## 1. Introduction

Convection over the ocean in cold-air outbreaks behind midlatitude cyclones exhibits various kinds of organization corresponding to the spatial variation of shear, surface fluxes, and the large-scale divergence of temperature and moisture. Downstream of the polar ice sheet, in the descending equatorward branch of the large-scale slope convection that characterizes a mature cyclone, there occurs a diversity of organization ranging from shear-parallel convection bands, open- and closed-cellular convection, to stratocumulus and fields of cumulonimbus and rainbands that occur in the neighborhood of the cold front. This structural diversity is illustrated in a cold-air outbreak over the Greenland Sea–Norwegian Sea in Fig. 1.

Similar kinds of organization occur over the northern Atlantic and nothern Pacific and extensively over the Southern Hemisphere oceans. This type of convection is also associated with cold outflow from the Siberian anticyclone during winter and early spring over the Sea of Japan, the East China Sea, the South China Sea, and eastern regions of the Pacific. Often referred to as “cold surges,” these outbreaks sometimes extend well into tropical regions and organize convection on a wide range of scales. Similar behavior is found over the eastern Atlantic during the cold season when outflow from the American continent encounters the Gulf Stream (Konrad 1998) and also over the Great Lakes. Although cold-air outbreak convection affects huge areas of the world’s oceans, it has received scant attention compared to marine stratocumulus associated with ocean upwelling located to the west of continents.

The most intense surface heat fluxes (order 1000 W m^{−2}) on earth occur in polar outbreaks. The attendant cooling is large enough to drive penetrative convection within the ocean and affect the deep oceanic circulation and the climate system on timescales of centuries (Marshall and Schott 1999). Convective momentum transport affects short temporal scales as well. For instance, in midlatitudes, the curl of the surface wind stress drives ocean currents, while the momentum flux divergence within the atmosphere can influence the life cycle and tracks of midlatitude cyclonic storms. Although convective momentum transport parameterization affects the accuracy of weather forecast models and is a factor in climate models (Gregory et al. 1997; Gregory 1997), no fundamental explanation is available. Cognizant of these issues, we search for a rigorous basis for momentum transport in cold-air outbreaks in terms of dynamical mechanisms to facilitate development of physically based parameterizations. In view of the above examples, the principles investigated in the context of cold-air outbreaks are envisaged to be relevant to strongly baroclinic conditions at large.

Several attempts have been made to parameterize convective momentum transport. Schneider and Lindzen (1976) assumed that in-cloud horizontal momentum is conserved, which is valid only if the horizontal pressure gradient is negligible. Using observations, Shapiro and Stevens (1980) and Flatau and Stevens (1987) included the pressure gradient in a parameterization scheme. The momentum transport parameterization based on Tiedtke (1989, hereafter T89) accounts for the pressure gradient by increasing the lateral entrainment rate in an empirical way. Guided by observational analyses of mesoscale convective systems, Wu and Yanai (1994) represented the pressure gradient using a linearized approximation of the diagnostic pressure equation. Kershaw and Gregory (1997, hereafter KG) and Gregory et al. (1997, hereafter GKI) derived an empirical relationship for the pressure gradient by employing a cloud-resolving model (CRM).

Recognizing the distinction from squall lines, we examine momentum transport by convection organized into open-cellular patterns. We employ finescale numerical simulations and an analytic model designed to reduce the numerical results to first principles. In the next section we describe numerical simulations of open-cellular convection in an idealized cold-air outbreak, a case originally presented in KG, and analyze the results. In section 3, using the simulation results, we evaluate two mass flux-based parameterizations implemented in global models [U.K. Meteorological Office Unified Model and the European Centre for Medium-Range Weather Forecasts (ECMWF) medium-range weather forecasting model]. An idealized dynamical model presented in section 4 explicates the physical assumptions underlying the above two schemes and provides a simple transport formulation. Finally, in section 5 conclusions are drawn and comments made regarding downgradient transport, countergradient transport, and mixing of momentum.

## 2. Numerical simulations

### a. Cloud-resolving model and experimental design

The cloud-resolving model (CRM), whose Eulerian variant is used in this study, is described in Smolarkiewicz and Margolin (1997). The prognostic variables are the three velocity components (*u, υ, w*); potential temperature *θ*; water substance mixing ratios *q*_{υ}, *q*_{c}, and *q*_{r} (vapor, cloud water, and rain water, respectively); and the turbulent kinetic energy on which the subgrid-scale turbulence model is based (Schumann 1991). We represent moist processes by a bulk parameterization (Kessler 1969; Grabowski and Smolarkiewicz 1996) and disregard the Coriolis acceleration.

The time integration of the discrete equations employs a regular unstaggered rectangular Cartesian mesh. Resolved-scale variables are treated implicitly, whereas subgrid-scale terms and slow phase-change tendencies, such as rain formation and evaporation, are treated explicitly. The elliptic equation for pressure, arising from the anelastic mass conservation constraint, is solved subject to appropriate boundary conditions using the generalized conjugate-residual approach (Smolarkiewicz and Margolin 1994; Smolarkiewicz et al. 1997). Simulations were carried out on a 64-processor CRAY T3D using a massively parallel version of the code (Anderson et al. 1997).

Simulations were initialized with the horizontally homogeneous profiles shown in Fig. 2. The *u* and *w* components of velocity and the microphysical variables were initially zero, and the *υ* component is an idealized shear flow. Convection was forced by spatially homogeneous and time-invariant surface fluxes of sensible heat (123 W m^{−2}) and latent heat (492 W m^{−2}). Convection was initiated by small-amplitude random perturbations of temperature (<0.002 K) and vertical velocity (<0.07 m s^{−1}) introduced in the lowest 1350 m of the domain. The initial profiles and the surface fluxes are identical to those in the idealized cold air outbreak case of KG.

The computational domain (50 km × 50 km × 15 km) was represented by a grid of 128 × 128 × 101 points, yielding horizontal and vertical grid intervals of ∼394 and 150 m, respectively. In order to test the robustness of results with respect to resolution, all experiments were repeated on a grid of 64 × 64 × 31 points (horizontal and vertical resolution of ∼794 and 500 m). The latter vertical resolution corresponds to that used in the majority of the KG experiments, whereas the larger of the two horizontal grid increments is intermediate to the 500 and 1000 m used by these authors. In order to prevent the undesirable reflection of vertically propagating, convectively generated gravity waves from the rigid upper boundary, the uppermost 5 km of the computational domain consists of a Rayleigh-like absorbing layer. Frictional drag with a drag coefficient *C*_{d} = 0.04 was imposed at the lower boundary to mimic a no-slip condition. The lateral boundary conditions are periodic.

The duration of each numerical experiment was 10 h. Convection typically became established in about an hour, and a statistically steady state was attained after a further 3–4 h. Unless otherwise specified, our statistical diagnostics are based on averages over the last 4 h of the control experiment. Besides the lower-resolution experiment, we have performed two more sensitivity tests: one with doubled shear and the other with doubled latent heat flux (Table 1).

### b. Analysis of the mean state

The vertical velocity field at a height of 150 m at the end of the second hour of simulation is shown in Fig. 3a. Near the lower boundary, convection is organized into intense, narrow updrafts along the edges and corners of hexagonal cells with weak, broad downdrafts in the middle—a typical open-cellular pattern. During the first few hours the cells broaden and their mean diameter increases from about 7 km at the end of the second hour to more than 30 km between hours 5 and 10. In the mature state, the mean diameter is in qualitative agreement with observations. However, when computer power permits, this result should be tested using a much larger computational domain in case it is an artifact of the periodic lateral boundary conditions.

The cellular pattern is coherent up to a height of about 1400 m. The strong updrafts at the corners of the hexagons (where horizontal convergence is largest) extend toward the middle of the convective layer. The average and maximum vertical velocities in the middle of the convective layer are about 4 and 15 m s^{−1}, respectively. The vertical velocity field at 4 km reveals an ensemble of isolated strong updrafts (in small cumulonimbi) surrounded by broad areas of weak subsidence (Fig. 3b). Figure 3c shows the vertical velocity field and condensate fields (clouds) in various stages in their life cycle in a *y–z* cross section through the plane at *x* = −18.7 km. Individual updrafts are organized by the environmental shear and tilt slightly downshear (Figs. 4a,b). The in-cloud component of the pressure gradient in the direction of the mean wind is negative (Fig. 4c). Gravity waves generated by convection are evident in the stably stratified layer overlying the convecting layer (Fig. 3c).

In the control experiment, the fraction of grid points occupied by clouds is less than 5% throughout the cloud layer, apart from the outflow (anvil) region between 5 and 7.5 km where it exceeds 50% (Fig. 5a). The maximum value of 95% is attained slightly above 6 km. However, the fractional area occupied by convective updrafts^{1} remains nearly constant with height (2.5%). The downdraft fractional area is larger than the updraft area below 4 km and almost constant with height (5%) in the lowest 2 km.

The intensity of convection, measured by the standard deviation of the vertical velocity (*w*^{2}

The pressure gradient force across updrafts and downdrafts is shown in Fig. 6a. The domain average *υ**υ*^{u}*υ*^{d}*υ*^{u}*υ**υ*^{d}*υ**υ*^{d}*υ**υ*^{d}*υ**υ*^{d}*υ*^{u}

In order to compare the magnitude of the convectively generated pressure gradient with the other terms of the momentum equation, we have calculated the full momentum budget within updrafts, shown in Fig. 7. Below 5 km, except for the small contribution from the time variability term,^{2} the pressure gradient approximately balances three-dimensional advection (Fig. 7a). Above this altitude, including the anvil outflow and the overshooting updrafts, the time variability term is significantly larger than the pressure gradient term. The subgrid-scale mixing is negligible everywhere, being an order of magnitude smaller than the other terms. Examination of the horizontal and vertical advection terms (Fig. 7b) reveals that the pressure gradient is everywhere almost completely balanced by vertical advection. It also shows that the horizontal advection parallel to the mean wind accounts for the temporal variability—a result consistent with statistically steady precipitating convection traveling at nearly the mean flow velocity. The contribution of advection perpendicular to the mean wind is small, showing that the transports are approximately two-dimensional (i.e., in the *y–z* plane).

We now summarize the sensitivity experiments. The ensemble-averaged statistics of the lower-resolution experiment reveal a sensitivity to resolution. While most of the averaged statistics change only weakly with resolution, the updraft mass flux in the lower-resolution experiment is about 25% smaller compared to the control experiment, whereas both the updraft and downdraft pressure gradient forces are about halved. Consequently, as in KG, the resulting cloud-environmental velocity difference and the total momentum forcing are larger in the lower-resolution experiment. Since a very high resolution experiment was not performed, we cannot conclusively say our results converge with increasing resolution, but we anticipate our highest-resolution simulations are reasonably accurate. However, our sensitivity is larger than that reported by Brown (1999a), who shows that many ensemble-averaged statistics of shallow-cumulus convection are comparatively insensitive to resolution.

For the remainder of the sensitivity experiments we found that the doubled shear leads to cumulonimbi with more extensive anvils resulting in 100% cloud cover around 6 km. Also, the momentum flux and the pressure gradient force in the doubled-shear experiment are about double those in the control experiment, whereas the mass flux remains almost unchanged. In the doubled latent heat flux experiment, the clouds reach 1 km higher. The updraft and downdraft cloud fraction, as well as the mass flux and the intensity of convection, are double that in the control experiment. The momentum flux is 50% larger, while the pressure gradients are about twice as strong. As a summary of sensitivity experiments, Fig. 8 shows the total change in the mean velocity profile (Δ*υ*

## 3. Evaluation of parameterized quantities

### a. Entrainment and detrainment

*M*

^{c}=

*σ*

^{c}

*ρ*

*w*

^{c}

^{c}represents averages over convective updrafts or downdrafts whose fractional area

*σ*

^{c}is assumed to be small compared to the grid area. The mass continuity equation averaged horizontally over the convective area (say, updraft) is

*E*and

*D*are entrainment and detrainment rates, respectively. It is usual to define a fractional entrainment rate

*ϵ*=

*E*/

*M*

^{u}and a fractional detrainment rate

*δ*=

*D*/

*M*

^{u}, in which case 1/

*M*

^{u}is an integrating factor for (1). It is typically assumed that convection can be represented as an entraining plume (Squires and Turner 1962); however, the values of

*ϵ*and

*δ*used in various convective parameterizations span a large range and are not necessarily consistent with those of Squires and Turner.

As part of the evaluation of the T89 scheme operative in the ECMWF forecasting system, we have computed the entrainment (*E*) and detrainment rates (*D*) for the ensemble of updrafts using the budget equations for scalar variables. Figure 5e shows the fractional rates *ϵ* and *δ.* In the middle of the cloud layer, entrainment and detrainment rates are about equal, which is consistent with the mass flux being approximately constant with height. The fractional rates we obtain in this deep convection case are *ϵ* ≈ *δ* ≈ 1.5 × 10^{−3} m^{−1}, which is the same order of magnitude as the rates determined for shallow convection (cf. Brown 1999b). These rates are also an order of magnitude larger than the turbulent entrainment and detrainment rates assumed in deep-convection parameterizations (T89; Gregory and Rowntree 1990). Our results indicate that the fractional rates depend on neither the strength of shear nor the latent heat flux. However, for the lower-resolution experiment we find somewhat larger values of *ϵ* ≈ *δ* ≈ 2 × 10^{−3} m^{−1}.

### b. Pressure gradient

*υ*′ =

*υ*−

*υ*

^{3}expressed as the product of the convective mass flux and the difference between the in-cloud horizontal momentum and the domain-average momentum.

*υ*

*M*

^{c}and the in-cloud wind field

*υ*

^{c}

*ρσ*

^{c}

*υw*

^{c}

*M*

^{c}

*υ*

^{c}

*υ*

^{c}

*C*= 0.7) was evaluated from their numerical simulations of convection in cold-air outbreaks.

*α*is an empirical constant. Values used by the ECMWF are

*α*= 2 (or 3 if

*ϵ*= 0) for deep convection, and

*α*= 0 (or 1 if

*ϵ*= 0) for shallow convection.

We now evaluate the above two pressure gradient approximations using the cloud-resolving model results. Figure 10a shows the two terms on the right-hand side of (5) and (6) computed from the averaged diagnostics by setting *C* and *α* equal to unity. The pressure gradient term is also shown for reference. Profiles of *C* and *α,* determined as the ratio of corresponding terms in (5) and (6), are shown in Fig. 10b. Throughout the equilibrium cloud layer, between 0.3 and 7 km, the value of *C* varies between 0.5 and 1.0 with the majority of points clustered near 0.75. Similar results are obtained for the doubled-shear and latent heat flux sensitivity experiments. The corresponding value computed from the lower-resolution diagnostics reveals more variation with height and a mean value closer to 0.5. Overall, the range of values is close to 0.7, the value estimated by GKI based on their numerical experiments.

The results also indicate *α* ≈ 0.25–0.5, which is an order of magnitude smaller than the value used for deep convection in the ECMWF scheme. Further inspection of (6) shows that this parameterization depends not only on the assumed value of *α* but also on the fractional rates *ϵ* and *δ,* namely, *ϵ*_{ECMWF} = *δ*_{ECMWF} = 1 × 10^{−4} m^{−1} for deep convection. Because the fractional rates derived from our simulations are an order of magnitude larger, it follows that *α*_{CRM} × *δ*_{CRM} ≈ *α*_{ECMWF} × *δ*_{ECMWF}. In other words, given the same mass fluxes for this case of deep convection, namely, *M*^{u} ∼ 0.06 kg m^{−2} s^{−1} for the ECMWF model (Fig. 1 in Gregory 1997) and *M*^{u} ∼ 0.1 kg m^{−2} s^{−1} from our results (Fig. 5d), the ECMWF parameterization with *α* = 2 (or 3) can produce reasonable results for deep convection (cf. Gregory 1997). The ECMWF scheme therefore produces reasonable results by virtue of a compensation of two errors.

More basically, the success of this scheme depends on the validity of the E/D parameterization of the horizontal advection and mixing terms. The ECMWF scheme is relatively successful in this case because the parameterized advection (albeit incorrectly) mimics the vertical variation of the convective pressure gradient (cf. Fig. 9b). However, this is probably not a general result but rather one that depends on convective organization. Indeed, Brown (1999b) documents a poor performance of the ECMWF parameterization for a case of shallow convection.

## 4. Analytic model

A turbulent entraining plume is a reasonable model to use for parameterizing convection in weakly sheared conditions where small-scale stochastic mixing with the environment (eddy diffusion) is the dominant process. It has long been known that vertical shear has a fundamental effect on convection by organizing the airflow into quasi-steady structures (Ludlam 1980). This concept was quantified in the steady-state nonlinear models of Moncrieff (1981) based on the conservation properties of the inviscid Lagrangian equations of motion and thermodynamics. These analytic models represent distinct regimes of organized airflow and accompanying transports that are fundamentally different from turbulent entraining plumes.

*p*is the horizontal pressure change across the cloud system,

*V*is an advective velocity scaling,

*L*and

*H*are characteristic horizontal and vertical scales,

*K*

_{e}is the eddy diffusion of momentum, and

*K*

_{e}/

*VL*is an eddy Reynolds number. Because the momentum budget in Fig. 7 shows turbulent mixing is negligible within simulated clouds,

We develop a variant of the Moncrieff (1981) steering-level regime of convective overturning in constant shear appropriate to the convection simulated herein. This is necessary because the original steering-level model tilts downshear and generates a countergradient transport^{4} of momentum (Moncrieff 1978), and the classical model of Moncrieff (1981), which consists of two antisymmetric updraft circulations, does not transport momentum (because the local flux at an arbitrary point in one overturning branch is canceled by the flux of equal magnitude but opposite sign at the image point in the other branch). Note that in Fig. 4 the simulated updrafts tilt slightly downshear, but the accompanying (countergradient) transport does not alter the overall downgradient transport.

The key feature identifying the model adopted here from the classical model is the single, vertically oriented, blocked overturning updraft (Fig. 11). This is an idealization of a simulated three-dimensional cumulonimbus where ambient air flows around the updraft. While three-dimensional transport is neglected in this two-dimensional idealization, the one-sided circulation [a limiting case of the blocked overturning regime of Liu and Moncrieff (1996)] provides the asymmetric pressure gradient responsible for the momentum transport. Kinematically this means that the one-sided (blocked) circulation causes deceleration of the horizontal in-cloud velocity of air parcels, while the vorticity of the overturning updraft, of the same (positive) sign as the ambient shear, maintains the far-field flow. The combined effect of these two mechanisms is to maintain the mean in-cloud momentum at each level at an intensity somewhat less than its domain-averaged value (see Fig. 6b).

*H*= 6.7 km and aspect ratio

*L*/

*H*≈ 1. In the analytic formulation, we assume constant density, constant shear, and neutrally stratified base state, and for convenience, we normalize the relative

^{5}coordinates and variables according to

*ẑ*→

*ẑ*/

*H,*

*ŷ*→

*ŷ*/

*H,*

*υ̂, ŵ*)

*υ̂, ŵ*)/

*V,*

*ψ̂*

*ψ*/(

*HV*), and

*p̂*=

*p*/(

*ρV*

^{2}). The finite-amplitude vorticity perturbation from the undisturbed shear represents potential flow described by solutions of Laplace’s equation. In rectangular domain 0 ⩽

*ŷ*⩽

*L̂,*0 ⩽

*ẑ*⩽ 1, where

*L̂*is the aspect ratio, the analytic solution is expressed in terms of the infinite series:

*υ̂*

*ψ̂*/∂

*ẑ*

*ŵ*= −

*ψ̂*/∂

*ŷ*

At the left-hand boundary at *ŷ* = 0 the horizontal velocity is zero in the full series solution and approximately zero in the truncated series. Therefore, the average in-cloud horizontal momentum is less than the inflow (environmental) momentum, and the momentum flux is downgradient.

### a. Physical basis of pressure gradient approximations

*û*

*υ̂*/∂

*x̂*

*relative*flow is steady. Integrating over the horizontal distance

*L̂,*and using the identity ∫ ( )

*dŷ*=

*L̂*(

*υ̂*

_{L̂}− ( )

_{0}].

*υ̂, ŵ*)

*L̂*≈ 1, we obtain

*υ̂*

^{2}

*L*

*ŵ*∂

*υ̂*/∂

*ẑ*

*L*

*ŵ*

*υ̂*

*ẑ*

*C*is an empirical constant determined from the cloud-resolving simulations. Comparing (13) and (15) this mass flux–type approximation is justified provided the following is satisfied. First, in-cloud shear and the (pointwise) mass flux (here

*M̂*

^{u}≡

*L̂*

*ŵ*

*υ̂*

^{2}

*C*should be a function of height:

*C*

_{min}=

*π*/(

*π*+ 2) ≈ 0.61 at

*ẑ*= ½ (Fig. 12b). The empirical value proposed by GKI is

*C*= 0.7, whereas our CRM results suggest a value closer to 0.75 (cf. Fig. 10). The assumption regarding horizontal advection is valid at the cloud layer midlevel, but its legitimacy deteriorates toward the upper and lower boundaries where

*C*becomes singular. The singularity and the concave shape of

*C*(

*ẑ*) in Fig. 12b stem from the neglect of the horizontal convergence, which increases from zero at the middle of the cloud to a maximum at the upper and lower boundaries— the value of

*C*must increase accordingly to compensate for this neglected term. In Fig. 10b, the analytic prediction for

*C*is compared to the

*C*profile based on the GKI parameterization. We note that the two

*C*profiles agree reasonably well in the middle of the cloud layer, but the agreement rapidly deteriorates toward the bottom and the top.

Wu and Yanai (1994) used a linearized approximation of the diagnostic pressure equation to determine the horizontal pressure gradient using a cumulus ensemble representation of Arakawa and Schubert (1974), in which each element is an entraining plume. They also assumed that horizontal advection does not contribute to the pressure gradient. It follows that their expression for the horizontal pressure gradient has the same form as that used by GKI [cf. Eq. (16) of Wu and Yanai and Eq. (15) herein], whose mass flux has a sinusoidal variation akin to our series solution. As Wu and Yanai point out, the calculation of *γ* (the counterpart of *C*) requires explicit knowledge of convection dynamics. In their study, a linearized analysis provides *γ* in terms of a ratio of the characteristic wavenumbers, whereas herein the pressure gradient is provided by the analytic model.

*C, α*should vary with height according to

*α*= 1 at the upper and lower boundaries. The profile of

*α*in Fig. 12b stems from the fact that the vertical advection of momentum, a maximum at the midlevel and zero at the boundaries, is neglected. In order to compensate,

*α*must increase toward the midlevel where it is unbounded.

Recall that the empirical value used in the ECMWF model for deep convection is *α* = 2 and that the value based on our numerically simulated data varies from 0.25 to 0.5 throughout the cloud layer. For reference, in Fig. 10b we have also included the analytic form for *α.* Following from the discussion in section 3b, the large discrepancy between the analytic and the simulation-based *α* is due to the overestimation of the horizontal advection and cloud interfacial mixing terms by the E/D parameterization at and below the midcloud level where these terms are nearly zero (cf. Fig. 9).

*C*

_{1}(

*z*) and

*C*

_{2}(

*z*), perhaps both constant, could be determined from numerical simulation results.

### b. Analytic representation of ensemble properties

In order to show the analytic model captures the salient dynamical properties of the numerically simulated cloud ensemble, we have computed the individual terms of (13) averaged over the updrafts. Since the steady-state analytic model is cast in system-relative coordinates, the horizontal advection term is given by *υ̂*∂*υ̂*/∂*ŷ**H*/*V*^{2})(*υ*∂*υ*/∂*y* − *υ*_{SL}∂*υ*/∂*y*), where *υ*_{SL} is the steering-level velocity of the cloud ensemble. By relating the horizontal average of (10) to the domain-average velocity of the numerically simulated cloud ensemble, it follows that during the last 4 h of the control experiment *υ*_{SL} = 5.5 m s^{−1}. This is approximately the initial mean flow velocity near the midcloud level at *z* = 3.65 km.

In Fig. 13 we show the pressure gradient term, the vertical and horizontal advection terms, and their ratios calculated using the approach adopted in section 3b with the parameterization terms. This analysis reinforces the point that the pressure gradient mimics vertical advection throughout most of the cloud layer (cf. Figs. 7 and 9b). The ratio of these two terms is approximately equal to a half below 5 km and is close to unity in the anvil region and within the overshooting updrafts. The corresponding analytic ratio for organized convection of vertical extent 0.3–7 km, also shown in Fig. 13b, starts from unity at the midcloud level and gradually becomes unbounded toward the top and bottom of the cloud layer. As predicted by the analytic model, the organized inflow and outflow due to horizontal advection contributes significantly to the pressure gradient term only in the top and bottom portions of the cloud layer. Consequently, the ratio of the latter two terms reflects the height dependence exhibited by the analytic model.

Based on the preceding analysis, as well as the analysis of momentum budget and entrainment and detrainment, we conclude that the vertical advection of momentum is the more accurate approximation of the in-cloud pressure gradient. For this reason, one can say the GKI parameterization is preferable to the T89 method, which is based solely on the horizontal advection.

In the following, we quantify the analytic transport dependence on the asymmetry of the blocked overturning updraft. Referring to (10) and (11), it is worthwhile to note that the overturning updraft has constant vorticity equal to the environmental shear. Also, the horizontal velocity, while decelerated in the updraft region, asymptotes to the undisturbed environment for *ŷ* ≫ *L.* Consequently, the downgradient momentum flux and the tendency to maintain shear are a hallmark of this regime of convection.

*υ̂*

*t̂*

*υ̂*′

*ŵ*′

^{u}

*ẑ*− 1) plus a nonlinear perturbation— the tendency, over a discrete time interval Δ

*t,*is

*t*used to obtain the results shown in Figs. 14a,b is the entire 10 h of the simulation. For this longer averaging period, the equilibrium cloud layer extends only up to 6 km (

*H*= 5.7 km), and the steering-level velocity is

*υ*

_{SL}= 5 m s

^{−1}.

The momentum flux provided by the analytic model is of correct sign and shape, but its amplitude is smaller than the numerically simulated results. We note that the analytic model represents neutral overturning in which kinetic energy is provided solely by the pressure gradient, whereas in the simulations at midlevels the conversion of convective available potential energy (CAPE) into kinetic energy is dominant. The effects of CAPE and the pressure field are estimated in the appendix. As shown in Fig. 14b, the inclusion of CAPE leads to a better agreement of the momentum flux profiles within the equilibrium cloud layer.

Finally, we reiterate that the analytic blocked overturning model applies only to the equilibrium state of the convecting layer. The momentum tendency near 6 km is associated with the overshooting updrafts between 6 and 8.5 km. These different regions are depicted in Fig. 15.

## 5. Conclusions

We evaluated two momentum parameterization schemes operational in global models (GKI and T89) using the cloud-resolving numerical simulations of open-cellular convection in an idealized midlatitude cold-air outbreak and the blocked overturning model. In spite of its simplicity, this two-dimensional analytic model captures the salient features of the three-dimensional numerical realizations. This success is due to (i) the blocked overturning updraft having the same vorticity as the ambient shear, and (ii) the cross-stream flow perturbations in unidirectional mean flow being second order, meaning the momentum transport is approximately two-dimensional.

The GKI and T89 schemes are based on the traditional mass-flux approximation to the eddy Reynolds stress where the formidable problem is the representation of the convectively generated pressure gradient. We established that the GKI scheme approximates the pressure gradient in terms of the vertical advection of momentum. The analytic model shows this approximation to be accurate at midlevels, but the accuracy deteriorates toward the upper and lower boundaries. For the numerically simulated convection, however, this approximation is evidently valid throughout almost the entire updraft depth. Conversely, the T89 scheme approximates the pressure gradient solely by the horizontal advection of momentum by way of enhanced entrainment and detrainment. In the analytic model, this is reasonable at upper and lower levels but not at midlevels where the vertical advection dominates. In the simulations, while the entrainment/detrainment parameterization of horizontal advection and cloud-interfacial mixing overestimates these terms, it mimics the vertical variation of the pressure gradient. On physical grounds, the representation of the pressure gradients by means of the vertical advection of momentum (GKI) is the better approximation.

The downgradient momentum transport associated with the blocked overturning updraft contrasts with the countergradient transport by squall lines, a property quantified in dynamical models (Moncrieff 1981, 1992) and identified in observations (LeMone 1983; Wu and Yanai 1994). Countergradient transport by *two-dimensional squall lines* is due to the upshear tilt of relative airflow. This tilt occurs if the sign of the ambient flow reverses with height (jetlike profiles) or if the shear is strong in low levels but weak aloft. The blocked overturning model is a simple representation of the salient aspects of *three-dimensional cumulonimbus* convection simulated herein. The mechanism was explained in terms of the pressure gradient or, alternatively, in terms of a blocked vertically oriented updraft having vorticity of the same sign as the mean flow shear.

GKI showed that the entraining plume works in practice when empirically tuned. In principle, however, for reasons already described it is not truly appropriate for organized convection. Ambient shear controls different regimes of convective momentum transport, yet shear is not part of any existing parameterization scheme. This situation must be rectified if momentum transports are to be properly represented in large-scale models.

## Acknowledgments

The research reported herein has been performed during the lead author’s affiliation with the Clouds in Climate Program at NCAR. V. G. acknowledges partial support from the Desert Research Institute in final stages of this work. We thank David Gregory for helpful discussions and all our reviewers for valuable comments and suggestions.

## REFERENCES

Anderson, W. D., V. Grubišić, and P. K. Smolarkiewicz, 1997: Performance of a massively parallel 3D non-hydrostatic atmospheric fluid model.

*Proc. Int. Conf. on Parallel and Distributed Processing Techniques and Applications PDPTA ’97,*Las Vegas, NV, Computer Science Research, Education, and Applications Tech, 645–651.Arakawa, A., and W. H. Schubert, 1974: Interaction of a cumulus ensemble with the large-scale environment, Part I.

*J. Atmos. Sci.,***31,**674–701.Brown, A. R., 1999a: The sensitivity of large-eddy simulations of shallow cumulus convection to resolution and subgrid model.

*Quart. J. Roy. Meteor. Soc.,***125,**469–482.——, 1999b: Large-eddy simulation and parametrization of the effects of shear on shallow cumulus convection.

*Bound.-Layer Meteor.,***91,**65–80.Brü≫er, B., 1999: Roll and cell convection in wintertime Artic cold-air outbreaks.

*J. Atmos. Sci.,***56,**2613–2636.Flatau, M., and D. E. Stevens, 1987: The effect of horizontal pressure gradients on the momentum transport in tropical convective lines. Part I: The results of the convective parameterization.

*J. Atmos. Sci.,***44,**2074–2087.Grabowski, W. W., and P. K. Smolarkiewicz, 1996: Two-time-level semi-Lagrangian modeling of precipitating clouds.

*Mon. Wea. Rev.,***124,**487–497.Gregory, D., 1997: Parametrization of convective momentum transports in the ECMWF model: Evaluation using cloud resolving models and impact upon model climate.

*Proc. New Insights and Approaches to Convective Parametrization, ECMWF Workshop,*Reading, UK, ECMWF, 208–227.——, and P. R. Rowntree, 1990: A mass flux convection scheme with representation of cloud ensemble characteristics and stability dependent closure.

*Mon. Wea. Rev.,***118,**1483–1506.——, R. Kershaw, and P. M. Inness, 1997: Parametrization of momentum transport by convection. II: Tests in single-column and general circulation models.

*Quart. J. Roy. Meteor. Soc.,***123,**1153–1183.Kershaw, R., and D. Gregory, 1997: Parametrization of momentum transport by convection. I: Theory and cloud modelling results.

*Quart. J. Roy. Meteor. Soc.,***123,**1133–1151.Kessler, E., 1969:

*On the Distribution and Continuity of Water Substance in Atmospheric Circulations. Meteor. Monogr.,*No. 32, Amer. Meteor. Soc., 84 pp.Konrad, C. E., II, 1998: Persistent planetary scale circulation patterns and their relationship with cold air outbreak activity over the eastern United States.

*Int. J. Climatol.,***18,**1209–1221.LeMone, M. A., 1983: Momentum transport by a line of cumulonimbus.

*J. Atmos. Sci.,***40,**1815–1834.Liu, C., and M. W. Moncrieff, 1996: A numerical study of the effects of ambient flow and shear on density currents.

*Mon. Wea. Rev.,***124,**2282–2303.Ludlam, F. H., 1980:

*Clouds and Storms.*The Pennsylvania State University Press, 405 pp.Marshall, J., and F. Schott, 1999: Open-ocean convection: Observations, theory, and models.

*Rev. Geophys.,***37,**1–64.Moncrieff, M. W., 1978: The dynamical structure of two-dimensional steady convection in constant vertical shear.

*Quart. J. Roy. Meteor. Soc.,***104,**543–567.——, 1981: A theory of organised steady convection and its transport properties.

*Quart. J. Roy. Meteor. Soc.,***107,**29–50.——, 1992: Organized convective systems: Archetypal dynamical models, mass and momentum flux theory, and parametrization.

*Quart. J. Roy. Meteor. Soc.,***118,**819–850.——, 1997: Momentum transport by organized convection.

*The Physics and Parametrization of Moist Atmospheric Convection,*R. K. Smith, Ed., NATO ASI Series, Vol. 505, Kluwer Academic, 231–253.——, and J. S. A. Green, 1972: The propagation and transfer properties of steady convective overturning in shear.

*Quart. J. Roy. Meteor. Soc.,***98,**336–352.Schneider, E. K., and R. S. Lindzen, 1976: A discussion of the parameterization of momentum exchange by cumulus convection.

*J. Geophys. Res.,***81,**3158–3160.Schumann, U., 1991: Subgrid length-scales for large-eddy simulation of stratified turbulence.

*Theor. Comput. Fluid Dyn.,***2,**279–290.Shapiro, L. J., and D. E. Stevens, 1980: Parameterization of convective effects on the momentum and vorticity budgets of synoptic-scale Atlantic tropical waves.

*Mon. Wea. Rev.,***108,**1816–1826.Smolarkiewicz, P. K., and L. G. Margolin, 1994: Variational solver for elliptic problems in atmospheric flows.

*Appl. Math. Comput. Sci.,***4,**527–551.——, and ——, 1997: On forward-in-time differencing for fluids: An Eulerian/semi-Lagrangian non-hydrostatic model for stratified flows.

*Atmos.–Ocean,***35**(special issue), 127–152.——, V. Grubišić, and L. G. Margolin, 1997: On forward-in-time differencing for fluids: Stopping criteria for iterative solutions of anelastic pressure equations.

*Mon. Wea. Rev.,***125,**647–654.Squires, P., and J. S. Turner, 1962: An entraining jet model for cumulonimbus updraughts.

*Tellus,***14,**422–434.Tiedtke, M., 1989: A comprehensive mass flux scheme for cumulus parameterization in large-scale models.

*Mon. Wea. Rev.,***117,**1779–1800.Wu, X., and M. Yanai, 1994: Effect of vertical wind shear on the cumulus transport of momentum: Observations and parameterization.

*J. Atmos. Sci.,***51,**1640–1660.

## APPENDIX

### Analytic Representation of the Effect of CAPE

*ŷ, ẑ*) = (0,

*ẑ*∗), where

*ẑ*∗ is the height of the steering level. In case of neutral overturning,

^{A1}

*ẑ*

_{0}∗ = ½, that is,

*z*

_{0}∗ =

*H*/2. In the buoyant updraft, the steering level is shifted toward the upper boundary so that

*H*/2 <

*z*

_{1}∗ <

*H.*The vertical velocity maxima can be determined by applying Bernoulli’s theorem along the lower-boundary streamline from inflow at (

*L̂,*0) to (0,

*ẑ*∗). By noting that the pressure perturbation vanishes ∀

*ẑ*at

*ŷ*=

*L̂,*

*γ*) and the static stability (

*B*) are constant, the integral on the right-hand side is equal to

*g*(

*γ*−

*B*)

*z*

^{2}

_{*}

*δp*∗/

*ρ*can be estimated from the steady-state momentum equation integrated along

*z*=

*z*∗ to yield

*δp*∗/

*ρ*≈

*V*

^{2}/4. Thus,

*g*(

*γ*−

*B*)

*H*

^{2}/2. In the neutral updraft,

*w*

^{2}

_{0*}

*V*

^{2}

_{0}

*V*

_{1}/

*z*

_{1}∗ = 2

*V*

_{0}/

*H,*and defining Δ

*V*≡

*V*

_{H}−

*V*

_{0}, it follows that

*V*

_{1}= Δ

*V*(

*z*

_{1}∗/

*H*). By substituting the solution of Moncrieff and Green (1972) for

*z*

_{1}∗/

*H,*the ratio of the buoyant solution to the neutral one is

*R*≡ 2 CAPE/(Δ

*V*)

^{2}is the convective Richardson number. It follows that the intensity of buoyant ascent compared to neutral overturning is a function of the convective Richardson number.

Based on the ten-hour averaged statistics from the control simulations, CAPE ≈ 30 J kg^{−1} and Δ*U* = 6.5 m s^{−1}. Thus, *R* = 1.4 and (*w*_{1}∗/*w*_{0}∗) = 2.5. The momentum flux and the total convective momentum forcing shown in Figs. 14c,d have been multiplied by (*w*_{1}∗/*w*_{0}∗)^{2} to approximate the effect of CAPE.

List of experiments discussed: control (COA1), lower-resolution (COA2), doubled-shear (COA3), and doubled latent heat flux (COA4) experiments. In all four experiments, sensible heat flux and surface drag coefficients are 123 W m^{−2} and *C _{d}* = 0.04.

^{1}

For the calculation of the partitioned diagnostics, we have adopted the definitions of KG, where an updraft at a grid point occurs if *w* > 1 m s^{−1} and the condensed water mixing ratio *q*_{c} > 0.1 g kg^{−1}. A downdraft is defined by *w* < 0 and the precipitating water mixing ratio *q*_{r} > 0.1 g kg^{−1}.

^{2}

The time variability term *σ*^{u}*υ*/∂*t*^{u}*σ*^{u}∂*υ*^{u}*t* + (*υ*^{u}*υ*_{b})∂*σ*^{u}/∂*t,* where (^{u} represents the average over the updrafts [*ϕ*^{u}_{u})_{Au}*dx* *dy*, where *A*_{u} is the total area occupied by updrafts within a grid box area of size *A.* The equivalent definition holds for downdrafts with the total area *A*_{d}], σ^{u} (≡*A*_{u}/*A*) is the updraft fractional coverage, and υ_{b} is the velocity of the updraft interfaces. In the statistically steady state, the first term on the right-hand side is very small. Consequently, the time variability reflects changes of cloud cover dominated by υ_{b}∂σ^{u}/∂*t.*

^{3}

Based on the partitioning *υ*′*w*′^{u}*υ*′*w*′^{u}^{d}*υ*′*w*′^{d}*σ*^{u} − *σ*^{d})*υ*′*w*′^{e}

^{4}

In classic terms “ countergradient transport” refers to a negative momentum transport coefficient, but in more generality, it is a mechanism by which the mean flow is accelerated by the action of subgrid-scale transport. Because of the integral constraint on momentum flux, this may not occur throughout the convecting layer but rather in discrete layers (Moncrieff 1992, section 7c and Fig. 12 therein).

^{5}

A simple coordinate transformation applies between the fixed Eulerian coordinates (*x, y, z*) and the relative coordinates of the analytic model (*x̂, ŷ, ẑ*), i.e., (*x, y* − *υ*_{SL}*t, z*) → (*x̂, ŷ, ẑ*) yielding *u, υ* − *υ*_{SL}, *w*)*û, υ̂, ŵ*)*υ*_{SL} is the steering-level velocity.

^{}

In the following, subscripts 0 and 1 denote the neutral and buoyant models, respectively.

^{+}

The National Center for Atmospheric Research is sponsored by the National Science Foundation.