Ocean Ultrasonic Shear and Compression Viscosities

The Open Acoustics Journal, 2009, 2, 87-94 87 1874-8376/09 2009 Bentham Open Open Access Ocean Ultrasonic Shear and Compression Viscosities Josè V. Alemán*,1, Pablo Sangrà1 and Rafael Carbó2 1Facultad de Ciencias del Mar, Campus Universitario de Tafira, 35017 Las Palmas de Gran Canaria, Spain 2Centro de Tecnologias Fisicas (CSIC),Serrano 144, 28006 Madrid, Spain Abstract: A comprehensive description of ocean molecular flow and deformation is provided with the help of hydrodynamic and ultrasonic principles. Hydrodynamic computation of true or natural viscosities shows that ocean shear viscosity (G), compression viscosity (K), and extensional viscosity (E) are interrelated. There are no experimental methods available for the in situ measurement of these viscosities. Sound absorption coefficients ( obs) allow to know the ultrasonic shear (UG), compression (UK), and longitudinal (L) viscosities, which decrease with increasing frequency and increase with increasing temperature, the flow activation energies having nearly equivalent values; pressure (depth) increase/decrease them at low/high frequencies. The viscosities * UG, * UK, * L are approached at about 1000 KHz. They decrease with temperature and pressure, and increase with salinity. The *UG becomes equal to the true shear viscosity  G at the viscosity ratio  = UK / UG = 0. Keywords: Chemical relaxation, ocean rheology, sound absorption, ultrasonic viscosity, viscosity ratio. 1. INTRODUCTION The continuum mechanic approach to ocean flow and deformation (ocean rheology) has been described elsewhere [1]. It may be summarized as follows: Turbulent diffusion prevails in the ocean over molecular diffusion. Molecular diffusion in simple fluids, as the ocean is assumed to be, is isotropic because it develops at those very small scales where stratification plays no role in inhibiting the vertical motions. The molecular diffusion mechanism of actual seawater is provided by the physical structural relaxation of clusters [2] which contain associations with weak (water-water) and strong (ion-water) bonds with their corresponding relaxation times [3]. The molecular horizontal momentum transport of seawater is the result of three contributions: shear ( G), extension (E), and compression ( K)(!) flows [4] which are interrelated by Eq. (4) :  G =  12  u  z =  12 1 z x t =  12   12 (1)  E =(  11   22 ) 1 L L t =(  11   22 )   =N 1   (2)  K =p 1    t =p 1 V V t =p   (3) *Address correspondence to this author at the Facultad de Ciencias del Mar, Campus Universitario de Tafira,35017 Las Palmas de Gran Canaria, Spain; E-mail: [email protected] h  E + i  G = j  K (4) (!) The term compression viscosity, internationally accepted [5], is used instead of the traditional bulk viscosity. being 12, (11- 22) = N1, and P, the shear, extensional, and compression stresses, respectively ;   12 ,   ,   , are the shear, extensional, and compression deformation rates ; h, i, j dimensionless parameters. The effect of the system variables on these viscosities is as follows [1]:  G (1.0 - 1.5 x 10 -7 dbar.s) is independent of shear rate   12 (Newton law, Eq. (1)), and  K (2 - 4 x 10 -7 dbar.s) decreases with compression rate   (s-1). Temperature (T) decreases G (Table 1) [1] and increases  K (Table 2) [1]. The Arrhenius equation flow activation energies are positive and negative respectively: EG = 4 kcal/mol, and EK = - 1 kcal/mol. Table 1. Ocean True Shear Viscosities Z (m) P dbar T ºC G a 10-7 dbar.s 10 1 23.58 0.92 100 10 16.69 1.01 500 50 11.87 1.25 1000 100 7.03 1.41 1500 150 5.48 1.48 20100 200 4.03 1.54 a Station B. Salinity (S) increases both  G and  K, but almost unnoticeably. 