Introduction to Physical and Chemical Oceanography
Introduction to Physical and Chemical Oceanography ATOC 5051
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ATOC 5051 INTRODUCTION TO PHYSICAL OCEANOGRAPHY class 25 Westward intensi cation western boundary current WBC Objectives of today s class 1 Scaling equations of motion in WB region 2 Solutions in WB region WBC Previous class Horizontal equations of motion du 1313 2 187quot a fbr a7AHVH aZv 1 z y dv ldP 16739 11 Av2 fu H Hvpaz7 By considering Ra ltlt 1 and EH 3 17 the equations of motic for the steadx ocean circulation becorne 18F 18739 fvig 75827 2 y 1613 18739 21 f 7 pag curbW 5V3 P0 0 Vs 2 vdZaTW 757 3 D hm curlT E 67 3 W 8513 8y Introducing stream function gt 8w CUT ZTW 3 33 90 7 3i 3 Us By VS 61 Flg 253 Schematic Surface Winds 92 30 N subtropfcaf deserts W n nheast tr f 4 equatoriau39 rain be I xmx WSW MM TRACKS Obs ervations Pedlosky 1987 a x 1 anus m4 curl w 6y Cg w CUTl TW IE P0 N curl39r 15N 261217 Idealized Winds 739 TOCDSZLQ T 70 at 15 N 7quot 7390 at 45 N So the windcurl is BT1 67 where an 70 3ON CUTZT To5in a05in 15N Til 261b curlTw The Sverdrup balance can be written as 37 050 7T3 2 836 posing L o By applying boundary condition 11135 LE O at the eastern boundary7 we obtain solution for 11 LE7 170 yL 2610 LE Scaling in the western boundary region p const steady state circulation R0 lt lt 1 Vertical 1 8P 2 1 U i EE 4a 16F 7 y 2 equatlon f u 567 AHVH 41339 Llnear vy 0 4c friction 4a 4mm we obtain 11 51 71quot uy AHv im uy pH Vertical integration 5Hv who Hum Aavmwm Hum a 1 T Denote Hu U7 and Ho V7 we have a y 3V w UH Aavm Uy p Because u vy 0 and thus U Vy 0 we introduce U w V 712117 97 VI UH warn 1 V21 80 the above equation can be written in term of w as rst Ww AHW 1339 5 L or7 1m 70 I wyy AHquotmcacc 21macyy I wyyyy 6 Note that equation 6 is the general vorticity equation in terms OflJ which applies to both the interior and the western boundary region N ow7 scaling the above equation for the western boundary current region using the scales Ly 1000km 106m7 Ll 100km 105m7 H 103m V H1 1000 gtlt 1m2s7 then me 10007 and 1 108171357 AH 105171257 7 10 55 1 p 1000 gm37 B fLy 104106 1010 739 01Nm27 Then we obtain the scales for each term in equation 6 iii 5121 1010 x 103 107 105 x 102 107 Way 105 x 104 109 Angn AH L39 105 x 1012 107 AHwyyyy AHLIL 105 x 1016 urn i 1i 0110 10 I I y 10 J 39 1 2AH1Jmmyy AH 105 X 10 14 10 9 Thus7 to the rst order7 the balance of terms in the western bound ary region are 5 1 Amermw i Stornrnel s solution AftJury 1 w AWE 7 Btu subject to boundary conditions 1 0 at r O7 and 1b int mm as a is large in nity By applying these boundary conditions we obtain w 1 lt9 The e folding decay scale for the boundary solution is x The WBC is VZHV 3 E 423 V 61 Yqibinteriare I 10 VmaX x0 Not very reasonable WB intensi cation ii Stommel s solution with It is symmetric in the east and west7 and thus there is no westward intensi cation See Figure v 263 No westward Intensi cation iii Munk s solution Munk chose the horizontal Viscosity to balance the ix term Boundary conditions are a At an 07 71 07 and 7413 0 no slip AS X is large in nity7 7 winterim The solutions are xgx 1i CE 1 f2f 12 7r wInmw T7 F C08 26m 87 26m d 2 a l39 J 12h U mrlIntenolzv 306 97 26m 7 where 6quot 49M is the Munk layer Width VmaX 12 6m Westward intensi cation 1 3 effect Rossby waves 2 Vorticity balance entire basin Interior Munk 1a er w Recirculation Munk layer 265 9 0 near the western boundary ATOC 5051 INTRODUCTION TO PHYSICAL OCEANOGRAPHY class 12 Objectives of today s Class 1 Wavesconcepts 2 Surface gravity waves 1 Concepts MEAN I A I R SURFACE quot L 1 CREST WATER h TROfUGH CL T 77777777 77 BOTTOM L WAVELENGTH T PERIOD C SPEED CELERITY RELATIVE TO WATER H WAVE HEIGHT 2 x AMPLITUDE A h DEPTH of WATER BELOW MEAN SURFACE LEVEL N B H 5 exaggerated relohve To A for clarity FIG 122 Terms related to ideal sine or cosine waves Radian wave number k 2i Radian frequency w 2i T Phase speed C E g T k 8w Group velocity Cg 8k 721 COSUL lfli cult A21 2 k Ak wl w Aw 772 008k21r mgt k2 k AAA L02 w Aw n m 772 coska wt Aka Am coskm wt Am Am Note that 008A B cosAcosB sinAsinB C08A B COSACOSB SinAsz39nB So 77 2003Aka Awtcoskx wt k1k2 AkT Aw 2 k1k2 m k 2 Akltltk Awltltw SUWFACE ELEVA HON g cnskawlccs1AklAul for 90 Ak H20 FIG 125 Surface elevationxar a group made up nf two pure casine waves Therefore Aw 9w p C g Ak dk Wave energy E P9A22 pyH2870ulesm2 Steepness ratio of HL is called steepness 2 Surface gravity waves Equations of motion 1 transient response 2 steady Circulation Recall that quotI L z39rt39iCLZ r 4 0 7 ltlt 110 393 on ol 2 s 73972L72g 3 quot f ltlt 1 10 quot C 0 ol s Which means we can ignore nonlinear inertial terms and mixing The equations of motion in an homogeneous constant density