Boundary-Layer Meteorol https://doi.org/10.1007/s10546-018-0356-4 RESEARCH ARTICLE

Turbulent Helicity in the Atmospheric Boundary Layer Otto G. Chkhetiani1 · Michael V. Kurgansky1 · Natalia V. Vazaeva1

Received: 21 April 2017 / Accepted: 3 May 2018 © Springer Science+Business Media B.V., part of Springer Nature 2018

Abstract We consider the assumption postulated by Deusebio and Lindborg (J Fluid Mech 755:654–671, 2014) that the helicity injected into the Ekman boundary layer undergoes a cascade, with preservation of its sign (right- or alternatively left-handedness), which is a signature of the system rotation, from large to small scales, down to the Kolmogorov microscale of turbulence. At the same time, recent direct field measurements of turbulent helicity in the steppe region of southern Russia near Tsimlyansk Reservoir show the opposite sign of helicity from that expected. A possible explanation for this phenomenon may be the joint action of different scales of atmospheric flows within the boundary layer, including the sea-breeze circulation over the test site. In this regard, we consider a superposition of the classic Ekman spiral solution and Prandtl’s jet-like slope-wind profile to describe the planetary boundary-layer wind structure. The latter solution mimics a hydrostatic shallow breeze circulation over a non-uniformly heated surface. A 180°-wide sector on the hodograph plane exists, within which the relative orientation of the Ekman and Prandtl velocity profiles favours the left rotation with height of the resulting wind velocity vector in the lowermost part of the boundary layer. This explains the negative (left-handed) helicity cascade toward small-scale turbulent motions, which agrees with the direct field measurements of turbulent helicity in Tsimlyansk. A simple turbulent relaxation model is proposed that explains the measured positive values of the relatively minor contribution to turbulent helicity from the vertical components of velocity and vorticity. Keywords Breeze circulation · Ekman boundary layer · Helicity · Thermal stratification

B 1

Michael V. Kurgansky [email protected] A.M. Obukhov Institute of Atmospheric Physics, Russian Academy of Sciences, Pyzhevsky 3, Moscow, Russian Federation 119017

123

O. G. Chkhetiani et al.

1 Introduction Flow helicity is a pseudo-scalar, the bulk density of which is defined mathematically by the dot product  H  u · ω of the Eulerian flow velocity u  (u, v, w) and vorticity ω ≡ ∇ × u  ωx , ω y , ωz at a point x  (x, y, z). 1 Its importance in dynamical meteorology and, in particular, in boundary-layer meteorology is being increasingly recognised, see, e.g., a recent review by Kurgansky (2017) and references therein. In basic fluid dynamics, Moreau (1961)  and Moffatt (1969) were the first to discover the conservation law of (total) helicity H  V u · ω dτ for  barotropic flows of an ideal fluid, in the presence of conservative external forces only; V dτ denotes the integration over volume V of the fluid. Since then, numerous papers have been devoted to the helicity concept and its applications, including within the geophysical and astrophysical fluid dynamics literature. Historically (Moffatt 1978; Krause and Raedler 1980) and up to the present the most successful application of helicity is to the ‘dynamo’ theory explaining the origination and amplification of a magnetic field. Etling (1985) and Lilly (1986) pioneered the use of helicity in meteorology and atmospheric physics: see also Bluestein (1992) and Kurgansky (1993), which were, eventually, the first meteorological monographs discussing helicity. In the atmospheric boundary layer (ABL), horizontal vortical structures, such as rolls and ‘cloud streets’ (the latter containing individual rotating clouds within them), indicate significant values of helicity (Etling 1985; Hauf 1985). The last property imparts stability and durability and, hence, predictability to these structures, cf. Lilly (1986) in the context of rotating thunderstorms. A prominent example of an atmospheric dynamical structure with non-zero helicity is the Ekman boundary layer. Different dynamical aspects of helicity in the Ekman layer have been studied by Hide (1989, 2002), Kurgansky (1989), Tan and Wu (1994), Chkhetiani (2001), Ponomarev et al. (2003), Ponomarev and Chkhetiani (2005), and Deusebio and Lindborg (2014). Kurgansky (1989) (see also Sect. 5.2 in Kurgansky 1993, 2002) and, more recently, Deusebio and Lindborg (2014) considered the helicity balance in the Ekman boundary layer and obtained the expression f ug2 for the time rate of helicity injection per unit square area, due to a pressure-related production term, into this layer. 2 Here, ug is the geostrophic velocity in the free atmosphere above the Ekman boundary layer; f is the Coriolis parameter, which is positive in the Northern Hemisphere and negative in the Southern Hemisphere. Kurgansky (1989) stated that this helicity injection is balanced exactly by the helicity turbulent viscous destruction within the Ekman boundary layer. Direct calculation (ibid) of the time rate of helicity turbulent viscous destruction, related to the so-called superhelicity S  ω · ∇ × ω (Hide 1989), 3 gave − f ug2 . The calculations were based on the explicit relations of Hide (1989) for the helicity bulk density in the classical Ekman spiral. Deusebio and Lindborg (2014) went further and, based also on direct numerical simulation results, conjectured that the injected helicity undergoes a cascade, with the preservation of its sign (right- or alternatively left-handedness) that is a sig1 Although H is commonly named the helicity density, or the helicity bulk density, and only the integral of H over an arbitrary volume is named the helicity, for the sake of brevity we use the word “helicity” for H throughout the paper, also keeping in mind that the latter term often means a property of the vortex flow to be helical. In any case, it should not lead to any confusion. 2 Tan and Wu (1994) took into account the effect of non-linearities in the governing equations and revealed the cyclonic-anticyclonic asymmetry that manifests itself in helicity values of the generalized Ekman spiral solution. They also considered the effect of baroclinicity (the thermal wind effect). However, these authors did not consider the helicity budget that constitutes an interesting future task in view of additional effects accounted for in Tan and Wu (1994). 3 Strictly speaking, a pseudo-scalar quantity S is the superhelicity (bulk) density, and only the integral of S over an arbitrary volume is named the superhelicity.

123

Turbulent Helicity in the Atmospheric Boundary Layer

nature of the system rotation, from large to small scales “all the way down to the Kolmogorov scale”. Deusebio and Lindborg stated: “By carrying out a measurement at millimetre scale of the helicity dissipation in the logarithmic range of the ABL, one would therefore be able to determine in which hemisphere the measurement is performed. In general, in a bottom Ekman boundary layer such a measurement would lead to a positive helicity dissipation in the Northern Hemisphere and a negative helicity dissipation in the Southern Hemisphere. The same quantity, measured at millimetre scale, would therefore show different signs in the two hemispheres.” Like Deusebio and Lindborg themselves, we find this result fascinating, but would like to draw attention to the discrepancy between it and the results of the direct measurements of turbulent helicity performed by Koprov et al. (2015) (hereafter, K2015) in the steppe region near Tsimlyansk Reservoir in southern Russia. It was experimentally found in K2015 that the w  ωz contribution to turbulent helicity was positive for all their measurements and during daytime convective conditions was on the order of 0.01 m s−2 . Here, a prime denotes a turbulent fluctuation and an overbar stands for the statistical averaging, approximated by the averaging over time; (x, y) are the horizontal coordinates and z is the height. Simultaneously, the u  ωx + v  ωy contribution is negative and ≈ − 0.03 m s−2 in the daytime, i.e., its magnitude is greater by approximately a factor of three than the w  ωz contribution. Therefore, the total turbulent helicity is negative. The indicated relationship between signs and magnitudes of the contributions to helicity remains unchanged during the night, when the boundary layer is stably stratified, but their absolute values are several times smaller. Here, we present certain fluid dynamical arguments to interpret possible reasons for the discrepancy between the theoretical results of Deusebio and Lindborg (2014) for an idealized barotropic Ekman boundary layer and the experimental results obtained in K2015 for a real baroclinic boundary layer in the presence of the sea breeze observed in Tsimlyansk. The paper is organized as follows: in Sect. 2, we give details of the experimental set-up in K2015 and describe the main results of the helicity measurements. In Sect. 3.1, a theoretical model is proposed, which explains why the velocity profile in the lower part of the ABL can have a mean helicity with a sign opposite to that of the helicity imposed by the Ekman spiral. It thus explains the cascade of negative helicity towards turbulent scales. In Sect. 3.2, we demonstrate that the model deductions, generally, agree with the helicity measurements in K2015. In Sect. 4, we derive expressions for the turbulent helicity, based on the helicity budget equation and simple relaxation theory application. These expressions are used to relate turbulent helicity to other turbulent quantities, particularly to the buoyant production rate of turbulent helicity. Section 5 summarizes the obtained results and presents concluding remarks.

