Spectral characteristics of velocity and vorticity fluxes in an unstratified turbulent boundary layer

Transcription

1 Spectral characteristics of velocity and vorticity fluxes in an unstratified turbulent boundary layer R.-C. Lien and T. B. Sanford Applied Physics Laboratory and School of Oceanography, College of Ocean and Fishery Sciences, University of Washington, Seattle Abstract. Wavenumber spectral characteristics of the velocity and vorticity fluxes in an unstratified turbulent boundary layer are presented. The observed vertical and streamwise velocity spectra agree with empirical forms found in the atmospheric boundary layer. Spectral ratios of 4/3 between the vertical and streamwise velocity spectra and the agreement between the observed vorticity flux quad spectrum and that of isotropic turbulence suggest local isotropy at scales smaller than Z. The normalized cospectrum of the momentum flux agrees remarkably well with the empirical form found in the atmospheric boundary layer. In the inertial subrange the momentum flux cospectrum shows a clear spectral slope of 7/3. The observed composite vorticity flux cospectrum has most of its variance at the streamwise wavenumber k x =(1 1) Z 1 and has a spectral slope of 7/3 in the inertial subrange. The 7/3 spectral slope is consistent with a dimensional argument, assuming that the vorticity flux cospectrum is proportional to the gradient of the mean vorticity, and depends on the turbulence kinetic energy dissipation rate ε and the wavenumber. A model turbulent vorticity flux cospectrum is constructed based on the shape of observed spectra, a 7/3 spectral slope in the inertial subrange, and the similarity scaling of the vorticity flux in an unstratified turbulent boundary layer. The turbulence vorticity flux is directly related to the divergence of turbulence momentum flux, the force exerted by turbulence on the mean flow. Therefore our proposed empirical cospectral form of the vorticity fluxes might be useful for turbulence parameterization in numerical models. 1. Introduction Extensive studies of turbulence spectral properties have been conducted in the atmospheric boundary layer, e.g., the 1968 Kansas experiment and the 1973 Minnesota experiment [Kaimal and Wyngaard, 199]. Understanding of the spectral and cospectral properties of turbulence is needed to interpret and correct measurements of turbulent energy and fluxes. These properties are also useful for parameterizations of turbulent fluxes in numerical models. Universal spectral forms of turbulence velocity, momentum flux, and heat flux have emerged from comprehensive measurements and theoretical studies of the atmospheric boundary layer. On the basis of dimensional arguments, Wyngaard and Co te [197] suggested a 7/3 power law for the turbulent momentum flux u w and heat flux w θ cospectrum in the inertial subrange. Evidence of the 7/3 slope has been found in the atmospheric surface layer [Kaimal et al., 197; Kader and Yaglom, 1991]. Antonia and Zhu [1994] found a large wavenumber range with a 5/3 slope for the heat flux cospectrum before the inertial subrange slope of 7/3. They suggested that the observed slope of 5/3 might be due to a strong anisotropy of their measured turbulence field. Horst [1997] presented a simple correction formula for turbulence fluxes estimated from a fastresponse anemometer. The formula depends on both the sensor response time and the true heat flux cospectrum, particularly the wavenumber of the cospectral 1

2 LIEN AND SANFORD: SPECTRA OF VELOCITY AND VORTICITY FLUXES peak. These formulas are needed to correct the observed flux cospectra so as to obtain correct estimates of fluxes. There have been relatively few studies of turbulence spectral properties in the oceanic boundary layer compared to studies in the atmospheric boundary layer. Gross and Nowell [1985] found a slope of 5/3 of velocity spectra in the inertial subrange in an unstratified oceanic tidal boundary layer. Their observed momentum flux cospectra have a significant fraction at wavelengths greater than the distance from the boundary, suggesting that significant portion of turbulence stress might be due to flattened-out eddies. Bowden and Ferguson [198] studied turbulence structure in the bottom boundary layer in the eastern Irish Sea. They also observed the -5/3 inertial-subrange spectral slope at high wavenumbers. The ratio of streamwise velocity spectra to vertical velocity spectra approaches 3/4 at k x πz, where k x is the streamwise wavenumber and Z is the height above the bottom, suggesting isotropic turbulence. McPhee [1994] proposed that the turbulence mixing length scale can be estimated as the inverse of the wavenumber of the peak of vertical velocity spectrum. Their velocity spectra observed beneath multiyear pack ice in the western Weddell Sea agree with the empirical spectrum in the atmospheric boundary layer formulated by Busch and Panofsky [1968]. In the turbulent boundary layer the turbulent advective flux of vorticity plays an important dynamical role. Laboratory experiments and numerical simulations of boundary layer turbulence have shown coherent vortical motions in turbulent boundary layers [Robinson, 1991]. These vortical motions are important for momentum transfer between inner and outer boundary layers [Kim et al., 1971; Willmarth and Lu, 197; Blackwelder and Eckelmann, 1979; Ranasoma and Sleath, 199]. The sweeping process of the momentum flux is related to the creation of new vortices at the boundary [Bernard et al., 1993]. Indeed, there exists a kinematic relation between the divergence of the momentum flux and the advective fluxes of vorticity. The divergence of the turbulence momentum flux represents the force that turbulence exerts on the mean flow. Therefore the turbulence vorticity flux is directly related to the turbulence force on the mean flow. Klewicki et al. [1994] have conducted extensive studies of advective vorticity fluxes. They found that the divergence of the momentum flux is balanced by two components of the advective