Tilting Isopycnals (the simplified version)

Sunset over Cumberland Sound

One of the things I’m attempting to determine is if the Baffin Island Current*, which passes outside of the mouth of Cumberland Sound, bends into the sound. Last summer, we were able to conduct two rounds of CTD** casts at regular intervals across the sound's mouth. Unfortunately, I wasn’t actually there as I was too pregnant to be at sea. I doubt I would have fit in the bunk as the ship we used is a particularly cramped research vessel (picture here).

Even though it was cramped, the ship had a hull-mounted current meter. Unfortunately, the instrument wasn’t turned on. No one on board had the knowledge to fiddle with it, so I missed out on that data (that’s the way it goes sometimes). Without measured currents, how does one infer water flow from CTD data?

The Baffin Island Current is geostrophic, that is, the pressure gradient force is balanced by the Coriolis force. In this case, friction and tides become unimportant and can be ignored when calculating current flows.

The pressure gradient force is the weight of water as the sea surface height is not at the same everywhere. This force is always directed from areas with high pressure to areas of low pressure. Without a balancing force, a parcel of water will move from the area of high pressure to the low one. But, there is another force out there to balance with - the Coriolis force.

Actually, the Coriolis force isn’t a real force; instead it is like an imaginary friend that shows up to solve a problem. It pops up when we treat our rotating planet as though it’s an inertial frame of reference to use Newton’s laws. Newton's laws form the base of ocean physics - and most other things that aren't moving too fast or are too small. Now, we’ll move on to pretending the Coriolis force is real. This force acts in different directions depending on the hemisphere, since I work in the northern hemisphere, I’ll take it as acting to the right.

As soon as the parcel of water from above starts to move because of the pressure gradient force, it will be acted upon by the Coriolis force and deflected to the right. The result will be a current that flows along an isobar (line of constant pressure) - a geostrophic flow.

In the ocean, pressure is difficult to measure. Fortunately, pressure is related to density and density depends on salinity and temperature which I measured. In the Arctic, where Cumberland Sound is, density depends mostly on the salinity, however, since I measured both I used both. I’ve calculated density and plotted up lines of constant density, which are called isopycnals. From plotting a cross-section of density, isopycnal slopes tell us if water flows in or out of the section, which is exactly what I’m looking for (note: isobars and isopycnals have opposite slopes).

From isopycnal slopes, a relative velocity can be calculated as currents move faster where the isopycnal slopes are steeper. Actual velocities would have been nice to get, leaving me wishing I had been on the ship to turn on the current meter. However, relative velocities still answer my question of whether the Baffin Island Current bends into my site. The answer is yes it does.

*The Baffin Island Current is the official name of this current which passes along the coast of Baffin Island (a nice diagram showing it can be found in this paper). Many currents have assigned names, the Gulf Stream and Kuroshio are perhaps more familiar examples. 

**CTD stands for Conductivity Temperature Depth. From conductivity, salinity is calculated. This instrument samples the water as it descends directly down from the ship resulting in profiles of these properties with depth.

Something on glass sponges...

Here I am building a mooring

For my masters work I looked at flow over a local glass sponge reef. It turns out that how the tides interact with a sub-surface ridge may influence the conditions the sponge reef lives in. I wrote a little about it for the UVic Ocean Student Society here.

What makes water wet?

Water is pretty fabulous stuff. It makes life possible and over enough time acts as a universal solvent. On earth, water is everywhere from the glass on my desk to covering over 70% of the surface. Oceans contain approximately 1,360,000 cubic kilometers worth of water – a lot of water even before land based, sub-surface, frozen or any other place water hides is considered.

Our current understanding of water came relatively recently. We held onto the ancient Greek concept that water was one of the four basic elements (along with fire, earth and air) up until the Renaissance. Water is very stable, making it difficult to break into its parts, but it was done at a time when most people considered it a fundamental element. The French chemist Lavoisier managed to break water apart into its components (two hydrogen molecules and one oxygen on) on 28 February 1785. In time it was determined that covalent bonds hold water together, that is, by sharing electrons between atoms.

A water molecule resembles Mickey Mouse's head. Two ears on top, which are hydrogen atoms, and the head would be the oxygen atom. The top, where the hydrogen is, is somewhat positive and the bottom, where the oxygen is, is somewhat negative resulting in a polar molecule. Since positive and negative are attracted to each other a hydrogen atom from one water molecule is attracted to the oxygen atom of a neighboring water molecule.

