Front Matter
 Report documentation page
 Title Page
 Table of Contents
 Theory and approximate expressions...
 Examples using known velocity...
 Expansion for weak vorticity
 Results for deep-water waves
 Comments on action flux conser...
 Appendix A: Action flux velocity...

Group Title: Technical report – University of Florida. Coastal and Oceanographic Engineering Program ; 77
Title: Surface waves on vertically sheared flows
Full Citation
Permanent Link: http://ufdc.ufl.edu/UF00075471/00001
 Material Information
Title: Surface waves on vertically sheared flows approximate dispersion relations
Series Title: UFLCOEL-TR
Physical Description: 56 p. : ill. ; 28 cm.
Language: English
Creator: Kirby, James Thornton
Chen, Tsung-Muh
University of Florida -- Coastal and Oceanographic Engineering Dept
Pacific Marine Environmental Laboratory (U.S.)
Publisher: Coastal and Oceanographic Engineering Dept.
Place of Publication: Gainesville Fla
Publication Date: 1988
Subject: Water waves   ( lcsh )
Genre: non-fiction   ( marcgt )
Bibliography: Bibliography: leaf 56.
Statement of Responsibility: by James T. Kirby and Tsung-Muh Chen.
General Note: "March 1988."
General Note: Work supported by National Oceanographic and Atmospheric Administration, Pacific Marine Laboratory, through contract 40ABNR67111.
Funding: Technical report (University of Florida. Coastal and Oceanographic Engineering Dept.) ;
 Record Information
Bibliographic ID: UF00075471
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved, Board of Trustees of the University of Florida
Resource Identifier: oclc - 19280479

Table of Contents
    Front Matter
        Front Matter 1
        Front Matter 2
    Report documentation page
    Title Page
        Page 1
    Table of Contents
        Page 2
        Page 3
        Page 4
        Page 5
    Theory and approximate expressions for the phase speed
        Page 6
        Page 7
        Page 8
        Page 9
        Page 10
        Page 11
        Page 12
    Examples using known velocity distributions
        Page 13
        Page 14
        Page 15
        Page 16
        Page 17
        Page 18
        Page 19
        Page 20
        Page 21
        Page 22
        Page 23
        Page 24
        Page 25
        Page 26
        Page 27
        Page 28
        Page 29
        Page 30
        Page 31
        Page 32
        Page 33
    Expansion for weak vorticity
        Page 34
        Page 35
    Results for deep-water waves
        Page 36
        Page 37
        Page 38
        Page 39
        Page 40
        Page 41
        Page 42
        Page 43
        Page 44
        Page 45
        Page 46
        Page 47
    Comments on action flux conservation
        Page 48
        Page 49
        Page 50
        Page 51
        Page 52
        Page 53
        Page 54
    Appendix A: Action flux velocity for linear shear current
        Page 55
        Page 56
Full Text




James T. Kirby
Tsung-Muh Chen

MARCH, 1988


Pacific Marine Environmental Laboratory
7600 Sand Point Way N.E.
Seattle, WA 98115


Gainesville, Florida 32611




James T. Kirby
Tsung-Muh Chen

MARCH, 1988


Pacific Marine Environmental Laboratory
7600 Sand Point Way N.E.
Seattle, WA 98115

1. Report No. 2. 3. Recipient's Accession No.

4. Title and Subtitle 5. Report Date
Surface Waves on Vertically Sheared Flows: March 1988
Approximate Dispersion Relations 6.

7. Author(s) 8. Performing Organization Report No.
James T. Kirby
Tsung-Muh Chen UFL/COEL-TR/077
9. Performing Organization Name and Address 10. Project/Task/Work Unit No.
Coastal and Oceanographic Engineering Department
University of Florida 11. Contract or Grant No.
336 Weil Hall 40ABNR6711
Gainesville, FL 32611 13. Type of Report
12. Sponsoring Organization Name and Address
U.S. Department of Commerce Final
National Oceanic and Atmospheric Administration
Pacific Marine Environmental Laboratory
7600 Sand Point Way N.E., Seattle, WA 98115 14.
15. Supplementary Notes

16. Abstract

Assuming linear wave theory for waves riding on a weak current of 0(E)

compared to the wave phase speed, an approximate dispersion relation is

developed to 0(E2) for arbitrary current U(z) in water of finite depth. The

0(e2) approximation is shown to be a significant improvement over the 0(s)
result, in comparison with numerical and analytic results. Various current
profiles in the full range of water depths are considered. Comments on

approximate action conservation and application to depth-averaged wave models
are included.

17. Originator's Key Words 18. Availability Statement

Linear wave theory
Perturbation methods
Wave-current interaction

19. U. S. Security Classif. of the Report 20. U. S. Security Classif. of This Page 21. No. of Pages 22. Price


Surface Waves on Vertically Sheared Flows:

Approximate Dispersion Relations


James T. Kirby and Tsung-Muh Chen

Coastal and Oceanographic Engineering Department

University of Florida, Gainesville, FL 32611

Work supported by the National Oceanographic and Atmospheric Administration,

Pacific Marine Environmental Laboratory, through Contract 40ABNR6711

March 1988



Abstract .................... .................................. 3

1 Introduction ..................... ............................ 4

2 Theory and Approximate Expressions for the Phase Speed.......... 6

2.1 Perturbation Method..................................... 7

2.2 Solutions to 0(E2) for Arbitrary U(z).................... 9

3 Examples Using Known Velocity Distributions................... 13

3.1 Linear Shear Current................................... 14

3.2 Cosine Profile......................................... 18

3.3 1/7-Power Law Profile.................................. 22

4 Expansion for Weak Vorticity................................. 34

5 Results for Deep-Water Waves.................................. 36

5.1 Comparison with Analytic Results........................ 37

5.2 Exponential and Linear Shear Profiles................... 45

6 Comments on Action Flux Conservation............................ 48

7 Conclusions ...................................................... 54


A Action Flux Velocity for Linear Shear Current.................. 55

References..................................................... .. 56


Assuming linear wave theory for waves riding on a weak current of 0(E)

compared to the wave phase speed, an approximate dispersion relation is

developed to 0(e2) for arbitrary current U(z) in water of finite depth. The

0(c2) approximation is shown to be a significant improvement over the 0(E)

result, in comparison with numerical and analytic results. Various current

profiles in the full range of water depths are considered. Comments on

approximate action conservation and application to depth-averaged wave models

are included.