2 Turbulent Helicity Measurements in K2015 In a field campaign during August 2012 in Tsimlyansk, Russia, see Fig. 1, 4 a set of four sonic anemometers positioned at the vertices of a rectangular tetrahedron, with an approximate 5-m distance between the anemometers and a 5.5-m elevation of the tetrahedron base above the ground surface, was applied to measure the helicity (Figs. 2, 3). The sampling rate of the four sonic anemometers was 32 Hz. The velocities measured at the tetrahedron vertices were used to calculate circulations along the edges of the faces perpendicular to the coordinate 4 This work by Koprov et al. (2015) continues the previous direct helicity measurements by Koprov et al. (2005) in Zvenigorod near Moscow, using new equipment and an updated methodology (see the main text).

123

O. G. Chkhetiani et al.

Fig. 1 A general view of the measurement site in Tsimlyansk. The Tsimlyansk polygon (Tsimlyansk Scientific Station of the A.M. Obukhov Institute of Atmospheric Physics in Moscow), where the measurements were taken, is labelled with a flag and positioned 2.6 km to the west of the coast of Tsimlyansk Reservoir (see also the zoomed image in the upper-left corner)

axes. The obtained circulations were, in turn, used to find the corresponding average vorticity components by means of division of the circulation by the face areas. These average values were taken as the vorticity components ωx , ω y and ωz , corresponding to the device spatial dimension, and used to calculate the helicity. An alternative calculation method uses finite differences for approximating the spatial derivatives in the relations for ωx , ω y and ωz at the point A (see Fig. 3). The helicity at this point was calculated from   w A u B − u A w B v A u D − u A v D w A vC − v A wC H A  u A (ωx ) A + v A ω y A + w A (ωz ) A ≈ + + , L AB L AD L AC

(1) where indices A, B, C and D denote a spatial point at which the velocity (vorticity) component is measured (calculated). Notations L AB , L AD and L AC stand for lengths of the corresponding edges, see Fig. 3. Both methods give similar results. The main results of these measurements are summarized in Sect. 1, and the turbulent helicity values measured in K2015 are given in more detail in Table 1. Turbulent fluctuations (denoted with a prime) are determined as the deviations of instantaneous values from 30min running mean values, calculated with the use of a -shaped rectangular window. An overbar denotes the time averaging over the whole measurement interval (the duration of a measurement session). 5 It has to be emphasized that even if the mean turbulent helicity has a negative sign, the measurements described in K2015 demonstrate large random fluctuations of both signs of the instantaneous turbulent helicity, such that the corresponding standard deviation largely exceeds the mean value. Therefore, one can only confidently infer the sign of the mean turbulent helicity. This is illustrated in Figs. 4 and 5. Figure 4 presents both the instantaneous turbulent helicity values H  u · ω averaged over 1 s (thin grey line) and the result of 5 The use of running-mean removal is equivalent to high-pass filtering of incoming data and is called ‘detrending’ in the literature (e.g., McMillen 1988). In practice, covariances calculated by Reynolds averaging and running-mean removal are essentially identical except during periods of non-stationarity, when calculations using a running-mean removal seem to produce results with less scatter (McMillen 1988).

123

Turbulent Helicity in the Atmospheric Boundary Layer

Fig. 2 Photograph of the tetrahedron composed of four sonic anemometers and used for helicity measurements in Koprov et al. (2015). An additional amplified image of the most elevated sonic anemometer installed at 10-m height over the ground together with a cup anemometer and a windvane (alongside temperature, pressure and humidity sensors) is shown in the upper-left corner. (Photograph copyright O. Chkhetiani)

Fig. 3 Schematic of the rectangular tetrahedron ABCD used for helicity measurements. Horizontal (x, y) axes are directed northward and westward, respectively. Three anemometers are positioned at the vertices of a rectangular isosceles triangle ABC with the length of AB and AC equal to 5 m, and the fourth anemometer at point D is located 5 m higher, exactly above the vertex of right angle BAC

the further averaging over 5 min (black line). The dominance of negative helicity values is visible. Figure 5 shows the histogram of the instantaneous turbulent helicity values shown in Fig. 4. The daytime measurement session (Series #21, see K2015) started at 1238 local time (UTC + 3 h) on 8 August 2012 and lasted 14,113 s, i.e., until 1630 local time; see also Table 1. Dividing the N values in Fig. 5 by 14,113, we obtain an estimate of the probability density function (p.d.f.) of the H values, which can be compared to the similar Fig. 7 in Deusebio and Lindborg (2014), but for positive mean turbulent helicity in the latter case.

123

O. G. Chkhetiani et al. Table 1 Covariances obtained experimentally in K2015 and used in this manuscript u  ωx (m s−2 )

v  ωy (m s−2 )

w  ωz (m s−2 )

Day

− 0.0170

− 0.0276

0.0143

0.1545

0.0456

Night

− 0.0138

− 0.0010

0.0085

− 0.0172

− 0.0123

w  θ  (m K s−1 )

ωz θ  (K s−1 )

A right-handed Cartesian coordinate system is used: the x-axis is directed to the north, the y-axis is directed to the west, and the z-axis is directed upward. Values in the middle row correspond to the measurement session from 1238 to 1630 local time (UTC+3 h) on 8 August 2012 (Series #21; Table 1 in K2015). The values in the lower row correspond to the measurement session from 2246 to 0034 local time, the end of the session occurring on 9 August 2012 (Series # 35; Table 2 in K2015)

Fig. 4 Instantaneous turbulent helicity values (in m s−2 ) measured in Tsimlyansk, southern Russia on 8 August 2012 at 1238–1630 local time (thin grey line); K2015, see the text for more details. On the horizontal axis the numbers show the time (in sec) passed from the beginning of the day. A thick black line corresponds to additional 5-min averaging of instantaneous turbulent helicity values

At this stage, it is tempting to make a tentative estimate of the turbulent helicity magnitude for the conditions roughly corresponding to the measurements in K2015, from the viewpoint of the idealized theory by Deusebio and Lindborg (2014). We consider the main bulk of the Ekman boundary layer and assume that the helicity injected with the time rate f ug2 is cascaded as an effectively passive scalar quantity to turbulent motions of gradually decreasing spatial scales l belonging to the inertial interval. The turbulent helicity H t ≡ u · ω  H t (l) on the given spatial scale l is defined by the dot product of the vectors of velocity and vorticity turbulent fluctuations on that scale. In the bottom part of the Ekman boundary layer, the turbulent helicity can be estimated from Brissaud et al. (1973) H t (l)  C H η ε −1/ 3 l 2/ 3 .

(2)

Equation 2 corresponds to the concept of the joint cascade of energy and helicity to small scales, as discussed in detail by Chen et al. (2003). In Eq. 2, ε is the specific kinetic energy

123

Turbulent Helicity in the Atmospheric Boundary Layer

Fig. 5 Histogram of instantaneous turbulent helicity values measured in Tsimlyansk, southern Russia on 8 August 2012 at 1238–1630 local time (K2015; see the text for more details). On the horizontal axis the turbulent helicity values (in m s−2 ) are shown; the width of each bin equals 0.05 m s−2 . On the vertical axis, the numbers show the number N of helicity values falling within the corresponding bin. The sum of all N values equals 14,113 (see the text). A vertical red line shows zero helicity values H = 0

dissipation rate and η is the time rate of helicity viscous destruction. A non-dimensional constant C H has been estimated in the literature to be either 2.3 (André and Lesieur 1977) or 2.5 (Avinash et al. 2006). In our case, we have η  2 f ug2



hE,

(3)

where h E is the Ekman boundary-layer thickness. Equation 3 with a factor of 2 follows from the expression for the height-dependent superhelicity bulk density for the Ekman spi      3 2 ral, SE (z) ≡ ω · ∇ × ω  2ug h E exp −2z h E (e.g., Kurgansky 1989), taken at   z h E → 0. Indeed, η  2K SE (0)  2 f ug2 h E , where K is the small-scale eddy vis  cosity, and by definition h E  2K f . The total turbulent viscous destruction of helicity ∞   reads 2K 0 SE z  dz   f ug2 (Kurgansky 1989). Taking h E  400 m, f  10−4 s−1 ,



ug  10 m s−1 , ε  10−3 m2 s−3 , CH  2.4 as a compromise, and l  10 m as a rough estimate for the tetrahedron size (cf. K2015), we have H t ≈ 5.5 × 10−3 m s−2 , which is several times smaller than the H t values measured in the daytime by K2015. It should be emphasized that we have actually obtained a lower estimate for H t , which does not account for other sources of turbulent helicity in a bottom Ekman boundary layer. These sources include contributions from other types of flows, which may not only lead to an increase in the magnitude of H t , but also explain the obtained reversal of the sign of H t . Therefore, we

