0.9.1 Deadlines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi0.9.2 Collaboration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

1 Dimensions 11.1 Buckingham Pi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1 Series 31.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.1 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.2 Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1.3 Small & Big . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1 Series 51.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.1.1 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.1.2 Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.1.3 Small & Big . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1 Series 71.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.1.1 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.1.2 Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.1.3 Small & Big . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1 Series 9

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CONTENTS CONTENTS

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.1.1 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.1.2 Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.1.3 Small & Big . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1 Series 111.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.1.1 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.1.2 Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.1.3 Small & Big . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1 Series 131.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.1.1 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.1.2 Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.1.3 Small & Big . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1 Series 151.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.1.1 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.1.2 Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.1.3 Small & Big . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1 Series 171.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.1.1 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.1.2 Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.1.3 Small & Big . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1 Series 191.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.1.1 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.1.2 Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.1.3 Small & Big . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1 Series 211.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.1.1 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.1.2 Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.1.3 Small & Big . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1 Series 231.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.1.1 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.1.2 Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.1.3 Small & Big . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

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CONTENTS CONTENTS

1 Series 251.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

1.1.1 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.1.2 Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.1.3 Small & Big . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

1 Series 271.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.1.1 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271.1.2 Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271.1.3 Small & Big . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1 Reference 291.1 Dimensionless Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

1.1.1 Vector and Vector Calculus Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291.1.2 Calculus Theorems (Green’s Theorems) . . . . . . . . . . . . . . . . . . . . . . . . . . 30

1.2 Tensor Identities and Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301.2.1 Cartesian Tensor Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

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Page vi, Section 0.0, January 30, 2015 Version

Syllabus

0.1 Course Description

Examines physical characteristics, processes, and dynamics of the global ocean to understand circulationpatterns and how they relate to ocean biology, chemistry, and climate change. Assignments address ocean’srole in the climate system; ocean observations and models; the origin, distribution, and dynamics of large-scale ocean circulation and water masses; energy and freshwater budgets; and variability of the coupledsystem on seasonal to centennial timescales e.g. El Nio. Intended for geological and physical sciencesundergraduate and graduate students with quantitative skills and an interest in oceans, climate, paleoclimate.Pre-requisite: GEOL0350 or PHYS0720 or APMA 0340. Offered alternate years, previously offered asGEOL1100.

0.2 Contacts

The professor for this class is: Baylor Fox-Kemperbaylor@brown.edu401-863-3979Office: GeoChem room 133http://fox-kemper.com/teaching, http://fox-kemper.com/po

Portions of the website are password-protected to ensure that fair use and copyrights are correctly obeyedas I share images from books, etc. You can access these by using:

username: io

password: ocean

0.3 Getting Help!

I am usually available by email. You can make an appointment other times. Just check my calendar athttp://fox-kemper.com/contact and suggest a time that works for you.

0.4 Goals

In this class you will:

• Learn about many of the physical processes that occur in the ocean.• Learn about how these physical processes are observed, budgeted, and quantified.• Learn about where these processes occur in the ocean.• Learn about and access recent ocean datasets.

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0.5. MEETINGS AND PLACES CHAPTER 0. SYLLABUS

• Get practice writing and thinking scientifically by focused study on particular processes.• You will also get a broader perspective and more practice by peer reviewing your colleagues’ efforts.• You will benefit from the reviews of your writing by your peers.

A list of the basic topics covered in this class is:

• Observations (1 Week)What instruments are common to oceanography?How are these measurements used?How do the types of measurements influence the theoretical developments?

• Fluids Mechanics (2 Weeks)The differential equations describing a fluidThe differential equations describing a Boussinesq fluidHow the differential equations relate to budgets & conservation of:

salt, freshwater, energy, entropy/potential temperature, vorticity, potential vorticityScale and Dimensional Analysis

How the differential equations may be reduced to a dominant balance for particular phenomenaCommon non-dimensional groupings may be used to identify which dominant balance is likely

• Rotation (2 Weeks)The equations for fluid motion in a rotating frame of referenceEkman layers and balanceGeostrophic balance

• Stratification (2 Weeks)The temperature, salinity, and density stratification of the oceanHydrostasyBaroclinic and BarotropicPotential temperature and potential densityThermal wind balance

• Vorticity balances (1 Week)Taylor-Proudman FlowSverdrup FlowThermal wind as vorticity budget

