a theoretical and numerical study of density currents in nonconstant shear...

1998 VOLUME 54J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S

q 1997 American Meteorological Society

A Theoretical and Numerical Study of Density Currents in Nonconstant Shear Flows

MING XUE

Center for Analysis and Prediction of Storms, University of Oklahoma, Norman, Oklahoma

QIN XU

Cooperative Institute for Mesoscale Meteorological Studies, University of Oklahoma, Norman, Oklahoma

KELVIN K. DROEGEMEIER

School of Meteorology and Center for Analysis and Prediction of Storms, University of Oklahoma, Norman, Oklahoma

(Manuscript received 14 June 1996, in final form 18 February 1997)

ABSTRACT

The previous idealized two-fluid model of a density current in constant shear is extended to the case wherethe inflow shear is confined to the low levels. The analytical solution is determined by the conservation of mass,momentum, vorticity, and energy. It is found that a low-level shear acts in a similar manner to a uniform verticalshear in controlling the depth of a steady-state density current. When the shear enhances the low-level flowagainst the density current propagation, the current is deeper than half of the domain depth. Time-dependentnumerical experiments are conducted for a variety of parameter settings, including various depths and strengthsof the shear layer. The numerical results agree closely with the theoretical analyses.

Numerical experiments are also performed for a case where the initial depth of the density current is set tobe comparable to the low-level shear, which is much shallower than that given by the steady-state solution. Thecirculation at the density current head remains shallow and is nonsteady in this case, whereas the time-averagedflow still exhibits a deep jump updraft pattern that is close to the theoretical solution, suggesting the applicabilityof the theoretical results to even more transient flows.

The simulated flow features are discussed in terms of balanced and unbalanced dynamics, and in the contextof forcing and uplifting at the frontal zone in long-lived convective systems. Here the term balance refers to aflow configuration that satisfies the steady-state solution of the idealized theoretical model.

1. Introduction

It is commonly accepted that the interaction betweenthe environmental shear and the cold pool of a thun-derstorm outflow (density current) may play an impor-tant role in producing long-lived squall lines (Moncrieff1978, 1992; Thorpe et al. 1982, hereafter TMM82; Xuand Chang 1987; Rotunno et al. 1988, hereafterRKW88; Fovell and Ogura 1988). Density currents alsooccur in the atmosphere in other forms, such as sea-breeze fronts (e.g., Simpson et al. 1977) and those as-sociated with cold frontal rainbands (e.g., Carbone1982). To improve our understanding of the interactionbetween density currents and their environment, simplenonlinear, two-fluid steady-state models were recentlydeveloped by Xu (1992, hereafter X92), Xu and Mon-crieff (1994, hereafter XM94), and Liu and Moncrieff

Corresponding author address: Dr. Ming Xue, CAPS, Universityof Oklahoma, Sarkeys Energy Center, Suite 1110, 100 East Boyd,Norman, OK 73019.E-mail: [email protected].

(1996a). Compared to the classic density current theoryof Benjamin (1968), the new ingredients in these modelsinclude: (i) the environmental shear, (ii) the internal coldpool circulation [based on earlier work by Moncrieffand So (1989)], (iii) negative vorticity generation as-sociated with energy loss along the interfacial layer be-tween the density current and its environment, and (iv)density stratification and latent heating. As archetypesof the physically more complex system characteristic ofquasi-two-dimensional organized convection, thesemodels allow for closed mathematical analyses showinghow the depth, propagation speed, and shape of thedensity current are controlled or influenced by the en-vironmental shear and cold pool strength.

Although very useful for improving our physical un-derstanding, the idealized solutions have their limita-tions. Laboratory experiments (e.g., Simpson 1969), ob-servations (e.g., Wakimoto 1982; Mueller and Carbone1987), and numerical simulations (Droegemeier 1985;Droegemeier and Wilhelmson 1987) of density currentsall indicate the development of vigorous Kelvin–Helm-holtz (KH) waves and strong turbulent mixing along the

1 AUGUST 1997 1999X U E E T A L .

FIG. 1. Schematic of the steady-state model of a density current circulation in an environmental flow with low-level shear. The remote system-relative inflow and outflow are indicated by u`(z) and u2`(z), respectively, and his the depth of density current. Other variables are defined in the text.

FIG. 2. The cold pool depth h plotted against inflow shear a fordifferent values of shear depth d0, according the steady-state theo-retical model.

layer between the denser and lighter fluids/air (we willuse fluid and air interchangeably, as well as density andbuoyancy). The effects of these transient features andrelated turbulence are not fully considered by the the-oretical models, and the extent to which the analyticsolutions are valid in their presence is unclear.

In Xu et al. (1996, hereafter XXD96), time-dependentnumerical simulations were performed to validate thetheoretical solutions of X92 and XM94. The numericalmodel was able to reproduce quasi-steady-state densitycurrents in uniform flow and constant shear. The prop-agation speed, depth, and gross shape of the simulateddensity current head agreed closely with the theoreticalresults. The dependence of the density current depth onthe environmental shear was also examined numericallyby Chen (1995), with a different setup of numerical

experiments than that in XXD96, and by Liu and Mon-crieff (1996b), with the effect of ambient flow included.Their results support the theoretical results of X92,XM94, and Liu and Moncrieff (1996a) in general.

In this paper, we extend the theoretical model of den-sity current in a constant shear flow given in X92 to acase where the inflow shear is confined to the lowerpart of a vertically bounded channel (Fig. 1). The inflowabove the shear layer is uniform. Such a configurationappears to be more relevant to meteorological appli-cations since most long-lived convective systems suchas squall lines occur in environments with most of theenvironmental shear confined to the lowest few kilo-meters of the troposphere (e.g., TMM82; RKW88; Blue-stein and Jain 1985). The top lid in our model may actlike the tropopause or a strong inversion layer.

In the following section, we present first a theoreticalmodel of density currents in a nonconstant shear. Thefar-field solutions away from the density current headare obtained by applying the conservation laws of mass,momentum, vorticity, and energy in a similar mannerto the way the solutions for constant shear case wereobtained in X92 and XM94. In sections 3 and 4, wedescribe the design and results of numerical experi-ments. In section 5, the numerical results are furtherdiscussed in the context of conservation principles andtheir relevance to long-lived convective systems. Fi-nally, conclusions are drawn in section 6.

2. Density current in a low-level shear flow

a. Flow configuration and scaling

In the idealized steady-state density current modelsof X92 and XM94, the environmental shear flow is con-stant throughout the depth of a two-dimensional verticalchannel and the density current front moves at a constantspeed into the sheared upstream inflow (see Fig. 2 ofX92). Here we include an additional degree of freedomby allowing the inflow shear to decrease from a constantvalue at the low levels to zero above a depth d0 (0 #


d0 # 1, where 1 is the nondimensional domain depth)(Fig. 1). The independent and dependent variables inthis model are nondimensionalized using the scaling

(x, z) ← (x, z)/H, (u, w) ← (u, w)/U,2p9 ← p9/(r U ), (2.1)0

where the variables on the left-hand side of the arroware dimensional.

In (2.1), x and z are the horizontal and vertical co-ordinates; u and w are the horizontal and vertical ve-locities, respectively; H is the depth of the domainbounded by two rigid boundaries; U [ (gHDr/r0)1/2 isthe velocity scale; g is the acceleration of gravity; Dr[ r1 2 r0 is the density difference between the denserfluid inside the cold pool (r1) and the lighter fluid out-side (r0); and p9 [ P 2 P0 is the perturbation pressure,which is the difference between total pressure P andreference pressure P0. The reference pressure P0 is theunperturbed pressure in the upstream inflow associatedwith constant density r0. Here r0 satisfies the hydrostaticrelation P0 5 gr0(H 2 z). The perturbation pressure p9is nonzero near, above, and inside the cold pool; it rep-resents the sum of the purely dynamic pressure pertur-bation due to the Bernoulli effect and the hydrostaticpressure perturbation due to the deviation of densityinside the cold pool from that of reference state.

In the following sections, steady-state far-field solu-tions are sought for this flow configuration. The resultsshow how the propagation speed and cold pool depthdepend on the inflow shear. The local structure of thefront is also briefly discussed.

b. Far-field solution

In Fig. 1, the shaded area indicates the cold pool ofan idealized density current. There exists an interfacebetween the two fluids that is assumed to be of infinitelysmall thickness. As was shown in XXD96 and by thenumerical results to be presented in this paper, this as-sumption does not affect the qualitative validity of thetheoretical solution. Viewed in a framework movingwith the density current front, the environmental flowmoves towards the head from the right. The circulationinside the cold pool is neglected in our case. Accordingto XM94, this feature usually has a much weaker effecton the density current than the shear in the environ-mental flow.

The flow illustrated in Fig. 1 can be fully describedby a set of six nondimensional parameters:

S5 5 {a, d0, c0, d1, c1, h}, (2.2)

where a is the (constant) low-level shear (vorticity) ofthe upstream environmental inflow, d0 (0 # d0 # 1) thedepth of this shear layer, and c0 (.0) the constant speedof inflow above the shear layer. Parameter d1 is the depthof the shear layer immediately above the cold pool, andc1 (.0) is the speed of outflow at the cold pool top.

Finally, h (0 , h , 1) is the depth of the density currenthead. According to vorticity conservation, the vorticityin the inflow shear layer and the shear layer immediatelyabove the cold pool should be the same; therefore, thereis no need for another shear parameter for the outflow,and vorticity conservation is automatically satisfied.

Following Benjamin (1968), the remote system-rel-ative environmental inflow and outflow are constrainedby mass continuity, energy conservation, and flow forcebalance. It can be shown that only two of the six pa-rameters in (2.2) are independent, and physical solutionscan be found given any two of these parameters. In thispaper, we will discuss only the cases where the depthand vorticity of the inflow shear are prescribed.

