Entrainment in dry and moist thermals

We study entrainment in dry thermals in neutrally and unstably stratified ambients, and moist thermals in dry-neutrally stratified ambients using direct numerical simulations. We find, in agreement with results of Lecoanet and Jeevanjee [J. Atmos. Sci. 76, 3785 (2019)], that turbulence plays a minor role in entrainment in dry thermals in a neutral ambient for Reynolds numbers $\text{Re}\lesssim {10}^4$. We then show that the net entrainment rate increases when the buoyancy of the thermals increases, either by condensation heating or because of an unstably stratified ambient. This is in contrast with the findings of Morrison et al. [J. Atmos. Sci. 78, 797 (2021)]. We also show that the role of turbulence is greater in these cases than in dry thermals and, significantly, that the combined action of condensation heating and turbulence creates intense small-scale vorticity, destroying the coherent vortex ring that is seen in dry and moist laminar thermals. These findings suggest that fully resolved simulations at Reynolds numbers significantly larger than the mixing transition Reynolds number $\text{Re}={10}^4$ are necessary to understand the role of turbulence in the entrainment in growing cumulus clouds, which consist of a series of thermals rising and decaying in succession.

This article appears in the following collection:

Figure 1

A typical (dry) thermal in our simulations, with the filled contours representing a typical distribution of the temperature $\theta$ coloured logarithmically. The radius of the thermal $R$ is defined as the distance measured from the axis of symmetry ($r=0$) to the widest location of the curve $\psi =0$. The location of the centroid of the thermal $z_{ct}$ is defined in Appendix pp1. The vertical locations of maximum radius and centroid of the thermal do not, in general, coincide. The thermal top height ($z_t$) is defined as the location of the steepest gradient of the azimuthally averaged temperature. The volume of the thermal is taken to be the volume bounded by the dividing streamline $\psi =0$ in a frame of reference moving with the velocity $w_{\mathrm{th}}$ of the thermal (see Sec. 2b). The maximum flow velocity in the region identified as the thermal is labeled $w_{\max }$. The trailing flow suggests detrainment, which is small for dry thermals (see LJ19), but larger for moist thermals.

Figure 2

The azimuthally averaged temperature $\theta$ in laminar ($\text{Re}=630$, left panels) and turbulent ($\text{Re}=6300$, right panels) dry thermals in a neutrally stratified ambient at the times indicated. The thermals were initialized with the same level of noise. The color maps are logarithmically scaled and the same for all three times shown. The laminar thermals have spun up into vortex rings as expected. The turbulent thermals are noticeably different in shape, with more diffuse cores and more prominent tails.

Figure 3

The (a) entrainment rate and (b) entrainment efficiency, defined in Eqs. (2.11) and (2.12), for laminar (blue, $\text{Re}=630$) and turbulent (red, $\text{Re}=6300$) dry thermals in a neutral ambient (${\mathrm{\Gamma }}_0={\mathrm{\Gamma }}_u$). The thin red lines are the five individual runs for the turbulent thermals with different instantiations of noise, while the average is plotted as a thick line. In (b), the curves from LJ19 are also plotted for comparison. Turbulent entrainment rates are marginally higher than laminar entrainment rates, and the difference is smaller than in LJ19. There is a dip in the value of $e$ after $z_{ct}>16$, more pronounced in the turbulent cases, which is also seen in LJ19.

Figure 4

Instantaneous 2D slices at $y=0$ of the temperature $\theta$ in laminar ($\text{Re}=630$, left panels) and turbulent ($\text{Re}=6300$, right panels) moist thermals at the times indicated. As in Fig. 2, a logarithmic color scale is used. Laminar moist thermals assume a distinct “arrowhead” shape where the vortex ring remains. See Fig. 8 for corresponding plots of the vertical velocity.

