Role of low-frequency
water waves on momentum
transfers between atmosphere
and ocean in the near
shore region
O C E A N O L O G IA , 26, 1988
PL ISSN 0078-3234
Atm osphere-ocean interaction
M om entum transfer
Low-frequency wave
J a n u s z K r z y s c in
Institute of Geophysics,
Polish Academy of Sciences,
W arszawa
M anuscript received F ebruary' 19, 1987, in final form December 28, 1987.
A bstract
Using the G ent and Taylor model, the calculations of modifications of an interaction between
atm osphere and ocean near coastal shore region by low frequency ocean surface waves were
made. M odel equations consist of the wave kinem atics equations and the wave energy equation.
In the latter equation sources of energy (wind, surface w ater current) and energy dissipation by
sea bottom were param eterized. The model equations were used to find changes of the long
waves in the near shore region. T he m ethod of num erical calculations is based on the m ethod of
characteristics.
Results show that significant changes of the drag coefficient for m om entum are caused by the
waves propagating opposite to the local wind. W hen waves are generated by local wind
(direction of the long wave is along the wind) the drag coefficient is mainly connected with high
frequency surface waves. A type of the sea bottom m aterial and surface current modify
propagation of the long wave. These induce appriopriate changes of the drag coefficient in the
near shore region.
1. Introduction
T he sea-atm o sp h ere interface consists o f m oving irregularities o f various
frequency. W ith in th e high-frequency ran g e o f th e w ind g enerated waves
p hase velocity is considerable low er th a n w ind velocity at a sta n d a rd level of
10 m. W hen th e w ind begins to blow , these co m p o n en ts o f th e wavy surface
are g en erated im m ediately a n d can be found in higher stages o f w ind wave
developm ent. C ap illary waves a n d sh o rt gravity waves belong to this wave
category. W hen th e w ind d u ra tio n is long enough, fast long gravity waves
ap p ear. P h ase velocities of these waves are co m p arab le to th e w ind velocity
so surface irregularities ca n n o t effectively slow dow n w ind velocity near
surface, as high frequency, nearly statio n ary , elem ents do. O n e can intuitively
say th a t am o n g various w ave com ponents, observed in developing surface
w ater waves, m ainly w aves from a high-frequency b a n d are responsible for
th e effective roughness of th e free surface of th e sea. An air flow over
ro u g h n ess elem ents is m odified by long surface gravity waves so these
frequency b an d s can play a specific role in the air-sea m o m en tu m transfer.
T his role is en h anced in processes w hich lead to flow sep aratio n , a n d during
swell p ro p a g a tio n against wind.
M u n k (1955) first recognized an im p o rtan ce o f th e high-frequency b an d
for sea-air m o m en tu m exchange processes. H e used w ater waves energy
density sp ectrum S(a>) to characterize ra n d o m w ave field. T he sta n d a rd
d ev iatio n er0 o f the ocean surface displacem ent is related to S(a>) as follow:
00
Co = f S(<y)dfc>.
o
(1)
M u n k suggested th a t waves steepness is an im p o rta n t p aram eter c h a ra c te ri­
zing th e in tera ctio n betw een atm o sp h ere an d ocean. F o r th e w ave o f an
am p litu d e a an d w ave nu m b er k steepness is equal to a - k . M ean steepness of
th e w avy surface V o is calculated as a sq u are ro o t from integral of wave
elem entary steepnesses a - k raised to th e second pow er. If we replace a 2 by
w ave energy sp ectru m we have finally:
(Vcr)2 = J S (co) 2 dco = g ~ 2 jS(a>)ci)4 dco,
o
o
(2)
w here:
co —wave frequency,
g —acceleration due to gravity,
k — w ave num ber.
In second in teg ral o n r.h.s. d ispersion relatio n for deep w ater zo n e is used.
H igh-frequency sp ectrum S(a>) can be p aram eterized as a pow er function
o) - p, w here p < 4, so th a t th e m ain c o n trib u tio n to th e integral com es from
higher frequencies. F ield ob serv atio n s gave for instance p = 3.7 at a n early
stage of w ave developing (D ruet, Siwecki, 1984) a n d p = 3.0 in the area, of
sm all relative d e p th (M assel, D ru e t, 1980).
In different regions o f th e sea (deep w ater a rea o r shallow w ater zone)
an d stages of th e w ave developm ent energy density sp ectrum S(a>) is ex p res­
sed by specific form ulas so th e sh o rt gravity waves m ay act differently in
these situ ations.
