Turbulent flow over craters on Mars: Vorticity dynamics reveal aeolian excavation mechanism

Impact craters are scattered across Mars. These craters exhibit geometric self-similarity over a spectrum of diameters, ranging from tens to thousands of kilometers. The late Noachian–early Hesperian boundary marks a dramatic shift in the role of mid-latitude craters, from depocenter sedimentary basins to aeolian source areas. At present day, many craters contain prominent layered sedimentary mounds with maximum elevations comparable to the rim height. The mounds are remnants of Noachian deposition and are surrounded by a radial moat. Large-eddy simulation has been used to model turbulent flows over synthetic craterlike geometries. Geometric attributes of the craters and the aloft flow have been carefully matched to resemble ambient conditions in the atmospheric boundary layer of Mars. Vorticity dynamics analysis within the crater basin reveals the presence of counterrotating helical vortices, verifying the efficacy of deflationary models put forth recently by Bennett and Bell [K. Bennett and J. Bell, Icarus 264, 331 (2016)] and Day et al. [M. Day et al., Geophys. Res. Lett. 43, 2473 (2016)]. We show how these helical counterrotating vortices spiral around the outer rim, gradually deflating the moat and carving the mound; excavation occurs faster on the upwind side, explaining the radial eccentricity of the mounds relative to the surrounding crater basin.

Figure 1

Digital elevation models of two mid-latitude caters on Mars with interior sedimentary mounds. Colors have been stretched to highlight the crater interior relief of 3.1 and 5.6 km for (a) Henry crater and (b) Gale crater, respectively.

Figure 2

Perspective images of synthetic craters considered in this study (see also Table 1): (a) case 1, (b) case 2, (c) case 3, (d) case 4, and (e) case 5.

Figure 3

Visualization of instantaneous flow above (a) and (b) case 2 and (c) and (d) case 4 craters. Color flood contour shows fluctuating streamwise velocity ${\tilde{u}}^{\prime }=\tilde{u}-{\langle \tilde{u}\rangle }_t$, while line contours denote positive (black) and negative (gray) streamwise vorticity ${\tilde{\omega }}_x$. Panels are retrieved from (a) and (c) $y/H=6$ and (b) and (d) $x/H=6$ (see Fig. 2 for reference).

Figure 4

Visualization of Reynolds-averaged flow above (a) and (b) case 2 and (c) and (d) case 4 craters. Contour shows streamwise velocity ${\langle \tilde{u}\rangle }_t$, while contours denote positive (orange) and negative (blue) streamwise vorticity ${\langle {\tilde{\omega }}_x\rangle }_t$. Panels are retrieved from (a) and (c) $y/H=6$ and (b) and (d) $x/H=6$.

Figure 5

Visualization of Reynolds-averaged differential helicity (7) for (a)–(h) case 1 and (i)–(p) case 4. Left and right panels are flow visualizations in the spanwise–wall-normal plane and the streamwise–wall-normal plane, respectively, for (a) and (i) $x/H\approx 5$, (c) and (k) $x/H=6$, (e) and (m) $x/H=6.94$, (g) and (o) $x/H=8.25$, (b) and (j) $y/H=3.75$, (d) and (l) $y/H=4.5$, (f) and (m) $y/H=4.875$, and (h) and (p) $y/H=6$. The color flood contour is $h_l$, while line contours illustrate Reynolds-averaged streamwise vorticity ${\langle {\tilde{\omega }}_x\rangle }_t\left(\mathit{x}\right)$, where black and gray denote ${\langle {\tilde{\omega }}_x\rangle }_t\left(\mathit{x}\right)>0$ and ${\langle {\tilde{\omega }}_x\rangle }_t\left(\mathit{x}\right)<0$, respectively.

Figure 6

Visualization of constituent terms from the Reynolds-averaged streamwise vorticity transport equation for flow over case 1 at streamwise positions $x/H\approx 5$ (left panels) and $x/H=6$ (right panels) (see Fig. 2 for reference), showing (a) and (b) stretching of ${\langle {\tilde{\omega }}_1\rangle }_t$ by ${\partial }_1{\langle {\tilde{u}}_1\rangle }_t$, (c) and (d) tilting of ${\langle {\tilde{\omega }}_2\rangle }_t$ by ${\partial }_2{\langle {\tilde{u}}_1\rangle }_t$, (e) and (f) tilting of ${\langle {\tilde{\omega }}_3\rangle }_t$ by ${\partial }_3{\langle {\tilde{u}}_1\rangle }_t$, (g) and (h) production of ${\langle {\tilde{\omega }}_1\rangle }_t$ by turbulent torque, (i) and (j) the sum of ${\langle {\tilde{\omega }}_1\rangle }_t$ gain and loss via mean-flow gradients, and (k) and (l) the sum of terms contributing to the gain and loss of ${\langle {\tilde{\omega }}_1\rangle }_t$.

