And are the kinematic viscosity and thermal diffusivity

J. Fluid Mech. (2022), vol. 946, A6, doi:10.1017/jfm.2022.554

2.554 Published online by Cambridge University Press

Budgets of turbulent kinetic energy (TKE) and turbulent potential energy (TPE) at different scales ℓ in sheared, stably stratified turbulence are analysed using a filtering approach. Competing effects in the flow are considered, along with the physical mechanisms governing the energy fluxes between scales, and the budgets are used to analyse data from direct numerical simulation at buoyancy Reynolds number Reb = O(100). The mean TKE exceeds the TPE by an order of magnitude at the large scales, with the difference reducing as ℓ is decreased. At larger scales, buoyancy is never observed to be positive, with buoyancy always converting TKE to TPE. As ℓ is decreased, the probability of locally convecting regions increases, though it remains small at scales down to the Ozmidov scale. The TKE and TPE fluxes between scales are both downscale on average, and their instantaneous values are correlated positively, but not strongly so, and this occurs due to the different physical mechanisms that govern these fluxes. Moreover, the contributions to these fluxes arising from the sub-grid fields are shown to be significant,

in addition to the filtered scale contributions associated with the processes of strain

https://doi.org/10.1017/jfm.202

946 A6-1

X. Zhang and others

1. Introduction

mean interscale energy transfer terms (Riley & de Bruyn Kops 2003; Lindborg 2006; Almalkie & de Bruyn Kops 2012). The study of Riley & de Bruyn Kops (2003) showed that the horizonal energy spectrum exhibits a k−5/3 h scaling for wavenumbers smaller than the Ozmidov wavenumber kO (corresponding to the wavenumber at which inertial and buoyancy forces are of the same order), where kh is the horizontal wavenumber, and the results indicated a downscale energy transfer of kinetic energy in the flow. This motivated (Lindborg 2006), who confirmed that strongly stratified turbulent flows exhibit a downscale cascade of turbulent kinetic and potential energy on average, and developed phenomenological predictions similar in spirit to Kolmogorov’s 1941 theory (Kolmogorov 1941). The observation of a downscale energy cascade was contrary to predictions that had been made by Gage (1979) and Lilly (1983) based on the assumption that in the limit of strong stratification, the flow should behave as two-dimensional turbulence. The basic reason why the predictions of Gage (1979) and Lilly (1983) failed is that, as shown in Billant & Chomaz (2001), as the strength of the stratification increases and the vertical velocity of the flow is suppressed, the vertical length scale of the flow also reduces in such a way that the terms in the dynamical equations associated with vertical motion always remain O(1), hence the flow never becomes two-dimensional.

Many questions remain, however, regarding the multiscale properties of SSST. For example, how do fluctuations of the SSST flow about its mean-field state behave? What are the mechanisms of the turbulent kinetic and potential energy transfers among

stratified turbulence across scales could aid in the development of appropriate sub-grid models.

and discuss areas for future investigation.

2. Theoretical considerations

https://doi.org/10.1017/jfm.2022.554 Published online by Cambridge University Press

Stokes equation coupled with an advection–diffusion equation for ρ′(Vallis 2006). Assuming that |ρ′|/ρ ≪ 1, the governing equation for u is then the Boussinesq–Navier–

946 A6-3

where Dt ≡ ∂t + ˜u · ∇,˜φ ≡ g˜ρ′/Nρr is the scaled density field (with dimensions of

a velocity), g ≡ ∥g∥, ν and κ are the kinematic viscosity and thermal diffusivity,

F = −γ (ez · x)(ex · ∇)u − γ ex(ez · u), (2.3)

f = −γ (ez · x)(ex · ∇)φ. (2.4)

In these equations, the scale-to-scale TKE flux is defined as ΠK ≡ − τ : ˜s, such that ΠK > 0 corresponds to a transfer of TKE from the large to the small scales. The

small-scale buoyancy term is B ≡ −N(�εK ≡ 2ν(�∥s∥2− ∥˜s∥2), the small-scale forcing is FK ≡�uzφ − ˜uz˜φ), the small-scale TKE dissipation rate is

2κ ∇2 ˜φ ˜φ − κ ∥∇ ˜φ∥2 + N ˜φ˜uz + ˜φ˜f,
∥u∥2) + (˜u · τ) + 2ν( �

φφ − ˜φ˜φ)/2. The equations
φφ/2, and this may
u · s − ˜u · ˜s). (2.7)

˜φf − ˜φ˜f, and the small-scale scalar transport involves

The equations for EK and EP shown above ((2.5) and (2.8)) were also derived in the

Energy source
(forcing)

EK	ΠK	eK	B	eP	ΠP	EP
		eK		∇∙TP
		∇∙TK		∇∙TP

Fr ≡ ⟨ϵK⟩NEK = (ℓO/L)2/3,

Reb ≡ ⟨ϵK⟩νN2 = (ℓO/η)4/3,

(2.12) (2.13)

Res ≡ ⟨ϵK⟩νγ2 = (ℓC/η)4/3.

