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# Introduction to Physical and Chemical Oceanography ATOC 5051

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This 270 page Class Notes was uploaded by Jon Johns on Thursday October 29, 2015. The Class Notes belongs to ATOC 5051 at University of Colorado at Boulder taught by Baylor Fox-Kemper in Fall. Since its upload, it has received 32 views. For similar materials see /class/232044/atoc-5051-university-of-colorado-at-boulder in Marine Science at University of Colorado at Boulder.

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Date Created: 10/29/15

Palm ooo Am C Variance Kzunit freq 0 o 8 s o Variance Kgunit ireq m u o o no sum 000 Am o Variance Kgunit freq 0 o 5 Power Spectrum Autocorrelaiion ENSO Variance K 30 1 2 a 4 5 Period yea s Power Spectrum 12 24 as Lag months Autocorrelaiion W 48 JFMAMJJASOND Month Variance K2 M 1 4 5 Period years Power Spectrum m 12 24 as 49 Lagmonihs Autocorrelaiion JFMAMJJASOND Momh Variance K2 Period years 0 i2 24 36 48 Lag months JFMAMJJASOND Month Nino3 SST Anomalies Observations Control NealeRichter El Nine and Southern Oscillation ENSO what is ENSO how does it work why can39t we predict it Markus J 0c The three stages of ENSO La Nquotra Condlllons Normal Corldlllons El Nl o Cond lons d TUNA A ENSO IMPACTS El NI D Weather Patterns December gt February quotgt m w NOAA The Observing Platform NOAAPMEL MOORING DESIGN NOAAPMEL The Joys of Oceanography T AA DETAILS Normal Condlllons El Nlquotno Condltlons 994797 l I v eDKSCHARGE 220 meters RECHARGE 9 x 3 1 4 2 LA NlNA SST c EL N NO gt Variance Kgunit ireq Variance Kzunii freq m Variance Kgunit freq 4 A Amupm Doggggg ooooooo Ammsm ooooooo ENSO frequency and variance Power Spectrum Autocorrelaiion Variance K 30 1 2 a 4 5 Period years Power Spectrum 12 24 as Lag months Autocorrelaiion 00 48 JFMAMJJASOND Month Variance K 30 1 4 5 Period years Power Spectrum m 12 24 as 45 Lagmonihs Autocorrelaiion JFMAMJJASOND Month Variance K2 Period years 0 i2 24 36 48 Lag months JFMAMJJASOND Month Nino3 SST Anomalies Observations old CCSM new CCSM ENSO spatial correlations Correlation 5 Nina 3 and SST Anomaly Timeseries ms was was no rww vznw nnw new sow u Conelalion 039 Nine 3 and SST Anoma y TTmeseries m m vsn vw unw usz w ww WW I m Covrelalion of Nmo 3 and SST Anoma y Timeselies sz saw saw ww Observations old CCSM new CCSM ENSO MECHANSIMS delayed oscillator or stochastically forced Correlation of anomalies of tau and 220 with 220 at 0N120W old CCSM W 0 ma WW 5 c wmd and 120 a cr 10 months d wind and 120 alter m months Correlation of anomalies of tau and 220 with 220 at 0N120W new CCSM memmm am an MW a mud and 120 after no months d wind and 120 after 13 mm Hovmoeller diagram of 220 and taux anomaly along equator 0095 0097 JASDNDJFMAMJJASDNDJFMAMJJASUNDJFNAMJ 0095 I d1 mm r w w as l E ng LAAAAA 0094 Correlation of highpass ltered tauX at ON 170E with 220 y m WWW MW w mm u M m a m c 2 months lag d 5 months lag Forecasts and Noise Model Forecasts of ENSO from Nnv 2007 NASA em my crs M me lt E w a o E 2 on L a E MaddenJulian Oscillation i M mquot m MJO mi lndum m biweekly snaphots of rain Wtsl Mnnhme Comm m Manuma thnenl Wesi Farm a semi paw Ocean am Paci c no 7 Western Hemisvher NOAACDC 130E BUL quot 40W iUCW 50W 20W 20f Tropical Instability Waves in SST Chelton et a1 2000 Today you learned the tropical Pacific is the most important part 01 the ocean El Nino is triggered by westerly windbursts the oceanic Kelvin wave carries the signal frorr the western Pacific warmpool to the east our forecast capabilities are hindered by ocean and atmosphere weather noise HOMEWORK go to the PMELTAO website download the temperature and fixed depth current data for l4OWON make a plot of their mean and standard deviation plot the frequency spectrum of SST and zonal velocity at 100m depth state the software you used and the approximations you made do SST and velocity have different spectral peaks Name your Equations Baylor FoxKemper ATOC 5051 Which will you balance Saliniy Advec rive Flux so Diffusive Flux BS 7 8Virtual salt ux Energy p01 I Temp Advec rivIe Flux Surface Flux Diffusive Flux 96 aheat ux at pcpaz Volume Whaf comes in a Do Volume Integrals of The Above Equa rion5 Turn in regmls of div ux in ro Fluxes in regm red over surfaces In Paper 2 You need lo a 1 Stole the differential equaiion you are rrea ring 2 Show how you39d in regm re i r ro Form a closed budge r 3 Discuss which rerms you will es rima re and which you will ignore Global Budgeting Baylor FoxKemper ATOC 5051 The S rory So Far 6 We have discussed conserva rion of Volume Sal r Energy and seen some of rhe equa rions e We have seen how po ren rial rempera rure compensa res For pressurelempera rure connec rion allowing po ren rial rempera rure ro rake rhe role of En rropyEnergy conserva rion We have seen how in es ruarine Flow in regra ring rhe budge rs over a xed volume leads Jro manageable equa rions From Es ruary ro Budge r PRECIPITATION IN MlXED RIVER AND PJ EVA88ANON RIVER FLOW SALT WATER E39f 1N OUT so p0 Vo SALT WATER 8pi Vquot gt IN Budgef ConsisfemL vah Jkgs 05 SOC COADS Direct AMIP nterquar1He Range Oort and Piexoto 1983 Sahn y MOIsture Transport 10 EnergyHea r Budge I I 200 gt150 100 o o 100 150 Mean heat fluxes Wm2 SOC How do we ge r This 1 Incomin ro A rm Absorbed Solar Radiation W mquot ZoTallM eor i SD 80 IOU 120 140 WED I30 200 220 240 260 280 300 320 340 W m Trenberfh amp Sfepaniak O3 2 Ou rgoing From A rm Outgoing Longwave Rodia an 1 I20 200 254 l l l I 200 220 240 260 250 W m 2 Trenberfh amp Sfepaniak O3 3 Ne r ro A rm lZO 100 60 60 40 FIG 2 Annualized mean TOA ERBE measurements for the period Feb 19857Apr 1989 for the ASR OLR and net in W m l The color key is under each plot and the contour interval is 20 W m l Zonal mean pro le panels are given at right Trenberfh amp Sfepaniak O3 Accoun r For A rm Transport w Reanalysis 0 0 O 20 30 40 0 EC 10 60 90 Latitude 39N FIG 1 TOA annualized ERBE zonal mean net radiation W m z for Feb 1985 Apr 1989 Trenber rh 84 Caron 01 RT ERBE AT NCEP AT ECMWF por l PW N b O Heal Trans N EQ Latitude FIG 2 The required total heat transport from the TOA radiation RT is given along with the estimates of the total atmospheric transport T from NCEP and ECMWF reanalyses PW Wh f39s LeH is NCEP Derived Hem Transport PW A qn c mum Hea39 Transpori Pw Fla 5 Imphed Zonal annual mean ocean heat transports based upon the surface uxes for Feb 19857Apr 1989 for the tocah Atlanuc man and Paeme basms for NCEP and ECMWF armor sphenc elds PW The 1 std en bars are mdxcated by the dashed curv 20 E0 20 39S Latimde 39N es Al rerna rively To ral Flux Can be mapped From Ocean Obs 200 150 100 50 O 50 100 150 200 Annual mean net heat flux Wmz SOC Beware Seasonal Variabili ry 180 90 W 180 90 W I 0 Monthly mean shortwave heat flux Wmz SOC Longwave 84 La ren r vary 00 180 90W 90E 180 180 90W 0 90E 180 180 90W 0 I I I quot l I 180 90w 0 90E 180 100 80 60 4O 20 0 Monthly mean Iongwave heat flux Wm2 lggngow 300 250 200 150 100 50 0 I latent heat flux Wm2 SOC d Oct 7 E I LII nun 9 Ionh Heat a 1 PW Uncertainty 39 j 05 PW Ganachaud amp Wunsch Bu r It doesn t sum up to no rhing There are internal Fluxes of heat Es rima red wi rh Inverse Methods39 130 120w 60W 60E 1205 Fig 523 Net southnorth heat transports PW from direct estimates superimposed on the map of annual average heat ux Fig 510 Black estimates from quotinverse modelsquot from many sources summaries in Bryden and Imawaki 2001 Talley 2003 Red Talley 2003 Positive transports are northward Ocean Heafed 84 cooled af fop Image Cour resy N BalmFor rh DiFFusivi ry Rayleigh Decreasing increasing as you go down Ocean Transpor rs Hea r Bu r A rmosphere is Hea r Engine rha r Drives he sys rem Sands rrom 1916 Wunsch amp Ferrari 04 LUNISOLAR GEOTHERMAL laps SURFACE WAVES u 2 Tu RBULENCE GENERAL 2Col zCULA I39ION 1 1EJ INTERNAL WAVES quot MEJ EDTTDM DRAG AB ISI S L F FSSDGES i Icss or naxanzen J I 5v MIXING I MAWTENANCE or AavssAL smLTiFicAi imi Thermoholine or Meridional Overfurning Circulation A bit like a conveyor bel r because energy is ex rernally supplied Winds amp Tides However who r39s on the conveyor aFFec rs how rhe conveyor moves Heating and cooling of Jrhe surface affect howwha r is rranpor red This is wha r we are aF rer now e 15 E 10 E 05 e 3 E 00E 3 o5 5 5 1 DE Total 0 15 Piucifio I v 1395l5 Atlantic 2 0 4 I i 25 I l I I 1 l l l I l 1 i 4 IN IA efvo DIANVquot O o HINDo PACIFIC I Surface Layer Water NADW North Atlantic Deep Water SAMW Subantarctic Mode Water UPIW Upper Intermediate Water 268 S o s 272 RSW Red Sea Water LOIW Lower Intermediate Water 272 3 5G 5 275 AABW Antarctic Bottom Water IODW Indian Ocean Deep Water NPDW North Pacific Deep Water BIW Banda Intermediate Water ACCS Antarctic Circumpolar Current System NIIW Northwest Indian Intermediate Water CDW Circumpolar Deep Water Plate I27 see p 22 A threedimensional schematic showing the meridional overturning circuIacion in each of the oceans and the horizontal connections in the Southern Ocean and the Indonesian Throughflowfhe surface layer circulations are in purple intermediate and SAMW are in red deep in green and near bottom in blue From Schmitz I996b i Differeni Ways To Consider Flow Vertical section with area AR Free Surface River End Ocean End Volume V Isohaline surface with area A Bottom Topography Figure 1 De nition sketch of the estuarine volume V whose seaward end is de ned by the moving curved isohaline surface with area A The landward end of the volume is de ned by the stationary vertical plane with area A R lntro to Matlab for ATOO 5051 Introduction to Physical Oceanography or7 Notions for the Motions of the Oceans Baylor Fox Kemper September 24 2008 1 Contacts The professor for this class is Baylor Fox Kem per bfk cooradoedu 303 492 0532 Office Ekeley room S2508 httpcirescoIoradoedusciencegroupsfoxkem perclasses 2 Getting Help The most important commands in matlab are help and lookfor The rst one allows to to get a description of any matlab function for example gt help plot7 tells you about the function named plot The second one allows to to search for keywords within a function description in case you don t know or can t remember the name of the function 3 The Basics Matlab is Matrix Laboratory Matlab is based on matrix algebra So when you think about data you think about making ar raysvectorsmatrices of data In that way it can be a lot like a spreadsheet program but it is much more powerful because 1 it can handle much larger quantities of data and 2 you can use pre programmed solver routines to get things done The following contains a number of examples Type them into matlab and see what happens Try changing them up a bit and see what happens then Good hunting 31 Scalars Some simple examples of matlab scalar that is one number arrays at work gtgt A1 gtgt sizeA ans 1 1 gtgt sizeB ans 1 1 gtgt