123

O. G. Chkhetiani et al.

Fig. 6 The direction of (1) the 850-hPa wind velocity and (2) the 10-m wind velocity (a 10-min running average has been applied) in Tsimlyansk, southern Russia. On the horizontal axis, elapsed time (in hours) is shown for 7–8 August 2012. The vertical axis shows the wind direction in degrees: 0° corresponds to the southward direction, 90° describes the westward direction, and so on

move to the discussion of these possible complicating factors and, first, describe the overall meteorological conditions during the field campaign described in K2015. 6 On the day of 8 August 2012 when the measurements were taken, two vast anticyclones dominated the fair-weather meteorological conditions. The axis connecting their centres was directed from the west-south-west to the east-north-east and the corresponding col was situated slightly northward of Tsimlyansk. As a result, the prevailing synoptic-scale westward flow in Tsimlyansk had a moderate wind speed (about 9 m s−1 at the 850-hPa level), but a remarkably intense local circulation was observed there, which is discussed in detail below. At the test site in Tsimlyansk parallel routine measurements of the meteorological parameters (air temperature, horizontal wind speed, wind direction, air pressure and air humidity) at heights of 2 and 10 m were performed; see also Fig. 2. The measurement complex was constructed based on Aanderaa Data Instruments’ sensors. In Fig. 6 we show for Tsimlyansk the wind direction at the 850-hPa level (approximately 1.5-km height), which is based on ECMWF re-analysis data and serves as a proxy for the geostrophic wind direction, and we depict the surface wind direction (at 10-m height) as well. Figure 7 shows the 10-m wind speed during an extended time period, by adding an extra day before the time range shown in Fig. 6. The diurnal variation in wind speed is apparent in Fig. 7, presumably related to the breeze circulation over the Tsimlyansk polygon. The westward oriented onshore (sea) breeze explains the wind-speed maxima in the noon and afternoon hours. Wind weakening occurs 

6 The scale λ ∼ ε 2/ 7 ν 3/ 7 2 f u 2 D g

 hE

−3/ 7

that emerges from balancing 2 f ug2



h E and the time rate   of helicity viscous destruction η  2ν ω · ∇ × ω corresponds to Ditlevsen’s scale (Ditlevsen and Giuliani

2001a, b; see also Ditlevsen 2011). Here, the Kolmogorov scaling for the turbulent velocity is used and ν is the (molecular) kinematic viscosity. For the above-indicated parameter values and ν ≈ 1.5×10−5 m2 s−1 (for air, under normal conditions) the scale λ D ≈ 1.1 × 10−1 m is two orders of magnitude greater than Kolmogorov’s microscale of turbulence λ K ≈ ν 3/ 4 ε −1/ 4 ≈ 1.4 × 10−3 m. The latter one is the true minimum scale of helical motions for a joint cascade of energy and helicity (Chen et al. 2003; see also, Briard and Gomez 2017). This difference between the two scales in question may be important when choosing the size of experimental equipment to measure helicity.

123

Turbulent Helicity in the Atmospheric Boundary Layer

Fig. 7 The 10-m wind speed V 10 (10-min running average has been applied) in Tsimlyansk, southern Russia. On the horizontal axis the local time (UTC+3, in hours) is shown for 6–8 August 2012. The vertical axis shows the wind speed (in m s−1 )

during the night hours, probably due to the contribution of the offshore (land) breeze, though the 10-m wind direction remains westward nearly all the time. It is worth mentioning in this regard that according to observations (e.g., Vorontsov 1955) the existent breeze circulation near Tsimlyansk has a complex and even fragile character related to the considerable activity of synoptic-scale pressure systems in this region. Maximum wind speeds in the sea breeze are observed at a height of 100 m and are up to 5–6 m s−1 ; for the land breeze the maximum wind speeds are about 2–3 m s−1 (Vorontsov 1955). Our hypothesis is that the joint action of different types of flows within the boundary layer, namely the Ekman spiral flow and the sea-breeze circulation, explains the negative sign of total turbulent helicity and its magnitude. To verify this hypothesis, a simple conceptual model of the planetary boundary-layer wind structure is considered in the next section.

3 Helicity and Superhelicity in the Ekman Boundary Layer Above a Non-uniformly Heated Surface 3.1 Theoretical Model In this sub-section, we propose a conceptual model that explains the negative sign of helicity in the bottom part of the ABL, even if positive helicity is injected into it. We consider a surfaceadjacent breeze circulation, caused by a non-uniformly heated surface, and superimpose it on the Ekman spiral flow. In such a way, we introduce a shallow baroclinic (buoyant) source of helicity, which is capable of explaining the negative sign of helicity in the bottom part of the boundary layer. In the presence of prescribed buoyancy b, the motion within the Ekman boundary layer is governed by the following set of equations, taken under the boundary-layer approximation Du ∂π ∂ 2u − fv  − +K 2, Dt ∂x ∂z ∂π ∂ 2v Dv + fu  − +K 2, Dt ∂y ∂z

(4a) (4b)

123

O. G. Chkhetiani et al.

∂π + b, ∂z ∂u ∂v ∂w + +  0. ∂ x ∂ y ∂z 0−

(4c) (4d)

Here (x, y, z) is the right-handed Cartesian coordinate system, (x, y) are the horizontal coordinates, (u, v) are the corresponding velocity components; w is the vertical velocity and z is the height; π is the hydrostatic pressure divided by the constant fluid density. The buoy ancy is defined as b  g (θ − θ∞ ) θ0 (g is the acceleration due to gravity, θ the potential temperature, θ∞ a height-dependent potential temperature,  environmental     θ0 a constant reference potential temperature); D Dt  ∂ ∂t + u∂ ∂ x + v∂ ∂ y + w∂ ∂z represents the total time derivative. Consistent with the boundary-layer approximation, the vertical accel eration, Dw Dt, is neglected in Eq. 4c. On the one hand, Eq. 4 constitute a particular case of Eqs. 2.1–2.3 analysed in Shapiro and Fedorovich (2013). On the other hand, the Earth’s rotation is accounted for in Eq. 4, and the considered problem is three-dimensional. Assume that at each level z   constant,fluid particles move with a constant speed along straight lines and, therefore, Du Dt  Dv Dt  0 in Eq. 4a, 4b. In this case, Eq. 4 reduce to a closed system of linear equations with the prescribed buoyant forcing b (x, y, z) ∂π ∂ 2u +K 2, ∂x ∂z ∂π ∂ 2v +fu  − +K 2, ∂y ∂z ∂π 0− + b. ∂z

−fv  −

(5a) (5b) (5c)

Following Gutman (1972), we assume that b  −F  (z) x − G  (z) y,

(6)

where functions F (z) and G (z), being otherwise arbitrary, are ‘concentrated’ in the vicinity of the Earth’s surface z  0. These two functions, along with their first derivatives, vanish at the top of the Ekman boundary layer at z → ∞. A prime in Eq. 6 denotes differentiation with respect to z. Equation 6 assumes that a relatively small area of the non-uniformly heated Earth’s surface is considered where, with good accuracy, the buoyancy is a linear function of the horizontal coordinates. This assumption makes the problem mathematically solvable. Owing to the geostrophic balance conditions in the free atmosphere, i.e., beyond the Ekman boundary layer, and in virtue of Eqs. 5c and 6, we have π  −F (z) x − G (z) y + f vg x − f u g y.

(7)

Here, by assumption, u g and vg are constant geostrophic wind components. By substituting Eq. 7 into Eq. 5a, 5b we obtain the resulting linear system of equations ∂ 2u + F (z) , ∂z 2 ∂ 2v f u  f u g + K 2 + G (z) , ∂z

− f v  − f vg + K

(8a) (8b)

which describes the Ekman boundary layer above an unevenly heated Earth surface. Equations 8 should be solved with the boundary conditions u  v  0 at z  0 and u → u g ,

123

Turbulent Helicity in the Atmospheric Boundary Layer

v → vg at z → ∞ We split the solution into the sum, u  u 0 + u 1 , v  v0 + v1 , of the solutions to the two systems of equations ∂ 2u0 , ∂z 2 ∂ 2 v0 + f u0  f ug + K 2 , ∂z

− f v0  − f vg + K

(9a) (9b)

and ∂ 2u1 + F (z) , ∂z 2 ∂ 2 v1 + f u 1  K 2 + G (z) , ∂z

− f v1  K

(10a) (10b)

correspondingly. Equations 9 correspond to the classical Ekman problem and are complemented by the boundary conditions u 0  v0  0 at z  0 and u 0 → u g , v0 → vg at z → ∞. The boundary conditions for Eq. 10 are, first, u 1  v1  0 at z  0, regardless of choices for F (z) and G (z). Second, the arbitrary functions F (z) and G (z) can always be chosen in such a way that u 1 → 0, v1 → 0 at z → ∞. Consequently, u  u 0 + u 1 , v  v0 + v1 satisfy both Eq. 8 and the relevant boundary conditions. Below we give a simple example of the implementation of this general scheme. To explain a possible reason for the sign reversal of the measured helicity compared to that in the main bulk of Ekman boundary layer, consider the classical Ekman solution (the geostrophic wind is directed along the x-axis and has a single component u g ) 