Figure 5

Azimuthally averaged liquid mixing ratio $r_l$ for laminar ($\text{Re}=630$, left panels) and turbulent ($\text{Re}=6300$, right panels) moist thermals in a dry-neutrally stratified ambient at the same times as in Fig. 4. (a) Initially, moist thermals behave similarly as dry thermals, forming vortex rings whose cores have large $r_l$ in both laminar and turbulent cases. (b) These vortex rings morph into the “arrowhead” shape with large $r_l$ in both the vortex cores as well as the tip of the arrow. (c) At $t=32$, the distribution of $r_l$ is very different in the laminar and turbulent cases; in the laminar thermal, large $r_l$ values occur in the vortex ring as well as the thermal top, whereas the vortex ring is destroyed in the turbulent thermal and the maximum in $r_l$ occurs only at the thermal top. Note that a logarithmic color scale is used.

Figure 6

Azimuthally averaged azimuthal vorticity ${\omega }_{\varphi }$ for laminar ($\text{Re}=630$, left panels) and turbulent ($\text{Re}=6300$, right panels) moist thermals in a dry-neutral ambient at the same times as in Fig. 4. (a) The vorticity is initially concentrated in the vortex ring that forms. (b) The vorticity in the ring increases in the laminar thermal, while the ring becomes more diffuse in the turbulent thermal. (c) Significant negative vorticity can be seen along the axis of symmetry in both laminar and turbulent thermals, and the vortex ring in the turbulent thermal is even more diffuse.

Figure 7

Vorticity magnitude ($\left|\omega \right|=\sqrt{{{\omega }_x^2+{\omega }_y^2+{\omega }_z^2}^{\phantom{0}}}$) in (a) DNT and (b) MNT thermals. The black circles in (a) denote the location of core of the vortex ring in the DNT thermal. In the MNT thermal in (b), the vortex ring has disintegrated and the vorticity is space-filling, and the maximum $\left|\omega \right|$ is $\approx 5$ times higher than in (a).

Figure 8

Instantaneous 2D slices at $y=0$ of the vertical velocity $w$ in laminar($\text{Re}=630$, left panels) and turbulent ($\text{Re}=6300$, right panels) moist thermals at the same times as in Fig. 4. The vertical velocities for moist thermals are significantly higher than for dry thermals. (a) The flow is initially axisymmetric, with maximum velocities at the vortex rings. (b) As the thermal is heated by condensation, the laminar case retains its symmetry but the turbulent case can be seen to show departures from symmetry. Large velocities occur both in the rings and in thermal center. (c) Velocities have increased significantly in both laminar and turbulent thermals. The laminar thermal continues to be axisymmetric and the vortex ring continues to become stronger, whereas the vortex ring is destroyed in the turbulent case. Cf. Figs. 4 and 6.

Figure 9

The (a) location $z_{ct}$, (b) maximum velocity $w_{\max }$ (c) radius $R$, and (d) volume $V$ of laminar ($\text{Re}=630$) and turbulent ($\text{Re}=6300$) moist thermals in a (dry-) neutrally stratified ambient.The $x$ axis starts from four for all the sub figures. The curves for dry thermals in the same ambient are shown for comparison. In each case, individual runs are plotted as thin lines while ensemble averages are plotted as thick lines. Moist thermals begin to accelerate for $z_{ct}\gtrsim 10$ due to condensation heating and, as a result, grow to larger radii, have greater volumes and rise faster than dry thermals. It is worth noting that, even though the volumes of the moist thermals grow much faster than their dry counterparts, the radius of the moist laminar thermal grows similarly to the dry cases. This is due to a change in the shape of the thermal (see, e.g., Fig. 19; see also Fig. 15).

Figure 10

As in Figs. 2 and 4, but for dry thermals in an unstably stratified ambient.

Figure 11

As in Fig. 8, but for dry thermals in an unstably stratified ambient.

