The Continuity Equation and Divergence
The Continuity Equation and Divergence
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]) (Fig.1); and (ii) even if density is everywhere constant, remove air from the air column (called three dimensional divergence) (Fig. 2).
Changes in Density
Within An Airare Density Velocity
Column Fixed due and/or
With Respect totoAdvection Divergence
the Earth
QUALITATIVE EQUATION OF CONTINUITY
The Equation of Continuity therefore is:
(1)
where DIV3d is the three dimensional divergence, which in rectangular coordinates is
(2)
Example 1.
and if positive, contributes to density rises, and states that the advection is the negative of the product of the wind speed and the (change in density in a unit amount of distance). Actually, the concept of advection is very important in many areas of synoptic analysis. Advection is always found by finding the product of the wind speed and the gradient of a quantity (say, temperature).
Figure 1: Schematic chart illustrating density advection (wind speed constant) locally increasing the density (mass) in a fixed air column.
Example 2:
Figure 2: Schematic chart illustrating wind divergence (density constant) locally decreasing density (mass) in a fixed air column.
DISCUSSION QUESTION:
Can you think of one important implication of the changes in density in an air column with fixed volume and fixed cross-sectional area?
2. Synoptic Scaling: Simplified 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 Velocity Divergence term. Also, at these scales, changes in density within an air column fixed with respect to the earth are also very small.
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.
The Equation of Continuity simplified for synoptic-scaling (called the Simplified Equation of Continuity) reduces to:
(3a)
(3b)
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 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.
(3c)
The importance of the equation may be more apparent now. It implies that we can say something about the field of vertical motion if we know something about the horizontal divergence.
DISCUSSION QUESTION:
Give three reasons why knowing something about Vertical Velocity is very important in meteorology?
3. The Principle of Dine's Compensation
Now, observations show that the vertical velocity at the Tropopause and at
the ground is nearly zero. Take a look at the picture on the next page. Let's use the Equation of Continuity to say something about the midtropospheric vertical velocity field.
L
Let's say that somehow you have calculated the DIVERGENCE in the layer from 500 mb to the Tropopause at 200 mb to be 1.5 X 10-5 sec -1.
You need to rewrite equation (3c) to solve for the vertical velocity at 500 mb. To do this you need to expand the term for the vertical divergence.
Now, solve ALGEBRAICALLY for the vertical velocity at 500 mb.
DISCUSSION QUESTION
What is the vertical velocity at 500 mb for the above example?
Using equation (3c) in a similar manner, what is the horizontal divergence at the ground for the above example?
You have conceptually developed one of the great principles applied in weather analysis and forecasting: DINE'S COMPENSATION.
DINE'S COMPENSATION
Upper tropospheric divergence tends to be "balanced" by mid-tropospheric upward vertical motion and lower tropospheric convergence.
Upper tropospheric convergence tends to be "balanced" by mid-tropospheric downward vertical motion (subsidence) and lower tropospheric divergence.
DISCUSSION QUESTION
What sort of weather pheonmena (clouds, fair, precipitation etc.) does each "half" of the principle of Dine's Compensation imply?
B. The Pressure Tendency Equation
The Equation of Continuity is
(1)
The hydrostatic equation is
(2)
Integrate (2) from a level z1 where the pressure is p1=p to the top of the atmosphere at level z2 where p2=0.
(3)
Equation (3) states that the pressure at any level is directly proportional to the density of the atmospheric layer of thickness dz or, in other words, the weight of the slab of atmosphere of thickness dz. If level 1 is sealevel and level 2 is the top of the atmosphere, then equation (3) simply states that sealevel pressure is really a function of the total weight of the atmospheric column.
The local pressure tendency can be determined from (3) for a given air column with dz thickness (dz is treated as constant) and is given as:
(4)
Substitute (1) into (4) to obtain the relationship between pressure tendency at a given level to the mass divergence with respect to and the mass advection in/out of the air column.
(5)
LocalMassMass
Pres due to In/Out andDivergence Or
TendAdvectionConvergence
At the synoptic scale, the advection of mass is small and may be dropped from this equation on an order of magnitude basis. (Note that this term is NOT small near fronts, outflow boundaries, sea-breeze fronts and other mesoscale features and MUST be retained in understanding pressure development with respect to those features.)
Thus, for synoptic scale flow, Equation (5) reduces to
(6)
Expanding the three dimensional divergence in (6) gives
(7)
which states that the pressure tendency at the base of an air column is a function of horizontal mass divergence into/out of the air column AND the vertical mass transport through the top and bottom of the air column.
The term represents difference in vertical wind through the air column defined by the depth dz. As such, the finite differencing yields
(8)
If the top of the air column is taken as always at the top of the atmosphere (or troposphere), then w2 = 0 which further simplifies the expression when substituted into (8) and if (8) is substituted into (7).
(9)
Equation (9) is the general pressure tendency equation. This states that the change in pressure observed at the bottom of a slab of air of thickness dz is due to the net horizontal divergence in/out of the slab modified by the amount of mass being brought in through the bottom of the slab. In this case, the top of the slab is also the top of the atmosphere. See Fig. 1.
Figure 1
To make this equation "relevant" to the surface weather map, remember that at the ground, w1=0. Hence, the SURFACE PRESSURE TENDENCY EQUATION is:
(10)
where is the NET HORIZONTAL DIVERGENCE[2] in the air column from the ground to the top of the atmosphere and can be approximated by the product of the NET DIVERGENCE (obtained by summing all the horizontal divergences through the layer) and the thickness of the layer. Remembering that 90% of the mass of the atmosphere lies beneath the tropopause and that we are treating the density as a mean density for the air column (a constant), then it can be seen that, at a synoptic-scale, surface pressure tendencies are directly related to horizontal divergence patterns in the troposphere.
Equation (10) can be used computationally to obtain qualitative estimates of surface pressure development. Substitution of equation (2) into the right side of (10) eliminates the density and gravity and allows one to compute the surface pressure tendency on the basis of the net divergence through the layer of dp thickness.
1
[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., -u (∆T/∆x)).
[2] Please note that the net divergence is not the mean divergence. The net divergence represents the arithmetic sum of the divergence from each layer from the bottom to the top of the slab considered.