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Numerical Simulations of Density Currents in Sheared Environments within a Vertically Confined Channel QIN XU Cooperative Institute for Mesoscale Meteorological Studies, University of Oklahoma, Norman, Oklahoma
MING XUE Center for Analysis and Prediction of Storms, 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 26 September 1994, in final form 3 October 1995) ABSTRACT Numerical simulations are performed to study the kinematics and dynamics of nearly inviscid, two-dimensional density currents propagating in a uniformly sheared environmental flow within a vertically confined channel. In order to study the physical properties of the numerical solutions relative to those of theoretical predictions, the initial cold pool depth and shear are chosen to be either similar to or significantly different than those prescribed by the theoretical steady-state model. The authors find that, regardless of the model initial condition, the density current front reaches nearly the same quasi-steady state. The propagation speed, depth, and gross shape of the density current head in the quasi-steady state agree closely with previously published theoretical results and are independent of the initial depth of the cold pool provided that the total volume of cold air is sufficiently large. Physical interpretation of the results is provided based on theoretical analyses and numerical diagnosis of the energy, vorticity, mass, and momentum conservation properties of the simulated flows.
1. Introduction It is now 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 (Moncrieff 1978, 1992; Thorpe et al. 1982; Xu and Chang 1987; Rotunno et al. 1988). To improve our physical understanding of this interaction, simple nonlinear, twofluid steady-state models were recently developed by Xu (1992, hereafter X92) and Xu and Moncrieff (1994, hereafter XM94). These models extend the classic density current theory of Benjamin (1968) by including three new ingredients: (i) environmental shear, (ii) internal cold pool circulation [based on earlier work by Moncrieff and So (1989)], and (iii) negative vorticity generation associated with energy loss along the interfacial layer between the density current and its environment. As archetypes of the physically more complex system characteristic of deep precipitating convection, these models allow closed mathematical analyses showing how the depth,
Corresponding author address: Dr. Qin Xu, CIMMS, University of Oklahoma, EC Rm. 1110, 100 E. Boyd, Norman, OK 73019-0628. E-mail:
[email protected]
propagation speed, and shape of the density current are controlled or influenced by the environmental shear and cold pool strength. Although these simple models have improved our physical understanding of the dynamics of density currents in shearing environmental flows, they more or less exhaust the useful application of inviscid, steady theory. Indeed, laboratory experiments (e.g., Simpson 1969), observations (Wakimoto (1982), Blumen 1984; Mueller and Carbone 1987), and numerical simulations (Droegemeier 1985; Droegemeier and Wilhelmson 1987) of density currents indicate the development of vigorous Kelvin–Helmholtz (KH) waves and strong turbulent mixing along the interfacial layer. These transient features and related turbulent mixing, however, were not fully considered in the above theoretical models, and thus it is not clear to what extent the analytic steady-state solutions and related properties obtained might be reproduced by numerical simulations with a full dynamic model. We examine this problem herein using a nonhydrostatic numerical model—the Advanced Regional Prediction System (ARPS) (CAPS 1992; Xue et al. 1995; Droegemeier et al. 1995). In the following section, we review the theoretical results of X92 and XM94 for idealized inviscid density
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currents in shear flows. In section 3, we describe the numerical model used and specification of the initial and boundary conditions. Section 4 presents the results of numerical simulations and shows that the simulated density currents can achieve quasi-steady states and that their gross flow structures and kinematic properties are well predicted by the theoretical models. In section 5, we examine the conservation properties (Bernoulli energy, vorticity, and flow force) of the simulated flows and show that the flow force is much better conserved than the Bernoulli energy and vorticity. This explains why the propagation speed of the simulated currents can be quite accurately predicted by the analytical theory. As in the theoretical models, the rigid upper boundary used in our numerical model plays an important role in keeping the simulated density currents quasi-steady, and this also sets a limitation for applications of our results to atmospheric density currents. A discussion about this limitation and suggestions for future research are offered together with conclusions in section 6. 2. Review of density current theory a. Flow configuration and scaling In the idealized density current models of X92 and XM94, the environmental shear flow is specified throughout the depth of the domain, with the density current front moving at a constant speed into the upstream inflow (Fig. 1). This scenario is described by dimensionless variables using the following scaling: (x, z) R (x, z)/H,
(u, w) R (u, w)/U,
p * R p * /( r0U 2 ),
(2.1)
where H is the depth of the domain between the lower and upper rigid boundaries; U å (gHDr / r0 ) 1 / 2 is the velocity scale; g is the acceleration of gravity; Dr å r1 0 r0 is the constant density difference between the two fluids, with r1 being the (constant) density of the cold pool and r0 the (constant) density of the environmental inflow; and p * å P 0 P0 , with P being the total pressure and P0 å g r0 (H 0 z) the reference hydrostatic pressure associated with density r0 on the upstream side. Thus, p * is the sum of the purely dynamic pressure perturbation caused by the Bernoulli effect and the hydrostatic pressure perturbation resulting from the deviation of the cold pool density ( Dr ) from the reference density ( r0 ). In the following subsections, the theoretical results of X92 and XM94 are briefly reviewed to facilitate the analysis of our simulations. In particular, section 2b shows how the depth and propagation speed of the density front depend on the inflow shear and cold pool circulation, while section 2c describes how the local and global structures of the front are controlled by the interface condition.
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FIG. 1. Schema of the steady-state model of a density current circulation in an sheared environmental flow. The remote system-relative sheared environmental inflow and outflow are indicated by uin(z) and uout(z), respectively, and h is the depth of the density current.
