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Rational approach to the bottom shear stress in tidal planetary boundary layer flow
Ocean £ngng, Vol. 19, No 4, pp. 395-404, 1t)92. Printed in Great Britain.
0029-801892 $5.00 * .01) Pergamon Press l,td
TECHNICAL
NOTE
RATIONAL APPROACH TO THE BOTTOM S T R E S S IN T I D A L P L A N E T A R Y BOUNDARY FLOW
SHEAR LAYER
D. MYRHAUG Division of Marine Hydrodynamics, Norwegian Institute of Technology, N-7034 Trondhcim-NTtt, Norway Abstract--The friction velocity associated with the m~ximum bottom shear stress in neutrally
stable tidal planetary boundary layer flow is presented. The directions of the bottom shear stresses for the anticlockwise and clockwise rotating components arc also presented. The results arc obtained by using similarity theory and arc given for flow conditions in the rough, smooth and transitional smooth-to-rough turbulent regime. An approximation for the maximum bottom shear stress by disregarding the rotation of the velocity in the boundary layer as the seabed is approached is also presented.
INTRODUCTION
THE FLUID motion in the tidal planetary boundary layer near the seabed controls and affects many phenomena in ocean engineering and oceanography. The bottom shear stress associated with the boundary layer flow represents the dominant mechanism governing sediment transport and erosion phenomena on continental shelves. Tidal planetary boundary layer flow also represents an important component in physical models for predicting ocean flow circulations. The vertical structure of the bottom boundary layer on the continental shelves is in the most general and complex case dominated by several interacting physical effects. Among these effects are the Earth's rotation; tidal effects; stratification due to salinity, temperature gradients and suspended sediments; internal friction in the fluid: and Iopographical effects. One important feature of the vertical structure of the bottom boundary layer is determined by the influence of planetary rotation on unsteady (due to tidal effects), horizontally uniform, unbounded, and unstratified flow. This idealized boundary-layer flow may occur in the ocean away from any coasts and in the region near the seabed where the internal friction in the fluid is significant. The details of the vertical structure of the current varies with the modelling of the momentum flux in the boundary layer. Various types of models have been represented in Sverdrup (1927), Long (1981), Prandle (1982), Davies (1985), King et al. (1985) and Myrhaug (1990) among others. Soulsby (1983, 1990) gives reviews of bottom boundary layers of shelf seas. Some of the idealized types of boundary layer situations were discussed using an eddy viscosity model which varies linearly with the distance from the bottom. In most cases the boundary layer flow is in the rough turbulent 395
39(~
D. M'~ ~I,~t ~(i
regime, but it also appears that the flow conditions might be in the smooth and the transitional smooth-to-rough turbulent flow rcgimes (Sternberg. lt~(~S. 1970: Untersteiner and Badgley, 1965). The purpose of this paper is to present si~nple formulae from which the maximum seabed shear stress in tidal planetary boundary layer flow can be determined. 3he semiempirical approach presented here gives rational formulae which should be useful for engineering purposes. TItEORY Following Gill (1982) the geostrophic velocity for steady homogeneous (neutrally stable) planetary flow can be written as R,_ =
~
In
,~?,,~,
-A
~1)
~
according to similarity theory. Here R~ ~ U~ * ivy, is the geostrophic velocity m complex form. U-~ and V~_ are the geostrophic velocity components along horizontal orthogonal x- and y-axis, respectively. The x-axis is taken along the shear stress at the surface, %, which has an angle ~ relative to the geostrophic velocity direction, that is. in counterclockwise rotation relative to the geostrophic velocity direction in the northern hemisphere, f = 2{~sin~ is the Coriolis parameter, where [~ is the Earth's angular frequency of rotation and • the latitude. Further, u~ = (~,/p)~n is the friction velocity, p is the density of the fluid, ~ is yon Karman's constant ( ~ 0.4), z~ is the roughness parameter of the seabed which will be discussed subsequently, A and B are dimensionless constants as required by similarity theory, and i = (-1)~/~. Since U ..... [R~ [cos~ and V~ = -{R~.[sin~ and by using Equation (1), A and B are given as
(' )_ K~,.,
u~,~ A = In, )~,,;
cos~.
B = ~R~tu sina
12a)(2b)
Thus Equations (2a) and (2b) determine A and B for a given data set. ~R~[ is the magnitude of R .... For known A and B, u, can be found from Equation (1), and a is determined from Equation (2b) as, respectively u
~=arcsin
(3)
~{R~l]
(4)
where c = e /~ is a dimensionless constant. An approximation to Equation (3) is obtained by using B = 0 and a modified value of c, that is, c'. In this approximation the rotation of the velocity in the boundary layer as the seabed is approached is disregarded, and the logarithmic boundary layer flow model, that is IR~I = ( u ~ / ~ ) ln(c'u./fzo¢.), is extended beyond its range of validity. The logarithmic boundary layer flow model is valid only in a region where the shear stress is constant. Here it is extended to a height c'u./f where the velocity is equal to the magnitude of the geostrophic velocity. Thus c' can be determined for a given data set.
Technical Note
397
This similarity theory will now be applied to oscillatory planetary boundary layer flow by using the results from the analysis of tidal currents. Some elements of this analysis are summarized in Appendix. However, more details of the analysis of tidal currents are given in Soulsby (1983, 1990). The result of separating a particular tidal constituent of angular frequency o~ to give the anticlockwise and clockwise rotating components is that the equations of motions for each rotating component are similar in form, and similar to that given for the steady planetary motion. This implies that when the result in Equation (3) is applied to oscillatory planetary flowfshould be replaced by (~o + f ) and (co - f ) for the anticlockwise and clockwise rotating components, respectively. In the following it is assumed that co > f in the northern hemisphere ( f > 0). However, the results can easily be generalized to cover situations where ~o < f and to the southern hemisphere ((f < 0) (see Soulsby, 1990). Thus, by using Equation (3), the friction velocities associated with the seabed shear stress for the anticlockwise ( + ) and clockwise ( - ) rotating components are given as
I
(51
where u,,, is the maximum friction velocity at the seabed as defined in Equation (A6). IRK +1 denotes the magnitude of the free stream velocity outside the boundary layer. By using Equations (5), (A6) and (A7), the maximum friction velocity at the seabed is given by ~'IR~. ~I-~ r [ cu,,,, ~]2 B e [lnk(~ V/)z,,t...JJ +
,,,:,,,=
+
J 11/2
K~-IR~ 12
r / cu,:,,, ,~ e B ~ [lnk(~ ~ f ) Z " , ) +
(6)
Equation (6) is an implicit equation for determination of u..,. For known similarity constants and for a given seabed roughness length, free stream current velocity components, frequency of oscillation and Coriolis parameter, U*m can be determined from Equation (6) by iteration. Further, the direction of the surface shear stress for the anticlockwise and clockwise rotating components are given from Equation (4) as c~÷ = ÷ arc -
s i n
KI/L±
.
