P18.5

*Ming Xue, **Qin Xu and †**Kelvin K. Droegemeier

*Center for Analysis and Prediction of Storms

**Cooperative Institute for Mesoscale Meteorological Studies

†School of Meteorology

University of Oklahoma, Norman, Oklahoma 73019

1. Introduction

It is commonly accepted that the interaction between the environmental shear and the cold pool of a thunderstorm outflow (density current) may play an important role in producing long-lived squall lines (e.g. Thorpe, at al. 1982, TMM82; Rotunno et al. 1988, RKW88). To improve our understanding of the interaction between density currents and their environment, simple nonlinear, two-fluid steady-state models were recently developed by Xu (1992, X92) and Xu and Moncrieff (1994, XM94). Compared to the classic density current theory of Benjamin (1968), these models include, among other new ingredients, the effect of environmental shear. In Xu et al. (1996, XXD96), time-dependent numerical simulations were performed to validate the theoretical solutions of X92 and XM94. The numerical model was able to reproduce with high accuracy quasi-steady-state density currents in uniform flow and constant shear.

In this paper, we extend the theoretical models of X92 to a case where the inflow shear is confined to the lower part of a vertically bounded channel. The inflow above the shear layer is uniform. This flow configuration is illustrated in Fig. 1 and is of more relevance to meteorological applications since most long-lived convective systems occur in environments with most of the shear confined to the lowest a few kilometers of the tropopause (e.g. Bluestein and Jain, 1985).

In the following sections, we present first the theoretical model, then describe the design and results of numerical experiments and finally some conclusions in the last section.

2. The theoretical model

Similar to X92 and XM94, we examine an idealized steady-state model of density current moving at a constant speed in an environmental flow with low-level shear. (Fig.1). The variables of this model are non-dimensionalized using the following scaling:

In a framework moving with the density current front, the steady-state flow in Fig.1 can be fully described by a set of six non-dimensional parameters:

According to Fig.1, the inflow profile is given by

and the outflow profile is given by

The mass continuity requires that the inflow mass flux be equal to the outflow mass flux for both shear and zero-shear layers. From these two conditions, we can obtain the following two constraints:

and (2.6)

Furthermore, for a steady-state flow, the Bernoulli energy () is constant along a streamline. Apply this theorem along the interfacial streamline from the frontal nose B to the downstream point D, we obtain . Inside the cold pool, the flow speed is zero, therefore . Integrating the hydrostatic relation for the denser fluid inside the cold pool gives . Equating and results in

which is another constraint among the control parameters.

By integrating the steady-state horizontal momentum equation over the entire domain and making use of the mass continuity, one can obtain the flow force balance (see Benjamin 1968):

Substituting velocity and pressure profiles into above, we obtain after certain algebra the fourth constraint:

3. The numerical experiments

The Advanced Regional Prediction System (Xue et al. 1995) in used in its simplest 2-D setting with minimum physics. A number of approximations, including Boussinesq approximation, are made to the model equations, which are then non-dimensionalized.

The initial flow at the far upstream and downstream boundaries are specified according to Eq.(2.3). The interior flow is obtained numerically by solving a Possion equation for streamfunction based on mass and vorticity conservation. A similar procedure is used in XXD96.

A set of six experiments were conducted and their initial and simulated parameter values are listed in Table 1. Limited by space we will only discuss results from selected experiments.

In fact, between T=12 and 18, the frontal nose in LS1 only retreated by 0.075, yielding a flow relative propagation speed of 0.40, corresponding to a mere 3% error compared to the theoretical value (see Table 1). Measured by the -0.5 buoyancy contours in the time averaged field (Fig.3), the height of the simulated current head is about 0.538, only slightly shallower than the theoretical value of 0.59. It is worth noting that, due to Kelvin-Helmholtz eddies, the time averaged density current shows a noticeable decrease in its depth to the rear of its head. This resembles the hydraulic jump during which the flow transitions from a supercritical to a subcritical state. This process was discussed in X92. Note also that the slope of the front at the nose is very close to 60Ú.

A similar agreement with the theory is found for LS2. Between T=12 and 18, the frontal nose remained exactly stationary. The simulated head height is 0.325 (Table 1) versus the theoretical value of 0.410. The accuracy in the depth prediction is not as good as the positive shear (deeper density current) case. A similar behavior is found in XXD96 for uniform shear cases.

Indeed, the simulated density current has a interface slope of more than 60° at the mid-levels (Fig.4). The depth of the current head and its propagation agree very well with the theoretical results (Table 1).

The flow at the density current head is highly unsteady. However, the time fields still exhibit an well organized flow pattern. The streamline field (not shown) depicts a predominant jump-type flow, and the buoyancy field (Fig.5) shows an envelope of the denser fluid extending to a height that is close to the theoretical head depth of 0.767. The shape of this envelope is not very different from that of the density current head in SLS. The results suggest that the basic dynamic control (pressure balance) on the flow interface is still working, despite the instantaneous flow is highly transient. This further confirms the role of the shear in the low-level inflow; positive shear in the low-level inflow does possess a power of strong vertical lifting at the frontal nose and stronger positive shear supports a deeper jump type or even an overturning flow.

4. Summary and Conclusions

The earlier two-fluid idealized model of density current models have been extended to include non-constant vertical shear. It is found that the shear confined to the low levels plays a similar role as the uniform vertical shear in controlling the depth of density currents. Time-dependent numerical experiments are conducted that supports the theoretical model solution and confirms the role of low-level inflow shear in maintaining a deep convective updraft. More details will be reported in a journal article.

Acknowledgments. This research was supported by NOAA Grant NA37RJ0203 and NSF Grant ATM-9417304 to CIMMS, by NSF Grant ATM91-20009 to CAPS, and by NSF Grant ATM92-22576 to the third author.

REFERENCES