Effective drift velocity from turbulent transport by vorticity

We highlight the differing roles of vorticity and strain in the transport of coarse-grained scalars at length scales larger than $\ell$ by smaller scale (subscale) turbulence. We use the first term in a multiscale gradient expansion due to Eyink [J. Fluid Mech. 549, 159 (2006)], which exhibits excellent correlation with the exact subscale physics when the partitioning length $\ell$ is any scale smaller than that of the spectral peak. We show that unlike subscale strain, which acts as an anisotropic diffusion/antidiffusion tensor, subscale vorticity's contribution is solely a conservative advection of coarse-grained quantities by an eddy-induced nondivergent velocity, ${\mathbf{v}}_{*}$, that is proportional to the curl of vorticity. Therefore, material (Lagrangian) advection of coarse-grained quantities is accomplished not by the coarse-grained flow velocity, ${\overline{\mathbf{u}}}_{\ell }$, but by the effective velocity, ${\overline{\mathbf{u}}}_{\ell }+{\mathbf{v}}_{*}$, the physics of which may improve commonly used LES models.

Figure 1

Schematic of how eddy-induced advection by a nondivergent velocity, ${\mathbf{v}}_{*}\sim \mathbf{\nabla }\times {\overline{\mathit{\omega }}}_{\ell }$, (blue arrow) from Eq. (43) can arise due to subscale vorticity (red). In this example, two symmetric but counterrotating eddies (red) are of a size smaller than that of the coarse-graining box (dashed). At length scales larger than that of the box, $\ell$, which may represent a grid cell in a simulation or the resolution limit of an observation, these eddies exert an eddy-induced advection with velocity ${\mathbf{v}}_{*}$ proportional to the curl of vorticity, depicted by the blue arrow at location $\mathbf{x}$.

Figure 2

Left to right: visualization of $x, y$, and $z$ components, respectively, of momentum, $\rho \mathbf{u}$, in physical space from the ${1024}^3$ DNS. These images are shown at an instant of time after the flow has reached steady state. Shocks can be seen as discontinuities. The spectrum of this flow is shown in Fig. 15 in the Appendix.

Figure 3

A 2D slice at $x=0$ from a snapshot of the 3D compressible turbulence DNS, comparing the (left) exact ${\overline{\tau }}_{\ell }(u_x,\rho )$ at $\ell =0.19635$ with (right) its approximation from Eq. (24c), ${\tau }_m=\frac{1}{3}{M}_2{\ell }^2{\partial }_k\overline{\rho }{\partial }_k\overline{u_x}$.

Figure 4

Comparison of ${\overline{\tau }}_{\ell }(\rho ,\mathbf{u})$ with its approximation ${\tau }_m=\frac{1}{3}{M}_2{\ell }^2{\partial }_k\overline{\rho }{\partial }_k\overline{\mathbf{u}}$ at $\ell =0.19635$. (a) $x$ component of $\tau$ and ${\tau }_m$ along a diagonal transect through the domain from a single snapshot. (b) Joint PDF between ${\tau }^{*}$ and ${\tau }_m^{*}$, where the asterisk superscript indicates z-scores, which emphasizes trends [see Eq. (32) and associated discussion in the text]. Color bar indicates the number of grid points on a logarithmic scale, proportional to the probability. Black line is the best linear fit, and the correlation squared, $r^2$, quantifies the fraction of variability in $\tau$ that is captured by ${\tau }_m$. (c), (d) Same as (a) and (b) but for the $y$ component. (e), (f) Same as (a) and (b) but for the $z$ component.

Figure 5

Correlation coefficient $r$ between ${\overline{\tau }}_{\ell }(\rho ,\mathbf{u})$ and its approximation ${\tau }_m=\frac{1}{3}{M}_2{\ell }^2{\partial }_k\overline{\rho }{\partial }_k\overline{\mathbf{u}}$ as a function of scale $\ell$. (a) $x$ component; (b) $y$ component; (c) $z$ component.

