4.1. Unconditional statistics

Table 2 lists various unconditional statistics characterizing the role of the pressure Hessian on vortex stretching, based on (2.7), at different ${R_\lambda }$. All quantities are appropriately non-dimensionalized by the Kolmogorov time scale $\tau _K$, and henceforth, should be interpreted as such (unless otherwise mentioned). We first consider the eigenvalues of the pressure Hessian. Owing to homogeneity $\langle H_{ii} \rangle = 0$; thus, $\sum _{i=1}^{3} \langle \lambda _i^{P} \rangle = 0$ and $\langle \lambda _i^{P} \rangle = \langle \lambda _i^{D} \rangle$. Table 2 reveals that the individual averages of the eigenvalues are approximately equal to $0.3: 0.03: -0.33$, without any appreciable dependence on ${R_\lambda }$. The intermediate eigenvalue is overall positive on average but its magnitude is substantially smaller than the other two, and essentially close to zero.

Table 2.

Unconditional averages of various quantities associated with correlation of vorticity and pressure Hessian, based on (2.7). Here $\lambda _i^{P}$, for $i=1,2,3$, are the eigenvalues of pressure Hessian, with corresponding eigenvectors $\boldsymbol {e}_i^{P}$. All quantities are appropriately normalized by the Kolmogorov time scale $\tau _K$.

Similarly, the mean square of alignment cosines between vorticity and eigenvectors of pressure Hessian, $\langle (\boldsymbol {e}_i^{P} \boldsymbol {\cdot } \hat {{\boldsymbol {\omega }}})^2 \rangle$, where $\hat {{\boldsymbol {\omega }}}$ denotes the unit vector parallel to ${\boldsymbol {\omega }}$, are also essentially independent of ${R_\lambda }$. Note that the square of cosines sum up to unity for all three directions. Additionally, they are bounded between 0 and 1 for each individual direction, respectively, for the case of perfect orthogonal and parallel alignment; whereas for a uniform distribution of the alignment cosine, the mean square average is $1/3$. From table 2, we observe that the measured alignments do not deviate significantly from $1/3$, with a weak preferential alignment of vorticity with the intermediate eigenvector of the pressure Hessian.

It is worth noting that while a uniform distribution of the alignment cosine implies the second moment is $1/3$, the reverse is not necessarily true. Thus, it is useful to also inspect the p.d.f. of the alignment cosines (Chevillard et al. Reference Chevillard, Meneveau, Biferale and Toschi2008), which are shown in figure 1. The solid and dashed curves correspond to ${R_\lambda }=1300$ and $140$, respectively, demonstrating that the alignments are independent of ${R_\lambda }$. The distributions for $|\boldsymbol {e}_1^{P} \boldsymbol {\cdot } \hat {{\boldsymbol {\omega }}}|$ and $|\boldsymbol {e}_1^{P} \boldsymbol {\cdot } \hat {{\boldsymbol {\omega }}}|$ conform with expectation from their second moments in table 2, indicating weak preferential orthogonal and parallel alignment, respectively. However, for $|\boldsymbol {e}_3^{P} \boldsymbol {\cdot } \hat {{\boldsymbol {\omega }}}|$, we observe an anomalous behaviour, showing simultaneous preferential orthogonal and parallel alignments, which cancel each other out when evaluating the second moment. Note that the limiting cases of 0 or 1 for the second moment of alignment cosines do not present such an anomaly for the p.d.f.s. In fact, we will see later that when considering conditional statistics, the alignments are substantially enhanced for extreme vorticity events (such that using the second moment only does not lead to any ambiguity).

Figure 1.

Probability density function (p.d.f.) of alignment cosines between vorticity and eigenvectors of pressure Hessian at ${R_\lambda }=1300$ (solid lines) and ${R_\lambda }=140$ (dashed lines).

