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WALL-BOUNDED TURBULENCE WITH STRATIFICATION
The literature on wall-bounded turbulence is vast, and, in this section, only a few selected elements, necessary for the rest of the thesis, are described. Included are also concepts that, arbitrarily, seem interesting for the study of turbulence on top of waves. More precisely, we focus on the description of the Atmospheric Surface Layer (ASL), which lies at the bottom of the atmospheric boundary layer. While, in the atmospheric boundary layer interior, the Coriolis force induces a rotation of the mean wind with height (the Ekman spiral), in the ASL the direction of the mean wind is constant with height and the Coriolis force can be neglected. This defines the streamwise (x) direction of a Cartesian coordinate system, aligned with the mean (ageostrophic) wind U. The flow is further assumed to be horizontally homogeneous, i.e. horizontal gradients, and in particular advection, cancel. All quantities are thus invariant with respect to the streamwise and spanwise (y) directions, and vary only with height above the surface z. The Reynolds-averaged streamwise momentum balance then reads ∂U ∂p ∂2U = − L − ∂u w ′ −ν (1.1)
We have introduced the (zero-mean) fluctuations of streamwise (u′) and vertical (w′) velocity, and • , the ensemble Reynolds average (the spanwise fluctuation of velocity is denoted by v′). We have also introduced the air viscosity ν, and a large scale pressure gradient (∂p/∂x)L resulting from a bulk atmospheric boundary layer forcing due to a large scale geostrophic equilibrium. Sufficiently far from the bottom boundary (see below), the viscous stress (the last term on the RHS) can be neglected with respect to the Reynolds stress (the penultimate term on the RHS).
The above equation shows that the Eulerian acceleration of the mean flow is related to turbulence through the vertical gradient of the anisotropic component of the Reynolds stress tensor u′w′. This bulk anisotropy results from a multiscale zoology of turbulent motions, which are described, in what follows, in terms of statistical eddies [Townsend, 1972]. This anisotropic component can be interpreted as a vertical flux of momentum, providing an incentive for its understanding in terms of events (or coherent motions) which transport momentum through the boundary layer. Some elements of the latter view will also be presented below.
We further consider a stationary ASL (∂U /∂t = 0), and neglect the large-scale pressure gradient, which leads to the following momentum balance du′w′ = 0. (1.2) dz
This defines the friction velocity u∗, as the square root of the (constant) turbulent momentum flux in the ASL (−u′w′)1/2 (following the meteorological convention, a negative momentum u′w′ flux is here a downward flux). 1
We now proceed by first presenting a broad overview of the ASL and its eddies for neutral conditions. We then focus on Monin-Obukhov Similarity Theory (MOST) for the description of a stratified ASL. This second part is motivated by MOST being at the heart of modern parameterizations of surface fluxes in the presence of waves.
This section is inspired from several reviews and books, which contain more details and references than what follows: (i) the reviews about smooth- and rough-wall flows of Raupach et al. [1991] and Jiménez [2004], (ii) the reviews on coherent structures in low-Reynolds number flows [Jiménez, 2012, 2018] and on top of (forest) canopies [Finnigan, 2000] and, (iii) the classic textbooks about turbulence of Pope [2000] and Wyngaard [2010].
FLAT WALL The description of a turbulent boundary layer in the vicinity of a flat wall relies on several basic quantities. These are (i) the friction velocity u∗, (ii) the air viscosity ν, and (iii) the height of the boundary layer δBL. If the atmosphere is neutral, those ingredients define two dimensionless scales: the inner scale lin = zu∗/ν and the outer scale lout = z/δBL. The inner scale can be interpreted as the Reynolds number for the attached, energetic eddies at a height z, while a friction Reynolds number can be introduced as Reν = δBLu∗/ν, i.e. as the Reynolds number of the largest eddies of the boundary layer. For the ASL, ν = O(10−5 m2 s−1), δBL = O(100 −1000 m) and u∗ ∼ 0.5 m s−1. The ASL is hence a high Reynolds-number flow, i.e. with Reν ranging from 106 to 108 (the latter being for a convective ASL).
