1. Introduction
The interplay between buoyancy and flow structures plays a crucial role in stratified turbulent mixing as it governs the exchange of kinetic energy with the background potential energy (Linden Reference Linden1979; Caulfield Reference Caulfield2020). Irreversible mixing involves various types of fluid motions that distort isopycnals, and shear and vortical motions, inherently linked to the vorticity of the flow, are particularly influential in enhancing mixing. Consequently, it is essential to discern the contributions of these components of the motion to the mixing of the scalar field.
The interaction between vortices and stratification has undergone extensive investigation in both laboratory experiments (e.g. Schowalter, Van Atta & Lasheras Reference Schowalter, Van Atta and Lasheras1994; Holford & Linden Reference Holford and Linden1999; Jiang et al. Reference Jiang, Lefauve, Dalziel and Linden2022a ) and numerical simulations (e.g. Enstad et al. Reference Enstad, Nagaosa and Alendal2006; Watanabe et al. Reference Watanabe, Riley, Nagata, Matsuda and Onishi2019; Lloyd, Dorrell & Caulfield Reference Lloyd, Dorrell and Caulfield2022). These studies revealed a close relationship between mixing efficiency and the dynamics of structures within the buoyancy field. The mixing process involves motions spanning from large to small scales which impact the transport of momentum and density. Specifically, the entrainment across stratified interfaces caused by vortices influences turbulent diffusion, leading to irreversible mixing (Linden Reference Linden1973; Cotel & Breidenthal Reference Cotel and Breidenthal1996; Caulfield Reference Caulfield2021). In turbulent shear layers, vorticity dynamics plays a significant role in vortex stretching (Corrsin & Kistler Reference Corrsin and Kistler1955; Watanabe, Riley & Nagata Reference Watanabe, Riley and Nagata2016). Moreover, the strength of the background shear and stratification affect the turbulence triggering process, thereby altering the mixing efficiency (Pham, Sarkar & Winters Reference Pham, Sarkar and Winters2012). In a shear-driven flow, shearing motions are directly associated with overturns (Mashayek, Caulfield & Peltier Reference Mashayek, Caulfield and Peltier2017) or Holmboe instability (Lefauve et al. Reference Lefauve, Partridge, Zhou, Dalziel, Caulfield and Linden2018), profoundly influencing the mixing characteristics of stably stratified turbulence (Arobone & Sarkar Reference Arobone and Sarkar2010; Yi & Koseff Reference Yi and Koseff2023). It has been observed that mixing efficiency is closely linked to shear intensity (see, e.g. VanDine, Pham & Sarkar Reference VanDine, Pham and Sarkar2021), while stratified turbulent dynamics is influenced by time scales or length scales associated with shear and buoyancy (Smyth & Moum Reference Smyth and Moum2000). Since rotational and shearing motions are fundamental components of flow vorticity (Tian et al. Reference Tian, Gao, Dong and Liu2018), decomposing vorticity into vortical and shear components in a stratified fluid provides a framework for examining their local interactions with the buoyancy field (Jiang et al. Reference Jiang, Lefauve, Dalziel and Linden2022a , Reference Jiang, Atoufi, Zhu, Lefauve, Taylor, Dalziel and Linden2023). However, the effect of the time scales associated with rotational component of the vorticity (hereafter referred to as rotation) and stratification is less known, as is the local relationship between shear and rotation motion within a stratified fluid. The present study will address these questions.
In addition to shear or rotation-stratification interactions, the interactions between straining motions and stratification also play a crucial role in the small-scale dynamics of stratified turbulence. This is particularly evident at the turbulent/non-turbulent interface within stably stratified mixing layers, where the interaction between vorticity and strain results in enstrophy production (Watanabe et al. Reference Watanabe, Riley and Nagata2016; Neamtu-Halic et al. Reference Neamtu-Halic, Krug, Mollicone, van Reeuwijk, Haller and Holzner2020). In this paper, we also consider the effect of straining motion and the enstrophy transport, comparing its role to rotational and shearing motions.
Previously, in an exchange flow within a stratified inclined duct (SID) (Meyer & Linden Reference Meyer and Linden2014), we (Jiang et al. Reference Jiang, Lefauve, Dalziel and Linden2022a ) described the evolution and role of vortical structures within increasingly turbulent, stratified shear layers. However, the role of shear was not completely analysed and the work primarily focused on the instantaneous flow behaviours. In a further study, a geometric analysis of turbulent mixing in SID flows was introduced (Jiang et al. Reference Jiang, Atoufi, Zhu, Lefauve, Taylor, Dalziel and Linden2023), focused on the alignment of the density gradient in vorticity–shear vectors and the principal axes of the viscous dissipation tensor. However, this study concentrated solely on the orientation of the vectors associated with isopycnal distortion, without exploring the correlation between structural strength and stratification. The present work builds on prior analyses of vorticity decomposition into rigidly rotational components (referred to as rortex vectors) and non-rotational components (shear vectors), as outlined by Tian et al. (Reference Tian, Gao, Dong and Liu2018) and Liu et al. (Reference Liu, Gao, Tian and Dong2018). This study specifically investigates these two deformation types, statistically evaluating the classes of motion least and most influenced by stratification. Furthermore, the interaction between the strain rate and these motions is analysed to provide a comprehensive understanding of the local dynamics between fluid deformation and stratification. The objective is to deepen our understanding of the statistical relationships between the strength of various components of local fluid motion and stratification, thereby shedding light on the mechanisms driving turbulent mixing in stratified shear flows.
Conventionally, a local gradient Richardson number has been used to estimate the local effects of stratification. For an unidirectional flow,
$u(z)$
, a function of the vertical coordinate
$z,$
the local gradient Richardson number (Turner Reference Turner1973),
\begin{eqnarray} Ri_g \equiv \frac {N^2}{(\partial _z u)^2} = \frac {-g/\rho _0\, \partial _z\rho }{(\partial _z u)^2}, \end{eqnarray}
measures the ratio of the time scales associated with the buoyancy frequency and shear, where
$N {=\sqrt {-({g}/{\rho _0})({{\textrm d}\rho }/{{\textrm d}z}}})$
is the (Boussinesq) buoyancy frequency associated with density
${\rho }(z)$
, the gravitational acceleration
$g$
and the reference density
$\rho _0$
. A necessary condition for inviscid linear instability in parallel stratified shear flows is that the minimum value of
$Ri_g(z)$
be less than 1/4 (Miles Reference Miles1961; Howard Reference Howard1961). Since the Miles–Howard theorem applies only to parallel shear flows, it cannot be applied directly to more complicated flows with non-zero rotational and shear components. Instead, we define in § 2.4 a ‘family’ of gradient Richardson numbers that incorporates time scales associated with the local stratification and the rotational and shearing motions to reveal the role of the flow components in triggering turbulence and mixing in stratified flows, which is the central theme of the current study.
In § 2, we introduce the experimental datasets and define the decomposition between rotational and shearing motions and further propose the concept of a coherent gradient Richardson number. Then, in § 3, we illustrate instantaneous snapshots of flow structures in both wave and turbulent regimes, and discuss the connection between coherent structures and the local measures of fluid deformation. In § 4, we investigate the correlation between stratification and shear/rotation in the two flow regimes. In § 5.1, we examine the stratification–vorticity relationship in a region without any rigid-body rotations. Additionally, we explore the relationship between stratification and strain rate, as well as the enstrophy transport in §§ 5.2 and 5.3. Then, in § 6, we summarise the overall correlation between fluid deformation and density distortion, and discuss the stratification effect on the rotation-shear distortion and associated length scales. Finally, we conclude in § 7.
2. Methodology
2.1. Data set of SID experiments
In this study, we use near-instantaneous three-dimensional (3D) velocity and density fields obtained in the SID using simultaneous stereoscopic particle image velocimetry and laser-induced fluorescence with a continuously scanning laser sheet (Partridge, Lefauve & Dalziel Reference Partridge, Lefauve and Dalziel2019). The SID configuration involves a long duct (L = 1350 mm) with square cross-section (
$H$
= 45 mm), tilted at a small angle
$\theta$
(see figure 1) to facilitate a salt-stratified shear flow (with Prandtl number
$\it{Pr}\approx 700$
) between two large reservoirs with different densities. Meyer & Linden (Reference Meyer and Linden2014) revealed that the flow state within the duct at these large
$\it{Pr}$
undergoes a transition from pre-turbulent Holmboe waves (H regime) to intermittent turbulence, eventually reaching full turbulence (T regime). For a given duct geometry and
$\it{Pr}$
, this transition is controlled by only two non-dimensional control parameters, the inclination angle
$\theta$
and the Reynolds number
$Re$
(Lefauve & Linden Reference Lefauve and Linden2020).
We examine two specific cases from the dataset described by Lefauve & Linden (Reference Lefauve and Linden2022a
,Reference Lefauve and Linden
c
), representing the Holmboe (H) regime and the turbulent (T) regime. Lefauve & Linden (Reference Lefauve and Linden2022a
) labeled these cases as H1 and T3, respectively, and the properties of these cases are outlined in table 1. For simplicity, they will be referred to as H and T, respectively, hereafter. The analysis uses the ‘shear-layer’ rescaling approach introduced by Lefauve & Linden (Reference Lefauve and Linden2022a
) and here, only a brief summary is given. The spatial coordinates are non-dimensionalised by half the shear layer height
$h_u$
, measured from the locations of minimum to maximum streamwise velocity
$\langle u^d(z^d,y=0)\rangle _{x,t}$
, as shown in figure 1. Here,
$\langle \boldsymbol{\cdot} \rangle _{x,t}$
indicates the average over
$x$
and
$t$
. The velocities are normalised by the peak-to-peak streamwise velocity difference
$\delta _u$
based on the
$\langle u^d(z^d,y=0)\rangle _{x,t}$
profile. The density is normalised by half the fixed maximum density jump
$\Delta \rho /2$
. The Reynolds number
$Re$
, bulk Richardson number
$Ri_b$
and data resolution (
$\Delta x$
,
$\Delta y$
,
$\Delta z$
,
$\Delta t$
) are scaled based on the shear layer thickness
$h_u$
. For example,
$Re \equiv ({h_u} \, {\delta _u})/4\nu$
and
$Ri_b \equiv (g \, {\Delta \rho \,h_u})/(\delta _u^2\,\rho _0)$
. The
$x$
,
$y$
and
$z$
coordinates denote the streamwise, spanwise and wall-normal directions within the duct, respectively. In this rescaling, the edges of the shear layer at
$z=\pm 1$
correspond to streamwise velocities of
$u=\mp 1$
. In the H case, the ratio
$R_h$
of the non-dimensional thickness of the shear layer (i.e. 2) to that of the density interface (defined as spacing between the points at which the non-dimensional density is at
$\pm \textrm {tanh(1)}$
) is relatively high (
$R_h=8.9$
). In contrast, the ratio is much smaller in the T case (
$R_h=1.9$
) where turbulence and mixing increase the width of the density interface.
Figure 1. Schematic showing SID set-up and non-dimensional ‘shear-layer’ dataset (the grey zone). Note that
$z$
is normal to the duct wall and inclined to gravity at an angle
$\theta$
.
