1. Introduction

Calculation of the surface wind vector is a key part of the planetary boundary layer (PBL) solution (e.g., Foster and Brown 1994). The surface fluxes of momentum, heat, and evaporation are directly related to the surface (10 m) wind speed, U₁₀, through the Monin–Obukhov similarity theory. The PBL vorticity, divergence, and associated Ekman pumping are related to both the speed and direction of the surface wind. Although the wind and flux profiles throughout the PBL depend on the geostrophic forcing, Coriolis force, thermal stratification, baroclinity, surface roughness, momentum transport by large eddies and local and advective accelerations, most PBL solutions assume that the PBL is horizontally homogeneous, in steady state, and barotropic. In this case the balance of forces produces a cross-isobar flow toward low pressure that is consistent with the classic Ekman layer solution. Parameterizations based on these solutions generally employ gradient transfer relationships with a stratification-dependent eddy viscosity characteristic of small to moderate size eddies in a homogeneous turbulent flow but cannot account for the transport by large (PBL scale) eddies (Brown 1981; Frech and Mahrt 1994).

The PBL frequently develops a persistent, organized secondary circulation in the form of counterrotating roll vortices (hereafter rolls) that organizes the smaller-scale turbulent eddies into linear patterns in near-neutral to moderately unstable stratification (e.g., Etling and Brown 1993). Under increasingly convective conditions observations and numerical simulations show that these shear-driven rolls undergo a transition to buoyancy-driven free convection thermals (e.g., Walter 1980; Chlond 1992). The latter are characterized by large buoyancy fluxes, smaller horizontal length scales, and reduced shear. In this paper we focus on the near-neutral boundary layer for which shear dominates buoyancy effects. Typically rolls are at a small angle relative to the surface isobars, span the depth of the PBL, and have horizontal wavelengths of 2–5 times its height (LeMone 1973; Mourad and Walter 1996). The roll-induced momentum fluxes efficiently transport high momentum air downward to produce a boundary layer with reduced shear in the midlevels, higher surface wind speed, and a changed surface wind direction. This important roll effect is not captured by most PBL schemes in large-scale and general circulation numerical models (Foster and Brown 1994; Brown and Foster 1994; Ayotte et al. 1996).

This note discusses how the combined effects of rolls and baroclinity affect the surface wind in near-neutral stratification. The nonlinear nature of the PBL should caution against the linear superposition of effects that has often been employed. We compare a standard model that linearly superposes baroclinic and roll parameterizations with a theoretical model that includes nonlinear interaction between these effects. The theory shows that the thermal wind not only affects the boundary layer mean flow, it also produces major modifications to the roll vortices that result in a significantly different mean flow from that of a baroclinic flow without rolls, a barotropic flow with rolls, or a linear superposition of the two. The implications to the general circulation are discussed.

2. Thermal wind effects

A typical midlatitude boundary layer mean horizontal temperature gradient of 0.75 K (100 km)⁻¹ acting on a 1-km-deep layer can change the geostrophic wind speed by ∼2.5 m s⁻¹. This is comparable to the frictional reduction of the geostrophic wind to its near-surface value. Observational studies by Bernstein (1973), Hoxit (1974), and Joffre (1982, 1985) show that relative to barotropic conditions the surface wind cross-isobar angle α increases (decreases) in cold (warm) advection and the ratio of the surface wind speed to the surface geostrophic wind speed (G), S = U₁₀/G, increases (decreases) when the thermal wind shear is in the same (opposite) direction as the surface wind. The cold (warm) advection regime relative to the surface geostrophic wind is 0° < β < 180° (180° < β < 360°), where β is the angle of the thermal wind from the surface isobars. These modifications are approximately sinusoidal in β although asymmetries in both α and S are observed depending on the sense of the thermal advection.

Aspects of these effects in near-neutral to moderately unstable stratification have been simulated in two-layer PBL models that match a Monin–Obukhov similarity surface layer model to a baroclinic Ekman layer (e.g., Arya and Wyngaard 1975; Brown and Liu 1982, hereafter BL82; Bannon and Salem 1995). Nondimensionalization of the equations governing the Ekman layer results in the following definitions for the eddy Reynolds number, Re, and the effective baroclinity (e.g., Levy 1989), B, that characterize the mean flow:

View Expanded

in which δ = δ(u*, f, λ, ϕ) is the Ekman depth scale, K is the characteristic eddy viscosity, ϕ is the nondimensional surface layer shear from Monin–Obukhov theory, u* is the friction velocity, λ = 0.15 (Brown 1982) is the nondimensional matching height, κ = 0.4 is von Karman’s constant, and the other symbols have their usual meteorological meanings.

The barotropic BL82 solutions have the expected decrease in α and increase in S as the PBL becomes more unstably stratified. The baroclinic modifications are larger in unstable stratification mainly due to the increase in the effective baroclinity (Levy 1989). For the chosen typical conditions the thermal wind modification of the surface wind is as important as that due solely to stratification and the combined effects can be larger than either acting alone. Such models have been used to explain the observed asymmetry in the surface wind between cold and warm advection since cold (warm) advection is generally associated with more unstably (stably) stratified conditions (Levy 1989).

3. PBL rolls

Rolls are caused by a rapidly growing mixed dynamic–thermal instability of the mean PBL flow. The finite amplitude perturbation’s fluxes of momentum and heat alter the original mean state sufficiently to cause a bifurcation into a new solution that includes the roll circulation as part of a horizontally periodic mean flow. A comprehensive review of observational, theoretical, and numerical investigations into roll dynamics is given in Etling and Brown (1993). The standard roll parameterization (Brown 1982) used in BL82 is based on Brown’s (1970) solution for 2D perturbations in the neutrally stratified, barotropic Ekman layer. Nonlinear solutions for modified mean flow were found by balancing the shear production of roll kinetic energy due to an estimated mean flow modification (a net loss term) with that of the original Ekman profile. The effect of thermal stratification was included in the barotropic linear stability analyses by Brown (1972) and equilibrium solutions were found in the same manner.

A parameterization of the roll effect as a function of Re and ΔT was developed from these calculations and superposed onto the two-layer similarity PBL model (Brown 1982; BL82). Its effect on the surface wind is shown in Table 1 and Fig. 3 (long dashes). In barotropic conditions and either neutral or unstable stratification, the surface wind speed increases by ∼10% and the cross-isobar angle by ∼5°. The amplitudes of the baroclinic modifications, A_α,S, increase by about 10% and the phases, γ_α,S, change little. Thus the linear superposition of rolls and baroclinity does not cause additional asymmetry in the surface wind for a given stratification. Rolls are damped out by stable stratification and there is little difference between solutions with or without rolls. This selective modification between unstable and stable stratification regimes can add to the asymmetry discussed in the previous section.

4. Nonlinear combination of roll and thermal wind effects

The model of PBL rolls used here is based on Foster (1996), who solved the complete (3D, eighth order) nonlinear instability equations for the near-neutral, baroclinic Ekman layer. Aspects of the linear stability problem associated with rapid transient growth may be found in Foster (1997). The nonlinear solution is a power series expansion around the most unstable normal mode for the given mean flow conditions. The perturbation energy density E is the small parameter in the expansion (Watson 1960; Herbert 1983). Solutions are found up to the 31st power of E depending on Re and B. Exact agreement is found between Foster’s theoretical solution and the Re = 150 fully nonlinear direct numerical simulation of Coleman et al. (1990). For comparison, in typical barotropic PBL conditions, the 2D solutions of Brown (1970, 1972) are qualitatively similar to Foster (1996). However, the additional nonlinear processes, forcings due to the perturbation temperature, the axial velocity component, and the baroclinic effects in Foster (1996) lead to solutions with significantly different roll structures and strengths and thus different mean flow profiles.

Rolls are significantly altered depending on the orientation of the thermal wind. Compared to the barotropic solution, rolls in cold (warm) advection are of smaller (larger) scale, lesser (greater) strength, and are at larger (smaller) angles to the surface isobars (Fig. 1). Consequently, the change in the near-surface wind between the roll-modified and no-roll baroclinic Ekman profile is an asymmetric function of the thermal wind orientation (Figs. 2a–c). We refer to the no-roll mean flow as the basic state. Roll-induced modifications to the mean flow parameterized without regard to these changes (Brown 1982) will miss much of the baroclinic and roll effects. For the mean climatological midlatitude conditions, the thermal wind is mostly zonal and the along-isotherm component augments the near-surface westerly wind while the cross-isotherm flow is responsible for most of the thermal and meridional mass advection.

The nondimensional near-surface mass flux (the integral of the mean flow from the surface up to z = 1.5δ) is a convenient measure of the baroclinic effect on the near-surface flow and allows a comparison between the baroclinic contributions of the basic state and that of the rolls. The contributions of both effects are secondary to that of the basic Ekman layer friction (Mahrt and Park 1976). Both the rolls and the basic state baroclinic effects increase (decrease) the cross-surface-isobar flux in cold (warm) advection (Fig. 2d). However, the roll contribution to the divergence in warm advection is larger than its contribution to convergence in cold advection, leading to a net asymmetry. In transition zones from one advection regime to the other, only the rolls contribute to the convergence. By definition there is no basic-state baroclinic contribution to the cross-isotherm flow and the along-isotherm contribution is 0.1125 in nondimensional units. The roll contribution to the cross-isotherm (mainly meridional) flux is significantly larger in warm advection compared to cold advection (Fig. 2e), aiding the near-surface poleward transport of warmer air. The combined (basic state and roll) along-isotherm (mainly zonal) flux is a maximum (minimum) near the transitions at β = 0° (β = 180°) and the roll flux is approximately equal to the basic-state flux near β = 0° (Fig. 2f).

Figure 3 shows the change from barotropic in the near-surface wind¹ as a function of β from the nonlinear solution at Re = 500 and B = 0.1 (solid lines). For reference, the BL82 solution including rolls (long dashes) and the pure Ekman solution (short dashes) are included in the plot. The Ekman and two-layer solutions are very similar and sinusoidal with β. However, the surface wind from the nonlinear solution is quite asymmetric with respect to the thermal advection. The change in the cross-isobar angle (Fig. 3a) has a maximum of +8.2° in cold advection and a minimum of −3.4° in warm advection while the change in S (Fig. 3b) has a maximum of +7.3% and a minimum of −9.3% near the transitions between cold and warm advection. The baroclinic modifications of the surface wind speed (Fig. 3b) are significantly larger than those of the Ekman layer without rolls or the two-layer model with rolls. Compared to the two-layer model with rolls the enhancement of α in cold advection is much larger, while the decrease in warm advection is less.

5. Discussion and summary

The results summarized in Figs. 2 and 3 clearly show that the simple superposition of PBL roll and baroclinic effects does not capture the full nonlinear interaction between these processes. It is notable that the presence of rolls in a neutrally stratified, baroclinic PBL causes an asymmetry in the cross-isobar angle with respect to thermal wind orientation that is consistent with observations. The roll-induced asymmetry is further enhanced by increased baroclinity or by the correlation between cold (warm) advection and more (less) unstable stratification. Without the rolls, only the correlation between the horizontal and vertical temperature gradients leads to such an asymmetry. The need for such asymmetry to satisfy global atmospheric momentum, energy, and mass requirements has been argued and observed in previous studies. The roll effect provides an effective mechanism that can help satisfy these requirements and explain observations.

Priestly (1967) argued that downward momentum transport in the midlatitudes must act to increase the westerly component of the surface wind in order to close the budget of angular momentum and that the thermal wind enhances the transport of momentum through the boundary layer. This argument was used by Hoxit (1974) to explain the bias in surface wind speed and cross-isobar angle he observed. Global mass and energy considerations require net poleward transport of water and heat. All of the momentum transfer to the earth and much of the meridional water vapor transport take place near the surface in the PBL. Much of this transport can be accomplished by the rolls as follows.

Ignoring land–sea contrast and local effects, the large-scale geostrophic forcing and the thermal wind in the midlatitudes above the nearly neutral, roll-bearing PBL are predominantly westerly. Consider a perturbed geostrophic flow whose poleward (equatorward) meridional component is associated with warm (cold) advection. Without the roll effect, baroclinity will modify the surface wind equally on either side of the surface low. With the rolls, the augmentation of the surface westerly and the reduction in the equatorward meridional flow in the cold advection region to the west of the low are larger than their counterparts in the warm advection region to the east. The net effect is to increase both the westerly momentum transfer to the earth and the poleward low-level meridional transport. At the same time, the larger, deeper, and warmer PBL rolls in warm advection (Fig. 1) will act to transport more mass and heat meridionally. Observations by Levy and Tiu (1990) have detected a consistent increase of the poleward component of the low-level wind in satellite data.

Asymmetry between cold and warm advection surface flows on a much smaller scale (i.e., synoptic-scale disturbances and fronts) was also observed and simulated (e.g., Hoxit 1974; Keyser and Anthes 1982; Levy 1989). In frontal regions, such asymmetries have been postulated to trigger feedback mechanisms that act to differentiate between surface cold and warm fronts (Levy 1989) through “ageostrophic divergence stretching” (Levy and Bretherton 1987). We have conducted our analysis for mean climatological baroclinic conditions where our weakly nonlinear model is valid. It is not clear whether or not our analysis can be extended to very baroclinic zones (i.e., sharp fronts) or to regimes where shear is not dominant. However, it is possible to assess the effects of the rolls at early stages of frontogenesis, where temperature gradients are small. The variation in the shear and turning (Fig. 3) can be combined into a single “ageostrophic parameter” (Levy and Bretherton 1987), which is an approximate measure of the strength of frontogenesis by divergence stretching. The baroclinic modification to the ageostrophic parameter due to the rolls is such that the difference between the “would-be” cold front (transition from cold to warm advection) and warm front (transition from warm to cold advection) values is increased by 30% in neutral conditions. The frontolytic effect of cross-isotherm mixing due to the rolls is at a minimum in the transition zones (Fig. 2e). The implication is that at the initial stage of frontogenesis, the differentiation between cold and warm surface fronts will be enhanced in the presence of rolls, although frontogenetic processes themselves may be slowed down due to the additional mixing by the rolls (cross-isotherm flux, Fig. 2e).

We note that our interpretation above considers somewhat idealized cases. The true atmospheric flow is more complicated and other processes are usually at play as well. Equatorial symmetric instability (Levy and Battisti 1995), varying thermal stratification effects, interaction between rolls and thermals in more convective conditions, orography, land–sea contrast, and other effects can all lead to asymmetry that acts to restore energy and momentum balance. Nonetheless, the PBL is generally baroclinic and is predominantly near-neutral over the global oceans (United States Weather Bureau 1965). Our analyses indicate that the normally neglected roll effect is significant. The baroclinic and thermal stratification modifications often work in concert, further amplifying this effect. Therefore the baroclinic effect should be properly parameterized in PBL models. We have no climatology of PBL rolls; however, the current research suggests that they are generally present except for either stably stratified or highly convective conditions (Etling and Brown 1993). Although they are large scale as far as PBL eddies are concerned, they are difficult to observe, cannot be resolved in operational models, and must be parameterized in the PBL schemes of such models. Our results show that the climatological effects of rolls and thermal wind are most likely underestimated in the standard parameterizations.