12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

PRODUCTION AND TRANSPORT OF TURBULENT HELICITY IN
WALL-NORMAL ROTATING CHANNEL FLOW

Maito Horie
Institute of Industrial Science
The University of Tokyo
4-6-1 Komaba Meguro-ku, Tokyo 153-8505, JAPANemail: maito@iis.u-tokyo.ac.jp

Fujihiro Hamba
Institute of Industrial Science
The University of Tokyo
4-6-1 Komaba Meguro-ku, Tokyo 153-8505, JAPANemail: hamba@iis.u-tokyo.ac.jp

for example, can be expressed by χhαα = Ch (ε /K)h where turbulent energy K, the dissipation rate ε , and helicity h, andCh is model constant. However, there is no verification that
discusses how accurate this model is, so we analyze the budget terms of helicity and discuss how the helicity is generated,transported, and dissipates.
In order to generate helicity, which is a pseudo-scalarquantity represented by the inner product of the axial and polar
vectors, it is necessary to break a mirror symmetry. One of themethods to provide a helicity is a rotation of the system. In thisstudy, we add a rotation of the system vertical to the wall of thechannel flow. It was shown by Deusebio et al.(2014) that positive turbulent helicity is generated near the wall by production
term and negative turbulent helicity is generated away from thewall by pressure diffusion term when the rotation vector of thesystem is vertically upward in the Ekman boundary layer.
In this study, we analyzed the production and transportmechanism of turbulent helicity by using the LES of turbulent channel flow with wall-normal rotation in order to improve the RANS turbulence model. We use LES to calculate in the case of high Reynolds numbers for future perspective. This is because to discuss the negative helicity awayfrom the wall, we should reduce the effect from the wall
for the accurate analysis. Since the turbulent helicity is distributed at low wavenumbers (Minnini, 2009), the turbulent
helicity is approximated only by grid-scale physical quantities
ω ′ ⟩+⟨uu”· ω ”⟩ ∼ ⟨ūu′ · ω̄ω ′ ⟩ where ⟨ ⟩ denotesh = ⟨uu′ · ω ′ ⟩ = ⟨ūu′ · ω̄′
ensemble averaging, u (= u − ⟨u⟩) is the fluctuation, and ū andu” are grid-scale and subgrid-scale velocities. Here, we further
decompose the helicity into three parts as ⟨ū′ · ω̄x ′ ⟩, ⟨v̄′ · ω̄y ′ ⟩,and ⟨w̄′ · ω̄z ′ ⟩ and examine the contribution of each term in thebudget equation for each term. In this study, we deal with thecase where both the rotation and Reynolds number are smaller
than the study reported by Deusebio et al. (2014).

ABSTRACT
Turbulent helicity is known to be generated in the boundary layer when the system is rotating on an axis perpendicular
to the wall. In order to investigate the production and transportmechanism, we used LES of wall-normal rotating turbulent
channel flow to examine the contribution of each term in thebudgets of the turbulent helicity and its three parts ⟨ū′ · ω̄x ′ ⟩,⟨v̄′ · ω̄y ′ ⟩, and ⟨w̄′ · ω̄z ′ ⟩. We found that the pressure-diffusion
term had the largest contribution to the generation of the turbulent helicity throughout the whole region. However, when
we decompose the turbulent helicity into three terms, the contribution of the production term is the largest in the budgets of⟨ū′ · ω̄x ′ ⟩ and ⟨w̄′ · ω̄z ′ ⟩, which are superior to ⟨v̄′ · ω̄y ′ ⟩. The
pressure term contributes to ⟨v̄′ · ω̄y ′ ⟩, and is redistributed to⟨w̄′ · ω̄z ′ ⟩ to decrease the absolute value of ⟨w̄′ · ω̄z ′ ⟩. Because⟨w̄′ · ω̄z ′ ⟩ has opposite sign to ⟨ū′ · ω̄x ′ ⟩, ⟨ū′ · ω̄x ′ ⟩ becomes thebiggest term of three and forms positive helicity near the walland negative helicity away from the wall. It was shown that a
subtle balance of ⟨ū′ · ω̄x ′ ⟩ and ⟨w̄′ · ω̄z ′ ⟩ accounts for the profileof the total helicity.