sqrtBexpA ans 41325 32 Vectors Now7 let s consider vectors You can make a horizontal vector gtgt A1 1 1 1 A 1 1 1 1 Or a vertical one gtgt B1111 B 1 1 1 1 You can t add together a horizontal and a vertical vector gtgt AB Error using gt plus Matrix dimensions must agree But you can use the transpose single quote 7 to transpose a matrix7 or in this case convert an horizontal vector to a vertical one gtgt A B ans ONION The operator is a vector or matrix multiply7 which in this case is the dot product of A and B gtgt AB ans 4 While the operator multiplies component by component if the vectors are the same shape 33 Matrices Matrices behave in much the same way as vectors7 except now there are both rows and columns gtgt A1 2 3 4 5 6 A 1 2 3 4 5 6 gtgt B1 42 5 3 6 B 1 4 2 5 3 6 gtgt AB Error using gt plus Matrix dimensions must agree gtgt AB ans 2 4 6 8 10 12 gtgt AB Error using gt times Matrix dimensions must agree gtgt A B ans 1 4 9 16 25 36 gtgt AB ans 14 32 32 77 There are many special commands for generating matrices The most important are gtgt ones5 ans 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 gtgt ones32 ans 1 1 1 1 gtgt zeros5 ans 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 gtgt zeros32 ans 0 0 0 0 0 0 gtgt eye5 ans 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 gtgt rand5 ans 08147 00975 01576 01419 06557 09058 02785 09706 04218 00357 01270 05469 09572 09157 08491 09134 09575 04854 07922 09340 06324 09649 08003 09595 06787 34 Accessing a Submatrix You don t have to use the whole matrix all at once The colon plays a special role in accessing a subset of the matrix Used alone7 it means 7that whole row Used with a number before A and afterl37 it means7 7that whole row from A to B Used with three numbers it means7 7that whole row from A to B jumping by C ACB For example gtgt C1 2 3 4 5 6 7 8 9 10 11 12 C 1 2 3 4 5 6 7 8 9 10 11 12 gtgt C1 ans 1 2 3 4 5 6 gtgt C2 ans 7 9 10 11 12 gtgt C2 ans 2 8 gtgt C245 ans 10 11 gtgt C212 ans 7 8 gtgt C212end ans 7 9 gtgt C213end ans 11 7 10 gtgt C211end ans 7 8 gtgt C2end11 ans 12 11 10 35 Higherorder Tensors Of course7 you can have more indices on your varialoles7 gtgt Aones345 ans1 1 1 1 1 1 1 1 1 1 1 1 1 ans2 1 1 1 1 1 1 1 1 1 1 1 1 ans3 1 1 1 1 1 1 1 1 1 1 1 1 ans4 1 1 1 1 1 1 1 1 1 1 1 1 ans5 1 1 1 1 1 1 1 1 1 1 1 1 But you won t be able to easily use the matrix multiply and other matrix based arithmetic is easy to convert a submatrix into a real matrix gtgt A2 ans1 However7 it I H ans2 1 1 1 ans3 1 1 1 ans4 1 1 1 ans5 1 1 1 gtgt squeezeA2 ans 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 36 Quiet Mode To stop matlab from spitting everything back at you7 end each statement with a semi colon gtgt Aones 3 A 1 1 1 1 1 1 1 1 1 gtgt Aones3 gtgt 4 Getting and Saving Data load save and loaddap Matlab is very good at loading and saving its own kind of les7 mat les Another useful function is 7who7 which tells you the names of de ned variables gtgt who Your variables are A B C ans endverbatime To make a mat file you just do the following beginverbatim gtgt save varsmat endverbatime To save only a few of the variables you list them after the desired filename beginverbatim gtgt save varsABmat A B Now to check what s in the les we rst clear the memory with 7clear then gtgt clear gtgt load varsABmat gtgt who Your variables are A B gtgt load varsmat gtgt who Your variables are A B C ans Load Save are also capable of handling text les with the ag fascii but you usually need to pretreat the text les eg by deleting headers on columns Furthermore standard unix and ms dos commands eg Is dir cd work as expected showing the contents of directories and changing the local directory 41 ncload and ncx The add on function ncload is available from the ncx toolbox linked from the course webpage It allows you to load a netcdf formatted dataset from a webpage almost as though it were a matlab le The website has a link to ncx called netcf on matlab Installing this software and adding the installed directories with subdirectories to the matlab path will allow you to browse load and save netcdf les Perhaps the easiest way to get data into matlab is using ncload gtgt ncload filenc This loads the entire contents of the le into matlab s memory buffer Your le may be too large to do it all at once so you can load subsets gtgt ncload filenc Varl Var2 Where Varl and Var2 are the names of variables in the dataset How do nd out the names Opening the dataset in the ncx browser works or using features from the netcdf toolbox THIS IS THE CRUCIAL STEPVJo o 0IT SAYS OPEN THE netcdf FILE WITH A HANDLE named nc here o o gtgt ncnetcdf file nc Now we can use this handle to manipulate the data in the file gtgt varsvarnc gtgt vars1 Gives output about variable 1M gtgt vars2 Gives output about variable 2 etcM The netcdf toolbox is particularly good about opening only a subset of the data in a le For example if by doing the varsvarnc trick above we nd out that there is a variable named T that we want to get the data from we can do gtgt nc T gt nc T 1 Z gives the first value of T gtgt nc T 15 Z gives the first 5 values of T gtgt nc T 15 10100 Z gives the 15 10100 block of T assuming T is a 2d matrix gtgt nc T Z gives all of T as a vector gtgt nc T Z gives all of T as a matrix assuming it s a 2d matrix V You can also nd lots of info about using the netcdf toolbox by typing gtgthelp netcdf There is also a user s guide with examples on the netcdf toolbox webpage 411 Trouble Installing Netcdf libraries are constantly changing as some of you have found If the netcdf libraries included with ncx don t work for you eg matlab complains about a missing mexcdf53 function you can try installing the mexnc binaries after installing ncx l have added a link to the binaries on the webpage 42 loaddap The add on function loaddap is available from the Matlab OPenDap toolbox linked from the course webpage It allows you to load an oceanographic dataset from a webpage almost as though it were a matlab le 43 ASCII les If all else fails above but you are able to download an ascii le version of your dataset matlab is fairly capable at interpreting such les The function load fascii will do very simply organized datasets but crashes if there are column titles missing values etc The more powerful function textscan can be instructed to ignore comments process nans etc but it takes a little TLC to get it working 5 Making Plots The important plot commands are use help 7command7 for more detail 0 plot plots scatter and line plots eg p0t110exp110 o plot3 plots 3d scatter and line plots eg plot23110exp110sin110 0 gure generates a new gure and window 0 subplot generates a subplot within a gure for making paneled gures 0 contour generates a contour plot eg contour110120sin12039110 o contourf generates a lled contour plot7 eg7 contourf110120sin12039110 c axis subselects the gure axes o saveas allows you to save a gure as a jpg or pdf7 etc o pcolor generates a shaded plot7 eg7 pco0r110120sin120391107 often used with shad ing interp or shading at o scatter allows you to make a map of scattered data7 with colored dots representing a dependent variables values the colorbar caxis allows you to change the limits of most color coded plots7 so you choose the max and min of o colorbar adds a colorbar to a gure based on the caxis values 6 Doing Stats and Calculations These are really easy Some examples A 07577 07431 03922 06555 gtgt meanA 06371 gtgt meanA1 ans 03348 gtgt meanA ans 06371 gtgt Arand4 5 01712 07060 00318 02769 02965 2 1 gtgt stdA1 01694 gtgt stdA ans 01694 gtgt varA 00287 02909 00846 00462 0 0971 08235 06948 04154 04009 0 1607 0 3171 09502 00344 04387 04351 03829 0 1466 7 Saving a Script or Function 0 3816 07655 07952 0 1869 05323 02975 0 0885 The last piece worth mentioning here is the m le format You can type a series of commands into a le named for example doitm Then7 while in the same directory as doitm7 if you type gtgt doit it will execute the commands If you de ne a function see help function then you can call that function from within the directory A function is written as a m le but has added capabilities eg input output arguments private variables etc In rro To the Meridional Over furning Circulation Baylor FoxKemper ATOCSOSl Numbered equa ons and gures From Vallis or Cushman Roisin amp Beckers The Sverdrup TranspomL mus1L close in fhe west in fhe Sfommel model it s by bof rom drag Srreamfunction Wind stress Subtlopicul Gyms Equam nl and Tm iml Circu a x n om Inmgyxe andu xnmrbusm Exchangcs Polar amp Subpolm Cunem Systems Hate LL Does wind do every rhing Wha r about the hea ring and cooling of The ocean surface Who r about Freshwater and salini ry forcing Wha r about ridal and rurbulen r mixing a Don t all of rhese change the baroclinic Flow 30 W H m l OOEAE V A C WMRGW Surface layer Walcl NADW North Allamlc Deep Water Subanlarclk Mode Watcv Upper Inlermedlale Walcr 25 8 lt a s 272 d Sea Wale lower llllermediale Welor 272 s a s 27 5 Antarctl Bottom wmel ne Wmcr Norm Paclhc Denp wellul BIW B da llllelmedlale Wde Amarulc clrcumpoler Cul39renl System Nuw Norlhwcsl Indian lntermedlnta Watul Cilcumpolar Deep Walcl39 Plate I21 see p 22 A thl39ecdllnensmnal schematlc showmg LI meridional ovel curnlng circulztlon in eazh n39i me means and me horizontal connectlons in the Southern Ocean and he lmmneslan Throughilow The wince layer Clrculmlons are in pul ple ll1tel mediare and SAMW are m led deep in green and Hearbottom in blue Fl om Schmlu l996b A Conveyor Warm smu Cumu PACIFIC Recilvmlul aquot 39 A 180 Schemlic diagram ofthe global ocean cimuhtion pathways 1he oonveyer39 belt aha W Emecker undifin by E Ma39 imzr Recall Hea r Transport Accoun r For A rm Firs r 0 0 O I 039 O p E 3 v E O In 2 u c o N 90 60 40 30 20 20 30 40 10 EC 10 60 90 S Latitude 39N FIG 1 TOA annualized ERBE zonal mean net radiation K m z for Feb l985 Apr 1989 Trenber rh amp Caron 01 RT ERBE AT NCEP AT scmwr por l PW O N I N u C D L U 0 I 0 E0 20 40 8 Lomude 39N FIG 2 The required total heat transport from the TOA radiation is given along with the estimates of the total atmospheric transport AT from NCEP and ECM reanalyses PW Hem Transport PW 25 W W l W I 5 x x 20 ECMWF Derived OCH Pa AH L Heal Transpon PW ci ON C it n n 7 20 E0 20 395 La imde 39N 5 hn hed Zonal annual mean ocean heat transports based upon the surface uxes for Feb 19857Apr 1989 for the tocah Atlanuc man and Paeme basms for NCEP and ECMWF atmor sphenc elds PW The 1 std en bars are mdxcated by the dashed curves IT is ThaT parT of The mean circulaTion which is driVen by densiTy differences as opposed To wind and Tides Because The ocean densiTy is a funcTion of TemperaTure i and saliniTy quot This circulaTion is referred To as The 39 circulaTion and indicaTes a driving mechanism These densiTy differences are primarily caused by surface fluxes of heaT and freshwaTer and subsequenT inferior