# What can wavy patterned sand tell about the Fluid that formed it?

Sand on the bottom of the ocean as well as sand on the low-tide beach often forms wavy patterns. Do the parameters of these wavy patterns have any relation to the water and waves that formed them? If yes, what information may be gleaned from them?

Sand in creek beds also forms wavy patterns occasionally.

 See below image of the beach: or the below image of a shallow beach (underwater): • ""bottom of the ocean "" What is that? down the Indian Ocean 3000 meters below surface? You should think of dunes formed by wind in arid regions. – Georg Oct 11 '11 at 8:37
• I don't know how deep in the ocean these patterns form, but I do know that they form at wading depths. I added pictures. – Dale Oct 11 '11 at 14:35
• @Georg Can sand dune dimensions tell us anything about the wind that formed them? (y/n) If yes I'll ask another question. – Dale Oct 12 '11 at 4:57
• Roughly, the answer lies in the interplay between the ripples in the sand and the ripples/turbulence in the water. It's a kind of instability, in that if you start with a smooth interface, then a random fluctuation will build on itself and cause a macroscopic "bump". Then various feedback processes kick back in and limit the size of the bump. One can also get abstract and talk about self-organised criticality. I might write this all up if time allows. – genneth Oct 12 '11 at 15:12
• By the way, the literal answer to the question is: not a huge amount. The formation is actually quite insensitive to the exact driving force, at least relative to the effects of varying the properties of the sand. The latter tends to be disproportionately effected by the exact minerals in it, and to a certain extent, what microscopic things the water is carrying. – genneth Oct 12 '11 at 15:14

Waves are modeled, depending on the wind and depth and shape of "container", and also density of water. Waves in a lake have a much shorter wavelength then the waves in open ocean under the same wind conditions because of the boundary conditions on all sides.

The first layer of sand is a liquid mixture of sand and water which will be much denser than water, that is why in the picture the wavelength is so short with respect to the waves that carried in the energy.

Boundary conditions are different for each region. I do not think the patterns of sand tell us anything more than "waves passed through here". The depth of the sand might be correlated with the height of the sea wave: for the same region one could observe whether this is true, i.e. if there are deeper grooves after a large storm.

• Hello Anna, do You think the dunes in Sahara etc are formed by "air waves"? – Georg Oct 11 '11 at 16:51
• @Georg Yes, the air and sand mixture, when the sand is lifted and whole dunes move, becomes like a liquid and follows liquid equations. It is less dense than the water and sand liquid and that is why the wavelengths are larger. Of course the boundary conditions and the strength of the wind are defining the whole. – anna v Oct 11 '11 at 17:08
• The water waves give the energy, the fluid, sand-water follows fluid equations which have harmonic vibrations as solutions depending on the boundary conditions and sand is deposited accordingly. In air-sand fluids the energy is given by the wind pressure and the air-sand fluid follows the solutions for its boundary conditions. – anna v Oct 11 '11 at 17:19
• The yes is extremely conditional to the boundary conditions that guide the winds. Certainly it can give the wind direction, and if one calibrates using data over time the wind strength for the specific region. – anna v Oct 12 '11 at 5:58
• Downvoted because I believe the answer fails to appreciate the question, then subsequently wrong. Of course everything is "simply" due to the laws of physics and boundary conditions. However, that is not the nature of the study of physics; we are aiming for concise and precise descriptions in certain regimes. In any case, the formation of sand dunes (and these count as mini-dunes) is a well studied subject. See, for instance: nature.com/nature/journal/v457/n7233/full/4571084a.html, and arxiv.org/abs/cond-mat/0101380 – genneth Oct 12 '11 at 15:08

I don't know any literature specifically on the mechanism of dune formation, but the physics subfield which studies sand is "granular media", a subfield of statistical condensed matter physics.

Granular media has only been active for a decade or two. The focus of granular flows is on the surface flow in sand or other particulate aggregates, which differs from fluid flow in that only the top-layers in sand move. But as far as I can make out, the detailed physics of granular flow isn't the significant factor here. The only important thing is that, unlike a fluid, sand will make a stable equilibrium when the surface is at a nonzero slope less than a critical slope $S_c$. This phenomenon leads to a memory effect, called hysteresis. If you make an indentation in sand, so that you set up a sloping surface, it will remain stable, so long as the slops is not too big. This allows patterns in sand which are sculpted at a finite wind velocity to remain stable even when the wind dies away, so that the sand equilibrates to a shape which is stable over the average speed of the wind.