INTROODUCTION
Fluid motions such as atmospheric and oceanic flows andconvection of molten iron in the outer core are accompanied bylarge-scale rotational motions such as the rotation of the Earth.
It has been pointed out that magnetic and fluid structures suchas geomagnetic fields and typhoons are often generated andmaintained by helical turbulent flow fields against eddy viscosity and anomalous resistance(Yokoi et al., 1993; Inagaki
et al., 2017). The mechanisms that maintain such large-scale
magnetic fields and vortices have been intensively studied especially in the field of magnetohydrodynamic flows. One of
the invariants that characterize the helical motion is helicity,which is a conserved quantity defined as the inner product ofvelocity u and vorticity ω = ∇ × u , and is known to play animportant role in the α effect, one of the mechanisms of magnetic field generation in magnetohydrodynamics.
The model of helicity was proposed by Yokoi et al.,(1993). In this model, the dissipation rate of helicity, χhαα ,METHODThe governing equations for the turbulent flow in a wallnormal rotating channel are grid-scale Navier-Stokes equations for an incompressible fluid. We use the SGS model1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
the viscous term to the Coriolis term. In this study, we discuss
how helicity is generated and transported in the case of weakrotation. The equation for ⟨u¯α ′ · ω¯α ′ ⟩ for grid-scale Reynoldsaveraged turbulent helicity in channel flow isfor approximating small scales. To normalize the equations,the√ friction velocity of the non-rotating channel flow uτ =| ν dU/dy | |y=wall is used as the velocity scale, and the halfheight of the channel δ as the length scale. Then the nondimensional equations are given as

∂ ūi
∂
∂
=−
ūi ū j −
p̄
∂t
∂xj
∂ xi
∂
+
[(νS + ν )S̄i j ] + 2εi jk ū j ΩFk
∂xj

(1)

∂
ūi = 0
∂ xi

(2)

where

S̄i j =∂ ūi ∂ ū j+
∂ x j ∂ xiD⟨u¯α ′ · ω¯α ′ ⟩=Pαα + Παα + χhαα − ∇ · t pressααDt− ∇ · tturbαα − ∇ · tviscαα + rαα

where Pαα : generation term, Παα : pressure strain correlation
term, χhαα : dissipation term, ∇ · t pressαα : pressure diffusion
term, ∇ · tturbαα : turbulent diffusion term, ∇ · tviscαα : viscous
diffusion term, rαα : transport term due to Coriolis force(3)

The sketch of the wall-normal rotating channel is shownin figure 1. The boundary conditions in the x- and z-directions
are periodic boundaries, and the boundary conditions in they-direction are solid wall boundary conditions with non-slip
conditions. The size of the computational domain is set toLx × Ly × Lz = 4πδ × 2δ × 2πδ . As shown in figure 1, thedirection which has a constant pressure gradient is the xdirection, the wall vertical direction is the y-direction, andthe span direction is the z-direction. The rotation axis direction is the y-direction. The channel half-width was set as δ,and the value at the center of the channel was set as the origin of the y-coordinate. Therefore, the mean pressure gradient−∂x ⟨p⟩ = u2τ /δ = 1.
To solve the incompressible Grid-Scale Navier-Stokes
equations (1) and (2), the second-order Adams-Bashforthmethod is used for the time integral, the second-order central
difference is used for the spatial difference, and to solve the
Poisson equation for pressure, FFT is used in the x-z directions.The number of computational grids is 64 × 64 × 64, andthe time width is 2.5 × 10−3 .
In the y-direction, non-uniform grid defined by equation(4) was used(Moin and Kim, 1982). For the wall function fW
attach to the eddy viscosity coefficient νS , the Van-Driest typeequation (8) is adopted. Equation (9) is used for the model
coefficients for νS .

yj =

1
2j
tanh(b(−1 + )) (0 ≥ j ≥ Ny ) = 0
aNya = 0.98346, b = tanh−1 (a)√S̄i j S̄i j2
νS = (CS fW ∆)2√∆ = ∆x∆y∆z(10)Pxx = −⟨ū′ v̄′ ⟩

′ ′
′ ′
∂ Ωx
∂U
− (⟨v̄ ω̄x ⟩ − ⟨ū ω̄y ⟩)
∂y
∂y

(11)

