# C. Calculating the Velocity of a Geostrophic Current

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## C. Calculating the Velocity of a Geostrophic Current

1. The importance of the geostrophic current velocity calculation is that it allows oceanographers to calculate current speeds in the upper ocean based on estimates of sea surface height

1. Traditionally sea surface heights were calculated from CTD measurements of depth profiles of T, S and P, as is discussed below

2. More recently sea surface height is estimated from depth profiles of T, S and P measured remotely by floats (ARGO float program)

3. Additionally satellites (TOPEX) can measure sea surface height remotely

4. The bottom line is that is becoming significantly easier to measure sea surface height and thus easier to estimate geostrophic current velocities in the ocean

5. Thus oceanographers can get a very good picture of the direction and speed of surface currents in the ocean (like that pictured in Fig. 1) based on maps of sea surface height (Figs. 9 or 14)

6. These calculated geostrophic current velocities are much easier to obtain than direct measurements of current speeds from current meters which are expensive to deploy everywhere in the ocean.

2. We calculate the geostrophic current velocity assuming there is a force balance between Coriolis Force and HPG force under steady wind conditions

3. The Coriolis Force = Mass * Current Velocity * f

-where f = 2* W*sinf and W is a constant that represents the rotational velocity of the earth (7.3x10-5 sec-1) and f represents the latitude at which the current is located

-the units for Coriolis Force are kg * m/sec * 1/sec = Newtons Δ

4. The Horizontal Pressure Gradient Force = mass * 1/ ρ * ΔP/Δx

-the units for HPG Force is kg * 1/(kg/m3) * Newtons/m3 = Newtons
4. Since Geostrophic currents are the result of a force balance between HPG and CF in the ocean,

 then Coriolis Force = HPG Force under a force balance condition

 Setting those two forces equal to each other at a force balance yields:

mass* velocity * f = mass * (1/ ρ) * ΔP/Δx

 The mass term on each side of the equation cancels. Rearranging to solve for velocity (v) yields:

v = (1/ ρ) * (1/f) * ΔP/Δx

v = 1 ΔP

ρ f Δx

• where v is geostrophic current velocity (m/s), W is the angular speed of rotation of earth (7.3x10-5 /sec) which is constant, f is latitude (°),ρ is density (kg/m3) and ΔP/Δx is the horizontal pressure gradient with P in Newtons/m2 and x in meters.

• units: velocity(V) = 1/(kg/m3) * (1/s-1) * (N/m2 /m) = m3/kg * s * kg*m/s2 * 1/m3 = m/s

(remember, Pressure=Force/area=mass*acceleration/area)

• Note: 1 atmosphere of pressure = 1 bar = 105 Newtons/m2

- Remember: Pressure is force per area

• 1 Newton/m2 = 1 Pascal (units of pressure)

4. Geostrophic current velocities depend on the magnitude of the sea surface height gradient between the two locations and the latitude (Fig. 11)

 since velocity = 1 ΔP

r f Δx
 assume, for now, that density (r) is equal between the sites and ΔP/Δx = g*r*z/x

 Substituting g*r*z/x for ΔP/Δx yields (after canceling the density term):

v = g z

f x
-one can express this geostrophic velocity in terms of the angle of the sea surface slope

v = g tan q, where tan θ = Δz/Δx

f
-units: m/s = (m/s2) / (1/s) * (m/m)
5. Characteristics of Geostrophic Surface Current Speeds

 The calculated geostrophic current speed represents the average surface current speed between the locations over which HPG is determined (between A and B in Fig 11).

 The calculated surface current speed represents only the component of current velocity perpendicular to the observed horizontal pressure gradient (e.g., if you measure a north-south HPG, then the component of geostrophic velocity calculated is in the east-west direction only)

7. Note: You should understand the force balance that causes geostrophic currents and the orientation between HPG force, Coriolis Force and geostrophic current direction. Quantitatively, you should be able to calculate the surface speed of a geostrophic current given a gradient in sea surface height and latitude.

## D. Estimating Surface Ocean Circulation from a Map of Sea Surface Height

1. Determining a global map of sea surface height provides a means to estimate the direction and speed of surface geostrophic currents on a global scale in the ocean. (Fig. 14)

2. Oceanographers use dynamic height or geopotential height to describe the sea surface height because changes in sea surface height represent potential energy.

-that is, if the wind stopped blowing, the surface water piled up in the "hills" would flow towards the "valleys" and, in doing so, would convert potential energy to kinetic energy.

