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I have seen this in a picture once online and thought it was very strange and wondered about the formation of this vortex and how could it happen.
Source
Not everyday do you see this happening. So how does this happen, and why does it not happen very often?
Another picture of this strange occurrence. This generally happens with thrust reversal.
Whenever there is under-pressure in air, the air flows along the gradient of this under-pressure, i.e. into the direction of the source of the under-pressure. (In first order.)
If there's a wall (or the ground) in the way, air will flow along the surface of this obstacle and will change its direction at some point, as shown in this sketch:
Now, imagine the whole air around the engine is rotating slowly for some reason around the symmetry axis in the sketch.
From physics, you may know the conservation of momentum, or in this case, of angular momentum. It says that the product of distance to rotation axis times RPM of this rotation is constant for a certain particle. If a particle reduces its distance to the axis, its RPM increases.
This is like the spinning ice skater, who rotates faster when he pulls his arms to his body (to the axis).
This law also applies to packets of air, which are sucked to the axis, though internal friction brakes the air to some extent.
The fast rotating air near the axis encounters a centrifugal force, i.e. the air is pulled away from the axis, forming a region of strong under-pressure. In this region, the air expands and so, cools down. This can cause condensation of water, if the humidity of the air is high enough. And that's what you can see on your pictures.
There's still one question unanswered: I wrote, this happens if the surrounding air is rotating slowly. This usually happens, because the wind is deflected from the aircraft itself, which means it changes its direction, which is some kind of rotation.
And of course, a strong, non-uniform air flow from the rear, as generated by the reverse thrusters, can cause more / stronger rotating air, which results into stronger vertices. So, it's no wonder when they appear (or better: become more visible) in this case.
Here is a nice picture of a simulation, which also shows that the fuselage can be the wall where the vortex is created.
Source: http://gallery.ensight.com
It seems the wind comes from the camera position, or a bit more right of it. It gets deflected by the fuselage and flows along it, which explains the straight lines coming from the rear and the curved ones coming from the front left of the aircraft. If you imagine the air flow along / under the fuselage, you may also understand why the two vertices are rotating in different orientations.
As the strong under-pressure inside the vortex, it may also suck in dust or even larger objects, which can damage the engine. For aircraft landing on unpaved airfields, unpaved strip kits may be available, which can contain a vortex dissipator:
Source: http://www.b737.org.uk
This is a nozzle blowing bleed air into the region where vortices could apper, reducing the under-pressure and so the risk of ingesting objects from the ground.
You can see something similar in the sink of your bathtub. As soon as there is a little rotation, it is amplified above the sink, the centrifugal force pushes the water aside, and forms this fast rotating vortex.
A tornado is a giant version of an inlet vortex. The air is moving along ground over far distances, until it moves upwards at some point, and a slight rotation leads to this fast rotating hose sticking to the ground and sucking in everything due to the strong under-pressure inside.
And a bit off-topic:
Sometimes, you can see vortices at the wing tips or at the edges of engaged flaps of aircraft flying through humid air. The reason is similar: The air is put into rotation due to the higher pressure below and the lower above the wings. (But that's not exactly the same as inlet vortices.) And sometimes, you also see fog forming above the wing just because of the under-pressure.
On this picture, you see both, and may imagine the rotation for the second vortex from the right.
Source: http://avioners.net
The inlet vortices shown form in the case of turbine engines and in propellers during operation near ground where there is a little or no crosswind. In standard atmospheric conditions, the vortex is invisible; however, with enough humidity, condensation occurs inside the vortex core (as temperature falls below the dew point temperature), and the vortex becomes invisible.
Source: wpi.edu
As the engine is started, the engine ingests air whose cross sectional area is greater than the engine frontal area. Near ground, this includes ground as well and induces a flow there. As air nears the engine, it is accelerated and if some amount of vorticity is already present, it forms an inlet vortex, which become visible under some conditions.
Image from patent application of Active ground vortex suppression system for an aircraft engine EP 1413721 B1
The inlet vortex forms between the engine (or propeller) and a stagnation point on the ground. This is usually not formed far away from the ground as the vortex lines cannot start or end in a fluid according to Helmholtz's theorems.
As a result, one end of the vortex is attached to the engine (or propeller) and the other end extends and attaches itself to the ground.
Two mechanisms for this inlet formation have been proposed,
The study of engine inlet vortex formation is an important research topic as the inlet vortex can cause a number of problems such as,
For the inlet vortices to form, there should be usually some sort of rotation flow in the far field, which is usually given by the crosswind as it interacts with the fuselage; In case there is no crosswind, inlet vortices still form, though a pair are formed, which rotate in opposite directions. This is probably the vortices in simulation which @sweber showed in answer..
Source: gallery.ensight.com
One reason for the presence of vortices when reverse thrusters are engaged is that they cause air flow with sufficient vorticity in the vicinity.
See An Experimental Study of the Formation and Unsteady Characteristics of Inlet Vortices by Wang and Gursul, accessed via enu.kz
Note: The vortex suppression system in the patent described above injects fluid near the inlet, which disrupts the flow structure of a ground vortex, thus mitigating ground vortex.
Image from patent application of Active ground vortex suppression system for an aircraft engine EP 1413721 B1
why does it not happen very often. Those vortices are always there. It's just that you only see them when they pick something up like dust or water. They are just a function of the engine sucking in large amounts of air. $\endgroup$