ISSN 0001-4370, Oceanology, 2007, Vol. 47, No. 1, pp. 33–41. © Pleiades Publishing, Inc., 2007. Original Russian Text © V.M. Kushnir, 2007, published in Okeanologiya, 2007, Vol. 47, No. 1, pp. 39–48.
MARINE PHYSICS
Bottom Boundary Layer in the Black Sea: Experimental Data, Turbulent Diffusion, and Fluxes V. M. Kushnir Marine Hydrophysical Institute, National Academy of Sciences of Ukraine, Sevastopol, Ukraine Received August 29, 2005
Abstract—Measurements of 14 vertical profiles of currents and hydrological parameters in the near-bottom layer with a depth resolution of 0.1 m were carried out in several regions of the Black Sea shelf, at five points over the continental slope, and in three deep water regions. The upper boundary of the benthic boundary layer (BBL) was reliably determined at a point at a distance from 5–7 to 35–40 m from the bottom where the gradients of the density and the velocity of the currents changed. The experimental data obtained were used to determine the coefficient of the bottom friction, the friction velocity, the coefficients of the vertical diffusion of momentum and density, and the vertical fluxes of temperature and salinity in the BBL. DOI: 10.1134/S0001437007010067
INTRODUCTION
genic and Technogenic Loads” (the code of the project is “Shelf”).
The bottom boundary layer (BBL) plays an important role in the dynamics of the near-bottom region since it determines the character and parameters of the bottom friction, significantly influences the processes of the formation of the sediments and bottom geomorphology, and the heat and matter exchange between the water and the surface of the bottom [1]. The data about the structure of the BBL are important for the solution of many problems of geological and geophysical research and prospecting and realization of many engineering projects in the near-bottom region and immediately at the sea floor. These factors determined the necessity of developing a special measuring complex at the Marine Hydrophysical Institute (MHI) of the National Academy of Sciences (NAS) of Ukraine for measurements in the BBL. Such a profiler was developed on the basis of the Hydrophysical complex for vertical profiling of currents and hydrological elements of the OLT type [2, 7, 8, 10, 23, 24]. Measurements of the vertical structure of the currents and the hydrological elements were performed in several regions of the Black Sea shelf, over the continental slope, and in the deep water region, which made it possible to obtain data about the parameters of the BBL [3, 4, 6, 20, 24]. Methods for distinguishing the characteristic particularities of the BBL structure and determining the parameters of the bottom friction, the characteristics of the turbulent diffusion, and estimates of the heat and matter fluxes were developed on the basis of the models of a boundary layer in a stratified fluid [11–16, 18, 24, 26, 35]. All these studies were carried out within the project of the National Academy of Sciences of Ukraine “Studies of the Regularities of Functioning of Shelf Ecosystems under the Conditions of Anthropo-
The objective of our study is systematization of the data obtained about the structure of the BBL in the Black Sea, about the estimates of the coefficients of the turbulent viscosity and diffusion calculated from the experimental data on the basis of different models, and about the temperature and salinity fluxes in the BBL. VERTICAL PROFILING OF THE BOTTOM BOUNDARY LAYER A modified hydrophysical profiler of the OLT-D type (Fig. 1) was used to measure the vertical profiles of the currents and the hydrophysical parameters in the BBL. By this instrument, the current velocity is measured using a three-component acoustic sensor whose measuring axes coincide with the axes of a ferroussound meter of angular deviations of the instrument coordinate system from the coordinate system of the earth [2, 7, 8]. The OLT-D complex is equipped with special means to measure the velocity of the profiler relative to the vessel as well as the interface with a navigation system to take into account the drift of the vessel. Metrological tests showed that the total error of the measurements of the current velocities does not exceed 0.018–0.025 m/s with a probability of 0.95. The hydrological parameters are measured by a CTD system with the following characteristics: the temperature range is from –2 to 35°ë, the sensitivity is 0.0004°C, and the measurement error does not exceed 0.01°C; the relative electric conductivity range is from 0.35 to 1.7, the sensitivity is 0.00002, and the measurement error does not exceed 0.0004; and the hydrostatic pressure range is from 0 to 20 MPa, the sensitivity is 0.0025 MPa, and the measurement error does not exceed 0.05 MPa. Dur33