ocean by ignoring nonlinear and mixing terms Rossby and Ekman numbers are small and ignoring Coriolis force this is crude are 8U 0P p0 9513 7 83 at P0 9197 l 9P V 11 t 00 dz 39 8U 8V 8quot O r dm 1 2 Assume background state U0 V0 Wb 0 Po 1303 P P0 UU0 U4 U4VV039UUWWUUJ w7PP0Zp7ppU Then we obtain equation for background state iOH po 02 g0 Equations that govern perturbation 9n 1 9p I E 8v 1 3p 5 8w 1 8 at 0 0 93 91L an 821 0 d9 y dz Write a single equation in p alone 92 82 82p 0 0133 dyz 252 Boundary conditions At the ocean bottom 2 D w 0 At the sea level 1739 97 0112 t it t Z OJ 009 Assume a wave form in X amp y direction but leave Zdirection structure undetermined 7 nocos cx ly out p pizcosk1 ly out Substituting these wavelike solutions to the single equation in p alone we obtain 821 2 2 r922 1022 Asthcz BcoshUzz Where 52 k2 2 By applying bottom boundary condition 3 lv1 u 0 and thus 0 92 3 and surface boundary condition 2 04 009 77 We obtain I gponocos xv 3 39 39tcoslmz D p coshhD 39 Since 9w 1 8p at i pig 02 We have gymsMUM ly wtsmlmz D w wcoshnD Now we have p w and 77 solutions gponococ kar y wtcosimz D P I coshUcD agnosin rrt ly 39 it3392n39usz D woos392HD 39 739 39r0c03kr y wt Let W satisfy boundary oonditiow at Z09 w2 ghzt1rzhD This is the dispersion relation for surface gravity waves 42 gnttmh aD w 1 When 11D N 07 tanhMD KB 092 gDH2 wg DH7 u Ii Surface long gravity waves wz gDICZ c cg xgD c cg 9D Nondispersive group velocity is not a function of frequency and wave number 5D ltlt 1 coshMD 1 Since in QP UC03I 5E M wt P is independent of depth z aw 8w 1 8p EO39 a p082 So long surface gravity waves are nondispersive their structure is barotropic and feel the bottom of the ocean HYDROSTATIC 2 Short surface gravity waves 1 L N ic ltlt D7 CD gt 1 42 gntanhnD gt w2 gr li CbdiqiE7 60w 9 H H K 9 8572w7 Short surface gravity waves are dispersive because c and Cg are functions of frequency and wave number P 90039770005 lkfr Z2 wt quot39za 39770S i nk17 y r t razz cc ZltO solutions exponentially decay Surface trapped and do not feel the bottom 03921 at 7 01 Nonhydrostatic board demo ATOC 5051 INTRODUCTION TO PHYSICAL OCEANOGRAPHY class 16 Objectives of today s class Effects of side boundaries coastally trapped waves Previous class f f0 i For high frequency waves with large 42 262 f3 ii For low frequency waves with smallu7 Long Short w 9 W k w 2 Crk 8w f Cquot 8k k7 w 317 Dispersion curves for 8 plane W Short Rossby free waves in rnidlatitude k IGW inertial gravity Existence ft PV explain on board Long Rossby Short Rossby waves are L V k hardly seen in the ocean interior because 1 they are too short to be effectively excited by largescale winds 2 mixing in the ocean acts strongly on short waves z Oceanic adjustment 77 3 Initial peiturbation 0 b Gravity wave radiation 0 Geostrophy d Rossby waves e Rossby waves V K f Equilibrium state Quiescent If there is wind forcing equilibrium state is Sverdrupblance Will be introduced later Observed midlatitude Rossby waves by TOPEX satellite altimetry Midlatitude Paci c mm M g 800 Jan As you can see they m M propagate westward with a decreasing speed with increasing latitude 4 72 o 2 4 Sealeveucm Global observations Today Effects of side boundaries coastallytrapped waves Coasts act as waveguides a Coastal kelvin wave A y v0 gtu Coast Vertical walls For simplicity we again use the shallow water equation with constant f f0 gt 0 Northern Hemisphere Find solutions for v0 subject to boundary condition quotI 03 y 0 9H an 9t J 917 97 quotU I f 1 8y l H u 0 7t 31 From the above equations we can obtain utt 111313 Assume wave form u quotLL 71 11 wt gt W ikic Where C 39 gH This is just like the dispersion relation for long gravity waves in nonrotating system n 7721e 39 7u uye 39 t7 The set of equations yields f 77y 777 and usingw kc7 f 712 i C777 Thus V 77 7106 yeikaiwt Since sea level increases as y is farther and farther away from the coast it is not a reasonable one becaus energy should decay away from the energy source for this case the source is the coast Choosing w kc we obtain 1 39 7 77 7067 Cyez cz 9107 and this solution obtains a maximum at the coast and decays away from it Reasonable Coastal Kelvin waves propagate with the coast to its right left in Northem Southern Hemisphere Solutions are trapped to the coast decaying away from it exponentially with an efolding scale 0f F the Rossby radius of deformation b Continental shelf waves A y deep shallow Coast Hy an 87 at 398 82 9739 z u 3 0t f 391 y 7 9 H d H t I p t I Z 0 t 91 y Analytic solution more complicated acts as l3 gt y 0 Topographic Rossby waves Propagation like coastal Kelvin waves Shelf waves dispersive Mechanism like Rossby waves Potential vorticity conservation P V 30729tasrzt H Near the coast scale is small 3072sta7392i L ltf y deep