Loss of Gaussianity in Oceanic Turbulent Flows with Internal Waves and Solitons

Transcription

1 Loss of Gaussianity in Oceanic Turbulent Flows with Internal Waves and Solitons Peter C. Chu Naval Ocean Analysis and Prediction Laboratory, Naval Postgraduate School Monterey, California

2 Abstract Structure functions of arbitrary order are analyzed using high-resolution upper layer (above 140 m depth) temperature data in the western Philippine Sea near Taiwan during July 28 August 7, The scaling features of these structure functions clearly discriminate between temperature fluctuations observed with internal waves and solitons. For internal wave propagation, Keywords: Gaussionity, internal waves, internal solitons, multifractal characteristics, structure function, turbulence 2

3 1. Introduction Investigations of the statistical properties of upper ocean isopycnal displacements from high resolution data have shown turbulent characteristics (Chu and Hsieh,. Because in situ measurements have been restricted thus far Eddy viscosity (K M ) and diffusivity (K H ) represent unresolved turbulent mixing and diffusion. For stable stratification, mixing and diffusion are conducted by two different processes: turbulence (vertical-horizontal isotropy) and internal waves (verticalhorizontal anisotropy) on similar scales. When averaging over all scales, there exists a range of the Richardson number (Ri), in which turbulence undergoes remarkably anisotropic (under the effect of internal waves); the vertical mixing becomes suppressed while the horizontal mixing is enhanced. Since the internal waves mix the momentum but do not mix a scalar variable such as temperature, K H decreases at large Ri and tends to its molecular value, K M remains finite and larger than its molecular value. Using physical variables space, the Reynolds stress models (commonly called turbulence closure models) were developed based on a hierarchy of turbulence correlations. Unknown correlations are related to the known ones via closure assumptions. In the Reynolds stress models, the anisotropic feature is difficult to represent (Sukoriansky et al., 2006). Recently, Sukoriansky et al. (2006) developed the quasi-normal scale elimination (QNSE) method to overcome such difficulty (anisotropic under stable stratification). The turbulence and internal waves are treated as one entity with the same spectrum. More specifically, the QNSE procedure is based upon the quasi-normal mapping of the velocity and temperature fields using the Langevan equations. It employs a recursive procedure of small-scale mode elimination that results in a coupled system of differential equations for 3

4 effective, horizontal and vertical, viscosities and diffusivities. With increasing stratification, the vertical viscosity and diffusivity are suppressed while their horizontal counterparts are enhanced thus explicitly accounting for the anisotropy introduced by stable stratification. Verified by several atmospheric boundary layer data sets such as BASE, SHEBA, and CASE99, this new method shows capability to model stable-stratification induced disparity between the transport processes in the horizontal and vertical directions and to account for combined effect of turbulence and internal waves. The anisotropic turbulent viscosities and viscosities obtained from the QNSE model are in good agreement with the experimental data. Questions arise: What is the necessary condition for the validity of quasinormality? Can such a condition be easily identified using observational data? To answer these questions, certain statistical characteristics from the same spectrum should be first discussed. These characteristics may serve as the necessary condition for the validity of the quasi-normality. In this study, preservation of multifractal characteristics in the structure function is used as criterion for the quasi-normality. The western Philippine Sea (WPS) near Taiwan (Fig. 1) was chosen for illustration. In WPS the Philippines Current flows northward from just north of Mindanao (the southernmost of the main island groups of the Philippines) at around (10 N, 128 o E) to Taiwan where it continues as the Kuroshio. A large warm eddy lies east of the Luzon Strait, which is a seemingly permanent feature (Nitani, 1972; Chu and Li, 2000; Chu and Fan, 2001). The internal waves are generated by the shallow ridges in the Luzon Strait (Liu et al., 1998, Chu and Hsieh, 2007). During 4