34
KUSHNIR
particularities of the profiles of the currents and hydrological parameters at a distance on the order of 10 m from the bottom [10, 13, 24]. Some characteristic examples of the profiles of the current velocity components [u(z) from the west to the east and v(z) from the south to the north] and of the profiles of the specific water density σt(z) in the near-bottom region are shown in Figs. 2a–d. The location of the upper boundary of the BBL in these figures is indicated by the arrows. Characteristic examples of the profiles of the current velocity components and the density in the near-bottom region, in which the water density increases with depth up to the contact with the bottom, are shown in Figs. 2c and 2d. An analysis of the profiling results showed that the profiles of the currents and density in the near-bottom region have the following general particularities: (i) variations in the vertical density gradient and the direction and current velocity at equal distances from the bottom equal to 10–30 m on the average and (ii) a monotonous change in the current velocity from 5−10 cm/s to zero at the bottom. Due to these general particularities of all the profiles, the upper boundary of the BBL is clearly manifested, which makes it possible to determine the location and thickness ç of the BBL as well as the velocity of the currents V at the upper boundary of this layer, the mean density gradient (the Brünt–Váisálá frequency), and the dynamical Richardson numbers Ri in the boundary layer. The main data related to the conditions of the measurements and the parameters of the BBL obtained are given in Table 1. ANALYSIS OF THE RESULTS OF THE VERTICAL PROFILING The thickness of the BBL is calculated from the following relation [35]: (1)
Z p = 0.4U * / f ,
Fig. 1. Photo of the OLT-D hydrophysical profiler. The suspension of the profiler is arranged with a special device in which the force pulley is combined with the meters of the length of the cable and the angle of its bitter end. The construction of the profiler is modular.
ing the motion of the profiler in the near-bottom boundary layer, the instrument uses a bottom echo sounder operating at a frequency of 120 kHz. Its range of operation is 30–50 m and the resolution is 0.1 m. The application of this echo sounder makes it possible to perform profiling from the surface to the bottom. The results of the vertical profiling in the BBL demonstrated the existence of a large variety of structural
where U* = τ/ρ = 〈u'v '〉 = C b V is the friction velocity; τ is the turbulent stress in the bottom region; ρ is the density; f is the Coriolis parameter; u' and v' are the pulsations of the horizontal and vertical current velocities; and C b = Cf is the coefficient of the bottom friction, which, according to the data of different authors, ranges within 0.026–0.056 with a mean value of 0.04 [24, 34, 35]. It follows from relation (1) that the thickness of the BBL is proportional to the velocity at its upper boundary, i.e., H = αV and α = 0.4Cf /f. The dependence ç = f(V) is shown in Fig. 3. It is seen that the thickness of the BBL linearly depends on the velocity at its upper boundary (the correlation coefficient is equal to 0.95). The proportionality coefficient obtained (α = 149.28) corresponds to the coefficient of the bottom friction Cf = 0.0373, which is OCEANOLOGY
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BOTTOM BOUNDARY LAYER IN THE BLACK SEA (a)
25 20 15 10
–0.06
–0.02
0.02
14.9
u, v, m/s 0.06 0.10
σt
40 35
15.0
–0.04
v
15.1
–0.02
0
u, v, m/s 0.02
σt
30
u
σt 15.2
(b)
σt 14.25 14.30 14.35 14.40 14.45 14.50 –0.10 40 35 30
35
25
u
v
20 15 10 5 0 z, m
5 0 z, m σt 15.1 15.2 15.3 15.4 15.5 15.6 15.7 (c)
σt 14.10
(d) 13.90
13.95
14.00
14.05
u, v, m/s –0.10 30
–0.06
25 20
–0.02 u
0.02
0.06
u, v, m/s 40
–0.04 –0.02
0
0.02 0.04 0.06 0.08
35
v
30
σt
25
σt
v u
20
15
15
10
10 5
5
0 z, m
0 z, m
Fig. 2. Profiles of the current velocity components u, v, and the specific density σt on the basis of the measurements at stations: (a) 14, (b) 69, (c) 30, and (d) 50. The arrows show the location of the upper BBL boundary.