N u m ero u s field an d th eo retical investigations have been carried o u t to
find how w avy surface m odifies a stru c tu re o f th e atm o sp h ere ad jacen t to the
w ater.
K itaig o ro d sk ii (1970) o b tain ed a n im p o rta n t result. H e found th a t the
log arith m ic profile o f th e w ind, ie:
U(z)
x
-In ( - ) ,
\ z 0J
(3)
where:
«„.—friction velocity,
x —K a rm a n ’s co n stan t,
z0 —ro u g h n ess p aram eter,
is a g o o d a p p ro x im a tio n for th e w ind blow ing over u n d u latin g sea.
T h e ro u g h n ess p ara m e te r z 0 in this form ula depends o n co n d itio n s u p o n
wavy surface. A ccording to K itaig o ro d sk ii (1970) z0 is determ ined m ainly by
th e high-frequency ban d . Special form ulas for different stages of w ave devel­
o p m en t an d aero d y n am ic stru c tu re o f the surface (rough or sm ooth) are
presented. These form ulas require field o b serv atio n to find a dim ensionless
fu n ctio n dep en d ing o n param eters connected w ith low a n d high-frequency
co m p o n en t o f th e surface irregularities.
C o n clu sio n s from field o b serv atio n s carried o u t by different a u th o rs w ere not
th e sam e. K u zn etso v (1978) found a re la tio n sh ip betw een th e friction velocity
an d th e ro o t m ean sq u are d ev iatio n o f the high-frequency irregularities, but
H asse et al (1978) p o in ted o u t th e dependence u* o n global energy, m ainly
d eterm in ed by long surface wave.
A th eo retical m odel p ro p o se d by G e n t a n d T ay lo r (1978) clarified the role
of low - a n d high-frequency b an d o f th e surface w ave sp ectrum in m o m en tu m
transfer. In this m odel th e a u th o rs calculated a ch aracteristic of th e flow over
rough, long, surface gravity waves. R oughness o f th e surface w as caused by
sm all irreg u larities (high-frequency waves) allocated along basic low -frequency wave. A p ara m e te r z0 w as p ro p o rtio n a l to th e m ean height o f the sm all
irregularities. In te ra c tio n betw een atm o sp h ere an d ocean the a u th o rs descri­
bed using a d rag coefficient C l 0 = ( u J U \ 0).
Som e o f th e results derived by G e n t a n d T ay lo r are presented in F igures 1
an d 2. F ro m these pictures one can d raw the follow ing conclusions:
—exchange processes betw een sea a n d air are intensified w hen direction
of th e w ave p ro p a g a tio n is o p p o site to th e near surface wind,
—steepness o f the long w ave (a- k) a n d ra tio o f its phase velocity to the
friction velocity (cr/ u *) are im p o rta n t param eters governing air-sea m o m en ­
tu m transfer,
—d efo rm atio n of th e air flow by long surface w ave plays a little role in
exchange process —curve Cs (z0, a - k ) in F igure 2.
—w hen long w ave is a w ind driven wave, m odel using only p aram eter z 0
to d eterm in e C 10 gives a g o o d accuracy.
A ccording to these conclusions th e d ra g coefficient in a region o f variable
p aram eters o f long low -frequency waves can also vary. A specific variability
of th e air-sea exchange processes in a n ear surface zone is expected, w here
p aram eters o f th e long w aves change rapidly. V ariability o f th e w ave
p aram eters a n d d ra g coefficient in this zo n e will be a subject o f th e present
paper.
ak
Fig. 1. V ariations of the drag coefficient with wave slope for z0 = 0.005 cm (after G ent,
Taylor, 1978)
Fig. 2. V ariation in C 10, Cs.(z0, 0) (part of the drag coefficient C ,0 caused by small scale
irregularities when gravity waves are absent), and Cs(z0, ak) (part of the drag coefficient caused
by small scale irregularities superim posed on long gravity waves) with Cj-/ut (after G ent, Taylor,
1978)
2. Basic assumptions and equations of the model
A ccording to G e n t a n d T aylor, th e drag coefficient depends o n friction
velocity and p aram eters o f th e long wave, ie w ave-length ( / o r w ave num ber
k), its am p litu d e (a) a n d g ro u p velocity (cg). A co n stan t roughness p aram eter
z 0 is p ro p o rtio n a l to th e sq u are ro o t of the energy o f high-frequency
co m p o n en ts o f wavy surface.