Figure 7

Visualization of constituent terms from the Reynolds-averaged streamwise vorticity transport equation for flow over case 4 at streamwise positions $x/H\approx 5$ (left panels) and $x/H=6$ (right panels) (see Fig. 1 for reference), showing (a) and (b) stretching of ${\langle {\tilde{\omega }}_1\rangle }_t$ by ${\partial }_1{\langle {\tilde{u}}_1\rangle }_t$, (c) and (d) tilting of ${\langle {\tilde{\omega }}_2\rangle }_t$ by ${\partial }_2{\langle {\tilde{u}}_1\rangle }_t$, (e) and (f) show tilting of ${\langle {\tilde{\omega }}_3\rangle }_t$ by ${\partial }_3{\langle {\tilde{u}}_1\rangle }_t$, (g) and (h) production of ${\langle {\tilde{\omega }}_1\rangle }_t$ by turbulent torque, (i) and (j) the sum of ${\langle {\tilde{\omega }}_1\rangle }_t$ gain and loss via mean-flow gradients, and (k) and (l) the sum of terms contributing to the gain and loss of ${\langle {\tilde{\omega }}_1\rangle }_t$.

Figure 8

Structural model of intracrater and extracrater secondary flows aloft an idealized synthetic crater, highlighting spatial locations wherein different mechanisms contribute towards gains and losses in mean streamwise vorticity. Preceding arguments lead us to conclude that tilting of vertical and spanwise vorticity is overwhelmingly responsible for streamwise vorticity upwind of the crater (denoted by black lines and annotations upwind of the rim), stretching of ambient streamwise vorticity is largest above the crater (red-blue contour), and turbulent torque is strongest within the crater (red-blue contour).

Figure 9

Probability density functions of streamwise and vertical velocity fluctuations ${\tilde{u}}^{\prime }({\mathit{x}}_l,t)$ (left panels) and ${\tilde{w}}^{\prime }({\mathit{x}}_l,t)$ (right panels), from local positions ${\mathit{x}}_l$ throughout the domain (see also Fig. 2 for reference): (a) and (b) ${\mathit{x}}_l={\mathit{x}}_2$, (c) and (d) ${\mathit{x}}_l={\mathit{x}}_3$, (e) and (f) ${\mathit{x}}_l={\mathit{x}}_4$, and (g) and (h) ${\mathit{x}}_l={\mathit{x}}_5$. Panel (a) shows conditional sampling thresholds used in present study. Colors correspond to case 1 (black), case 2 (dark gray), case 3 (gray), case 4 (light gray), and case 5 (lightest gray). The vertical blue line in (a) illustrates the threshold used to conditionally sampled the flow.

Figure 10

Conditionally sampled vorticity from flow over case 2 (see also Fig. 2), with the conditional sampling threshold defined based on streamwise velocity fluctuations at location ${\mathit{x}}_2$, where a probability density function of this quantity has been provided in Fig. 9. (a) Spanwise vorticity ${\widehat{\tilde{\omega }}}_y$ and (b) streamwise vorticity ${\widehat{\tilde{\omega }}}_x$ are shown, with $U_0$ added to denote the main atmospheric flow direction (note also that the horizontal ranges of the panels have been varied to show as much information as possible on flow structures associated with the crater). Shown in (b) are ${\widehat{\tilde{\omega }}}_x$ contours for equal-and-opposite values of ${\widehat{\tilde{\omega }}}_x$ (red and blue are positive and negative, respectively), in order to reveal the counterrotating extracrater vortices flanking the rim.

Figure 11

Conditionally sampled spanwise vorticity from flow over case 4 (see also Fig. 2), with the conditional sampling threshold defined based on streamwise velocity fluctuations at location ${\mathit{x}}_2$, where a probability density function of this quantity has been provided in Fig. 9. (a) Spanwise vorticity ${\widehat{\tilde{\omega }}}_y$ and (b) streamwise vorticity ${\widehat{\tilde{\omega }}}_x$ are shown from (b) and (c) two visualization perspectives, with $U_0$ added to denote the main atmospheric flow direction (note also that the horizontal ranges of the panels have been varied to show as much information as possible on flow structures associated with the crater). Shown in (b) and (c) are ${\widehat{\tilde{\omega }}}_x$ contours for equal-and-opposite values of ${\widehat{\tilde{\omega }}}_x$ (red and blue are positive and negative, respectively), in order to reveal the counterrotating extracrater vortices flanking the rim.

Figure 12

Conditionally sampled spanwise vorticity from flow over case 5 (see also Fig. 2), with the conditional sampling threshold defined based on streamwise velocity fluctuations at location ${\mathit{x}}_2$, where a probability density function of this quantity has been provided in Fig. 9. (a) Spanwise vorticity ${\widehat{\tilde{\omega }}}_y$ and (b) streamwise vorticity ${\widehat{\tilde{\omega }}}_x$ are shown from (b) and (c) two visualization perspectives, with $U_0$ added to denote the main atmospheric flow direction (note also that the horizontal ranges of the panels have been varied to show as much information as possible on flow structures associated with the crater). Shown in (b) and (c) are ${\widehat{\tilde{\omega }}}_x$ contours for equal-and-opposite values of ${\widehat{\tilde{\omega }}}_x$ (red and blue are positive and negative, respectively), in order to reveal the counterrotating extracrater vortices flanking the rim.

