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UFL/COEL-TR/077
SURFACE WAVES ON VERTICALLY SHEARED
FLOWS: APPROXIMATE DISPERSION RELATIONS
by
James T. Kirby
Tsung-Muh Chen
MARCH, 1988
Sponsor:
NOAA
Pacific Marine Environmental Laboratory
7600 Sand Point Way N.E.
Seattle, WA 98115
COASTAl & OCEANOqRAphic ENCqiNEERiNq dEpARTMENT
UNIVERSITY OF FLORIDA
Gainesville, Florida 32611
UFL/COEL-TR/077
SURFACE WAVES ON VERTICALLY SHEARED
FLOWS: APPROXIMATE DISPERSION RELATIONS
by
James T. Kirby
Tsung-Muh Chen
MARCH, 1988
Sponsor:
NOAA
Pacific Marine Environmental Laboratory
7600 Sand Point Way N.E.
Seattle, WA 98115
REPORT DOCUMENTATION PAGE
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
UNCLASSIFIED UNCLASSIFIED 56
UFL/COEL-TR/077
Surface Waves on Vertically Sheared Flows:
Approximate Dispersion Relations
by
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
TABLE OF CONTENTS
SECTION PAGE
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
Appendix
A Action Flux Velocity for Linear Shear Current.................. 55
References..................................................... .. 56
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(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)
(2.1)
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
equation
[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
(2.2)
(2.3)
(2.4)
77(xt)
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
(2.5)
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)
(2.6)
where we introduce e as an apparent ordering and retain dimensional variables.
We then introduce the expansions
c = e n (2.7)
n=O
CO
f = I nfn (2.8)
n=O
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)
n
n=0,1,...
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)
-h
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.
0(1):
We have
F0 = SO = 0
(2.13)
and the homogeneous solution (with amplitude arbitrarily taken as 1) is
f0(z) = sinh k(h+z)
with
2 = g tanh kh
CO k
This is the usual result for linear waves on a stationary domain.
0():
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
(2.14)
(2.15)
(2.16)
(2.17)
c 2k f U(z) cosh 2k(h+z)dz U (2.18)
1 sinh 2kh h
-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
current:
~2
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)
S-h
+ {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
where
z
I (z) = f U(S)sinh 2k(h+d)dS (2.25)
-h
z
2I(z) = f U(E)cosh 2k(h+S)dS (2.26)
-h
0(e2):
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:
I
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)
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
0
k2 0
+ 2k /f U2(z)(I + 2cosh2k(h+z))dz
2gf0 (0) -h
2k3
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
required.
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
(3.5)
(3.6)
(3.7)
(3.8)
(3.9)
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)
gives
2
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)
2
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)
2
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
wh
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.
0
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
16
1.15
C-u
Co
Figure 2.
F=I.
0.00 0.20 0.40 0.60 080 100
o<
Phase speed corrections (c U)/cO for long waves on a linear shear
current. exact solution; --- 0() approximation; _._. 0(E2)
approximation.
---
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
n=0
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
0(E)
U + *) (3.19)
s (1 + 82)
where
S a sina (3.20)
S2kh sinh 2kh
I
The particular solution fl(z) is given by
U
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
2
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)
and
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)
where
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
condition
q(-1) = 0 (3.33)
and with kh, a (and hence 6) and 6 specified. The surface boundary condition
becomes
2
T= q(0)/(6-1)2 (3.34)
gh
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
0
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
and
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
apparent.
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
2.00
1.50 -5
1.00 -
0.50
0.00
-0.50
-1.00
-1.00 -0.50 0.00 0.50 1.00
Us
SFia
Figure 3. Cosine profile. a=i/2, numerical results for dispersion relation.
.25
.20
.15-
ICn-CEI
n=1
.10
.05-
.00
-1.5 -1.0 -0.5 0.0 0.5 1.0
Us
Figure 4. Absolute error Icn- CEI for cosine profile. kh=0.
a) n=l, first order approximation, b) n=2, second order
approximation.
1.5
-1.0 -0.5 0.0 0.5 1.0 1.5
Us
vgTT
Figure 4.
.08
.06
ICn- CEI
n=2
.04
.02-
00 -
-1.5
Continued
.25
.20-
kh=O
.15 0.5
Cn- CEI
n=1
.10 1
.05 2
3
.00
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
Us
Figure 5. Absolute error jcn- CEI for cosine profile. a=n/2. a) n=l, first
order approximation. b) n=2 second order approximation.