b. Depth and propagation speed The system-relative upstream inflow and downstream outflow above the density current head (see Fig. 1) can be described by the following set of five dimensionless parameters: S5 å { a0 , a1 , h, c0 , c1 },
(2.2)
where a0 is the (constant) vertical shear of the upstream environmental inflow, a1 is the (constant) vorticity of the cold pool circulation, h is the depth of the density current head, c0 ( ú0) is the propagation speed of the density current relative to the surface environmental flow, and c1 ( ú0) is the outflow speed (absolute value) immediately above the density current (i.e., at z Å h in Fig. 1). Because the latter has the same vertical shear a0 as the upstream environmental inflow, the vorticity constraint is automatically satisfied. Further, since the remote system-relative environmental inflow and outflow are also constrained by mass continuity, energy conservation, and flow force balance (Benjamin 1968), only two of the five parameters in (2.2) are independent. To find a relationship among the five parameters in (2.2), XM94 found it convenient to specify the depth (h; 0 õ h õ 1) and vorticity ( a1 ) within the density current. Mass continuity and energy conservation yield the following expressions for c0 and c1 [see (3.1) and (3.3) of XM94]: c0 Å c1he / a0 (1 0 h 2e )/2
(2.3)
c 21 Å 2h / a 21 h 2 /4,
(2.4)
where he å 1 0 h. The flow force balance is obtained by integrating the horizontal momentum equation over the control domain (Fig. 1) and applying mass continuity:
*u
2 in
dz Å
* (p *
out
/ u 2out )dz,
(2.5)
where the integral is taken over the entire nondimensional domain depth [0, 1], p *in Å 0 is assumed for the upstream pressure perturbation, and p *out (z) is the downstream pressure perturbation (which is the sum of the hydrostatic pressure perturbation association with
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the lower-layer density difference and the Bernoulli pressure). Substituting the explicit expressions for p *out (z), uin , and uout [see (3.1) and (3.5) in XM94] yields a quadratic equation for a0 . As shown by (3.10) of XM94 or (2.10) of X92 (for the case of a1 Å 0), this quadratic equation has only one physical root that ensures c0 ú 0 in accordance with the assumed flow structure. After a0 is obtained, c0 can be determined from (2.3). It is physically meaningful to choose S2 å { a0 , a1 }
(2.6)
as independent external control parameters, with the remaining three parameters S3 å {c0 , c1 , h}
(2.7)
being internal. Note that, because c1 is related to a 21 in (2.4), it is independent of the sign of a1 . XM94 showed that all three internal parameters in S3 are independent of the sign of a1 and, therefore, the direction of the cold pool circulation. Additionally, in comparison with the environmental inflow shear, the strength of the density current’s internal circulation was shown to play only a secondary role in controlling the depth and propagation speed of the density front. For simplicity, all experiments presented herein are performed with zero initial vorticity within the density current. Although weak internal circulations can eventually be generated by mixing, their effect on the depth and propagation of the density current can be neglected, at least for the cases studied here due to the use of freeslip upper and lower boundary conditions. According to the theoretical results of XM94 (see their Fig. 2), when the internal circulation is moderately strong (as often observed in thunderstorm outflows), the effect of the internal circulation on the depth and propagation of the density current may not be neglected but may still be much weaker than the impact of the vertical shear of the environmental inflow. The dependencies of h and c0 on a0 are plotted in Fig. 2, where a1 Å 0. Clearly, the density current becomes deeper and propagates faster relative to the surface environmental flow as the inflow shear increases from negative through positive values. When both the cold pool shear and upstream inflow shear vanish (i.e., a1 Å a0 Å 0), the model degenerates to the classic density current solution (h Å 0.5) of Benjamin (1968). c. Local and global structures of the front The previous section reviewed the flow solutions away from the density front; however, the flow structure near the front is of equal importance. In the case of a thunderstorm outflow, the shape of the front has direct relevance on the orientation of low-level updrafts. An upshear-tilted updraft plays a key role in the development and maintenance of long-lived, squall-
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FIG. 2. (a) Depth h and (b) propagation speed c0 of an idealized density current vs environmental shear a0 with a1 Å 0. Crosses indicate the depths (measured by the maximum height of the contour of u* Å 0Du/2 Å 01.5 K above the density current head), and dots indicate the estimated propagation speeds of the simulated density current heads.
line-type convection (e.g., Thorpe et al. 1982; Rotunno et al. 1988). Von Ka´rma´n (1940) and Benjamin (1968) showed analytically that the angle at the stagnation point between the density front and the horizontal is 607 for an idealized inviscid density current in a uniform (nonsheared) environmental inflow with a free-slip boundary condition. It was also shown in X92 and XM94 that this 607 angle is independent of the inflow shear and cold pool circulation. Because the cross-interface continuity of pressure represents a balance primarily between the lower-layer hydrostatic perturbation pressure and the upper-layer Bernoulli pressure, the shape of the interface between an inviscid density current and its environment is constrained by the following dynamic condition: D(u 2 / w 2 ) Å 2z
along the interface,
(2.8)
where D( ) represents the jump of ( ) across the interface. Using this interface condition along with the vorticity equation and boundary conditions, the interface shape, together with the flow fields on the two sides of the interface, can be solved numerically (Xu et al. 1992). According to the results of X92 and XM94, when the inflow shear is weak or negative (moderately strong), the interface slope rapidly (slowly) decreases with height away from the stagnation point at the surface. The geometric shape of the frontal interface and its slope at the middepth (z Å h/2) of the density current are moderately sensitive to the inflow shear. However, when the inflow shear is strongly positive and the
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internal density current circulation is weak or zero, the interface slope becomes steeper than 607 at the middepth location. Thus, the middepth interface slope can be either smaller or larger than 607, depending upon the inflow shear and the intensity of the density current internal circulation (see Fig. 9 of XM94). 3. The numerical model and experiment design a. The numerical model The numerical model used in this study is a modified version of ARPS developed at the Center for Analysis and Prediction of Storms (CAPS) (CAPS 1992; Xue et al. 1995). ARPS is a general purpose, nonhydrostatic, compressible model designed for storm- and mesoscale atmospheric simulation and real-time prediction on both conventional scalar/vector as well as parallel computers (e.g., Johnson et al. 1994; Droegemeier et al. 1995; Janish et al. 1995). The dynamic framework consists of prognostic equations for momentum, potential temperature, pressure, water substance, and subgrid-scale turbulent energy, all of which are solved on an Arakawa C grid using a split-explicit time integration scheme (Klemp and Wilhelmson 1978) and second-order spatial and temporal (leapfrog) differencing. For the purposes of this study, ARPS is used in its simplest two-dimensional setting with all physical processes switched off. The equations solved by the model are ut Å 0 (uux / wuz ) 0 p *x / r0 / Kd divx / Du wt Å 0 (uwx / wwz ) 0 p *z / r0 / g
(3.1a)
u* / Kd divz / Dw uU (3.1b)
p t* Å 0 r0 c 2s (ux / wz )
(3.1c)
u t* Å 0 (u u*x / w u*z ) 0 w uU z / Du ,
(3.1d)