5 the Asian Seas International Acoustics Experiment for the South China Sea, a moored array of current, temperature, conductivity, and pressure sensors was deployed across the Chinese continental shelf and slope. The dominant oceanographic signal by far was in fact the highly nonlinear internal waves which were generated near the Batan Islands in the Luzon Strait and propagated 485 km across deep water to the observation region (i.e., northern South China Sea) (Ramp et al., 2004). Recently, a coastal monitoring buoy (CMB) with attached 15 thermistors at various depths was deployed by the U.S. Naval Oceanographic Office in WPS (Fig. 2) during July 28 August 7, The sampling rate of the thermistors is one per 15 s. These observations provide useful data to identify the quasi-normality, and in turn the validity of the QNSE method. The outline of this paper is as follows: A description of the QNSE method is given in Sections 2. Multifractal characteristics of the structure function are discussed in Section 3. The dataset is presented in Section 4. Indentification of quasinormality on three periods is depicted in Section 5. Section 6 presents the conclusions. 2. QNSE Method Let [x = (x 1, x 2, x 3 ), t] be the position vector and time and [u = (u 1, u 2, u 3 ), T] be the velocity vector and temperature, which are satisfied the momentum, heat, and continuity equations with the Boussinesq approximation. Here, the subscripts (1, 2) indicate horizontal direction, and the subscript (3) indicates the vertical direction. The spectral theory is developed for a fully three dimensional turbulent flow field (u) with imposed vertical, stabilizing temperature gradient with the buoyancy frequency (N) defined by 5

6 N dt g dz 1/2 = ( α ), (1) with α the thermal expansion coefficient, g the gravitational acceleration, and T the horizontally mean temperature. The spectral domain is bounded by the viscous dissipation wave number, kd = ( ε / ν ), where ε is the dissipation rate. On the base of 3 1/4 0 quasi-normal assumption, turbulence and internal-waves are represented by the same Fourier transforms in the velocity and temperature fields (Sukoriansky et al., 2006), 1 u ( x, ) ˆ i t = d dωui(, ω)exp[ i( ωt)] (2 π ) k k k x, (2a) 4 k kd 1 T( x, t) = d dωtˆ (, ω)exp[ i( ωt)] (2 π ) k k k x. (2b) 4 k k d Here, uˆ ( k, ω), Tˆ ( k, ω) are the Fourier transformations of u ( x, t), T( x, t). The i i QNSE method uses the renormalized perturbation theory of turbulence to get effective or eddy viscosity and diffusivity first for an infinitesimal band of smallest scale modes through ensemble averaging on perturbation solutions derived at the Reynolds number (Re) of the order of O(1). Then, this procedure is repeated for the next band of the remaining smallest scales, etc. In this process, the small-scale modes are successively purged from the governing equations which thus undergo a gradual coarsening while the effective, based upon the eddy viscosity, Reynolds number Re remains O(1) (Sukoriansky et al., 2006). With the spectral representation (2a) and (2b), one can explicitly control the effects of scaling and transition in the Fourier domain. Here, the data are spatially discrete since k d the highest wave number. 3. Structural Function 6

7 Self-similarity and scaling in fully developed turbulent flow are shown in the q-th order structure function, Srq (, ) = ux ( + r,) t ux (,) t q. (3) Here, r = r L, and L is the length scale of energy input. The angular brackets refer to an average over different realizations of the flow field. The first-order structure function S(r, 1) was initially introduced to study small-scale turbulence where it appeared to be an effective tool in linking turbulence physics with statistics. The scale-invariance is represented by Sr (,1) ~ r H, (4) where H is the scaling exponent, called the Hurst exponent. In 1941, Kolmogorov suggested that the velocity increment in high-reynolds number turbulent flows should scale with the mean (time-averaged) energy dissipation and the separation length scale. The Hurst exponent H is equal to 1/3. For the q-th order structural function, simple selfsimilarity implies ux ( + r, t) ux (, t) q ux ( + r, t) ux (, t) q, which is Srq (, ) ~ r qh. (5) Multifractal behavior may be generalized from (5) into Srq (, ) ( q) = r ζ, (6) 7

