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Derivation of Stokes drift for a deep-water random gravity wave field
Deep-Sea Research, 1971, Vol. 18, pp. 255 to 259. Pergamon Press. Printed in Great Britain.
SHORTER
CONTRIBUTION
Derivation of Stokes drift for a deep-water random gravity wave field NORDEN E. H U A N O *
(Received 28 July 1969; in revised form 8 July 1970; accepted 5 August 1970) Abstraet--A new method was used to derive the Stokes drift. Full non-linear terms were included in the equations which were transformed into integral equations by Fourier-Stieltjes representations. Solution was obtained by the method of successive approximations. To the second order, the result was similar to that of KENYON(1969). INTRODUCTION
THE PROPERTIESof wave induced mass transport have received increasing attention ever since STOKES (1847) calculated the second order mean velocity of a single train irrotational wave in an inviscid fluid. Though the works of LONCUFr-HIGGINS (1953), RUSSELLand Osomo (1957), and HUANO (1970), have modified Stokes' results considerably, especially for the shallow-water cases, Stokes' classical results still remain valid for deep-water waves. Recently, attempts have been made to extend the study of Stokes drift to a random wave field, for example, HOANG (1967), BYE (1967), CrtAN~ (1969), CHIN and PIERSON(1969), and KENYON (1969, 1970). As a result, the expression of Stokes drift has been obtained correctly, both by CHAN~ (1969), using one-dimensional Lagrangian equations of motion for deep water, and by KENYON(1969) using Stokes expansion in a two-dimensional wave field represented by the sum of an infinite number of discrete components. Subsequent applications of these expressions were made by CHIN and PIERSON(1969), and KInd,YON (1970). In the present note, a method similar to that used by PHILLIPS (1960, 1961) in the study of nonlinear random wave interactions is employed to derive the Stokes drift in a deep water random gravity wave field. The random wave field is represented by a Fourier-Stieltjes integral and the full non-linear terms in the kinematic and dynamical surface boundary condition equations are included. Through the Fourier-Stieltjes integral, the non-linear equations become integral equations which can be solved by successive approximations to any order desired. To the second order, the same result as that of K~r,rvoN (1969) is obtained. The concept of this approach is more general and the method used is more straightforward and simple in principle than that used by Kenyon. SPECIFICATION
FOR
THE
RANDOM
SEA
Consider a rectangular coordinate system with the z-axis vertically downward, and z = 0 at the equilibrium surface. Then, any point in this coordinate system can be denoted by a horizontal position vector, x, and z. Under the assumption of incompressible and inviscid fluid and small-amplitude waves, ~ lkl < 1 ; the motion can be approximated as irrotational and the governing equation of the motion everywhere below the free surface can be given by V2 ~ (x,z, t) = 0
(1)
where ~(x,z,t) is the velocity potential. If we furtber assumethat tbe random wave field is statistically stationary with respect to both space and time, and that there is no motion far below the surface (i.e. that the fluid is at rest when z -~ ~ ) , the solution to equation (1) is (x, z, t) = f f dA (k, n) e-[klz et(k.x -- n,) ,J k n
*Department of Geosciences, North Carolina State University, Raleigh, N.C. 27607, U.S.A.
255
(2)
256
Shorter Contributions
where dA (k, n) is a complex random function of the horizontal wave-number vector, k and frequency, n. The integration of equation (2) is to be taken over the whole wave-number and frequency space. From this expression the Eulerian velocity can be easily calculated from q (x, z, t) -~ V~ (x, z, t) /
1
tl
qd
k
n
where ea is a unit vector in the vertical direction. Under the same assumptions, the surface elevation (x, t) can be expressed as
t
~(x,t) = , ! k
-5
d B ( k , n ) e i ( k ' × m)
(4)
n
where dB (k, n) is another complex function. Since g (x, t) is real,
~(x,t)= f f dB(k,n)ei~kx "')= f f dB*(k,n)e i~kx- ~t) k
n
k
n
=ffdB*(--k,--n) ei(k'x--nt) k
(5)
n
where dB* (k, n) is the complex-conjugate of dB (k, n). It follows immediately that dB (k, n) -- dB* (-- k, - n).