fluxes of vorticity, one associated with the gradient of vorticity and the other associated with the change of eddy scales from boundaries. This balance is consistent with Tennekes and Lumley s [197] dimensional argument. The advective flux of vorticity is also important in the surface boundary layer, where the turbulence vorticity advected by Stokes drift is the main driving mechanism of Langmuir circulation [Li and Garrett, 1995]. Recently, Sanford and Lien [1999] reported velocity and vorticity measurements in an energetic tidal channel and estimated the turbulent enstrophy, momentum flux, and advective vorticity flux. They proposed a new method to estimate the friction velocity based on the turbulence vorticity flux. Their sensor scale is.9 m. Wallace and Foss [1995] reviewed attenuation problems in vorticity measurements by numerous sensors and concluded that in order to have accurate measurements of enstrophy, sensor scales must be of the order of the Kolmogorov scale. In oceanic environments and tidal channels this requires sensors on millimeter to centimeter scales. Sanford and Lien [1999] defined an eddy diffusivity of vorticity A ζ based on the ratio of the advective vorticity flux and the gradient of the mean vorticity in analogy to the definition of eddy viscosity A ν. Because the estimate of A ζ is equal to A ν in the boundary layer, Sanford and Lien [1999] argued that both the momentum and vorticity fluxes are not significantly attenuated by the finite sensor scale. To confirm this assertion, one needs to know the spectral forms of the momentum flux and the vorticity flux. Here we present spectral properties of turbulence velocity and vorticity fluxes observed in an oceanic tidal boundary layer. We begin with a brief review of classical theory of the turbulent velocity spectra and flux cospectra for isotropic turbulence and empirical velocity spectra and flux cospectra observed in atmospheric boundary layers (section ). In section 3, spectra of the velocity and the momentum flux calculated from measurements in an unstratified tidal channel are presented and compared with theoretical spectral forms and empirical forms found in the atmospheric boundary layer. Our observed vorticity flux spectrum is discussed in section 4. The spectral slope of the vorticity flux cospectrum in the inertial subrange is consistent with the prediction from the dimensional argument. An empirical model for the vorticity flux spectrum is proposed. In section 5 we discuss the quad spectrum of the vorticity fluxes, scaling of the vorticity flux, and effect of viscous dissipation and summarize our results. Sensor response functions for the momentum and vorticity fluxes are described in Appendix A. Fractions of resolved momentum fluxes and vorticity fluxes as a func-

3 LIEN AND SANFORD: SPECTRA OF VELOCITY AND VORTICITY FLUXES 3 tion of wavenumber and distance from the boundary are discussed in Appendix B.. Review of Turbulence Velocity and Flux Spectra.1. Isotropic Turbulence For isotropic turbulence the velocity wavenumber spectra and the momentum flux cross spectra in the inertial subrange are described by Batchelor [1959]. In the inertial subrange the turbulence energy flux is constant and spectral properties are independent of the external forcing at large scales and viscous dissipation at Kolmogorov microscales. In this wavenumber range, turbulence energy spectrum is well described by the Kolmogorov spectrum with a -5/3 spectral slope. The vorticity flux cross spectra in the inertial subrange of isotropic turbulence can be obtained based on Batchelor s formula. In this analysis we are interested mostly in the streamwise wavenumber k x spectra. After Batchelor s three-dimensional spectra are integrated over the vertical and spanwise wavenumbers, the streamwise wavenumber spectra of the streamwise velocity Φ uu (k x ), the vertical velocity Φ ww (k x ), the momentum flux Φ uw (k x ), and the vorticity flux Φ wζy (k x ) have the following forms: Φ uu (k x ) = αε/3 k 5/3 x, (1) Φ ww (k x ) = αε/3 kx 5/3, () Φ uw (k x ) =, (3) Φ wζy (k x ) = iα 3 1 ε/3 k /3 x. (4) The Kolmogorov constant α is 1.5 [Sreenivasan, 1995]; ε is the turbulence kinetic energy dissipation rate; u and w are the streamwise velocity and vertical velocity components, respectively; and ζ y is the spanwise vorticity. For isotropic turbulence the streamwise wavenumber spectrum of vertical velocity is 4/3 of the streamwise velocity spectrum in the inertial subrange. This has been used to test for local isotropy [McPhee and Smith, 1976]. Another salient feature is the vanishing turbulence fluxes, because the isotropic turbulence has no preferred flux direction. Note that the nonzero quad spectrum of the vorticity flux (4) for isotropic turbulence does not involve dynamic reasons but, rather, is a result of pure kinematics. The reality condition of the vorticity flux implies that Φ wζy (k x ) = Φ wζ y ( k x ), where the asterisk indicates the complex conjugate. Therefore the quad spectrum in the positive streamwise wavenumber domain equals the minus of the quad spectrum in the negative streamwise wavenumber domain, and the integration of quad spectrum over the whole streamwise wavenumber domain vanishes... Atmospheric Boundary Layer Observations The expressions for the spectra discussed in the previous section exclude effects of the boundary and stratification, which introduce anisotropy into the turbulence field. In the presence of a boundary and stratification, these spectral forms are modified. On the basis of data taken in the 1968 Kansas turbulent boundary layer experiment, Kaimal et al. [197] obtained empirical spectral forms for turbulent velocity and momentum flux in different stability conditions. Under the neutral stability condition the streamwise velocity spectrum Φ uu, the vertical velocity spectrum Φ ww, and the momentum flux cospectrum P uw are expressed as Φ uu (k x ) = 1 ( π u Z k ) 5/3 xz, (5) π [ Φ ww (k x ) =.1 ( ) ] 5/3 1 kx Z π u Z , (6) π P uw (k x ) = 1 ( π u Z ) 7/3 π k xz, (7) where the factor of π is used to convert their expressions from cyclic wavenumber to radian wavenumber and u is the friction velocity, u = (τ/ρ) 1/, where τ is the bottom stress and ρ is the density. The vertical velocities of turbulence are strongly affected by the boundary, so that most of their variance is near the scale of the height above the bottom Z. The horizontal turbulent velocities are less affected by the presence of the boundary, and most of their variance is at scales significantly greater than Z. Turbulence velocity spectra observed in the atmospheric boundary layer have been found to be in good agreement with the spectra predicted by the empirical formulas of Kaimal et al. [197] [Lee, 1996; Young, 1987]. These empirical velocity spectra were also reproduced based on a theoretical argument of the turbulence

