A theory is presented for the vertical motion induced by LLJs over the gently sloping terrain of the Great Plains. The theory does not address CI per se, but makes a case for how LLJs associated with spatially varying geostrophic winds (a special case of east-west variation of a southerly geostrophic wind is considered) can force broad zones of mesoscale ascent at the top of the atmospheric boundary layer (ABL), thus creating a favorable environment for elevated CI. The basic framework of the theory can also be applied to study the impact of spatial variations of other parameters (e.g., surface buoyancy, surface drag coefficient, and surface heat exchange coefficient) on LLJ-associated mesoscale ascent. By seeking the vertical motion field at the top of the ABL as an integral of the horizontal divergence over the ABL, we need to contend only with layer-mean estimates of the wind and buoyancy fields. The mechanism we explore is based on the conflation of two hypotheses concerning primarily southerly LLJ phenomena over the Great Plains during the warm season:

(i) The LLJ arises as an inertia-gravity oscillation originating from the sudden relaxation of the frictional constraint in the ABL around sunset [mechanism proposed by Blackadar (1957), but augmented with a buoyancy effect arising from diurnal heating of the slope, as suggested by Holton (1967)]. Simple analytical solutions for the wind and buoyancy in such an oscillation were derived in Shapiro and Fedorovich (2009). Integrating the divergence of the wind field provided by these solutions from the slope up to the top of the ABL yields an expression for the evolution of the vertical velocity above the ABL. Integrating the latter equation in time then yields the vertical displacement of an air parcel originating (at sunset) above the ABL. In this approach, spatial variations in the geostrophic wind and other governing parameters drive LLJs with spatially variable strengths; these variations are accounted for through initial conditions. The approach is analogous to the estimation of vertical motion via Ekman pumping resulting from the aggregate of one-dimensional (1D) Ekman spiraling motions around a low pressure system, where each spiral is associated with a different geostrophic wind vector aloft. In our case, though, the building block is the (unsteady) solution for a 1D inertia-gravity oscillation rather than the (steady) 1D solution for wind in the Ekman layer.

(ii) Mixed-layer-mean estimates of late afternoon wind and buoyancy in the sloping dry convective ABL are used as initial conditions for the layer-mean flow in the solutions for the winds associated with the inertia-gravity oscillations described above. These initial conditions are derived from a new zero-order model for a quasi-stationary late-afternoon convectively mixed layer above a heated slope. The model is based on a mixed-layer analysis of the layer-mean equations of motion and thermal energy in the (quasi) steady state. Among the governing parameters are the slope buoyancy, geostrophic wind at the layer top, slope angle, Coriolis parameter, free-atmosphere Brunt-Vaisala frequency, boundary layer depth, surface drag coefficient, surface heat transfer coefficient, and an entrainment parameter.

Preliminary results are obtained using imposed southerly geostrophic winds at the ABL top in the form of a lateral jet, that is, geostrophic winds that increase in intensity toward the east, attain a peak magnitude, and then decrease in intensity toward the east. To the west of the geostrophic wind maximum, the first half of the inertia-gravity oscillation is associated with ascent, while the second half is associated with descent. The timing of the ascent/descent is reversed to the east of the geostrophic wind maximum. Solution sensitivity is explored with respect to the main input parameters of the zero-order model. Vertical parcel displacements in the cases of strong geostrophic shear can be as large as 500 m to 1 km over the course of the ascent phase of the oscillation.