Figure 6

Initial condition for two-layer stacked shallow water simulation following [45]. The top layer has zonal (east-west) velocity with a Gaussian profile peaking at $\approx 0.35\mathrm{m}$/s (positive to the east) while the bottom layer is initially quiescent. Bottom topography is flat. The surface has initial height ${\eta }_{1/2}=0\mathrm{m}$. The interface height ${\eta }_{3/2}$ between the two layers is initialized to the reference ${\eta }_{3/2,\mathrm{ref}}$ shown to satisfy geostrophic balance. ${\eta }_{3/2}$ is allowed to evolve freely while being relaxed back to ${\eta }_{3/2,\mathrm{ref}}$ at the rate $\gamma$.

Figure 7

Visualization of the relative vorticity normalized by the Coriolis paramater $f$ at the domain's midlatitude from our simulation of an eastward flowing ocean current that is baroclinically unstable, resulting in eddy generation and turbulence. The simulation solves the two-layer shallow water equations, with the top and bottom panels showing the respective top and bottom layers. Time increases from left to right: panels (a1) and (a2) are from day 60, (b1) and (b2) from day 93, and (c1) and (c3) from day 180. Our analysis uses model data after day 169, when the flow has become statistically steady.

Figure 8

Comparing the (left) exact ${\overline{\tau }}_{\ell }(\mathbf{u},h)$ at $\ell =100\mathrm{km}$ with (right) its approximation from Eq. (24c), ${\tau }_m=\frac{1}{2}{M}_2{\ell }^2{\partial }_k\overline{h}{\partial }_k\overline{\mathbf{u}}$, from a single snapshot. Top panels are for the zonal (eastward) component. Bottom panels are for the meridional (northward) component. Gray bands at northern and southern boundaries are excluded from our analysis to avoid details of the coarse graining next to boundaries.

Figure 9

Comparison of ${\overline{\tau }}_{\ell }(h,\mathbf{u})$ with its approximation ${\tau }_m=\frac{1}{2}{M}_2{\ell }^2{\partial }_k\overline{h}{\partial }_k\overline{\mathbf{u}}$ at $\ell =100\mathrm{km}$. (a) Zonal (eastward) component of $\tau$ and ${\tau }_m$ along the transect $y=802.5\mathrm{km}$ from a snapshot on day 171 of the simulation. (b) Joint PDF for all space-time points between ${\tau }^{*}$ and ${\tau }_m^{*}$, where the asterisk superscript indicates Z scores, which emphasizes trends [see Eq. (32) and associated discussion in the text]. Color bar indicates the number of grid points on a logarithmic scale, proportional to the probability. Red line is the best linear fit, and the correlation squared, $r^2$, quantifies the fraction of variability in $\tau$ that is captured by ${\tau }_m$.

Figure 10

Correlation coefficient $r$ between ${\overline{\tau }}_{\ell }(h,\mathbf{u})$ and its approximation ${\tau }_m=\frac{1}{2}{M}_2{\ell }^2{\partial }_k\overline{h}{\partial }_k\overline{\mathbf{u}}$ as a function of scale $\ell$. (a) Zonal (eastward) direction; (b) meridional (northward) direction.

Figure 11

Same as in Fig. 4 for $\ell =0.3927$.

Figure 12

Same as Fig. 4 for $\ell =0.7854$.

Figure 13

Same as Fig. 4 for $\ell =1.5708$.

Figure 14

Similar to the joint PDFs of the z-scores in Fig. 4, here we show the joint PDFs of the actual ${\overline{\tau }}_{\ell }(\rho ,\mathbf{u})$ and its approximation ${\tau }_m=\frac{1}{3}{M}_2{\ell }^2{\partial }_k\overline{\rho }{\partial }_k\overline{\mathbf{u}}$. Top row (a)–(c) shows the joint PDFs between $\tau$ and ${\tau }_m$ at at $\ell =0.19635$. Second row (d)–(f) is for $\ell =0.3927$. Third row (g)–(i) is for $\ell =0.7854$. Bottom row (j)–(l) is for $\ell =1.5708$. Left, middle, and right columns show the $x, y$, and $z$ components, respectively.

