WAVES and FLOW PAST MOUNTAINS, BUMPS, PENINUSULAS AND IN CHANNELS

    In both oceans and atmosphere, fluid flow can be concentrated as it is forced to flow around or over obstacles, or through narrow passages. When this occurs, various waves and instabilities are excited, and these can put limits on the speed of the flow, as well making it mix with its fluid environment. The sometimes complex topography of the rim of the ocean and its islands and estuaries can press on the fluid, providing essential 'form drag' and frictional drag that keep its fluid circulations in balance. Many chemical and biological effects arise from such regions of intense velocity and mixing.    Similar kinds of phenomena occur at many different scales: from a few mm. to many km., as you can see in the images below. At the largest scales new physical effects become important, particularly the rotation and spherical shape of the Earth. Then there are both 'rotating' and 'Rossby' hydraulics: consult the Rossby-wave chapter on this website.  For more detail, read on.

    Background ideas. Waves occur in air and water, due to gravity acting on the water surface, or acting on fluid with differing mass density. Waves on the sea surface are familiar; they are known as 'external' gravity waves. The most fundamental physical property of these fluids is that they are layered, with increasing density (and effective or 'potential' density) as one goes downward.   This layering resists upward or downward movement of the water and, if disturbed, the layers will undulate in 'internal' gravity waves.  The buoyancy of a small element of fluid is the upward force of pressure exerted by surrounding fluid, and this may be less than or greater than the downward pull of gravity.  The 'buoyancy frequency', N, measures the strength of this gravitational stability.  2π/N is the natural frequency of a column of fluid rocking up and down: ranging from 10  minutes or so in the upper ocean and the lower atmosphere, to several hours in the deep ocean.

   Gravity waves, both internal and external, have a crucially important transition when their wavelength exceeds the depth of the fluid (or exceeds their own natural vertical scale if it is different from the total fluid depth).  Short-wavelength waves are dispersive, meaning that the speed of propagation varies with wavelength.  When a pebble is tossed into a pond, it produces a beautiful pattern of concentric rings, moving outward.  If it were not for wave dispersion there would be just one, or very few rings.  However the natural modes of motion of the fluid surface are sinusoidal waves ....  'wavetrains' .... and when we express a single disturbance of the surface mathematically it unfolds into an infinity of sinusoidal waves. These each have their natural speed, and in time they fall out of step with one another, producing more and more individual wave-crests.  Dispersive surface waves can be far more extensive and elegant than you might expect, when viewed from far above. Below is a speed-boat wake in Puget Sound. Notice the other subtle wave patterns, and try to imagine what generates them (click the image for very large 2.1MB version).


The vee-wake of a fast boat is different than the transverse wake of a slow (or large) boat or ship; the key parameter is U²/gL where U is the boat speed, g is gravity acceleration and L is the boat length. The sailboat below generates the transverse wake which is visible surprisingly far behind. Again, click for a 6 Mpixel large version

Surfaces waves have been tracked across major oceans, for example the waves along southern California surfing beaches can be traced back to storms in the Southern Ocean or Indian Ocean, half a world away. This is in northern-hemisphere summer, when the strongest winds are in the southern hemisphere; during the other half of the year, northern-hemisphere winter, the wave sources shift to the north Pacific.

    Very long gravity waves, by contrast, are often 'non-dispersive'. The propagation speed is the same for all wavelengths, except for effects of their amplitude, which can make them slightly faster or slower. In this case a pebble thrown into a (very shallow) pond, will make a disturbance that propagates outward as a single wave-crest, or just a few. In practice this would have to be done with a 'rock' rather than a pebble to avoid exciting small ripples. It is easy to see these effects in a pan of water in the sink or the laboratory: if you make waves by vibrating a spoon back and forth, you will see many wave-crests when the water is deep. With a bigger paddle like a pancake spatula, you can fill the pan with waves and look for 'resonance' which is a locked pattern of waves formed as they bounce back and forth.    When the water is very shallow (say 2.5 cm or 1 inch deep), make waves by tilting the pan up and down slightly. Then a dominant pulse propagates back and forth, but you will have a hard time making a sine-wave shape. 

