METEO 300
Fundamentals of Atmospheric Science

9.5 How is the horizontal divergence/convergence related to vertical motion?

Composite of satellite infrared image overlaid with surface analysis and radar for united states
Composite of satellite infrared image overlaid with surface analysis and radar. Note that the precipitation is generally associated with the low pressure regions and the high pressure areas are generally free of significant clouds. As we will soon see, low-level air converges into low pressure regions, causing uplift, which causes clouds and precipitation. It diverges from high pressure regions, causing downwelling air and warming and drying by adiabatic descent. Take a look at the latest composite image.
Credit: Unisys

Our goal here is to relate horizontal convergence and divergence to vertical motion. If vertical motion is upward, then the uplifted air will cool, clouds will form, and it might rain or snow. If vertical motion is downward, then the downwelling air will warm by adiabatic descent, clouds will evaporate, and it will become clear.

To find out what will happen, we need to go back to a fundamental law of mass conservation, which we will derive in detail in Lesson 10. Here we simply quote the result:

1 ρ Dρ Dt + U =0This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.

where ρThis equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. is the density and D/Dt is the total derivative.

For divergence, U >0 This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. , volume increases and density must decrease to conserve mass.

For convergence, U <0 This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. , volume decreases and density must increase to conserve mass.

However, to good approximation, density does not change with time for any given horizontal surface. Sure, density decreases exponentially with height, but for each height level, the density at that level is fairly constant.

So, to a good approximation:

· U =0This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.

and because we can separate out the horizontal and vertical components of divergence:

U = H · U + w z This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.

we see that:

H U +  w z =0,  or   w z =  H U
[9.5]

Thus, horizontal divergence is compensated by vertical convergence and horizontal convergence is compensated by vertical divergence.

H · U >0 means   w z <0 This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.

Horizontal divergence gives a decrease in vertical velocity with height.

H U <0 means   w z >0This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.

Now, in the troposphere, the vertical velocity is close to zero (w~0) at two altitudes. The first is Earth’s surface, which forms a solid boundary that stops the vertical wind. The second is the tropopause, above which the rapid increase in stratospheric potential temperature strongly inhibits vertical motion from the troposphere (see two figures below), so much so, that we can say that the vertical wind must be ~0 at the tropopause.

diagram of surface convergence/divergence and vertical velocity as described in the text
Surface convergence or divergence and the resulting vertical velocity (note: v used in this figure is the same as U in the text.).
Credit: H.N. Shirer
diagram of tropopause convergence/divergence and vertical velocity as described in the text
Tropopause convergence or divergence and the resulting vertical velocity (note: v used in this figure is the same as U in the text.).
Credit: H.N. Shirer

These processes can be summarized in the following table:

plane process surface area change ∂w/∂z w
surface convergence decrease + up
surface divergence increase - down
aloft convergence decrease + down
aloft divergence increase - up

Let’s now consider the effect that divergence/convergence aloft has on surface convergence/divergence (see figure below).

Divergence aloft is associated with rising air throughout the troposphere, which is associated with low pressure and convergence at the surface.

Convergence aloft is associated with sinking air throughout the troposphere, which is associated with high pressure at the surface and thus divergence at the surface.

So, starting at the surface, the vertical velocity becomes more positive with height when there is surface convergence, reaches some maximum vertical velocity, and then becomes less positive with height again toward the divergence aloft.

Similarly, starting again at the surface, the vertical velocity becomes more negative with height when there is surface divergence, reaches some maximum negative velocity, and then becomes less negative with height again near convergence aloft.

diagram to show how divergence aloft causes surface low pressure and convergence and how convergence aloft causes surface high pressure and divergence, as described in the text
How divergence aloft connects to surface low pressure and convergence and how convergence aloft connects to surface high pressure and divergence.
Credit: H.N. Shirer

Now watch this video (3:52) on horizontal divergence:

Click here for a transcript for Horizontal Divergence Vertical Motion Video

 
Surface map of weather for February 2, 2015
Surface map of weather for February 2, 2015. Fronts, lows, and highs are indicated. Radar shows where precipitation is falling.
Credit: Unisys
Weather map of the United States for February 2, 2015. Analysis for the 500 hPa pressure surface (about 5 km altitude), with isopleths, wind vectors, and enhanced relative humidity.
500 mb weather map for February 2, 2015. Black lines are surface heights and wind bars show the direction and magnitude of wind.
Credit: NOAA.

We have shown that convergence and divergence aloft near the tropopause is related to surface highs and lows. Now it's your turn to find some examples. Go to a source of information about surface pressure and upper air winds and pick out some regions that show this relationship. One good source is the Penn State e-Wall, for which you can use the "U.S. Satellite Overlays".