1. Introduction

The characteristic metastability of the planetary boundary layer means that air has to be lifted through a finite distance before convection is initiated. Density currents are a very effective lifting mechanism and were they less common (i.e., if the boundary layer were typically saturated), the distribution of precipitating convection would almost certainly depart from the observed state. This statement follows from the linear perturbation theory of an unstably stratified shear flow, where the preferred orientation of convection is parallel to the ambient flow, whereas squall lines are commonly aligned perpendicular to the low-level flow. The alignment issue led to the use of localized cooling as a finite-amplitude convection initiation mechanism in numerical models (Moncrieff and Miller 1976) because the orientation of density currents generated by the cooling is consistent with the propensity for squall lines to be flow perpendicular.

A primary example of an atmospheric density current is the outflow of evaporatively cooled downdrafts from thunderstorms (Charba 1974; Goff 1976; Wakimoto 1982; Mueller and Carbone 1987). The interaction of downdraft outflows with ambient shear also affects the life cycle of convection, notably the longevity and dynamical structure of squall lines (Thorpe et al. 1982, hereafter TMM; Rotunno et al. 1988, hereafter RKW). These outflows can trigger convection at great distances from the source if they travel into a favorable environment (Carbone et al. 1990), are subjected to mesoscale oscillations (Crook et al. 1990), or collide with other outflows and convergence lines (Wilson and Schreiber 1986).

Sea breezes and land breezes are also described by density current dynamics (Simpson 1987). Although horizontal convergence due to differential heating is the ultimate cause of a sea breeze, the horizontal gradient of heating must exceed a critical value before it attains density current characteristics (Reible et al. 1993). Further examples are sharp cold fronts (Clarke 1961; Nielsen and Neilley 1990) and cold-frontal rainbands (James and Browning 1979; Carbone 1982; Hobbs and Persson 1982).

Theoretical and numerical models of density currents are reviewed in Liu and Moncrieff (1996a, hereafter LM96), who focused on the dynamical effects of ambient shear. Density currents in a shear flow have been studied analytically (Moncrieff and So 1989; Xu 1992;Xu and Moncrieff 1994; Liu and Moncrieff 1996b; Xue et al. 1997) and numerically simulated (Chen 1995; Xu et al. 1996).

Although convection initiation by density currents is primarily a function of horizontal convergence, the dynamical role of shear is not fully understood. An issue we emphasize is how shear affects lifting through its dynamical effect on organization. Because we consider unidirectional mean flow, there is little loss of generality in using a two-dimensional framework.

2. Relative effects of surface convergence and shear

The elements of density current dynamics have long been known (von Karman 1940; Benjamin 1968) in terms of the Froude number, which for atmospheric density currents is F = U₀[g(Δθ_υ/θ_υ)h]^−1/2, where h is the far-field depth of the current, U₀ the surface inflow, and −Δθ_υ the virtual potential temperature deficit in the density current. While the basic dynamics are captured by this classical theory, ambient shear has an influence over and above the bulk effect of the Froude number.

In a motionless environment, density current circulations are symmetric about a vertical plane centered on the cold air source. Uniform flow affects the dynamics through the Froude number. LM96 showed if the ambient flow opposes the system movement (headwind), the density current head is raised compared to calm surroundings while a following wind (tailwind) has the opposite effect.

The horizontal pressure gradient due to the hydrostatic effect of the cold air is basically why a density current propagates. The system-relative flow is strongly affected by the direction of the propagation velocity relative to the shear vector. Because, in our study, the low-level ambient shear is positive, we can unambiguously refer to the density currents as propagating either downshear (the direction of the shear vector) or upshear. LM96 and Xu et al. (1996) showed that weak or moderate shear elevates the head of the downshear-propagating current. When the current propagates at the speed of a sheared ambient flow at a steering level the airflow organization is markedly different. The upshear-propagating density current is steadier and less sensitive to shear than its downshear counterpart and its structure is basically described by classical theory.

a. Downshear propagation

The downshear-propagating model sketched in Fig. 1a has three components: (i) an overturning updraft that moves along with the ambient flow at z = H/2, (ii) a density current circulation characterized by a forward-directed surface flow, and (iii) a stagnant layer overlying the density current. Application of the Bernoulli equation to the surface flow relates U_L, U_R and the mean buoyancy in the density current, −gΔθ/θ. Assuming that the remote pressure field is hydrostatic, we obtain

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Because Δθ is nonnegative, G_R = U_L/U_R ⩽ 1 and the strength of the circulation in the cold air inflow is limited by the surface inflow to the density current. If G_R = 1, the temperature deficit is zero and the model represents either a gust front or a collision between two currents of equal density in an unstratified environment. Defining the Froude number as F²_R = U²_R/[g(Δθ/θ)h], we obtain

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showing F_R must not be less than 2.