Finally, we consider the net contributions to the budget of vortex stretching (as per (2.7)). Table 2 shows separately the mean contributions from the deviatoric and isotropic components of the pressure Hessian and their sum $\langle \omega _i \omega _j H_{ij} \rangle$, contrasted with the nonlinear term $\langle W_i W_i \rangle$. Remarkably, the deviatoric and isotropic contributions are comparable in magnitude, but opposite in sign. Taking into account the negative sign before the pressure Hessian term in (2.7), it follows that ${\boldsymbol {H}}^{D}$ favours vortex stretching, whereas ${\boldsymbol {H}}^{I}$ inhibits it. This essentially establishes that non-local effects of the pressure field enable vortex stretching (Ohkitani & Kishiba Reference Ohkitani and Kishiba1995). This also conforms with non-locality of vortex stretching as highlighted in recent studies (Hamlington et al. Reference Hamlington, Schumacher and Dahm2008; Buaria & Pumir Reference Buaria and Pumir2021); however, it is worth noting that the non-locality in these studies was investigated in the context of the Biot–Savart relation between vorticity and strain. We will further elaborate on this connection later.

The strong cancellation between the deviatoric and isotropic contributions results in a weakly positive value of $\langle \omega _i \omega _j H_{ij} \rangle$, implying that pressure Hessian overall weakly opposes vortex stretching – which is primarily a local effect stemming from the dominant isotropic contribution of pressure Hessian. Both the deviatoric and isotropic contributions become stronger with ${R_\lambda }$, but the latter is always slightly stronger. In contrast, the term $\langle W_i W_i\rangle$, also given in table 2, is positive by definition, and thus always enables vortex stretching. This term increases with ${R_\lambda }$ and is noticeably larger than $\langle \omega _i \omega _j H_{ij} \rangle$ (at all ${R_\lambda }$). This leads to the (anticipated) result that the nonlinear effects in turbulence predominantly enable vortex stretching, with a net positive energy cascade from large to small scales (Batchelor Reference Batchelor1953; Betchov Reference Betchov1956; Kerr Reference Kerr1985). Although not shown, the ${R_\lambda }$ dependence of these quantities conforms with an approximate power law of $R_\lambda ^{0.39}$ in nominal agreement with fourth-order moment of velocity derivatives (Gylfason, Ayyalasomayjula & Warhaft Reference Gylfason, Ayyalasomayjula and Warhaft2004; Buaria & Sreenivasan Reference Buaria and Sreenivasan2022b).

4.2. Conditional statistics

In the previous subsection, we considered unconditional statistics, which provide an overall global perspective of the flow. To specifically characterize the extreme events or regions of intense vorticity, we condition the statistics on the magnitude of vorticity; specifically, we will use $\varOmega \tau _K^2$ (or equivalently, $\varOmega /\langle \varOmega \rangle$), to quantify the extremeness of an event with respect to the mean field.

Figure 2 shows the conditional mean square of alignment cosines between vorticity and the eigenvectors of pressure Hessian, at various ${R_\lambda }$. For weak enstrophy, all the curves are at $1/3$, consistent with a uniform distribution of cosines. For $\varOmega \tau _K^2 \approx 1$, we notice that vorticity has a weak preferential alignment with $\boldsymbol {e}^{P}_2$, in agreement with the unconditional results in table 2. However, for extreme events, a different picture emerges and vorticity almost perfectly aligns with $\boldsymbol {e}^{P}_3$ (becoming orthogonal to both first and second eigenvectors). This alignment can be explained by considering the familiar picture of intense vorticity being arranged in tube-like structures (Jiménez et al. Reference Jiménez, Wray, Saffman and Rogallo1993; Ishihara et al. Reference Ishihara, Kaneda, Yokokawa, Itakura and Uno2007; Buaria et al. Reference Buaria, Pumir, Bodenschatz and Yeung2019), as discussed at the end of the present subsection.

Figure 2.

Conditional expectation (given enstrophy $\varOmega$) of second moment of alignment cosines between vorticity unit vector ($\hat {{\boldsymbol {\omega }}}$) and eigenvectors of pressure Hessian ($\boldsymbol {e}_i^{P}$), at various ${R_\lambda }$.

