PRESENTER: The key concepts that allow horizontal divergence to be converted into vertical motion are that mass is conserved, but the air density and density vertical structure are fairly constant with time, and that the vertical wind at Earth's surface and at the tropopause is effectively 0. This means that the total divergence must be approximately 0 so that the air parcel volume remains constant. Thus, changes in the horizontal area cause changes in the vertical height to maintain the air quality. A key to remember is that the vertical velocity w is partial derivative with respect to height z do not always have the same sign. The second key point to remember is that the partial derivative of w with respect to z is a negative divergence.

We would look first at diverging mirror surface. But if there is horizontal convergence, then the air must go somewhere, and it cannot go down, so it goes up. The equation actually says that the partial derivative of w, the vertical velocity with respect to z, must be positive.

But if w equals 0 at Earth's surface and w is increasingly with altitude, than w must be positive. For divergence near Earth's surface, we see that the partial derivative of w with respect to z is negative, which means that w must be negative above the surface since w equals 0 at earth's surface. So the air velocity w must be downward.

But the tropopause, the rapid increase in stress for potential temperature acts like a lid on the troposphere. Effectively makes w go to 0 at the tropopause. There is a horizontal divergence aloft then w must be upward to maintain the air parcel volume as the air parcel spreads out horizontally near the tropopause.

Mathematically, this means that w must be positive. But we know that it must go to 0 tropopause. Therefore, the partial of w with respect to z must be negative as it approaches the tropopause, i.e. w is decreasing with increasing height to 0 at the tropopause from a positive value in the troposphere.

On the other hand, if there is convergence in the air near the tropopause then, the air must go down and the vertical velocity w must be negative. If we look at the changes in w with respect to height above the level of non-divergence, as z increases, w goes from more negative to less negative, which is a positive change in w, with a positive change in z. So the partial derivative is positive even though w is negative.

Putting these pieces together, we see that if we have convergence at Earth's surface, which occurs in low pressure areas for reasons we will see in lesson 10, then at the tropopause, there's divergence. In between the two surfaces the velocity is upward-- that is, w is positive. If we have a divergence near Earth's surface, which occurs in high pressure areas, then there is convergence at the tropopause.

In between, the vertical velocity is downward. That is, w is negative. The outward moving air above low pressure creates cooling, which leads to clouds and precipitation. The downward moving air above high pressure region causes warming and drying, resulting in clear conditions.