H u H n H 4 HS shallow X NH coast Midterm Review 1 Basins and Properties of Sea water You do not need to remember the basin Widths etc You only need to remember the major features of each basin For example The Pacific is the largest of all oceans Because of its vast size airsea coupling is an important process for its climate such as ENSO Salinity distinguishes the sea water from fresh water Light in the sea understand light penetration law Sound in the sea Physical oceanography appliation 2 Water masses amp Observational methods Five deep sources two from Southern Hemisphere AAIW amp AABW Three from Northern Hemisphere MOW LSW NSOW Concentration amp dilution basin Measurements CTD satellite etc Geostrophic method Important to understand the geostrophic method For given TS distributions for two stations you should know how to infer the direction of geostrophic currents Maj or current systems WBCs ACC currents that close the STGs in the Atlantic and Paci c 3 Scaling For given scales in coastal or interior oceans I ll provide the scales one should know how to perform scale analysis to obtain the most important terms This is an important approach for your future research if you Wish to understand dynamics VERY important 4 Wave dynamics rnidlatitude Midlatitude waves Gravity waves Rossby waves Dispersion relation w A Inertial gravity waves f0 Short Rossby 4 Long Rossby Require Dispersion relation understand and draw Energy propagation group velocity phase speed Oceanic adjustments to equilibrium states see Assignment 3 for detail Wave refraction and breaking 5 Coastal Kelvin waves Understand dispersion relation direction of propagation and what is the cause for the coastal Kelvin waves to be trapped to the coast The efolding scale for coastal Kelvin waves is the Rossby radius ATOC 5051 INTRODUCTION TO PHYSICAL OCEANOGRAPHY class 5 Observational method Measurement of temperature Old Bathythermograph a liquidinmetal thermometergt metal pint to move in one direction over a smoked or gold plated glass slide move at right angle by a pressure sensitive bellow Continuous Tz but less accurate Expendable Bathythermograph XBT Widely used Uses a thermistor as T sensitive element which is in a small streamlined weighted casing Fine wire connected to a recorder on shipgtTz O2001800m CTDConductivity Temperature and Depth actually Pressure T is measured by a thermistor mounted close to the conductivity sensor Accuracy 0005K S CarolinaSt Lucia NOAASMp CTD bottles Measurements of temperature Protected reversing mercury thermometer N egretti and Zamba 1874 Accuracy 0004C Thermistors chains The best quality one Accuracy 0002C Buoy drifting buoy Measurements of temperature Satellite AVHRR Advanced very high resolution radiometer on NOAA satellite 1985present accuracy of 0103K Multi channel Visible amp infrared Problem Cloud vapor absorption and cloud re ection Satellite TRMM Tropical Rainfall Measuring Mission TRMMMicrowaye Imager TMI 1997 present Penetrate clouds but affected by strong rainfall measuring SST 02C difference with buoy httpwwwssmicom Acoustic tomography httpatocdbucsdedu Potential temperature Potential temperature is the temperature a water parcel has when it is moved adiabatically to the sea surface Remove T change due to adiabatic compression and expansion effects Measurement of salinity Laboratory Old Evaporate and weigh residual Lab Classical Knudsen method 1957 Determine amount of chlorine bromine and iodine gt chlorinity Via silver nitrate titration Sl80655Cl Accuracy 0025 Measure conductivity Accurary0001 0004 Conductivity salinity measurement Conductivity depends strongly on T then on S 1 Sea water sample lab autosalinometer gt ratio of sea water conductivity against sea water standard KCl control lab T important 2 CTD Electronic instrument inductive or capacitance cells are used For known Conductivity and Temperature at certain Depth gt salinity Accuracy 0005 Satellite NASA mission Still underway Measurement of Density 0 Standard laboratory method weighing bottle Not practical 0 Calculated from equation of state ppTySyP TP SP For more detailed equation see Gill s book Page 599 At one standard atmosphere effectively p0 is ms t 0 pm S0824493 40899 x 10 3t 76438 x 1r5t2 82467 x 103 53875 x 109t4 S324572466 X 10 3 10227 x 10 41 16546 x 10 48314 x 10452 Where pm is the density of pure water with 80 see Gill appendix 3 for the equation at pressure p Measurement of currents 0 Goals 3dimensional circulation gt heat amp salt transport gt climate 0 Typical horizontal currents vary from a fraction of l cms deep ocean and much of near surface 10100 cms equatorial current system ZOOCms western boundary current 0 Vertical speed loi r cmS Why Prompt class Measurements of currents Lagrangian methods follows uid parcels Eulerian methods