(2.14)

	We now turn to consider the contributions to the equations governing the mean energy fields ⟨eK⟩ and ⟨eP⟩. For SSST, the mean transport equations for these quantities reduce to
	0 = ⟨B⟩ − ⟨εK⟩ + ⟨ΠK⟩ + ⟨FK⟩, 0 = −⟨B⟩ − ⟨εP⟩ + ⟨ΠP⟩.

	⟨FK⟩∞∼ −⟨B⟩∞+ ⟨ϵK⟩,	(2.17)
	For FSST, ⟨ΠK⟩ ∼ −⟨B⟩ + ⟨ϵK⟩ when ℓF ≫ ℓ ≫ η, and for ℓO/ℓF → ∞ (neutrally buoyant), this would correspond to a TKE cascade ⟨ΠK⟩ ∼ ⟨ϵK⟩. However, for a stably 946 A6-6

Analysis of scale-dependent kinetic and potential energy

For SSST, the behaviour of ⟨ΠK⟩ is quite different. In this case, ⟨FK⟩ operates down

to the scale ℓC, so because ℓC < ℓO, the behaviour ⟨ΠK⟩ ∼ −⟨B⟩ + ⟨ϵK⟩ never emerges, and the regime that emerges instead for Ri ≪ 1 is ⟨ΠK⟩ ∼ −⟨FK⟩ + ⟨ϵK⟩ for ℓ ≫ η. In this regime, the TKE flux is again not in the form of a cascade since ⟨FK⟩ depends on ℓ. Indeed, based on the behaviour of the co-spectrum (Katul et al. 2013), we might expect

is true for isotropic turbulence, it will not apply in general in strongly stratified flows In the above discussion, it has been assumed that ⟨εK⟩ ∼ ⟨ϵK⟩ for ℓ ≫ η. While this

with Fr ≪ 1. This is because when Fr ≪ 1, the flow structures are highly anisotropic, so while the horizontal length scale may be large enough for viscous effects to be negligible,

scale dependency of ⟨B⟩ at these scales leads to a non-constant flux of TPE at these scales. This observation seems to be in conflict with the well-known Bolgiano–Obukhov (BO)

scaling (Bolgiano 1959; Obukhov 1959) that has been proposed for stratified turbulent

stratified’ regime Re ≫ 1 and Fr ≈ 1, for which it has been argued that BO should apply

946 A6-7

dimensions; however, the underlying mechanisms driving the energy transfers in HIT and SSST might be quite different. In the context of HIT, it has long been thought that the key mechanism driving the TKE cascade is that of vortex stretching (VS) (Taylor 1922, 1938; Tennekes & Lumley 1972; Davidson 2004; Doan et al. 2018). However, recent studies have demonstrated quantitatively that while VS plays an important role, the largest contribution to the TKE cascade comes from the dynamical process of the self-amplification of the strain-rate field (Carbone & Bragg 2020; Johnson 2020, 2021). An important question is how this understanding applies to SSST, where effects such as internal waves and mean-shear can play a role (whether directly or indirectly) in how TKE is transferred between scales, as well as the question of the mechanism driving the TPE transfer. In Johnson (2020, 2021), a powerful, exact relationship was derived for ΠK that assumes only that the filtering kernel Gℓ used in constructing ˜u is Gaussian. The result is

https://doi.org/10.1017/jfm.2022.554 Published online by Cambridge University Press	ΠK = ΠF,SSA		+ ΠSG,C	,	(2.22)
					(2.23)
					(2.24)
	superscript SG denotes that the quantity depends on the sub-grid fields as well as the filtered fields. Explicit integral formulae for the SG terms can be found in Johnson (2020, 2021); we do not quote them here as they will not be considered in detail in the present paper. Note that in Johnson (2020, 2021), the F contributions are referred

derived using filtering and an asymptotic expansion (Carbone & Bragg 2020). Moreover,

due to the relation of Betchov (Betchov 1956; Eyink 2006), ⟨ΠF,SSA⟩ = 3⟨ΠF,VS⟩ for