Figure 3 shows the conditional expectation of the eigenvalues of pressure Hessian. To focus on extreme events, the abscissa is set to $\varOmega \tau _K^2 \geqslant 1$. The identity $\sum _i \lambda _i^{P} = \nabla ^2 P = (\varOmega - \varSigma )/2$ implies that $\sum _i \langle \lambda _i^{P} | \varOmega \rangle = (\varOmega - \langle \varSigma |\varOmega \rangle )/2$. Recent numerical works (Buaria et al. Reference Buaria, Pumir, Bodenschatz and Yeung2019, Reference Buaria, Bodenschatz and Pumir2020a; Buaria & Pumir Reference Buaria and Pumir2022) have shown that for large enstrophy, $\langle \varSigma |\varOmega \rangle \sim \varOmega ^\gamma$, where the exponent $\gamma < 1$, but weakly increases with ${R_\lambda }$ (with $\gamma \to 1$ being the asymptotic limit). Thus, while the average sum of eigenvalues is zero for the mean field, it is strongly positive in regions of intense enstrophy. In fact, to the leading order, one can anticipate that $\langle \lambda _i^{P} | \varOmega \rangle \sim \varOmega$ when $\varOmega \tau _K^2 \gg 1$. Figure 3(a) shows the quantity $\langle \lambda _i^{P} | \varOmega \rangle / \varOmega$. The behaviour of the averaged eigenvalues at $\varOmega \tau _K^2 =1$ is consistent with the results in table 2. However, in regions of intense enstrophy, the first two eigenvalues are strongly positive, whereas the third eigenvalue is essentially zero. Once again, no appreciable dependence on ${R_\lambda }$ is seen for this case.

Figure 3.

(a) Conditional expectations (given enstrophy $\varOmega$) of the eigenvalues of pressure Hessian, for various ${R_\lambda }$. (b) Conditional expectations of the eigenvalues of the deviatoric part. The legend in panel (a) also applies to panel (b).

Given that vorticity is almost perfectly aligned with third eigenvector of the pressure Hessian and the corresponding eigenvalue is close to zero, the behaviour of the term $\omega _i \omega _j H_{ij}$ cannot be easily inferred. Figure 4(a) shows the conditional expectation $\langle \omega _i \omega _j H_{ij} |\varOmega \rangle /\varOmega ^2$, along with the individual contributions in each eigendirection. The normalization by $\varOmega ^2$ comes from simple dimensional grounds. For $\varOmega \tau _K^2 \simeq 1$, the contributions from the first and third eigendirections are expectedly positive and negative, respectively, largely cancelling each other to give a weakly positive net contribution (whereas the contribution from second direction is essentially negligible). However, for $\varOmega \tau _K^2 \gg 1$, the contributions from all directions become positive and comparable. Thus, in regions of intense vorticity, the role of pressure (Hessian) is to oppose vortex stretching.

Figure 4.

(a) Conditional expectation (given enstrophy $\varOmega$) of the term $\omega _i \omega _j H_{ij}$ (marked as sum) and individual contributions from each eigendirection of pressure Hessian (in solid lines). The quantities are normalized by $\varOmega ^2$ to reveal a plateau-like behaviour for $\varOmega \tau _K^2 \gg 1$. (b) Same as panel (a), but for $\omega _i \omega _j H_{ij}^{D}$, (the contribution from the deviatoric part of pressure Hessian). (c) The net contributions from a and b are further contrasted with $\omega _i \omega _j H_{ij}^{I}$ (the contribution from isotropic part of pressure Hessian).

To better understand this result, we consider next the contributions from the deviatoric and the isotropic components of the pressure Hessian. The sum of eigenvalues of the deviatoric part ${\boldsymbol {H}}^{D}$ is always constrained to be zero, i.e. $\sum _i \lambda _i^D = 0$, and thus, $\sum _i \langle \lambda _i^{D} | \varOmega \rangle = 0$. Figure 3(b) shows the conditional expectation $\langle \lambda _i^{D} | \varOmega \rangle / \varOmega$. While the results shown in figure 3(b) for $\varOmega \tau _K^2 \simeq 1$ are essentially identical to $\langle \lambda _i^P | \varOmega \rangle$, the behaviour for $\varOmega \tau _K \gg 1$ is different, with the first and second eigenvalues being always positive (with the second being noticeably weaker), whereas third eigenvalue is always negative (perfectly cancelling the contribution from other two). Thus, from the results in figures 2 and 3(b), it can be anticipated that $\langle \omega _i \omega _j H^{D}_{ij} |\varOmega \rangle$ is negative in regions of intense enstrophy. The contribution of the isotropic component is also easy to understand, since $\langle \omega _i \omega _j H^{I}_{ij} |\varOmega \rangle = \langle \varOmega (\varOmega - \varSigma ) |\varOmega \rangle /6 = (\varOmega ^2 - \varOmega \langle \varSigma |\varOmega \rangle )/6$. Using $\langle \varSigma |\varOmega \rangle \sim \varOmega ^\gamma$ (with $\gamma < 1$) implies that $\langle \omega _i \omega _j H^{I}_{ij} |\varOmega \rangle$ is positive in regions of intense enstrophy. We verify these expectations in figures 4(b) and 4(c).