Table 1. Properties of the Holmboe (H) and turbulent (T) data sets primarily used in the paper, adapted from Lefauve & Linden (Reference Lefauve and Linden2022a )’s tables 1 and 3.
2.2. Decomposition based on rigid-body rotation
2.2.1. Criterion of rigid-body rotation
The analysis in this paper builds on the kinematic analysis of vortical motions and their interaction with density fields using the rortex–shear decomposition discussed by Tian et al. (Reference Tian, Gao, Dong and Liu2018) and Jiang et al. (Reference Jiang, Lefauve, Dalziel and Linden2022a ,Reference Jiang, Lefauve, Dalziel and Linden b ). This framework is briefly reviewed here.
The rortex, or rortex vector, represents the rigid-body rotational component of the velocity gradient tensor (i.e. the pure rotational component in deformations of fluid elements). Unlike traditional vorticity, which includes both rotational and shearing components, the rortex isolates the true solid body rotational motion of fluid elements. This decomposition enables a clearer understanding of coherent vortical structures.
-
(i) Direction: the rortex aligns with the axis about which rigid-body rotation of the fluid elements occurs. This direction corresponds to the unit eigenvector associated with the velocity gradient tensor’s rotational component.
-
(ii) Magnitude: the magnitude of the rortex, the rorticity, quantifies the strength of rigid-body rotation, distinct from shear deformation, thereby isolating rigid-body rotation and providing a more refined and accurate measure of vortical activity compared with traditional vorticity.
Mathematically, the rortex is defined by decomposing the vorticity vector
$\boldsymbol{\omega }$
into a rigid-body rotation vector
$\boldsymbol{R}$
(rortex) and a residual-shear vector
$\boldsymbol{S}$
:
\begin{equation} \boldsymbol{\omega } = \boldsymbol{R} + \boldsymbol{S}. \end{equation}
The rortex vector
$\boldsymbol{R}$
is derived using eigenvalue-based techniques, as described by Liu et al. (Reference Liu, Gao, Tian and Dong2018) and Tian et al. (Reference Tian, Gao, Dong and Liu2018).
When the velocity gradient tensor (VGT)
$\boldsymbol{\nabla }\boldsymbol{u}$
at a specific point in the flow has only real eigenvalues, fluid parcels in a small region surrounding that point stretch or compress along dominant real eigenvectors while, when the VGT has one real and a pair of complex conjugate eigenvalues, fluid parcels rotate around the axis given by the eigenvector corresponding to the real eigenvalue (Chong, Perry & Cantwell Reference Chong, Perry and Cantwell1990; Tian et al. Reference Tian, Gao, Dong and Liu2018). In incompressible flow, fluid parcels undergo straining or rotating motions for
$\varDelta \geqslant$
or < 0, respectively, where
\begin{eqnarray} \varDelta =\tfrac {1}{2}[\textrm {tr}{([\boldsymbol{\nabla }{\boldsymbol{u}}]^2)}]^3-27 \left [\det {(\boldsymbol{\nabla }{\boldsymbol{u}}})\right ]^2, \end{eqnarray}
which prescribes the form of solution (all real, or one real and two complex conjugate, respectively) to the characteristic equation
\begin{eqnarray} \lambda ^3-\tfrac {1}{2} [\textrm {tr}{([\boldsymbol{\nabla }{\boldsymbol{u}}]^2)} ] \lambda - \det {(\boldsymbol{\nabla }{\boldsymbol{u}}})=0, \end{eqnarray}
where
$\lambda$
denotes the eigenvalues of
$\boldsymbol{\nabla }\boldsymbol{u}$
. When
$\varDelta \geqslant 0$
, the flow undergoes irrotational straining or pure shearing motions since there is no unique axis for rotation. Local rigid-body rotation is possible only when
$\varDelta \lt 0$
, in which case, the unique rotation axis is aligned with the only real eigenvector (Tian et al. Reference Tian, Gao, Dong and Liu2018). We refer to regions of the flow where
$\varDelta \lt 0$
and rigid-body rotation exists as ‘rortical’, and in these regions, the VGT is denoted as
$ ({(\partial u_k)}/{(\partial x_j)})_r$
. Conversely, in regions where
$\varDelta \geqslant 0$
and rigid-body rotation does not exist, we refer to them as ‘non-rortical’, with the VGT there denoted as
$ ({(\partial u_k)}/{(\partial x_j)} )_n$
. Similarly, we refer to the vorticity vector in the rortical region as
$\boldsymbol \omega _r$
; and vice versa in the non-rortical region as
$\boldsymbol \omega _n$
.
2.2.2. Original definition of rorticity
In this section, we introduce the original definition of rorticity, i.e. the magnitude of the rortex vector. Where the flow has a component of rigid-body rotation, the vorticity is rortical and the vorticity
$\boldsymbol \omega =\boldsymbol \omega _r$
can be further decomposed into rigid-body rotation
$\boldsymbol R$
(the rortex vector) and the residual motions containing mainly shearing deformation
$\boldsymbol S$
(the shear vector), i.e.
$\boldsymbol \omega _r=\boldsymbol R + \boldsymbol S$
(Tian et al. Reference Tian, Gao, Dong and Liu2018). Here, the rortex vector is defined as
\begin{equation} \boldsymbol R = \left (1 - \sqrt {1 - \frac {4 \lambda _{ci}^2}{(\boldsymbol \omega \boldsymbol{\cdot} \boldsymbol r)^2}}\right ) (\boldsymbol \omega \boldsymbol{\cdot} \boldsymbol r)\boldsymbol r, \end{equation}
where
$\boldsymbol \omega =\boldsymbol{\nabla }\times \boldsymbol u$
is the vorticity field,
$\boldsymbol{r}$
is the local unit real eigenvector of
$\boldsymbol{\nabla }\boldsymbol{u}$
chosen such that
$\boldsymbol \omega \boldsymbol{\cdot} \boldsymbol{r}\geqslant 0$
and
$\lambda _{ci}$
is the imaginary part of its complex conjugate eigenvalues (Xu et al. Reference Xu, Gao, Deng, Liu and Liu2019). Hereafter, the magnitude
$R=|\boldsymbol{R}|$
of the rortex vector is called rorticity and the rotational structures identified by
$R$
are called rortices. For the non-rortical regions (
$\varDelta \geqslant 0$
), we note that
$\boldsymbol R= \boldsymbol 0$
to facilitate the analysis.
The vortex vector identification method proposed by Tian et al. (Reference Tian, Gao, Dong and Liu2018) and Xu et al. (Reference Xu, Gao, Deng, Liu and Liu2019) identifies regions in the flow with minimal angular velocity (or spin). This approach involves two consecutive rotations of the coordinate system. The first rotation transforms the coordinate system to make its
$\hat {z}$
axis aligned with the local rotation axis
$\boldsymbol{r}$
. The second coordinate transformation introduces a rotation angle around
$\boldsymbol{r}$
. As velocity gradients depend on the coordinate system, it is possible to find this angle such that the angular velocity of the flow is minimised (see Liu et al. Reference Liu, Gao, Tian and Dong2018 for further details). Although this minimisation captures rortical motions with the slowest rotation through the second (optimal) coordinate transformation, local interactions persist between the vortical motions and the residual shearing motions. As illustrated in figure 2, increasing the magnitude of the rortex vector, from solid red to dashed red lines in the sketch, leads to a decrease in the strength of the residual shear vector
$\boldsymbol{S}$
. Note that the orientation of the shear vector also depends on the magnitude of the rortex vector since the vorticity,
$\boldsymbol \omega$
, remains fixed.
Figure 2. (a) Schematic showing variation of the
$\boldsymbol{S}$
vector (direction and strength) with changing strength of the rortex vector
$\boldsymbol{R}$
. Since
$\boldsymbol{\omega }=\boldsymbol{S}+\boldsymbol{R}$
, an increase in the strength of the rortex from minimal (solid red line) to maximal strength (dashed red line) results in attenuation of the shear strength and the orientation of the residual shear vector (same line style as rortex vector but in blue). (b) Geometry of local
$\boldsymbol S^*$
,
$\boldsymbol R^*$
and
$\boldsymbol{\nabla }\rho$
for
$\phi =0$
based on Jiang et al. (Reference Jiang, Atoufi, Zhu, Lefauve, Taylor, Dalziel and Linden2023).
2.2.3. Extending the definition of rorticity
In this paper, we aim to identify rotational and shearing motions that are affected by stratification. Considering rorticity with locally minimal and maximal strengths allows us to differentiate motions affected by density stratification from those that are not. In this section, we extend the definition of rorticity. The minimal and maximal rorticity can be defined as
\begin{eqnarray} \boldsymbol{\mathring {R}}_\phi \equiv \boldsymbol{\mathring {R}}(\boldsymbol{u}, \phi ) = (1 - \phi ) \, (\boldsymbol{\omega } \boldsymbol{\cdot} \boldsymbol{r}) \boldsymbol{r}, \quad \phi = \left \{ \!\!\begin{array}{ll} f_m & \text{(minimal rorticity)} \\ 0 & \text{(maximal rorticity)} \end{array} \right . \end{eqnarray}
where
$f_m = \sqrt {1-({4\lambda _{ci}^2}/{(\boldsymbol{\omega} \boldsymbol{\cdot} \boldsymbol{r})^2})}$
and
$\phi$
is a parameter that regulates the two special magnitudes of rortical motions. The
${\boldsymbol{{\mathring {S}}}}_\phi =\boldsymbol \omega _r-{\boldsymbol{{\mathring {R}}}}_\phi$
thus gives the corresponding residual shear.
The first case with
$\phi = f_m$
refers to the minimal strength rortex (or slow-spin rortex)
${\boldsymbol{{\mathring {R}}}}_1\equiv \boldsymbol{R}$
, identical to the local rotation defined based on the minimum angular velocity (Liu et al. Reference Liu, Gao, Tian and Dong2018; Tian et al. Reference Tian, Gao, Dong and Liu2018) with strength
${\mathring {R}}_1=|{\boldsymbol{{\mathring {R}}}}_1|=R$
. The second case
$\phi =0$
, giving
${\boldsymbol{{\mathring {R}}}}_0\equiv \boldsymbol{R}^*$
, is a special rigid-body rotation that maximises the strength of the rorticity
${\mathring {R}}_0=|{\boldsymbol{{\mathring {R}}}}_0|=R^*$
. Note that the case when the rortex strength is large enough such that
$\boldsymbol{S}$
and
$\boldsymbol{R}$
approach orthogonality is of particular importance since the rortex becomes shear-free in this case, as is shown in figure 2. This paper emphasises the latter case, where
$\phi =0$
and a comparison with the minimal case (
$\phi =f_m$
) is given in Appendix B. As
$\boldsymbol{R}^*$
and
$\boldsymbol{S}^*$
are orthogonal, it is worth mentioning that the plane in which
$\boldsymbol{R}^*$
and
$\boldsymbol{S}^*$
are located is referred to as the ‘
$\boldsymbol{\hat {n}{-}\hat {r}}$
plane’ by Jiang et al. (Reference Jiang, Atoufi, Zhu, Lefauve, Taylor, Dalziel and Linden2023). This plane was found to be preferentially normal to the density gradient (as shown in figure 2
b), particularly in regions of high stratification.