(6)
From the definition of the spectrum, we also have dB(k,n) dB~kllnD = 0
......
and
dB (k,
n) dB* (ka, nl)
--
X
(k, n) dk dn
if
k v~ ka,
n # m
if
k = ka,
n = nl.
]
)
(7)
Here and hereafter the overbar denotes the ensemble average, and x (k, n) the three-dimensional wave-number frequency spectrum. In order to determine the velocity from this spectrum function, a relation between dA (k, n) and dB (k, n) is required. Using the kinematic surface boundary condition ~t
~
(v~)_~.v~
~
(8)
we have ..... k
n
k
+
ffff k
n
kl
"
n
k' kldA
(k,n)dB(kl, nDe!klgei[(k t kO.x
O, + n l ) t ]
(9)
Ill
In this expression, [k[ g is the surface slope of the waves. It is always small as required by the smallwave-amplitude assumption. After expanding elkl g in the Taylor series near z = 0, we can solve (9) by successive approximations. The first-order approximation reduces to in
dA (k, n) = -- ~-~ dB (k, n) and the second-order approximation becomes
(10)
257
Shorter Contributions
in
d/l (k, n) -- -- v~ dB (k, n) + i
f f k'(k--k0. ~_~l[tn--nOdB(k--kl, [k I
kl
n--nlldB(kt, nl).
(11)
nl
Using this latter relationship in (3), we can now relate the Eulerian velocity to the spectrum function. DETERMINATION
OF
THE
STOKES
DRIFT
The Stokes drift is the mean velocity following a fluid particle and hence, by definition, is a Lagrangian property• Let Q (a, c, t) be the Lagrangian velocity of a particle with the initial position a, c. The particle position at subsequent time will be t
X (a, c, t) + Z (a, c, t) ea = (a + cea) + f Q (a,
c, t') dt'.
(12)
0
Then, through a formal relationship between the Eulerian velocity q (x, z, t) and the I.~gr~mgian velocity Q (a, c, t), we have Q (a, c, t) = q [X (a, c, t), Z (a, c, t), t] t
0
=q(a,c,t)+ [/Q(a,c,t')dt']Vo.q(a,c,t)
(13)
0
expressed to the second order, where Vo is the del-opomtor with respect to the position variables at t = 0. Solving (13) for Q (a, c, t) to the second order, we obtain t
Q(a,c,t)=q(a,c,t)+[fq(a,c, tt)dtt]'Voq(a,c,t).
(14)
0
The combination of equations (3) and (14) now yields
Q (a, c, t) = f f n (ik -- ,k[ es) da (k, n) e - lk'c e ' o ' ' - "o k
n
fff('
+
k
n
kl
- ( - k• kx + n
Ikl Ikll) (ikl +
nl
(Ikl + IkxDc ei(kl, a--m t). [el(k. a --
• e-
nt)
__ elk.
[kll es) ~
(k, n) da (kl, 1/1)
a].
(15)
Further, if one substitutes (11) in (15), one obtains
Q(a,c,t) = f f n (~k~+ ie~)dB(k,n)e-lklcdO"a-nt) k
n
i~ = ~ k
n
kl
-- nil dB (k -- kx, n -- nl) dB (kl, nl)
nl
• e--Iklceitk'a--nt)
+ If f f ns(kl+ilklle)[1-k'k'1 I~Il~J dB ,
k
n
kl
nl
• e - (Ikl+lkxl)c {elr(k+kD. a--
ok, n)dn (kl,m)
(n+nDt]__el[(k+kD.a--nl t]}.
(16)
258
Shorter Contributions
Taking the average of (16) and using equations (6) and (7) to simplify the result, we obtain
2nkX(k,n) e-21klrdkdn.
