Labs

The Lab Experiments are linked below. Please note that Lab Experiment IX was skipped (as shown in the calendar) and Lab Experiment XV is not available to OCW users.

GFD0: Rotation stiffens fluids
GFDI: Cloud formation on adiabatic expansion
GFDII: Convection
GFDIII: Radial inflow
GFDIV: Parabolic table
GFDV: Inertial circles
GFDVI: Perrot’s bathtub experiment
GFDVII: Taylor columns
GFDVIII: Thermal wind and Hadley circulation
GFDIX: Slope of a frontal surface
GFDX: Ekman layers
GFDXI: Atmospheric general circulation
GFDXII: Ekman pumping and suction
GFDXIII: Ocean gyres
GFDXIV: Thermohaline circulation

GFD0: Rotation stiffens fluids

GFDI: Cloud formation on adiabatic expansion

GFDII: Convection

GFDIII: Radial inflow

GFDIV: Parabolic table

GFDV: Inertial circles

GFDVI: Perrot’s bathtub experiment

GFDVII: Taylor columns

GFDVIII: Thermal wind and Hadley circulation

GFDIX: Slope of a frontal surface

GFDX: Ekman layers

GFDXI: Atmospheric general circulation

GFDXII: Ekman pumping and suction

GFDXIII: Ocean gyres

GFDXIV: Thermohaline circulation

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We take two tanks (of diameter roughly 50 cm) and place one on a rotating table and the other on a desk. We fill them with water to a depth of 20 cm or so, and set the rotating table turning anticlockwise (looking down from the top) at a speed of order 10 rpm (revolutions per minute, i.e. a period of 6 seconds). We allow the water in the rotating tank to settle down into ‘solid body’ rotation - 20 minutes or so. Then we gently stir the fluid - our hand is best, but try not to introduce a systematic swirl - to generate motion, and wait a minute or so for things to settle down a little, but not long enough for the currents to die away. We observe the motion by introducing dye (food coloring).

Here’s a shot of the rotating tank some minutes after the coloring has been introduced. It’s also fun to use two different colors and watch the interleaving of fluid.

Non-Rotating

In the non-rotating tank the dye disperses much as we might intuitively expect - have a look at the pictures and movie loop below. Note that we can also see a side view on the left-hand side of the images below obtained using a mirror sloped at 45 degrees.

See the movie

Experiment 0: Non-Rotating Fluids

Rotating

In the rotating body of water, by contrast, something glorious happens - have a look at the images and the movie loop below. We see beautiful streaks of dye falling vertically; the vertical streaks become drawn out by horizontal fluid motion into vertical ‘curtains’ which wrap around one another. The vertical columns - called ‘Taylor columns’ after G.I. Taylor, who discovered them - are a result of the rigidity imparted to the fluid by the rotation of the tank. The water moves around in columns which are aligned parallel to the rotation vector. Since the rotation vector is directed upward, the columns are vertical. Thus we see that rotating fluids are not really like fluids at all!

See the movie

Experiment 0: Rotating Fluids

Note that the movie is recorded in the frame of reference of the tank - i.e. by a camera mounted above the rotating table, rotating at exactly the same speed.

From the movie loop of the dye in the rotating frame try and estimate the Rossby number - see the introduction and chapter 6 of our notes. Is the Rossby number large or small?

Have a look at more Taylor columns here:

See the movie

Experiment 0: Rotating Fluids 2

Have a look here to learn about the life of G I Taylor.

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The sensitive dependence of saturation vapor pressure on temperature can be readily demonstrated by taking a carboy and pouring warm water into it to a depth of a few cms or so, as shown in the photograph. We leave it for a few minutes to allow the air above the warm water to become saturated with water vapor. We rapidly reduce the pressure in the bottle by sucking at the top of the carboy. One’s lungs can provide the `suck’ or, more elegantly, the hose of a vacuum cleaner can be inserted at the top for a second or two. One might expect that the rapid adiabatic expansion of the air would reduce its temperature and hence the saturated vapor pressure sufficiently that the vapor would condense out to form water droplets, a `cloud in a jar’. To one’s disappointment, this does not happen.
The process of condensation of vapor → water to form a water droplet requires condensation nuclei - small particles on which the vapor can condense. We can introduce such particles into the carboy by dropping in a lighted match and repeating the experiment. Now on decompression we do indeed observe a thick cloud forming which disappears again when the pressure returns to normal - see photograph (right).