Pzz = −⟨v̄′ w̄′ ⟩

′ ′
′ ′
∂ Ωz
∂W
− (⟨v̄ ω̄z ⟩ − ⟨w̄ ω̄y ⟩)
∂y
∂y

(12)

Παα = ⟨ p̄′

′

∂ ω̄α
⟩
∂ xα
′

χhαα = 2⟨[(νS + ν )S̄α j ]∇ · tviscαα =

′

∂ ω̄α
⟩
∂xj

(14)

∂ ′ ′
⟨ p̄ ω̄y ⟩
∂y

(15)

′
′
′ ′
′
∂
[⟨ūα ω̄α v̄ ⟩ − ⟨ūα2 ω̄y ⟩]
∂y

(16)

∇ · t pressyy =∇ · tturbαα =

(13)

′ ′
∂
[−2⟨{(νS + ν )S̄α y } ω̄α ⟩]
∂y
′
′ ∂
∂
+ [−⟨ū{(νS + ν )S̄zα } ⟩
∂y
∂ xα
′ ∂
′
+ ⟨w̄
{(νS + ν )S̄xα } ⟩]∂ xαryy = 2ΩF

∂ ′2
⟨v̄ ⟩
∂y

(17)

(18)

Here, ⟨ ⟩ represents the average in the x-z plane and in time.
α = x, y, z and no summation is taken for αα . All terms notshown in (11)-(18) are zero.

(4)
RESULT
Mean Velocity(5)

The profiles of the mean velocity in the streamwise andspanwise directions, U and W , are shown in figures 2(a) and(b). As the rotation is imposed, the spanwise velocity W is produced and maintained. When the rotation increases, the spanwise mean velocity increases while the streamwise mean velocity decreases. Compared with Liu’s DNS(Liu et.al. 2005),
our LES calculations qualitatively reproduce the DNS results.

(6)
(7)

y+
fW = 1 − exp(− )Ap(8)CS = 0.1, A p = 25

(9)Helicity
Figure 3 shows the distribution of turbulent helicity andthe intensities of turbulent helicity ⟨ū′ · ω̄x ′ ⟩, ⟨v̄′ · ω̄y ′ ⟩, ⟨w̄′ ·ω̄z ′ ⟩ for Nτ = 0, 0.01, 0.02. Since all of the distributions wereantisymmetric across the channel, we will discuss about thelower-half of the channel. Turbulent helicity generated andmaintained positive values for −1 < y < −0.9 and negativevalues for −0.9 < y < 0. The relationship between the sign

Table 1 shows the non-dimensional parameters used inthis study. The Rossby number Roτ represents the ratio of theadvection term to the Coriolis term, and the smaller it is, themore dominant the Coriolis force becomes. Since the Rossby
number is quite large in this study, the Coriolis force is expected to contribute only to the mean velocity with a large
spatial scale. The Ekman number Ekτ represents the ratio of2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
on ⟨v̄′ ω̄y′ ⟩ is almost balanced by the pressure strain correlation
term. The pressure strain correlation term represents the redistribution from ⟨v̄′ ω̄y′ ⟩ to the other terms, and since almost
all of them are redistributed, ⟨v̄′ ω̄y′ ⟩ itself was small and thedissipation term was also small. The redistribution was biased toward ⟨w̄′ ω̄z′ ⟩ rather than ⟨ū′ ω̄x′ ⟩ (Fig. 4 and 5 (b,d)).
The redistributed helicity acted to reduce the absolute value of⟨w̄′ ω̄z′ ⟩ because it was opposite to the sign of ⟨w̄′ ω̄z′ ⟩. On theother hand, ⟨ū′ ω̄x′ ⟩ had no effect of redistribution and its absolute value was not reduced, so the resulting turbulent helicity
h represented the same sign as ⟨ū′ ω̄x′ ⟩.
In the case of Nτ = 0.01 and Nτ = 0.02, where the rotation is relatively small, the pressure-diffusion term is significantly related to the turbulent helicity generation mechanismin the whole region. Near the wall, the production term wasalso effective. The pressure diffusion term transports the positive helicity towards the wall. Viscous dissipation and viscous diffusion generally contributed in the direction of reducing the absolute value of helicity. Looking at each of the threedecomposition terms of the turbulent helicity h, ⟨ū′ ω̄x′ ⟩ and⟨w̄′ ω̄z′ ⟩, so that the magnitudes of these two terms are larger
than the sum of all three terms h. Near the center of the channel, −0.5 < y < 0, the ⟨v̄′ ω̄y′ ⟩ is generated by the pressure diffusion term and the ⟨w̄′ ω̄z′ ⟩ is generated by the production term
were redistributed and canceled by the pressure redistribution
term. The ⟨ū′ ω̄x′ ⟩ was generated by the production term without any contribution from the other helicity by the pressure
redistribution term and dissipated by the term. In addition, because of the weak rotation in this study, the transport term dueto the Coriolis force had little effect on the helicity.
The variation in turbulent helicity with increasing the rotation parameter in figure 5 shows that the negative turbulent
helicity decreases in the region −0.9 < y < −0.8. Comparing
the contribution of each term in the budgets of turbulent helicity in figures 4(a) and 5(a), most of the budgets doubled whenrotation parameter ΩF doubled, however, the effect of the negative magnitude of the turbulent diffusion term decreases inthis region, indicating that this change is due to the turbulent
diffusion term. As for the reason for the decrease in the turbulent diffusion term, from the study of Liu et al. (2005), it isfound that the turbulent energy decreases when the rotation isincreased, while equation (16) shows that the turbulent energy
has a positive effect on the turbulent diffusion term.