In th e m odel presented th e results derived by G e n t an d T ay lo r are
a d o p ted to th e shallow w ater zone. O th e r p aram eters to express precisely
variability o f a n ear shore region are also included. T he changes in long
w ave p aram eters are d u e to variatio n s o f surface cu rren t, w ater depth, and
energy d issip atio n by b o tto m in tera ctio n effects (dissipation by friction,
perco latio n , an d b o tto m m otion). P ro p a g a tio n o f long surface waves is
d eterm in ed by w ave k inem atics e q u a tio n in the form :
- k + V(a) + k - V p) = 0,
ot
(4)
w here k is w ave n u m b er vector (|/c| = k), Vp is velocity o f th e surface w ater
cu rren t.
E q u a tio n (4) describes refraction o f th e long surface waves. U sing dispersion
re latio n a>2 = gk ta n h (kd) — w here d is w ater d ep th —an d wave g ro u p
d efinition cg = V k a>, after som e re arran g em en ts we have:
d
d
8
d
, 8
8
^ K + (cgx+ Up) — kx + (cgy+ Vp) — k x + k x
Up + ky
Vp + —
8 , „
d - 0,
oj —
0
^
0
3
&
3
3
— k y + (cgx + Up) — ky 4-(cgy + Vp)-fy,k y + k x-fy,u p + k y-fy, Vi, + ~dd(°~dy(i = ° '
(5)
W ave energy (E) is defined as a sum of po ten tial an d kinetic energy o f all
w ater wavy fluctu ations (o rb ital m o tio n o f elem entary particles in the water)
below an d o n th e surface. F o r infinitesim al waves, p o ten tial th eo ry o f the
surface waves gives E = 0.5 gwg <a2 > either for deep w ater region o r shallow
w ater zone, w here gw is a w ater density, <a2 > is a m ean sq u are height of
surface irregularities. W ave energy e q u a tio n in th e form first given by
L o nguet-H iggins an d S tew art (1961) is:
^ E + - ^ ( c , x + Ur) E + - ^ ( c „ + V r) E = Z - D ,
(6)
w here Z an d D d en o te sources a n d sinks o f energy of w ater wavy fluctua­
tions.
E nergy tran sferred to th e w aves as a result o f w ind actio n in a unit o f tim e is
ch aracterized by a dim ensionless coefficient a,
T dE
(7)
n E dr ’
w here T is w ave period o f a d o m in an t w ave in a w ave energy spectrum .
A ccording to G e n t (1977) a can be p aram eterized as:
« =
—*— G en ( R , oc/c),
QwQ
(8)
w here:
qx — air density,
G en —ta b u la te d function,
R
=
— l n ( / c z 0)·
U sing eq u a tio n s (7) a n d (8) final expression o f energy in p u t from the w ind is
given by form ula:
Zwind= 0.5— ( —
Qw \ c f J
a>Vdnh(kd)Gen(R, oik)E.
(9)
In o rd e r to find values of G e n in in term ed iate points we used a linear
in terp o latio n betw een th e values given by G e n t (1977). An angle betw een
d irectio n o f long waves p ro p a g a tio n a n d w ind is not co n stan t. T his effect is
caused either by w ind variability o r wave refraction, so in e q u a tio n (9) u*
rep resen ts projection o f surface R eynolds stress (in the air) o n directio n of
w ave p ro p a g atio n .
V alues o f Z winJ need not be positive. In th e case Z wind< 0 wave energy
d im inishes as a result of w ind-w ave in teractio n* . T his situ a tio n is m et w hen
swell p ro p a g ates o p p o site to the local w ind o r d u rin g a b ru p t changes of
m eteo ro lo gical fields (waves ca n n o t a d a p t im m ediately to th e local wind).
D issip atio n m echanism s o f th e wave energy in a near sh o re region are
con n ected w ith b o tto m friction, p erco latio n w ithin the san d layer, a n d wave
m o tio n in th e m u d layer induced by hyd ro d y n am ic forces acting at th e m ud
line. A decay ra te induced by these m echanism s depends strongly on the type
o f b o tto m sedim ents a n d g ra in diam eter. P erc o la tio n is m ost effective in
coarse sand (m ean d iam eter > 0.5 mm). In fine san d (m ean d iam eter < 0.4
mm) b o tto m friction becom es m ore im p o rta n t th a n percolation. B ottom
friction d epends strongly on th e presence o f stable san d ripples on the
b o tto m . W hen th e b o tto m is com po sed o f silt, clay o r soft org an ic m atter,
b o tto m m o tio n becom es a d o m in an t dissip atio n m echanism (Shem din et al,
1978). In the m odel presented one can param eterize th e p erco latio n effect as
follows:
ykE
D = —
c o s h 2 (kd)'
w here y is a coefficient of perm eability.