The scales lin and lout define different layers, in which the mean wind shear scales differently (see Fig. 1.1) . The inner layer, where the mean wind shear is independent of δBL and of the free stream velocity (U(z = δBL)), is defined for lout ≤ 0.1. At the bottom of the inner layer lies the viscous sublayer (lin < 5) where the Reynolds stress is negligible with respect to the viscous stress, and the mean wind shear is constant2 dU = u∗2 and U(0) = 0. (1.3) dz
Figure 1.1: Schematic of the different layers and dimensionless quantities on which the mean wind profile of a wall-bounded flow depends. For a smooth wall, the definition of the layers depends on lin and lout , and the values given here follow Pope [2000]. For a rough wall, lin is replaced by z/hr where h r is the height of the roughness sublayer (bottom right axis). The value U /u∗ is an order of magnitude, corresponding to pipe flows over a smooth wall. The height of the logarithmic layer (rightmost axis) is estimated for an ASL of height 100 m and u∗ ∼ 0.5 m s−1.
As height increases, for lin > 50, effects of viscosity become negligible, which marks the bottom of the outer layer. At the overlap between the inner and outer layers (for lin > 30 −80 and lout < 0.2 −0.3, according to Pope [2000] and Jiménez [2012]), the only relevant quantity is the height z, and the mean wind follows a logarithmic profile [Townsend, 1976] U(z) = u∗ log(z/z0), dU = u∗ . (1.4a, b).
Following the meteorological convention, we have introduced the roughness height z0 which, for a flat surface, is 0.14ν/u∗. The Von Kármán constant κ ∼ 0.4 is a universal constant (here, we will not discuss its variations). The logarithmic region exists for sufficiently turbulent flows, for which the separation between the largest turbulent eddies and the smallest viscous eddies is broad [e.g. Reν > 750, Jiménez, 2012]. For an ASL of height 100 m, the logarithmic sublayer range 30 < lin and lout < 0.3 corresponds to 6 × 10−4 m < z < 30 m. Note that above the logarithmic region lies the wake region, which won’t be discussed here, and where the effects of the boundary layer height cause a deflect of the mean wind profile from the logarithmic law.
Kraus [1967] proposed an interesting interpretation of the Von Kármán constant. By requiring continuity of the wind shear on top of the viscous sublayer (between the linear and logarithmic profiles), the inner Reynolds number at the top of the viscous sublayer reads linν = 1/κ. Note that with the usual value κ ∼ 0.4, this yields linν ∼ 2.5, of the same order of magnitude than the value lin = 5 of Pope [2000]. This simplified picture neglects however the presence of the buffer layer, located in between the viscous and logarithmic sublayers (5 < lin < 50) and where the linear wind velocity profile gradually merges into the logarithmic profile. The buffer layer is especially important in low-Reynolds number flows [Reν = O(103) or lower, Smits et al., 2011], where it is home of a « viscous cycle » in which low momentum streaks destabilize periodically [Jiménez and Moin, 1991] and act as a TKE source for the boundary layer aloft [Jiménez, 1999].