of the turbulent helicity and the direction of the given rotationis consistent with the results of Deusebio et al. (2014). By
decomposing the turbulent helicity into three terms, we cansee in which terms the turbulent helicity is dominant. ⟨ū′ · ω̄x ′ ⟩and ⟨w̄′ · ω̄z ′ ⟩ are about the same size and are 20-30 times larger
than ⟨v̄′ · ω̄y ′ ⟩. Since ⟨ū′ · ω̄x ′ ⟩ and ⟨w̄′ · ω̄z ′ ⟩ are in opposite
phases to each other, h is about 1/6 to 1/3 times as large as ⟨ū′ ·ω̄x ′ ⟩, but still larger than ⟨v̄′ · ω̄y ′ ⟩. Since ⟨ū′ · ω̄x ′ ⟩ is slightly
larger than ⟨w̄′ · ω̄z ′ ⟩, the phase of h and ⟨ū′ · ω̄x ′ ⟩ are the same.
Comparing the distributions of ⟨ū′ · ω̄x ′ ⟩ and ⟨w̄′ · ω̄z ′ ⟩ for eachrotational parameter, both values are roughly proportional to
the rotational parameter for −1 < y < −0.9, which is close
to the wall. In contrast, for −0.9 < y < 0, the variation in⟨w̄′ · ω̄z ′ ⟩ is larger than that in ⟨ū′ · ω̄x ′ ⟩. In the case of Nτ =0.01, further away from the wall, at −0.7 < y < 0, ⟨w̄′ · ω̄z ′ ⟩had a much smaller value than ⟨ū′ · ω̄x ′ ⟩. Since the value of⟨ū′ · ω̄x ′ ⟩ did not cancel out with the other terms, h was large
enough to be compared to the peak value even near the centerof the channel.

