3 Results

We first discuss the results for $Sc=700$, and figure 2 shows the spatial distribution of kinetic and potential energy at three different times ($t=4,20$ and $40$). At the earliest time ($t=4$, figure 2a,b), the buoyancy time is still small ($t_{N}=N^{\ast }t^{\ast }/(2\unicode[STIX]{x03C0})=0.64({<}1)$. cf. figure 1a), and the buoyancy effect would be generally insignificant. The Ozmidov scale ($L_{O}=1/k_{O}=0.26$) is smaller than the horizontal and vertical integral scales ($L_{H}=1.4$ and $L_{V}=0.77$), but is still larger than the Kolmogorov scale ($L_{K}=1/k_{K}=0.11$), and the eddies with the Kolmogorov scale would not be strongly affected by stratification. Here, the horizontal and vertical integral scales are defined by

(3.1a,b)$$\begin{eqnarray}L_{H}=\frac{\displaystyle \unicode[STIX]{x03C0}\int \text{d}k_{y}\int \text{d}k_{z}|\hat{u} (0,k_{y},k_{z})|^{2}}{\overline{u^{2}}}\quad \text{and}\quad L_{V}=\frac{\displaystyle \unicode[STIX]{x03C0}\int \text{d}k_{x}\int \text{d}k_{y}|{\hat{w}}(k_{x},k_{y},0)|^{2}}{\overline{w^{2}}},\end{eqnarray}$$

(e.g. Bradshaw Reference Bradshaw1971), where $\boldsymbol{k}=(k_{x},k_{y},k_{z})$ is the wavevector, hat ($\hat{~}$) denotes the Fourier component and overlines denote the spatial average.

Since the Batchelor scale for a scalar of high-Schmidt-number ($Sc=700$) is much smaller than the Kolmogorov scale ($L_{B}=L_{K}/Sc^{1/2}=0.004$), density has fluctuations of much smaller scales than the velocity fluctuations. It has actually a thin sheet-like structure with a thickness approximately equal to the Batchelor scale (figure 2b), as observed in isotropic turbulence for a passive scalar with mean gradient (e.g. Brethouwer, Hunt & Nieuwstadt Reference Brethouwer, Hunt and Nieuwstadt2003).

Later at $t=20$ (figure 2c,d), the Ozmidov scale $L_{O}(=\,0.045)$ becomes much smaller than the Kolmogorov scale $L_{K}(=0.25)$, and the smallest-scale motion at $L_{K}$ would be significantly affected by stratification. The kinetic-energy distribution at this time shows large-scale flat structures with the horizontal and vertical integral scales of velocity given by $L_{H}(=2.2)$ and $L_{V}(=1.2)$. On the other hand, the potential-energy distribution shows large-scale clouds (figure 2d) composed of small-scale horizontal streaks, as can be identified in the enlarged view (figure 2e).

At $t=40$ (figure 2f), most of the small-scale fluctuations of density have already dissipated, leaving only the large-scale flat structures but with smaller-scale horizontal wrinkles. Similar structures have been observed at $Sc=70$ (Okino & Hanazaki Reference Okino and Hanazaki2019) since a vertical scale smaller than the Kolmogorov scale is generated by a vertically sheared horizontal flow which appears in the final period of decay.

Figure 2.

Spatial distributions of the kinetic and potential energy for $Sc=700$ at $t=4,20$ and $40$. The isosurfaces of (a) $u_{i}^{2}/2=3KE$ ($t=4$), (b) $\unicode[STIX]{x1D70C}^{\prime 2}/(2Fr_{0}^{2})=8PE$ ($t=4$), (c) $u_{i}^{2}/2=3KE$ ($t=20$), (d) $\unicode[STIX]{x1D70C}^{\prime 2}/(2Fr_{0}^{2})=8PE$ ($t=20$) and (f) $\unicode[STIX]{x1D70C}^{\prime 2}/(2Fr_{0}^{2})=6PE$ ($t=40$), where $KE$ and $PE$ are the spatially averaged kinetic and potential energy. (e) The red cube in (d) with side $\unicode[STIX]{x03C0}$, which is four-times enlarged. The horizontal and vertical sizes of a yellow box at the origin $(0,0,0)$ in each figure indicate the horizontal and vertical integral scales.

To investigate the large- and small-scale structures of density fluctuations, we decompose the spectral components into low- and high-wavenumber components relative to the Kolmogorov wavenumber, i.e.

(3.2a,b)$$\begin{eqnarray}\unicode[STIX]{x1D70C}_{L}^{\prime }(\boldsymbol{x})=\mathop{\sum }_{|\boldsymbol{k}|\leqslant k_{K}}\hat{\unicode[STIX]{x1D70C}^{\prime }}(\boldsymbol{k})\exp (\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{x})\quad \text{and}\quad \unicode[STIX]{x1D70C}_{H}^{\prime }(\boldsymbol{x})=\mathop{\sum }_{|\boldsymbol{k}|>k_{K}}\hat{\unicode[STIX]{x1D70C}^{\prime }}(\boldsymbol{k})\exp (\text{i}\boldsymbol{k}\boldsymbol{\cdot }\boldsymbol{x}),\end{eqnarray}$$

where

(3.3)$$\begin{eqnarray}\unicode[STIX]{x1D70C}^{\prime }(\boldsymbol{x})=\unicode[STIX]{x1D70C}_{L}^{\prime }(\boldsymbol{x})+\unicode[STIX]{x1D70C}_{H}^{\prime }(\boldsymbol{x}).\end{eqnarray}$$

In figure 3(a,b), we show the distributions of potential energy at $t=20$, composed only of low or high wavenumbers. The low-wavenumber components constitute round structures (figure 3a), while the high-wavenumber components constitute long horizontal streaks, whose cross-section is vertically thin (figure 3b,c). We note that neither the low-wavenumber components (figure 3a) nor the high-wavenumber components (figure 3b,c) have a noticeable structure of the Kolmogorov scale ($L_{K}=0.25$), which is indicated by the side of a pink box. The absence of potential energy near the Kolmogorov scale will be demonstrated later, using premultiplied energy spectra (cf. figures 6d and 11b).