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

2c for z $ d0 0u (z) 5 (2.3)` 52c 1 a(z 2 d ) for z , d0 0 0

or by

2c (z 2 d ) for z $ d0 0 0c (z) 5` 252c (z 2 d ) 1 a(z 2 d ) /2 for z , d ,0 0 0 0

(2.4)

where c is the streamfunction defined by u [ ]c/]z andis obtained by integrating (2.3) with respect to z. Weassume the streamline separating the shear and zero-shear layers to be zero, that is, c`(d0) 5 0 at the inflowboundary and c2` (h 1 d1) 5 0 at the outflow boundary.

The outflow profile is given by

2c 1 ad for z $ h 1 d1 1 1u (z) 5 (2.5)2` 52c 1 a(z 2 h) for z , h 1 d ,1 1

or by streamfunction

c (z)2`

2(c 1 ad )(z 2 h 2 d )1 1 1

for z $ h 1 d1522(c 1 ad )(z 2 h 2 d ) 1 a(z 2 h 2 d ) /21 1 1 15

for z , h 1 d .1

(2.6)

Mass continuity requires that the inflow mass flux beequal to the outflow mass flux, for both shear and zero-shear layers. Given that cmid 5 c2`(h 1 d1) 5 c`(d0)5 0, where cmid defines the streamline separating theshear and zero-shear flows, we have

ctop 5 c2`(1) 5 c`(1) (2.7)

and

cbot 5 c2`(h) 5 c`(0). (2.8)

Substituting (2.4) and (2.6) into (2.7) and (2.8) yieldstwo relations between the nondimensional parameters:

c0(1 2 d0) 5 (c1 2 ad1)(1 2 h 2 d1) (2.9)

1 AUGUST 1997 2001X U E E T A L .

and

d0(c0 1 ad0/2) 5 d1(c1 2 ad1/2). (2.10)

Furthermore, for a steady-state flow, the Bernoullienergy (p9 1 u2/2) is constant along a streamline. Ap-plying this theorem along the interfacial streamline fromthe frontal nose B to the downstream point D and notingthat the flow speed is zero at B, we obtain

5 1 /2.2p9 p9 cB D 1 (2.11)

Inside the cold pool, the flow speed is zero; therefore,5 . Integrating the hydrostatic relation,p9 p9B C

]p9/]z 5 21, (2.12)

for the denser fluid inside the cold pool, we have

5 1 h.p9 p9C D (2.13)

Equating in (2.11) and (2.13) results inp9 p9B C

5 2h,2c1 (2.14)

which is another constraint among the control param-eters.

Integrating the steady-state horizontal momentumequation over the entire domain and making use of masscontinuity, one can obtain the flow force balance [pres-sure-momentum force integral, see Benjamin (1968)]:

1 1

2 2(p9 1 u ) dz 5 (p9 1 u ) dz. (2.15)E ` ` E 2` 2`

0 0

By applying the Bernoulli theorem again along the low-er boundary streamline through the upstream point Aand the frontal nose B, one can obtain

5 1 (c0 1 ad0)2/2 5 (c0 1 ad0)2/2, (2.16)p9 p9B A

where we have, without loss of generality, assumed theperturbation pressure at the upstream point A to be zero( 5 0). Given that Dr` 5 0, we have, according top9Athe hydrostatic relation ]p9/]z 5 0:

(z) 5 5 0.p9 p9` A (2.17)

On the outflow boundary, p9 is again obtained by in-tegrating vertically the hydrostatic relation, making useof (2.16):

2(c 1 ad ) /2 2 z for 0 # z # h0 0p9 (z) 5 (2.18)2` 25(c 1 ad ) /2 2 h for h , z # 1.0 0

Substituting velocity profiles in (2.3), (2.5) and pressureprofiles in (2.17)–(2.18) into (2.15), we obtain the flowforce balance constraint among the parameters:

2 2 2 3c (1 2 h) 1 c ad 1 a d /32 2 1 1

2 21 (c 1 ad ) /2 1 h /2 2 h0 1

2 2 2 35 c 1 c ad 1 a d /3. (2.19)0 0 0 0

Now that we have four constraints given in (2.9), (2.10),(2.14), and (2.19) for six control parameters, only two

of them are free parameters. Note that c0 can be elim-inated by performing d0 (2.9) 2 (1 2 d0) (2.10), so that

2 2ad (1 2 d )/2 5 (1 2 d )(c d 2 ad /2)0 0 0 1 1 1

2d (c 2 ad )(1 2 h 2 d ),0 1 1 1

which can be written as a quadratic equation for d1:

A2 1 A1d1 1 A0 5 0,2d1 (2.20a)

where

A [ a(1 1 d )/2, (2.20b)2 0

A [ 2ad (1 2 h) 2 c , (2.20c)1 0 1

2A [ ad (1 2 d )/2 1 c d (1 2 h). (2.20d)0 0 0 1 0

We seek to find sets of values of the six parameters thatsatisfy all four constraints. We choose a and d0, that is,the vorticity and depth of the low-level shear layer inthe inflow as the independent external control param-eters, and then try to find the solutions of the other fournumerically. The procedure is as follows.

For given values of a and d0 (a . 0, 1 . d0 . 0),we solve for (c0, c1, h, d1):

1) Loop over all physical values of h (1 . h . 0);2) Compute c1 from (2.14);3) Solve for d1 from (2.20a–d),

d16 5 20.5 A1/A2 6 0.5( 2 4A2A0)1/2/A2, (2.21)2A1

and keep only the root(s) in the range of 1 2 h $d1 $ 0;

4) Compute c0 from (2.9) (for d0 , 1) or (2.10) (for d0

. 0); and5) Check to see if (2.19) is satisfied.

Only one set of physical solutions that satisfies theabove four constraints is found using the procedure giv-en above. This set corresponds to the second root of d1

(d12) in (2.21) and reduces to that of Benjamin (1968)in the limit of constant inflow and to that of X92 in thelimit of constant inflow shear. In Fig. 2, the cold pooldepth h is plotted and in Fig. 3 the system-relative inflowspeed at the ground level (or the propagation speed ofdensity current relative to inflow) c0 1 ad0 is plottedagainst the low-level inflow shear a for different valuesof shear depth d0. It can be seen that for d0 5 1, a specialcase of constant inflow shear, the solution reduces tothat of X92 (see Fig. 2 of X92). When the inflow shearis zero (a 5 0), the model degenerates to the classicdensity current solution (h 5 0.5, c0 1 ad0 5 0.5) ofBenjamin (1968).

From Figs. 2–3, it is clear that the density currentbecomes deeper and propagates faster relative to thesurface environmental flow as the inflow shear increasesfrom negative through positive values. The positiveshear raises the density current head above the 0.5 valueof the constant flow case while the negative shear con-tributes in the opposite direction. Correspondingly, thedeeper density current propagates faster than the shal-


FIG. 3. The system-relative inflow speed at the ground level or thedensity current propagation speed c0 1 ad0 plotted against inflowshear a for different values of shear depth d0, according to the steady-state theoretical model.

FIG. 4. The system-relative inflow speed at the top boundary orthe speed of the uniform inflow at upper levels c0 plotted againstinflow shear a for different values of shear depth d0, according tothe steady-state theoretical model.

lower ones. The effect of shear is more dramatic for adeep inflow shear and the solution tends to the limit ofconstant shear (X92) when d0 approaches 1 for smallvalues of a. For example, for a 5 1, a d0 5 0.1 deepinflow shear raises the density current head to 0.547 andgives a propagation speed of 0.546, while a shear ofdepth d0 5 0.9 supports a solution with h 5 0.715 andc0 1 ad0 5 0.752.

It is worth noting that in Figs. 2 and 3, the solutioncurves terminate in the middle of the figure for largepositive values of a and relatively large values of d0,except for d0 5 1. This means that a solution satisfyingall constraints discussed earlier cannot be found for cer-tain combinations of a and d0. It is clear from Fig. 4that the curves for d0 , 1 all terminate as c0 → 0 andthus h 1 d1 → 1 (not shown), indicating that no solutioncan be found for c0 , 0 unless a discontinuity is allowedat z 5 h 1 d1 for the outflow profile in (2.5). As shownin Fig. 1, 2c0 is the upper-level flow velocity on theright boundary and 2(c1 2 ad1) is the upper-level flowvelocity on the left boundary. According to (2.9), thesetwo velocities have the same sign because (1 2 d0) and(1 2 h 2 d1) are both positive. Since (1 2 h 2 d1) ismuch smaller than (1 2 d0), the absolute value of theflow speed zc1 2 ad1z at the left boundary is much largerthan the absolute value of the flow speed zc0z at the rightboundary. If the upper-level flow reverses (c0 , 0), thenthe vertical profile of the horizontal velocity will be-come discontinuous on the left boundary between theupper-level zero shear flow (from left to right) and theshear flow below (from right to left). Since such a dis-continuity is not realistic for atmospheric flows, we willnot pursue it further here.

Without velocity discontinuity, upper-level flow re-versal at the right boundary is possible only when theupper level flow has nonzero shear. It should be possibleto obtain a steady-state far-field solution for a flow con-taining two shear layers, and the solution should liebetween the solutions from the constant shear case ofX92 and the low-level shear case discussed herein. Evenfor this more general case, we will find based on thesame mass continuity and vorticity conservation argu-ment used above that the flow at the bottom of the upperlayer cannot be directed outward (from left to right);that is, u`(d0) cannot be positive in our case. This sameconclusion is true of the constant upper-level flow casediscussed in this paper. We can draw a general conclu-sion that is applicable for both cases: to obtain a steady-state solution in which the velocity field is continuousbetween the two shear layers, the inviscid density cur-rent must travel at least as fast as the flow at some levelin the upper layer.