close to the mean value of Cf for the oceanic conditions (Cf = 0.04). Assuming that the thickness of the boundary layer is Zp = ç, we can estimate the friction velocity in the BBL from the relation U* = 2.5Hf. The values of U* for the data of the vertical profiling in the near-bottom region considered above are given in Table 2. Despite the variety of current velocity and density profiles in the bottom layer, it is interesting to compare the data obtained with the model of the logarithmic boundary layer over a smooth surface [16, 18]. It is known that the current velocity V log at the upper boundary of the BBL is equal to HU * V log = U * ⎛ C 1 log ------------ + C 2⎞ , ⎝ ⎠ ν OCEANOLOGY
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(2)
where ν = 1.4 × 10–6, m2/s is the coefficient of kinematic viscosity, and C1 and C2 are the constant parameters. The maximal agreement between the calculated and actual current velocities at the upper boundary of the BBL corresponds to the parameters C1 = 4.272 and C2 = 4.568 with a correlation coefficient of 0.93. According to the data of the measurements in pipes with smooth sides, these parameters vary in the following ranges: 5.52 < C1 < 5.75 and 5.50 < C2 < 5.84 for the range of variations of the dimensionless variable U*H/ν from 50 to 1000 [16]. The obtained discrepancy in the estimates of these parameters by approximately 20–25% is explained by the fact that, in the case of the boundary layer over the sea bottom considered here, the mean value of U*H/ν is equal to (4…5) × 104; i.e., it is more than one order of magnitude greater than the upper limit of the universal logarithmic profile for
36
KUSHNIR
Table 1. Conditions of vertical profiling in the near-bottom region and the main parameters of the BBL Station no.
Depth, m
Bottom sloping
Coordinates, N, E
V, m/s
H, m
N × 103, rad/s
Ri
1
P1
84
0.167
41°26',29°12'
0.144
18
6.51
0.66
2
85
500
0.180
44°27', 33°21'
0.2
30
5.12
0.59
3
7
250
0.092
14°40', 31°30'
0.075
12
8.01
1.64
4
21
150
0.126
44°50', 32°00'
0.23
35
6.85
0.92
5
55
450
0.183
44°15',33°27'
0.11
20
6.87
1.56
6
30
160
0.091
44°10', 33°45'
0.061
12
17.82
10.12
7
31
460
0.208
44°08', 33°45'
0.214
20
1.96
0.033
8
50
120
0.091
44°15', 34°00'
0.065
17
8.52
5.64
9
69
125
0.088
44°50', 32°45'
0.051
27
2.04
0.32
10
14
120
0.153
44°30', 33°15'
0.11
28
5.49
1.69
11
15
760
0.185
44°25', 33°15'
0.085
14
1.93
0.12
12
49
685
0.109
44°15', 34°01'
0.041
12
1.62
0.22
13
71
70
0.061
44°27', 33°38'
0.053
12
6.02
1.84
14
72
70
0.061
44°27', 33°38'
0.073
12
4.91
0.64
15
73
70
0.061
44°27', 33°38'
0.075
14
5.12
0.91
16
74
70
0.061
44°27', 33°38'
0.013
5
4.61
3.24
17
77
70
0.061
44°27', 33°38'
0.025
12
3.61
3.03
18
78
70
0.061
44°27', 33°38'
0.024
5
5.39
1.32
19
16
1758
0.007
44°20', 33°14'
0.069
20
9.8
8.7
20
28
148
0.095
44°15', 33°30'
0.2
37
22.0
16.6
21
32
1175
0.012
44°05', 33°45'
0.231
35
5.2
0.62
22
19
1930
0.005
44°05', 33°15'
18
4.4
0.33
No
13.8
The results of the calculations of V log and the values of the current velocity V at the upper boundary of the BBL obtained from the data of measurements are given in Table 2, while the comparison of these values is visualized in Fig. 4. The relatively high value of the correlation coefficients between the calculated value V log and the actual current velocity V testifies to the correspondence of the BBL structure to the logarithmic profile with the parameters given above despite the deviation of individual experimental points from the line of equal values.
where κ = 0.41 is the Carman constant, zb is the distance to the bottom, and z0 is the parameter of the bottom roughness. On the basis of the experimental data presented above, we get zb = 〈ç〉 = 18.7 m, 〈Cf〉 = 0.0373, and z0 = 0.00033 m. In this case, zb = Hmax = 37 m and Cf, min = 0.0353 at zb = çmin = 5 m and ëf, max = 0.0426, which insignificantly differs from the mean value of the bottom friction coefficient for marine conditions, which is equal to 0.04. Since the mean size of the inhomogeneities of the bottom fractions is ks ≈ 30z0 and the mean value of the friction velocity on the basis of the experimental data presented above is equal to 0.0169 m/s, the mean Reynolds number is Re = (ks 〈U*〉)/ν = 3.3, which corresponds to the prevailing regime of smooth flowing over the sea bottom.