(7)
Here c~, is the angle relative to the free stream velocity direction of the anticlockwise rotation component ( R ~ ) , that is, in counterclockwise rotation relative to the free stream velocity direction in the northern hemisphere. Similarly, o~ is the angle relative to the free stream velocity direction of the clockwise rotating component (R~_), that is, in clockwise rotation relative to the free stream velocity direction in the northern hemisphere. For transitional smooth-to-rough turbulent flow the roughness parameter is given by
~!)S
I)
:,,~
~,~ -
M~ R~I,\~ <,
( 3Oz,,u,,,) 1 -
exp
-~v ,_ ~
L, ~-
l'
(S'~ t)ll
,.,,
"
where z~,u.,,,.v is the ratio of the rough t~ smooth surface lengthscalcs, and > is lhc kinematic viscosity of the fluid. Equation (8) is obtained from a fil to thc data points in Schlichting (1979, fig. 20.21. p. 620) (Christoffersen and .lonsson. lt)SSt. Here e. = k/30 is the seabed roughness length and k the Nikuradse's equivalcm sand r~ughness of the seabed (Schlichting, 1979). Soulsby (1983) has given a detailed discussion on how k is determined for various seabed conditions. For large and small xalues of z,~u :,,, /v, Equation (8) reduces to ;~,~ ~: g,~ .: k 3 0 and e~, ::- v/gu ,,~ fo~ rough and smooth surfaces, respectively (the smooth surface value z,~,/(v/u ,,,) :-~ 1/9 is consistent with the assumption that the smooth and rough layers are equivalent when zo ~ 0.1 v/u ,,,). It should be noted that according to Soulsby (1983). it is m>t yel sufficient evidence to confirm the use of the faclors 30 and 9 in these expressions for the sea. However. since no definite conclusions have been drawn about this. these factors will be used herein. According to Schlichting (1979) the h)llowmg turbulent flow regimes arc e~ven by: smooth transitional
0 <: e,,tt ,,, t, .- 0.1 0 . 1 7 < z,,u,,~ t,.
rough
(~
2.3
(1~
z~,u,,,/v ~ 2 . 3
(11
Measurements in the sea by Sternberg (1968, 1970) indicated that the value 2.3 in Equation (11), which is based upon laboratory data, should be replaced by 5.5 as the critical value for rough turbulent flow (see also Soulsby, 1983). However, this latter value is based upon limited data. Thus. in consistency with Equation (8) the criteria in Equations ( 9 ) - ( 1 1 ) will be used in this approach. It should be noted that the present approach covers flow over a seabed wilh a given constant roughness; that is, either smooth, rough or transitional smooth-to-rough, and not flow over a seabed with a step change in the roughness. Myrhaug (1989) determined the constants A and B in Equation (1) according to Equations (2a) and (2b) for time-independent planetary boundary layer flow by using two data sets from measurements of the rough turbulent flow structure under Inultiyear drifting Arctic pack ice in the Beaufort Sea reported by McPhee and Smith (1976). The data represent neutrally stable conditions. An inverted boundary layer similar to that under the sea ice is applicable at the seabed. The values of A and B and the corresponding value of c in Equation (3) for (steady) planetary boundary layer flow were determined as A - 2.9,
B = 3.6.
c = 0.055 .
(12)
Also c' = 0.12 was determined for (steady) planetary boundary layer flow by disregarding the rotation of the velocity in the boundary layer as the surface is approached. Thus this approximation corresponds to the use of Equation (3) with
B=O,
c=0.12
for (steady) planetary boundary layer flow.
(13}
Icchmcal Note
399
For oscillatory planetary boundary layer flow it is also suggested to use Equations (12) and 113) in Equation (61. It should be noted that Equation (6) can be applied with other values of B, c and c' if in the future other very precise experiments should give other values. RESULTS AND DISCUSSION Example of results Table 1 gives an example of results of the similarity theory for the following conditions: IR~ I = 10 cm/sec, ]R~ ] = 20 cm/sec, o~ = 0.00014 rad/sec, f = 0.00011 tad/ sec, v = 0.01 cm2/sec and with three values of zn as given in Table 1. z~ = 0.04 and 0.005 cm can be taken as representative for seabeds with unrippled sand and silt/sand, respectively (Soulsby, 1990). zo = 0.001 cm can be taken as representative for a relatively flat and uniform seabed, consisting mainly of fine sand (Slaattelid et al., 1990). It appears that these three flow conditions represent rough, transitional and smooth turbulent flow regimes according to Equations ( 9 ) - ( 1 1 ) . For this example it appears from Table 1 that Equation (6) using Equation (13) is a good approximation to Equation (6) using Equation (12), both for rough, transitional and smooth turbulent flow. It is reasonable to believe that this will also be valid for other flow conditions, since this appeared to be the case for (steady) planetary boundary layer flow for a wide range of flow conditions (Myrhaug, 19891. Further, it appears from Table 1 that the predictions show the correct qualitative behaviour, that is, the values of u ~ , o~ and (-~__) decreases as the seabed roughness decreases. Comparison with measurements The present similarity theory predictions will now be compared with two sets of field measurements reported by Pingree and Griffiths (1974) and King et al. (1985), respectively. Pingree and Griffiths (19741 made measurements on the continental shelf southwest of Lands End, England. These measurements were taken under completely neutral
T A B L E 1.
INPUT
EXAMPLE
OF SIMILARITY THEORY R E S U L T S FOR R O U G H , T R A N S I T I O N A l . A N D S M O O T H T U R B U L E N t =
v = 0.01
cm2/sec
A N D WITH THREE V A L U E S OF Z o AS T A B U k A T E D
Variables
Similarity theory predictions Equations (6) and (7) with (a) Equation 1121
.
FLOW.
IR~+]~ 10 cm/sec, ]R~ I - 20 cm/sec, co 0.00014 rad/sec, f = 0.00011 rad/sec,
VARIABLES:
.
.
.
.
.
(b) Equation (I3)
.
z~ (cm)
1t.04
11.(X15
0.001
0.04
O.(X)5
O.(X)I
u,,, (cm/sec)
0.839
0.694
0.665
0.829
0.681
t).653
z~.u,,/v Flow regime
3.4 Rough
0.35 Transitional
0.067 Smooth
3.3 Rough
0.34 Transitional
0.065 Smooth
a+ (deg)
23.1
18.5
17.6
0
0
0
o~ (deg)
-18.8
-15.6
15.0
0
0
0
401)
l)
MvRII \t~{;
TABLE 2. COMPARISON BEFWEEN PINGREE \ N D GRI|:FIItlS' ( 1 9 7 4 ) I ) A I \ \ N D PREDICIIONS. [NPUI ~. \RIABI IS: {R, +I = 9 . 9 c m / s e c , JR. ! " 35.2 c m / s c c , t,) 0.00(}143 t a d scc, l 0.0(101(}8 r a d , ' s c c . :~, {~ ~ c m Variables
Predictions
Data
S i , n i l a r i l \ theory: 1 2 q u a l l t m s ((~1 m~d ( 7 ! aith ( a ) [ i q u a l i o n {12) (b) Ettu:~tion ( l ? ) u ,,, ( c m / s e c )
zou,,,,/v* c~+ ( d e g ) c~ ( d e g )
1.5 4.5 -
i 2v ~:', 2l.; is,,!
i 25 ~ ,', i! ~,
M'~ }d* x!
i i~Ui} i
i -*: 4 3, II,< l-! i
~v - 0.01 cm"/scc.
moored in water depths of 180 m, more than 21)(/km from the mainland. For mooring No. 1 the measurements were made at 3.5, 7.5, 33.5 and 98 m above the bottom. The semidiurnal frequency of oscillation and the Coriolis parameter were o~ := 0.000143 tad/ sec and f = 0.000108 rad/sec, respectively. The free stream amplitudes of counterclockwise and clockwise motion were IR~., i :- 9.9 cm/sec and IR~ i = 35.2 cm/sec, respectively, 33.5 m above the bottom. The logarithmic boundary layer flow model was matched to the data for the lowest 33.5 m, and the logarithmic flow model friction velocity (u,) and roughness (z,) parameters were determined. The flow was in the rough turbulent regime. The seabed consisted of fine sand and broken and abraded shells. For comparison model predictions according to Myrhaug (1990) are also given in Table 2. The Myrhaug (1990) model is based on a simple theory that describes the motion in a neutrally stable turbulent oscillatory planetary boundary layer near a rough seabed by using a two-layer, time-inw~riant eddy viscosity approach. As shown in Table 2 the estimate of u,:,,, from the measurements is underpredicted by the present similarity theory, while it is very well predicted by the Myrhaug (199(/) model. The estimated values of e~ and ~ from the measurements were not very accurate since they were close to the limits of the absolute accurac} obtainable from the instrumentation ( Pingrec and Griffiths, 1974). However, it appears that the similarity theory and the Myrhaug (1990) model predict values of o~, and (~ which are fairly close to each other. King el al. (1985) made measurements on the continental shelf in the Celtic Sea wcs~ of the Scilly Isles in 120 m water depth in March 1983. The measuremems wcrc taken over a fiat, horizontal, uniform bed under completely neutral conditions well away from the influence of local coastlines, and there was no report on any movement of sediments. Measurements of two orthogonal velocity components were made at l, 2.5. 15, 30, 50, 7(/ and 90 m above the seabed. The x- and y-velocity components were identified with the measured easterly and northerly components, respectively. This data set was analyzed in two ways. Firstly~ from the measurements 11 tidal cycles, over a 6-day period during the peak of the spring tides, were used to obtain ensemble-averaged velocity profiles over one tidal cycle. The 10-rain values thus obtained were further averaged to give profiles every 30 min. A harmonic analysis of each series showed that there was a non-zero mean about 2 cm/sec at each level. These mean wdues, which were assumed to be wind-induced, were subtracted from each velocity value. More
T e c h n i c a l Note
401
details about the measurements and the analysis of the data are given in King et al. (1985), together with a comparison with the data and model predictions by a K - e model. As in King et al. (1985), the conditions at 70 m above the seabed will be used as the free stream condition. They also tried the conditions at 90 m as the free stream condition, but the best agreement was found by using the data at 70 m. A second analysis of the measurements is given in Soulsby (1990), who presents the results of a harmonic analysis of 29 days of the same data. The semidiurnal frequency of oscillation and the Coriolis parameter were ~ = 0.0001425 tad/see and f - 0.0001112 rad/sec, respectively. The free stream amplitudes of the Me constituent counterclockwise and clockwise motion, taken from Soulsby (1990, Fig. 5), were IRaqi = 10.1 cm/sec and IR,~ I = 28.8 cm/sec, respectively (referred to 70 m above the bottom). The zo values estimated from the data ranged between 0.1 and 1.0 cm, with zo = 0.4 cm as an average during the tidal cycle. By fitting a logarithmic profile through the lowermost two current meters for both wtating velocity components, Soulsby (1990) obtained the estimates o f u , , , = 1.74 cm/sec and z, = 0.62 cm. Comparison between measurements and predictions for zo - [).4 and 0.62 cm are given in Table 3. From Table 3 it appears that the estimate of u,,,, from the measurements is underpredicted by the similarity theory for both roughnesses. The Myrhaug (1990) model also underpredicts the u , , values for both roughnesses. The directions estimated from the measurements are the relative differences of the directions between the lowest current meter (at 1 m) and that at 70 m. It should be noted that c~+ as given in Equation (7) is the relative difference between the direction of the shear stress (or velocity) at the seabed and the free stream velocity direction of the rotating velocity components. Thus, if the 70 m elevation matches that of the free stream velocity, then kc~+ will coincide with e~+. From Table 3 it appears that for the Myrhaug (1990) model c~, and kc~ are very close to each other, which suggests that the 70 m elevation can be taken as the height where R , = R .... This is also in agreement with the data (see King et al., 1985; Soulsby, 1990). Further, it appears from Table 3 that the Myrhaug (1990) model suggests that TABLE 3, COMPARISON BETWEEN filE DAIA OF KIN(; el al. (1985) AND PREDI(TIONS, INPUJ VARIABLES IR, ~] 1(!.1 c m / s e c , IR~ 1 = 28.8 c m / s c c , ~ 0.0001425 r a d / s e c , . / = 0.0001112 r a d / s e c AND WIfH 1WO VAI.UES ()~ 2t~ AS [ABUI.A1EI) Variables
Data
Predictions S i m i l a r i t y theory: E q u a t i o n s (6) and (7) with
z, (cm)
(t. 1-1.0
u,, (cm/s) z,,u ,,,/v* (x, ( d e g ) ~,a, ( d e g ) + c~ ( d e g ) Aa ( d e g ) +
1.74 17-170 -13.0 -11.9
M y r h a u g (199(I)
(a) K q u a t i o n (12)
(b) E q u a t i o n (13)
0.4
0.4
0.62
0.4
1.34 54 0
1.411 87 0
1.58 63 21.4 19.6 17.3 8.8
1.34 54 28.7 . -22.5 .
(I.62
. .
1.40 87 30.2 . -23.5 .
. 0
0
.
'% - 0.(11 crn-~/sec. ~Aa. is thc relative diffcrcncc of thc dircction bctwccn the I m and lhc 70 m elevation.
0.62 1.64 102 22.4 211.5 - 18.(/ -8.8
402
I). M', Rtl,\tx,
the free stream value of R is not reached at 70 m, which also is in agreement with the data. It appears that the Myrhaug (1990) model gives larger and smaller values of ke~+ and ( - A c ~ ) , respectively, compared to the measured values. The similarity theory gives larger values of c~: and ( - ¢ x ) than those obtained by Myrhaug (1990) For all the results given in Tables 2 and 3 it appears that Equation (6) by using Equation (13) is a very good approximation to Equation (6) by using Equation (12). Some final remarks should be given to the approach presented here. The similarity constants used herein are based on rough turbulent flow data. However, data from other flow regimes are also needed in order to get reliable formulae valid in all flow regimes. Since the present formulae are based on very limited data, sensitivity analysis on the results may be useful. However, for engineering purposes, it is useful to have working formulae as long as possible limitations are kept in mind. SUMMARY AND ('ONCLI,!S1ONS The friction velocity associated with the maximum bottom shear stress in neutrally stable tidal planetary boundary layer flow is presented. The directions of the bottom shear stresses for the anticlockwise and clockwise rotating components are also presented. The results are obtained by using similarity theory and are given for flow conditions in the rough, smooth and transitional smooth-to-rough turbulent regime. An approximation for the maximum bottom shear stress by disregarding the rotation of the velocity in the boundary layer as lhe seabed is approached is also presented. Comparisons are made with results from fields measurements of rough turbulent tidal planetary boundary layer flow reported by Pingree and Griffiths ( 1974} and King et al. (1985), and a generally acceptable agreement is found. Comparisons are also made with results from a simple eddy viscosity approach (Myrhaug, 1990) and the agreement is reasonably good with the results from similarity theory. The semi-empirical approach presented here gives formulae which should be useful for engineering purposes. REFF.REN('E~ CHRISTOFFERSEN, J.B. and JONSSON,I.G. 1985. Bed lriction and dissipation in a combined currenl ~md ~ave motion. Ocean Engng 12, 38% 423. DAVIES, A.M. 1985. On determining current prolilc~ m oscillatory flox~s. Appl. Mathematical Uodclling 9,
419-42K GHx, A.E. 1982. Almosphere-Ocean Dvt~amav. Academic Prcs~. Nc~ York, KING, H.L., DAv~s, A.G. and SOULSB~,R.L. 1985. A numerical model of lhc turbulcnl boundary layer beneath surface waves and tides. Institute ~f Oceanographic Sciences. Rcporl No 1¢~6. Wormlcx. Oodalming, U.K. (90 pp,). LONG,C.E. 1981. A simple model for timc-depcndcn~ slably slratilied turbulent boundary taycrs. Departmen~ of Oceanography, Spec. Report 95, University of Seaulc. Seattle. Washington, U.S.A ( 170 pp. L McPttEE. M.O. and SMrlH, J.D. 1976. Measurements ~ff the lurbulenl boundary layer under pack ice J phys. Oceanogr. 6, 696-711. MYRttAUG, D. 1989. Simple approach ~o air and water drag on ~ea ice. J. WalWav Por~ (oasml Oceatt Engng. Am. Soc. cir. Engrs 115, 466-476. MYRHAUG, D. 1990. A simple specification for depth-varying eddy viscosity in lidal flows. In Encyclopedia of Fluid Mechanics, CHEREMISINOFF,N.P. (cd.), VOI. t0, pp. 371-390. Gulf Publishing, Houston, Texas. PINOREE, R.D. and ORIFFIrHS,D.K. 1974. The turbulem boundary layer on ~hc continental shelf. Nature 250, ?20-?22. PRANDLE,D. 1982. The verUcal structure of tidal currents. (;eophys. Aslrophy~s. Fluid l)vnatnics 22, 29-49. SCHL1CHTING,H. 1979. Boundary-Layer Theory (Tth EdnL McGraw-Hill, New York. SLn~TrEt.m, O.H., MYRHAUG,D. and LAMBRAKOS,K.F. 1990. North Sea bottom steady boundary layer measurements. J. WatWay Porl Coaslal Ocean Engn~,, Am. Soc. cir. Engrs 116, 614-633.
Technical Note
4~}3
S(}UI_SBY, R.L. 1983. The bottom boundary layer of shelf seas. In Physical Oceanography qt Coastal and Shelf Seas. JOHNS, B. (ed. L pp. 189-266. Elsevier, New York. S{}t:LsuY, R.L. 1990. Tidal-current boundary layers. In The Sea, LE MI~HAUI'~,B. and HAYFS, D.M. {cds). Vol. 9, Part A. Ocean Engineering Science, pp. 523 566. John Wiley, New York. STEINBERG,R.W. 1968. Friction factors in tidal channels with diffcring bed roughness. Mar. Geol. 6,243-260. STERNBERG,R.W. 1970. Field measurements of the hydrodynamic roughness of the deep-sea boundary. Deep Sea Res. 17, 413-420. SVERI)RUP, H.U. 1927. Dynamic of tides on the North Siberian Shelf. Geo6,siske Publikas~oner 4. (75 pp). L!NFERS[EINER,N. and BADGLEY,F.I. 1965. The rongbncss parameters of sea ice. Geophys. Res. 70, 4573 4577. APPENDIX Some elements of the analysis of tidal currents will now be summarized. Details of the full analysis of tidal currents can be found in Soulsby (1983, 1990) (see also Myrhaug. 199(}). By using complex notation a particular tidal velocity constituent (R) of angular frequency to is separated to give the anticlockwise (R~) and clockwise ( R ) rotating components as R = R~e i ' ° ' + R ~ e i.,, =fR,[e~t% '~')+[R
le'(+,~
,o,,
(all
Here R. represents a velocity vector with magnitude iR, I and phase ,5~-, which rotates counterclockwise with frequency to, viewed from above. R represents a velocity vector with magnitude IR ! and phase d)n , which rotates clockwise with frequency to. Thus the combination of the two rotations in Equation (A1) represents a velocity vector that describes an ellipse. The magnitude of the velocity has its maximum for 1
0,~ = .~(Oon - +te~)
(A2)
for which its maximum value is
R ......
(IR,[ + [R I)e'
'
(bt¢
~
(A3)
Here IR[...... - ] R ~ [ + ]R_[ corresponds to the semi-major axis of the ellipse, and (PR,, - (+,~ + 4,~ )/2 gives the orientation of the semi-major axis of the ellipse relative to the x-axis (east in the northern hemisphere) and positive in counterclockwise rotation. Similarly, the frictional shear stress ( r ) can be divided into counterclockwise (7",) and clockwise ( T ) rotating components in the same m a n n e r as the velocity in Equation (A1), with m a g n i t u d e s (IT~ I, iT-I) and phases ( ~ r ~ . +~-). Thus the magnitude of the shear stress has its maximum for l
O t - ~(+~
- +-r+)
(A4)
for which its maximum value is d)/~
L ..... - ( ~ r ~ l
+~r-~): ~
+ (b 7
~
~ •
(as)
The maximum friction velocity at the seabed is defined as -
..,.,
~
'
+
.
.
,-
.
.
~{}/
.
(a6)
Here the terms in the parenthesis represent the friction velocities associated with the seabed shear stress for the anticlockwise and clockwise rotating components, that is
404
D. M~ RL~aU(~
It s h o u l d be n o t e d that at the s e a b e d the direction of the rotating c o m p o n e n t s ot the velocity, and the s h e a r stress are the same, that is (6~,)~
:,,,- ( ( ~ ) =
.~,,and(d)~
)
:,,, ~ (d~
). . . . . .
0029-801892 $5.00 * .01) Pergamon Press l,td
TECHNICAL
NOTE
RATIONAL APPROACH TO THE BOTTOM S T R E S S IN T I D A L P L A N E T A R Y BOUNDARY FLOW
SHEAR LAYER
D. MYRHAUG Division of Marine Hydrodynamics, Norwegian Institute of Technology, N-7034 Trondhcim-NTtt, Norway Abstract--The friction velocity associated with the m~ximum bottom shear stress in neutrally
stable tidal planetary boundary layer flow is presented. The directions of the bottom shear stresses for the anticlockwise and clockwise rotating components arc also presented. The results arc obtained by using similarity theory and arc given for flow conditions in the rough, smooth and transitional smooth-to-rough turbulent regime. An approximation for the maximum bottom shear stress by disregarding the rotation of the velocity in the boundary layer as the seabed is approached is also presented.
INTRODUCTION
THE FLUID motion in the tidal planetary boundary layer near the seabed controls and affects many phenomena in ocean engineering and oceanography. The bottom shear stress associated with the boundary layer flow represents the dominant mechanism governing sediment transport and erosion phenomena on continental shelves. Tidal planetary boundary layer flow also represents an important component in physical models for predicting ocean flow circulations. The vertical structure of the bottom boundary layer on the continental shelves is in the most general and complex case dominated by several interacting physical effects. Among these effects are the Earth's rotation; tidal effects; stratification due to salinity, temperature gradients and suspended sediments; internal friction in the fluid: and Iopographical effects. One important feature of the vertical structure of the bottom boundary layer is determined by the influence of planetary rotation on unsteady (due to tidal effects), horizontally uniform, unbounded, and unstratified flow. This idealized boundary-layer flow may occur in the ocean away from any coasts and in the region near the seabed where the internal friction in the fluid is significant. The details of the vertical structure of the current varies with the modelling of the momentum flux in the boundary layer. Various types of models have been represented in Sverdrup (1927), Long (1981), Prandle (1982), Davies (1985), King et al. (1985) and Myrhaug (1990) among others. Soulsby (1983, 1990) gives reviews of bottom boundary layers of shelf seas. Some of the idealized types of boundary layer situations were discussed using an eddy viscosity model which varies linearly with the distance from the bottom. In most cases the boundary layer flow is in the rough turbulent 395
39(~
D. M'~ ~I,~t ~(i
regime, but it also appears that the flow conditions might be in the smooth and the transitional smooth-to-rough turbulent flow rcgimes (Sternberg. lt~(~S. 1970: Untersteiner and Badgley, 1965). The purpose of this paper is to present si~nple formulae from which the maximum seabed shear stress in tidal planetary boundary layer flow can be determined. 3he semiempirical approach presented here gives rational formulae which should be useful for engineering purposes. TItEORY Following Gill (1982) the geostrophic velocity for steady homogeneous (neutrally stable) planetary flow can be written as R,_ =
~
In
,~?,,~,
-A
~1)
~
according to similarity theory. Here R~ ~ U~ * ivy, is the geostrophic velocity m complex form. U-~ and V~_ are the geostrophic velocity components along horizontal orthogonal x- and y-axis, respectively. The x-axis is taken along the shear stress at the surface, %, which has an angle ~ relative to the geostrophic velocity direction, that is. in counterclockwise rotation relative to the geostrophic velocity direction in the northern hemisphere, f = 2{~sin~ is the Coriolis parameter, where [~ is the Earth's angular frequency of rotation and • the latitude. Further, u~ = (~,/p)~n is the friction velocity, p is the density of the fluid, ~ is yon Karman's constant ( ~ 0.4), z~ is the roughness parameter of the seabed which will be discussed subsequently, A and B are dimensionless constants as required by similarity theory, and i = (-1)~/~. Since U ..... [R~ [cos~ and V~ = -{R~.[sin~ and by using Equation (1), A and B are given as
(' )_ K~,.,
u~,~ A = In, )~,,;
cos~.
B = ~R~tu sina
12a)(2b)
Thus Equations (2a) and (2b) determine A and B for a given data set. ~R~[ is the magnitude of R .... For known A and B, u, can be found from Equation (1), and a is determined from Equation (2b) as, respectively u
~=arcsin
(3)
~{R~l]
(4)
where c = e /~ is a dimensionless constant. An approximation to Equation (3) is obtained by using B = 0 and a modified value of c, that is, c'. In this approximation the rotation of the velocity in the boundary layer as the seabed is approached is disregarded, and the logarithmic boundary layer flow model, that is IR~I = ( u ~ / ~ ) ln(c'u./fzo¢.), is extended beyond its range of validity. The logarithmic boundary layer flow model is valid only in a region where the shear stress is constant. Here it is extended to a height c'u./f where the velocity is equal to the magnitude of the geostrophic velocity. Thus c' can be determined for a given data set.
Technical Note
397
This similarity theory will now be applied to oscillatory planetary boundary layer flow by using the results from the analysis of tidal currents. Some elements of this analysis are summarized in Appendix. However, more details of the analysis of tidal currents are given in Soulsby (1983, 1990). The result of separating a particular tidal constituent of angular frequency o~ to give the anticlockwise and clockwise rotating components is that the equations of motions for each rotating component are similar in form, and similar to that given for the steady planetary motion. This implies that when the result in Equation (3) is applied to oscillatory planetary flowfshould be replaced by (~o + f ) and (co - f ) for the anticlockwise and clockwise rotating components, respectively. In the following it is assumed that co > f in the northern hemisphere ( f > 0). However, the results can easily be generalized to cover situations where ~o < f and to the southern hemisphere ((f < 0) (see Soulsby, 1990). Thus, by using Equation (3), the friction velocities associated with the seabed shear stress for the anticlockwise ( + ) and clockwise ( - ) rotating components are given as
I
(51
where u,,, is the maximum friction velocity at the seabed as defined in Equation (A6). IRK +1 denotes the magnitude of the free stream velocity outside the boundary layer. By using Equations (5), (A6) and (A7), the maximum friction velocity at the seabed is given by ~'IR~. ~I-~ r [ cu,,,, ~]2 B e [lnk(~ V/)z,,t...JJ +
,,,:,,,=
+
J 11/2
K~-IR~ 12
r / cu,:,,, ,~ e B ~ [lnk(~ ~ f ) Z " , ) +
(6)
Equation (6) is an implicit equation for determination of u..,. For known similarity constants and for a given seabed roughness length, free stream current velocity components, frequency of oscillation and Coriolis parameter, U*m can be determined from Equation (6) by iteration. Further, the direction of the surface shear stress for the anticlockwise and clockwise rotating components are given from Equation (4) as c~÷ = ÷ arc -
s i n
KI/L±
.
(7)
Here c~, is the angle relative to the free stream velocity direction of the anticlockwise rotation component ( R ~ ) , that is, in counterclockwise rotation relative to the free stream velocity direction in the northern hemisphere. Similarly, o~ is the angle relative to the free stream velocity direction of the clockwise rotating component (R~_), that is, in clockwise rotation relative to the free stream velocity direction in the northern hemisphere. For transitional smooth-to-rough turbulent flow the roughness parameter is given by
~!)S
I)
:,,~
~,~ -
M~ R~I,\~ <,
( 3Oz,,u,,,) 1 -
exp
-~v ,_ ~
L, ~-
l'
(S'~ t)ll
,.,,
"
where z~,u.,,,.v is the ratio of the rough t~ smooth surface lengthscalcs, and > is lhc kinematic viscosity of the fluid. Equation (8) is obtained from a fil to thc data points in Schlichting (1979, fig. 20.21. p. 620) (Christoffersen and .lonsson. lt)SSt. Here e. = k/30 is the seabed roughness length and k the Nikuradse's equivalcm sand r~ughness of the seabed (Schlichting, 1979). Soulsby (1983) has given a detailed discussion on how k is determined for various seabed conditions. For large and small xalues of z,~u :,,, /v, Equation (8) reduces to ;~,~ ~: g,~ .: k 3 0 and e~, ::- v/gu ,,~ fo~ rough and smooth surfaces, respectively (the smooth surface value z,~,/(v/u ,,,) :-~ 1/9 is consistent with the assumption that the smooth and rough layers are equivalent when zo ~ 0.1 v/u ,,,). It should be noted that according to Soulsby (1983). it is m>t yel sufficient evidence to confirm the use of the faclors 30 and 9 in these expressions for the sea. However. since no definite conclusions have been drawn about this. these factors will be used herein. According to Schlichting (1979) the h)llowmg turbulent flow regimes arc e~ven by: smooth transitional
0 <: e,,tt ,,, t, .- 0.1 0 . 1 7 < z,,u,,~ t,.
rough
(~
2.3
(1~
z~,u,,,/v ~ 2 . 3
(11
Measurements in the sea by Sternberg (1968, 1970) indicated that the value 2.3 in Equation (11), which is based upon laboratory data, should be replaced by 5.5 as the critical value for rough turbulent flow (see also Soulsby, 1983). However, this latter value is based upon limited data. Thus. in consistency with Equation (8) the criteria in Equations ( 9 ) - ( 1 1 ) will be used in this approach. It should be noted that the present approach covers flow over a seabed wilh a given constant roughness; that is, either smooth, rough or transitional smooth-to-rough, and not flow over a seabed with a step change in the roughness. Myrhaug (1989) determined the constants A and B in Equation (1) according to Equations (2a) and (2b) for time-independent planetary boundary layer flow by using two data sets from measurements of the rough turbulent flow structure under Inultiyear drifting Arctic pack ice in the Beaufort Sea reported by McPhee and Smith (1976). The data represent neutrally stable conditions. An inverted boundary layer similar to that under the sea ice is applicable at the seabed. The values of A and B and the corresponding value of c in Equation (3) for (steady) planetary boundary layer flow were determined as A - 2.9,
B = 3.6.
c = 0.055 .
(12)
Also c' = 0.12 was determined for (steady) planetary boundary layer flow by disregarding the rotation of the velocity in the boundary layer as the surface is approached. Thus this approximation corresponds to the use of Equation (3) with
B=O,
c=0.12
for (steady) planetary boundary layer flow.
(13}
Icchmcal Note
399
For oscillatory planetary boundary layer flow it is also suggested to use Equations (12) and 113) in Equation (61. It should be noted that Equation (6) can be applied with other values of B, c and c' if in the future other very precise experiments should give other values. RESULTS AND DISCUSSION Example of results Table 1 gives an example of results of the similarity theory for the following conditions: IR~ I = 10 cm/sec, ]R~ ] = 20 cm/sec, o~ = 0.00014 rad/sec, f = 0.00011 tad/ sec, v = 0.01 cm2/sec and with three values of zn as given in Table 1. z~ = 0.04 and 0.005 cm can be taken as representative for seabeds with unrippled sand and silt/sand, respectively (Soulsby, 1990). zo = 0.001 cm can be taken as representative for a relatively flat and uniform seabed, consisting mainly of fine sand (Slaattelid et al., 1990). It appears that these three flow conditions represent rough, transitional and smooth turbulent flow regimes according to Equations ( 9 ) - ( 1 1 ) . For this example it appears from Table 1 that Equation (6) using Equation (13) is a good approximation to Equation (6) using Equation (12), both for rough, transitional and smooth turbulent flow. It is reasonable to believe that this will also be valid for other flow conditions, since this appeared to be the case for (steady) planetary boundary layer flow for a wide range of flow conditions (Myrhaug, 19891. Further, it appears from Table 1 that the predictions show the correct qualitative behaviour, that is, the values of u ~ , o~ and (-~__) decreases as the seabed roughness decreases. Comparison with measurements The present similarity theory predictions will now be compared with two sets of field measurements reported by Pingree and Griffiths (1974) and King et al. (1985), respectively. Pingree and Griffiths (19741 made measurements on the continental shelf southwest of Lands End, England. These measurements were taken under completely neutral
T A B L E 1.
INPUT
EXAMPLE
OF SIMILARITY THEORY R E S U L T S FOR R O U G H , T R A N S I T I O N A l . A N D S M O O T H T U R B U L E N t =
v = 0.01
cm2/sec
A N D WITH THREE V A L U E S OF Z o AS T A B U k A T E D
Variables
Similarity theory predictions Equations (6) and (7) with (a) Equation 1121
.
FLOW.
IR~+]~ 10 cm/sec, ]R~ I - 20 cm/sec, co 0.00014 rad/sec, f = 0.00011 rad/sec,
VARIABLES:
.
.
.
.
.
(b) Equation (I3)
.
z~ (cm)
1t.04
11.(X15
0.001
0.04
O.(X)5
O.(X)I
u,,, (cm/sec)
0.839
0.694
0.665
0.829
0.681
t).653
z~.u,,/v Flow regime
3.4 Rough
0.35 Transitional
0.067 Smooth
3.3 Rough
0.34 Transitional
0.065 Smooth
a+ (deg)
23.1
18.5
17.6
0
0
0
o~ (deg)
-18.8
-15.6
15.0
0
0
0
401)
l)
MvRII \t~{;
TABLE 2. COMPARISON BEFWEEN PINGREE \ N D GRI|:FIItlS' ( 1 9 7 4 ) I ) A I \ \ N D PREDICIIONS. [NPUI ~. \RIABI IS: {R, +I = 9 . 9 c m / s e c , JR. ! " 35.2 c m / s c c , t,) 0.00(}143 t a d scc, l 0.0(101(}8 r a d , ' s c c . :~, {~ ~ c m Variables
Predictions
Data
S i , n i l a r i l \ theory: 1 2 q u a l l t m s ((~1 m~d ( 7 ! aith ( a ) [ i q u a l i o n {12) (b) Ettu:~tion ( l ? ) u ,,, ( c m / s e c )
zou,,,,/v* c~+ ( d e g ) c~ ( d e g )
1.5 4.5 -
i 2v ~:', 2l.; is,,!
i 25 ~ ,', i! ~,
M'~ }d* x!
i i~Ui} i
i -*: 4 3, II,< l-! i
~v - 0.01 cm"/scc.
moored in water depths of 180 m, more than 21)(/km from the mainland. For mooring No. 1 the measurements were made at 3.5, 7.5, 33.5 and 98 m above the bottom. The semidiurnal frequency of oscillation and the Coriolis parameter were o~ := 0.000143 tad/ sec and f = 0.000108 rad/sec, respectively. The free stream amplitudes of counterclockwise and clockwise motion were IR~., i :- 9.9 cm/sec and IR~ i = 35.2 cm/sec, respectively, 33.5 m above the bottom. The logarithmic boundary layer flow model was matched to the data for the lowest 33.5 m, and the logarithmic flow model friction velocity (u,) and roughness (z,) parameters were determined. The flow was in the rough turbulent regime. The seabed consisted of fine sand and broken and abraded shells. For comparison model predictions according to Myrhaug (1990) are also given in Table 2. The Myrhaug (1990) model is based on a simple theory that describes the motion in a neutrally stable turbulent oscillatory planetary boundary layer near a rough seabed by using a two-layer, time-inw~riant eddy viscosity approach. As shown in Table 2 the estimate of u,:,,, from the measurements is underpredicted by the present similarity theory, while it is very well predicted by the Myrhaug (199(/) model. The estimated values of e~ and ~ from the measurements were not very accurate since they were close to the limits of the absolute accurac} obtainable from the instrumentation ( Pingrec and Griffiths, 1974). However, it appears that the similarity theory and the Myrhaug (1990) model predict values of o~, and (~ which are fairly close to each other. King el al. (1985) made measurements on the continental shelf in the Celtic Sea wcs~ of the Scilly Isles in 120 m water depth in March 1983. The measuremems wcrc taken over a fiat, horizontal, uniform bed under completely neutral conditions well away from the influence of local coastlines, and there was no report on any movement of sediments. Measurements of two orthogonal velocity components were made at l, 2.5. 15, 30, 50, 7(/ and 90 m above the seabed. The x- and y-velocity components were identified with the measured easterly and northerly components, respectively. This data set was analyzed in two ways. Firstly~ from the measurements 11 tidal cycles, over a 6-day period during the peak of the spring tides, were used to obtain ensemble-averaged velocity profiles over one tidal cycle. The 10-rain values thus obtained were further averaged to give profiles every 30 min. A harmonic analysis of each series showed that there was a non-zero mean about 2 cm/sec at each level. These mean wdues, which were assumed to be wind-induced, were subtracted from each velocity value. More
T e c h n i c a l Note
401
details about the measurements and the analysis of the data are given in King et al. (1985), together with a comparison with the data and model predictions by a K - e model. As in King et al. (1985), the conditions at 70 m above the seabed will be used as the free stream condition. They also tried the conditions at 90 m as the free stream condition, but the best agreement was found by using the data at 70 m. A second analysis of the measurements is given in Soulsby (1990), who presents the results of a harmonic analysis of 29 days of the same data. The semidiurnal frequency of oscillation and the Coriolis parameter were ~ = 0.0001425 tad/see and f - 0.0001112 rad/sec, respectively. The free stream amplitudes of the Me constituent counterclockwise and clockwise motion, taken from Soulsby (1990, Fig. 5), were IRaqi = 10.1 cm/sec and IR,~ I = 28.8 cm/sec, respectively (referred to 70 m above the bottom). The zo values estimated from the data ranged between 0.1 and 1.0 cm, with zo = 0.4 cm as an average during the tidal cycle. By fitting a logarithmic profile through the lowermost two current meters for both wtating velocity components, Soulsby (1990) obtained the estimates o f u , , , = 1.74 cm/sec and z, = 0.62 cm. Comparison between measurements and predictions for zo - [).4 and 0.62 cm are given in Table 3. From Table 3 it appears that the estimate of u,,,, from the measurements is underpredicted by the similarity theory for both roughnesses. The Myrhaug (1990) model also underpredicts the u , , values for both roughnesses. The directions estimated from the measurements are the relative differences of the directions between the lowest current meter (at 1 m) and that at 70 m. It should be noted that c~+ as given in Equation (7) is the relative difference between the direction of the shear stress (or velocity) at the seabed and the free stream velocity direction of the rotating velocity components. Thus, if the 70 m elevation matches that of the free stream velocity, then kc~+ will coincide with e~+. From Table 3 it appears that for the Myrhaug (1990) model c~, and kc~ are very close to each other, which suggests that the 70 m elevation can be taken as the height where R , = R .... This is also in agreement with the data (see King et al., 1985; Soulsby, 1990). Further, it appears from Table 3 that the Myrhaug (1990) model suggests that TABLE 3, COMPARISON BETWEEN filE DAIA OF KIN(; el al. (1985) AND PREDI(TIONS, INPUJ VARIABLES IR, ~] 1(!.1 c m / s e c , IR~ 1 = 28.8 c m / s c c , ~ 0.0001425 r a d / s e c , . / = 0.0001112 r a d / s e c AND WIfH 1WO VAI.UES ()~ 2t~ AS [ABUI.A1EI) Variables
Data
Predictions S i m i l a r i t y theory: E q u a t i o n s (6) and (7) with
z, (cm)
(t. 1-1.0
u,, (cm/s) z,,u ,,,/v* (x, ( d e g ) ~,a, ( d e g ) + c~ ( d e g ) Aa ( d e g ) +
1.74 17-170 -13.0 -11.9
M y r h a u g (199(I)
(a) K q u a t i o n (12)
(b) E q u a t i o n (13)
0.4
0.4
0.62
0.4
1.34 54 0
1.411 87 0
1.58 63 21.4 19.6 17.3 8.8
1.34 54 28.7 . -22.5 .
(I.62
. .
1.40 87 30.2 . -23.5 .
. 0
0
.
'% - 0.(11 crn-~/sec. ~Aa. is thc relative diffcrcncc of thc dircction bctwccn the I m and lhc 70 m elevation.