Budget of Helicity
The budgets of turbulent helicity h and its intensities ineach direction are shown in figures 4 and 5. First, As it isshown in figures 3(a), 4(a) and 5(a), the positive helicity inthe region around −1 < y < −0.9 is produced by the pressurediffusion term, and production term with almost the same contribution. The largest contribution of the pressure-diffusion
term is different from the result of Deusebio et al.(2014),
where the contribution of the generation term was the largest
when the rotation was stronger. On the contrary, the viscous
diffusion and viscous dissipation terms contributed to attenuation of the positive helicity in this region. The viscous diffusion terms transported the positive helicity away from the walland helped to equalize the helical structure. The viscous dissipation also generally contributed to the dissipation of the positive helicity. However, there is a gap between the point where
the positive and negative helicity switched and the point wherethe viscous dissipation and viscous diffusion terms switched,so that around y = −0.9, where the helicity switched, the contribution was not in the direction of positive helicity dissipation. There isn’t enough evidence to say that without looking
at the spectrum of helicity, but it is thought that viscous dissipation is not simply proportional to helicity like the modelof Yokoi et al., (1993) because it works on a small scale. The
turbulent diffusion term works in this region to transport thepositive helicity from the region a little farther from the wall
to the vicinity of the wall. On the other hand, if we focus on themechanism of negative helicity generation in the region awayfrom the wall (y > −0.9), we found that the turbulent diffusion
term contributes the most to the generation of negative helicity
from figure 5(a) to about −0.9 < y < −0.8 in the same section,and the pressure diffusion term contributes the most to the generation of negative helicity at y > −0.8. The fact that the pressure diffusion term was effective farthest from the wall wasin agreement with the study by Deusebio et al. (2014). The
turbulent transport term transports the positive helicity fromthe lower side to the upper side of the channel and negative
helicity from the upper side to the lower side of the channel.
Viscous dissipation and viscous diffusion generally contributeto the reduction of the absolute value of helicity.
Next, we consider the budgets of three terms of the turbulent helicity ⟨ū′ ω̄x′ ⟩, ⟨v̄′ ω̄y′ ⟩, and ⟨w̄′ ω̄z′ ⟩. First, we discuss⟨v̄′ ω̄y′ ⟩, where the pressure diffusion term was the most important contributor to the generation of turbulent helicity. Figures
4 (c) and 5 (c) show that the pressure diffusion term acting

CONCLUSIONSTo discuss the generation mechanism of turbulent helicity,
we evaluate the distribution of turbulent helicity and its threeparts by using LES of channel turbulence rotating normal to
the wall. The positive turbulent helicity is generated near thewall by the production term and pressure diffusion term andthe negative turbulent helicity is generated away from the wallby the turbulent transport term and pressure diffusion term. In
addition, we found that ⟨v̄′ · ω̄y ′ ⟩ is generated by pressure diffusion term and is redistributed to ⟨w̄′ · ω̄z ′ ⟩, which makes its
absolute value smaller. Because the magnitude of ⟨ū′ · ω̄x ′ ⟩ isslightly greater than that of ⟨w̄′ · ω̄z ′ ⟩, the total helicity h becomes positive near the wall, −1 < y < −0.9, and negative
away from the wall, −0.9 < y < 0. It was shown that a subtle
balance of ⟨ū′ · ω̄x ′ ⟩ and ⟨w̄′ · ω̄z ′ ⟩ accounts for the profile oftotal helicity h. If we increase the parameter of the rotation to
double, most of the budgets get doubled but the effect of turbulent diffusion decreases, this is because the turbulent diffusion
probably monotonically decreases when turbulent energy decreases as rotation increases.

3

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
REFERENCES

boundary layer, J.Fluid Mech, 755, p.654, 2014.
[4] P. Mininni, A. Pouquet, Helicity cascades in rotating
turbulence, Physical Review E 79, p.026304, 2009[5] P.Moin, J.Kim, ”Numerical investigation of turbulent
channel flow”J.Fluid Mech,118, pp.341-377, 1982
[6] N.-D. Liu, X.-Y. Lu, Direct numerical simulation of turbulent flows in a wall-normal rotating channel, Journalof Turbulence, 6, No.34, 2005.

[1] N. Yokoi, A. Yoshizawa, Statistical analysis of the effects of helicity in inhomogeneous turbulence, Physics
of Fluids A: Fluid Dynamics 5,p.464, 1993.
[2] K. Inagaki, N. Yokoi, F. Hamba, Mechanism of mean
flow generation in rotating turbulence through inhomogeneous helicity, Physical Review Fluids 2,p.114605,
2017.
[3] E. Deusebio, E. Lindborg,
Helicity in the Ekman

412th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

Table 1: Parameters of simulations

Reτ
194
194
194

ΩF
0
0.005
0.01

Nτ
0
0.01
0.02

Roτ
∞
200
100

(a)

Ekτ
∞
1.03
0.515

4

𝘕 = 0.00
𝘕 = 0.01
𝘕 = 0.02

2\it{U}(\it{y})

𝘩

0

−2

y Ω0

x

−4
−1.0 −0.8 −0.6 −0.4 −0.2 0.0𝘺- ∂p/∂x
U(y)
W(y)