The logarithmic law is usually used to determine the coefficient of the bottom friction from the relation
TURBULENT DIFFUSION AND FLUXES
pipes. In addition, unknown irregularities of the bottom, which also influence the parameters C1 and ë2, were not taken into account.
κ C f = ---------------, z ln ⎛ ----b⎞ ⎝ z 0⎠
(3)
Turbulent diffusion in the near-bottom region determines the parameters of the propagating particulate matter and its fluxes to the overlying layers of the sea. Different methods for determining the coefficient of OCEANOLOGY
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BOTTOM BOUNDARY LAYER IN THE BLACK SEA H, m 40
37
Vlog, m/s 0.25 R = 0.93
R = 0.95 0.20
30
0.15 20 0.10 10 0.05
0
0.05
0.10
0.15
0.20
0.25 V, m/s
vertical diffusion of momentum Kz (the coefficient of turbulent viscosity) and the scalar parameter (density) Ks in the BBL are known [24]. In the general form, it is written as Rf ε K s = --------------- ------2 , 1 – Rf N
(4)
where Rf is the flux Richardson number, ε is the dissipation rate of the turbulent energy in a unit volume, S2 is the squared vertical shear of the velocity, and N2 is the squared Brünt–Váisálá frequency [22]. It follows from relations (4) that K Ri ------z = ------ . K s Rf
(5)
The simplest method for estimating the value of Kz is by using parameterization of the following form: 2
U* H K z = ------------, V
(6)
where the value of the friction velocity U* is given in Table 2 for all the BBL profiles observed. The calculated values of äz (m/s) are also given in Table 2. Lykosov and Gutman [14] suggested a model of the atmospheric boundary layer with a nonuniform density and inclined underlying surface. This model was used in [35] to analyze the BBL under marine conditions. OCEANOLOGY
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0.05
0.10
0.15
0.20
0.25 V, m/s
Fig. 4. Comparison of the current velocity at the upper boundary of the BBL calculated from relation (2) with its actual value from the data of vertical profiling. R is the correlation coefficient.
Fig. 3. Thickness of the BBL versus the velocity of the currents at its upper boundary (individual points) and the regression line ç = 147.45V + 1.227.
1 ε K z = --------------- -----2 , 1 – Rf S
0
The thickness of the BBL depends on the vertical diffusion Kz, LG and other parameters such as [24] 2 0.25
H = 2K z, LG f ( 1 + α N / f ) –1
2
2
0.5
,
(7)
where α is the slope of the bottom. The coefficients of the vertical momentum diffusion, which take into account the slope of the bottom and the mean density gradient in the BBL (Kz, LG = 0.5 H2f[1 + (αN/f)2]–0.25), are given in Table 2. In the paper by Eremeev and Kushnir [5], the authors suggested a parameterization of the vertical diffusion of momentum on the basis of the empirical relation ln S = Ä + B ln N between the Brünt–Váisálá frequency N and the vertical shear of the current velocity S. This approach leads to the following relation for the coefficient of the vertical diffusion of momentum: K z, EK = 15C 1 ν10 N 2A
2( B – 1)
(8)
,
where ë1 = 0.1 is the Ozmidov constant, ν = 1.4 × 10−6 m2/s is the coefficient of the kinematic viscosity, and Ä = – 0.045 and Ç = 0.31 are the empirical constants. The results of the calculation of Kz, EK from relation (5) are presented in Table 2. Nabatov and Ozmidov [15] developed a model of the turbulent boundary level on the basis of an energy balance analysis. In this model, the thickness of the BBL is determined from the following relation [24] H = C ( 7.5K z, NO ) ( RiN ) 0.5
–0.5
,
(9)
38
KUSHNIR
Table 2. Parameters of the boundary layer No
V, m/s
U∗ × 103, m/s
V log , m/s
Kz × 103, m2/s Kz, LG × 103, m2/s Kz, EK × 103, m2/s Kz, NO × 103, m2/s
1
0.144
4.5
0.104
2.53
4.90
1.78
1.89
2
0.2
7.5
0.197
8.43
14.77
2.47
3.68
3
0.075
3.0
0.061
1.44
2.64
1.33
2.56
4
0.23
8.75
0.237
11.65
20.78
1.65
10.46
5
0.11
5.0
0.119
4.55
5.63
1.65
5.81
6
0.061
3.0
0.061
1.77
1.88
0.51
1.50
7
0.214
5.0
0.119
2.33
13.18
3.05
0.79
8
0.065
4.25
0.097
4.72
5.47
1.35
17.45
9
0.036
2.5
0.047
1.74
2.12
1.90
0.27
10
0.11
6.5
0.165
9.98