1. Introduction

The problem of describing wave propagation through regions containing

tidal, ocean or discharge currents is fundamental in describing the nearshore

wave climate. Great strides have been made in extending wave propagation

models to include the effect of irrotational, large currents (assumed to be

uniform over water depth). However, currents typically do not possess so

simple a form, but instead have variations over depth and associated

vorticity, which renders the assumption of irrotationality invalid. The

resulting problem for wave motion on arbitrarily varying currents remains

unsolved, even in the linearized, uniform domain extreme.

The purpose of this study is to describe a perturbation method for the

special case of U(z)/c << 1, where c is the absolute phase speed, which allows

for the solution of the linearized problem for arbitrarily varying current

U(z). This case is restrictive in the general context of the study of wave-

current interaction, where U/c is taken to have no restriction on size.

However, in the context of coastal wave propagation, where we typically

consider waves propagating from a generation region into regions of varying

current, current fields of practical interest typically satisfy the scaling

restriction considered here except possibly in special situations such as

strong flows in inlets.

We proceed in section 2 by establishing the problem for a linear wave in

a uniform domain with arbitrary U(z). We then outline a perturbation

expansion based on small parameter e = 0(U/c) << 1, following the method

employed by Stewart and Joy (1974) for deep water. We then obtain solutions

to the general problem to 0(e) (reproducing the result of Skop (1987)) and to

0(e ). In section 3, we apply the method to linear, cosine, and 1/7 power

current variations, and compare results to analytically or numerically

obtained exact solutions.

The results of the analysis show that the solutions are valid in the

regime (maxlU-UI/U) << 1, where U is the depth-averaged current, leading to

the conjecture that the expansions are valid for arbitrarily large currents

having weak vorticity. In section 4, we outline an expansion for weak

vorticity and obtain the results for a linear shear current, proving

equivalence of the expansions for this restricted case.

In section 5, various results for the second-order approximation in the

deep-water limit are described.

In section 6, we end with some analysis of action flux formulations and

some cautionary notes on the direct use of the 0(e) depth-averaged velocity in

wave propagation models.

2. Theory and Approximate Expressions for the Phase Speed

We consider here the inviscid motion of a linear wave propagating on a

stream of velocity U(z), where water depth and current speed are assumed to be

uniform in the x-direction (Figure 1). Associated with the wave-induced

motion is a stream function of the form

4(x,z,t) = f(z)eik(x-ct)


After eliminating pressure from the Euler equations and linearized free-

surface boundary conditions, and using the continuity equation, the boundary

value problem for f(z) is given by the Rayleigh or inviscid Orr-Sommerfeld


[c-U(z)] (f" k2f) + U"(z) f = 0

; -h < z < 0

together with the boundary conditions

(U-c)2f' = [g + U'(U-c)] f

f = 0

; z= 0

; z = -h

Here, primes denote differentiation with respect to z, g is the gravitational

constant, h is the water depth in the absence of waves and k is the wavenumber

given by 2r/X, where A is the physical wavelength. This model has been used

in a number of studies of waves on arbitrary or particular current





Figure 1. Definition sketch

distributions; reference may be made to Peregrine

Jonsson (1983) for a review of existing results.

analysis is to obtain approximate expressions for

w = kc

(1976) or Peregrine and

The goal of the present

c in the dispersion relation


where w is the wave frequency with respect to a stationary observer. The

approximations are based on the assumption of U(z) arbitrary but IU(z) <
will thus assume the current to be weak and then evaluate the extent of this

restriction after obtaining the solutions.

2.1 Perturbation Method

A perturbation expansion for weak currents {U/c < 0(1)} is employed,

following the analysis by Stewart and Joy (1974) for deep water. We take

U(z) ~ eU(z)


where we introduce e as an apparent ordering and retain dimensional variables.

We then introduce the expansions

c = e n (2.7)

f = I nfn (2.8)

Equations (2.6-2.8) are substituted in (2.2-2.4) and the resulting expansions

are ordered in powers of e, which gives

f" k2f = F -h < z < 0 (2.9)
n n n

c 2f' gf = S z = 0 (2.10)

f = 0 z = -h (2.11)

where the F and S are inhomogeneous forcing terms involving information at
n n

lower order than n. For n>l, the homogeneous solution of (2.9-11) is the

lowest order solution fo(z), as derived below, and it is necessary to

construct a solvability condition according to the Fredholm alternative.

Using Green's formula on the quantities f0 and fn leads to the condition

Sf0 ndz = f0(0) S n > 1 (2.12)

This relation is used below to solve for the phase speed corrections cl and c2

due to the presence of a weak, arbitrary current profile.

2.2 Solutions to 0(c2) for Arbitrary U(z)

The perturbation problems obtained above are now solved in sequence.


We have

F0 = SO = 0


and the homogeneous solution (with amplitude arbitrarily taken as 1) is

f0(z) = sinh k(h+z)


2 = g tanh kh
CO k

This is the usual result for linear waves on a stationary domain.


At this order we have

FI = U''(z) f0(z)/c0 = U'sinh k(h+z)/c0

S1 = 2c0(U(0) c1) f'(0) coU'(0) f0(0)

= 2kc0(U(0) c ) cosh kh c U'(0) sinh kh

Substituting (2.16-17) in (2.12) gives





c 2k f U(z) cosh 2k(h+z)dz U (2.18)
1 sinh 2kh h

This is the finite-depth extension of the result of Stewart and Joy (1974),

who obtained the result

0 2kz
U = 2k f U(z)e dz (2.19)

in the limit kh+-. The result (2.18) has also been obtained by Skop (1987).

