CONCEPTUAL IMPLICATIONS OF THE SIMPLIFIED EQUATION OF CONTINUITY

 

A.       THE PRINCIPLE OF DINE'S COMPENSATION

 

Now, observations show that the vertical velocity at the Tropopause and at  

 

 

the ground is nearly zero.  Take a look at the above picture.  Let's use the Equation of Continuity to say something about the midtropospheric vertical velocity field. 

 

Inclass Exercise: Let's say that somehow you have calculated the DIVERGENCE  in the layer from 500 mb to the Tropopause at 200 mb to be 1.5 X 10-5 sec -1 and were asked to compute the vertical motion that would occur at 500 mb in "compensation" to this divergence. We'll do it in class, since you will also be doing something like this on the Homework.

 

You need to rewrite the equation to solve for the vertical velocity at 500 mb.  To do this you need to expand the term for the vertical divergence (right hand side of the expression) "in your mind".  How would you do it? 

 

Finite

 

 

 

Just like you would determine a temperature gradient between two points:  the temperature at Point 2 minus the temperature at Point 1 divided by the distance between the two points.  So, the term is expanded in the following manner

 

                       

 

Now, solve ALGEBRAICALLY for the vertical velocity at 500 mb.

 

 

 

 

DISCUSSION QUESTION

 

What is the vertical velocity at 500 mb for the above example?

 

 

But the equation works for the lower half of the diagram too!!

 

 

           

 

DISCUSSION QUESTION

 

What is the horizontal divergence at the ground for the above example?

 

You have conceptually developed one of the great principles applied in weather analysis and forecasting:  DINE'S COMPENSATION.

 

DINE'S COMPENSATION

 

Upper tropospheric divergence tends to be "balanced" by mid-tropospheric upward vertical motion and lower tropospheric convergence.

 

Upper tropospheric convergence tends to be "balanced" by mid-tropospheric downward vertical motion (subsidence) and lower tropospheric divergence.

 

 

DISCUSSION QUESTION

 

What sort of weather pheonmena (clouds, fair, precipitation etc.) does each "half" of the principle of Dine's Compensation imply?

 

 

 

 

 

 


 

B.  THE PRESSURE TENDENCY EQUATION

 

In reality, the balance implied by Dine's Compensation is never exact.  Let's examine this notion in a simple-minded way, retaining our model of the troposphere which has a "divergent" upper half and a "convergent" lower half.

 

 

DISCUSSION QUESTION

 

Can you think of two reasons why the upper tropospheric divergence (or convergence) almost always exceeds that  in the lower troposphere?

 

 

 

 

Because of this imbalance, upper tropospheric divergence is not quite balanced by lower level convergence and more mass leaves the top of the air column than enters the bottom.  Substitution of the Equation of State into Equation of Continuity, and making synoptic-scaling assumptions yields another very important equation. This equation essentially provides the answer to the question:   How do pressure changes (in this context) at sea-level occur (of course, the general expression relates to the bottom of an air column no matter where the base of the air column is found).