We use
$\boldsymbol R^*$
and
$\boldsymbol S^*$
hereafter for
$\phi =0$
to indicate the locally shear-free rortex and the corresponding shear. We define the magnitudes
$S^*=|\boldsymbol{S}^*|$
and
$R^*=|\boldsymbol{R}^*|$
. Where there is no rigid-body rotation, the vorticity is non-rortical
$\boldsymbol \omega =\boldsymbol \omega _n$
and solely dependent on the shearing part
$\boldsymbol S$
, i.e.
$\boldsymbol \omega _n = \boldsymbol S$
.
2.3. Applications to stratified shear flows
The rortex and rorticity have been applied to study coherent vortical structures in stratified shear flows, such as those observed in the SID experiment by Jiang et al. (Reference Jiang, Lefauve, Dalziel and Linden2022a , Reference Jiang, Atoufi, Zhu, Lefauve, Taylor, Dalziel and Linden2023). These methods provide new insights into the interaction of vortices with density interfaces in increasingly turbulent flows where the evolution of coherent vortical structures, including hairpin-like vortices, was analysed across Holmboe, intermittent and turbulent regimes. Key insights include the following.
-
(i) Morphological insights: the rortex revealed ubiquitous hairpin-like vortices, similar to those observed in boundary-layer turbulence, and showed their geometry evolution in increasingly turbulent regimes.
-
(ii) Mixing mechanisms: hairpin vortices play a crucial role in turbulent mixing within stratified shear layers by entraining fluid both laterally and vertically. These vortices interact dynamically with density interfaces, lifting denser fluid into lighter regions and sweeping lighter fluid downward. This motion overturns the density interface and enhances stirring of fluid layers, promoting irreversible mixing of the density field. The use of the rortex-shear decomposition has provided deeper insight into these processes, allowing for the precise identification of vortical contributions to density field modifications and stratified turbulence dynamics (Jiang et al. Reference Jiang, Lefauve, Dalziel and Linden2022a , Reference Jiang, Atoufi, Zhu, Lefauve, Taylor, Dalziel and Linden2023; Riley Reference Riley2022).
However, when applied to stratified shear flows to identify structures responsible for density interface deformation, it becomes clear that rotation is not the only form of fluid deformation that can cause such distortions. The role of other types of fluid deformation, such as pure shear and straining motions, in processes like overturning and scouring, remains an open question. Furthermore, stratified flows exhibit a range of characteristic time scales, where some regions may experience faster rigid-body rotation than others. This raises an important question: how does the distortion of a density interface relate to the local rorticity, pure shear and straining motions? Addressing this question requires the identification of measures that can account for and disentangle the contributions of different types of fluid deformation to the observed density field dynamics which will be introduced next.
2.4. Fluid deformation versus buoyancy frequency
As discussed in § 1, to examine the interplay between coherent structures in the flow and the stratification, we need to define a parameter that characterises the strength of the interaction. The denominator of local gradient Richardson number (1.1) is the conventional definition of ‘shear’, which can be attributed to both rotational (
$\boldsymbol R$
) and shearing (
$\boldsymbol S$
) structures. In a similar spirit, we define a set of gradient Richardson numbers
$Ri_C$
, in terms of the ratios of the time scales associated with the local rotation and shear rates, and the time scale associated with the buoyancy frequency,
\begin{eqnarray} Ri_{C} = \dfrac {N^2}{{\mathscr{F}}^2}, \end{eqnarray}
where
$\mathscr{F}$
refers to the deformation rate due to local motions. Choosing
$\mathscr{F}=\partial u/\partial z$
gives the gradient Richardson number. The vorticity decomposition allows us to further analyse the relationship among stratification, residual shear and rigid-body rotation. Choosing
$\mathscr{F}= \mathring{S}$
and
$\mathscr{F} = \mathring{R}$
gives the following shear and rortex Richardson numbers, respectively:
\begin{eqnarray} Ri_{S} = \dfrac {N^2}{\mathring{S}^2}, \qquad Ri_{R} = \dfrac {N^2}{\mathring{R}^2}. \end{eqnarray}
When
$\phi =0$
(
$\mathscr{F}=S^*$
and
$\mathscr{F}=R^*$
), the Richardson numbers defined previously are denoted as
$Ri^*_{S}$
and
$Ri^*_{R}$
, respectively.
3. Structures in instantaneous snapshots
Before examining the correlation between flow structures and stratification, we present visualisations of the flow structures and density gradients in the Holmboe and turbulent cases. To visualise the contrasting characteristics of the maximal and minimal rortices within stratified shear layers, 3D iso-surfaces of rorticity are depicted in figures 3 and 4, representing the Holmboe (H) and turbulent (T) cases, respectively.
Figure 3. Instantaneous structures in the H case (
$t_n=30$
). (a) Instantaneous streamlines coloured with maximal shear (
$S$
in
$\phi =f_m$
, in purple), juxtaposed on the right with a
$y$
–
$z$
contour (at plane A) of vertical velocity
$w$
(superimposed with velocity vectors); (b) for
$\phi =f_m$
, isosurfaces of rorticity
$R=1.2R_{rms}$
(light transparent red) and
$R=1.8R_{rms}$
(non-transparent red), where
$R_{rms}$
indicates the root-mean-square value of
$R$
; the
$x{-}z$
slice shows the shear
$S$
, sharing the same colour legend as panel (a); (c) for
$\phi =0$
, isosurfaces of rortices
$R^* = 1.2R^*_{rms}$
and
$R^*=1.8R^*_{rms}$
, with
$x{-}z$
slice showing the shear
$S^*$
, sharing the same colour legend as panels (a) and (b); (d) vertical density gradient
$\partial \rho /\partial z$
. The
$y$
–
$z$
slices are at
$x=-22.7$
(plane A); the
$x$
–
$z$
slice is at
$y=0$
.
Figure 4. Instantaneous structures in the T case (
$t_n=40$
). (a–d) Same plot types as in figure 3(a–d). The
$y$
–
$z$
slice is at
$x=$
−12.6 (plane A); the
$x$
–
$z$
slice is at
$y=- 0.4$
. For better visualisation, the region
$z\lt - 0.5$
is not shown in panels (b) and (c).
3.1. Holmboe flow
Figure 3 shows the H case in which a strong density interface is embedded in the shear layer. The streamlines in panel (a) show a sequence of swirling flow motions (e.g. labelled I) coloured with shear (
$S$
in purple) that occurs when
$\phi =f_m$
. These swirling structures, with a small vertical-to-horizontal aspect ratio (sheet-like), are found in the high-shear region (dark purple), which shows that coherent, elongated vortices can form in the high-shear and strong-stratification environment. Between pairs of neighbouring elongated vortices, we find cross-interface ejection as shown by the vertical velocity (
$w$
) contour (see plane A) shown on the right-hand side of panel (a). Note that the vectors on plane A also show a strong spanwise movement at the top of the layer, which highlights the spanwise stirring motions discussed by Jiang et al. (Reference Jiang, Lefauve, Dalziel and Linden2022a
).
Panels (b) and (c) compare the structures of slow- and fast-spin rorticies when
$\phi =f_m$
(i.e.
$\boldsymbol{R}$
) and
$\phi =0$
(i.e.
$\boldsymbol{R}^*$
), respectively. The isosurfaces of rorticity are qualitatively different when
$\phi =f_m$
and
$\phi =0$
. The slow-spin rortices identified using
$\phi =f_m$
(panel b) are tube-like or bulb-like and typically occur above or below the interface. However, fast-spin rortices with
$\phi =0$
(panel c) are sheet-like (see, e.g. the structure labelled E in panel c), coincident with the elongated vortices in panel (a). This visualisation demonstrates that the coherent elongated vortices coincide with high shear using the
$\phi =f_m$
criterion of Tian et al. (Reference Tian, Gao, Dong and Liu2018) and Liu et al. (Reference Liu, Gao, Tian and Dong2018), while the same structures coincide with high rorticity when
$\phi =0$
. The vertical density gradient (panel d) shows that the density interface is mainly two-dimensional, with little interface distortion along
$y$
.
3.2. Turbulent flow
In figure 4, we illustrate the turbulent case by plotting the same quantities shown for the Holmboe case in figure 3. Panel (a) shows streamlines colour coded with the strength of the shear
$S$
(which corresponds to
$\phi =f_m$
, shown in purple). The streamlines show strong twists marked by L
$'$
and H
$'$
which significantly distort isopycnals. However, these streamline twists mainly appear in the middle region where the shear is relatively weak (light purple, see e.g. regions labelled L
$'$
).
The vertical velocity contour on Plane A (shown on the right-hand side of panel a) highlights the lift-up and sweep-down motions induced by the vortices. In the upper layer, the vortices on either side of the central plane (
$y=0$
) rotating in opposite directions cause the fluid in the middle to lift upward and the fluid at the flanks (between the vortex leg and sidewalls) to sweep downward. The spanwise movement is also apparent in the middle of the shear layer. These structures are similar to the hairpin-like vortices found by Jiang et al. (Reference Jiang, Lefauve, Dalziel and Linden2022a
) (their figure 14
c).
The instantaneous isosurfaces of slow- and fast-spin structures in the T case are compared in figures 4(b) and 4(c). The weak stratification at the centre is bounded by two density interfaces, one above and one below
$z=0$
(see panel d). Comparison of the rortices identified using the two criteria reveals the following.
-
(i) The tube-like rortical structures representing shear-free rotation in panel (c) are similar to the rortices identified by
$\phi =f_m$
shown in panel (b) (e.g. the hairpin-like rortices labelled K and L), but the heads of some hairpin rortices are not identified in panel (b) (e.g. the hairpin with the leg L). -
(ii) There are no clear sheet-like rortices observed for either
$\phi =0$
or
$\phi =f_m$
, in contrast to the Holmboe (H) case.
These observations imply that the
$\phi =0$
criterion is able to distinguish spanwise rortices, especially close to the density interface. Such structures are usually classified as spanwise shear if the local rotation strength is minimised, as in the case
$\phi =f_m$
(Gao & Liu Reference Gao and Liu2018).
As shown by the contours of density gradient in panel (d), the statically unstable regions (where
$\partial \rho / \partial z \gt 0$
) are in the middle of the shear layer and where the interface rolls up and overturns (e.g. the region labelled P). The overturning of the interfaces is closely related to neighbouring rortices that interact with the density interface by lifting inner (pre-mixed) denser fluid away from the upper interface and entraining outer lighter fluid into the shear layer (Jiang et al. Reference Jiang, Lefauve, Dalziel and Linden2022a
).
3.3. Comparison of gradient Richardson numbers in T regime
The contours of buoyancy frequency (
$N^2$
, figure 5
a) and shear (
$S^2$
for
$\phi =f_m$
, figure 5
b) show that strong stratification and high shear follow roughly similar patterns near the two density interfaces on either side of the shear layer and slightly differ in the middle layer. The contours of rortices (red lines superimposed in figure 5
b) appear mainly within the middle layer and in the unstable regions (negative
$N^2$
) consistent with the discussion regarding figure 4.
