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:

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:

Equation [9.6] gives the **kinematic vertical velocity**.

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

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

The trick is to find b using some other information. To find b, note that $\frac{\partial w}{\partial z}$ 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**, z_{LND, }

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

So, typically, for surface divergence:

At $z\text{}=\text{}{z}_{LND}=5000m,w({z}_{LND})=\text{}-2.5\text{}cm\text{}{s}^{-1},\text{}or\text{}-2.5x{10}^{-2}x86400\text{}s/day\text{}=\text{}-2.2\text{}km\text{}da{y}^{-1}$ (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.

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

- 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. - When you feel you are ready, take
**Quiz 9-2**. You will be allowed to take this quiz only**once**. Good luck!