11.62
2.24
8.50
11
0.085
3.5
0.075
2.01
4.04
4.94
0.10
12
0.041
3.0
0.061
2.63
5.05
12.10
0.07
13
0.053
3.0
0.061
2.04
3.69
1.98
2.16
14
0.073
3.0
0.061
1.48
4.05
2.62
0.61
15
0.075
3.5
0.075
2.28
5.41
2.47
1.24
16
0.013
1.25
0.015
0.41
0.72
2.85
0.51
17
0.025
3.0
0.061
4.32
4.63
4.00
2.13
18
0.024
1.25
0.015
0.33
0.67
2.30
0.24
19
0.069
5.0
0.119
7.25
7.39
1.01
2.8
20
0.2
9.25
0.252
15.80
20.00
0.33
27.7
21
0.231
8.75
0.237
11.60
29.60
2.42
5.35
22
0.138
4.5
0.104
2.64
11.40
3.04
0.64
where ë = 9.92 is the dimensionless constant. The results of the calculation of the coefficient of vertical momentum diffusion Kz, NO = 1.335 × 10–3 RiNH2 from relation (6) are given in Table 2. A comparison of the estimate of the vertical diffusion of momentum Kz with similar estimates calculated from the models mentioned above is shown in Fig. 5 on a logarithmic scale. It is seen that, despite the significant scattering of individual points, the main array of the estimates forms a group around the line of equal values. The best correspondence is provided by the model of Lykosov–Gutman, and the corresponding correlation coefficient is equal to 0.89. Different empirical relations are used to estimate the flux Richardson number Rf, for example, Rf = f(Pi, Ri), where Pi are the constants [3, 4, 19, 20, 25, 27]. An analysis of these relations and the experimental estimates of Rf available demonstrated their significant
ambiguity. Thus, in our study, we used the data of the experimental relations äs = f(Ri) (cm2/s) and their approximations published in [21,33] log K s = 0.4257 ( log Ri )
2
(10)
– 1.5724 ( log Ri ) – 0.2637,
plotted in Fig. 6 at a reliability level of R2 = 0.81 (the correlation coefficient is equal to 0.9). The results of the comparison of the values of äs are given in Table 3. The vertical density flux is calculated from the relation K s ρ0 N dρ -, 〈 ρ'w'〉 = K s ------ = – ----------------dz g 2
(11)
where ρ0 is the mean density and g = 9.81 m/s2 is the acceleration due to gravity. OCEANOLOGY
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BOTTOM BOUNDARY LAYER IN THE BLACK SEA log(103 Kz, LG), log(103 Kz, EK), log(103 Kz, NO) Kz, LG
39
logKs 2 R = 0.9
1 1
0 Kz, EK
–1
Kz, NO
–1
–1
0 log(103
0
1 Kz)
–2 –2.0
–1.5
–1.0
–0.5
0
0.5
1.0 logRi
3
Fig. 5. Comparison of the values of log ( 10 K z ) and those 3
3
of log ( 10 K z, LG ) (circles), log ( 10 K z, EK ) (squares), 3
and log ( 10 K z, NO ) (crosses).
On the other hand, 〈 ρ'w'〉 = β 〈 s'w'〉 – a 〈 t'w'〉 = βF s – αF t = αF t ( Rf – 1 ),
These estimates correspond to the values obtained from the measurements given in Table 3. (12)
where α is the coefficient of the temperature expansion, β is the coefficient of the salinity compression, Ft is the vertical temperature flux, and Fs is the vertical salinity flux; Rf = 0.15 at Rρ ≥ 2 and Rf = 1.85 – 0.85Rρ at Rρ < 2, where Rρ = (β∆s)/(α∆t) is the density relation and ∆s and ∆t are the salinity and temperature jumps in the BBL [17]. The values of the density ratio Rρ and the estimates of the vertical fluxes of the temperature Ft and salinity Fs in the BBL calculated from the relations K s ρ0 N F t = ---------------------------, αg ( 1 – Rf ) 2
αF t Rf -. F s = --------------β
(13)
are given in Table 3. The mean value of the vertical temperature flux in the BBL calculated from the data obtained is equal to 4.32 × 10–6 °ë m/s; the standard deviation of the estimate is 7.5 × l0–6 °C m/s. The corresponding values for the vertical salinity flux are equal to 0.58 × 10–7‰ m/s and 0.68 × 10–7‰ m/s. The vertical temperature flux in the BBL can be compared with the geothermal heat flux Fw (mW/m2) in the Black Sea. Its values range from 5−10 to 50–120 mW/m2 [4, 6]. Since Fw = 4.21 × 106Ft [9], the corresponding temperature fluxes range from (1.18–2.37) × 10–6 to (11.87…28.50) × l0–6 °C m/s. OCEANOLOGY
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Fig. 6. Comparison of the experimental estimates of the vertical diffusion Ks (crosses) and the approximating powerlaw dependence (black squares); R is the correlation coefficient.
