Measuring ocean currents
Wikipedia. Ocean currents can be measured directly or indirectly. The indirect method is based on the concept of geostrophy.
1 Balances in the vertical and in the horizontal
An important balance in the vertical is the hydrostatic equilibrium.
An important equilibrium in the horizontal is the geostrophic equilibrium. Hydrostatic balance
퐺푟푎푣𝑖푡푎푡𝑖표푛푎푙 푓표푟푐푒 퐺 = 푔 ∙ 휌 ∙ 훿ℎ ∙ 퐴 Density r
Pressure, p, force on upper surface: p(h) A h p Area A z Force by pressure on lower surface (from below): p(h) p(h dh)(A) ( p(h) dp)(A) dh p(h+dh) 휕푝 = (푝 ℎ + 훿ℎ)∙ (−퐴) 휕푧
Sum of both: 휕푝 푃푟푒푠푠푢푟푒 푔푟푎푑𝑖푒푛푡 푓표푟푐푒 = − 훿ℎ ∙ 퐴 휕푧 1 휕푝 hydrostatic = 푔 휌 휕푧 balance 휕푝 At equilibrium, net force = 0 = 휌푔 휕푧
4 Conservation of momentum in the ocean – Navier-stokes equation dv F / m (Newton) dt
dv - 1/r 흏p/흏z + horizontal component dt + f v horizontal vector + g vertical vector + tidal forces + friction
Horizontal pressure gradient force = - 1/r 흏p/ 흏x or - 1/r 흏p/ 흏y
5 Horizontal pressure gradient
constant density r
p h z
p g r h x
6 Horizontal pressure gradient
h
p h h
A B x pA gr h pB gr (h h)
p pB pA gr (h h) gr h gr h
7 Horizontal pressure gradient
x h
Pressure gradient
p h Negative pressure h gradient
A B x
p h m kg _ _ gr force /volume 2 3 x x s m
8 Horizontal pressure gradient
x h
Pressure gradient
p h Negative pressure h gradient
acceleration A B x
p h du 1 p h gr g x x dt r x x
9 Conservation of momentum in the ocean dv F / m (Newton) dt
Acceleration of a water parcel = dv - 1/r 흏p/흏z - 1/r 흏p/흏x horizontal vector dt + f v vertical vector + g + tidal forces + friction
Horizontal pressure gradient acceleration = - 1/r dp/dx or - 1/r dp/dy
10 FP FC
Horizontal plane y h h h
Coriolis force FC u fu
Pressure gradient force FP
x 11 Equilibrium between pressure
gradient force FhP h and Coriolis force FC
Horizontal plane = Geostrophy
Velocity u constant parallel to FP FC isobars (=lines of equal pressure) y h
Coriolis force FC u fu
Pressure gradient force FP
x 12 Geostrophic current Northern hemisphere Southern Hemisphere N N High pressure High pressure p4 p4
p3 p3 -u FC F u y g y C g
p2 F p2 FP P
p1 p1 Low pressure Low pressure
x x
1 dp fu geostrophic velocity (ug, vg) g r dy can be measured through pressure!
13 But how to measure pressure differences in the ocean?
Can it be done with bottom-mounted high-precision pressure sensors?
Measurement of the distance sea surface – sea floor is good for determining the sea floor topography (accuracy 1-10 m) but not for dynamic sea level changes.
Measurement of dynamic sea surface elevations only possible since satellite altimetry.
14 Measuring sea surface deviation from Geoid via satellites
Since 1975 Radar at carrier frequency ca. 14 GHz Send frequency modulated puls in a narrow angle Capture reflected signal
• determine elapsed time -> distance to sea surface
(Increase of echo signal -> Measure of wave height and thus wind velocity)
- Need to know exactly the geoid. - Need to know exactly the satellite orbit.
15 Sea surface elevation relative to geoid through satellite observation
Geoid is known down to 1 mm on 100 km-Scale. Changing elevations can be recognized!
http://sealevel.jpl.nasa.gov/technology/technology.html 16 Sea surface elevation relative to geoid through satellite observation
Radar altimetry Satellite orbit relative to reference ellipsoid
Dynamic height of ocean surface
From GRACE mission
Pugh, 1987 17 http://oceanmotion.org/html/resources/ssedv.htm 18 Horizontal sea surface elevation gradient and currents
Sea surface height anomalies from altimetry …
topex-www.jpl.nasa.gov/ (TOPEX Poseidon)
… reflects ocean circulation.