velocity is stated at every point in the uid instrument is xed in a space Direct amp indirect current measurements Direct Surface drifters Subsurface oats Current meters Acoustic Doppler Current Pro ling ADCP Indirect geostropic method Surface drifters Lagrangian method Ship drift earliest circulation map Drift pole near current B end land mark Drift buoy sh1p extend the pole idea to open ocean A begin radio transrnitor or tracked by satellite Subsurface drogues TOGA WOCE drifter 15m Subsurface oats SOFAR oats sound source rnoored receiver RAFOS oats sound receiver rnoored sound source Pop up oats pop up regularly to communicate with the satellite bladder WOCE subsurface oats 8001000m S at Eulerian methods 1 Current meter Rotor current Meter RCM Aeeuraeya few 4 O erns Acoustic Doppler Current Pro ling ADCP Using sound Wave s Doppler Shifting effect Indirect current measurements Geostrophic method TSP gt density gt vertical shear of geostrophic currents between two station pair Coriolis force Pressure gradient force Geostrophic balance balance between the Coriolis force and pressure gradient force w l T p V X 2 Q 871 p 0 1 ATOC 5051 INTRODUCTION TO PHYSICAL OCEANOGRAPHY class 22 Objectives of today s Class 1 Heat ux 2 Mixed layer temperature equation Previous class oceanic boundary layer Ekman layer often referred to as surface mixed layer T S well mixed represent sea surface temperature sst and sea surface salinity sss The mixed layer direct contact with atmosphere subject to forcing by winds heat uxes salinity uxes Winds momentum ux drive oceanic circulation Heat uxes directly affect sst Importance for understanding sst change drives atmospheric circulation air sea interaction Hadley and Walkder circulation global climate Observations Meiidional distribution of net surface heat ux 300 7300 2507 7 250 E 2007 200 E m Surp us Heat Energy Transferred m 5 5 By Atmosphere And Oceans 50 1g E To Higher Lautudes g me we 50 A 50 9 7D 50 40 3390 20 10 i0 20 30 40 50 709 Nurlh lt Lalilude gt South Processes that determine the temperature change of a water parcel Lagrangian in the surface mixed layer are 1 Net surface radiative ux Qnr Q71 u QZTLU Solar shortwave Out oin lon wave rad1at10n g g g radiation 2 Surface turbulent sensible heat ux Q s 3 Surface turbulent latent heat ux Q1 4 Heat transfer directly by precipitation usually small for long time scale Qpr 5 Oceanic processes entrainment cooling th Mixed layer temperature equation The rst law of thermodynamics says heat absorbed by a system is used to increase its internal energy and do external work heating Example For oceanic mixed layer energy absorbed is used mainly to increase its internal energy temperature Apply the rst law of thermodynamics to the mixed layer With depth hm for a unit area For a water column of hm with an area of Am gtlt Ay internal energy increase is 77 denSIty 1011wa 1771A5EA3 C Where CPU Is the specific heat of water fkgOC dTWI For a un1t area 1t 1s pm 1pm 711m JSmz Wmz This energy increase is caused by net heat uxes from both surface and bottom of the mixed layer dTm Whm Qnr l Qs Q Qp39r l Qantv Here we assume the shortwave radiation is completely absorbed within the surface mixed layer without shortwave penetration into the deeper ocean pm 61710 Because 1T 8T T Td 7T1 r 71 V O 771 I 771 gt Tm Qn r Q5 Q1 Qpr V vj Vm ij TdHw t plucpu39hm hm Tm Td went hm Qent Paameterize to pwcpwhm This is the mixed layer temperature equation Next we ll discuss each term in detail 61 Qnr Q7 050 8w 00 Top of atmosphere 8 1 atm radiative transfer 39 Tu Maw Q Positive into the ocean b Q57 Q1 Q8 Pand w9l07 Q1 pale J1007 Potentialgtemperature Top of PBL sst Board demo for detall Bulk formulae Q5 PandCDHVa V0 To T07 Ql paleCDEVa V0 1210 qv07 0 Qp39r law prP 397 Tum d VVT I Tm 6 39went I m f Upwelling cooling T T QljZZ 9 JEIQlj IZ7TI TAO data in the eastern Paci c Color SST black arrow WindsWhite arrow Ekrnan transport 15398 30 S 140Wr 120 W 100W SDquot 60quot Equatorial upwelling meridional View 5 N a Ekman transport surface wind equator w surface wind Ekman transport 5 S b 5 equator 508 I l I a r m c 0 I d w a rm thermocline Zero I Vel Zonal View a STRONG wind T SeaSurface Zero 5Com Em level A 1 4quot lt o lt 30m 230m A g J Momma x cold wa er b WEAK wind 39 CAT 3 marwl 156 ATOC 5051 INTRODUCTION TO PHYSICAL OCEANOGRAPHY class 13 Objectives of today s Class 1 Adjustment under gravity in nonrotating uid 2 Wave refraction and breaking in shallow water 3 Swells and tsunamis 4 Internal gravity waves Previous class if gstrrnlisD Dispersion w relation surface gravity waves 1 1 Long waves L N gt D When RD N 0tanhnb RD wz Drag Surface long gravity waves 3 cg gD7 c cg gD7 Nondispersive group velocity is not a function of frequency and wave number p gponocosk3 1211 wt Barotropic