To 0(e), the dispersion relation is given by

c = cO + U (2.20)

or, equivalently,

w = a + kU (2.21)

where w is the absolute frequency in stationary coordinates and a is the

frequency relative to a frame moving with velocity U, the weighted-mean


a = gktanh kh (2.22)

The particular solution flp(z) may be obtained by the method of variation of

parameters; the entire solution fl(z) is then

fl(z) = {A1 ( U(S)sinh 2k(h+S)dS}sinh k(h+z)


+ {B + f- U''(E)[cosh 2k(h+E) -l]d(} cosh k(h+z) (2.23)
0 -h

where B1=0 in order to satisfy the bottom boundary condition and A1 may be set

to zero arbitrarily. Note that fl(z) is identically zero for a flow with

constant or zero vorticity (U''=0). For flows for which U'' is known, fl(z)

may be evaluated directly from (2.23). For general U(z), repeated integration

by parts is applied to (2.23) to obtain the expression

2kIl (z) 21 (z)
{(U(z) + U(-h)) 2k11 (z) 212(z)
f (z) = {+ } f (z) + fl(z) (2.24)
1c c 0 c 0


I (z) = f U(S)sinh 2k(h+d)dS (2.25)

2I(z) = f U(E)cosh 2k(h+S)dS (2.26)


We obtain

U"(z)f,(z) U"(z)(U-U)
F = co + 2 f0(z) (2.27)
0 0c

S = 2c (U(0) U)f'(O) c0U'(0)fl(0)

[2c0c2 + [U(0) ']2]f(0) + U'(0)[U(0) T]f0(0) (2.28)

Substituting (2.27-2.28) in (2.12) gives an expression for c2:


c.2 =U'(0)[U(0) N] [U(0) U]2
c 2g 2c02

+(U(0) U) f1(O) U'(0) f1(0)
c f'(0) 2c0 f(0)

+ 2 I U"'(z) {c0f0(z)fl(z) + (U(z) i)f02(z)}dz (2.29)
2gf0 (0) -h

where we have used the fact that f0(0)/f0(0) = c02/g. As above, several

options are possible here. For flows with zero or constant vorticity, we have

fl(z) = U"(z) = 0, and we obtain directly

= U'()[U() ]c0 [U(O) 2(2.30)
c2 (2.30)
2 2g 20

For flows where U'' is known, we may evaluate fl(z) from (2.23) and substitute

for U"(z) in (2.29) and then solve for c2. For general U(z) (especially for

tabulated U(z) where higher derivatives are not known), we use (2.24) in

(2.29) and integrate by parts to obtain the expression

c 2
c-2 = 2 [4kll(0) (1 + 2cosh 2kh)U]
0 2c0

k2 0
+ 2k /f U2(z)(I + 2cosh2k(h+z))dz
2gf0 (0) -h

2 1 [I2(z)I'(z) I (z)I'(z)]dz (2.31)
gf0 (0) -h 1 1

The remaining integrals are expressed completely in terms of U(z), and may not

be tractable for a general U(z); numerical approximation will then be


3. Examples Using Known Velocity Distributions

We now investigate the accuracy of the approximations derived above,

considering the results both to 0(e) and 0(s2). We consider three examples;

1.) Linear Shear Current

U(z) = Us (1 + a ) (3.1)

where a is the normalized constant vorticity w0 given by

a = 0h/Us (3.2)

2.) Cosine Profile

U(z) = Uscos (a ) (3.3)

3.) Power-Law Profile

U(z) = Us (1 + )1/7 (3.4)

These examples represent a hierarchy in increasing difficulty. For the linear

shear profile, analytic solutions may be obtained for both the full problem

and the perturbation solutions. (In fact, the perturbation may be carried to

any order with no difficulty, as will be seen below.) For the cosine profile,

analytic results may be obtained in the case of c=0 (i.e., a stationary

wave). For c*O, the problem must be solved numerically. In this case,

however, the approximate results U and c2 are obtained explicitly. For the

third case of a power-law profile, analytic results again have been obtained

only for the case c=0 (Lighthill, 1953). Fenton (1973) has provided a

numerical scheme for cases with cO# using a shooting method; this scheme is

utilized below. For the case of a power-law profile, evaluation of U and c2

also requires numerical approximation or approximation of the analytic result

by means of truncated series expansions.

3.1 Linear Shear Current

For the case of a current with uniform vorticity, the stability problem

posed in (2.2) (2.4) reduces to

f" k2f = 0

(Us- c)2 f' = [g + w0(Us- c)]f

f = 0

; -h z 0

; z= 0

S z= -h

The solution to (3.5) and (3.7) is simply

f(z) = sinh k(h+z)

Substituting (3.8) in (3.6) then gives directly

Watanh kh gtanh kh tanh kh 1/2
cE = Us 02k + kgtanh kh+ 4gk
E s --2k k 4gk

which is the exact solution which we will denote by subscript E. Turning to

the approximate solutions, we substitute (3.1) in (2.18) and obtain






w tanh kh
U = u (3.10)
s 2k

The exact solution may then be written as

2 12
0 tanh kh 2
E = + c {1 + gk (3.11)

where co is taken from (2.15). Evaluating the 0(e2) correction c2 from (2.30)


m tanh kh
c2 = c 8gk } (3.12)

and the phase speeds to 0(c) and 0(e2) are given by

c(e) = co + U ; 0(c) (3.13)

20 tanh kh
c(ec) = co {1 + 8gk + ; 0(2) (3.14)

Comparing (3.11), (3.13) and (3.14), it is apparent that the 0(s)
approximation is fairly weak unless normalized w02 is in some sense << 1. The

approximation to 0(e2) adds the first small term in the binomial expansion of

the square root appearing in the exact solution. Referring to (2.30), it is

apparent that c2 also provides the leading-order correction for non-zero

current shear.

The approximations obtained above are least accurate in shallow water.

To inspect this limit, we normalize the phase speeds by (gh) 1/2 and define

1/, tanh kh 2 2
F Us/(gh) 2 = kh = 1 + 0(kh)

Further, the weighted mean current U is given by

U= U -+ 0(kh)2 = U + 0(kh) (3.15)
s 2

where U is the unweighted, depth-averaged current. (U and U tend to converge

as kh+O for all velocity profiles as the wave motion loses its vertical

structure; this may be verified by inspection of (2.18).) Introducing a

defined in (3.2) and taking the limit kh+O, we may express the phase speeds

relative to i/(gh) 1/2 or U/(gh) /2 as

2F2 1/2
cRE = { a+ ; exact (3.16a)

cR = 1 ; 0() (3.16b)

cR 1 + ; 0(e)2 (3.16c)
R2 8

We may take a>0, where a=l reduces the velocity to zero at the bed and a>1

indicates reversed flow at the bed relative to the surface. Ignoring the

latter possibility, the worst case is for a=l, and we require F<<2 to employ

the truncated binomial expansion implied by (3.16c). The approximation to

0(e)2 is thus quite good for any subcritical flow. Plots of results for

various choices of a and F are given in Figure 2 for c' = 1.
We remark that higher-order approximations to the problem posed by (3.5)

- (3.7) may be obtained quite simply using a modification of the technique of

section 2. Substituting the expansions for c in (3.6) and using (3.8) gives





Figure 2.


