METEO 300
Fundamentals of Atmospheric Science

9.6 How fast is the vertical wind and which way does it blow?

What are typical values of the vertical velocity caused by convergence or divergence and how do they vary with height? The vertical velocity, w, is too small to measure by a radiosonde. It is so small that it is hard to visualize. But we can estimate w from the convergence/divergence patterns:

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

Note that this equation just gives the derivative of the velocity, not the velocity. So to find the velocity, we must integrate both sides of the equation over height, z.

Integrate this equation from the surface (z=0) to some height z:

0 z w z' dz'=  0 z H U  dz' This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
w( z )=  0 z H U  dz' This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
[9.6]

Equation [9.6] gives the kinematic vertical velocity.

To a good approximation, it has been determined that the divergence/convergence varies linearly with altitude.

H U = ( H U ) z=0 +bz This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
[9.7]

where b is a constant. Substituting this expression for the horizontal divergence into Equation [9.6], we get:

w( z )=  0 z H U  d z =  ( H U ) z=0 z 1 2 b z 2 This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
[9.8]

The trick is to find b using some other information. To find b, note that 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. must equal 0 at some level because w must be 0 at both Earth's surface and the tropopause, so the derivative is positive near Earth's surface. It must become negative at the tropopause in order for w to go to zero, and so somewhere in between, being positive and negative, it must be zero. We call this level the level of nondivergence, zLND,

0= ( H U ) s b z LND ,  or  b=  ( H U ) s z LND   This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
[9.9]
w( z )=  ( H U ) s z+ ( H U ) s 2 z LND z 2 This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
[9.10]

The surface divergence typically has a value of 10-5 s-1.  The level of nonconvergence is typically about 5000 m. So,

b=  ( H U ) s z LND = 10 5 s -1 5000m =2x 10 9   m -1 s -1 This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.

So, typically, for surface divergence:

w( z )=  10 5 z+ 10 9 z 2 This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.

At z =  z LND  = 5000m, w( z LND ) = 2.5 cm  s 1 , or 2.5x 10 2 x86400 s/day = 2.2 km da y 1   (see figure below).

The result is that w is only a few cm s-1.  In a day, the air mass can rise or fall only a few kilometers. Compare this vertical motion dictated by convergence and divergence to the vertical motion in the core of a powerful thunderstorm, where the vertical velocities can be many m s-1. This simple model is called the bowstring model because the shape of the vertical velocity looks like a bowstring that is fixed at two points but can vary as a parabola in between.

graph showing Divergence and vertical wind
Divergence (left) and vertical wind (right). For convergence aloft (negative divergence), the vertical wind is negative with a maximum value near 5000m and remains negative as it decreases toward zero at the surface, where there is divergence (positive divergence) near the surface.
Credit: W. Brune

Quiz 9-2: Connecting the dots with vertical motion.

  1. Find Practice Quiz 9-2 in Canvas. You may complete this practice quiz as many times as you want. It is not graded, but it allows you to check your level of preparedness before taking the graded quiz.
  2. When you feel you are ready, take Quiz 9-2. You will be allowed to take this quiz only once. Good luck!