1. Geostrophic Currents Physical oceanography Instructor: Dr. Cheng-Chien LiuCheng-Chien Liu Department of Earth Sciences National Cheng Kung University Last updated: 24 November 2003 Chapter 10

2. Introduction  Geostrophic approximation Place:  within the ocean's interior away from the top and bottom Ekman layers Scale:   x,  y > a few tens of kilometers  t > a few days Balance:  Horizontal: u, v  F c   P  Vertical: F g   P

3. Introduction (cont.)  Geostrophic equations The typical size of each term in the N-S eq.  L  10 6 m  f  10 -4 s -1  U  10 -1 m/s  g  10 m/s 2  H  10 3 m   10 3 kg/m 3

4. Introduction (cont.)  Geostrophic equations (cont.) N-S eq.  Negligible viscosity and nonlinear terms  Our intuition  this is a reasonable assumption  Vertical velocity  Horizontal velocity 00 000 00 000 00 00 00 00

5. Hydrostatic Equilibrium  Stationary u = v = w = 0  No friction N-S eq.  Isobaric surface: a surface of constant pressure  Pressure at any depth h For many purposes, g and  are constant  p =  g h Table 10.1: Units of Pressure

6. Geostrophic Equations  Assumptions no acceleration, du / dt = dv / dt = dw / dt = 0 w << u, v The only external force is gravity  Friction is small  Geostrophic Equations N-S eq.

7. Geostrophic Equations (cont.)  Geostrophic Equations (cont.) Equation for u Equation for v

8. Geostrophic Equations (cont.)  Barotropic flow Homogeneous  = constant g = constant 0 0 0 0 0

9. Geostrophic Equations (cont.)  Stratified flow Pressure  Due to horizontal density differences  Due to the slope at the sea surface Velocity  Due to variations in density  (z): relative velocity  Thus calculation of geostrophic currents from the density distribution requires the velocity ( u 0, v 0 ) at the sea surface or at some other depth 0 0

10. Surface Geostrophic Currents From Altimetry  Level surface (geoid) Constant gravity potential Fig 10.1: p =  g (  + r )   : the height of the sea surface above a level surface (geoid)  Geoid  The level surface that coincided with ocean surface at rest  A surface of constant geopotential  surfaces of constant geopotential  = gh Surface geostrophic currents  surface slope

11. Surface Geostrophic Currents From Altimetry (cont.)  The oceanic topography (Fig 10.2) Surface geostrophic currents (u s, v s )  surface slope  surface topography (SSH)  satellite altimeter Dynamic processes  dynamic topography Accuracy (Fig 10.3)  Geoid:  50cm  Topography:  15cm

12. Surface Geostrophic Currents From Altimetry (cont.)  Satellite altimetry Systems  Seasat, Geosat, ERS–1, and ERS–2  Topex/Poseidon (1992), Jason (2001) Measurements  Changes in the mean volume of the ocean  Seasonal heating and cooling of the ocean  Tides  The permanent surface geostrophic current system (Fig 10.5)  Changes in surface geostrophic currents on all scales (Fig 10.4)  Variations in topography of equatorial current systems such as those associated with El Nino (Fig 10.6)

13. Surface Geostrophic Currents From Altimetry (cont.)  Altimeter Errors (Topex/Poseidon) Instrument noise, ocean waves, water vapor, free electrons in the ionosphere, and mass of the atmosphere Tracking errors Sampling error Geoid error  GRACE, CHAMP Resulting error  4.7cm

14. Geostrophic Currents From Hydrography  Basic idea Measurements of T, S, C, p  equation of state     p   the relative velocity field u, v at depth Question: why measure p to determine p? Answer: small  z  large  p

15. Geostrophic Currents From Hydrography (cont.)  Geopotential Surfaces Within the Ocean Geopotential Atmosphere  meteorology community  Dynamic meter D =  /10  Geopotential meter (gpm) D =  /9.8 Ocean  oceanography community  The difference between depths of constant vertical distance and constant potential can be relatively large  P(z = 1m) = P(z = 1gpm) = 1.007 decibars

16. Geostrophic Currents From Hydrography (cont.)  Equations for Geostrophic Currents Within the Ocean (Fig 10.7) To calculate geostrophic currents at depth  calculate the horizontal  P within the ocean  calculate the slope of a constant P surface relative to a constant  surface

