Large-scale intermittency and rare events boosted at dimensional crossover in anisotropic turbulence

Understanding rare events in turbulence provides a basis for the science of extreme weather, for which the atmosphere is modeled by Navier-Stokes equations (NSEs). In solutions of NSEs for isotropic fluids, various quantities, such as fluid velocities, roughly follow Gaussian distributions, where extreme events are prominent only in small-scale quantities associated with the dissipation-dominating length scale or anomalous scaling regime. Using numerical simulations, this study reveals another universal promotion mechanism at much larger scales if three-dimensional fluids accompany strong two-dimensional anisotropies, as is the case in the atmosphere. The dimensional crossover between two and three dimensions generates prominent fat-tailed non-Gaussian distributions with intermittency accompanied by colossal chainlike structures with densely populated self-organized vortices (serpentinely organized vortices). The promotion is caused by a sudden increase of the available phase space at the crossover length scale. Since the discovered intermittency can involve much larger energies than those in the conventional intermittency in small spatial scales, it governs extreme events and chaotic unpredictability in the synoptic weather system.

Figure 1

Illustrative sketch of large chainlike structure named the serpentinely organized vortices (SOVs). A SOV structure contains a mass of coherently assembled elementary vortices. Sizes of these element vortices are comparable or smaller than the CLS. The SOV structure looks like many densely distributed peas (vortices) contained in a pod (chain structure).

Figure 2

Energy spectra of horizontal velocity, $E_h\left(k_h\right)=0.5\left[\right|{\check{u}}_x\left(k_h\right){|}^2+|{\check{u}}_y\left(k_h\right){|}^2]$, normalized by $E_h$ at $k_h=k_{\mathrm{in}}$. Here, ${\check{u}}_x\left(k_h\right)={\left[{\sum }_{k_z}{\widehat{u}}_x^2\left(k_h,k_z\right)\right]}^{1/2}$ for (a) N4096-$\lambda 64$ and (b) N4096-$\lambda 128$. The vertical downward (horizontal) blue arrows indicate the center (spread) of energy injection. The red upward indicate half of the CLS wave number, i.e., $0.5k_{\mathrm{cr}}$.

Figure 3

PDF of horizontal velocity $u_y$ (red curve with red squares) and $\mathrm{\Delta }{u}_y\left(L_z/2,0\right)$ (blue curve with blue circles) for N4096-$\lambda 64$, with the Gaussian distribution shown for reference (gray without symbols). Here and in the following figures, the abscissae of the PDF and the color contours are normalized by each standard deviation $\left(\sigma \right)$. $\mathrm{\Delta }{u}_y\left(L_z/2,0\right)$ shows a wider tail, meaning stronger intermittency.

Figure 4

Typical in-plane contour snapshots of horizontal velocities and bird's-eye views for (a) $|{\mathit{u}}_h|$ at $z=0$ and (b) $|\mathrm{\Delta }{\mathit{u}}_h|$ for N4096-λ64. The color contours are normalized by the standard deviations of $u_y,\sigma \left(u_y\right)$ [or equivalently $\sigma \left(u_x\right)]$ in (a) and $\sigma \left(\mathrm{\Delta }{u}_y\right)$ [or $\sigma \left(\mathrm{\Delta }{u}_x\right)]$ in (b). The mean and standard deviaitons are 0.899 and 0.737, respectively, in (a) and 0.0374 and 0.0327, respectively, in (b). Note that $|{\mathit{u}}_h|=\sqrt{u_x^2+u_y^2\phantom{{}^2}}$ and $|\mathrm{\Delta }{\mathit{u}}_h|=\sqrt{\mathrm{\Delta }{u}_x^2+\mathrm{\Delta }{u}_y^2\phantom{{}^2}}$. Large values appear in (b), corresponding to the wide tail in the PDF of $\mathrm{\Delta }{u}_y\left(L_z/2,0\right).$

Figure 5

Flatness factors of the spherical-shell (band-pass)-filtered horizontal velocity ${\tilde{u}}_h^K$ with filter scale $K$ (see Appendix pp3a) for N4096-λ64, N4096-λ128, and N1024-λ32. Arrows indicate $k_{\mathrm{cr}}$ for each case (red circles, blue squares, and green triangles, respectively). The prominent intermittency is always observed at the CLS.

Figure 6

$xy$-plane cross section of (a) $|\mathrm{\Delta }{\mathit{u}}_{\mathit{h}}\left(L_z/2,0\right)|$ (see also Movie S1 [26]), (b) enstrophy $\zeta$, and (c) finite-time Lyapunov exponents $\mathrm{\Lambda }$ at $z=0$ for N4096-λ64. The region shown in (c) corresponds to the close-up of the dotted square areas in (a) and (b). The color contours are normalized by standard deviations. The mean and standard deviaitons are 2.28 and 6.55, respectively, in (b) and 0.603 and 0.378, respectively, in (c). The distributions of extreme values indicated by red or dark yellow areas form a SOV (chain structure) and the shapes are very similar among the three panels.

Figure 7

$xy$-plane cross section of low-pass-filtered horizontally oriented vorticity ${\omega }_y$ (see Appendix pp3b) at $z=0$ with threshold wave numbers of (a) 10, (b) 20, (c) 40, (d) 80, and (e) 4096 for N4096-λ64. The color contours are normalized by the standard deviations. The characteristic large-scale SOV feature is observed only when the crossover wave number is included, i.e., only in (d), (e).

