Direct Numerical Simulation of the Moist Stably Stratified Surface Layer: Turbulence and Fog Formation

Abstract

We investigate the effects of condensation and liquid water loading on the stably stratified surface layer, with an eye towards understanding the influence of turbulent mixing on fog formation. Direct numerical simulations of dry and moist open-channel flows are conducted, where in both a constant cooling rate is applied at the ground to mimic longwave radiative cooling. Depending on the cooling rate, this can lead to either turbulent (weakly stable) or laminar (very stable) flows. Compared to the completely dry case, the condensation of liquid water in the moist case enables slightly higher cooling rates to be achieved before leading to turbulence collapse. In the very stable cases, runaway cooling leads to the substantial condensation of liquid water close to the ground and fog (visibility less than 1 km) results over much of the domain. In the weakly stable cases, turbulent mixing narrowly yields visibilities of 1 km close to the ground over a similar time period. However, despite the idealized nature of the system, the present results suggest that turbulence impedes, although will not necessarily inhibit, fog formation. A possible mechanism for fog formation within turbulent flows is identified, wherein regions of increased liquid water content form within the low-speed streaks of the near-wall cycle. These streaks are energized in the moist cases due to reduced dissipation of turbulence kinetic energy compared to the dry case, although in both cases the streaks are less energetic and persistent than in neutrally-stratified flow.

Appendices

Appendix 1: Effect of Computational Domain Size

In the present study, the domain size in the streamwise and spanwise directions is $2\uppi {h}\times \uppi {h}$, which is standard for neutrally-stratified dry simulations (Lozano-Durán and Jiménez 2014; Munters et al. 2016). However, there can exist patches of laminar and turbulent flow in stably-stratified flows close to laminarization, which are of size 10h (that is, the present domain size). A small computational domain can therefore result in turbulent structures that get ‘locked’ in place due to the periodic boundary conditions (Flores and Riley 2011; García-Villalba and del Álamo 2011). This can require substantial time for the turbulence to repopulate the domain and does not represent a physically realistic scenario.

We double the domain size to $4\uppi {h}\times 2\uppi {h}$ in the streamwise and spanwise directions, with the results shown in Fig. 12 for cooling with $h/L=0.6$ and $Re_\star =395$. We see a similar behaviour for both dry (solid) and moist (dashed) cases compared to the regular domain size. The peak around $tU_\star /h\approx 10$ for the dry case temperature fluctuations (Fig. 12b) is due to the turbulence becoming locked in place. Visual inspection of the velocity field of the dry case (not shown) reveals a similar pattern to Fig. 5 of Flores and Riley (2011), in which there exist stripes of turbulent and laminar flow extending down the entire streamwise length of the domain. The present larger domain dry case also exhibits this effect, where García-Villalba and del Álamo (2011) notes domains of at least $8\uppi {h}\times 3\uppi {h}$ are required to observe intermittency in the streamwise direction for statistically steady flows. This behaviour is not observed in any of the moist case, as presumably the mixing induced by condensation is sufficient to completely avoid these large laminar patches at these cooling rates. In this work, as in Flores and Riley (2011), we ignore data from when the turbulent flow is locked in place, as it is artificial. This is primarily of concern for the dry cases, and was not observed for the moist cases.

Fig. 12

Time series of a r.m.s. vertical velocity fluctuations, and b temperature fluctuations at $z^+=15$. Line styles are: red, $h/L=0.6$ with $T_0=279\hbox { K}$ (base case); grey, base case except with enlarged domain; blue, base case except with $T_0=285\hbox { K}$. Solid lines denote dry cases, dashed denote moist cases

Fig. 13

Same as Fig. 6b, showing time series of the height where $\langle q_l\rangle$ is equal to a specified threshold, $q_{l,t}$: dotted, $q_{l,t}=5\times 10^{-6}$; dashed, $q_{l,t}=1\times 10^{-6}$; and solid, $q_{l,t}=10^{-8}$ (saturation interface) $\hbox {kg kg}^{-1}$, for the weakly stable case 395M06 with $h/L=0.6$. Line colours: red, $T_0=279\hbox { K}$ (base case); grey, $T_0 = 285\hbox { K}$

Appendix 2: Effect of Initial Temperature

Given that the liquid water mixing ratio depends non-linearly on temperature, we also investigate the effect of the temperature at which the simulations are initialized, $T_0$. For the cases studied herein, we set $T_0=279.15\hbox { K}$, while here we increase the initial temperature to 285.15 K, for matched cooling rates of $h/L=0.6$. Figure 13 shows the time series of fog height, as in Fig. 6b for these two initial temperatures. The two cases are initially similar although they start to diverge after approximately $tU_\star /h\gtrsim 15$, at which point the interface between saturated and unsaturated water has reached the top boundary. The vertical velocity and temperature fluctuations shown in Fig. 12 also show good agreement between the base case (red dashed line) and increased initial temperature case (blue dashed line). This suggests that, while the temperature will be important for specific meteorological events, it is likely not relevant here given the idealized system used in the study.