Tag Archives: Rossby waves

Planetary Rossby waves on Beta-plane. A super easy tank experiment!

This is seriously one of the easiest tank experiments I have ever run! And I have been completely overthinking it for the last couple of weeks.

Quick reminder: This is what we think hope will happen: On a slope, melt water from a dyed ice cube will sink, creating a Taylor column that will be driven down the slope by gravity and back up the slope by vorticity conservation, leading to a “westward” movement in a stretched, cyclonic trajectory.

We are using the DIYnamics setup: A LEGO-driven Lazy Susan. And as a tank, we are using a transparent plastic storage box that I have had for many years, and the sloping bottom is made out of two breakfast boards that happened to be a good size.

Water is filled to “just below the edge of the white clips when they are in the lower position” (forgot to take measurements, this is seriously what I wrote down in my notes. We didn’t really think this experiment would work…)

The tank is then rotated at the LEGO motor’s speed (one rotation approximately every 3 seconds) and spun into solid body rotation. We waited for approximately 10 minutes, although I think we had reached solid body rotation a lot faster. But we had a lot of surface waves that were induced by some rotation that we couldn’t track down and fix. But in the end they turned out to not matter.

To start the experiment, Torge released a blue ice cube in the eastern corner of the shallow end. As the ice cube started melting, the cold melt water sank down towards the ground, where it started flowing towards the bottom of the tank. That increased the water column’s positive relative vorticity, which drove it back up the slope.

This was super cool to watch, especially since the ice cube started spinning cyclonically itself, too, so was moving in the same direction and faster than the rotating tank.

You see this rotation quite well in the movie below (if you manage to watch without getting seasick. We have a co-rotating setup coming up, it’s just not ready yet…)

Very soon, these amazing meandering structures appear: Rossby waves! :-)

And over time it becomes clear that the eddies that are being shed from the column rotating with the ice cubes are constant throughout the whole water depth.

It is a little difficult to observe that the structure is really the same throughout the whole water column since the color in the eddies that were shed is very faint, especially compared to the ice cube and the melt water, but below you might be able to spot it for the big eddy on the left.

Or maybe here? (And note the surface waves that become visible in the reflection of the joint between the two breakfast boards that make up the sloping bottom. Why is there so much vibration in the system???)

You can definitely see the surface-to-bottom structures in the following movie if you don’t let yourself be distracted by a little #HamburgLove on the back of the breakfast boards. Watching this makes you feel really dizzy, and we’ve been starting at this for more than the 8 seconds of the clip below ;-)

After a while, the Taylor column with the ice cube floating on top starts visibly moving towards the west, too. See how it has almost reached the edge of the first breakfast board already?

And because this was so cool, we obviously had to repeat the experiment. New water, new ice cube.

But: This time with an audience of excited oceanographers :-)

This time round, we also added a second ice cube after the first one had moved almost all the way towards the west (btw, do you see how that one has this really cool eddy around it, whereas the one in the east is only just starting to rotate and create its own Taylor column?)

And last not least: Happy selfie because I realized that there are way too few pictures like this on my blog, where you see what things look like (in this case in the GEOMAR seminar room) and who I am playing with (left to right: Torge, Franzi, Joke, Jan) :-)

Rossby waves in a rotating tank — three different demonstrations

For both of my tank experiment projects, in Bergen and in Kiel, we want to develop a Rossby wave demonstration. So here are my notes on three setups we are considering, but before actually having tried any of the experiments.

Background on Rossby waves

I recently showed that rotating fluids behave fundamentally differently from non-rotating ones, in that they mainly occur in the horizontal and thus are “only” 2 dimensional. This works really well as long as several conditions are met, namely the water depth can’t change, nor can the rotation of the fluid. But this is not always the case, so when either the water depth or the rotation does change, the flow still tries to conserve potential vorticity and stay 2 dimensional, but now displays so-called Rossby waves.

Here are different setups for Rossby wave demonstrations I am currently considering.

