2 Pancake vorticity structures

We analyse the evolution of high-vorticity regions with numerical simulations of the Euler equations (in the vorticity formulation),

in the periodic box $\boldsymbol{r}=(x,y,z)\in [-\unicode[STIX]{x03C0},\unicode[STIX]{x03C0}]^{3}$ . The pseudospectral Runge–Kutta fourth-order method is used, together with the Fourier cutoff function suggested in Hou & Li (Reference Hou and Li2007) to avoid the bottle-neck instability. The inverse of the curl operator and all spatial derivatives are calculated in the Fourier space. We start from initial condition $I_{1}$ of Agafontsev et al. (Reference Agafontsev, Kuznetsov and Mailybaev2015), which represents a perturbation of the shear flow $\unicode[STIX]{x1D714}_{x}=\sin z$ , $\unicode[STIX]{x1D714}_{y}=\cos z$ , $\unicode[STIX]{x1D714}_{z}=0$ . Taking advantage of the anisotropy of the vorticity field, we use an adaptive anisotropic rectangular grid, which is uniform along each direction and adapted independently along each coordinate. For more details of the numerical scheme, see Agafontsev et al. (Reference Agafontsev, Kuznetsov and Mailybaev2015), where it was verified that the accuracy within the simulation time interval is very high and is not affected by the Fourier cutoff filter. The simulation previously stopped at time $t=6.89$ with the grid $486\times 1024\times 2048$ is continued here until $t=7.75$ with the eight times larger final grid $972\times 2048\times 4096$ , when the thinnest pancake structure is resolved with $10$ grid points at the level of vorticity half-maximum. The results of the two simulations converge perfectly at the same times. Both the energy $E=(1/2)\int \boldsymbol{v}^{2}\,\text{d}^{3}\boldsymbol{r}$ and the helicity $\unicode[STIX]{x1D6FA}=\int (\boldsymbol{v}\boldsymbol{\cdot }\unicode[STIX]{x1D74E})\,\text{d}^{3}\boldsymbol{r}$ are conserved, with a relative error smaller than $10^{-11}$ .

Figure 1.

(a) Global vorticity maximum as a function of time (logarithmic vertical scale). The red dashed line indicates the slope $\propto \text{e}^{\unicode[STIX]{x1D6FD}_{2}t}$ , with $\unicode[STIX]{x1D6FD}_{2}=0.5$ . The thin vertical line marks the final simulation time $t=6.89$ in Agafontsev et al. (Reference Agafontsev, Kuznetsov and Mailybaev2015). (b) Regions of the largest vorticity, $\unicode[STIX]{x1D714}\geqslant 0.85\unicode[STIX]{x1D714}_{max}(t)$ , at different times. The colour represents the vorticity in the midplane of the pancake, from blue for $0.85\unicode[STIX]{x1D714}_{max}(t)$ to yellow for $\unicode[STIX]{x1D714}_{max}(t)$ . The structures are shifted vertically for better visualization. (c) Characteristic spatial scales $\ell _{1}$ (black), $\ell _{2}$ (blue) and $\ell _{3}$ (red) of the pancake structure. The red dashed line indicates the slope $\propto \text{e}^{-\unicode[STIX]{x1D6FD}_{1}t}$ , with $\unicode[STIX]{x1D6FD}_{1}=0.74$ .

Figure 1(a) shows the evolution of the global vorticity maximum $\unicode[STIX]{x1D714}_{max}(t)=\max _{\boldsymbol{r}}|\unicode[STIX]{x1D74E}(\boldsymbol{r},t)|$ , demonstrating the vorticity increase from $\unicode[STIX]{x1D714}_{max}(0)=1.5$ up to $18.4$ at $t=7.75$ and the asymptotically exponential vorticity growth at late times. Figure 1(b) shows the three-dimensional regions containing the vorticity $\unicode[STIX]{x1D714}=|\unicode[STIX]{x1D74E}|\geqslant 0.85\unicode[STIX]{x1D714}_{max}(t)$ at different times, and one can clearly see the formation of a thin pancake structure. It is convenient to introduce the pancake midplane as a surface, where the vorticity attains a maximum within the pancake thickness. The colour in figure 1(b) describes the midplane vorticity, from blue for $0.85\unicode[STIX]{x1D714}_{max}(t)$ to yellow for $\unicode[STIX]{x1D714}_{max}(t)$ . At $t=3$ and $4$ , the pancake spans the whole periodic domain in the $x$ -direction; for larger times, its lateral sizes decrease, but eventually stabilize and remain almost constant at $t\geqslant 6$ . On the contrary, the thickness keeps decreasing rapidly. Thus, at late times, vorticity variations become large (small) in transversal (tangential) directions to the pancake.