Figure 3.

Spatial distributions of potential energy for $Sc=700$ at $t=20$, composed only of low ($|\boldsymbol{k}|\leqslant k_{K}$) or high ($|\boldsymbol{k}|>k_{K}$) wavenumbers. Isosurfaces of (a) $\unicode[STIX]{x1D70C}_{L}^{\prime 2}/(2Fr_{0}^{2})=3PE$ and (b) $\unicode[STIX]{x1D70C}_{H}^{\prime 2}/(2Fr_{0}^{2})=3PE$. (c) The red cube in (b) with side $\unicode[STIX]{x03C0}$, which is four-times enlarged. The sum of (a,b) does not recover the local distribution of $\unicode[STIX]{x1D70C}^{\prime 2}/(2Fr_{0}^{2})$ like figure 2(d), since $\unicode[STIX]{x1D70C}^{\prime 2}=(\unicode[STIX]{x1D70C}_{L}^{\prime }+\unicode[STIX]{x1D70C}_{H}^{\prime })^{2}=\unicode[STIX]{x1D70C}_{L}^{\prime 2}+\unicode[STIX]{x1D70C}_{H}^{\prime 2}+2\unicode[STIX]{x1D70C}_{L}^{\prime }\unicode[STIX]{x1D70C}_{H}^{\prime }$, and contribution from $2\unicode[STIX]{x1D70C}_{L}^{\prime }\unicode[STIX]{x1D70C}_{H}^{\prime }/(2Fr_{0}^{2})$ is lacking. However, $\unicode[STIX]{x1D70C}_{L}^{\prime }\unicode[STIX]{x1D70C}_{H}^{\prime }$ does not contribute to $PE$ since $\overline{\unicode[STIX]{x1D70C}_{L}^{\prime }\unicode[STIX]{x1D70C}_{H}^{\prime }}=0$, and $PE=\overline{\unicode[STIX]{x1D70C}^{\prime 2}}/(2Fr_{0}^{2})=(\overline{\unicode[STIX]{x1D70C}_{L}^{\prime 2}}+\overline{\unicode[STIX]{x1D70C}_{H}^{\prime 2}})/(2Fr_{0}^{2})$. The size of the pink box at the bottom represents the Kolmogorov scale $L_{K}=1/k_{K}=0.25$.

Figure 4(a,b) shows a time development of the radial spectra of potential energy and kinetic energy, $E_{P}(k)$ and $E_{K}(k)$, for $Sc=700$. Since the potential energy is zero at $t=0$, kinetic energy is initially converted to potential energy through the vertical density flux due to the buoyancy (cf. figure 8). The potential energy is then cascaded to high wavenumbers at the same time as the energy is converted from the kinetic energy, and a potential-energy spectrum proportional to $k^{-1}$ is generated near the Kolmogorov wavenumber before the vertical density flux oscillates once ($t=4$). This initial spectrum is similar to a variance spectrum of a passive scalar of $Sc\gg 1$ in isotropic turbulence (Batchelor Reference Batchelor1959).

However, the potential energy decays with time, and bending of the spectrum appears at $t=8$. At the same time, the spectrum becomes flat at wavenumbers higher than the wavenumber at which the spectrum bends, which is approximately $k=7$ for the period of $8\leqslant t\leqslant 20$.

In decaying turbulence where $\unicode[STIX]{x1D716}_{K}$ decreases with time, the Kolmogorov and Batchelor wavenumbers decrease with time, while the Ozmidov wavenumber increases. Thus, the Kolmogorov wavenumber and the Ozmidov wavenumber coincide ($k_{K}^{\ast }=k_{O}^{\ast }$) at the primitive wavenumber of stratified turbulence $k_{P}^{\ast }=\sqrt{N^{\ast }/\unicode[STIX]{x1D708}^{\ast }}$ (Gibson Reference Gibson and Nihoul1980) in due course. In the present numerical simulations, its non-dimensional value is $k_{P}=\sqrt{N^{\ast }/\unicode[STIX]{x1D708}^{\ast }}L_{0}^{\ast }=(Re_{0}/Fr_{0})^{1/2}=(50/1)^{1/2}=7.1$. Therefore, as far as we can identify from figure 4(a), the bend of the spectrum appears near the primitive wavenumber.

The primitive wavenumber $k_{P}$ is constant in time since it is determined by the stratification $N^{\ast }$ and the viscosity $\unicode[STIX]{x1D708}^{\ast }$. Also, $k_{P}$ does not change with the Schmidt number $Sc$ (figure 5) since it is independent of the diffusion coefficient of density $\unicode[STIX]{x1D705}^{\ast }$. Furthermore, the definition of this wavenumber contains neither the total amount of kinetic energy nor the kinetic-energy dissipation rate $\unicode[STIX]{x1D716}_{K}^{\ast }$ which is related to the rate of the energy cascade per unit time, so it is not affected by nonlinear effects.

The corresponding kinetic-energy spectrum $E_{K}(k)$ also decreases with time (figure 4b), and bends near $k_{P}$ after a long time ($t\geqslant 16$). This occurs since the local decrease of potential energy near $k_{P}$ leads to the reduction of energy conversion into kinetic energy via the vertical density flux also near $k_{P}$.

Figure 4.

Temporal variation of (a) potential-energy spectrum $E_{P}(k)$ and (b) kinetic-energy spectrum $E_{K}(k)$ for $Sc=700$. Symbols on each spectral line show Kolmogorov wavenumber (○), Ozmidov wavenumber (●) and Batchelor wavenumber (▵).