ROUGH WALL Smooth walls are almost never encountered in geophysical flows. More precisely, the presence of surface roughness alters the above picture by introducing at least an additional scale, the height of the roughness elements hr , which defines the roughness Reynolds number Rer = hr u∗/ν [see the review Raupach et al., 1991]. From this, three regimes can be defined: aerodynamically smooth flows for Rer < 5, transitional for 5 < Rer < 70 and aerodynamically rough flows for Rer > 70 [from the sand-grain experiments of Nikuradse, 1933]3
We hence see that, for an ABL flow not to be considered as aerodynamically rough, hr should be of the order of 10−3 m. As an example, for wheat, typical Rer are of 104, and up to 106 for 23-m high forest canopies such as the Kondo forest [Jarvis, 1976]. In the aerodynamically rough regime (Rer → ∞), the flow becomes independent of Rer (i.e. of viscosity), and the relevant parameters are then those of the surface geometry (streamwise and spanwise aspect ratios, density of roughness elements ….). Their determination is strongly dependent on the nature of the roughness elements, and, as will be seen in the rest of the thesis, is remarkably difficult for the ocean surface. 4
With respect to the mean flow scaling discussed above, the viscous and buffer sublayers are, for an aerodynamically rough flow, replaced by the roughness sublayer, in which the mean wind speed does not depend on viscosity but on the geometry of the roughness elements. Its height is generally between 2hr and 5hr [see Raupach et al., 1991, and Fig. 1.1]. In this layer, the mean wind profile deviates from the logarithmic law reflecting, among others, the effect of airflow separation behind roughness elements, and the presence of dispersive fluxes, resulting from the spatio-temporal heterogeneity of the surface (see Sec. 1.3). For flow over forest canopies, it is generally accepted that this deviation results in an exponential law [Finnigan, 2000]. It can be derived theoretically from the momentum balance, by including an additional form drag, quadratic with wind speed [e.g. Finnigan and Belcher, 2004]. Note that this is similar to the description of the impact of breaking waves on the wind profile (as described in Sec. 1.3) even though, for wind-waves, additional sources of form drag are also present that are not quadratic with wind speed.
In the logarithmic layer, the essential question, posed here following the meteorological conventions, is then to determine which geometrical parameters control z0. Here we only mention the study of Lettau [1969] which has been influential in theoretical works about wind-wave interactions [Csanady, 1985, Kitaigorodskii et al., 1995]. By studying the flow over three-dimensional bushel baskets on a frozen lake [Kutzbach, 1961], Lettau [1969] showed that z0/hr = 0.5hr hy /D2, where the roughness frontal area par unit surface (or roughness density) hr hy /D2 depends on hr , on the mean separation distance between roughness elements (D) and on their spanwise extension (hy ). Note however that, as mentioned by Raupach et al. [1991], for roughness densities high enough, the ratio z0/hr decreases with roughness density. This reflects the mutual sheltering of roughness elements. An extreme case of such a sheltering, first highlighted by Perry et al. [1969], are d-type surfaces, for which z0 depends only on the boundary layer height δBL.
TKE BALANCE IN THE LOGARITHMIC LAYER Townsend [1961] suggested that the logarith-mic layer (above flat or rough surfaces alike) is an equilibrium layer, in which the dissipation of Turbulence Kinetic Energy (TKE) ǫ is balanced by mechanical production u2 dU = ǫ. (1.5) ∗ dz
The above equation assumes that the flow is stationary and horizontally homogeneous (which are the ASL hypotheses), but neglects in addition the vertical transport of TKE (due to vertical gradients of skewness and of velocity-pressure covariance). Numerous experiments in a near-neutral ASL have shown that this balance is generally valid in the logarithmic sublayer [e.g Bradley et al., 1981].
For a low Reynolds-number flow above a smooth surface, Jiménez [1999] further suggested that this equilibrium layer, in which the energy flux is constant, is fed by the buffer sublayer, which acts as a source of energy from below, and that this energy in then dissipated in the overlying wake sublayer (for lout > 0.2). This analysis was quantified by Cimarelli et al. [2016] using a budget for the second order structure function (interpreted as energy of the dominant eddies) on DNS data. It hence suggests an upward inverse energy cascade, as energy goes from small viscous motions to large wake motions.
This picture reversed for ASL flows in the presence of a roughness sublayer. Within the roughness sublayer, transport terms were shown to be significant, moving TKE from the top of the roughness sublayer down to the surface where energy is dissipated in the wake of the roughness elements [Raupach et al., 1991]. This mechanism requires the presence of an additional term in the TKE budget, the so-called wake production term, resulting from dispersive fluxes. It will be discussed at length in Sec. 1.3, in the case of the sea surface.