Topographic Rossby wave

For a demonstration of topographic Rossby waves, we want the Coriolis parameter f to stay constant but have the depth H change. We use the instructions by geosci.uchicago.edu as inspiration for our experiment and

  • build a shallow ridge into the tank. They use an annulus and introduce the ridge at a random longitude, we could also build one across the center of the tank all the way to both sides to avoid weird things happening in the middle (or introduce a cylinder in the middle to mimic their annulus)
  • spin up the tank to approximately 26 rpm (that seems very fast! But that’s probably needed in order to create a parabolic surface with large height differences)
  • wait for it to reach solid body rotation (ca 10 min)
  • reduce rotation slightly, to approximately 23 rpm so the water inside the tank moves relative to the tank itself, and thus has to cross the ridge which is fixed to the tank
  • introduce dye upstream of the ridge, watch it change from laminar flow to eddies downstream of the ridge (they introduced dye at the inner wall of their annulus when the water was in solid body rotation, before slowing down the tank).

What are we expecting to see?

In case A, we assume that the rotation of the tank is slow enough that the surface is more or less flat. This will certainly not be the case if we rotate at 26rpm, but let’s discuss this case first, anyway. If we inject dye upstream of the obstacle, the dye will show that the current is being deflected as it crosses the ridge, to one direction as the water columns are getting shorter as they move up the ridge, then to the other direction when the columns are stretched going down the obstacle again. Afterwards, since the water depth stays constant, they would just resume a circular path.

In case B, however, we assume a parabolic surface of the tank, which we will have for any kind of fast-ish rotation. Initially, the current will move similarly to case A. But once it leaves the ridge, if it has any momentum in radial direction at all, it will overshoot its circular path, moving into water with a different depth. This will then again expand or compress the columns, inducing relative vorticity, leading to a meandering current and eddies downstream of the obstacle (probably a lot more chaotic than drawn in my sketch).

So in both cases we initially force the Rossby wave by topography at the bottom of the tank, but then in case B we sustain it by the changes in water depth due to the sloping surface.

My assessment before actually having run the experiment: The ridge seems fairly easy to construct and the experiment easy enough to run. However what I am worried about is the change in rotation rate and the turbulence and Ekman layers that it will introduce. After all, slowing down the tank is what we do create both turbulence and Ekman layers in demonstrations, and then we don’t even have an obstacle stuck in the tank. The instructions suggest a very slight reduction in rotation, so we’ll see how that goes…

Planetary Rossby waves on beta-plane

If we want to have more dramatic changes in water depth H than relying on the parabolic shape of the surface, another option is to use a rectangular tank and insert a sloping bottom as suggested by the Weather in a Tank group here. We are now operating on a Beta plane with the Coriolis parameter f being the sum of the tank’s rotation and the slope of the bottom.

Following the Weather in a Tank instructions, we plan to

  • fill a tank with a sloping bottom (slope approximately 0.5)
  • spin it at approximately 15 rpm until it reaches solid body rotation (15-20 minutes later)
  • place a dyed ice cube (diameter approximately 5 cm) in the north-eastern corner of the tank

What do we expect to see?

Ice cube and its trajectory (in red) on a sloping bottom in a rotating tank. Note: This sketch does not include the melt water water column!

Above is a simplified sketch of what will (hopefully!) happen. As the ice cube starts melting, melt water is going to sink down towards the sloping bottom, stretching the water column. This induces positive relative vorticity, making the water column spin cyclonically. As the meltwater reaches the sloping bottom, it will flow downhill, further stretching the water column. This induces more positive relative vorticity still, so the water column, and with it the ice cube, will start moving back up the slope until they reach the “latitude” at which the ice cube initially started. Having moved up the slope into shallower water, the additional positive vorticity induced by the stretching as the water was flowing down the slope has now been lost again, so rather than spinning cyclonically in one spot, the trajectory is an extended cycloid.

My assessment here (before having run it): I find this experiment a little more unintuitive because there are the different components of stretching contributing to the changes in relative vorticity. And from the videos I’ve seen, we don’t really get a clear column moving, but there are cyclonic eddies in the boundary layer that are shed. So I think this might be more difficult to observe and interpret. But I am excited to try!

Planetary Rossby wave on a cone (cyclical beta-plane?)

Following the Weather in a Tank instructions, we plan to also do the experiment as above but with cyclical boundary conditions, by using a cone in a cylindrical tank instead of a sloping bottom in a rectangular one.

The experiment is run in the same way as the one above (except they suggest a slightly slower rotation of 10 rpm). Physics are the same as before, except that now the transfer to reality should be a little easier, since we now have Rossby waves that can really run all the way around the pole. Also the experiment can be run for a longer time, since we don’t run into a boundary in the west if we are moving around and around the pole.