Figure 4(b) shows the conditional expectation $\langle \omega _i \omega _j H_{ij}^{D} |\varOmega \rangle /\varOmega ^2$, along with the individual contributions in each eigendirection. It can be clearly seen that for large enstrophy, the contributions from both first and second eigendirections are essentially zero and the third eigendirection completely dominates the overall sum (as explained earlier). This again establishes that the non-local portion of the pressure Hessian actually enables vortex stretching. In figure 4(c), we compare the contributions from the deviatoric and isotropic components, together with the overall average. We notice that the isotropic contribution is strongly positive, cancelling the deviatoric contribution to give a weakly net positive average. Once again, this shows that the depletion of vortex stretching by pressure Hessian is local.

It is worth noting that many qualitative aspects of the previously discussed results can be explained by noting that intense enstrophy is found in tube-like vortices (Jiménez et al. Reference Jiménez, Wray, Saffman and Rogallo1993; Ishihara et al. Reference Ishihara, Kaneda, Yokokawa, Itakura and Uno2007; Buaria et al. Reference Buaria, Pumir, Bodenschatz and Yeung2019), for which the Burgers vortex model is a good first-order approximation (Burgers Reference Burgers1948; Jiménez et al. Reference Jiménez, Wray, Saffman and Rogallo1993). For the simple case of a Burgers vortex, pressure is minimum and constant along the axis of the vortex. This implies that the smallest (third) eigenvalue of the pressure Hessian is zero and the corresponding eigenvector is perfectly aligned with vorticity; whereas the first two eigenvalues are positive and the corresponding eigenvectors are perfectly orthogonal to vorticity (Andreotti Reference Andreotti1997). Indeed, these expectations are qualitatively consistent with the results shown in figures 2 and 3. However, we note that the precise structure of vortices in turbulence is different than the Burgers vortex in some very crucial aspects. For instance, due to the structural properties of the Burgers vortex, the term $\omega _i \omega _j H_{ij}$ is essentially zero, which is not the case in turbulence. Additionally, vorticity is perfectly axial in the Burgers vortex, but not real turbulent vortices, and there is some noticeable degree of Beltramization (Choi, Kim & Lee Reference Choi, Kim and Lee2009; Buaria et al. Reference Buaria, Pumir and Bodenschatz2020b) – an effect that is essential to the self-attenuation mechanism analysed in Buaria et al. (Reference Buaria, Pumir and Bodenschatz2020b) and Buaria & Pumir (Reference Buaria and Pumir2021). In fact, we will discuss in the next subsection that the net positive contribution from the pressure Hessian as observed in figure 4 is in fact connected to the self-attenuation mechanism observed in Buaria et al. (Reference Buaria, Pumir and Bodenschatz2020b) and Buaria & Pumir (Reference Buaria and Pumir2021).

4.3. Contrasting nonlinear and pressure Hessian contributions

Table 2 demonstrates that the (unconditional) contribution $\langle W_i W_i \rangle$ is substantially larger than $\langle \omega _i \omega _j H_{ij}\rangle$ at all ${R_\lambda }$. A comparison of the conditional result is first made in figure 5(a). Evidently, the nonlinear term prevails over the pressure Hessian term at $\varOmega \tau _K^2 \simeq 1$, in essential agreement with the observation in table 2. However, for intense enstrophy, we notice a substantial reduction in the nonlinear term $\langle W_i W_i | \varOmega \rangle /\varOmega ^2$ relative to the pressure Hessian term $\langle \omega _i \omega _j H_{ij} | \varOmega \rangle /\varOmega ^2$, with the latter being approximately constant. Since $W_i = \omega _j S_{ij}$, it follows that $\langle W_i W_i |\varOmega \rangle \sim \langle \varOmega \varSigma |\varOmega \rangle \sim \varOmega ^{1+\gamma }$. In contrast, we find $\langle \omega _i \omega _j H_{ij} |\varOmega \rangle \sim \varOmega ^2$. Thus, the observation in figure 5(a) essentially shows that the pressure Hessian term grows significantly faster than the nonlinear term as $\varOmega$ increases, such that for large enough $\varOmega$, the pressure Hessian contribution eventually becomes stronger than the nonlinear contribution (as indeed seen in figure 5a). To show this more clearly, figure 5(b) shows the ratio between the two terms, which is close to zero at $\varOmega \tau _K^2 =1$, but steadily increases and becomes greater than unity at large $\varOmega$ for every ${R_\lambda }$. Hence, the attenuating effect of pressure Hessian eventually prevails over the nonlinear term $W_i W_i$. Interestingly, the crossover point in $\varOmega$ increases with ${R_\lambda }$, which essentially is a reflection of intermittency and $\gamma$ slowly increasing with ${R_\lambda }$.