The local geometry of the thin structure can be studied using the Hessian matrix $\unicode[STIX]{x2202}_{i}\unicode[STIX]{x2202}_{j}\unicode[STIX]{x1D714}^{2}$ of second derivatives of $\unicode[STIX]{x1D714}^{2}$ with respect to $(x,y,z)$ , computed at the point of maximum vorticity; the location of the latter in between the grid nodes is approximated with the second-order finite-difference scheme. The (unit) normal vector $\boldsymbol{n}_{1}$ to the pancake is defined as the eigenvector corresponding to the largest of the three eigenvalues $|\unicode[STIX]{x1D706}_{1}|\geqslant |\unicode[STIX]{x1D706}_{2}|\geqslant |\unicode[STIX]{x1D706}_{3}|$ of the Hessian. The pancake thickness $\ell _{1}$ , as well as its lateral scales $\ell _{2}$ and $\ell _{3}$ , can be estimated with the local second-order approximation as $\ell _{i}=\sqrt{2\unicode[STIX]{x1D714}_{max}^{2}/|\unicode[STIX]{x1D706}_{i}|}$ ; see Agafontsev et al. (Reference Agafontsev, Kuznetsov and Mailybaev2015). Figure 1(c) shows these characteristic scales as functions of time. At late times, the pancake thickness $\ell _{1}$ is exponentially decreasing, while the dimensions $\ell _{2}$ and $\ell _{3}$ do not change substantially, in agreement with the three-dimensional picture in figure 1(b). Thus, the pancake develops only one small scale corresponding to its transversal direction, and the vorticity growth is locally one-dimensional. The span-to-thickness ratio grows exponentially, reaching $\ell _{2}/\ell _{1}\sim \ell _{3}/\ell _{1}\gtrsim 100$ at the final time $t=7.75$ . The vorticity vector within the pancake structure is tangent to the pancake midplane and oriented roughly anti-parallel to the $y$ -axis.

3 Exact solution of the Euler equations as a model for pancake evolution

Following the numerical results, we suggest an analytical model for the vorticity growth. Assuming that in Cartesian coordinates $\boldsymbol{a}=a_{1}\boldsymbol{n}_{1}+a_{2}\boldsymbol{n}_{2}+a_{3}\boldsymbol{n}_{3}$ the vorticity changes only along the $a_{1}$ -axis and is oriented along the $a_{2}$ -axis, we write

where $\unicode[STIX]{x1D6FA}(t)$ is the characteristic vorticity amplitude and $\ell _{1}(t)$ is the pancake thickness. The ansatz (3.1) contains a derivative (denoted by a prime) of an arbitrary function $f(\unicode[STIX]{x1D709})$ taken at $\unicode[STIX]{x1D709}=a_{1}/\ell _{1}(t)$ . One can check that (3.1) together with the velocity field

represents an exact solution of the Euler equations (2.1), where $\unicode[STIX]{x1D6FD}_{1}(t)$ , $\unicode[STIX]{x1D6FD}_{2}(t)$ and $\unicode[STIX]{x1D6FD}_{3}(t)$ are given by

with the dots denoting time derivatives. The velocity (3.2) is a superposition of the shear flow $-\unicode[STIX]{x1D6FA}\ell _{1}f(a_{1}/\ell _{1})\boldsymbol{n}_{3}$ and the asymmetric irrotational straining flow $(-\unicode[STIX]{x1D6FD}_{1}a_{1},\unicode[STIX]{x1D6FD}_{2}a_{2},\unicode[STIX]{x1D6FD}_{3}a_{3})$ , and satisfies the Euler equations $\dot{\boldsymbol{v}}+\boldsymbol{v}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{v}=-\unicode[STIX]{x1D735}p$ with the pressure

