Problem 1

The pressure tendency at the bottom of an air column that extends from the level where the pressure is 850 mb to the tropopause where the pressure is 250 mb is given as

(1)

where the Pressure Tendency Equation has been simplified by the boundary condition w2 = 0 at the tropopause.

Equation (1)  states that  the change in pressure observed at the bottom of a slab of air of thickness dp is due to the net horizontal divergence in/out of the slab modified by the amount of mass being brought in through the bottom of the slab.  In this case, the top of the slab is the top of the troposphere and the bottom is at the surface, at which level the pressure is 850 mb. See Fig. 1.

Figure 1:  Conceptual setup of problem

states that the pressure tendency at level 1 (where the pressure is 850 mb) is due to the net divergence of pressure thickness Ęp=(200 mb – 850 mb) = -650 mb.

The net divergence in this layer is (3.99 – 3.05) X 10 –5 s-1.  Substitution of these into the finite difference form of (4) and reworking the units to mb 1h-1 yields the correct answer of –22.0 mb 1h-1.

Although the answer is probably one order of magnitude too large, it suggests that the system would be deepening rapidly. The error in estimating the magnitude of the change is due to the fact that the net divergence really must be estimating the contribution to the divergence made by each layer, say at 25 mb layer intervals from the surface to the tropopause.

Fig 2:  Observed 3 h pressure tendencies, 12 UTC 22 October 2004