Such bending of the spectrum has not been observed in the previous numerical simulations of stratified turbulence at lower Schmidt numbers ($Sc\lesssim 7$). We show in figure 5 the Schmidt number dependence of the potential-energy spectrum ($Sc=1,7$ and $70$). When $Sc=1$ (figure 5a,b), $E_{P}(k)$ and $E_{K}(k)$ dissipate at the same wavenumber since the Kolmogorov and Batchelor scales are the same, and their spectral shapes are very similar at wavenumbers larger than $k_{K}$.

When $Sc=7({>}1)$ (figure 5c,d), $k_{K}<k_{B}$ holds, and the dissipation of potential energy occurs at wavenumbers larger than $k_{K}$, and $E_{P}(k)$ differs from $E_{K}(k)$ at high wavenumbers. However, neither the bending nor the plateau of the spectrum are observed yet for $Sc\leqslant 7$. Since the potential energy initially cascades to wavenumbers larger than the Kolmogorov wavenumber, the kinetic energy at high wavenumbers also increases through the energy conversion by the vertical density flux, compared to the case of $Sc=1$ (cf. figure 5b,d).

In contrast, when $Sc$ increases to 70 (figure 5e), the potential energy spectrum $E_{P}(k)$ shows the $k^{-1}$ spectrum near the Kolmogorov wavenumber at $t=4$, due to the initial energy cascade (Batchelor Reference Batchelor1959). After this time, a weak bending of $E_{P}(k)$ is observed as time progresses. The same trend could be observed in the numerical simulation for $Re_{0}=100$ (Okino & Hanazaki Reference Okino and Hanazaki2019, figure 13d), although the bending of the spectrum was not discussed there. The potential-energy spectrum bends at $k\sim 7$, approximately the same value as the case of $Sc=700$. The independence of this wavenumber from the Schmidt number $Sc$ supports the possibility that this wavenumber is equal to the primitive wavenumber $k_{P}$ determined only by $N^{\ast }$ and $\unicode[STIX]{x1D708}^{\ast }$ (or $Fr_{0}$ and $Re_{0}$). However, the bending at $Sc=70$ is weaker than $Sc=700$, and thus the spectrum does not become flat at high wavenumbers, since the initial potential-energy transfer to high wavenumbers is smaller due to larger molecular diffusion of density.

Figure 5.

Temporal variation of potential-energy spectrum $E_{P}(k)$ and kinetic-energy spectrum $E_{K}(k)$. (a) $E_{P}(k)$ for $Sc=1$; (b) $E_{K}(k)$ for $Sc=1$; (c) $E_{P}(k)$ for $Sc=7$; (d) $E_{K}(k)$ for $Sc=7$; (e) $E_{P}(k)$ for $Sc=70$; (f) $E_{K}(k)$ for $Sc=70$. Symbols on each spectral line show Kolmogorov wavenumber (○), Ozmidov wavenumber (●) and Batchelor wavenumber (▵).

Figure 6.

Temporal variation of the premultiplied potential-energy spectrum $k\times E_{P}(k)$ for (a) $Sc=1$, (b) $Sc=7$, (c) $Sc=70$ and (d) $Sc=700$. Symbols on each spectral line show Kolmogorov wavenumber (○), Ozmidov wavenumber (●) and Batchelor wavenumber (▵). The vertical dotted lines indicate the primitive wavenumber $k_{P}=\sqrt{50}\simeq 7.1$.

When the energy spectrum $E_{P}(k)$ is multiplied by wavenumber $k$ and plotted against $\log k$, the area below the line of $kE_{P}(k)$ indicates the amount of potential energy. Such a spectrum is often called ‘premultiplied spectrum’ (e.g. Perry, Henbest & Chong Reference Perry, Henbest and Chong1986), and we use it here to visualise the amount of energy in a specific range of wavenumbers. We then show the time development of the premultiplied spectrum of potential energy $kE_{P}(k)$ in figure 6 to discuss the variation of potential energy at specific wavenumbers.

When $Sc=1$ (figure 6a), i.e. $\unicode[STIX]{x1D705}^{\ast }=\unicode[STIX]{x1D708}^{\ast }$, the Kolmogorov scale is equal to the Batchelor scale, and the potential energy also dissipates near the Kolmogorov wavenumber (○). The total amount of potential energy is small compared to the other cases of $Sc(=7,70\text{ and }700)$ shown in figure 6(b–d), since the molecular diffusion of density is large.

When the Schmidt number increases to $Sc=7$ (figure 6b), a certain amount of potential energy exists below the Kolmogorov scale since $k_{B}/k_{K}=\sqrt{7}\simeq 2.6$.

When $Sc=70$ (figure 6c), the viscous-convective subrange becomes broader ($k_{B}/k_{K}=\sqrt{70}\simeq 8.4$), and a relatively flat spectrum or a plateau appears initially ($t\sim 4$), corresponding to the appearance of the $k^{-1}$ spectrum in $E_{P}(k)$ (figure 5e). Later, a significant decrease is observed near the primitive wavenumber $k_{P}$ (vertical dotted line), where $k_{O}$ and $k_{K}$ agree.

When the Schmidt number further increases up to $Sc=700$ (figure 6d), a plateau appears initially ($t=3{-}4$), again corresponding to the appearance of the $k^{-1}$ law in $E_{P}(k)$ (figure 4a). Subsequently, the potential energy decreases locally near the primitive wavenumber, and two peaks are formed at low and high wavenumbers. Unlike the case of $Sc=70$, the peak at the higher wavenumber is large since the Batchelor wavenumber ($k_{B}$) is 26 times larger than $k_{K}$ and the significant portion of potential energy initially exists below the Kolmogorov scale. The peak at the lower wavenumber ($k\sim 1.5$) corresponds to the large-scale round structures observed in figure 3(a), while the peak at the higher wavenumber ($k\sim 40$) corresponds to the small-scale horizontal streaks observed in figure 3(b,c).