Figure 5. Snapshots of the turbulent stratified shear layer (T) in an
$x$
–
$z$
plane (
$- 1\leqslant z\leqslant 1$
): (a) localised
$N^2={- (g/\rho _0)\partial _z{\rho }}$
; (b) localised squared shear
$S^2$
(contour) and rigid-body rotation
$R$
(red contour lines for
$R=0.5$
and 1.5); (c) localised
$Ri_g$
(black contour lines for
$Ri_g = Ri_b$
); (d) localised
$Ri_S$
, (black contour lines for
$Ri_S = Ri_b$
).
The bulk Richardson number for the T case is
$Ri_b \approx 0.15$
as reported in table 1, which provides a point of comparison with the gradient Richardson number,
$Ri_g$
. The contours of
$Ri_g$
(figure 5
c) show that both regions where
$Ri_g \gtrsim Ri_b$
(red) and overturned regions where
$Ri_g \lt 0$
(blue) appear in the middle layer. Some regions where
$Ri_g \gt Ri_b$
have
$Ri_S \lt Ri_b$
, suggesting that the shear there may be more significant (and potentially destabilising) than captured purely by
$Ri_g$
. This is readily apparent in figure 5, where regions of red in panel (c) are blue in panel (d). This is especially visible in the middle of the layer (see region labelled by arrows), highlighting the effect of rortical motions in the ‘shear’ of
$\partial u/\partial z$
. This qualitative analysis motivates us to examine the correlation between the components of the local fluid motion (e.g.
$R$
or
$S$
) and buoyancy in the next section.
A comparison of the vertical profiles of the
$x$
-,
$y$
- and
$t$
-averages of the set of the gradient Richardson numbers
$Ri_C(z)$
for the T regime is shown in figure 6. This figure confirms that, on average,
$Ri_S\lt Ri_g$
in the middle layer, as observed in the local instantaneous contours. The figure also illuminates the distinction between
$\phi =0$
and
$\phi =f_m$
. Due to slow-spin rortices (
$\phi =f_m$
),
$Ri_R$
is significantly larger than the other
$Ri_C$
measures. However, according to the fast-spin criterion
$\phi =0$
, the average profile of
$Ri^*_{R}$
is comparable to both
$Ri_g$
and
$Ri^*_{S}$
. However, in the centre of the mixing layer (
$N^2 \lt 0.1$
), the gradient Richardson number associated to fast-spin rortices is smaller than that corresponding to the shear, indicating a stronger averaged rortex strength in the middle. Moreover, the conventional profile
$ Ri_g$
, with its shear contaminated by rotational motions, yields smaller values at the interface (and larger values in the middle) compared with
$ Ri^*_{S}$
, thereby underestimating (or overestimating) the time scales of the local shear relative to the stratification. We further examine the localised correlation between stratification, shear and rortices in the following sections.
Figure 6. Comparison of the vertical profile of averaged gradient Richardson numbers for the T regime.
4. Correlation between stratification and shear/rortex
In § 3, we discussed the instantaneous structures for the H and T regimes, and distinguished the topological difference between rortex/shear families based on
$\phi =0$
and
$\phi =f_m$
(i.e. fast and slow spinning rortices) in these two regimes. This section concentrates on the statistical correlation between fluid motions and stratification. We focus on the correlation between stratification and slow-shear/fast-rortex when
$\phi =0$
, while their correlation when
$\phi =f_m$
is presented in Appendix B for completeness.
4.1. Correlation between stratification and shear
Figure 7(a) shows the joint probability density function (p.d.f.) of
$N^2$
and
$S^{*2}$
using all space–time data points. In the H flow, the maximum of the p.d.f. occurs for weak stratification and weak shear. In the T case, the stratification is overall weaker and the shear strength is almost twice that of the H case as shown in figure 7(b). Moreover, the joint p.d.f. is more symmetric about
$N^2=0$
in the T case since regions with
$N^2\lt 0$
are more common than in the H case.
Figure 7. Joint probability distribution function (p.d.f.) of stratification
$N^2$
and squared shear
$S^{*2}$
when
$\phi =0$
: (a) Holmboe regime; (b) turbulent regime. The colour denotes the joint p.d.f. value. The red, green, blue and black dashed lines are for
$Ri^*_{S}$
= 1, 0.25, 0.1 and 0, respectively. The black contour lines represent the joint p.d.f. with a value of 0.2, 0.04, 0.008 and 0.0016 for panel (a) and 0.04 for panel (b).
The slope of the straight dashed lines in figure 7 indicate specific values of
$Ri^*_{S}$
(
$Ri^*_{S}=1$
,
$0.25$
,
$0.1$
and
$0$
for the red, green, blue and black lines, respectively). To examine the impact of shear strength on the coherent gradient Richardson number, we use a p.d.f., conditioned on the magnitude of
$S^*$
, similar to the statistics conditioned on enstrophy presented by Buaria et al. (Reference Buaria, Bodenschatz and Pumir2020a
). Figure 8 shows time-averaged p.d.f.s for
$Ri^*_{S}$
conditioned on
$S^*/S_{rms}^*\geqslant k_{th}$
, where
$S_{rms}^*$
is the root mean square (r.m.s.) of
$S^*$
and
$k_{th}$
is a threshold value below which the data are discarded. The vertical locations of the
$x\hbox{-}$
and
$t\hbox{-}$
averaged shear strength under the condition settings are shown in figure 19 in Appendix A. The average temporal length is 232 advective time units (A.T.U) for the H case, corresponding to approximately 18 periods of the Holmboe wave, and 552 A.T.U for the T case, which is more than 50 times the turbulent dissipation time scale. The fraction of the domain satisfying the condition for the
$k_{th}$
values from 0.5 to 3 as indicated in the figure, is 0.677, 0.318, 0.122, 0.040, 0.011 and 0.003, respectively. Though strong shear constitutes only a small portion of the flow, it remains important as it indicates extreme events within the flow.
Figure 8. Time-averaged conditional p.d.f. of
$Ri^*_{S}$
: (a) Holmboe regime for
$\phi =0$
; (b) turbulent regime for
$\phi =0$
. Legends are
$k_{th}$
for shear.
Figure 8(a) shows that in the H regime the peak of the conditionally sampled p.d.f. shifts when the threshold value
$k_{th}$
is varied. In general, the peaks of the conditional p.d.f. occur at higher values of
$Ri^*_{S}$
for larger values of
$k_{th}$
. This shift in these peaks means that the time scale of buoyancy is reduced relative to the time scale of shear in regions with strong shear. However, this shift is non-monotonic such that the location of the peak
$Ri^*_{S}$
first increases with
$k_{th}$
(see peaks labelled D–F), before decreasing to a smaller
$Ri^*_{S}$
(
$\approx 0.35$
, labelled G). The shift of the location of the peak from D to F as shear strength increases corresponds to the transition from the blue dashed line to the green dashed line in figure 7(a). This indicates that stratification increases more rapidly than shear as we filter out the motions with smaller
$S^*$
, moving the focus of the conditional average closer to the centre of the shear layer where the sharp density interface is located. Conversely, the leftward shift of the peak location from F to G suggests a saturation or a relative decrease in stratification in comparison to the increasing shear when focusing on the regions of extremely high shear (
$k_{th}\gt 2$
). These regions, comprising 4 % of the domain, are where extreme shearing events occur. This shift typically occurs where the shear-free rortices are found near the density interface, as illustrated in figure 3(c). It is worth noting that this peak shift is a robust characteristic of all Holmboe wave regimes although we only showed it for one H case.
Figure 8(b) shows conditional p.d.f.s in the turbulent case, with peaks occurring at
$Ri^*_{S}$
$\approx 0.04$
at low-shear regions and
$Ri^*_{S}$
$\approx 0.02$
at extremely high-shear regions. For larger
$k_{th}$
, the variance in
$Ri^*_{S}$
decreases and the probability associated with the mode of the p.d.f. increases (see the dashed arrow). The time scale of shearing motion is approximately five times smaller than the buoyancy time scale (
$N/S^*\approx 0.2$
), though the ratio varies slightly with the shear strength. This suggests that the length scale related to high shear in turbulence is likely smaller than the Ozmidov scale, the scale at which the eddy turnover time scale is similar to buoyancy time scale (Atoufi, Scott & Waite Reference Atoufi, Scott and Waite2020). We will further discuss the length scales in § 6.3. The conditional p.d.f.s for other turbulent cases where we have comparable measurements look similar to the T case and so are not shown here for brevity.
So far, we have focused on the statistical correlation between
$S^{*2}$
and
$N^2$
. To help understand the value of this correlation, the statistical correlation between
$S^2$
and
$N^2$
is briefly discussed here. Figures 20(a) and 20(b) in Appendix B.1 presents the joint p.d.f. between
$N^2$
and
$S^2$
in both H and T cases. The main difference for the H case by comparing figures 7(a) and 20(a) lies in the region labelled P in figure 7 (the region with
$Ri^*_{S} \gt 1$
). The correlation between strong
$N^2$
and weak residual shearing motions identified by
$\phi =f_m$
is smaller compared to the correlation when residual shearing motions are identified by
$\phi =0$
. We observe a similar pattern for the T case by comparing figures 7(b
) and 20(b) where the joint p.d.f computed from
$\phi =0$
shows that the
$N^2$
–
$S^{*2}$
correlation is stronger than the
$N^2$
–
$S^2$
correlation when stratification is strong. Therefore, the shear-free rortex (identified by
$\phi =0$
) unmasks residual shearing motion that can coexist with stratification where stratification is strong (i.e. at the interface). Additionally, in the case of
$\phi =f_m$
, as seen in figures 20(c) and 20(d) in Appendix B.1, the conditional p.d.f.s of
$Ri_{S}$
in both H and T cases closely resemble those of
$Ri^*_{S}$
shown in figure 8. This similarity suggests that the variation trend of the ratio remains consistent regardless of the definition used for shear strength.
4.2. Correlation between stratification and rortex
Here, we examine the statistical correlation between rorticity and stratification. Figure 9(a) shows the joint p.d.f. between
$N^2$
and
$R^{*2}$
in the Holmboe regime. It shows a similar probability distribution to that of
$N^2$
–
$S^{*2}$
in figure 7(a), indicating that in the case of
$\phi =0$
, stratification has a similar relationship to rortical and shearing motion. Additionally, the magnitude of the shear-free rortex
$R^*$
is comparable to the residual shear
$S^*$
, indicating they are equally critical in distorting isopycnals. In the case of
$\phi =f_m$
, only weak rortices form on the density interface where the stratification is strong (
$N^2\gt 2$
), and strong rortices mainly appear in the regions with weak stratification, as shown in figure 21(a) in Appendix B.2. This observation is consistent with the snapshot presented in figure 3. It demonstrates the superiority of the criteria
$\phi =0$
over
$\phi =f_m$
for diagnosing the flow state in flows with strong stratification as it reveals rortices that can coexist with strong stratification. In the T case, the joint probability distributions of
$N^2$
with
$S^*$
and
$R^*$
are similar. The general trend is that strong vortical motions, particularly those identified using the
$\phi = 0$
criterion, prevail in weakly stratified regions but are also observed at interfaces with high stratification, albeit with a relatively low frequency of co-occurrence (as also shown in figure 5
b). As the orientations of
$\boldsymbol{R}$
and
$\boldsymbol{R}^*$
are the same and they differ only in magnitude as we change
$\phi$
from
$0$
to
$f_m$
(with the ratio
$R^*/R$
varying but typically approximately 5, as shown in figure 21
b), it is of no surprise that either
$R$
or
$R^*$
are good at identifying vortical motions in both the H and T regimes.