How many water molecules must we have before it can be called a liquid? In the field of fluid mechanics, they have what is called a 'continuum hypothesis'. This hypothesis assumes that to be a fluid there must be over a million water molecules present within a reasonable volume (by reasonable, I mean some where between packing them in so tightly they become like a black hole or spreading them out so much it looks like a vacuum).

So, assuming we have enough water molecules to make a liquid, let's think about putting them all into a glass of water. In the middle of the glass, each water molecule attracts its neighboring molecules (remember the polar trait from above). This attraction occurs from all direction at once, resulting in a balance and no net force. Things don't work out so nicely at the surface. Here, the attraction doesn't balance because there is only attraction from the molecules to the sides and below as there are no water molecules above the surface. This causes surface tension. Molecules at the surface are pulled towards the center of the liquid, minimizing the surface area. A very small drop of water pulls itself into a sphere because a sphere has the smallest ratio of surface area to volume. Water in a glass will form a flat surface (ignoring the meniscus) as it is the minimum space it can take up.

Since wet is defined as 'consisting of, containing, covered with, or soaked with liquid (as water)'. Water acts to makes something else wet (as opposed to being wet itself) – so if I go walking outside on a rainy day, I end up wet from the rain.

Chaos

Back in the mid 90's, when I was on a winter army exercise, I was lent a copy of James Gleick's Chaos. We were in the middle of Alberta and it was cold – so cold we had moved our accommodations out of tents and into a heated H-hut (H-huts were built as temporary army barracks during WWII that were still in use). Our exercise was shut down until the temperature increased, so I had plenty of time to read.

I have always read about math and science. On one command post exercise, the brigade commander caught me reading during a lull in the action. In jest, he made a big deal of it. I suspect he thought I was reading a trashy romance novel, but instead I was reading a book on math. Shocked to discover the topic of my reading material, he told me that if math was what I was reading about then I was welcome to read on any of his exercises.

The book Chaos was the first time I had heard of chaos theory. I loved it. The fact that seemingly simple processes could generate such complexity was fascinating. Now I saw dripping faucets and swinging pendulums as gateways to observing chaos. To me, the most fascinating chaotic idea was turbulence. Water is complex! Move it just a bit and all sorts of phenomenon spring up including eddies and whorls.

Chaotic turbulent motions are found within the surface of stars, in combustion, in the ocean - even in water flowing from a faucet. Leonardo da Vinci included turbulence in his extensive studies, and he probably wasn't the first to study it. In all the centuries turbulence has been studied, we still haven't come up with a precise definition of what turbulence is. We know turbulence is what takes over when smooth fluid motion breaks into a complex network of eddy-like structures at all scales. Da Vinci's sketches of tiny eddies within small eddies within larger eddies and so on demonstrates the different size scales through which turbulent flow breaks up.

In Chaos, James Gleick describes turbulence this way: 'It is a mess of disorder at all scales, small eddies within large ones. It is unstable. It is highly dissipative, meaning that turbulence drains energy and creates drag. It is motion turned random.'

Okay, a picture of waves breaking on a beach isn't exactly a picture of turbulence, but it is the best I have.

An essay on turbulence

An essay of mine on one of my favorite topics (that is turbulence) has been published in my university grad journal. Check it out here.

Foamy frothy fun

Most of my days end with a long soak in a deliciously hot bubble bath. I don't need fancy scents, instead I enjoy the heat of the water and playful texture of the bubbles. Bubbles rise up as a mound ringing the stream of water from the faucet. Further away, bubbles slide into irregular shapes reminiscent of fictitious moon bases and futuristic homes. If it's really quiet, the muffled pops as multiple bubbles end their existence is audible. In addition to a relaxing end to a day, my bubble bath is an example of a foam.

Foams form when billions of tiny bubbles are packed together within a solid or a liquid. Irregular sized bubbles are common in all foams except the most idealized ones. Like what happens for individual bubbles (check out my bubble post), it's surface tension that helps keep a foam stable. Liquid foams break down eventually – there are even chemicals on the market to make this process go faster. Gas can diffuse from small bubbles into large ones and eventually out of the foam, or gravity can drain liquid out the bottom making bubbles so weak they pop on top.