where all symbols have their conventional meaning and, as discussed further below, cS is a pseudoacoustic wave speed. Subscripts are used to represent partial differentiation, for example, ut å Ìu/ Ìt. To facilitate the comparison of numerical results with analytic solutions, several approximations are made to the original equations used in the ARPS. First, the Boussinesq approximation is imposed by setting the base-state density r0 to a constant. The base-state pressure is a function only of height and is in hydrostatic balance. Second, the effects of compressibility are neglected in the buoyancy term such that the thermal buoyancy (g u* / uU ) is identical to 0 g r* / r0 , as in the theoretical model of X92. Third, an isentropic base state is specified ( uU Å constant) to represent the neutral environment typically found outside the cold pool of a rain-cooled outflow propagating in a well-mixed boundary layer. Additionally, the supercompressibility approximation is made
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to the pressure equation (Anderson et al. 1985; Droegemeier and Davies-Jones 1987; Droegemeier and Wilhelmson 1987). In doing so, we replace the full pressure equation, which can be obtained by differentiating the integral form of the first law of thermodynamics for an adiabatic flow, by the anelastic continuity equation in which the time derivative of pressure is retained on the right-hand side with a pseudoacoustic wave speed, cS , multiplying the mass divergence. Though not entirely physical because it neglects advection and diabatic effects, this approximation is now generally accepted and used (e.g., Straka 1989; Tripoli 1992). With a value of cS smaller than the true acoustic wave speed, the fluid is made supercompressible, resulting in an artificial enhancement of the coupling between the gravitational and acoustic modes, as shown in the linear analysis of Droegemeier and Davies-Jones (1987). However, as long as cS is more than about twice the speed of the fastest meteorologically significant waves, this enhanced coupling has no deleterious impact on the solution, and convergence toward an anelastic state is achieved. It is therefore appropriate to regard (3.1c) as a time-iterative solver of the anelastic continuity equation. In our experiments, we set cS Å 150 m s 01 . The only advantage of using a reduced acoustic wave speed is the provision of a larger small time step. With the above approximations, (3.1) describes a flow that is very similar to that used in the theoretical model of X92. Further, neglecting the diffusion and divergence damping terms, and substituting p * / r0 for p [p å Cp u /(p * /p0 )], (3.1) is identical to the equations used in the 2D simulations of Skamarock and Klemp (1993). In (3.1), Df å Kxfxxxx / Kzfzzzz represents a fourthorder numerical diffusion term for u, w, and u, where Kx and Kz are the diffusion coefficients in the x and z directions, respectively. We set Kx Å 12 500 m4 s 01 and Kz Å 780 m4 s 01 . This highly selective fourth-order diffusion ensures that grid-scale noise is effectively controlled, while sharp gradients in both the temperature (density) and velocity fields at the density current interface are well maintained. The terms involving div å ux / wz in the momentum equations are ‘‘divergence damping’’ terms designed to suppress acoustic waves. A coefficient of Kd Å 0.05[min( Dz, D x)] 2 / Dt is used, where Dt is the small time step. Skamarock and Klemp (1992) showed that divergence damping can effectively suppress unstable acoustic modes associated with split-explicit time integration. Finally, the top and bottom boundaries of the model domain are rigid, free-slip plates, while the lateral boundaries employ Orlanski’s (1976) wave radiation condition. b. Scaling parameters and boundary conditions To facilitate comparison of our simulation results with the analytic models discussed earlier, we find it convenient to express all variables in nondimensional form according to the following scaling parameters:
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FIG. 3. Initial configuration of cold pool described by (3.3) with h0 Å 0.7. The right and left portions of the initial cold pool are shown to their physical scale, with the midportion of the cold pool being omitted. Note that the domain is nondimensional. The x-axis origin is shifted from the domain center to the right front nose location in the picture. The horizontal and vertical velocity scales (W Å U Å 10 m s01) are shown by the arrows to the lower left corner.
length scale (domain depth): H Å 1 km, 1/2
velocity scale: U å (gHDr/r0 )
Å (10 1 1000 1 0.01)1/2 Å 10 m s01,
pressure scale: P å r0U 2 Å gHDr Å 120 Pa, timescale: T å H/U Å 100 s, where Dr / r0 å Du / u0 Å 3 K/300 K Å 0.01 is the basic density (or potential temperature) difference. The dimensionless unperturbed environmental flow at the far upstream and downstream lateral boundaries is specified as uin (z) Å 0 c0 / a0z,
(3.2)
where the surface inflow speed c0 is related to the inflow shear a0 as shown in Fig. 2. Since c0 is the propagation speed of the idealized density current relative to the upstream environmental flow, the numerically modeled density front is expected to remain quasistationary in the computational domain when the upstream inflow is given by (3.2). The density current in the ARPS is generated by placing an initially static block of cold air (uniform temperature deficit of 3 K) in the middle of the computational domain. A computational grid of 801 1 41 points with D x Å 50 m and Dz Å 25 m is used in all experiments. Because the computational domain is sufficiently long, the velocity near the inflow boundary (in reference to the flow at low levels) remains unperturbed throughout the integration period. Cold air that reaches the downstream model boundary is allowed to propagate out freely, owing to the radiation boundary condition. The above initial setting allows the cold pool and associated circulations to evolve freely in time without arbitrary specification of cold pool properties on the lateral boundaries, for example, as in Droegemeier and Wilhelmson (1987). Furthermore, since both the initial upstream and downstream flows are unperturbed, the far-field solutions satisfy the flow–force balance. This treatment is similar to that of Rotunno et
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al. (1988) (see their Fig. 19); however, our computational domain is much wider (L/H Å 40) than theirs (L/H Å 4), and more importantly as shown in (3.3) and Fig. 3, our initial cold pools are also much wider (2L0 Å 8) and deeper ( h 0 ú 0.5 for a positive shear) than theirs (2L0 Å 1 and h 0 Å 0.2) in dimensionless space. Because their initial pool was quite small, their simulated density currents could not achieve quasisteady states in the two cases of positive shear (see their Fig. 20). As will be shown later, the initial cold pools specified in this paper can provide a sufficient cold air supply necessary for achieving and maintaining quasisteady density currents.
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c. Initial conditions and classification of experiments We report on three sets of numerical experiments that differ only in the specification of the initial states. In the first set (B0–B3), the initial flows on the upstream side of the density currents are prescribed to closely resemble those of the idealized steady-state models discussed in the previous section (hereafter we refer to these as idealized initial states). In the second set (UB1–UB2), the initial depth of the cold pool deviates from that of the idealized model for a given environmental flow (hereafter, nonidealized initial states) to allow for an examination of the dependence of the final steady state (if it can be achieved) on the initial condition. The third set consists of a single experiment that uses a deep cold pool in strong shear. First, we consider the idealized initial states, the shape of the density current front, and the overrunning flow field. The theoretical results reviewed in section 2 concern the system-relative flow in the vicinity of the density current front. With our numerical experiment design, the cold pool has two density fronts, one propagating upstream and the other downstream. We focus on the density current front on the upstream side where, for idealized initial states, the front is expected to remain quasi-stationary relative to the model grid. The density current front on the downstream side tends to
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XU ET AL. TABLE 1. Initial settings and simulated values of the kinematic parameters.
