## The Straight Jet Four-quadrant Model

If the centripetal accelerations owing to flow curvature are small, then we can use the "straight" jet streak model. The schematic figure directly below shows a straight jet streak at the base of a trough in the height field. The core of maximum winds defining the jet streak is divided into four quadrants composed of the upstream (entrance) and downstream (exit) regions and the left and right quadrants, which are defined facing downwind.

Isotachs are shaded in blue for a westerly jet streak (single large arrow). Thick red lines denote geopotential height contours. Thick black vectors represent cross-stream (transverse) ageostrophic winds with magnitudes given by arrow length. Vertical cross sections transverse to the flow in the entrance and exit regions of the jet (J) are shown in the bottom panels along A-A' and B-B', respectively. Convergence and divergence at the jet level are denoted by "CON" and "DIV". "COLD" and "WARM" refer to the air masses defined by the green isentropes.

a. The Equation of Motion Argument
It is well known that air moves through wind speed maxima in the upper troposphere. Accordingly, air accelerates within the entrance region as it approaches the core of the jet streak, with the greatest acceleration being along the axis of the jet. Likewise, air decelerates within the exit region as it leaves the core of the jet streak, with the greatest deceleration also being along the axis of the jet. According to the momentum equations, the ageostrophic wind is perpendicular and to the left of the parcel acceleration vector (in the Northern Hemisphere), as shown mathematically:

Therefore, as the top panel of the diagram shows, northward (southward)-directed ageostrophic winds are found in the entrance (exit) region of a westerly jet, with maximum values along the jet axis where the accelerations are greatest. This pattern of ageostrophic winds results in a 4-cell pattern of convergence and divergence.

The same 4 cells are shown in the vertical cross sections taken transverse to the jet axis within the entrance and exit regions, as seen in the bottom panel of the figure. If vertical motions vanish at the tropopause and the jet steak is located near the tropopause level, then Dines' compensation (mass continuity) argues that there must be ascent under regions of upper-level divergence, i.e., within the right entrance and left exit regions of the jet. Likewise, where the convergence overlies divergence, we find descent, namely, within the left entrance and right exit regions of the jet. Therefore, a thermally direct vertical circulation results in the entrance region, which is to say that the warm air rises and the colder air sinks; conversely, a thermally indirect vertical circulation results in the exit region. These systems are referred to as the "transverse ageostrophic circulations".

b. The Quasi-geostrophic Omega Equation Argument
We can also arrive at the same answer for the vertical circulation pattern from the quasi-geostrophic omega equation. For the straight jet streak situation, a single vorticity maximum occurs on the left side of the jet axis (looking downwind), and a vorticity minimum occurs on the right-hand side, as shown below.

Isotachs are shaded in blue for a westerly jet streak (single large arrow). Thick red lines denote vorticity isopleths, with the vorticity maximum (VORT MAX) and minimum (VORT MIN) shown on the left and right sides of the jet core, respectively. Resultant patterns of positive vorticity advection (PVA) and negative vorticity advection (NVA) are also depicted.

This result follows from the definition of vorticity for a simplified two-dimensional wind in which there is no v-component (curvature) - thus, the vorticity is determined by the y-variation of the zonal wind (y being positive to the north here):

Given the westerly geostrophic flow present for the straight jet, it immediately follows that positive vorticity advection (PVA) will be present downstream of the vorticity maximum in the left exit region of the jet, as well as upstream of the vorticity minimum in the right entrance region. Conversely, we find NVA patterns in the left entrance and right exit regions. Since both the vorticity and the zonal geostrophic wind component increase with height between the ground and the level of the jet, then PVA increases with height in the left exit and right entrance regions; similarly, NVA increases with height in the opposite quadrants. Assuming no thermal advection (i.e., the isotherms are parallel to the winds), then the traditional form of the quasi-geostrophic omega equation

states that rising motion will occur in the two quadrants in which there is a vertical increase of PVA and subsidence will be present in the opposite two quadrants. Clearly, this result is in the same sense that we found from the momentum argument presented earlier.

c. The Energy Conservation Argument
Namias and Clapp (1949) presented a third argument based on the principle of energy conservation for the 4-cell pattern of vertical motions surrounding straight jet streaks. Now, the acceleration of flow in the entrance region results in an increase in the parcel kinetic energy at the expense of its potential energy (which decreases as air flows "down the hill" toward lower heights). Conversely, in the exit region, the parcel gains potential energy as it flows "up the hill" toward higher heights and loses kinetic energy. The kinetic plus potential energy (K+P ) is conserved in the process.

The transverse circulations associated with a straight upper-level jet streak also couple with the winds at low levels to complete the circulations. Uccellini and Johnson (1979) showed that pressure will fall (rise) underneath the rising (sinking) branch of the circulation. This pressure change couplet induces an isallobaric ageostrophic wind, which is necessarily perpendicular to the flow in the jet streak, as depicted in the cross sections.

Before leaving this section, it is noteworthy that coupling between the transverse circulations of the polar, subtropical, and low-level jets can create some very interesting effects on the generation or suppression of deep convection. For example, when the right entrance region of the polar jet is juxtaposed with the left exit region of the subtropical jet, the ascent is strongly enhanced (Uccellini and Kocin 1987). One can also imagine that when the rising branch of the thermally indirect circulation associated with the exit region of an upper-level jet streak becomes situated precisely over the rising branch of the thermally direct circulation associated with a low-level front, focused ascent conducive to convection will occur. These topics are not treated further here.

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