A Belgian physicist, Joseph Plateau (1801-1883), figured out the basis of what we know about soap films and foams. His diverse interests also included spending time with moving image illusions like the action one sees when using a "flip-book". Back to soap film, Plateau came up with a series of laws to describe stable foam structures (foams that don't follow these laws tend to rearrange themselves until they do). They are:
1.Soap film surfaces are smooth.
2.The soap film curvature is continuous and constant along the entire a surface.
3.When three or more bubbles connect together, they will shift around until each line only contains three bubble-wall intersections – called a Plateau Border. With matching surface tensions all three angles are 120 degrees, the most efficient option.
4.Only four Plateau Borders can meet at a point.

Foams form in nature. Examples include: sea and river foams, and the foaming at the mouth of a rabid dog. There are fish, such as gourami and Siamese fighting fish, that blow a mass of bubbles coated in saliva to house their eggs. In the same spirit, some species of frogs make foam nests to lay their eggs in. These nests may be constructed in crevices, on the surface of water, or on forest floors.

Beyond bubble baths, soaps can be whipped to form a lather with bubbles so small they hardly can be seen. This idea was extended into modern shaving foams where a compressed gas is rapidly decompressed to expand a cream into a foam. Foams can also be hardened into permanent structures like insulation and flotation devices. Ceramic can be made into foams useful for acoustic insulation, absorption of environmental pollutants, and the filtration of molten metal alloys among other applications. Cement foams are used as a light-weight building material with good insulating capability. Even metals can be manipulated into becoming a foam. Metal foams are used in exhaust mufflers as they a great at dampening noise. They also make efficient materials for heaters and heat exchangers since they have so much surface area.

We eat a lot of foams: they can add a light texture to an angel food cake, or be tasty in the form of whipped cream and meringues. Breads are often a foam as the yeast produces tiny bubbles of gas which causes the dough to rise. One of the best foams is the head that forms when a beer is poured into a glass; this foam is made by which is made by carbon dioxide bubble rising to the surface. For the best head on your glass of beer, chose a wheat beer instead of a barley beer.

Note: there is an optimum foam combination of wheat beer consumed in a bubble bath – use with caution.

Tiny bubbles

Anyone remember the show “That's Incredible!”? It was on in the early 80's. I watched it as a kid (and yes I'm dating myself as the show went off the air in 1984). One episode I remember vividly: The hosts hyped how this man could make a square bubble, which I suppose what hosts are supposed to do. Since all the bubbles I had ever seen were round, I was really curious how a square bubble could be made (and I wasn't yet jaded about TV). I was expecting the square bubble to be free-floating by itself, so I was kinda disappointed in how it was done. The “performance artist” blew a bunch of connected bubbles (I forget how many). Next he took a long inhale from a cigarette, then stuck a straw between the connected bubbles and filled the space with the smoke. The filled space was in the form of a cube, and I felt tricked.

A short lived creation, soap bubbles hold a sphere of air with a thin film of soapy water which is formed by surface tension. Spherical shapes are preferred (really large bubbles can end up forming elongated shapes from air currents) because a sphere is the smallest surface area possible to contain a specific volume of air. The soap film surface tension is strong and flexible enough that waves can travel along the surface and is so thin the surface appears iridescent.

Surprisingly, soapy water has less surface tension that water alone and is needed to keep the bubble stable. As a bubble is formed, the soap film stretches decreasing the concentration of soap which increases the surface tension. This mechanism is called the 'Marangoni Effect' and occurs when a surface tension gradient (that is regions of greater and lesser surface tension) causes liquid to move away from areas of low surface tension. The soap acts as a stabilizer by letting the thinnest parts of the film to have the strongest surface tension thus keeping the bubble together.

What happens when two bubbles stick together? Well, they will arrange themselves in such a way that minimizes the surface area. Bubbles of different sizes will end up with a bulging internal wall into the larger on as smaller bubbles have higher internal pressure. If they are the same size, the internal wall will be flat - a phenomenon exploited by the cube-making bubble performer.

So sneaky internal bubble-wall cubes aren't so impressive. How about antibubbles ….

Sticky Honey

At first honey flows as a thick glop that evolves into an seemingly infinite stream. I never bother to wait long enough for it all to come off my measuring spoon, instead I just lick it – actually I could just lick spoons of honey without putting it into anything (like everyone else, I'm hardwired to like sweet things). Honey is sweet in a complex way I find intriguing. I'm always on the lookout for different types of honey to try. Recently, I bought a big plastic tub of clover honey on my trip east. Years ago, while wandering around downtown Munich, I found an entire shop devoted to different types of honey – I was amazed by the sheer number of types of honey: lavender, clover, buckwheat, avocado, heather and the list could go on. I think people really like honey.