88 The Open Acoustics Journal, 2009, Volume 2 Alemán et al. Decreasing pressures (P) (depth z) reduce both G (Table 1) and  K (Table 2) making the clusters to move vertically from a compressed (spheroidal) to an elongated (ellipsoidal) state. Table 2. Ocean True Compression Viscosities Z (m)  K a vs z 10-7 dbar.s  K b vs T 10-7 dbar.s T ºC 10 0.20 4.12 25 100 0.40 4.06 20 500 1.20 3.99 15 1000 2.10 3.90 10 1500 3.00 3.80 5 2000 4.20 ------ --- aT = 15ºC,   = 4.5 x 109 s-1,  = 3.5. b = 6.5 x 108 s-1, z = 250 m,  = 3.5. There are no experimental techniques available to date to corroborate these results, but the needed suplementary experimental information may be obtained with in situ measurements of ocean sound absorption. Recent developments of measurement techniques on molecular viscosity by ultrasound may be found in the literature [6-8]. Main studies on the ocean where devoted to oil slicks [9], red algae blooms [10] or the marine microlayer [11]. However to our knowledge there are no studies on the behavior of ultrasonic viscosity on the whole water column of the ocean. In section 3-a we formulate the expression of the ocean sound absorption coefficients, next in section 3-c we discuss the effect of temperature on the seawater viscosity ratio, we follow discussing the effects of pressure (depth), temperature and salinity on ultrasonic viscosities, finally we conclude summarizing the main findings. 2. EXPERIMENTAL PROCEDURES Sound absorption coefficients (abs) were measured with the method of sofar shots detonated at different ocean locations by [12] and references there on. Viscosity ratios () were deduced [1] from standard seawater shear viscosities [13] and ocean compressibility data [14]. 3. RESULTS AND DISCUSSION 3.1. Ocean Sound Absorption Coefficients The sound absorption coefficient (i) for a single relaxation process obeys the equation [15]:  i=  i  0c fif2 fi 2+f2 (5) where o is the isothermal compressibility ( is the symbol commonly used for compressibility in ultrasonics), i the chemical compressibility (Table 3) [16], c the sound velocity, f the acoustic frequency, and fi the relaxation frequency which according to [17] is about 1 KHz for boric acid (B(OH)3) (in this frequency range the contribution of scattering of inhomogeneities in the index of refraction of the medium is a constant independent of frequency, and amounts to 10-6 dB/m [18], about 10 KHz for magnesium sulfate (MgSO4), and about 1,000 KHz for water (H20). Ocean sound absortion coefficients (abs, dB/m) may be obtained from data of [12] (which are also used to asses the potential effects of suspended particles and gas bubbles) with the following equations [17]: At salinity S = 35 [4] (A, B, C, f1, f2, being temperature, pressure, and salinity dependent parameters):  obs =Af 1 f 2 f 1 2 +f 2 +Bf 2 f 2 f 2 2 +f 2 +Cf 2 (6) The first, second, and third terms of the right -hand side of Eq. (6) are related to the relaxation mechanisms of the B(OH)3 (carbonate modify it [4]), MgSO4, and H20, respectively. See Table 3 [16] in which are also shown the partial molal volume change of the solutes in sewawater ( V * ), and the partial molal compressibility of the solutes: Table 3. Chemical Relaxation Mechanisms V * cm3 mol-1   * cm3 mol-1 bar-1 Reaction 25ºC 5ºC 25ºC 5ºC Acid-base equilibrium (non-electrolites): Water H2OH++OH 18.0 19.5 3.15 4.73 Boric acid B(OH) 3 +H 2 0H + +B(OH) 4  25.7 28.0 1.93 2.79 Carbonic acid CO 2 +H 2 OCO 3 H 2 22.6 24.7 1.90 3.51 CO 3 H  H  CO 3 2 ------ ------ ----- ------ Ion-pair equilibrium: MgSO 4 (aq) Mg 2+ (aq) +SO 4 2 (aq) 7.2 ------- 