Figure 8

Distributions of the horizontally oriented vorticity (color contour) at three vertical walls whose top view coordinates are indicated by black dashed lines in Fig. 6 together with the color contour at the bottom plane at $z=0$ for N4096-λ64. The enstrophy is also plotted by the white 3D isosurface. The threshold of the white isosurface was set at $m+3\sigma$, where $m$ and σ are the mean and standard deviation, respectively. The vertical real-space scale is doubled. For vorticity, calm regions are colored green in the planes, and active regions are colored red or blue (as in the color scale bar). Large structures (SOVs) consist of a mass of assembled tiny active spots and specks with sizes comparable to or smaller than the CLS. The bottom plane cross section corresponds to the cross section illustrated in Fig. 6 and the active SOV structure has one-to-one correspondence. See also Movie S2 [26] for a temporal evolution of the structures.

Figure 9

Flatness of the time series of $u_y$ estimated after frequency-resolved band-pass filtering at frequency $\omega$ (see Appendix pp3d) for N4096-λ64. The abscissa shows $k_h^{*}(\equiv \omega /u_{\mathrm{rms}}$, where $u_{\mathrm{rms}}$ is the root mean square of velocity in the whole simulation). The red arrow indicates the CLS wave number, $k_{\mathrm{cr}}$.

Figure 10

Results from global atmospheric simulation. (a) Energy spectra of horizontal wind at 200 hPa. (b) Flatness of horizontal velocities at 200 hPa estimated by frequency-resolved band-pass filter. The graphs were obtained by averaging the data at 1024 measurement points, which were dispersed uniformly on the whole globe. Here, $k_h^{*}[=2\pi {\left(Ut\right)}^{-1}$, with $U$ chosen to be the root mean square of the horizontal velocity at 200 hPa for the entire global domain] is the effective wave number. The red arrows indicate the CLS wave number.

Figure 11

(a) Absolute value of the difference between the horizontal winds at 700 and 850 hPa $|\mathrm{\Delta }{\mathbf{u}}_h|=|{\mathbf{u}}_h\left(700\mathrm{hPa}\right)-{\mathbf{u}}_h\left(850\mathrm{hPa}\right)|$ (see also Movie S3 [26]). (b) Finite-time Lyapunov exponent $\mathrm{\Lambda }$ at 700 hPa. The color contours are normalized by $\sigma \left(\mathrm{\Delta }{u}_h\right)$ in (a) and $\sigma \left(\mathrm{\Lambda }\right)$ in (b). The mean and standard deviations are 4.27 and 3.07, respectively, in (a) and $2.42\times {10}^{-5}{\mathrm{s}}^{-1}$ and $2.15\times {10}^{-5}{\mathrm{s}}^{-1}$, respectively, in (b). Active regions form similar structures in (a), (b).

Figure 12

Schematic illustration of the formation mechanism of the colossal structure. Vector ${\mathit{k}}_3$, whose horizontal wave number is small, is generated via mode coupling that satisfies momentum conservation.

Figure 13

Schematic illustration of the emergence of the intermittency by phase space expansion at the 2D-3D crossover. In the 2D cascade, the self-similarity is approximately satisfied and the broken-up smaller vortices fill the space again. However, at the 2D-3D crossover, the active area is unable to fill the whole phase space anymore because of the expansion of the phase space in the $z$ direction. This generates active and calm regions with intermittency.

Figure 14

Properties of isotropic homogeneous 3D turbulence. (a) PDFs of horizontal velocity $u_y$ (red curve with red squares) and $\mathrm{\Delta }{u}_y\left(L_z/2,0\right)$ (blue curve with blue circles). (b) Flatness factors of the spherical-shell (band-pass)-filtered horizontal velocity ${\tilde{u}}_h^K$ with filter scale $K$. Unlike Fig. 5, which shows the flatness factors in F3DF, strong intermittency is limited only in the dissipation scales. It should be noted that the Taylor microscale corresponds to $K=27$ in (b).

Figure 15

Flatness of Gaussian-filtered velocity ${\tilde{\mathit{u}}}^g\left(k_v\right)$ for F3DF with vertical wave number $k_v=0$ and $k_v=k_z^{\min }$ for (a) N4096-λ64 and (b) N4096-λ128. The intermittency is pronounced only for $k_v=k_z^{\min }$ around $1/\alpha =0.5k_{\mathrm{cr}}$, indicating that it originated from the dimensional crossover.

Figure 16

PDFs of mode-selected velocities. Horizontal (red curve with red squares)/vertical (blue curve with blue circles) mode-extracted velocities $u_{y|z}\left(\mathit{k};\mathit{x},t\right)={\sum }_{k^{\prime }=k}{\widehat{u}}_{y|z}\left({\mathit{k}}^{\prime },t\right)exp\left(i{\mathit{k}}^{\prime }·\mathit{x}\right)$ consist of several modes (defined below). These PDFs are compared with the Gaussian distribution (gray curves without symbols). (a) $\mathit{k}=\left(\pm k_{\mathrm{cr}},0,0\right)$, (b) $\mathit{k}=\left(0,0,\pm k_{\mathrm{cr}}\right)$, and (c) $\mathit{k}=\left(k_x,k_y,\pm k_{\mathrm{cr}}\right)$ with $k_{\mathrm{cr}}-1/2\le \left(k_x^2+k_y^2+k_z^2\right)<k_{\mathrm{cr}}+1/2$. $u_z\left(\mathit{k};\mathit{x},t\right)=0$ is satisfied due to the divergence-free condition in (b). Only (c) shows clear deviations from the Gaussian distribution, i.e., clear intermittency.