**The Continuity
Equation: Dines Compensation **

**and the
Pressure Tendency Equation**

**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 Air are
Density Velocity

Column Fixed due
and/or

With Respect to to Advection Divergence

the Earth

The Equation of Continuity therefore is:

(1)

where DIV_{3d} 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)

Local Mass Mass

Pres due
to In/Out
and Divergence Or

Tend Advection Convergence

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]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 (ÆT/Æx)).

[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.