It should be noted that the pressure is determined entirely by the coefficients $\unicode[STIX]{x1D6FD}_{1}$ , $\unicode[STIX]{x1D6FD}_{2}$ and $\unicode[STIX]{x1D6FD}_{3}$ of the straining flow. A uniform velocity field with an arbitrary time dependence $(0,v_{b2}(t),v_{b3}(t))$ can be added to (3.2). This field, describing a drift of the pancake structure, leads to the change of pressure as $p\rightarrow p-a_{2}(\unicode[STIX]{x1D6FD}_{2}v_{b2}+\dot{v}_{b2})-a_{3}(\unicode[STIX]{x1D6FD}_{3}v_{b3}+\dot{v}_{b3})$ .

The suggested solution is an essential generalization of the pancake model of Brachet et al. (Reference Brachet, Meneguzzi, Vincent, Politano and Sulem1992); the latter is obtained as a special case with $\unicode[STIX]{x1D6FA}=\ell _{1}^{-1}=\text{e}^{t/T}$ , $\unicode[STIX]{x1D6FD}_{1,2}=1/T$ , $\unicode[STIX]{x1D6FD}_{3}=0$ . The solution (3.1)–(3.4) has infinite energy in $\mathbb{R}^{3}$ and allows for an arbitrary time dependence of $\unicode[STIX]{x1D6FA}(t)$ and $\ell _{1}(t)$ , in particular, the one leading to a finite-time blowup. In addition to an arbitrary function $f(\unicode[STIX]{x1D709})$ , the new solution is characterized by a single dimensionless parameter

describing the asymmetry of the straining flow in (3.2). In our numerical simulations, nearly exponential behaviour for $\unicode[STIX]{x1D6FA}(t)$ (vorticity maximum) and $\ell _{1}(t)$ is observed, see figure 1(a,c), which corresponds to constant numbers for $\unicode[STIX]{x1D6FD}_{1}$ , $\unicode[STIX]{x1D6FD}_{2}$ and $\unicode[STIX]{x1D6FD}_{3}$ in (3.3). Then, the asymmetry parameter defines the exponent in the power-law relation

between the vorticity amplitude and the pancake thickness. Consequently, in the transversal direction, the velocity has variation $\unicode[STIX]{x1D6FF}v_{3}\propto \unicode[STIX]{x1D6FA}\ell _{1}\propto \ell _{1}^{1-\unicode[STIX]{x1D701}}$ at the scale of the pancake thickness $\unicode[STIX]{x1D6FF}a_{1}\sim \ell _{1}$ , and this variation vanishes for $\unicode[STIX]{x1D701}<1$ (i.e.  $\unicode[STIX]{x1D70E}<1$ ) as $\ell _{1}\rightarrow 0$ .

The solution (3.1)–(3.4) can be extended for the Navier–Stokes equations with kinematic viscosity $\unicode[STIX]{x1D708}$ , if the function $f(\unicode[STIX]{x1D709},t)$ changes with time as $f_{t}-(\unicode[STIX]{x1D708}/\ell _{1}^{2})f_{\unicode[STIX]{x1D709}\unicode[STIX]{x1D709}}=0$ . The latter becomes the heat equation after a simple transformation of time, $\unicode[STIX]{x1D70F}=\int \,\text{d}t/\ell _{1}^{2}(t)$ . For the axisymmetric straining flow with $\unicode[STIX]{x1D6FD}_{2}=\unicode[STIX]{x1D6FD}_{3}=\unicode[STIX]{x1D6FD}_{1}/2$ , the suggested solution becomes the special case of the Lundgren stretched-spiral vortex; see Lundgren (Reference Lundgren1982). It should be noted that solutions of the Navier–Stokes equations in the form of stretched vortices embedded in a uniform straining flow were first studied by Burgers (Reference Burgers1948); see also Prochazka & Pullin (Reference Prochazka and Pullin1998), Pullin & Saffman (Reference Pullin and Saffman1998), Gibbon, Fokas & Doering (Reference Gibbon, Fokas and Doering1999) and Maekawa (Reference Maekawa2009).