   In hydraulic  flows, nearly non-dispersive waves act to control the rate of flow.  Typically there is a single wave speed on average (for example, √(gh), where g is the acceleration due to gravity and h is the average fluid depth).  Then variations in depth, h, occur in the flow and these lead to variations in wave-speed. One of the most common such flows occurs when water is flowing out of a large reservoir, over a dam.  Imagine the process of gradually lowering the water level downstream of the dam, and watching the flow that it creates.  The fluid accelerates as it falls downhill, yet its speed is limited by the difference in height.  If the downstream level is lowered the flow will reach a speed equal to the value of √(gh) where it is smallest: over the 'sill' of the dam. Further lowering of the downstream water surface will not cause any more water to flow over the dam, or its velocity to increase.  Gravity waves carry information upstream, 'telling' the reservoir the level of the downstream water surface, but they cannot do so if the downstream velocity exceeds their propagation speed. This is called hydraulic control,  with flow speed U = √(gh) at the narrowest, or most constricted part of the flow.

    You might carry out a similar experiment by having no fluid at all outside of the reservoir, and simply lowering one wall of the reservoir slightly below the mean water level, letting the water pour out.  Let's calculate the rate of water flow if the wall (the 'dam') is a distance H lower than the water surface in the reservoir.  When the outflow event has begun, the water depth at the dam will be h which is less than H:  again we predict that the rate of outflow of water over that wall will be such that U =√(gh), and hence the volume flow will be √(gh³). This can be established from the conservation equations for mass and momentum, given a smoothly rounded dam (which show that for smooth flows the critical flow speed is reached at the point where the dam is shallowest). The prediction is not yet complete, however because we don't yet know either U or h.  These are found by using conservation of momentum in the form of Bernoulli's equation.  Simply put, a dot of fluid moving along the surface will accelerate as it flows downhill, so that its velocity will be √(2gΔz),   Δz being the dip in the free surface relative to the upstream water level in the reservoir.   Comparing the above two expressions we see that

                                                2Δz =  h = 1/3 H

                                                U = √(gH/3),  hU =√(gH³/27)

The prediction, then is that the outflowing water will have a depth equal to 1/3 of H at its shallowest point over the sill (the dam), and speed equal to 0.577 times the gravity wave speed for undisturbed water of depth H (recall that H is the depth of the sill measured from the initial, undisturbed water surface). This can be tested by a careful experiment in a kitchen sink!

    These ideas suggest that the √(gh) is a 'critical flow speed' analogous to Mach 1 in airplane flight, where a body moves through the air at the speed of sound. It is a very good analogy.

    Hydraulic jumps.  Shock waves form when flow around an airplane exceeds Mach 1.  Similarly, in hydraulic flows these ideas predict something catastrophic: if indeed the waves have just a single propagation speed for a given water depth, then wave energy will tend to focus at special points in the flow. For example, if a flow is gradually slowing down as it spreads out from its source of origin (as in tap water flowing outward along the base of a kitchen sink), it will pass through the critical flow speed and there upstream propagating waves will be stopped:  a strong hydraulic jump forms.  The idea is that fluid flow cannot smoothly decelerate through the critical flow speed: it must do so abruptly by jumping from a supercritical (faster) to subcritical (slower) speed.  You can readily observe the hydraulic jump in a kitchen sink, because the water is very much shallower inside, and deeper outside the jump. It is a dramatically sharp transition, though it is somewhat complicated by its ripples on the water surface.  The hydraulic jump, like a wave, has a propagation speed. In the kitchen sink it is just sitting there looking at you, but it has motion relative to the fluid flow. In the limit of a very weak hydraulic jump, where the upstream and downstream velocity are only slight different, the jump is just a small difference in water height which propagates as a gravity wave, with speed c₀. Yet as its amplitude increases, and the upstream and downstream fluid velocities become very different, the propagation speed is given by the expression   c₀(1 + ½β)^½(1 + β)^½ where β is the 'strength' of the hydraulic jump, β =(h₁ - h₀)/h₀ expressed in terms of the fluid depths downstream and upstream of the jump.

    Isaac Newton, among his many successful scientific works, made a mistake or two.  Though he had little practical reason to be interested in high-speed airflow, he did an analysis of a shock wave and predicted its speed of propagation. Unfortunately he made the assumption that the air temperature would stay constant in the flow through the shock region, which does not turn out to be correct. Instead the entropy of the air is more nearly conserved.