Further physical understanding of these solutions canbe gained by analyzing the behavior of flow force bal-ance under the conditions of mass, vorticity, and energyconservation. Effectively, steps 1 to 4 of the solutionprocedure provide solutions that satisfy all conservationconditions except for the flow force balance. With theincrease of h, several components of the flow force ad-just themselves so that a balance is reached for a givenh. The response of the components of flow force to thechange in h and other parameters can be examined ina similar manner to X92.

1 AUGUST 1997 2003X U E E T A L .

c. Local and global structures of the front

The previous sections discussed the flow solutionsaway from the density front; however, the flow structurenear the front is of equal importance. In the case of athunderstorm outflow, the shape of the front has a directinfluence on the vertical orientation of low-level up-drafts. An upshear-tilted updraft allows condensed waterto be downloaded underneath the updraft without in-terrupting the low-level inflow, thus long-lived squallline–type convection can be maintained (e.g., TMM82;RKW88).

Von Karman (1940) and Benjamin (1968) showedanalytically that the angle at the stagnation point be-tween the density front and the horizontal is 608 for anidealized inviscid density current in a uniform (non-sheared) environmental inflow with a free-slip lowerboundary condition. It was further shown in X92 andXM94 that this 608 angle is independent of the inflowshear and cold-pool circulation.

Because the cross-interface continuity of pressurerepresents a balance primarily between the lower-layerhydrostatic perturbation pressure and the upper-layerBernoulli pressure, the shape of the interface betweenan inviscid density current and its environment is con-strained by the dynamic condition

D(u2 1 w2) 5 2z along the interface, (2.22)

where D( ) represents the jump of ( ) across the interface.Using this interface condition along with the vorticityequation and boundary conditions, the interface shape,together with the flow fields on the two sides of theinterface, can be solved numerically (Xu et al. 1992).

According to the results of X92 and XM94, when thepositive inflow shear is very strong, the frontal interfaceslope can become steeper than 608 at the middepth (z5 h/2) location. Thus, the middepth interface slope canbe either smaller or larger than 608 depending upon theinflow shear and the intensity of the density currentinternal circulation (see Fig. 9 of XM94). These findingscan be extended to our case of nonconstant verticalshear.

In the next section, we present and discuss the resultsfrom numerical experiments that are designed to vali-date these theoretical results.

3. The numerical model and experiment design

a. The numerical model

The numerical model used in this study is a modifiedversion of the Advanced Regional Prediction System(ARPS) developed at the Center for Analysis and Pre-diction of Storms (CAPS) (Xue et al. 1995a; Xue et al.1995b). The ARPS is a general-purpose, nonhydrostatic,compressible model designed for storm- and mesoscaleatmospheric simulation and real-time numerical weatherprediction (Droegemeier et al. 1996; Xue et al. 1996).The model is designed to run on both conventional and

massively parallel processors (Johnson et al. 1994;Droegemeier et al. 1995). The dynamic framework con-sists of prognostic equations for momentum, potentialtemperature, pressure, water substances, and subgrid-scale turbulent energy. The equations are solved on anArakawa C-grid using a split-explicit time integrationscheme (Klemp and Wilhelmson 1978).

For the purposes of this study, the ARPS is used inits simplest two-dimensional setting with minimumphysics. To facilitate direct comparison of numericalresults with the theoretical solutions, several approxi-mations are made to the original model equations. First,the Boussinesq approximation is assumed by setting thebase-state density r0 to a constant. Second, the effectof compressibility is neglected in the buoyancy termsuch that the thermal buoyancy (gu9/u0) is in its formequivalent to 2gr9/r0. The latter appears in the theo-retical models discussed earlier. The base-state potentialtemperature u0 is also set to a constant, representing aneutrally stable air/fluid outside the cold pool.

The fully compressible pressure equation is also sim-plified by retaining only the local time tendency of pres-sure and the velocity divergence term. The sound wavespeed is assumed to be a constant. To improve modelefficiency, the supercompressibility approximation (e.g.,Droegemeier and Davies-Jones 1987) is made by settingthe sound wave speed cs to a value (150 m s21) less thanthe true speed of sound. The justification and effect ofthe above approximations are discussed in XXD96.

At the top and bottom boundaries, rigid free-slip con-ditions are applied. Wave-radiation conditions similarto those of Klemp and Wilhelmson (1978) are used onthe lateral boundaries. A 1D wave equation [ut 1 (u 1C)ux 5 0] is applied to the normal velocity component.Different from Klemp and Wilhelmson (1978), thisequation is applied within each small time step and theconstant speed C is chosen to represent the reducedsound wave speed. Since the flow is neutrally stable atthe upstream and downstream boundaries and the pri-mary signals there are acoustic waves, such an imple-mentation yields the most transparent boundary con-dition with minimum domain-wide pressure fluctua-tions.

Since we attempt to explicitly simulate turbulent ed-dies, we choose not to use the subgrid-scale turbulenceparameterization available in the model. This is in con-trast to some other studies of density currents (e.g. Par-ker 1996; Klemp et al. 1994) but should not make asignificant difference in the solutions when the spatialresolution is high. In the simulations, only a very weakfourth-order horizontal diffusion is applied to the ve-locity and temperature equations. This highly selectivediffusion ensures that grid-scale noise is effectively con-trolled, while sharp gradients in both the temperature(density) and velocity fields at the density current in-terface are well maintained. Different from XXD96, wedo not include any diffusion in the vertical direction soas to ensure that the upstream inflow profile is preserved


throughout the simulation. This is even more desirablewhen the inflow shear is not constant.

b. Model scaling and initial conditions

The equations actually solved by the model are givenin nondimensional form as

u 5 2(uu 1 wu ) 2 p9 1 K div 1 D , (3.1a)t x z x d x u

w 5 2(uw 1 ww ) 2 p9 1 b 1 K div 1 D , (3.1b)t x z z d z w

2p9 5 2(u 1 w )/M , (3.1c)t x z

b 5 2(ub 1 wb ) 1 D . (3.1d)t x z u

Here u, w, x, z, and p9 have their meanings as definedin the previous section and are all nondimensional. Vari-able b is the nondimensional buoyancy (scaled by gDu/u0). Parameter M is the Mach number defined as cs/U,that is, the ratio of sound wave speed to the flow speed.Terms represented by Df [ 2Kfxxxx are the fourth-orderhorizontal diffusion terms mentioned earlier and K thediffusion coefficient. The terms involving div [ ux 1wz in the momentum equations are ‘‘divergence-damp-ing’’ terms that are included to attenuate acoustic waves(Skamarock and Klemp 1992). Subscripts x, z, and t areused to represent partial differentiations, for example,ut [ ]u/]t.

The scaling parameters used for nondimensionaliza-tion in the above equations are

Length scale: H (the domain depth)1/2Velocity scale: U [ (gHDr/r )0

2Pressure scale: P [ r U 5 gHDr (3.2)0

Time scale: T [ H/U

Buoyancy scale: B [ gDr/r 5 gDu/u .0 0

In the dimensional ARPS model, we choose H 5 1 km,Du 5 3 K, u0 5 300 K, and g 5 10 m s22. The otherderived scaling parameters are U 5 10 m s21, P 5 120Pa, and T 5 100 s. The chosen domain depth is typicalof the atmospheric boundary layer but is about an orderof magnitude smaller than the depth of the troposphere.Since the results will be presented in nondimensionalspace, their interpretation can remain general.

The density current in the ARPS is generated by plac-ing an initially static block of cold air (b 5 21) in themiddle portion of an elongated computational domainof size 40 3 1 nondimensional units consisting of 8013 41 grid points. Therefore the grid spacings are Dx 50.05 and Dz 5 0.025 in the horizontal and vertical di-rections, respectively. This domain is sufficiently longso that the upstream boundary flow remains undisturbedthroughout the integration period, while disturbancesthat do reach the downstream boundary can propagatefreely out of the domain. As is pointed out in XXD96,the cold pool thus specified is, in most cases, able to

supply sufficient cold air for achieving and maintaininga quasi-steady density current. This initial setup is alsosimilar to that of Rotunno et al. (1988) (see their Fig.19). However, their domain size and their initial coldpool size and depth were too small for their simulateddensity current to achieve a quasi steady state in thetwo cases of positive shear (see their Fig. 20). The re-sultant flow at the cold pool front was, we believe, in-herently transient.

As in XXD96, the initial symmetric cold pool has theshape (described by the interface height z0)

0h for 0 , j , L0

0 0z (j) 5 h 2a(j 2 L ) for L , j , L 1 L0 0 0 150 for L 1 L , j,0 1

(3.3)

where j [ zx 2 x0z, L0 5 4, a 5 3/(4h0), L1 5 2/ 3,Ïx0 is the center location, and h0 the depth of the coldpool. Here, the interface has a slope of 608 at the front,in agreement with the theoretical solution discussed ear-lier.

For all numerical experiments reported in this paper,the initial flow at the far upstream and downstreamboundaries is specified according to (2.3). This auto-matically ensures the flow force balance between thetwo boundaries at the initial time. The low-level sheara and shear layer depth d0 are specified as the experi-ment control parameters. The top boundary flow speed(c0) and the balanced cold pool depth (h) correspondingto the steady-state far-field solutions are defined as inFig. 3 and Fig. 2, respectively.