A common atmospheric example of the phenomenon studied in our bottle is the formation of fog due to radiational cooling of a shallow, moist layer of air near the surface. On clear, calm nights, cooling due to radiation can drop the temperature to the dew point, at which point fog forms.

The sonic boom pictured below is a particularly spectacular consequence of the sensitive dependence of saturation vapor pressure on temperature - just as in our bottle, condensation of water is caused by the rapid expansion and consequent adiabatic cooling of air parcels induced by the shock waves resulting from the jet going through the sound barrier.

A photograph of the sound barrier being broken by a US Navy Jet as it crosses the Pacific Ocean at the speed of sound just 75 feet above the water. Condensation of water is caused by the rapid expansion and consequent adiabatic cooling of air parcels induced by the shock (expansion/compression) waves caused by the plane outrunning the sound waves in front of it. Shot by John Gay from the top of an aircraft carrier. The photo won First Prize in the science and technology division of the World Press Photo 2000 contest.

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We can study convection in a laboratory setting using the apparatus sketched in the diagram below. A stable stratification can be set up in a 50cm square tank by slowly filling it up with water whose temperature is slowly increased with time. This is done using (i) a mixer that mixes hot and cold water together and (ii) a diffuser, which floats on top of the rising water. The diffuser ensures that the warming water floats on the top without generating turbulence.

For our diffuser we use a light metal box with an open top and holes drilled through its bottom. The box is then tightly packed with sponges and a buoyant ‘filler’ so that the whole device floats on water. After placing it into the tank, we take a hose with a stopper in the end and several holes along its length, and place it on the sponges, coiling the hose around itself once or twice. After slowly filling up the tank with cold/hot water we remove our diffuser as carefully as possible and let the tank settle for a few minutes.

Using the hot and cold water supply in the laboratory we can achieve a temperature difference of 20 C or more over the depth of the tank. The temperature profile can be measured and recorded using the thermocouples.

Heating at the base triggers convection from the bottom. It is supplied by a heating pad, whose power can be controlled with a transformer.

The motion of the fluid is made visible by sprinkling a VERY SMALL amount of potassium permanganate evenly over the base of the tank after the stable stratification has been set up. After switching on the heating at the base, thermals will be seen to rise from the base, overshoot the level at which they have zero buoyancy (the level where the T of the thermals is equal to that the environment, Te), and sink back. Successive thermals rise higher as the layer deepens.

Convection into a stratified layer heated from below:

See the movie:

Experiment II: Stratified Convection

On the left above we see the convection layer early in the experiment; on the right some time later. The convection ‘burrows’ into the stratified layer, eroding it to create a well-mixed convection layer. Notice the thermals rising up and brushing the stratified layer above, generating gravity waves on the inversion.

Convection carries heat from the heating pad into the body of the fluid, distributing it over the convection layer.

Convection into an unstratified layer heated from below:

If the initial fluid is unstratified, then buoyant fluid rises from the heating pad right up to the free surface. There is no ambient stratification to constrain vertical motion. The picture on the right zooms in on the convective plumes.

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For a detailed description of this experiment, click PDF.

Everyone is familiar with the swirl and gurgling sound of water flowing down a drain. In this laboratory experiment we explore this problem quantitatively, and draw out the strong parallels between it and the large scale flow in the atmosphere and ocean. We rotate a cylinder about its vertical axis. The cylinder has a circular drain hole in the center of its bottom. Water enters through a diffuser on the outer wall of the cylinder. The water is supplied by the diffuser at a constant rate so that a steady state can be set up in which the flow down the central drain exactly balances the inflow from the outer edge.