mixing The oceanic densiTy disTribuTion is iTself affecTed by The currenTs and associaTed mixing Thermohaline and wind driven currenTs inTeracT wiTh each oTher and Therefore cannoT be Truly separaTed From Danabasoglu s Talk for ATOC 6020 WHAT IS MERIDIONAL OVERTURNING CIRCULATION MOC If is a rela red field referring lo a sireamfunc rion on The depfh la ri rude plane If can be obfained from 0 east 1Pyzt fdz fVxyztdx Z H E Sf where X longi rudinal zonal direclion ve eas rwards y la rifudinal meridional direc rion ve nor rhwards 2 height ve upwards T Time V meridional velocity componenT This field is ofTen used in The modeling communiTy because if is easy To diagnose MOC INCLUDES WINDDRIVEN CIRCULATION From Danabasoglu39s Talk For ATOC 6020 3 WIND DRIVEN GLOBAL E T E N NORTH ATLANTIC DEEP WATER ATLANTIC E NADw E ANTARCTIC Q WATER Units Sverdrup Sv 106 rn3 s From Danabasoglu39s ralk For ATOC 6020 MERIDIONAL OVERTURNING CIRCULATION DepTh La iiiqde Plane 0 Densi ry La ri rude Plane From Danabasoglu39s ralk For ATOC 6020 NOTE Po ren rial Densi ry transformations involve heatingcooling or mixing S rommelArons Faller Fig 1510 The experimental setup in the StommeliAronsiFaller rotating tank experr imemv a A plan View of the apparatus The fluid is contained in the sector at left b Side view The free surface of the uid slopes up with increasing radius giving a balance in the rotating frame between the centrifugal force painting outwards and the pressure farte painting inwards Small pipes may be introduced into the uid to provide mass sources and sinks S rommelArons Faller Fig l5 ldealized examples of the ow In the re ning senor experiments with various ocations of a source S or sink 5 of mass The S rommelArons Model Warm upper ocean llllll Thermocline I I I I I I uniform upwelling localized Abyss mass source convection w 0 Equator Pole Fig 1512 The structure of simple StommeleArons ocean model of the abyssal circula tion Convection at high latitudes provides a localized masssource to the lower layer and upwelling through the thermocline provides a more uniform mass sink i Rotating dish ii Winddriven ow iii Abyssal ow W Conservation D Dr H This leads to l U icurlz39r B S is localized curlir is wind stress 5H is upwelling mass source curl mass loss Meridional mass flow away from boundaries is thus determined by sign and not location of sign of windstress curl upwelling and sign of localized mass source cur 1139 r so polewards if S lt 0 upwelling The S rommel Arons ow is typically polewards in inferior wi rh de ci rs WWW made up by WBC Latitude Upwelling S is upwelling Fig l5 Mass budget in an Idealized mass loss abyssal ocean Polewards of some lat Long tUde itude y the mass source Sn plus Khe S determined by poleward mass ux across 339 T1 are upwelling and Sign of f so polewards if rent Tu and the integrated loss due 5 lt 0 upwelling to upwelling U poleward of y 1 569 For Example Fig ISIS Schematic ofa StommeleArons circue lation in a single sector The transport of the western boundary current is greater than that provided by the source at the apex illustrating the property of recirculation The transport in the western bcun ary current T decreases in intensity equatorwards as it loses mass to the poleward interior flow and thence to upwelling The integrated sink due to upwelling U exactly matches the strength of the source S 30 W H m l OOEAE V A C WMRGW Surface layer Walcl NADW North Allamlc Deep Water Subanlarclk Mode Watcv Upper Inlermedlale Walcr 25 8 lt a s 272 d Sea Wale lower llllermediale Welor 272 s a s 27 5 Antarctl Bottom wmel ne Wmcr Norm Paclhc Denp wellul BIW B da llllelmedlale Wde Amarulc clrcumpoler Cul39renl System Nuw Norlhwcsl Indian lntermedlnta Watul Cilcumpolar Deep Walcl39 Plate I21 see p 22 A thl39ecdllnensmnal schematlc showmg LI meridional ovel curnlng circulztlon in eazh n39i me means and me horizontal connectlons in the Southern Ocean and he lmmneslan Throughilow The wince layer Clrculmlons are in pul ple ll1tel mediare and SAMW are m led deep in green and Hearbottom in blue Fl om Schmlu l996b Fig l518 The ocean currents at a depth of 2300m in he North Atlantic obtained using a ombination of ubsewations and model as in Fig 142 Note the southwards owing deep western boundary wrrent V OXY en 1139 2500 From NODCWOA05 Fig l51 The ocean currents at a depth of 2500 m in he Nonh Atlantic obtained using a umbination of observations and model as in Fig 142 Note the southwards owing deep western boundary currents Fig ISJG Schematic of a Stommel n the Northern Hemisphere there is still a western boundary unent and a recir ulation The integrated sinks due to upwe ing exactly match the strength of the source Example P Oxygen of 4000m From NODC WOAOS 30 W H m l OOEAE V A C WMRGW Surface layer Walcl NADW North Allamlc Deep Water Subanlarclk Mode Watcv Upper Inlermedlale Walcr 25 8 lt a s 272 d Sea Wale lower llllermediale Welor 272 s a s 27 5 Antarctl Bottom wmel ne Wmcr Norm Paclhc Denp wellul BIW B da llllelmedlale Wde Amarulc clrcumpoler Cul39renl System Nuw Norlhwcsl Indian lntermedlnta Watul Cilcumpolar Deep Walcl39 Plate I21 see p 22 A thl39ecdllnensmnal schematlc showmg LI meridional ovel curnlng circulztlon in eazh n39i me means and me horizontal connectlons in the Southern Ocean and he lmmneslan Throughilow The wince layer Clrculmlons are in pul ple ll1tel mediare and SAMW are m led deep in green and Hearbottom in blue Fl om Schmlu l996b Go Vo re Equa rions Scaling Nondimensionaliza rion Eulerian vs Lagrangian Baylor FoxKemper ATOCSOSl FYI Numbered equa rions are From Vallis Chp 1 Figures roo And now on lo lhe underlying principles 6 We ve been Talking about observations and his rory For rhe pas r rwo weeks Now i r s Jrime ro s rar r ralking abou r wha r has been learned I r39s lime to become quan ri ra rive as well a 0 wha r are the equa rions For oceanography BeFore we ge r s rar red wi rh equa rions how do we consider the problem o Wha r migh r we have equa rions For 6 Should we consider the uid as many bodies or some other way a Wha r o rher rheories migh r we compare ro Wha r ques rions do we wan r ro ask Eulerian vs Lagrangian IF you were a physics major you39ve probably seen most equa rions oF mo rion For a body like Excep r For Elec rrici ry and Magne rism or Quan rum Mechanics which aren39t equa rions of mo rion rhey39re equa rions For elds VBJ 8E V X B 63 4 way Eulerian Fluid Dynamics is a Veloci ry Field39 Theory Conservafion of Momem um No ce fhaf fhe velocify here appears ju51L like fhe elecfric eld or probabilify disfribu on in EampM or Quam um Mechanics Why n01L Lagrangian Fluid mechanics 6 How many wa rer parcels would you need To consider ro capture the Full Fluid mo rion How do you model rhe in reracfion wi rh boundaries moving rela rive ro rhe parcels As soon as we ge r To 3 in rerac ring bodies we39re in rrouble Fluid Dynamics is a Veloci ry Field39 Theory with other conserved quan ri ries roo Tracer Concem ra om arfsvolume l 382ib Where does rhis come From Bpu A A A pu BX x y z xAx Fig 1 Mass conservation in a Cubic Eulenan con rol volume Fluid 105 J JV39 d3 5 J V 4 pv dV Figurem Mass ccn V servaticn in an arbitrary d5 Eulerian control volume swfa e elemem ur V palms Outward Mass increase op JV 01 dv 39 K I r39s rhe Ma rerial Derivative rhal39 le rs you conver r be rween a Lagrangian conservation law and an Eulerian eld a When we have some rhing rha r rracks wi rh rhe Fluid eg salt hea r momen rum we39d like lo budgeiL For it 6 However we don t wan r To keep Track of individual parcels a Wri ring rhe conserva rion law in lerms of the malerial deriva rive allows us ro produce Eulerian equalions For Lagrangian budge rs On Thurs we39ll see speci c examples Material and Eulerian Derivatives The material derivative of a SCillill 1 and a vccmr b licld am given by D4 7 mp Db Jl DI a 39V D1lt Iv39vb39 Various material derivatives of integrals are D L45 EVlwv flDl v vdVVm v vdV 2deVvdI D V y D D4 dV idV DI V N V 390 Dz These fomlulac also hold if is a vector The Eulcrizm dcrivulivc ufun imcgrul ix d a 11V dV dI Ld fy III so hill I I 9 i dV0 and L wave KN av d V d1 y y m D2 D3 D4 D5 Rela red ro Leibniz39s J31 5 111 3quot bulk J 1 E f nigh i 39 I If The maferial view confrol mass gives infegra on bounds ha move in fime wi rh he Fluid so you need o diFFeren afe The infegral bounds The Eulerian view confrol volume gives infegra on bounds ha are xed in me so you don need o diFFeren afe he infegral bounds Material and Eulerian Derivatives The material derivative of a SCillill 1 and a vccmr b licld am given by D4 7 mp Db Jl DI a 39V D1lt Iv39vb39 Various material derivatives of integrals are D L45 EVlwv flDl v vdVVm v vdV 2deVvdI D V y D D4 dV idV DI V N V 390 Dz These fomlulac also hold if is a vector The Eulcrizm dcrivulivc ufun imcgrul ix d a 11V dV dI Ld fy III so hill I I 9 i dV0 and L wave KN av d V d1 y y m D2 D3 D4 D5 These are complex equa rions how can we possibly solve rhem We es rima re rhe size of each rerm by estimating the relevan r scales of the phenomena we wan r ro s rudy wi rh rhe equa rions or This is called scaling the equa rions Le r s see how This goes For a simple example A Simple Scaling Exercise Q 1unitsday CD lunit QM Q lunitsday units 7 7 lunit 00lms 39 39 106m units So wifh fhese LTVQ scales if seems Nondimensionalizaiion a The physics can39i possibly care what units we use only rha r The dimensions work out Thai is we could use me rers or Fee r or Frac rions of an ocean basin wid rh and if doesn39i ma r rer It you think about it a bit rhis means we should always be able to wri re rhe equa rions with only nondimensional rerms to Now le r39s revisi r our simple example and see A Simple Nondimensional Exercise 39I39 864005 me r l y 6 v M QQM L 10001071 10 m T U 001ms Q 1unitsday Cb lunit NM 50 wifh fhese LTVQ scales if seems Some rimes we give rhe special names a 7 v Vv v HVW 120221 1202b LargeScale WindDriven I Sverdrup Balance Baylor FoxKemper ATOC5051 Numbered Eq rns amp Figures from Vallis Chp 14 or CushmanRoisin amp Beckers Chp 20 So whaf did we learn abou1L PV Conserva rion of PV is a powerful cons rmin r on the motion 6 Once you have the PV and also the densi ry you may be able ro inver r back For The geos rrophic veloci ry Thus PV cap rures rhe essence of ro ra rionally cons rmined mo rion Ano rher Example hic Contours gtbBaVOllOpicinerlialcase D g f H F Dt h 039 Figure From Jackson ef al 2006 Haw Ano rher Example Wind CfQ Figure Fnom Jackson ef al 2006 Ano rher Example Wind 3 Barolropic fnctionai case H Western boundary layer Wind stress W39nds Do Wha r NOr u x CIRFSECIEIW I mr39lt1 3 Gamer Mean Winds From CushmanRosin amp Beckers W39nds Do Wha r NOr u x CIRFSECIEIW I mr39lt1 3 Gamer Mean Winds From CushmanRosin amp Beckers Equa rorial UPwelling Winds Do Wha r Mean Winds From CushmanRosin amp Beckers Equa rorial Sub rropical Upwelling Winds DO Convergence Mean Winds