Experiment Idealized initial states (h0 Å h) B0 B1 B2 Nonidealized initial states (h0 x h) UB1 UB2 High shear case (h0 Å h ú 0.8) B3
a0
c0
h0
Simulated h
Simulated c0
0.00 00.84 0.88
0.50 0.326 0.755
0.5 0.3 0.7
0.405 0.214 0.667
0.486 0.318 0.722
00.84 0.88
0.326 0.755
0.5 0.5
0.215 0.643
0.322 0.760
1.25
0.9
0.857
1.30
2.26
propagate quickly, removing large quantities of air from the initial cold block. In order for the upstream density front to reach a quasi-steady state, the cold air supply must be sufficient during the integration period, which in turn requires the initial volume of the cold pool to be sufficiently large. Recalling the discussion in section 2c, we construct a symmetric cold pool centered at x Å x0 at the initial time whose shape is described by
5
h
0
z 0 ( j ) Å h 0 0 a( j 0 L0 ) 2 0
for
0 õ j õ L0
for
L0 õ j õ L0 / L1 ,
for
L0 / L1 õ j
0
where z ( 0 j ) Å zq_( j ), j å x 0 x0 , L0 Å 4, a Å 3/ (4h), and L1 Å 2/ 3 in dimensionless space. Here, the interface slope at the front is ensured to be 607. For an idealized initial state, the initial depth (h 0 ) of the cold pool should be determined from the theoretical h curve in Fig. 2 in accordance with the given inflow shear ( a0 ), while the surface inflow speed ( c0 ) is determined from the theoretical c0 curve in Fig. 2. In this case, as shown in Fig. 3, the upstream density front described by z 0 ( j ) with h 0 Å h Å 0.7 has approximately the same shape as that in Fig. 4c of X92. The initial flow field outside the cold pool is obtained by solving the following vorticity equation with given boundary conditions: Ç2c 0 Å a0
0
c Å
5
c0 0 a0 /2 0
along upper boundary z Å 1, 0
Ìc / Ìx Å 0
at lateral boundaries,
(3.4)
where c 0 is the initial streamfunction that defines the initial velocity: u 0 å 0Ìc 0 / Ìz and w 0 å Ìc 0 / Ìx , and z 0 is given by (3.3). Mass continuity is satisfied by this initial flow, and vorticity is conserved (constant a0 ) along the streamlines. When the initial depth of the cold
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h 0 £ z £ 1,
(3.5)
0
where c å 0.5a0 (1 0 h ) / ( c0 0 0.5a0 )/(1 0 h 0 ). If h 0 Å h, then c 01 Å c1 , and u 0out (z) is the same as in the idealized solution [see (2.3) of XM94]. In this case, the initial outflow u 0out (z) in (3.5) carries the same mass flux, energy, and vorticity as the upstream inflow, while the flow force balance is initially satisfied not only between the upstream and downstream lateral boundaries of the computational domain but also between the upstream boundary and the vertical section across the flat top of the cold pool (e.g., at x Å 20). The initial shape of the cold pool in (3.3) and the associated initial flow in (3.4) can also depart significantly from the theoretical steady-state model if the cold pool is significantly deeper or shallower than the theoretical depth (h) determined in accordance with the given shear ( a0 ). In this case, the initial flow described by (3.5) above the flat top (e.g., at x Å x0 ) of the cold pool carries the same mass flux and vorticity, but not the same energy, as the upstream inflow. Again, because the upper and lower boundaries are rigid and free slip, the flow force balance should be satisfied initially between the upstream and downstream lateral boundaries. 4. Quasi-steady state and kinematic properties
along lower boundary (interface) z Å z 0
u 0out (z) Å u 0 (x0, z) Å 0 c 01 / a0 (z 0 h 0 ) 0 1
(3.3) 0
pool is determined from the theoretical h curve in Fig. 2, that is h 0 Å h, the flow field described by (3.4) is quite similar to the idealized flow ahead of and above the density current in Fig. 4 of X92. The initial outflow above the flat top of the cold pool is horizontal and can be expressed by
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Six numerical experiments (B0–B3, UB1, and UB2) are performed to examine the evolution and quasi-steadiness of the upstream density current front for different initial settings of inflow shear and cold pool depth. As shown in Table 1, the inflow shears are zero in experiment B0, negative in B1 and UB1, moderately strong and positive in B2 and UB2, and strong and positive in B3. The initial cold pool depths are specified according to theoretical steady-state predictions only in cases B0, B1, B2, and B3.
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a. Idealized initial state with no inflow shear Consider first experiment B0, which consists of a density current propagating in a uniform environmental flow. In this case, the theoretical model in section 2 reduces to the classic density current model of Benjamin (1968). Because the cold pool depth is initially set to the theoretical value (h Å 0.5), the upstream front remains nearly stationary (exhibiting a slow rearward movement of about 00.14 m s 01 Å 00.014U) until the cold air mass becomes insufficient to maintain the cold pool depth during the later part (about 40 min Å 24T ) of the simulation. [Because the front on the downstream side (see Fig. 3) is initially out of balance with the forward flow, it surges ahead and propagates at a speed faster than the downstream retreating flow, causing the cold pool to collapse rapidly on this side and drain the cold air from the source region.] The frontal structure maintains a quasi-steady configuration in the system-relative framework, which is made clear when we compare the wind vector and perturbation potential temperature fields averaged over a period of 10 min ( Å6T ) and sampled at 30-s ( Å0.3T ) intervals (Fig. 4b) with the instantaneous fields at 20 min ( Å12T ) (Fig. 4a). The shape and the sharpness of the front are retained in the time-averaged fields, suggesting very little transient activity in the frontal zone. The upper density interface is subject to KH instability due to the presence of strong shear, and smallamplitude KH billows are indeed generated near the front in a quasi-periodic fashion. As these features propagate downstream along the interface, they grow to large amplitude and eventually break down, producing a mixed layer consisting of both light and heavy fluid [see Fig. 4a and Droegemeier and Wilhelmson (1987)]. Note that the billows are completely smoothed out in the time-averaged fields, leaving a layer of well-mixed air having an overall negative vorticity. Since the major interest of this paper is to examine the time-averaged structures and kinematic properties of the simulated density currents, a two-dimensional framework seems adequate for the goals sought in this paper, though it is not adequate for simulating the detailed turbulent decay processes of KH billows due to the incorrect treatment of turbulence energy cascade in two dimensions. The time-averaged fields in Fig. 4b show a noticeable decrease in the depth of the heavier-fluid layer on the back side of the head. As the depth of this layer decreases abruptly, the lighter-fluid layer undergoes a turbulent hydraulic jump, and the upper-layer flow switches from supercritical to subcritical (Benjamin 1968; X92). The simulated flow structure and its evolution bear a close resemblance to that of laboratory experiments conducted by Britter and Simpson (1978) and the numerical simulations of Droegemeier and Wilhelmson (1987). The front portion of the simulated density current, shown in Fig. 4c, is close to the inviscid