 1+i w0 ≡ u 0 + iv0  u g 1 − exp − z . (11) hE To simplify further calculations, we introduce a complex velocity w. Well-known relations describing the Ekman spiral follow by taking the real and imaginary part of Eq. 11. Superpose Eq. 11 with the surface-adjacent wind jet, which is assumed to be directed along the x-axis, similar to the geostrophic velocity above the Ekman boundary layer, i.e., v1 ≡ 0 in Eq. 10. This assumption is only made to simplify the mathematics and present the main idea in the clearest way. Afterwards, the general result for an arbitrary relative orientation of the geostrophic wind and surface-adjacent jet will be presented; see Eq. 17. It follows from  Eq. 10 that F (z)  −K ∂ 2 u 1 ∂z 2 and G (z)  f u 1 , i.e., F (z)  −K f −1 G  (z). For mathematical convenience, the vertical profile of the jet is described by the Prandtl (1942) classical relation for the slope wind (in the recent literature, see, e.g., Shapiro and Fedorovich 2007) 





 π 1+i 1−i z z U π exp i − sin w1 ≡ u 1 + iv1  U exp − ≡ z − exp i − z , h1 h1 2 2 h1 2 h1

(12) where h 1 is the characteristic vertical scale and U is the velocity magnitude, both to be specified further on. Equation 12 corresponds to   choice   of functions  the  following −2 F (z) and G in Eq. 10: F  2K U h cos −z h 1 and G (z)  exp −z h (z) (z) 1 1       f U exp −z h 1 sin −z h 1 . In our case the velocity profile described by Eq. 12 may have a different physical origin, which we broadly attribute to the hydrostatic breeze circulation developing heated surface. A known analytical solution over a non-uniformly    w1  U z h 1 exp −z h 1 to the problem of breezecirculation (Gutman 1972) is qualitatively similar to Eq. 12 and coincides with it at z h 1 << 1, but is less convenient

123

O. G. Chkhetiani et al.

mathematically because it does not allow the representation as a sum of two complex exponents as in Eq. 12. The hydrostatic equation, Eq. 5c, explains the lower/higher pressure above the more/less-heated ground surface. The resulting horizontal pressure gradient causes the breeze-like surface-adjacent circulation. Our approach is to focus on a simplified model of the kinematic structure of the boundarylayer flow, which is capable of explaining the effect of sign reversal of both helicity and superhelicity in the lowermost part of the Ekman boundary layer when the surface-adjacent baroclinic forcing is taken into account. To explain this kinematic flow, the baroclinic forcing may have a rather complex, though realistic, structure and may need the support of the corresponding diabatic forcing. Kurgansky (2005) applied a conceptually similar approach in his study of dry convective helical vortices. It is assumed hereafter  that the breeze circulation is very shallow, i.e., h 1 h E , and therefore λ ≡ h E h 1  1. In addition, we define a ratio δ ≡ U u g . Following the general  procedure outlined above, we take the dimensionless linear superposition W  (w0 + w1 ) u g of the complex velocities described by Eqs. 11 and 12, which depends  on the non-dimensional height ζ  z h E . Using u g as the velocity scale and h E as ˜ the height scale, we pass to the dimensionless helicity H˜ and superhelicity    S, which   are related to the original dimensional quantities H  u · ω  −u dv dz + v du dz           and S  ω · ∇ × ω  d2 u dz 2 dv dz − d2 v dz 2 du dz via the expressions −2 3 ˜ ˜ H˜  H u −2 g h E and S  Su g h E . It is most convenient mathematically to calculate H and ˜S with the use of   dW ∗ , (13a) H˜  Im W dζ   dW d2 W ∗ , (13b) S˜  Im dζ dζ 2 where the asterisk denotes complex conjugation. We obtain correspondingly H˜  exp (−2ζ ) + exp (−ζ ) (sin ζ − cos ζ ) + δ exp [− (1 + λ) ζ ] (sin ζ − cos ζ ) sin λζ − δ λ exp [− (1 + λ) ζ ] (sin λζ − cos λζ ) sin ζ

,

(14) S˜  2 exp (−2ζ ) − 2δλ exp [− (1 + λ) ζ ] (sin λζ − cos λζ ) cos ζ + 2δ λ2 exp [− (1 + λ) ζ ] (sin ζ − cos ζ ) cos λζ.

(15)

The first terms in Eqs. 14 and 15 (lacking the factor of δ) correspond to the classic Ekman solution (Hide 1989), while the other terms describe an ‘interference effect’ between the Ekman and Prandtl solutions (see Eqs. 11, 12). The helicity (superhelicity) obviously vanishes for the Prandtl solution described by Eq. 12. Near the ground surface ζ  0, the superhelicity S˜ (0)  2 + 2δλ − 2δ λ2 is negative when δ > λ−1 (λ − 1)−1 , i.e., the jet (see Eq. 12) has the same orientation as the geostrophic velocity (see Eq. 11). Forthe opposite orientation, S˜ (0) >





2 2 ˜ ˜ ˜ ˜ 0. Because H (0)  d H dζ

 0 and S (0)  d H dζ

(compare, Chkhetiani ζ 0

ζ 0

2001; Ponomarev and Chkhetiani 2005), in the former case the helicity H˜ is negative near the ground. Consequently, the negative helicity is transferred to turbulent motions within the surface-adjacent boundary layer. In dimensional units, the rate of injection of helicity to turbulent motions near the ground reads η (0) 

123

2 f u 2g hE

[1 − δλ (λ − 1)] .

(16)

Turbulent Helicity in the Atmospheric Boundary Layer

Fig. 8 Dependence of the dimensionless helicity on the non-dimensional height ζ  z/ h E , see Eq. 14, for three parameter sets: (A) δ  0 (solid line); (B) δ  1/3, λ  3 (dashed line); (C) δ  1, λ  3 (dash-dotted line)

 For δ  1 3, i.e., in the case of velocities with identical orientation described by Eqs. 11 and 12, and λ 3, which satisfactorily fits atmospheric conditions (see, below), we have η (0)  −2 f u 2g h E ( f > 0). Therefore, the estimates of Sect. 2 are equally applicable in our case by taking into account only the sign change of helicity. If we increase the δ value by a factor of three, in order to obtain more realistic values of the intensity of the breeze  circulation (see Sect. 3.2), and take λ  3, then we have η (0)  −5 f u 2g h E . This increases the magnitude of (negative) turbulent helicity by a factor of 2.5 compared to the previous case. ˜ as In Figs. 8 and 9 we show the vertical profiles of the helicity H˜ and superhelicity S,  functions of ζ , see Eqs. 14 and 15, for three parameter sets: (A) δ  0; (B) δ  1 3, λ  3; (C) δ  1, λ  3. Case A corresponds to the classic Ekman spiral flow, with positive helicity and superhelicity in the main bulk of the boundary layer. In cases B and C, negative values of H˜ and S˜ are evident near the ground. In addition, Fig. 10 presents the velocity hodograph for the parameter values indicated above; the left rotation of the wind velocity with increasing height in the lower part of the boundary layer can be seen for case B and, most clearly, for case C. Consider the case when the direction of the surface-adjacent wind jet constitutes an angle ϑ with the x-axis. Then the factor exp (iϑ) has to be introduced in the right-hand side of Eq. 12 and the resulting relations (not shown here) give η (0) 

2 f u 2g   1 + δλ cos ϑ − δ λ2 cos ϑ + δ λ2 sin ϑ . hE

(17)

When the jet and geostrophic velocity have the same orientation ϑ  0, and this angle grows in the counter-clockwise direction; when ϑ  π the jet and the geostrophic velocity are directed oppositely. Equation 16 follows from Eq. 17 at ϑ  0. According to Eq. 17, the only

123

O. G. Chkhetiani et al.

Fig. 9 Dependence of the dimensionless superhelicity on the non-dimensional height ζ  z/h E , see Eq. 15, for three parameter sets: (A) δ  0 (solid line); (B) δ  1/3, λ  3 (dashed line); (C) δ  1, λ  3 (dash-dotted line)

Fig. 10 Velocity hodograph, corresponding to the dimensionless linear superposition W  (w0 + w1 ) /u g of complex velocities (see Eqs. 11, 12) as a function of the non-dimensional height ζ  z/ h E , for three parameter sets: (A) δ  0 (solid line); (B) δ  1/3, λ  3 (dashed line); (C) δ  1, λ  3 (dash-dotted line); U and V correspond to u and v notations in the main text: see Eq. 2

123

Turbulent Helicity in the Atmospheric Boundary Layer

Fig. 11 Schematic of the classic Ekman spiral (U and V correspond to u and v notations in the main text: see Eq. 2). Net helicity and superhelicity can be negative if the velocity vector (see Eq. 12) outgoing from the origin of coordinates (U, V ) is directed to the right from the line AB

  possible range of negative values of turbulent helicity corresponds to −3π 4 < ϑ < π 4 (Fig. 11), with δ exceeding a critical value that depends on both λ and ϑ.