4 Comparison with the numerical simulations

In simulations, we define the local coordinate system $(a_{1},a_{2},a_{3})$ for the pancake structure in the following way. The origin is chosen at the point of the global vorticity maximum, where we also compute the pancake normal vector $\boldsymbol{n}_{1}$ of the $a_{1}$ -axis, as described in § 2. According to the exact solution (3.1), the $a_{2}$ -axis should be parallel to the vorticity vector $\unicode[STIX]{x1D74E}$ . However, in simulations, the angle between $\boldsymbol{n}_{1}$ and $\unicode[STIX]{x1D74E}$ may differ from $90^{\circ }$ . We checked that this difference, in fact, is tiny, reaching $0.02^{\circ }$ at the final time. Thereby, we define the $a_{2}$ -axis with the direction $\boldsymbol{n}_{2}=c[\unicode[STIX]{x1D74E}-(\unicode[STIX]{x1D74E}\boldsymbol{\cdot }\boldsymbol{n}_{1})\boldsymbol{n}_{1}]$ , where $(\unicode[STIX]{x1D74E}\boldsymbol{\cdot }\boldsymbol{n}_{1})\boldsymbol{n}_{1}$ is a small correction and the prefactor $c$ is chosen such that $\Vert \boldsymbol{n}_{2}\Vert =1$ . Finally, the $a_{3}$ -axis has the direction $\boldsymbol{n}_{3}=\boldsymbol{n}_{1}\times \boldsymbol{n}_{2}$ . It should be noted that this coordinate system is computed at each moment of time, which is necessary to account for a possible drift of the whole structure. We choose the vorticity amplitude as the maximum vorticity, $\unicode[STIX]{x1D6FA}=\unicode[STIX]{x1D714}_{max}$ , and compute coefficients $\unicode[STIX]{x1D6FD}_{1}$ , $\unicode[STIX]{x1D6FD}_{2}$ and $\unicode[STIX]{x1D6FD}_{3}$ according to (3.3).

Figure 2.

(a) Normalized vorticity component $\unicode[STIX]{x1D714}_{2}/\unicode[STIX]{x1D714}_{max}$ as a function of $\unicode[STIX]{x1D709}=a_{1}/\ell _{1}$ at different times. (b) Comparison of the logarithmic derivatives $\unicode[STIX]{x1D6FD}_{1}=-\dot{\ell }_{1}/\ell _{1}$ and $\unicode[STIX]{x1D6FD}_{2}=\dot{\unicode[STIX]{x1D714}}_{max}/\unicode[STIX]{x1D714}_{max}$ with the velocity gradients $-\unicode[STIX]{x2202}v_{1}/\unicode[STIX]{x2202}a_{1}$ and $\unicode[STIX]{x2202}v_{2}/\unicode[STIX]{x2202}a_{2}$ computed at the global vorticity maximum; see (4.2). After prior computation of the time derivatives, $\ell _{1}$ and $\unicode[STIX]{x1D714}_{max}$ are smoothed with the weighted local regression (lowess filter); see Cleveland, Devlin & Grosse (Reference Cleveland, Devlin and Grosse1988).

In this section, we provide several numerical tests supporting the pancake model proposed in § 3. The first test is related to self-similarity of the transversal vorticity profile, which should be kept, according to (3.1), as

This is confirmed in figure 2(a), where the vorticity profile $\unicode[STIX]{x1D714}_{2}/\unicode[STIX]{x1D714}_{max}$ along the $a_{1}$ -axis is shown at different times. The self-similarity region grows with time in the $\unicode[STIX]{x1D709}$ -coordinate, remaining finite in physical space, where it matches with the background flow. It should be noted that the function $f(\unicode[STIX]{x1D709})$ may be arbitrary; in our simulations, different vorticity profiles are obtained for different pancakes and initial conditions.