-because of this ability for a fluid to convert potential to kinetic energy, oceanographers refer to the sea surface height as dymanic height (DH).
3. In practice, oceanographers calculate the dynamic height of the sea surface at any location by measuring the vertical density distribution down to a chosen isobar or pressure level, e.g., 1000 or 2000 dbar

• since pressure (P) = g*ρ*z, one can calculate z (the height of the water column above a constant isobar surface) from CTD measured values of P and calculated values of ρ (from measured temperature and salinity)

• one repeats this calculation at many sites in the ocean always calculating the depth to a constant chosen isobar (e.g. typically 1000, 1500 or 2000 dbar)

• since the depth distribution of density varies from site to site in the ocean (g does not vary), then the depth z of the water column above a constant isobar will vary

• thus sea surface height will vary from site to site in the ocean

4. Class Problem: Calculate the height of the sea surface above the 1000 dbar isobar from measured pressure and calculated density.

• Assume the average density above the 1000dbar pressure is 1026 kg/m3

• (Note: 1bar = 105 N/m2 or 1 dbar = 104 N/m2 and 1 N/m2 = 1 m/s2 * kg /m2 =kg/(m s2 ))

• Since P = g * ρ * Z, therefore

• Z = P/(g*ρ)

• Z= 1000 dbar * (104 kg/(m s2) / dbar) / (9.8m/s2 * 1026 kg/m3)

• Z= 994.6 m

• this means the sea surface is 994.6 m above the 1000dbar surface at this location

• if the mean density above the 1000 dbar surface was higher at another location, then Z would be lower and the sea surface height above 1000 dbar surface would be lower

• in the above example, if the mean density was 1027 kg/m3 (rather than 1026 kg/m3), then Z would be 993.6 m

• thus the difference in height between these two sites would be 1.0m (994.6m – 993.6m)

5. Dynamic Height distribution in surface ocean (Fig. 14)

1. Dynamic Height (DH) is often presented as an anomaly, that is, the lowest measured DH for the surface of the world ocean is subtracted from the measured DH at any location. This way all the DH values are positive and represent the height difference between any location and the lowest value for the sea surface.

2. The total range in the ocean is about 2.5 dynamic meters or 250 dynamic centimeters

6. The highest dynamic heights are found in the western portions of the subtropical gyres (20-30° latitude bands) because this is where the Ekman transport from the Trade Winds and Westerlies converge and “pile up” water.

7. The lowest DHs are found in the regions of cold waters (near Antarctica and Greenland)
Question: Why would the lowest dynamic heights occur in the regions of coldest water?

-Hint: look at the Pressure equation (P = g*ρ*z) and consider the effect of temperature on density.

8. The steepest horizontal gradients in DH occur in the regions where the DH contours are bunched together (Fig. 14)

- a contour map of DH is like a contour map of topography. Where the contours of DH are bunched together, this represents maximum in horizontal gradients of DH (maximum Δz/Δx).

- DH contours are closest (maximum Δz/Δx) where the western boundary currents, like the Gulf Stream and Kuroshio Currents, are located.
9. The steeper the horizontal DH gradient (DH/x, which is the same as Δz/Δx), the greater the horizontal pressure gradient and the faster the geostrophic current.

-remember: geostrophic velocity = g/f*Δz/Δx

10. The direction of geostrophic current flow follows the contours of dynamic height.

• at steady-state, a force balance between Coriolis force and horizontal pressure gradient force implies that the current flows in a direction perpendicular to the HPG (Fig. 14)

• the HPG is perpendicular to the contours of dynamic height

• thus the geostrophic currents flow along the contours of DH in order to have the Coriolis Force oppose the HPG Force. Look at Fig. 13 to help visualize this situation.

11. The pattern of geostrophic current flow based on the map of dynamic height shows the major features of surface circulation, like the subtropical gyres, western boundary currents (like the Gulf Stream and Kuroshio Current) and the Antarctic Circumpolar Current (Fig. 14)

- another map of sea surface height (Fig. 9), from ARGO float measurements, shows essentially the same pattern of surface currents

- compare Figures 1, 9 and 14 to see how the generalized surface ocean circulation is derived from maps of sea surface height
12. By following the DH contours in Fig. 14, you can see the pathway of surface geostrophic currents lead away from regions of Ekman convergence (at ~ 30°) towards regions of Ekman divergence (latitudes of 0° and 60°)

• this current pathway results in flow away from the subtropical gyres and towards the equator and subpolar gyres

13.The direction of surface geostrophic current flow opposes the direction of Ekman transport in the surface layer (<100m) which moves water towards the subtropical gyres and away from 0º and 60º.