.08
ICn- CEI
n=2
.005-05 00 05 10
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
Us
ig
Continued
Figure 5.
2kU h -/7 0
= sinh 2kh (h+z) cosh2k(h+z)dz (3.37)
-h
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)
n=0
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
accuracy.
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
F(0)
[=FO) q(0) (3.45)
[1 + F(O)F'(O)]
where F = (U-c)/(gh) 1/2. This result leads to the expression
2
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 *"
2.00
kh O
1.50
0
C 1.00 -
0.50
0.00
-0.50
-1.00
-1.00 -0.50 0.00 0.50 1.00
Us
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
expansion
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)
2744(kh)
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.
1.00
0.95
U
SI I
/*I
u*i
0.90 *-
0.85 I I I
0 3 6 9 12 15
kh
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
n=O
and solving sequentially for the cn then gives (to 0(e2))
= (g tanh kh 1/2 +U
WOh w tanh kh
S2k
1 2 2k
g tanh kh
c2 = (g k
c = (g tanh kh 1/2
k
2
1/2 (tanh kh
8gk
2
2 tanh kh
+8gk
0oh
2
m tanh kh
2 + 0(e )
2k
(4.6c)
(4.7)
Using (4.2) and (3.10), (4.7) may be written as
2
m tanh kh
c = (g tanh kh) 1/2 + } +
k 8gk
(4.8)
where U is given by (3.10). (4.8) is identical to (3.14), and the two
expansions are seen to be equivalent.
(4.3)
(4.4)
(4.5)
(4.6a)
(4.6b)
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)
where
c = (g/k) /2 (5.2)
0 2kz
I = U = 2k / U(z)ez dz (5.3)
-00
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
0
z < d
In order to facilitate comparison with
dimensionless variables according to
kde
0 =-
U
s
(gd) 1/2
U
s
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
(5.7)
(5.8)
(5.9)
(5.6)
(5.5)
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 ) }
1
(5.10)
(5.11)
(5.12)
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)
K+O
.010
.005
0.0 5
-.005 /
0 123 //
E Dep //l
C: -.010 "
\\
-.015 \\ /
aI
-.020 \2/
-.025
0 1 2 3 4 5 6
K
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)
K+O
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
.010
.005
0.0
-.005
-.010
-.015
-.020
-.025
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
-2K
S = 2(1 (1 (5.20)
and
1 + [1 2 1 -4K -2K /2(5.21)
S1
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
2.5
0.0
-.005 -
-.010 -
-.015 %' --
S\ \ 4 -O
\\ 4
-.020 3.5
dO .
-.025 \ -I
1 2.5
-.030 5 I
S--e
-.025 --- -
0 1 2 3 4 5 6
K
Figure 10. As in Figure 8 for depth limited, opposing linear shear current in
deep water.
.050
.025
0.0
-.025
-.050
-.075
-.010
-.125
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
contribution.)
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+
(2K+1)
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
K
-8
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 (
2Z
n = Re{aei} 2- (6.2)
2g
where
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
(6.5)
(6.6)
(6.7)
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
(6.8)
(6.9)
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
where
and
(6.10)
is a proper estimate of action flux to the level of approximation considered
here.
Jonsson et al (1978) give an exact expression for action flux on a linear
shear current in one direction, which we write here as
(6.11)
F =E
a a
where
a= o kU
(6.12)
C = Crs+ U = (kCrs) + Us
(6.13)
where Crs is the phase speed relative to the surface current, given by (3.9)
as
rs-n k ) 1/ ( 2 tanh kh 1/2
C = g anh k 1(1+ 4gk
rs k 4gk
0 tanh kh
2k
Considering terms only to 0(e), it is apparent that
2
C + U = c + U + 0(s)
Crs s 0
(6.14)
(6.15)
S E E
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
a
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)
s
where
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
2
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
2
w kc0
a g (6.22)
2g
Factoring out a = kc0 then gives
0C0
s = l ( ) (6.23)
s 2g
Substituting (6.22) in (6.20) finally gives
2
+ 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
domains.
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
a
co m0wtanh kh
(1 + G) + U + 2k (1 G) (A.1)
where
G = 2kh/sinh 2kh (A.2)
From Jonsson et al we have
C =a (kC ) + U
g 8k rs s
tC
C [(1 + G) 0 ]
=- + U (A.3)
2 MCrs s
[1 rs]
[1 2g J
To the required order, we have
2
~0c0
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
~0
C =C + (- (A.5)
ga ga =gh
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