THE ATTACHED EDDY HYPOTHESIS By using the logarithmic wind profile (1.4) in equation (1.5), the TKE dissipation reads ǫ(z) = u∗3/κz. This scaling is consistent with the seminal attached eddy hypothesis formulated by Townsend [1976] which states that, in the logarithmic layer, the characteristic size of the dominant momentum-transporting eddies scales with distance from the surface. This length is defined by considering that ǫ is the ratio of the cube of an eddy turnover velocity (u∗) and of its size (z). This corresponds to the size of so-called « energy-containing eddies », which extract energy from the mean shear at a given height z, before it cascades down to the Kolmogorov scale η = (ν3/ǫ)1/4 through a 3D isotropic cascade.5 Across this spectral cascade, the energy flux is constant, and the dissipation rate ǫ of η-scale eddies is hence equal to the rate at which energy-containing eddies inject energy in the cascade (see Fig. 1.2a).
Eddies can hence be separated into two kinds: attached (or energy-containing, or active) eddies, which interact with the mean shear and whose size scales with height z, and small-scale eddies which transfer energy down to the scale η where it is dissipated. Bradshaw [1967] further proposed that eddies larger than z act, at a given height, as inactive motions. Those eddies, unlike energy-containing eddies, do not contribute to vertical motions at a height z, but only to horizontal motions. Several works have proposed methods for the identification of attached eddies and their separation from inactive motions, e.g. using wavelet transform [Katul and Vidakovic, 1996], a watershed algorithm [Srinath et al., 2018], conditional sampling [Lozano-Durán et al., 2012] or a clustering method [Cheng et al., 2020].
This separation of motions was used by Perry and Abell [1977] to explain the behaviour of the longitudinal turbulent spectrum, which exhibits two main regimes, shown in Fig. 1.2b, for the related TKE spectrum, as a function of the streamwise wavenumber k (solid line): a low-wavenumber regime (the energy-containing subrange) with a k−1 dependence, at the overlap between the scales of inactive eddies and energy-containing eddies, and a k−5/3 high-wavenumber regime, indicative of 3D cascade where motions are isotropic6.
The separation between the two regimes occurs at a wavenumber proportional to 1/z, representative of the size of energy-containing eddies. Perry and Abell [1977] further used this model to recover the scaling behaviour of the horizontal velocity variances, as also done by Banerjee and Katul [2013] using a phenomenological model similar to the one presented in Sec. 1.2 . The presence of a k−1 regime has been observed in a variety of ABL studies [e.g. Drobinski et al., 2007, and references therein], but it is still unclear under which conditions it emerges. Note that, since energy-containing motions do not contribute to vertical motions, the spectrum of vertical velocity (dashed line) lacks the k−1 overlap regime, which is instead replaced by a k0 regime [see Wyngaard, 2010, p. 234 for a theoretical argument]. The latter scaling has been more consistently observed than the k−1 regime of the TKE spectrum, indicative of its robustness to variations in external parameters.
Panchev [1971] (pages 219-223) proposed an interesting argument for the emergence of a k−1 regime in the TKE spectrum. He considered the budget for the spectrum of TKE at a given wavenumber, assumed to be a balance between mechanical injection of energy, non-linear transfer of energy from large to small scales [modeled following Heisenberg, 1948], and viscous dissipation. With respect to its scale-averaged counterpart (1.5), the spectral balance contains an additional non-linear transfer of energy term, whose integrated contribution to the averaged balance is zero. Panchev [1971] showed that the k−1 regime emerged in the limit where, following Tchen [1953, 1954] the mechanical injection of energy occurs only by interaction of small-scale turbulent vorticity with the mean wind vorticity. This limit is opposed to the case where mechanical injection of energy results from the interaction between small and large required for a Direct Numerical Simulation (DNS) of the full boundary layer, and is also related to the number of degrees of freedom of the flow [Landau and Lifshitz, 1959]. scale turbulent vorticity [Tchen, 1953]. The former limit occurs when the ratio between the large- and Kolmogorov-scale vorticity is small (dU /dx)(ν/ǫ)1/2 ≪ 1. In the logarithmic layer, this condition is valid, and can be expressed equivalently as lin1/2 ≫ 1 or as (z/η)2/3 ≫ 1. A suggested interpretation of this analysis is that energy-containing eddies in the logarithmic sublayer should be viewed as an imprint of the mean flow on turbulence statistics, giving a justification for their scaling with mean-flow variables.