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solution of Fig. 4b in X92, while the slope of the front is very close to 607. The depth of the head is slightly shallower (h Å 0.405 as measured by the contour of u * Å 0Du /2 Å 01.5 K) than the theoretical value (h Å 0.5). Since the simulated density front moves rearward very slowly ( 00.14 m s 01 Å 00.014U) in the computational domain, the propagation speed (c0 Å 0.486) is very close to that of the idealized front (c0 Å 0.5). These values are compared against the theoretical curves in Fig. 2. b. Idealized initial states with inflow shear Turning now to experiment B1, we examine a density current propagating in a negative shear flow ( a0 Å 00.84). Because the initial depth of the cold pool is set to the theoretical value (h Å 0.3), the upstream front exhibits a very slow rearward movement of about 00.08 m s 01 ( Å 00.008U). After about 40 min ( Å 24T ), the cold air mass becomes insufficient to maintain this state. The KH waves develop in a manner similar to those in experiment B0, except that the amplitude is smaller due to the reduced depth of the density current (Fig. 5a). A qualitatively similar relationship between KH billow size and cold pool depth was noted by Droegemeier and Wilhelmson (1986) for density currents propagating in a stratified flow. The front portion of the density current maintains a quasi-steady configuration in the system-relative framework, while the frontal structure, as shown in Figs. 5b and 5c, is very close to the inviscid solution of Fig. 4b in X92. In particular, the frontal slope is very close to 607, though the depth of the head is shallower (h Å 0.214 as measured by the 01.5 K contour of u * ) than the theoretical value (h Å 0.3), and the propagation speed (c0 Å 0.318) is very close to the theoretical prediction (c0 Å 0.326) in Fig. 2. In experiment B2, the inflow now has positive shear with a0 Å 0.88 and an initial cold pool depth of h Å 0.7. A nearly stationary (with a very slow forward movement of about 0.17 m s 01 Å 0.017U) density front is again obtained, and it remains quasi-steady during the first 40 min ( Å24T ) of the simulation. The KH waves develop along the sloping front and break down behind the head in a manner similar to those in experiments B0 and B1, though the amplitude is larger due to the greater depth of the density current (Fig. 6a). The time-averaged flow (Figs. 6b and 6c) is very close to the inviscid solution of Fig. 4c in X92, and the front has an approximately 607 slope. The depth of the density current head is slightly shallower (h Å 0.667) than the theoretical value (h Å 0.7), while the propagation speed (c0 Å 0.772) is again very close to the theoretical prediction (c0 Å 0.755, see Fig. 2). These three experiments, in which the initial depths of the cold pool are set to the theoretical values in Fig. 2, clearly demonstrate that density currents remain
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FIG. 4. Wind and perturbation potential temperature (density) fields (a) at 20 min (Å12T ), (b) averaged over the period from 20 to 30 min, and (c) streamline field averaged as in (b) for experiment B0. When the averages are performed in (b) and (c), the fields are sampled at a 0.5-min interval with the front locations shifted to coincide with the location at 20 min. The flows are system relative (with a small grid-relative propagation speed of the front subtracted) in the nondimensional coordinate framework of the domain. The annotated x-coordinate origin is located at the initial position of the density current front on the right side to facilitate the comparison among experiments. The horizontal and vertical velocity scales (both 10 m s01) are shown by the arrows in the lower-left corner. The temperature contours are plotted every 1 K (Å Du/3 corresponding to 1/30 of the buoyancy scale:
quasi-stationary and quasi-steady from the onset of the integration and that the theoretically predicted depth is maintained for a considerable period of time (approximately 40 min or 24T, which could nominally be extended if the supply of cold air were continuous). c. Nonidealized initial states with inflow shear We now explore two experiments, UB1 and UB2, designed to show that the achievement of a quasi-
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steady-state density current does not depend upon the initial state provided that the initial cold pool has sufficient mass to undergo the necessary transient adjustments. As shown in Table 1, both UB1 and UB2 have an initial cold pool depths of h Å 0.5; however, UB1 has an inflow shear of a0 Å 00.84 (the same as in B1), while UB2 has an inflow shear of a0 Å 0.88 (the same as in B2). These shears correspond to theoretically predicted depths of 0.3 and 0.7, respectively, rather than
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gDu/u0), and the streamlines are every 400 m2 s01 (Å 0.04HU ).
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FIG. 5. As in Fig. 4 but for experiment B1.
the given 0.5. This initially overspecified (underspecified) cold pool depth causes the density current to surge ahead (retreat backward) during the first 5 min of integration, thus dropping (raising) the depth of the upstream head. Figure 7 (Fig. 8) shows that the location of the front is farther to the right (left) than in experiment B1 (B2) as a result of the initial surge (retreat). (The origin x Å 0 in the plots is placed at the initial position of the front nose in all cases.) By 15 min ( Å9T ), the density front has evolved into a quasisteady state that is very similar to that in experiment B1 (B2), with the average propagation speed [c0 Å 0.322 (0.760) in UB1 (UB2)] being very close to but slightly slower (faster) than that of the idealized front. The similarity between the counterpart solutions
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demonstrates that the quasi-steady solutions are controlled solely by the inflow shear, lending further support to the prediction of the theoretical models of X92 and XM94. d. Very strong inflow shear Finally, experiment B3 simulates a density current propagating into a very strong positive shear ( a0 Å 2.26). In this case, the initial cold pool [see (3.3)] has the same depth as the theoretical prediction (h Å 0.9), but its frontal slope is not sufficiently steep in the middle depths (z É h/2). The density current undergoes about 10 min ( Å0.6T ) of initial adjustment, with the upstream front steepened toward the vertical
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FIG. 6. As in Fig. 4 but for experiment B2.
in the middle depths (z É h/2). Thereafter, the front evolves into a quasi-steady configuration with a very slow forward movement (about 0.50 m s 01 Å 0.05U ). This quasi-steady frontal structure, shown in Fig. 9a, is quite different from its initial shape and those in experiments B0, B1, and B2. In particular, the interface is locally steeper than 607 and is concave (viewed from top and upstream) at the middle depth, even though the slope at the front remains 607. This configuration (see Figs. 9b and 9c) is similar to that in Fig. 8a of XM94 except that here the front is more vertical at the middle depth in accordance with the very strong inflow shear and that the flow is not exactly steady but quasi-steady. In this case, the light fluid layer above the density current head is very shallow, so the KH billows are se-
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verely restricted by the upper rigid boundary and become significantly shallower than the depth of the density current. As shown in Fig. 9b, the depth of the density current head (h Å 0.857) is slightly shallower than the theoretical value (h Å 0.9), while the propagation speed (c0 Å 1.30) is very close to the theoretical prediction (c0 Å 1.25) in Fig. 2. The proximity of the model’s rigid upper boundary to the top of the density current and circulations forced by it clearly limit the applicability of this particular experiment to atmospheric flows. Indeed, these results are more germane to laboratory flows, where we believe that a quasi-steady state in strong shear is achievable only with a rigid upper lid and a density current depth comparable to that of the containing vessel. As
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FIG. 7. As in Fig. 4a but for experiment UB1.