CONCLUSIONS 1. The results of vertical profiling of the near-bottom region in the Black Sea with a vertical resolution equal to 0.1 m demonstrated the great variety of the profiles of the bottom currents and density. However, regardless of this diversity, the general particularity of the BBL lies in the clearly manifested variations in the vertical density gradients and the vertical gradients of the direction of the current velocity at the upper boundary of the BBL. This boundary is usually reliably detected at a distance from 37–40 to 5–7 m from the floor and corresponds to the thickness of the BBL (Table 1). Determination of this boundary can also be made on the basis of the location of the extremum of the second derivative of these parameters [13]. 2. The dependence of the BBL thickness on the current velocity at its upper boundary is close to linear with a correlation coefficient of 0.95. The coefficient of proportionality between these parameters corresponds to the coefficient of bottom friction ëf = 0.0373 m, which is close to the mean value ëf = 0.04 for different regions of the World Ocean [24, 34, 35]. However, regardless of the strong variability among the profiles of the bottom currents, their general particularity is the correspondence of the actual current velocity at the upper boundary of the BBL to the value calculated for the logarithmic law of the velocity variation in the boundary layer over an even underlying surface. The
40
KUSHNIR
Table 3. Coefficients of turbulent diffusion and fluxes of temperature and salinity in the BBL Ri
Ks × 104, m2/s
Rρ
N × 103, rad/s
1
0.66
1.08
4.63
1.07
4.99
6.51
2
0.59
1.32
3.52
0.81
3.72
5.12
3
1.64
0.26
1.69
0.39
2.68
8.01
4
0.92
0.62
2.94
0.68
4.13
6.85
5
1.56
0.28
1.33
0.31
8.13
6.87
6
10.12
0.04
1.28
0.29
8.93
17.82
No
Ft × 106, °C m/s Fs × 107, ‰ m/s
7
0.033
3.91
2.34
0.35
1.65
1.96
8
5.64
0.06
0.44
0.10
5.34
8.52
9
0.32
4.16
2.62
0.40
1.67
2.04
10
1.69
0.25
0.76
0.17
2.64
5.49
11
0.12
35.09
30.76
3.06
1.43
1.93
12
0.22
9.00
8.85
0.55
1.27
1.62
13
1.84
0.22
0.81
0.19
3.12
6.02
14
0.64
1.14
2.78
0.64
1.92
4.91
15
0.91
0.63
1.67
0.39
2.78
5.12
16
3.24
0.11
0.24
0.06
3.85
4.61
17
3.03
0.12
0.16
0.04
4.14
3.61
18
1.32
0.36
1.06
0.24
2.38
5.39
19
8.7
0.05
0.49
0.11
9.23
9.8
20
16.6
0.03
1.47
0.34
16.73
22.0
21
0.62
1.21
3.31
0.76
1.98
5.2
22
0.33
3.91
21.95
1.77
1.35
4.4
parameters of the logarithmic profile calculated for the sea bottom differ by 20–25% from the parameters of the universal logarithmic profiles for pipes [16]. The possible causes for this discrepancy are the following: (1) the dimensionless quantity U*H/ν in the BBL over the sea bottom is, on the average, equal to (4… 5) × 104, which is more than one order of magnitude higher than the corresponding value for the logarithmic profile in pipes; (2) the unknown roughness of the bottom is not taken into account, which also influences the parameters of the logarithmic velocity profile. 3. The coefficients of the vertical diffusion of the momentum (turbulent viscosity) äz were calculated using different independent methods. A simple parameterization in which the value of äz is determined by the thickness of the BBL, by the current velocity at its upper boundary, and by the friction velocity is used as the basic estimate. Other estimates of äz were obtained on the basis of application of the models of Lykosov–Gutman [14], Eremeev–Kushnir [5], and Nabatov–Ozmidov [15]. A comparison of these estimates with the basic one