The swirling flow set up exhibits a number of important principles of rotating fluid dynamics - conservation of angular momentum, geostrophic and cyclostrophic balance, and Ekman layers.

The experiment described here was designed by Jack Whitehead of the Woods Hole Oceanographic Institution. For more details refer to:

Whitehead, J. A. and D.L. Potter. “Axisymmetric critical withdrawal of a rotating fluid.” Dynamics of Atmospheres and Oceans 2 (1977) 1-18.

The apparatus

Take a roughly 40 cm cylindrical tank of approximately 15 cm depth with a drain hole in the center of the bottom. A diffuser with 30 cm inside diameter is constructed of a wire screen, placed in the tank, and stones approximately 1cm in size used to fill the space between the screen and the outer wall of the tank. Water is then fed into the bottom of the rock bed through a loop hose with numerous holes (roughly 0.5 cm diameter), so that water is fed evenly into the bottom of the diffuser. The diffuser is effective at producing an axially-symmetric, inward velocity at the screen.

Place a large catch basin below the tank, partially filled with water and containing a submersible pump whose purpose is to feed water to the diffuser in the upper tank.

The entire apparatus is then mounted on a turntable.

Side view:

Top view:

Typical flow patterns

When the apparatus is not rotating, water flows radially inward from the diffuser to the drain in the middle. When the apparatus is rotating, however, water in the interior conserves angular momentum and, in so doing, acquires a vigorous swirling motion. The water spirals inward, as sketched in the diagram. The centrifugal force directed radially outward is balanced by a pressure gradient force directed radially inward. This pressure gradient force is set up by the free surface tilting relative to the horizontal.

SLOW rotation movie:

Experiment III: Slow Radial Inflow

FAST rotation movie:

Experiment III: Fast Radial Inflow

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If our cylindrical tank is filled with water, set turning, and left until it comes into solid body rotation, then the free-surface of the water will not be flat - it will be depressed in the middle and rise up slightly to its highest point along the rim of the tank (see fig. below).

The shape of the free surface is given by

h(r) = h(0) + ((Ω²r²)/(2g)),

where h is the local depth, r is the distance from the axis or rotation, and h(0) is the depth at r = 0. Thus the free surface takes on a parabolic shape.

Let’s put in some numbers for our tank. We can obtain rotation rates of up to 10 rpm (which is an Ω of 1 /s). The radius of the tank is 0.30 m and g = 9.81 m/s**2, giving

((Ω²r²)/(2g)) ~ 5mm,

a small fraction of the depth to which the tank is typically filled.

It is very instructive to make the surface of our turntable parabolic. This can readily be achieved by filling a large flat-bottomed pan with resin on a turntable and letting the resin harden while the turntable is left running (10 rpm works well) for several hours (see instructions). The resulting parabolic surface can then be polished to create a low friction surface.

Place a ball-bearing on the rotating parabolic surface - make sure that the table is rotating at the same speed as was used to create the parabola! Note that it does not fall in to the center, but instead finds a state of rest in which the component of gravitational force resolved along the parabolic surface is exactly balanced by the outward-directed horizontal component of the centrifugal force.

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We place a large tank on our rotating table and fill it with water to a depth of 10 cm or so. We then take a hollow metal cylinder, generously rub petroleum jelly around its lower rim, and place it in the center of the tank so that it protrudes slightly above the surface (see picture below). A suitable cylinder, roughly 10cm in diameter, can readily be made by taking the lid off both ends of a coffee bean can.

The table is set into rapid rotation at a speed of about 14 rpm (f = 3) and allowed to settle down for 20 minutes or so. Whilst the table is rotating we carefully and slowly remove the water within the cylinder and replace it with dyed, saturated salty (and hence dense) water delivered from a small beaker. When the hollow cylinder is full of colored saline water, it is rapidly removed in a manner which causes the least disturbance possible. Practice may be necessary.

The column of dense salty water slumps under gravity but is `held up’ by rotation forming a cone whose sides have a distinct slope. The cone acquires a definite sense of rotation, swirling cyclonically at its top in the same sense of rotation as the table. We measure typical speeds through the use of paper dots, measure the density of the dyed water and the slope of the side of the cone (the front) and interpret them in terms of the theory of thermal wind.