### Mechanism

The force responsible for forming the dunes is the same force which pulls up your umbrella on a windy day. When a wind is blowing in a certain direction, where the surface is curved downward, the wind will deflect down, creating a negative pressure which pulls the umbrella up, and where the surface is curved upward, the wind will deflect upward, creating positive pressure which pushes an indentation. When impacting a soft hysteretic surface like sand, the wind pressure will dig holes in places which curve upward, and it will move the sand locally to settle in places which curve downwards. The result of this is an instability which forms a pattern of dunes whose peaks and troughs are perpendicular to the wind velocity, and whose characteristic wavenumber is determined by the average velocity of the flow, the density of the fluid, and the height of the fluid.

I assume the mechanism of forming sand dunes in wind is the same, but for air-sculpted dunes, you will get additional complications from turbulence. The tops of sand dunes can be have further instabilities which go on to deform the shape periodically along the peak of the dunes.

It is also possible that a similar instability is responsible for setting up water waves in deep ocean in response to a constant wind.

### Estimating the instability wavenumber

First, the equilbrium law. When sand is placed at a constant pressure, it will statistically settle down so that the surface is horizontal. If the sand is jostled and allowed to resettle, the surface of the sand will be flat.

When the same sand is exposed to a slowly spatially varying hydrostatic pressure P(x) on the bottom of the water, the sand will settle locally so that the force profile is the same as flat at any point. This means that the internal sand-o-static pressure from the changing sand depth needs to be enough to cancel out the horizontal variations of pressure. This happens when the depth of the depression produced is proportional to the applied pressure, or when

$$P(x) = -(\rho_S - \rho_W) g h(x)$$

This equation describes a pressure equilibrium in a circumstances where the pressure is varying from place to place. $\rho_S$ is the density of sand, $\rho_W$ is the (necessarily smaller) density of water, $g$ is the acceleration of gravity, and $h(x)$ is the height of the sand on the bottom.

Now supposing that the water is flowing with velocity v, and the bottom h(x) is sinusoidally varying

$$h(x) = H \cos(kx)$$

then the water velocity must curve with the bottom (note that this is not easy to solve exactly, the sand violates no slip to some extent, since the first layer of sand will move along with the water a little, but the essential fluid flow follows the pattern of the sand). The estimate for the pressure differential established by a flow with velocity v is the centripetal acceleration for a given mass of fluid to go round the bends.

The height of the bends is H over a distance 1/k, so that the inverse radius of curvature is $Hk^2$ (this is correct for more than dimensionally, it is expanding cos in a second order Taylor series). This gives a centripetal acceleration

$$a_c = H k^2 v^2$$

Where $v$ is the constant horizontal fluid velocity. This gets multiplied the the fluid density $\rho_W$ times the height of the part of the fluid that curves along the bends, which is given by the decay of the vertical components of the velocity, which is by a Laplace equation, and so the height is approximately 1/k. The result for the pressure is the maximum pressure, modulated by the height variations, to account for where the pressure is least and greatest, according to the curve on the bottom, or

$$P(x) = - H k \rho_W v^2 cos(kx)$$

This goes into the equilibrium condition

$$g(\rho_S-\rho_W) h(x) = - P(x) = k \rho_W v^2 h(x)$$

Which is solved by $h=0$, reflecting the obvious fact that if there are no sinusoidal variations, the pressure is constant, and this is an equilibrium. The condition for stable equilibrium is that the net linear coefficient in the equation determining h is positive. This can be understood physically--- a small perturbation of h at wavenumber k will produce either a smaller pressure than required to sustain the perturbation, or a greater pressure that will deepen the perturbation. If the pressure is more than is required, it will deepen the perturbation, and if the pressure is less, the perturbation will die away.