For the analysis of the velocity field, one should take into account a drift of the pancake structure by a background flow with non-trivial time dependence. However, this difficulty can be avoided if we examine the velocity gradient, which for the pancake model solution (3.1)–(3.4) has the form

In figure 2(b), we compare at different times the logarithmic derivatives $\unicode[STIX]{x1D6FD}_{1}=-\dot{\ell }_{1}/\ell _{1}$ and $\unicode[STIX]{x1D6FD}_{2}=\dot{\unicode[STIX]{x1D714}}_{max}/\unicode[STIX]{x1D714}_{max}$ with the velocity gradients $-\unicode[STIX]{x2202}v_{1}/\unicode[STIX]{x2202}a_{1}$ and $\unicode[STIX]{x2202}v_{2}/\unicode[STIX]{x2202}a_{2}$ evaluated at the global vorticity maximum. One can see very good overall agreement, with a larger deviation for $-\unicode[STIX]{x2202}v_{1}/\unicode[STIX]{x2202}a_{1}$ at $t=7$ . This deviation can be attributed to the variation (up to 20 %) of the velocity derivative within the pancake thickness. At the final simulation time $t=7.75$ and at the global vorticity maximum, the numerical velocity gradient is given by

confirming that there is a single large $(3,1)$ -component, $\unicode[STIX]{x2202}v_{3}/\unicode[STIX]{x2202}a_{1}\approx -\unicode[STIX]{x1D714}_{max}$ . The diagonal components corresponding to the straining flow are in very good agreement with the coefficients $-\unicode[STIX]{x1D6FD}_{1}=-0.74$ , $\unicode[STIX]{x1D6FD}_{2}=0.53$ and $\unicode[STIX]{x1D6FD}_{3}=0.21$ , while the remaining components, $(1,2)$ , $(1,3)$ , $(2,1)$ , $(2,3)$ and $(3,2)$ corresponding to vanishing elements in (4.2) are small.

Figure 3.

(a) Pancake midplane (vorticity above 70 % of the maximum value) in the projection to the $(x,y)$ -plane. The arrows show projections of the vorticity vector, scaled by a factor of $0.025$ . The coordinates are shifted to place the vorticity maximum at the origin. (b) Velocity derivatives $\unicode[STIX]{x2202}v_{i}/\unicode[STIX]{x2202}a_{j}$ evaluated at different points of the pancake midplane; marked with $(i,j)$ . (c) Normalized vorticity profile $\unicode[STIX]{x1D714}_{2}/\unicode[STIX]{x1D6FA}$ in the direction perpendicular to the midplane, at five different points corresponding to the red dots in (a). The red line corresponding to the centre of the pancake is the same as in figure 2(a).

The previous tests confirmed that the pancake model solution agrees with the flow in the vicinity of the global vorticity maximum. However, this model cannot describe the whole region of high vorticity, since the pancake is not completely flat, with deviations from the plane $a_{1}=0$ much larger than the pancake thickness; see figure 1(b). Nevertheless, we can check whether the model approximates the pancake locally, at every nearly flat pancake segment. For this purpose, we consider the final time and isolate the principal part of the pancake with vorticity $\unicode[STIX]{x1D714}\geqslant 0.7\unicode[STIX]{x1D714}_{max}$ ; the isolated region is roughly parallel to the $(x,y)$ -plane. Despite the fact that this region is very thin, $\ell _{1}\sim 0.01$ , its lateral dimensions are comparable to the size of the numerical box, as shown by the projection to the $(x,y)$ -plane in figure 3(a). Within this region, we define the pancake midplane $z=z_{m}(x,y)$ , chosen as points of maximum vorticity, $\max _{z}\unicode[STIX]{x1D714}$ , for the given values of $x$ and $y$ . Then, at each point $\boldsymbol{r}_{m}=(x,y,z_{m}(x,y))$ of the midplane, we define the new local coordinates $(a_{1},a_{2},a_{3})$ , using the first eigenvector of the Hessian matrix and the vorticity vector, as described above in this section. It should be noted that $(x,y)$ serve as parameters in this representation, while the coordinates $(a_{1},a_{2},a_{3})$ explore the neighbourhood of $\boldsymbol{r}_{m}$ .