• Ocean Circulation (2.5 Weeks)The Wind-Driven Ocean CirculationWestern Boundary CurrentsAntarctic Circumpolar CurrentThe Meridional Overturning CirculationTheories of the ThermoclineThe pressure distribution of the ocean

• Forcing (1 Week)Air-sea exchanges of sensible & latent heat, momentum, energyRadiation forcing (shortwave=solar and longwave)

• Wave Basics (2 Weeks)Dispersive Wave Kinematics: Phase & Group velocityImportant kinds of ocean waves: surface gravity, internal gravity, Rossby, KelvinNonlinear waves, eddies, vortices, and coherent structuresEl Nino

0.5 Meetings and Places

We will meet Tuesdays and Thursdays from 1 to 2:20PM in GeoChem 039. Office hours will be by appoint-ment (see my schedule at http://fox-kemper.com/contact).

The student computer lab in GeoChem should have the software we’ll be using (primarily matlab) loaded.You can download a copy of matlab for yourself from http://software.brown.edu/dist/index.html for

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CHAPTER 0. SYLLABUS 0.6. CANVAS AND WEBSITES

free. The website has many links to datasets that are freely available. I will provide local access to some ofthe most convenient ones at http://fox-kemper.com/data.

0.5.1 Calendar

The main webpage for the class http://fox-kemper.com/1520 will have the calendar with all assignmentdeadlines, readings, etc., set up by the first class session. The three big paper assignments will be due nearthe last day of February, March, and April, respectively.

0.6 Canvas and Websites

The primary resource for this class is the webpage: http://fox-kemper.com/1520. The class webpage iswhere all of your assignments will be announced, links to reading and textbooks will be posted, etc. Thesecond web resource is the canvas page for the class. All homeworks, papers, and peer reviews will be turnedin through http://canvas.brown.edu.

You will want to familiarize yourself with Google Scholar (http://scholar.google.com) and the Web ofScience (http://apps.webofknowledge.com/). Both are free to you, and they will help you with your paperpreparations.

0.7 Textbooks and Software

We will use three primary textbooks: Ocean Circulation and Climate (?), Ocean Circulation (?), andAtmospheric and Oceanic Fluid Dynamics (?). Electronic versions of all three can be accessed electronicallythrough the Brown site/vpn access at http://www.sciencedirect.com/science/bookseries/00746142/103 and http://site.ebrary.com/lib/brown/docDetail.action?docID=10190755 and http://bit.ly/SDSMSK (you must be on campus or hooked in with VPN or the library’s EZproxy to access). You can alsofind links to them on Josiah. More mathematical detail will be found in ?, which is a required textbook forthis and other climate courses in the department. Two good online textbooks are (??), which are linkedfrom the course webpage.

All of the required readings will be posted on the website. Hard copies of many of these books, as well assome others you should become familiar with are in the Science Library. I recommend browsing a bit andseeing what there is (Library of Congress Call Numbers GC11, GC150, GC228, and QC809 are good placesto start).

The class will require use of software capable of opening ocean datasets. I strongly recommend matlab, butyou could use Java Ocean Atlas, IDL, Ocean Data View, python, or ferret (although I will be less able tooffer help!). All of these are available to you free for PC, Mac, and Linux.

0.8 Assignments and (lack of) Exams

There will be four major assignments for this class, and all of them will be in the form of short scientificreports. You will often be working on two assignments at a time, reviewing and revising the last one (a littlework), and preparing the next one (more work). There will also be short homework assignments based onlectures and reading. There will be no additional exams or midterms. The final revision of the final paperis presently scheduled to be turned in at the end of exam week, although I have tried to schedule sufficienttime the preceding week so that it can be completed before exams. The weighting of the assignments willbe:

• 15% Reading/Paper Prep HWs.

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0.8. ASSIGNMENTS AND (LACK OF) EXAMS CHAPTER 0. SYLLABUS

• 15% Plans for papers.• 40% Original submission of papers.• 15% Revised submission of papers.• 15% Reviews of other students’ papers.

What can help me get a good grade? Turn all of the assignments in on time. This is more important thanturning in complete assignments. For the peer-reviewing format of the course to work, ON TIME matters,so that I can give timely feedback on paper plans, and so we can get to the reviewing.

Assignments are to be in pdf file format and created according to American Geophysical Union GeophysicalResearch Letters templates. If you want to use LaTeX, I can help. If you want to use something else, youcan download the AGU templates, but you’re on your own if things go wrong!