Figure 12

The (a) location $z_{ct}$, (b) vertical velocity $w_{\mathrm{th}}$ (c) radius $R$, and (d) volume $V$ of laminar ($\text{Re}=630$) and turbulent ($\text{Re}=6300$) dry thermals in an unstably stratified ambient with ${\mathrm{\Gamma }}_0-{\mathrm{\Gamma }}_u=0.02$ compared with dry thermals in a neutrally stratified ambient. Similarly to moist thermals in a (dry-) neutrally stratified ambient, the accelerating thermals here rise faster, grow to greater radii, and have larger volumes than dry thermals in a neutrally stratified ambient. Also note that the laminar thermals rise faster but have smaller radii and volumes than the turbulent thermals. The changing shape of the thermal accounts for the increase in thermal volume in the laminar case even though the thermal radius evolves similarly to the dry thermals in a neutral ambient (cf. Fig. 9).

Figure 13

The evolution of a dry turbulent thermal at $\text{Re}=12600$. The contour plots shows the aximuthally averaged temperature $\theta$ and as with other cases, the black line represent the thermal boundary. Cf. the laminar and turbulent ($\text{Re}=6300$) cases in Fig. 2.

Figure 14

The (a) location $z_{ct}$, (b) maximum velocity $w_{\max }$, (c) radius $R$, and (d) volume $V$ of a laminar dry thermal ($\text{Re}=630$) and turbulent dry thermals ($\text{Re}=6300$ and $\text{Re}=12600$) in a neutrally stratified ambient. In the turbulent case, curves for individual runs at $\text{Re}=6300$ (thin gray lines) and $\text{Re}=12600$ are plotted along with the ensemble average.

Figure 15

As thermals rise, the volume scales with the thermal radius as $V\propto R^3$, suggesting self-similarity.

Figure 16

(a) The net entrainment rate ${\epsilon }_{\mathrm{net}}$ as a function of the thermal radius $R$ for the cases in Secs. 4a–4d. The dotted line represents the scaling relation Eq. (2.11). The moist laminar and turbulent cases seem to have slightly higher entrainment rates. (b) ${\epsilon }_{\mathrm{net}}$ as a function of the thermal location $z_{ct}$. (c) The entrainment efficiency $e$ [Eq. (2.12)] plotted as a function of $z_{ct}$. The entrainment efficiency in the moist cases is larger than the entrainment efficiency in the corresponding dry cases. (d) The correlation ${\epsilon }_{\mathrm{net}}\sim B_{\mathrm{avg}}{w}_{ct}^{-2}$ (see text), showing a reasonable collapse. The points are plotted for every 0.3 flow time unit.

Figure 17

The terms in the impulse budget, Eq. (3.4), as a function of the thermal height $z_{ct}$ for (a) dry laminar (b) dry turbulent, (c) moist laminar, and (d) moist turbulent thermals. Terms calculated using the radius of the thermal are plotted with solid lines, while terms calculated using the vortex ring radius (in plots a–c) are plotted with dashed lines. In moist thermals, the buoyancy, the circulation and entrainment terms are all an order of magnitude higher than in dry thermals. Turbulence also leads to a greater entrainment rate in moist thermals. Note that $\rho =1$ under the Boussinesq approximation.

Figure 18

(a) The average buoyancy $B_{\mathrm{avg}}$, and (b) the average value of the passive scalar $C$ (see text) in dry and moist thermals as they rise in dry-neutral ambient.

Figure 19

Identification of coherent thermals using the steps from Appendix pp1. The parameters are the same as in Fig. 8. The dividing streamline ($\psi =0$, black curve) delineates the volume of the thermal from the ambient. The dividing streamline is superimposed on filled contours of the azimuthally averaged temperature, showing that the using the streamfunction allows for nearly autonomous boundary detection, with minimal manual intervention. The pink curve shows a spherical volume drawn with the thermal centroid as its center and radius $R$ (defined in Sec. 2b). The green curve is a spheroidal volume with the distance from thermal top to the centroid as its major axis and $R$ as its minor axis. These three methods, variously used in the literature (e.g., [12, 13, 23, 26]) predict similar thermal volumes for the turbulent case while the differences are noticeable in the laminar case.