The low-level mean vertical motion caused by the horizontal convergence and the effect of shear on its strength is the key issue in convection initiation. Referring to Fig. 1a and integrating the mass continuity equation in the domain −x_L ⩽ x ⩽ x_R; 0 ⩽ z ⩽ H gives the mean vertical velocity

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in which case

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where U_L, U_R, S_L, and S_R are nonnegative. The two terms on the right-hand side of Eq. (4) are the contributions by surface convergence and shear, respectively. Inspection of Eq. (4) shows that shear decreases the mean ascent caused by downshear-propagating density currents because horizontal convergence decreases with height.

No vorticity is generated within the overturning updraft; therefore, the height of no relative motion defining the steering level is exactly half the depth of the overturning circulation in which case U_R = HS_R/2. For similar reasons U_L = hS_L/2. Using Eq. (4) it follows on defining ζ = z/H and α = h/H that the ratio of the horizontally averaged updraft mass flux to the total inflow mass flux is

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The mean vertical motion is a maximum at z_m = (U_R + U_L)/(S_L + S_R); that is, ζ_m/α = (1 + G_R)/2(α + G_R). The left-hand side of Eq. (5) represents the total vertical mass flux normalized by the averaged inflow mass flux. Figure 2a shows the solution of Eq. (5) for ζ = ζ_m (z = z_m). The dashed line denotes the special case ζ_m = α (z_m = h) and divides the solution domain into two parts. The region on the right-hand side corresponds to ζ_m ⩽ α (z_m ⩽ h), whereas that on the left corresponds to ζ_m > α. In the latter, the maximum mass flux is obtained by setting ζ = α instead of ζ = ζ_m. The results indicate that while the maximum mass flux is proportional to the cold pool depth, it is insensitive to the variation of G_R, provided the cold pool is shallow.

Figure 2b presents the profiles of the mass transports for G_R = ½. Note that the solution for ζ > α (z > h) is, in any case, the same as in the absence of a cold pool (i.e., α = 0). Three kinds of distribution occur corresponding to a single maximum at ζ = ½ when α < ¼, dual maxima at ζ = ζ_m and ζ = ½ when ¼ < ζ < ½, and a single maximum at ζ = ζ_m when ζ > ½, respectively.

If R is the ratio of the shear term and the horizontal convergence term in Eq. (4), it follows that

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for 0 ⩽ z ⩽ h. Note that R vanishes at z = 0 where the horizontal convergence is a maximum.

Integrating from z = 0 to an arbitary height gives the average effect of shear on average ascent,

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and the average effect of horizontal convergence on average ascent,

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Defining the β = S_L/S_R, it follows that

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Figure 3 shows the solution of Eq. (9) in terms of the variation of R with α expressed in isopleths of β. The case of ζ = α/2 corresponds to the midlevel of the density current. The maximum value R_max = ⅓ occurs when the mean vertical motion is a maximum.

In summary, while shear decreases the horizontal convergence due to the downshear-propagating gravity current, it is nevertheless fundamental to the dynamical organization. In particular, the overturning branch lifts boundary layer air to much higher levels than the density current head alone can achieve.

b. Upshear propagation

The flow relative to the upshear propagating density current is shown in Fig. 1b. A kinetic energy discontinuity Δ[½u²] occurs at z = h. This value is obtained by applying the Bernoulli equation along the lower boundary and the two streamlines on either side of the density current interface to give

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where Δ[ ] is the difference operator. The interface separating the jump updraft from the density current is a vortex sheet (which may or may not be dynamically stable).

It can be shown that in this case

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for 0 ⩽ z ⩽ h. The effect of shear on convergence depends on the sign of (S_L − S_R). In contrast to the downshear regime, shear can increase the horizontal convergence and mean ascent in the upshear-propagating density current. For the special case S_R = S_L, Eq. (11) shows mean ascent increases linearly with height.However, no overturning branch exists to accentuate lifting—further evidence of shear-induced dynamical organization.

3. Numerical simulations

Consider a time-invariant, earth-stationary temperature perturbation in a column of height d and width l. The maximum temperature deficit is Δθ = −8 K at the lower boundary, decreasing linearly to zero at z = d. Scaling length, velocity, and pressure by l, U_s, and the averaged quantity DCAPE = g(Δθ/θ)d defines the convective Froude number (or inverse Richardson number) for the cold pool F_d = U_s(DCAPE)^−1/2. This is a simple way to relate our results to evaporative cooling in downdrafts.

Since the ambient flow profile is uniquely determined by the surface flow and the shear, the shear flow relative to the cold source is

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where S is the shear; h_s and H are the depth of the shear layer and that of the model, respectively; and U_s is the surface wind speed. The two profiles used are plotted in Fig. 4.