F o r th e co arse sands an d fine san d s y is equal to 1.15 cm /s a n d 0.124 cm /s,
respectively (Shem din et al, 1978).
W hen o cean waves en co u n ter a cu rren t in w hich the surface velocity
varies, an in terch an g e energy betw een w ave a n d cu rren t appears. A ra te oi'
w o rk in g by surface w ave m o tio n against th e m ean ra te of the cu rren t shear
we param eterize by a form ula given by L onguet-H iggins a n d S tew art (1961):
'
'c u r r e n t
= _ e
‘J a/i
___ y
( 11)
pP*
w here VpP is th e /? co m p o n en t o f th e ocean cu rren t velocity (a = x , y ;
= x , y), S^p is a ra d ia tio n stress eq u al to:
S l. Î ^ . £ +
cf k k
o
Æ
y cf
i
M
( 12)
w here: 5xp —K ro n e ck er’s delta.
F o rm u la (11) neglected a p art of th e global energy included in high-frequency
b a n d o f w avy flu ctuations in th e w ater. A ccording to Phillips (1961) this
assu m p tio n is acceptable except for ocean regions w ith intensive wave
breaking.
P ro p a g a tio n s o f surface waves are exam ined in th e region depicted in
F ig u re 3. W e sta rt o u r calcu latio n in the region w here w ater d ep th is equal
to half o f th e w avelength. V alues w ith index zero refer to th e b o u n d a ry of
shallow w ater zone. W ave p aram eters at this point, ie g ro u p velocity and
w avelength (or w ave num ber), define a scale of velocity an d length, respecti­
vely. W e assum e th a t all p aram eters describing a near shore region (bottom
to p o g rap h y , surface current) a n d surface wave param eters do no t vary along
an axis p arallel to th e shore (axis y in o u r m odel) an d depend only on the
d istan ce from th e shore (variable x in o u r model).
W e assum e th a t in a n ear shore region w ater d ep th dim inishes according
to a fo rm ula: d(x) = d 0 e x p ( —Bx), w here B characterizes a steepness of the
b o tto m to p o g ra p h y a n d the velocity o f w ater cu rren t has a p arab o lic profile
w ith a m axim um value at th e b o u n d a ry x = 0.
W e sto p o u r calcu latio n s at the p o in t w here w ater d e p th is sm all, ie:
d
— =
0. 1.
In shallow w ater new effects due to n o n lin ear in tera ctio n betw een w ave and
b o tto m sh o u ld be included in m odel equ atio n s. A m eth o d o f num erical
calcu latio n does no t depend o n o u r assu m p tio n s for profiles of b o tto m an d
cu rren t. D ifferent profiles can also be exam ined. It seems th a t in a real near
sh o re region w ave p aram eters should be m ore variable th a n in m odel
presented. Especially effects o f b o tto m an d cu rren t n o n -hom ogeneity along a
line parallel to th e sh o re should be im p o rtan t. T h e present m odel gives only
an answ er to th e q u estio n : to w hat extent d o long gravity waves m odify
exchange processes betw een atm o sp h ere an d ocean in a near shore region?
O u r results suggest th a t m ore realistic co n fig u ratio n o f b o tto m irregularities
and w ater cu rren t sh o u ld be analysed.
3. Method of calculation
W e solve o u r problem using a m ethod of th e characteristics. L et’s start
from th e sh o rt d escription o f th e m ethod because it is no t widely used in
surface w ave m odelling. E q u atio n s (5) co n tain tw o un k n o w n s k x , k y, an d we
present this m ethod for tw o eq u a tio n s for tw o un k n o w n variables. M ore
general system s can also be studied (for details see Berezin, Z idkov, 1962). In
a m eth o d o f characteristics instead solving a hyperbolic system o f quasi-linear p artial equ atio n s, ie:
aij ~ + bu ~ 1- = q ; (i = 1, 2; j = 1, 2),
1 ox
cy
(13)
w here ai}, bij are given function o f x , y , K x, K 2, we solve so-called equ atio n s
o f the d irectio n of th e ch a rac te ristic in the form :
dv
/ = A,; (i = 1 ,2 )
dx
w ith th e differential co n d itio n o n th e characteristic:
(A; · A + B )d K j + C d ^ 2 + M d x + Ndj> —0; (i = 1, 2),
w here
(14)
(15)
are the ro o ts o f th e follow ing d eterm in a n t:
b i i /-Cli i b\ y
~
,
.