17. Geostrophic Currents From Hydrography (cont.)  Slope of a constant P surface Vertical pressure gradient  The specific volume  =  (S, t, p) Differentiating with respect to x Evaluating  /  x using hydrographic data

18. Geostrophic Currents From Hydrography (cont.)  Slope of a constant P surface (cont.) (  1 -  2 ) std  The standard geopotential distance between two constant P surfaces P1 and P2  A  The anomaly of the geopotential distance between the surfaces The slope of the upper surface The geostrophic velocity at the upper geopotential surface  Direction Why use  not  ?  It is the common practice in oceanography  Tables of specific volume anomalies and computer code to calculate the anomalies are widely available

19. Geostrophic Currents From Hydrography (cont.)  Barotropic flow  = constant or  =  (z) but    (x, y) Constant P surface // sea surface // isopycnal surface The geostrophic velocity V  fn(z)  Baroclinic Flow  =  (x, y, z) Constant P surface is inclined to isopycnal surface Example: Fig 10.8 Constant-density surfaces cannot be inclined to constant- pressure surfaces for a fluid at rest  Decomposition of the variation of vertical flow A barotropic component that is independent of depth A baroclinic component that varies with depth

20. An Example Using Hydrographic Data  Hydrographic Data  geostrophic velocity General procedure  Processing of Oceanographic Station Data Data  Collected on Cruise 88 along 71 0 W across the Gulf Stream south of Cape Cod, Massachusetts at stations 61 and 64 Instrument: Mark III CTD/02 Processing  Sampled 22 times per second  Averaged over 2 dbar intervals  Tabulated at 2 dbar pressure intervals  centered on odd values of pressure because the first observation is at the surface  the first averaging interval extends to 2 dbar, centered at 1 dbar  Smoothed with a binomial filter  Table 10.2, 10.3

21. An Example Using Hydrographic Data (cont.)  Hydrographic Data Processing (cont.)  t, S, p   (S, t, p)   the average value of specific volume anomaly for the layer between standard pressure levels  the average of the values of  (S, t, p) at the top and bottom of the layer  10 -5   the product of the average specific volume anomaly of the layer times the thickness of the layer in decibars  The distance between the stations is L = 110,935m  The average Coriolis parameter is f = 0.88104 × 10 -4  Table 10.4: the relative geostrophic currents  Fig 10.8

22. Comments on Geostrophic Currents  Converting Relative Velocity to Velocity Assume a Level of no Motion  Reference surface  2,000m below the surface  Table 10.4  V(z=2000) = 0  V = V(z)  Some experimental evidence support this surface exists  Defant’s recommendation: at z where T vh (z)  minimum   2,000m  Figure 10.9  The geopotential anomaly and surface currents in the Pacific relative to the 1,000dbar pressure level  If V(p= 1,000dbar) = zero, the map would be the surface topography of the Pacific Use Conservation Equations  Lines of hydrographic stations across a strait or an ocean basin may be used with conservation of mass and salt to calculate currents  inverse problem

23. Comments on Geostrophic Currents (cont.)  Converting Relative Velocity to Velocity (cont.) Use known currents  The known currents  current meters or by satellite altimetry  Problems  time domains are not consistent  The hydrographic data may have been collected over a period of months to decades, while the currents may have been measured over a period of only a few months  Fig 10.10  Nearly the same time domain  Solid line  buoy data + assuming no motion surface  Dashed line  match measurements from the current meter  Disadvantage of Calculating Currents from Hydrographic Data Only the current relative a current at another level The assumption of no-motion level is not valid over the continental shelf Stations must be tens of kilometers apart  no information in-between

24. Comments on Geostrophic Currents (cont.)  Limitations of the Geostrophic Equations No acceleration  Geostrophic currents cannot evolve with time  Scale   x,  y > a few tens of kilometers  t > a few days Not apply near the equator where f  0  However, the geostrophic balance is still valid even within a few degrees of the Equator The only external force is gravity  friction is small  Ignores the influence of friction

25. Currents From Hydrographic Sections  Fig 10.11 hydrographic data in a vertical section along ship track  Sharply dipping density surfaces with a large contrast in density on either side of the current Margules’ equation  Estimate the speed and direction of currents perpendicular to the section by a quick look at the section Assumptions  Homogeneous layers of density  1 <  2  Both are in geostrophic equilibrium

