 The key concepts that allow horizontal divergence to be converted into vertical motion are that mass is conserved, that 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 zero. This means that the total divergence must be approximately zero, so that the air parcel volume remains constant. Thus, changes in the horizontal area cause changes in the vertical height to maintain the air volume. A key to remember is that the vertical velocity w and its 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 the negative of the divergence. We will look first at divergent near Earth's surface. If there is horizontal divergence, convergence, then the air must go somewhere and it can not 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 zero at Earth's surface and w is increasing with altitude, then w must be positive. For a 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 zero at Earth's surface, so the air velocity w must be downward. At the tropopause, the rapid increase in stratospheric potential temperature acts like a lid on the troposphere and effectively makes w go to zero at the tropopause. If 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 zero at the 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 zero 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 is divergence. In between the two surfaces, the velocity is upward, i.e. 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, i.e. w is negative. The upward 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.