Q(a,c,t) = k
(17)
n
This equation yields a general relation applicable to any fluid particle with a given initial position a, c.
However, if we used the dynamic surface-boundary condition with constant atmospheric pressure,
5
+ ½ (q2)_~ + g~ = 0.
(18)
Combining equation (18) with equations (2), (3), (4) and (11), we obtain the dispersive relation glk [=n z=~2
(19)
as a second-order approximation. This allows the rewriting of equation (17) as /
O (a, t, t) = / 2c~k~b(k) e - 2 Ik[ c dk
(20)
I/ k
where ~ (k) is the two-dimensional wave-number spectrum. Consider a single wave train with amplitude ~, wave number k0, and frequency ~r0. The spectrum function in this case reduces to ¢2 x(k, n) = ~- 8 (k -- k0) 8 (n -- or0)
(21)
where 8 (k -- k0) and 8 (n -- ~r0) are the Dirac delta functions. It then follows that Q (a, c, t) ~
-~ 15(k -- k0) ~ (n -- oo) • 2nk e 2 Ikl c dk dn k
n
= ¢2 o0 k0 e - 2 Ik01c
(22)
which is precisely the Stokes drift. DISCUSSION
The present result is similar to that of KENYON (1969), but may be applied to deep water waves only. For shallow water cases, the influence of the bottom friction and possibly of the surface contamination are much more complicated (HuANG 1970). In the light of the studies of LONGUETHIGGINS (1953), RUSSELL and OSORIO (1957) and HUANG (1970), it is doubtful that KENYON'S (1969) result could be applied to arbitrary depth. The approach used in this study can also be used in other problems of random wave study, especially in the cases in which higher order interactions may become important. A n example, is the derivation of the dispersive relationship in a random wave field. By using equations (2), (3), (4) and (9), a complete relationship could be obtained inchtding all possible contributions from interactions. This problem will be discussed in another paper.
Acknowledgements--This research was supported in part by the U.S. Atomic Energy Commission under contract AT (45-1)-1725 (Ref. RLO-1725-145) to the Department of Oceanography, University of Washington where the author held a post-doctoral research position. The author wishes to thank Professors C. A. Barnes and A. C. Duxbury for their criticism. REFERENCES
BYE J. A. T. (1967) The wave-drift current. J. mar. Res. 25, 95-102. CHANG M. S. (1969) Mass transport in deep-water, long-crested gravity waves. J. geophys. Res. 74, 1515-1536.
Shorter Contributions
259
CmN H. and W. J. PmRSOU fiR. (1969) Mass transport and thermal undulations caused by large and small scale variability in the Stokes mass transport due to random waves. Proc. Mil. Oceanogr. Syrup. Seattle, Washington, May 1969. HUANG N. E. (1967) Particle diffusion in a random gravity wave field. Ph.D. Thesis, John Hopkins University, 1967. HUANG N. E. (1970) Mass transport induced by wave motion. J. mar. Res. 28, 35-50. KENYON K. E. (1969) Stokes drift for random gravity waves. J. geophys. Res. 74, 6991-6994. KENYON K. E. (1970) Stokes transport. J. geophys Res. 75, 1133-1135. LONGUET-HIOGINSM. S. (1953) Mass transport in water waves. Phil. Trans. R. Soc. (A), 245, 535-581. PHILLIPS O. M. (1960) On the dynamics of unsteady gravity waves of finite amplitude. Part 1. J. fluid Mech. 9, 193-217. PHILLIPS O. i . (1961) On the dynamics of unsteady gravity waves of finite amplitude. Part 2. J. fluid Mech. 11, 143-155. RUSSELL R. C. H. and I. D. C. OSORIO (1957) All experimental investigation of drift profiles in a closed channel. Proc. 6th Conf. on Coastal Engr. Council on Wave Research, Univ. of Calif., Berkeley, 171-183. STOKESG. (~. (1847) On the theory of oscillatory waves. Trans. Camb. Phil. Soe. 8, 441--445.
