# Presentation: 060042

## The Sverdrup balance

- Science Olympiad Resource Page
- Oceanography B
- Anand Gnanadesikan

## Purpose fo this presentation

- This presentation describes the Sverdrup balance, which plays an important role in determining the large-scale flow of the ocean.
- To understand this fully, you may want to  look at the companion presentations on Coriolis force, Ekman flow, and geostrophic flow first.

## Consider a spinning earth...

- Animation courtesy of NASA
- http://visibleearth.nasa.gov

- If you look down on a column of water at the north pole, it will seem to spin *counterclockwise*.

- But a column of water at the equator won’t seem to spin at all, it will just move eastward.

- And a column of water at the south pole will spin *clockwise*.

## Sverdrup

- Harald U. Sverdrup was a Norwegian
- oceanographer who served as director of the Scripps Institution of Oceanography during the Second World War. As such he helped train many American naval officers in oceanography and paved the way for the expansion of the field in the following years. One of his main accomplishments during the war was the development of a system for predicting waves, without which many American servicemen would have died in the Pacific surf. He also wrote the first modern textbook on oceanography with colleagues Martin Johnson and Richard Fleming.

- http://scilib.ucsd.edu/sio/archives/siohstry/sverdrup-biog.html

## Sverdrup balance

- Sverdrup’s key insight was an equation that showed that if  a column of fluid were to move north or south it would tend to develop a strong spin relative to the earth underneath it.
- But in general this does not happen, flows in the ocean are generally quite small in comparison to the rotation of the earth underneath them.
- So something must act to balance this tendency to spin.

## Sverdrup proposed that this “something” was the surface winds. The red vectors in the plot above show surface wind stresses over the North Pacific, while the colors and contours show an estimate of the sea surface height.  Look at the center of the basin between 40N and 55N, in the box shown.

- Which way is the wind causing the water at these latitudes to spin?

## The answer is counterclockwise, or more like the spin at the North Pole.

- Now consider the box to the south, between 20 and 40 degrees North.
- Which way is the wind causing the water to spin?

## The answer is clockwise, or more like the spin at the South Pole.

- So the winds in the northern part of the basin are making the column spin more like the earth to the north- so they are consistent with a *northward flow*.
- The winds in the southern part of the basin are making the column spin more like the earth to the south, so they are consistent with a *southward flow*.
- This is exactly what the flow lines associated with the surface height predict!

## Spin added by the wind

- The plot above shows the spin added by the wind, in units of 10-10 m/s2. The regions with dotted contours are negative, those with solid contours are positive. The zero line marks the boundary between regions where the main motion in the interior is northward and those where it is southward.

## How well does it predict the actual structure of the ocean?

- Actually it does work pretty well in many locations. The red line shows what I’d predict for the surface height at 30N in the Pacific if Sverdrup balance held and all I knew was the wind stress and that the velocities went to zero at 1500m. The black line shows what I compute from the density. So it works pretty well, at least between about 40N and 40S.

## Key points

- If the winds are causing the water to spin counterclockwise they will drive a northward flow.
- If the winds are causing the water to spin clockwise they will drive a southward flow.
- This flow does a pretty good job at explaining why there are gyre circulations.
- If you want to understand a bit more about how spin balance is used to make quantitative predictions, read on.

## Spin balance

- If we look down on a spinning column of water, the spin is defined as that associated with the north-south velocities, plus that associated with the east-west velocities. This ends up being twice the rate of rotation. So a column at the north pole has a spin rate of 2 cycles per day.
- This is a rate of 1 cycle/43200 seconds.
- Physicists measure spin in radians, of which there are 2 per cycle. Think of it as the velocity around the edge. So the spin is actually around 1/7000 sec
- A column 1000m thick will thus have a total spin of 1/7 m/s at the North Pole and 0 at the equator.

## More on spin balance

- At 50N the spin
- added by the wind is 1.5 x 10-10 (m/s)/s
- So it would take about 109 (one billion) seconds of this spin for a 1000m thick column to move from 0N to 90N.
- This sounds like a lot, but there are 60 seconds in a minute, 60 minutes in an hour, 24 hours in a day and 365 days in a year! How many seconds is this?

## More on spin balance

- The answer 31.6 million seconds!
- So a billion seconds is only about 30 years.
- So the rough speed of a column of water is about 90 degrees in 30 years, or 3 degrees per year. 1 degree of latitude is about 111 km, so this is about 300 km every year. How many cm/s is this?
- The answer is pretty surprising, it’s actually about 1 cm/s! Very slow, but over time it adds up!