In the viscous sublayer, the scale of energy containing-eddies and the scale at which viscous dissipation occurs are similar, and the above picture does not hold. Note that this is used by Jiménez [2018] to determine the bottom of the logarithmic sublayer (see its figure 3a).
What is more interesting for the following is what happens in the roughness sublayer, where the scale of energy-containing eddies is fixed and depends solely on the roughness sublayer height [Raupach et al., 1991, Gioia et al., 2010, Bonetti et al., 2017]. This is at the core of Chapter 4 of this thesis. A possible explanation for this scaling is reviewed below. Finally, the presence of airflow separation events in the roughness sublayer has a last important impact, as it creates small-scale eddies, shortcutting the Kolmogorov cascade. Large-scale energy is then directly transported at the dissipation scale, which can have impact on the shape of the inertial-subrange spectra [Finnigan, 2000].
COHERENT STRUCTURES So far we have described the properties of statistical eddies in the roughness and logarithmic sublayers, which represent the most likely state of the flow with respect to some of its properties. The flow can also be described in terms of coherent structures, which are representative structures of the flow with intrinsic dynamics [Jiménez, 2018].7 This gives an interesting insight on the dynamics of turbulence, with the hope that the properties of coherent structures are related to the statistical properties of the flow, and hence to the eddies described above. Here we discuss some of these coherent structures, summarized in Fig. 1.3.
Energy-containing eddies have been introduced above as « momentum-transporting eddies », which support most of the turbulent momentum flux −u ∗2 in the logarithmic and roughness sublayers. A widely-used tool to go beyond this scaling description is quadrant analysis, in which instantaneous contributions to the momentum flux are separated into four quadrants (see Fig. 1.3a). The first (Q1, u′ > 0 and w′ > 0) and third (Q3, u′ < 0 and w′ < 0) quadrants correspond to outward and inward interactions respectively, interpreted as a slow interaction induced by a momentum-transporting vertical motion w′. The second (Q2, u′ < 0 and w′ > 0) and fourth (Q4, u′ > 0 and w′ > 0) quadrants correspond respectively to ejections (or bursts) of low momentum fluid upwards and to sweeps of high momentum fluid downwards. In wall-bounded flows, since the momentum flux u′w ′ is downwards (negative), most of the contributions to its intensity arise from ejections and sweeps.
In the roughness sublayer above forest canopies, the distribution is skewed towards sweeps (Fig. 1.3a). As mentioned above, this indicates that the dominant events in the roughness sublayer of canopies are downward intrusions of high momentum fluid, which transport TKE down in the roughness elements where it is dissipated. It is interesting to note that, even though the contribution of sweeps to u′w′ is higher than that of bursts, they are less frequent [Finnigan, 2000]. This is an example of the statistical properties of turbulence being set by intense and intermittent events (another one is breaking of ocean surface waves, discussed in Sec. 1.3).
Low-Reynolds-number studies give an interesting insight on the coherent structures associ-ated with these events. In the logarithmic sublayer, the DNS of Lozano-Durán et al. [2012], Lozano-Durán and Jiménez [2014] showed that Q2 and Q4 events come in pairs of counter-rotating vortices, with a statistically significant asymmetry (see Fig. 1.3b) and a long lifetime in which they grow in a self-similar manner. Interestingly, these pairs have a size and a velocity similar to Townsend’s attached eddies (z and u∗ respectively). Alternative candidates to these events are hairpin vortices (see Fig. 1.3c) which originate from a near-wall instability and grow in packets, transporting momentum and TKE upwards in the logarithmic layer [Adrian, 2007]. This is a much more organized scenario than Lozano-Durán and Jiménez [2014], where Q2-Q4 pairs do not necessarily originate near the wall, but has been very useful in building theoretical models of near-wall turbulence [Marusic and Monty, 2019].