26. Currents From Hydrographic Sections (cont.)  Deriving Margules’ equation BC:  p 1 =  p 2 on the boundary   : the slope of the sea surface   : the slope of the boundary between the two water masses

27. Currents From Hydrographic Sections (cont.)  Example An application of Margules’ equation to the Gulf Stream  = 36 0 At a depth of 500 decibars   1 = 1026.7kg/m 3   2 = 1027.5kg/m 3  t = 27.1 surface  Changes from a depth of 350m to a depth of 650m over a distance of 70km  tan  = 4300 × 10 -6 = 0.0043   v = v 2 - v 1 = -0.38m/s  v 1 = 0.38m/s  comparable with estimations from Table 10.4 assuming a level of no motion at 2,000 decibars

28. Currents From Hydrographic Sections (cont.)  Example (cont.) Table 10.4  The slope of the sea surface is 8.4 × 10 -6 or 0.84m in 100km Note  Constant-density surfaces in the Gulf Stream slope downward to the east  Sea-surface topography slopes upward to the east  Constant pressure and constant density surfaces have opposite slope Oceanic front:  The sharp interface between two water masses reaches the surface  Such fronts have properties that are very similar to atmospheric fronts Eddies in the vicinity of the Gulf Stream (Fig 10.12)  Warm-core ring  center high  anticyclone

29. Lagrangean Measurements of Currents  Lagrangean  Eulerian  Basic Technique Track the position of a drifter that follows a water parcel Average velocity = distance / time Error sources  Determining the position of the drifter  The failure of the drifter to follow a parcel of water  Sampling errors  Convergent zones > divergent zones 

30. Lagrangean Measurements of Currents (cont.)  Satellite Tracked Surface Drifters (Figure 10.13) Surface drifters  A drogue plus a float  radio transmitter  stable frequency F 0  A receiver on the satellite receives the signal Principle  Determines the Doppler shift F as a function of time t  The Doppler frequency:  where R is the distance to the buoy  c is the velocity of light  F = F 0  R is a minimum  the closest approach  V satellite  the line from the satellite to the buoy  The time of closest approach and the time rate of change of Doppler frequency at that time gives the buoy’s position relative to the orbit with a 180° ambiguity  Because the orbit is accurately known, and because the buoy can be observed many times, its position can be determined without ambiguity Accuracy: 1cm/s  1km/day

31. Lagrangean Measurements of Currents (cont.)  Holey-Sock Drifters Culmination of surface drifters  holey-sock drifter  A circular, cylindrical drogue of cloth 1m in diameter by 15m long with 14 large holes cut in the sides  The weight of the drogue is supported by a submerged float set 3m below the surface  The submerged float is tethered to a partially submerged surface float carrying the Argos transmitter Niiler et al. (1995): measured the rate at which wind blowing on the surface float pulls the drogue through the water The buoy moves 12 ± 9 0 to the right of the wind at a speed  DAR is the drag area ratio defined as the drogue’s drag area divided by the sum of the tether’s drag area and the surface float’s drag area  D is the difference in velocity of the water between the top of the cylindrical drogue and the bottom  If DAR > 40, then the drift U s < 1cm/s for U 10 < 10m/s

32. Lagrangean Measurements of Currents (cont.)  Subsurface Drifters Swallow and Richardson Floats For measuring currents below the mixed layer Neutrally buoyant chambers Tracked by sonar using the SOFAR (Sound Fixing and Ranging) system for listening to sounds in the sound channel Aluminum tubing containing electronics and carefully weighed  the same density as water at a predetermined depth The only important error is due to tracking accuracy  The failure to stay within the same water mass causes small error Primary disadvantage  not available throughout the ocean

33. Lagrangean Measurements of Currents (cont.)  Pop-Up Drifters ALACE (Figure 10.14)  Autonomous Lagrangean Circulation Explorer (ALACE) drifters  Cycle between the surface and some predetermined depth  Spends roughly 10 days at depth, and periodically returns to the surface to report it’s position and other information using the Argos system  Track deep currents, it is autonomous of acoustic tracking systems, and it can be tracked anywhere in the ocean by satellite  The maximum depth is near 2km, and the drifter carries sufficient power to complete 70 to 1,000m or 50 cycles to 2,000m PALACE  Profiling ALACE drifters  Measure the T(z) and S(z) between the drifting depth and the surface when the drifter pops up to the surface  ARGO Drifters: the latest version of PALACE