The initial flow in the interior is obtained numericallyunder the constraints of mass and vorticity conservation.The vorticity equation describing the flow outside thecold pool is

10 for z $ z (x)2 0¹ c 5 (3.4)

15a for z , z (x),

where c0 is the initial streamfunction that defines theinitial velocity:

u0 [ ]c0/]z and w0 [ 2]c0/]x. (3.5)

Equation (3.4) is solved numerically using successiveoverrelation (SOR) method subject to the boundary con-ditions

0 2 0c 5 c d 1 ad /2 along flow interface z 5 z ,0 0 0

0c 5 2c (1 2 d ) along upper boundary z 5 1,0 0

0]c /]x 5 0 at lateral boundaries, (3.6)

where z0 is defined in (3.3).It should be noted that, in (3.4), z1(x), the height of

the interface separating the low-level shear layer andthe uniform flow above, depends on the solution of c0.We perform additional iterations during which the rhsof (3.4) is updated using new solutions of the stream-

1 AUGUST 1997 2005X U E E T A L .

FIG. 5. Initial configuration of cold pool and outside flow obtained from Eq. (3.4) for a 5 1, d0 5 0.2, and h0

5 0.59. The far-field flow satisfies the theoretical steady-state solution for the given set of parameters. The rightand left portions of the initial cold pool are shown to their physical scale, with the midportion of the cold poolomitted. Note that the domain is nondimensional. The x axis origin is located at the right frontal nose at the initialtime. The horizontal and vertical velocity scales (w 5 u 5 1) are shown by the arrow key at the lower left-handcorner.

TABLE 1. Initial settings and simulated values of the kinematicparameters of the numerical experiments.

Expt

Model specified

a d0 h0

Theoretical/model simulated

h c0 1 ad0

LS1LS2LS1A

1.021.0

1.0

0.20.20.2

0.5900.4100.410

0.590/0.5380.410/0.3250.590/0.531

0.412/0.4000.318/0.3180.412/0.412

DSSLSSLSA

1.03.03.0

0.50.20.2

0.6830.7670.20

0.683/0.6380.767/0.7630.767/0.763

0.696/0.7090.777/0.7890.777/0.635

function. The initial wind is then calculated from (3.5)using finite differences. An example of the initial flowis given in Fig. 5 for one of the numerical experiments.

The theoretical results discussed in section 2 concernthe system-relative flow in the vicinity of the densitycurrent front. In our case, the cold pool has two densityfronts; one propagates upstream and the other down-stream. We focus on the upstream side. When the up-stream flow and cold pool depth are specified accordingto the theoretical solution with respect to this front, thefront is expected to remain quasi stationary relative tothe model grid. The density current front on the down-stream side will surge ahead in the absence of a head-wind, removing significant amounts of air from the ini-tial cold block. In order for the upstream density frontto become quasi steady, the cold air supply must besufficient during the integration period. We thereforespecify the initial cold block to be more than eight non-dimensional units in width, which is larger than thatused by most other researchers (e.g., RKW88; Chen1995).

c. Experimental design

As was discussed in the introduction, the main pur-pose of our numerical experiments is to examine thevalidity of the theoretical results obtained in the pre-vious section. Towards this end, we designed six ex-

periments with initial and simulated parameter valuesto be listed in Table 1.

In experiments LS1 and LS2 (LS stands for low-levelshear), the inflow shear is confined to the lower levels(d0 5 0.2) and the shear is moderately strong. Thesetwo experiments differ only in the sign of the low-levelshear. The inflow profile is specified according to (2.3)and the initial cold pool height (h0) is in balance withthis profile (in agreement with the steady-state theoret-ical solution). The initial flow is set up using the pro-cedure discussed earlier so that it closely matches thetheoretical model solution (the dependence of the modelsolution to the initial condition will be examined insection 4b). The initial flow adjustment is therefore min-imized and we expect that the upstream front remainquasi stationary and the cold pool depth maintain itsinitial height throughout the simulation. This was showntrue by XXD96 for density currents in a constant inflowshear, even though significant transient Kelvin–Helm-holtz waves are observed toward the rear of the currenthead.

Experiments LS1 and LS2 are the nonconstant shearcounterparts of experiments B3 and B2 in XXD96. Itis worth noting that we did not attempt to construct apressure field that matches the initial flow; rather, weset the initial pressure perturbation to zero. The initialperturbation pressure field can be obtained by integrat-ing the hydrostatic equation or, more elaborately, bysolving an elliptic pressure equation as is commonlydone in anelastic type numerical models (e.g., Clark1977; Xue and Thorpe 1991). However, this was feltunnecessary since the pressure field responds to the flowfield very quickly. A consistent pressure field can usu-ally be established within a few time steps into the mod-el integration. In fact, a sensitivity experiment was con-ducted using a hydrostatically balanced initial pressurefield, and no significant difference was found in theresults.

Experiment LS1A is the same as LS1, except that theinitial cold pool depth is set to 0.41, which is signifi-


cantly lower than the theoretically predicted value of0.59. The value 0.41 corresponds to that of negativeshear in case LS2. In this case, the initial flow abovethe flat top of the cold pool satisfies mass and vorticitycontinuity with the upstream inflow but does not carrythe same amount of energy. Further, the flow force bal-ance is not satisfied. Significant initial flow adjustmentis expected for this configuration. Similar to experimentUB2 reported in XXD96, LS1A is designed to test thedependency of the final solution on the initial config-uration of the cold pool. In constant shear case(XXD96), the final depth and propagation speed werefound to be solely controlled by the environmental flow,provided that the cold air supply was sufficient to es-tablish a deep cold pool. As will be shown later, thesame is true for the nonconstant shear case discussedin this paper.

The next experiment (in which DS stands for deepshear) has a relatively deep shear layer (d0 5 0.5) anda moderately strong positive shear (a 5 1.0). The initialdepth of cold pool is set equal to the prediction of thetheoretical model (h0 5 0.683). This experiment extendsthe parameter space covered by earlier experiments.

Finally, experiment SLS represents a case with stronglow-level shear (a 5 3.0, d0 5 0.2). This configurationis expected to support the deepest cold pool (h 5 0.767)among all the experiments. It was shown by X92 andXM94 that, for the constant shear inflow case, the frontalsurface at the midlevels can have a slope of more than608 in the case of very strong shear, even though theslope at the frontal nose is independent of shear. Thisbehavior was verified by the numerical experiments ofXXD96, and a similar behavior can be expected forstrong nonconstant shear. The slope of the midlevelfrontal surface and the associated flow are of practicalimportance since they may help us understand the be-havior of updrafts in convective systems, for example,squall lines. The orientation of the updraft is believedto have a controlling effect on the longevity of suchsystems (TMM82; RKW88). An additional run (SLSA,which stands for strong low-level shear) is performedwith all parameters the same as in SLS except for theinitial depth of cold pool, which is set to 0.2. This iswell below the theoretical value of 0.767. It is, however,this last case that is most directly relevant to squall linedynamics.

In the next section, we will discuss the results of thesesix numerical experiments (see Table 1).

4. Kinematic properties and quasi-steady-statebehavior of simulated density currents

a. Cases with shallow, moderate shear

All numerical simulations reported in this paper arecarried out to a nondimensional time T 5 36. Ensembletime averages are produced for simulated fields over theperiod T 5 12 to T 5 18 at a sampling interval of 0.5.

To avoid spatial smoothing due to the current propa-gation, the instantaneous fields are translated in x beforeaveraging so that their frontal noses collocate with thenose at time T 5 12. The instantaneous field at T 5 12and the time averaged fields will be shown.

The results from experiments LS1, LS2, and LS1Aare discussed in this section. In LS1, positive shear ex-ists in the lower levels (d0 5 0.2) of the environmentalflow, while the flow above is uniform. According to Fig.2, such an inflow supports a steady-state density currentof depth h 5 0.59 (see also Table 1). Since the initialflow and density current are configured to reflect thetheoretical far-field solution, we expect the simulateddensity current to remain quasi stationary relative to themodel grid. This expectation is well supported by thenumerical solution. Between T 5 12 and 18, the frontalnose retreated by only 0.075, yielding a phase speed of21.25 3 1022 relative to the model grid. The flow-relative propagation speed is therefore 0.40 instead ofthe theoretical value of 0.412, corresponding to a 3%error (see also Table 1). The front remains quasi sta-tionary until T 5 30, when most of the cold pool air isdepleted on the downstream side and the density currentstarts to retreat.

Measured by the 20.5 buoyancy contours (boundaryof shaded areas in Fig. 6b) in the time-averaged field,the height of the simulated density current head is about0.538, slightly shallower than the theoretical value of0.59. Similar to the results of XXD96, the simulateddensity current head is in general slightly shallower thanthe theoretical value. The presence of transient turbu-lence and the associated energy loss are believed to bethe cause.

Due to the presence of strong shear at the densitycurrent interface, KH billows are pronounced featuresin the instantaneous fields. In fact, these billows are shedperiodically at the back edge of the elevated densitycurrent head and move rearward. They entrain lighterair from above into the denser air below, and vice versa(Fig. 6a), and produce a mixed transition layer that isroughly half the depth of the head (Fig. 6b). Such bil-lows are characteristic of numerically simulated densitycurrents with high resolution models (e.g., Droegemeierand Wilhelmson 1987; XXD96) as well as laboratoryexperiments (e.g., Benjamin 1968; Britter and Simpson1978). Because of their transience, the billows are com-pletely smoothed out in the time-averaged field (Fig.6b), and the negative vorticity that is otherwise con-centrated within a vortex sheet is now spread over afinite depth. In this layer, it can be shown that the shearinstability is released by the mixing and the turbulentRichardson number is above the commonly quoted crit-ical value of 0.25.