(b)

z

𝘕 ̄ 0.00
𝘕 ̄ 0.01
𝘕 ̄ 0.02

20

Figure 1: Configuration of wall-normal rotating channel

flow⟨𝘶′̄ 𝘹̄ ′⟩0−20−1.0 −0.8 −0.6 −0.4 −0.2 0.0

𝘺

(c)

0.4

𝘕 ̄ 0.00
𝘕 ̄ 0.01
𝘕 ̄ 0.02

0.2⟨𝘷′̄ 𝘺̄ ′⟩

0.0

−0.2
−0.4
−1.0 −0.8 −0.6 −0.4 −0.2 0.0𝘺(d)𝘕 ̄ 0.00
𝘕 ̄ 0.01
𝘕 ̄ 0.02

20⟨𝘸̄ ′𝘻̄ ′⟩0−20−1.0 −0.8 −0.6 −0.4 −0.2 0.0
Figure 2: Profiles of the mean velocities: (a) streamwise

component; (b) spanwise component𝘺Figure 3: (a) Turbulent helicity h and its three parts, (b)⟨ū′ · ω̄x ′ ⟩, (c)⟨v̄′ · ω̄y ′ ⟩ and (d)⟨w̄′ · ω̄z ′ ⟩512th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

(a)

50
𝘩

(a)

Production :𝘗
Viscous dissipation :𝘩Turbulent diffusion :−∂𝘺𝘛𝘵𝘶𝘳𝘣𝘺
Pressure diffusion : −∂𝘺𝘛𝘱𝘳𝘦𝘴𝘺
Viscous diffusion :−∂𝘺𝘛𝘷𝘪𝘴𝘤𝘺
Coriolis transport : 𝘳

0

50
𝘩

𝘩 2.5
0.0
−2.5

−50

0

′

𝘺

(b)

Production : 𝘗𝘹𝘹
Pressure strain correlation : 𝘹𝘹
Viscous dissipation : 𝘩Turbulent diffusion : −⟩𝘺𝘛𝘵𝘶𝘳𝘣 𝘺
Pressure diffusion : −⟩𝘺𝘛𝘱𝘳𝘦𝘴 𝘺
Viscous diffusion : −⟩𝘺𝘛𝘷𝘪𝘴𝘤 𝘺Coriolis transport : 𝘳𝘹𝘹

′

𝘹𝘹

𝘹𝘹

0

′

−100

′

′

50

−200

𝘷 𝘺̄ ⟩
′

0

(c)

−25

50

Production : 𝘗
Pressure strain correlation : 
Viscous dissipation : 𝘩Turbulent diffusion : −⟩𝘺𝘛𝘵𝘶𝘳𝘣𝘺
Pressure diffusion : −⟩𝘺𝘛𝘱𝘳𝘦𝘴𝘺
Viscous diffusion : −⟩𝘺𝘛𝘷𝘪𝘴𝘤𝘺Coriolis transport : 𝘳𝘺𝘺25𝘷 𝘺̄ ⟩′′⟨ ̄⟨𝘷 ̄ 𝘺̄ ⟩ 2.5
0.0
−2.5
′

′

0

−25

𝘷 𝘺̄ ⟩ 2.5
′

⟨ ̄

′

0.0
−2.5

−0.5 𝘺0.0−50−1.0 −0.8 −0.6 −0.4 −0.2 0.0−0.5 𝘺0.0−50−1.0 −0.8 −0.6 −0.4 −0.2 0.0

(d)

(d)

𝘺

𝘺

Production : 𝘗𝘻𝘻
Pressure strain correlation : 𝘻𝘻
Viscous dissipation : 𝘩Turbulent diffusion : −⟩𝘺𝘛𝘵𝘶𝘳𝘣 𝘺
Pressure diffusion : −⟩𝘺𝘛𝘱𝘳𝘦𝘴 𝘺
Viscous diffusion : −⟩𝘺𝘛𝘷𝘪𝘴𝘤 𝘺Coriolis transport : 𝘳𝘻𝘻

