a. Basic flow analysis

We refer to a Cartesian coordinate system in which the x axis is parallel to the mean wind direction DD in the boundary layer and the y axis is perpendicular to it. This coordinate system has the advantage that the x axis is nearly aligned with the rotation axis in case of roll convection.

In order to determine the underlying thermal and dynamical mechanisms forcing organized convection we assume that the basic flow is sufficiently characterized by the following parameters: the height h of the boundary layer; the depth Δh_c of the cloud layer; the vertical averages of potential temperature θ̃; specific humidity q̃; longitudinal wind component ũ and lateral wind component υ̃ over the depth h of the boundary layer; the air–sea differences Δθ_as, Δq_as, Δu_as, and Δυ_as and the vertical differences Δθ_bl, Δq_bl, Δu_bl, and Δυ_bl across the boundary layer; the vertical differences Δθ_ct, Δq_ct, Δu_ct, and Δυ_ct over a 300-m-deep layer above h; the curvature ∂²u/∂z² and ∂²υ/∂z² within the boundary layer; and an eventually existing inflection point in the u- or the υ-wind profile with corresponding values for the height h^u_ip and h^υ_ip.

These parameters were taken from the vertical profiles measured at both sides of the crosswind vertical planes (see Fig. 3). As an example, Fig. 4 shows a profile measured on 24 March 1993 at 1209 UTC at a distance Δx of 160 km from the ice edge and illustrates how the characteristic parameters were specified in this case.

Cloud base h_cb and cloud top h_ct were observed by eye by the aircraft scientist (the author was the aircraft scientist on board the research aircraft Falcon during all flights). Here, we assume that h and h_ct are identical. The air–sea differences were taken between the sea surface and the lowest permitted flight level at 90 m. The air immediately at the sea surface is assumed to be saturated with respect to the sea surface temperature T_s, that is, q_s = q_sat(T_s), and to have zero velocity, that is, u_s = υ_s = 0.

b. Secondary flow analysis

Parameters representing the characteristics of the secondary flow are analyzed using the data from the crosswind horizontal flight legs (see Fig. 3). These parameters are the horizontal wavelength λ of the rolls (or diameter in case of cells), the variances a²_R (a = θ, q, u, υ, w) in the wavelength range Δλ representing the secondary flow, and the vertical fluxes w_Ra_R (a = θ, q, u, υ), where w is the vertical wind component and the index R refers to the secondary flow scale.

As an example, Fig. 5 shows the time series of the downwelling longwave radiation L↓, the temperature T, the water vapor mixing ratio m, and the three wind components u, υ, and w measured during a crosswind flight leg at 92 m on 24 March 1993. The leg was situated directly to the west of the position of the profile displayed in Fig. 4. The aircraft flew from east to west in the direction of the negative y axis at a speed of 100 m s⁻¹ so that the time series covers a total distance of about 42 km. The maxima of L↓ mark the overlying cloud streets, which have wavelengths between 3.2 and 5.0 km. The corresponding secondary flow is clearly reflected in the local wind and thermodynamic field, particularly in the lateral wind component υ and in the mixing ratio m. Sections with confluent lateral flow, ∂υ/∂y < 0, coincide with higher values of m and T and with clouds above. The longitudinal wind component u exhibits also a regular pattern and is lower underneath and higher between the cloud streets. The secondary flow vertical motion w is not that clearly visible because its amplitude is relatively small at this low level, and turbulent as well as cloud-scale updrafts and downdrafts are superimposed. The maximum amplitude of w_R occurs normally in the middle of the boundary layer, as will be shown below.

Since the secondary flow pattern is clearly reflected in the cloud field, measurements of the longwave radiation are used to determine the wavelength range Δλ of the secondary flow. Figure 6 shows four variance spectra of longwave radiation calculated from crosswind flight legs flown at 92, 272, 468, and 694 m in the same vertical plane. The spectrum for 92-m height belongs to the time series shown in Fig. 5. Since the cloud pattern is independent of the measuring level in the vertical plane and does not change significantly during the typical duration of the measurements in the same vertical plane (in this case from 1135 to 1208 UTC for four legs), the wavelength range Δλ is fixed by taking all spectra in the same vertical plane into consideration. The limiting frequencies f₁ and f₂ are chosen as to contain the relevant variance maxima of downwelling (L↓) and upwelling (L↑) longwave radiation. Of course, the choice of f₁ and f₂ is somewhat subjective, but we have tested that eligible shifts of f₁ and f₂ do not change the results for the secondary flow-scale variances a²_R and fluxes w_Ra_R essentially.