The density current exhibits a noticeable decrease inthe depth of the heavier-fluid layer on the rear side ofthe head. It was shown in X92 that, as the depth of thislayer decreases abruptly, the lighter-fluid layer can un-dergo a transition similar to a turbulent hydraulic jump

1 AUGUST 1997 2007X U E E T A L .

FIG. 6. Nondimensional velocity and buoyancy (density) fields (a) at T 5 12 (20 min model time), (b) ensembleaveraged fields over a period of T 5 6 from T 5 12 to 18, and (c) streamlines corresponding to the flow in (b),for experiment LS1. For the ensemble average, the fields are sampled at an interval of T 5 0.5, and the frontalnose at all times is shifted to the nose location at T 5 12 before averaging. The depicted flows are system relativein a nondimensional coordinate. The x origin indicates the initial position of the frontal nose. The horizontal andvertical velocity scales (w 5 u 5 1) are shown by the arrow keys to the lower left-hand corner. The contourintervals are 0.1 for buoyancy and 0.05 for streamlines. The areas with buoyancy less than 20.5 are shaded.

with the upper-layer flow switching from supercriticalto subcritical. Below the subcritical flow, the denserfluid layer is shallower, in resemblance to the presentexperiment.

Near the frontal nose of the simulated density current,the fluid interface is actually quite smooth. Strong bar-oclinicity exists along the interface, and thus large pos-itive vorticity is generated there. Animations of the flowfield show clearly that small eddies with such positivevorticity are carried away by the larger-scale flow, inthis case toward the rear. These eddies grow in size asthey move rearward, as in Droegemeier and Wilhelmson(1987), and significantly enhance the KH billows form-ing in situ along the fluid interface. The smoothness of

the interface at the frontal zone is due to the effectiveremoval of vorticity by the strong rearward flow. Thesize of the billows is positively correlated with the mag-nitude of the shear across the interface, while a negativecorrelation exists with the speed of the rearward ad-vective flow above the fluid interface (cf. this case withFigs. 6a and 8). A similar behavior was found by Chen(1995), who used the ‘‘vorticity ventilation’’ mechanismto explain the local characteristics of simulated densitycurrents.

It should be noted that the density current front hasalmost the same shape and sharpness in the time av-eraged field (Fig. 6b) as in the instantaneous field (Fig.6a). This indicates very little transient activity in the


frontal zone and thus the front is essentially steady. Alsonote that the slope of the front at the nose is very closeto 608 (the figures are plotted to the physical scale),which agrees with the theory.

The transient and time-averaged behavior of the sim-ulated density current in this case is similar to that ofexperiment B2 of XXD96, where the inflow has a mod-erate positive shear that extends over the entire depthof the model domain. The positive shear in both casessupports a density current that is deeper than 0.5—thedepth in a uniform shear.

A similar agreement is found between the numericalsolutions of LS2 and the corresponding theoretical re-sults (Fig. 7). In LS2, the low-level (d0 5 0.2) inflowshear is negative (a 5 21.0) and the expected frontalheight is less than 0.5 (h 5 0.41, see Table 1 or Figs.2 and 3). Again, the initial cold pool is configured ac-cording to the theoretical solution, and thus it is ex-pected to remain quasi stationary. In fact, between T 512 and 18, the frontal nose remains exactly stationary,resulting in a flow-relative phase speed that is equal tothe theoretical value of 0.318 (see Table 1). The depthof the current head measured from the time-averagedfields (Fig. 6b) is 0.325 compared to the theoreticalvalue of 0.410. Note that the agreement is better for thepositive shear (deeper density current) case. A similarbehavior is found in XXD96 for uniform shear cases(compare their experiment B1 and B2). The accuracyis actually slightly better here. The increased verticalresolution and the absence of vertical numerical smooth-ing in the current cases is believed to be responsible forthe improvement. The behavior of the KH billows issimilar to that in LS1, except that the size (or amplitude)is smaller. This, we believe, is partly due to the stronger‘‘ventilating’’ flow in LS2 and partly due to relativelyless baroclinic vorticity generation at the current head.

b. The dependency of simulated density currents oninitial parameters

In most previous numerical studies of density currentsand thunderstorm outflows, the cold air is introducedinto the model by either specifying a time-dependentheat sink that simulates the evaporative cooling in athunderstorm downdraft (e.g., Mitchell and Hovermale1977; Thorpe et al. 1980) or by prescribing a time andspace invariant distribution of cold air source either atthe model boundary (e.g., Crook and Miller 1985; Droe-gemeier and Wilhelmson 1987) or in the interior domain(Chen 1995). These methods, although not totally un-realistic, do have their shortcomings. Since the temporaland/or spatial distribution of the source is prescribed, itis hard to separate the internal dynamics of the densitycurrents from the initiating mechanisms. In the simu-lations reported here, as well as in XXD96, an initialcold pool is specified under the guidance of a theoreticalsolution and is then allowed to evolve freely by itself.Such a cold pool has a sufficiently large volume so that

the resulting behavior of the density current will notdepend upon the source. The previous two experimentshave shown that quasi steady states can indeed be es-tablished after the system is allowed to evolve freely.

To further test the dependency of the final behaviorof the current on the initial model configuration, werepeat experiment LS2 with the initial depth of coldpool set at 0.41 instead of the theoretical value of 0.59.Interestingly, the initially low current head is quicklyraised to a level similar to that in LS2. By T 5 6, itshead has a depth of about 0.538 (not shown) and thefront remains almost completely stationary from T 5 6onward. The time-averaged fields shown in Fig. 8 arevery close to those in Fig. 6b for LS2. The depth of thehead is about 0.531, and the propagation speed is about0.412. These agree extremely well with the theoreticalvalues (see Table 1). This experiment, together withsimilar experiments performed for cases with differentshears (not shown), further confirms that the shape,depth, and propagation of a density current in low-levelshear flow is primarily governed by the internal flowdynamics (mass and vorticity conservation and flowforce balance). This extends our findings for the constantshear case reported in XXD96.

c. The deep, moderate shear case

In LS1 and LS2, the inflow shear was confined to thelower 20% of the model domain. In experiment DS, weexamine a case in which the shear extends to half (d0

5 0.5) the height of the domain (see Table 1) and theinflow shear is set to a 5 1. The instantaneous fieldsat T 5 12 are shown in Fig. 9, together with the time-averaged fields. In this case, theory predicts a headheight of 0.683 and a (flow relative) propagation speedof 0.696. The simulated depth and propagation are, re-spectively, 0.638 and 0.709, representing 5% and 2%errors, respectively. The depth is again determined bythe 0.5 buoyancy contour. The current front movesslightly faster than the theoretical prediction in this case.

The KH waves develop in a manner similar to thoseseen previously, except with a larger amplitude. Again,this can be explained by the production of more vorticityat the much deeper frontal interface. This vorticity, car-ried rearward, contributes to the intensification of theKH billows. As a result, the flow behind the head ismuch more turbulent and transient. These experimentssuggest that larger amplitude billows tend to be asso-ciated with a deeper head.

d. Cases with strong low-level shear

We now explore cases in which a strong shear (a 53) is confined to the lowest 0.2 of the domain. Such aconfiguration is more typical of a squall line environ-ment (e.g., Bluestein and Jain 1985; TMM82). If weassume the model top represents a tropopause of 10-kmheight, then the scaling parameter is H 5 10 km. Using

1 AUGUST 1997 2009X U E E T A L .

FIG. 7. As in Fig. 6 but for experiment LS2, a case with a negative low-level shear (a 5 21 and d0 5 0.2).

FIG. 8. As in Fig. 6b but for experiment LS1A, in which the initial cold pool is lower (h0 5 0.41) than thetheoretical depth (h0 5 0.59) corresponding to the specified inflow (a 5 1 and d0 5 0.2).

the relations in Eq. (3.2), this gives U ø 31 ms21, andtherefore a 5 3 gives a speed change of 18.6 m s21

over a 2-km depth. This is close to the magnitude ofshare that comprises the ‘‘optimal’’ condition for long-

lived squall lines (TMM82; RKW88). On the otherhand, if we apply our density current model to a 1-kmdepth domain, that is, H 5 1 km, we have a speeddifferential of 6 m s21 over a 200-m layer, which is also


FIG. 9. As in Fig. 6 but for experiment DS, a case with deep inflow shear (a 5 1 and d0 5 0.5).

typical of the shear observed in the atmospheric bound-ary layer.

In the first experiment (SLS) with this strong low-level shear, the initial cold pool depth is specified ac-cording to the theoretical solution (h0 5 0.767). Similarto the previous experiments, the density current is ex-pected to remain quasi stationary and maintain its initialdepth. However, the frontal slope at the midlevels maysteepen to more than 608. This behavior was shown byX92 and XM94 for the constant shear inflow case andverified by XXD96 using numerical experiments. Al-though no vigorous solution is sought for the presentnonconstant shear case, we expect a similar behaviorand wait for the verification by numerical experiments.

Figure 10a shows the velocity and buoyancy fields atT 5 12. It is clear that the front is steeper at the mid-levels than at the frontal nose, where the slope is closeto 608. The front is also steeper than at the initial timegiven in Eq. (3.3). The steepening occurred very quickly

during the first few time units. By T 5 6, the flow pattern(not shown) is already very similar to that at T 5 12.As discussed earlier, the slope of the front is determinedby the pressure balance across the interface, and themidlevel slope has a significant bearing on the dynamicsof squall-line-type convection.

Similar to what was found earlier, KH billows areevident and do not develop significant amplitudes untilone unit behind (x 5 21) the forward edge of the currenthead. The time-averaged fields in Fig. 10b show thatthe fluid interface is hardly mixed for x $ 21, indicatingthat most of the eddies exist to the left of this point.Significant mixing occurs to the left of x 5 21 andproduces a mixed layer of about 0.3 in depth. The vortexsheet between the two fluids becomes a zone of nearlyconstant moderate shear. The density current depth, asindicated by the shading, drops from above 0.75 to about0.5 as one moves from the head rearward. Again, onecan explain this using hydraulic jump theory in the pres-

1 AUGUST 1997 2011X U E E T A L .

FIG. 10. As in Fig. 6 but for experiment SLS, a case with strong low-level shear (a 5 3 and d0 5 0.2).

ence of energy loss (X92). Finally, the depth of thecurrent head as well as the propagation of the front agreevery well with the theoretical results (0.763 vs 0.767and 0.777 vs 0.789, respectively).

