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12Vertical Fluxes Induced by Weak-Nonlinear Internal Waves in a Baroclinic Flow
A.A. Slepyshev
Marine Hydrophysical Institute, Russian Academy of Sciences, Sevastopol,
Russian Federation
In a Bussinesq approximation free internal waves are considered at the account of turbulent viscosity and diffusion in a vertically-non-uniform flow. In linear approximation the dispersive relation and decrement of wave attenuation are found. The equation for amplitude of the vertical velocity, containing small parameter at the higher derivative, proportional to turbulent viscosity, solved by Ljusternik-Vishik asymptotic method. Boundary-layer solution on a vicinity of a bottom and a free surface are found. The non-viscous boundary value problem of the second order is solved numerically according to an Adams's implicit scheme of the third order accuracy. The wave number at the fixed frequency of a wave is found by a shooting method. Stokes drift velocity and vertical wave flux of salt are determined in the second order on wave amplitude. Shift of phases between fluctuations of salinity and vertical velocity with regard to turbulent viscosity and diffusion differs from n / 2, therefore the vertical wave flux of salt differs from zero. The dispersive parity, decrement of wave attenuation and wave fluxes are calculated for internal waves observed during the 3rd stage of the 44 cruise of R/V "Mikhail Lomonosov" to the northwest shelf of the Black sea. Critical layers for a current profile were absent at the test site (where the measurements were carried out), i.e. phase rate of internal waves exceeded the current velocity. It is shown that out of a layer with the maximum gradient of salinity, i.e. out of surface layer, the wave flux of salt is comparable in absolute value with the turbulent one. In a surface layer turbulent flux of salt exceeds the wave one. The consideration of current results in slight decrease of the wave flux. Horizontal component of Stokes drift velocity, which is transversal to the wave propagation direction, differs from zero and is one order less than longitudinal one when taking into account the current.
Keywords: internal waves, turbulence, Stokes drift.
DOI: 10.22449/1573-160X-2015-1-59-72
© 2015, A.A. Slepyshev © 2015, Physical Oceanography
Introduction. Internal waves are presented everywhere in the ocean due to effect of sources which generate them. Internal waves can exist in stratified medium when water density increases with the depth. Below the upper mixed layer such situation is typical for the World Ocean. Atmospheric pressure perturbations, wind stresses on the sea surface, interaction of tides and currents with bottom topography inhomogeneities [1], eddy currents may be referred to internal wave energy sources.
Actuality of the problems is related to the fact that internal waves can contribute to vertical transfer in the ocean. Usually, vertical transfer in marine environment is connected with small-scaled turbulence that has an intermittent character, i. e. the turbulence is presented in the form of "patches", generated by hydrodynamic instability of currents and breaking of internal waves. Vertical transfer plays an important role in admixture transport, oxygen diffusion to the deep layers of the sea and hydrogen sulphide diffusion from deep layers of the Black Sea.
The internal waves with regard for eddy viscosity and diffusion were considered in a number of works [1 - 3], where a decrement of wave attenuation on
PHYSICAL OCEANOGRAPHY NO. 1 (2015) 59
turbulence was determined. Non-linear effects at wave propagation with no regard for eddy viscosity and diffusion were considered in [4, 5]. In these works a mean wave-induced current and non-oscillatory correction to the average density were determined.
Internal waves with regard for nonlinearity and viscosity were considered in [6], but vertical wave fluxes of heat and salt were not taken into account.
Vertical transfer in stratified seawater column is associated with breaking of small-scale internal wave [7], and turbulent energy dissipation rate and a coefficient of vertical eddy diffusivity are estimated. It is shown that in the area of continental slope at the edge of the Black Sea shelf the intensification of vertical transfer, related with internal wave amplitude increase at their propagation to the shallow water area [8], takes place.
Vertical flows, determined by weak-nonlinear internal waves, were considered in [9]. Such flows exist due to vertical velocity and temperature (salinity) oscillation phase shift, which differs from n/2 when diffusion and eddy viscosity are taken into account.