8 where ζ ( q) is the spectral exponent. In other words, simple self-similarity may be described with a single scaling exponent H, while multifractal scale-invariance requires a spectrum of scaling exponents with ζ ( q) qh. The QNSE algorithm explicitly accounts for the combined effect of turbulence and internal waves and derives a modified dispersion relation for internal waves, a relationship for internal wave frequency shift, and a threshold criterion for internal wave generation in the presence of turbulent scrambling. This indicates that basic spectral characteristics such as multifractal features of the structure function (6) should preserve within the QNSE system, which is the necessary condition for the validity of quasinormality (i.e., validity for QNSE). 4. Data Description The original design of CMB is to collect the data every 10 min near the air-ocean interface (Chu and Hsieh, 2007). Above the ocean surface, the surface winds, air temperature, and air pressure are measured. Below the ocean surface, the temperature is observed at 1, 3, 5, 10, 15, and 20 m. During the observational period (July 28 August 7, 2005) the CMB travels km along the track (Fig.2) with an average speed of 3.82 m per 15 sec. The surface winds are weak (around 4 m/s) and the surface air temperature is close to the water temperature at 1 m depth (implying weak buoyancy flux across the air-ocean interface). Weak surface winds and buoyancy fluxes make shallow surface mixed layer. Fifteen thermistors are attached to a wire rope extending from the code of CMB (20 m deep) to 140 m with high frequency sampling rate (every 15 sec) (Fig. 3). Time- 8

9 depth cross section of temperature along the CMB track shows a multi-scale variability with highly irregular nature (Fig. 4). The surface mixed layer is very thin (depth around 5 m) on July 28 with temperature about 28.5 o C. Below the surface mixed layer, two thermoclines appear with the first thermocline to the depth of 50 m. The vertical gradient in the first thermocline is around 0.1 o C/m. A relatively uniform sublayer (24 o C) exists below the first thermocline from 50 to 130 m. Below the uniform sublayer, there is a second thermocline with a vertical temperature gradient around 0.04 o C/m. As time approaches, the surface mixed layer deepens and the first thermocline descends. These processes contain small scale fluctuations. The surface mixed layer reaches 70 m on August 5, when an evident cooling occurs with the mixed layer temperature reducing to 25.5 o C. At the same time, the two thermoclines merge to a single themocline with the vertical gradient of 0.06 o C/m. Theories of turbulence as applied to point measurements in ocean drifting buoy concern the scaling properties, in a statistical sense, of differenced time series, where the Taylor hypothesis is invoked so that the difference between measurements at some time t and a later time t +τ acts as a proxy for the difference between measurements made at two points in the fluid separated by length scale l. For a time series at a certain depth z, T(x i, z) with an evenly spaced interval l and a total length L, an increment series x = il, i = 0,1,..., Λ, L =Λ l, i Δ T ( x, z) = T( x, z) T( x, z), i = 0,1,..., Λ r, (7) r i i+ r i can be constructed with respect to the increment rl (r is an integer). 9

10 During the temperature sampling by CMB thermistors (Fig. 4), temperature perturbation in the upper layer can be calculated by T'( t, z) = T( t, z) T( z), (8) which is the deviation of the observed temperature from the time-mean temperature. Without salinity data, the isopycnal displacement is calculated by (Desaubles and Gregg, 1981) which fluctuates with various amplitudes ( A η ). T'( t, z) η (, tz) =, (9) dt / dz Studies of scaling in upper ocean turbulence have focused on the power spectra and the structure functions (see e.g. Chu 2004). The q-th order structure function (3) is calculated Λ r q 1 q Srq (, ) Δ T(, xz) = ΔT( x, z). (10) Λ r r i r i= 0 Here, r is inversely proportional to the wavenumber l, 1 r. l Obviously, for small r, Δ T( x, rl) represents the small-scale fluctuations (usually i turbulence). 5. Quasinormalilty Identification Three 5-hr observational periods are used for identifying the quasinormality: (1) GMT August 1, (2) GMT July 29, and (3) GMT July 30. To do so, the structure functions are computed for the three periods using (3). Linear dependence of Log 2 [S(r, q)] on Log 2 (r) shows the existence of the multifractal 10