Have a look at the sequence of pictures below. Note how we use a mirror (at the top) inclined at 45 degrees to reveal the vertical structure.

See the movie:

Experiment IX: Cylinder Collapse

Why does the dense fluid remain contained in a cone, rather than flow over the bottom of the tank? Consider the figure below. On the left we see a cylinder of dyed salty (and hence dense) water collapsing under gravity in to a rotating tank of fresh water. Its final state is not the intuitive one, with resting light fluid over dense separated by a horizontal interface. The schematic on the right indicates how the slumping column `concentrates’ and `dilutes’ the angular momentum of the rotating tank to generate horizontal swirling motions that prevent the column from collapsing.

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Visualizing the Coriolis force

Detailed description of this experiment, along with attendant theory (PDF).

The Coriolis force can be visualized by making use of the parabolic surface constructed in Lab IV. If a ball, initially at rest in the rotating frame, is given a push, it is deflected to the right.

We can experiment further by forming a `puck’ from a 1/2 inch diameter disc cut from rods of `dry ice’. The gas sublimating off the bottom almost eliminates frictional coupling between the puck and the surface of the parabolic dish.

Play games with the puck and study its trajectory on the parabolic turntable, both in the rotating and laboratory frames. Notice that when the puck moves in the rotating frame it is deflected to the right. The following are useful reference experiments:

Launch the puck so that it is motionless in the rotating frame of reference - it will follow a circular orbit around the center of the dish in the laboratory frame.
Launch the puck on a trajectory that lies within a fixed vertical plane containing the axis of rotation of the parabolic dish. Viewed from the laboratory, the puck moves backwards and forwards along a straight line. The straight line will expand out into an ellipse if the frictional coupling between the puck and the rotating disc is not negligible. When viewed in the rotating frame, however, the trajectory appears as a circle tangent to the straight line. See the movie loop below.
The puck is again launched so that it appears stationary in the rotating frame, but is then slightly perturbed. In the rotating frame the puck undergoes oscillations consisting of small circular orbits passing through the initial position of the unperturbed puck. These circles are called `inertial circles'.

On the left we see the puck in the non-rotating frame. On the right we see the same puck but in the rotating frame.

See the movie:

Experiment V: Visualizing the Coriolis Force

Trajectory of the puck on the rotating parabolic surface in (a) the inertial frame and (b) the rotating frame of reference. The parabola is rotating in an anticlockwise (cyclonic) sense.

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An experiment in the Earth’s rotation

A classic experiment was carried out by Perrot in 1859 which closely paralleled Foucault’s pendulum. He filled a barrel with water (the barrel had a hole in the middle of its base plugged with a cork) and left it standing for two days. He returned and released the plug. As fluid flowed in toward the drain-hole it conserved angular momentum, thus `concentrating’ the rotation of the Earth, and acquired a `spin’ that was cyclonic (in the same sense of rotation as the earth) - the experiment was carried out in Paris in the northern hemisphere.

We will repeat Perrot’s experiment now. The experimental setup is shown in the diagram above. It comprises a leveled cylindrical container of radius 30cm filled to a depth of 10 cm with water (to prevent convection the container is covered with a lid until the time of the experiment). At the center of the cylinder is a small hole, 1.5 mm in diameter. Attached to the hole is a hose (also filled with water and stopped by a rubber bung) which hangs down in to a pail of water.

The water is left for a day or so (why?). When the rubber bung sealing the hose is released at the time of the experiment, water in the cylinder is sucked out in to a pail (the pressure in the hose is less than atmospheric pressure). The cylinder takes about 5 minutes to drain. Toward the end of the draining process a small polystyrene ball - the ball has a cross marked on it to ease observation - is gently floated on the water, and its rate and sense of rotation observed.

According to theory, we expect to see the ball turn in the same sense of rotation as the Earth.

This experiment is very tricky - you may have to carry it out many times and do some statistics!