,
. 12 = 0 .
b2 \ —^ 2 \ b22 — ^ 2 2
(16)
F u n ctio n s A, B, C , M , N are defined as the follow ing d eterm in an ts:
(17)
T ak in g in to acco u n t w ave k inem atics eq u a tio n s a n d using o u r assu m p tio n s
for th e profile of the b o tto m a n d w ater cu rren t, e q u a tio n (15) can be
rew ritten as:
dk
dk
x d x X+{Cgy+Vp)d x y ~
dp
S x
dio dd
d d'dx’
(18)
o n th e first ch aracteristic curve y = const an d
& - 0
dx
(18a)
on second ch aracteristic j'(x ) = fA2 d x .
For
one
v ariable
problem
a — + £ — = c, eq u a tio n s
(14), (15), reduce
to eq u atio n s:
dy
b
dx
a
dE _ c
dx
(19)
a
A m eth o d o f solving w ave energy e q u a tio n is based on form ulae (19). In o u r
p ro b lem e q u a tio n (19) can be w ritten as:
dE
C” T X - Z ~ D ·
( 20)
F o r a system o f eq u a tio n s (5) ro o ts o f th e d eterm in a n t (10) are equal to
= 0 an d / 2 = (i'9,,+ Vp)/(cgx+ Up). T he existence o f tw o different ro o ts is
a p re co n d itio n for using a m ethod o f characteristics.
F ro m o ne b o u n d a ry point o f th e calculated region, tw o curves (ch ara cteri­
stics) given by e q u a tio n (14) em erge. In a m eth o d o f characteristics we find a
so lu tio n of th e system (5), (6) at th e grid points w hich we find by intersection
o f these curves. T he calcu latio n erro rs dep en d on th e grid length chosen on
th e b o u n d a ry x = x 0.
T his m eth o d has an ad v an tag e over o th e r num erical m ethods in th a t we
have to put initial values k x , ky, E at only one b o u n d ary .
C alcu latio n s are based o n th e M asso m ethod. T h e differentials in (14) and
(15) are ch an g ed in to finite differences an d next four linear eq u a tio n s for one
grid p o in ts are solved. S uppose th a t we kn o w w ave p aram eters at the tw o
n eig h b o u ring grid p o in ts (x l5 j j ) a n d (x 2, y 2) we can find w ave p aram eters at
th e p o in t o f in tersectio n (x3, y 3) o f the characteristics w ith these end points.
T h e first tw o form ulae, according to th e M asso m ethod, we derive from
e q u a tio n (14):
y 3 ~ y 2 = ^ i(* 3 -* 2 ),
y s - y i = a „ (x 3- x i) .
(21)
E q u atio n s (21) describe intersection o f characteristics at th e p oint (x3, y 3).
T he last tw o form ulae are given by e q u a tio n (15). W e rew rite (15) for
n eig h b o u ring p o in ts ( x l5 y t) a n d (x 2, _y2) a n d different families o f the c h a ra c ­
teristics:
( ^ A j + B j) (k
- k
) + C , (/c - k ) + M , (x 3 - x x) + N t (>>3 - y j = 0,
~
*
*
(A „ A 2 + B 2) (k X3 - k X2) + C 2 (ky3 - ky2) + M 2 (x 3 - x 2) + N 2 (y3 - y 2) = 0,
(22)
w here: A l5 A 2, ... are values of the d eterm in a n ts (17) for po in ts (x l5 y x) an d
(x2, ^ 2), respectively.
A system of eq u a tio n s (21), (22) allow s us to find values kX3, k y3 at th e point
(x3, j>3) an d co -o rd in ates o f these points. In fact A b An depend on
k X3, ky3, x 3, y 3 so the system (21) a n d (22) are n o t linear a n d the iterative
m eth o d is used to avoid this co m p licatio n (for details see Berezin, Z idkov,
1962). In area w here X2 changes sign (/12 ~ 0) a system of the e q u a tio n s (5) is
n o t hyperbolic, a n d in o u r calcu latio n p ro ced u re we change a system of
eq u a tio n s w hen X2 c ( —0.1, 0.1). If X2 ^ 0, term s of the eq u a tio n s (5) m u ltip ­
lied by cgy + V p are sm aller th a n th e term s m ultiplied by cgx+ U p, so after
ad d in g first o f tw o eq u atio n s (5) to the second one we have:
(23)
T his e q u a tio n w ith irro ta tio n a l c o n d itio n for wave vector num ber, ie
Skx
dy
dkv
= —^ co n stitu te a new system o f th e eq u a tio n s w hich we solve by a m ethod
ox
o f ch aracteristics. P rec o n d itio n w hich enables to use this m eth o d is fulfilled
because ro o ts of th e d eterm in a n t (16) are equal to 0 a n d 1. Fam ilies o f the
ch aracteristic curves are given by eq u atio n s: y = co n st, y = x 4-const j .