independent of 2 Long surface gravity waves are nondispersive their structure is barotropic and feel the bottom of the ocean HYDROSTATIC 2 Short surface gravity waves 1 LNEltltD 42 gntanhMD gt Short surface gravity waves are dispersive because c and cg are functions of frequency and wave number P Jp 77003k Z Wile Ago7 708i72kr y r 1km L4 ZltO solutions exponentially decay Surface trapped and do not feel the bottom 811 at 013 Nonhydrostatic 1 Adjustment of ocean under gravity in nonrotating system Free waves in this system Long amp short surface gravity waves 1 Initial state quiescent ocean or pond Math U0 V0 W0 0 P Po 1 8P0 77 0 P0 32 g 2 Perturbation 8a 1 8p Math 9v 1 810 at 8w 1 8p 9t 00 3239 8U 9U 8w 8m8y82 3 Energy dispersion W2 gmtanhMD w Long amp short surface waves 4 Equilibrium state Quiescent ocean or pond again Refer equations 2 Refraction and breaking in shallow water Above Long surface gravity waves C V913 Cg ng y HO Set up by forcing Refraction long surface gravity waves approach the coast DFEP WATEP gt z P A 31chst 1 lt A V 39 x L ngzC39g8 constant7 Conserved Break m DEU WATS v m MA 15 r L NS vycp HA ACREASLa cw1m PEACH H v a V 39 775 7 p 7 7 w amcH m 2mm no IZ Shapanfwnwv m mu qey waterb m Sim11mg mm g mm 0 mm 3 Swells and Tsunamis Swells long surface gravity waves Tsunamis Long surface gravity waves 1960 Chilean Tsunami Aerial View of coastal area on Isla Chiloe Chile showing tsunami damage and wave extent 200 deaths were reported here from the tsunami generated just off Chile39s coast by the magnitude 86 earthquake Energy speed cg x 9D Oceankpt g 98ms2 The Paci c Continental shelves 2m1 Evooksr CUBe r mumson Lsammq Aftermath of the Chilean tsunami in the Waiakea area of Hilo Hawaii 10000 km from the generation area Along the Peru Chile coast the estimated lost of life from the tsunami ranged from 330 to 2000 people A city along the western coast of the United States which received notable run up was Crescent City California where the run up reached 17 m and the rst wave arrived 155 hrs after the tsunami was triggered Numerical Models For 1960 Chilean Tsunami play the movie This animation 23 MB produced by Professor Nobuo Shuto of the Disaster Control Research Center Tohoku University Japan shows the propagation the earthquake generated 1960 Chilean tsunami across the Paci c Note the vastness of the area across which the tsunami travels Japan which is over 17000 km away from the tsunami39s source off the coast of Chile lost 200 lives to this tsunami Also note how the wave crests bend as the tsunami travels this is called refraction Wave refraction is caused by segments of the wave moving at different speeds as the water depth along the crest varies The QuickTime movie presented here was digitized from a video tape produced from the original computer generated animation Model for 1960 Chilean Tsunami also pay attention 0 wave refraqgion December 26 2004 Indian Ocean Tsunami Tsunami warning NOAA West Coast amp Alaska Tsunami warning center Alaska Pacific Tsunami Warning Center Hawaii Predict earthquake 4 Internal gravity waves in density strati ed ocean a A 21ayer model H1 100200m H 9 Ad Pyenoeline 3800m Barotropic and baroclinic modes The 2layer system has 2 vertical modes Total Solution is the superposition of the two modes Barotropic mode independent of 2 which represents verticallyaveraged motion Restoring force gravity g Baroclinic mode vertical shear flow and verticallyintegrated transport is zero Restoring force is reduced grav1ty g m N 003ms2 P2 Barotropic mode 39 1 H2 W ul p1 gtU 2 P2 Pycnocline D h quotm 2 CO 9H1 H2 xgD N ZOOms H u 11i1 u 1 Baroclinic mode ff39 20 QM p1 39 Pycnoclme 1L24 3 p2 h D g 902 P1 E I H1H2 biean C1 g N 2 3m3 density 11 g I N 3 internal gravity waves 7 A 1 ayer model 7111 Pl lt h T Pycnocline P2 1 h Deep Ocean infinitely deep and V1 O The system has only one baroclinic mode The barotropic mode is ltered out by the assumption VP 0 C 1 g h1 ATOC 5051 INTRODUCTION TO PHYSICAL OCEANOGRAPHY class 21 Objectives of today s class 1 Ekman spiral 2 Bottom boundary layer 3 Concepts turbulent mixed layer 4 Thermocline halocline pycnocline Previous class Ekman transport amp pumping TAO data in the eastern Paci c Color SST black arrow WindsWhite arrow Ekman transport Coastal amp Equatorial Upwelling 14D W lBO39W IOO39W 80quot SDquot l Ekman Spiral Vertical structure of ow in Ekman layer Ekman 1905 a simple Winddriven ocean model Assumptions frictional laminar boundary layer Nonturbulent Ekman assumed internal stress is balanced by Viscosity 1 97 1 82 X Y A7 r plt gt ramp Recall that in the Ekman layer fUE fv pg 2757 qu m My Applying X7yAz7271 We have fIUE AZ7 qu AZ 821 922 39 Using boundary conditions uEvEO as zw oo an 1 AZ r 77 02 p Equations 82UE Z 7 82m 922 f UE A yield qu AZ xfQAZZ uE WW