discussed further in section 6, these results form an important underpinning that must be established and understood before more physically realistic and complex simulations (e.g., without rigid upper boundaries) can be undertaken. Summarizing the suite of experiments in section 4, we have shown that the propagation speed of simulated density currents can be quite accurately predicted by inviscid theory even though the simulated steady-state head is always shallower than the inviscid limit due to turbulence generated by transient KH eddies. Further, the propagation speed of the front is slightly faster than theoretical predictions for a positive inflow shear (experiments B2, UB2, and B3) and slightly slower for a zero or negative inflow shear (experiments B0, B1, and UB1). To understand these kinematic similarities and differences, we must examine the related dynamic properties of the simulated flows and compare them with inviscid theory as well as with theory that considers the effects of energy loss and vorticity generation due to internal mixing and friction (Benjamin 1968; X92). We do so in the next section. 5. Conservation properties and physical interpretation of the simulations The kinematic properties of idealized inviscid flow are uniquely determined by the Lagrangian conserva-
tion of vorticity and Bernoulli energy, along with the flow force balance and mass continuity. To understand why inviscid theory can predict the propagation speed better than the depth of the simulated density currents, we must examine to what extent these conservation properties can be satisfied in the simulated flows. In general, mass continuity is satisfied, while the Lagrangian conservation of vorticity and Bernoulli energy may not be. With the free-slip boundary conditions assumed in the model, the flow force balance should be quite well satisfied for the time-averaged flow between the upstream and downstream vertical cross sections of the control volume as long as the time-averaged flow is quasi-steady at the latter cross section. These speculations are examined in detail for experiment B0, the results of which are representative of B1 and B2 as well. The dimensional Bernoulli energy, E å u 2 /2 / p * / r0 , and vorticity, z å uz 0 wx , are computed for the time-averaged flow (Figs. 4b and 4c) in experiment B0. Their vertical profiles are shown (thick curves) in Figs. 10a and 10b at four vertical cross sections: the right boundary (RB, x Å 1.6 (upstream side), x Å 00.8 (density current head), and x Å 01.6 (behind the head, as shown in Fig. 4b). (Note that x Å 0 represents the initial position of the front.) The idealized profiles obtained from inviscid theory are shown by thin-dashed curves (upstream cross section) and thin solid curves
FIG. 8. As in Fig. 4a but for experiment UB2.
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FIG. 9. As in Fig. 4 but for experiment B3.
(downstream cross section in Fig. 1). Because the inflow is uniform, Bernoulli energy has a constant vertical profile, and vorticity is zero at both the right boundary and at x Å 1.6. Note that the two thick-dashed curves (for RB and x Å 1.6) both coincide with idealized profiles (thin dashed) in Figs. 10a and 10b; thus, the idealized upstream inflow profile is well maintained in the model, at least from x Å 1.6 to the upstream lateral boundary. By comparing the thick solid lines (at x Å 00.8, crossing the density current head) with the thin solid (idealized) profiles in Figs. 10a and 10b, we see that Bernoulli energy is nearly conserved and vorticity is roughly conserved above the density current head and within the cold pool but neither is conserved in the
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mixed interfacial layer. The negative vorticity spike in this layer is generated by the cross-frontal buoyancy difference (baroclinic generation), and in the idealized inviscid profile, this spike is sharpened into a delta function with infinitely large amplitude as shown schematically by the thin horizontal line in Fig. 10b. Farther downstream behind the density current head (at x Å 01.6 in Fig. 4b), as shown in Figs. 10a and 10b, Bernoulli energy is still nearly conserved, and vorticity is still roughly conserved above the density current, while the mixed interfacial layer becomes deeper, and the negative vorticity spike is further smeared and reduced in amplitude. Note also that Bernoulli energy has the same value (E Å c 20 /2) along the lower boundary (z Å 0), consistent with theoretical analyses (Benjamin
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FIG. 10. Vertical profiles of (a) dimensional Bernoulli energy E å u2/2 / p*/r0 (m2 s02) and (b) vorticity z å uz 0 wx (s01) for the timeaveraged flow (Fig. 4) in experiment B0. The thick curves are computed at four vertical cross sections: the right boundary (RB), x Å 1.6 (upstream side), x Å 00.8 (density current head), and x Å 01.6 (behind the head). The thin-dashed and solid curves are the idealized profiles (obtained from the inviscid theory) at the upstream and downstream cross sections (see Fig. 1), respectively. The nondimensional values for E and z can be obtained by dividing their dimensional values with the scaling parameters: U2 Å 100 m2 s02 and U/H Å 1002 s01, respectively.
1968; X92). The Bernoulli energy directly above the cold pool at x Å 00.8 is roughly equal to that at z Å 0 at the upstream boundary (RB), suggesting that Bernoulli energy is indeed conserved along the streamline that starts at the z Å 0 level and runs over the cold pool. Similar behavior can be seen in Fig. 11 and Fig. 12 (experiments B1 and B2). Although mass continuity is well satisfied in the model, the flow force balance needs to be examined for the time-averaged flows. The flow force balance in (2.5) is derived by integrating the horizontal momentum equation for inviscid steady-state flow over the control domain (see Fig. 1) while making use of mass continuity. The same flow force balance can be derived for steady viscous flows as long as the boundary condition is free slip and the diffusion term is of the Fickian type. The flow force balance (2.5) can be put in a convenient form in association with the following definition of flow force F : Få
* (p * / u ) dz Å const 2
for all values of x. (5.1)
Because the diffusion used in the numerical model (see section 3a) consists of fourth-order differentials of the non-Fickian type, we derive the flow force balance for the time-averaged flow by integrating the timeaveraged version of (3.1) over the domain to obtain
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Fd å F0 / F 9 Å const
for all x,
* ( » p *… / » u … )dz É const, F 9 å * » u 9… dz / ** » u 0 D … dzdx, 2
F0 å
2
t
u
(5.2a) (5.2b) (5.2c)
where »·… is the time-average operator covering the period from 20 (12T ) to 30 min (18T ) in our case and u 9 å u 0 » u … is the transient part of u. The horizontal integral in (5.2c) is from the upstream lateral boundary to the concerned vertical cross section at x. As shown in Fig. 13, F0 is indeed nearly independent of x for the time-averaged flow (experiments B0, B1, and B2) and is very close to the theoretical value of F in the analytical solution. In the upstream region, the flow is exactly steady and horizontally uniform so that F 9 Å 0 and thus F0 Å F. In the vicinity of the front, F 9 is small but not zero so that F0 becomes slightly different from F. The last term in the second integral in (5.2c) is related to the numerical diffusion and is found to be negligibly small, and thus the difference F 9 Å F 0 F0 is mainly caused by transient motion. Among the two terms related to transient motion, the contribution from the time-averaged local tendency should be small given the quasi-steady nature of the solution. However, the first term can be large, especially in the layer where strong KH billows exist. This explains why F 0 F0 is nearly zero in experiment B1 but relatively large in B2.
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FIG. 11. As in Fig. 10 but for experiment B1.
The above analyses show that the Lagrangian conservation of vorticity and Bernoulli energy are approximately satisfied above the density current head but not in the mixed interfacial layer. Further, the flow force balance is quite accurately satisfied for the time-averaged flow, explaining why inviscid theory is valid in describing the gross flow structures. However, it is not
obvious why this theory can better predict the propagation speed than the depth of the simulated density current. To examine this problem, we need to consider the effects of energy loss and negative vorticity generation. According to Benjamin (1968) and X92, when the effects of energy loss and negative vorticity generation
FIG. 12. As in Fig. 10 but for experiment B2.