From CushmanRosin amp Beckers Consider cons ran r densi ry Geos rrophic Flow 1 3991 20139 p d l 01 30 lb 0 8y On fhe Befa Plane Consider Curl 0F Geos rrophic Momen rum ie Vor rici ry 20121 201b PI 011 39u I LU39I39 0 206 011 On fhe F plane our rofa ng funk Consider Curl 0F Geos rrophic Momen rum ie Vor rici ry 1 0p 2012 0 l 20113 PO 0 39u i luv 7 0 206 J 0 On fhe befa plane on a rofa ng planef 39u I39 V i y ful U r y Massaging rhe Vor rici ry Equation 20 21 9 Given Incompressibilifyz l 111 H Massaging rhe Vor rici ry Equa rion Infegrafe From fop 0 boHom of ocean inferior 74 3 30 1391 w z I 1L39 HV 711 Thus Norfh Soufh Flow requires vor rex sfrefching my 7171L39 II Nonll 111 L12 lt w um Figure 204 Melidinnul Inigmlinn Slrelching I 2 2X 14 v s5 5 V Squeezing t 0139 fluid parcels 39n um by vcrucn muhing or s ucczlng in me lurgc cale nceumu urculnuun I quotMean Win om Vs hmnRbsi quot 4 14 Ixv tr 120E 150 new 1 i T J um f 7 f 2010 0 1 j y f The Sver f w 11 u39 711 I 1 a 7quot a rquot 1 i 397 i Lk 70 aT f 01 f Neglec1L boHom w and le1L V be depfh infegrafed volume h ansporf 7 2011 pad 139fgt l The Sver u39 1 f 39Ek l 701139 Pmin HI f 39y f 39 4 add the Eknmulayer contribution given by 834b 201 lb f T V Ek n mel VVEk 1 37 0 11C 1031 01 i The Sver Vtmnl 7 1 f 201 1c pm 139 3 To make fhe surface rela vely Flat we need Umml F ovmml 1 0y 7 2012 1 J 3 I f IT 11 2013 mm 1 0y 1 0 n The Sverdrup Transport will not naturally close Western boundary layer Wind stress Eastern boundary layer Fig l43 Two possible Sverdrup flows 1 for the wind stress shown in the centre Each solution satis es the no ow condition at either the eastern or western boundary and a boundary layer is therefore required at the other boundaryl Both ows have the same equatorward meridional ow in the interior Only the ow with the western boundary current is physically realizable however because only then can friction produce a curl that opposes that of the wind stress so allowing the flow to equilibrate The Sverdrup Transpor r may have mul riple gyres Summary 6 The Sverdrup rela rion balances rhe change of vor rici ry due ro NorfhSou rh displacements on the Sphere wi rh wind s rress curl The resulting circula rion doesn f na rurally close but an eas rern or wes rern boundary curren r migh r do the job N X139 Time Wes rward Inlensi calion Nex r rime we39ll consider Fric rion and iner ria A Bo r rom Fric rion Slommel Model will allow For a wes rern boundary curren r to close rhe Sverdrup Flow 6 A La reral Fric rion Munk Model will also do if e We ll also discuss rhe na rure of Friction and consider some inviscid solu rions and rhe boundary viscosi ry problem What and How To Measure Baylor FoxKemper ATOC 5051 LiF efime 0F Scien risfs Durafion of E xperimem P mc ml use of resuls ie needs 01 shipping shing Reproducibiliy Aliasing Mos energe c Mos r amenable 0 exis n ins rrumen rs Pruc cnl consfruinis 23 how long is he ship in he rea Wha r Do You Mean Mean Earlh A e The joke a One observa rion de rermines rhe mean so A second one gives you rhe variance There is variabili ry on every rimescale From seconds ro billennia Mean over Dura rion oF Experimen r Wha r abou r Spa rial Mean How many observa rions Are They synop ric Wha r are The rimescales associa red wi rh rhe spa rial scales Are There physical scales To consider Wha r Do You Mean Mean Huybers 8 Curry 06 037 t 005 Energy 0 ds 5D 5150 ice Al kenones 3139 50 calcite Fauna counts SrCa CAC CRU Forcing39 The joke 10395 10392 Frml Innnv math vr l a One observa rion de rermines rhe mean e A second one gives you rhe variance Lal39evarves 12903913 6 There is variabili ry on every l39imescale From seconds ro billennia Mean over Dura rion oF Experimen r Wha r abou r Spa rial Mean How many observa rions Are They synop ric Wha r are The rimescales associa red wi rh rhe spa rial scales Are There physical scales To consider Physical Scale Examples 6 Time a Solar Day Sidereal Day Pendulum Day sidereal daysinlaf Year Flushing Time Basin Turnover Time Eddies Viscous Time For migh r moon 6 6 6 Space BasinScale Curren r Wid rh DeForma rion Radius Eddy Scale Seamoun r Scale Plume Size Viscous Scale Transi r Dis rance Waveleng rh Where we will Focus mos rly Time In rervals Grea rer rhan 12 Pendulum Day the iner rial period Spa rial Scales Less than BasinScale Grea rer rhan Eddy Scale For Fiddler on rhe Roof Fans TRADITIONTRADITION ShipBased Ins rrumen rs eg ReidManfyla eWOCE a CTD Conduc rivi ry Tempera rure Dep rh really pressure a Reversing Thermome rer Pressure amp Temp 6 ADCP Acous ric Doppler Curren r Pro ler LADCP Moun red BoH les Bucke r Nansen Niskin Rose r res T S 02 Phospha re Ni rra re Chlorophyll e rc 3 Ne rs Plank ron Nek ron Fishing Ca rch Records 6 Deploymen r Running TowYo Cas rs Surface Tracer Release 3 More De rail King e r al in WOCE Vol Chp 31 Emery Chp 6 Also Sverdrup e r al MooringBased Inslrumenfs eg TOGATAO CTD Thermis rors Fixed ADCP mul riple e Meteorology if surface radialion winds e rc Curren r Me rers propeller o rher Eulerian Ins rrumen r Timeseries Crawlers in developmen r s See Emery ef al Chp 6 For more FloafBased Insirumenis a Lagrangian Ins rrumen r Floa rer DriF r Can be pressure xed by being less compressible lhan seawafer 6 Can be con rrollable like a submarine e CTDs eg P ALACE Used ro be acous rically rracked now sa relli reGPS more common 6 Emery el al 6 and WOCE 32 Q amp SatelliteBased Ins rrumen rs Surface Al rime rry Wave Heigh r Wind Ac rive RADAR or LIDAR Backsca r rer Surface Tempera rure Passive Infrared AVHRR e rc Sea Ice Chlorophyll e rc Ocean Color Gravi ry GRACE Also Humidi ry Sea Level Pressure Rainfall Clouds Radia rion more a rmospheric bu r can be used ro infer ocean values Surface Salini ry in res ring Emery e r al 6 WOCE 33 NASA Tu rorial Oiher PlaiForms FLIP Floa ring Labora rory Ins rrumen r Pla rForm e Slocum Glider A Winged Floa r Acous ric Tomography Radar Video Nearshore surface moni roring Residual oF Me reorology Can be useful For Fluxes of hea r e rc Tide Gauges Shoreline Piers e rc moni roring Emery el al Chp 6 Instrumental Error quotEhble 311 Onemime WHP standards for CTD sensorsThe data quality goals given here and in Table 312 are regarded as attainabie in low gradient oceanic domainstThey represent an ideal that can be achieved by appropriate methodologies such as those speci ed in the WHP Operations and Methods manual WHPO 99421 Parameter Standard Accuracy of 0002 C preciSiOn 040005 C lTSgo Accuracy of 0002 PSS78 depending on frequency and technique of calibration precision OtOOI PSS78 depending on processing techniquesquot Accuraey 3 decibar dber with carefui iaboratory calibration precision 05 dbar dependent on processingb Reproducibilityi 1 same for precision if no absolute standards are available for a measurement then accuracy should be taken to mean the reproducibility presently obtainable in the better iaboratories quot Although conductivity is measuredsz analyses require it to be expressed as salinity Conversion and calibration techniques from conductivity to salinity should be stated quotDil cuities in CTD salinity data processing occasionaliy attributed to conductivity sensor problems or shortcomings in processing may actually be due to difficulties in accounting for pressure sensor iimitations Existing polarographic sensors have been found to meet these requirements but better sensors need to be developed particularly for use in high latitudes e WOCE Emery at al Talley Website Aliasing spectral energy above the Nyquist or folding frequency that is folded back into the frequencies below the Nyquist creating higher spectral energy at these frequencies than is actually in the time series Covariance and correlation a measure of the covariability of two variables computed as the averaged sum of the cross product of the variations from the respective means of the two variables When normalized by the product of the standard deviations the covariance becomes the correlation which ranges from 1 to 1 Determination the actual direct measurement of a variable eg the length of a piece of wood using a ruler Other words used to represent this same thing are observation measurement or sample Estimation a value for one variable derived from one or more determinations either of the variable of interest or of other related variables e g the estimation of salinity from the determination of conductivity and From Emery et al 7 temperature This also refers to the use of repeated determinations to define a statistical parameter such as the mean or standard deviation Thus we can speak of an estimate of the mean Gaussian population or distribution this probability density function PDF is a symmetric bell curve with a mean and a standard deviationlt is also known as a normal population or distribution Mean the average of a series of measurements over a fixed time interval such as a week a month a year etc or over a specific spatial interval square kilometer a 1 degree square a five degree square etc Precision the difference between one estimate and the mean of several obtained by the same method ie reproducibility includes random errors only Probability Density Function PDF the sampling population from which the data are collected This is best depicted by a histogram showing the frequency of occurrence of the data as a function of data value Random error one that results from basic limitations in the method eg the limit to the accuracy with which one can read the temperature of a thermometer It is possible to determine a value for this type of error by statistical analysis of a sufficient number of measurements it affects precision Truly random errors have a Gaussian probability density function Standard deviation variance the square root of the Systematic error or bias one that results from a basic but unrealized fault in the method that causes values to be consistently different from the true value Systematic error cannot be detected by statistical analysis of values obtained and affects Accuracy Variance a measure of the variability of a probability density function PDF computed as the mean square difference between a sample value and the sample mean Re resenfafiveness a Constant SOS I 39 quot 39b Isopycnal mdepth 1200 m 39 322 51 Wha r s a mean Mean a r dep rh Mean a r Densi ry Aliasing 6 Noise e Other par rs of the Spec rr um e Who r is rhe signal Who r are rhe physics you seek ro s rudy What are The proc ricol consequences Ocean S ready Circula rion Deduced Largely From Wa rer Masses n PACIFIC Deduced Secondarin From Pressure Field inferred Veloci ries w fjsztatz39iiicizxWe a mum Antarclir Qrcumpu mr Currenl System NIIW cucumpmar DEer Wan NADw North Auannc D1 Mater p at BIW Band 1 1th mediale Water lehwost Indian lmermedmt Water Plate 11 see p 11 A Lhrecrdimensiunal schema Showng the ne xonas overmrnmg clrculatmn in each of the Oceans and me honzunm connections in z e