Nov 4: Geostrophic currents from hydrography

 

 

Surface geostrophic currents from altimetry

 

Sea surface elevation gradients are balanced by geostrophic surface currents given by

 

 

 

 

 

 

 

Satellite altimeter observations are now precise enough to measure:

·        Changes in mean volume of the ocean (sea level rise and global warming)

·        Seasonal heating and cooling

·        Tides

·        Mean dynamic topography (on long length scales)

·        Variability in surface geostrophic currents

·        Variations in the topography of the equatorial current systems associated with El Nino (but geostrophy can’t be used at the equator)

 

 

 

 

 

Figure 10.4 Global distribution of variance of topography from Topex/ Poseidon altimeter data from 10/3/92 to 10/6/94. The height variance is an indicator of variability of currents. (From Center for Space Research, University of Texas).

 

 

Figure 10.5 Global distribution of time-averaged topography of the ocean from Topex/ Poseidon altimeter data from 10/3/92 to 10/6/99 relative to the JGM-3 geoid. Geostrophic currents at the ocean surface are parallel to the contours. Compare with Figure 2.8 calculated from hydrographic data. (From Center for Space Research, University of Texas).

 

From Stewart Chapter 10

 

The pressure gradient associated with the departure of sea surface height from a geopotential is felt throughout the water column, and these currents are often referred to as barotropic currents.

 

They are often said to be the part of the flow that does not vary with depth, but strictly speaking barotropic processes are associated with pressure surfaces that are parallel to density surfaces.

 

Baroclinic flows, on the other hand, are the result of pressure surfaces that are not parallel to density surfaces.

 

Geostrophic currents from hydrography

 

To use geostrophy to infer currents at depth we need to determine not only the pressure gradient due to the sloping sea surface, but also the subsurface pressure gradients due to variable density stratification.

 

 

Though we can measure water pressure with a pressure transducer lowered from a ship, we can’t simply use this observation because we seldom have an independent way of measuring depth.

 

Even if we could measure depth independently, it would have to be a very precise measurement:

 

A 10 cm/s current corresponds to a pressure gradient of

 

  Pa m-1  or   1000 Pa  in  100 km

 

From the hydrostatic relation we know that 1000 Pa is equivalent to the pressure change due to 10 cm of water.

 

We would need to know the depth of the pressure gauge to accuracy much better than 10 cm to make an observation adequate for calculating geostrophic currents, and we would still need to deal with the issue of the slope of the geoid.

 

 

Geopotential surfaces in the ocean

 

In practice, what we do in oceanography is to estimate the slope of the geopotential surface at one depth compared to another, and this tells us the relative strength of the current at the two depths.

 

This is a complimentary approach to that used in satellite altimetry which calculated the slope of a constant pressure surface (p = patmosphere). 

 

Stewart Figure 10.7: Sketch of geometry used for calculating geostrophic current from hydrography.

 

The steps taken are:

 

1.     Calculate the differences in geopotential between two different pressure surfaces 1 and 2

2.     Calculate the slope of the upper surface relative to the lower from observations at two locations A and B

3.     Calculate current at the upper surface relative to the lower – this is the current shear

4.     Integrate vertically the shear in the current assuming some knowledge of the current at a reference depth

 

We use a modified form of the hydrostatic equation, which for historical reasons is written:

 

 

so that  d is the change in potential energy associated with raising 1 kg through a distance of dz.  Units are energy per unit mass, J/kg, or m2/s2.

 

The geostrophic balance is written:

 

         

 

where  is the geopotential along a constant pressure surface.

 

Now consider how hydrographic data can be used to evaluate

 

At station A, the difference in geopotential between surfaces P1 and P2 is:

 

 

where the specific volume anomaly is written as the sum of two parts:

 

         

 

where  is the specific volume of a standard ocean of S=35, T=0 at pressure p.

 

The term   is the specific volume anomaly, and tables and computer programs exist for easily calculating this for any observed hydrographic data.

 

which is the sum of:

·        the standard geopotential distance between the pressure surfaces
and in meters would be approximately 
- this is what a SCUBA depth gauge measures

·        the geopotential anomaly  
- usually about 0.1% of the geopotential distance

 

The standard geopotential distance is the same at any horizontal location in the ocean because there is no variation in the vertical profile of T or S, so this is not going to enter into the calculation of pressure gradients.

 

Consider now the geopotential anomaly between P1 and P2 at two different stations A and B:

 

For simplicity, assume the lower surface is a level surface i.e. the constant pressure and geopotential surfaces coincide.

 

The slope of the upper surface is:

 

           slope of constant pressure surface P2

 

because the standard geopotential distance is the same at stations A and B.

 

The geostrophic velocity at the upper surface is calculated from:

 

similar to the way that we calculated surface velocity from altimetry from the slope of the sea surface (also a constant pressure surface).

 

Units:  s.(m2/s2)/m = m/s  (geopotential anomaly has units of m2/s2)

 

Geopotential anomaly is often referred to as dynamic height.

 

Oceanographers also often scale by 1/g, and call this steric height, h, with units of meters.