The scheduling of the assignments are listed on the webpage, and other than the exceptional week aroundfall break will be as follows.

• HW assignment #n will be due on a Saturday at 11:59 PM two weeks before the paper. They will besubmitted electronically, and they establish basic tools for the upcoming paper.

• Plans #n will be due on a Wednesday at 11:59 PM one and a half weeks before the paper. They willbe submitted electronically, and I will comment on them ASAP. I encourage you to have generatedsome results and looked for some references. I will respond by email with suggestions for the followingweek.

• Paper assignment #n will be due on a Saturday at 11:59 PM. They will be submitted electronically.• By Sunday at noon, you will receive two of your colleagues’ #n papers to review electronically.• On Tuesday, we will have class and office hours. You can discuss anything with me or the class about

your paper or the papers you are reviewing (#n) or paper #n+1 that you have already begun thinkingabout.

• On Wednesday by 5 PM, your reviews of your colleagues’ two papers (#n) are done and submittedelectronically.

• By Friday midnight, I will have your paper #n back to you, with my reviews and your peer reviews.Discussion during office hours is encouraged of returned paper #n

• The following Wednesday at 5 PM, a revised version of your paper is due.• That Saturday, the next HW #n+ 1 is due.• The process restarts.

All of this will be charted out on the calendar on the website and in canvas.

Before you get worried about writing five papers for one class, let me explain the goals of the paper writing.These are not supposed to be polished, ready to submit papers detailing years of research. Instead, they aresupposed to be practice in writing drafts for your real research. The idea is to get used to pounding out aworking draft in only a couple of hours, so that when the time comes for you to do it for real, that part willbe easy.

Since you all have different preparation, you will all be able to take advantage of what you know. However,we are working to develop elements in all of the following:

• Quantitative Skills and Equations for the Ocean (Theory/Modeling Component)• Descriptive Skills and Geography of Ocean Currents (Descriptive Oceanography)• Understanding of Ocean Observations and Techniques (Observations/Engineering Component)• Physical Intuition and Dynamical Understanding (Theory/Dynamics Component)• Implications for Climate, Society, etc. (Policy & Climate Component)

By the end of the semester, you should have addressed each element somewhat in some of your papers.

I hope you will get inspired along the way, and you can revise one of these little papers into a real paper ora prelims or honors project, but don’t get upset because you have to hit the ground running. You’ll have todo it sometime, and this will be in a friendly environment!

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CHAPTER 0. SYLLABUS 0.9. POLICIES

0.8.1 Why papers?

I think that this approach helps to balance the class. Folks with a lot of quantitative experience will be ableto use that to their advantage while folks with more substantial preparation in writing, argument, and logicalstructure will be able to use those abilities. We will be using the most up-to-date oceanographic datasetsavailable, and so the work you are doing is potentially cutting edge research (but that’s up to you!). Also, inthe future you will be expected to write much more complicated papers in a more tightly constrained timeframe, you might as well get some practice now.

0.8.2 Peer review

In addition to writing the papers, you will each be performing anonymous reviews of each others work. Thiswill give you an opportunity to read closely about topics other than the one you chose, and hopefully youwill be able to learn about science writing more quickly. Also, there are a lot of quandaries that arise inpeer-reviewing (e.g., one reviewer loves it and one hates it, or a reviewer makes incorrect statements), soyou’ll get some experience with those issues by practice in a friendly environment.

We will be using a rubric based on the AGU guidelines for review. They are a useful guideline to go by, andwhen you do reviews of your fellow students, I’ll expect to get a A1 or B2 or B1 score, etc.

There are a few lessons to be learned here, that will help you write your own papers and will help you provideeffective and useful reviews in your career.

• Learning to spot unfounded claims• Learning how to properly support claims• Learning to distinguish poor writing from poor thinking• Learning to label equations, graphs, and numerical information understandably• Learning about a broader swath of oceanography than those isolated topics you choose for your own

papers.

0.8.3 What will we do with all these peer-reviewed papers?

As the semester continues, there will be some papers of yours that you like quite a bit, or that were verypositively reviewed. Either you or me can designate them as ’publishable’, so that the whole class can seethem and subsequent classes can, too. At the end of the semester, all of you will be able to access theassembled published papers: ’Proceedings of GEOL1520: Notions for the Motions of the Oceans, Spring2013’. You can use this proceedings for your future reference, and it will be available for future GEOL1520inductees. It is up to you which papers you want published (if any).

