On the Nature of the Transition Between Roll and Cellular Organization in the Convective Boundary Layer

Abstract

Both observational and numerical studies of the convective boundary layer (CBL) have demonstrated that when surface heat fluxes are small and mean wind shear is strong, convective updrafts tend to organize into horizontal rolls aligned within 10–20$^\circ$ of the geostrophic wind direction. However, under large surface heat fluxes and weak to negligible shear, convection tends to organize into open cells, similar to turbulent Rayleigh-Bénard convection. Using a suite of 14 large-eddy simulations (LES) spanning a range of $-z_i/L$ between zero (neutral) and 1041 (highly convective), where $z_i$ is the CBL depth and L is the Obukhov length, the transition between roll- and cellular-type convection is investigated systematically for the first time using LES. Mean vertical profiles including velocity variances and turbulent transport efficiencies, as well the “roll factor,” which characterizes the rotational symmetry of the vertical velocity field, indicate the transition occurs gradually over a range of $-z_i/L$; however, the most significant changes in vertical profiles and CBL organization occur from near-neutral conditions up to about $-z_i/L \approx$ 15–20. Turbulent transport efficiencies and quadrant analysis are used to characterize the turbulent transport of momentum and heat with increasing $-z_i/L$. It is found that turbulence transports heat efficiently from weakly to highly convective conditions; however, turbulent momentum transport becomes increasingly inefficient as $-z_i/L$ increases.

Appendix 1: Grid convergence tests

In order to investigate the effects of grid resolution on results from our LES, we performed several additional simulations on a $12\times 12\times 2\, \text {km}^3$ domain at six different grid resolutions ranging from $96^3$ to $320^3$. The simulations were configured following the case described in Sullivan and Patton (2011), where $U_g = 1$ m s$^{-1}$ and $Q_0 = 0.24$ K m s$^{-1}$. Simulations were run for approximately 25 large-eddy turnover times $T_{\ell } = z_i/w_*$, and averages calculated over the time interval $10T_\ell$–$25T_\ell$. Parameters of the simulations for the grid convergence tests can be found in Table 2.

Table 2 Properties of simulations for grid convergence study, including x- and z- filter widths ($\varDelta _x$ and $\varDelta _z$), characteristic filter width $\varDelta _f = (\varDelta _x \varDelta _y \varDelta _z)^{1/3}$, timestep ($\varDelta _t$) and characteristic length and velocity scales

Fig. 13

Convergence of mean profiles for highly convective simulations with $U_g = 1.0$ m s$^{-1}$ and $Q_0 = 0.24$ K m s$^{-1}$ [the test case described in Sullivan and Patton (2011)] with grid resolutions ranging from 96$^3$ to 320$^3$. a Mean velocity profile, b mean temperature profile, c total (resolved $+$ SGS) heat flux profile, d total (resolved $+$ SGS) vertical velocity variance, e total (resolved $+$ SGS) horizontal velocity variance, f resolved vertical velocity skewness

Mean profiles from each grid resolution can be found in Fig 13, where the mean velocity profile is displayed in Fig 13a, the mean temperature profile in Fig 13b, the total (resolved +SGS) heat flux in Fig 13c, the total vertical velocity variance $\langle w_\mathrm{tot}^2 \rangle /w_*^2 = \langle \widetilde{w}^{\prime 2} + 2e/3 \rangle /w_*^2$ in Fig 13d (where e is the SGS energy), the total horizontal velocity variance $\langle u_h^2 \rangle /w_*^2 = \langle \widetilde{u}^{\prime 2} + \widetilde{v}^{\prime 2} + 4e/3 \rangle /w_*^2$ in Fig 13e, and the resolved vertical velocity skewness $\langle w^3 \rangle / \langle w^2 \rangle ^{3/2}$ in Fig 13f. The SGS energy was estimated using the similarity model of Knaepen et al. (2002), via $e = \frac{1}{2} C_\mathrm{sim} L_{ii}$, where $L_{ii}$ is the trace of the Leonard stress tensor $L_{ij} = \overline{\widetilde{u}_i \widetilde{u}_j} - \overline{\widetilde{u}}_i \overline{\widetilde{u}}_j$ and $C_\mathrm{sim} = [ (\overline{\varDelta }/\widetilde{\varDelta })^{2/3} -1 ]^{-1}$ is a similarity coefficient, where $\overline{\varDelta }$ and $\widetilde{\varDelta }$ denote the test and grid filters, respectively. Note that $C_\mathrm{sim} = 1.7$ when $\overline{\varDelta } = 2 \widetilde{\varDelta }$. The more sophisticated model of Salesky and Chamecki (2012) that accounts for the shape of the energy spectrum in the ABL yields similar results, since the assumption of a $k^{-5/3}$ spectrum at scale $\overline{\varDelta }$ is reasonable under highly convective conditions.

As one can see from Fig. 13a, b, mean vertical profiles of first moments ($\langle U \rangle$ and $\langle \varTheta \rangle$) are in good agreement between all resolutions considered here. Profiles of the total heat flux, seen in Fig. 13c are similar for all resolutions in the lower part of the CBL; the differences become most noticeable near the inversion. Profiles of velocity variances and vertical velocity skewness are displayed in Fig. 13d–f. Here it should be emphasized that while the LES directly provides the resolved velocity and temperature fields, and the SGS heat flux (which is used in the numerical integration) through an SGS model, an a posteriori model for the SGS energy must be used to estimate the SGS contribution to $\langle w^2\rangle _\mathrm{tot}$ and $\langle u_h^2 \rangle _\mathrm{tot}$, since the SGS energy is included in the modified pressure, and is not necessarily modeled in LES. Thus the convergence of the total horizontal and vertical velocity variances will depend both on grid resolution and on the model for e employed. The SGS contribution to the w-skewness is unknown, due to the lack of a model to estimate it.

Regardless of the uncertainty in the true SGS contribution to the velocity variances and vertical velocity skewness, we find that, using the present approach to estimate e, all moments are nearly grid-independent for the $256^3$ grid, or a characteristic filter width of $\varDelta _f = 25.8$ m. Note that we find convergence at a larger value of $\varDelta _f$ than Sullivan and Patton (2011) (they found that most mean profiles were well converged for their $256^3$ case on a $5120\times 5120\times 2048$ m domain, or $\varDelta _f = 19.3$ m). We attribute the convergence at larger values of $\varDelta _f$ in our case to the difference in the subgrid model employed in our simulations vs. Sullivan and Patton (2011). We use the Lagrangian scale-dependent dynamic model (Bou-Zeid et al. 2005), whereas Sullivan and Patton (2011) used a one-equation model for the SGS energy (e.g. Moeng 1984).