Tycos f2Azz TI Tampa2AM emz y I i y vE p mw 7 608f2Az2 7 T szntmz Important features a The Ekman layer thickness is H 7711117 f 7 is the efolding scale The stronger the Viscosity the thicker the mixed layer b The ow in the Ekman layer is not in geostrophic balance because Viscosity is so important 0 Ekman transport is the vertical integral of Ekman spiral Ekman spiral Depth W UH BCUOD 01 r l1 tkman I a 3qu f V 4 a Ekmon Spiral a f 2 Bottom Ekman layer A W geostrophy Bottom Ekman Bottom drag due to roughness and torques due to bathymetry can affect uid motion Currents slow down or use 11 v O No slip boundary condition For a at bottom ocean the interior geostrophic ow plays the role of surface windstress When the geostrophic ows approach the bottom they are slowed down by the bottom drag Following similar procedure we can obtain bottom Ekman layer thickness 2A2 E 7 f Validity of Ekman Spiral solution It is very difficult to observe or not observed Why 3 Turbulent mixed layer Turbulent S T Mixed layer not laminar De nition The depth at where T decreases by 05C from the SST or density increases by a value that is equivalent of 05C decrease Mixing processes can be affected by Wind mechanical stirring KrauXTurner Mixed layer physics KrauX and Turner 1967 Surface cooling that weakens strati cation Shear instabilities 4 Thermocline halocline pycnocline 1 024 O 500 r 1000 1 500 2000 2500 3000 3500 4000 Density glcma 1 025 Temperature quot0 Salinity IL 1028 0 5 10 15 32 33 34 35 Pycnoctlno I Surface zone Deep zone I I a Thernlocline 19 heating T Thermocline theory Pedlosky 1987 Winddriven ocean circulation Vertical diffusion can not balance vertical advection because of diffent timescales Seasonal thermocline T I Winter Springssts29C seasonal ATOC 5051 INTRODUCTION TO PHYSICAL OCEANOGRAPHY class 15 Objectives of today s class Waves in an ocean with a varying f Rossby waves Previous classes f0 amp fconstant b fconstant 42 H202 f27 H a f0 wz gD 1 g gravity waves Notew2 gntanhMD Short waves are eliminated by hydrostatic approx Previous classes f0 adjustment 42 C2162 w ik39c 7 a Initial perturbation 0 X L 7 0 b Gravity waves 0 X 77 0 c Equilibrium state X state of rest Previous class fconstant adjustment Z 77 3 Initial peiturbation 0 b Gravity wave radiation 0 Z 0 Equilibrium state 0 C Jab a f Geostrophic balance Waves in an ocean with a varying f Rossby waves In real Oceanv 298272va varies With latitude f f0 319 Midlatitude plane approximation Concepts f Planetary vorticity u lt 9U Relative vorticity vertical component Absolute vorticity 69y Potential vorticity Where H is the depth of water column Equations of motion Shallow water equation as for the foontant case exceptthathoref fb 3y 811 d e 22 at f 1 91 92 77 U 8 f quotI 0y 97 an 912 4 HL 0 dt 1 y Write a single equation in V alone amp assume periodic waves N eiUCIHZiwt we obtain Dispersion relation 2 2 2 W f 0 5 k 2l2 39 2w 2 4w2 where C 1 gH To simplify the case let 1 0 and look at ldimensional situation We have 2 2 2 w k B 2 fl 5 3 49112 i For high frequency waves with large m the dispersion relation can be approximated by 2 2 W f0 2 a or t l r This is long gravity wave under the in uence of f we obtained in the previous Class It is also called inertial gravity wave ii For low frequency waves with smallw g J2 f3 k 2 2wgt 2 4w2 5 2 52 f k 2w 4012 62 or dispersion relation 2 2 f 0 W C2 2 These waves do not exist in fconstant case their existence is due to the introduction of They are called Rossby or planetary waves 32 f3 3 7 4 71 1 4 w2 2 2w 2F To further understand the wave property we simplify the above equation 3 f fwz k i 1 l 439 4 0252 2w f2w2 Slnce 0 1s small for Rossby waves because of their low frequency gt fng k 1 1 2 2w q 3202 Choose sign 5 f w2 mfg k 1 1 2 2w 3202 3127 2 C w 32 i M 0 w CT is independent of frequency SinceC 7 g 8k and wavenumber they are nondispersive 332 They are long Rossby waves and CH CI f2 propagate westward speed 0 decreases as 1at1tude 1ncreases Choose the sign 3 w E 8w w CPE E They are short Rossby waves Group velocity propagates eastward but phase propagates westward They are dispersive Dispersion curves for free waves in midlatitude 8 plane W f0 Short Rossby Long Rossby L V 1 Short Rossby waves are hardly seen in the ocean interior because 1 they are too short to be effectively excited by largescale winds 2 mixing in the ocean acts strongly on short waves Rossby waves in midlatitude 8 plane i Existence of Rossby wave the variation of Planetary vorticity f with latitude p l 91 iiThey are low frequency waves z Oceanic adjustment 77 a Initial peiturbation 0 b Gravity wave radiation c Geostrophy d Rossby waves e Rossby waves V K f Equilibrium state Quiescent If there is wind forcing equilibrium state is Sverdrupblance Will be introduced later Sea