Snuthe n Ocean and the Indonesian ThroughilowThe surface aer39 Cvl39culnnons are in pur plcine n1ediate and SAMW are In red deep in green and Hoarbottom in lue mm Schrmu 1995b Wa rer Masses NORTH ATLANTIC DEEP WATER w Id 0 PACIFIC DEEP Ol CEO lt 34 50 4 34 4o ATLA NTm MEDITERRANEAN R15 BALTIC ANTARCTIC BOTTOM 50 77 WATER T SOUTHERN INDIAN From Wor rhing ron 81 via Emery e r al Using o rher rracers makes For even be r rer dis rinc rions Obsgt Pressure gt Velocity a S T p gt densi ry Equa rion of S ra re known by lab work for seawa rer Hydrosl39al39ic Balance A r any dep rh he pressure Force area is rhe accumula red weigh rarea of rhe wa rer above you plus a rmospheric pressure The densi ry de rermines rhe weight 0 e Geos rrophic Balance Horizon ral pressure anomalies lead ro mo rion In s ready s ra re rhe Coriolis force due l39o rhis mo rion may balance pressure anomalies lt9 Thermal Wind Balance In reali ry don f know pressure jus r densi ry surface displacemen rs are a problem Thus assume level of no mo rion or use veloci ry or sa relli res or bo r rom pressure e rc ro leverage dudzgtu Ocean S read Circulation Subuopioul Gyms E ualorinl and Tm ul Cuculanons Imexgyxe andor xmerbusm Exchaan Polar SrSubpblax Current Systems Tips on Paper Writing for ATOC5051 1 Contacts The professor for this class is Baylor Fox Kem per bfk cooradoedu 303 492 0532 Office Ekeley room S2508 httpcirescoIoradoedusciencegroupsfoxkem perclasses 2 Getting Help I am usually available by email You can also come to me from 2 5 Tuesday or Thursday or by appointment other times If you are having trouble with the writing there are lots of places to nd help You can make an appointment at the writing center1 You also might ask last year s students for tips You can see examples of their best work in the proceedings I ve put my favorite writing style guides in the bibliography Turabz39an 2007 Strunk et al 2005 Montgomery 2003 3 General Comments on Papers Before you get worried about writing ve 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 are supposed to be practice in writing drafts for your real research The idea is to get used to pounding out a working draft in only a couple of hours so that when the time comes for you to do it for real that part will be easy 31 AGU formats and templates We will be writing all of the papers according to the style page length and guidelines of the American Geophysical Union AGU journal Geophysical Research Letters GRL GRL is geophysics7 own version of Nature or Science and it contains only very short focused articles 4 pages usually 4 gures or fewer I chose this journal because it has very clear guidelines for formatting and reviews as well as an online article length checker that we can borrow Regardless of your specialization you will probably have an opportunity to write a GRL paper in the near future so this will be good practice 32 Abstracts What are they and do you need them You need to have an abstract on every paper It is a summary of what you ve done with enough detail that a reader can decide whether your paper has what they need or not in it and they can quickly refresh their memory as to which paper of yours it is too It s the rst thing after the title and authors7 names Imagine doing a google scholar search for a keyword when you are working on one of these papers For example 77North Atlantic Deep Water77 input to scholargooglecom just got 6590 hits so how do you sort through them 1 The number of citations generally tells you if it is a useful andor a controversial treatment 2 You read the titles 3 You skim the abstract 4 you skim the gures reading the captions only You should write your title abstract and captions for this audience someone skimming a mess of papers on a related topic trying to nd the particular treatment or fact that they need without reading all of the papers 1httpwwwltooradoeduPWRwritingcenterhtml 2httpciresltooradoedusciencegroupsfoxkemperclassesATOC505107proceedings 33 Graphics A gure should be included inside the text just after the gure is mentioned in the text It makes for easier reading with gures on same page as discussion3 Note every gure deserves at least one sentence of explanation in the text Every gure should have a caption which should be short but detailed enough Just like writing the title and abstract for the skimmer write the gure caption so that by reading just the title abstract and gure captions gains an outline of the work 34 Acknowledgements Pay and Friends Over time the acknowledgements has become a place to state who paid for the research you ll notice journal articles with acknowledgments that begin This research was funded under NSF So you can begin with this if you like eg This research was funded under a GRA sponsored by insert advisordept here More importantly if you talked to classmates or other teachers and they gave you a good idea it is good to mention them here for two reasons 1 It is a nice way to recognize their help and 2 it closes the door on plagiarism What I mean by 2 is if you state that someone helped you in some regard then they can t say that you stole the idea from them Instead you just borrowed it with adequate acknowledgement 35 A Special Role for Facts Because of the special role for facts in the scienti c method scienti c papers must be very careful when dealing with statements of fact There are three ways to make a factual statement in a paper You can 1 Prove it in data or analysis 2 Cite it and pass the buck to another source 3 Speculate it and clearly indicate you re doing so If you aren t sure which one you re doing you aren t allowed to make the statement at least not in a scienti c paper For example if you are trying to make a point like The oxygen content of NADW is anomalously high You can 1 make a gure 2 cite a source or 3 hypothesize that it should be high because the NADW was recently near the surface where it equilibrated with the atmosphere and then sunk quickly below the depths of important biological activity Or if it is an important point you can do all three 36 Citations When and Why Citations are a bit like the acknowledgements in that they shield you from plagiarism but they also serve another equally important role they allow you to pass the buck to another authorwork who has proven it elsewhere 361 How to do citations in IATEXwith BibTeX DATEX has a sister program called BiBTeX which processes a database of bib les to extract and label the citations you use within a particular paper In the atocsampletex le that I provided there is also an atocbibliographybib le that has some useful references for the class If you want to add another one just cut and paste one of the existing ones and edit it For example the rst reference in the database bib le is 3In ETEX use preprint rather than draft BOOKPedlosky87 AUTHOR quotPedlosky Jquot TITLE quotGeophysical Fluid Dynamicsquot PUBLISHER quotSpringerquot ADDRESS quotBerlinquot YEAR quot1987quot EDITION quot2ndquot PAGES quot710quot Most of the listings are obvious but perhaps the most important is not It is the reference name Pedlosky87 If you want to cite this book anywhere in a LaTeX le you can use citetPedlosky87 When LaTeX sees that it will automatically pull the reference and put it in the bibliography The citet command gives you an in sentence form eg 7As we can see from Pedlosky 1987 Another useful command is citep It gives you a parenthetical reference eg 7The ocean is big Pedlosky 198777 There are other examples in the atocsampletex le Depending on the kind of LaTeX program you are using you might have to do the following to get this to work eg Mac TeXShop Run LaTeX run BiBTeX run LaTeX run LaTeX again The rst call sets up the reference the second one gets the info from the bib le the third one completes the reference and the last call gets the numberinglabelling right 37 Acronyms De ne upon rst use eg The Gulf of Mexico GOM is warm The surface of the GOM is even warmer 38 First Person No rst person or at least very infrequently So instead of 7I downloaded ODV 7ODV was downloaded 39 Dataset versus Plotting Program This may be due to the nature of the rst assignment but be careful about where the data comes from For example the data is not from Matlab or ODV It was from Reid and Mantyla Matlab was just used to plot it Often you will not need to say how you plot something but you will always need to say which dataset it is usually including a citation so that someone else can understand what you re showing or look it up 310 Where in the World On a similar note where are your gures located in the world Any gure you show should be labeled or captioned telling the geographic location Latitude and Longitude may be quickest but including a map of the sectiondata point may be nicer depending on the point you re making To say that a CTD cast is located in the Atlantic is not really speci c enough 311 Piggyback off of reading or lectures Many of you already realize that starting from a statement made in one of the readings makes for an easy start to the paper This is generally true because all of the references you need and all of the terminology is probably right there However you may end up with less exciting conclusions eg 7Just as Pickard and Emery said it would be NADW was there Many of you will take the bolder route of just plotting something up and trying to make sense of it This is harder because it s not easy to gure out where to nd help You can ask me or use google scholar and that may help but more importantly be circumspect about what you say If you say the temperature is warmer at the top than the bottom and that is what your gure shows then great If you say the temperature is warmer at the top because of solar heating you either need a citation or need to be obviously speculative eg 7Presumably the temperature is warmer at the top due to solar heating7 It is best to be both interesting and correct If one must choose it is better to be correct and boring rather than interesting and wrong at least for the purposes of this class 312 Where can you make a paper more interesting The introduction and conclusion are a good place to stimulate broader interest In the introduction you can motivate the research with whatever you like including appropriate citations of course In the conclusion you can often speculate as to the importance of what you ve done or directions for potential future investigations In the middle don t try to push too hard just state what s in front of you and add more gures if you want to show something else 313 Use Google Scholar and Web of Science You will learn quickly the importance of writing a good abstract because you can scan the abstracts of the papers that you nd to see if they will answer the question you ve got in mind Also a good citation will save you pages of discussion and hours of ddling with gures 314 Who is your audience