0.00 0.20 0.40 0.60 080 100

Phase speed corrections (c U)/cO for long waves on a linear shear
current. exact solution; --- 0() approximation; _._. 0(E2)


00 00
(cUs I cn)2 ={g + 0[EU I E th kh
n=0 n=0

= 2 1 +--- I enc ]} (3.17)
0 g s n= n

The next several approximations beyond the level attained in section 2 are

2 2
1 O tanh kh2
C3 = = 8 { 4gk C0

2 3
m 2tanh kh
c5 = 0 c6 = 1 4gk } c (3.18)
5 = 6 16 4gk 0

c4 and c6 are consistent with the next two smaller terms in the binomial

expansion of (3.11), and it is apparent that the perturbation solution will

eventually converge to the exact solution when w02/gk is suitably small.

(Note that in the limit of m0 + 0, the solution to 0(e) is exact for all

current velocities, including stationary waves for which U/c = c.)

3.2 Cosine Profile

We now consider the cosine profile (3.3). Values of 0<4a
unidirectional flow, with a uniform velocity for a=0 and velocity reduced to

zero at the bed for a=r/2. Using the results of section 2.2, we obtain at


U + *) (3.19)
s (1 + 82)


S a sina (3.20)
S2kh sinh 2kh


The particular solution fl(z) is given by

fl(z) = s {B[sina + sin(z)] coshk(h+z)
c0(1 + B2)

2[cose + cos( -)] sinhk(h+z)} (3.21)

Also, the depth-mean velocity U is given by

U = U sina/a (3.22)

and U+U as kh+O for arbitrary a. At 0(e2), we obtain

U 2 2
2 = { 2 snkh + 08*cosa[ + 2 1
2 c +2 sinh 2kh + 2 2(1 + 02)2
1 + 8 + 4 2(1 + 8

( 8 2 cosh 2kh + [ + 22 + )2]} (3.23)
1 + 8 2 (1 + 48 ) 2(1 + 2 )

The asymptotic validity of the approximate solution in the limit of weak

vorticity (8,8* << 1) may be investigated by comparison with the exact

solution for the special case of c = 0 (stationary waves). Using this

condition in (2.2-3) and then solving (2.2-4) for the cosine profile (3.3)

leads to the results

f(z) = sinh[kh(l 482) 1/2 ] (3.24)


U2 = g tanh[kh(l 482) 1/2 ] (3.25)
k(1 482)1/2

Expanding (3.25) for the case 82 << 1 leads to the expression

U2 g tanhkh (1 288*) g tanhkh (1 + 2)2 (3.26)
= = + 0(s ) (3.26)
s k (1 22) k (1 + 88*)2

The second form of the right hand side is equivalent to the result of the

approximate solution to 0(e), given by

c + U = => 2 = c (3.27)

We see again that the approximate solution (for weak Us) converges to the

exact solution under the condition of weak vorticity with no restriction on

U as in the constant vorticity case. In particular, U/c + o for this case

and the perturbation method is seemingly inapplicable.

For cases where c#O, the full solution must be obtained numerically. We

have obtained solutions here using a modification of Fenton's (1973) shooting

method. Referring back to (2.2-4), we define a Froude number F according to

F = U(z) c (3.28)

(Note the difference from F in section 3.1.) We non-dimensionalize z

according to z' = z/h and define the variable transformation

q(z') = f(z)/(hf'(z)) (3.29)

The problem is then reduced to a Riccati equation

q' = 1 y22 ; 1 < z' < 0 (3.30)


2 (2 ( 42) 6cos(az') 1I (31)
S= (kh) { cos(z') 1 } (3.31)
S 4 ) cos(az') 1

6 = Us/c (3.32)

Equation (3.30) is solved over the interval 1 < z' < 0 with initial


q(-1) = 0 (3.33)

and with kh, a (and hence 6) and 6 specified. The surface boundary condition


T= q(0)/(6-1)2 (3.34)

which determines c. Us is then determined using (3.32). The numerical scheme

may be tested for accuracy against the long wave result (kh+0), which is

determined directly from the expression

1 = g f z (3.35)
-h (U(z) c)

(Burns, 1953) and gives

2 ta (1+6) tan()
c 6sina 2 -1
S- =- 22 3+ tan 1 ; 6<1 (3.36a)
gh a(1-62)(1-6cosa) a(1-623/2 )(1- 1/2
(i-_2) I/2


c2 6sin 1 (1+6) tan( (62-1) 2 (3.36b)
h= 2 -1 n 1 6>1
g a(1-6 2)(1-6cosa) (6 21)/2 (1+6) tan(!) + (6 2-1) /2

which is implicit in c and where 6 is given by (3.32). Five decimal place

accuracy was achieved straightforwardly in the numerical solution.

In Figure 3, we show the numerically determined exact dispersion relation

for the cosine profile for the case a=i/2 (velocity = 0 at bottom). The

Froude number chosen is based on the surface current speed. Consideration is

restricted to subcritical mean flow conditions in the long wave limit. In

Figure 4, we display the absolute value of the absolute error cn cE for the

long wave limit and a range of a values, with subscript n denoting the order

of approximation and cn representing the perturbation solution. Figures 4a

and 4b display results for the first and second order approximations,

respectively. The second order approximation is seen to provide an order of

magnitude reduction in error in comparison to the first-order approximation

(indeed, for the range of parameters considered it proves inconvenient to plot

the two results on equivalent scales).