In the given work the vertical flows caused by internal waves on the baroclinic flow are considered. It is of interesting to compare wave fluxes with the corresponding turbulent fluxes both in the presence of current and in the absence of it. Moreover, the Stokes drift speed in both these cases is found.
Formulation of the problem. Free internal waves on the baroclinic flow with account for eddy viscosity and diffusion in the Boussinesq approximation are considered. Internal wave amplitude distribution, dispersive relation and wave attenuation decrement are calculated in linear approximation. The Stokes drift speed and wave fluxes of salt are found in the second order of the wave amplitude.
We introduce dimensionless variables by the following formulas (dimensional physical quantities are denoted by wavy lines above the symbols):
t ~ k
t = —, Ik = —, cc = ú)„Ú) , U~1 = u1Hc, u2 = u2Hc u~3 = u3Ha„,
C H
~ H n H n ~
P = p0H VP, p = p0V-, p0 = p0V-0, xi = Hxi, Kt = ,
g g
Mi = Mm (i = 1,2,3), K = K2, Ml = M2, M =^2,
where /u1, ju3 - are the characteristic values of horizontal and vertical eddy viscosity; g - is an acceleration of gravity; x1, x2, x3 - are two horizontal and one vertical coordinates, vertical axis x3 is directed vertically upwards; p and P - are the wave perturbations of the density and pressure; p0 - is an unperturbed average water density; p0 - is a depth-averaged density; u1, u2, u3 - are two horizontal and one vertical components of velocity wave disturbances; K1, K3, M1, M3 - are horizontal and vertical coefficients of eddy viscosity and
diffusion, respectively; H - is a sea depth; v - is a characteristic wave
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PHYSICAL OCEANOGRAPHY NO. 1 (2015)
frequency. The coefficients of vertical eddy diffusivity and viscosity, and two components of the mean current velocity U0, V0 are assumed to be dependent on
the vertical coordinate. The coefficients of horizontal eddy diffusivity and viscosity are assumed to be constant. The system of hydrodynamic equations for the wave disturbances in the Boussinesq approximation has the following form:
Cki, Cki,
_1__L i i _1
dt
u
' * U0
ox.
du, ^ du,
____L T/ __
dx_ 0 cX2
dUn
u
dP dx1
s2 Kii+d
dx_ cX2
d
^ dx3 ^ 3 dx3 ,
du1
(la)
du-, du,, - u
dt
' dx.
-U
du2 dx.
du2 dVr,
- V+ u
dx.
dx.
dx.
dP 2 ( d2u^ d2u,, - + s K.
dxx
+ s2p2 Mk3 ——2
dx, I dx,
d
(_b)
du3 ~dt
du, ^ du, ^ du, u. —3 + U0 —3 + V0 —-
d
dx1
s2 p2 -L
dx.
dx,.
K du3
dP dx.
■+s2 Ki
i ~i2 ~,2 \ du, du,
--H--3
v dx_ dx\ j
dx.
3 j
s2p2 àK— du3
dx3 dx3
P,
(1c)
dp dp dp dp 2-Kt
+ u —— + Un—+ = s M,
dt ' dx. 0 dx 0 dxn ' 1 2
fd2p d2p ^ y Sx_2 5x2 j
2 o2 d (dp "\ dp,
+ s p --- u 3 ——
dx3 ^ dx3 j dx3
du,
dx.
= 0.
, (1d)
(_e)
Here s2 = M - is a small parameter proportional to the value of the horizontal
H 2a,
eddy viscosity coefficient p2 = M-, while << s2.
Mi
Boundary conditions on the surface are the solid top condition and the absence of tangential stresses [2]:
u-(0) = 0, p2K- du_ + Kl du- = 0
ox3 ox1
p2 K3 ^u2+K_ ^u- = 0
ôx dx.
(2)
x3 =0
x3 =0 3 2
Boundary conditions on the bottom are the adhesion conditions:
u_( -1) = 0, u2(-1) = 0, u3(-1) = 0. (3)
PHYSICAL OCEANOGRAPHY NO. 1 (2015)
61
Linear approximation. We seek the solutions of linear approximation in the following form:
U = u10(x3)Ae'e + c.c., u0 = u20(x3)Ae'e + c.c., ul = u30(x3)Ae'e + c.c.,
p = Po(X3)Aew + c.c., p = pio(X3)Aew + c.c., (4)
where c.c. - are complex conjugate terms; A - amplitude factor; d - wave phase;
= k; = —v (k - wave number, V - frequency). It is assumed that the
dx1 dt
wave propagates along the x1 axis.