ΠSG,VS

inertial range of isotropic turbulence, so that overall, the SSA mechanism contributes more

flow dynamics. As such, in SSST, SSA and VS will still be the key dynamical processes

governing the TKE flux. Moreover, the relation of Betchov (1956) still applies (since it

since, for example, in SSST, internal waves can contribute to the behaviour of ˜s and ˜ω,

and anisotropy in the flow can modify the alignments between ˜s and ˜ω (see Sujovolsky & Mininni 2020) which affects the VS process. Some of these more involved questions

ΠP = ΠF P+ ΠSG,S + ΠSG,V ,

(2.27)

(2.28)

although it does affect ΠF Pimplicitly since rotation in the fluid affects the alignment between ˜s and ∇˜φ. The contribution from the filtered field ΠF Pdescribes the flux of TPE associated with the amplification (if ˜s : (∇˜φ ∇˜φ) < 0) or suppression (if ˜s : (∇˜φ ∇˜φ) > 0) of∥∇˜φ∥ due to the filtered strain rate ˜s. The contribution ΠF ΠF,VS another dynamical field, rather than SSA in which the strain-rate field amplifies or K , in that both depend upon the strain-rate field ˜s acting to amplify or suppress Pis similar in form to

suppresses itself. However, the negative sign appearing in ΠF Pbut not in ΠF,VS leads to a significant difference in how the strain rate contributes to these energy

amplification of ∇˜φ due to compressional straining motions in the flow. Similar behaviour was also suggested previously based on an LES model for scalar gradients in turbulence

(Leonard 1997; Higgins, Parlange & Meneveau 2004).

(2.6) and (2.9) to understand the processes and mechanisms controlling the behaviour

946 A6-10

Reb	Ri	Res	Fr	L/η	ℓO/η	ℓC/η
160	0.157	25.12	0.48	126.35	43.90	10.36

of the TKE and TPE across scales in SSST. The DNS data used are from the data set presented in Portwood et al. (2019) and Portwood, de Bruyn Kops & Caulfield (2022), which we summarize here. In the DNS, the unfiltered versions of (2.1) and (2.2) are solved with constant mean velocity gradient γ and mean density gradient ζ using the Fourier pseudospectral scheme described in de Bruyn Kops (2015) and Almalkie & de Bruyn

https://doi.org/10.1017/jfm.2022.554 Published online by Cambridge University Press

normalized by
for a two-dimensional plane in the streamwise and vertical directions. The snapshots

illustrate clearly both the strong anisotropy in the flow and the inclination of the

We begin by considering the behaviour of ⟨eK⟩, ⟨eP⟩ and the diagonal components of⟨τ⟩/2 (which correspond to the TKE associated with different components of u) as a 946 A6-11

X. Zhang and others

(b)

(d )

blue), which correspond to (−3, 3) for the normalized quantities, centred at white = 0. Velocities are�2EK/3, (c) vertical component uz/�

2EK/3, (d) fluctuating density ρ′/√⟨ρ′ρ′⟩. Values go from (red,�2EK/3, (b) streamwise component

⟨eK⟩/⟨eP⟩ = O(10) at the large scales of the flow. The flow is therefore far from a state of equipartition of large-scale energy among the TKE and TPE fields, unlike the behaviour

that is thought to emerge for strongly stratified flows with Fr ≪ 1 (Billant & Chomaz

https://doi.org/10.1017/jfm.2022.554 Published online by Cambridge University Press	(a)
	(b)
	(c)
	(d )
	⟨τ⟩/2, it is seen that although the total TKE is much larger than the TPE, the TKE associated with particular components of the velocity field is comparable to the TPE. In particular, ⟨τzz⟩/2 ≈ 0.8⟨eP⟩ and ⟨τyy⟩/2 ≈ 1.3⟨eP⟩ at the large scales. As such, even though the energy contained in the TPE field is small compared to that in the total TKE

X. Zhang and others

(a)	⟨eK⟩/⟨ET⟩	(b)	⟨εK⟩/⟨ϵT⟩
100		100
10–1		10–1
10–2	⟨τxx⟩/[2⟨ET⟩]	10–2
10–2
10–3			⟨εP⟩/⟨ϵT⟩
10–3	101 ℓ/η		⟨εP⟩/⟨ϵT⟩