The deep density currents studied in this paper aremore germane to laboratory flows, where the supply ofthe denser fluid can be sufficient. While it has beenshown that the eventual quasi steady state of the sim-ulated density currents is independent of the initial coldpool configuration, the realization of the theoreticaldepth does depend on a sufficient supply of cold air. Inour numerical experiments, this implies that the initialdepth of cold pool cannot be too shallow. In the at-mosphere, where the troposphere is usually about 10km deep, thunderstorm outflow density currents rarelyexceed 2 km in depth due to the vertical thermodynamicprofile, among other factors. They usually occur onlyin the subcloud layer, where evaporatively cooled airfeeds the outflow. It is of more practical importance in

understanding the behavior of the density current thatcannot reach the balanced steady-state depth. In suchsystems, energy loss occurs in order to maintain asteady-state solution (Benjamin 1968; X92); in a nu-merical model, the energy loss may exhibit considerableunsteady transient activity.

In experiment SLSA, an initial cold pool of depth 0.2is specified within the same strong low-level inflow asin SLS. In the inviscid limit, the balanced depth of thedensity current is about 0.77 (i.e., to counteract the spec-ified shear). The given initial cold pool obviously cannotresist the inflow, nor does it have enough mass toachieve the balanced depth in the same way that thecold pool in experiment LS1A does. As a result, thedensity current head recedes quickly. By T 5 6, thefrontal nose has receded by about 0.8 units (Fig. 11a)and, at T 5 12, the nose is located near x 5 21.5 (Fig.11b). This propagation occurs in the form of head ero-sion at the front rather than a retreat of the entire cold


FIG. 11. Nondimensional velocity and buoyancy (density) fields at T 5 6 (a) and T 5 12 (b), for experimentSLSA. This case is the same as SLS except that the initial cold pool is at a depth of 0.2 instead of the balanceddepth of 0.767.

pool body; as a result, the flow-relative speed of thefront is reduced so that a different quasi-balanced statecan be established. The propagation of the front is rathersteady between T 5 6 and 24, with an average speedof 20.142. The corresponding flow-relative speed istherefore 0.635 (see Table 1).

The flow at the density current head in case SLSAis, however, far from steady throughout the entire pe-riod. When the low-level inflow of high shear approach-es the frontal nose, it is deflected upwards. Most of thisinflow turns into a nearly vertical, though unsteady, up-draft and brings with it some of the denser air from thecold pool (Fig. 11). Animations show that discrete‘‘patches’’ of cold air at the frontal interface are peri-odically ejected upward and that most of them reach theupper third of the model domain. This air generallycarries negative vorticity, owing to the presence of neg-ative vorticity at the flow interface as well as that gen-erated by baroclinic processes. As a result of this pro-cess, counterclockwise-rotating eddies are created andcarried up to near the model top, where they propagatemostly rearward. In a way, their behavior is similar tothe KH billows observed previously, except that theyare less organized due to the absence of support froma steady shear interface. In spite of significant mixingand interaction with denser fluid, the upward-deflectedinflow retains most of its positive vorticity. Most of thisflow turns rearward as it reaches the model top, while

a small portion turns to the right, forming an overturningbranch of the draft (Fig. 11).

The time-averaged system-relative flow, buoyancy,vorticity, and streamlines for experiment SLSA areshown in Fig. 12. The streamline field depicts a pre-dominantly jump-type flow, with the surface streamlinein the inflow reaching a height of about 0.8. The over-turning branch is also evident to the right of the primaryupdraft.

The buoyancy field shows an envelope of denserfluid extending to a height of about 0.76. The shapeof this envelope is, interestingly, not very differentfrom that of the density current head in SLS, wherethe cold pool is specified to be close to this shape atthe beginning. If we take the buoyancy b 5 20.1 con-tour as the outline of the density current head, it showsa slope close to 608 at the nose and a steeper slope atthe midlevels, a behavior also found in SLS. Theseresults indicate that the basic dynamic control (pressurebalance) on the flow interface works to a large extentfor the averaged flow, despite that the instantaneousflow is very transient and nonhydrostatic and oursteady-state theory for the inviscid two-fluid densitycurrent is not strictly applicable. The results also sug-gest the important role of the low-level shear in con-trolling the global flow pattern.

Positive shear in the low-level inflow does makepossible strong vertical lifting at the frontal nose;

1 AUGUST 1997 2013X U E E T A L .

FIG. 12. (a) Time-averaged velocity and buoyancy (density) fields, (b) vorticity, and (c) streamlines of simulateddensity current in experiment SLSA. Figure conventions for (a) and (c) are the same as in Fig. 6.

stronger positive shear supports a deeper jump-typeor even an overturning flow. An additional experimentwas conducted (figures not shown here), in which theinflow shear magnitude is 2 instead. In this case, theouter contour of the cold fluid is about 0.7 deep, closeto the balanced solution of 0.68. Associated with itis a jump-type flow, but the front slope is less than608 at the midlevels and the overturning branch isabsent.

In summary, we have described the transient andtime-averaged behavior of simulated density currentspropagating in an environment with low levels. Thedepth and propagation in the simulations agree very well(typically to within a few percent) with theoretical pre-dictions from an inviscid two-fluid model in spite of thepresence of transient eddies in the former. In next sec-tion, we will further examine the dynamic properties ofthe simulated flows.

5. Conservation properties of simulated densitycurrents

The kinematic properties of idealized inviscid floware determined by the Lagrangian conservation of vor-ticity and Bernoulli energy, along with the flow forcebalance and mass continuity in the theoretical modelpresented in section 2. It is worthwhile to examine theextent to which these conservation properties are sat-isfied in the simulated flows. In general, mass continuityis satisfied, while the Lagrangian conservation of vor-ticity and Bernoulli energy may not be. With the free-slip boundary conditions assumed in the model, the flowforce balance between the upstream and downstreamcross sections should also be well satisfied if the flowis quasi steady. We check the conservation propertiesfor LS1, LS2, SLSA, and SLS in what follows.

The nondimensional Bernoulli energy E [ u2/2 1 p9


FIG. 13. (a) Vertical profiles of nondimensional Bernoulli energyE [ u2/2 1 p9 and (b) vorticity z [ uz 2 wx, for the time-averagedflow in LS1 (Fig. 6). The simulated profiles are plotted for x 5 2.4(ahead of front), x 5 21.2 (density current head), and x 5 22.4(behind the head), together with the profiles at the upstream anddownstream boundaries from the theoretical solution.

and vorticity z [ uz 2 wx are computed for the time-averaged flow of LS1 (Fig. 6b). Their vertical profilesare plotted in Fig. 13 for three representative cross sec-tions, that is, at x 5 2.4 (ahead of front), x 5 21.2(density current head), and x 5 22.4 (behind the head).The upstream and downstream profiles based on thetheoretical solutions are also plotted as a reference.

On the upstream side, the contribution to E comes

from the u velocity only. The low-level shear profileresults in a maximum in E at the surface, which de-creases quadratically upward until z 5 0.2 and thenremains constant above (Fig. 13a). The fact that the Eprofile at x 5 2.4 is nearly the same as the theoreticalone indicates that the inflow is modified little up to thispoint.

The simulated E profiles to the left of the frontal noseare similar to those at the upstream location at z 5 0.This indicates that along the surface, the E on the right,which is due only to the influence of u, is balanced bythe E inside the cold pool, where the pressure pertur-bation is the sole contribution. It is not surprising thatE at x 5 22.4 agrees with the theoretical value lessthan that at x 5 21.2, because the former is behind thehead where eddy mixing is significant. The theoreticalprofile of E decreases linearly with high values insidethe cold pool as the hydrostatic pressure decreases lin-early. Then, E jumps across the flow interface to thevalue of the upstream flow at the ground level. Thesimulated profile has a similar behavior. The peak valueand the shape of the profile near z 5 0.6 indicates thatthe fluid above the cold pool indeed comes from thelow levels and conserves Bernoulli energy. The peak ofthe minimum is less sharp, apparently because of theweak mixing at the interface. At x 5 22.4, the flowbetween z 5 0.4 and 0.6 is well mixed (see Fig. 6b)and, as a result, there is significant energy loss there(Fig. 13a).

Figure 13b shows that vorticity is roughly conservedabove the density current head and within the cold poolbut is not conserved in the mixed interfacial layer. Theupstream profiles are characterized by a constant posi-tive vorticity of 1 below z 5 0.2, and the downstreamanalysis profile has a negative spike at the fluid interfacelevel and positive constant value of 1 in the layer im-mediately above the interface. In the simulated flow, thenegative spike is spread out over a depth of about 0.1at x 5 21.2 and over a much deeper layer at x 5 22.4.The simulated vorticity profile at x 5 21.2 has a pos-itive peak at z ; 0.65, which agrees with the theoreticalprofile since the fluid there actually comes from theshear layer in the inflow. The magnitude of the peak is,however, about 4 instead of 1. This excess of vorticityis believed to be due to the enhancement of rearwardflow immediately below this layer and caused by ro-tating eddies whose negative vorticity comes mostlyfrom baroclinic generation at the frontal interface. Asimilar excess of vorticity in this layer is also observedin other cases reported in this paper and in XXD96.

Similar profiles of the Bernoulli energy (E) and vor-ticity (z) are plotted in Fig. 14 for LS2, a case withnegative low-level shear. The qualitative agreement withthe theoretical profiles is similar to LS1. Both profilesupstream of the front are almost the same as the theo-retical ones, while the profiles at the head region (x 521.2) are less similar. The transition zone between the

1 AUGUST 1997 2015X U E E T A L .

FIG. 14. Same as in Fig. 13 but for experiment LS2.