Think about how to make the classroom assignment extend beyond the classroom Can you address the underlying questions in the assignment but do so in a way that reads like an article for the general oceanographic community Or at the very least will any student taking any version of an Intro to Physical Oceanography class get something out of it 315 Cut off debate When you7re out of timeroomideas It is not necessary or possible to address everything in one short paper or even one very long paper So zealously assert your right to stop somewhere and try to choose that somewhere on a logical basis eg the dataset doesn t extend there or this paper focuses on Oxygen not Nitrate or we don t yet know how to calculate the velocity elds from the data we have etc It is OK to stop anywhere that is convenient but be clear about why you stop there The reader may be interested in following up on your work and it helps to state what you d need to go further in that direction References Montgomery S L The Chicago guide to communicating science University of Chicago Press Chicago 2003 Strunk W E B White and M Kalman The elements of style Penguin Press New York 2005 Turabian K L A manual for writers of research papers theses and dissertations Chicago style for students and researchers 7th ed ed University of Chicago Press Chicago 2007 vaii riy Waves Baylor FOX Kemper AT OC SO 51 Numbered Equa rions and Figures From Kundu The basic sefup For surface gravify waves Fig 74 Wave nomenclature Surface Gravi ry Waves are irroi39a rional This is usually jus1L assumed buiL if s rems From he vor cify equa on barofropic inviscid ihaiL is Any 2d velocify can be wriHen as w 81 62 w 82 81 Surface Gravi ry Waves The Equa rions Irrofa onal Surface Gravity Waves The Equations The irrotational incomressible Flow obe s Solid Bottom Pressure Matching dynamic Velocity Matching kinematic Surface Gravi ry Waves The Equa rions Bo r rom Pressure Ma rching dynamic Velocify Mafching kinema c Surface Gravi ry Waves The Equa rions me has Bo r rom Pressure 3 f Ma rching 7 Hgn dynamic H Velocify 8 Mafching quot7 13 kinema c d 3 Surface Gravity Waves The Solu rion The irrofa onal incompressible Flow solns are fhus n a 205k wt ee L A v I 1 k sinth mm m velocity components are found as cosh kiz 111 sinh kH sinh kz H sinh 11 u aw cosk39 wt w L a sinkx 1 Sur ace am y aves Sfreamlines The Sha Fig 77 Insmnnmcous streamline puucrn in 1 surface gravity wzwc propugmmg m the I lghl Par ricle mo rions The u v decay exponem ially foward fhe boHom wifh decay scale propor onal b x c nuullu c 10 fhe wavelengfh Thus kH is a measure of nondimensional dep rh u IXcp39 39 I c snulw Surface Gravity Waves The Solu rion DE er on P w fgk Lanh kH 40 rda on Phase Speed Group c E SPeed 2 sinh 2ki1 These waves are dispersive Surface Gravi ry Waves Shallow Wa rer Case Dispersion w gktanth 7L 7 rela rion Phase Speed C 8H Group Speed These waves are NOT dispersive Surface Gravi ry Waves Deep Wa rer gtgt H 0 C052 tanh kH 1 Dispersion w gktantha w rela rion U Phase Speed Group Speed These waves are dispersive Though rs For rhe Beach Larger Amplifude We assumed small waves Le 77 lt H When waves ge139 bigger or dep rh ge139s shallower 139hey s reepen and break essenfially 139he equa139ion is 9H 77 Though rs For the Beach Parallel to shore IF anes approach beach at an angle C I ensures wavesihaf reach shallow wafer rmL slow and leiL waves in deeper wafer cafch up In rernal waves Reduced Gravil39y gum Waves over a mo rionless abyss 41729 Phunomcnnn of quotdead water in Norwegian ords As we would expec r From our layered shallow wa rer eq rns Shallow wa rer waves Infernal Waves in Coniinuous Siraii caiion 2101 Gravity waves and convection In a Bousslnesq uid Let us consider a lloussinesq fluid initially at rest in which the buoyancy aries linearly m th height and the buoyancy frequency N is a constant Linearizing the equations of motion about this basic state gives the linear momentum equations B u39 a at r a 39 1 4 17 124331 a 6X 5 62 the mass continuity and thermodynamic equations 2 38 3 w39N 0 2216mm where for simplicity we assume that the flow is a r39unCtion only of X and L A little algebra gives a single equation for uquot Seeking solutions of the form tu Re l1 explith m2 ml I where lie denotes the real part yields the dispersion relationship for gravity waves Some s rrange perpendiculari ry in In rernal waves Incompressibilify means Dispersion rela on Form means Some frange perpendiculari ry in Internal wves wave group II 1 4 5amp7 p P P wnv gmup n I gt r Fig 735 Illustration or phase and group propagation in inlcrmxl vm39cs Positions of a wave group at Iwo limes urc shown The phase line PP H Him I propagums m NY at 2 Some s rmnge perpendiculari ry in Infernal waves 3 c c 12 736 WW gencmlcd in a slrulilicd uid or uniform buoyancy l39rcqucn y l rads The mm Agent hc pnprr 5 With lhc y n n Ilm39izomul cylinder wih im axis perpendicular 0 Ike pl or mhng mmnu m frequenty w o7x ds wim mN rum 20L le Hm agxce 39 elc ul39 4 r by Ihc humus n he hurimmal The rcnical dark lulu in the may lmlfnl me plmmgmph is me cylmdcr Mlppult and gthnukl be ignored The gm and dark 4393quot 39 39 Thcschu ic pllomgrnpll shows me umuom of mm c ror hr ram 13mm Pliomglaph N vacnsou Univcnily 0F Manchester e yambuluw me xupplim by 1 T Internal Waves Obscura or Essential Recall that we needed deep mixing to provide energy to drive the Meridional Overturning Circulation by mixing up deep abyssal water with buoyant surface waters 0 This energy is partly From winds via Ekman suction esp in S Ocean Q 6 However much of the energy is From internal wave tides and radiated internal waves From the surface and topography In Fact internal waves are strangely ubiquitous throughout the world ocean with a common spectrum the GarrettMunk Spectrum Coriolis and Pressure Geos rrophic Balance Baylor FoxKemper ATOCSOSl LamL Time 6 We ve discussed wha r happens when rhe Coriolis Force is unopposediner rial oscilla rions and TaylorProudman Flows Las r rime we had Coriolis and Fric rion Turbulence roge rher Ekman Boundary Layers o This rime we will ex rend our unders randing of TaylorProudman Flows and Ekman Flows to include Baro rropic Geos rrophic Flows 6 Consider a Small Ro O 1 balance so a Coriolis over a layer fgtltvF as Fric rion over a lay r TOP VIEW Wind stress d Seasuri39ace Corlolls Surface current Ekmdn d Ekman H W Ekman transport Interim onSI er a balance so Coriolis a r any dep rh a Pressure Gradien r a r any dep rh TOP VIEW Wind stress 2 Sea surface Inlet nr Where does The pressure come From l34 Hydrostatic balance The vertical component 7 the component parallel to the gravitational force g g of the momentum equation 15 DIL l Dt p82 939 where w is the Ql tital component of the Velocity and g gk If the uid is static the gravitational term is balanced by the pressure term and we have a 7 r a pg HUB Where does rhe pressure come From cons ran r densi ry Ocean f Surface of 77 isobars In a consfamL densify ocean p pognygt Z Physical Geos rrophy cons ran r densi ry becomes f X V gV77 no re F ver rical so horiz vel onl Wind stress 2 Sea surface Ekman Interior Then V Physical Geos rrophy f x v p cons ran r densi ry becomes 3900 Pressure 7 39 quot Grudien r 39 39 Geos rophic 7 Flow Cons ran r Densi ry Geos rrophic Flow is in Taylor columns Where does rhe pressure come From baro rropic Ocean 1 Surface 17 isobars isopycnals In a barofropic ocean 771 p pp p pgdz z hysica Geos rophy baro rropic becomes no re f ver rical horiz vel 39 no dudz Then f X V fXV gvquot Wind stress 2 Interior 39i5 Physical Geos rrophy Then f baro rropic becomes 2 SH Taylor Proudma 7 39 11 N 3 quotr Pressure Gradien r erso Geos rophic Flow and Thus Barofropic Geosfrophic quotQ Flow is in 392 4 39 Taylor columns ysical Geos rrophy onSIer a balance so a Coriolis a r any dep rh Pressure Gradien r a r any dep rh to F TOP VIEW Wind stress 2 Sea surface Interim Review of Balances Full Momem um 88 V39VVfXV gilEV2V p Review of Balances Inerlial Oscillalions 88 V39VVfXV gilEV2V p Adveclive Rossby Number Small Temporal Rossby Number 01 Ekman Number Small VE U E16 W ROG 1 if ROtifiT Review of Balances Ekman Layer 88 VVvf gt1V A H gz 1112V39v Adveclive Rossby Number Small Temporal Rossby Number Small Ekman Number 01 VE U E16 W ROG 1 if ROtifiT Review of Balances Geos rrophic Balance 8 a V39VVfXV gi7EV2V p Advec ve Rossby Number Small Temporal Rossby Number Small Ekman Number Small VE U E16 W ROa 1 if ROtifiT Review of Balances Hydrostatic Balance 8 VV39VVfXVZII giIEV2V 875 p AspemL Ra o Small HL2ltlt1 See Vallis 211 212 I r39s good ro be linear 6 Consider a small advec rive Re How 8 V7 V fgtltV p g2 I VEV2V 915 p Ekinan f X VE VEV2VE Wind stress 2 Sea surface J39qu add up superposi rion v VE IF if were s rrongly nonlinear 6 Consider a 01 advec rive R0 ow 9V V A gI V39VVI fXVZ l gZI VEV2V p a i Nonlinear Inerlial quota vi W f gtlt vi 0 Nonlinear Ekman vE VvE f x vE IEVZVE V1 Nonlinear Geos rrophic Vg va f 1x vg 1 They don 1L add up V 75 Vi l V Vg VVv viVvivE VVE IVgVVg Because of JrVi 39 VVE VE Vvi vi va red cross ferms Vg Vvi Ir vg VVE V va I r39s good ro be linear 6 Consider a small advec rive Re How 8 V7 V fgtltV p g2 I VEV2V 915 p Ekinan f X VE VEV2VE Wind stress 2 Sea surface J39qu add up superposi rion v VE Visualizing Geos rrophy in NH An ricyclone clockwise Flg 25 Schematic of geosrrophic ow with a DOSiIIVe value of the Coriolis paramerer Flow is parallel to the lines of conslanr pressure isobars Cyclonic ow is Cyclone Coun rerclockwise j anticlockwise around a low pressure region and amicyclonic ow is clockwise around 2 negative as in the Southern Hemisphere antilcydonii ow would a high If j39wer be anriklockwrs Visualizing Geos rrophy in SH An ricyclone coun rerclockwise Cyclone Clockwise 1515m515q 2il0i103 arh 10 aulsv svuizoq 5 mm wo iriqonzosg io immsrhz 25 98 2 won Jinobv lemdozi 51u2251q manna 30 zanil 9th 01 lsllsisq 2i on I bnuou eaivnbob 2i won ainobvaims hm noist Biuzzsxq wol 5 bnums eaiw aobims bluow won inolv3ims 91sriq2irn5H msrhuoa 5d ni 25 svilsgsn 915w il rigid 5 sz erwbiimE 5d Thermal Wind Hl her ressure Vp Lower ressure g P gt P O ugt0 L M lt 0 OWBI pressure V p Higher pressure Fl l 26 The mechanism of rhermal wInd A cold uid IS denser than a warm uid SD bv hydrosrasy the vemcal pressure gradient IS greaterwhere the uid Is old Thu ressure gradients form as shown where hxgher39 nd Iower mean relauve to the average at rhar heighL The horizontal pressure gradlenrs are balanced by the Coriolis farm produrmg for f gt 0 the horizontal winds shown 2 inm rhe paper and 1 cut of he paper Only the wmd Shear S given by the thermal windr Ro ra ring Coordina res Coriolis and Cen rrifugal Forces Baylor FoxKemper ATOCSOSl Numbered Equa ons and Figures From Vallis We ve discussed 6 Energy Sal r Volume budge rs Advec rion Diffusion Material and Eulerian Deriva rives Po ren rial