Figure 3.1 Schematic illustration contrasting deep and shallow atmospheric circulations. The depth corresponds to the vertical extent of the circulation. In the Earth’s atmosphere, .

3.1.1 Deep Continuity Equation

To examine the necessity for retaining individual terms in Eq. 3.5, it is customary to examine their ratio relative to one of the terms that is expected to remain. In the first case to be examined, the terms are divided by the order of magnitude estimate for , resulting in

Since , the terms , and are much less than , and and can be neglected in Eq. 3.5, provided

(3.7)

Then Eq. 3.5 can be written as

(3.8)

Since is approximately equal to , conditions (ii) and (iii) in Eq. 3.7 require that

(3.9)

If therefore, is one or two orders of magnitude larger than , it is expected that the vertical velocity should be one or two orders of magnitude less than and (e.g., for , then ). This relation between velocities and the horizontal and vertical scales of atmospheric circulations results from the condition that . Because of this constraint, velocities in a longer, horizontal leg of an atmospheric circulation must be proportionately stronger to preserve mass continuity without creating large fluctuations in specific volume.

The last requirement that needs to be justified in Eq. 3.7 is condition (iv). The scaled variable represents a time period in which significant variations in specific volume occur on the mesoscale and can be approximated by

(3.10)

where is the wavelength over which variations occur and the rate of movement of these variations. If the changes are caused by advection, then , or is used to represent , whereas a characteristic group velocity will be utilized if wave propagation is dominant. The wavelength is estimated as and for horizontal motion and for vertical movement. When changes in specific volume are assumed primarily to be caused by advection or when the wave group velocity has approximately the same speed as the wind velocity,3 then

where conditions (i)-(iii) in Eq. 3.7 have been used. Therefore,

so that local variations in density can be neglected in the conservation of mass relation if .

Finally, if , mesoscale variations in the direction are expected to be dominant and the derivatives in Eq. 3.8 can be ignored. In this case the equation reduces to the two-dimensional form given by

Equation 3.8 is customarily written to include the terms and and is given by

or, returning to the use of density instead of specific volume,

(3.11)

where .

Dutton and Fichtl (1969) refer to this relation as the deep convection continuity equation because the vertical depth of the circulation is on the same order as the density scale depth. As originally shown by Ogura and Phillips (1962), and discussed by Lipps and Hemler (1982), and as will be shown in Section 5.2.2, the use of this form of the conservation-of-mass relation eliminates sound waves as a possible solution, which led Ogura and Phillips to refer to Eq. 3.11 as the anelastic, or soundproof assumption. Because such waves are of no direct interest in most applications of mesoscale meteorology, this equation is often employed to represent mass conservation in mesoscale models in lieu of the more complete prognostic conservation-of-mass equation given by Eq. 3.1. Moreover, as discussed in Section 9.3, the elimination of sound waves permits more economical use of certain numerical solution techniques because their computational stability is dependent on having a time step less than or equal to the time it takes a wave to travel between grid points. Sound waves are the fastest nonelectromagnetic waveform in the atmosphere.

3.1.2 Shallow Continuity Equation

A more restrictive mass-conservation relation is derived by dividing the scaled terms in Eq. 3.6 by the scaled form of resulting in

As in the previous analysis, since , and can be neglected in Eq. 3.5, provided

(3.12)

then Eq. 3.5 can be written as

(3.13)

The first condition in Eq. 3.12 is easier to satisfy than the equivalent requirement in Eq. 3.7 because . This requirement implies that the neglect of specific volume variations in Eq. 3.13 is even less important than in Eq. 3.11. The second condition in Eq. 3.12 requires that the depth of the circulation be much less than the scale depth of the atmosphere. For this reason Dutton and Fichtl refer to Eq. 3.13 as the shallow convection continuity equation. Written in tensor...