4 LIEN AND SANFORD: SPECTRA OF VELOCITY AND VORTICITY FLUXES 4 energy budget which assumed that turbulence in the Kansas experiment was isotropic [Moraes and Goedert, 1988]. Turbulent momentum fluxes are carried mostly by large eddies. In the inertial subrange, momentum flux cospectra P uw do not vanish as required by the local isotropy condition (3) but, rather, decrease rapidly with a k 7/3 dependence [Wyngaard and Co te, 197]. The 7/3 spectral slope was predicted by the dimensional argument in which the cospectrum of momentum flux depends only on ε and k and is linearly proportional to the mean shear z U. Note that the integration of the momentum flux cospectrum over the entire wavenumber domain should yield the total momentum flux, i.e., u. The integration of the empirical momentum flux cospectrum proposed by Kaimal et al. [197], however, yields a slightly underestimated momentum flux,.94 u. A similar empirical spectral form for the turbulent vertical velocity was proposed earlier by Busch and Panofsky [1968]; that is, Φ BP ww (k x ) =.54u Z [ ( ) ] 5/3 1 kx Z (8) At low wavenumbers k x Z the spectral level of Busch and Panofsky s [1968] model is greater than that of Kaimal et al. s [197] model. McPhee [1994] found that his vertical velocity spectra taken in the ocean under the ice agree with Busch and Panofsky s empirical form. 3. Observed Spectra 3.1. Measurements, Experiment, and Data Selection The measurements discussed in this analysis were taken with an electromagnetic vorticity meter (EMVM) [Sanford et al., 1999]. The sensor scale l is.9 m. The data were taken in Pickering Passage, Washington, in a strong ebb tide,.8 m s 1 peak surface tidal flow, during October 3 6, During this experiment the flow was unstratified. Details of the experiment and measurements and a summary of the turbulence properties are given by Sanford and Lien [1999]. In this paper, spectra will be calculated using data taken when the EMVM was held at fixed depths to eliminate possible contamination of the vertical velocity by sensor motion and yield a clean estimate of the streamwise wavenumber spectrum when using Taylor s hypothesis to convert frequency to wavenumber. There are 15 min of data taken at a -Hz sampling rate divided into 79 time segments. Of these, 74 segments were taken within 8 meters above the bottom (mab). Because the EMVM has two sensors on opposite sides of the instrument, 158 velocity spectra and flux cospectra are computed in this analysis. The two sensors are separated by.5 m spanwise. Therefore these two sensors observe the same eddies if the eddy scales are >.5 m. 3.. Velocity Spectra The streamwise and vertical velocity spectra are calculated using a multitaper spectral analysis with two tapers [Percival and Walden, 1993]. The observed velocity spectra are attenuated at small scales by the sensor response [Sanford et al., 1999]. To correct for the sensor effect, the observed velocity spectra have been compensated by dividing by the sensor response functions. The observed velocity spectra are normalized by the measured turbulence kinetic energy dissipation rate ε and averaged in several depth bins (Figure 1). This normalization (equations (1) and ()) is done to collapse the spectra in the inertial subrange for isotropic turbulence. The normalized streamwise velocity spectra are nearly independent of depth and show a spectral slope of 5/3. The normalized vertical velocity spectra are clearly dependent on Z. At scales greater than Z, i.e., k x Z 1, the vertical velocity spectra are nearly white. At scales smaller than Z, i.e., k x Z 1, the vertical velocity spectra exhibit a 5/3 inertial-subrange spectral slope consistent with (). According to the empirical spectral form observed in the atmospheric boundary layer (equations (5) and (6)), velocity spectra normalized by Z should depend only on the nondimensional wavenumber k x Z and the friction velocity u. The friction velocity was calculated as.4 m s 1 by Sanford and Lien [1999]. Normalized wavenumber spectra at various depth bins indeed collapse to a universal form (Figure 1b). The ratios between observed streamwise and vertical velocity spectra converge to the value of 3/4 expected for isotropic turbulence at k x Z 5 (Figure 1c). Saddoughi and Veeravalli [1994] tested the local isotropy in the turbulent boundary layer at high Reynolds numbers in a wind tunnel and found that the energy spectra reached the local isotropy at k x ε 1/ S 3/ 3. If we assume that ε = u 3 κ 1 Z 1 and that the mean shear S = u κ 1 Z 1 in the inertial sublayer, the energy spectra should exhibit a local isotropy condition at k x Z 7.5, close to our observed range of k x Z 5. The composites of normalized spectra of the observed