First, we verified numerically that, within the pancake, the vorticity vector is tangent to the midplane and almost unidirectional. This is illustrated in figure 3(a), where projections to the $(x,y)$ -plane of the vorticity vector are shown by arrows, which are scaled (multiplied by $0.025$ ) to fit the plot. At several distant points on the midplane, we considered the vorticity profile $\unicode[STIX]{x1D714}_{2}/\unicode[STIX]{x1D6FA}$ as a function of $a_{1}/\ell _{1}$ , where $\unicode[STIX]{x1D6FA}=\max _{z}\unicode[STIX]{x1D714}$ is the vorticity at the corresponding point on the midplane. At every point, a convergence similar to figure 2(a) is observed, showing that the vorticity profile changes with time self-similarly according to (4.1); the convergence gets worse near the pancake border. The vorticity profile varies from point to point; this means that the function $f(\unicode[STIX]{x1D709})$ of the model (3.1) changes along the pancake. The magnitude of this variation can be seen in figure 3(c), which presents the final-time vorticity profiles $\unicode[STIX]{x1D714}_{2}/\unicode[STIX]{x1D6FA}$ at five different points marked with red dots in panel (a) of the same figure. It should be noted that a specific form of $f(\unicode[STIX]{x1D709})$ is not important in the exact solution (3.1)–(3.2). Thus, this function accounts for the local self-similarity only, while its variations along the pancake may be captured by a next-order correction to our model.

Figure 4.

(a) Relation between the vorticity local maxima $\unicode[STIX]{x1D714}_{max}(t)$ and the respective characteristic lengths $\ell _{1}(t)$ during the evolution of the pancake structures. The lines represent the evolution of the maxima with increasing time, with the red dots corresponding to the final simulation time. The dashed red line indicates the power-law scaling $\unicode[STIX]{x1D714}_{max}\propto \ell _{1}^{-2/3}$ . (b) The same graph for a different simulation with a generic initial condition and the final grid $1152\times 972\times 864$ .

The structure of the gradient (4.2) is confirmed in figure 3(b) along the whole pancake midplane. The components $(1,2)$ , $(1,3)$ , $(2,1)$ , $(2,3)$ and $(3,2)$ are concentrated in the middle of the figure: they are approximately one order of magnitude smaller than the diagonal components and more than two orders of magnitude smaller than the $(3,1)$ -component related to the vorticity. The large $(3,1)$ -component does not fit to the vertical range of the figure, but it is in excellent agreement with the vorticity, $\unicode[STIX]{x2202}v_{3}/\unicode[STIX]{x2202}a_{1}\approx -\unicode[STIX]{x1D714}$ , with the difference below 0.6 %. The diagonal components $\unicode[STIX]{x2202}v_{1}/\unicode[STIX]{x2202}a_{1}$ (blue) and $\unicode[STIX]{x2202}v_{2}/\unicode[STIX]{x2202}a_{2}$ (red) vary significantly along the midplane. We do not show $\unicode[STIX]{x2202}v_{3}/\unicode[STIX]{x2202}a_{3}$ due to its exact relation to $(1,1)$ - and $(2,2)$ -components, originating from incompressibility of the flow, $\text{div}\,\boldsymbol{v}=0$ . With these observations, we confirm that, for every nearly flat pancake segment, the flow can be approximated by the pancake model solution suggested in § 3.

As was noticed in Agafontsev et al. (Reference Agafontsev, Kuznetsov and Mailybaev2015), at late times the pancakes develop according to the Kolmogorov-like power law

This tendency is clearly seen in figure 4(a), where all other local vorticity maxima are shown and the $2/3$ -slope is represented by the dashed line. We stress that the pancake model solution (3.1)–(3.4) allows for an arbitrary power-law exponent $\unicode[STIX]{x1D701}$ . Thus, the universality of the asymptotic value $\unicode[STIX]{x1D701}=2/3$ , which corresponds to the asymmetry parameter $\unicode[STIX]{x1D70E}=1/3$ of the straining flow, goes beyond this solution. We think that restrictions on the power law may come from the non-local interactions of the pancakes and the background flow.

All of the results presented so far relate to the main pancake structure from the simulation with the $I_{1}$ initial condition. We confirmed that several other pancake structures, associated with some of the local vorticity maxima shown in figure 4(a), also agree with the pancake model solution. We performed a series of simulations in grids with a total number of nodes of $1024^{3}$ , starting from fully generic (large-scale) initial conditions. The regions of high vorticity, emerging from such initial flows, have arbitrary orientation, which does not allow us to use anisotropic grids effectively. Thereby, such simulations yield considerably smaller overall vorticity enhancement. However, these regions represent pancake-like structures developing close to the model solution (3.1)–(3.4), and the same relation (4.4) between the vorticity maximum and the pancake thickness is observed; see the example of one such simulation in figure 4(b).