Reading #6: The Continuity Equation, Divergence. Dine's Compensation and Associated "Lofting"

1.  General

The Continuity Equation is a restatement of the principle of Conservation of Mass applied to the atmosphere.  The principle simply states that matter can neither be created or destroyed and implies for the atmosphere that its mass may be redistributed but can never be "disappeared".

The Equation of Continuity restates this by telling us that there are TWO basic ways to change the amount  of air (density ) within fixed volume of air (an air column, for the sake of this discussion, fixed with respect to the earth and extending from the ground to the top of the atmosphere):  (i)  if air is flowing laterally through the air column, have the upstream air more (less) dense than the downstream air, hence, air of different density replaces the air in the air column (density advection[1]);  and (ii) even if density is everywhere constant, remove air from the air column (called three dimensional divergence).

Note: in the geosciences the concept of some property of the geosophere, hydrosphere or atmosphere/ocean system being transported from place to place is very important...the general name given to that is advection.

Changes in Density

Within An Air         are          Density                 Velocity

Column Fixed        due                           and/or

With Respect to      to            Advection             Divergence

the Earth

EQUATION OF CONTINUITY

In-class Exploration of Continuity Equation

2.  Synoptic Scaling of Continuity Equation

In this class, we are concentrating on the larger "scales" of atmospheric phenomena--the Macroscale (10000 km or so) and the Synoptic Scale (1000 km or so).  At these scales, the density advection is negligible compared to the Three Dimensional Velocity Divergence term. That term is often referred to as "divergence".

By the way, the algebraic expression for velocity divergence is relatively simple:  it is the change in wind speed over distance or, for example, horizontal speed divergence and vertical speed divergence in the rectangular coordinate system can be written:

At the scale of a weather map and larger (synoptic scale and macroscale), changes in density within an air column fixed with respect to the earth are also very small.

Hence the continuity equation applied at the synoptic scale reduces to:

It turns out that the horizontal divergence is just a measure of the lateral expansion or contraction of an air column or layer.

Note:  both density advection and local changes in density experienced locally are NOT negligible at smaller scales. For example, the circulations near fronts (mesoscale) and outflow boundaries (thunderstorms) are strongly influenced by advection of density.

 IN-CLASS DISCUSSION: Horizontal divergence is a measure of the percentage or fractional rate of change of the horizontal cross sectional area of an air or water column. In fact, its basic mathematical definition can be traced to the philosophical question...if the top of (a thunderstorm, or a water column) expands from an initial area to a larger area in two hours, what was the percentage (or fractional) change in area over that time interval. Using the algebraic expression above, provide the UNITS of divergence.  However, in terms of significance in understanding the atmosphere, ocean or earth, if a geoscientist can compute the divergence that occurs at the top of a (thunderstorm, water column adjacent to the coast, mantle column), that geoscientist can compute the vertical motion that is producing (clouds, upwelling, upwelling in the mantle) and vice versa.

The Equation of Continuity simplified for synoptic-scaling (called the Simplified Equation of Continuity) reduces to:

(1)

3.        THE SIMPLIFIED CONTINUITY EQUATION

Equation (1) is important. It states that horizontal divergence (lateral spreading or contraction) is paired with changes in vertical motion. We'll see how important this is below. Notice that horizontal divergence is proportional to the gradient of the wind speeds, for example ?u/?x in the rectangular coordinate system, and ?V/?s in the natural coordinate system.

Cross sections of the atmosphere, or the wind profiles shown on the right margin of soundings, show that in the middle latitudes wind speeds generally increase with height. In other words, the higher you go, the greater the horizontal wind speeds, typically. Thus, the jet stream is most marked at the 200 mb level. Often times we see strong jet streams coming across the Pacific aimed at California.

For the sake of argument, let's say that the winds at all levels of the atmosphere are initially calm at San Francisco, but that a strong jet stream is approaching us from the west.

Let's say that 1000 km off the coast the wind speed at 400 mb is 60 km/h, at 300 mb is 100 km/h and at 200 mb is 180 km/h, whereas the wind is calm at each of those levels over San Francisco. Evaluating the left hand side of the expression using Delta V/Delta s you can see that divergence (in this case the equation returns a negative result, indicating convergence) is maximum at the level where the jet stream is the strongest. This is generally true in the atmosphere, the ocean and the mantle of the earth. Horizontal divergence or convergence is the greatest at the top of the layer.

The importance of the simplified continuity equation (Equation (1) (above) may be more apparent now.  If the left hand side says something about whether an air column is laterally spreading or contracing, the right hand side relates that spreading or contraction (commonly referred to as divergence and convergence) to the vertical motion that occurs through the layer. Thus, we can say something about the field of vertical motion in a layer of the atmosphere if we know something about the horizontal divergence in that same layer.

DISCUSSION QUESTION:

Give one reason why knowing something about Vertical Velocity is very important in meteorology?

[1]Advection is defined as the transport of an atmospheric property soley by the velocity field (i.e., temperature advection, moisture advection etc.) and in scalar form is given as the product of the wind velocity component and the gradient of the property along the respective coordinate axis (e.g., -V (∆T/∆s)).