Substituting (4) into the system of equations (1), we obtain the relation of amplitude functions u10, Pw, u30and equation for u20(x3), p10(x3):
i du
u10 =--—, Q = v- kU0, (5a)
k dx3
P10 = (s2p2K3 ^ + s2p2 dK3— + i'Q-s2K1k2)^^ + LdU.u30, (5b) dx^ d^3 d^3 k d^3 k d^3
(s202M3 ddxY + s2p2 Mdr s2Mk2 + i^)A0 = u30 dp, (5c)
CIJ\'3 dJ\'3 CUx'3 dJ\'3
s2p2K3 ^ + s2P2 ^ ^ + (i^s2Klk2)u20 =u30 dpi. (5d)
CIJ\'3 dJ\'3 dJ\'3 dJ\'3
(5d) equation should be supplemented by boundary conditions arising from (2), (3):
du.
20
dx,
3 x3=0
The function u30 satisfies the equation
-N2u30 = (s2^2M3 ^ + s1p1 -e1Mlk2 + iQ) x
= 0; U20 X=_j = 0. (6)
x <j iQu30 —— I dx3
dx> 3 dx3 dX3
K3 d2- + s2?2 - s2 K1k2 + iQ)^- du30 + u3{
dx 3 dx3 dx3 k dx3 k dx3
(7)
-s2K1k2u30 + s2$2K3 ^ + 2s2^2 dK3 du30
dX3 dX3 A3
on the surface at x3 = 0
From the boundary conditions (2), (3) we obtain the conditions for u30 :
d 2u3
dx32
62 PHYSICAL OCEANOGRAPHY NO. 1 (2015)
u30 = 0, —f- = 0, (8a)
on the bottom at x3 = -1
u3o = ^ = 0. (8b)
dx3
Equation (7) has a small parameter at the higher derivative ~(s3)4 . We solve this equation, following [10, 11], by the Lusternik - Vishik asymptotical method, expanding u30, < in series by s? small parameter:
U30 (X3) = X (spy ui (X3, s) + s?Z (s//) v + (s/?)2X (s//)'vi, (9a)
< = <01(s) + s/<11(s) + (s/)2<21(s) + ..., (9b)
where v1 ((1 + x3) / sp) - boundary layer solution in the neighborhood of the bottom, v°(x3/sp) - in the neighborhood of free surface. Boundary layer
corrections are the functions, rapidly decreasing with the distance from the boundary, which provide the execution of the boundary conditions. In the general case the functions u3i(x3,s) depend on the parameter s , because it is contained
in the equations for these functions.
Substituting the expansions (9) in (7), we obtain the boundary value problem for the function u3°0 in the first order on s3 parameter:
(s2Mik2 -/Qoi)J/CV0o
I dx,
(/Q01 -^Kt)+~dJT"0o -^KU k dx k dx
3 3
:#2Ml,(io)
where N2 = - - square of the Brent-Vaissala frequency; Qoi = co1 - kUo -dx3
wave frequency with Doppler shift.
Boundary conditions for u°o are the following:
uox=o = o, = o. (11)
The equation (1o) has a small parameters and following the method, described in [1o], we seek the solution and frequency co1 in a form of asymptotic series by this parameter:
u3o(x3,s) = wo(x3) + ew1( x3) + s2w2(x3) +..., (12a)
co1 = co +sc1 + s2c2 +.... (12b)
PHYSICAL OCEANOGRAPHY NO. 1 (2o15)
63
In a zero order on parameter s function w0 satisfy boundary value problem
(N2 -«2).
d2w0 2(N2 -n2) k d2Ut ' + k
dx,
a
w0 = 0.
0 1x3 =0
2 W0 +
a0 dx3
w = 0.