showed that, in the presence of a significant scattering of the individual values, their majority is concentrated in the vicinity of the line of equal values. The characteristic values of äz are within 10−100 cm2/s, and the mean value is approximately equal to 50 cm2/s. The closest estimate to the basic dependence is the one calculated on the basis of the Lykosov–Gutman model. 4. The coefficients of the vertical density diffusion Ks were calculated on the basis of an analysis of published results of experimental studies in the form of a power-law relation of the type log K s = f( log Ri ). The estimates of Ks obtained vary from 0.028 to 35.1 cm2/s, while the mean value is equal to 2.9 cm2/s. The estimates of the density ratio Rρ were performed on the basis of the analysis of the temperature and salinity jumps in the BBL. The range of the Rρ variation is 1.27–16.73, while the mean value is 4.27. 5. The vertical fluxes of the temperature Ft and salinity Fs were calculated using a universal relation between these fluxes in a stratified near-bottom layer. The values OCEANOLOGY
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of Ft and Fs vary within (0.16–30.76) × 10−6 °ë m/s and (0.04–3.06) × 10–7‰ m/s. The estimates of the temperature fluxes obtained correspond to the values of the geothermal heat flux characteristic of the Black Sea. REFERENCES 1. V. K. Voitov, A. V. Solov’ev, and V. S. Yastrebov, “Hydrophysical Properties of the Near-Bottom Layer of the Ocean (A Review),” Okeanologiya 24 (6), 885–898 (1989). 2. A. E. Drozdov, V. M. Kushnir, A. V. Nikitin, et al., “A Hydrophysical Sounding Set for Oceanographic Research Vessels,” Zap. Gidrogr., No. 226, 49–57 (1991). 3. V. K. Eremeev and V. M. Kushnir, “Near-Bottom Boundary Layer in the Shelf Zone of the Black Sea,” Izv. Akad. Nauk, Fiz. Atmos. Okeana 30 (6), 848–853 (1994). 4. V. N. Eremeev and V. M. Kushnir, “Thermodynamical Parameters of the the Deep Near-Bottom Layer in the Black Sea,” Dokl. NAN Ukrainy, No. 6, 104–108 (1994). 5. V. K. Eremeev and V. M. Kushnir, “The Layered Structure of Currents and Vertical Exchange in the Black Sea,” Okeanologiya 36 (1), 13–19 (1996) [Oceanology 36 (1), 9–15 (1996)]. 6. V. N. Eremeev and V. M. Kushnir, “Near-Bottom Boundary Layer in the Black Sea: Results and Perspectives of Studies,” Morsk. Gidrofiz. Zh., No. 1, 50–69 (1998). 7. V. M. Kushnir, Vertical Structure of Oceanic Currents: Methods, Facilities, and Results of Experimental Studies (Sevastopol, 1988) [in Russian]. 8. V. M. Kushnir, “A Probe for Vertical Profiling of Currents and Hydrological Elements,” Okeanologiya 34 (2), 303–305 (1994). 9. V. M. Kushnir, “Heat Fluxes and Stability of the Upper Boundary of the Near-Bottom Boundary Layer in the Black Sea,” Morsk. Gidrofiz. Zh., No. 1, 35–45 (1996). 10. V. M. Kushnir, “Instrumental and Methodological Provision of the Measurements of the Parameters of the NearBottom Boundary Layer,” Navigatsiya i Gidrografiya, No. 2, 123–128 (1996). 11. V. M. Kushnir, “Structure of the Wave Near-Bottom Boundary Layer over Flat and Irregular Bottoms,” Morsk. Gidrofiz. Zh., No. 6, 54–67 (2005). 12. V. M. Kushnir, T. P. Lebedeva, and E. B. Perepada, “Near-Bottom Boundary Layer in the Black Sea and Parameterization of Turbulent Fluxes,” Dokl. NAN Ukrainy, No. 12, 129–134 (1998). 13. V. M. Kushnir, T. P. Lebedeva, and E. B. Perepada, “Layered Structure of the Near-Bottom Boundary Layer in the Black Sea,” Morsk. Gidrofiz. Zh., No. 6, 40–55 (2000). 14. V. N. Lykosov and L. N. Gutman, “Turbulent Boundary Layer over an Inclined Bottom Surface,” Izv. AN SSSR. Fiz. Atm. I Okeana 8 (8), 799–809 (1972). 15. V. N. Nabatov and R. V. Ozmidov, “Studies of the Turbulent Near-Bottom Boundary Layer in the Ocean,” Okeanologiya 27 (1), 5–11 (1987). 16. L. Prandtl, Essentials of Fluid Dynamics (Blakie, London, 1952; Inostr. Liter., Moscow, 1949). OCEANOLOGY