    The undular bore: fairly-long gravity waves. Momentum is conserved in the fluid flow through waves or hydraulic jumps (because it cannot be readily destroyed, even by complicated fluid motion. Viewed from the side, there is a pressure difference from the higher water level to the lower; this establishes (by the vertical hydrostatic balance in the regions of nearly level water surface) a pressure difference from one side to the other, which drives the acceleration of the flow through the jump. Mechanical energy, however, is much easier to lose into heat. It turns out that fluid flowing through a hydraulic jump indeed loses energy at a rate ρgU(h₁ - h₀)³/4h₁ where  ρ is the density of the water.  The description here of a jump from one flow velocity to another conceals a small region where this energy loss has to occur.  Complicated turbulence, or else very strong viscous effects, are present 'inside' the jump.  There is, however another way to get rid of energy in this flow: waves.   We know they carry mechanical (potential + kinetic) energy, and it is conceivable that they, instead of fluid turbulence, might provide for the (rather mysterious) disappearance of mechanical energy from the jump itself.

    The undular bore is the result of all this rather complicated argument.  It appears in many visible flows, and most dramatically as an external wave on the surface of some estuaries when the incoming tide enters a narrowing channel.  Upriver of the advancing tidal wave the water flow is nearly zero, yet seaward of the wave it is up-river.  The amplitude increases as the river narrows, and the result can be an undular bore of considerable height (see Lighthill, 1978, Figure 48).  

    Internal hydraulics in a stratified fluid. These ideas carry over to density stratified fluids, with additional complexity in the rich vertical structure which there may occur. In the simplest case of a deep, dense layer with a shallow buoyant layer on top, the ideas are very much the same, after replacing g by the 'reduced gravity', g multiplied by the fractional change in density between the layers. The waves then exist on the shallow thermocline, the interface between the upper and lower layers. The pattern in the figure above is an undular internal bore propagating south (left) past the southern tip of Whidbey Island, in Puget Sound, Washington. The sound is about 5 miles wide at this point. The density front and its following train of internal gravity waves moved about 1/2 meter per second southward, and may have originated at the strong ebb tide, from the tidal flats to the north (further description is given below). This undular bore occurs when waves are available, and when the hydraulic jump is not too strong.  There is a problem however: if the waves are truly non-dispersive then they carry energy along with their wavecrests, and if the hydraulic jump flow is steady (the form of the water surface is fixed in time, while fluid flows through it), then non-dispersive waves could not carry energy away.  We find that, to do the work, the waves must be 'almost-long' waves which have a small amount of dispersion and this gives them the ability to carry energy even their wave crests are sitting still. We mentioned at the beginning that large-scale flows can feel the Earth's rotation. This happens typically when the time it takes the flow to evolve exceeds 4 hours or so, because in that fraction of a day the Earth has rotated significantly. The advancing front of the undular bore above may be slightly tilted, faster on the far side of the Port Susan channel, as a result (the Coriolis force presses the outflowing river water to the right-hand side of its channel). The effect is easily recreated in the lab, in fluid experiments on a rotating table.

These internal waves change the roughness of the sea surface, their convergent horizontal currents 'squeezing' short surface gravity waves, and hence become visible in reflected skylight. Radar beams from 'SAR' satellites sense this roughness change and make remarkable images from space, where undular bores can be see throughout the world-ocean www.ifm.uni-hamburg.de/ers-sar/Sdata/oceanic/intwaves. The waves are often generated where the tide accelerates over a shallow ridge or sill on the seafloor, and they tend to 'break' and disappear when they enter water so shallow that the depth of the lower layer (below the thermocline) is less than about twice the depth of the upper layer.  

    Gravity currents are a related phenomonon, in which a strong surge of fluid, either denser or more buoyant than its surrounding fluid, moves horizontally like a large-amplitude gravity wave. It has a sharp front, rather like a hydraulic jump with no fluid in front of it! Satellite images show such dynamics of the atmosphere, sometimes with impressively large scale. A remarkable animation of such a gravity current in the lower atmosphere over Georgia is seen, at a GOES satellite gallery at University of Wisconsin, cimss.ssec.wisc.edu/goes/misc/030418/030418.html.

A surge of wind offshore in Baja California sends with it a gravity wavetrain: cimss.ssec.wisc.edu/goes/misc/010930/010930.html.