3.2 Comparison with Data in K2015 For the Ekman spiral matching

Ethe

logarithmic wind profile within the

surface-adjacent

u and the geostrophic wind speed ug are related via layer, the 10-m wind speed 10

E



u  ug (cos φ − sin φ). Here, φ is the angle between the vectors u E and ug , which 10 10 is measured (in the Northern Hemisphere) in the counter-clockwise direction from the vecE -value can tentatively be estimated for the tor ug (e.g., Etling 1996; Sect. 20.10). The u10

E

◦ −1



daytime measurement period

by using−1φ ≈ 30 , see Fig. 6; this gives u10 ≈ 3.3 m s for a geostrophic wind speed ug ≈ 9 m s observed on 8 August 2012. The exact contribution of the breeze circulation to the measured 10-m wind speed is in fact unknown. However, even

E estimate provides a lower value than in reality, the contribution from the if the above u10 daytime sea breeze is important in explaining the observed 10-m wind-speed maximum ≈ 8 m s−1 (Fig. 7). Figure 7 shows that the recorded 10-m wind speed exceeded the above value of 3.3 m s−1 for nearly the whole day of 8 August 2012, when the helicity measurements were made. From our perspective, this serves as an indication of a persistent breeze-like local flow that day. A pronounced diurnal cycle in the 10-m wind speed apparent in Fig. 7 can be partially explained by the effect of stratification of the nocturnal boundary layer (Van de Wiel et al. 2012; their Fig. 13), but in the light of the previous arguments, it is mainly attributed to the effect of the existent breeze circulation. Figure 6 indicates that in the main portion of the observation time period, the angle ϑ between the surface-adjacent wind jet contributing to the 10-m wind speed, and the geostrophic velocity is less than 45◦ . According to the theoretical analyses of Sect. 3.1, this favours the negative sign of helicity and superhelicity in the surface boundary layer. 7 7 This interpretation is most clear for the daytime sea breeze. There is reason to assume that, because of the thermal inertia, the breeze, albeit weaker in intensity, is still westward oriented during the measurement period late in the evening, which favours the negative helicity. Indeed, just before the beginning of the measurement period, at 2200 local time on 8 August 2012, the surface air temperature at the polygon site was 31 °C, and just after the end of the measurement period at 0100 local time on 9 August 2012 it was 29.3 °C. However, the water temperature in the Tsimlyansk Reservoir did not exceed 27 °C. That is, the land remained warmer than the water, although a stable thermal stratification was established over the measurement site. Under these conditions, the negative buoyant production term of helicity makes an important contribution too, which helps explain the resultant negative sign of turbulent helicity (see more in Sect. 4.2).

123

O. G. Chkhetiani et al. Table 2 Pressure records PA (in hPa) at the main measurement site in K2015; temperature TB (in °C) and pressure PB records (converted from mmHg to hPa) at the stationary weather station in Tsimlyansk Local time

PA (hPa)

TB (°C)

PB (hPa)

PB (hPa)

PB − PA (hPa)

0300

1000.54

28.1

1003.78

1000.61

0.07

0600

1000.50

27.6

1003.92

1000.74

0.24

0900

1000.24

30.9

1003.78

1000.64

0.40

1200

999.72

34.3

1002.85

999.74

0.02

1500

998.32

36.2

1001.65

998.56

0.24

1800

997.38

33.3

1000.45

997.33

− 0.05

2100

997.82

30.7

1001.25

998.10

0.28

2400

997.48

28.3

1000.72

997.54

0.06

The temperatures TB were used to reduce the pressure PB to a greater height of the main measurement site and calculate the reduced pressure PB . In the last column the pressure difference PB − PA is shown, whereas the first column shows the local time of pressure/temperature measurements

When making a further comparison with the theoretical model of Sect. 3.1 we note that the  wind  speed in the Prandtl jet, see Eq. 12, attains its maximum value at z  z max  π 4 h 1 . Therefore, by adopting zmax ≈ 100 m (Vorontsov 1955) and using hE ≈ 400 m, we infer that λ ≈ 3.15, which is close to the value λ  3 cited in Sect. 3.1 and used in plotting Figs. 8, 9 and 10. The choice of realistic values of the parameter   δ is more  delicate. The maximum wind speed of the Prandtl jet is u max  U exp −π 4 sin π 4 ≈ 0.323 U , and the value δ  1 cited in Sect. 3.1 and used in Figs. 8, 9 and 10 corresponds to a breeze circulation with the wind-speed maximum ≈ 3 m s−1 if the geostrophic wind speed is ≈ 9 m s−1 (cf. Vorontsov 1955). The arguments of Sect. 3.1do not impose an upper limit on the values of the parameter δ. For example, the choice δ  5 3 matches the sea-breeze velocity maximum ≈ 5 m s−1 reported in Vorontsov (1955), and presumably observed in our case. As follows from Eq. 16 and the arguments in Sect. 2, it would proportionately increase the predicted negative values of turbulent helicity in the bottom part of the boundary layer. To check the overall applicability of solution described by Eq. 12 to our problem, we have computed the difference between, (i) the pressure records at 2-m height using a Davies sensor at the main measurement site in K2015 (site A) and, (ii) routine pressure measurements at the stationary (network) weather station in the downtown area of Tsimlyansk, which is located closer to Tsimlyansk Reservoir, approximately 2700 m to the south-east of the main measurement site (site B). The difference in elevation above sea level between sites A and B is 28 m, with site A being at a greater altitude. Note that the slope of the terrain near Tsimlyansk Reservoir is too small and irregular to provide a noticeable direct slope-wind effect. In Table 2 we show both the measured pressure data (in hPa) for 8 August 2012 and the pressure values at site B reduced to the elevation of site A via Babinet’s relation (cf. Forsythe 2003). During the whole day of 8 August 2012, except at 1800 local time, one observes a southeast component of the local horizontal pressure-gradient force, i.e., the reduced pressure at site B is higher than the pressure at site A. At 1500 local time, this force component equals ≈ 7.9 × 10−3 m s−2 , if normalized by the air density. In the absence of the component of the pressure gradient force in the direction perpendicular to the line connecting sites A and B, it would result in the sea-breeze circulation having a wind direction of 135°. In fact, as it follows from the data shown in Fig. 6, the recorded 10-m wind speed, with an important

123

Turbulent Helicity in the Atmospheric Boundary Layer

contribution from the sea-breeze circulation, coincides with a wind direction of approximately 075° at 1500 local time on 8 August 2012. The last figure also characterizes the orientation of the local horizontal pressure-gradient force that explains the existent breeze circulation. A factor of approximately cos (135◦ −75◦ )  cos 60◦  1/2 consequently determines the non-zero projection of this force onto the line connecting sites A and B. Therefore, the measurements of the component of the horizontal pressure-gradient force along this line are expected to give an order-of-magnitude estimate of the actual value of the horizontal pressure-gradient force. The theoretical model of Sect. 3.1 predicts that the magnitude of the horizontal pressure-gradient force (in the same units), related to solution described by   2 Eq. 12, is equal to F (0)  2K U h −2 ≡ f U λ2 . For U ≈ 9 m s−1 (i.e., 1  f U h E h1 −3 −2 δ  1) and λ ≈ 3 we obtain F (0) ≈ 8.1 × 10 m s , in agreement with the previous estimate. A contribution from the pressure-gradient force related to the geostrophic velocity above the Ekman boundary layer is an order of magnitude smaller. Though the obtained close coincidence of estimates may be accidental, our considerations are in favour of the plausibility of the theoretical model of Sect. 3.1 and its applicability to the studied problem.