So the stability condition is found by equating the coefficients of the left and right side of the equation above for h:

$${g\over v^2} ({\rho_S\over\rho_W} - 1) = k$$

This determines the critical k for instability. For MKS units, $g=10$, $v=.5$, sand twice as massive as water, and assuming the water is deeper than the wavelength predicted by this relation, k is 40, corresponding to a wavelength

$$\lambda = {2\pi \over 40}\approx .15$$

For larger k, there is an instability, while smaller k are stable. The critical k will be the wavelength of the dunes that are formed. So from the approximately 5cm wavelength of the waves in the sand, I would say that the typical velocity of the water at that spot is approximately 20cm/s, perpendicular to the wavefronts.

This is only slightly better than an order of magnitude dimensional analysis estimate, but since the picture is given, it assures one that the correction factors are not too big, at worst case a factor of 2 or 3. A more precise calculation is certainly possible.

Note that the sand is constantly shifted by the moving water, and this is important, since otherwise it would not find the correct dynamical equilibrium with the waves.

### Restoring equilibrium

In the approximate description above, the loss of equilibrium condition is from the first order in H condition. This means that once the wave starts to grow, its growth will not be checked.

The condition which checks the growth is the critical slope for sand. Once the critical slope is exceeded, the sand will flow downhill, washing out both higher order in k perturbations, and leading to a sawtooth profile for the equilibrium height, with a slope equal to the critical slope for sand flow.

• I don't think your explanation of the mechanism is quite correct. The pressure effect you describe is part of the mechanism for forming water waves (probably) but for a bulk mass of sand a pressure difference of that magnitude isn't going to budge anything. Wind or water flow transports sand grains via a process called "saltation", where single grains are picked up and moved in the flow direction. This occurs due to viscous effects, not pressure differences. Saltation depends on the slope of the sand, speed of of flow, turbulence, grain size etc., and it's this that leads to pattern formation. – Nathaniel Apr 3 '12 at 13:05
• At least, the above is true for sand dunes, which I've looked into a bit. For wet sand on the sea floor I don't know - perhaps your explanation is the dominant one in that case. – Nathaniel Apr 3 '12 at 13:08
• @Nathaniel: The mechanism is correct, but you are right that it isn't how the sand moves--- the sand moves by some fluid pick-up and deposit mechanism that is irrelevant. The pressure calculations only give the self-consistent equilibrium slope, and the only assumption is that, after being picked up, the sand is deposited in a slope-pattern that is most stable given the average fluid flow pattern in the enviroment. The pressure isn't picking up the sand, it is just determining statistically how it settles, but nice comments, thank you. – Ron Maimon Apr 3 '12 at 20:17
• @Nathaniel: I don't know whether I persuaded you or not--- the way I derive the self-consistency is by assuming that the sand settles flat when there is no pressure gradient. This is a statistical statement, the sand could settle in any stable way, it is just most likely to settle flat. This means that if there is a pressure gradient, it will settle at a slope so that it is flat relative in the effective gravity provided by the pressure gradient plus the ordinary gravity. This leads to the self-consistency equation, and it does not describe the statistical mechanism of removal and deposit. – Ron Maimon Apr 5 '12 at 5:56
• It's an interesting argument. I'm neither convinced nor unconvinced yet ;). My doubt comes from thinking that the sand level should rise where the transport converges and lower where it diverges (essentially the continuity equation applied to the sand), and therefore variations in the height of the sand ought to be very strongly dependent on the factors that determine the transport rate. – Nathaniel Apr 5 '12 at 10:59

The relationship is not unique, because bedform patterns typically evolve even with a fixed flow field. Some pattern-scale statistics are robust, for example in the images shown the bedform crestline orientation is very likely nearly perpendicular to the flow direction on average (in space-time). Other features are much less robust: For example in many cases of migrating bedforms, average height and wavelength increase through time as smaller bedforms merge together.

• technically a "dune" is a particular type of bedform. (In dunes height scales with flow thickness, they are a "depth-limited" bedform).