Figure 9. Joint p.d.f. of stratification
$N^2$
and
$R^{*2}$
for
$\phi =0$
: (a) Holmboe regime; (b) turbulent regime. The colour denotes the joint p.d.f. value. The red, green, blue and black dashed lines are for
$Ri^*_{R}$
= 1, 0.25, 0.1 and 0, respectively. The black contour lines represent the joint p.d.f. with a value of 0.04, 0.008 and 0.0016 for panel (a) and 0.04 for panel (b).
As expected from the joint p.d.f between
$N^2$
and
$R^{*2}$
, the p.d.f.s of
$Ri^*_{R}$
conditioned by
$R^*/R^*_{rms}\geqslant k_{th}$
(see figure 19
b in Appendix B for the vertical distribution of the normalised
$ R^*$
) is similar to figure 8 and thus are not shown. Instead, here in figure 10, we show a comparison between the p.d.f.s of
$Ri^*_{R}$
and
$Ri^*_{S}$
for
$k_{th} = 0.5$
. We first observe that
$Ri^*_{R}$
conditioned on
$R^*/R^*_{rms}$
(red solid line) exhibits a distribution similar to that conditioned on
$S^*/S^*_{rms}$
(red dotted line). This observation indicates that, although
$Ri^*_{R}$
is defined with respect to
$R^*$
, the dependence of its probability density distribution on the intensity of
$R^*$
appears analogous to its dependence on the intensity of
$S^*$
. A comparison between the
$Ri^*_R$
and
$Ri^*_S$
(dotted lines, under the same condition) further illustrates that the gradient Richardson number based on
$R^*$
is statistically smaller than that associated with
$S^*$
, particularly in turbulent regimes. This observation agrees with the behaviour of the averaged
$Ri_C(z)$
profiles shown in figure 6, where
$Ri^*_{R} \lt Ri^*_{S}$
near the density interface. Since high stratification is not always aligned with high shear, we further investigate the distribution of shear and rotational motions, along with their corresponding gradient Richardson numbers, under varying stratification intensities.
Figure 10. Comparison of p.d.f.s between
$Ri^*_{R}$
and
$Ri^*_{S}$
in (a) the Holmboe regime and (b) the turbulent regime, conditioned by shear and rorticity strengths (
$k_{th}=0.5$
).
4.3.
Stratification effect on the
$Ri_C$
distribution
The stratification strength is quantified by the normalised buoyancy frequency,
$ N^2/N^2_{{rms}}$
, where
$ N^2_{{rms}}$
denotes the r.m.s. value of the non-negative
$ N^2$
. The streamwise and temporally averaged profiles along
$ z$
are shown in figure 19(c) in Appendix B, which highlights the approximate vertical location of different stratification regions. Similar to the shear conditions determined using the threshold
$ k_{th}$
, only regions with
$ N^2/N^2_{{rms}} \geqslant k_{th}$
are considered for statistical analysis, with
$ k_{th}$
varying from 0.5 to 1.5.
Figures 11 and 12 show the stratification effect on the p.d.f.s of normalised shear and rorticity (each normalised by its own mean and r.m.s. values), as well as the probability distibution of corresponding gradient Richardson numbers for the H and T cases, respectively. In the H case, the distributions are distinctly non-Gaussian. The normalised shear exhibits a wider distribution than the rorticity, but both display a bimodal distribution with two local maxima on the left and right sides of their mean. This bimodal feature will be further discussed in § 6.2. With increasing stratification, both shear and rorticity exhibit a reduced probability of extreme values occurring (e.g. the tails tend to be closer to the mean).
Figure 11. Conditioned p.d.f. of (a) normalised
$R^*$
and
$S^*$
, and (b)
$Ri^*_{R}$
and
$Ri^*_{S}$
for the H case. Red lines represent
$R^*$
-related quantities and blue lines correspond to
$S^*$
-related quantities. The line colour transitions from light to dark, indicating increasing
$k_{th}$
values from 0.25 to 1.5, in increments of 0.25 for
$N^2/N^2_{{rms}}$
. The
$+$
symbols mark the standard Gaussian.
Figure 12. Conditioned p.d.f. of (a) normalised
$R^*$
and
$S^*$
, and (b)
$Ri^*_{R}$
and
$Ri^*_{S}$
for the T case. Red lines represent
$R^*$
-related quantities and blue lines correspond to
$S^*$
-related quantities. The line colour transitions from light to dark, indicating increasing
$k_{th}$
values from 0.25 to 1.5, in increments of 0.25 for
$N^2/N^2_{{rms}}$
.
Stratification significantly influences the distributions of
$Ri^*_{R}$
and
$Ri^*_{S}$
, particularly the left tail, as shown in figure 11(b). As expected, stronger stratification (closer to the interface) corresponds to higher values of
$ Ri^*_{R}$
and
$ Ri^*_{S}$
, with the peak mode shifting from 0.2 to 0.5 as
$ k_{th}$
increases from 0.5 to 1.5. However, the relationship
$ Ri^*_{R} \lt Ri^*_{S}$
remains unchanged with increasing
$ k_{th}$
. This implies that the time scale of rotational motions is statistically smaller than that of shearing motions. Although stronger stratification causes a rightward shift in the distribution, signalling a stabilising effect on the localised flow region, the right tail reveals a consistent power-law scaling for both
$ Ri^*_{R}$
and
$ Ri^*_{S}$
(with the exponent being approximately −1.57).
For turbulence, the normalised
$ R^*$
and
$ S^*$
exhibit skewed p.d.f.s in figure 12, with the peak p.d.f. mode smaller than the mean. Much stronger extreme events are observed compared with the wave regime, and these are more likely to occur in regions of low stratification due to reduced suppression from the density field. The right tails appear to follow an exponential distribution, highlighting the intermittent nature of turbulence (Jiménez Reference Jiménez1998; de Bruyn Kops Reference de Bruyn Kops2015).
The distribution of gradient Richardson numbers in turbulence resembles that in the wave regime, except that the peak mode is smaller (
${\simeq}0.1$
). Once again, the right tails exhibit power-law scaling, with the exponent slightly smaller than in the wave regime (
${\simeq}{-}1.74$
). The left tails indicate the unstable regions where the time scale of flow motions is much smaller than that associated with the buoyancy frequency and corresponds to the large-scale flow structures observed in the right tails of figure 12(a).
5. Effect of non-rortical vorticity and strain on the stratification
As indicated in § 2.2.1, certain areas within the flow exhibit non-rortical behaviour, indicating the absence of rigid-body rotation (yielding
$R=R^*=0$
). While the preceding sections have focused on the rortical regions characterised by
$R\gt 0$
, the current section aims to investigate the correlation between stratification and non-rortical vorticity. A complement to the vorticity, the strain, which represents the symmetric aspect of the VGT, is an important component of the flow deformation. This section also examines the statistical correlation between stratification and symmetric straining motions.
5.1. Correlation between stratification and non-rortical vorticity
In our previous investigations (Jiang et al. Reference Jiang, Lefauve, Dalziel and Linden2022a
, Reference Jiang, Atoufi, Zhu, Lefauve, Taylor, Dalziel and Linden2023), the non-rortical regions (
$R=0$
) were excluded from consideration. Consequently, the specific influence of these regions on stratification remains unexplored. According to Jiang et al. (Reference Jiang, Lefauve, Dalziel and Linden2022a
), rortical flow regions (
$\varDelta \lt 0$
and
$R\neq 0$
) dominate in both flows, but in almost a quarter of all points (in space–time for both H and T cases), the flow is non-rortical (
$\varDelta \geqslant 0$
and
$R=0$
).
A comparison study reveals that, for
$\varDelta \lt 0$
, the joint p.d.f. of
$N^2 {-} \omega _n^2$
closely resembles that of
$N^2 {-} {S^*}^2$
. Similarly, the conditional p.d.f. of
${N^2}/{\omega _n^2}$
is nearly identical to that of
$Ri^*_{S}$
. This finding suggests
$\omega _n$
has a similar role to the residual shearing motions in the distortion of stable stratification. This observation also suggests that local rortex and shear appear to be independent, such that the presence of rortices does not influence the stratification–shear correlation. However, this does not imply that rortices are unimportant; rather, it can be attributed to the time scale of local rortical motions being shorter than those of both shear and the buoyancy frequency. We will further discuss the correlation between shear and rortex, along with their associated length scales, in §§ 6.2 and 6.3.
5.2. Correlation between stratification and strain
In our analysis of the interplay between stratification and fluid deformation so far, we have considered the antisymmetric part of the VGT (
$\unicode{x1D63D}$
, i.e. the vorticity tensor consisting of rigid-body rotation and asymmetric shearing motion). To complete our analysis, we now investigate the interplay between strain (
$\unicode{x1D63C}$
, i.e. the symmetric part of the VGT) and stratification.
By definition,
$ (\lVert {\unicode{x1D63C}}\rVert ^2+\lVert {\unicode{x1D63D}}\rVert ^2 )/\lVert {\boldsymbol{\nabla }{u}}\rVert ^2=1$
. The p.d.f. of
$\lVert {\unicode{x1D63C}}\rVert ^2/\lVert {\boldsymbol{\nabla }{u}}\rVert ^2$
shown in figure 13(a) is concentrated around 0.5, which indicates that
$\boldsymbol{\nabla }{u}$
is equipartitioned between
$\unicode{x1D63C}$
and
$\unicode{x1D63D}$
, regardless of flow regime.
Figure 13. (a) Probability distribution function of
$\lVert {\unicode{x1D63C}}\rVert ^2/\lVert {\boldsymbol{\nabla }{u}}\rVert ^2$
and (b,c) joint p.d.f. of
$N^2$
and
$\lVert {\unicode{x1D63C}}\rVert ^2$
for (b) H case and (c) T case. The red, green and blue dashed lines are for
$Ri^*_{R}$
= 1, 0.25, 0.1, respectively.
The joint p.d.f. between buoyancy frequency and the norm of the strain rate tensor
$\lVert {\unicode{x1D63C}}\rVert ^2$
is shown in figures 13(b) and 13(c) for Holmboe flow (H) and turbulent flow (T), respectively. We observe that the straining motions are weaker than both the shearing and vortical motions. Furthermore, regions characterised by very weak strain are unlikely to exhibit high stratification, which differs from
$R^*$
and
$S^*$
. This indicates that straining motions are less likely to coexist with strong stratification compared with motions arising from the antisymmetric part of VGT, unless the strain reaches a certain magnitude. A physical explanation can be provided as follows: a straining motion in an
$x$
–
$z$
plane requires
$({\partial w}/{\partial x}) ({\partial u}/{\partial z}) \gt 0$
(Tian et al. Reference Tian, Gao, Dong and Liu2018), which implies a vertical lift of particles in SID flow with a shearing background, where
$ {\partial u}/{\partial z} \lt 0$
. However, strong stratification suppresses the upward movement of heavier fluids, reducing the likelihood of straining motions within the density interface. It is worth noting that the joint p.d.f.s between
$N^2$
and
$\lVert {\unicode{x1D63C}}\rVert ^2$
, conditioned by rortical and non-rortical regions, resemble the overall joint p.d.f. shown here in figure 13, indicating that the relationship between buoyancy frequency and the strain rate is not dependent on the strength of rortices.