Figure 4. (a) Plot of mean small-scale TKE ⟨eK⟩, TPE ⟨eP⟩, and diagonal components of ⟨τ⟩/2, normalized by total energy ⟨ET⟩ ≡ limℓ/η→∞[⟨eK⟩ + ⟨eP⟩], as a function of filter scale ℓ. The thick dotted line indicates scaling ∝ ℓ2. (b) Plot of mean small-scale TKE ⟨εK⟩ and TPE ⟨εP⟩ dissipation rates, normalized by the total turbulent energy dissipation rate ⟨ϵT⟩ ≡ limℓ/η→∞[⟨εK⟩ + ⟨εP⟩]. The thin vertical dotted lines from right to left are L/η, ℓO/η, ℓC/η = 126.3, 43.9, 10.4, respectively.

In figure 4(b) we consider the mean small-scale turbulent kinetic ⟨εK⟩ and potential⟨εP⟩ energy dissipation rates. For ℓ/η → ∞, these satisfy ⟨εK⟩ → ⟨ϵK⟩ and ⟨εP⟩ → ⟨ϵP⟩, and the results show that these are in the ratio ⟨ϵK⟩/⟨ϵP⟩ ≈ 5. Both ⟨εK⟩ and ⟨εP⟩ are approximately independent of ℓ only down to ℓ/η = O(10), consistent with the usual observation that the Kolmogorov scale underestimates the scale at which viscous forces

become important (Pope 2000).

does not give rise to an inertial TKE cascade since at the scales where ⟨ΠK⟩ ∼ ⟨εK⟩, ⟨εK⟩is a decreasing function of ℓ. To observe an inertial TKE cascade with ⟨ΠK⟩ ∼ ⟨ϵK⟩ would require considering a flow possessing a range of scales ℓO > ℓC ≫ ℓ ≫ η. The results in figure 5 show that the contribution to the TKE flux coming from the filtered field dynamics,

i.e. ⟨ΠF role in the small-scale TKE budget equation, and also that the contribution involving the K⟩, makes a significant contribution to ⟨ΠK⟩ at scales where ⟨ΠK⟩ plays a significant

100
100
10–1			⟨ΠK F ⟩/(ϵK⟩
10–2			–⟨εK⟩/⟨ϵK⟩
10–2
10–3
10–3	100	102

sub-grid fields ⟨ΠSG K⟩ is also significant, just as for isotropic turbulence (Johnson 2020, 2021). At sufficiently small ℓ, ⟨ΠK⟩ ≈ ⟨ΠF K⟩, consistent with the exact limiting behaviour limℓ/η→0 ΠF K→ ΠK.

The mean buoyancy term ⟨B⟩ is negative at all scales, indicating a mean transfer of TKE in figure 5. Nevertheless, buoyancy plays a key role in the flow because its magnitude to TPE, though the magnitude of ⟨B⟩ is sub-leading compared to the other terms shown

In order to understand the energetics of the flow beyond its mean-field behaviour, we will

now consider the probability density functions (PDFs) of various quantities. We begin in

100
10–1			⟨ΠP⟩/⟨ϵP⟩
			⟨ΠP F⟩/⟨ϵP⟩

10–2
10–2	100	102

lengths ℓ/η. At larger scales, the PDFs have a large variance, showing that at these scales, In figure 7(c), we show the PDFs of ΠK/⟨ΠK⟩ and ΠP/⟨ΠP⟩ for different filtering

the fluxes of TKE and TPE can exceed their mean values significantly. The PDFs are also

	ΠK/⟨ΠK⟩ and ΠP/⟨ΠP⟩ as ℓ/η increases, and for B/⟨B⟩ as ℓ/η decreases, is due mainly to the fact that the standard deviations of the variables are much larger than their mean 946 A6-16

(a)	(b)
PDF	100	1	2 eK/⟨eK⟩	ℓ/η = 0.25		5	100
	100						100
	10–2						10–2
	10–2						10–2
	10–4				4		10–4
	10–4				4		10–4	0	1	2 eP/⟨eP⟩	4	5
(c)	(d )
PDF	–5 0 5 ΠK/⟨ΠK⟩, ΠP/⟨ΠP⟩						B/⟨B⟩
Figure 7. Plots of PDFs of (a) eK/⟨eK⟩, (b) eP/⟨eP⟩ and (d) B/⟨B⟩ for different filter lengths ℓ. In (c), the solid lines correspond to the PDFs of ΠK/⟨ΠK⟩, and the dashed lines correspond to the PDFs of ΠP/⟨ΠP⟩. Different colours/symbols correspond to different ℓ/η as indicated by the legend in (a). 946 A6-17