Some of these structures have been shown to follow a self-sustaining cycle, i.e. they are observed in DNS without the presence of larger scale flow, and are sufficient to reproduce the bulk flow statistics. While it is well known that, in buffer-layer, coherent structures are modulated by the outer flow [e.g. Marusic et al., 2010, Squire et al., 2016], this point is less clear in the logarithmic layer. As Jiménez [2012] mentions, « we can expect some modulation of the logarithmic layer, including possibly long-range ordering, from the global modes above it ». This modulation by outer flow eddies is of major importance, since this interaction governs the general response of wall-bounded flows to external influences. In particular, in the presence of a roughness sublayer, it has been recently shown that these modulations are particularly strong [Anderson, 2016].
It should be noted that, besides this bottom-up view of the logarithmic sublayer, other interpretations exist, such as the top-down approach presented in Hunt and Morrison [2000] for high-Reynolds number flows. This approach draws on observations of cat’s paws [e.g. Dorman and Mollo-Christensen, 1973, on top of water], and describes the downward advection of upper layer eddies down to the surface, which generates ejections and logarithmic-layer vortices through various mechanisms (an example of such a mechanism is given in Fig. 1.3d). It is however out of the scope of this introduction to compare those two views in more detail.
We end this discussion on coherent structures by describing a model, first introduced by [Raupach et al., 1996], to explain the scaling of energy-containing eddies in the roughness sublayer. The model draws on the analogy between the roughness sublayer top and a plane mixing layer, supported by data on top of forest canopies. In both cases, the presence of an inflection point in the mean wind speed generates streamwise rolls by Kelvin-Helmoltz instability. Secondary instability around these rolls lead to coherent structures of similar spanwise and streamwise extensions (see Fig. 1.3e). Their extension is dictated by the scale of the mean wind shear, which is similar to the roughness-sublayer height. Hence this « mixing layer analogy » describes the scale (2 − 5hr ) of energy-containing eddies in the roughness sublayer as being set by instabilities near the roughness sublayer top.
Table of contents :
Introduction
1 Turbulence and waves: a literature review
1.1 Wall-bounded turbulence with stratification
1.1.1 General considerations for neutral conditions
1.1.2 Effect of stability: Monin-Obukhov Similarity Theory
1.2 A phenomenological spectral link
1.3 The dynamical interaction between near-surface turbulence and waves
1.3.1 Momentum balance in the wave boundary layer
1.3.2 Theoretical models of wave-induced stress
1.3.3 Turbulence in the wave boundary layer
1.3.4 Concluding remarks
1.4 Objectives of the present work
2 Towards a « spectral link » for the vertical velocity spectrum?
2.1 Introduction
2.2 Article: « Scalewise return-to-isotropy in stratified boundary layer flows »
2.3 Article: « Scaling laws for the Length Scale of Energy-containing Eddies in a Sheared and Thermally Stratified Atmospheric Surface Layer »
2.4 Conclusion
3 Geometrical impact of wind-waves on energy-containing eddies
3.1 Introduction
3.2 Article: « On the impact of long wind-waves on near-surface turbulence and momentum fluxes »
3.3 Conclusion
4 Dynamical impact of wind-waves on energy-containing eddies
4.1 Introduction
4.2 Article: « Revisiting Beaufort scale: the dynamical coupling between turbulence and breaking waves »
4.2.1 Main text
4.2.2 Supplementary material
4.3 Conclusion
Conclusion
Appendices
Annex A: Some steps for the derivation of the spectral budget
Annex B: Comparison of several return-to-isotropy models
Annex C: Details on the preliminary numerical simulation