The total amount of potential energy $PE(=\overline{\unicode[STIX]{x1D70C}^{\prime 2}}/(2Fr_{0}^{2}))$ increases significantly with $Sc$ since the molecular dissipation becomes smaller for a smaller diffusion coefficient (cf. figures 2a and 5 of Okino & Hanazaki (Reference Okino and Hanazaki2019) for $Sc=1,7$ and $70$). This can be identified here also for $Sc=700$, by the increase of the area below the curves of $kE_{P}(k)$. On the other hand, the total amount of kinetic energy $KE(=\overline{u_{i}^{2}}/2)$ is rather insensitive to $Sc$, since the effects of a large Schmidt number are limited mainly to small scales below the Kolmogorov scale (figures 4b and 5b,d,f). Therefore, the energy ratio $PE/KE$ becomes larger for larger $Sc$.

To investigate the origin of the significant decrease of potential energy near the primitive wavenumber, we consider the budget for the potential-energy spectrum, i.e.

(3.4)$$\begin{eqnarray}\frac{\unicode[STIX]{x2202}E_{P}(k)}{\unicode[STIX]{x2202}t}=-D_{P}(k)+C_{\unicode[STIX]{x1D70C}^{\prime }w}(k)+T_{P}(k),\end{eqnarray}$$

where $D_{P}(k)$ is the potential-energy dissipation spectrum,

(3.5)$$\begin{eqnarray}D_{P}(k)=\mathop{\sum }_{\left|\sqrt{k_{x}^{2}+k_{y}^{2}+k_{z}^{2}}-k\right|<k_{min}/2}\frac{k_{j}^{2}}{Re_{0}ScFr_{0}^{2}}\big|\hat{\unicode[STIX]{x1D70C}^{\prime }}(k_{x},k_{y},k_{z})\big|^{2}\frac{1}{k_{min}},\end{eqnarray}$$

$C_{\unicode[STIX]{x1D70C}^{\prime }w}(k)$ is the cospectrum of the vertical density flux responsible for the exchange between the kinetic and potential energies,

(3.6)$$\begin{eqnarray}C_{\unicode[STIX]{x1D70C}^{\prime }w}(k)=\mathop{\sum }_{\left|\sqrt{k_{x}^{2}+k_{y}^{2}+k_{z}^{2}}-k\right|<k_{min}/2}\frac{1}{Fr_{0}^{2}}\text{ Re}[\hat{\unicode[STIX]{x1D70C}^{\prime }}(k_{x},k_{y},k_{z}){\hat{w}}^{\ast }(k_{x},k_{y},k_{z})]\frac{1}{k_{min}},\end{eqnarray}$$

and $T_{P}(k)$ is the nonlinear potential-energy transfer spectrum,

(3.7)$$\begin{eqnarray}T_{P}(k)=\mathop{\sum }_{\left|\sqrt{k_{x}^{2}+k_{y}^{2}+k_{z}^{2}}-k\right|<k_{min}/2}\frac{k_{j}}{Fr_{0}^{2}}\text{ Im}[\hat{\unicode[STIX]{x1D70C}^{\prime }}^{\ast }(k_{x},k_{y},k_{z})\widehat{\unicode[STIX]{x1D70C}^{\prime }u_{j}}(k_{x},k_{y},k_{z})]\frac{1}{k_{min}}.\end{eqnarray}$$

Its counterpart for the kinetic-energy spectrum is

(3.8)$$\begin{eqnarray}\frac{\unicode[STIX]{x2202}E_{K}(k)}{\unicode[STIX]{x2202}t}=-D_{K}(k)-C_{\unicode[STIX]{x1D70C}^{\prime }w}(k)+T_{K}(k),\end{eqnarray}$$

where $D_{K}(k)$ is the kinetic-energy dissipation spectrum,

(3.9)$$\begin{eqnarray}D_{K}(k)=\mathop{\sum }_{\left|\sqrt{k_{x}^{2}+k_{y}^{2}+k_{z}^{2}}-k\right|<k_{min}/2}\frac{k_{j}^{2}}{Re_{0}}|\hat{u} _{i}(k_{x},k_{y},k_{z})|^{2}\frac{1}{k_{min}},\end{eqnarray}$$

and $T_{K}(k)$ is the kinetic-energy transfer spectrum,

(3.10)$$\begin{eqnarray}T_{K}(k)=\mathop{\sum }_{\left|\sqrt{k_{x}^{2}+k_{y}^{2}+k_{z}^{2}}-k\right|<k_{min}/2}k_{j}\text{ Im}[\hat{u} _{i}^{\ast }(k_{x},k_{y},k_{z})\widehat{u_{i}u_{j}}(k_{x},k_{y},k_{z})]\frac{1}{k_{min}}.\end{eqnarray}$$

Here, the asterisk in (3.6), (3.7) and (3.10) denotes the complex conjugate.

We first consider the budget for the potential energy. The three terms on the right-hand side of (3.4) are presented in their premultiplied forms in figure 7(a) for $Sc=700$ at the time of $t=5$, i.e. when the local minimum near the primitive wavenumber $k_{P}$ is observed in the potential energy spectrum (cf. figure 6d). The potential-energy dissipation $-kD_{P}(k)$ has a negative peak near the Batchelor wavenumber ($k\sim 10^{2}$), and it counterbalances with the potential-energy transfer $kT_{P}(k)$. Near the primitive wavenumber $k_{P}(=7.1)$, vertical density flux $kC_{\unicode[STIX]{x1D70C}^{\prime }w}(k)$ is negative and counter-gradient, and this causes the local reduction of potential energy observed in figure 6(d). Indeed, the density flux is persistently counter-gradient near $k_{P}$, while it oscillates in time and changes sign at low wavenumbers. This behaviour will be discussed later in figure 9(d).