You can access the previous year’s proceedings through the course website with the password I’ve emailedto you. You will find many good ideas and interesting papers there to help you find your way in writingyour own.

0.9 Policies

0.9.1 Deadlines

Because of the reviewing process, the scheduling of assignments is tight. Thus, I will have to insist thatall papers be turned in on time. If they are late, they will drop a letter grade. If they are really late (sothat they mess up the next step in the reviewing process) they will be counted as missed and can not bemade up. If you foresee that there are big problems coming up (medical, family, etc.) let me know before anassignment is due and we can figure something out.

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0.9. POLICIES CHAPTER 0. SYLLABUS

0.9.2 Collaboration

I encourage you to work together, and I do not mind at all if you have similar papers or share figures ormatlab scripts. However, in this case, I want you to list all of your study group as co-authors or put themin the acknowledgments section of your paper. You are all required to submit a version of each assignmentas first author (that is, one that you wrote yourself). You need to be careful to cite your colleagues or thetextbooks or papers you might be working from. You can use as much of these resources as is convenient inyour version of the paper, but you need to properly cite the sources. We will discuss this topic more as theclass (and the inevitable trouble) ensues. These issues of plagiarism and proper sourcing are a big part ofwhat is to be learned in this method of assignments.

A few other items.

• Attendance is expected. If you will miss a class, please let me know when and why so I can be sureyou’ll get any announcements, etc.

• Clothing and behavior (e.g., cell phone use) should be appropriate for a learning environment.• Discrimination and harassment will not be tolerated.• Please contact me if you have any disabilities that require accommodation.

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Chapter 1

Dimensional Analysis

Introduction

1.1 Buckingham Pi

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1.1. BUCKINGHAM PI CHAPTER 1. DIMENSIONS

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Chapter 1

Series Expansions

1.1 Introduction

1.1.1 Convergence

1.1.2 Accuracy

1.1.3 Small & Big

1.2

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1.2. CHAPTER 1. SERIES

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Chapter 1

Series Expansions

1.1 Introduction

1.1.1 Convergence

1.1.2 Accuracy

1.1.3 Small & Big

1.2

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1.2. CHAPTER 1. SERIES

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Chapter 1

Series Expansions

1.1 Introduction

1.1.1 Convergence

1.1.2 Accuracy

1.1.3 Small & Big

1.2

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1.2. CHAPTER 1. SERIES

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Chapter 1

Series Expansions

1.1 Introduction

1.1.1 Convergence

1.1.2 Accuracy

1.1.3 Small & Big

1.2

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1.2. CHAPTER 1. SERIES

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Chapter 1

Series Expansions

1.1 Introduction

1.1.1 Convergence

1.1.2 Accuracy

1.1.3 Small & Big

1.2

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1.2. CHAPTER 1. SERIES

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Chapter 1

Series Expansions

1.1 Introduction

1.1.1 Convergence

1.1.2 Accuracy

1.1.3 Small & Big

1.2

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1.2. CHAPTER 1. SERIES

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Chapter 1

Series Expansions

1.1 Introduction

1.1.1 Convergence

1.1.2 Accuracy

1.1.3 Small & Big

1.2

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1.2. CHAPTER 1. SERIES

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Chapter 1

Series Expansions

1.1 Introduction

1.1.1 Convergence

1.1.2 Accuracy

1.1.3 Small & Big

1.2

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1.2. CHAPTER 1. SERIES

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Chapter 1

Series Expansions

1.1 Introduction

1.1.1 Convergence

1.1.2 Accuracy

1.1.3 Small & Big

1.2

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1.2. CHAPTER 1. SERIES

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Chapter 1

Series Expansions

1.1 Introduction

1.1.1 Convergence

1.1.2 Accuracy

1.1.3 Small & Big

1.2

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1.2. CHAPTER 1. SERIES

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Chapter 1

Series Expansions

1.1 Introduction

1.1.1 Convergence

1.1.2 Accuracy

1.1.3 Small & Big

1.2

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1.2. CHAPTER 1. SERIES

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Chapter 1

Series Expansions

1.1 Introduction

1.1.1 Convergence

1.1.2 Accuracy

1.1.3 Small & Big

1.2

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1.2. CHAPTER 1. SERIES

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Chapter 1

Series Expansions

1.1 Introduction

1.1.1 Convergence

1.1.2 Accuracy

1.1.3 Small & Big

1.2

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1.2. CHAPTER 1. SERIES

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Chapter 1

Reference

1.1 Dimensionless Numbers

1.1.1 Vector and Vector Calculus Rules

In vector notation, the Cartesian Tensor rules are equivalent to a list of lemmas. They are:

Triple Products

~A · ( ~B × ~C) = ~B · (~C × ~A) = ~C · ( ~A× ~B) (1.1)~A× ( ~B × ~C) = ~B( ~A · ~C)− ~C( ~A · ~B) (1.2)

Product Rules

∇(fg) = f(∇g) + g(∇f) (1.3)∇( ~A · ~B) = ~A× (∇× ~B) + ~B × (∇× ~A) + ( ~A · ∇) ~B + ( ~B · ∇) ~A (1.4)∇ · (f ~A) = f(∇ · ~A) + ~A · (∇f) (1.5)

∇ · ( ~A× ~B) = ~B · (∇× ~A)−A · (∇× ~B) (1.6)∇× (f ~A) = f(∇× ~A)− ~A× (∇f) (1.7)

∇× ( ~A× ~B) = ( ~B · ∇) ~A− ( ~A · ∇) ~B + ~A(∇ · ~B)− ~B(∇ · ~A) (1.8)

Second Derivatives

∇ · (∇× ~A) = 0 (1.9)∇× (∇f) = 0 (1.10)

∇× (∇× ~A) = ∇(∇ · ~A)−∇2 ~A (1.11)(1.12)

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1.2. TENSOR IDENTITIES AND CALCULUS CHAPTER 1. REFERENCE

1.1.2 Calculus Theorems (Green’s Theorems)

Gradient Theorem (Fundamental Theorem of Calculus)

ˆ ba

(∇f) · d~l = f |b − f|a (1.13)

Divergence Theorem (Gauss’s Theorem)

ˆˆˆ

vol

(∇ · ~A)dV =˛˛

surf

( ~A · n̂)dA (1.14)

Curl Theorem (Stoke’s Theorem)

ˆˆ

surf

(∇× ~A)dA =˛cont

~A · d~l (1.15)

1.2 Tensor Identities and Calculus

In this section, I reference some of the basic tensor identities, and prove them using Cartesian tensors. I alsorecord some standard calculus identities in a tensor-invariant form.

1.2.1 Cartesian Tensor Rules

There are only three important rules in using Cartesian tensors: the Epsilon-Delta rule, the swapping ofdummy indices, and the commutation of partial derivatives.

Epsilon-Delta Rule

The following identity is extremely useful.

�ijk�klm = δilδjm − δimδjl (1.16)

Swapping of Dummy Indices

The names of indices are irrelevant, therefore, if an valid equation is identical to another equation if thenames of the indices are changed, the second equation is also valid. For example, xixi = xjxj .

Commutation of Partial Derivatives

In Cartesian tensors, all of the spatial derivatives commute, so indices may be freely traded among nestedderivatives. For example, ∂i∂j = ∂j∂i.

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SyllabusCourse DescriptionContactsGetting Help!GoalsMeetings and PlacesCalendar

Canvas and WebsitesTextbooks and SoftwareAssignments and (lack of) ExamsWhy papers?Peer reviewWhat will we do with all these peer-reviewed papers?

PoliciesDeadlinesCollaboration

DimensionsBuckingham Pi

SeriesIntroductionConvergenceAccuracySmall & Big

SeriesIntroductionConvergenceAccuracySmall & Big

SeriesIntroductionConvergenceAccuracySmall & Big

SeriesIntroductionConvergenceAccuracySmall & Big

SeriesIntroductionConvergenceAccuracySmall & Big

SeriesIntroductionConvergenceAccuracySmall & Big

SeriesIntroductionConvergenceAccuracySmall & Big

SeriesIntroductionConvergenceAccuracySmall & Big

SeriesIntroductionConvergenceAccuracySmall & Big

SeriesIntroductionConvergenceAccuracySmall & Big

SeriesIntroductionConvergenceAccuracySmall & Big

SeriesIntroductionConvergenceAccuracySmall & Big

SeriesIntroductionConvergenceAccuracySmall & Big

ReferenceDimensionless NumbersVector and Vector Calculus RulesCalculus Theorems (Green's Theorems)

Tensor Identities and CalculusCartesian Tensor Rules