An example of frequency-weighted variance spectra, S_aaf, and secondary flow-scale variances, a²_R, (a = θ, m, u, υ, w), is presented in Fig. 7. It is calculated by FFT analysis methods from the time series in Fig. 5 and the secondary flow wavelength range Δλ is taken from the limiting frequencies f₁ and f₂ in Fig. 6. The secondary flow signal appears clearly in all spectra, except in w for the above-mentioned reasons.

Variance and covariance spectra have been calculated for all crosswind flight legs and then integrated both over the entire and the secondary flow-scale frequency range to obtain the total (all frequencies) and secondary flow-scale variances and covariances. If the variations of the secondary flow in the crosswind direction are assumed to be sinusoidal, a = A₀ sin[(2π/λ)y + ϕ] (what is close to reality), the following relation between the secondary flow-scale variance a²_R and the amplitude A₀ of the sinusoidal variation holds:

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From (1) the average amplitude of the secondary flow-scale variations is estimated.

a. Geometrical dimensions of the secondaryflow patterns

Wavelength λ and vertical depth h of the secondary flow patterns were measured, as mentioned above, in 13 aircraft missions during the field experiments ARKTIS 1991 and ARKTIS 1993. During seven flight missions measurements within a vertical crosswind plane were made at two different downwind distances Δx from the ice edge. All 20 cases are listed in Table 1, showing day, Δx, the geometrical dimensions h, λ, and the aspect ratio λ/h of the secondary flow patterns, as well as the geometrical dimensions of the cloud field, that is, cloud base h_cb, cloud layer depth Δh_c, and cloud coverage N. Based on the eye observations of the aircraft scientist, on a cockpit video camera, and on satellite images, the cases are classified into four categories with respect to the convective pattern. It is distinguished between rolls (R), transitional plan forms between rolls and cells (R/C), and cells (C). The “roll” class is subdivided further with respect to ice edge distance Δx into two classes named “small” rolls (RS) and “normal” rolls (RN).

The relation between λ and h is shown in Fig. 8a. As also found by other authors (e.g., Kelly 1984; Miura 1986), rolls occur at small λ and h, predominantly with λ < 5 km and h < 1 km. In our cases (with one exception discussed below), clear cellular patterns occurred with λ > 15 km and h > 1.4 km. The aspect ratio λ/h in the roll cases was between 2.6 and 6.5 and thus clearly smaller than in the cell cases, which had values up to λ/h = 12.6. On all seven days when crosswind vertical sections were flown at two different Δx, the downwind increase in h is accompanied by an increase in λ in such a way that λ/h increases, too.

This is more evident from Fig. 8b where the aspect ratio λ/h is displayed as function of the downwind distance Δx from the ice edge. However, there is a large variability in the λ/h(Δx) relation from case to case both in the level of the λ/h values and in the rate it increases with Δx. The variability of λ/h appears to be larger in the region of the cellular patterns than in the roll region. Regarding all 20 cases as independent measurements the best logarithmic fit of λ/h for Δx between 25 and 1200 km is given by λ/h = 5.5 + 1.7 lnΔx, where Δx is taken in units of 100 km.

In Figs. 8a,b, a somewhat outstanding role is played by the two observations on 7 March 1991. On this day cells were observed having typical h values but untypically short wavelengths λ so that very small λ/h values result. The reason for this is not clear. This day was the only case when cells were not observed in an off-ice airflow but to the north of the Lofoten Islands in an easterly airflow coming over the mountains of northern Norway. In this case lee effects were involved. Cloud development started at a distance of about 150 km off the coast whereas in all other cases it started right at the ice edge. Table 1 shows that the cloud depth Δh_c on 7 March 1991 at the first crosswind vertical section at Δx = 200 km is only 315 m and thus is rather small in comparison to the depth (h = 1115 m) of the boundary layer and also in comparison to the other cases. The weak cloud activity as a consequence of the lee effect may be one reason for the small aspect ratios on this day, which amounted to only λ/h = 3.0 at Δx = 200 km and λ/h = 3.7 at Δx = 450 km. This conclusion is based on model simulations of convection in cold-air outbreaks performed by Müller and Chlond (1996). With a convection-resolving large eddy simulation model they showed that a downwind increase of λ/h can only be achieved if sufficient cloud development is present. Without clouds the convective patterns retain a small aspect ratio around λ/h = 3.0 similar to that known from the linear Rayleigh instability theory (λ/h = 2.8).