11 chracteristics in structure functions, and therefore in turn indicates the satisfaction of the quasinormality condition. Non-existence of multifractal chracteristics in structure functions means non-existence of quasinormality. From the following analysis, existence of quasinormality is identified during Period-1 and Period-2, and non-existence of quasinormality is identified during Period Period-1 (no Internal Waves and Solitons) During this period ( GMT August 1), time-depth cross-section of temperature (Fig. 5a) and temperature perturbation (Fig. 5b) do not show evident internal wave propagation. The maximum temperature fluctuation decreases with depth from ± 1.3 o C at the surface to ± 0 o C at m depth, and then increases with depth to ± 0.9 o C at 140 m. The isopycnal displacement η ( tz, ) at four different depths (40 m, 45 m, 55 m, and 60 m) is used for illustration. Fig. 6a shows that η ( tz, ) is very small with maximum amplitude around 4-6 m. Fig. 6b shows time evolution of the isopycnal displacement profiles with 7 min apart from 0100 to 0135 GMT August 1, The vertical profiles of η ( tz, ) does not show oscillation pattern, which implies no evident internal wave propagation. Near-linear dependence of Log 2 [S(r, q)] on Log 2 (r) is found with different q- values from 1 to 6 for all depths (Fig. 7), i.e., the structure function S(r, q) satisfies the power law (6) with the exponent ζ ( q) only depending on q. The power of the structure function ζ ( q) is monotonically and near-linearly increasing with q. Fig. 8 shows the dependence of the structure function s power, ζ ( q), on q and depth. The power exponent, ζ ( q), increases with q for all depths when q is equal or less than 2. It decreases with q for q > 2 at z = 25 m and z = 80 m, and otherwise (i.e., increases with q) 11

12 at other depths. For the latter, the increase of ζ ( q) with q is more evident at the depths away from the thermocline (65 80 m). For q = 6, the power ζ ( q) varies from 0.5 to Period-2 (with Internal Waves) During this period ( GMT July 29), time-depth cross-section of temperature (Fig. 9a) and temperature perturbation (Fig. 9b) show evident internal wave propagation. The maximum temperature fluctuation decreases with depth from ± 1.6 o C at the surface to ± 0.4 o C at 100 m deep and then increases with depth to ± 1 o C at 140 m deep. The isopycnal displacement η (, tz) at four different depths (40 m, 45 m, 55 m, and 60 m) is used to illustrate the strength of the internal waves. Fig. 10a shows that η (, tz) oscillates with time and has maximum downwelling (20 m) at all the four depths at 1200 GMT July 29, Fig. 10b shows time evolution of the isopycnal displacement profiles with 7 min apart from 1052 to 1155 GMT July 29. Evident oscillation is noted in the upper layer above 50 m with maximum amplitude around 20 m. Near-linear dependence of Log 2 [S(r, q)] on Log 2 (r) is found with different q- values from 1 to 6 for all depths (Fig. 11), i.e., the structure function S(r, q) satisfies the power law (6) with the exponent ζ ( q) only depending on q. The power of the structure function ζ ( q) is monotonically and near-linearly increasing with q. Fig. 12 shows the dependence of the structure function s power, ζ ( q), on q and depth. For q 1, there is not much difference between Fig. 12 and Fig. 8. The difference becomes evident as q increases. For q = 6, the power ζ ( q) varies from 0.5 to 2.1 for Period-1 (no internal waves), but from 1.2 to 3.0 for Period-2 (internal waves). This indicates that the internal 12

13 waves increase the power of the structure function especially for high moments. This may be related to the higher energy in each mode during Period-2 (turbulence and internal waves) than during Period-1 (turbulence only) Period-3 (with Internal Solitons) During this period ( GMT July 30), time-depth cross-section of temperature (Fig. 13a) and temperature perturbation (Fig. 13b) show evident internalsoliton propagation. The maximum temperature fluctuation increases with depth from ± 1.8 o C at the surface to ± 2.9 o C at 60 m deep, then decreases with depth to ± 0.8 o C at 100 m deep, and finally increases with depth to ± 2.1 o C at 140 m deep. The observed temperature profile oscillates evidently almost in all depths from 25 to 140 m (Fig. 13b). Different descending of the first thermocline is found between Period-2 (50 m, Fig. 9) and Period-3 (80 m, Fig. 13). The amplitude of the oscillation is much larger during Period-3 (internal solitons) with maximum amplitude around 4 o C than during Period-2 (internal waves) with maximum amplitude around 2 o C. The isopycnal displacement η (, tz) at four different depths (40 m, 45 m, 55 m, and 60 m) is used to illustrate the strength of the internal solitons. Fig. 14a shows that the amplitude of isopycnal displacement η(, tz) is around 50 m at 45 and 50 m depths and increases to near 100 m at 60 m depth with frequency around 4 cycles per hour (CPH). Fig. 14b shows time evolution of the isopycnal displacement profiles with 7 min apart from 0752 to 0827 GMT July 30, On 0752 GMT (blue curve), the profile represents upward displacement with maximum value of 20 m at 30 m depth. Seven min later (0757 GMT), the profile (green curve) represents downward displacement with 13