2 "0
w0 = 0,
0 lx3 =-1
(13a)
(13b)
Where Q0 = <0 - kU0. In the absence of singularity Q0 = 0 the boundary value problem (13) has a numerable set of eigenfunctions - a set of modes and for any value of frequency <0 < max(N) corresponds a certain wave number k for given mode.
The next term in the expansion (12a) is determined from the following equation
d2w ,2(N2 -a2) k d2U0 m
-2- + k --w1 +--20 w1 = —
dx3 a2 a0 ^^0
Boundary conditions for w1 function
2kw0 - 2 ^.2
d w kwn d U,
2
= f ( X3 )(14)
W-
1L =0
= 0,
W-
1L =-1
= 0.
The solvability condition of the boundary value problem (14), (15)
u
J f1W0dx3 = 0.
(15)
(16)
For < ^ 0 this condition is not generally fulfilled and the boundary value problem (14), (15) has no solutions.
The next approximation w2 in the parameter s satisfies the following equation
a2
dxl
w2 + N 2k2 w2 +ka0 d-u20 w2 = (®2 +iMk2 )|k 2a0w0 -a0 -kd-U0 w„
dx, V 'I dx
-a
®2ik 2W0 \iKt2 dx0+Kk 4W0
= o.
(17)
Boundary conditions for w2 have the following form
w.
2 lx3 =0
= 0,
w.
2 Ix, =-1
= 0.
Solvability condition of the boundary value problem (17), (18)
u
J Ow0 dx3 = 0 .
(18)
(19)
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PHYSICAL OCEANOGRAPHY NO. 1 (2015)
Hence we find the expression for®2 :
a)2 = -i
2
0 f Mk\r2 ^ , 2 d2w0 ,4 ^
/[m^ - K'k *t+kKw ,
a
dx,
N2 2 k d U0 ^
— 2k2 + —-^
a3 a dx2
w0 dx3
3 J
Let us find a boundary layer solution v0 in the expansion (9a) to satisfy the boundary conditions (8a), (8b) on the surface. Substituting the expansion (9a) in the equation (7), and expanding K3 ,M3 and U0, V0 in Taylor series in the
neighborhood x3 = 0 we obtain the equation for v0(n), in the zero order on parameter sf :
K3 (0)M3 (0) d6v0 + /Qo(0)(K3(0) + M3(0)) ^ = Q0(0) d2v0, n = ^ (21)
dn dn dn sf
The solution of the equation (21), attenuating with the distance from the surface, has the following form:
V00 = C0 exp(^°n) + F,0 exp(^2n), (22)
where
,0 _ d2Wo(0)_1 _
dx23 ®2 - (40)2'
Cu = w- T0V»y_^_ Fu = _ c0 (93)
Here A,0 , 4 are calculated according to the formulas
40 = tofa_,), 4 = (WLf0_0. (24)
" 1 2M:,(0) I V ' 2 I 2^3(0). V ' K '
'J
The equation for the boundary layer solution v0((1 + x3)/s2) has the following form:
dV+ffl0( _i) Mi±KL dV dV=0, (25)
dtf M3K: dtf M3K: dtf
where n0 = ~-). The solution of the given equation, attenuating with the
sj3
distance from the bottom and satisfying the boundary conditions (8b), is determined by formula:
V0 = D° exp(_4n) + G exp(_4ni), (26a)
where
D0 = ^ -_-, G0 =_ D0. (26b)
caj^: /to
PHYSICAL OCEANOGRAPHY NO. 1 (2015)
65
The values \ , 12 are calculated according to the formulas (24), only the eddy
transfer coefficients and wave frequency with the Doppler shift are calculated at the lower boundary.
The equation for salinity wave perturbations 5 has the form
ds
ds
ds
ds
dSn
d
f
— + (u1 + U0)— + (u2 + V0)-+ u3-+ u3—0 = s—
dt
dx,
M,
Sx,
ds
1 dx,
dx.
Sx
dx,
dx,
ds
\
M1 — . dx1 y
+ (sfifé
2 y
dx,
M
ds
dx.