5 LIEN AND SANFORD: SPECTRA OF VELOCITY AND VORTICITY FLUXES 5 vertical and streamwise velocity are compared with empirical spectra suggested by Kaimal et al. [197] and Busch and Panofsky [1968] (Figure ). Because the momentum flux u w is nearly constant in the bottom 9 m [Sanford and Lien, 1999], only spectra at Z < 9 m are used for constructing the composite spectra. The streamwise spectrum agrees with the empirical form of Kaimal et al. [197] within the 95% confidence interval, except at the highest wavenumbers, where the observed spectrum falls below the empirical spectrum. The discrepancy at high wavenumbers might be due to sensor attenuation. Although the sensor effect has been corrected by the response function, it is possible that the correction was not adequate owing to the smearing effect of the varying mean streamwise velocity. Our composite vertical velocity spectrum agrees with the empirical form suggested by Busch and Panofsky [1968] within the confidence interval, similar to McPhee s [1994] results. At low wavenumbers the observed spectrum is slightly greater than the empirical spectrum found by Kaimal et al. [197] in the atmospheric boundary layer Momentum Flux Cospectra We compute the momentum flux cospectra from observations, normalize them by Z, and form a composite cospectrum as a function of the normalized wavenumber k x Z (Figure 3). Again, only spectra calculated at Z < 9 m are used to construct the composite spectrum. The observed composite momentum flux cospectrum agrees, within a 95% confidence interval, with the empirical spectrum [Kaimal et al., 197] based on the estimated friction velocity of.4 m s 1 [Sanford and Lien, 1999]. At small scales, i.e., k x Z, the observed cospectrum falls below the empirical cospectrum, although they still agree with each other within the confidence intervals. The sensor response attenuates the observed cospectra at small scales. The sensor response function for the momentum flux cospectrum of the EMVM measurements is derived in Appendix A. We correct the observed spectra by the sensor response function and recalculate the composite momentum flux cospectrum. The corrected momentum flux cospectrum is slightly greater than the uncorrected cospectra at large k x Z, but the difference is insignificant. In other words, our estimates of the momentum flux are not significantly affected by the finite size of the EMVM sensor in the high- Reynolds-number flow of an unstratified tidal channel. Indeed, on the basis of the empirical momentum flux cospectrum and the EMVM response function, we conclude that the EMVM estimates capture more than 94% of the total momentum flux at.5 mab (Appendix B). Φ u Z 1 & Φ w Z 1 (m s ) Φ u (ε / ε) /3 (m 3 s ) m (a) k x Z 5.3 m 1.9 m.5 m (b) k x (m 1 ) Φ u / Φ w Figure 1. (a) Observed spectra of streamwise velocity (solid curves) and vertical velocity (dashed curves) averaged in various depth bins as a function of streamwise wavenumber k x. The labels indicate the height above the bottom Z of each depth bin. The shading represents the 95% confidence interval. These spectra have been normalized by the observed turbulence kinetic energy dissipation rate ε. (b) Observed spectra and wavenumbers normalized by Z (using equations (5) and (6)). The horizontal and vertical velocity spectra are collapsed to their corresponding universal forms. The thick solid and dashed curves represent the mean normalized streamwise and vertical velocity spectra, respectively. (c) Ratios between observed streamwise velocity spectra and vertical velocity spectra. Dots are ratios of spectra at different depth bins. Crosses are ratios of mean spectra. Vertical bars across symbols denote the 95% confidence interval of the F test. The horizontal line in Figure 1c indicates the 3/4 value expected for isotropic turbulence (equations (1) and ()). k x Z (c)

6 LIEN AND SANFORD: SPECTRA OF VELOCITY AND VORTICITY FLUXES 6 Fractions of the resolved momentum flux as a function of wavenumber are discussed in Appendix B to provide other experimentalists with references for interpreting their observations. Φ u / (u * Z) & Φ w / (u * Z) k x Z Figure. Comparison of the observed streamwise velocity spectrum and vertical velocity spectrum with empirical spectra found in the atmosphere boundary layer. The thick solid curve is the composite observed streamwise velocity spectrum. The thin solid curve is the empirical streamwise velocity spectrum suggested by Kaimal et al. [197]. The thick dashed curve is the composite observed vertical velocity spectrum, and the thin dashed curve is Kaimal et al. s [197] empirical vertical velocity spectrum. The thin solid curve with open circles is the empirical vertical velocity spectrum proposed by Busch and Panofsky [1968]. Shadings denote the 95% confidence interval ofthe observed spectra. P uw / (u * Z) /3 (a) k x Z 1 k x Z P uw / (u * Z) (b) k x Z 1 Figure 3. Comparison of the observed momentum flux cospectrum with the empirical cospectral form found by Kaimal et al. [197] in the atmospheric boundary layer. (a) Comparison of observed and model spectra. (b) Variance-preserving form of Figure 3a. The thick solid curves are observed spectra. The thin solid curves are the empirical spectra suggested by Kaimal et al. [197]. The thick dashed curves, which are nearly overlapped by the thick solid curves, are observed spectra corrected for the sensor response function. In the inertial subrange the empirical momentum flux cospectrum has a spectral slope of 7/3. Saddoughi and Veeravalli [1994] found that the momentum flux cospectrum dropped to zero, an indication of local isotropy, at k x Z 5. The falloff of our observed spectrum from the model spectrum at k x Z could also be because k x Z is in the wavenumber range where the viscous dissipation becomes important. 4. Vorticity Flux Spectrum 4.1. Observed Vorticity Flux Cospectrum The observed vorticity flux cospectra averaged in three depth bins are shown in Figure 4 as a variancepreserving plot. Most of the vorticity flux is at wavenumbers of. 5 m 1. In this wavenumber range the vorticity flux is mostly positive. Below and beyond this range the vorticity flux cospectra are sometimes less than zero. Because of their large error bars, the vorticity flux cospectra in these three depth bins are not significantly different, except that the vorticity flux in the depth bin 4 8 m seems to be slightly larger at k x of.4.5 m 1. Sanford and Lien [1999] observed a simi-