5.3. Enstrophy evolution in the stratified shear layers
Nonlinear interaction between vorticity and the strain rate tensor plays a significant role in the formation of extreme events (Buaria et al. Reference Buaria, Pumir and Bodenschatz2020b
). We have analysed the statistical correlations between stratification and flow motions embedded in both the vorticity tensor and the strain rate tensor. Here, we investigate the contributions of rortex and shear to their interaction with the strain rate tensor, which drives enstrophy (
$\omega ^2/2$
) production. The vertical shape of the enstrophy profile is crucial for understanding the maintenance of coherent structures, as it arises from the equilibrium among inertial, viscous and buoyancy forces (Neamtu-Halic et al. Reference Neamtu-Halic, Mollicone, van Reeuwijk and Holzner2021). We compare the vertical profiles of the terms in the dimensionless enstrophy transport equation under the Boussinesq assumption
\begin{equation} \begin{aligned} \frac {D\omega ^2/2}{Dt} =\omega _i\omega _jA_{\it{ij}} + Re^{-1}\boldsymbol{\nabla }\boldsymbol{\cdot} (\boldsymbol{\nabla }\omega ^2/2) -2\ Re^{-1}\boldsymbol{\nabla }\omega _i:\boldsymbol{\nabla }\omega _i + \epsilon _{ijk} \omega _i \frac {\partial g'_k}{\partial x_j}, \end{aligned} \end{equation}
where
$St = \omega _i \omega _j A_{\it{ij}}$
is the enstrophy production through vortex stretching and tilting,
$D_f = {Re^{-1}} \nabla \boldsymbol{\cdot} ( \nabla \omega ^2 / 2 )$
is the viscous diffusion term,
$D_p = -2 {Re^{-1}} \nabla \omega _i:\nabla \omega _i$
is the viscous dissipation term and
$B_a = \epsilon _{ijk} \omega _i ({(\partial g'_k)}/{(\partial x_j)})$
is the baroclinic torque, where
$\boldsymbol{g}' = \rho Ri_b (\sin {\theta }, 0, {-}\cos {\theta })$
is the buoyancy in the coordinates attached to the inclined duct (figure 1). Based on the decomposition of the vorticity into orthogonal components
$\boldsymbol{R}^*$
and
$\boldsymbol{S}^*$
, the vortex stretching and tilting term
$St$
consists of
$St^*_{R} = R^*_i R^*_j A_{\it{ij}}$
,
$St^*_{S} = S^*_i S^*_j A_{\it{ij}}$
, with a small residual term
$St-(St^*_{R}+St^*_{S})$
, as we see shortly. Therefore, the interaction of rigid-body rotation and shear with respect to the strain rate tensor can be measured independently.
We first discuss the H case. Figure 14(a) shows that the inequalities
$\overline {St^*_{S}} \gt \overline {St^*_{R}} \gt 0$
hold at the density interface, indicating that
$S^*$
contributes more to stretching than
$R^*$
at the interface. Moreover, the baroclinic term
$B_a$
is significantly larger than the production terms at the interface due to strong stratification, implying that baroclinic torque has a non-negligible contribution to enstrophy evolution in the wave regime.
Figure 14. Vertical profiles (averaged in
$x, y$
and
$t$
) of the enstrophy transport terms in (5.1) for the (a) H case and (b) T case. Labels on the top axis correspond to
$ \overline {\rho 'w'}$
.
Both viscous terms
$\overline {D_f}$
and
$\overline {D_p}$
are negative, on average, at the centre of the interface. The diffusion term,
$\overline {D_f}$
, becomes positive on either side of the interface, approximately at the same location where the dissipation term,
$D_p$
, displays two local maxima (in magnitude) at the flanks (indicated by the arrows). This sign change in
$\overline {D_f}$
occurs in accordance with the change in sign of
$\overline {\rho ' w'}$
, the vertical mass flux, as seen in figure 14(a). In the conventional eddy diffusivity model for mass flux,
$ \overline {\rho 'w'} \approx \kappa _\rho N^2 \rho _0/g$
, the places where
$ \overline {\rho 'w'} \lt 0$
while
$N^2 \gt 0$
requires negative eddy diffusivity
$\kappa _\rho \lt 0$
. Consequently, this implies upgradient mass fluxes driven by eddies, a phenomenon referred to as anti-diffusion, which involves scouring-type mixing that preserves the sharpness of the density interface in the H case (Caulfield & Peltier Reference Caulfield and Peltier2000; Salehipour, Caulfield & Peltier Reference Salehipour, Caulfield and Peltier2016). As
$\overline {St^*_{S}} \gt \overline {St^*_{R}} \gt 0$
in the centre of the shear layer where
$\overline {\rho 'w'} \lt 0$
and
$\overline {N^2} \gt 0$
, we may conclude that scouring of the density interface is predominated by the shearing of fluid parcels. The rigid-body rotation of fluid parcels contributes only slightly to the anti-diffusion and scouring mixing process in the H case.
For the T case shown in figure 14(b), the enstrophy production by rotational and shearing motions is comparable at the centre of the shear layer,
$St^*_{S} \approx St^*_{R} \gt 0$
on average, where stratification is reduced and stirring is enhanced due to the interaction between vortical structures and the density field. Near the two density interfaces above and below the mixed layer at
$z \approx \pm 0.5$
, the enstrophy is produced more by the shearing motions than rigid-body motions similar to the H case (i.e.
$St^*_{S} \gt St^*_{R} \gt 0$
). However, unlike the H case, the viscous diffusion and the baroclinic generation of enstrophy are negligible compared with the other terms in the T case, particularly at the centre of the mixing layer. Although not explicitly presented here, we find that the ratios of the baroclinic terms to the stretching terms, represented as
$(\overline {B_a/St^*_{R}})^{2/3}$
and
$(\overline {B_a/St^*_{S}})^{2/3}$
, exhibit distributions closely following that of
$Ri^*_{R}$
and
$Ri^*_{S}$
, respectively, shown in figure 6. This observed similarity indicates that
$Ri_C$
serves as a useful parameter also for identifying coherent structures that significantly contribute to enstrophy production in relation to the baroclinic torque.
Note that the residual production term is considerably small in both H and T cases suggesting that the
$R^*$
and
$S^*$
structures are almost statistically independent, as will be further quantified in § 6.2.
6. Discussion
6.1. Comparison between different motions
We first summarise the key findings from § 4 to § 5. To do so, we draw together the joint p.d.f in these two sections in figure 15 for simplicity of comparing the correlations between stratification and the different components of the velocity gradient tensor. The density interface is approximated as the region above the green dashed lines where
$N^2\gt 1$
for H and
$N^2\gt 0.2$
for T. Regions
$\unicode{x2460}$
–
$\unicode{x2462}$
are within the interface and
$\unicode{x2463}$
–
$\unicode{x2464}$
refer to neighbouring regions on either side of the interface.
Figure 15. Comparison of joint p.d.f. for different motions with respect to stratification in the (a) H case and (b) T case. The four quadrants show the joint p.d.f. of
$N^2$
with one of
$S^{*2}$
,
$\lVert {A}^{2}\rVert$
,
$\omega _n^{2}$
and
$R^{*2}$
, as indicated in labels in panel (a). The quadrants are not labelled in panel (b), but correspond to the same joint p.d.f.s as in panel (a). The plots are for
$N^2\geqslant 0$
. Light blue and red dashed lines indicate
$Ri_C=0.1$
and
$1$
, respectively.
$\unicode{x2460}$
to
$\unicode{x2464}$
denote the regions within each quadrant divided by nearby dashed lines, applied to all quarters. The region within the black contour lines represents the joint p.d.f. with a value of 0.02.
In the H case and within the interface where the stratification is strong (beyond the green dashed lines), the symmetric part of the VGT
$\unicode{x1D63C}$
seems to have the highest possibility of coexisting with
$N^2$
compared with the other motions. However, the joint p.d.f. for the strain is large only in the range
$0.25 \leqslant N^2/\lVert {\unicode{x1D63C}}\rVert ^2 \leqslant 1$
. The light-red region appears to follow the trend of the red dashed line (
$N^2/\lVert {\unicode{x1D63C}}\rVert ^2 = 1$
), indicating that the time scale associated with straining motions in the H case is more comparable to that associated with the local buoyancy frequency than the other motions. Based on the correlation comparison, one reason behind the relatively small enstrophy production in the wave regime (figure 14
a) is that high strain is less likely to coexist with high shear and high rorticity, as shown in region
$\unicode{x2461}$
, leading to weak interaction under strong stratification. Away from the interface (regions
$\unicode{x2463}$
–
$\unicode{x2464}$
), quadrants 1 and 4 look similar, except that the contour of the joint p.d.f for
$R^*$
is slightly closer to the horizontal axis compared with the shear (see the black contour lines where the p.d.f is 0.02). This observation is consistent with figure 10. The main difference between the non-rortical vorticity
$\omega _n$
and the shear
$S^*$
or rorticity
$R^*$
is that much fewer points satisfy
$N^2/\omega _n^2 \gt 1$
(more blue within the
$\unicode{x2460}$
region), implying that flow motions in these non-rortical regions are less likely to form large-scale structures with frequencies lower than the buoyancy frequency. Rortical regions are thus more strongly associated with coherent structures that disrupt the density field.
In the T case, straining motions are generally smaller than other motions, consistent with the H case. A significant distinction from the H case is observed in region
$\unicode{x2462}$
for all the compared motions, which represents mixing behaviours within the interfaces (where all choices of
$Ri_C$
are less than
$0.1$
). Away from the interfaces (e.g.
$\unicode{x2463}$
, within the middle mixing layer), rotational motions are more likely to cause locally unstable regions than shear, which agrees with the observation in figures 6 and 12. Regions
$\unicode{x2462}$
and
$\unicode{x2463}$
represent locations where extreme events and large-scale coherent structures occur. It is worth noting that these extreme events occur in both the interfaces and the premixed middle layer, although extreme rotational motions are more likely to arise away from the interface (i.e. longer tails in quadrant 4 within region
$\unicode{x2463}$
). This observation aligns with the left tail of the p.d.f.s of
$Ri^*_{R}$
in figure 12(b), where weakly stratified regions facilitate strong rortices, leading to smaller
$Ri^*_{R}$
. The intermittency of turbulence is primarily driven by motions within the left tails of the p.d.f. of
$Ri^*_{R}$
in stratified turbulence. As the stratification strength increases, the rightward shift of the p.d.f.s for both
$Ri^*_{R}$
and
$Ri^*_{S}$
slows and appears to saturate at a critical value, with their right tails collapsing to follow a power-law distribution (figure 12
b). This behaviour implies that localised flow motions become self-organised under relatively high stratification (Salehipour et al. Reference Salehipour, Caulfield and Peltier2016).
6.2.