This is driven by the winds of tropical depression Juliette, 30 September 2001. In the same animation you will see delicate vortices in the air flowing round the island of Guadalupe. Gravitiy waves and vortex motions interact in many interesting ways in atmosphere/ocean dynamics.

    Solitons.  Watch the surf ride up a gently sloping beach and you will see waves, which in the deep sea are perfectly good sine-waves, turn into 'lumps', or long ridges of water advancing toward shore. The waves are dispersive (short-, deep-water-) waves while out at sea, but they become closer and closer to non-dispersive (long-, shallow-water-) waves on the beach.  As they make this transition, the also can focus into what are called solitions or more classically, 'solitary waves'. They are easy to make in the laboratory, in a channel a few meters long.   They are the result of a wonderful focusing of energy, with dispersion seeking to spread them apart, making the surface of the water less steeply sloped, while their sizable height tends to make them steepen. While it seemed to me a precarious idea, these two different tendencies in perfect balance, when I listened to T.Brooke-Benjamin lecture on them in Cambridge long ago, the power of the idea comes from time-dependent solutions of the equation describing them (the Korteweg-deVries equation), which push the solutions toward just this balance. 

    Our undular bore, above, is in fact a train of solitons.

  These flows introduce us to a remarkable and historic area of science. John Scott-Russell, an English scientist (and gentleman) of the 19th Century, noticed waves on ship-canals in England that seemed to propagate very long distances without changing their shape, breaking into many wavecrests or otherwise disappearing. He timed their speed by riding on horseback along the canals. He wrote, in 1834,

``I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped - not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well-defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed. I followed it on horseback, and overtook it still rolling on at a rate of some eight or nine miles an hour, preserving its original figure some thirty feet long and a foot to a foot and a half in height. Its height gradually diminished, and after a chase of one or two miles I lost it in the windings of the channel. Such, in the month of August 1834, was my first chance interview with that singular and beautiful phenomenon which I have called the Wave of Translation''.

                            (http://www.ma.hw.ac.uk/~chris/scott_russell.html)

   Now, by the 21st Century, solitions have been found to describe 'strong' waves of many kinds, from plasma physics to fibre-optics.  Indeed, the long-life of solitons makes them ideal for transmitting information, and some practical use is being explored in light pulses moving along optical fibres.  We see Physics learning from oceanography, which shows that the water can flow in both directions.

    What waves? The waves shown in images above are from a variety of observations and experiements.  All are generated by fluid flow encountering some kind of topography that tends to accelerate the flow. We see ripples, which are waves on a water surface, due to surface tension.  Ripples are in fact dispersive, and they can produce a 'forward' undular bore on a thin water layer (easily done in the sink).  In the upper ocean in spring and summer there is a layer of water warmed by the sun, and thus made buoyant.  Internal waves can ride along the interface between this layer and the colder, denser water beneath, as seen in the figure above from Puget Sound..  In doing so they roil the water and create 'slicks' on the sea surface.  These slicks are made visible as the horizontal flow in the wave tends to accordian the small surface waves, at one point squeezing them and at another stretching them out.  This is visible as a contrast in surface roughness. As far away as an orbiting satellite, they are clearly visible. With wavelengths of 10 to 100 m and propagation speed of typically 1/2 meter per second, they are often produced by the change of the tide, in the presence of a ridge on the seafloor, or just an interesting lateral boundary like a tidal flat.

    Downslope winds.  A special aspect of hydraulics is worth some discussion.  We spoke earlier of flow over a dam.  In atmosphere and ocean a very similar effect occurs as flow spills over a mountain (or sea-floor mountain).  On the downwind side of the mountain the flow concentrates into thin, very rapidly flowing layer: a sort of 2-dimensional jet.  If you live downwind you will be very aware of such a wind. They are so distinctive as to have names: the Chinook (Boulder Colorado), the Bora (Trieste), the Santa Anna (Southern California), perhaps the Mistral (Rhone Valley, France).   Of course these are internal-gravity flows rather than the external gravity flow over a dam, but there are great similarities, and many potential science fair projects to be found in this analogy.  The supercritical downslope flow is particularly interesting in that it can occur even when dispersive waves (internal gravity waves) are available to disperse it.