Tempem rure and Po ren ridl Densi ry Now i r39s Time To ge r down to rhe No rions For The Mo rions of the Oceans rhe Momen rum Equd rion Fundamental Equations of Motlon of a Fluld The following consiiiutes in principle a complete set of equations for an inviscid uid heatcd at a rate Q and whose composition S changes at a ram 5 Evolution equations for velocity density and composition D V I7 m pi Dp DS a 0 i Dr v 39 D 5 Internal energy equation or entropy equation p Dr l awv g 7 p U Q1 TQ g D D where Q 45 is the total rate of energy inpull Fundamental equation of State I l pS 7 Diagnostic equation for temperature and pressure 6 rlt an 5 9 Variables Momentum Equa rion Local Advec rion Ra re of Of momen rum Change Pressure Ma rerial Deriva rive O rher of Momen rum Body DvD r Forces This is Newfon II In an inerfial Frame fhe raie of change of momem um is equal 0 fhe Forces applied Iner rial Frame Fig 2l A vector C rotatlng at an angular velomy 1 I an pears to be a constan vecmr In the roraung hame whereas In me Inemal frame I evolves according IO IdCFdl I nxC The change in C a xed vecfor in r01L Frame 6C ICIC0595m 21 The change in C The change in C a vecfor xed in r01L Frame dC 7 9 dt 1 Qx C 3 Thus For any vecfor B The posifion nd velocify in a rofafing r 21 A venw c muung ltmvdmg m mmu T nxC Frame Thus For any vecfor B Q dt dB d 25 Thus he posi on change gives he velocify dr 1 dr 3 mvRan QgtB R er R m 21 A venw ruutlng The accelerafion in a rotating Frame 7 QXB 11 R dun chm nka I R df dt d v d dtw QXH dtRQXquot Rquuxrnx mxvgaxmxry R dt I 2 I A vertw c rutallng The accelerafion in a rofafing Frame dt g mxvgaxmxrr K I it Coriolis Cenlrifugal Coriolis Force Ffquot 20 x UR 213 It plays a central role in much of geophysical fluid dynamics and will be considered exten sively later on For now we just note three basic properties ii There is no Coriolis force on bodies that are stationary in the rotating frame rm The Coriolis force acts to deflect moving bodies at right angles to their direcrion of travel Hill The Coriolis force does no work on a body because it is perpendicular to the velocity and so u a x uquot 0 A venw r muung The accelerafion in a rofafing Frame mxvrnxmxri K I it dt Coriolis Cem rifual r Then using me L39UOI39 idunlil the rst term is zero we sue mm Centrifugal farce If rA is the perpendicular distance from me axis 0i romiion see Fig 21 21nd Slthlllul r or C lhOl39l because a is perpendicular to ri n x r n x 0 x rl i I r 7 I Url and noting hm Ihn cemrifugal force poi unit mass is just given I Fuv7 xtnxr21r 2131 This may usefully be written as the gradient of a scalar polamial E V Pm r 21 A venw r muung The accelerafion A in a rotating mn7 1 Frame 22xvgax2xr R I dt d Corlo IS Cem ri ugal In an iner rial Frame he momen rum eua rion is In Ro ra ring Frame wi rh F Vq and absorbing cen rrifugal con CentrifugalSchmen rrifugal Of which way is up Cemmugal Figure 22 Ha me mlenhm of me 39 3E2 led in 1 m cambme into d ih 119 local Cl m i rth From Cushman Roisin amp Beckers Chp 2 r 21 A venw r muung The fracers in a rofafing Hume In an iner rial Frame rhe rracer equa rion is D EF Vv O e2m In a ro ra ring Frame he Jrracer equa rion is rhe same Because rhe change ro rendency cancels rhe change in advec rion mm W vVltIgtvRQgtltrV CenfriFugalu Schmen rrifugal quot Haw m attening of m cal so mat equilib nm is It hei From Cushman Roisin amp Beckers Chp 2 CenfriFugaln Schmen rr ifugal r Centrifugal Figure 23 Equillbxium surface of a ing uid in an 0an Cnntdlnel The such mm gruvimtinndl farce mmbmc int 11 1 h iuned n ma lncul normal u he su From Cushman Roisin amp Beckers Chp 2 Figure 28 Orblls full quot 39 39 V From Cushman apparent unjecmry Each row shows the siualion for h differenl inmle candllion and after 134 ml 12 and h mll pelind 2m Orbits differ accoldlng to the initial velocily the rst mw a shows ROISI n amp BeCkerS l l K n V Kant the third row C conesponds O the Opposite inilial velacily UD 1 w exam and C h 2 the lust OV d corresponds k an lithium initial velocity P LargeScale WindDriven II Wes rward In rensi ca rion Baylor FoxKemper ATOC5051 Numbered Eq rns amp Figures from Vallis Chp 14 or CushmanRoisin amp Beckers Chp 8 The Sverdrup Transpor r could close to rhe east or rhe wes r The vor cify equa on For fhe inferior geosfrophic Flow d fin7 139 I w 1 m II Surface w was wind driven Ekman I l 6 T i T39T u a k r Bk Du 01 f all Wha r about a bo r rom Ekman L yer Inferior Geosfrophic Figure 85 Divcrgcncc in he bollom Ekmam layer and compensating duwnwclling in the interior Such u siluuliun arises in the presence of an unlicyclnnic gyre in the inlerinr as depicted by the urge horiznmul iurnws Similarly inicnnr cyclnnic motion causes convergence in the Ekmun laycr and upwelling in the interior Boundary Wha r abou r a bo r rom Ekman Layer Inferior Geos rro hic 1 013 0 0139 8 2021 820b w l d l fx A Wha r abou r a bo r rom Ekman Layer Solu rion in BL 3917 1 eizf39l l cos 7153 sin 1 82321 1 39d 3 V gquotd 1 n 5111 d k z 1 c on I 8sb Ekman Volume Flow in BL U quot 11 17M v f1 W 8241 u 2 824b Wha r abou r a bo r rom Ekman Ekman Volume Flow in BL l r 7 r If I 2 R The S rommel Model Ekman Pumping From wind I l 9 T E 7quot 3 r a r r 7 Lk 0 HT f 01 Ekman Pumping From boHom drug 36 7L1 0139 1 OF 717 N 39 d 39 1 01 y 0139 y I v27 1 V 1 31f 1 0 The S rommel Model 539 r A y I I J vJl J p A I h I 1 C C 1 77 I i 77 r 7 WT Curl V lt o Cu 139Clu lt O Screamfunction Wind Stress The S rommel Model Z A O dhlllllll 39 iiiiii vorfex sfrefching inWBC vorfex squashing inSverdrup d H lt ltlt n The S rommel Model can39t close in Eas r f 30 2fo dVXV Hd pof 2 AZ VXTlt0 dhlllllll iiiiiiiiii e e e G vorfex squashing in Sverdrup vorfex squashing T Tin wsc d H I I ll lt lt 39Fl V X v gt 0 But Vor rex Squashing everywhere means sou rhward Flow everywhere The Munk Model The Munk model is a 01L like Siommel39s excepiL insfead of bo om drag if uses laferal Fric on which fends 0 remove rela ve vor ci ry near bndy Streamfunction gt Wind stress The Munk Model doesn39f close in rhe Eas r Now if rela ve vor cify near bndy is removed insfead of increasing vor cify 0 move norfhward Fric on decreases vor cifyuINCONSISTENT Streamfunction gt Wind stress Iner ria IF we blow ihe wind harder he Flow and hence he Rossby number increases which makes advecfion oF voriiciiy Similarly iF we decrease ihe boHom drag he Flow and hence he Rossby number increases Lef s discuss a Few aspecfs of wha1L we see in he presence of inerfia P rely inerlial boundary layers or Charney Boundary Layers are limi red Some Friction seems required l gt0 I decreasing g39 quot535 9 increase I t E idecreesmg I dacrsaslng immeasan I I I I decreasing I quotmusing Efcraaslr lincreesing 39 I gmcyeaging i ncroasngl I I1 Itso ltO I o Flg 1410 Putative inertial boundary layers on netted to a westwards owing interior ow left panel or eastwards flowmg interior flow right panel in the Northern Hemisphere Westwar w into the western boundary layer or ow emerging from an eastern boundary layer is able to conserve its potential vorticity through a balance between changes in relative vorticity and Coriolis parame er But ow can or emerge smoothly from a western boundary layer Into an eastwards owing interior and still tonserve its potential vorticity The right panel thus has inconsistent dynamics Purely iner rial solu rions exis bu r r y can39f be Forced because can39f be dissipa red Fig la The Fofonoff solutionl PlOl red are contours streamlines of 1487 in the plane 0 x XI 0 r y yw with U 1 1 1 03 and 5 01 The interior ow is westwards every where and w O at y m n addixion boundary layers of thickness 51 U7 bring Ihe solurion 0 zero atx 10 XI and y l0 ynl except ing small regions a the corners V As Fric rion decreases the Fric rional layer ge rs smaller and smaller As Fric rion decreases the Flow ge rs Fas rer and Fas rer As Fric rion decreases the iner rial boundary curren rs s rar r ro Fail in some locations Wha1L lhen ddies Coriolis and FrictionTurbulence rhe Ekman Layer Baylor FoxKemper ATOCSOSl Numbered Equa ons From Vallis La51L Time Las r rime we discussed wha r happens when rhe Coriolis Force is unopposed o If i r combines wilh rime deriva rive we see iner rial oscilla rions 6 IF Coriolis s rands alone we see rhe Taylor Proudman ows columns39 oF Fluid where veloci ries do not change in the direc rion of the axis oF ro ra rion This lime we ll see wha r happens when we have Coriolis and Fric rionTurbulence roge rher First a huge Kg The F Plane a rangen r approx Dv 1W W Au 7 a V Centrifugal 2 3 v Wu 4mm r v 5v Bl lvTvzmur2quotw1 g w Wuwzr v 42quotquot fig 17 Bl 702 where me rotation vector 0 12 12quot 4 03k and 2 0 12quot QC S sin 9n If we make me ImdiIional appl oximatiun and so ignore me camp in me direction of mu local m u ml than Du 13p l L 839 or quot U pax A n D1 1 3p i Wi 8b D qu paquot 1 l Du39 lap A 2 8 D p 32 H I c where 390 20 Min 90 l e ning the horiwnml ulocx39yvecmr u u tum the firs 0 equations may also be wri39uen as Du i I mu Vn 7 f0 2 sin 90k Second a big simli ca rion The be raplane linearized abou angen approx DU 1 720gtltu7AV VltIJ Dt p F7 F Du 1 7 c x l V 1 D1 f 39 p J mere f U Will in component form this equation becomes Du lap i v u 7i 3 Dr fl p ax 3a f Ji1 B So TaylorProudman is approxima rely e No change of u v in direc rion of F local ver rical a What could break such a symmetry 6 Any other rerm in the equations Le r s consider Frictionturbulence 6 Consider a balance be rween Coriolis over a layer U fgtltvF a Fric rion over a layer a The viscosi ry eddy or molecular dIs rrIbu res In rhe ver rlcal TOP VIEW Wind stress E Sea surface 1 Ekman transport Interim The Molecular Viscosi ry Divergence Viscous Flux nearly cons rah r viscosity Fig 8 quot Illustration of molecular proee quot ow 10 1y Laplacian is deviation from neighbors LL molecules U is the speed of the molecules Random molecule motions early information about larg ale ow to other 1 nr 115 ting ous stresses scous stress depends on the mean molecular sp c mean molecular free path lL39l The Reynolds S rresses and eddy viscosi ry FV VVV Divergence Viscous Flux I Fe v Vv V1 vivj Advec rion ferm Divergence Eddy Flux IF eddies are like molecules lhen I Tha r is we jus r need a bigger rhan molecular viscosi ry rhe eddy viscosi ry Scaling rhe Momentum Equafion Following Vallis 2121 8311 fx u V 182 Or For jus r rhe per rurba rion veloci ry due ro viscosi ryz Cons ran r Eddy Viscosi ry Solu rion For per rurba rion u v From CushmaniRoisin amp Backers 2 139 v 7 m rm 0 f u 17 pg 7 8321 539 amp32a 11 39 739 Surface 0 mlE rquot ning 832C 1 r Toward interim 7 7 6 832d The solution 10 this problem is 9 V 1T1 7r 83311 In391 9 I F WV T 7 v 83313 IQl 7 Resul rs in The Ekman The solution to this problem is 14quot 7quot 035 v T siu3 l 8331 CZd Tr sini 739 053 7 H 833b Resul rs in The Ekman Spiral TOP VIEW Depfh infegmfed Flow s ll f0 fhe righ1L of