Figure 5a and 5b show first and second order results, respectively, for

Icn CEl for the case of a = r/2 and a range of kh values. Here, the

comparison is between perturbation solutions and numerical solutions. The

dramatic increase in accuracy given by the second approximation is again


3.3 1/7-Power Law Profile

We now turn to the 1/7-power profile given by (3.4). This profile

differs from the previous examples both in analytic complexity and in the fact

that a weak-vorticity range is not available through choice of parameters, due

to the form of the profile near the bed. Using the results of section 2.2,

the expression U at 0(e) is given by


1.50 -5

1.00 -




-1.00 -0.50 0.00 0.50 1.00

Figure 3. Cosine profile. a=i/2, numerical results for dispersion relation.







-1.5 -1.0 -0.5 0.0 0.5 1.0


Figure 4. Absolute error Icn- CEI for cosine profile. kh=0.
a) n=l, first order approximation, b) n=2, second order


-1.0 -0.5 0.0 0.5 1.0 1.5


Figure 4.






00 -




.15 0.5


.10 1

.05 2


-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5


Figure 5. Absolute error jcn- CEI for cosine profile. a=n/2. a) n=l, first
order approximation. b) n=2 second order approximation.



.005-05 00 05 10
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5



Figure 5.

2kU h -/7 0
= sinh 2kh (h+z) cosh2k(h+z)dz (3.37)

This expression is of little direct use. Two integration by parts yields

SUs tanh kh 7 6 f t7 sinh kht7dt] (3.38)
S 14kh 7kh sinh 2kh

where t = (1 + z/h)1/7. This expression has been given previously by Hunt

(1955). The remaining integral in (3.38) vanishes in the limit kh+- but

contributes significantly at finite values of kh (see below). Approximation

is thus required in order to evaluate the integral. Numerical experiments

with quadrature indicated that 32 weighting points were required in order to

obtain three decimal place accuracy for kh = 1, with the number increasing for

increasing kh. For this reason, we chose to treat the integral by taking the

Taylor series expansion for cosh2k(h+z) about h+z = 0

co 2n 2n
cosh2k(h+z)= I (2k) ( (3.39)
n=0 (2

and then summing the resulting expression for the integral up to the required

number of terms. The resulting expression is given by

14kh U N 2n
S s (2kh)
U sinh 2kh (2n)!(14n + 8) (3.40)

Values of N required to obtain three-decimal place accuracy are given for

various kh in Table 1. The expansion procedure is most appropriate for

shallow water, with convergence being obtained more slowly as depth increases.

kh N

0.5 3

1.0 4

2.5 8

5.0 14

Table 1. Rate of convergence of U (3.38) for varying kh. N is number of
required terms in series expansion in order to obtain 3 place

As kh+0, we obtain the shallow water limit

U -- U = U (3.41)
8 s

Exact numerical results indicate that higher-order effects are not as

important in this case as in previous examples, and so the solution is not

carried to O(c2) here. This result is due to the weak vorticity of the

current profile near the surface; only the longest waves are strongly affected

by the current shear. Reference may be made to the results of Thomas (1981),

who conducted experiments on nearly deep-water waves over a turbulent shear

flow of nearly 1/7-power form, and found the waves to respond only to the

surface current speed.

Numerical results were obtained following Fenton (1973) and the procedure

outlined in section 3.2. Using 6 defined by (3.32) and q(z') defined by

(3.29), we obtain the problem (as in Fenton (1973))

q' = 1 y2(z')q2 1 < z < 0 (3.42)

q(-l) = 0 (3.43)

where now

, -13/7

L = (kh)L + oo z-J (3.44)
49(1 6(1 + z')1/7)

The surface boundary condition yields the relation

[=FO) q(0) (3.45)
[1 + F(O)F'(O)]

where F = (U-c)/(gh) 1/2. This result leads to the expression

c q(0) (3.46)
gh (6 1)[(6 1) 6q(0)/7]

Equations (3.42-43) are solved using specified values of kh and 6. We then

use (3.46) to determine c and then (3.32) to determine Us. Numerical results

were checked against plots given by Fenton and also against the long-wave

analytic result, given by

U 2 3 4
= 7[T + )+ (c + 2(-) + 5(c)
gh 2 U U
s s s s

5 c-U 4
+ 6( -) n( + c(- c )] (3.47)
s s s

Note that (3.47) corrects an error appearing in Fenton's unnumbered expression

in the logarithmic term.

Plots of normalized phase speed vs. normalized Us are given in Figure 6

as solid curves for kh values ranging from 0 to (long to short waves). Also

plotted in Figure 6 are results of the 0(e) perturbation solution given as

dashed lines. The perturbation solution is seen to be in close agreement with

the full solution except for kh small and IUs/(gh) /2 large, representing

long waves on strong currents.

r *"

kh O


C 1.00 -




-1.00 -0.50 0.00 0.50 1.00

Figure 6. Dispersion relation for 1/7 power profile. -- numerical results;
--- first-order approximation.

The expression (3.40) used to determine U was found to be useful at all

water depths tested but converges very slowly for large values of kh. An

alternate expansion procedure for (3.37) could be based on using the binomial


1/7 2 3
( + 1/ = 1 + 3 Z) 13 ( + ... (3.48)

Substituting (3.48) in (3.37) leads to an expression for U which

increases in accuracy as more terms in the expansion are retained. The first

several expressions obtained in this manner are given by

Two terms: U = Us(1 tnh (3.49a)

~ tanh kh 3 1 1
Three terms: U = l tanhkh + { 2kh}) (3.49b)
s 14kh 49kh sinh 2kh 2kh

Stanh kh 81 3
Four terms: = Us(l tanhkh 81 +
U 14kh 686kh sinh 2kh 98(kh)2

78tanh kh) (349)

These expressions represent an ascending series in powers of (kh)-1 as kh+-,

and convergence is rapid. As kh+0, however, all terms beyond the first two in

a truncated expansion cancel identically, and the value of U+0.929 U Note

that the third term in (3.38) contributes all the higher-order terms in the

expansion; however, the 0(1) contribution as kh+0 is not obtained in a

truncated expansion due to the fact that the expression (1 + z/h)1/7 is not

differentiable at z=-h. Convergence of the truncated series is thus limited

to a range excluding the neighborhood of kh=0.