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FIG. 13. Simulated flow force F0 (plotted in m3 s02 as functions of x for the time-averaged flows in experiments B0, B1, and B2) vs analytical flow force F (obtained from the inviscid solutions). The nondimensional value for flow force can be obtained by dividing the dimensional value with the scaling parameter: HU2 Å 105 m3 s02.
are taken into account for the entire or physically constrained fractional depth of the upper-layer outflow, multiple solutions are found for two possible flow states: supercritical and subcritical. The supercritical (subcritical) state is characterized by a large (small) Froude number for the downstream upper-layer outflow, in which case the density current is shallower (much shallower) and propagates slightly faster (significantly slower) than its idealized inviscid counterpart. Figure 2 shows that the depths of the simulated density current heads are quite close to their inviscid limits. By comparing these depths with the theoretical curves in Figs. 7a–10a of X92 (for wide ranges of energy loss and negative vorticity generation), we find that the depths of the simulated density currents are all in the supercritical ranges. Note that the supercritical propagation speeds with energy loss, as predicted by the theoretical (solid) curves in Figs. 7b–10b of X92, are slightly higher than their respective inviscid propagation speeds. This may explain why the simulated density currents propagate slightly faster than the inviscid theoretical prediction for a positive inflow shear (experiment B2, UB2, and B3), even though the density current is shallower than its inviscid limit. Further, according to XM94 (see their Fig. 2), the weak circulation within the cold pool may also enhance the propagation speed, especially when the cold pool is deep (corresponding to a positive inflow shear as in experiments B2, UB2, and B3). When the inflow shear is uniform or negative, and thus the cold pool is shallow (experiments B0, B1, and UB1),
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the above two factors are overshadowed by the effect of internal friction between the cool pool air and the flow above. This may explain why the simulated density currents propagate slightly slower than the theoretical prediction for a zero or negative inflow shear (experiments B0, B1, and UB1). Since the flow is supercritical, the frontal propagation speed is insensitive to energy loss and negative vorticity generation, which explains why the propagation speed of the current is better predicted by theory than its depth. The density current propagation is apparently not sensitive to the mixing of density (or potential temperature) along the interfacial layer, a process that was not considered in Benjamin (1968) nor X92. The energy loss considered in X92 was purely due to friction (no mixing of mass), while the Bernoulli energy in the mixed interfacial layer combines the energy carried by the warm air (with a certain energy loss) from upstream and the energy possessed by the cold air (in proportion to their respective mass). Because of these differences, the Bernoulli energy profiles within the mixed interfacial layer in Fig. 10 cannot be quantitatively compared with the results in X92. Qualitatively, however, our results suggest that the vertical distribution of energy loss derived in X92 is valid and should occur, together with negative vorticity generation, over a finite depth above the density current and with a decrease in intensity with height. This vertical distribution of energy loss is at least more realistic than the uniform distribution of energy loss assumed by Benjamin (1968). Our numerical results confirm the theoretical analyses of X92 and XM94 with regard to the roles and relative importance of the above conservation properties in controlling the structure and propagation of the density current in a dissipative system. Clearly, energy conservation is a weak condition because interfacial dynamical instability is pervasive in density currents and leads to turbulence and kinetic energy dissipation. Vorticity conservation depends on energy conservation (see the appendix of XM94) and is also a weak constraint for the global flow compared to mass continuity and flow force balance that apply even in dissipative flows. The theoretical analyses of X92 and XM94 suggested that the flow force balance is the principal global property that controls the flow structure and the interaction between the environmental shear and the cold pool circulation under the strong constraint of mass continuity and weak constraints of conservation of energy and vorticity. Thus, the well-maintained flow force balance in our numerical experiments is the key factor that explains why the simulated density currents propagate at nearly the same speed as predicted by inviscid theory, while the nonconservative nature of Bernoulli energy and vorticity is responsible for the reduced depth of the simulated flow compared to its inviscid limit.
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6. Summary and conclusions Numerical simulations were performed to study the kinematics and dynamics of two-dimensional density currents propagating in a uniformly sheared environmental flow. The results show that the density current theory of X92 and XM94 is not only valid in describing the gross flow structures but also quite accurate in predicting the frontal propagation speeds. The initial depths of the cold pool in the model were specified in two ways relative to the environmental shear: closely resembling or deviating significantly from those of theoretical steady-state models. In both cases, the density front always reached roughly the same quasisteady state, and this state was maintained as long as a sufficient supply of cold air was available. Detailed diagnoses of the numerical results showed that Bernoulli energy and vorticity are approximately conserved above and below the mixed interfacial layer near the front but are not conserved in the mixed interfacial layer where dynamical instability leads to turbulence and kinetic energy dissipation. In general, energy and vorticity conservation are interdependent and are weak constraints for the global flow compared to mass conservation and flow force balance. Mass continuity is always satisfied, while the flow force is found to be nearly balanced for the simulated flows, especially in the time-averaged sense (Fig. 13). These results confirm the theoretical analyses of X92 and XM94 with regard to the roles and relative importance of the above conservation properties in controlling the structure and propagation of density currents. The quasi-steady density currents simulated in this paper are subject to an important limitation: the flow is restricted by a rigid upper boundary. When this boundary is replaced by an inversion layer and a deep region of stratified fluid aloft (like in the real atmosphere), a moderately steep front might still be maintained to a certain degree of steadiness (but less or much less steady than obtained in this paper) if the shear is not overly strong and if the jump updraft can be deflected by and beneath the inversion layer to flow rearward at a supercritical speed. However, if the shear is too strong, the jump updraft tends to be nearly vertical and becomes too strong to be deflected by the inversion layer. In this case, the density current may become grossly unsteady with a collapsed subcritical head (as speculated in Fig. 11b of X92). An obvious condition necessary for maintaining a quasi-steady state is that the initial cold pool must have sufficient mass to provide an essentially continuous source of dense air. A limited cold air mass may also cause a flow-state transition as the positive shear becomes very strong, and thus, the required depth for a quasi-steady density current cannot be sustained by the limited cold air mass. The collapsed density current head may then behave subcritically with a significant drop in depth [as sketched in Fig. 11b of X92 in con-
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nection with the numerical result in Fig. 20c of Rotunno et al. (1988)]. Until the low-level environmental shear becomes sufficiently strong, the density current head may behave supercritically, and its depth will then increase with the jump inflow shear. In this sense, an optimal inflow shear might exist (in terms of producing the deepest density current head and maximum lifting of the low-level inflow to maintain a long-lived squall line), though further investigation is needed. Furthermore, in the real atmosphere the cold pool is produced by thunderstorm outflow, while the latter is controlled by the environmental flow, so the effects of the environmental shear on the availability or continued supply of cold air to the density current may also indirectly control the realization of a quasi-steady state. The numerical experiments presented in this paper are intended to establish an underpinning for more physically realistic and complex simulations that are now being conducted or are planned for the future (e.g., including moist process and surface friction without rigid upper boundary). It is our belief that basic solution characteristics in simple model configurations, such as those used here, must be clearly understood before moving on to more complete scenarios where the complexities of the flow are much more difficult to unravel. Acknowledgments. The numerical simulations were performed on the Cray-C90 at the Pittsburgh Supercomputer Center, with postprocessing and analysis conducted on the CAPS cluster of IBM RISC System6000 workstations at the University of Oklahoma. Figures were produced using the ZXPLOT graphics package written by the second author. The authors gratefully acknowledge Rit Carbone and two anonymous reviewers for providing comments that improved the quality of the manuscript. This research was supported by NOAA Grant NA37RJ0203 and NSF Grant ATM9113906 to the Cooperative Institute for Mesoscale Meteorological Studies (CIMMS), by NSF Grant ATM91-20009 to the Center for Analysis and Prediction of Storms (CAPS), and by NSF Grant ATM9222576 to the third author. Additional information about this research as well as related projects and organizations can be obtained electronically through a World Wide Web server at URL address http:// wwwcaps.gcn.uoknor.edu/. An animated movie of the simulated density currents can be obtained from the authors. REFERENCES Anderson, J. K., K. K. Droegemeier, and R. B. Wilhelmson, 1985: Simulation of the thunderstorm subcloud environment. Preprints, 14th Conf. on Severe Storms, Indianapolis, IN, Amer. Meteor. Soc., 147–150. Benjamin, B. T., 1968: Gravity current and related phenomena. J. Fluid Mech., 31, 209–248. Blumen, W., 1984: An observational study of instability and turbulence in nighttime drainage winds. Bound.-Layer. Meteor., 28, 245–269.