Honey has been consumed since ancient times, likely as far back as 10,000 years ago or more. At one time honey was often the only sweetener available. When the tomb of King Tutankhamun was found in 1922, a pot of honey was discovered inside that was still edible (I'm not sure I'd go for potentially cursed mummy honey). On top of sweetening, honey has been used medicinally for eons. Modern studies have shown that honey helps healing wounds, it even provides a soothing effect when applied to burns. It also is a possible treatment for gingivitis, cataracts, ulcers and more.

According to 'The Flavor Bible', honey is a moderately loud flavor considered 'rustic' that goes with both the savory and the sweet. Every honey made tastes unique because the flavour of honey is determined by the flower the nectar came from, and there are almost an infinite number of possible flower combinations. Bees can be picky about what flowers they use. In one study, beehives were situated in the middle of avocado orchards while the avocado trees were blooming. Avocados produce a lot of nectar, so it should have been a win-win for the trees and bees. It turned out the bees preferred nectar from flowers surrounding the orchard.

Once nectar is brought back to the hive, the bees ingest and regurgitate the nectar multiple times until it is partially digested. At this point, it's stored in the honey comb while worker bees fan it with their wings until enough water evaporates. When the water content is low enough, honey will never spoil. When finished, honey has the following composition: 17.1% water, 82.4% total carbohydrate and 0.5% proteins, amino acids, vitamins and minerals.

At 64 calories per tablespoon, honey is a good source of nutrition. Honey colour is an indication of how strong the honey will taste. Light honeys are mild and dark honeys are stronger. It turns out that the darker honeys often have more antioxidants and potassium, but beware, honey can darken during shipping and storage.

What I think is cool about honey (beyond snacking possibilities) is that it's a non-Newtonian fluid. A non-Newtonian fluid is a fancy way to say that it doesn't respond evenly when poured in contrast to a Newtonian fluid like water. If you take a spoon of water and turn it over the water will flow out at an even rate. If you take a spoon of honey and turn it over, at first it will just bulge then a thick stream will slowly descend downwards. Over time the stream will speed up and thin.

In a non-Newtonian fluid the relationship between shear stress (pushing parallel to flow) and strain rate (how fast it deforms) is non-linear. That is, a simple number, usually viscosity (resistance or thickness of the fluid), can't be used to relate shear stress and strain rate together. To add complexity, this effect can even vary with time. The large molecules of the honey form links with each other that have an elasticity to them and can actually counteract for a while forces like gravity. Eventually, the flow overcomes this resistance and links will break. Other links will form and this non-linear effect will persist.

I'm going to keep looking for different types of honey when I travel and I won't expect it to come out of the jar easily.

Flow over bumps

As Fluid flows over a bump surface disturbances, such as waves, develop potentially extending both up and downstream. The form these disturbances take depends on fluid velocity and fluid depth which can be combined together in the Froude (F) number. The Froude number equals the fluid velocity over the square root of gravity times fluid depth. Because it has no dimensions, the Froude number allows flow in dramatically different circumstances to be compared, for example, atmospheric flow over a mountain range could be compared to tap water flow from your kitchen sink.

Naval architecture provided the original application for the Froude number. Here the hull length of a ship is used instead of fluid depth. It's important because, the Froude number relates to the ship's drag or resistance to moving through the water. This number is named after William Froude (1810-1879) who experimented on ship's hulls in his large fluid tank. William developed towing tank techniques in his efforts to model frictional drag on ships. However, he didn't discover the dynamical relationship between the fluid velocity and hull length. It was Ferdinand Reech (1805-1880) who first described this relationship and used it for testing ships and propellers around 1852. It's likely that this relationship's roots reside with even earlier French mathematicians.

So what does the Froude number tell us? When F is smaller than one, flow over the bump is 'subcritical'. Waves on the surface can travel upstream, meaning that downstream conditions affect the flow upstream. For example, when a pebble is tossed into the water of a flowing stream, the resulting ripples propagate both upstream and downstream. When F is larger than 1, flow is 'supercritical'. In this case, no surface disturbance can travel upstream. The ripples created by a pebble tossed in downstream cannot overcome the speed of the water. The flow upstream is not changed. When F is equal to one the flow is 'critical'. This is the point of transition from subcritical to supercritical effects.