Figure 7 shows several expressions for U. The solid curve represents the

series (3.40) with a sufficient number of terms retained to obtain

convergence. Each of the three truncated expansions (3.49a-c) are also

included, and show the increase in accuracy with each included term as well as

the failure of the series at kh=0. Finally, a plot of (3.40) truncated to 10

terms is included, and shows the extreme sensitivity of the series to the

number of retained terms as kh increases. It is noted also that the

expression for U given by (3.49a) has a maximum relative error of only 5.8% in

the shallow water limit.





0.90 *-

0.85 I I I
0 3 6 9 12 15

Figure 7. Various estimates of U for the 1/7 power profile. full series
(3.40). ... (3.49a), --- (3.49b), -- (3.49c) --- 10 term
expansion (3.40).

4. Expansion for Weak Vorticity

The results of sections 3.1 and 3.2 have indicated that the approximate

solutions for the regime of weak currents with arbitrary vorticity are

seemingly valid for the regime of arbitrarily strong currents having weak

vorticity. This result may be expected in hindsight due to the form of the

approximate solutions. In particular, the weighted mean current U deviates

from the true mean current U by an amount proportional to the first power of

some vorticity parameter, while the expressions for c2/c0 are proportional to

some (current parameter x vorticity parameter)2. Thus, in the limit of small

vorticity, U + U and c2/cO represents a consistently small correction for

arbitrarily large U values. In order to further support these results and

claims, we provide the schematic for an expansion for weak vorticity and

examine the case of a linear shear current.

We proceed by defining a reference current U* according to

U(z) = U* + U(z) (4.1)

where JU/U*| << 1 owing to the weak vorticity assumption. From the results of

section 3, a natural choice for U* would be U; however, U must be regarded as

undetermined in the present context since it was found for the case jU*/c <<

1, which doesn't hold here. We thus take U* = U. For the case of a linear

shear current, we obtain

0 h
SUs 2 ; U(z) = 0o(z + 2) (4.2)

from (3.1). (2.2-4) may then be solved to obtain

f(z) = sinh k(h+z)

subject to the condition

2( ^'tanh kh
(U + EC(O) c = [g + (0)( + ) c)] tan kh

where e << 1. Introducing the expansion

c = enc

and solving sequentially for the cn then gives (to 0(e2))

= (g tanh kh 1/2 +U

WOh w tanh kh
1 2 2k

g tanh kh
c2 = (g k

c = (g tanh kh 1/2

1/2 (tanh kh

2 tanh kh


m tanh kh
2 + 0(e )



Using (4.2) and (3.10), (4.7) may be written as

m tanh kh
c = (g tanh kh) 1/2 + } +
k 8gk


where U is given by (3.10). (4.8) is identical to (3.14), and the two

expansions are seen to be equivalent.






5. Results for Deep-Water Waves

Skop (1987) has investigated the application of the first-order

approximation of Stewart and Joy (1974) in the deep-water limit to several

cases for which analytic results exist, including the general case for depth-

limited current profiles of zero and constant vorticity (Taylor, 1955) and the

case of stationary waves on these profiles as well as cosine and exponential

profiles (tabulated by Peregrine and Smith, 1975). Skop has shown that the

first-order approximation provides generally good estimates of the wave

parameters. However, the second-order approximation for deep-water waves is

particularly simple in form, and we include it here for comparison.

The deep-water formulation follows from (2.2-4) by replacing the bottom

boundary condition with a boundedness condition on f; in the remainder of the

formulation, lower limits of integration become --. The results for the wave

phase speed become

c = co + cl + c2 (5.1)


c = (g/k) /2 (5.2)

0 2kz
I = U = 2k / U(z)ez dz (5.3)

k 2 2kz 2
c2 = U (z)e dz c1 /2c0 (5.4)
0O -

where cl was given by Stewart and Joy (1974) and c2 is new here. These

results may also be obtained by taking the appropriate limits of (2.15),

(2.18) and (2.31) directly.

5.1 Comparison with Analytic Results

We consider first the case of waves propagating on opposing depth limited

currents. The case of a uniform current is given by

; d < z 0

(z) = -Us
U ( z ) = 0


z < d

In order to facilitate comparison with

dimensionless variables according to

0 =-

(gd) 1/2

Skop's results, we introduce

K = kd

where Q is dimensionless frequency and S is an inverse Froude number. From

Taylor (1955), the exact solution gives the dispersion relation

[(nE + K)4 Sn2S2 K tanh K + (nE + K2 (nE2 S2) = 0

where subscript E denotes the exact value of n.

The first-order approximation is given by (Skop, 1987)

S1/2 S + (i-e2K
Q1 = K S + K(1-e )

while the second approximation is given by

1 K2 '-2/2 + 2-2 )
12 = /S + K(1-e -2)(1 + 2 e- K






Skop has presented plots of RE and S1 vs. K for a range of values of S. In

Figure 8, we provide plots of absolute error n E(n = 1,2) vs. K for a

range of S values. For S < 2, the trend of increased accuracy for the second-

order approximation breaks down, indicating divergence of the expansion for

the case of strong currents in deep water.

For the case of stationary waves on a current, n = 0 and we determine the

value of S(K). The exact results is given by

SE = (ktanh K) /2

and the two approximations are given by

SI = 1/2(1-e-2K)

S 2(K 1/2- S) /2
S2 2 11 +(1 S ) }




Each form is asymptotic to the limit S = K2 as K+0. The results here also

mirror problems with the expansion for strong currents, in that the term

inside the square root in (5.12) has a zero at positive K given by KCR =

- n(-) = 0.549, for which the corresponding exact stopping inverse Froude

number SE = 0.524. For K < K CR the second order solution cannot predict the

value of the stopping current. In contrast, the first-order solution predicts

the value reasonably well for the entire range of K, with increasing error as

K+0. Note that

Jim S_ = K (5.13)



0.0 5

-.005 /

0 123 //
E Dep //l
C: -.010 "

-.015 \\ /


-.020 \2/

0 1 2 3 4 5 6


Figure 8. Absolute frequency error nn E- E Depth limited, opposing uniform
current in deep water. -- O(E2) approximation (n=2); -- 0(c)
approximation (n=l). Curve labels are values of S.

tim S = 2K3/2 (5.14)

A plot of absolute error for Sn SE; n=1,2 is given in Figure 9. The

first-order approximation is essentially accurate for K > 2 (SE > 1.39) while

accuracy in the second-order approximation is deferred to K > 5 (SE > 2.24).