7 LIEN AND SANFORD: SPECTRA OF VELOCITY AND VORTICITY FLUXES 7 lar feature in a vertical profile of vorticity flux that was calculated from data taken while the EMVM was vertically profiled. The averaged vorticity flux in the depth range m is slightly greater than that in the 4 m depth bin but not significantly so relative to the statistical confidence interval. The variation of the estimated vorticity flux within each depth bin is greater than the variation among different depth bins. To our knowledge, no empirical spectral form has been suggested for the vorticity flux in the atmospheric or oceanic turbulent boundary layer. Despite the great variability of the observed vorticity flux cospectra, they do have a common spectral shape, i.e., large and positive at intermediate wavenumbers and smaller and sometimes negative at small and large wavenumbers. This commonly observed shape suggests the existence of a universal spectral form. Following the dimensional analysis used by Wyngaard and Coté [197] for deriving the momentum flux and heat flux cospectra in the inertial subrange, we assume that the vorticity flux cospectrum in the inertial subrange depends on only three parameters: ε, k, and the vertical gradient of mean spanwise vorticity z ζ y. We further assume that the cospectrum is linearly proportional to the gradient of the mean vorticity but with an opposite sign, consistent with the concept of a downgradient flux. The dimensional analysis yields a form for the vorticity flux cospectrum of k x P wζy (m s ) 3.5 x m.8 m 1.1 m k x (m 1 ) Figure 4. Observed vorticity flux cospectra averaged in three depth bins: m (solid curve), 4 m (dashed curve), and 4 8 m (dotted curve). The shading denotes the 95% confidence interval calculated from the bootstrap method. Labels indicate the averaged heights above the bottom of the three depth bins. 4.. Model Vorticity Flux Cospectrum P wζy (k x ) ε 1/3 z ζ y k 7/3 x u (k x Z) 7/3. (9) This expression is similar to the empirical forms for the momentum and heat flux cospectra. In the inertial sublayer, ε = u 3 κ 1 Z 1 and z ζ y u κ 1 Z. These identities have been used in (9). The vorticity flux cospectrum does not vanish in the inertial subrange, but it decays rapidly with a spectral slope of 7/3. In the turbulent boundary layer, large eddies are affected by the presence of the boundary. Following the empirical momentum flux cospectral form [Kaimal et al., 197], we propose that the vorticity cospectrum has the form P model wζ y (k x ) = au [1 + b(k x Z)] r. (1) This form is consistent with our observed vorticity spectra; that is, most of the variance is in the intermediate wavenumbers and decays at lower and higher wavenumbers. The dimensional argument in (9) suggests that the spectral slope in the inertial subrange r = 7/3. Parameters a and b depend on the magnitude of the vorticity flux and the wavenumber of the dominant vorticity flux, respectively; that is, w ζ y = dk x Pwζ model a y (k x ) = b(r 1) u /Z, (11) (k x Z) peak = (r 1) 1 b 1, (1) where (k x Z) peak is the wavenumber of the maximum vorticity flux variance k x P model wζ y. An averaged normalized vorticity flux cospectrum in normalized wavenumber bins k x Z is shown in Figure 5. This spectrum is constructed by first normalizing each of the total 148 vorticity flux cospectra by u and the corresponding wavenumbers by Z 1 and then averaging all the 148 normalized cospectra in the normalized

8 LIEN AND SANFORD: SPECTRA OF VELOCITY AND VORTICITY FLUXES 8 wavenumber bins. Note that 8% of spectra are observed in the bottom 4 m. The averaged, normalized vorticity flux cospectrum shows a wavenumber-bandlimited spectral shape similar to that of the momentum flux cospectrum. At small, normalized wavenumbers the lower bounds of the vorticity flux estimates are negative and show large error bars. The variancepreserving plot k x ZP wζy u shows a peak at k x Z. The corresponding estimate of b is 3/8 based on (1). P wζy / u * (a) 7/ k x Z k x Z P wζy /u * (b) k x Z Figure 5. (a) Comparison of the observed vorticity flux cospectrum with our proposed model vorticity flux cospectrum. (b) The variance-preserving form of Figure 5a. The thick solid curve is the observed vorticity flux cospectrum calculated using spectra obtained in the bottom 8 m, and the thin curve is the model spectrum (equation (13)). The thick dashed curve is the observed vorticity flux cospectrum corrected for the sensor response function. A spectral slope of 7/3 is shown for comparison with the model spectrum in the inertial subrange. Open circles represent observed vorticity flux cospectra averaged using data in the bottom 5 m, where the observed vorticity flux follows the similarity scaling (Figure 7). In high-reynolds-number flows one might expect the mixing coefficients of momentum and vorticity to have a similar magnitude. Indeed, Sanford and Lien [1999] found that the magnitudes of the estimated eddy diffusivity of vorticity [A ζ = w ζ y ( z ζ y ) 1 ] and the eddy viscosity of momentum [A ν = u w ( z U) 1 ] were similar within 6 mab. In the energetic tidal boundary layer the primary contribution to the mean spanwise vorticity is the vertical shear of streamwise velocity; that is, ζ y = z U [Sanford and Lien, 1999]. In the inertial sublayer the eddy viscosity of momentum A ν and the eddy diffusivity of vorticity A ζ can be approximated as u κz, and the vertical gradient of the mean spanwise vorticity z ζ y = u κ 1 Z. Therefore the turbulent vorticity flux in the inertial sublayer can be expressed as w ζ y = A ζ z ζ y = u Z 1. Tennekes and Lumley [197] similarly scaled the vorticity flux w ζ y as ul z ζ y, where u = u is the turbulent velocity scale and l = κz is the turbulent length scale in the boundary layer. Therefore the scaling of w ζ y = u Z 1 is also suggested by Tennekes and Lumley [197]. If we accept the foregoing scaling argument for the turbulent vorticity flux, (11) can be expressed as a/[b(r 1)] = 1. Given r = 7/3 and b = 3/8, the estimate of a is 1/. Therefore we have constructed the following form for the model vorticity flux cospectrum: P model wζ y (k x ) = 1/u [1 + 3/8(k