surface height SSH anomaly Observed midlatitude Rossby waves by TOPEX satellite altimetry Midlatitude Paci c mm M g 800 Jan As you can see they m M propagate westward with a decreasing speed with increasing latitude 4 72 o 2 4 Sealeveucm ATOC 5051 INTRODUCTION TO PHYSICAL OCEANOGRAPHY class 7 Objectives of todays classs 1 Geostrophic method previous class 2 Observed ocean circulation 1 Geostrophic method indirect current measurement M n 1 1 DB DA Vic V0 f 91 V f Arr 1 A B M25 1 adj A517 137 J I l p f 208237qu P 1quot P3 gb is latitude Q 729 X 10 55 1 Calculation using hydrographic observations Step 1 Use observed T S P derive density p and thus Oz at stations A and B Step 2 Calculate I from Pr to P 2 at stations A and B by inte grating equation Geostrophic current Step 3 Calculate geostrophic current 181 1qJBp4 V2 V39quot Vquot ltzgt ltzgtf 0 V0 0 Level of no motion ii VDquot 75 0 Direct observations Reference level of 500m 1000m and lSOOm are often used Dynamic meter Dynamic meter dyn m I Dynamic height D IJg 1dynamicmD1 D Because 9 ism32 D lOJ k D i g 98m52 Unit conversion white board 1 dyn meter is very close to the depth of 1m is close to 1 Pressure level ldb is Close to 1m P p92 1dbaxr 104PCLSCCLZ p lOOOkgm3 g 9877132 z is very Close to be 1 m Geostrophic current example 0d 500 B DA 1000 aAdp TSP Air TasaP 500 500 2 DB 7 1000 lde pA PGF CF pB W ag QB 10 gg P3 1 1 Northern Hemisphere p Geostrophic method Advantage use all useful hydrograhic data to infer general ocean circulation Disadvantage i level of no motion ii lter out barotropic mode Surface geostrophic current sea level measurements i satellite altimetry TOPEXPOSEIDONJASON ii coastal sea level stations Both barotropic and baroclinic modes Total motion B arotmpic Baroolinio y V0 V 500db V 1000 db Shear 2 Observed ocean circulation The Paci c L 4133 393quot n r ar TX x 391quot ED 100 95quot in amen suf ce rel41m 39 hc Pacn COW ansem HG My Main umud nam apogfgphj cf by 355 omnnumwm 1 lo JCT dcar i39n yn cm 539 v 1U L Flow 0 db Adjusted steric height at 0 db IO m s 2 or lOJkg I Geostrophic current the Atlantic The Pacific iEQ 1 anstern Pacific Indonesmn gzghome of E1 Nim Through ow Equatorial Undercurrent EUC 110 O 39JFtG39CIUfFEFtS 13m D39Ea39 63 Fquot 3 7 A km 3931 lm wuuu quott 393 quot quot1l39310fi1quot Urrcnt svstcm u the lzcenlrul Paci c ct an WCNW EJstward How is shadLrl unmg cu39mm aw hMEh d F39wud5 395 3quotquot right r033 alwud HOW th CI its hatched upwards m ll u lrl All weslwud How nurd39l 05 5quot 1 congrjmlm 1m 30 hauzlurml nrrcnt WCWNd ow mth M 5quot mnsidc the EIC Hummm 1h Szmth lfq1ntorial Current EL Equatorial Umlvruurremr Eli a Fquguuil lm mtdmm Current NIKE and SECC North and South eqlaz rrial Com lllerrurmm ygcc and 55m Mmh Imi Sunlll Sumurfacc Counlcrcmmnm l runxporu in swalmm are gwm 01 lg rw botd gures Inner on nhaewariong from Apri 11 mirth 933 and 5553 iilahis burl rm luuuur NHL J ml 1355 The Atlantic Indian Ocean 7V r qfni H a a r ATOC 5051 INTRODUCTION TO PHYSICAL OCEANOGRAPHY class 10 Objectives of today s class 1 Forces and equations of motion 2 Effects of friction Previous class ampere an m sooo n at nee H gr Aquot exmpm m a TV as qth mgtnw w semannns at man awuc cnaw Eu 3 ms jvatrum m m mute vr Mn N amnvzs Dersmmcale av x Mrqu W ma nr Mner v51 AEW Ammm Sultan mm NADW 7 Narh A m m gt15 mm 50 n n mm Ocean forced from above 1 Forces i Pressure gradient force PGF iiGraVity iiiFriction wind forcing buoyancy forcing etc y I ZA Equations of motion Newton s second law of motion 2 773 acceleration Sum of all forces Mass of uid element ml i Fixed frame nonrotating X zAk Z i PGF term i y Xdirection i p6y63i Similarly we obtain PGF in y and z directions as 8p 8y j V616y6z 9p k 6w6y62 Therefore the total PGF acts on the uid element is Vp5176 y6 2 Where 8 V 1 IL a 39 k J8z ii The gravitational force kpg56y6z 7f66y62 Acceleration of gravity relative to a fixed frame iii Effects of friction and other forces F The Newton s second law of motion gt velocity F i dV relative to a a i 7 m dt fixed frame gt 1 1 gt 1 iv Vp0r6y62pg5616y62F vpgfF dt p zr y z p where F F p6175y6z frictionalwind etc force per unit mass Equations of motion in a rotational frame The earth is rotating It is more convenient for us to observe motions relative to the earth Convert xed frame to rotating frame For any vector R here we denote it as distance d R d R Elf V V gtltR QgtltR gt Rotational frame So gt dV dV dt quot dt f Recall that dV Vp6iv6y62 ng01501965 F 1Vp gf F dt p x y z 0 Then E 1Vp 2QXVgf gtltQXRFr it pi i PGF Coriolis Gravity gentripetalFric orce force em Acceleration of unit mass Q X Q X R 03gf the two terms usually conbine to give g We obtain dV 1 C 7Vp 29gtltVgF This is the eqn of motion on earth in vector form Eqn of motion component form V ui Uj wk F Fmi ij Bk 