100

Production : 𝘗𝘻𝘻
Pressure strain correlation : 𝘻𝘻
Viscous dissipation : 𝘩Turbulent diffusion : −⟩𝘺𝘛𝘵𝘶𝘳𝘣 𝘺
Pressure diffusion : −⟩𝘺𝘛𝘱𝘳𝘦𝘴 𝘺
Viscous diffusion : −⟩𝘺𝘛𝘷𝘪𝘴𝘤 𝘺Coriolis transport : 𝘳𝘻𝘻

200

𝘻𝘻

𝘻𝘻

𝘻𝘻

𝘻𝘻

𝘻𝘻

′

𝘻𝘻

𝘻𝘻

𝘸̄ 𝘻̄ ⟩ 0

⟨

𝘺

𝘺

𝘺𝘺

⟨ ̄

′

0.0
−1.0 −0.8 −0.6 −0.4 −0.2 0.0

Production : 𝘗
Pressure strain correlation : 
Viscous dissipation : 𝘩Turbulent diffusion : −⟩𝘺𝘛𝘵𝘶𝘳𝘣𝘺
Pressure diffusion : −⟩𝘺𝘛𝘱𝘳𝘦𝘴𝘺
Viscous diffusion : −⟩𝘺𝘛𝘷𝘪𝘴𝘤𝘺Coriolis transport : 𝘳

25

5.0
0.0
−2.5
−5.0 −0.5

𝘶 𝘹̄ ⟩ 2.5

⟨ ̄

𝘺

′

𝘹𝘹

𝘹𝘹

0

′

−0.5 𝘺0.0−1.0 −0.8 −0.6 −0.4 −0.2 0.0

(c)

𝘹𝘹

𝘹𝘹

𝘶 𝘹̄ ⟩⟨ ̄2⟨𝘶 ̄ 𝘹̄ ⟩0−2′

Production : 𝘗𝘹𝘹
Pressure strain correlation : 𝘹𝘹
Viscous dissipation : 𝘩Turbulent diffusion : −⟩𝘺𝘛𝘵𝘶𝘳𝘣 𝘺
Pressure diffusion : −⟩𝘺𝘛𝘱𝘳𝘦𝘴 𝘺
Viscous diffusion : −⟩𝘺𝘛𝘷𝘪𝘴𝘤 𝘺Coriolis transport : 𝘳𝘹𝘹

200

𝘹𝘹

𝘹𝘹

𝘶 𝘹̄ ⟩

⟨ ̄

2.5
0.0
−2.5

−0.5 𝘺0.0−1.0 −0.8 −0.6 −0.4 −0.2 0.0

𝘺

100

𝘩

−50

−0.5 𝘺0.0−1.0 −0.8 −0.6 −0.4 −0.2 0.0(b)Production :𝘗
Viscous dissipation :𝘩Turbulent diffusion :−∂𝘺𝘛𝘵𝘶𝘳𝘣𝘺
Pressure diffusion : −∂𝘺𝘛𝘱𝘳𝘦𝘴𝘺
Viscous diffusion :−∂𝘺𝘛𝘷𝘪𝘴𝘤𝘺
Coriolis transport : 𝘳

′

2.5
0.0
−2.5

′

′

𝘸̄ 𝘻̄ ⟩

⟨

−100

′

𝘻𝘻

𝘸̄ 𝘻̄ ⟩ 0⟨′𝘸̄ 𝘻̄ ⟩ 2.5

⟨

−200

−0.5 𝘺0.0−1.0 −0.8 −0.6 −0.4 −0.2 0.0

′

′

0.0
−2.5

−0.5 𝘺0.0−1.0 −0.8 −0.6 −0.4 −0.2 0.0𝘺𝘺Figure 4: Budgets of (a) turbulent helicity h and its three

Figure 5: Budgets of (a) turbulent helicity h and its threeparts (b) ⟨ū′ · ω̄x ′ ⟩, (c)⟨v̄′ · ω̄y ′ ⟩ and (d)⟨w̄′ · ω̄z ′ ⟩, when Nτ =0.02. The inset is the magnified view around the center ofthe channel.

parts (b) ⟨ū′ · ω̄x ′ ⟩, (c)⟨v̄′ · ω̄y ′ ⟩ and (d)⟨w̄′ · ω̄z ′ ⟩, when Nτ =0.01. The inset is the magnified view around the center ofthe channel.

6