x Z)] 7/3. (13) The model spectrum has the shape of observed spectra, the slope of 7/3 in the inertial subrange predicted by the dimensional analysis, and the dominant peak displayed by the observed spectra, and the magnitude of the vorticity flux is consistent with flows in the inertial sublayer. Note that when constructing the model spectrum, we have used only one property of the observed spectrum, the wavenumber of the maximum vorticity flux variance. We did not force the level and shape of the model spectrum to agree with those of the observed spectrum. Surprisingly, the model spectrum clearly reproduces the observed vorticity flux cospectrum (Figure 5) in the wavenumber range from the peak of the vorticity flux variance extending more than 1 decade into the inertial subrange where a spectral slope of 7/3 exists. At low wavenumbers, k x Z.9, the observed vorticity flux spectrum drops much more rapidly than the proposed model spectrum. Although the vorticity flux cospectrum and the momentum flux cospectrum have a very similar spectral shape, the wavenumber of the maximum flux is higher for the vorticity flux than for the momentum flux. This is partially because most of the turbulent momentum is at large scales and the variance of vorticity is mostly contributed by smaller-scale eddies. To correct for the finite-size-sensor effect, we modify the observed vorticity flux cospectra by their response functions (Appendix A) and form the composite spectrum. The corrected cospectrum is not significantly different from the uncorrected one, except at k x Z 4, where the corrected spectrum shows a small bump. We suspect that this small bump is unreal and is caused by dividing the observed spectrum by a response func-

9 LIEN AND SANFORD: SPECTRA OF VELOCITY AND VORTICITY FLUXES 9 tion that is small. On the basis of the EMVM response function for the vorticity flux cospectrum (Appendix A) and our model spectrum, the EMVM resolves > 9% of the total vorticity flux at Z > 1 m (Appendix B). 5. Discussion and Summary 5.1. Vorticity Flux Quad Spectrum of Isotropic Turbulence Note that the vorticity flux quad spectrum Q wζy of the isotropic turbulence does not vanish [Batchelor, 1959]. We calculate the composite quad spectrum of the measured vorticity flux, correct for the sensor response function, and compare the result with the expected quad spectrum in the inertial subrange, as predicted by (4) (Figure 6). The response function of the quad spectrum is based on the theoretical quad-spectral form of Batchelor [1959] for isotropic turbulence. The observed vorticity flux quad spectrum shows a peak near k x Z =, similar to its cospectrum, and rolls off at a slope of /3, as expected from (4). At k x Z the observed spectral level is one half the model level before the correction for the sensor effect. After correction the observed spectrum agrees much better with the expected spectrum at k x Z >. This confirms the derived response function and the local isotropy at large k x Z. 5.. Scaling of Vorticity Flux On the basis of the turbulent properties in the inertial sublayer of a turbulent boundary layer, the turbulent vertical transport of spanwise vorticity w ζ y should scale as u Z 1. The variation of our observed vorticity flux cospectra at individual depths is greater than the variation among cospectra at different depths. Longer stationary time series of vorticity flux in the inertial sublayer are needed to confirm this scaling. Using data taken during the period when the EMVM was profiled through the water column in the same experiment, Sanford and Lien [1999] found that the vertical flux of the spanwise vorticity scaled with Z 1 in the bottom 3 m. To reveal the details of the vorticity flux in the bottom few meters, we calculated the average vorticity flux in depth bins proportional to Z 1 using the data used by Sanford and Lien [1999]. The vorticity flux in the bottom 3 m indeed shows a u Z 1 scaling, with a fitted friction velocity u of.6 m s 1 and a 95% confidence interval of. m s 1 (Figure 7). The estimated u agrees with the value of.4 m s 1 calculated by the profile method, the eddy correlation method, and the dissipation method [Sanford and Lien, 1999]. This result supports the scaling of vorticity flux we used for constructing the model vorticity flux cospectrum. At Z > 5 m the vorticity flux does not scale as u Z 1. Indeed, the scaling should work only in the inertial sublayer. Sanford and Lien [1999] found two log layers in their observed streamwise velocity profile. The lower log layer exists in the bottom 3 m with an estimated friction velocity of.4 m s 1, a transition layer between 3 and 5 mab, and an upper log layer between 5 and 1 m with a friction velocity of.43 m s 1. The averaged, normalized vorticity flux cospectrum calculated using spectra in the bottom 5 m is not significantly different from that calculated using spectra in the bottom 8 m (Figure 5) because more than 8% of observed spectra are in the bottom 5 m. Q wζy (m s ) /3 1 1 k x Z Figure 6. Comparison of the observed composite quad spectrum of the vorticity flux and the theoretical quad spectrum of isotropic turbulence [Batchelor, 1959]. The observed quad spectrum is denoted by the thick solid curve. The thin solid curve is the theoretical quad spectrum for isotropic turbulence in the inertial subrange. The thick dashed curve is the observed quad spectrum corrected for the sensor response function. The shading denotes the 95% confidence interval Viscous Effects The empirical turbulence spectra and cospectra discussed in this analysis were aimed at explaining the observed spectra in the inertial subrange. Beyond the inertial subrange, molecular viscosity becomes impor-

