2. Linearize the conservation of potential vorticity equation Dgq = 0 Dt about a background state in which the heights of the pressure surfaces tilt downward toward the poles. The geopotential can be partitioned into this background state (?) plus subsequent deviations to it ('): (x,y,p,t) = F(y,p) + '(x,y,p,t) In thermal wind balance with the background part of the height field, there is a background jet that will also be a function of y and p. The full wind will be the background jet plus any perturbations to it (note that all parts of the wind field are geostrophically balanced): ug(x,y,p,t) = u(y,p) + u'(x,y,p,t) vg(x,y,p,t) = v'(x,y,p,t) Thus, you can partition q into its background part (a) plus deviations that arise from later, smaller disturbances to it (q'). Assume that all perturbations (to q, to , to the wind) are small compared to their background values. As usual, you may treat the stratification parameter as constant.