• The opposition of Ekman transport and geostrophic current flow results in a balance between Ekman flow towards regions of convergence (‘hills’) and geostrophic flow away from regions of convergence. Such a balance must exist or otherwise the ‘hills’ would continue to grow in size over time, which isn’t happening.

- an analogous balance exists between Ekman transport away from regions of divergence (‘valleys’) and geostrophic current towards these regions (0 and 60º)

14. Class Problem: Estimate the surface velocity of the eastward flowing Kuroshio Current Extension in the N. Pacific at ~35°N and 180ºW (use Fig. 14 to estimate Δz/Δy)

• the northward change in DH is from about 180 to 150cm between 30° and 40°N

• use the geostrophic equation to calculate the surface current speed

• assume surface density is constant in this region

Velocity = g/f * [Δz/Δy]

= 9.8 m/s2 / (2*7.3x10-5 /s * sin(35°)) * [0.30m/(10°*110km*1000m/km)]

= 0.028 m/sec

= 2.8 cm/sec

• where f = 2 Ω sin (latitude), and Ω = 7.3x10-5 /sec.

• Note: This calculation represents the component of the surface current velocity oriented perpendicular the direction of the HPG. Since we estimated the northward sea surface height gradient (Δz/Δy) and thus the HPG was directed northward, we have calculated the eastward component of the surface current velocity.

E. Barotropic and Baroclinic Conditions

1. Barotropic conditions exist when the shape (or slope) of the isopycnals (contours of constant density) are parallel to the shape (or slope) of the sea surface height (Fig. 15)

- this causes the shape (or slope) of contours of equal pressure (isobars) to be parallel (follow) the shape (or slope) of the sea surface height

2. Under barotropic conditions, the horizontal pressure gradient at depth is equal to the HPG occurring at the surface of the ocean caused by changes in sea surface height (Δz/Δx).

-the slope of the isobars remains constant with increasing depth

3. Importantly, the geostrophic current speed remains constant with increasing depth under barotropic conditions because the HPG remains constant with increasing depth. (Fig. 15)
4. Baroclinic conditions exist when shape (or slope) of the isopycnals oppose the shape (or slope) of the sea surface height (Fig. 15)

 thus isopycnals are inclined to the isobars (rather than parallel to isobars as they are under barotropic conditions)

 thus there is a horizontal gradient in density along an isobars under baroclinic conditions

5. Under baroclinic conditions the horizontal pressure gradient decreases with depth (Fig 15)

• this results from the slope of isobars decreasing with increases depth

• this decrease in HPG and isobar slope with depth results from having denser water underlying regions of low sea surface height (SSH) and less dense water underlying regions of high sea surface height

• since pressure depends on the density of the water column, the pressure increases at a greater rate with depth (ΔP/Δz) where the water is denser (under regions of low SSH) than where the water is less dense (under regions of high SSH) (Remember: ΔP = g*ρ*Δz)

6. Geostrophic current speeds decrease with increasing depth under baroclinic conditions (Fig 16)

-this is a result of the HPG force decreasing with increasing depth

7. If ones goes deep enough under baroclinic conditions, then the density distribution ultimately causes the horizontal pressure gradient to approach zero (Fig. 16)

• at the depth where the HPG approaches zero (ΔP/Δx → 0), the geostrophic current speed approaches zero (Remember: Geostrophic Velocity = 1/(ρ*f) * ΔP/Δx)

• this depth is called the level of no motion and is typically located at around 1000-2000m

• if, at a specific location, a measurable current is present at the assumed level of no motion, this current speed is added to the calculated geostrophic current speed (see Fig. 16b)

• although there are currents in the deep sea (below 2000m) their speeds are much slower than surface current speeds (see Fig. 16), so assuming a level of no motion exists is a reasonable assumption for calculating the surface speeds of geostrophic currents and usually adds only a small error in the calculated current speed

8. Baroclinic conditions are much more prevalent in the open ocean than barotropic conditions

• thus the ocean adjusts its density distribution (the slope of the isopycnals) to reduce horizontal pressure gradients with increasing depth

9. Isopycnals dip downward under regions of high sea surface height (e.g. under subtropical gyres) and upward under regions of low sea surface height (e.g. equator and poleward of 40-50º) (Fig. 17)

• thus the meridional trend in the depth of an isopycnal surface is the mirror image of the meridional trend in sea surface height

10. The density (isopycnal depth) adjustment under baroclinic conditions occurs in part by having water move downward (downwell) in regions of high sea surface height and having water move upward (upwell) under regions of low sea surface height

-these vertical water motions help yield the depth trend in isopycnals required to maintain baroclinic conditions