The increasing importance of clouds in downwind direction from the roll pattern to the cellular pattern region is also indicated in Table 1. Cloud depth in the roll region is between 200 and 700 m, whereas in the cell region it is between 700 and 2000 m. Although there is a gradual decrease of cloud coverage N in downwind direction, the average cloud layer thickness if flattened homogeneously over the horizontal area, that is, Δh_cN, is nearly twice as large in the cell (660 m) than in the roll region (250–340 m). However, the relative area F_cl,rel of a vertical y–z cross section through the secondary flow pattern that is filled with clouds, that is,

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decreases from about 0.5 to 0.3 from the roll to the cell region. Thus, taken together, when moving in downwind direction the depth of the layer filled by clouds increases absolutely, but decreases in relation to the depth of the layer involved in the secondary flow.

b. Amplitudes of the secondary flow

Figure 9 represents a sketch of the secondary flow in the vertical y–z plane perpendicular to the mean wind. The sketch illustrates that the largest variations of the secondary flow variables in the crosswind y direction are to be expected at low levels for the lateral (υ_R) and longitudinal (u_R) wind component as well as for temperature (θ_R) and water vapor mixing ratio m_R. The largest variations of the vertical wind component w_R are expected to occur at levels near the middle of the boundary layer. The latter holds also for the vertical fluxes: w_Rθ_R, w_Rm_R, w_Ru_R, and w_Rυ_R.

The amplitudes A_a (a = θ_R, m_R, u_R, υ_R, w_R) of the secondary flow variables calculated according to (1) and classified with respect to the convective pattern according to Table 1 are listed in Table 2. They refer to the lowest permitted flight level of 90 m for θ_R, m_R, and u_R, υ_R. The amplitudes of w_R refer to the level z(w_max) where the maximum of the secondary flow–scale vertical wind variance w²_R was measured. Unfortunately, in many cases—especially in the roll cases with a very shallow boundary layer—there were only flights near the bottom and top of the boundary layer (see Fig. 3) so that the real maximum of A_wR was not measured. In those cases the “90-m” values are quoted in brackets in Table 2.

Table 2 shows that the secondary flow temperature amplitude A_θR is only a few tenths of a degree. The same order of magnitude holds for the water vapor amplitude A_mR. However, A_mR exhibits a clear increase in downwind direction from the roll to the cell patterns. The ratio of the heat content involved with temperature (A_θR) and moisture fluctuations (A_mR), that is, (L/c_p)(A_mR/A_θR), where L is the latent heat of vaporation and c_p the specific heat at constant pressure, increases from 0.65 for small rolls to 2.10 for “cells.” This underlines the importance of moisture processes as was derived by Brümmer (1997) from boundary layer heat budget calculations for the same cold-air outbreaks as discussed in this paper. He found that in the cell pattern region the latent heat release by condensation in clouds is the dominating term in the heat budget equation. The heating of the boundary layer by the surface sensible heat flux plays only a secondary role in this region but is the primary heat source in the roll pattern region.

