text: These baroclinic instabilities may be initiated by the process of sloping convection or slanted thermal convection. To understand this, consider a fluid in steady state and under the thermal-wind balance. Initially, a fluid parcel is at location A. The fluid parcel is slightly perturbed to location B, while still retaining its original density. Therefore, the fluid parcel is now in a location with a lower density than itself and the parcel will just sink down to its original position; the fluid parcel is now stable. However, when a parcel displaced to location C, it is surrounded by fluid with a higher density than the parcel itself. Due to its relatively low density with respect to its surroundings, the parcel will float up even further. Now a small perturbation grows into a larger one, which implies a baroclinic instability.
  A criterion for an instability to occur can be defined. As stated before, in a baroclinic fluid, the thermal-wind balance holds, which implies the following two relations:
   and ,
  where  is the density and ,  and  are the spatial coordinates in the horizontal (latitudinal and longitudinal) and vertical direction, respectively.  and  represent the horizontal (zonal and meridional) components of the velocity vector  in the - and -direction, respectively. Now thus  and  are the two horizontal density gradients.  is the gravitational acceleration at the surface of the Earth and  the Coriolis parameter.
  Therefore a horizontal density gradient in the -direction  leads to a gradient in horizontal flow velocity  over depth . 
  The slope of the displacement is defined as 
  where  and  are the horizontal and vertical velocities of the perturbation, respectively. 
  An instability now occurs when the slope of the displacement is smaller than the slope of the isopycnals. The isopycnals can be mathematically described as . Now this results in an instability when: 
  From now on, only a two layer system with  and  the slopes of the top and bottom layer, respectively, is considered to simplify the problem. This is now similar to the classic Philips model.  From the thermal-wind balance it now follows that 
  where  is the reduced gravity and  the Coriolis-parameter at the equator according to the beta-plane approximation. 
  Performing a scale analysis on the slope of the perturbation allows to assign physical quantities to this mathematical problem. This now results in 
  where  is the scale height,  the horizontal length scale, and  is the Rossby-parameter. 
  From this it can be stated that an instability occurs when 
   or , 
  where  is the reduced gravity and   is the velocity difference between the lower and upper layer. This criterion can be used to identify whether a small perturbation will grow into a larger one and thus whether an instability is expected to occur. From this it follows that you need some kind of shear  to obtain an instability, it is easier to get an instability for long waves (perturbations) with large , and the  and therefore the beta-effect is stabilizing.  
  Furthermore, for the baroclinic Rossby radius of deformation it holds that . Now the instability criteria simplify to 
   or . 
  From this analysis it also follows that baroclinic instabilities are important for small Rossby numbers, where .