Figure 7.

The spectral budget of (a) the potential energy and (b) the kinetic energy for $Sc=700$ at $t=5$. The three arrows at the top of the figure indicate the Ozmidov, Kolmogorov and Batchelor wavenumbers.

We next consider the budget for the kinetic energy. The three terms on the right-hand side of (3.8) are presented in their premultiplied forms in figure 7(b). Above the primitive wavenumber, the counter-gradient density flux $-kC_{\unicode[STIX]{x1D70C}^{\prime }w}(k)>0$ cancels with the kinetic-energy dissipation $-kD_{K}(k)$, and prevents the decrease of kinetic energy.

Figure 8.

Temporal variation of the vertical density flux $\overline{\unicode[STIX]{x1D70C}^{\prime }w}/Fr_{0}^{2}$. The upper abscissa indicates the buoyancy time $t_{N}(=N^{\ast }t^{\ast }/(2\unicode[STIX]{x03C0}))$.

Figure 8 shows the temporal variation of the vertical density flux $\overline{\unicode[STIX]{x1D70C}^{\prime }w}/Fr_{0}^{2}(=\int _{0}^{\infty }C_{\unicode[STIX]{x1D70C}^{\prime }w}(k)\,\text{d}k)$. Since the density perturbation is initially absent in this study, kinetic energy is converted to potential energy at early times ($t\lesssim 2$), through the positive vertical density flux. After that, the vertical density flux oscillates due to the buoyancy effect, at a frequency approximately twice the Brunt–Väisälä frequency (Métais & Herring Reference Métais and Herring1989; Gerz & Yamazaki Reference Gerz and Yamazaki1993), which is fundamentally a linear effect (Hanazaki & Hunt Reference Hanazaki and Hunt1996).

Since it takes some time before the effect of molecular diffusion appears, the initial temporal development of the vertical density flux does not depend on $Sc$. However, after a while ($t\gtrsim 2$), stronger counter-gradient fluxes ($\overline{\unicode[STIX]{x1D70C}^{\prime }w}/Fr_{0}^{2}<0$) appear at larger Schmidt numbers. This is in agreement with the previous wind tunnel experiments for temperature stratification ($Sc=0.7$, Lienhard & Van Atta (Reference Lienhard and Van Atta1990)), water tank experiments for temperature stratification and salt stratification ($Sc=6$ and $600$, Komori & Nagata (Reference Komori and Nagata1996)), and the prediction by the rapid distortion theory (Hanazaki & Hunt Reference Hanazaki and Hunt1996). The vertical density fluxes at $Sc=70$ and $Sc=700$ are almost the same since they are determined mainly by the large-scale density perturbation and the vertical velocity, and are hardly affected by the spectral difference below the Kolmogorov scale (figures 4a and 5e).

The initial behaviour of the vertical density flux depends largely on the initial conditions such as the initial energy ratio $PE/KE$, as investigated previously by the rapid distortion theory (Hanazaki & Hunt Reference Hanazaki and Hunt1996). For example, the initial amplitude of the vertical density flux becomes maximum when $PE/KE=0$ or $\infty$ since the initial energy conversion between $KE$ and $PE$ becomes largest, while it becomes zero when $PE/KE=1/2$. However, after a sufficient time necessary for the energy exchange (i.e. $t_{N}=N^{\ast }t^{\ast }/(2\unicode[STIX]{x03C0})\gtrsim 1$) and the energy cascade (i.e. $t\gtrsim 1$) has passed, we can expect spectral behaviours similar to those observed in this study, at least at small scales.

Figure 9.

Temporal variation of the cospectrum of the vertical density flux in the premultiplied form $k\times C_{\unicode[STIX]{x1D70C}^{\prime }w}(k)$ for (a) $Sc=1$, (b) $Sc=7$, (c) $Sc=70$ and (d) $Sc=700$. Open circle shows the Kolmogorov wavenumber, and closed circle shows the Ozmidov wavenumber at $t=6,8$ and $12$. The vertical dotted line indicates the primitive wavenumber $k_{P}=\sqrt{50}\simeq 7.1$.

Figure 9 shows the temporal variation of the premultiplied cospectra of the vertical density flux ($kC_{\unicode[STIX]{x1D70C}^{\prime }w}(k)$). Since kinetic energy is converted to potential energy at early times, $kC_{\unicode[STIX]{x1D70C}^{\prime }w}(k)$ initially ($t\sim 1$) takes a positive maximum near $k=2$, corresponding to the initial peak wavenumber or the integral scale of the kinetic-energy spectrum. The effects of molecular viscosity and molecular diffusion are small at low wavenumbers, and it takes some time before they begin to affect the high-wavenumber components. Therefore, $kC_{\unicode[STIX]{x1D70C}^{\prime }w}$ at $t=1$ exhibits almost the same behaviour for all the Schmidt numbers except for high wavenumbers ($k\lesssim 10$). As a result, $\overline{\unicode[STIX]{x1D70C}^{\prime }w}/Fr_{0}^{2}({>}0)$ does not depend largely on $Sc$ at $t\lesssim 1$ (figure 8).

Subsequently, the low wavenumber components of the cospectrum ($k\lesssim 4$) oscillate around zero along with the exchange of kinetic energy and potential energy, corresponding to the oscillation of $\overline{\unicode[STIX]{x1D70C}^{\prime }w}/Fr_{0}^{2}$ (figure 8). The low-wavenumber components of the cospectra are not affected by $Sc$, and no significant difference is found among $Sc=1,7,70$ and $700$, since the viscosity and the diffusivity are not effective at low wavenumbers. Nevertheless, the amplitude at low wavenumbers decreases with time, not only because the energy is transferred to higher wavenumbers, but also because the components other than the horizontal wavevector cease to contribute to the cospectrum (Hanazaki & Hunt Reference Hanazaki and Hunt1996). The latter mechanism explains why the vertical density flux $\overline{\unicode[STIX]{x1D70C}^{\prime }w}/Fr_{0}^{2}$ decays with oscillation, even in the absence of viscosity, diffusivity and nonlinearity.