This range of validity is likely to still be representative of relevant field

conditions (note that for a surface current of speed 1 m/sec, S = 2.25 implies

a depth of flow of 51.7 cm; increased depth of flow further increases S and

strengthens the validity of the second-order expansion).

A second case for which analytic results are available is the case of a

linearly-sheared jet

Us( + d) d < z < 0

U(z) = (5.15)

0 z < -d

The exact dispersion relation (Taylor, 1955) is given by

(SE + K){1 e + E )[2 E + e2 1}

+ KS [2E + 1- e-2K = 0 (5.16)

and first and second approximations are given by

1 -2K
I = S K{I1 (1 2K) (5.17)

1 2K
Ki2 1 -4K e-2K}
9 = Q + K I-L (1 e )- e (5.18)
2 1 2S 4









2 4 6 8

Absolute error Sn SE for predicted stopping current. Opposing
depth limited current in deep water. -- O(2) approximation
(n=2); --- 0(e) approximation (n=l).

Figure 9.

Plots of absolute error E Si ; n = 1,2 are given in Figure 10 for 0 < K < 6

and 2.5 < S < 4. For this case, the improvement afforded by the second-order

approximation is dramatic. This result is most likely due to enhanced

representation of the effect of surface shear, as was noted in the results for

linear shear currents in section 3.

The prediction of stopping currents leads to the exact formula

SE = (Kcoth K 1) /2 (5.19)

and the approximations

S = 2(1 (1 (5.20)


1 + [1 2 1 -4K -2K /2(5.21)

The expression under the square root in (5.21) again has a zero at KCR

0.78633, which corresponds to a stopping current SE = 0.44506. The second-

order approximation gives no prediction of S below this value of KCR. Plots

of absolute error SE Sn are included in Figure 11. For this case, the error

in predicted stopping current obtained from the second-order approximation is

reduced essentially to zero for K > 5, which corresponds to S = 2.0001 from

(5.19). The approach of the first-order prediction to the exact solution is

deferred to much higher values of K and is not as qualitatively satisfactory,

possibly due again to inadequate representation of the effects of surface

current shear.



-.005 -

-.010 -

-.015 %' --
S\ \ 4 -O
\\ 4

-.020 3.5

dO .
-.025 \ -I

1 2.5
-.030 5 I
-.025 --- -

0 1 2 3 4 5 6


Figure 10. As in Figure 8 for depth limited, opposing linear shear current in
deep water.








0 2 4 6 8

Figure 11.

As in Figure 9 for depth limited opposing linear shear current in
deep water.

5.2 Exponential and Linear Shear Profiles

Wave-current interaction assumes a role of great importance in the theory

of generation of wind waves. Waves generated by wind action interact with a

wind-driven, sheared current profile with a thickness of the order of the

wavelength. Several recent studies have shown that the dispersion properties

of the initial wavelets are not strongly dependent on the form of the current

chosen as long as the current profile reproduces the value of the current and

shear existing at the surface. (See Gastel et al, 1985, for a recent


In this section, we compare the dispersion relation for an exponential

current profile

U(z) = Usez/d z < 0 (5.22)

to the dispersion relation for a depth limited profile having the same

velocity and surface shear, namely

U ( 1 + -) d 4 z < 0
s d)
U(z) = (5.23)

0 z < -d

These profiles have total mass flux rates differing by a factor of 2 but have

very similar structure close to the surface, where the linear profile neglects

terms of 0(z/d)2 in (5.22) where d is the e-folding length scale for decay of

the exponential profile. To the second order of approximation in the present

theory, the dispersion relations for the two profiles are given by

S= 2 2K 2 3/2 2K } (5.24)
exp (2K+1) S 2(K+1) (2K+
S1 -2K K'2 1 -4K -2K
n = S 2+ K (1 e ) + 2- {e (1 e e } (5.25)

where Us is defined positive for a following current (k>0) and where the

notation of the previous section is retained. Figure 12 shows a plot of 0 vs.

K for a range of S values. There is close agreement between the two

approximate dispersion relationships. This result suggests that a velocity

potential solution based on a depth-limited linear shear profile could be used

to some advantage in the study of initial wave growth, since three-dimensional

effects could be handled more simply than is possible when the analysis is

based on a stream function.

We remark that agreement between the first-order approximations for the

two profiles considered are also close, but that there is a general overall

deviation between the dispersion curves for the first and second

approximations, reflecting the reduced accuracy of the first-order

approximation. In reference to the discussion of the limited range of

validity of the second-order approximation, we consider the basic no-wave

state described in Figure 1 of Gastel et al. For this case, d is

approximately 5 mm with Us = 0.08 ms-1, yielding a value S = 2.77, which is

well up into the range of validity of the present approximations. Capillarity

is neglected here and would significantly alter the expressions (5.24-25) at

the length scale for this particular example.

Figure 12.

--8 4 4 8 12



Comparison of dispersion relations for exponential profile (5.22)
and linear profile (5.23) having equal speed and shear at
z=0 exponential; --- linear.

6. Comments on Action Flux Conservation

One of the chief applications for approximate dispersion relations for

wave-current interaction is in the construction of models for waves in slowly

varying domains. Such an application deserves a detailed analysis in its own

respect and will be the subject of further work, which in any case is

necessitated by the findings below; here, we can provide some initial results,

using the results of the 0(e) problem in the context of irrotational wave

theory. In particular, Skop (1987) suggests that the velocity U obtained in

section 2.2 may be used as the basis for the wave-current interaction in

propagation models, but provides no further analysis or support. Here, we

proceed using such an assumption and then analyze the results for the special

case of a linear shear current, using the results of Jonsson et al (1978) as

the basis for analytic comparisons.