We conducted a pair of two-dimensional numerical simulations using a dry version of the Clark et al. (1996) numerical model, initialized with a horizontally homogeneous wind profile and a uniform potential temperature of θ₀ = 300 K. For simplicity we considered a fully mixed (neutrally stratified) planetary boundary layer. The two-dimensional domain is 160 km × 10 km. Constant 100-m grid intervals are used in both vertical and horizontal directions. As shown in Fig. 4, we set h_s = 3 km, d = 2 km, l = 4 km, and S = 4 m s⁻¹ km⁻¹. We assume S is a positive constant, so the combined effect of shear and surface flow is measured by U_s = ±6 m s⁻¹. Note that LM96 examined the impact of varying h_s and S (see their Fig. 13).

Density currents propagate rightward and leftward from the buoyancy source. Although the same shear affects both, the flows relative to the left-moving and right-moving currents are very different. LM96 showed how the ambient shear and the surface velocity affect the height of the density current head. Referring to the left-moving current in Fig. 5b, a headwind (ambient flow in the opposite direction to the direction of travel of the density current, which is synonomous to upstream movement) raises the head height compared to calm surroundings in Fig. 5a. On the other hand, examination of the right-moving current in Fig. 5b shows that a tailwind lowers the head height.

When the shear and surface flow vectors are in the same direction (experiment U1), the potential temperature perturbation (Fig. 6a) and the vertical velocity (Fig. 7a) show that the upshear current is deeper and stronger. This behavior is broadly consistent with the Froude number effect of a constant ambient flow (Fig. 5). However, when the shear and surface wind oppose each other (experiment U2), the opposite response occurs: the downshear current is deeper and more intense than its upshear counterpart (Figs. 6b and 7b). This is due to the combined effect of convergence (cf. Figs. 5a and 5b) and shear (cf. Fig. 4 and Fig. 9a of LM96) on density current dynamics.

The flow relative to the upshear and downshear head of U1 is shown in Figs. 8a and 8b, where the propagation speeds are approximately 6 and 17 m s⁻¹, respectively. The corresponding relative flows for U2 are shown in Figs. 8c and 8d, where the propagation speeds are 15 and 9.5 m s⁻¹, respectively. Figure 9 shows that the most obvious difference in the maximum vertical velocity between the upshear and downshear currents occurs for the U2 profile; the relative flows in the upstream and downstream currents are very different (section 4).

We examined two-dimensional density currents. Because the cross-stream effects of unidirectional ambient flow on three-dimensional density currents are second order, useful inferences can be made without invoking three-dimensional simulations. For instance, Fig. 5 approximates the transverse dynamical structure and Fig. 6 the streamwise structure.

4. Convection initiation

By definition convection will be initiated if air is lifted to the level of free convection (LFC). The strongest lifting occurs in organized flow of the type sketched in Fig. 1a, which is realized in Figs. 11b and 17b of LM96 and Fig. 8b herein.

5. Conclusions

It may be a surprise that basic issues remain in density current dynamics and convection initiation in view of the copious literature on these subjects but then the effects of shear on density currents have been little studied. We used analytic models and numerical realizations to put the relative roles of horizontal convergence, shear, and dynamical organization into perspective. Ambient shear reduces the mean ascent in the downshear-propagating case and affects the dynamical organization in a fundamental way. The organizing role of shear is therefore paramount.

Regarding the downshear-propagating regime and convection initiation over peninsulas/islands, (i) when of the same sign, low-level shear and the surface wind (Fig. 10a) are counteractive on both leeward and windward coasts. (ii) When of opposite sign, as in Fig. 10b, the dynamic lifting is a minimum on the windward coast and a maximum on the leeward one. Thus the widely held opinion that sea breezes are more intense in offshore flow holds only if shear and surface flow are in the opposite direction or if the low-level flow is unsheared.

Dynamical organization is also a key issue in convection initiation by downdraft outflows. In downshear-propagating outflows, although shear decreases the mean ascent, the overturning updraft provides deep lifting. This kind of updraft requires that the density current propagates at the speed of the ambient flow at a critical (steering) level. Equation (4) shows the mean ascent in unsheared flow is larger by ½S_Rz² (e.g., by 4 m s⁻¹ at 1 km for the profile in Fig. 4) than in ambient flow of shear S_R. However, unsheared inflow is associated with a front-to-rear (jump) updraft, similar to that in the upshear-propagating current, which is not conducive to deep lifting. Density currents in unsheared flow and upstream-propagating currents in a sheared flow therefore have a similar potential for convection initiation.

Finally, Moncrieff (1989) drew attention to the stagnant (degenerate overturning circulation) overlying the inflow to narrow cold-frontal rainbands, which are intermediate between squall lines and density currents in terms of intensity. The effect of shear on narrow cold-frontal rainbands should be investigated in order to evaluate our findings in terms of a broad class of organized small-scale structures in a sheared flow.