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The Taylor-Proudman theorem demands that vertical columns of fluid move along contours of constant fluid depth. Suppose a rotating, homogeneous fluid flows over a bump on a bottom boundary, as shown here.

Near the boundary, the flow must of course go around the bump. But the Taylor-Proudman theorem says that the flow must be the same at all heights: so, at all heights, the flow must be deflected as if the bump on the boundary extended all the way through the fluid! Thus, fluid columns act as if they were rigid columns and move along contours of constant fluid depth. We can demonstrate this behavior in the laboratory.

We place a cylindrical tank of water on a rotating turntable. A few obstacles, none of which are taller than a small fraction of the depth of the water, are on the base of the tank. With f = 3 /s and h = 10 cm we wait for the fluid to settle down and come in to solid body rotation and then carefully drop a few crystals of dye in to the water. Each crystal leaves a vertical dye streak as it falls. Note the vertical ‘rigidity’ of the fluid.

We sprinkle black dots over the surface to mark the fluid and reduce f to 2.9 /s. Until a new equilibrium is established (the “spin-down” process takes several minutes, depending on rotation rate and water depth) the water will be moving relative to the tank. We should be able to see the dots being diverted around the obstacles in a vertically coherent way (as shown schematically below) as if the obstacles extended all the way through the water, thus creating stagnant “Taylor columns” above the obstacles.

Below are a snapshot and a movie to show you what actually happens. It’s a pretty tricky experiment - you have to practice hard to get it to work.

Note that the black dots below are floating on the surface. The cylinder is submerged (see photograph above).

See the movie:

Experiment VII: Taylor-Proudman Theorem

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The Thermal Wind Relation

It is straightforward to obtain a steady, axially-symmetric circulation driven by radial temperature gradients in our laboratory tank, which provides an ideal opportunity to study the thermal wind relation.

We fill the cylindrical tank with water to a depth of 15 cm, and rotate it very slowly - at no more than 0.3 f (or even less) in a cyclonic sense (anticlockwise looking down). At the center of the tank we place a 15 cm diameter tin can full of ice and water (and a few lumps of metal to prevent the can floating away). The cold sides of the can cool the water adjacent to it and induce a substantial radial temperature gradient. The experiment is left for twenty minutes or so, for the circulation to develop.

First we drop in a few permanganate crystals which streak the vertical column and settle on the bottom. The streaks do not remain vertical: rather they tilt over in an azimuthal direction, carried along by the currents which increase in strength with height in a cyclonic sense (anticlockwise looking down) relative to the tank. We sprinkle black paper dots over the surface and note that they move in the same sense as, but more swiftly than, the rotating table - we have generated westerly (to the east) winds! And these currents tilt over the Taylor column (see above schematic).

Green dye is injected and evolves in to a tightly wound spiral.

In this circulation, relatively warm water rises at the outer wall, moves inward in the upper layers (more-or-less conserving angular momentum as it does so), rubs against the cold inner wall, and becomes cold. This is how the fluid carries heat to offset cooling at the ‘pole’ induced by melting ice.

The permanganate crystals on the bottom give an indication of the flow in the bottom boundary layer. We see flow moving radially outwards and being deflected to the right - note the pink streamers moving outward and clockwise. This flow is directly analogous to the trade winds of the atmosphere.

See the movie:

Experiment VIII: Hadley Circulation

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Ekman layers: frictionally-induced cross-isobaric flow

We bring the cylindrical tank, filled to a depth of 10 cm or so with water at a uniform temperature, up to solid-body rotation at a speed of 5 rpm, say. We sprinkle a few small crystals of potassium permanganate in to the tank. Note the Taylor columns.

Now we reduce the rotation rate to, say, 3 rpm. The fluid continues in solid rotation like a cyclonic vortex (same sense of rotation as the table) with lower pressure in the center and higher pressure near the rim of the tank. Note that the plumes of dye from the crystals on the bottom of the tank flow inward to the center of the tank at about 45 degrees relative to the geostrophic current (see fig. below, left panel).