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17. J. S. Turner, Buoyancy Effects in Fluids (Cambridge Univ. Press, Cambridge, 1973; Mir, Moscow, 1977). 18. H. Schlichting, Boundary Layer Theory, 6th ed. (McGraw-Hill, New York, 1968; Nauka, Moscow,1974). 19. K. F. Bowden, “Turbulence,” in The Sea (Wiley and Sons, New York−London, 1962), Vol. 1. Physical Oceanography, pp. 802−825. 20. V. N. Eremeev and V. M. Kushnir, “The Bottom Boundary Layer in the Black Sea Deep-Water Part,” in International Conference “Dynamics of Oceans and Atmosphere” (Shirshov Institute of Oceanology, Russian Academy of Sciences, Moscow, 1995), p 122. 21. M. C. Gregg, “Diapycnal Mixing in the Thermocline: A Review,” J. Geophys. Res. 92 (C5), 5249−5286 (1987). 22. E. C. Itsweire, J. R. Koseff, D. A. Briggs, and J. H. Ferziger, “Turbulence on Stratified Shear Flows: Implications for Interpreting Shear-Induced Mixing in the Ocean,” J. Phys. Oceanogr. 23, 1508−1522 (1993). 23. V. M. Kushnir, “Vertical Profiling of the Bottom Boundary Layer,” in Proc.IEEE Fifth Working Conference on Current Measurement” (St. Petersburg, Flo., United States, 1995), pp. 255−258. 24. V. M. Kushnir, “Turbulent Diffusion in the Near-Bottom Boundary Layer of the Black Sea Shelf Zone,” J. Mar. Systems 21 (1−4), 243−253 (1999). 25. V. M. Kushnir, R. Khanbilvardi, and B. S. Shteinman, Turbulent Diffusion in Lake Kinneret under Conditions of Pronounced Density Stratification,” in 6th International Conference on Hydro-Science and Engineering. Abstracts of Papers (Brisbane, Australia, 2004). 26. V. M. Kushnir, T. P. Lebedeva, and E. B. Perepada, “Vertical Structure of Marine Bottom Boundary Layer and Turbulent Fluxes Parameterization,” in Proc. ISFMTM 98 “Flow Modeling and Turbulence Measurements-VII (Tainan, Taiwan, 1998), pp. 19−26. 27. O. I. Mamayev, “The Influence of Stratification on Vertical Turbulent Mixing in the Sea,” Bull. Acad. Sci. USSR, Geophys. Ser. 7, pp. 484−492. 28. Marine Turbulence Revisited (J. Mar. Systems Special Volume), Ed. by J.-M. Beckers (1999). 29. G. L. Mellor and N. A. Yamada, “A Hierarchy of Turbulent Closure Model for Planetary Boundary Layer,” J. Atmos. Sci. 31, 1791−1806 (1974). 30. “Mixing in Stratified Fluids,” J. Gephys. Res., Spec. Section 92 (C5), 5229−5495 (1987). 31. W. H. Munk and E. R. Anderson, “Notes on a Theory of the Thermocline,” J. Mar. Res. 7, 276−295 (1948). 32. R. C. Pacanowski and S. G. H. Philander, “Parameterization of Vertical Mixing in Numerical Models of Tropical Oceans,” J. Phys. Oceanogr. 11, 1443−1451 (1981). 33. S. A. Thorpe, “Turbulence and Mixing in a Scottish Loch,” Philos. Trans. Roy. Soc. London A286, 125−181 (1977). 34. G. L. Weatherly, “A Study of the Bottom Boundary Layer of the Florida Current,” J. Phys. Oceanogr. 2 (1), 54−72 (1972). 35. G. L. Weatherly and P. J. Martin, “On the Structure and Dynamics of the Oceanic Bottom Boundary Layer,” J. Phys. Oceanogr. 8 (4), 557−570 (1978).