Stratification effect on the
$R^*$
–
$S^*$
distortion
We observe a bimodal distribution of both normalised
$R^*$
and
$S^*$
for the H case in § 4.3, which motivates us to further investigate their local interactions under different stratified conditions. The correlation coefficient between the motions, defined by
$\gamma = {(\text{Cov}(R^{*2}, S^{*2}))}/{(\sigma _{R^*} \sigma _{S^*})}$
, where
$\text{Cov}$
represents the covariance, and
$\sigma _{R^*}$
and
$\sigma _{S^*}$
are the standard deviations of
$R^{*2}$
and
$S^{*2}$
, respectively, is 0.17 for the H case and 0.06 for the T case. This suggests that local rortex and shear are weakly correlated and statistically independent in both the H and T cases. Here, we examine the stratification effect on the local correlation between the two motions.
In figure 16, the joint p.d.f. of
$R^{*2}$
and
$S^{*2}$
is conditioned at different stratifion strengths for both the H and T cases. The conditioning is based on stratification ratio
$N^2/N^2_{rms}$
(see Appendix A for its mean profile), where
$N^2_{rms}$
is the r.m.s. value of all non-negative
$N^2$
. In the weak stratification region (
$0\lt N^2/N^2_{rms}\lt 0.5$
), the two motions are most likely to be weak, with a low correlation coefficient between
$R^{*2}$
and
$S^{*2}$
(approximately 0.10). However, the joint p.d.f.s of the H case show two peaks for larger values of
$N^2/N^2_{rms}$
. This can be seen in the regions labelled by A and B in figure 16(b). The bimodal distribution indicates a discernible preference for
$R^{*2}$
and
$S^{*2}$
in high-stratification regions. For the T case, the preference of
$R^*$
mode appears earlier than the
$S^*$
mode, see the region labelled C. The two-peak distribution also appears for the T case at higher stratification, although not shown here. The relationship between the two motions is also found to intensify for both the cases as the stratification increases. It seems that stratification acts as a modulator between shear and rotation, not only in their direction, as found by Jiang et al. (Reference Jiang, Atoufi, Zhu, Lefauve, Taylor, Dalziel and Linden2023), where the
$\boldsymbol{R}^{*} \!\boldsymbol{-} \!\boldsymbol{S}^{*}$
plane was shown to be more normal to the density gradient at higher stratification, but also in their relative strength. However, the mechanism by which stratification allocates energy to different motions across various spatial regions is not yet clear.
Figure 16. Joint p.d.f. of
$R^{*2}$
and
$S^{*2}$
in regions of different stratification for the (a,b) H case and (c,d) T case at (a,c)
$0 \lt N^2/N^2_{rms} \lt 0.5$
and (b,d)
$1 \lt N^2/N^2_{rms} \lt 1.5$
.
To further link these correlations to physical space and study the dominance of the two distortions (i.e. A and B) in the shear layer, we define a distortion parameter
$D = {(R^{*2} - S^{*2})}/{(R^{*2} + S^{*2})}$
. The value of
$D$
is positive if
$R^*$
is dominant, approaching 1 when
$R^{*2}\gg S^{*2}$
. Conversely, if
$S^*$
is dominant,
$D$
tends to be negative. For a non-rortical region (
$R^*=0$
),
$D$
equals −1. The physical distribution of
$D$
in rortical regions provides insights into the relative strengths of shear and rigid-body rotation within the shear layer.
In figure 17, we present vertical profile of averaged distortion parameter for both the H and T cases in rortical regions (
$R^* \neq 0$
), i.e.
$\overline {D}= {(\langle R^{*2}\rangle - \langle S^{*2}\rangle)}/{(\langle R^{*2}\rangle + \langle S^{*2}\rangle)}$
with
$\langle \boldsymbol{\cdot} \rangle$
representing
$\langle \boldsymbol{\cdot} \rangle _{x,y,t,R^*\neq 0}$
. The vertical profiles based on
$N^2/N^2_{rms}$
are also superimposed with dotted lines to indicate the vertical variation of stratification. Figure 17(a) shows that
$\overline {D}$
is negative within the density interface and positive away from it, demonstrating a bimodal distribution similar to that in figure 16(b). This indicates that the shear-dominated zone A (see figure 16
b) primarily appears in the density interface for the H case and shear is more relevant to the enstrophy dissipation, as shown in figure 14(a). However, rortices are generally more intense than shear within the turbulent shear layer, as shown in figure 17(b), where
$\overline {D}$
is mainly positive along the
$z$
direction. This further supports the finding that the time scales of vortical motions are statistically shorter than those of shearing motions as discussed in § 4. Therefore, the larger contribution of enstrophy production by shear in figure 14(b) may be due to the stronger strain rate correlated with the localised shear, causing greater interaction. It is interesting to note that a small portion at the bottom of the turbulent shear layer shows
$\overline {D}$
< 0. This might be due to the higher density gradient at the lower interface (see the dotted line of
$N^2/N^2_{rms}$
), making the rortices less intense and thus more similar to the H case. This asymmetry is caused by the non-periodicity of the flow along
$x$
and the asymmetrical location of the measuring volume with respect to the duct length (Lefauve & Linden Reference Lefauve and Linden2022a
).
Figure 17. Vertical profile of averaged distortion parameter
$\overline {D}$
for the (a) H case and (b) T case. The dotted line indicates averaged
$\langle N^2/N^2_{rms}\rangle _{x,y,t}$
.
6.3. Length scales
Thus far, our comparisons have primarily been based on a time scale perspective, focusing on the relative time scales associated with rotational and shearing motions versus stratification, characterised by the buoyancy frequency. This approach led to the definition of coherent Richardson numbers in § 2.4. A similar analysis can be conducted from a length scale perspective by comparing the length scales of these motions to those associated with stratification.
Stratification predominantly influences scales larger than the Ozmidov scale,
$L_o = ({\overline {\varepsilon }/}{N^3} )^{1/2}$
, which represents the size of overturning eddies before being suppressed by stratification. Here,
$\varepsilon$
denotes the turbulent dissipation rate, defined as
$ \varepsilon = {2 s'_{\it{ij}}s'_{\it{ij}}/}{{Re}},$
where
$s'_{\it{ij}}$
is the strain-rate tensor based on fluctuating velocity, expressed as
$ 2s'_{\it{ij}} = {\partial u'_j/\partial x_i + \partial u'_i/\partial x_j}.$
For consistency, we adopt the same averaging process as Lefauve & Linden (Reference Lefauve and Linden2022b
). Figure 18 illustrates the profiles of the Ozmidov scale (green lines) for both the H and T cases. The smallest Ozmidov scales are located approximately at the density interfaces:
$|z| \approx 0.2$
for the H case and
$|z| \approx 0.6$
for the T case. These locations coincide with the points where
$N^2$
reaches its local maximum.
Figure 18. Vertical profile of averaged length scales
$L$
for the (a) H case and (b) T case.
Similarly, mean shear predominantly acts to deform the largest eddies, converting kinetic energy into dissipation, while smaller eddies remain unaffected. The smallest scale at which eddies are significantly deformed by the mean shear is estimated using the Corrsin scale, defined as
$L_c = ( \overline {\varepsilon } / (\partial \overline {u}/ \partial z )^3 )^{1/2}$
(Smyth & Moum Reference Smyth and Moum2000). Figure 18 reveals that the Corrsin scale is consistently smaller than the Ozmidov scale within the shear layer, with their ratio being approximately 0.5 for the H case and 0.2 for the T case. Notably, this ratio is larger at the density interface. The smaller
$L_c/L_o$
in the T case suggests that, under weak stratification such as in the middle of the turbulent shear layer, with wider scale separation, a broader range of scales influenced by mean shear exists. These scales remain unaffected by stratification leading to enhanced overturnings, as shown in figures 4 and 5.
Analogous to the definitions of the Corrsin scale
$L_c$
, we define two additional length scales: the rortex length scale,
$L_r$
, and the residual-shear length scale,
$L_s$
, given by
\begin{eqnarray} L_r = \left ( \frac {\overline {\epsilon }}{\overline {R^*}^3} \right )^{1/2} \!, \quad L_s = \left ( \frac {\overline {\epsilon }}{\overline {S^*}^3} \right )^{1/2} \!. \end{eqnarray}
Unlike
$L_c$
, which accounts only for the vertical shear,
$L_r$
and
$L_s$
include all directions of shear and rotation. These scales provide a more general framework for evaluating the relative contributions of rortex- and shear-induced motions to turbulent dissipation. For example, for scales larger than
$L_r$
and smaller than
$L_s$
, assuming
$L_r \lt L_s$
, turbulent dissipation is expected to be largely due to rigid-body motions. Conversely, for scales smaller than
$L_o$
and larger than
$L_s$
, turbulent dissipation arises from residual shearing or non-rortical motions.
Figure 18(a) shows that, in the H case, the
$L_r \approx L_s \approx L_o$
at the interface, and
$L_r \lt L_s \lt L_o$
almost everywherelse for both the T and H cases. Figure 18 shows remarkable agreement in the data for the following relations:
\begin{equation} Ri^*_{R} \approx \left (\frac {L_r}{L_o}\right )^{4/3} \approx \left (\frac {B_a}{St^*_{R}}\right )^{2/3}, \quad Ri^*_{S} \approx \left (\frac {L_s}{L_o}\right )^{4/3} \approx \left (\frac {B_a}{St^*_{S}}\right )^{2/3}, \end{equation}
for both the H and T cases. The
$Ri^*_{R} \approx (B_a/St^*_{R})^{2/3}$
and
$Ri^*_{S} \approx (B_a/St^*_{S})^{2/3}$
components of this relation were discussed in § 5.3. Consequently, (6.2) implies
\begin{eqnarray} L_r \approx \left (\frac {B_a}{St^*_{R}}\right )^{1/2} L_o, \quad L_s \approx \left (\frac {B_a}{St^*_{S}}\right )^{1/2} L_o. \end{eqnarray}
Combining
$L_r \lt L_s \lt L_o$
, as observed in most locations in figure 18, we can infer that rortex scales, which may be an order of magnitude smaller than the Ozmidov scale, are also affected by stratification through baroclinic vortex generation as described by (6.3). Therefore, Ozmidov scales are not necessarily the smallest scales influenced by stratification in stratified shear layers.
To wrap up this section, we can conclude that the coherent gradient Richardson number is not only useful for comparing the time scales of rotational and residual-shearing motions to the buoyancy frequency, but it is also a valuable measure of the relative size of these scales compared with the Ozmidov scale when their length scales are defined using
$L_r$
and
$L_s$
. Furthermore, this definition of length scales and the concept of the coherent Richardson number reveal that rortex structures, particularly in the centre of the stratified shear layer, are an order of magnitude smaller than the Ozmidov scale. However, they are still influenced by stratification through baroclinic generation of vorticity. This argument revisits the concept of Ozmidov scales being the smallest scales affected by stratification, which is rooted in the classical theory of homogeneous stratified turbulence, particularly in the context of buoyancy-driven exchange flows at high Prandtl numbers.