• granular physics has been studied for much longer than decades, e.g. The Physics of Blown Sand and Desert Dunes (1941), mostly under the label "sediment transport" or "morphodynamics", and mostly by civil engineers (see e.g. Albert Einstein's son Hans )

A fluid (gas or liquid) flowing along a flat surface at increasing laminar flow rates develops simple harmonic (SH) long-crested oscillations (a.k.a., “vibrations”) in the fluid along the boundary just before a turbulent flow pattern erupts. An oscillation in a fluid creates a sound wave; simple harmonic long-crested fluttering oscillations (boundary layer flutter, or BLF waves) flowing in the boundary layer along a flat surface create SH sound waves. These BLF waves flow along SH wavy paths; these paths slide slowly along a smooth shiny flat surface – more slowly than the rate of flow. The sliding is arrested along a compliant particulate flat surface like sand or snow, where a standing wave SH long-crested pattern is revealed. The fluttering handbreadths student demonstration reveals a “travelling” standing wave nature to BLF waves as well (“Simple Harmonics”, Aylmer Express 2015, pp. 11, 12).

When the boundary is rigid and smooth, the orderly wavy pattern suddenly breaks up into an apparent irregular pattern termed turbulent flow, with the resistance to flow suddenly rising as the square of the velocity, instead of a linear relationship of laminar flow.

Along flat particulate boundaries, like sand on beaches or the beds of flowing water, long crested SH stationary waves develop during transition. Bagnold studied these sand waves in the desert and then in wind tunnels. His photographs (“The Physics of Blown Sand and Desert Dunes”, 1941) revealed sand grains ejected perpendicularly, creating troughs and deposited at shallow angles on the crests of long crested SH sand waves. This mechanism might occur if a planar variety of a standing wave sound field were being created by the SH boundary layer oscillations of transition, analogous to the standing wave particulate accumulations in the Kundt’s tube experiment.

Bagnold proved a similar mechanism prevailed in water flow along sand as in his wind tunnel air flow experiments. In shallow water flow, the SH long crested sand waves are reflected in the similar boundary layer water waves that extend through all laminae to display similar water surface waves (Order in Chaos, Aylmer Express, 2011, pp. 53-55). Surprisingly, as air flow, or water flow rates extend well into turbulent flow rates, these compliant boundary sand waves persist – and grow, leading Bagnold to wonder: “instead of finding chaos and disorder, the observer never fails to be amazed at a simplicity of form, an exactitude of repetition and a geometric order” (“Simple Harmonics” 2015).

The pattern of long crested SH standing waves along stretches of sand in air and water flows seems to reveal the influence of coherent sound energy in the physics of transition to turbulence.

The important parameters to observe from these ripple patterns are the distance between each peak and the depth of each valley.

The phenomenon of sand at the top of the peak not being disturbed is similar to a guitar string which is vibrating at a harmonic frequency. During such motion there are always a number of points along the string (called nodes) which do not move.

These ripple patterns suggest that the water in the creek was moving/vibrating at a frequencies which left the high points in the sand as nodes.

• This is not correct, but it is a bold idea, so no downvote. The mechanism is not nodes of the velocity at the top (or bottom) of the dunes, but an instability of the sand/wind system. – Ron Maimon Dec 14 '11 at 5:52

## Anna V stated that the sand-water mixture is more dense than the water above. This seems like an ideal situation for internal waves to form. en.wikipedia.org/wiki/Internal_wave

It is important to distinguish between liquids and gasses since liquids may be considered incompressible, while gases are compressible and therefore the density of gases cannot be treated as constant at high velocity.

It is also important to distinguish unidirectional flow from bi-directional flow.

There are four basic forms of wavy patterns known as "bedforms" that arise from linear flow: Ripples, Sand waves, Dunes, and Antidunes. There is a table in the wikipedia page for bedforms, which describes qualitatively the conditions associated with each feature.

There are five types of ripples: straight, sinuous, catenary, linguoid, and lunate. The significance of these different types of ripples is not fully understood. More complex forms generally form in shallower water. (Blatt p.151-152) "In natural environments it has been observed that as water becomes shallow, the following sequences occur: for ripples, strait to sinuous to symmetric linguoid to asymmetric linguoid; for dunes, strait to sinuous to catenary to lunate."

I intend to add more on bi-directional flow when I have more time.

Blatt, Middleton, and Murray, "Origin of Sedimentary Rocks," 2nd Ed. p.90-104, 136-160