San Francisco State University Department of Earth & Climate Sciences. Homework 7 Key: Pressure Tendency Equation

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Monica Hart
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1 San Francisco State University Department of Earth & Climate Sciences Name Homework 7 Key: Pressure Tendency Equation 1. On November 11, 2012 at 12 UTC, the 300 mb divergence over extreme southcentral Iowa was 3.30 X 10-5 s- 1. At the same spot, the 850 mb divergence was X 10-5 s -1. Compute the one hour surface pressure tendency for this point using the pressure tendency equation. You may make the following assumptions which will simplify the procedure: (a) (b) (c) the 850 mb divergence is representative of the net divergence BELOW the level of nondivergence; the 300 mb divergence is representative of the net divergence ABOVE the level of nondivergence; the tropopause is at the 200 mb level and the surface pressure at station A is 900 mb (this station is not at sea level); p t = V h dp (1) 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. 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. 1

2 Tropopause 200 mb DIV h = 3.30 X 10-5 s -1 LND 500 mb p DIV h = X 10-5 s -1 Surface 900 mb Figure 1: Conceptual setup of problem Equation 1 states that the pressure tendency at level 1 (where the pressure is 850 mb) is due to the net divergence of pressure thickness p=(200 mb 900 mb) = -700 mb. The net divergence in this layer is ( ) X 10 5 s -1 = 9.4 X X 10 6 s -1 Substitution of these into the finite difference form of (1) and reworking the units to mb 1h -1 yields the correct answer of 23.7 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. 2. The 300 mb height analysis with companion 300 mb convergence chart are provided below. a. Discuss how your answer in 1 and the patterns shown on the two charts are generally consistent with the principle embodied in Dines Compensation but 2

3 that in the real atmosphere there is not quite the balance (upper divergence and lower convergence). (25 pts) Dines Compensation strictly states that at the synoptic scale upper tropospheric divergence tends to be balanced by lower level convergence and vice versa. If this were strictly true, there would be no pressure change at the bottom of the air column. In this case, there is NET divergence in the air column (because the upper tropospheric divergence exceeds lower level convergence) so there MUST be pressure falls at the ground. b. Qualitatively discuss the contributions of the two terms that make up divergence in the natural coordinate system. Limit your discussion to Iowa. (25 pts) An examination of Fig. 1 indices that over Iowa there is clear spreading of the streamlines. Thus, there is diffluence over that area. However, there is also speed convergence along the streamlines. Thus, the diffluence term is opposed by the speed divergence term. Fig. 2 indicates that there is divergence over Iowa, hence the diffluence term must exceed the speed convergence evident in Fig. 1. 3

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Changes in Density Within An Air are Density Velocity Column Fixed due and/or With Respect to to Advection Divergence the Earth

Changes in Density Within An Air are Density Velocity Column Fixed due and/or With Respect to to Advection Divergence the Earth The Continuity Equation: Dines Compensation and the Pressure Tendency Equation 1. General The Continuity Equation is a restatement of the principle of Conservation of Mass applied to the atmosphere. The