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Britter, R. E., and J. E. Simpson, 1978: Experiments on the dynamics of a gravity current head. J. Fluid Mech., 88, 223–240. CAPS, 1992: Advanced Regional Prediction System (ARPS) Version 3.0 User’s Guide. Center for Analysis and Prediction of Storms, University of Oklahoma, 183 pp. [Available from Center for Analysis and Prediction of Storms, University of Oklahoma, Room 1110, 100 E. Boyd St., Norman, OK 730190628.] Droegemeier, K. K., 1985: The numerical simulation of thunderstorm outflow dynamics. Ph.D. dissertation, University of Illinois, 695 pp. [Available from Dept. of Atmospheric Science, University of Illinois, 105 S. Gregory Ave., Urbana, IL 81801.] , and R. B. Wilhelmson, 1986: Kelvin–Helmholtz instability in a numerically simulated thunderstorm outflow. Bull. Amer. Meteor. Soc., 67, 416–417. , and R. P. Davies-Jones, 1987: Simulation of thunderstorm microbursts with a super-compressible numerical model. 5th International Conference on Numerical Methods in Laminar and Turbulent Flow, Montreal, PQ, Canada, Concordia University, Pratt and Whitney, Canada, and the National Science and Engineering Research Council of Canada, 1386–1397. , and R. B. Wilhelmson, 1987: Numerical simulation of thunderstorm outflow dynamics. Part I: Outflow sensitivity experiments and 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: A scalable storm-scale model. High Performance Computing, G. Sabot Ed., Addison-Wesley, 45–92. Janish, P. R., K. K. Droegemeier, M. Xue, K. Brewster, and J. Levit, 1995: Evaluation of the Advanced Regional Prediction System (ARPS) for storm-scale modeling applications in operational forecasting. Proc., 14th Conf. on Weather and Forecasting, Dallas, TX, Amer. Meteor. Soc., 224–229. Johnson, K. W., J. Bauer, G. A. Riccardi, K. K. Droegemeier, and M. Xue, 1994: Distributed processing of a regional prediction model. Mon. Wea. Rev., 122, 2558–2572. Klemp, J. B., and R. Wilhelmson, 1978: The simulation of threedimensional convective storm dynamics. J. Atmos. Sci., 35, 1070–1096. Moncrieff, M. W., 1978: The dynamic structure of two-dimensional steady convection in constant shear. Quart. J. Roy. Meteor. Soc., 104, 543–567. , 1992: Organized convective systems: Archetypal dynamical models, momentum flux theory, and parameterization. Quart. J. Roy. Meteor. Soc., 118, 819–850.
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, and D. K. So, 1989: A hydrodynamic theory of conservative bounded density currents. J. Fluid Mech., 198, 177–197. Mueller, C., and R. Carbone, 1987: Dynamics of a thunderstorm outflow. J. Atmos. Sci., 44, 1879–1898. Orlanski, I., 1976: A simple boundary condition for unbounded hyperbolic flows. J. Comput. Phys., 21, 251–269. Rotunno, R., J. B. Klemp, and M. L. Weisman, 1988: A theory for strong long-lived squall lines. J. Atmos. Sci., 45, 463–485. Simpson, J. E., 1969: A comparison between laboratory and atmospheric density currents. Quart. J. Roy. Meteor. Soc., 95, 758– 765. Skamarock, W. C., and J. B. Klemp, 1992: The stability of timesplitting numerical methods for the hydrostatic and the nonhydrostatic elastic equations. Mon. Wea. Rev., 120, 2109–2127. , and , 1993: Adaptive grid refinement for two-dimensional and three-dimensional nonhydrostatic atmospheric flow. Mon. Wea. Rev., 121, 788–804. Straka, J. M., 1989: Hail growth in a highly glaciated central high plains multi-cellular hailstorm. Ph.D. dissertation, University of Wisconsin, 413 pp. [Available from the Department of Meteorology, University of Wisconsin, 1225 W. Dayton St., Madison, WI 53706.] Thorpe, A. J., M. J. Miller, and M. W. Moncrieff, 1982: Two-dimensional convection in non-constant shear: A model of midlatitude squall lines. Quart. J. Roy. Meteor. Soc., 108, 739–762. Tripoli, G. J., 1992: A nonhydrostatic mesoscale model designed to simulate scale interaction. Mon. Wea. Rev., 120, 1342–1359. von Ka´rma´n, T., 1940: The engineer grapples with non-linear problems. Bull. Amer. Math. Soc., 46, 615–683. Wakimoto, R. M., 1982: The life cycle of thunderstorm gust fronts as 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 in shear flows. J. Atmos. Sci., 51, 434–446. , F. S. Zhang, and G. P. Lou, 1992: Finite-element solutions of free-interface density currents. Mon. Wea. Rev., 120, 230–233. Xue, M., K. K. Droegemeier, V. Wong, A. Shapiro, and K. Brewster, 1995: Advanced Regional Prediction System (ARPS) Version 4.0 User’s Guide, Center for Analysis and Prediction of Storms, 320 pp. [Available from Center for Analysis and Prediction of Storms, University of Oklahoma, Room 1110, 100 E. Boyd St., Norman, OK 73019-0628.]
AMS: J At Sci (March 1 96) 1311