We consider a linear wave riding on a flow of uniform-over-depth

velocity U, given by

S= Re{- iga cosh k(h+z) + dx (6.1)
2a -~ cosh kh -d (

n = Re{aei} 2- (6.2)


k= Vh = Pt (6.3)

subject to the dispersion relation

o = a + kU ; a = (gk tanh kh) 1/2 (6.4)

which is a simple extension of (2.21) to two dimensions. Following Kirby

(1984), (6.1) may be used as a trial function in a variational principle due

to Luke (1967), leading to a wave equation

2~ ~
D' D2 2
2 + (Vh D) Vh (CC Vh ) + (a kCC) = 0
Dt2 h Dt h gh
Dt cosh k(h)/cosh kh

= $ cosh k(h+z)/cosh kh

S c k

D _
S+ U*Vh
Dt t h




C --
g 3k

We allow U and h to have slow spatial derivatives and a (the amplitude) to

vary slowly in space and time. Taking a and I to be real functions allows

(6.5) to be reduced to an eikonal equation for and a transport equation

given by

( + Vh'9(
() + v ( (c + U)) = 0
a t a

where E is given by the simple expression

~ 1 2
E = pga



and where C = C k/k. The quantity E/o is an estimate of the wave action
~g g~
density, and (6.8) expresses the conservation of flux of wave action. The

question to be addressed is whether




is a proper estimate of action flux to the level of approximation considered


Jonsson et al (1978) give an exact expression for action flux on a linear

shear current in one direction, which we write here as


F =E
a a


a= o kU


C = Crs+ U = (kCrs) + Us


where Crs is the phase speed relative to the surface current, given by (3.9)


rs-n k ) 1/ ( 2 tanh kh 1/2
C = g anh k 1(1+ 4gk
rs k 4gk

0 tanh kh

Considering terms only to 0(e), it is apparent that

C + U = c + U + 0(s)
Crs s 0



f (C + U) =- C
crg ~ ga

and hence Cga, given by

C = {kcO + k = C + 0(E2) (6.16)
g 9k 0 g
a a

is the correct advection velocity to 0(e). However, note that

g -^ v -^ (6.17)
C + U + k (6.17)
g 3k 3k

is not equivalent to the simple estimate obtained from irrotational theory,

where U is entered simply as a local estimate of depth-uniform velocity, and

hence is not apparently a function of k. It is necessary to take this

dependence into account explicitly in arriving at the correct expression for

the group velocity in the absolute reference frame. Details of a comparison

of the expressions for Cga and Cga are given in Appendix A.

Turning to the expression for wave action, we may write the exact

expression for E for a linear shear current (from Jonsson et al) as

1 2 O rs.
E = pga (- 2g (6.18)


as = w kUs (6.19)

is the frequency relative to the surface current. To 0(e), we may then write

E as

c Wgh
E= (1-a--)- +0 (6.20)
s Jgh

The expression for wave action density is E/a which then gives

E (1 ) + 0 (0 (6.21)
a s Vgh

Examining as, we have

a = m kU = m kU k(U U)

w tanh kh
~ "0

w kc0
a g (6.22)

Factoring out a = kc0 then gives

s = l ( ) (6.23)
s 2g

Substituting (6.22) in (6.20) finally gives

+ 0 (0 (6.24)
a a gh

and we see that E/I is a proper estimate of action density to the required

order. The final expression for wave action flux is then given by

2 2
co 0cO o0h
= { (1 + G) + U + (1 G)} + 0 ) (6.25)
2 2g
where we have used (A1) and where G is defined in (A2)gh

where we have used (A.1) and where G is defined in (A.2).

It is apparent that the derivation of a wave propagation model based on

irrotational theory and using U as the local uniform-over-depth velocity does

not produce a consistent model at the order of the expansion considered. The

construction of proper wave equations or evolution equations depends on

further investigation of the full rotational problem in the context of a

slowly-varying, one- or two-dimensional (in plan) domain. Direct use of U as

a depth-averaged velocity in existing evolution equation models based on

irrotational theory will incur an error of 0(e) in action flux conservation,

thus rendering the models invalid over accumulated distances of 0(e- ).

However, the expression (6.25) (or alternate forms of (6.16) for non-constant

shear) may be used in eikonal-transport models for refraction calculations,

with consistency maintained up to 0(e2).

7. Conclusions

This study has provided approximate dispersion relations to 0(e2) for

waves propagating on weak currents U(z) = 0(sc). In contrast to approximate

results for deepwater, where 0(e) approximations are quite sufficient (Stewart

and Joy, 1974, Skop, 1987), the results here indicate that approximations to

0(e2) are required for any degree of accuracy to be obtained in finite water

depth, except for very weak current conditions or for cases where vorticity is

confined near the bed and waves are relatively short. The 0(e2) results

provide the next correction to the results of Skop (1987) and provide

significant improvements for cases where vorticity is distributed more or less

evenly over the depth. Additional analysis indicates that an expansion

procedure for arbitrarily strong currents with weak vorticity yields

equivalent results to the weak current case; this conjecture is proven here

only for the case of a linear shear current.

A consideration of the formulae for action flux resulting from the 0(e)

approximation and the exact solution for a linear shear current indicates that

the use of the 0(e) average velocity U as an estimate of depth-averaged

velocity in existing wave models incurs an error of 0(e) in the action flux,

rendering existing models invalid for length scales of 0(e ). Correction of

this problem awaits further research on rotational waves in slowly-varying


Appendix A: Action Flux Velocity for Linear Shear Current

Based on the results in (6.14) (6.17), we consider the equivalence of

the advection velocity Cga between the 0(e) solution and the expansion of

Jonsson et al's exact solution to that order. (Here, we take the viewpoint of

the large current, small vorticity expansion so that e = 0(o0h/(gh) /2)

<< 1.) Evaluating Cga from (6.17) gives

c (0tanh kh
g 2( + G)+U 2k G

co m0wtanh kh
(1 + G) + U + 2k (1 G) (A.1)


G = 2kh/sinh 2kh (A.2)

From Jonsson et al we have

C =a (kC ) + U
g 8k rs s

C [(1 + G) 0 ]
=- + U (A.3)
2 MCrs s
[1 rs]
[1 2g J
To the required order, we have
Cs = + U U = 2g (A.4)
rs 0 s 0 2g

Using (A.4) in (A.3) and retaining terms only to first order in MO gives back

(A.1), indicating the desired result

Wh 2
C =C + (- (A.5)
ga ga =gh


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