19 Measuring the surface elevation of the Gulf Stream
Wells, 1998
Surface temperature (NASA)
See https://www.ospo.noaa.gov/data/sst/contour/global_small.cf.gif for a global SST picture 21 Horizontal pressure gradients through temperature and salinity
Horizontal pressure gradients in the ocean are caused • by a tilt of the surface • by internal density differences
h Hydrostatic equilibrium p(x) g r(x)h(x) r 1 r2 r x p1 p2
22 Gulf stream – surface tilt AND density front
Sea surface temperature Wells, 1998
23 Gulf stream - pressure gradient through density gradient
Salinity Temperature
0 200 400 600 km
1026.6 kg/m3
1027.6 kg/m3
Horizontal density difference in 600 m depth ca. 1 kg/m3 over 100 km Density sT
0 200 400 600 km 24 Gulf stream - pressure gradient through density gradient
Salzgehalt Temperatur
0 200 400 600 km
Horizontal density Horizontal density gradient difference in 600 m depth ca. horizontal pressure gradient 1 kg/m3 over 100 km Pressure gradient force directed southeastward
density sT
0 200 400 600 km Only density distribution known?
In case we have no surface tilt information but only temperature/salinity profiles - what can we do?
We can get information about the vertical shear of the geostrophic velocity 흏풖품 흏풖품 (or ) 흏풛 흏풑 Combination of surface tilt and density gradient
Temperature, salinity and density stratification P0
r11 X less dense Pressure T1 S1 r1 gradient P1 Pressure r z gradient force 22 less dense r2 X
P2 r3 r3
r1 > r11 x r2 > r22
Density smaller Geostrophic current due to specific volume larger, X horizontal density difference i.e. layer is thicker.
The mass is the same on both sides.
27 X Velocity points into the screen – away from us
Velocity points out of the screen – towards us Combination of surface tilt and density gradient
Temperature, salinity and density stratification P0
r11 X less dense Pressure T1 S1 r1 gradient P1 Pressure r z gradient force 22 less dense r2 X
P2 r3 r3
r1 > r11 x r2 > r22
Density smaller specific volume larger, i.e. layer is thicker.
The mass is the same on both sides.
29 Combination of surface tilt and density gradient
Temperature, salinity and thus density stratification
P0
X r11 Pressure gradient less dense r1
z P1
r2 r22 less dense
P2 r3 r3
x
30 Combination of surface tilt and density gradient
Temperature, salinity and thus density stratification
P0
r1 Pressure gradient r11 less dense z z
P r2 1
r22 less dense
r3 P2 r3
Horizontal pressure differences due to density differences in one level can only be determined relative to those of other levels. Consequently, also the geostrophic velocity can be determined only relatively.
Yet, if the geostrophic velocity is known in one level, it can be determined also in all other levels by integrating the vertical shear. 31 Combination of density derived geostrophic profiles and direct current measurements
Absolute: Relative: Baroclinic current profile Baroclinic current profile from adjusted to direct current two CTD profiles. measurements. x
x
x Direct current observations. x
x
Stewart (2002) 32 Horizontal pressure gradient due to tilt of sea surface:
„barotropic“ geostrophic current
Horizontal pressure gradient due to horizontal density gradient:
baroclinic geostrophic current shear
Which gradients cause which velocity?
33 Pressure gradient in the Gulf stream due to surface tilt
Satellite altimetry: ca. 1 m over 100 km
p h g r rfv x x geostr _ barotrop
푚 푘푔 1푚 = 9.81 ∙ 1030 ∙ 푠2 푚3 100푘푚 kg 104 ms 2 100km
2 10 Pa / km Wells, 1998 Pressure gradient force towards west
Can you please derive how large the geostrophic velocity then is?
34 푘푔 1푚 Relative pressure gradient in the Gulf stream due to density gradient
Density difference in 600 m depth over 100 km
p r g h x x
푚 푘푔 = 9.81 ∙ (26.4 − 27.4) /100푘푚 ∙ 600푚 푠2 푚3
kg 6103 /100km 0.6102 Pa / km ms 2
I.e. in the Gulfstream in about 600 m depth, the pressure gradient through surface tilt is decreased by 60% due to the horizontal density difference. The density gradient causes a pressure gradient opposite to the pressure gradient due to the surface tilt. The geostrophic current diminishes to that depth respectively. 35 Horizontally variable density and surface tilt
In most oceans there is spatially differential warming and/or salinification (or vice versa). As well as surface tilt due to wind stress curl.
p(h) g r h
dp dh dr gr gh dx dx dx
Determine vertical velocity shear from r temperature and salinity stratification h 1 through CTD measurements z (thermal wind equation). P Pro: high spatial resolution Con: valid only for observation time r2 x Determine absolute velocity either p from surface tilt (satellite altimetrie) ref or by velocity mesurements through current meters. Pro: values are absolute x Con: only available in few levels
36 Little exercise:
The figure on the next slide shows the mean sea surface height in the Atlantic Ocean in meters.
Please draw schematically geostrophic velocity vectors at the 4 indicated locations onto the map. To indicate faster currents, draw longer vectors, for slower currents draw shorter vectors.
Make sure that the direction of the vectors is correct!
39 Mean sea surface height in the Atlantic Ocean in meters.