Pressure distribution of ideal gas under non-uniform temperature

[Margalit]

Suppose there is a square box with an ideal gas inside at standard temperature and pressure. Now one side of the box is heated up while the other opposite side remains at room temperature (assume a large heat sink). It is clear the temperature distribution of the gas inside the chamber will have a gradient. Assume that the temperature difference is not larger (a few degrees Kelvin), so that convection is not a major driver. If the ideal gas law is followed locally in the space between the walls, the difference in temperature means that either one or both of the pressure or gas density must non-uniform to compensate the variable temperature.

My question is as follows: In this situation is the pressure or density or both non-uniform between the walls?

 

[cjl]

Generally, in a container, the pressure will be uniform throughout. In this case, the density is what will be non-uniform.