FIG. 15. Simulated nondimensional flow force plotted as functionsof x for the time-averaged flows and the flows at T 5 18, for ex-periments LS1 and LS2. Also plotted are the analytical flow force Fobtained from the inviscid solutions.

cold pool and the upper-level flow is broader and lowerin altitude than predicted by theory.

Next we examine the validity of flow force balancein the simulated flow. As in XXD96, we rewrite theflow force balance in Eq. (2.15) as

1

2F [ (p9 1 u ) dzE0

5 constant for all values of x. (5.1a)

For a time-averaged flow described by a time-averagedversion of Eqs. (3.1), the flow force can be written as

F [ F0 1 F0 5 constant for all x, (5.1b)

where1

2F [ (^p9& 1 û &) dz0 E0

and1 x 1

0 2F 0 [ û & dz 1 û & dz dx9 (5.1c)E E E t

0 x 00

and represent the flow force of the time-averaged flowand the contribution from the transient parts, respec-tively.

Here ^ & is the time-average operator covering theperiod from T 5 12 to T 5 18 in our case and u0 [ u2 û& is the transient part of u. The horizontal integralin (5.1c) is from the upstream lateral boundary to theparticular vertical cross section of interest at x. Notethat in the simulations reported here, no vertical com-putational mixing is included, and thus a term due toits presence in XXD96 does not appear in (5.1c).

The flow force for the averaged flow, F0, is plottedas a function of x in Fig. 15 for experiments LS1 andLS2, together with F from the theoretical solution andfor the instantaneous flow at T 5 18. It is clear that F0

for LS2 is close (maximum error is less than 3%) to thetheoretical constant value for all x values shown but isin good agreement for LS1 only for x $ 21. At x 523, the difference is about 12%. On the other hand, Ffor the instantaneous flow at T 5 18 is in much betteragreement for all x in both cases. For the instantaneousflow, the discrepancy comes from the local time ten-


FIG. 16. Vertical profiles of nondimensional Bernoulli energy E [u2/2 1 p9 for the time-averaged flow in SLS.

FIG. 17. Vertical profiles of nondimensional vorticity z [ uz 2 wx

for the time-averaged flow in SLS.

dency term only. The fact that this term is very smalleven for instantaneous flow suggests that the main errorin F0 comes from u0, the transient part of u, since thetime tendency term for the time-averaged flow shouldbe even smaller (it is not identically zero because thetime average is performed on a set of data sampled atdiscrete times; see section 4 for more details on the timeaveraging procedure). This analysis is consistent withthe results in Fig. 15; most of the transient activity, theKH eddies, exist to the left of x 5 21.0, and theseeddies are much stronger in LS1 than in LS2 (Figs. 7aand 6a).

The above analyses show that the Lagrangian con-servation of vorticity and Bernoulli energy are approx-imately satisfied above the density current head but toa lesser extent in the mixed interfacial layer. Further,the flow force balance is very well satisfied across thedomain for the instantaneous flows but is less so inregions of strong transient eddies for the time-averagedfields. The discrepancy in the latter is attributed to thetransient part of the flow force (u0 term in F0) that isnot accounted for in the force calculation [F0 in (5.1a)is calculated and F0 is neglected]. These results explainwhy inviscid theory is largely successful in describingthe numerically simulated density currents.

Bernoulli energy (E) profiles are plotted for SLS inFig. 16, showing a similar degree of conservation as inthe previous cases (e.g., LS1). It can be seen from Fig.16 that E is well conserved along the lower boundary.The profile at the left side of the front exhibits an iden-tifiable minimum at about z 5 0.7, which correspondsto the minimum at z 5 0.77 in the theoretical profile.The minimum is a result of the linear decrease withheight of hydrostatic pressure and the zero flow speed

inside the cold pool. Above z 5 0.8, E is close to thatof the upper-level inflow, although the positive peakassociated with air originating at z 5 0 is reduced.

The vorticity profiles given in Fig. 17 show interest-ing features that agree with the theoretical profiles.Above and below z 5 0.77, the positive and negativeanomalies have a clear correlation with those in thetheoretical profiles, indicating that the positive vorticityat the low-level inflow did make its way to the upperlevels and that the negative vorticity associated with theflow interface is primarily responsible for the negativeanomaly near z 5 0.7. The flow force profiles for SLSin Fig. 18 show that the conservation for the instanta-neous flow is much better than for the time-averagedflow, for the same the reason discussed earlier.

In SLSA, the initial depth of the cold pool deviatessignificantly from the theoretical solution, and thus themain body of the cold pool remains shallow and thelocal flow near the front is much more transient. Thecold pool front in this case retreats at a roughly constantspeed, effectively reducing the system-relative flowspeed. As was shown earlier, the time-average flow hasa broad-scale feature that is similar to the balanced flowpredicted by the inviscid stead-state model (Fig. 12).Since the flow is highly turbulent and nonsteady, Ber-noulli energy and vorticity are far from conserved andcannot be closely compared with the theoretical profiles.Nevertheless, the flow force profiles for SLSA in Fig.18 show a similar degree of conservation as in the pre-vious cases.

In general, the numerical model predicts the propa-gation speed more accurately than it does the depth.XXD96 explains this by considering the contribution ofenergy loss and circulation inside the cold pool. Ac-

1 AUGUST 1997 2017X U E E T A L .

FIG. 18. Same as Fig. 15 but for experiments SLSA and SLS.

cording to Benjamin (1968) and X92, the energy lossand circulation inside the cold pool tend to increaseslightly the propagation speed, while reducing the headdepth of the supercritical state of a density current. Thesame argument can be applied to cases reported here.

The theoretical analyses of X92 and XM94 suggestedthat the flow force balance is the principal global prop-erty that controls the flow structure and the interactionbetween the environmental shear and the cold pool cir-culation. The flow is at the same time subject to thestrong constraint of mass continuity and weak con-straints of energy and vorticity conservation. Thus, thewell-maintained flow force balance in our numericalexperiments is the key factor that explains why the sim-ulated density currents propagate at nearly the samespeed as predicted by inviscid theory, while the less-conservative nature of the Bernoulli energy and vorticityis responsible for the reduced depth of the simulatedflow compared to its inviscid limit.

6. Summary and conclusions

The two-fluid idealized density current model in con-stant shear developed by Xu (1992) and Xu and Mon-crieff (1994) has been extended to the case of noncon-stant vertical shear. Theoretical solutions are determinedby the conservation of mass, momentum, vorticity, andenergy. It is found that shear confined to the low levelsplays a role similar to uniform vertical shear in con-trolling the depth of steady-state density currents. Whenthe shear enhances the low-level flow against the densitycurrent propagation, the current is deeper than the bal-anced depth of a current in a uniform flow, the halfdepth of a vertically bound channel.

Time-dependent numerical experiments are conduct-ed for a variety of parameter settings, including varying

depths and strengths of the shear layer. It was foundthat the depth and propagation speed of the simulateddensity currents in the simulations agree very well (typ-ically to within a few percent) with predictions by theidealized theoretical model in spite of the presence oftransient eddies behind the head of the former. The quasisteady state of the simulated density currents is inde-pendent of the specification of the initial cold pool, aslong as the initial depth is reasonably close to the bal-anced solution.

When the initial cold pool is much shallower than thebalanced depth, the upstream front of the cold pool re-cedes at a nearly constant speed but the flow is highlyturbulent and nonsteady in the region above the densitycurrent head. The primary body of the density currentremains shallow, but the broad-scale jump flow canreach a height similar to that predicted by the theoreticalmodel. The main density current body behaves morelike the subcritical flow that exists in the presence ofenergy loss, as discussed in X92 and Benjamin (1968).Similar to the numerical experiments in RKW88, theseexperiments show that the slope of the frontal interface,the depth of the density current, and the associated jumpflow are controlled to a large extent by the shear in thelow-level inflow. This information is significant for ourunderstanding of the forcing and cell regeneration mech-anisms along an outflow gust front in convective sys-tems. Using a hydrodynamical model of shear flow oversemi-infinite barriers to represent a hypothetical coldpool, Shapiro (1992) also found that the vertical dis-placement of the low-level inflow has a direct relationwith the inflow shear.

Detailed diagnoses of the numerical results showedthat the Lagrangian Bernoulli energy and vorticity areapproximately conserved at the current head (except forexperiment SLSA) but are not conserved in the mixedinterfacial layer behind where dynamical instabilityleads to turbulence and kinetic energy dissipation. Ingeneral, energy and vorticity conservation are interde-pendent (XM94) and are weak constraints for the globalflow compared to mass conservation and flow force bal-ance. Mass continuity is always satisfied, and the flowforce balance is also satisfied to a high degree. Thelarger error in the flow force balance calculated for time-averaged fields is due to the transient eddies, which arenot accounted for in the flow force calculations. Theseresults further confirm the theoretical analyses of X92and XM94 with regard to the roles and relative impor-tance of the conservation properties.

As in XXD96, the density currents studied in thispaper are subject to an important limitation: the flow isrestricted by a rigid upper boundary. The presence ofthis rigid lid forces the flow above the density currentto run through a narrow channel, therefore it reducesthe pressure (based Bernoulli energy conservation) inthe region. As a result, a deep density current is sup-ported. In the real atmosphere, we speculate that a stronginversion layer and the tropopause may act to some


extent like a rigid lid in channeling the flow below. Thevalidity of this speculation is the subject of future re-search. In the real atmosphere, a relatively shallow den-sity current like the one in SLSA with strong low-levelshear is more common. This type of solution typicallyinvolves energy loss and transientness. It is interesting,however, that the time-averaged flow can be well or-ganized and can exhibit a predominant jump feature thatresembles the jump updraft in a steady squall line (Mon-creiff 1978; 1992). The transient eddies at the frontalzone can be related to the regenerating convective cells.Furthermore, in the real atmosphere, a cold pool is main-tained by downdraft cooling in a convective storm,which itself is significantly controlled and/or modulatedby the cold pool strength. It is not the purpose of thispaper to study the complete convective system like squallline, but rather to establish a firm understanding of theinteraction between the density current (thunderstormoutflow boundary) and ambient shear flow in a moreidealized setting for which theoretical solutions can befound. The effect of open upper boundary and the pres-ence of stable layer and stratification, as well as theeffect of local heating, will be studied in future research.