7. Conclusions
In this study, we have examined the correlation between stratification and flow motions in a stratified inclined duct, using experimental datasets consisting of Holmboe wave and turbulent regimes, two key regimes in stratified shear layers. Employing the rigid-body rotation criterion to determine the existence of a unique local rotation axis, we partitioned the flow into two regions: rortical (
$\boldsymbol{R} \ne \boldsymbol{0}$
) and non-rortical (
$\boldsymbol{R} = \boldsymbol{0}$
). Vorticity is decomposed into a rortical motion and a residual shearing motion.
First, we examined the minimal strength rortex (
$R$
when
$\phi =f_m$
) and the shear-free rortex (
$R^*$
when
$\phi =0$
) in § 3, with a specific emphasis on the latter. We investigated the instantaneous rortices and the corresponding residual shear in both the Holmboe wave and turbulent regimes. Our findings indicate that employing the shear-free criteria allows for the identification of sheet-like rortices or spanwise rortices that are constrained by intense stratification.
We found that
$\partial u/\partial z$
alone does not adequately capture the influence of shear, as it incorporates both shear and rotational motions. Therefore, we proposed investigating the interplay between buoyancy and flow motions by using a family of gradient Richardson numbers
$Ri_C$
, exploring
$Ri^*_{S}$
and
$Ri^*_{R}$
as well as the traditional
$Ri_g$
, to establish a more comprehensive measure. The various choices of
$Ri_C$
allowed us to analyse the statistical correlation between fluid deformations and stratification using their joint probability distributions.
In the Holmboe case, the conditional probability distribution of
$Ri^*_{S}$
reveals that higher shear does not always correspond to an unstable region. We highlighted a non-monotonic shift of the peak of the p.d.f. as the shear strength increases, which is a characteristic of Holmboe flow. In contrast, the correlation between stratification and shear in a turbulent regime exhibits a more consistent behaviour. Stronger shear overcomes stratification, resulting in a locally more unstable state for the flow (§ 4.1). However, rortices possess significantly shorter time scales compared with the local buoyancy frequency and shear, making them more destructive than shear to stratification (§ 4.2). An investigation into the effect of stratification on
$Ri_C$
reveals that it has a significant influence on the left tail of the
$Ri_C$
p.d.f., while the right tail remains nearly unchanged, following a power-law scaling. Extreme events associated with the left tail are observed to occur more frequently in regions with low stratification (§ 4.3).
Next, we found that non-rotational vorticity and irrotational strain play a role similar to residual shearing motions in the distortion of stable stratification. However, straining motions are generally weaker than both shearing and rortical motions, and are less likely to have time scales larger than those associated with the buoyancy frequency. We further investigated the vertical enstrophy evolution in both wave and turbulence cases, decomposing the stretching term into rortex–strain and shear–strain interactions. Our analysis reveals that shear contributes more significantly to stretching than rortex, particularly near the density interface. In the wave region, the scouring of the density interface is predominantly driven by shearing motions, while baroclinic torque provides a non-negligible contribution to enstrophy transport (§ 5.3).
Lastly, we found that shear and rigid-body rotation are usually independent of each other, but their interconnectedness increases in high-stratification regions. In the turbulent shear layer, rortices more strongly distort the density interface than shear. A comparison of length scales along the shear layers highlights the coherent gradient Richardson number as a valuable measure of the relative sizes of rortex and shear motions compared with the Ozmidov scale. Notably, despite being an order of magnitude smaller than the Ozmidov scale, rortex structures are still influenced by stratification through baroclinic torque, underscoring the intricate interplay between stratification and small-scale turbulence at sub-Ozmidov scales in high-Prandtl-number stratified exchange flows.
In conclusion, we have demonstrated that the interplay between flow motions and stratification, through their statistical correlation, provides valuable insights into the characteristics of different flow motions in stratified shear layers. While this study focuses on SID flows, the methodology we introduced – using the coherent gradient Richardson number and examining length scales of various motions – can serve as a framework for exploring a wide range of stratified flows.
Acknowledgements
The authors would like to thank Dr Adrien Lefauve for valuable discussions and comments on the manuscript.
Funding
The authors acknowledge support from the European Research Council (ERC) under the European Union Horizon 2020 research and innovation Grant No. 742480 ‘Stratified Turbulence And Mixing Processes’ (STAMP). For the purpose of open access, the authors have applied a Creative Commons Attribution (CC BY) licence to any Author Accepted Manuscript version arising from this submission.
Declaration of interests
The authors report no conflict of interest
Appendix A. Condition based on shear, rotation and stratification strength
Conditional statistics provide a powerful tool to examine flow correlations within specific spatial regions (Jiang et al. Reference Jiang, Lefauve, Dalziel and Linden2022a
; Buaria et al. Reference Buaria, Pumir and Bodenschatz2020b
). In this study, these statistics are evaluated based on the conditions of normalised shear, rotation and stratification strengths, enabling a detailed investigation of their interplay and influence on turbulent mixing dynamics. The streamwise- and spanwise-averaged profile of the normalised shear strength
$\langle S^*/S^*_{rms}\rangle _{x,y}$
for
$\phi =0$
is shown in figure 19(a). The shear strength is normalised by
$S_{rms}^*$
, r.m.s. of
$S^*$
, and
$k_{th}$
is a threshold value below which the data are discarded. To condition the statistics for different shear strengths, we use
$k_{th} = [0.5, 1, 1.5, 2, 2.5, 3]$
for all data points. Note that regions with
$k_{th} \gt 2$
correspond to extremely high shear, with only a few points (< 1 %) meeting this condition.
Figure 19. Streamwise- and spanwise-averaged profiles of (a) normalised shear strength
$\langle S^*/S^*_{rms} \rangle _{x,y}$
, (b) normalised rorticity strength
$\langle r^*/r^*_{rms} \rangle _{x,y}$
and (c) normalised buoyancy strength
$\langle N^2/S^2_{rms} \rangle _{x,y}$
for the H and T cases at
$\phi = 0$
. Solid lines represent the T case, while dashed lines correspond to the H case. Light and transparent lines indicate profiles without streamwise and time averaging at the midplane of the duct.
Similarly, the conditions for rorticity and stratification are also determined based on their respective threshold values,
$k_{th}$
. The average profiles of the normalised rorticity and buoyancy frequency are presented in figures 19(b) and 19(c), respectively. Note that in figure 19, the background grey and light blue lines represent the non-averaged profiles in the midplane of the duct. Additionally, we observe that the variation of
$ R^*$
is larger than that of
$ S^*$
and
$ N^2$
.
Appendix B. Correlation between stratification and shear/rortex in rortical regions when
$\boldsymbol{\phi} = \boldsymbol{f}_\boldsymbol{m}$
B.1.
Correlation between stratification and shear when
$\phi =f_m$
To compare the stratification–shear interplay with that of strong rortices (
$\phi =0$
) in § 4, we present both joint p.d.f.s and conditional p.d.f.s for the H and T cases in figure 20 for
$\phi =f_m$
. The joint p.d.f.s for both cases (figures 20
a and 20
b) exhibit a distribution similar to that shown in figure 7, except in the weak shear region where
$Ri_S\gt 1$
(labelled as P in figure 7). This suggests that the presence of weak shear in highly stratified regions, such as in the density interface, is unlikely.
Figure 20. Probability distribution of stratification and shear in the (a,c) Holmboe regime and (b,d) turbulent regime when
$\phi =f_m$
. (a,b) Joint p.d.f. of
$N^2$
versus
$S^2$
, (c,d) conditional p.d.f. of
$Ri_S=N^2/S^2$
. The colour in panels (a,b) denotes the p.d.f. value. The red, green, blue and black dashed lines in panels (a,b) are for
$Ri_S$
= 1, 0.25 0.1 and 0, respectively. Legends in panels (c,d) are
$k_{th}$
for conditional shear strength. Note that
$k_{th}=2.5$
is not shown in panel (d) due to too few points satisfying this condition.
The conditional p.d.f.s depicted in figures 20(c) and 20(d) closely resemble the distribution observed for
$\phi =0$
in figure 8. For instance, we note the shifting of the peak location (
$\textrm{D}\rightarrow \textrm{E}\rightarrow \textrm{F}\rightarrow \textrm{G}$
) in the H case, as well as the tendency of the p.d.f. mode to increase with an increase in the threshold,
$k_{th}$
, in the T case.
B.2.
Correlation between stratification and rortex when
$\phi =f_m$
The stratification–rotation interplay for
$\phi =f_m$
critera is shown in figure 21 for both the H and T cases. For the H case, figure 21(a) shows that strong rortices appear at weak stratification (region H), but weak rorticies coexist with weak to strong stratification (region G). Most of the fluid is in the latter case (with
$Ri_R\gt 1$
). This distribution suggests that it is highly unlikely to have a co-occurrence of strong rortices and strong stratification in the H case. Figure 21(b) shows the joint p.d.f.s in the T case, which is similar to the H case, except that the distribution is narrower as the stratification gets weaker in the turbulent regime and the rorticity gets stronger.
Figure 21. Probability distribution of (a,b)
$N^2$
versus
$R^2$
and (c,d)
$Ri_R$
for (a,c) Holmboe regime and (b,d) turbulent regime when
$\phi =f_m$
. The red, green and blue dashed lines in panels (a,b) are for
$Ri_R$
= 1, 0.25 and 0.1, respectively. Legend in panels (c,d) is
$k_{th}$
for rorticity.
Figures 21(c) and 21(d) show the p.d.f.s of
$Ri_R$
conditioned by
$R/R_{rms}\geqslant k_{th}$
for the H and T cases, respectively. Figure 21(c) shows that an increase of rorticity in the H case will increase the p.d.f. mode of the small
$Ri_R$
, but it does not change the peak location (
$\approx 0.025$
). This small peak value in the Holmboe wave regime indicates that rortical motions have statistically much smaller time scales compared with the local buoyancy frequency.
Figure 21(d) shows the corresponding p.d.f.s in the turbulent regime, with a similar trend, but much smaller peak location (close to zero), indicating turbulent rortices are more destructive than rortices in the H case. A comparison between
$Ri_R$
and
$Ri_S$
(figure 20
d) reveals that rortices have smaller time scales than shear, which is consistent with the criteria
$\phi =0$
. The peaks of
$Ri_R$
move slightly to the left as the
$k_{th}$
increases, indicating stronger rorticity appears at the region with weaker stratification, which agrees with the observation of joint p.d.f.s of figure 21(b).
Appendix C. Angle between rortex vector and shear vector when
$\boldsymbol{\phi} = \boldsymbol{f}_\boldsymbol{m}$
The angle
$\beta$
between the shear vector
$\boldsymbol{S}$
and the rortex vector
$\boldsymbol{R}$
in the T case is illustrated in figure 22, conditioned in regions with varying shear strengths. Generally, these vectors are not perpendicular, with their average angle measuring
$50^\circ$
. As the shear strength increases (from the blue to red lines), the angle
$\beta$
tends to decrease, indicating the shear and rortex vectors tend to be more aligned in high shear regions.
Figure 22. Conditional p.d.f. of the angle between the rortex vector
$\boldsymbol R$
and the shear vector
$\boldsymbol S$
for the T case, conditioned on the shear strength
$S$
. The shear strength increases from the minimal to the maximal, as indicated by the blue to red lines (trend denoted by dashed arrows). The position of the green line represents the averaged angle without any conditions.
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