If X2 g> 1 we use the sam e p ro ced u re as w ritten above, only term s m ultiplied
by cgx+ U p a r elim inated. E q u atio n s of the characteristics are as follows: y
= x + c o n st2, x = c o n st3 . In a case o f asy m p to tic b eh av io u r (A2 ~ 0 o r
A2
1) calcu latio n is also based o n th e system (21), (22).
O n e of th e ch aracteristic curves given by th e system (5) (y (x) = J X2 d x)
coincides w ith a ch aracteristic curve for e q u a tio n (6). U sing e q u a tio n (20),
w hich is satisfied for points on this curve, after solving e q u a tio n (5), o n e can
easily find energy o f th e long gravity waves using th e m eth o d described
below . If differential in e q u a tio n (20) is replaced by a finite difference, we
have:
cgx
E ”Aa v 11 ~ En,i — Z — D,
*x n + l , n , i
(24)
A x n + 1 ,n,i
where:
i —ch aracteristic curve w ith the end point at x = 0, y 0i = (i — 1) A y,
Ay —a rb itra rily chosen grid length along y axis at th e b o u n d ary ,
£ n-i- w a v e energy at the p oint (xBii, y ni),
£ n+i.i —w ave energy at th e p oint (x„+ 1>(·, yn+1>i),
Ax„
= Axn+lii —x n>i.
S ig n 'm a r k s a m ean value o f the coefficients in e q u a tio n (20) calculated as an
arith m etic m ean o f the values at the points (xn i, y n i) a n d (xn+ u , yn+ lti). P oints
(x n,h y n,i) w ere found earlier by solving wave k inem atics eq u a tio n s using a
m eth o d o f ch aracteristics. T hese points are given as intersection points o f the
ch aracteristics. S ta rtin g from b o u n d a ry points (x = 0, y 0i) for w hich wave
energy is eq u al to E 0, w ave energy for the follow ing grid points (on th e sam e
ch aracteristic curve labeled by index /') is found from a linear e q u a tio n (24).
4. Results
G e n t a n d T ay lo r (1978) p ro p o se an in terp o latio n form ula to a p p ro x im a te
th eir m odel p red ictions of th e drag coefficient C 10 = { u J U 10)2 in the form :
Ci o = Cs(z0, 0) + ( a k ) 2 F ( R , Cf /uJ,
(25)
w here:
Cs (z0, 0) —d ra g coefficient over a plane surface w ith the roughness p a ra m e ­
ter z Q,
ak = steepness of long gravity surface waves,
R = —In (z0 -k),
F ( R , Cf/u^) — th e com bined effects o f direct in tera ctio n betw een long waves
a n d atm o sp h ere, a n d sm all scale exchange process d u e to flow m odification
by long wave.
In th e active wave g en eratio n region (in w hich 8 < c f /u^ < 1 2 ) an d for
lim ited range o f w ind speed (3 m - s ~ ‘ < U l0 < 12 m - s _1) F can be ta k e n as
R — 2.5· 10~ 3. F o r higher values o f Cf/u^ F is m uch low er a n d for negative
values of cf / u w hen the w ind is against the waves, m uch higher value F is
a p p ro p ria te (F = 10“ 2 —10“ 3). W e use form ula (25) to find variability o f C 10
in a n ear sh o re region. In presented m odel in p u t values are given at the
3 — O ceanologia 26.
b o u n d ary o f region w here w ave param eters depend o n w ater depth. These
values are as follows: Vp0/ C g0 = {0, 0.1, 0.2, 0.5], (ak)0 = 0.1, C90/w* = ± 3 0
and L//.Q = 100. T he last p aram eter expresses dim ensionless w idth o f the
near sh o re region.
F o r a w ave 100 m long th e value o f Vp0/ C g0 — 0.2 gives a m axim um
velocity of the w ater cu rren t equal to 1.25 m - s - 1 . T his is a resonable value.