(D. Peterson, GFD Lab UW, Nikon Coolpix image with fish-flake particle tracer in fluid)

Something has to trap the wave energy in the lower atmosphere or ocean for this to work.  It requires some mechanism to prevent the waves from radiating upward, and that mechanism can involve a 'critical layer' where the fluid flow reverses direction, or a wave-breaking region, both of which can trap energy at the lower levels. In the image above, the flow is right to left over the mountain, yet left to right above the mid-line of the channel. The strong downslope flow rises into a train of lee waves downstream.

    Forces. Viewing the flow from the side, it is easy to imagine the strong pressure difference between the upstream and downstream faces of the mountain, especially so in the single layer fluid just below. The vertical balance of forces on a nearly horizontal flow is simply that the weight of the fluid is balanced by an increase in pressure downward. This in turn establishes the greater pressure in the region of deeper fluid. The pressure difference acting on the sloping topography is a force to the right (downstream) on the mountain and a reaction force of the mountain on the fluid to the left (upstream): this is 'form drag'. The experiment can also be carried out nicely with cold CO2 gas generated by dry ice. It is made visble by condensation of water droplets.


(UW GFD Lab 2003, P.Rhines)

    Ocean estuaries. Downslope winds and lee waves occur in the ocean as internal hydraulics. When a river flows into the sea along a 'drowned valley' which forms an extended bay, it is known as an estuary. Some have been formed by glacial scouring and melting, such as Puget Sound, Washington. The entrance to Puget Sound is known as Admiralty Inlet. In the image below, you can see signs of rapid flow in the form of banded slicks on the sea surface. Spanning between the visible points of land (Port Townsend above and Keystone ferry slip on Whidbey Island below) is an undersea ridge, the site of a cascading 'waterful' of water entering from the Pacific Ocean. This inflow is another example of a downslope jet.

(7 June 2003 P. Rhines)

    A beautiful side view can be constructed from high-frequency acoustic surveys carried out by oceanographic ships. A much studied example is Knight Inlet, British Columbia where the 'downslope wind' and undular bore structures can be seen vividly. David Farmer and Lawrence Armi describe these images www.whoi.edu/science/AOPE/people/tduda/isww/text/farmer/farmer.htm. . A computer model showing the development of the flow, from Kevin Lamb of Universtiy of Waterloo, Canada is www.moisie.math.uwaterloo.ca/~kglamb/Knight_Inlet_animations.html.. This work shows how viscous/turbulent effects can cause flow separation at the sill, and how hydraulic jumps are created near the ocean surface at the turn of the tide, and subsequently become undular bores as seen above.

    Peninusulas (with sloping sides underwater) exert form drag on passing flows, and these are being investigated in Puget Sound by Parker MacCready, www.ocean.washington.edu/people/faculty/parker/index.html. This adds a new wrinkle to the waves and hydraulics story: the shedding of eddies in the lee of an obstacle (whether mountain or peninsula). Flow separation, as seen in the 2-dimensional simulations referenced above, or in 3-dimensional flows where vertical vorticity can be produced, is intimately connected with drag and vortex shedding. There is once again a strong historical connection with the aerodynamics of flow around a wing, its lift and its drag.

In a simple model of an estuary, constructed by Parker MacCready in the GFD lab, one sees hydraulic flows in the context of both the river-driven overturning circulation (left to right at the bottom, right to left above) and the tides. Lee waves and downslope jets both occur, as well as rotors of mixing. The model is driven by salinity differences, as in real estuaries, and as is typical in the ocean, seafloor ridges break the estuary into basins over which the saline ocean water spills to renew the nutrients and oxygen of this ecosystem.

Reference: Waves in Fluids, M.J. Lighthill, Cambridge University Press, published 1978.

Imaging surface waves in the lab (or a swimming pool):

  Another form of imaging of waves comes from the refraction of light passing through a water surface. In the figures linked below, a point source of light 8m above the water project on the floor of the GFD lab, showing a range of interesting wave patterns, some 'well-resolved' and some 'over-modulated' (that is, too much refraction).

The interplay of gravity and surface-tension dynamics makes undular bores two-sided (ripples radiating forward and gravity waves radiating backward from the front). These are evident in the short and longer waves in the image below, on a layer of water less than 1 cm deep. Of course the sun is the ultimate compact light source, and a swimming pool can be the site of wonderful images without any GFD Lab at all.

For scale, the overhead plexiglas cylinder has a diameter of 1.4m. Images using Canon EOS-10D digital still camera, P. Rhines, Dec. 2003. Click on image below.