Now we increase the rotation rate. The relative flow is now anticyclonic, with high pressure in the center and low pressure on the rim. Note how the plumes of dye sweep around to point outward.

In each case we see that the rough bottom of the tank slows the currents down there, and induces a cross-isobaric flow from high to low pressure. Above the frictional layer, however, the flow remains close to geostrophic.

See the movie:

Experiment X: Ekman Layers

The figure below shows schematics of a cyclone (low pressure system) and an anticyclone (high pressure system). In the free atmosphere, where the flow is geostrophic, the flow just rotates around the system - cyclonically around the low, anticyclonically around the high. Near the surface, however, the wind deviates toward low pressure, inward in the low, outward from the high. The boundary layer within which this happens is known as the Ekman layer. Because the horizontal flow is convergent into the low, mass continuity demands a compensating vertical outflow. This Ekman pumping produces ascent - and, in consequence, clouds and rain - in low pressure systems. In the high, the divergence of the Ekman layer flow demands subsidence (Ekman suction) and are dry, which is why high pressure systems tend to be characterized by clear skies.

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Baroclinic Instability of the Thermal Wind

The purpose of this experiment is to observe `Hadley’ and `eddying’ regimes in a differentially heated, rotating fluid annulus. The experimental arrangement is the same as in GFD Lab VIII (see figure opposite). It consists of a cylindrical plexiglass tank filled with water to a depth of about 15cm and placed on a rotating table. Initially, the water is of uniform temperature. But centered on the rotation axis, we place a metal bucket filled with ice. This sets up a radial temperature gradient (decreasing “poleward”) that will drive motions in the tank. Temperature variations in the tank are monitored at strategic positions using thermistors attached to data loggers. Currents are observed using paper dots, potassium permanganate crystals, etc.

‘Hadley’ and ’eddying’ turbulent regimes can be set up in the tank by adjusting the rotation rate, Ω, of the tank - Ω can be ranged from zero to 10 rpm.

We observe the following:

When the table is not rotating, a simple overturning circulation is observed, with sinking at the edges of the cold can, outflow near the bottom boundary layer, and inflow aloft.
When weakly rotating, an Ω of 0.3 rpm or less, we see the development of the thermal wind and a Hadley circulation as described in GFD Lab VIII with a single meridional overturning cell.
If the rotation rate is increased in small increments (0.2 rpm or so) we can map the transition from axisymmetric flow through regular waves to turbulent eddies.
When the table is rotating rapidly, an Ω of 1 rpm or more, we see the development of turbulent eddies in the tank, through baroclinic instability. The baroclinic eddies carry heat from the ’equator’ of the tank to the ‘pole’ in the center.

Here is a sequence of images showing the evolution of the flow in the fully turbulent regime:

See the movie:

Experiment XI: Baroclinic Instability of the Thermal Wind

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Here we study the mechanism by which the wind stress drives ocean circulation. We induce circulation by rotating a disc at the surface of a tank of water which is itself rotating. The laboratory setup is as follows.

We rotate a disc at rate ω on the surface of a cylindrical tank of water (in fact, the disc is just submerged beneath the surface). The tank of water and the disc driving it is then rotated at rate Ω using our turntable and left for about 30 minutes to come to equilibrium. Once equilibrium is reached, dye crystals are dropped in to the water to trace the motions. The whole system is viewed from above in the rotating frame. Mirrors can be used to capture a side view, as shown below.

Have a look at the pictures below showing the experiment in action.

Note the following:

In the interior (away from the bottom boundary) the horizontal flow is independent of height. Why? Since the water has uniform density, there is no `thermal wind’ shear.
Near the bottom boundary, there is inflow when ω has the same sign as Ω (cyclonic flow) and outflow when ω has the opposite sign (anticyclonic) - think about our Ekman layer experiment, GFD X.
There is also an Ekman layer at the top (beneath the rotating lid), in which the radial component of the flow is opposite to that at the bottom boundary. (Can you figure out why? Can you picture the overall meridional (radial/vertical) circulation in the tank?) To help, have a look at the pictures below showing dyed water upwelling beneath a cyclonically rotating disc:

See the movie:

Experiment XII: Ekman Pumping and Suction

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Wind-driven ocean circulation

It is relatively straightforward to demonstrate the essential mechanism behind wind-driven ocean circulation in a laboratory experiment.