The secondary flow wind amplitudes A_uR, A_υR, and A_wR are on the order of 0.5–3 m s⁻¹. All three amplitudes increase in a downwind direction and are roughly twice as large in cell patterns as in roll patterns. In general, the lateral wind component exhibits the largest amplitude of all three wind components and the vertical wind component the smallest one. However, there is a clear downwind trend concerning the ratio A_uR/A_υR. In accordance with the geometrical planform of the convective pattern, the ratio is about 0.66 in roll circulations and increases to about 1.0 in cell circulations, where there is no preferred space direction.

c. Total and secondary flow-scale fluxes

In this section the vertical fluxes of heat, moisture, and momentum by the secondary flow w_Ra_R and by all scales of motion w′a′ (a = θ, m, u, υ) are investigated with respect to the pattern form of the secondary flow and to their height dependence. The flux profiles are presented in Figs. 10a–d in a normalized form. The normalized height z_N is calculated using cloud-base height h_cb and cloud layer depth Δh_c in the following way:

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In this way the normalized height z_N is between 0 ⩽ z_N ⩽ 2. Here z_N = 1 represents the cloud base and z_N = 2 the top of the boundary layer. However, in order to take into account the extreme differences in h_cb and Δh_c between the roll and cell pattern regions, the height is again scaled by the mean values h̃_cb and Δh̃_c for the respective category of the secondary flow pattern, namely, the classes RS, RN, R/C, and C. Thus, the normalized but rescaled height z_sc is defined as

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In an analogous way, the flux at any height is first normalized by the total surface flux (w′a′)_o,

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and is then, in order to retain the basic flux differences between the different secondary flow patterns, scaled by the respective mean surface flux (denoted by a tilde) for each convective pattern class, that is, RS, RN, R/C, and C, as follows:

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The values used to normalize and rescale the fluxes are listed in Table 3.

To normalize the sensible and latent heat flux with the corresponding surface value is a meaningful and usual procedure, because the surface flux is the largest value and the decisive parameter in thermal convection. However, to normalize the momentum flux with its surface value may be questionable, because it is, in contrast to the heat and moisture flux, not always the maximum flux value and not the only decisive parameter. The momentum flux profile can depend on many factors, for example, the thermal wind.

Figures 10a–d show the total w′a′ and the secondary flow–scale vertical fluxes w_Ra_R of the variables a = θ, m, u, and υ. The results are grouped with respect to the four classes of convective patterns: RS, RN, R/C, and C. The curves within each class show some degree of variability from day to day depending on the actual conditions of air–sea temperature difference, wind speed, etc. The height variations within each curve represent systematic differences with height as well as sampling uncertainties resulting, for example, from a somewhat unrepresentative flight leg. In spite of possible uncertainties in the individual curves, the scatter of the curves within each category in Figs. 10a–d shows that the general differences between the secondary flow and the total flow as well as the general differences between the individual categories are significant. Only these general differences will be discussed in the following.

To point out the systematic differences between the total and secondary flow fluxes and the differences between the individual categories more clearly, the average sensible and latent heat flux profiles are displayed in Fig. 11. The total latent heat flux, m′w′, shows, concerning its profile shape, a systematic variation from the roll pattern region to the cell pattern region. Whereas the flux profile between the maximum surface value and the nearly vanishing top value is only slightly curved in the roll region, where a shallow cloud layer is present, it is strongly curved and even exhibits a maximum that is larger than the surface flux in the cell region, where deeper clouds are present.

Similar systematic variations of the moisture flux profiles were found in roll convection by Brümmer (1985) and in cell convection by Brümmer et al. (1986). Since the divergent subcloud moisture flux in the cell region would finally dry out the subcloud layer (what is not observed), there must be processes acting in the opposite direction. Possible candidates are a large-scale horizontal moisture convergence or the evaporation of precipitation. Assuming precipitation rates of about 4 mm day⁻¹ in the cell convection region, as was estimated by budget analyses by Brümmer (1997) for the same cold-air outbreaks, evaporation rates needed to balance the moisture budget of the subcloud layer would be far beyond plausibility. Thus the large-scale moisture convergence remains as a possible balance process. This conjecture is supported (a) by the presence of large-scale convergence, which was analyzed from the mass budget estimates in Brümmer (1997); and (b) by a systematic downwind transition from an anticyclonically curved flow at distances Δx < 100 km from the ice edge to a cyclonically curved flow at distances Δx > 200 km, which was observed in the ARKTIS 1993 cold-air outbreaks (Brümmer 1996).