34. Lagrangean Measurements of Currents (cont.)  Lagrangean Measurements Using Tracers Rare molecules  tag the water parcel  follow the water parcel  Atomic bomb tests in the 1950s  Recent exponential increase of chlorofluorocarbons (CFC) in the atmosphere  List of tracers used in oceanography (§13.3) Applications  Infer the movement of the water  Calculating velocity of deep water masses averaged over decades  Calculating eddy diffusivities. Figure 10.15  Two maps of the distribution of tritium in the North Atlantic collected in 1972– 1973 by the Geosecs Program and in 1981  Tritium penetrated to depths below 4 km only north of 40°N by 1971 and to 35°N by 1981  This shows that deep currents are very slow, about 1.6mm/s in this example  Mean currents in the deep Atlantic, or the turbulent diffusion of tritium?

35. Lagrangean Measurements of Currents (cont.)  Lagrangean Measurements Using Tracers (cont.) T and S  Land  reference points Limitations  Cloud cover  Track the motion of small eddies embedded in the flow near the front and not the position of the front

36. Lagrangean Measurements of Currents (cont.)  The Rubber Duckie Spill Two accidents  29,000 bathtub toys at 44.7°N, 178.1°E  80,000 Nikebrand shoes at 48°N, 161°W A good test for numerical models  Fig 10.17

37. Eulerian Measurements  Moorings (Figure 10.18) Deployed by ships Last for months to longer than a year Expensive  deploy and recovery Submerged moorings are preferred  The surface float is not forced by high frequency, strong, surface currents  The mooring is out of sight and it does not attract the attention of fishermen  The floatation is usually deep enough to avoid being caught by fishing nets Errors  Mooring motion  Inadequate Sampling  Not to last long enough  Fouling of the sensors by marine organisms

38. Eulerian Measurements (cont.)  Moored Current Meters The most commonly used Eulerian technique Examples include:  Aanderaa current meters which uses a vane and a Savonius rotor (Figure 10.19).  Vector Averaging Current Meters, which uses a vane and propellers.  Vector Measuring Current Meters, which uses a vane and specially designed pairs of propellers oriented at right angles to each other

39. Eulerian Measurements (cont.)  Acoustic Tomography Acoustic signals transmitted through the sound channel to and from a few moorings spread out across oceanic regions Expensive  many deep moorings and loud sound sources The number of acoustic paths across a region rises as the square of the number of moorings  many modes  give the vertical temperature structure in the ocean, and the spatial distribution of temperature

40. Important Concepts  Pressure distribution is almost precisely the hydrostatic pressure obtained by assuming the ocean is at rest. Pressure is therefore calculated very accurately from measurements of temperature and conductivity as a function of pressure using the equation of state of seawater. Hydrographic data give the relative, internal pressure field of the ocean.  Flow in the ocean is in almost exact geostrophic balance except for flow in the upper and lower boundary layers. Coriolis force almost exactly balances the horizontal pressure gradient.  Satellite altimetric observations of the oceanic topography give the surface geostrophic current. The calculation of topography requires an accurate geoid, which is known with sufficient accuracy only over distances exceeding a few thousand kilometers. If the geoid is not known, altimeters can measure the change in topography as a function of time, which gives the change in surface geostrophic currents.

41. Important Concepts (cont.)  Topex/Poseidon is the most accurate altimeter system, and it can measure the topography or changes in topography with an accuracy of ± 4.7cm.  Hydrographic data are used to calculate the internal geostrophic currents in the ocean relative to known currents at some level. The level can be surface currents measured by altimetry or an assumed level of no motion at depths below 1–2m.  Flow in the ocean that is independent of depth is called barotropic flow, flow that depends on depth is called baroclinic flow. Hydrographic data give only the baroclinic flow.  Geostrophic flow cannot change with time, so the flow in the ocean is not exactly geostrophic. The geostrophic method does not apply to flows at the equator where the Coriolis force vanishes.

42. Important Concepts (cont.)  Slopes of constant density or temperature surfaces seen in a cross-section of the ocean can be used to estimate the speed of flow through the section.  Lagrangean techniques measure the position of a parcel of water in the ocean. The position can be determined using surface or subsurface drifters, or chemical tracers such as tritium.  Eulerian techniques measure the velocity of flow past a point in the ocean. The velocity of the flow can be measured using moored current meters or acoustic velocity profilers on ships, CTDs or moorings.