4 Turbulent Helicity 4.1 General Analysis In this sub-section we study in more detail the balance conditions of turbulent helicity in the thermally-stratified, stable and unstable boundary layer. We decompose the velocity, vorticity and (potential) temperature into a sum of average (mean) values and turbulent fluctuations. Under the Boussinesq approximation, from the linearized momentum and vorticity equations   ∂u + (u · ∇) u + u · ∇ u + 2Ω k × u  −∇π  + αgθ  k + ν ∇ 2 u , (18) ∂t     ∂ω + (u · ∇) ω + u · ∇ ω  (2Ωk · ∇) u + (ω · ∇) u + ω · ∇ u + αg∇θ  × k + ν ∇ 2 ω , ∂t

u



u,

v ,

w











(19)

where u  (U (z) , V (z) , 0), , ω  −dV dz, dU dz, 0 , ω      ωx , ωy , ωz , α  θ0−1 , Ω ≡ f 2 and ν is the (molecular) kinematic viscosity, we derive the balance equation of turbulent helicity H t  u · ω

 dU   dV ∂ Ht    d2 V d2 U + w ωx − u  ωz + w ωy − v  ωz − u  w 2 + v  w 2 − 2 αgωz θ  ∂t dz dz dz dz   2 2 2   dv dw du d −Ω −Ω  ν ω · ∇ 2 u + u · ∇ 2 ω − ωz π  . + 2Ω u  ωy − v  ωx − Ω dz dz dz dz

(20) The homogeneity of turbulent fluctuations in the horizontal plane isassumed, with a statistically-steady equilibrium turbulent regime considered when ∂ H t ∂t  0. Upon neglection of the molecular terms (the first right-hand-side term in Eq. 20) and in the spirit  of the Rotta (1951) hypothesis, the right-hand side of Eq. 20 can be written as −H t τ , where, more generally, τ ∼ E ε is the turbulence relaxation time (Wyngaard et al. 1975; see also, Wyngaard 2010; Chapter 10) analogous to the Rotta return-to-isotropy time scale (cf. Zilitinkevich et al. 1999). Here, E is the turbulent velocity variance and ε the specific kinetic energy dissipation rate (see Sect. 2). According to Kurien et al. (2004), the relaxation

123

O. G. Chkhetiani et al.

time scale for helicity can be given by this estimate. The next step is to obtain an expression for the turbulent helicity H t in terms of the turbulent second moments and their derivatives with respect to height. By using the identities u  ωy − v  ωx 

  ∂w ∂w ∂w 1 d  2 1 d  2 u + v 2 + w2 − u  u + v 2 − w2 , − v − w  2 dz ∂x ∂y ∂z 2 dz

(21a) w ωx − u  ωz  −

dv  w

dz  w du w ωy − v  ωz  , dz

,

(21b) (21c)

we have from Eq. 20   du  w  dV dv  w  dU d w 2 d2 V d2 U t       + +u w −v w + 2αg ωz θ + 2Ω H  − τ. dz dz dz dz dz 2 dz 2 dz (22) Consider first the case of a neutral thermal stratification. Since the mean potential temperature is now constant with height, the turbulent fluctuations of potential temperature vanish and the term 2αg ωz θ  in Eq. 22 can be neglected. By using a representation of Reynolds stresses in the form dU , dz dV v  w   −K , dz

u  w   −K

(23a) (23b)

where K is the turbulent viscosity that is height-dependent generally speaking, we obtain from Eq. 22 8 2

d U dV d2 V dU d w 2 t H  2K τ − + 2Ω τ , (24) dz 2 dz dz 2 dz dz  since the terms with dK dz are cancelled out. The combination in parentheses in the first right-hand-side term in Eq. 24 stands for the superhelicity of the mean flow discussed in Sect. 3.1. In agreement with the arguments of Sect. 3.1, this term denotes the negative turbulent helicity for the negative superhelicity of the mean flow. The second, generally positive in the Northern Hemisphere, right-hand-side term in Eq. 24 is two to three orders of magnitude smaller than the first right-hand-side term for the neutrally stratified near-surface layer in the presence of a mean spiral flow. 8 A generalized representation of Reynolds stresses, taking into account the contribution of helicity, and       specifically of the last right-hand-side term in Eq. 20, reads u  w   −K dU dz + K h dV dz , v  w         −K d V dz − K h dU dz (Chkhetiani 2001), where the ‘helical viscosity’ K h constitutes ≈ 10–15% of the turbulent viscosity K and the alternating sign before K h reflects its relation to such a presudoscalar quantity as helicity. Accounting for the Earth’s rotation gives a similar structure for the Reynolds stresses in the atmospheric boundary layer (Stubley  and Riopelle 1988). When K h  constant, then Eq. 24 is again obtained; otherwise, a term containing dK h d z and presenting an additional, generally negative, source of turbulent helicity appears in Eq. 24. However, we do not accentuate this point in the main text, because it lies beyond the scope of the manuscript.

123

Turbulent Helicity in the Atmospheric Boundary Layer

When the boundary layer is thermally stratified, then the term 2αgωz θ  matters and Eq. 22 yields H t  2K τ

d2 V dU d2 U dV − dz 2 dz dz 2 dz

+ 2Ω

dw 2 τ + 2αg ωz θ  τ. dz

(25)

There are three sources of turbulent helicity in the thermally-stratified ABL. The first source has already been discussed in connection with Eq. 24, and predominantly determines the contribution to helicity u  ωx + v  ωy from the horizontal components of velocity and vorticity. As in the case of neutral stratification, the second source describes the direct turbulent helicity injection into the near-surface layer under the influence of the Coriolis force and, generally, only makes a minor contribution to the helicity. The third source is due to the thermal stratification, and according to the data in Table 1, it is positive in daytime conditions and is negative in night-time conditions, respectively. In order to make an independent estimate of the first right-hand-side term in Eq. 25 and thus to make a comparison with the results of Sects. 2 and 3.1, we use sodar measurements from the Tsimlyansk Scientific Station during the period 2–26 August 2012 (Vazaeva et al. 2017) simultaneously with the velocity and temperature measurements described in detail in K2015. The measurements by Vazaeva et al. (2017) were performed with a Doppler minisodar LATAN-3 m developed and manufactured at the A.M. Obukhov Institute of Atmospheric Physics in Moscow. The minisodar has an improved vertical resolution of 10 m, a pulse emission interval of 5 s, an altitudinal range of 400 m, and a basic carrier frequency of 3.5 kHz (see, e.g., Kouznetsov 2009). The device was positioned at a distance of about 100 m west of the tetrahedron and used to measure vertical profiles of the three velocity components. The time-averaged velocity components were used to calculate helicity and superhelicity, and to this end, a rectangular filter was used. The averaging interval was chosen empirically and, in this case, amounted to 20 min, which value ensured the adequate reproduction of the spatiotemporal velocity-field structure. Since velocity profiles obtained with the Doppler sodar remain quite jagged, even after averaging, in order to estimate the vertical derivatives of velocity, these profiles were approximated by cubic splines with the subsequent calculation profiles. To calculate superhelicity    of the  derivatives   of the smoothed  S  − dU dz d2 V dz 2 + dV dz d2 U dz 2 , the profiles of the first derivatives of the velocity were repeatedly smoothed with cubic splines, with the subsequent calculation of the second derivatives of the obtained profile. The vertical profile of the resulting superhelicity values for the time period from 1238 to 1630 local time on 8 August 2012, corresponding to the daytime measurement session in Table 1 and afterwards averaged over the indicated period, is shown in Fig. 12, starting from a 20-m height. As a whole, the predominance of the negative superhelicity values within the lowest 100m layer is evident, except for a swing towards zero at 40 m. For instance, at a height of 30 m, by using K ≈ 5 m2 s−1 and τ ≈ 20 s, from Eq. 24 we obtain the contribution to turbulent helicity, which is ≈ −0.02 m s−2 . In view of all the complicating factors and approximations, the agreement between this result and those in K2015 is satisfactory. The third source term in Eq. 25 contributes to all the three helicity components, including the w  ωz component that is due to the vertical velocity and vorticity. The following procedure is proposed to determine the relationship between 2αg ωz θ  τ and w  ωz . We make use of the equation for the time rate-of-change of the covariance ωz θ  , ∂   ω θ  −w  Π  , ∂t z

(26)

123

O. G. Chkhetiani et al.

Fig. 12 The vertical profile of superhelicity S of the mean flow obtained from sodar measurements in Tsimlyansk for the time period from 1238 to 1630 local time on 8 August 2012; the average S values over the indicated period are shown

   where Π   ωz dθ dz + ω · ∇θ  + ω · ∇θ  is the turbulent Ertel potential vorticity under the Boussinesq approximation and all notations are explained above. Equation 26 follows from the vorticity equation and the thermodynamic equation taken under the Boussinesq approximation. The homogeneity of turbulent fluctuations in the horizontal plane is assumed, while also we take into account that the turbulent vorticity vector ω is divergence-free. Similar to Deardorff (1972), who considered the corresponding equation for the kinematic heat flux w  θ  , we have neglected the molecular terms in Eq. 26 and consider the turbulent flow over nearly homogeneous level terrain. In Eq. 26 we have also neglected a minor term    2Ω ∂w  ∂z θ  related to the vertical vorticity production due to the stretching of the vortex tubes of planetary vorticity.   A steady statistically equilibrium turbulent regime is considered when ∂ ωz θ  ∂t  0 in Eq. 26 and consequently −w  ωz

dθ  w  (ω · ∇θ  ) + w  (ω · ∇θ  ). dz

(27)

The next step is to parametrize the right-hand side of Eq. 27 in terms of ωz θ  . In Fig. 13   we the time dependence   of two turbulent quantities: (i) P ≡ ωz θ , and (ii) Q ≡   show     w ω · ∇θ + w ω · ∇θ , see the right-hand side of Eq. 27, for the daytime measurement session in K2015 at 1238–1630 local time (see Sect. 2). An additional 10-min running average was applied to plot P and Q. The correlation between the two curves in Fig. 13 is evident, and we set the right-hand side of Eq. 27 equal to τ −1 ωz θ  , thus broadly following a relaxation hypothesis, outlined at the beginning of this section. We remind the reader that overbar quantities are the averages over time for the whole measurement period of about 4 h.