Furthermore, similar conditions are observed in situations of deep convection in the Tropics. Brümmer (1979) found that the vertical moisture flux in the subcloud layer was divergent (convergent) under disturbed (undisturbed) weather conditions with large-scale horizontal flow convergence (divergence) and deep (shallow) convection over the tropical Atlantic Ocean during the 1974 Global Atmospheric Research Program Atlantic Tropical Experiment. The relation between large-scale horizontal moisture convergence and deep convection is a repeatedly observed fact. It is also taken into account in the parameterization schemes for deep convection (e.g., Tiedtke 1989).

The systematic changes of the moisture flux profiles from the roll region to the cell region and the above-mentioned facts on the large-scale moisture convergence suggest the following hypothesis: a necessary condition for the transition from roll convection to mesoscale cell convection is the formation of deep clouds as a result of large-scale moisture convergence. This hypothesis is further supported in section 5a where it is found that the ratio of the condensational heating in clouds to the surface heating increases from less than 1 in the roll regions to more than 1 in the cell regions.

We return to Fig. 10 and the discussion of the fluxes of longitudinal (w′u′) and lateral momentum (w′υ′). The momentum flux profiles vary considerably from case to case within each category and a systematic trend from the roll region to the cell region is not present. This is a consequence of the different shapes of the wind profiles of both the longitudinal and the lateral wind component. Thus, the average flux profiles are not of general meaning. However, independent of the category of the convective pattern, the total momentum flux profiles exhibit the typical known features.

1. The total w′u′ flux decreases with height between the surface and at least cloud base, where the flux may be negative or positive. This is equivalent to an export of momentum and thus to a reduction of momentum in this layer, which of course has to be compensated by the mean pressure gradient force. These average conditions are in agreement with a profile where the u-wind component increases with height to a maximum value at or below cloud base.

2. The total w′υ′ flux is around zero at the surface and has more frequently positive values than negative ones at the levels above. Positive w′υ′ fluxes suggest decreasing υ values with height. This is equivalent to a turning of the wind direction with height to the right as it is the typical case under the action of surface friction. Continuing this interpretation, negative w′υ′ fluxes would suggest increasing υ values with height and a backing of wind direction. This occurs in situations when the cold-air advection is so strong that the thermal wind effect overcompensates the frictional wind veering.

a. Thermodynamic features of the basic flow

Table 4 summarizes the thermodynamic conditions in the surface, boundary, cloud, and inversion layers in the regions of RS, RN, R/C, and C. In the surface layer (termed here as the layer between the sea surface and the lowest flight level of 90-m height), the stratification is rather unstable. The air–sea temperature difference Δθ_as gradually changes from extremely unstable values in the area of small rolls near the ice edge to moderately unstable values in the area of cells. This goes parallel with the general downstream increase of the mean potential air temperature θ. Depending on the synoptic conditions, the wind speed at 90-m height U₉₀ shows a large degree of variability from case to case in each convective category. Thus the mean wind differences from one convection category to the other cannot be regarded as systematic for the pattern transition. A similarly large degree of variability is present for the Monin–Obukhov length L_M, which was calculated according to

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In (9) g is the acceleration of gravity, k = 0.4 is the von Kármán constant, and the bulk formulas are used for calculation of momentum and heat flux. For simplicity the corresponding transfer coefficients c_D and c_H are assumed to be equal as c_D = c_H = 1.3 × 10⁻³. This assumption does not influence the results for L_M significantly. In all convection categories L_M is clearly unstable but shows no clear trend from the roll to the cell region. The same is true for h/L_M, regarded as a stability parameter for the entire boundary layer. Thus, both L_M and h/L_M do not appear to be useful measures in distinguishing between roll and cell convection.

The stratification in the overlying part of the boundary layer, between 90 m and h, is slightly stable as represented by the corresponding temperature difference Δθ_bl. On the average, the stability Δθ_bl/(h − 90 m) is about twice as large in the cell region as in the roll region.

The thermal stability conditions in the boundary layer are usually described by the Rayleigh number Ra. It represents the ratio between viscous and buoyancy force and is determined here from the relation

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where K_M and K_H are the turbulent diffusion coeficients for momentum and heat, respectively. Since K_M cannot be measured directly, it was calculated according to Louis (1979) as

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for the surface layer with depth Δz = 90 m. In (10) the mixing length, l, is given by the relation

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with z = 45 m and λ = 100 m; the Richardson number Ri in its bulk version as

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assuming Δv = v₉₀ − 0; and the stability function F(Ri) is defined as

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with the coefficient b = 9.4 and c = 7.4bl²[(z + Δz/z)^1/3 − 1]^3/2/(z^1/2Δz^3/2). Because of the uncertainties that are inherent in an approximation of K_M like in (11), and for the sake of simplicity, we assume that K_M = K_H in (10) and that the surface layer values of K_M are roughly representative of the average values in the boundary layer.