Figure 5.

(a) Conditional expectations (given enstrophy $\varOmega$) of the nonlinear and pressure Hessian contributions to the dynamics of vortex stretching vector, as given in (2.7). (a) The ratio of two terms, revealing that pressure Hessian contribution overtakes the nonlinear contribution at large $\varOmega$.

Further insight on the attenuation induced by the pressure Hessian in regions of intense vorticity can be obtained by rewriting (2.7) for the conditional field,

(4.1)\begin{equation} \frac{{\rm D} \langle \omega_i W_i |\varOmega \rangle }{{\rm D}t} = \langle W_i W_i | \varOmega \rangle + (-\langle \omega_i \omega_j H_{ij}^{D} |\varOmega \rangle) - (\langle \omega_i \omega_j H_{ij}^{I} |\varOmega \rangle) + \text{viscous terms} . \end{equation}

Note that the left-hand side is not zero even in stationary turbulence. We essentially observe that the first two terms on the right-hand side are positive, whereas the third term is negative, i.e. vortex stretching is enabled by the nonlinear term (which is local) and the deviatoric pressure Hessian (which is non-local), whereas the isotropic pressure Hessian (which is local) strongly opposes it. For weak or moderate $\varOmega$, the positive contribution prevails, resulting in a net positive rate of change of vortex stretching leading to increased vorticity amplification. However, for large $\varOmega$, the negative contribution from $H_{ij}^{I}$ prevails, leading to a negative rate of change of vortex stretching. In all cases, the viscous terms are ignored, which are negligibly small at large ${R_\lambda }$ (although not shown, this can be anticipated).

We stress that, although the pressure Hessian is known to attenuate vortex stretching (Tsinober et al. Reference Tsinober, Ortenberg and Shtilman1999), the results in figure 5 indicate that this attenuation overwhelms even the nonlinear terms in regions of most intense vorticity. This points to an inviscid regularizing mechanism, which can be traced back to the prevalence of the contribution of $H^I$, which can be understood as local (Ohkitani & Kishiba Reference Ohkitani and Kishiba1995). A very similar observation for attenuation of vorticity amplification was also recently uncovered in Buaria et al. (Reference Buaria, Pumir and Bodenschatz2020b) and Buaria & Pumir (Reference Buaria and Pumir2021). In these works, the non-locality of vortex stretching was analysed by writing strain as the Biot–Savart integral of vorticity and decomposing it into local and non-local contributions. The local contribution is obtained by integrating in a sphere of radius $R$, whereas the remaining integral is the non-local contribution. Thereafter, it was observed that vortex stretching is engendered by the non-local contribution and remarkably, the local contribution acts to attenuate intense vorticity, also representing an inviscid mechanism to counter vorticity amplification. It stands to reason that the self-attenuation mechanism identified in Buaria et al. (Reference Buaria, Pumir and Bodenschatz2020b) and Buaria & Pumir (Reference Buaria and Pumir2021) is essentially related to the local pressure mechanism identified in this work. An important underlying connection between them is that they both act only when enstrophy becomes sufficiently strong and this critical value increases with ${R_\lambda }$ (Buaria et al. Reference Buaria, Pumir and Bodenschatz2020b). Nevertheless, we note that precisely underpinning the causality between the two mechanisms requires further analysis, particularly by considering Lagrangian particle trajectories, as evident from (4.1). Such an analysis will be considered in a future work.