SHORTER
CONTRIBUTION
Derivation of Stokes drift for a deep-water random gravity wave field NORDEN E. H U A N O *
(Received 28 July 1969; in revised form 8 July 1970; accepted 5 August 1970) Abstraet--A new method was used to derive the Stokes drift. Full non-linear terms were included in the equations which were transformed into integral equations by Fourier-Stieltjes representations. Solution was obtained by the method of successive approximations. To the second order, the result was similar to that of KENYON(1969). INTRODUCTION
THE PROPERTIESof wave induced mass transport have received increasing attention ever since STOKES (1847) calculated the second order mean velocity of a single train irrotational wave in an inviscid fluid. Though the works of LONCUFr-HIGGINS (1953), RUSSELLand Osomo (1957), and HUANO (1970), have modified Stokes' results considerably, especially for the shallow-water cases, Stokes' classical results still remain valid for deep-water waves. Recently, attempts have been made to extend the study of Stokes drift to a random wave field, for example, HOANG (1967), BYE (1967), CrtAN~ (1969), CHIN and PIERSON(1969), and KENYON (1969, 1970). As a result, the expression of Stokes drift has been obtained correctly, both by CHAN~ (1969), using one-dimensional Lagrangian equations of motion for deep water, and by KENYON(1969) using Stokes expansion in a two-dimensional wave field represented by the sum of an infinite number of discrete components. Subsequent applications of these expressions were made by CHIN and PIERSON(1969), and KInd,YON (1970). In the present note, a method similar to that used by PHILLIPS (1960, 1961) in the study of nonlinear random wave interactions is employed to derive the Stokes drift in a deep water random gravity wave field. The random wave field is represented by a Fourier-Stieltjes integral and the full non-linear terms in the kinematic and dynamical surface boundary condition equations are included. Through the Fourier-Stieltjes integral, the non-linear equations become integral equations which can be solved by successive approximations to any order desired. To the second order, the same result as that of K~r,rvoN (1969) is obtained. The concept of this approach is more general and the method used is more straightforward and simple in principle than that used by Kenyon. SPECIFICATION
FOR
THE
RANDOM
SEA
Consider a rectangular coordinate system with the z-axis vertically downward, and z = 0 at the equilibrium surface. Then, any point in this coordinate system can be denoted by a horizontal position vector, x, and z. Under the assumption of incompressible and inviscid fluid and small-amplitude waves, ~ lkl < 1 ; the motion can be approximated as irrotational and the governing equation of the motion everywhere below the free surface can be given by V2 ~ (x,z, t) = 0
(1)
where ~(x,z,t) is the velocity potential. If we furtber assumethat tbe random wave field is statistically stationary with respect to both space and time, and that there is no motion far below the surface (i.e. that the fluid is at rest when z -~ ~ ) , the solution to equation (1) is (x, z, t) = f f dA (k, n) e-[klz et(k.x -- n,) ,J k n
*Department of Geosciences, North Carolina State University, Raleigh, N.C. 27607, U.S.A.
255
(2)
256
Shorter Contributions
where dA (k, n) is a complex random function of the horizontal wave-number vector, k and frequency, n. The integration of equation (2) is to be taken over the whole wave-number and frequency space. From this expression the Eulerian velocity can be easily calculated from q (x, z, t) -~ V~ (x, z, t) /
1
tl
qd
k
n
where ea is a unit vector in the vertical direction. Under the same assumptions, the surface elevation (x, t) can be expressed as
t
~(x,t) = , ! k
-5
d B ( k , n ) e i ( k ' × m)
(4)
n
where dB (k, n) is another complex function. Since g (x, t) is real,
~(x,t)= f f dB(k,n)ei~kx "')= f f dB*(k,n)e i~kx- ~t) k
n
k
n
=ffdB*(--k,--n) ei(k'x--nt) k
(5)
n
where dB* (k, n) is the complex-conjugate of dB (k, n). It follows immediately that dB (k, n) -- dB* (-- k, - n).