Figure 13

(a), (b), (e), and (f) Interpolated isoelevation velocity and (c), (d), (g), and (h) corresponding surface stress for (a), (c), (e), and (g) Reynolds-averaged and (b), (d), (f), and (h) conditionally sampled flow over (a)–(d) case 1 and (e)–(h) case 2 craters, showing (b) and (f) fluctuation about the Reynolds average $\widehat{\tilde{u}}-{\langle \tilde{u}\rangle }_t$, (c) and (g) surface stress computed with Eq. (13), and (d) and (h) surface stress computed with Eq. (15). Note that Panels (c) and (g) are $log\left[\right|{\langle {\tau }_{yz}^{\zeta ,w}\rangle }_t(x,y)/\rho u_{\tau }^2(x,y)\left|\right]$ and (d) and (h) are $log\left[\right|\widehat{\tau }{}_{yz}^{\zeta ,w\prime }(x,y)/\rho u_{\tau }^2(x,y)\left|\right]$, where the logarithm helps to reveal spatial variability of surface stress. Solid red lines denote $h(x,y)$. The sampling point ${\mathit{x}}_2$, illustrated in Fig. 2, is marked with a blue circle; the PDF of $\tilde{u}({\mathit{x}}_2,t)$ and the sampling threshold are shown in Fig. 9.

Figure 14

Illustration of the deflationary trajectory of once-filled craters, showing (a), (c), and (e) perspective visualization, (b), (d), and (f) planform visualization, and (a) and (b) inferred deflation from case 1, representing (c) and (d) an inferred intermediate case and (e) and (f) configurations most similar to cases 3 and 4. Circle denotes the conditional sampling location ${\mathit{x}}_2$ (see also Fig. 2) and arrows show vortex lines, colored by the local streamwise vorticity (where blue and red correspond to negative and positive ${\langle {\tilde{\omega }}_x\rangle }_t$, respectively).

Figure 15

Time-series processing of fluctuating (a)–(d) streamwise and (e)–(h) vertical velocity for case 1 (left panels) and case 3 (right panels), recorded at location ${\mathit{x}}_3$ [see Figs. 2 and 2 for reference], showing (c), (d), (g), and (h) input time series and (a), (b), (e), and (f) the corresponding wavelet power spectrum contours $E_{{\tilde{u}}^{\prime }{\tilde{u}}^{\prime }}({\mathit{x}}_3,t)f{U}_0^{-3}$ and $E_{{\tilde{w}}^{\prime }{\tilde{w}}^{\prime }}({\mathit{x}}_3,t)f{U}_0^{-3}$, with outer-normalized frequency to facilitate introduction of Strouhal number $\text{St}$ (i.e., $fH{U}_0^{-1}=\text{St}$). Note that time-series colors are consistent with the profile colors used in Figs. 9 and 16 (i.e., black and gray for cases 1 and 3, respectively).

Figure 16

Global wavelet power spectrum of (a) streamwise ${\langle E_{{\tilde{u}}^{\prime }{\tilde{u}}^{\prime }}({\mathit{x}}_2,t)\rangle }_{t}f{U}_0^{-3}$ and (b) vertical ${\langle E_{{\tilde{w}}^{\prime }{\tilde{w}}^{\prime }}({\mathit{x}}_2,t)\rangle }_{t}f{U}_0^{-3}$ velocity. The blue band denotes the shear-normalized frequency $fH{U}_0^{-1}=StH/L$, where $L\approx h$. The orange line denotes $fH{U}_0^{-1}=StH/L$, where $L\approx D/2$. Line colors correspond to case 1, black; case 2, dark gray; case 3, gray; case 4, light gray; and case 5, lightest gray.

Figure 17

Profiles of require time ${\delta }_T$ (years) to excavate the sedimentary fill of a once-filled crater under ambient planetary boundary layer forcing $u_{\tau }\left(\mathrm{m}{\mathrm{s}}^{-1}\right)$. Profiles correspond to varying atmospheric density: black, $\rho =1\times {10}^{-3}\mathrm{kg}{\mathrm{m}}^{-3}$; dark gray, $\rho =3\times {10}^{-3}\mathrm{kg}{\mathrm{m}}^{-3}$; light gray, $\rho =1\times {10}^{-2}\mathrm{kg}{\mathrm{m}}^{-3}$; and lightest gray, $\rho =3\times {10}^{-2}\mathrm{kg}{\mathrm{m}}^{-3}$. The horizontal orange line at ${\delta }_T=3.5\times {10}^9$ yr denotes the approximate Noachian-Hesperian boundary.