wind NH 8331 833b Nor rh Hem gtRigh r Sou rh HemgtLef r The solution to this problel 5 Perfurba on Follows f Coas ral Upwelling alumni Vi quotm Intel39qu upxwlling uJ Iv Fig From Cushman Roisin amp Beckers Figure 155 Schematic development of coastal upwelling I l l l upwzllmg cold Figure 156 Other types of upwelling a equatorial upwelling b upwelling alnng the ice edgei Ekman PumpingSuction Application of Volume 84 Salt Conservation Estuarine Circulation Baylor Fox Kemper September 26 2007 Here we will H derive the budgets for volume and salt conserva tion M then do examples from the Black and Meditere ranean Sea from Pickard and Emery 1990 0 consider pollutants along with salt 7 and finally solve a timeedependent problem to demonstrate the role of the flushing timescale The most essential use of the equations of motion of a fluid is to calculate budgets over a known volume or mass of fluid Perhaps the nicest example of this in oceanog raphy is the flow into and out of estuaries This estuarine circulation is a useful example of the use of volume and salt conservation attributed to Knudsen 1900 We begin with the conservation of massquot cl Dp iv o 7 v 0 1 chmquot Dtp V U Which is replaced by conservation of volume for an incompressible fluid Vv0t 2 Where we recall our notation that Vt follows a material surface Conservation of salt is neglecting molecular diffusion d DS 3Vtps dV vtp dV 039 3 Or since it didn t matter which parcel of fluid we followed and p gt 0 DS E We note that salinity can change eg by evapprecip but it is the quantity of water that changes not the mass of salt 0 4 VV O 5 applies everywhere so OVvdVjgtv dA 6 Where j dV is by Gauss s identity is equiv alent to 35 dA over the surrounding surface We break up the surface integral 0 jiv dA 7 Alv1dAA2v2dA 8 EdAi PdA vrdA AE AP AR 6 The following figure schematizes the situation Pickard and Emery 1990 There is a two layer flow over the sill runoff evaporation and precipitation PRECIPITATION lN EVAESTRATION RWIE R FLOW V1KS1 p1 V S p 2 2 2 BASlN We proceed by integrating the differential equations over the estuary Renaming using volume fluxes V L3T o dA dA 9 A1V1 A2V2 E dAi P dA dA AE AP ARVr o V1V2AEE7APP7R 10 E V1V27F 11 Where F is the freshwater supplied to the estuary Recall conservation of salt gives the equa tion on salinity neglecting mixingdiffusion DS BS 0 7 7 VS D7 67 V We use Vv 0 to find BS 0 7 Si 12 at V v Integrate to find 0VVS dVjltSv dAi 13 Little salt in rivers evap amp precip carry no salt so ova dA Sv dA Sv dAi A1 A2 8 So we have two equations so far S1V1 752V2 V1 V2 F We can eliminate either V1 or V2 using V1 S2V27 V2 S1V1 S1 52 To give F Vi y F Vl 14 S1 52 Formally we can define velocity weighted av erage salinities so 0 Sv dA Sv dA A1 A2 S1V1 S2Vz Where we define 7 fAlSv f1 dA fAl v f1 dA But more loosely there is a typical incoming salinity and a typical outgoing salinity which may make approximate values for 81 and 5392 obvious 51 We will now do two classic examples from Pickard and Emery 1990 the Mediter ranean and the Black Sea The Mediterranean has a sill depth at the Strait of Gibraltar of 330m It is observed that 81 363 psu 5392 378 psu V1 7175106 m3s2 7175 Svi Where the Sverdrup 106 m3s E 1 Sv is a useful oceanographic unit The Black Sea has a sill depth at the Bospho rus of 70m It is observed that 81 17 psu 5392 35 psu V1 13103 m3si Where the Sverdrup 106 m3s E 1 Sv is a useful oceanographic unit So we can infer that V2 76103 m3s F 7 103 m3si So we can infer from 81 363 psu 82318 psu V1 7175 Sv is V S is V2 71 1 FV172 1 52 52 that V2 1 68 Sv F 7007 Svi So for a tiny amount of net freshwater loss through evaporation exceeding precipitation and runoff a huge exchange flow is re quired with an outflow of salty water exiting at depth and fresher Atlantic water entering at the surface 13 Com are the two basins Mediterranean 31 36 3 psuSg 37 8 psu V1 71 75106 m3sV2 1 68 106 m3s Black S1 17 psu32 35 Dsu V1 13 103 m3sV2 76103 m3s F 7 Mediterranean has outflow at depth 25x the Volume of fresh water Black has inflow at depth 1x the volume of freshwater Key differences the amount of mixing in basin Med a S1 z SQ while Black has S1 lt S and inflowoutflow at surface governed by freshwater deficitsupply assuming 31 lt 2 15 We could also treat pollutants notjust salt DP 7 z 0 15 Dt The pollutants will have river sources and potentially an exchange at sill so the steady equation is V1P1 V2132 RP2 If we follow the Dollumnt thrOLJgh the Bosphorus out into the Mediterranean we have Black Sea sources P3 z EPR V3 13103 m3s And the Med budget Will be V1231 my VBPB Vg PR Using our Med numbers V1 71 75106 m3sV2 1 68 106 m3s 1 0 We find the Med outflow lS Very dilute l P miP 2 260 R Reinforcing our notion that the Med lS better migtlted than the Black Sea 18 Suppose we have constant pollutant concen tration in river discharge in Black Sea then FZR7103 m3s V1 22R P2 z 0 So the steady state result will be V1P1 V2132 RP2 P ziP 1 2 R Thus the incoming pollution is diluted very little in the Black Sea One trick we haven39t used is time depen dence While volume conservation didn39t have a time derivative pollutantsalt did Suppose we impusively started polluting the rivers then EOaiPdV7Pv ds Dt clt We know the steady state solution P55 we just did it so let39s see how long it takes to get there How row to reach steady sate fltP7 PM W 7 P7 am e as W ltP 7 PM 7 VRltPR 7 PM 7 mom 7 Pour d V Referehces P Pa OJmi PM Ahgre brackets are VO UWQ averages ahd Vor rs the VO UWQ If KWdSQH M r E hydmgraph smer ehrsatzr An HW OW e assume a weH7rhrgtlted bash the out ow COHCQHU aUOU Pour MEquot Mereomr 28 3167320 1900 MM be hear the average Pr so 03 7 P z 7371703 7 P PrCKard G L ahd W J Emery Descrptye Physca Owe7097 Thus aphy Ah Introductoh 5th ed Butterworth Herherh d V V Oxford 1990 7 g i 7 g 463 dtus PM W P P a P P As P m PM 1 754 Whrch rhtroduces the ush77g tme Vol Wm 1h the Med or 3 8 0 5 m3 rt sabout 70 yrs f rthe BraCK Vor 6 101 m3 rt s about 1500 yrs 20 Baqur FiaxeK egmper AT OC SOSI Wha r is a wave Vaves arc nor asy to de ne Whilliam 9741 de nes a wave as a rccogniy39ihlc signal that is Lran red from one part of a medium to another with recognizable vvlocrty of propagation 1 his is a very broad de nition and encompasses an enormous range of dynamical systems as well as physical procrsscs That is waves ran occur in many lill39erent media and Lake on many different forum We often think of ravos as simple sinusoidal undulations of some substance but this 39I is loo restricted and often not my ust l ull Chapman and Rizzoli Waves are The means by which informafion is rransmi r red be rween rwo poin rs in space and rime wi rhou r movemen r of The medium across The Two poin rs Kundu Some Examples 6 Surface wa rer wavewaves a r rhe beach Many of the processes leading ro El Nino Sound 6 Phone calls 6 Radio Epilepsy Traf c Jams 6 Aurora Borealis Linear Nondispersive Waves The m051L basic wave equa on Solved by 7701770 773j Ct Where n is some wave properfy eg pressure velocify Linear Nondispersive symme rrical Waves The second basic wave equa on 9277 2 8277 C 8752 8362 Solved by 77937 75 77zv 675 77 56 675 Where n is some wave properfy eg pressure velocify Plane Waves 7796t 2 ER nkeikxmdx So we consider W t 9 77k mwtgt eg plug info 8277 2 8277 C O 752 8132 yields the disPersion rela rion w2 CZkZ generally the dispersion rela rion rela res Freq ro wavenumber Phase 84 Ampli rude nk ikvlct I nk i The ampli rude of the wave is 77k The phase of rhe wave is G Radian Frequency and Period Frequency Si r ring a r a xed loca rion 77Ut 3 77k6ikac wt Repea rs i rsehC every out 2 27m 80 period is T 27Tw 80 real Frequency is f w27T Radian Frequency and Period Frequency Si r ring a r a xed rime 77Ut 3 77k6ikac wt Repea rs i rselF every 0 waveleng rh is A ZWk and k is called wavenumber Dispersive and Nondispersive Waves IF C twk Then 770B R77k zkx wt mnk zkxlct Which means lha r every wave fravels lhe same speed c 50 lhe a superposilion of diFFeremL waves will slay logelher But iF lhis isn l frue lhen diFFer emL wavenumbers will have diFFer emL speeds and will separale out Example Linear Wave Equa rions Linearized Equation Plane wave Dispersion Relation a a 0 a m 1 W cwquot 0 gin m C lt25 E Vqs o CiEi iaz d 413 03V2 0 aim m 0 via m z 0 aimwt a ikU2 Chapman amp Rizzoli Phase e rc in 2d Wave Equa rions Chapman amp Rizzoli S randing Waves The form xle l f f is called a travelling plane wave The superposition of oppositely travelling plane waves 1 1ci li Ac 0 216 c0503 1 is called a standing wave because the crests and troughs do not propagate with lime Chapman amp Rizzoli Basin Modes Even iF slanding waves are r101L possible basin modes are likely to exist These modes have no energy propagalion buiL lhere may be phase propaga on Resonance OFlen a wave will roulinely lransiiL an en rire basin IF lhe limescale 0F ihis lransiiL is close lo lhe limescale 0F Forcing eg year lunar monlh e rc lhen lhe response of lhaiL parlicular wave will be very large This eFFecl is parlicularly pronounced in lides eg lhe Bay of Fundy Wave Packef of stationary phase As a preview let us consider a one dimensional exzi nple with the special initial condition g25z 0 z aze E O The modulated plane wave is said to be a narrow band signal Group Veloci ry We can evaluate r i by expanding in a Taylor series amul k0 4411 f Alcc quot quot 39l l ll MWilkz ntkoiz lk kn lmon M a Akequ micai pka ykwqekn xm M Cilknz mka Akcik kallI lkkull I That is 89 n3 4iku139r kal 7V1 w e I M M gt The modulating envelope moves at a velocity 7901 him defined by the dispersion I Cl lon 739 This velocity is called the group velocity 09 Cg alkali Chapman amp Rizzoli GroutP amp Phase Velocity 773 75 Rnk ikc wt nkeikmict w C E phase velocity 8w 8k cg c gt nondispersive waves 09 group velocity Group 84 Phase Veloci ry In a slowly Varying medium fhe dispersion rela on is a gem le Funcfion oF39 space Phase ie cresfs and roughs propagafe wi rh C E phase velocity Groups ie wave packefsnpropagafe wifh Cg 2 group velocity w 8k LargeScale Waves Rossby amp Topographic Baylor FoxKemper ATOCSOSl Numbered equa rions From CushmanRoisin amp Beckers Chp 9 amp 12 Vallis Chp 3 amp 8 or Chapman amp Rizzoli Chp 5 Fluid s ur face I Flg 3l A shallow waxer svsmmr h is the tmckness of a water column H IIS mean hickness I the height nf the free surfata and m is he height of he lower rigid surface above some arburary origin typically chosen such that the average of qr is zero J1 Is he dEVIEKiOn free surface haighx in we have I 17 II I t Jr MomeMum Vdume Rossby or Plane rary Waves Consider very large scale waves fhus Befa plane so f fl 439 30 d fn where the dimensionless ratio d can be called lhc plulwluijv MINI76 lt1 The governing equations having become u Eh Tr ft dayquot 7 i V 439 Hullquot 1 7H 0139 H f 71 4 0y

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