On the other hand, the behaviour of the components with $k\gtrsim 4$ is highly affected by $Sc$. When $Sc=1$ (figure 9a), the sign of the cospectrum changes almost simultaneously at all the wavenumbers (Gerz & Yamazaki Reference Gerz and Yamazaki1993), but when $Sc\geqslant 7$ (figure 9b–d), the high wavenumber components do not oscillate and are persistently negative (i.e. counter-gradient flux). The amplitude of the counter-gradient flux increases with $Sc$ in agreement with $\overline{\unicode[STIX]{x1D70C}^{\prime }w}/Fr_{0}^{2}$, although it decreases monotonically in its time development. The difference between $kC_{\unicode[STIX]{x1D70C}^{\prime }w}(k)$ at $Sc=70$ and that at $Sc=700$ is small, reflecting that $E_{K}(k)$ and $E_{P}(k)$ at low wavenumbers ($k\lesssim 10$) do not change with high Schmidt numbers (figures 4a,b and 5e,f).

The wavenumber at which the counter-gradient flux becomes strongest is almost steady, and is close to the primitive wavenumber of stratified turbulence ($k_{P}=7.1$). At small times when $t\lesssim 7$, the Ozmidov wavenumber $k_{O}$ is smaller than $k_{P}$, and the nonlinear effects still exist at wavenumbers where the counter-gradient flux occurs. However, we consider here tentatively why the counter-gradient flux is strong near $k_{P}$, using the linear rapid distortion theory. Using the three-dimensional spectrum $\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D70C}^{\prime }w}(\boldsymbol{k})$, the cospectrum can be written as

(3.11)$$\begin{eqnarray}C_{\unicode[STIX]{x1D70C}^{\prime }w}(k)=2\unicode[STIX]{x03C0}k^{2}\int _{0}^{\unicode[STIX]{x03C0}}\text{d}\unicode[STIX]{x1D703}\sin \unicode[STIX]{x1D703}\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D70C}^{\prime }w}(k,\unicode[STIX]{x1D703}),\end{eqnarray}$$

where $\unicode[STIX]{x1D703}$ is the angle between the wavevector and the vertical axis. Although $\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D70C}^{\prime }w}$ has the time oscillating terms in the form of $\sin \unicode[STIX]{x1D6FC}^{\ast }t^{\ast }$ or $\cos \unicode[STIX]{x1D6FC}^{\ast }t^{\ast }$ where

(3.12)$$\begin{eqnarray}\unicode[STIX]{x1D6FC}^{\ast }=(4N^{\ast 2}\sin ^{2}\unicode[STIX]{x1D703}-(\unicode[STIX]{x1D708}^{\ast }-\unicode[STIX]{x1D705}^{\ast })^{2}k^{\ast 4})^{1/2},\end{eqnarray}$$

only the components of $\unicode[STIX]{x1D703}\sim \unicode[STIX]{x03C0}/2$ tend to contribute to the $\unicode[STIX]{x1D703}$-integral in (3.11) as time progresses (Hanazaki & Hunt Reference Hanazaki and Hunt1996). Since $\sin \unicode[STIX]{x1D6FC}^{\ast }t^{\ast }$ approaches $\sin ((4N^{\ast 2}-(\unicode[STIX]{x1D708}^{\ast }-\unicode[STIX]{x1D705}^{\ast })^{2}k^{\ast 4})^{1/2}t^{\ast })$ in the limit of $\unicode[STIX]{x1D703}\rightarrow \unicode[STIX]{x03C0}/2$, the integral (3.11) oscillates in time when $4N^{\ast 2}-(\unicode[STIX]{x1D708}^{\ast }-\unicode[STIX]{x1D705}^{\ast })^{2}k^{\ast 4}>0$, i.e. in the wavenumber range of $k^{\ast }<\sqrt{2N^{\ast }/|\unicode[STIX]{x1D708}^{\ast }-\unicode[STIX]{x1D705}^{\ast }|}$. Therefore, $k_{c}^{\ast }=\sqrt{2N^{\ast }/|\unicode[STIX]{x1D708}^{\ast }-\unicode[STIX]{x1D705}^{\ast }|}$ is the critical wavenumber between the decaying oscillation and the monotonic decay. In non-dimensional form, the wavenumber becomes

(3.13)$$\begin{eqnarray}k_{c}=\left(\frac{2Re_{0}}{Fr_{0}}\frac{1}{\left|1-{\displaystyle \frac{1}{Sc}}\right|}\right)^{1/2}.\end{eqnarray}$$

This expression shows that, when $Sc=1$, $k_{c}$ becomes infinity and the cospectrum oscillates in time at all the wavenumbers. On the other hand, when $Sc\gg 1$, the critical wavenumber can be approximated by

(3.14)$$\begin{eqnarray}k_{c}\simeq \left(\frac{2Re_{0}}{Fr_{0}}\right)^{1/2}=\sqrt{2}k_{P}.\end{eqnarray}$$

This value is close to the primitive wavenumber $k_{P}$, and $k_{c}\simeq 10$ in the present study. Figure 9 shows that the zero of $kC_{\unicode[STIX]{x1D70C}^{\prime }w}(k)$ is at around $k=4$, and is smaller than $k_{c}=10$. However, the graph of cospectrum will move downward due to the persistently negative components which are significant at high wavenumbers (cf. (4.9) of Hanazaki & Hunt (Reference Hanazaki and Hunt1996)). Thus, the point of zero can move toward a lower wavenumber, and will partly explain the difference between $k=4$ and $k_{c}=10$.

