Undulatiform clouds

motion, at certain conditions, are responsible for undulatiform cloud formation.

The undulatiform clouds are seen as long banks, bands, streets, and plates. The sizes of these formations are: 10³-10² km in horizontal directions, and 101-10² meters in vertical, rarely up to 2 km.

atmosphere
The air involved into the cloud formation should be humid enough to become saturated at a small ascent.

Reasons for the wave-like motion to arise

Two layers ( one above another) with different density and speed of motion.
Inversion (barrier layer) must exist to prevent the layer to mix up.

According to Helmholtz formula, the waves spread with the speed

not change its value. Unstable wave amplitude value increases.
The wave stability depends on the wave length at a given conditions, i. e. on the radicand value in the Helmholtz’ formula.

Setting this value to zero, we obtain

Critical wavelength

Value varies from a few meter up to 8 km

The waves are stable. Clouds may appear, but not develop

The waves are unstable. Clouds appear and develop

observed wave , this wave is stable.
If at these conditions the observed wave , this wave is unstable.

wave arise on the lee side of mountains. That is why they are known as lee waves

Lee wave structure depends on:
Velocity of air currents
Stratification of the atmosphere (unstable stratification results in formation of lee vortexes and chaotic cloudiness appearance)
Size of the mountains.

Martner photos

Cascades and Rocky Mtns of MT and Alberta
Wave clouds induced by South Sandwich Islands on 27 January 2004, MODIS satellite

exist in form of mesoscale objects that are called cloud cells. These cells have a form of not fully regular systems – hexagons. The cells are not the only systems; there are also bands and banks of clouds.
Benar (1900) and Rayleigh (1916) were the first to discover this kind of cloud field structure. According to Rayleigh’s investigations, regime of motion causing waves and cloud formation depends on non-dimensional number (Rayleigh’s number)

Depth of convective layer

Molecular viscosity

Temperature conductivity

Open cell

Closed cell

Over warm surface

Over cold surface

Wavelike motion of the cell type; unstable waves

Any type of motion; waves (if any) are stable

Team, NASA/GSFC
http://visibleearth.nasa.gov/view_rec.php?id=6394

In "closed" cells, seen primarily in the top right corner of the image, warm air is rising in the center, and sinking around the edges, so clouds appear in cell centers, but evaporate around cell edges.
Closed and open cell convection represent two stable atmospheric configurations—two sides of the convection coin.

12 April 2006

12:30 UTC - resolution 1 km; Image courtesy of MODIS Rapid Response Project at NASA/GSFC,
[Van Delden, A., 1998, On the flow-pattern of shallow atmospheric convection,
Beitr.Phys.Atmos., 61, 169-186])