The values of K_M estimated in this way are listed in Table 4 and vary between 29 and 71 m² s⁻¹. They differ only by a factor of about 2 so that a constant K_M, as it is often assumed (e.g., Walter 1986) for the calculation of Ra, is not too much of a simplification. Our Ra values increase in downstream direction from the roll to the cell region. This holds not only for the category averages, but also for each of those days when measurements at two distances Δx from the ice edge were made. The Rayleigh number Ra is about 10 to 20 times larger in the cell area than in the roll area. It appears from Table 4 that the transition from roll convection to cell convection occurs in the range between 0.8 × 10⁶ < Ra < 1.0 × 10⁶.

Walter (1986) also investigated the relation between Ra and convection type. He scaled Ra with the critical Rayleigh number Ra_c from the linear theory (Ra_c = 1708) and observed rolls for 20 ⩽ Ra/Ra_c ⩽ 100 and cells for Ra/Ra_c ≥ 250. In 15 out of 16 cases in the convection categories RS, RN, and C, our values of Ra/Ra_c do not contradict Walter’s classification. However, the range of Ra/Ra_c values in our transitional category R/C is much broader (88 ⩽ Ra/Ra_c ⩽ 533) than that given by Walter (100 < Ra/Ra_c < 250).

Beside surface heating in promoting thermal instability, the latent heat release by condensation in clouds is another important process. In connection with boundary layer moisture and heat budget calculations for cold-air outbreaks from the Arctic sea ice, Brümmer (1997) estimated the net condensation/evaporation, c − e, in clouds. For those cases where the budget-study days are identical to the days investigated here, the c − e values are listed in Table 4. The heating effect of net condensation is converted in Table 4 to an equivalent heat flux, (L/c_p)h(c − e), at the bottom of the boundary layer and can thus be directly compared with the surface buoyancy flux (θ^′_υw′)₉₀. Table 4 shows that the condensational heating increases gradually by a factor of 2 to 3 from the roll to the cell convection region and that it is as large or even larger than the surface buoyancy flux in the cell region. It is thus hypothezised that condensation is a necessary process for the generation of cell convection and that its heating effect must at least reach the magnitude of the surface heating.

Numerical simulations by Müller (1995) show that the condensation process is a necessary condition for the transition from convection with small aspect ratios to convection with large aspect ratios as it is observed in mesoscale cellular convection. Model results of Rao and Agee (1996) show that even the mode of precipitation (water or ice) and the associated microphysical processes have a significant effect on the structure of the convective boundary layer. A broadening of the aspect ratio is also known to be caused by increasing stability in the cloud layer (e.g., Bjerknes 1938; Chlond 1988). In agreement with this idea are the above-mentioned increase of Δθ_bl (Table 4) and the corresponding increase of aspect ratio λ/h (Table 1). Except for 8 March 1991, this relation holds on all days when measurements at two distances Δx were made (11, 19, 20, 24, and 25 March 1993; 7 March 1991). Since both condensation and stability are working or are present simultaneously, their individual contribution to the observed aspect ratio broadening cannot be seperated. Furthermore, condensation and stability are interacting in a sense that increasing stability reduces condensation and vice versa.