123

Turbulent Helicity in the Atmospheric Boundary Layer

Fig. 13 Time dependence of P (in K s−1 , red line) and Q (in K s−2 , blue line) values, which are calculated based on the measurements in Tsimlyansk, southern Russia on 8 August 2012 at 1238–1630 local time; K2015, see the text for notations and more details. On the horizontal axis, the initial time corresponds to 1248 local time

Qualitatively similar results are obtained for the night-time measurement session in K2015 (not shown). In the framework of Eq. 27 the above-made closure assumption guarantees that the covariance ωz θ  achieves a steady-state magnitude at any given height. Despite its simplicity, this approach leads to results that are internally consistent (see below) and we postpone to future studies further elaboration and justification of this or related assumptions. Now, it follows from Eq. 27 that 9 ωz θ   −w  ωz

dθ τ, dz

(28)

or identically 1 α g ωz θ  τ, ξ dθ ξ ≡ −αg τ 2 , dz

w  ωz 

(29a) (29b)

where ξ is a non-dimensional parameter, which is positive/negative for an unstably/stablystratified boundary layer.

4.2 Application of Eqs. 28 and 29 to Experimental Data in K2015 First, a comparison with the experimental results for daytime convective conditions (daytime on 8 August 2012) reproduced in Table 1 is made. Given that the Obukhov length L corresponding to the data of K2015 equals −19.8 m and the measurement height z ≈ 5 m, we have 9 For the sake of simplicity, the same τ -value as in Eq. 24 is used here, although, strictly speaking, the turbulence relaxation time in Eq. 24 is somewhat different from that in Eq. 28. This difference is, however, not essential in the context of this paper and might play a role only in arguments of Sect. 4.2 related to a “semi-quantitative” estimate of the horizontal component of turbulent helicity due to the buoyant forcing.

123

O. G. Chkhetiani et al.

     w  θ   −κ u ∗ zϕh−1 z L dθ dz , (30)      −1/ 2 where ϕh z L  0.95 1 − 11.6 z L is the Dyer–Businger form of the Monin–Obukhov universal function for the sensible heat exchange according to Businger et al. (1971), recalculated by Högström (1988), and recommended for use in Foken (2008). Here, w  θ  is the kinematic heat flux given in Table 1, κ  0.4 is the von Karman constant, and u ∗  0.34 m s−1 isthe friction velocity (this value follows from data in K2015). From Eq. 30, we obtain dθ dz ≈ −0.1089 K m−1 , and by taking ωz θ  and w  ωz from Table 1, one  has from Eq. 28 that τ ≈ 15.2 s. For α  1 310 K−1 and g  9.81 m s−2 it now follows that ξ ≈ 0.80, and Eq. 29 gives w  ωz ≈ 1.25α g ωz θ  τ . Because the buoyant production of turbulent helicity is 2αg ωz θ  τ , see Eq. 25, it follows that the buoyant forcing contributes to u  ωx +v  ωy according to u  ωx +v  ωy ≈ 0.75α g ωz θ  τ . In daytime conditions, corresponding to the measurements in K2015, the last positive term makes a noticeably smaller contribution than the negative first right-hand-side term in Eq. 25. Second, for a comparison of Eqs. 28 and 29 with experimental results in K2015 for nocturnal stable stratification on 8 August 2012) we again make use  conditions   (night-time   of Eq. 30, where now ϕh z L  1+5 z L (e.g., Brutsaert 1984). The kinematic heat flux w  θ  is given in the lower row of Table 1; the friction velocity u ∗ ≈ 0.22 m s −1 and the Obukhov length L ≈ 46.2 m can be inferred from K2015. For height z ≈ 5 m, we obtain from Eq. 30 that dθ dz ≈ 0.060 K m−1 , and by taking ωz θ  and w  ωz from Table 1, we derive  from Eq. 28 that τ ≈ 24.1 s. In this case, when α  1 300 K−1 , it follows from Eq. 29 that ξ ≈ −1.14, w  ωz ≈ −0.88α g ωz θ  τ , and (since the buoyant production of turbulent helicity is 2α g ωz θ  τ ) the buoyant forcing contributes to u  ωx + v  ωy according to u  ωx + v  ωy ≈ 2.88α g ωz θ  τ . During nocturnal conditions, corresponding to the measurements in K2015, the last negative term is added to the negative (now relatively small by magnitude) first right-hand-side term in Eq. 25; see footnote 7. In summary, Eqs. 28 and 29 explain correctly the positive sign of w  ωz for both unstable and stable conditions, and the corresponding reversal of sign of 2α gθ  ωz τ . The buoyant production term of helicity 2α g ωz θ  τ in Eq. 25 mainly contributes to the vertical component of helicity w  ωz in daytime conditions. The opposite is true in night-time conditions where this term mainly contributes to the horizontal components of helicity u  ωx + v  ωy . The obtained turbulence relaxation time τ valuesare close to those that result from a simple estimate, based on dimensional arguments, τ ∼ l u ∗ , where l is the characteristic spatial scale of turbulent motions under consideration. 10 Indeed, by taking l  5 m and u ∗  0.34, 0.22 m s−1 , we have τ ≈ 14.7, 22.7 s, correspondingly.

5 Summary In the Introduction, we reviewed a theoretical possibility (Deusebio and Lindborg 2014) that for a typical magnitude of helicity injection into the Ekman boundary layer, the signature of the Earth’s rotation can reach the Kolmogorov microscale of turbulence within the surfaceadjacent logarithmic layer. 

10 It is equivalent to the definition of the turbulence relaxation time as τ ∼ E ε (Wyngaard et al. 1975; 2 Wyngaard  2010) if the turbulent velocity variance E ∼ u ∗ and the specific kinetic energy dissipation rate ε ∼ u 3∗ l.

123

Turbulent Helicity in the Atmospheric Boundary Layer

At the same time, the recent direct field measurements of turbulent helicity by Koprov et al. (2015), overviewed in Sect. 2, show the opposite sign of helicity from that expected. A possible explanation for this phenomenon may be the joint action of different types of flows within the boundary layer. Correspondingly, we considered the Ekman layer over a non-uniformly heated surface and applied a formal superposition of the classic Ekman spiral solution and the Prandtl (1942) jet-like slope wind solution, which we used to mimic a hydrostatic breeze circulation over a non-uniformly heated flat surface. Finally, we were able to show that a 180°-wide sector on the hodograph plane exists, within which the relative orientation of the Ekman and Prandtl velocity profiles favours the left rotation of the resulting velocity vector with increasing height in the atmospheric surface layer. Therefore, this explains, for the Northern Hemisphere, the negative (left-handed) helicity cascade toward small-scale turbulent motions, the latter generally agreeing with the direct field measurements of turbulent helicity by K2015. Based on the helicity balance equation in the thermally-stratified ABL, we have shown that there are three sources of turbulent helicity, see Eq. 25. The first source relates to the superhelicity of the mean flow that in our case is the superposition of the classical Ekman flow and the breeze-like near-surface circulation (see Sect. 3.1). This source predominantly determines the negative contribution to turbulent helicity arising from the horizontal components of velocity and vorticity. The second source, which is positive in the Northern Hemisphere, is the direct turbulent helicity injection into the near-surface layer under the influence of the Coriolis force, which generally makes only a minor contribution to the helicity. The third source is related to the thermal stratification: it is positive in daytime conditions and negative in night-time conditions. By using simultaneous sodar measurements of the three velocity components at a location close to that used in K2015, we have obtained an independent corroboration of the predominately negative sign of superhelicity of the main flow in the lower part of the boundary layer during the measurement period in K2015. This explains a negative contribution to the turbulent helicity arising from the horizontal components of velocity and vorticity, in accord with direct field measurements in K2015 (Sect. 2) and theoretical arguments (Sects. 3.1 and 3.2). A simple turbulent relaxation theory has been developed to determine the relationship between the contribution to turbulent helicity from the vertical components of velocity and vorticity, w  ωz , and the rate of production of turbulent helicity due to the thermal stratification (buoyancy), 2α gωz θ  . As follows from comparison with the results by K2015, this theory correctly predicts the sign of w  ωz and 2α gωz θ  both for convectively unstable daytime and for stable night-time conditions. Our results support the idea that the sign of net turbulent helicity observed in direct field measurements of helicity within the ABL depends strongly on local physical/meteorological conditions at the observational site and on the geographical position of the site. Therefore, further research in this direction is highly desirable, particularly concerning direct field measurements of helicity in the Southern Hemisphere. Acknowledgements We thank L.O. Maximenkov and V.A. Bezverkhniy for help with organizing Figs. 5 and 6, and are sincerely grateful to B.M. Koprov, V.M. Koprov and M.E. Gorbunov for helpful discussions and kind collaboration. We also thank three anonymous reviewers whose critical review and useful suggestions contributed tremendously to improvements in the content and in the style of this article. The work is supported by the Russian Foundation for Basic Research, Project Nos. 15-05-02407-a and 17-05-01116-a.