14 maximum value of -100 m at 60 m depth. Fourteen min later, the profile (red curve) represent weak displacement. Most evident difference between Period-3 and Periods-1, 2 is in the structure functions. the power-law (6) does not exist during Period-3 (Fig. 15). It is broken approximately at r = 8 min, which is nearly half period of the internal solitons (with frequency of 4 CPH). This phenomenon occurs at all depths. The internal solitary waves are a class of nonsinusoidal, nonlinear, more-or-less isolated waves of complex shape that maintain their coherence. The internal solitons are usually composed of several oscillations confined to limited region of space, and their energy spectrum is totally different from the turbulence and internal wave spectrum (2a) and (2b). This might be the reason for the multifractal characteristics of the structural function break at near half of the oscillation period (8 min) for Period-3 (internal solitons) with coherent structures. 6. Conclusions (1) Quasinormality is the necessary condition for the QNSE algorithm. In this study, we propose a use structure function to identify the quasinormality. Since the internal waves and turbulence are represented by the same Fourier spectrum (2), the structure function should have similar characteristics. It is well known that for a turbulent flow, the structure function has multifractal characteristics with power law. The power exponent ζ ( q) only depends on the order of the structure function. Therefore existence of power law in the structure function can be treated as the necessary condition of the quasinormality. (2) Temperature of the Western Philippine Sea upper layer (to 140 m deep) was sampled with high frequency from July 28 to August 7, 2005 from the coastal monitoring 14

15 buoy with attached 15 thermistors. The isopycnal displacement is calculated from the temperature field, and the internal waves and solitons were observed. Three periods with different types of thermal variability are identified: Period-1 (( GMT August 1) with no evident internal waves and solitons, Period-2 ( GMT July 29) with evident internal waves, and Period-3 ( GMT July 30) with internal solitons. (3) During Period-1 the temperature fluctuation has maximum values at the surface, decreases with depth to mid-depths (60-65 m deep), and then increases with depth to 140 m deep. Such depth dependent (decreasing then increasing) pattern preserves during the internal wave propagation during GMT July 29, 2005 (Period-2). However, this was altered during the internal soliton propagation (Period-3) to a pattern that increases with depth from the surface to 60 m deep, decreases with depth from 60 m deep to 100 m deep, and increases again with depth from 100 m to 140 m deep. The temperature fluctuation enhances with the internal wave and soliton propagation. Between the two, the internal solitons bring larger fluctuations. (4) The structure function satisfies the power law with multifractal characteristics for Period-1 and Period-2, but not for Period-3. The internal waves increase the power of the structure function especially for high moments. The internal solitons destroy the multifractal characteristics of the structure function. The power law is broken approximately at the lag of 8 min, which is nearly half period of the IS (with frequency of 4 CPH). This indicates that the QNSE method can not be applied to turbulence with internal solitons. 15

16 Acknowledgments. This work was funded by the Office of Naval Research, Naval Oceanographic Office, and the Naval Postgraduate School. 16

17 References Chu, P.C., Multifractal thermal characteristics of the southwestern GIN Sea upper layer. Chaos, Solitons and Fractals, 19 (2), Chu, P.C., Li, R.F., South China Sea isopycnal surface circulation. Journal of Physical Oceanography, 30, Chu, P.C., Fan, C.W., Low salinity, cool-core cyclonic eddy detected northwest of Luzon during the South China Sea Monsoon Experiment (SCSMEX) in July Journal of Oceanography, 57, Chu, P.C., Ivanov, L.M., Kantha, L., Melnichenko, O., Poberezhny, Y., Power law decay in model predictability skill. Geophysical Research Letters, 29 (15), /2002GLO Chu, P.C., Hsieh, C.-P., Multifractal thermal characteristics of the western Philippine Sea upper layer. Indian Journal of Marine Sciences, 36 (2), Desaubles Y., Gregg, M.C., Reversible and irreversible finestructure. Journal of Physical Oceanography, 11,