The apparatus is shown in the figure below. A tank with a false sloping bottom is filled with water so that the water depth varies between 5cm at the shallow end and 15 cm at the deep end. A Perspex disc is rotated very slowly at the surface of the water in a (in the rotating frame) clockwise sense - a rate of 0.4 rpm works well. To minimize irregularities at the surface, the disc can be submerged so that its upper surface is a millimeter or so underneath the surface. The whole apparatus, disc and all, is then rotated in an anticlockwise sense at a speed of f = 2 (~ 10 rpm).

It is left to settle down for 1/2 hour or so. Dye can then be used to help map out the circulation: holes bored in the lid can be used to inject dye and visualize the circulation beneath, as in the picture below.

The varying depth - in the direction parallel to the rotation vector - mimics the variation of the ocean depth measured in the direction parallel to the rotation vector on the sphere (see figure below). The shallow end of the tank is analogous to the poleward side of the ocean basin (why?) and the deep end to the tropical side.

The stress applied by the lid to the water is analogous to the wind stress on the ocean surface. With clockwise differential rotation of the lid, fluid is drawn inwards in the Ekman layer just under the lid and pumped downwards into the interior, mimicking the pumping down of water in subtropical gyres by the action of the winds, as sketched below.

We see a clockwise (anticyclonic) gyre in the water with interior flow towards the deep end of the tank (‘southwards’), as the Ekman pumping drives fluid columns to the deeper end of the tank. This flow (except near the lid) will be independent of depth. A strong (’northward’) return flow forms at the ``western’’ boundary; this is the tank’s equivalent of the Gulf Stream.

Have a look at the sequence of pictures and the movie loop below:

See the movie:

Experiment XIII: Ocean Gyres

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Here we illustrate the dynamical principles that underlie the abyssal circulation of the ocean, driven by the sinking of dense fluid formed by surface cooling at polar latitudes. As in Lab XIII, we represent the sphericity of the Earth with a sloping false bottom. The sinking of water at polar latitudes is represented by a source of fluid in the top right-hand-corner of the tank.

We set a tank of water rotating at speed f = 2 (~ 10 rpm), as sketched in the diagram below. The square tank, of side 60 cm, whose bottom is inclined to the horizontal, is filled with water at constant temperature to a depth of 20 cm or so. The shallow end of the tank represents ‘North’, the deep end ‘South’, as in Lab XIII.

Dyed water at the same temperature is then slowly introduced in to the top right-hand corner of the tank through a hose (fitted with a diffuser) at a rate 100ml/minute. The photograph on the right below shows the funnel device used to introduce the dyed (red) fluid. The funnel is in the rotating frame; a reservoir of dyed fluid stands on the top of ladder in the laboratory (non-rotating) frame.

The rotating tank fills up with water in a manner that is very different from that in which a non-rotating tank would fill up. Instead of the dyed fluid ‘diffusing’ in to the interior, it runs along the ’northern’ boundary of the tank and feeds in to a ‘western’ boundary current, as sketched in the diagram above.

Have a look at the sequence of the pictures below illustrating how the tank fills up with water.

See the movie:

Experiment XIV: Thermohaline Circulation

Theory and interpretation

Let ‘h’ be the depth of the fluid, α the slope of the bottom, ‘L’ the side of the square tank, and ‘S’ the rate at which water is introduced through the diffuser.

The free surface of the water rises at a rate given by:

Because the fluid is rotating and is in steady, slow, frictionless motion, then, by the Taylor-Proudman theorem, columns must remain of constant length. Hence Taylor columns in the interior of the tank must move toward the shallow end of the tank to retain their length as the free surface rises. The northward speed at which they move is given by:

It is useful to estimate typical interior speeds from the formulae above. Typical experimental set-ups have α = 0.2, L= 60 cm, Ω= 10 rpm and S= 100ml/minute.