10 LIEN AND SANFORD: SPECTRA OF VELOCITY AND VORTICITY FLUXES 1 tant, and the inertial subrange scaling should not work. The inertial subrange lies in the wavenumber range of k(ν 3 /ε) 1/4 1, where ν is the molecular viscosity. In Pickering Passage, ε is 1 6 m s 3 in the bottom boundary layer and ν is m s 1. Accordingly, the inertial subrange lies at k x 4 1 m 1. Corrsin [1964] and Pao [1965] showed that the peak of the shear spectrum and the roll-off of the velocity spectrum occur at a wavenumber of.(ν 3 /ε) 1/4. Therefore the inertial subrange of our observed spectra in Pickering Passage probably lies below 1 m 1. Z 1 (m 1 ) u * =.6 (±.) m s <w ζ y > (m s ) x 1 3 Figure 7. Vertical profile of w ζ y averaged in depth bins as a function of Z 1. For reference, the equivalent depth scale is shown on the right margin. The thin solid curve is the observed mean w ζ y, and the shading denotes its 95% confidence interval. The thick line is the fitted vorticity flux as a function of Z 1 ; that is, w ζ y = u Z 1. The estimated u is.6 m s 1, with the 95% confidence interval of. m s 1 denoted by the two thick dashed curves. This estimate of u agrees with the value of.4 m s 1 calculated by conventional methods [Sanford and Lien, 1999]. 1 Z (m) 6. Summary The present analysis shows that the spectral and cospectral properties of the velocity and the momentum flux observed in an unstratified oceanic tidal boundary layer are in good agreement with the universal forms found in the atmospheric boundary layer. The momentum flux is carried by large-scale eddies, which are fully resolved by the EMVM sensor. The vorticity flux cospectrum shows a wavenumber-band-limited spectral shape, similar to that of the momentum flux cospectrum, but centered at smaller scales. A model spectrum of the turbulent vorticity flux is proposed. The spectral slope in the inertial subrange is determined by a dimensional analysis following that of Wyngaard and Co te [197]. The wavenumber of the maximum variance of the turbulent vorticity flux is determined by the observed spectrum. The spectral level is determined by classical theories of turbulent boundary flows. The observed vorticity flux is dominantly at k x from 1 to 1 Z 1. The model spectrum agrees with the observed spectrum at most wavenumbers, except at the lowest wavenumbers where the observed spectrum falls below the model. The response functions of the vorticity and momentum flux cospectra characterizing the sensor effect are obtained, and the observed spectra are corrected by the response functions. The correction does not produce significant changes in the observed momentum and vorticity flux cospectra. This indicates that the scale of the EMVM sensor is sufficiently small to resolve turbulent momentum and vorticity flux cospectra, especially at Z m. The turbulence vorticity flux plays an important dynamic role in the turbulence boundary layer. The similarity scaling of the turbulence vorticity flux presented in this analysis can be used to estimate the bed stress. The turbulence vorticity flux is related to the divergence of turbulence momentum flux, which represents

11 LIEN AND SANFORD: SPECTRA OF VELOCITY AND VORTICITY FLUXES 11 the turbulence force on the mean flow. Therefore understanding the spectral properties of the turbulence vorticity flux will help improve turbulence parameterization schemes in numerical models. The advective flux of turbulence vorticity by Stokes drift is the primary driving mechanism for the surface Langmuir circulation. Therefore studying properties of the advective vorticity flux in the Langmuir circulation will lead to better understanding of its dynamics. Further investigation of the vorticity flux cospectrum is necessary to confirm or revise our proposed model spectrum. Appendix A: Appendix A: Sensor Response Functions for Momentum and Vorticity Flux Response functions depend not only on the sensor characteristics but also on the true spectral forms. To derive the response functions of the momentum flux and vorticity flux, we need to know their spectral forms. For the momentum flux we use the empirical cospectral form. For the vorticity flux cospectra we derive the response function based on our model spectrum. For the vorticity flux quad spectrum we use the spectral form for the isotropic turbulence [Batchelor, 1959]. The response functions for the momentum flux cospectrum and vorticity flux cospectrum are derived numerically (Figure A1). The response function of the momentum flux cospectrum shows a sharp attenuation at k x l.5, where l is the EMVM sensor scale. A negative response function also exists at k x l. 1.8 Response Function.6.4. Q wζ P uw P wζ k x l Figure A1. Response functions of the momentum flux cospectrum (solid curve), vorticity flux cospectra (dashed curve), and vorticity flux quad spectra (dotted curve). Appendix B: Appendix B: Resolving Momentum and Vorticity Flux Measurements of turbulent and vorticity fluxes are taken mostly from sensors of finite size, implying a low-pass filtering of the turbulence field. To ensure that measurements capture most of the turbulent fluxes,

12 LIEN AND SANFORD: SPECTRA OF VELOCITY AND VORTICITY FLUXES 1 the sensor size must be smaller than the scales of the fluxes. Because the scale of the dominant vorticity flux is smaller than the scale of the dominant momentum flux, a sensor s scale may be small enough to measure the momentum flux, but not sufficiently small to measure the vorticity flux. We define the fractions of momentum flux and vorticity flux that are resolved as a function of wavenumbers as R uw (k x ) = R wζy (k x ) = kx dk x P uw dk x P uw, (B1) kx dk x P wζy dk x P wζy. (B) On the basis of the empirical momentum flux spectrum of Kaimal et al. [197] and our proposed vorticity flux cospectrum, R uw and R wζy are calculated as function of Z. At a fixed wavenumber the fraction of the resolved fluxes is smaller closer to the bottom (smaller Z) and the fraction of the resolved vorticity flux is smaller than that of the resolved momentum flux (Figure B1). To measure more than 95% of the vorticity flux, the sensor scale has to be smaller than. m when the measurement is taken at 1 mab and smaller than.1 m when the measurement is taken at more than mab. To measure more than 95% of the momentum flux, the sensor scale has to be smaller than.1 m when the measurement is taken above 