F o r instance, o n shelf near P o rt E d w ard (S outh Africa) w ater reaches a
d ep th o f 50 m at 5 —6 km from the shore and w ater cu rren t velocity is ab o u t
1.5 m - s -1 there, an d decreases rapidly to w ard s the shore (Pearce, 1977). In
this case value L/A0 is also set correctly.
Values Vp0/ C g0 close to 0.5 are representative for a sh o rter wave, an d 0.5 is
an exact value o f Vp0/ C g0 for wave 23 m long o n the shelf n ear P o rt E dw ard.
T he value (ak)0 = 0.1 can be observed in a near shore region. F o r waves 100
m an d 20 m long this steepness im plies a wave am p litu d e of 1.6 m a n d 0.32
m, respectively.
D im ensionless phase velocity equal to + 3 0 characterizes w ave (swell)
w hich p ro p ag ates freely at the b o u n d ary of calculated region. E nergy input
from w ind to this wave an d the o th er sources an d sinks o f the w ave energy
are practically negligible in this area. W aves w ith a value of cf / u * = —30 can
ap p e a r in a n ear shore region as a residual o f d istan t storm . T hese waves
com e to the shore w here w ind blow s op p o site to a direction o f long waves
p ro p a g a tio n (off-shore) a n d waves generated by th e w ind d o no t interact
w ith swell.
F ig u re 4 show s v ariatio n s of w ave -num ber in a near shore region. In the
case w ith o ut w ater cu rren t w ave nu m b er grow s as result o f red u ctio n of
w ater d epth. As one can see from F igure 4, th e role o f the w ater cu rren t in
changing w ave nu m b er is secondary. A relative g ro w th (when angle betw een
wave d irection an d cu rren t is acute) and red u ctio n of wave n u m b er (when
Fig. 4. C om puted variations of the wave num ber (curves 1, 2, 2', 3, 3') and angle between k and
line perpendicular to the beach (curves a, b, c) with dimensionless velocity of the water current
Ko/Cgo
1- w ith o u t water current, 2 - Vp0/C g0 = 0.2, 2 '- V p0/C g0 = - 0 .2 , 3 - Vp0/C g0 = 0.3, 3 ' - Vp0/C t0
= - 0 .3 , a —w ithout w ater current, b — Vp0/C g0 = 0.5, c — Vp0/C g 0 = —0.5
th e angle is obtuse) reaches a m axim um o f 10°/o for cu rren t w ith Vp0/C g0
= 0.1. C o rre ctio n to form ula (25) depends on k 2, so in tera ctio n betw een
surface wave an d w ater cu rren t can be im p o rta n t for th e analysis o f an
exchange procesess o cean -atm o sp h ere, especially for stro n g current.
In F igure 4 is also present a change of angle betw een w ave nu m b er
vector an d d irectio n perp en d icu lar to th e shore. T he angle decreases w ith the
red u ctio n o f w ater d epth, w ave becom ing m ore nearly perp en d icu lar to the
beach. W ater cu rren t plays a little role to change this angle. F o r strong
c u rren t th e angle decreases to ab o u t 2°.
In F ig u re (5) we show variability of wave energy an d drag coefficient as a
function o f dim ensionless w ater d ep th (k0 d). W ater energy decreases in the
O
k Qd
Fig. 5. C om puted variations of the wave energy (curves a, b, c) and drag coefficient (curves 1, 2,
2', 3, 3', 4) with Vp0/C g0 and wave-to-wind angle
W hen direction of the wave propagation is opposite to the wind, n otation is the sam e as in
Figure 4. C urve 4 refers to the case when wind blows along waves and Vp0/C g0 — 0.2
calcu lated region. In the case w ithout sources a n d sinks of th e w ave energy
this is caused by changes in wave g ro u p velocity. T he g ro u p velocity at first
increases slightly th en rapidly decreases as th e w ater becom es m o re shallow ,
so th e energy first decreases slightly before its ra p id increase. W hen we
include d issip atio n m echanism - p erco latio n described by eq. (10)- i n the
w ave energy tra n sp o rt eq u atio n , w ave energy decreases also in a region
w here th e g ro u p velocity increases. As th e ra tio of w ave am p litu d e to the
local w ater d ep th increases, the w aves are deform ed a n d ap p e a r to behave as
series o f so litary waves o r cnoidal waves. S oon they becom e u n stab le an d
b reak. Swell ten d s to b re ak w hen th e c rest-to -tro u g h height is co m p arab le to
th e local w ater d epth. In o u r calcu latio n s this zone is not tak en in to account.