(6)
From the definition of the spectrum, we also have dB(k,n) dB~kllnD = 0
......
and
dB (k,
n) dB* (ka, nl)
--
X
(k, n) dk dn
if
k v~ ka,
n # m
if
k = ka,
n = nl.
]
)
(7)
Here and hereafter the overbar denotes the ensemble average, and x (k, n) the three-dimensional wave-number frequency spectrum. In order to determine the velocity from this spectrum function, a relation between dA (k, n) and dB (k, n) is required. Using the kinematic surface boundary condition ~t
~
(v~)_~.v~
~
(8)
we have ..... k
n
k
+
ffff k
n
kl
"
n
k' kldA
(k,n)dB(kl, nDe!klgei[(k t kO.x
O, + n l ) t ]
(9)
Ill
In this expression, [k[ g is the surface slope of the waves. It is always small as required by the smallwave-amplitude assumption. After expanding elkl g in the Taylor series near z = 0, we can solve (9) by successive approximations. The first-order approximation reduces to in
dA (k, n) = -- ~-~ dB (k, n) and the second-order approximation becomes
(10)
257
Shorter Contributions
in
d/l (k, n) -- -- v~ dB (k, n) + i
f f k'(k--k0. ~_~l[tn--nOdB(k--kl, [k I
kl
n--nlldB(kt, nl).
(11)
nl
Using this latter relationship in (3), we can now relate the Eulerian velocity to the spectrum function. DETERMINATION
OF
THE
STOKES
DRIFT
The Stokes drift is the mean velocity following a fluid particle and hence, by definition, is a Lagrangian property• Let Q (a, c, t) be the Lagrangian velocity of a particle with the initial position a, c. The particle position at subsequent time will be t
X (a, c, t) + Z (a, c, t) ea = (a + cea) + f Q (a,
c, t') dt'.
(12)
0
Then, through a formal relationship between the Eulerian velocity q (x, z, t) and the I.~gr~mgian velocity Q (a, c, t), we have Q (a, c, t) = q [X (a, c, t), Z (a, c, t), t] t
0
=q(a,c,t)+ [/Q(a,c,t')dt']Vo.q(a,c,t)
(13)
0
expressed to the second order, where Vo is the del-opomtor with respect to the position variables at t = 0. Solving (13) for Q (a, c, t) to the second order, we obtain t
Q(a,c,t)=q(a,c,t)+[fq(a,c, tt)dtt]'Voq(a,c,t).
(14)
0
The combination of equations (3) and (14) now yields
Q (a, c, t) = f f n (ik -- ,k[ es) da (k, n) e - lk'c e ' o ' ' - "o k
n
fff('
+
k
n
kl
- ( - k• kx + n
Ikl Ikll) (ikl +
nl
(Ikl + IkxDc ei(kl, a--m t). [el(k. a --
• e-
nt)
__ elk.
[kll es) ~
(k, n) da (kl, 1/1)
a].
(15)
Further, if one substitutes (11) in (15), one obtains
Q(a,c,t) = f f n (~k~+ ie~)dB(k,n)e-lklcdO"a-nt) k
n
i~ = ~ k
n
kl
-- nil dB (k -- kx, n -- nl) dB (kl, nl)
nl
• e--Iklceitk'a--nt)
+ If f f ns(kl+ilklle)[1-k'k'1 I~Il~J dB ,
k
n
kl
nl
• e - (Ikl+lkxl)c {elr(k+kD. a--
ok, n)dn (kl,m)
(n+nDt]__el[(k+kD.a--nl t]}.
(16)
258
Shorter Contributions
Taking the average of (16) and using equations (6) and (7) to simplify the result, we obtain
2nkX(k,n) e-21klrdkdn.