Acknowledgments. The numerical simulations wereperformed on the Cray-C90 at the Pittsburgh Super-computing Center (PSC), and on the Cray-J90 of theEnvironmental Computing Applications System(ECAS), University of Oklahoma. Figures were pro-duced using the ZXPLOT graphics package developedby the first author. This research was supported byNOAA Grant NA37RJ0203 and NSF Grant ATM-9417304 to the Cooperative Institute for Mesoscale Me-teorological Studies (CIMMS), by NSF Grant ATM91-20009 to the Center for Analysis and Prediction ofStorms (CAPS), and by NSF Grants ATM92-22576 andEAR95-12145 to the third author. Animations of thesimulations reported here can be found on the WorldWide Web at http://wwwcaps.ou.edu/ARPS/ARPS.pubs.html.

REFERENCES

Benjamin, B. T., 1968: Gravity current and related phenomena. J.Fluid Mech., 31, 209–248.

Bluestien, H. B., and M. H. Jain, 1985: Formation of mesoscale linesof precipitation: Severe squall lines in Oklahoma during thespring. J. Atmos. Sci., 42, 1711–1732.

Britter, R. E., and J. E. Simpson, 1978: Experiments on the dynamicsof a gravity current head. J. Fluid Mech., 88, 223–240.

Carbone, R. E., 1982: A severe frontal rainband. Part I: Stormwidehydrodynamic structure. J. Atmos. Sci., 39, 258–279.

Chen, C., 1995: Numerical simulations of gravity currents in uniformshear flows. Mon. Wea. Rev., 123, 3240–3253.

Clark, T. L., 1977: A small-scale dynamic model using a terrain-following coordinate transformation. J. Comput. Phys., 24, 186–215.

Crook, N. A., and M. J. Miller, 1985: A numerical and analyticalstudy of atmospheric undular bores. Quart. J. Roy. Meteor. Soc.,111, 225–242.

Droegemeier, K. K., 1985: The numerical simulation of thunderstorm

outflow dynamics. Ph.D. dissertation, University of Illinois atUrbana–Champaign, 695 pp. [Available from Dept. of Atmo-spheric Science, University of Illinois at Urbana–Champaign,105 S. Gregory Avenue, Urbana, IL 81801.], and R. P. Davies-Jones, 1987: Simulation of thunderstorm mi-crobursts with a super-compressible numerical model. Preprint,Fifth Int. Conf. on Numerical Methods in Laminar and TurbulentFlow, Montreal, Quebec, Canada, Concordia University, Prattand Whitney, Canada, and the National Science and EngineeringResearch Council of Canada, 1386–1397., and R. Wilhelmson, 1987: Numerical simulation of thunder-storm outflow dynamics. Part I: Outflow sensitivity experimentsand turbulence dynamics. J. Atmos. Sci., 44, 1180–1210., M. Xue, K. Johnson, M. O’Keefe, A. Sawdey, G. Sabot, S.Wholey, N. T. Lin, and K. Mills, 1995: Weather prediction: Ascalable storm-scale model. High Performance Computing, G.Sabot, Ed., Addison-Wesley, 45–92., and , A. Sathye, K. Brewster, G. Bassett, J. Zhang, Y. Lui,M. Zou, A. Crook, V. Wong, R. Carpenter, and C. Mattocks,1996: Real-time numerical prediction of storm-scale weatherduring VORTEX-95. Part I: Goals and methodology. Preprints,18th Conf. on Severe Local Storms, San Francisco, CA, Amer.Meteor. Soc., 6–10.

Fovell, R. G., and Y. Ogura, 1988: Numerical simulation of a mid-latitude squall line in two dimensions. J. Atmos. Sci., 45, 3846–3879.

Johnson, K. W., J. Bauer, G. A. Riccardi, K. K. Droegemeier, andM. Xue, 1994: Distributed processing of a regional predictionmodel. Mon. Wea. Rev., 122, 2558–2572.

Klemp, J. B., and R. Wilhelmson, 1978: The simulation of three-dimensional convective storm dynamics. J. Atmos. Sci., 35,1070–1096., R. Rotunno, and W. C. Skamarock, 1994: One the dynamicsof gravity currents in a channel. J. Fluid Mech., 269, 169–198.

Liu, C., and M. W. Moncrieff, 1996a: An analytical study of densitycurrents in sheared, stratified flow and the effects of latent heat-ing. J. Atmos. Sci., 53, 3303–3312., and , 1996b: A numerical study of the effects of ambientflow and shear on density currents. Mon. Wea. Rev., 124, 2282–2303.

Mitchell, K. E., and J. B. Hovermale, 1977: A numerical investigationof the severe thunderstorm gust front. Mon. Wea. Rev., 105, 657–675.

Moncrieff, M. W., 1978: The dynamic structure of two-dimensionalsteady convection in constant shear. Quart. J. Roy. Meteor. Soc.,104, 543–567., 1992: Organized convective systems: Archetypal dynamicalmodels, momentum flux theory, and parameterization. Quart. J.Roy. Meteor. Soc., 118, 819–850., and D. K. So, 1989: A hydrodynamic theory of conservativebounded density currents. J. Fluid Mech., 198, 177–197.

Mueller, C., and R. Carbone, 1987: Dynamics of a thunderstormoutflow. J. Atmos. Sci., 44, 1879–1898.

Parker, D. J., 1996: Cold pools in shear. Quart. J. Roy. Meteor. Soc.,122, 1655–1674.

Rotunno, R., J. B. Klemp, and M. L. Weisman, 1988: A theory forstrong long-lived squall lines. J. Atmos. Sci., 45, 463–485.

Shapiro, A., 1992: A hydrodynamical model of shear flow over semi-infinite barriers with application to density currents. J. Atmos.Sci., 49, 2293–2305.

Simpson, J. E., 1969: A comparison between laboratory and atmo-spheric density currents. Quart. J. Roy. Meteor. Soc., 95, 758–765., D. A. Mansfield, and J. R. Milford, 1977: Inland penetrationof sea-breeze fronts. Quart. J. Roy. Meteor. Soc., 103, 47–76.

Skamarock, W. C., and J. B. Klemp, 1992: The stability of time-splitting numerical methods for the hydrostatic and the nonhy-drostatic elastic equations. Mon. Wea. Rev., 120, 2109–2127., and , 1993: Adaptive grid refinement for two-dimensional

1 AUGUST 1997 2019X U E E T A L .

and three-dimensional nonhydrostatic atmospheric flow. Mon.Wea. Rev., 121, 788–804.

Thorpe, A. J., M. J. Miller, and M. W. Moncrieff, 1980: Dynamicmodels of two-dimensional downdraft. Quart. J. Roy. Meteor.Soc., 106, 463–484., , and , 1982: Two-dimensional convection in non-constant shear: A model of mid-latitude squall lines. Quart. J.Roy. Meteor. Soc., 108, 739–762.

von Karman, T., 1940: The engineer grapples with non-linear prob-lems. Bull. Amer. Math. Soc., 46, 615–683.

Wakimoto, R. M., 1982: The life cycle of thunderstorm gust frontsas viewed with Doppler radar and rawinsonde data. Mon. Wea.Rev., 110, 1060–1082.

Xu, Q., 1992: Density currents in shear flows—A two-fluid model.J. Atmos. Sci., 49, 511–524., and L. P. Chang, 1987: On the two-dimensional steady upshear-sloping convection. Quart. J. Roy. Meteor. Soc., 113, 1065–1088., and M. W. Moncrieff, 1994: Density current circulations inshear flows. J. Atmos. Sci., 51, 434–446., F. S. Zhang, and G. P. Lou, 1992: Finite-element solutions offree-interface density currents. Mon. Wea. Rev., 120, 230–233.

, M. Xue, and K. K. Droegemeier, 1996: Numerical simulationof density currents in sheared environments within a verticallyconfined channel. J. Atmos. Sci., 53, 770–786.

Xue, M., and A. J. Thorpe, 1991: A mesoscale numerical model usingthe nonhydrostatic sigma coordinate equations: Model experi-ments with dry mountain flows. Mon. Wea. Rev., 119, 1168–1185., K. K. Droegemeier, and V. Wong, 1995a: Advanced RegionalPrediction System (ARPS) and real-time storm prediction, Pre-prints, Int. Workshop on Limited-Area and Variable ResolutionModels, Beijing, China, World Meteorological Organization, 23–27., , , A. Shapiro, and K. Brewster, 1995b: AdvancedRegional Prediction System (ARPS) Version 4.0 User’s Guide,380 pp. [Available from Center for Analysis and Prediction ofStorms, University of Oklahoma, 100 E. Boyd, Norman, OK73019.], K. Brewster, K. K. Droegemeier, F. Carr, V. Wong, Y. Liu, A.Sathye, G. Bassett, P. Janish, J. Levit, and P. Bothwell, 1996:Real time prediction of storm-scale weather during VORTEX-95, Part II: Operation summary and example cases. Preprints,18th Conf. on Severe Local Storms, San Francisco, CA, Amer.Meteor. Soc., 178–182.

a theoretical and numerical study of density currents in nonconstant shear...

Documents