(27)
3 y
where S0(x3) - is an average salinity profile. The solutions of linear approximation will be sought in the following form:
= j10(x3)Ae'e + c.c., (28)
where 510 satisfies the equation
dSn
d _ , ds,,
-1^10 + = s2 (-k2M^o +P2^- (M3 . (29)
dx,
d^3 (^3
Equation (29) has a small parameter at the higher derivative, and we are to seek a solution at the form similar to (9a):
s10 = s/o + (s/)2 sjj +
x + 1
(s/3)2 sfo +s/32 s^^/S +(s/)2 2 . (30)
s3
s3
From (29) it follows that where
j 2
s10 = s0 + S s2 + ... :
iw0 dS0
Q0
s2 =-(®2i + k 2M1 )
w dSn dSn
££2 <^3 ££0 d^3
(31) (32a) (32b)
0
here a>2i = — . Boundary layer solutions s n 0 and s n0 in the neighborhood of the
bottom and free surface satisfy the following equations, respectively
d (M (0)d )/„„ = dS0(0)-dn
(0)sn0 +d (M3(0)d)s°n0 = «S^1 v0(n)-.
dn
d
d
i£0(-1)sn0 + (M3(-1)—yn0 =
dS,(-1) ,
v0(n).
Boundary conditions have the form
s10(0) = s10(-1) = 0.
(33 a)
(33b)
(34)
i=0
i=0
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PHYSICAL OCEANOGRAPHY NO. 1 (2015)
Solution of the equation (33a) in the neighborhood of the sea surface is the following:
•C (n) = C0S exp(^0n) + Q0 exp(^0n) + nP0 exp(A), (35)
where
P = C dSM, (36a)
S 2A!0M3 (0) dx3
Q0 =_C_iS(0>, (36b)
Q 0(0)[1 - M3(0)/ K3(0)] dx3
C0 =-Q0. (36c)
Boundary layer solution of the equation (33b) in the neighborhood of the bottom has the following form:
S n 0 (ni ) = C12 S
exp(-A, ni) + Qsi exp(-A2n,) + n, PSi exp(-A, n, ), (37)
where
Ps,=__dS0£=i), (38a)
Si 2^M3( -i) dx3
Qs, =_D_dSai-i> . (38b)
-Sl Q,(-1)[1 -M,(-1)/Kj(-1)] dx,
From the boundary condition (34) it follows
CL = -Qsi. (39)
Nonlinear effects. The velocity of the Stokes drift of the fluid particles is determined by the formula [i2]:
u s = J udrVu, (40)
0
where u - is a field of Eulerian velocities, a horizontal bar means the averaging over the wave period. Horizontal velocity component of the Stokes drift (directed along the wave vector) in the second order of amplitude, has the following form:
„ = AiAi i 200 du30 du30 + «30 d U30 + «30 d U30\ (4i)
uis =~1 ( 1 + + * A 2 ) ' ^ ^
k oo dx3 dx3 6) dx3 6) dx3
Horizontal velocity component of the Stokes drift (transversal to the direction of wave propagation) is calculated according to the formula
u2s = d(U30U30) + c.c., A, = Ae6 . (42)
s o dx3
At the presence of the mean current, whose velocity component V0 (transversal
to the direction of wave propagation) depends on vertical coordinate, the value u2s differs from zero and in the non-viscous case is calculated as follows:
PHYSICAL OCEANOGRAPHY NO. i (20,5) 67
2 A A* d ,
U2s =--("
wL dVf)
dx^ ^^ q dx^
(43)
Vertical wave flux of salt u3S taking into account the expansions (9a), (3Q), (31) determined by formula:
U3s /| Ai2| = Wq(s2 s 2 +SJ3S n Q + (sJ3)2 O* + s Q)(sJ3v10 + (sj3)2 vQ) + s2 W2 sQ + c.c. (44)
Calculation results. Let us calculate salt wave fluxes for the internal waves, which had been observed southwestward of Yevpatoria during the experiment, performed at the third stage of the 44th R/V Mikhail Lomonosov cruise.