 

Steric height measures variations in the vertical distance between two surfaces of constant pressure, and should be stated as the steric height of surface p1 relative to po, e.g., the height of the sea surface (p1 = 0) relative to 1000 decibars (approximately 1000 m).

 

The velocity v is perpendicular to the plane of the two hydrographic stations and directed into the plane the way the figure is sketched.

 

Light on the right

 

A useful rule of thumb is that the flow is such that lighter (less dense, warmer) water is on the right looking in the downstream direction in the northern hemisphere – light on the right.

 

This only works if the level surface is below.

 

 

The notions of geopotential anomaly, steric height, and pressure are somewhat interchangeable. All can be used to visualize the pressure gradients that give rise to geostrophic currents.

 

Tomczak and Godfrey, Figure 2.7 – Schematic steric height and pressure section across a cold core eddy:

http://gyre.umeoce.maine.edu/physicalocean/Tomczak/regoc/text/2steric.html

 

Since the weight of water above z0 must be the same at all locations, the sea surface must be lower over the region of higher density.

 

·        pressure surfaces parallel the sea surface, but become flat at depth

·        some density surfaces outcrop at the surface, and this feature would have a cold center (possibly visible in satellite imagery)

 

The distance between the surface p=0 and reference level p(z0) is the steric height h(0,z0).

 

If we plot the pressure map at constant depth, z = zr, we get:

 

[See Tomczak and Godfrey, fig. 2.7b]

 

If we plot the steric height at constant pressure, p = p1, we get:

 

[See Tomczak and Godfrey, fig. 2.7c]

 

At any depth level, contours of steric height coincide with contours of pressure. Since the shape of the sea surface is so difficult to map accurately, oceanographers instead use maps of the steric height relative to a reference level of no motion to map the "dynamic topography" of the ocean.

 

This is the oceanographic equivalent of a meteorologists pressure map, and is an effective way of visualizing geostrophic currents.

 

We’ve already seen in the case of satellite altimetry that we can relatively easily measure the variability of geostrophic currents, but we are left with uncertainty in the mean circulation.

 

Similarly, the dynamic method used to compute geostrophic velocity of one depth relative to another leaves us uncertain about the absolute velocity.

 

Q: Is it possible to find a flat surface in the ocean ... one where horizontal pressure variations vanish?

 

Yes ... at depth in most ocean basins the density field is so uniform horizontally that, for example, steric height of the 1500 m relative to 2000 m varies by only a cm or so.

[Tomczak and Godfrey, fig. 2.8 - dyn hgt 1500/2000 db and 0/2000 db]

 

Steric height of the surface relative to 2000 m shows differences of order 0.5 m in a single basin.

 

The Southern Ocean is a marked exception to this – here strong geostrophic currents extend to the bottom.

 

Since maps of dynamic height and pressure are similar, we can sketch the pattern of geostrophic currents on a dynamic height or steric height map.

 

This method relies on the assumption that there is little or no flow at, say, 2000 db. Assuming this, we can compute the dynamic topography at other depths w.r.t. 2000 db (including deeper depths).

 

This gives ocean currents at any depth we select, so we get u(z), v(z).

 

You won’t go too far wrong in the gyre centers, but there are regions where this won’t work (notably coastal, and boundary currents).

 

There are other observational clues we can use to make a more informed choice of reference level (such as O2 minimum, or breaks in tracer property distributions).

 

Oceanographers play fast and loose with the terms

·        dynamic height/topography

·        steric height

 

Units are your friend

·        use them to check how to calculate geostrophic currents (m/s)

·        f is 10-4 s-1,   g is 10  m/s2

·        think about whether you have a sensible answer …
* 0.1 m/s is a moderate to brisk flow
* 1 m/s/ is hauling
* 10 m/s is ballistic

 

 

Thermal wind

 

The “light on the right” rule can be derived another way by reconsidering the geostrophic balance:

 

 

Multiply through by r and differentiate w.r.t. z

 

         

 

Recalling the hydrostatic relation, we can replace the vertical pressure gradient:

 

 

to get

 

         

 

We can simplify this further by approximating the left-hand-side terms.

 

             (the chain rule)

 

Consider the magnitude of the two terms in the expansion.

 

             3 kg/m3 / 1000 m * 1 m/s  :    1000 kg/m3 * 1 m/s / 1000 m

                      3/1000                       :                1

 

Since  varies so little (3 kg/m3) compared to its mean value, this first term is negligible, and we can take  (and f) outside the vertical derivative, leaving

 

         

 

The slopes of the density field are large (100 to 300 times greater than the surface) and readily measured from data. So these thermal wind equations give us a straightforward way to compute the velocity shear. If we have direct observations of the velocity at some depth, such as from current meters, drifters, or by assuming a level of no motion, we can compute flow at all other depths.

 

Rule of thumb: “light on the right” (in the northern hemisphere)

 

Thermal wind: If in the northern (southern) hemisphere isopycnals slope upward to the left (right) across a current when looking in the direction of flow, current speed decreases with depth; if they slope downward, current speed increases with depth.