Now, back to flow over a bump. As subcritical water is pushed over the bump, squeezing takes place because the water is now shallower and the same amount of water is flowing through. This forces the water to speed up over the bump and transition to supercritical. This faster water crosses over to the other side of the bump, where it's again deeper and slower moving. When the fast flowing water reaches the slower water it abruptly slows and waves form. Since the water is moving too quickly to allow waves to propagate upstream, (because it is supercritical) these waves build up, forming a sudden water level increase that can be standing still in the flowing water. This is called a hydraulic jump, a non-linear effect and can be observed in a kitchen sink or in water passing over a weir. Mathematically, a hydraulic jump is a discontinuity, however in the real world viscosity makes it a region of rapid change instead.

The greater the Froude number is, the more pronounced the jump will be. For initial flow speeds slightly above the critical speed, the transition appears as an undulating wave. As flow speed increases, the Froude number also increases and the transition becomes stronger eventually developing a more abrupt shape. When the speed is high enough, the transition front will break and curl back upon itself. At this point, the jump may contain violent turbulence, eddying, air entrainment, and surface waves. Turbulence removes the extra energy, allowing the flow to transition from supercritical back to subcritical.

My glass of water

A glass of water sits on my desk. The glass is clear and the water is clear, therefore the boundaries between them vanish. Instead of pondering if my glass is half-full or half-empty, I'm thinking about the water. In my glass, sitting still on my desk, the water looks like such a simple thing, but it is composed of zillions of tiny molecules of two atoms of hydrogen and one of oxygen pinned together like Micky-Mouse ears.

A single molecule of water is quite dull to the non-chemist like myself. But, put together a bucketfull of these molecules and an exciting thing happens: liquidity. The molecules slosh as one, they fill the nooks and cranies of their container completely and can disolve almost anything given enough time. Along with gases, liquids are considered a fluid. Their physical properties (like density and pressure) are continuous, there are no gaps or instantaneous changes occuring thoughout their volume. Like projectiles and pendulums, fluids follow Newton's laws. Mass, energy and momentum are all still conserved but, the equations describing their motion are exceptionally complex for even the simplest case.

If I drop a stone in the center of my glass a circular ripple will move outwards, taking the information about the pebble's disturbance to the edge of the glass, where it will be reflected back. This ripple is a type of wave, and though it looks like the water is moving outwards, it isn't. Instead, each water particle moves up and down, then its neighbor moves up and down next – exactly like what happens in a stadium when the audience does the wave. If I blew on the surface of the water, I would create similar waves, except this time they would be moving in one direction instead of radially away from a point. In the ocean, you get waves created this way by the wind. The waves formed can move across entire ocean basins and crash against distant shores long after the wind that formed them has stopped blowing.

If, instead of a stone, I released a drop of dye into my completely still glass of water, a different effect would occur. The momentum of the dye dropping would take the dye down into the middle of the glass. From there it would spread out in random tendrils, the edges between the dye and the water would become blurred and ultimately I would have a homogeneous (evenly mixed) solution of dye and water. But I haven't stirred or disturbed the water in the glass, so how does the dye get evenly mixed in? It is because I keep my office well above absolute zero, so the water and dye particles in the glass all have thermal energy, and this energy causes the particles to move. At the dye-water boundary, some of the dye will move out into the water and some of the water will move into the dye, blurring this boundary. With enough time this effect, diffusion, will completely mix the dye and water.

To mix in the dye faster I could stir the water in a constant circle with a chopstick. When I pull out the chopstick, the water would continue moving in a circle, which is also called an eddy, whirlpool, or vortex. The edges would be higher than the center because of centrifugal forces, that is, each water particle, once set into motion, wants to move in a straight line until it encounters the glass edge. At the edge, there is no choice except to move in a circle. A similar effect would occur if you spun the glass, or in the case of the oceans, spun the globe beneath. The effect of the spinning earth is called the coriolis force (which isn't really a force) and creates eddies in the ocean basins that spin in different directions in the north and south hemisphere. When you flush your toilet, a whirlpool often forms, but its direction doesn't depend on the hemisphere, instead it is dependent on the conditions within the toilet itself.

Perhaps I want to mix my dye and water even faster, so I put a lid on the glass and shake it. The moving water would stretch the boundary with the dye in random directions, creating more boundary surface area allowing for faster diffusion. The sloshing water can be described as turbulent, that is, the flow interacts with itself creating an element of randomness and becomes chaotic.

What I find most interesting about water, especially in the ocean, is that all of these phenomenon can happen at the same time and at all scales. On a tiny scale there can be turbulence, on a slightly larger scale there can be eddies, larger scales still may contain waves. Ultimately we reach the scale of tides, but that is a topic for another day.