1 mab. Fraction of Resolved Flux % R uw Z = 5 m. (a) k x (m 1 ) R wζy Z = 5 m R uw Z = 1 m R wζy Z = 1 m Z (m) (b) k 95% (m 1 ) Figure B1. (a) Fraction of fluxes resolved (equations (B1) and (B)) at 1 and 5 meters above bottom (mab). (b) Cutoff wavenumber at which more than 95% of fluxes are resolved as a function of Z. Solid curves are for momentum fluxes, and dashed curves are for vorticity fluxes. On the basis of the empirical momentum flux cospectrum, our model vorticity flux spectrum, and the EMVM response functions (Figure A1), we calculate the resolved momentum and vorticity fluxes as a function of height above the bottom (Figure B). The EMVM captures more than 94% of the momentum flux and 9% of the vorticity flux above 1 mab. Z (m) <u w > EMVM / <u w > & <w ζ y > EMVM /<w ζ y > Figure B. Fraction of momentum flux (solid curve) and vorticity flux (dashed curve) resolved by the electro-magnetic vorticity meter (EMVM) sensor as a function of Z. Fractions were computed using the empirical momentum flux cospectrum, our model vorticity flux cospectrum, and the EMVM sensor response functions. Acknowledgments. The successful observations taken in Pickering Passage were achieved with the vital help of John Dunlap, James Carlson, Eric Boget, and Gordon Welsh. Discussions with Eric Kunze and Eric D Asaro have been very helpful. This research was supported by the Office of Naval Research. References Antonia, R. A., and Y. Zhu, Inertial range behaviour of the longitudinal heat flux cospectrum, Boundary Layer Meteorol., 7, , Batchelor, G. K., The Theory of Homogeneous Turbulence, Cambridge Univ. Press, New York, Bernard, P. S., J. M. Thomas, and R. A. Handler, Vortex dynamics and the production of Reynolds stress, J. Fluid Mech., 53, , Blackwelder, R. F., and H. Eckelmann, Streamwise vortices associated with the bursting phenomenon, J. Fluid Mech., 94, , Bowden, K. F., and S. R. Ferguson, Variation with height

13 LIEN AND SANFORD: SPECTRA OF VELOCITY AND VORTICITY FLUXES 13 of the turbulence in a tidally-induced bottom boundary layer, in Marine Turbulence, Proceedings of the 11th International Liege Colloquium on Ocean Hydrodynamics, edited by J. Nihoul, pp , Elsevier Sci., New York, 198. Busch, N. E., and H. A. Panofsky, Recent spectra of atmospheric turbulence, Q. J. R. Meteorol. Soc., 94, , Corrsin, S., Further generalizations of Onsager s cascade model for turbulent spectra, Phys. Fluids, 7, , Gross, T. F., and A. R. M. Nowell, Spectral scaling in a tidal boundary layer, J. Phys. Oceanogr., 15, , Horst, T. W., A simple formula for attenuation of eddy fluxes measured with first-order-response scalar sensors, Boundary Layer Meteorol., 8, 19 33, Kader, B. A., and A. M. Yaglom, Spectra and correlation functions of surface layer atmospheric turbulence in unstable thermal stratification, in Turbulence and Coherent Structures, edited by O. Metais and M. Lesieur, pp , Kluwer Acad., Norwell, Mass., Kaimal, J. C., and J. C. Wyngaard, The Kansas and Minnesota experiments, Boundary Layer Meteorol., 34, 31 47, 199. Kaimal, J. C., J. C. Wyngaard, Y. Izumi, and O. R. Coté, Spectral characteristics of surface-layer turbulence, Q. J. R. Meteorol. Soc., 98, , 197. Kim, H. T., S. J. Kline, and W. C. Reynolds, The production of turbulence near a smooth wall in a turbulent boundary layer, J. Fluid Mech., 5, , Klewicki, J. C., J. A. Murray, and R. E. Falco, Vortical motion contributions to stress transport in turbulent boundary layers, Phys. Fluids, 6, 77 86, Lee, X., Turbulence spectra and eddy diffusivity over forests, J. Appl. Meteorol., 35, , Li, M., and C. Garrett, Is Langmuir circulation driven by surface waves or surface cooling?, J. Phys. Oceanogr., 5, 64 76, McPhee, M. G., On the turbulent mixing length in the oceanic boundary layer, J. Phys. Oceanogr., 4, 14 31, McPhee, M. G., and J. D. Smith, Measurements of the turbulent boundary layer under pack ice, J. Phys. Oceanogr., 6, , Moraes, O. L. L., and J. Goedert, Kaimal s isopleths from a closure model, Boundary Layer Meteorol., 45, 83 9, Pao, Y. H., Structure of turbulent velocity and scalar fields at large wave numbers, Phys. Fluids, 8, , Percival, D. B., and A. T. Walden, Spectral Analysis for Physical Applications, Cambridge Univ. Press, New York, Ranasoma, K. I. M., and J. F. A. Sleath, Velocity measurements close to rippled beds, in Proceedings of the 3rd International Conference on Coastal Engineering, pp , Am. Soc. of Civ. Eng., Reston, Va., 199. Robinson, S. K., Coherent motions in the turbulent boundary layer, Annu. Rev. Fluid Mech., 3, , Saddoughi, S. G., and S. V. Veeravalli, Local isotropy in turbulent boundary layers at high Reynolds number, J. Fluid Mech., 68, , Sanford, T. B., and R.-C. Lien, Turbulent properties in a homogeneous tidal bottom boundary layer, J. Geophys. Res., 14, , Sanford, T. B., J. A. Carlson, J. H. Dunlap, M. D. Prater, and R.-C. Lien, An electromagnetic vorticity and velocity sensor for observing finescale kinetic fluctuations in the ocean, J. Atmos. Oceanic Technol., 16, , Sreenivasan, K. R., On the universality of the Kolmogorov constant, Phys. Fluids, 7, , Tennekes, H., and J. L. Lumley, A First Course in Turbulence, 3 pp., MIT Press, Cambridge, Mass., 197. Wallace, J. M., and J. F. Foss, The measurement of vorticity in turbulent flows, Annu. Rev. Fluid Mech., 7, , Willmarth, W. W., and S. S. Lu, Structure of the Reynolds stress near the wall, J. Fluid Mech., 55, 65 9, 197. Wyngaard, J. C., and O. R. Co te, Cospectral similarity in the atmospheric surface layer, Q. J. R. Meteorol. Soc., 98, 59 63, 197. Young, G. S., Mixed layer spectra from aircraft measurements, J. Atmos. Sci., 44, , R.-C. Lien and T. B. Sanford, Applied Physics Laboratory and School of Oceanography, College of Ocean and Fishery Sciences, University of Washington, Seattle, WA (lien@apl.washington.edu; sanford@apl.washington.ed June 7, 1999; revised December 13, 1999; accepted January 3,. This preprint was prepared with AGU s LATEX macros v4, with the extension package AGU ++ by P. W. Daly, version 1.6b from 1999/8/19.