1.7 -------- Ocean Ultrasonic Shear and Compression Viscosities The Open Acoustics Journal, 2009, Volume 2 89   * =V * /P (7) At S = 35 - 37 and frequency f = 2 - 25 KHz [19]:  obs =8.6866 2.34 106Sf Tf2 fT 2+f2+3.38 106f2 fT     16.54 104P () (6a) with the frequencies f and fT = 21.9 x 106 - 1520/(T+273) in KHz, TºC, P in atmosphere, and the factor 8.6866 to convert nepers/m into dB/m. Seawater obs (Fig. 1) behaves differently at frequencies of the order of the H20 relaxation one (500-1,000 KHz), and shows a change of slope at about 15ºC. The molecular structure of pure water is [20] (Fig. 1c): hexagonal (ice crystals) below 0ºC, tetrahedral (pentamer) above 5ºC, chain like (trimer) at about 27ºC, more than 10% of the water molecules have changed their tetrahedral structure to the chain one above 30ºC. The ocean sound absorption coefficient is related to the ultrasonic shear ( UG) and compression ( UK) viscosities by the equation [17]:  theor =16  2f2 3  c3  UG +3 4  UK     =16  2f2 3  c3  UG +4  2f2  c3  UK (8) Fig. (1). Ocean sound absorption coefficient (obs) at S = 35, depth z = 1,000 m,  = 3.2, (a) as a function of temperature at () 10 KHz, () 100 KHz, () 500 KHz, () 1,000 KHz, () 3,000 KHz. (b) as a function of frequency at () 5ºC, () 10ºC, () 15ºC, () 20ºC, () 25ºC. (c) Pure water structure: (i) pentamer (() hydrogen atom, () oxygen atom, from [24]), (ii) trimer (from [10].), (iii) dimer (from [25]), (iv) monomer (from [26]), (v) ion cluster illustrating one water molecule in the second layer (from [2]). α obs(dB/m) 0 200 400 600 800 1000 f (Khz) 10-5 10-4 10-3 10-2 10-1 10 1 α obs(dB/m) (a) (b) (c) 0 5 10 15 20 25 30 T ºC 10-5 10-4 10-3 10-2 10-1 10 1 90 The Open Acoustics Journal, 2009, Volume 2 Alemán et al. in which f is the frequency in Hz, and the viscosities expressed in poises, the density in g/cm3, and the sound velocity c in cm/s calculated with the algorithm described by [17]. The first term of Eq. (8) is the absorption coefficient class in the low frequency limit, i.e., the shear viscous loss [4]:  class =16  2f2 3  c3  UG (9) 3.2. Viscosity Ratio The shear and compression viscosities are interrelated by the equation [1]:  =  UK /  UG (10) Measurement of this dimensionless ratio provide  = 2.81 for seawater [18]. Seawater  values at S = 35 - 37, T = 5ºC25ºC, P = 1 - 2000 dbar, may be deduced as in [1]. Fig. (2a) is a typical example at S = 35 since: i) the G used are for standard seawater [13], ii) the effect of S on both G and K are almost unnoticeable [1], iii) Eq. (6) used to compute the sound absorption coefficients ( obs) is valid only for S=35 [17]. As may be observed the plots are nearly straight lines and  increases with increasing depth (z) and decreasing compression rate (   ). The effect of temperature on the seawater viscosity ratio (Fig. 2b) shows a different slope below and above 15ºC [1], what possibly is due to the fact that seawater is a dilute polymeric system consisting of 6-60 water molecules with an ion in the center (solvation shell) (Fig. 1c). Under the pressure of sound waves the relaxation mechanism is a chemical association-dissociation reaction [15] giving rise to a large volume change upon ionization (Table 3, [16]). Most seawater cations are in a 99% dissociated state, while magnesium sulfate (MgSO4) is only 78.5% dissociated (Table 4) [21]. As the temperature increases a decrease of both the partial molal volume change ( V * ) and the partial molal compressibility (   * ) occurs (Table 3). The corresponding cluster structure expands decreasing the volume deformation ( = ( V / V)) [1]. The final result is an increase in compression viscosity and viscosity ratio (Eqs. (3) and 10)). The viscosity ratio  of pure water has been measured [4] to be 2.68 -2.75 and is almost independent of depth (pressure), while it decreases slightly at all temperatures [4, 22] and, at frequencies of the order of GHz, decreases steeply below 15ºC (Figure 8 of [23]); compare with Fig. (1a) and Eq. (11). 