Finally, think about the relevance of the experiment to the thermohaline circulation of the ocean.

What do the parameters S, α, L and Ω correspond to oceanographically? Insert some oceanographic values for S, α, L and Ω, and hence estimate typical current speeds associated with thermohaline flow in the ocean.

Look at a simulation of CFC’s invading the ocean here.

The rotation rate of our table can be quoted in various units. The following are often used - period, revolutions per minute, or units of ‘f’, as described below:

The angular velocity of the tank, Ω , in radians per second
The Coriolis parameter (f) defined as f = 2Ω
The period of one revolution of the tank, τ, in seconds is τ = 2π/Ω
Revolutions per minute, rpm = 60/τ

Conversion table

Ω (rad/s)	0	0.25	0.5	1	1.5	2
f=2Ω(rad/s)	0	0.5	1	2	3	4
τ(s)	∞	8π = 25. 1	4π = 12. 6	2π = 6. 3	4π/3 = 4. 2	π = 3.14
rpm	0	2. 4	4. 7	9.5	14.3	19.1

Note that on the ‘big’ table pictured above, the digital readout is the rotation rate in `milli-f’: thus 1000 is a rotation rate of 1× f: the period of rotation is thus 12.6 s.

Fabricating a parabola

A ‘geoid’ can be constructed for our rotating table by pouring resin in to a mold on the rotating table. The surface of the resin takes up a parobolic shape and solidifies. If care is taken the resulting surface can be made to be very smooth and then used in experiments on the Coriolis force.

A simple form in to which resin is poured can be constructed as follows. A 1/4" circular aluminum plate is used as a base. A piece of flexible, vinyl, “base” molding (available in building supply houses) is wrapped around the base and attached with a small bead of 5 minute epoxy. The building materials and glue should be tested to check that there are no adverse reactions to the resin that is poured in to the form. The form is not treated with a mold release.

The base should be constructed of metal and plastic or other non-moisture absorbent material because the resin is moisture sensitive. We chose a metal base to add stiffness, preventing the variable thickness resin from warping when curing. Measuring/mixing containers and stirrers should also be metal or plastic.

The resin is poured into the form while it is rotating at the desired speed. The pot life, or working time, should be about 20 minutes, minimum. Shorter working times risk the possibility of the resin starting to harden before it has finished flowing into the parabola shape. Carefully measure the two parts according to resin instructions. Care should also be taken while pouring the two part resin into measuring/mixing containers and when pouring into the form to prevent creating and entrapping air bubbles. Any bubbles in the resin will rise to the surface and create small cavities in the hardened resin surface. Stir the combined parts carefully also, using the manufacturer’s guidelines. Pouring the resin in a single pour seemed to reduce the amount of bubbles and eliminated the interaction between subsequent pours.

A total of 3,000ml of resin (combined total of part A and part B) was enough for a 42" parabola for a rotation rate of f = 2. This included a little extra as insurance. Strict adherence to all manufacturer guidelines is a good investment and will prevent wasted time and material.

Smooth-On resin was chosen because of its availability, minimum odor and because mixing was done by volume, eliminating the need for expensive scales. Odor is a major factor as the mixing, pouring and curing is done in the lab and on the rotating turntable. When adding color tint or any additive, it is possible to create problems, such as inability to harden. Excessive amounts or incompatible additives can prevent curing. Do sample tests when using additives. We used the manufacturer’s own color tint, using only a few drops and no other additives. The resin may be purchased on the web at www.smooth-on.com. Order Smooth-Cast 322, liquid plastic casting resin, slow setting off-white resin in the gallon Unit at $58.45 each. Each unit has a gallon of part A and a gallon of part B. Two units are required for a 42" diameter parabola and allows extra for experimentation.

Sand the cured surface lightly with a medium grade sandpaper to remove irregularities and finish sand with a 300 or 400 grade wet or dry sandpaper until the surface is very smooth to the touch. Sand uniformly over the surface to preserve the parabola shape. The surface can be used as is, or automotive primer and paint applied.