ξ = ΠP
ξ = –B

100

101

102

(a)

Corr = 0.22695

(b)

Corr = 0.1474

–6

–1

–3

–10

–12

(c)

Corr = 0.23027

Corr = 0.14807

–3

–2

–5

–5

–10

–5

–9

ΠP/⟨ΠP⟩

Figure 9. Contour plot of the logarithm of the joint PDF of ΠP/⟨ΠP⟩ and ΠK/⟨ΠK⟩ for (a) ℓ/η = 0.25, (b) ℓ/η = 6, (c) ℓ/η = 16, (d) ℓ/η = 60. Colours correspond to the logarithm of the PDF, and the correlation coefficient is shown at the top of each plot.

(a)	5	Corr = 0.99352						0	5	Corr = 0.74019
(a)	5								5
3									3
									1										–5
								–10	–1										–5
–3								–15	–3										–10
–5		–3		–1	1	3	5	–15	–5	–3		–1	1		3		5		–10
(c)		Corr = 0.55206							5	Corr = 0.32478
5		Corr = 0.55206								Corr = 0.32478
5
3									3										–4
ΠK/⟨ΠK⟩ 1									3
									1
									1
									–1
									–1
–3								–10	–3
–3								–10	–3
–5			–3	–1	1	3	5	–5			–3	–1		1		3		5
ΠK F /⟨ΠK F⟩								ΠK F /⟨ΠK F⟩
Figure 10. Contour plot of the logarithm of the joint PDF of ΠK/⟨ΠK⟩ and ΠF K/⟨ΠF K⟩ for (a) ℓ/η = 0.25, (b) ℓ/η = 6, (c) ℓ/η = 16, (d) ℓ/η = 60. Colours correspond to the logarithm of the PDF, and the correlation coefficient is shown at the top of each plot. amplification) due to nonlinearity yielding ΠSG K < 0, and if this is strong enough, then ΠK = ΠF the relative contribution of ΠF K+ ΠSG < 0. The spread of the PDF about the line ΠK = ΠF Kto ΠK is similar both during events where ΠK ∼ ⟨ΠK⟩Kalso implies that results show that as ℓ is decreased, the PDF reorients from being extended along the ΠK In figure 12, we consider the joint PDF of B and ΠK for different filter lengths ℓ. The axis, to being extended along the B axis. Furthermore, the peak of the PDF (indicating the mode) is along ΠK/⟨ΠK⟩ ≈ 0 for small scales and B/⟨B⟩ > 0 for larger scales, which

(a)	5	Corr = 0.99554						(b)	5	Corr = 0.80967					0
(a)	5							(b)	5
3								3
ΠP/⟨ΠP⟩ 1								1
							–10	1
							–10	–1
–3							–15	–3							–15
–5		–3	–1	1	3	5	–5			–3	–1	1	3	5
(c)	5	Corr = 0.65245								Corr = 0.287
(c)	5
3								3
–1							–5	1
							–5	–1
							–10	–3
–3							–10	–3
–5		–3	–1	1	3	5	–5			–3	–1	1	3
ΠP F /⟨ΠP F⟩
Figure 11. Contour plot of the logarithm of the joint PDF of ΠP/⟨ΠP⟩ and ΠF P/⟨ΠF P⟩ for (a) ℓ/η = 0.25, (b) ℓ/η = 6, (c) ℓ/η = 16, (d) ℓ/η = 60. Colours correspond to the logarithm of the PDF, and the correlation coefficient is shown at the top of each plot. of observing B/⟨B⟩ < 0 is very low, suggesting that convective motion is very rare at these scales. As ℓ decreases, however, the probability of observing B/⟨B⟩ < 0 increases significantly. At ℓ/η = 16, −⟨B⟩ ≈ ⟨ΠK⟩, and the results in figure 12 show that at this scale, convective motion B/⟨B⟩ < 0 can occur, but the PDF is strongly skewed towards stably stratified regions that have B/⟨B⟩ > 0. Moreover, for this scale the PDF is stretched significantly along the ΠK axis, showing that fluctuations of ΠK are considerably is varied over the range considered, while other statistical characterizations of the velocity gradients also reveal behaviour that is qualitatively similar as the filter scale is increased