3.3. Ultrasonic Viscosities The ultrasonic compression viscosity ( UK) is [4]:  UK =4 3  UG  obs   class  class      (11) Substituting in Eq.(11) the Eqs.(6-6a, 9, 10) results:  UK =    UG =4 3  UG  obs  class 1      (12) which allows to find the ultrasonic shear viscosity (UG) as follows (the factor 0.25 comes from 0.25 = (4/3) / (16/3)):    UG =4 3  UG   obs 16  2 f 2 3  c 3  UG 4 3  UG =0.25   obs   c 3  2 f 2 4 3  UG (13) Fig. (2). Seawater viscosity ratio () (a) at T = 15ºC, S = 35, as a function of depth (z) and different compression rates (   ): () 4.2 x 108 s-1, () 1.2 109 s-1, () 4.25 x 109 s-1, ( = 25ºC,  = 5ºC), () 1.1 x 1010 s-1. (b) as a function of temperature at S=35, z = 2,000 m, ()   = 4.2 x 109 s-1, ()   = 1.1 x 1010 s-1, () 1.16 x 109 s-1 (z = 500 m). 0123 -2000 -1800 -1600 -1400 -1200 -1000 -800 -600 -400 -200 0 ( ) δ 012345 0 5 10 15 20 25 30 T ºC δ (a) (b) Ocean Ultrasonic Shear and Compression Viscosities The Open Acoustics Journal, 2009, Volume 2 91 Therefore    UG +4 3  UG =  +4 3       UG =0.25   obs   c3  2f2 (14) and (using the factor 10-2 to have the results with units 10-7 dbar.s) the ultrasonic shear viscosity is:  UG =0.25 10 2   obs   c 3  2 f 2 1  +4 3 (15) which at  = 0 is equivalent to Eq. (9) (see also Fig. 3d). The ultrasonic compression viscosity ( UK) is obtained then from Eq. (12):  UK =    UG =0.25 10 2   obs   c 3  2 f 2    +4 3 (16) which at  = 0 becomes  UK = 0 (Fig. 3f). The longitudinal viscosity (L) may be estimated as [18]: L = 4 3 UG + UK (17) Fig. (3a, c, e, g), show that the three viscosities  UG,  UK,  L decrease with increasing frequency, becoming independent of it at about 20 KHz (Fig. 3b). The viscosities * UG, * UK, * L, shown in Fig. (3d, f, h), are approached at the H20 relaxation frequency (1,000 KHz), as shown in Fig. (3c, e, g). As  decreases: * UG increases towards the true viscosity  G (Table 1) [1]. * UK decreases towards * UK = 0. It also decreases with temperature: note that this is contrary to what happens with continuum mechanics  K values (Eq. 3) which increase with temperature (Table 2) [1]. It is due to fact that the effect of temperature is different on the chemical asociationdissociation relaxation than on the physical cluster structure relaxation. The activationn energies (E) computed with Arrhenius equation [1] have nearly equivalent values (shear EUG = 2.46 kcal/mol, compression EUK = 1.80 kcal/mol, longitudinal EL = 2.10 kcal/mol). * L is almost constant suggesting that (Eqs. (17) and (4)):  L *(  0) =4 3  UG *(  0) +  UK *(  0) h  E =i  G +j  K (18) It shows the effect on  E of the strength of the cluster hydrogen bonds (which are, for example, stronger for Na+ and weaker for Cl-), and that the clusters are more compacted at large depths. Fig. (4a) shows that as the temperature increases  UG,  UK,  L increase at low frequencies up to 15ºC and decrease at higher temperatures for all frequencies. It is the result of the combined effects of the sound velocity c (increase with increasing temperature [17]), the sound absorption coefficient obs (Fig. 1), and the viscosity ratio  (Fig. 2b). * UG, * UK, * L (Fig. 4b) decreases with temperature. The effect of pressure (depth) is to increase  UG,  UK, and  L (Fig. 4c). The viscosities * UG, * UK, and * L above 1,000 KHz (Fig. 4d) decrease