**6. Conclusions**

In this work we have reviewed the influence of horizontal temperature gradients on convective instabilities, focusing on results with geophysical interest.

10 Will-be-set-by-IN-TECH

Regarding the numerical solutions found in this case, the horizontal temperature gradient generates convective states and tends to concentrate motion near the warmer wall. This fact coincides with experiments in Refs. (38; 40; 41) and is consistent with previous numerical results reported in (27; 45). The temperature dependent viscosity localizes motion near the region of lower viscosity, i.e., the bottom plate. This also coincides with experiments in Refs. (36; 46; 49) and numerical results in Ref. (48). It is remarkable that the horizontal temperature gradient favours a threedimensional structure after the bifurcation, while the pattern continues being axisymmetric after the bifurcation in the only vertical gradient case.

As detailed in Ref. (48) we found qualitative similarities between the vortical structures computed numerically and some meteorological phenomena such as dust devils and cyclones. One of the main characteristics of dust devils is a low-pressure region in the center of the dust devil which coincides with the dust devil's warm core (33). This is also observed in our numerical vortices (see figure 5 c). Regarding temperature, in dust devils, the maximum temperature deviation from the environment temperature (i.e. the temperature furthest from the dust devil center) occurs at the lowest levels. This feature is observed in the temperature

The experimental measures provided in Ref. (33) show that there is radial inflow at the lower levels of the dust devil and radial outflow in the upper levels. It is also shown that the vertical velocity reaches highest values and then falls off rapidly as the radius is increased. These

The trajectory of particles around the inner cylinder described in this section appears to be very similar to the trajectory of particles of air (or dust) in a dust devil, characterized by a

Other more complex meteorological phenomena such as cyclones also present these structural characteristics. It is known that the center (eye) of a cyclone is the area of lowest atmospheric pressure in the region, which corresponds to a warm core in some kind of cyclones (e.g. tropical and mesoscale) (31; 34). This coincides with that observed in figures 5 a) and 5 b). Regarding the motion in cyclones, it is observed the inward flow next to the surface, strong upward motion in the eyewall and outflow in a layer near the top of the storm (31; 34). This characteristic is described in the combined effect of the radial and vertical velocity components observed in our vortices as pointed out above (see figures figures 5 c, d) and e)). In cyclones, a counter-clockwise motion (clockwise in the southern hemisphere) is observed around the center of the storm, stronger just above the surface in a ring around the center and sligther as we go up from the surface (31; 34). That coincides with the effect of the azimuthal velocity component observed in the vortices we have computed numerically responsible for

In this work we have reviewed the influence of horizontal temperature gradients on

features are appreciated in the profile of *ur* and *uz* shown in figures 5 d) and e).

the movement of the particles around the inner cylinder.

convective instabilities, focusing on results with geophysical interest.

**5. Discusion**

**5.1 Contained fluid**

**5.2 Not contained fluid**

profile of our vortices.

spiral up motion (33).

**6. Conclusions**

We have distinguished two cases, a first one where the fluid is simply contained in a domain, and a second one where the fluid can flow throughout the boundaries.

In the first case three subcases can be grouped. The case corresponding to small cells and small *Pr* number displays stationary and oscillatory instabilities depending on the multiple parameters present in the problem: properties of the fluid, surface tension effects, heat exchange with the atmosphere, aspect ratio, dependence of viscosity with temperature, etc. This problem has been treated from experimental, theoretical and numerical points of view. The cases corresponding to small cells and large or infinite *Pr* number are closer to mantle convection. Boundary layer waves are observed in experiments and 3D stationary patterns of rolls perpendicular to the temperature gradient appear numerically. Finally for the case of infinite *Pr* number with temperature dependent viscosity, the closest to mantle convection, 3D stationary patterns concentrated in the region of lower viscosity and waves for larger values of the *R* number appear. Summarizing, horizontal temperature gradients favour the presence of waves and the totally three dimensional patterns.

The problem where the fluid can flow throughout the boundaries has been treated usually as direct numerical simulations. For the first time it has been studied under the perspective of instabilities or bifurcations in Ref. (29). In this reference vortical solutions, very similar to those found for some atmospheric phenomena such as dust devils or hurricanes, appear after a stationary bifurcation. This is a powerfull and simple explanation of those atmospheric phenomena as an instability.