9 Qcosqu Qsinqbk board demo Q X V chosz stiw u cos k We obtain the eqn of motion for each Component du 1 8P f ZQsingbv 29009ng Era dt 0 6117 A d 1 8P Iquot i 1 dt p ay emgbu 11 v 1 3P 2960896sz g Fr Writing Change from Lagrangian form to Eulerian form dt 915 81 03 182 We have 113 Uzi w 293mm ZQcos w F7 9v 91 I 91 I 917 1 9P 7 7 7 if i 2Q F at Liam may waz pay smq u i y 0m 0w 810 810 1 OP 29 F 11 12 39w 7 cos u Bi 817 By 82 p 8 p g k f z smw is Coriolis parameter 2 Effects of f ction 8 8n 8 9n 8 Bu FLU 7 z 7 7 7 27 01A 813 8y Ag By 82 82 8 82 8 82 8 81 F 7 I 7 7 7 7 z 7 y 893 87 8y y y 82 82 8 911 8 8w 8 8w 2 i a 87 ya 3amp8 Ax Ay 10 105m2s A2 105 1071m2s North Pole A Earth skinnier up here a mmargm Path of Buffalo in one day Earth fat here Path of Quito in one day ozom monksCote A Yhnmsnn Learning South Pole Quito moves at 1668 kmlhr 1036 milhr Note Quito s longer distance through space in 1 hour is still 15 Buffalo moves at 1260 kmhr 783 milhr Note Buffalo s shorter distance through space in 1 hour is still 15quot a 2mm Bronksrcoie rThomson Leavmng Buffa o I 1260 kmlhr east 783 milhr Lands off course Cannonball 1 mom EmmaCola Tnumson Lsammg ATOC 5051 INTRODUCTION TO PHYSICAL OCEANOGRAPHY class 17 Objectives of today s Class Equatorial waves Previous classes Oceanic bore China Oceanic bore France muncth anmunu Surface gravity waves M Hume CHANSGN 2000 52 532 521 253 1225quot 232 2355 Emmnoa m ea 053365 Observed equatorial Rossby waves Cycle 21 Aer 13 1993 5 um um Observed equatorial Rossby waves Cycle 21 April 13 1993 7 70 n 4 Sea level cm Equatorial waves Math Hermite functions Therefore provide general description 1 Energy dispersion Phase speed and group velocity 2 Solution character 3 Symmetric property about the equator 4 Forcing 1 The equatorial Kevin wave a Dispelsion relation to kc barotropic or baroclinic mode speed Barotropic mode 3 0 xgH ZOOms First baroclinic mode cl 2 N 3ms Phase speed and group velocity of Kelvin waves Cp Cg C Nondispersive Both phase and energy propagate eastward I A u b Solution EQygt0 NH 716 26 Cm ct EQy0 3 2 u geig C GIE Ct EQyltOSH c v 0 Efolding scalea 2 6 Equatoriallytrapped due to 3 c Symmetric about the equator d Forcing Changing winds with time Westerly Wind burst WWB forcing Tt1 Tt2 r KJ Linear 15layer model 20 Bplane1 lorclng m 1m conlaurs and currents arrows 2 Equatorial Rossby waves a Dispersion relation 72 2971 1 n123 order number of Hermite function Long Rossby waves 3k 271 1 c 2n l Cg Cp 39 Long Rossby waves nondispersive Propagates westward The largest speed is for n1 rst meridional mode Rossby wave 1 CHZ l C 3 Which is 13 of Kelvin wave speed So Rossby waves propagate slower than Kelvin waves Short Rossby waves 3 U k A quot3 3 9 Phase propagates westward energy eastward Dispersion relation Ara W Rossby Kelvin short long b Solution 7 D 7 n2 i j U77 i n39 C ycoslm wt 7 2 Hn we 2C 6030615 wt EL Dn D717 393 wi t Wm M wismw m Nut kcw ktcw D39n 1V nDn l u l sinkrr wt kc w kcw C LC Vi Equatorial Rossby radius of deformation D1 C Symmetry can be both symmetric and antisymmetric d Forcing Winds Eastern and western boundary re ection Rossby wave Kelvin Wave E t Coastal Kelvin Kelvin Short Rossby Rossby wave Off equatorial in uence Western boundary l Coastal Kelvin EQ Kelvin EQ i Coastal Kelvin ATOC 5051 INTRODUCTION TO PHYSICAL OCEANOGRAPHY class 26Winddriven ocean circulation Coastal ocean circulation Objectives of today s class 1 Coastal ocean circulation 2 Final review Previous class vorticity equation steady circulation Roltlt1 EHltlt1 2 4 WI7V AHV 7 5 L 0139 6 Note that equation 6 is the general vorticity equation in terms ofil which applies to both the interior and the western boundary region 3 lug li Scale analysis for ocean interior 1 7 y m 51 p Scale analysis for the western boundary region Fundamental difference interiorWestern boundary Interior Wind forcing Western boundary remote response to interior forcing Two questions from the previous class a What does stredam function starch for physgcjtlly Hui a Ufa I Stream function contours are the lines that currents or Winds in the atmosphere flow along If the currents are geostrophic stream lines are parallel to the pressure contours 3P LQ Pof 327 U9 Pof 35 Assume H and f are constants for geostrophic ow H 1 po p Streamlines are parallel to p lines Near the surface Where Ekman drift is strong not Geostrophic streamlines are not parallel to p lines b Why do you choose 1 0 at eastern rather than western boundary curl39r lt 07m3 lt 0 E hmiX Eastern boundary LIlCI IIlULlll le 1 4p 3 continuous Wind forcing 7w UTITW 17 curlr gt w W r3 u drrl we L 073 I00 905 k6 Exlglanation using vorticity balance 3 r 45N 3ON 1D Oz 2 36 y0 v L15N Cfmf E if 10 4 O 1 Z Z 7 vx ugmL 106 10 curlT T lt O 3y 0 m0 x0 Cfzf 1310 41 V 01 39 7 cursz T lt0 3y