since increasing depth (P) decrease   * (equation 7). The effect of salinity (S) on the ultrasonic viscosities can be known replacing Eq. (6) by Eq. (6a) in Eq. (15). Fig. (4e, f) show that the salinity increases the three viscosities since increasing salinity decreases V * and   * . Table 4. Dissociation States of Seawater Salts Free Ions % Associated With Cation g/kg (15) % SO4 2+ CO3HSodium Magnesium Calcium Potasium Strontium 10.733 1.294 0.412 0.339 0.0079 99 87 91 99 1.2 11.0 8.0 0.01 1.0 1.0 Anion Na+ Mg2+ Ca2+ K + Chloride 19.354 ---------------------- reference ion---------------------- Sulfate Bicarbonate Carbonate Borate Bromide Fluoride 2.712 0.142 -------- 0.0045 0.067 0.0013 54 69 9 0.01 21 8 17 21.5 19.0 67.0 2 4 4 0.5 ----- 92 The Open Acoustics Journal, 2009, Volume 2 Alemán et al. Fig. (3). Ultrasonic viscosities at T = 15ºC, S = 35,  =1.52, as a function of frequency: (a) ((b) decimal plot) () UG, () UK, () L, and at different viscosity ratios  ((o) 80.23, () 1.52, () 3.03, (c) UG, (e) *UK, (g) *L. As a function of  at frequency fi = 1,000 KHz and different temperatures () 5ºC, () 15ºC, () 25ºC); (d) *UG, (f) *UK, (h) *L. 1 10 100 1000 10000 0.01 0.1 1 10 100 η 10 -7 (dbar.s) f ( KHz) 0 100020003000 0 20 40 60 η 10 -7 (dbar.s) f ( KHz) 1 10 100 1000 10000 0.01 0.1 1 10 100 η UG 10 -7 (dbar.s) f ( KHz) 0.1 1 10 0.01 0.1 1 η∗ UG 10 -7 (dbar.s) δ (a) (b) (c) (d) 1 10 100 1000 10000 0.01 0.1 1 10 100 η UK 10 -7 (dbar.s) f (KHz) 0.1 1 10 0.01 0.1 1 η∗ UK 10 -7 (dbar.s) δ 1 10 100 1000 10000 0.1 1 10 100 η L 10 -7 (dbar.s) f ( KHz ) 0.1 1 10 0.1 1 η∗ L 10 -7 (dbar.s) δ (e) (f) (g) (h) Ocean Ultrasonic Shear and Compression Viscosities The Open Acoustics Journal, 2009, Volume 2 93 4. CONCLUSIONS Seawater is a system which contains compressed hydrogen-bonded reversible ion networks (clusters) suspended in water oligomers. The molecular behavior of this system may be interpreted with two different approaches. The continuum mechanics with which the shear, extensional, and compression viscosities may be evaluated [1]. Ultrasonic measurements which show that ultrasonic shear, ultrasonic compression, and longitudinal viscosities decrease with frequency, increase/decrease with temperature, Fig. (4). Effect of temperature at S = 35, z = 1,000 m,  = 3.2, on (a) ()  UG, ()  UK, ()  L, at frequency (white symbols) 1KHz, and () 700 KHz. (b) () * UG, () *UK, () * L (fi = 1,000 KHz). Effect of depth (z) at T = 15ºC, S = 35,  = 3.2, and frequency (white symbols) 1 KHz, and () 700 KHz: on (c) ()  UG, ()  UK, ()  L ((), , at 700 KHz); (d) () * UG, () * UK, ()  L (f i = 1,000 KHZ). Effect of salinity (S) at T = 15ºC, z = 1,000 m,  = 3.21, and f = 10 KHz: (e) ()  UG, ()  UK, ()  L ((), , , at 700 KHz) ; (f) at fi = 1,000 KHz () *UG, () *UK, () *L. 051015202530 T ( C) 0 1 2 3 4 5 6 7 η 10 -7 (dbar.s) η ∗ 10 -7 (dbar.s) 0 102030405060 -2000 -1500 -1000 -500 0 z(m) 0123456 -2000 -1500 -1000 -500 0 z(m) η 10 -7 (dbar.s) η ∗ 10 -7 (dbar.s) (a) (b) (c) (d) 0 5 10 15 20 25 30 T ( C) 0 10 20 30 40 50 60 94 The Open Acoustics Journal, 2009, Volume 2 Alemán et al. and the pressure (depth) increase/decrease them at low and high frequencies, respectively. At the lower viscosity ratio () the ultrasonic shear viscosity is similar to the continuum mechanics one. ACKNOWLEDGMENTS We thanks to J.L. Pelegri, co-author of the Ocean rheology (2006) article, for useful discussions. Partial support for this work has been provided by the Spanish government through project COUPLING (CTM200806343C02-01/ANT). REFERENCES [1] Aleman JV, Pelegrí JL, Sangrà P. Ocean rheology. 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