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

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:

$\frac{1}{\rho} \frac{D \rho}{D t}+\vec{\nabla} \bullet \vec{U}=0$

where ρρ is the density and D/Dt is the total derivative.

For divergence, $$\vec{\nabla} \bullet \vec{U}>0$$, 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:

$\vec{\nabla} \bullet \vec{U}=0$

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

$\vec{\nabla} \bullet \vec{U}=\vec{\nabla}_{H} \bullet \vec{U}_{H}+\frac{\partial w}{\partial z}$

we see that:

$\vec{\nabla}_{H} \bullet \vec{U}_{H}+\frac{\partial w}{\partial z}=0, \quad\) or $$\quad \frac{\partial w}{\partial z}=-\vec{\nabla}_{H} \bullet \vec{U}_{H}$ Thus, horizontal divergence is compensated by vertical convergence and horizontal convergence is compensated by vertical divergence. $\vec{\nabla}_{H} \bullet \vec{U}_{H}>0 \quad$$ means $$\quad \frac{\partial w}{\partial z}<0$ Horizontal divergence gives a decrease in vertical velocity with height. $\vec{\nabla}_{H} \bullet \vec{U}_{H}<0 \quad$$ means \(\quad \frac{\partial w}{\partial z}>0$

Horizontal convergence gives an increase in vertical velocity with height.

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.

Credit: H.N. Shirer

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.

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:

Horizontal Divergence Vertical Motion