5.1. Unconditional statistics

We first analyse the alignment cosines between eigenvectors of strain and the pressure Hessian, as measured by $\langle (\boldsymbol {e}_i^{P} \boldsymbol {\cdot } \boldsymbol {e}_j)^2 \rangle$ for $i,j=1,2,3$. The (unconditional) average of eigenvalues of the pressure Hessian can be found in table 2, whereas those of the strain tensor were previously discussed in Buaria et al. (Reference Buaria, Bodenschatz and Pumir2020a). Table 3 lists all the nine individual terms for various ${R_\lambda }$, revealing no particularly strong alignment between the eigenvectors of strain and the pressure Hessian. The strongest alignment corresponds to $\langle (\boldsymbol {e}_1^{P} \boldsymbol {\cdot } \boldsymbol {e}_2)^2 \rangle \approx 0.4$, which is only marginally larger than $1/3$. Moreover, all alignment results are virtually independent of ${R_\lambda }$ as it was earlier the case for the alignment between vorticity and eigenvectors of pressure Hessian.

Table 3.

Second moment of alignment cosines between the eigenvectors of pressure Hessian ($\boldsymbol {e}_i^{P}$) and strain ($\boldsymbol {e}_j$), at various ${R_\lambda }$.

To rule out any anomalous behaviour, figure 6 shows the p.d.f.s of the alignment cosines. While the distributions are not exactly uniform, it can be seen that they are essentially consistent with the behaviour anticipated from their second-order moments in table 3, i.e. demonstrating some weak preferential alignment for moments larger than $1/3$ (and vice versa). The most notable alignments are between $\boldsymbol {e}_2$ and $\boldsymbol {e}^P_1$, $\boldsymbol {e}^P_2$, which can be loosely understood by also considering alignment of vorticity with the eigenvectors of strain and pressure Hessian. Nevertheless, it is evident that none of the alignments are particularly strong. In all the panels, the solid and dashed lines at ${R_\lambda }=1300$ and $140$, respectively, near-perfectly coincide, showing that the alignment results are independent of ${R_\lambda }$.

Figure 6.

The p.d.f. of alignment cosines between eigenvectors of pressure Hessian ($\boldsymbol {e}_i^{P}$) and strain ($\boldsymbol {e}_j$), at ${R_\lambda }=1300$ (solid lines) and ${R_\lambda }=140$ (dashed lines).

5.2. Conditional statistics

To analyse the extreme strain events, we now consider various statistics conditioned on $\varSigma \tau _K^2$ (which equals $\varSigma /\langle \varSigma \rangle$). Figure 7 shows the conditional alignment between eigenvectors of strain and pressure Hessian as measured by $\langle (\boldsymbol {e}_i^{P} \boldsymbol {\cdot } \boldsymbol {e}_j)^2 | \varSigma \rangle$. For clarity, we only show ${R_\lambda }=1300$ (solid lines) and ${R_\lambda }=650$ (dashed lines). The dependence on ${R_\lambda }$ is very weak, and results at lower ${R_\lambda }$ (not shown) essentially follow the same trends. Overall, the alignment results indicate that there is no strong preferential alignment between strain and the pressure Hessian, even when extreme events are considered. The strongest alignments, parallel and orthogonal, are observed for $j=3$ and $j=2$, respectively, both with $i=3$, but the deviations from $1/3$ remain weak.

Figure 7.

Conditional expectation (given $\varSigma$) of second moment of alignment cosines between eigenvectors of pressure Hessian ($\boldsymbol {e}_i^{P}$) and strain ($\boldsymbol {e}_j$), at ${R_\lambda }=1300$ (solid lines) and ${R_\lambda }=650$ (dashed lines).

The conditional expectations of the eigenvalues of the pressure Hessian, and of its deviatoric part, are shown in figures 8(a) and 8(b), respectively. In both cases, we observe that the first and third eigenvalues are strongly positive and negative, respectively, and the second eigenvalue is very close to zero. The eigenvalues of the pressure Hessian satisfy $\langle (\lambda _1^{P} + \lambda _2^{P} + \lambda _3^{P}) | \varSigma \rangle = \langle (\varOmega - \varSigma ) | \varSigma \rangle$. Since $\langle \varOmega |\varSigma \rangle \sim \varSigma$, but with a prefactor which is slightly smaller than unity (Buaria & Pumir Reference Buaria and Pumir2022; Buaria et al. Reference Buaria, Pumir and Bodenschatz2022), it follows that the sum of eigenvalues $\lambda _i^P$ divided by $\varSigma$ is a small, negative constant. Indeed, this is the observation in figure 8(a), which shows that $\langle -\lambda _3^{P} |\varSigma \rangle \gtrsim \langle \lambda _1^{P} |\varSigma \rangle$, whereas $\langle \lambda _2^{P} |\varSigma \rangle \approx 0$. On the contrary, the sum of eigenvalues of the deviatoric part is exactly zero. Indeed, figure 8(b) conforms with this expectation, with $\langle -\lambda _3^{D} |\varSigma \rangle \gtrsim \langle \lambda _1^{D} |\varSigma \rangle$ being still true, but $\langle \lambda _2^{D} |\varSigma \rangle$ is weakly positive, ensuring that the sum of the eigenvalues is zero.