Ice cube and its trajectory (in red) on a cone in a rotating tank. Note: This sketch does not include the melt water column!

My assessment before actually having run the experiment: This shouldn’t be any more difficult to run, observe or interpret than the one above (at least once we’ve gotten our hands on a cone). Definitely want to try this!

Vorticity and Rossby waves

Usually when we talk about waves on this blog, we talk about surface- or sometimes internal waves, but my waves almost always oscillate vertically. Today, we’ll mix things up a little: Rossby waves are waves in the horizontal plane. They exist for example as oscillations on the atmosphere’s jet stream. In order to understand what causes them, we need the concept of vorticity, which I will go over first before giving examples for Rossby waves.

Vorticity

Vorticity is a measure of how much a fluid is rotating. Generally speaking, once a fluid is in rotation, it wants to keep rotating (as we saw for example with the bottom Ekman layers in a rotating tank, where the water inside kept on rotating after the tank was stopped, until it was slowed down by friction). There are several components that are at play here — the rotation that we see when looking at eddies, the rotation of the Earth, and others — which I will go over in the following.

Relative vorticity

The easiest way to imagine what “vorticity” is (and the only one that I’ve talked about on this blog, see here), is to think of a little float in a flow. In a vorticity-free flow, that little float will always keep its orientation (see below). However if there is shear in the flow, i.e. the flow field carries vorticity, it will start to turn.

vorticity1

Flow fields without vorticity (top) and with vorticity (bottom). 

Relative vorticity is what we see for example when looking al leaves swirling in rivers.

Planetary vorticity

Since the Earth is turning (and all the water on it with it), the water also carries planetary vorticity, i.e. the rotation of the Earth, which is the Coriolis parameter. The Coriolis parameter is largest at the poles and zero at the equator, meaning the rotation changes with latitude.

The rotation of the Earth is clearly important enough for us to want to spin our tanks to simulate its effects on ocean currents.

Absolute vorticity

The sum of relative and planetary vorticity is called absolute vorticity: This is how much any fluid column is rotating in total, including all possible components of rotation (which are only the two mentioned above, but still…).

Potential vorticity

One more factor that can influence the rotation of a fluid parcel is the water depth. When water depth increases, columns of water get expanded vertically (since, for continuity reasons, they still have to go all the way from surface to bottom, even if the distance is now larger) or, if water depth decreases, squished. Similarly to figure skaters that stretch or crouch to increase or decrease their rotation, the expansion of a column of water leads to a change in its rotation.

Potential vorticity is defined as absolute vorticity over water depth.

Conservation of potential vorticity leads to waves

Potential vorticity is conserved, so if water depth, planetary vorticity changes or relative vorticity changes, something else has to change to compensate. And if water changes how much it is rotating, this leads to meanders in currents, i.e. waves.

Depth is constant, but latitude changes: Planetary Rossby waves

Planetary vorticity changes with latitude, therefore if a water parcel moves in north-south direction over constant water depth, its relative vorticity needs to change in order for potential vorticity to be conserved. This leads to so-called planetary Rossby waves, where currents in the ocean or the atmosphere start oscillating in north-south direction (see figure below).

At position 1, a fluid parcel gets for any random reason pushed northward. As it moves north, its planetary vorticity increases and its relative vorticity therefore has to decrease to compensate (2). This leads to southward movement, but the initial latitude (3) is overshot a little (4). This again leads to a change in relative vorticity (4), which brings the water parcel back to its initial latitude (5), but it overshoots again… So this mechanism leads to a wave-like motion in the horizontal plane, with the phase of the wave propagating westward.

This can happen at any latitude, even at the equator where “equatorial planetary Rossby waves” occur. At the equator f=0, but as soon as the water column has moved slightly north or south from the equator, f kicks in and drives the water column back to the equator (where it then overshoots, is turned back, overshoots again…….).

Latitude is constant but depth changes: Topographic waves

If a current encountered a ridge, the water depth changes and the current thus gets deflected. This motion is called topographic wave: When a water column gets stretched, it gains relative vorticity, making it rotate cyclonically. When it runs into shallower water, it loses relative vorticity and starts turning the other way.

I’m hoping to set up demonstrations for both types of Rossby waves soon. Stay tuned! :-)