Waves bedforms.

Waves bedforms generated only by current.

The different bedforms are classified from low velocity current to high velocity current.

Ripples:

In the case of fine particles, when the shear bed stress are are important enough for sediment transport start, the bedform is instable, small triangular sand waves happen: the ripples. They are usually less tha 60 cm-length, and less than 60mm-heigth.

Dunes:

There are larger sand waves, more or less regular, made in natural streams.It's the more common bedform. Dunes have mainly triangular forms. The dune front move in the same direction than the current stream. Indeed, current takes sediment from the first sideof the sand wave (in stream direction) of the dune, and then the sediment are deposited on the lee side of the dune (the side backward in the diretcion of flow).

Transition and plane bed:

When the current force still rise, the dunes arer more and planed and finally disappeared.

Antidunes:

This phenomenum appears when the flow force rise again. The bedform is sinusoidal. This bedform is called antidunes because it moves in the opposite direction of the flow. The current velocity decrease when it arrives on the bedform, and so the sediments deposit. Then, by the turbulence recirculation, sediments are taken up on the other side.

In order to go further in the mecanisms of bedforms, we are going to make the hypothesis that the sand waves are moving with a constant velocity a, without any bed shape change.

The bedform is set by the equation:

where h is the heigth of the dune above the x-abscis, placed at the level of bed troughs.

Considering the sedimentary flow qT through 2 consecutive sections separated with an unit space in the x-direction (horizontal). The change in sediment amount must be equal to the change in bed heigth (n is the porosity of the bed, the suspension load is neglected). The continuity equation is:

The combination of these 2 equations gives us:

qT(x)=q0+a(1-n)h

where q0 is a constant corresponding to qT for h=0.

In the small shear stress case, q0 equal 0 so:

qB=a(1-n)h

which indicates that the intensity of bed load is proportionnal to the local bed heigth.

The amount of sediment deposited on the wave front is called qD.This amount characterise the velocity of the bed wave a by:

When the shear stress rise, the amount of sediment deposited is not only due to bed load transport, but also to suspension load.

But considering mainly the bed load transport, we can mix together the 2 last equations (considering that qD is the amount of sediment transported by bed load (quoted qB,top for h=HD):

Considering only bed load transport, sedimentary flow can be calculated by the Meyer-Peter formula, which, introducing in the previous equation gives (equation 1):

where is the friction angle, and the adimensionnal variation of friction on bed (shear stress variation).We so have to know the shear stress on the bed. For instance, Bradshaw and Wong (1972) expressed the local variation of shear stress by:

where Hs is the heigth step, and the shear stress on the top of the dune.

The local water depth changes because of the presence of the dune, which involve a local variation in velocity:

(q is the discharge per width)

In the case of dune, this variation can be included in the spatial variation of the bed shear stress , because the shear stress is proportionnal to the square of the velocity:

Mixing the two last equations:

Using this final equation in the equation 1, we can now calculate the bed shape:

Dune heigth:

It's possible to know the dune heigth without resolving all the equations above:

Including this equation in the continuity equation:

Replacing q by the dimensionless sediment transport rate (which is a fonction of the Shields parameter):

By making an approximation of on the crest dune:

Finally:

Dune length:

In the case of dominant bed load transport (i.e. when shear stress is small):

LD=16HD

For higher shear stress , the formula below estimate the length of the dune: