Comparison of a simple planetary boundary layer model with measurements of a turbulent boundary layer under pack ice

ContinentalShelfResearch,Vol. 7, No. 2, pp. 135--148,1987.

0278-4343/87 $3.00 + 0.00 PergamonJournaLsLtd.

Printed in Great Britain.

Comparison of a simple planetary boundary layer model with measurements of a turbulent boundary layer under pack ice DAGMYRHAUG*

(Received 25 March 1985; in revised form 21 October 1985; accepted 6 February 1986) Abstract--A simple analytical theory which describes the motion in a turbulent planetary boundary layer near a rough sea bed by using a two-layer eddy viscosity model is presented. The vertical structure of the current in the boundary layer is presented, and comparisons are made with data from MCPHEE and SMITH (1976, Journal of Physical Oceanography, 6, 696-711) obtained from measurements of the turbulent boundary layer under drifting Arctic ice.

INTRODUCTION

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 and temperature gradients and suspended sediments, internal friction in the fluid and topographical effects. One important feature of the vertical structure of the bottom boundary layer is determined by the influence of planetary rotation on steady, horizontally uniform, unbounded and unstratified flow. In this case there is a balance between the driving pressure gradient, the momentum flux gradient and the Coriolis force. Outside the boundary layer the flow is geostrophic, in which the friction is negligible and the pressure gradient is balanced by the Coriolis force. This idealized boundary layer flow may occur in the ocean away from any coasts and in the region between the sea bed and currents which are steady over times comparable with the inertial period, 2~/f, where f = 2f~ sin x¢ is the Coriolis parameter. Here f~ is the earth's angular frequency of rotation and V the latitude. In this case the vertical structure of the current in the boundary layer is characterized by velocity and shear stress vectors that increase and decrease, respectively, in magnitude and rotate clockwise (in the northern hemisphere) with increasing distance from the sea bed. This main feature was found by EKMAN(1905) using a constant eddy viscosity to model the momentum flux in the boundary layer. However, the details of the vertical structure of the current in the boundary layer will vary with changes in the vertical eddy viscosity distribution which is used. Various types of models are treated in SVERDRUP (1927), LONG (1981) and PRANDLE (1982). SOULSBY(1983) gives a review of bottom

* Division of Marine Hydrodynamics, Norwegian Institute of Technology, N-7034 Trondheim-NTH, Norway. 135

136

D. MYRHAUG

boundary layers of shelf seas. Some of the idealized types of boundary layer situations are presented by using an eddy viscosity model varying linearly with the distance from the bottom. This paper presents an analytical theory which describes the fluid motion in a turbulent boundary layer near a rough sea bed where the flow is determined by a balance between the driving pressure gradient (water surface slope), the shear stress gradients and the Coriolis force. A two-layer eddy viscosity model is used to model the shear stress. The eddy viscosity in the inner layer increases quadratically with the distance from the bottom. In the outer layer the eddy viscosity is taken as a constant. The present eddy viscosity model is a reasonable compromise between accuracy and simplicity and has the benefit over the linear profile of having one disposable constant which admits for some adjustment of the model to data. The choice of this constant determines the magnitude of the eddy viscosity in the outer layer, as well as the height of the overlap point. The vertical structure of the eddy viscosity and the velocity and shear stress profiles are presented together with the drag coefficient at the sea bed as a function of the Rossby number based on the geostrophic velocity, the roughness length of the sea bed and the Coriolis parameter. Comparisons are also made with data from McP~tEE and SMm~ (1976) obtained from measurements of the turbulent boundary layer under drifting Arctic pack ice. An inverted boundary layer similar to that at the sea bed is applicable under the pack ice. The theoretical model predictions seem to predict the essential features of the experimental results reasonably well. THE PLANETARY

BOUNDARY

LAYER

MODEL

B o u n d a r y layer e q u a t i o n s a n d e d d y viscosity m o d e l

In studying the first-order marine boundary layer problem it is common to neglect vertical components of velocity, convective acceleration terms and density effects. The time-independent equations of motion along horizontal orthogonal axes x and y (east and north, respectively, in the northern hemisphere) can be combined to be written in complex notation by (SouLsBY, 1983) lOT i f R = - US + - 90z

,

(1)

where i = (-1) ½and the following complex notation has been introduced for the velocity, the water surface slope and the frictional shear stress, respectively R = U + iV

(2)

s = -o;- + i -o; -

(3)

T = Txz + i Xyz,

(4)

Ox

Oy

where z is the vertical axis measured positive upwards from the sea bed, U and V are velocities along x and y, g the acceleration of gravity, { the water surface elevation, p the density of the fluid, Xxz and Xyz the x- and y-component of the frictional shear stress, respectively. The flow is driven by the horizontal pressure gradient due to the horizontal steady water surface slope.

Measurements of a turbulent boundary layer

137

The frictional shear stress will be modelled as OR r = p

(5)

Oz

where the kinematic eddy viscosity e is assumed to be the same in both horizontal directions. In this case equation (1) takes the form

ifR = - gS + uz

(oR)

.

(6)

At the sea bed the velocity is zero and outside the boundary layer the frictional shear stress is negligible, i.e. the flow is geostrophic. Thus the boundary conditions are given by R = 0

at

z = Zo

(7) R---* R~ =--ig s

forz--->~,

f

where z0 is the sea bed roughness length and R= is the geostrophic velocity. The flow is assumed to be in the rough turbulent regime, that is (see SouesBv, 1983) u,k --

>

165.

(S)

v

In the roughness Reynolds number, u , k / v , v is the coefficient of kinematic viscosity of the fluid k is the Nikuradse's equivalent sand roughness, that is, the characteristic dimension of the physical roughness of the sea bed, and u, is the friction velocity at the sea bed defined by u, = \P/

=

pyz,z z,, ,

(9)

where % = ] T[ z=z,) is the magnitude of the shear stress at the sea bed. In the ocean k may be very different than the local physical roughness would suggest. Equation (8) is valid provided that the height of the roughness elements replaces k if the bed is not flat [see SOULSBY (1983) for a closer discussion]. SOULSBY(1983) gives a detailed discussion on how k is determined for various sea bed conditions. When the flow is assumed to be in the rough turbulent regime the sea bed roughness elements are higher than the (hypothetical) viscous sublayer. In that case the eddy viscosity is of greater magnitude than the molecular viscosity close to the sea bed. Thus the shear stress due to the molecular viscosity has been neglected in the equations of motion. Equation (6) can be simplified by substituting for S from (7) and introducing the defect velocity Rd = R - R~,

(10)

138

D. MYRHAUG

i.e. since R~ is independent of z, -Oz

8

Oz /

- i f R d =0

(11)

subject to the boundary conditions k =Zo-30

Rd=-R=atz

(12)

Rd'-'> 0 for Z---~ oo.

The boundary condition at the sea bed is taken in analogous form to that used for steady, unidirectional rough, fully turbulent flow based on laboratory experiments. If 8 is taken as a constant with the height above the sea bed the solution is the familiar Ekman spiral (EKMAN, 1905). However, it is more realistic to take the eddy viscosity to vary with z. In this study the following two-layer eddy viscosity model is proposed 8i - ½ - ½ ~:u,8 G o -

-

KU,8

-1

- ! --

2

forz~<8

(13)

for z > 8,

(14)

where K is von Karman's constant, 8 is the distance from the sea bed where the overlap takes place. Note that 8 in this context is an overlap height and that the boundary layer thickness will be greater than 6. The indexes i and o denotes inner and outer boundary layer, respectively. A Taylor expansion of (13) for small z/8 reduces to ei = ~ u , z , which automatically gives the correct logarithmic profile near the bed. Different values are assigned to von Karman's constant dependent on the problem which is considered. In laboratory experiments • = 0.40 or 0.41 is generally used, while atmospheric workers often use ~: = 0.35 (SouLSBY,1983). In this study ~ = 0.40 will be used. 8 is taken as 8 = c --u*

f

(15)

since u , / f i s a characteristic length of the thickness of the boundary layer. C is a constant which will be discussed later in the comparison with data. Then the eddy viscosity model will be discussed closer and compared with data. The choice of C determines the magnitude of the eddy viscosity in the outer layer, as well as the height of the overlap point. The proposed model for e seems to have a reasonable realistic behaviour, e reaches a maximum at z = 8 and keeps this constant value throughout the boundary layer. An even more realistic behaviour'would be to let e decrease to zero as z ~ ~ as the BUSINGER and ARYA (1974) and the LONG (1981) model did. BUSINGERand ARYA (1974) used an eddy viscosity of the form e Ku,ze -z/h, where h is the boundary layer scale height. This model was suggested for planetary boundary layer and was later used by LONG (1981) for unsteady flow. SOULSBY (1983) has discussed a solution based on the eddy viscosity distribution e = r~u,z. However, the present eddy viscosity model seems to be a reasonable compromise between accuracy and simplicity and has the benefit over =

M e a s u r e m e n t s of a turbulent boundary layer

139

the linear profile of having one disposable constant which admits for some adjustment of the model to data. The same type of eddy viscosity profile was used in a different context by JOHNS and DYKE (1971). Simple eddy viscosity models have often proved to predict the overall structure of measurements reasonably well. More refined turbulence modelling by using higher-order closure models is necessary to do a better prediction of the detailed turbulent structure. However, in order to do better modelling of turbulence one should also have access to high-quality data from detailed measurements of the turbulent structure as well as of the mean fluid velocities in idealized neutrally stable boundary layer flow.

Velocity and shear stress profiles The solution of equation (11) with the same eddy viscosity model as in (13) and (14) subject to the boundary condition in (12) was given in a different context in MYRHAUG (1982), and the details will not be.repeated here. In the present notation and by using equation (10) the solution is given by inner layer (z0 ~< z ~< 6):

(p - cos ~.,Oe~ (~) + e ~ (-~) ] R(~) = R~ 1

(p---c-~s ~-)P-~-(~ + P~ (----~o)]

(16)

outer layer (z > 6):

[ . . . . (P-C°Skn+l)P~(O) e-'l(l+i)(z-~)] R(z) = R~ 1 (p-cos )~n)Pz (~o) + Pz(-~o)

(17)

P)~ (~) is the Legendre function of first kind given in terms of the normalized variables z

=--1 5

and

~o-

zo

5

1

(18)

Z, p and 1] are given in equations (A1), (A2) and (A3), respectively, in the Appendix. Z, p and (T15) are all constants which depend on C/K only. In MYRHAUG(1982) is shown that the asymptotic expression for R(~) (equation 16) for small arguments, i.e. for ~ close to ~o, has a logarithmic behaviour close to the sea bed. The magnitude and the direction of the velocity vector, respectively, in the boundary layer are now given by IRI = (U 2 + V2)½= {[Re(R)] 2 + [Im(R)]2} '-'

qbn = arctg

= arctg

r'm( 'l LR--~J"

(19)

Here Re and Im represent the real and imaginary parts, respectively, of a complex number. By using the results in equations (5), (13), (14), (16) and (17) the shear stress is given by inner layer (Zo ~< z ~< 6): T = - -~ p~ u , (1 - ~2) Z (~; u , )

(20)

140

D. MYRHAUG

outer layer (z > 6): T = - ~ p~; u,Sr I (1 + i) Ra(z),

(21)

where Rd(z) is given according to equations (10) and (17). Z is a complex quantity given in equation (A4) in the Appendix and has dimension of velocity. The magnitude and the direction of the shear stress vector, respectively, in the boundary layer are now given by IT[ = (~z + Z2z)~ = {[Re(T)] 2 + [Im(T)]2) -~ qbr = arctg (~y__5t = arctg j i m ( T ) ] \ Zx~/ LRe(T)J"

(22)

The friction velocity at the sea bed has to be known to calculate the velocity according to equations (16) and (17) and the shear stress according to equations (20) and (21). By using equations (9), (20) and the first in (22) the friction velocity at the sea bed is given by u, = ½ ~: (1 - ~2) [Z(~0 ; u,) • 9~ (G0 ; u,)] ~,

(23)

where 2 denotes the complex conjugate of Z. Equation (23) is an implicit equation for determination of u,. For a given sea bed roughness length, geostrophic current velocity and Coriolis parameter, u, can be determined from equation (23) by iteration. A friction or drag coefficient associated with the friction velocity at the sea bed can be defined by

% CD --

(u,~2 --

(24)

where equation (9) has been used and ] R~ I is the magnitude of the geostrophic velocity. Figures 1--4 show an example of results of the model for the following condition: f = 0.0001 s-1, z0 = 0.3 cm (can be taken as representative for gravel) and IRoo [ = 30 cm s-1. The results are given for both C = 0.05 and C = 0.10 given by full and dotted lines, respectively. These two values of C were chosen based on comparison of the eddy viscosity profile with data from McPHEE and SMITH (1976) (see Fig. 9 and discussions in the next section). These sets of data give u, = 1.27 and 1.33 cm s-1, respectively. In this case u,k/v is about 850, i.e. the flow is in the rough turbulent regime according to (8). Figure 1 shows the magnitude and the direction of the velocity as functions of the height above the sea bed in the boundary layer according to equations (16), (17) and (19). The direction is relative to the geostrophic flow direction. The velocity vector rotates clockwise (in the northern hemisphere) with increasing distance from the sea bed. A slight overshoot in both magnitude and direction is seen in the model predictions. As seen from the figure the solution shows some sensitivity to various values of C. The rotation is larger for C -- 0.05 than for 0.10. With this eddy viscosity distribution the velocities just above the logarithmic layer are greater than the logarithmic value, whereas for a linear e-profile they are less. Figure 2 shows an alternative presentation of the results in Fig. 1 for C = 0.05. Figure 3 shows the magnitude and direction of the shear stress as functions of the height above the sea bed in the boundary layer according to equations (20)-(22). The

141

Measurements of a turbulent boundary layer

1:2I

10 2 ,

I

i

5 %%N

lO

'-\

10

6

5

5

N

o // LU

/

-

N

£

| t

o 5 -

5

m

L9 UJ "1-

w i

101

I0-' :

s!

10.2

II

i

5

I

I

10 15 20 25 30 MAGNITUDE OF VELOCITY,IRI (crn/s)

Fig. 1.

35

10-2 -5

L

I

II

5

10

15

210

25

DIRECTION OF VELOCITY. $Rldeg)

Magnitude and direction of the velocity. Example of results from model predictions for

f=O.OOOls-l,z.=O.3cm, lR~[=3Ocms-l:

C = 0.05;---

C = 0.10 in (15).

direction is relative to the geostrophic flow direction of the velocity vector. The direction of the shear stress is the same as the direction of the velocity at the sea bed, but the shear stress vector shows a much stronger rotation clockwise than the velocity vector. The figure shows a "constant-stress" layer up to about 1-2 m, and the direction is constant also. As for the velocity the shear stress shows some sensitivity to various values of C and the rotation is larger for C = 0.05 than for 0.10. Figure 4 shows an alternative presentation of the results in Fig. 3 for C = 0.05.

DIRECTION AT SEA BED . ° R = 193

GEOSTROPHIC FLOW I R~ I =30 cm/s

Fig. 2.

Alternative presentation of the results in Fig, 1 for C = 0.05.

142

D. MYRHAUG 10 2

10 2

\%\\ 5

5 \\\\

I0

--

\\\ \N% \N\

-6

10 5

5

5 IJJ 1 om <

5

5 mr

IJJ "1-

I-

10-I

10"~

5

5

10-2

I

I

I

0 0.5 1.0 1.5 2.0 MAGNITUDE OF SHEAR STRESS.ITI/p(crn21s 2)

Fig. 3.

10-300

i

I

I

I

I

L

I

-2/.0 -180 =120 -60 DIRECTION OF SHEAR STRESS,OT(deg}

0

30

Magnitude and direction of the shear stress. Example and symbols as in Fig. 1.

~ ~ . - - ~ /

Fig. 4.

SEA BED

Alternative presentation of the results in Fig. 3 for C = 0.05.

COMPARISON

WITH EXPERIMENTS

Now comparisons will be made with the measurements of the turbulent flow structure under drifting Arctic pack ice reported by MCPHEE and SMITH (1976). The main parameters and results from the experiment are given in Table 1. Two sets of measurements of three velocity components were made at 2, 4, 8, 12, 16, 20, 26 and 32 m below the pack ice. The geostrophic velocity given in Table 1 is measured in a reference system fixed to the moving pack ice with the x-axis in the direction of surface stress, the z-axis positive in the upwards direction and the y-axis determined as in a right-handed system.

143

Measurements of a turbulent boundary layer

Table 1.

Main parameters and results from MCPHEE and SMITH (1976)

Data Set No. Coriolis parameter Magnitude of geostrophic velocity Direction* of geostrophic velocity Direction of pack ice velocity Roughness parameter Friction velocity Drag coefficient Roughness Reynolds number Flow regime

f(s l) IR~ I (cm s-l) 0R~ (deg) (deg) k = 30zo (cm) u, (cm s 1) Co u,k/v

Rossby number

IR~llfzo

1

2

0.00014 23.62 96.7 73.0 6 1.0 1.79 × 10 -3 -600 Rough turbulent 8.44 X 105

0.00014 16.32 96.2 70.4 6 0.84 2.65 x 11) 3 -500 Rough turbulent 5.83 x 105

* The angle is measured clockwise from north.

The two data sets were generated by selecting time segments considered best for current measurements in terms of maximum speed and steadiness. Mean flow, turbulent energy and shear stress components were calculated for each of 20 min intervals during the selected time segments, and then these results were averaged to arrive at the final data sets. Data sets Nos 1 and 2 were based on 5 and 8 h of data, respectively. In the data the shear is calculated directly from the measured velocities, while the eddy viscosity concept is used to relate the shear stress to the velocity gradient in the present model. More details about the experiment and analysis of the data are given in McPHEE and SMIXH(1976). The Rossby number based on the independent variables IRo~ l, f and z0 and the roughness Reynolds number are given in the table. According to (8) the flow is in the rough turbulent regime. Comparisons of the model predictions with the data are presented in Figs 5-9. The results are shown for two values of C in equation (15). The full and dotted lines represent curves for C = 0.05 and C = 0.10, respectively. The crosses represent the data. The value C = 0.05 corresponds to the value ~ = 73 c m 2 s -1 calculated from the usual definition of the Ekman depth in the simple constant -e treatment, ~ foZ/2~ 2, with D = 32 m as the reference depth (McPHEE and SMITH, 1976), fitting reasonably well with the values of ~ for data set No. 1 at depths larger than 4 m (see Fig. 9). The value of C = 0.10 is chosen more or less arbitrary just to check the sensitivity in the solution for different values of C. According to equation (23) the friction velocity at the sea bed is given by u, = 1.02 and 0.73 cm s-1 for C = 0.05 for data sets Nos 1 and 2, respectively. The corresponding values for C = 0.10 are u, = 1.07 and 0.76 cm s-t, respectively. The model predictions seem to be in fair agreement with the values estimated from the data, see Table 1. Figures 5 and 6 show the magnitudes and the directions of the velocity vector as functions of the distance below the pack ice in the boundary layer according to equations (16), (17) and (19) for data sets Nos 1 and 2, respectively. The angle OR is given relative to north and is positive in clockwise direction. As seen from the figures the magnitude of the velocity vector is reasonably well predicted, while the direction of the velocity vector shows larger differences between the model predictions and the data. The data shows only little changes in direction the first 15 m. The total rotation of the velocity vector is underpredicted by about 10° for C = 0.05 compared with what the data shows. The reason for this difference between the measurements and the model predictions could be related to the influence of the large-scale topography of the underside of the ice (see McPHEE and SMITH, 1976). =

144

D. MYRHAUG MAGNITUDE OF VELOCITY,IRI 5 L

o r-

10

15

(crn/s)

OR(deg)

20

25 '

70 80 90 100

-s

I

"E -10 N

~- -15.

-15

xI

~-2o

o~ -20.

"

i

~ it

oJ

I

~ -25-

~

-25

,,

z

m

o -30

-30 Data set no 1

-35

-35

Fig. 5. Magnitude and direction of velocity, 0R is given relative to north and is positive in clockwise direction: x data set No. 1 (McPHEE and SMITH, 1976); - - - present theory with C = 0.05 and C = 0.10, respectively. OR (deg) 70 80 90 100

MAGNITUDE OF VELOCITY,IRI (cmls) 5 10 15 20 0

o 6

-5

\\

-10

.~ -10

-15

°uJ" -15

N

_o

N

~

-2O LU 130

-20

en

o= z

-25

E3

-30

-25

<[

to

5

-30

Doto set no 2

-35

Fig. 6.

-35

Magnitude and direction of velocity: xdata set No. 2 (MCPHEE and SMITH, 1976); as Fig. 5, i.e. - - - - present theory etc.

Figures 7 and 8 show the magnitudes and the directions of the shear stress vector as functions of the distance below the pack ice in the boundary layer according to equations (20)-(22) for data sets Nos 1 and 2, respectively. The angle 0T is defined as in Figs 5 and 6. As seen from the figures the model predictions are in reasonable good agreement with the magnitudes of the shear stress vector estimated from the measurements. The direction of the shear stress vector shows larger discrepancies between the model

Measurements of a turbulent boundary layer ITI/p [cm2/s z) 0

05

1.0

70

1.5

i

i

i

i

.....

"~ 4o

,

-15-

i

h

i

i

L

\\

-~ 4o /

--

DIRECTION OF SHEAR STRE$$,e T (deg) 130 160 190 220 250

100

-50~/~x/t////// l/ l/ X' -50 x

]45

~

i

i

i

i

[

~

i

280 i

J

J

•

\\\\

,.. 45

/

t

o~ -20

J,

~: -20 •

m

en

|j

z

-35 ] 1 1

-35

Fig. 7. Magnitude and direction of shear stress, 0T is given relative to north and is positive in clockwise direction: x data set No. 1 (MCPHEEand SMITH, 1976); - present theory with C = 0.05 and C = 0.10, respectively. ITI/p {cm2/s2) 0 05 1.0 0

-5

~

-

70

~

100 ~

I

I

DIRECTION OF SHEAR STRESS.e T (deg) 160 190 220 250 280

130 I

I

I

I

I

I

J

I

I

I

I

t

I

I

I

I

I

I

310 I

I

I

340 L

I

' -5

"~ -10

E -10

• -20

o~ -20

" \ \ ~

DetQset no2 -35

-35 ]

"',

Fig. 8. Magnitude and direction of shear stress: x data set No. 2 (McPHEEand S M I T H , . . . . present theory with C = 0.05 and C = 0.10, respectively.

1976);

p r e d i c t i o n s a n d the data. I n data set No. 1 the angle is almost c o n s t a n t from 12 to 20 m b e l o w the pack ice. I n data set No. 2 the total angle of r o t a t i o n is overpredict'ed c o m p a r e d with what the data shows. This m a y be associated with effects r e l a t e d to the t o p o g r a p h y of the u n d e r s i d e of the ice. H o w e v e r , this is n o t essential since the difference is most significant b e l o w 20 m w h e r e the m a g n i t u d e of the shear stress is very small. T h e gross b e h a v i o u r of the d i r e c t i o n of the s h e a r stress vector seems to be r e a s o n a b l y well

I

I

146

D. MYRHAUG

predicted in regions within the boundary layer where the magnitude of the shear stress is significantly different from zero. Figure 9 shows the eddy viscosity as function of the distance below the pack ice in the boundary layer according to equations (13) and (14). As mentioned above, the figure shows that with C = 0.05 the present eddy viscosity model is in fairly good agreement with the eddy viscosity estimated from the data by using peaks in spectra of vertical velocity (see McPHEE and SMITH, 1976). C = 0.10 gives too large values of the eddy viscosity compared with the data. Based on this result and the results in Figs 5-8 it seems reasonable to recommend to use C = 0.05 in equation (15). However, this does not seem to be essential since the model predictions were only slightly affected by a change in C from 0.05 to 0.10. The model should be calibrated with more data before a final conclusion is stated about this. Figure 10 shows the drag coefficient defined in equation (24) as a function of the Rossby number Ro = IR~ l/fzo for C = 0.05. The values estimated from data sets Nos 1 and 2 are also given, showing very good agreement with the model prediction for data set No. 1 and poorer agreement for data set No. 2. This suggests that Fig. 10 can be used to predict the drag coefficient at the sea bed for a given sea bed roughness length, geostrophic current velocity and Coriolis parameter. The Rossby number given in Fig. 10 is used to cover the range likely to be encountered in shelf seas. The drag coefficient given here is valid in the rough turbulent flow regime and therefore the criteria in (8) has to be kept in mind. MCPHEE and SMITH(1976) also give a detailed comparison between their measurements and the BUSINGERand ARYA (1974) model [which is the same as the LONG (1981) model]. The general impression from a comparison on a qualitative basis is that the overall predictions are equally well done by their model and the present model. The differences seem only to be marginal, However, the BUSINGERand ARYA (1974) model seems to give a somewhat thinner boundary layer than the present model. EDDY VISCOSITY, ~ (cm2/s) 0

50

100

150

200

i

I

I

l

-5

\\\ it

-10N C.) ~C O

-15-

x

-20-

-25 Z

O

-30D~tQ sef n o l

-35

Fig. 9.

K i n e m a t i c e d d y viscosity: x d a t a set No. 1 (McPHEE and SMITH, 1976); m o d e l w i t h C = 0.05 and C = 0.10, respectively.

, --- present

147

Measurements of a turbulent boundary layer

10 -2

J Z t~ ¢J

q

E

11ill

i

f

i

~

ill~l

I

~

i

i

IJTll

~

~ i I~1 5

I

I

I

I

I

l

I

ITq

10-3 5

K_ it

10-z'

, 105

~ t l ~ l 5

~ 106

~ ~ ~ 1 5

107

10 8

Llll

5

10g

ROSSBY N U M B E R , Ro=I R.ol I f z o

Fig. 10,

Drag coefficient as a function of Rossby number: x data sets Nos 1 and 2 (MCPHEE and SMITH, 1976); present theory with C = 0.05.

The comparisons of model predictions and experimental results in Figs 5-10 seem to indicate that the proposed eddy viscosity model predicts the observations reasonably well. At present it seems justified to use C = 0.05 in equation (15). CONCLUSIONS

In this paper an analytical theory describing the fluid motion in a turbulent boundary layer near a rough sea bed is presented. The flow is determined by a balance between the driving pressure gradient (water surface slope), the shear stress gradients and the Coriolis force. A two-layer eddy viscosity is used to model the shear stress. The eddy viscosity in the inner layer increases quadratically with the height above the sea bed. In the outer layer the eddy viscosity is taken as a constant. The present eddy viscosity model is a reasonable compromise between accuracy and simplicity and has the benefit over the linear profile of having one disposable constant which determines the magnitude of the eddy viscosity in the outer layer, as well as the height of the overlap point. This admits for some adjustment of the model to data. The vertical structure of the eddy viscosity and the velocity and shear stress profiles are presented together with the drag coefficient at the sea bed as a function of the Rossby number Ro = I R ~ I/fzo. Comparisons are also made with two sets of data obtained from measurements of the turbulent boundary layer under drifting Arctic pack ice. The solution shows some sensitivity to where the boundary between the two layers in the boundary layer is taken, but the results were only slightly affected. The vertical structure of the eddy viscosity was most influenced by this. Some of the differences between the measurements and the model predictions are probably related to the influence of the large-scale topography of the underside of the ice. However, there seems to be a fairly good agreement between the essential features of the experimental results and the predictions by this simple eddy viscosity model. Finally, there seems to be a lack of quantitative comparisons between different theoretical models. However, to do this in a proper way one should also have access to high-quality data from detailed measurements of the turbulent structure as well as of the

148

D. MYRHAUO

mean fluid velocities in idealized neutrally stable boundary layer flow. A data set like this will also be crucial in order to do more refined and improved turbulence modelling by using higher-order

closure models. REFERENCES

BUSlNGERJ. A. and S. P. S. ARYA (1974) Height of the mixed layer in the stably stratified planetary boundary layer. In: Advances in geophysics, Academic Press, New York, pp. 73-92. EKMANV. W. (1905) On the influence of the Earth's rotation on ocean currents. Ark.Mat., Astron. & Fys., 2, 1-53. ,IOHNSB. and P. DYKE (1971) On the determination of the structure of an off-shore tidal stream. Geophysical Journal of the Royal Astronomical Society, 23, 287-297. LONGC. E. (1981) A simple model for time-dependent stably stratified turbulent boundary layers. Department of Oceanography, Special Report 95, University of Seattle, Seattle, Washington, 170 pp. M(;PHEE M. G. and J. D. SMITH(1976) Measurements of the turbulent boundary layer under pack ice. Journal of Physical Oceanography, 6, 696-711. MYRHAUOD. (1982) On a theoretical model of rough turbulent wave boundary layers. Ocean Engineering, 9, 547-565. PRANDLED. (1982) The vertical structure of tidal currents. Geophysical and Astrophysical Fluid Dynamics, 22, 2%49. SOULSBYR. L. (1983) The bottom boundary layer of shelf seas. In: Physical oceanography of coastal and shelf seas, B. JOHNS, editor, Elsevier, New York, pp. 189-266. SVERDRUP H. U. (1927) Dynamic of tides on the North Siberian Shelf. Gcofys. Publ. Vol. 4, 75 pp. APPENDIX

k is given by

= - .', (~ - ~,) - i -13,

(A1)

where c~ = (_',(1 + (1 + 16132)'))~ 26f

13--

Kll.

p is given by p =

q ( l + i)8 (cos k n - l) + (cos ~,n + 1)q

n(1 + 06 + q

(A2)

where (A3)

1 ~,2sin~-/~ [l-" ( ~ ) F ( 1 - ~ ) ] q = 5 74~

2,

where F is the Gamma-function. ~( is given by ( p - c o s ~ . n ) P ~ ( ~ ) - P-~ (-~) Z(~; u,) = R~ (P _ cos ~n)Px (~l)) + Px (-~l))'

where P~.(x)

--

dPx (x)

k + l - ~ [P~+ ,(x) -xP~.(x)].

(A4)