E nergy d issip atio n due to p erco latio n effectively changes w ave energy
(Fig. 6). W hen b o tto m m aterial consists o f coarse sand, ch aracterized by
coefficient o f p erm eability y = 2 c m - s - 1 , w ave energy decays ab ru p tly in
shallow w ater, so n ear th e beach the co rrectio n factor in form ula (25) is
ab o u t 2 °/0 of th e value o f Cs (z0, 0). In this case th e o th er m odel param eters
are as follow: Vp0/C g0 = 0.1 an d L //.0 = 100. W hen b o tto m m aterial consists
Fig. 6. The role of bo tto m m aterial in variations of the wave energy (curves a, b, c) and drag
coefficient (curves 1, 2, 3); for Vp0/C g0 = 0.1 a and 1 —y = 0.1 cm s ~ ‘ (coefficient of perm eabili­
ty), b and 2 —y = 0.5 c m - s '1, c and 3 —y — 2 cm s ' 1
o f fine san d a n d y = 0.1 c m - s -1 energy d issip atio n is quite sm all. W ave
energy decreases slowly an d th e co rrectio n factor is m uch greater. As we can
see from F igures (5) an d (6), angle betw een w ave an d w ind plays a crucial
role in m o m en tu m exchange processes. In th e case w ith o u t p erco latio n an d
w ater cu rren t, w hen th e w ind blow s against long surface wave, th e drag
coefficient increased ab o u t 60°/o, but w hen th e w ind blow s along direction of
th e w ave p ro p a g a tio n the co rrectio n factor fall do w n to 1 0 % ·
W hen th e angle betw een waves a n d w ater cu rren t is acute, g ro w th of the
w ave n u m b er is g reater th a n in the case w hen the angle is obtuse. In th e first
V
case —k v— > 0 (in th e second case < 0) an d the ra te of g ro w th kx is larger
x
th a n in th e second case — see eq u a tio n (18).
A ccording to form ula (25), the drag coefficient in the first case is g reater th a n
in th e second case. F o r Vp0/ C g0 = - 0 . 2 an d y = 0 m axim um co rrectio n s are
40°/o ° f th e value o f Cs (z0, 0). It is ab o u t 50°/o of th e co rrectio n for the case
w hen Vp0/ C g0 = 0.2. F o r g re ater values of (ak)0 a n d \VpJ C g0\ this difference is
m ore distinctive.
A n energy flux to the w ave due to w ater current-w ave in teractio n
practically do not change w ave energy (Fig. 5). M ean sh ear of w ater cu rren t
in o u r m odel is quite sm all. A coefficient ch aracterizing th e efficiency of
sources (sinks) ie f T 1 d E / d t is o f th e o rd e r o f 10- 4 . In shallow w ater zo n e it
is co m p arab le to a dum p in g coefficient d u e to p erco latio n in fine sand. As we
can see, only a change in the w ave k inem atics caused by w ater cu rren t is
im p o rta n t for m o m en tu m exchange process.
In o u r m odel energy input from the w ind is n o t decisive for w ave energy.
A w id th of the calculated p a rt o f th e near shore region is sm all ie L / a 0
= 100, w hereas a length scale for change of w ind driven w ave is ab o u t
th o u san d s of w avelengths. F o r a w ider near shore region a n d for a sm aller
value o f \cf / u j th e w ind-w ave in tera ctio n should becom e m ore im p o rtan t.
5. Conclusions
T he follow ing conclusions can be d raw n from o u r study:
(i) C han g es o f long surface w ave p aram eters in a near shore region
influence m o m en tu m tran sfer betw een the atm o sp h ere an d ocean.
(ii) In a n ear shore region, w hich is hom ogeneous along an axis parallel
to th e beach, th e sea-atm o sp h ere in tera ctio n increases as th e w ater becom es
m ore shallow .
(iii) W hen th e w ind blow s op p o site to the direction o f w ave p ro p a g atio n ,
th e sea-atm o sp h ere in tera ctio n intensifies. F o r steep gravity w aves th e role of
low -frequency b a n d o f w ave sp ectrum in m o m en tu m tran sfer betw een the
atm o sp h ere a n d o cean (A-O) is co m p arab le to th e role o f high-frequency
surface irreg u larities (these co m p o n en ts define roughness o f the sea surface)
(iv) W a te r cu rren t an d type o f th e b o tto m m aterial influence exchange
process A -O . T h e first factor changes m ainly kinem atics o f th e w ave train ,
th e second dynam ics.
(v) A n u m erical m eth o d o f characteristics applied to kinem atics a n d wave
energy eq u a tio n s is fast a n d accurate.
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