In Fig. 1 we have plotted four realizations of the temperature contour elevations calculated according to the data of the GRAD instruments, that is to say, the gradient-distributed temperature gauges [13]. The first device was situated in 5 - 15 m layer, the second - in 15 - 25 m layer, the third - in 25 - 35 m layer and the fourth - in 35 - 6Q m layer. It is obvious that strong oscillations with 15 m period in 25 - 6Q m layer are in against phase with the oscillations in 25 - 6Q m layer, which indicates the oscillations of the second mode. The maximum amplitude by swells made up Q.5 m. In Fig. 2, a, b the vertical profiles of the Brent-Vaissala frequency and two components of the current velocity are represented. The boundary value problem (13) for internal waves is solved numerically by the implicit Adams scheme of the third order accuracy. Wave number is found by the shooting method from the necessity of satisfying the boundary conditions (13b). The eigenfunction of 15 min internal waves of the second mode is shown in Fig. 2, b.
Fig. 1. Time variation of the vertical displacements temperature contours
This implies
max^3
A =
1 2 max |wQ / Qq i
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PHYSICAL OCEANOGRAPHY NO. 1 (2Q15)
Thus, the amplitude of vertical displacements is proportional to w0. According to the experiment data (Fig. 1; 2, b), the maximum of w0 function correspond to the maximum vertical displacements, i.e. in the experiment the second mode was observed. We calculate a vertical eddy diffusion coefficient M3 by the empirical formula, which is valid in the area of continental slope at the Black sea north-west shelf [7]: M3 = 8.4•10-4Ni-1 m2/s, Nc corresponds to Brent-Vaissala frequency in cph. Wavelength of the second mode 15-min internal waves is 195.6 m, typical value of horizontal eddy diffusion coefficient M1 is 1 m2/s. The dispersion curves of the first two modes in the presence and absence of current are shown in Fig. 3.
Fig. 2. Vertical profile of Brent-Vaissala frequency - a, an eigenfunction of 15 min internal waves -b, vertical profiles of current velocity components jj (■■■), y (-)-c
PHYSICAL OCEANOGRAPHY NO. 1 (2015)
69
л о.
к, rad/m
Fig. 3. Dispersion curves: the first mode without current (-) and with current (---); the second
mode without current (.........) and with current (-----)
Рис. 4. Stokes drift velocity in the presence of current (-) and in the absence of current (.........)
Boundary value problem (17), (18) on the determination of w2 function is solved numerically by the implicit Adams scheme of the third accuracy order at K3 = 2M3, Kj = 2Mj. We find the only solution, orthogonal wQ and wave
attenuation decrement 8(0 = (i . In case of 15 min internal waves of the second
mode 8( = -1.46 -1Q-3 rad/s. Vertical profiles of Stokes drift velocity horizontal component in the presence and absence (dashed curve) of current are shown in Fig. 4. In the presence of current the Stokes drift velocity in modulus is lesser. Stokes drift velocity component, transversal to the direction of wave propagation, differs from zero only with regard to the current (Fig. 5).
Fig. 5. Stokes drift velocity component, transversal to the direction of wave propagation Fig. 6. Vertical salinity profile
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PHYSICAL OCEANOGRAPHY NO. 1 (2015)
Fig. 7. Profiles of wave and turbulent salt fluxes: wave flux in the presence of current (-) and in
the absence of current (.........); turbulent salt flux (---)
Wave salt flux is calculated at the vertical salinity profile, shown in Fig. 6.
dS
Turbulent salt flux is calculated according to the formula sf = -M3 —0. Vertical
f 3 dz
wave salt flux u3S for 15 min internal waves of the second mode and a turbulent
flux are represented in Fig. 7. In the presence of current the wave salt flux is somewhat lesser than in its absence. In the absence of current a wave flux (dashed line) is comparable in modulus with the turbulent flux at a depth more than 25 m, i.e. outside the surface layer of the maximum salinity gradients.
Conclusions.
1. With account of eddy viscosity and diffusion the wave salt flux u3S
differs from zero, and in the presence of current it differs somewhat smaller than in its absence.
2. In the absence of current the wave salt flux is comparable in absolute value with the turbulent flux out of the subsurface layer (where the salinity gradients are maximal).
3. The component of Stokes drift velocity (transversal to the wave propagation direction) differs from zero when the current is taken into account.
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