Q(a,c,t) = k
(17)
n
This equation yields a general relation applicable to any fluid particle with a given initial position a, c.
However, if we used the dynamic surface-boundary condition with constant atmospheric pressure,
5
+ ½ (q2)_~ + g~ = 0.
(18)
Combining equation (18) with equations (2), (3), (4) and (11), we obtain the dispersive relation glk [=n z=~2
(19)
as a second-order approximation. This allows the rewriting of equation (17) as /
O (a, t, t) = / 2c~k~b(k) e - 2 Ik[ c dk
(20)
I/ k
where ~ (k) is the two-dimensional wave-number spectrum. Consider a single wave train with amplitude ~, wave number k0, and frequency ~r0. The spectrum function in this case reduces to ¢2 x(k, n) = ~- 8 (k -- k0) 8 (n -- or0)
(21)
where 8 (k -- k0) and 8 (n -- ~r0) are the Dirac delta functions. It then follows that Q (a, c, t) ~
-~ 15(k -- k0) ~ (n -- oo) • 2nk e 2 Ikl c dk dn k
n
= ¢2 o0 k0 e - 2 Ik01c
(22)
which is precisely the Stokes drift. DISCUSSION
The present result is similar to that of KENYON (1969), but may be applied to deep water waves only. For shallow water cases, the influence of the bottom friction and possibly of the surface contamination are much more complicated (HuANG 1970). In the light of the studies of LONGUETHIGGINS (1953), RUSSELL and OSORIO (1957) and HUANG (1970), it is doubtful that KENYON'S (1969) result could be applied to arbitrary depth. The approach used in this study can also be used in other problems of random wave study, especially in the cases in which higher order interactions may become important. A n example, is the derivation of the dispersive relationship in a random wave field. By using equations (2), (3), (4) and (9), a complete relationship could be obtained inchtding all possible contributions from interactions. This problem will be discussed in another paper.
Acknowledgements--This research was supported in part by the U.S. Atomic Energy Commission under contract AT (45-1)-1725 (Ref. RLO-1725-145) to the Department of Oceanography, University of Washington where the author held a post-doctoral research position. The author wishes to thank Professors C. A. Barnes and A. C. Duxbury for their criticism. REFERENCES
BYE J. A. T. (1967) The wave-drift current. J. mar. Res. 25, 95-102. CHANG M. S. (1969) Mass transport in deep-water, long-crested gravity waves. J. geophys. Res. 74, 1515-1536.
Shorter Contributions
259
CmN H. and W. J. PmRSOU fiR. (1969) Mass transport and thermal undulations caused by large and small scale variability in the Stokes mass transport due to random waves. Proc. Mil. Oceanogr. Syrup. Seattle, Washington, May 1969. HUANG N. E. (1967) Particle diffusion in a random gravity wave field. Ph.D. Thesis, John Hopkins University, 1967. HUANG N. E. (1970) Mass transport induced by wave motion. J. mar. Res. 28, 35-50. KENYON K. E. (1969) Stokes drift for random gravity waves. J. geophys. Res. 74, 6991-6994. KENYON K. E. (1970) Stokes transport. J. geophys Res. 75, 1133-1135. LONGUET-HIOGINSM. S. (1953) Mass transport in water waves. Phil. Trans. R. Soc. (A), 245, 535-581. PHILLIPS O. M. (1960) On the dynamics of unsteady gravity waves of finite amplitude. Part 1. J. fluid Mech. 9, 193-217. PHILLIPS O. i . (1961) On the dynamics of unsteady gravity waves of finite amplitude. Part 2. J. fluid Mech. 11, 143-155. RUSSELL R. C. H. and I. D. C. OSORIO (1957) All experimental investigation of drift profiles in a closed channel. Proc. 6th Conf. on Coastal Engr. Council on Wave Research, Univ. of Calif., Berkeley, 171-183. STOKESG. (~. (1847) On the theory of oscillatory waves. Trans. Camb. Phil. Soe. 8, 441--445.