123

O. G. Chkhetiani et al.

References André JC, Lesieur M (1977) Influence of helicity on the evolution of isotropic turbulence at high Reynolds number. J Fluid Mech 81(01):187–207 Avinash V, Verma MK, Chandra AV (2006) Field-theoretic calculation of kinetic helicity flux. Pramana J Phys 66(2):447–453 Bluestein H (1992) Synoptic-dynamic meteorology in midlatitudes: principles of kinematics and dynamics, vol 1. Oxford University Press, Oxford Briard A, Gomez T (2017) Dynamics of helicity in homogeneous skew-isotropic turbulence. J Fluid Mech 821:539–581 Brissaud A, Frisch U, Leorat J, Lesieur M, Mazure A (1973) Helicity cascades in fully developed isotropic turbulence. Phys Fluids 16:1366–1377 Brutsaert W (1984) Evaporation into the atmosphere. D. Reidel Publ. Company, Dordrecht Businger JA, Wyngaard JC, Izumi I, Bradley EF (1971) Flux-profile relationships in the atmospheric surface layer. J Atmos Sci 28:181–189 Chen Q, Chen S, Eyink GL (2003) The joint cascade of energy and helicity in three-dimensional turbulence. Phys Fluids 15(2):361–374 Chkhetiani OG (2001) On the helical structure of the Ekman boundary layer. Izv Atmos Ocean Phys 37:569–575 Deardorff JW (1972) Theoretical expression for the counter-gradient vertical heat flux. J Geophys Res 77:5900–5904 Deusebio E, Lindborg E (2014) Helicity in the Ekman boundary layer. J Fluid Mech 755:654–671 Ditlevsen PD (2011) Turbulence and shell models. Cambridge University Press, Cambridge Ditlevsen PD, Giuliani P (2001a) Dissipation in helical turbulence. Phys Fluids 13:3508–3509 Ditlevsen PD, Giuliani P (2001b) Cascades in helical turbulence. Phys Rev E 63(3):036304 Etling D (1985) Some aspects of helicity in atmospheric flows. Beitr Phys Atmos 58(1):88–100 Etling D (1996) Theoretische Meteorologie. Eine Einführung. Vieweg & Sohn, Braunschweig, Wiesbaden Foken T (2008) Micrometeorology. Springer, Berlin Forsythe WE (2003) Smithsonian physical tables, 9th edn. Smithsonian Institution, Washington Gutman LN (1972) Introduction to the nonlinear theory of mesoscale meteorological processes. Edited by G.I. Marchuk. Translated from Russian. Israel Program for Scientific Translations, Jerusalem (Russian original: Gutman LN (1969) Vvedenie v nelineinuyu teoriyu mesometeorologicheskikh protsessov. Gidrometeoizdat, Leningrad) Hauf T (1985) Rotating clouds within cloud streets. Beitr Phys Atmos 58(3):380–398 Hide R (1989) Superhelicity, helicity and potential vorticity. Geophys Astrophys Fluid Dyn 48:69–79 Hide R (2002) Helicity, superhelicity and weighted relative potential vorticity. Useful diagnostic pseudoscalars? Q J R Meteorol Soc 128:1759–1762 Högström U (1988) Non-dimensional wind and temperature profiles in the atmospheric surface layer: a reevaluation. Boundary-Layer Meteorol 42:55–78 Koprov BM, Koprov VM, Ponomarev VM, Chkhetiani OG (2005) Experimental studies of turbulent helicity and its spectrum in the atmospheric boundary layer. Dokl Phys 50:419–422 Koprov BM, Koprov VM, Kurgansky MV, Chkhetiani OG (2015) Helicity and potential vorticity in surface turbulence. Izv Atmos Ocean Phys 51:565–575 Kouznetsov RD (2009) The multi-frequency sodar with high temporal resolution. Meteorol Z 18(2):169–173 Krause F, Raedler K-H (1980) Mean-field magnetohydrodynamics and dynamo theory. Academie, Berlin Kurgansky MV (1989) Relationship between helicity and potential vorticity in a compressible rotating fluid. Izv Atmos Ocean Phys 25(12):979–981 Kurgansky MV (1993) Introduction to large-scale atmospheric dynamics (adiabatic invariants and their use). Gidrometeoizdat, Saint Petersburg (in Russian) Kurgansky MV (2002) Adiabatic invariants in large-scale atmospheric dynamics. Taylor & Francis, London Kurgansky MV (2005) A simple model of dry convective helical vortices (with applications to the atmospheric dust devil). Dyn Atmos Ocean 40:151–162 Kurgansky MV (2017) Helicity in dynamic atmospheric processes. Izv Atmos Ocean Phys 52(2):127–141 Kurien S, Taylor MA, Matsumoto T (2004) Cascade time scales for energy and helicity in homogeneous isotropic turbulence. Phys Rev E 69:066313 Lilly DK (1986) The structure, energetics and propagation of rotating convective storms. Part II. Helicity and storm stabilization. J Atmos Sci 42(2):126–140 McMillen RT (1988) An eddy correlation technique with extended applicability to non-simple terrain. Boundary-Layer Meteorol 43:231–245 Moffatt HK (1969) The degree of knottedness of tangled vortex lines. J Fluid Mech 35(1):117–129

123

Turbulent Helicity in the Atmospheric Boundary Layer Moffatt HK (1978) Magnetic field generation in electrically conducting fluids. Cambridge University Press, Cambridge Moreau J-J (1961) Constantes d’un ilot tourbillonnaire en fluide parfait barotrope. C R Acad Sci Paris 252:2810–2812 Ponomarev VM, Chkhetiani OG (2005) Semiempirical model of the atmospheric boundary layer with parametrization of turbulent helicity effect. Izv Atmos Ocean Phys 41:418–432 Ponomarev VM, Khapaev AA, Chkhetiani OG (2003) Role of helicity in the formation of secondary structures in the Ekman boundary layer. Izv Atmos Ocean Phys 39(4):391–400 Prandtl L (1942) Führer durch die Strömungslehre. Vieweg & Sohn, Braunschweig Rotta JC (1951) Statistische Theorie nichthomogener Turbulenz. Z Phys 129:547–572 Shapiro A, Fedorovich E (2007) Katabatic flow along a differentially cooled sloping surface. J Fluid Mech 571:149–175 Shapiro A, Fedorovich E (2013) Similarity model for unsteady free convection flows along a differentially cooled horizontal surface. J Fluid Mech 736:444–463 Stubley GD, Riopelle G (1988) The influence of the earth’s rotation on planetary boundary-layer turbulence. Boundary-Layer Meteorol 45(4):307–324 Tan Z, Wu R (1994) Helicity dynamics of atmospheric flow. Adv Atmos Sci 11(2):175–188 Van de Wiel BJH, Moene AF, Jonker HJJ (2012) The cessation of continuous turbulence as precursor of the very stable nocturnal boundary layer. J Atmos Sci 69:3097–3115 Vazaeva NV, Chkhetiani OG, Kuznetsov RD, Kallistratova MA, Kramar VF, Lyulyukin VS, Kuznetsov DD (2017) Estimate of helicity in the atmospheric boundary layer based on acoustic sounding data. Izv Atmos Ocean Phys 53(2):174–186 Vorontsov PA (1955) Breezes of Tsimlyansk Reservoir. Trudy Glav Geofiz Obs 54(116):81–95 (in Russian) Wyngaard JC (2010) Turbulence in the atmosphere. Cambridge University Press, Cambridge Wyngaard JC, Coté OR, Rao KS (1975) Modeling the atmospheric boundary layer. Adv Geophys 18:193–211 Zilitinkevich S, Gryanik VM, Lykossov VN, Mironov DV (1999) Third-order transport and nonlocal turbulence closures for convective boundary layers. J Atmos Sci 56:3463–3477

123