The stability in the inversion topping the cloud layer decreases systematically from the roll to the cell region as it is represented by the potential temperature difference Δθ³⁰⁰_ct over a 300-m-deep layer above cloud top (Table 4). This is a consequence of the continuous warming of the boundary layer air over the open water, while the air above the boundary layer remains nearly unaffected. The weaker stability of the inversion in the cell region allows more variability in the heights of the cloud tops than in the roll region. One exception was on 25 February 1991 when closed cellular convection and a large value of Δθ³⁰⁰_ct were observed in a situation of high pressure subsidence with flat cloud tops and high cloud coverage N (see Table 1). All other cell cases presented here refer to open cellular convection.

b. Kinematic features of the basic flow

The vertical profile of the horizontal wind plays an important role for the organization of the convection. Wind shear (e.g., Asai 1970), curvature (e.g., Kuettner 1971), and inflection point (e.g., Lilly 1966) may cause longitudinal organization. In case of thermal instability with wind shear and curvature the convection is organized in rolls that are parallel to the shear vector or to the vertical plane in which the curvature is present. In case of inflection point instability the roll axes are oriented perpendicular to the plane in which the inflection point is observed.

In order to characterize the vertical wind structure in a simple (but sufficient) manner the wind vector has been taken at four levels only: 90 m, h/2, h, and h + 300 m. The resulting wind profiles relative to the corresponding wind at 90 m in each convection case are shown in Fig. 12 while Fig. 13 shows the mean wind values for each convection category in absolute wind units. Based on the wind at these levels, the layer-averaged boundary layer wind, the shear across the boundary layer and across the inversion layer, as well as the curvature in the boundary layer were calculated and the results are listed in Table 5. The curvature was calculated according to

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The layer-averaged boundary layer wind does not appear to be useful measure to distinguish between rolls and cells. It varies from small to large values within each category and simply reflects the various gradients of the large-scale pressure field. The wind shear, however, is significantly smaller in the cell than in the roll regions. Taking the category averages of the shear magnitude the difference between rolls and cells is on the order of a factor of 2–3. This holds for the shear in the boundary layer as well as in the inversion layer. Independent of the convective pattern form, the shear in the inversion layer is always larger than that in the boundary layer, which reflects the different turbulence states within both layers.

Assuming 10⁻⁵ m⁻¹ s⁻¹ as a critical magnitude of the curvature above which convection is organized in rolls (Kuettner 1971), the curvature ∂²u/∂z² in the x direction along the mean wind is larger than the critical one in nearly all roll convection cases. This is in accordance with the observation that the cloud streets were aligned in the downwind direction. On the other hand the curvature in all cell cases is clearly below the critical value. With only one exception (11 March 1993 in the RS category) our observations support the critical value of the wind curvature that was suggested by Kuettner (1971) and was also based on observational evidence.

In several cases of roll convection the author (as aircraft scientist on board the research aircraft Falcon) observed by eye periodic variations of the cloud field in the along-roll direction giving the impression of a second instability mechanism overlying more or less perpendicularly the primary rolls (Fig. 14). It is supposed that the overlying variations originate from the inflection point instability mechanism operating additionally at levels near the top of the boundary layer. In Table 5 the occurrence of an inflection point in the wind profile is indicated by its height h_ip relative to h (two values for the same case refer to the two profiles at both sides of the vertical plane in which the horizontal flight legs were flown; see Fig. 3). In the roll categories RS and RN, an inflection point in the u component occurs in 75% of the cases; it is situated relatively close above the top of the boundary layer. On the other hand, an inflection point in the υ component is less frequent and situated at a much higher level above the top. These figures at least do not contradict the presumption on the origin of the secondary periodic cloud variations. In the transition region R/C and the cell region C an inflection point is less frequent but occurs in both wind components and relatively close above h. In this case secondary periodic variations may have also occurred but are more difficult to be recognized by eye in the cloud field because of the underlying cellular pattern, the larger wavelength of the cell pattern, and the smaller cloud coverage as compared to an underlying roll pattern.

c. Relation between the basic flow and the kinetic energy of the secondary flow

One aim of the investigations on the secondary flow, besides the study of its properties, is to relate these properties to parameters of the basic flow, in order to account for the secondary flow effects in large-scale weather and climate models that are not able to resolve the secondary flow explicitly. Since in this study parameters of both the basic and the secondary flow have been measured or derived from the measurements, a first attempt is made to relate the layer-integrated secondary flow–scale kinetic energy,

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to the Rayleigh number Ra, regarded as a measure of the stability conditions of the basic flow in the boundary layer. The secondary flow–scale kinetic energy as well as the individual contributions by the three wind components are listed in Table 6 and the relation between E_kin,R and Ra (taken from Table 4) is presented in Fig. 15.