The Continuity Equation


Let us consider a thin layer defined by a pressure interval, Δp, and the area, A, of its projection on the horizontal plane. Then, for any transformation of the layer under the conservation of mass, the product of Δp and A does not change:

Δp1 * A1 = Δp2 * A2

where subscript 1 refers to the initial state and subscript 2 to the final state. This is a form of the equation of continuity.

By the hydrostatic equation, we know that

Δp = -ρgh

where ρ is density, g is gravity, and h is the thickness of a layer (in feet or meters).

Substituting for Δp

1gh1 A1 = 2gh2A2

If we divide both sides by g, we arrive at

1h1 A1 = 2h2A2

We may write this equation in ratio form as

equation

When convergence occurs, the area shrinks, so A2/A1 is less than one. Conversely, when divergence occurs A2/A1 is greater than one. There are innumerable combinations of the three parameters ρ, h, and A that satisfy the above relationship.

Now, let us look at the case of the vertical motion of a layer in the absence of horizontal divergence. If there is no horizontal divergence, then A1 = A2. Therefore,

ρ12 = h2/h1

That is, without horizontal convergence or divergence (meaning A does not change), the density and thickness of a layer vary inversely with one another.

If the layer ascends, then ρ1 > ρ2, since the density decreases with height in the atmosphere. From the last equation it follows that if ρ1 > ρ2, then h2 > h1; in other words, the layer expands vertically. For an entirely dry-adiabatic process, as the layer expands, the temperatures within the layer will change at the dry adiabatic lapse rate. Because of this, as the layer progressively expands, its lapse rate gets closer and closer to the adiabatic lapse rate. In other words, as the unsaturated (or saturated) layer expands, the lapse rate will tend to become less stable, approaching the dry (or moist) adiabatic rate.

On the other hand, if, again without horizontal divergence, the layer descends, ρ1< ρ2 and h2< h1; i.e., the layer shrinks vertically. In this case, the lapse rate becomes more stable, tending to depart further from the adiabatic.