18 Liu, A. K., Chang, Y. S., Hsu, M.-K., Liang, N. K., 1998 Evolution of nonlinear internal waves in the East and South China Seas. Journal of Geophysical Research, 103, Nitani, H., 1972: Beginning of Kuroshio. In Kuroshio: Its Physical Aspects, edited by H. Stommel and K. Yoshida, University of Tokyo Press, Tokyo, Ramp, S. R., Tang, T.-Y., Duda, T. F., Lynch, J. F., Liu, A. K., Chiu, C.-S., Bahr, F. L., Kim, H.-R., Yang, Y.-J., Internal Solitons in the Northeastern South China Sea Part I: Sources and Deep Water Propagation, IEEE Journal of Oceanic Engineering, 29, Sukoriansky, S., Galperin, B., Perov, V., 2006, A quasi-normal scale elimination model of turbulence and its application to stably stratified flows. Nonlinear Progress in Geophysics, 13,

19 Figure1. Topography of the western Philippine Sea and surrounding areas. 19

20 Figure 2. Track of CMB (from July 28 to August 7, 2005) deployed by the Naval Oceanographic Office. 20

21 Figure 3. Coastal Monitoring Buoy used in the WPS survey. Fifteen thermistors are attached to a wire rope extending from the code of CMB (20 m deep) to 140 m with high frequency sampling rate (every 15 sec). 21

22 Figure 4. Time-depth cross section of temperature obtained from the CMB data collected along the track shown in Figure 2. 22

23 (a) (b) Figure 5. Time-depth cross section on GMT August 1 of (a) temperature, and (b) temperature perturbation obtained from the CMB data collected along the track marked as wave-c in Figure 2. There are no evident internal waves. 23

24 (a) (b) Figure 6. Isopycnal displacements measured by thermistors with maximum value less than 5 m on GMT August 1, 2005: (a) versus time at 45 m, 50 m, 55 m, and 60 m depths, and (b) versus depth at different time. It is noted that the maximum displacement is less than 5 m. 24

25 (a) (b) (c) (d) (e) (f) Figure 7. Structure functions of temperature for GMT August 1, 2005 (no evident internal waves) at (a) 25 m, (b) 40 m, (c) 60 m, (d) 75 m, (e) 100 m, and (f) 140 m depths. 25

26 Figure 8. Dependence of the structure function s power, ζ ( q), on q and depth for GMT August 1 (no evident internal waves),

27 (a) (b) Figure 9. Time-depth cross section on GMT July 29, 2005 of (a) temperature, and (b) temperature perturbation obtained from the CMB data collected along the track marked as wave-a in Figure 2. Both figures show evident internal wave propagation. 27

28 (a) (b) Figure 10. Isopycnal displacements measured by thermistors with maximum value less than 5 m on GMT July 29, 2005: (a) versus time at 45 m, 50 m, 55 m, and 60 m depths, and (b) versus depth at different time. 28

29 (a) (b) (c) (d) (e) (f) Figure 11. Structure functions of temperature for GMT July 29, 2005 (internal waves) at (a) 25 m, (b) 40 m, (c) 60 m, (d) 75 m, (e) 100 m, and (f) 140 m depths. 29

30 Figure 12. Dependence of the structure function s power, ζ ( q), on q and depth for GMT July 29 (internal waves). 30

31 (a) (b) Figure 13. Time-depth cross section on GMT July 30, 2005 of (a) temperature, and (b) temperature perturbation obtained from the CMB data collected along the track marked as wave-b in Figure 2. Both figures show evident internal soliton propagation. 31

32 (a) (b) Figure 14. Isopycnal displacements measured by thermistors with maximum value less than 5 m on GMT July 30, 2005: (a) versus time at 45 m, 50 m, 55 m, and 60 m depths, and (b) versus depth at different time. 32

33 (a) (b) (c) (d) (e) (f) Figure 15. Structure functions of temperature for GMT July 30, 2005 (internal solitons) at (a) 25 m, (b) 40 m, (c) 60 m, (d) 75 m, (e) 100 m, and (f) 140 m depths. It is noted that the power law is broken at around r = 8 min, which is nearly half period of the internal solitons (with frequency of 4 CPH). 33