Another explanation for the possible importance of $k_{P}$ is that it is near the Kolmogorov wavenumber. In decaying turbulence, $k_{K}$ decreases with time while $k_{O}$ increases with time, and they agree at $k_{P}$ when $t\simeq 7$. Then, the Kolmogorov wavenumber is not far from the primitive wavenumber at which the strong counter-gradient flux is observed. On the other hand, since the vertical density flux is the product of $\unicode[STIX]{x1D70C}^{\prime }$ and $w$, it decreases significantly for $k\gtrsim k_{K}$, where $w$ decreases. Then, a strong counter-gradient flux could be observed just above the Kolmogorov scale. This may also explain why the strong counter-gradient flux has been observed near $k_{P}$.

Figure 10.

Time development of the one-dimensional spectra of potential energy, i.e. $\unicode[STIX]{x1D6E9}_{P}(k_{x})$ and $\unicode[STIX]{x1D6E9}_{P}(k_{z})$, for (a) $Sc=1$, (b) $Sc=7$, (c) $Sc=70$ and (d) $Sc=700$.

The one-dimensional spectra of potential energy are shown in figure 10 to examine the anisotropy of turbulence. The horizontal spectrum $\unicode[STIX]{x1D6E9}_{P}(k_{x})$ has been measured in most of the experiments (e.g. Lienhard & Van Atta Reference Lienhard and Van Atta1990), and the vertical spectrum $\unicode[STIX]{x1D6E9}_{P}(k_{z})$ has sometimes been measured (Keller & Van Atta Reference Keller and Van Atta2000), and these would be preferable for comparison with the experiments. They are computed in this study by

(3.15)$$\begin{eqnarray}\unicode[STIX]{x1D6E9}_{P}(k_{x})=\mathop{\sum }_{k_{y},k_{z}}\frac{1}{2Fr_{0}^{2}}(|\hat{\unicode[STIX]{x1D70C}^{\prime }}(k_{x},k_{y},k_{z})|^{2}+|\hat{\unicode[STIX]{x1D70C}^{\prime }}(-k_{x},k_{y},k_{z})|^{2})\frac{1}{k_{min}}\end{eqnarray}$$

and

(3.16)$$\begin{eqnarray}\unicode[STIX]{x1D6E9}_{P}(k_{z})=\mathop{\sum }_{k_{x},k_{y}}\frac{1}{2Fr_{0}^{2}}(|\hat{\unicode[STIX]{x1D70C}^{\prime }}(k_{x},k_{y},k_{z})|^{2}+|\hat{\unicode[STIX]{x1D70C}^{\prime }}(k_{x},k_{y},-k_{z})|^{2})\frac{1}{k_{min}}.\end{eqnarray}$$

For all the Schmidt numbers, we note that $\unicode[STIX]{x1D6E9}_{P}(k_{z})>\unicode[STIX]{x1D6E9}_{P}(k_{x})$ when $k_{x}=k_{z}$ except for the lowest wavenumbers ($k_{x},k_{z}\sim 1$), showing that the density fluctuations have a vertically thin structure. The weak anisotropy exists already at $t=4$, and the anisotropy becomes more significant with time as the Ozmidov scale becomes smaller, and the difference between $\unicode[STIX]{x1D6E9}_{P}(k_{x})$ and $\unicode[STIX]{x1D6E9}_{P}(k_{z})$ increases. This corresponds to the appearance of small-scale flat structures of density perturbations (figure 2e,f) or the flow dominated by the vertical shear at small scales (Okino & Hanazaki Reference Okino and Hanazaki2019). In a wind tunnel experiment for stratified shear flow, Keller & Van Atta (Reference Keller and Van Atta2000) have compared the horizontal and the vertical spectrum and found that the vertical spectrum is larger than the horizontal spectrum at small scales.

Comparison with the radial spectra $E_{P}(k)$ in figures 4(a) and 5(a,c,e) shows that the vertical spectra are very similar to the radial spectra (i.e. $\unicode[STIX]{x1D6E9}_{P}(k_{z})\simeq E_{P}(k)$ for $k_{z}=k$), while the horizontal spectra are generally smaller, especially at small scales (i.e. $\unicode[STIX]{x1D6E9}_{P}(k_{x})<E_{P}(k)$ for $k_{x}=k$).

The Schmidt number does not affect the anisotropy significantly, but the one-dimensional spectra ($\unicode[STIX]{x1D6E9}_{P}(k_{x})$, $\unicode[STIX]{x1D6E9}_{P}(k_{z})$) and the radial spectra $E_{P}(k)$ all extend to the Batchelor wavenumber $k_{B}$, with a noticeable flat spectrum only when $Sc=700$.

Figure 11.

Premultiplied two-dimensional spectra of potential energy $k_{H}k_{V}e_{P}(k_{H},k_{V})$ for $Sc=700$ at (a) $t=4$ and (b) $t=20$, where contour values are $n\times 10^{-3}$ ($n=1,2,\ldots$) in (a), and $n\times 10^{-4}$ ($n=1,2,\ldots$) in (b). Figure 11(c) is the premultiplied two-dimensional cospectrum of vertical density flux $k_{H}k_{V}c_{\unicode[STIX]{x1D70C}^{\prime }w}(k_{H},k_{V})$ for $Sc=700$ at $t=5$, with contour values of $2n\times 10^{-3}$ ($n=0,\pm 1,\pm 2,\ldots$). The positive and null values are indicated by solid lines, while negative values are indicated by dotted lines. Dashed lines indicate $(k_{H}^{2}+k_{V}^{2})^{1/2}=k_{K}$, dashed-dotted lines indicate $(k_{H}^{2}+k_{V}^{2})^{1/2}=k_{B}$, and dashed two-dotted line indicates $\unicode[STIX]{x1D6FC}=0$, where $\unicode[STIX]{x1D6FC}$ is defined by (3.12) in dimensional form.