Figure 15

Spectra of (left) velocity, $E^u\left(k\right)$, and (right) density, $E^{\rho }\left(k\right)$. Both decay with wave number $k$ sufficiently faster than the $k^{-1}$ required for the multiscale gradient expansion to converge. Here $E^u\left(k\right)={\sum }_{\left|\right|\mathbf{k}|-k|<0.5}\frac{1}{2}{\left|\widehat{\mathbf{u}}\left(\mathbf{k}\right)\right|}^2$ and $E^{\rho }\left(k\right)={\sum }_{\left|\right|\mathbf{k}|-k|<0.5}{\left|\widehat{\rho }\left(\mathbf{k}\right)\right|}^2$, where $\widehat{\mathbf{u}}\left(\mathbf{k}\right)$ and $\widehat{\rho }\left(\mathbf{k}\right)$ are Fourier coefficients.

Figure 16

Same as in Fig. 3, comparing the (left column) exact ${\overline{\tau }}_{\ell }(\rho ,\mathbf{u})$ with (right column) its approximation for the (top row) $y$ components and (bottom row) $z$ components.

Figure 17

Joint PDFs of ${\mathbf{v}}_{*}$ and $\overline{\mathbf{u}}$. Color bar indicates the number of grid points on a logarithmic scale, proportional to the probability. Slope of best linear fit and correlation squared, $r^2$, are shown in the top left of each panel. The best linear fits are omitted in the the plots due to small slopes. The rms values are shown in the bottom right of each panel. Top row (a)–(c) shows the joint PDFs between ${\mathbf{v}}_{*}$ and $\overline{\mathbf{u}}$ at at $\ell =0.19635$. Second row (d)–(f) is for $\ell =0.3927$. Third row (g)–(i) is for $\ell =0.7854$. Bottom row (j)–(l) is for $\ell =1.5708$. Left, middle, and right columns show the $x, y$, and $z$ components, respectively. This figure highlights that the magnitudes of ${\mathbf{v}}_{*}$ and the coarse velocity ${\overline{\mathbf{u}}}_{\ell }$ are often comparable.

Figure 18

Same as in Fig. 9 for $\ell =50\text{km}$.

Figure 19

Same as in Fig. 9 for $\ell =200\text{km}$.

Figure 20

Same as in Fig. 9 for $\ell =400\text{km}$.

Figure 21

Spectra of (left) $\mathbf{u}$ and (right) $h$ computed from the two-layer stacked shallow water simulation. These are filtering spectra, calculated by filtering at successive length scales as described in [58, 59].

Figure 22

Same as in Fig. 8 for $\ell =50\mathrm{km}$.

Figure 23

Same as in Fig. 8 for $\ell =200\text{km}$.

Figure 24

Same as in Fig. 8 for $\ell =400\text{km}$.

Figure 25

Similar to the joint PDFs of the z-scores in Fig. 9, here we show the joint PDFs of the actual ${\overline{\tau }}_{\ell }(h,\mathbf{u})$ and its approximation ${\tau }_m=\frac{1}{2}{M}_2{\ell }^2{\partial }_k\overline{h}{\partial }_k\overline{\mathbf{u}}$, without normalization. Top and bottom panels show zonal (eastward) and meridional (northward) components, respectively. (a1) and (a2) are for $\ell =50\mathrm{km}$, (b1) and (b2) for $100\mathrm{km}$, (c1) and (c2) for $200\mathrm{km}$, (d1) and (d2) for $400\mathrm{km}$.

Figure 26

Joint probability density of ${\mathbf{v}}_{*}$ and $\overline{\mathbf{u}}$. The colorbars are in log scale. The top panels show zonal (eastward) component and the bottom panels show meridional (northward) component. (a1) and (a2) are for $\ell =50$ km, (b1) and (b2) are for $\ell =100$ km, (c1) and (c2) are for $\ell =200$ km, (d1) and (d2) are for $\ell =400$ km. The red lines are linear regression fits of which the slope and goodness of fit($r^2$) are in the text in each panel. The rms values are shown in the bottom right of each panel. This figure highlights that ${\mathbf{v}}_{*}$ is often comparable in magnitude relative to the coarse velocity ${\overline{\mathbf{u}}}_{\ell }$.