0.62 1.64 102 22.4 211.5 - 18.(/ -8.8
402
I). M', Rtl,\tx,
the free stream value of R is not reached at 70 m, which also is in agreement with the data. It appears that the Myrhaug (1990) model gives larger and smaller values of ke~+ and ( - A c ~ ) , respectively, compared to the measured values. The similarity theory gives larger values of c~: and ( - ¢ x ) than those obtained by Myrhaug (1990) For all the results given in Tables 2 and 3 it appears that Equation (6) by using Equation (13) is a very good approximation to Equation (6) by using Equation (12). Some final remarks should be given to the approach presented here. The similarity constants used herein are based on rough turbulent flow data. However, data from other flow regimes are also needed in order to get reliable formulae valid in all flow regimes. Since the present formulae are based on very limited data, sensitivity analysis on the results may be useful. However, for engineering purposes, it is useful to have working formulae as long as possible limitations are kept in mind. SUMMARY AND ('ONCLI,!S1ONS The friction velocity associated with the maximum bottom shear stress in neutrally stable tidal planetary boundary layer flow is presented. The directions of the bottom shear stresses for the anticlockwise and clockwise rotating components are also presented. The results are obtained by using similarity theory and are given for flow conditions in the rough, smooth and transitional smooth-to-rough turbulent regime. An approximation for the maximum bottom shear stress by disregarding the rotation of the velocity in the boundary layer as lhe seabed is approached is also presented. Comparisons are made with results from fields measurements of rough turbulent tidal planetary boundary layer flow reported by Pingree and Griffiths ( 1974} and King et al. (1985), and a generally acceptable agreement is found. Comparisons are also made with results from a simple eddy viscosity approach (Myrhaug, 1990) and the agreement is reasonably good with the results from similarity theory. The semi-empirical approach presented here gives formulae which should be useful for engineering purposes. REFF.REN('E~ CHRISTOFFERSEN, J.B. and JONSSON,I.G. 1985. Bed lriction and dissipation in a combined currenl ~md ~ave motion. Ocean Engng 12, 38% 423. DAVIES, A.M. 1985. On determining current prolilc~ m oscillatory flox~s. Appl. Mathematical Uodclling 9,
419-42K GHx, A.E. 1982. Almosphere-Ocean Dvt~amav. Academic Prcs~. Nc~ York, KING, H.L., DAv~s, A.G. and SOULSB~,R.L. 1985. A numerical model of lhc turbulcnl boundary layer beneath surface waves and tides. Institute ~f Oceanographic Sciences. Rcporl No 1¢~6. Wormlcx. Oodalming, U.K. (90 pp,). LONG,C.E. 1981. A simple model for timc-depcndcn~ slably slratilied turbulent boundary taycrs. Departmen~ of Oceanography, Spec. Report 95, University of Seaulc. Seattle. Washington, U.S.A ( 170 pp. L McPttEE. M.O. and SMrlH, J.D. 1976. Measurements ~ff the lurbulenl boundary layer under pack ice J phys. Oceanogr. 6, 696-711. MYRttAUG, D. 1989. Simple approach ~o air and water drag on ~ea ice. J. WalWav Por~ (oasml Oceatt Engng. Am. Soc. cir. Engrs 115, 466-476. MYRHAUG, D. 1990. A simple specification for depth-varying eddy viscosity in lidal flows. In Encyclopedia of Fluid Mechanics, CHEREMISINOFF,N.P. (cd.), VOI. t0, pp. 371-390. Gulf Publishing, Houston, Texas. PINOREE, R.D. and ORIFFIrHS,D.K. 1974. The turbulem boundary layer on ~hc continental shelf. Nature 250, ?20-?22. PRANDLE,D. 1982. The verUcal structure of tidal currents. (;eophys. Aslrophy~s. Fluid l)vnatnics 22, 29-49. SCHL1CHTING,H. 1979. Boundary-Layer Theory (Tth EdnL McGraw-Hill, New York. SLn~TrEt.m, O.H., MYRHAUG,D. and LAMBRAKOS,K.F. 1990. North Sea bottom steady boundary layer measurements. J. WatWay Porl Coaslal Ocean Engn~,, Am. Soc. cir. Engrs 116, 614-633.
Technical Note
4~}3
S(}UI_SBY, R.L. 1983. The bottom boundary layer of shelf seas. In Physical Oceanography qt Coastal and Shelf Seas. JOHNS, B. (ed. L pp. 189-266. Elsevier, New York. S{}t:LsuY, R.L. 1990. Tidal-current boundary layers. In The Sea, LE MI~HAUI'~,B. and HAYFS, D.M. {cds). Vol. 9, Part A. Ocean Engineering Science, pp. 523 566. John Wiley, New York. STEINBERG,R.W. 1968. Friction factors in tidal channels with diffcring bed roughness. Mar. Geol. 6,243-260. STERNBERG,R.W. 1970. Field measurements of the hydrodynamic roughness of the deep-sea boundary. Deep Sea Res. 17, 413-420. SVERI)RUP, H.U. 1927. Dynamic of tides on the North Siberian Shelf. Geo6,siske Publikas~oner 4. (75 pp). L!NFERS[EINER,N. and BADGLEY,F.I. 1965. The rongbncss parameters of sea ice. Geophys. Res. 70, 4573 4577. APPENDIX Some elements of the analysis of tidal currents will now be summarized. Details of the full analysis of tidal currents can be found in Soulsby (1983, 1990) (see also Myrhaug. 199(}). By using complex notation a particular tidal velocity constituent (R) of angular frequency to is separated to give the anticlockwise (R~) and clockwise ( R ) rotating components as R = R~e i ' ° ' + R ~ e i.,, =fR,[e~t% '~')+[R
le'(+,~
,o,,
(all
Here R. represents a velocity vector with magnitude iR, I and phase ,5~-, which rotates counterclockwise with frequency to, viewed from above. R represents a velocity vector with magnitude IR ! and phase d)n , which rotates clockwise with frequency to. Thus the combination of the two rotations in Equation (A1) represents a velocity vector that describes an ellipse. The magnitude of the velocity has its maximum for 1
0,~ = .~(Oon - +te~)
(A2)
for which its maximum value is
R ......
(IR,[ + [R I)e'
'
(bt¢
~
(A3)
Here IR[...... - ] R ~ [ + ]R_[ corresponds to the semi-major axis of the ellipse, and (PR,, - (+,~ + 4,~ )/2 gives the orientation of the semi-major axis of the ellipse relative to the x-axis (east in the northern hemisphere) and positive in counterclockwise rotation. Similarly, the frictional shear stress ( r ) can be divided into counterclockwise (7",) and clockwise ( T ) rotating components in the same m a n n e r as the velocity in Equation (A1), with m a g n i t u d e s (IT~ I, iT-I) and phases ( ~ r ~ . +~-). Thus the magnitude of the shear stress has its maximum for l
O t - ~(+~
- +-r+)
(A4)
for which its maximum value is d)/~
L ..... - ( ~ r ~ l
+~r-~): ~
+ (b 7
~
~ •
(as)
The maximum friction velocity at the seabed is defined as -
..,.,
~
'
+
.
.
,-
.
.
~{}/
.
(a6)
Here the terms in the parenthesis represent the friction velocities associated with the seabed shear stress for the anticlockwise and clockwise rotating components, that is
404
D. M~ RL~aU(~
It s h o u l d be n o t e d that at the s e a b e d the direction of the rotating c o m p o n e n t s ot the velocity, and the s h e a r stress are the same, that is (6~,)~
:,,,- ( ( ~ ) =
.~,,and(d)~
)
:,,, ~ (d~
). . . . . .