To further elucidate the anisotropy of potential energy, we consider the two-dimensional spectrum of potential energy and vertical density flux. For example, the two-dimensional spectrum of potential energy $e_{P}(k_{H},k_{V})$ satisfies

(3.17)$$\begin{eqnarray}PE=\int _{0}^{\infty }\text{d}k_{H}\int _{0}^{\infty }\text{d}k_{V}\,e_{P}(k_{H},k_{V}),\end{eqnarray}$$

and is defined by

(3.18)$$\begin{eqnarray}e_{P}(k_{H},k_{V})=\frac{k_{H}}{2Fr_{0}^{2}}\int _{0}^{2\unicode[STIX]{x03C0}}\text{d}\unicode[STIX]{x1D719}\,(|\hat{\unicode[STIX]{x1D70C}^{\prime }}(k_{H},\unicode[STIX]{x1D719},k_{V})|^{2}+|\hat{\unicode[STIX]{x1D70C}^{\prime }}(k_{H},\unicode[STIX]{x1D719},-k_{V})|^{2}),\end{eqnarray}$$

where wavevectors are represented in cylindrical coordinates $\boldsymbol{k}=(k_{H},\unicode[STIX]{x1D719},k_{z})$, with $k_{z}$ being the axial coordinate, $\unicode[STIX]{x1D719}$ the azimuthal angle and $k_{H}=(k_{x}^{2}+k_{y}^{2})^{1/2}$. The two-dimensional cospectrum of vertical density flux $c_{\unicode[STIX]{x1D70C}^{\prime }w}(k_{H},k_{V})$ is defined similarly.

At an early time of $t=4$, figure 11(a) shows that the premultiplied two-dimensional spectrum of potential energy $k_{H}k_{V}e_{P}(k_{H},k_{V})$ is symmetric about the line of $k_{H}=k_{V}$, indicating an approximate isotropy at all scales when $k_{O}<k_{K}$ still holds and $k^{-1}$ spectrum is observed in the viscous-convective subrange (figure 4a).

As time proceeds, anisotropy appears in the spectrum, and later at $t=20$ (figure 11b), the spectrum has two peaks at $(k_{H},k_{V})\sim (1,1)$ and $(20,30)$, corresponding to the two peaks at $k=(k_{H}^{2}+k_{V}^{2})^{1/2}\simeq 1.4$ and 36 in the radial spectrum (figure 6d). The peak at $(20,30)$ is in the viscous-convective subrange and satisfies $k_{V}>k_{H}$, consistent with the vertically thin structure at small scales observed in figures 2(e) and 3(c).

The two peaks in figure 11(b) appear since the vertical density flux persistently converts the potential energy into kinetic energy near the primitive wavenumber ($k_{P}=7.1$). The premultiplied two-dimensional cospectrum between density and vertical velocity, i.e. $k_{H}k_{V}c_{\unicode[STIX]{x1D70C}^{\prime }w}(k_{H},k_{V})$ (figure 11c), is indeed negative near the Kolmogorov wavenumber or the wavenumbers satisfying $\unicode[STIX]{x1D6FC}=0$, which determines the threshold between the decaying oscillation ($\unicode[STIX]{x1D6FC}^{2}>0$) and the monotonic decay ($\unicode[STIX]{x1D6FC}^{2}<0$) of linear fluctuations (Hanazaki & Hunt Reference Hanazaki and Hunt1996). We note again that $\unicode[STIX]{x1D6FC}=0$ in the limit of $\unicode[STIX]{x1D703}\rightarrow \unicode[STIX]{x03C0}/2$ (i.e. $k_{V}\rightarrow 0$) leads to $k=k_{c}\simeq \sqrt{2}k_{P}=10$. The area of negative $k_{H}k_{V}c_{\unicode[STIX]{x1D70C}^{\prime }w}$ is off the diagonal ($k_{H}=k_{V}$), and mostly satisfies $k_{H}>k_{V}$, indicating the anisotropy of the counter-gradient vertical density flux at small scales.

Figure 12.

Normalised potential-energy spectrum for (a) $Sc=1$, (b) $Sc=7$, (c) $Sc=70$, (d) $Sc=700$. The vertical dashed line indicates $k_{K}^{\ast }/k_{B}^{\ast }(=Sc^{-1/2})$. Closed circle shows $k_{O}^{\ast }/k_{B}^{\ast }$, which increases with time.

Figure 12 shows the time development of the normalised potential-energy spectra. We note that the spectra at different times agree in the decaying process of stratified turbulence. The agreement is incomplete, however, particularly at scales larger than the Ozmidov scale above which the stratification effect is significant. Since the Ozmidov wavenumber increases with time while the Batchelor wavenumber decreases, the normalised wavenumber $k_{O}^{\ast }/k_{B}^{\ast }$ increases with time. Then, the wavenumber range with unequal normalised spectra extends to higher wavenumbers. On the other hand, since $\unicode[STIX]{x1D716}_{K}^{\ast }$ does not change significantly with $Sc$, $k_{O}^{\ast }/k_{K}^{\ast }=Re_{b}^{-3/4}=(\unicode[STIX]{x1D716}_{K}^{\ast }/(\unicode[STIX]{x1D708}^{\ast }N^{\ast 2}))^{-3/4}$ is rather independent of $Sc$. Then, $k_{O}^{\ast }/k_{B}^{\ast }(=k_{O}^{\ast }/(k_{K}^{\ast }Sc^{1/2}))$ is smaller for larger $Sc$ in general so that the range of unequal spectra becomes narrower for larger $Sc$.