Figure 8.

(a) Conditional expectation (given $\varSigma$) of the eigenvalues of pressure Hessian, at various ${R_\lambda }$. (b) Conditional expectation of the eigenvalues of the deviatoric part.

Figure 9 shows the conditional expectation of the correlation $S_{ij} H_{ij}$, together with the individual contributions from the eigendirections of pressure Hessian; figure 9(a) shows the result for pressure Hessian and its eigenvalues and figure 9(b) shows the corresponding result for the deviatoric part. Note that despite $S_{ij} H_{ij} = S_{ij} H_{ij}^{D}$, the individual contributions from their respective eigendirections differ. Given the lack of any strong alignment between strain and the pressure Hessian eigenvectors, it can be anticipated that the largest contribution to $S_{ij} H_{ij}$ would arise from their largest eigenvalues, i.e. the product $\lambda _3^{P} \lambda _3$ (or $\lambda _3^{D} \lambda _3$). Additionally, this contribution would be positive, since both these eigenvalues are negative. Indeed, figure 9(a,b) confirms this expectation. For both figure 9(a) and figure 9(b), the largest contribution is positive and corresponds to $i=3$. In contrast, the contribution for $i=1$ is negative, since it is dominated by the product $\lambda _1^{P} \lambda _3$, (or $\lambda _1^{D} \lambda _3$ for figure 9b). The behaviour for $i=2$ is not as straightforward to predict, since the alignments are non-trivial for this case. Interestingly, we observe that the contribution from $i=2$ is negative in figure 9(a), but weakly positive in figure 9(b). For large $\varSigma$, all the contributions (as divided by $\varSigma ^{3/2}$) appear approximately constant, implying a simple scaling (Buaria et al. Reference Buaria, Pumir and Bodenschatz2022).

Figure 9.

Individual contributions to (a) $\langle S_{ij} H_{ij} |\varSigma \rangle = \langle \lambda _i^{P} \lambda _j ( \boldsymbol {e}_i^{P} \boldsymbol {\cdot } \boldsymbol {e}_j )^2 | \varSigma \rangle$, and (b) $\langle S_{ij} H_{ij}^{D} |\varSigma \rangle = \langle \lambda _i^{D} \lambda _j ( \boldsymbol {e}_i^{D} \boldsymbol {\cdot } \boldsymbol {e}_j )^2 | \varSigma \rangle$, from their eigendirections. Note, the summation is only implied over $j$, and results for each $i$ are shown. The quantities have been rescaled by $\varSigma ^{3/2}$. The solid lines are for ${R_\lambda }=1300$ and dashed lines for ${R_\lambda }=650$. The black lines given the total sum over all eigendirections.

The main observation from figure 9 is that the pressure Hessian term attenuates strong strain and amplifies weak strain (since the net effect has to be zero). Nevertheless, as evident from (2.6), strain amplification involves other nonlinear mechanisms. Their relative amplitudes were already compared in Buaria et al. (Reference Buaria, Pumir and Bodenschatz2022). For completeness, figure 10 shows the conditional expectations of the various inviscid terms involved in the budget equation for strain (together with their corresponding signs). All terms are once again normalized by $\varSigma ^{3/2}$. It can be seen that the self-amplification term is the largest contribution and the primary mechanism for generating intense strain. In contrast, the vortex stretching contribution always opposes strain amplification, whereas the pressure Hessian term opposes intense strain but amplifies weak strain. When considering strong strain events ($\varSigma \tau _K^2 > 1$), the pressure Hessian term is noticeably weaker than vortex stretching term, and both are considerably weaker than the strain self-amplification term.

Figure 10.

Conditional expectations (given $\varSigma$) of various inviscid terms on the right-hand side of (2.6). All quantities are normalized by $\varSigma ^{3/2}$ revealing a plateau like behaviour for $\varSigma \tau _K^2 > 1$.